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dissertation entitled

AN EXPLORATORY STUDY OF THE USE OF MUSIC THERAPY
IN TEACHING MATHEMATICAL SKILLS
TO INDIVIDUALS WITH WILLIAMS SYNDROME

presented by

Eunmi Emily Kwak

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AN EXPLORATORY STUDY OF THE USE OF MUSIC THERAPY
IN TEACHING MATHEMATICAL SKILLS
TO INDIVIDUALS WITH WILLIAMS SYNDROME
By

Eunmi Emily Kwak

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY

Music Education

I
I
t
I
[I

2008

 

ABSTRACT
AN EXPLORATORY STUDY OF THE USE OF MUSIC THERAPY
IN TEACHING MATHEMATICAL SKILLS
TO INDIVIDUALS WITH WILLIAMS SYNDROME
By

Eunmi Emily Kwak

This study was designed to develop effective music therapy strategies to teach
mathematical concepts to individuals with Williams Syndrome (WS) by using their
affinity and abilities in music as instructional aids. WS is caused by a microdeletion of 16
— 23 genes on chromosome 7q11.23. These missing genes cause common facial
appearance, similar physical characteristics, common cognitive and behavioral
characteristics, and physiological difficulties, including cardiovascular abnormalities. A
well-known trait of people with WS is a pronounced difficulty with mathematics. In
addition, individuals with WS are known as music lovers. An “affinity to music” is often
used to describe this characteristic. The basic premise for the effectiveness of music
therapy intervention for individuals with WS in this study was that, by combining
arithmetic with musicing, individuals with WS could be motivated to attend to
mathematical tasks and their attention could be held for longer periods, which might
enhance their ability to learn and retain the lessons.

A collective case study was the methodological framework for the study. The
participants in the study were three individuals diagnosed with WS, ranging 10 to 21
years of age. Sessions took place over a 10-week period, once per week for 40 to 60
minutes, in each participant’s home environment. All thirty sessions were digitally

recorded and transcribed.

The analysis and interpretation of the sessions identified the participants’
roadblocks in their mathematical development in metacognition, visuospatial ability,
attention, processing speed, and fine motor skills. The severity of these difficulties was
shown to vary among the three participants. While many of the effects of these
difficulties overlap and are interrelated, the findings suggest that each difficulty requires a
different form of intervention. These findings also suggest that music therapy could offer
various strategies that assist in coping with the difficulties. Three primary functions in
music were identified as being particularity beneficial for mathematical development:
music used to advance cognitive development; music used to encourage and motivate;
and music used to provide structure for academic learning interventions.

Based on the findings of the study, several suggestions are made for music
therapists working with individuals with WS as well as individuals with other genetic
disorders. These suggestions include the importance of specific research on the
differences in the roadblocks among the various types of genetic disorders and an
informed clinical practice based on knowledge, theory, and research with the particular
type of genetic disorder. This study suggests that music therapists might need to develop
three sets of music therapy interventions: 1) individuals with WS who do not develop the
comprehension of magnitude and cardinality; 2) individuals with WS who develop the
basic concepts referred to above, but still struggle with Operating numbers such as
composition and decomposition; and 3) individuals with WS who develop a basic
understanding of numbers and operations and are ready to move on to addition and
subtraction. This Study demonstrated the strong possibility that music therapy can assist

individuals with WS in their mathematical development.

COPyright by
Eunmi Emily Kwak

2008

Dedication
This dissertation is dedicated to my parents, Changgun Kwak and Youngja Chun,
in gratitude for their prayers, support, and trust. I also dedicate this to my Heavenly
FATHER for keeping me strong, focused, and seeing the beauty in individuals who are

not just like me. Without HIM I could not have completed this project nor come this far.

 

Acknowledgements

I am sincerely indebted to the faculty, family, friends, and my three participants
and their family members. During the 10 weeks of sessions, Angela, Roxy, and Big Red
were so sincere and tried to make each session, even through Thanksgiving, Christmas,
and family engagements. Thank you Angela, even though you had a cold, even though
you just returned fi'om the dental appointment and you still felt the pain and numbness in
your mouth, you came to the session. Thank you Roxy, even though you expected your
best fiiend to arrive any minute, you were able to focus on the task. Thank you Big Red. 1
know I made you uncomfortable at times, but you did not complain. You were so patient
and generous with me. The Williams Syndrome Association and parents of individuals
with WS were so eager to find a way to help their children and their support and
enthusiasm continues to be inspirational.

Next, I want to express my gratitude to my extraordinary committee: to Dr.
Frederick Tims, my advisor and mentor, for his thoughtfirl and genuine care, guidance,
and enthusiasm about the subject matter, and for his trust in me during the most difficult
times; to Dr. Mitchell Robinson, for his endless guidance with the qualitative research
methodology, and for his genuine care about not only the academic aspects but also
‘Emily’ as a person; to Dr. Cynthia Taggart, for her productive and constructive
comments on my dissertation and her high expectations for my project; to Dr. John
Kratus, for his facilitating critical thinking and inspiration. I am also thankful to the
College of Music, and the Graduate School for supporting my research through several
generous fellowships and assistance for participation in conferences which shaped my

dissertation.

vi

 

Thank you to the Williams Syndrome Association and Terry Monkaba, their
executive director, Tia Sager, the program director of the Association, and the Whispering
Trail Camp, music camp for individuals with WS, for their‘enthusiasm, support, and
encouragement to make an impact on WS research. The idea for this research came to me
at the 2006 WSA convention, and Ifinished the very last page of my dissertation at the
2008 music camp. It was very meaningful to finish the paper while I was surrounded by
66 individuals with WS; no other place could have been better to do this. Thank you to
the Camp Staff for understanding and encouraging me to finish and for checking on me.

Thank you to my family and my friends. I also appreciate Hayng Eun Lee, who is
my dear friend who drew numerous illustrations whenever the need arose; Jae Heung
Kwak, my brother, who became my computer consultant; and Jiyeun Lee, my dear friend
and fellow music therapist for helping with music notation and tedious behind the scenes
jobs. Their knowledge, skills, and time ensured the completion of this dissertation. I want
to say a special word of thanks to my friends, Rick Mulligan, Jessica Schaub, and Wendy
van Gent. They shared their time, and energy, and providing encouragement through on
the actual writing process. I would like to thank my MOTHER and FATHER for helping,
supporting, and being patient with me as I completed my long journey. Finally, I want to

thank my heavenly FATHER who made me, guides me, and is with me always.

In the beginning God created the heavens and the earth. Genesis 1:1

vii

TABLE OF CONTENTS

LIST OF TABLE ............................................................................................................ xiv
LIST OF FIGURES .......................................................................................................... xv
PROLOGUE
“Buying a Cup of Coffee” ................................................................................................... 1
CHAPTER I
INTRODUCTION .............................................................................................................. 2
CHAPTER H
REVIEW OF RELATED RESEARCH ............................................................................ 13
Descriptions of Williams Syndrome ................................................................... 13
Physical Appearances and Medical Issues ................................................ 16
Language Characteristic ............................................................................ 19
Social Characteristics ............................................................................. 21
Behavioral Domain ................................................................................... 23
Visuospatial Abilities ................................................................................ 27
Affinity to Music ....................................................................................... 29
Neurobiological Factors ...................................................................................... 33
Mathematical Development ................................................................................ 43
Music as a Therapeutic Medium ......................................................................... 48
The Learning Process ................................................................................ 49
Memory Process ........................................................................................ 50
Dyslexia and Music ................................................................................... 52
Mnemonic and Strategic songs ................................................................. 54
Issues in Attention and Motivation ........................................................... 56
Wish List from Parents and Music Therapists .......................................... 57
Reason for Study, Initial Theory, and Research Questions ................................. 60
Research Questions ................................................................................... 62
CHAPTER III
METHOD ......................................................................................................................... 64
Methodological Framework ...................................................................... 65
Researcher’s Lens ..................................................................................... 66
Participant Selection ................................................................................. 68
Sessions Procedures .................................................................................. 69
Data Collection ......................................................................................... 7O

viii

 

Video Recording ....................................................................... 70

Parent Interviews. ..................................................................... 7O

Worksheets (Artifacts). ............................................................ 70

Personal Journals and Field Notes ............................................ 71

. Peer Debriefing ......................................................................... 71

Data Analysis ............................................................................................ 71

Trustworthiness ......................................................................................... 72
CHAPTER IV

DESCRIPTION OF MATHEMATICAL INTERVENTIONS, SONGS, AND PROPS. 74

Mathematical Interventions ................................................................................. 75

Coin Related Intervention ......................................................................... 75

Written Addition ....................................................................................... 76

Clock Related Interventions ...................................................................... 77

Lexicon of Mathematical Terms and Concepts ........................................ 77

Songs and Props .................................................................................................. 79

Songs ......................................................................................................... 79

How Many Pennies in a Quarter? ............................................. 8O

Nickel, dime, and quarter ......................................................... 81

Give Me a Quarter! The song ................................................... 82

Finding the bigger number ....................................................... 82

Nobody’s perfect by Hanna Montana and You can do it ......... 84

Props ....... '. ................................................................................................. 85

Coins ......................................................................................... 86

Coin hand props ........................................................................ 86

The coin-classification-chart .................................................... 87

Number flashcard. .................................................................... 89

Analog clock ............................................................................. 90

Address numbers for address plaque ........................................ 91

Summary ............................................................................................................. 91

CHAPTER V

THREE CASE STUDIES ................................................................................................. 92

Big Red Clown .................................................................................................... 93

Language ................................................................................................... 95

Finger Dexterity ........................................................................................ 97

Attention ................................................................................................... 98

Mathematics Exercise ............................................................................. 100

Coin discrimination. ............................................................... 101

Coin calculation ...................................................................... 103

Written addition ...................................................................... 110

Magnitude ............................................................................... 112

Address numbers for address plaque ...................................... l 14

Clock. ..................................................................................... 114

ix

 

 

 

 

Music ....................................................................................................... 1 16

Emotional Rollercoaster .......................................................................... 121

Roxie Montana .................................................................................................. 124

Finger Dexterity ...................................................................................... 126

Attention ................................................................................................. 126

Mathematical Exercises .......................................................................... 127

Coin discrimination ................................................................ 127

Coin calculation ...................................................................... 128

Written addition ...................................................................... 132

Sense of Numbers ................................................................... 134

2 fingers in right and 5 fingers in left for 25. ......................... 136

Clock. ..................................................................................... 137

Music ....................................................................................................... 138

Emotional Support .................................................................................. 143

Angela Hill ........................................................................................................ 145

Language ................................................................................................. 147

Finger Dexterity ...................................................................................... 148

Attention ................................................................................................. 149

Mathematical Exercises .......................................................................... 150

Coin discrimination ................................................................ 150

Coin calculation ...................................................................... 15 1

Coin versus written ................................................................. 155

Witten addition. ...................................................................... 157

Number buddies song ............................................................. 161

Sense of Numbers ................................................................... 163

Lack of strategies .................................................................... 164

Clock ...................................................................................... 165

Music ....................................................................................................... 166

Learned hopelessness and emotional support ......................................... 169
CHAPTER VI

ANALYSIS ..................................................................................................................... 171

Case Study Summary ........................................................................................ 172

Mathematical Interventions ............................................................................... 173

Coins ....................................................................................................... 1 74

Written addition ...................................................................................... 176

Magnitude and Cardinality ..................................................................... 178

Clock ....................................................................................................... 1 84

Fine Motor skills ............................................................................................... 187

Music ................................................................................................................. 1 8 8

 

CHAPTER VII

INTERPRETATION ....................................................................................................... 191
Roadblocks to Mathematical Development ...................................................... 192
Metacognition ......................................................................................... 192
Visuospatial Perception Difficulties ....................................................... 197
Attention Deficits and Slower Processing Speed .................................... 199
Fine Motor Skills .................................................................................... 200
Missing Developmental Concepts .......................................................... 201
Learned hopelessness .............................................................................. 205
Judgment: Music as a Therapeutic Medium Based on the Results of this Study’s
Interventions ...................................................................................................... 208
Music for Cognitive Development .......................................................... 208
Music as a Motivator and Encourager .................................................... 213
Music as a Structure Provider ................................................................. 216
Summary of Interpretation ................................................................................ 220
CHAPTER VIII
CONCLUSION ............................................................................................................... 222
Suggestions for Music Therapy Practice ........................................................... 222
Think Like Individuals with WS ............................................................. 222
What is the Most Crucial “Key Element” in Mathematical Development?
................................................................................................................. 225
Music Score Reading and Ear Training .................................................. 227
Choose Your Battles ............................................................................... 229
Miscellaneous Thoughts ......................................................................... 231
Recorded as opposed to live music ........................................ 231
Managing with coins. ............................................................. 231
Importance of props ................................................................ 233
Transcription of a session as an educational tool for music
therapy students whose first language is not English ............. 235
Attention and study room environment .................................. 236
Limitations and Other Considerations ............................................................... 236
Mathematical Difficulties are the Tip of the Iceberg .............................. 236
Fish is Fish .............................................................................................. 238
Future Research ................................................................................................. 240
The Mental Number Line ........................................................................ 240
Music Aptitude Tests and Musical Abilities ........................................... 242
Attention Deficit, Hyperacusis, and Odynacusis .................................... 244
Research Opportunities in the WS population ........................................ 246
Final Thought .................................................................................................... 247
EPILOGUE ..................................................................................................................... 252

xi

 

Appendix A

Song and Lyrics .............................................................................................................. 254
Figure A-l. Increments of 5 .................................................................... 254
Figure A-2. Number Buddy. ................................................................... 254
Figure A-3. Thumb and pinky meet each other. ..................................... 255

Appendix B

Final list of codes and themes ......................................................................................... 256

Appendix C

Table for Abbreviation .................................................................................................... 258

Appendix D

Example of Session Transcripts with Codes ................................................................... 259

Appendix E

Modified score for Angela and Roxy .............................................................................. 263

Appendix F

Handwriting Samples for Angela .................................................................................... 264

Appendix G

Example of Session Progress Chart ................................................................................ 265

Appendix H

Developmental Guidelines for Number and Operations* ............................................... 266

Appendix I

Props and Visual Material ............................................................................................... 278
Figure I-l. Number Chart ....................................................................... 279
Figure I-2. Coin classification chart ........................................................ 280
Figure I-3. Nickel Hand Prop .................................................................. 281
Figure I-4. Dime Hand Prop ................................................................... 281
Figure I-5. Quarter Hand Prop Pattern 1 ................................................. 282
Figure I-6. Quarter Hand Prop Pattern 2 ................................................. 283

xii

 

Appendix J

Pre-test of Angela, Roxy, and Big Red ........................................................................... 284
Angela ..................................................................................................... 284
Roxy ........................................................................................................ 285
Big Red ................................................................................................... 286

Appendix K

Mathematical Mistakes and Interesting Incidents ........................................................... 287
Big Red ................................................................................................... 287
Roxy ........................................................................................................ 292
Angela ..................................................................................................... 295

REFERENCES ............................................................................................................... 298

xiii

 

LIST OF TABLES

Table 1. Example of task breakdown for two-digit by two-digit multiplication .............. 46
Table 2. Major building blocks on the way to division ..................................................... 47
Table 3. Lexicon of mathematical terms and concepts ..................................................... 78
Table 4. The comparison between Angela and Big Red ................................................. 181
Table 5. Links between genetic syndromes and behavioral phenotypes ........................ 224

xiv

 

LIST OF FIGURES

Figure 1. Surmnary of major WS phenotypic features. .................................................... 15
Figure 2. Drawing based on characteristics of children with WS ..................................... 16
Figure 3. Drawing of two bicycles .................................................................................... 27
Figure 4. Model of phenotype to genotype map of Williams Syndrome. ......................... 36
Figure 5. Anatomy of WS’ Brain. ..................................................................................... 39
Figure 6. Limbic System in WS’ Brain. ............................................................................ 42
Figure 7. A simplified dual-store model of memory. ........................................................ 51
Figure 8. The initial cognitive map linking mathematics and music. ............................... 63
Figure 9. How Many Pennies in a Quarter? ...................................................................... 80
Figure 10. Nickel, Dime, and Quarter. .............................................................................. 81
Figure 11. Finding the Bigger Number. ............................................................................ 83
Figure 12. You Can Do It. ................................................................................................. 85
Figure 13. Nickel-hand-prop and Dime-hand-prop. ......................................................... 86
Figure 14. Quarter-hand-prop Pattern 1 & 2 ................................................ 87
Figure 15. Coin-classification-chart .................................................................................. 88
Figure 16. Example of number flashcard .......................................................................... 89
Figure 17. Example of analog clock. ................................................................................ 90
Figure 18. Example of address numbers. .......................................................................... 91
Figure 19. Normal, Tiny, Big. ......................................................................................... 102
Figure 20. Visual prop for five pennies make a nickel. .................................................. 104
Figure 21. Five Pennies Make a Nickel. ......................................................................... 105

XV

 

Figure 22. Three dime-hand-props and Big Red’s answers. ........................................... 108
Figure 23. Two dirne-hand-props and one nickel-hand-prop, and Big Red’s answers... 109

Figure 24. Two dime-hand-props and one nickel-hand-prop, and Big Red’s answers. 1 1 0

Figure 25. Big Red’s worksheet. ...................................................................................... 112
Figure 26. Address numbers. ........................................................................................... 114
Figure 27. Clock with tactile cues .................................................................................... 115
Figure 28. Four nickel-hand—props and one dime-hand-props and Roxie’s answer. ...... 129
Figure 29. One dime-hand-props and two nickel-hand-prop, and Roxie’s answer ......... 130
Figure 30. Roxie’s worksheet. ........................................................................................ 133
Figure 31. Quarter, dime, nickel, and penny improvisation. .......................................... 139
Figure 32. Roxie’s version Mary had a little lamb .......................................................... 140
Figure 33. Correct version in the key of C, Mary had a little lamb. ............................... 141
Figure 34. Intonation of Roxie’s “I keep for getting.” .................................................... 142
Figure 35. Hand gestures. ............................................................................................... 149
Figure 36. Two-digit addition formation ......................................................................... 153
Figure 37. Two-digit addition in horizontal and vertical formation. .............................. 156
Figure 38. Addition in the vertical formation. ................................................................ 159
Figure 39. 25 + 25 in the vertical format. ....................................................................... 160
Figure 40. “75 + 25” and “50+25” in the vertical format. .............................................. 161
Figure 41. Angela’s subtraction. ..................................................................................... 163
Figure 42. Number flash card examples. ........................................................................ 164
Figure 43. 25 + 5 in two different formats. ..................................................................... 195

Figure 44. Expected mathematical foundation with the imagination of the game J enga.
................................................................................................................................. 204

xvi

Figure 45. Perceived mathematical foundations after the project completion ................ 205

Figure 46. Music Therapy Theory Diagram. .................................................................. 208
Figure 47. Flowchart for Finding the Bigger Number Song ............................................ 211
Figure 48. Flowchart for coin calculation. ...................................................................... 212
Figure 49. Diagram of interpretation. ............................................................................. 221

Figure 50. A pentagon—shape building versus a triangular-pyramid-shaped building. 226

Figure 51. Big Red One More Dollar .............................................................................. 233
Figure 52. Hopscotch in a vertical line and a horizontal line. ........................................ 241
Figure 53. A keyboard instrument with numbers. ........................................................... 242
Figure 54. Think Before You Answer. ............................................................................ 244
Figure 55. The Fox and the Stork ................................................................................... 248

xvii

 

PROLOGUE

“Buying a Cup of Coffee”

July 2006: Richmond, Virginia

I attended the Williams Syndrome Conference in Richmond, Virginia, to gain a
better understanding of Williams Syndrome (WS) and to learn what current treatments
and interventions were available for individuals with WS. The Williams Syndrome
Association (WSA) National Conference is a family-oriented conference that was
designed to share information with parents as well as those who work with WS.
Individuals with WS, their family members, researchers, educators, therapists, and
volunteers for the conference inundated the hotel. The atmosphere was friendly, warm,
and busy, like a huge family reunion. I enjoyed talking with individuals with WS, and
meeting their parents and other therapists. Then it happened; I was waiting to buy a cup
of coffee at a Starbuck's coffee shop in the conference hotel just 10 minutes before an
afiemoon session began, and the man in front of me had WS and loose change. His cup of
coffee came to $3.72 and he was trying to count coins to pay for it. Anyone else would
have handed the cashier $4.00 and received change. Although I felt sympathetic to his
situation, I was in a rush and stuck in line behind him. I could sense the frustration of the
others waiting. It was at that moment that I discovered a way I might help individuals
with WS. As a music therapist, perhaps I can help them with one of the little life skills
that can make a difference in their acceptance in society. And so began my journey on the

road to discovering the real face behind Williams Syndrome.

CHAPTER I

INTRODUCTION

My journey with Williams Syndrome (WS) actually began 6 years before my
experience in Richmond, Virginia. My interest in WS was stimulated after watching a
television documentary about an individual with WS. Her name was Suzie, and she was
still non-verbal at the age of seven, had very limited cognitive function, and a feeding
tube directly connected to her stomach due to her inability to swallow. In the
documentary, her brother tapped a large can with a stick while he worked on his
homework near her. While be tapped, Suzie rolled in her bed with a euphoric smile on her
face; the narrator commented on how amazing it was to see someone this happy from
such a simple sound. From a music therapist’s point of view, this incident clearly
illustrated a sound-based fixation. Suzie’s reaction made me wonder if it might be
possible to use this fixation for therapeutic purposes by employing different sounds. It
also made me wish for a chance to work with her. Although I had no particular
knowledge about WS when I watched the documentary, that moment inspired me to learn
more about this disorder, and started me on my long journey of studying WS and music
therapy.

Williams Syndrome (WS) is caused by a microdeletion of at least 16 — 23 genes
on chromosome 7ql 1 .23. These missing genes cause similar physical characteristics and
physiological difficulties, including cardiovascular abnormalities. This subset of genes is

thought to be partially responsible for cognitive ability, linguistics, social interaction,

behavior, and visuospatial perception. The combined variances within these domains are
recognized as the phenotype of WS. While the medical description of this phenotype may
seem clear on paper, the reality is much more complicated when observing the related
phenomena. Those with WS have been a puzzle because of the inconsistencies in the
manifestation of their individual abilities and limitations. Semel and Rosner (2003)
illustrate this difficulty: “How is it possible to conceptualize a group of children who test
as though retarded, speak as though gifted, behave sometimes as though emotionally
disturbed, and fimction like the learning disabled?” (p. 1). Individuals with WS are
initially perceived to have standard or higher intelligence because of their exceptional
verbal skills. The well-known example in WS literature is “the elephant description” by a
15 years old girl with WS. “It has a long trunk that can pick up grass or pick up hay...
you don’t want an elephant as a pet, you want a cat or a dog or a bird.” (Full Scale IQ of
49, Verbal 1Q of 52, and performance IQ of 54) (Bellugi & George, 2001, p. xii). This
communication skill is illusory to some degree, but the apparent communication abilities
in individuals with WS can mislead individuals who are the parents or work with
individuals with WS into assuming a higher level of function than may actually be
present. In one moment, they seem competent, but in the next they reveal an astonishing
inability to handle mundane tasks, especially those involving mathematics or visuospatial
cognition. It is a profound experience to witness a 21 year old male, who communicates
well and appears to be “normal,” calculate the value of a quarter and two pennies as three
cents, rather than twenty-seven cents, and who cannot put the power plug into the wall
outlet. When children with WS are young—those who have not yet started elementary

school—their strength in language and communication give their family members hope

 

that they may be capable of functioning independently later in life. Unfortunately, this
strength in verbal skills is one small peak in the topographical map of their functional
domains. Their cognitive ability is usually more limited than first perceived and can lead
to strain and disappointment for families and caregivers. Individuals with WS exhibit
strengths in language, music, storytelling, sociability, and face processing, while they
exhibit weaknesses in mathematics, abstract concepts, and visuospatial cognition.

A well-known trait of people with WS is a pronounced difficulty with
mathematics. However, there is little detailed information about, and research into the
specific causes of these difficulties, including specific problem areas, average functional
levels, and possible coping mechanisms. Speculation on mathematical impairment
suggests that the difficulties are due to visuospatial deficits and abnormalities in the
structure of the parietal lobe (Ansari & Kannilofi-Smith, 2002; O'Heam & Landau, 2007;
Paterson, Girelli, Butterworth, & Karmiloff—Smith, 2006). Research on 28 subjects (14
with WS and 14 mental-age matched participants) using a standardized test showed that,
while individuals with WS have relatively strong verbal math skills, such as reading
numbers, the participants exhibited weaknesses related to spatial aspects such as the
mental number line and representations of magnitude. While the participants’
Chronological mean age was 17 years, 9 months (the oldest 38 years 10 months and the
youngest 10 years 5 months), their age-equivalency based on Test of Early Mathematical
Ability (TEMA-s) scores were 6.1 to 8.6). One participant was outside the age range for
the test i.e., his equivalent age was older than 8.11 (O'Hearn & Landau, 2007).

During the preparation of this study, I had the opportunity to interact with children with

special needs, individuals with WS, and typically developed individuals (TDI). During

 

these preparatory interactions, I experimented using various musical interventions to get a
sense of the range of responses for different functions and ages. While working with
Zach, an 11 year old who is developmentally delayed, we sang a song meant to promote

the use of addition.

How much is +
ldont,|dont

What should I do?
What should I do?
Let’s count together.
1, 2, 3, 4, 5,
That’s right. You are right!

 

 

Zach demonstrated proficiency with simple addition: 1 + 2, 3 + 5, 4 + 4, etc. After
working on these specific questions, I thought that “1 + 1” would be the easiest equation
for him to solver. However, when I held up two index fingers, Zach proudly announced,
“Eleven!” His answer demonstrated confusion between quantity and the actual symbol of
the Arabic numeral for “1 This incident indicated that the difficulties in mathematics for
individuals with learning disabilities, including WS, might be much more complex and
may be rooted at a deeper level than mere performance of addition and subtraction.
Information about the mathematical development of WS is limited and it was difficult to
predict what issues might manifest during experimental interaction. The incident with
Zach strongly suggested the need to study the mathematical development of young
children and review techniques used to teach mathematics to children with special needs.

Individuals with WS enjoy music, possess an affinity to music, or are described as
music lovers. Not all individuals with WS can play instruments or sing well, but each has

an affinity for music. A majority of individuals with WS show distinguished abilities in

music, such as melody memorization or rhythmic ability. Additionally, a higher
percentage of individuals with WS have demonstrated absolute pitch than in the general
population (Lenhoff, Perales, & Hickok, 2001; Semel & Rosner, 2003). Although many

individuals with WS cannot learn to read and write music because of their cognitive

 

limitations and visuospatial difficulties, they can sing, compose, improvise, and play
musical instruments, especially percussion and keyboard-based instruments. At music
camps for individuals with WS, such as Whispering Trails Camp in Grand Rapids,
Michigan, and Berkshire Hills Music Academy Summer Camp in South Hadley,
Massachusetts, there are usually more than enough dmmmers, keyboard players,
guitarists, and singers to organize two or three bands. Because of this musical
proficiency, the recreational and occasionally vocational use of these abilities has been
strongly supported by parents and advocates. However, limited attention is given to their
affection for, and affinity to music as a means for assisting in the intervention of specific
limitations.

There are a variety of resources and materials used to teach academic concepts to
children with special needs, the greatest concentration of which are from preschool
through second grade. However, few materials exist to help children with cognitive
limitations when they reach their preteen and teen years. While many children with WS
do not progress beyond a certain level of cognitive function, their physical and emotional
development continues, and age-appropriate educational material may be the first key for
successful intervention. At a mathematics workshop for children with WS at the WSA
conference, one presenter introduced hands-on materials to teach mathematical concepts,

specifically the “touch” method for counting. These materials included illustrations that

 

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were “cute and cuddly”—directed towards younger users. One mother in attendance
responded that her son would not use these materials because he would perceive them as
insulting: they were designed for younger children.1 This strongly suggests that the use of
age-appropriate learning materials is imperative in order to respect individuals with WS’
sense of self-esteem and to reinforce cognitive development without jeopardizing
confidence. Music therapy literature emphasizes the use of age-appropriate songs and
materials, but this is often difficult to put into practice in cases with cognitively
challenged populations. Individuals with WS are especially sensitive to others and keenly
aware of their limitations, and reactions from the public to their limitations. Developing
age—appropriate materials for individuals with WS within their limited cognitive function
would be challenging but highly rewarding.
A number of examples exist that demonstrate the use of sounds and music to

stimulate action or as an educational aid, especially with pre-Kindergarten children.

J ingling keys have been used to attract a toddler’s attention and can promote crawling or
walking; modified song lyrics enhance the memorization of the spelling of last names,
phone numbers, and addresses; and nursery rhymes are a way to advance language and
speech development (Sforza, 2006). Unfortunately, the use of music as a teaching tool
decreases or stops during the early elementary school years, somewhere between 8 and
10 years of age, when academic concepts became more complicated and children start to
develop their own strategies. The utility of music as a mnemonic and teaching tool

decreases as children advance through elementary school. Because of the lack of

 

I The mother who sat next me said to herself, “Charley wouldn ’t even touch those things. " After the
workshop, I asked her how old her son was and the reason for her statement. She explained that Charley
was 12 years old, and even if he could not do simple addition, he did not want to use workbooks with
‘childish’ illustrations or kids stuff, and he was highly sensitive about using the same material as his
younger sister.

 

research, theory, and practice with regard to the WS population, music therapists, as well
as educators, are not quite certain of the most effective approach to facilitate cognitive
development for individuals with WS.

The efforts and practice of music therapy deal with a broad range of
psychological, emotional, and physical issues. However, the existing body ,of research
used for children with WS was initially developed for children with special needs. The
current methods and practices used for WS are taken directly from this prior research, but
they do not address the specific needs of individuals with WS in relation to their
cognitive development and neurology. According to Duerksen (1968), research, theory,
and practice should stand together like a tripod. Theory is needed to explain existing
phenomena, research must validate or disprove theory, and methods must be developed
from validated theory to be used in practice. The outcomes of practice suggest
modifications and revisions to existing theory and the development of new ideas
(Wheeler, 2005). Thus far, methods used with the WS population are based on music
therapy directed towards individuals with cognitive challenges. Work with individuals
with WS indicates that the traditional methods used for cognitive impairment do not
correspond to the expected outcomes for clients with WS. Because of the unique
characteristics of individuals with WS, music therapy interventions require new research.
in order to develop more effective methods to increase positive outcomes.

WS research has expanded and intensified in the past two decades, especially
since the diagnostic method for WS was discovered in 1993. However, there remain
many questions regarding this syndrome. Information about WS has continued to evolve

and grow in direct relation to the technological developments of neurology and genetic

research (Martens, Wilson, & Reutens, 2008; Mervis & Morris, 2007). Theories of the
various characteristics of WS have been refined, challenging previous theories and
provoking more debate and discussion. A review of the current literature on WS focuses
primarily on a single characteristic or domain such as particular language characteristics,
visuospatial characteristics, or attention difficulties. The literature seldom relates these
characteristics across domains. While it is indispensable to have focused research on each
of the domains, it is essential to connect these issues because individuals with WS are
more than the sum of these individual parts. Results of this diverse research must be
integrated and a holistic approach taken when considering the experiences of those with
WS. The problems those with WS experience with mathematics may not be solely based
on cognitive difficulties. They may be the result of interwoven factors that include
fimctional areas such as language, visuospatial relations, or the interplay fiom additional
emotional and psychological issues.

My undergraduate training in mathematics led me to work for several years as a
computer programmer. The logic and structure of algorithms and code drew me to
programming in much the same way that the structure of musical notation and sequences
underlie my love of music. Eventually I discovered that music blended with my tendency
to want to help others and I began to pursue an education in the field of music therapy.

Before I learned details about WS, I was already falling in love with these
individuals because of their affection for music. I was amazed by their fascination with
sound and the way I could use music to communicate with them. During the earliest
sessions with my first WS client, I felt like a music teacher who had stumbled upon a

musical prodigy. That client only had to listen to a song or selection a couple of times

before he could repeat it back exactly. He could imitate complicated rhythms and patterns
and maintain a musical dialogue with me, something of which few adult musicians are
capable—and he was only 5 years old.

Several other clients with WS have demonstrated similar talents in various
degrees. They were entranced by new sounds and inStruments and mesmerized by the
sounds of finger cymbals, a gong, a rain stick, and a thumb piano. My limited piano
playing was enough to enthrall them just like a world-famous pianist engages their
audiences. Music makes these individuals happy. In fact, when music is involved, their
behavior significantly improves. When I became more deeply involved with individuals
with WS and spent more time with them outside of sessions, such as working at the music
camp for individuals with WS, and attending their conferences and gatherings, I
witnessed the desperation, frustration, and agony that are often part of their daily lives.
Those behaviors do not typically emerge while individuals are happy and content. During
musical therapy sessions when the clients are enveloped by music, they seldom showed
fi'ustrations and limitations such as uncontrollable tamper-tantrums, emotional roller-
coasters, and obsessive-compulsive behaviors.

“They are smart enough to know that they are different,” said Dr. Barbara Pober,
director of Williams Syndrome Clinic at Massachusetts General Hospital in a TV
documentary about WS (CBS, 1997). One of the statements repeated during the
documentary was that persons with WS did not feel that they “fit in.” When they are with
other individuals with WS, they feel a sense of belonging and can relate to and
understand each other. There is a sense of hopelessness, ‘there is nothing I can do to

improve this condition,’ or, ‘I tried this and that, and nothing has been working.’ These

10

 

 

are devastating feelings. Perhaps the love of music in individuals with WS can be used to
augment their ability in mathematics, and be a significant breakthrough that could make
their lives and their parents’ lives more manageable.

Today’s materialistic and product-oriented society does not always comprehend
individuals with WS’ inabilities in certain areas. I can empathize with the day-to-day
interactions and prejudices someone with WS might experience. English is not my
mother tongue and I have found over the past ten years in the United States that this can
cause a range of negative responses. I am not perceived as foreign, necessarily, but as
“language-impaired” and this sometimes translates into being treated as ignorant or as
less than competent, something a person with WS experiences intimately. The fact that I
am normal in my native Korean society does not help me feel any better. Shifting fiom
my linguistic challenges to individuals with WS’ mathematic obstacles, I might claim that
I have experienced something similar because of the perceived lack of ability, allowing
me to relate to how they might feel in their daily lives.

This study’s major focus was to observe the relationship between teaching
mathematical concepts and music-related activities. The mathematical difficulties of
clients with WS were compared with a range of ages provided through the literature
review. No specific research exists, but the literature suggests that roadblocks for the
mathematical development of individuals with WS may include language limitations,
difficulties with abstract concepts, attention deficits, visuospatial difficulties, and fine
motor control problems. During the course of the study, these factors were considered for
each of the participants and compared to existing literature. The purpose of this study is

to better understand mathematical difficulties within individuals with Williams Syndrome

11

 

and to find ways to use their affinity for music to compensate for mathematical deficits.

An instrumental collective case study method was chosen for these purposes. The
close examination was performed with three participants to provide detailed and in-depth
information of specific problems with mathematics. During the course of this study, three
questions guided my investigation: What are the roadblocks in the mathematical
development of individuals with WS? What can they do and not do mathematically?;
Based on existing music therapy theories for individuals with special needs as described
in music as a therapeutic medium, are there some possible music therapy interventions
that can serve as strategies to help individuals with WS in terms of their mathematical
development?; and How do individuals with WS respond to music therapy interventions
to overcome their mathematical learning obstacles? Are these interventions useful or

effective for their needs?

12

CHAPTER II

REVIEW OF RELATED RESEARCH

In the previous chapter, the general descriptions of individuals with Williams
Syndrome (W S) and reasons for conducting research on this issue were discussed. In this
chapter, the discussion focuses on more detailed information about WS characteristics,
especially those related to genetic and neurobiological factors, mathematical development
in the general population and WS population, current music therapy theory for

individuals with cognitive challenges, and parents’ concerns in WS education.

Descriptions of Williams Syndrome

WS was first described by Fanconi, Giradet, Schlesinger, Butler, and Blade (1952)
based on their observations of patients with infantile hypercalcaemia with short stature
and congenital malformations (Bellugi, Lichtenberger, Jones, 2., & St. George, 2001;
Mervis & Morris, 2007; Morris, 2006; Semel & Rosner, 2003). WS is named after J. C.
R. Williams, a British cardiologist in New Zealand, who described a group of children
with heart defects (supravalvular aortic stenosis, SAVS), cognitive impairment, and
“pixie-like” or “elfin-like” dysmorphic facial features in 1961 (Bellugi et a1., 2001;
Mervis & Morris, 2007; Morris, 2006). In Germany, Beuren and colleagues also
separately identified the syndrome based on SAVS, peripheral pulmonary stenosis,
mental retardation, and similar facial appearance in 1962. The label Williams-Beuren

Syndrome is the term primarily used in Europe (Morris, 2006). The etiology and

13

 

 

epidemiology of WS has puzzled experts. The causes of WS range from a vitamin D
deficiency or calcium teratogenesis in early-years (Friedman & Roberts, 1966), to
chromosome anomalies (Burn, 1986), and finally to the missing elastin gene (ELN) in
chromosome 7 identified as the causative gene for SVAS (Ewart et al., 1993; Lowery et
al., 1995; Morris, Loker, Ensing, & Stock, 1993). When the diagnostic method for WS
was defined in 1993, scientists were excited to be able to end the many years of
speculation about the etiology of WS and about the possibility of focusing on this genetic
disorder to study molecular genetics profiles and neuropathology (Bellugi et al., 2001;
Semel & Rosner, 2003). Information about WS has significantly and rapidly changed
based on increasing research.

WS is currently defined as a disorder that arises from a hemizygous deletion on
the long arm of chromosome 7q.11.23 (Donnai & Karmiloff-Smith, 2000; Morris, 2006).
The deletion is approximately 1.3 Mb and includes at least twenty-three genes (Partsch et
al., 1999). WS is associated with “elfin-like” facial features and physical appearance,
heart and blood vessel problems, infantile hypercalcaemia, growth and developmental
retardation, and difficulties in gross motor and fine motor skills to various degrees
(Bellugi et al., 2001; Mervis & Morris, 2007; Semel & Rosner, 2003). The cognitive

profile of individuals with WS exhibits complex patterns. WS literature uses terms such

,, l.‘ 9, ‘6

as “strengths and weaknesses, peaks and valleys, puzzle,” and “uneven” to describe
the WS’ cognitive profile (Lacro & Smoot, 2006; Semel & Rosner, 2003). Individuals
with WS show salient abilities in phonological skills, storytelling, music, face
recognition, and sociability. In contrast, individuals with WS display uneven skills within

language ability, difficulties in visuospatial abilities, deviations in the sensory system,

14

 

and unusual social behavior (Mervis & Becerra, 2007). Figure 1 illustrates an overview
of major characteristics of individuals with WS. These characteristics will be discussed in

detail in the following chapter.

 

 

 

Neurological Neurocognitive F acies
0 Average IQ o Friendly, loquacious o Medial eyebrow flare

(range 40'80) personality 0 Short nasal palpebral fissures:
0 Mild neurological dysfunction 0 Enhanced musical ability epicanthal folds

:Ecirili'tclbiicficfiraciion o Relatively spared language 0 Flat nasal bridge\

. development, enhanced . St ellate iris
0 Hyperacusrs vocabulary, and socral use
0 Harsh, brassy or hoarse voice of language compared to . Long philtrum
Visual-spatial perception o Prominent lips with open mouth
Cardiovascular Musculoskeletal
0 Supravalvular aortic stenosis 0 Joint limitations
0 Peripheral pulmonary artery stenosis 0 Kyphoscoliosis
0 Pulmonic valvular stenosis 0 Hallux valgus
0 Ventricular/atrial septal defects 0 Hypoplastic nails
Genitourinary Endocrine
0 Nephrocalcinosis 0 Transient infantile
hypercalcemia

0 Small, solitary, and /or pelvic kidneys

o Vesicouretal reflus

 

Figure 1. Summary of major WS phenotypic features. *

*From “Genome Structure and Cognitive Map of Williams Syndrome” by Kornberg et al., Journal of
Cognitive Neuroscience 12(90001), p. 90. Copyright 2000 by the MIT. Reproduced with permission.

The prevalence of individuals with WS in the general population was changed
from 125,000 to 1:50,000 during the early 19805, and to 127,500 in 2007 (Morris, 2006;
Semel & Rosner, 2003). Journal articles published in 2007 and 2008 raised questions and
provoked discussions about previous articles, sometimes suggesting findings

contradictory to previous research. Articles about WS published in the 19905 are now

15

 

considered out of date, although those from as early as the 19803 are presented here as

part of the history of WS research.

Physical Appearances and Medical Issues

The characteristic facial appearances of those with WS have often been described
as “pixie-like” or “elfin-like.” This is typically characterized by a broad forehead,
puffiness around the eyes, full cheeks, a wide mouth, full lips, small chin, a small
upturned nose, long philtrurn (upper lip length) and large ears across different races
(Monis, 2006; Semel & Rosner, 2003). The drawings of young children with WS (see
Figure 2) are constructed based on 12 pictures of children with WS and emphasize these

characteristics.

 

 

 

 

 

Figure 2. Drawing based on characteristics of children with WS.

Illustration by Hyang Eun Lee

16

The majority of individuals with WS are subject to numerous health problems
including cardiovascular abnormality in various degree, hypercalcaemia, hernias, kidney
abnormalities, and musculoskeletal abnormalities (wain & Yule, 1990). The spectrum
of cardiovascular manifestations plays an important role in the recognition of a group of
individuals with similar condition and leads physicians to the etiology of WS as a
molecular genetic abnormality. Approximately 50 % to 75% of individuals with WS are
clinically detected with supravalvular aortic and peripheral pulrnonic artery stenosis
(Semel & Rosner, 2003).

Children with WS appear younger than they are because of their short stature and
infantile facial characters, such as full cheeks. As they age, adults with WS look older
than their chronological age because of a lack of subcutaneous tissues, which gives them
lax facial skin and a prematurely aged appearance. Their body posture also contributes to
this appearance; long necks, sloping shoulders, deformity of the spine (kyphosis and
lordosis), and limitations of the joints (Semel & Rosner, 2003). When a group of
individuals with WS gathers, they seem to be missing those in their 205 or early 308. The
premature looks and the mean adult height (165.2 i 10.9 cm/5 .41 i- 0.35 feet in males
and 152.4 i 5.7 cur/4.98 i 0.18 feet in females) contributes to this appearance (Bellugi et
al., 2001). There has been some speculation that individuals with WS may have been the
inspiration or prototype for elves and pixie in folktales because of these characteristics
and because folklore presents elves as energetic, as well as music lovers (Lenhoff et al.,
2001; Lenhoff, Wang, Greenberg, & Bellugi, 1997). While there is no way to substantiate

such speculation, it does lend itself to an interesting hypothesis.

l7

Several medical conditions are common among individuals with WS. Eighty
percent of children with WS have cardiovascular disease, including supravalvular aortic
stenosis (SVAS), and/ or high blood pressure (Mervis & Morris, 2007). Fifieen percent of
infants were diagnosed with hypercalcaemia, and 40% of infants needed to have surgery
for inguinal hernia. Thirty-five percent of individuals with WS have been reported to
have structural urinary anomalies by renal ultrasound examination (Mervis & Klein-
Tasman, 2000). Hypercalcaemia and hernias improve during childhood, but
cardiovascular and kidney abnormalities become more problematic as they age (Bellugi
et al., 2001; Semel & Rosner, 2003). Many of these medical difficulties have mild
manifestations. However, in some cases they become life threatening, especially with
SVAS, which occasionally requires open heart bypass surgery (Bellugi et al., 2001;
Mervis & Monis, 2007; Semel & Rosner, 2003).

While young children with WS often have low muscle tone and joint laxity, as
they age, contractures (joint stiffness) may develop and physical therapy is recommended
to improve muscle tone, strength, and range of motion. When these individuals walk, they
raise their shoulders, throw out their chests, and have stiff movement in their arms and
legs. As toddlers, their movements are cute and adorable; however, as they grow, the
awkwardness in their movements becomes more noticeable. This awkward gait is typical
of older children and adults with WS. Hypertonic and hyperactive deep tendon reflexes,
kyphosis, and lordosis are typical symptoms in individuals with WS. Combined with the
spatial cognition defect, children with WS often have difficulties with fine motor skills.
As a result, self-care skills involving fine motor control are affected. Personal grooming,

manipulating zippers, snaps, and buttons, tool-use (scissors, knife, utensils), and writing

18

 

and drawing skills are all limited to some extent. These difficulties become more obvious

as they grow and attempt to take care of themselves (Semel & Rosner, 2003).

Language Characteristic

Despite their cognitive challenges, the linguistic ability of many individuals with
WS tends toward an appearance of precociousness. While researchers generally agree
with this statement, there are different abilities within various language areas, e. g.,
semantic, syntax, and pragmatic, as well as within different age groups (Bellugi et al.,
2001; Mervis & Becerra, 2007; Semel & Rosner, 2003). Some researchers strongly
believe that individuals with WS’ extraordinary language ability is evidence of the
separation of language and cognition (Bellugi et al., 2001; Semel & Rosner, 2003).
Meanwhile, others, more recently, claim that even though WS individuals showed better
semantic skills, their language ability is affected by cognitive limitations such as
difficulties in abstract concepts and the apprehension of spatially related propositions
such as “between,” “in,” “on”; “above,” “below” (Brock, 2007; Mervis & Becerra, 2007;
Peregrine, Rowe, & Mervis, 2006).

Adults with WS typically can achieve advanced language development, which is
in contrast to the majority of toddlers with WS, who show difficulties in the initial stages
of language development (Semel & Rosner, 2003). While typically developing children
(TDC) exhibit two-word utterances at age two, children with WS usually exhibit two-
word utterances at age three (Semel & Rosner, 2003). While children with Down
Syndrome (DS) tend to plateau after the onset of their first words, children with WS

improve dramatically in language development as grammar awareness emerges (Dykens

l9

& Rosner, 2006). Vocabulary acquisition and grammatical development of individuals
with WS appear to be positively correlated to auditory rote memorization ability (Rice,
Warren, & Betz, 2005).

Considering their level of cognitive ability, children with WS seem to have
impressive linguistic capabilities, especially with vocabulary. When interacting with
individuals with WS, one’s initial impression is that the individual with WS has average
or greater abilities with spoken language. However, there are peaks and valleys among
adolescents and adults with WS with regard to language. Adolescents and adults with WS
are usually talkative and have superficial sociability described as “cocktail party speech”
(F idler, Lawson, & Hodapp, 2003). However, when people talk with individuals with WS
long enough, it is noticeable that there is something awkward in their speech (Semel &
Rosner, 2003). Affected individuals have a difficult time maintaining the topic of the
conversation. They float in and out of the topic or jump to other topics often overusing
word fillers, cliches, and stereotyped phrases (Reiss et al., 2001; Semel & Rosner, 2003).
Little data exists on the pragmatic skills of individuals with WS (Howlin, Davies, &
wain, 1998; Mervis & Becerra, 2007; Mervis & Klein-Tasman, 2000; Semel & Rosner,
2003). Studies have shown they have various limitations in comprehending semantics,
syntax, abstract concepts, spatial relations, and figurative language (Levitin et al., 2003).
In general, it can be concluded that individuals with WS are proficient in imitating
sounds, such as those used in articulation, storytelling, narration, prosody, rhythm, and
lexical stress in speech, but they have difficulties with the semantic relational
vocabularies, complex syntax, abstract semantic concepts, and the social, communicative

use of language.

20

Semel and Rosner (2003) suggest that mediational strategies can promote a more
balanced language deve10pment. Mediational strategies include the subcategories of
labeling, rehearsal, demonstration, dramatization, and self-instruction. Mediational
strategies may help to develop more adequate language skills by increasing attention
span, assisting in the comprehension of abstract concepts, and promoting self-regulation

procedures.

Social Characteristics

Individuals with WS are friendly, outgoing, and sensitive to the feelings of others
(Semel & Rosner, 2003). Their behavior is often described as “highly sociable,” “never
going unnoticed in a group,” “highly approachable,” and “overly friendly, highly
empathic” (OT-learn & Landau, 2007). Therapists and medical personnel commonly
describe individuals with WS as overly friendly, which is considered typical WS
problematic behavior (Levitin & Bellugi, 2006). A therapist who worked with children
with WS reported an anecdote about a boy who was scolded because of his conversation
with a stranger. The boy replied to his mother, “But he is not a stranger; because he said
‘Hi ’to me.” Greeting everybody in a hallway at school or other settings is not uncommon
in children with WS. However, it can be problematic, especially in public settings. This
characteristic increases risks of exploitation, abuse, difficulty initiating and maintaining
fiiendships, and may result in older individuals with WS losing a job. The strong desire to
be everyone’s fiiend might lead to poor social judgment and make them vulnerable to
abuse. It was reported that over 90% of children with WS showed “no fear of strangers”

(Semel & Rosner, 2003) and 100% of employers of adolescents with WS (n=21)

21

mentioned the problem of overfiiendliness as a major concern (Davis, Howlin, & wain,
1997). When I attended the Williams Syndrome Conference for families in Richmond,
Virginia, I knew no one from the association, but I never felt lost, isolated, or out of place
because of the warm welcomes, greetings, and open conversations from those with WS.
They acted like they knew me and introduced me to their parents, in spite of parents’
wishes to keep them out of conversations with strangers. Parents are not likely to be
successful with such a request.

Empathy is the other salient social characteristic of individuals with WS. Affected
individuals typically tune into others’ discomfort or distress making it difficult to conceal
our feelings. When they feel the discomfort of others, affected individuals become
nervous and uncomfortable. They often want to talk about it. In society, even if we
suspect someone’s discomfort, unless the person specifically asks help, TDC do not
intervene. However, individuals vvith WS, because of their fearless nature and
overfriendliness, often approach persons in distress and ask what is wrong with them. It is
not difficult to find an anecdote about their empathy: When a ten year old boy with WS at
the William Syndrome Conference found somebody in the lobby at the hotel who looked
depressed and agitated, he approached her and asked, “Are you OK? Are you OK?” with
filll expressions and physical gestures. Individuals with WS’ naturally empathetic virtues
can be socially appropriate and in many cases need to be praised and reinforced. At the
same time, their expressions of empathy need to be shaped to avoid unnecessary or
unwanted interference in public settings and limit potential exploitation by others.

The intensity of fear and anxiety in individuals with WS vary from being on edge,

uneasy, or worried, to phobias or panic (Scheiber, 2000; Semel & Rosner, 2003). I

22

 

observed a 21 year old camper at the Williams Syndrome summer music camp in Grand
Rapids, Michigan who could not go to sleep by himself in his cabin when other cabin
members were not there. Even though there was another cabin a few steps away, a staff
member had to stay with him to calm him down. The boy was in and out of his cabin
every few minutes looking for signs of his cabin-mates’ return. He was unable to calm
down and go to sleep. Even though fears and anxieties during childhood are common in
the human developmental process, individuals with WS exhibit unusually persistent and

intense reactions far beyond their childhood years.

Behavioral Domain

Observations and research reports on the behavioral domain in individuals with
WS suggest there are six major problematic areas: fears and anxieties, distractibility and
attention problems, impulsiveness, poor adaptability, low frustration tolerance, and
atypical activity (Dykens, 2003; Dykens & Rosner, 1999; Dykens & Rosner, 2006; Semel
& Rosner, 2003). The results from the studies using the Child Behavior Checklist or
similar measures for adults have indicated that a high number of individuals with WS
have significantly elevated problematic scores (Mervis & Klein-Tasman, 2000).
Comparison research with TDC, children with Prader-Willi Syndrome (PWS), or children
with mental retardation of mixed etiology suggest that children with WS exhibit greater
activity, intensity, and distractibility (Dykens & Rosner, 2006; Mervis & Klein-Tasman,
2000). Additionally, children with WS display increased negative mood and self-image,

and decreased persistence, adaptability, and threshold to arousal.

23

When compared with other types of mental retardation, severe aggressive
behaviors (destroying objects, attacking others, and extreme temper tantrums) are less
common in children with WS (Gosch & Pankau 1994; wain & Yule, 1990; Dykens &
Rosner, 2006). According to Gosch and Pankau (1997), while 19% of adults with WS
(twenty-seven adults, aged twenty years and older) were unable to sit still, 62% of
children with WS showed restlessness and difficulty sitting still. The fiequency and
intensity of hyperactivity and aggression decrease as they age. However, distractibility
remains the same across ages (90% of informants) (Davies et a1. 1998). Einfeld, Tonge,
and Rees (2001) conducted a longitudinal study of maladaptive behavior in children with
WS and concluded that observation of “overactivity” or “tantrums” indicates reduction in
those areas over a five-year period. This is an indication of gradual reduction of
aggressive behaviors during the childhood to adolescent years.

Semel and Rosner (2003) concluded that high reactivity and low self—regulation
are the two major elements underlying those symptoms. They claimed that high reactivity
caused overreaction to various stimuli. As a result, individuals with WS’ fears, anxieties,
distractibility, and irnpulsivity were manifested. Low self-regulation tendencies cause
individuals with WS to have difficulties handling or overcoming unexpected and
challenging situations. Consequently, individuals with WS showed poor adaptability and
low frustration tolerance. Atypical activities could be confused with obsessive
compulsive disorder. For example, afflicted individuals may have a fixation about cars,
an insect, or a TV show, and they need to know everything about it. Every conversation
will revolve around that topic. Their atypical activity is the manifestation of a lack of

self-regulation that hinders their ability to restrain or withhold unessential forms of

24

 

activities. In addition, these two elements are interwoven in their impact on individuals
with WS’ behavior phenotype.

When Big Red, a 20-year old male with Williams Syndrome, did a presentation
about WS with his music therapist for an audience of music therapist at the Great Lakes
Region Conference of the American Music Therapy Association (AMTA) in Novi,
Michigan in 2006, he was supposed to talk about his journey with music and how music
helped him overcome his difficulties. Outside of the window, there was a dove that made
noise and walked along with the window sill. Big Red could focus his attention on the
task. He visually followed the dove and tried to talk at the same time, but could not do
both. When the music therapist whispered his name, he told us, “I’m supposed to pay
attention to my task.” He could keep to his speech for a few seconds but then he started
following the dove again. This incident demonstrated how the two elements are
connected. High reactivity caused him to focus on the dove, which was an irrelevant and
unessential stimulus at that moment. He could not take the necessary steps to control his
reactions to the dove because of low self-regulation.

Distractibility and attention issues are persistent throughout life and a major cause
of behavioral disturbance in individuals with WS. These characteristics can interfere with
the learning process and hinder achievement. Distractibility is speculated to be caused by
different perceptual processes such as auditory hypersensitivity, visuospatial difficulties,
and tactile defensiveness (Bellugi & George, 2001; Mervis & Morris, 2007; Semel &
Rosner, 2003). As illustrated in the aforementioned case, 97% of parents with children
with WS report difficulty in their children’s ability to pay attention, and 88% of parents

report their child pays attention to irrelevant stimuli. However, individuals with WS’

25

abilities to concentrate vary depending on the task. If the task is of special interest , their
attention span and concentration level is higher. It is possible that if we can use their
abilities in music and combine it with other difficult areas like mathematics; they might
be able to focus better. This ability to focus is a major premises of this project.

Semel and Rosner (2003) interpret individuals with WS’ impulsivity as, “an
outward manifestation of the internal, uncontrolled distractions and the uninhibited,
unrestrained patterns of WS behavior” (p.259). In TDC, the ability to engage impulse
control usually improves gradually during childhood. However, in individuals with WS,
something hinders this process, and their ability to control impulses is seriously delayed
or impaired.

Individuals with WS demonstrate great difficulty adjusting to unanticipated
changes in various aspects of their lives such as schedules at school, routines at home,
cancellations of appointments, and running out of supplies can lead to uncontrollable
temper tantrums. As mentioned above, the difficulties seem to stem from high reactivity
and low self-regulation. Dykens and Rosner (1999) suggest that reducing environmental
stimuli and introducing behavioral modification techniques, such as ignoring and
redirecting off—task behaviors, and rewarding on-task behaviors, can be usefirl
interventions. A verbal mediation strategy was found to be very helpful in managing fear
and anxiety as well as distractibility. The strategy involves a discussion of the strategies
to overcome difficulties by using labels, rehearsal, demonstration, dramatization, and

self-instruction (Semel & Rosner, 2003).

26

Visuospatial Abilities

Weakness in visuospatial ability is repeatedly reported in WS literature (Bellugi
& George, 2001; Mervis & Morris, 2007; Semel & Rosner, 2003). Individuals with WS
display difficulties in visuospatial construction, which denotes the ability to perceive
visual and spatial information of an object or picture and to perform a reproducing task
(e. g., a pattern reconstruction based on visual representation, drawing, or completing a
jigsaw puzzle). Early research by Bellugi and her colleagues (Bellugi et al., 2001;
Bellugi, Marks, Bihrle, & Sabo, 1988; Bellugi, Wang, & Jemigan, 1994) reported that
individuals with WS have difficulties drawing an object. Instead of drawing an outline of
familiar objects, such as a flower, bicycle, or elephant, they draw a part of the object

without particular spatial relationship of the object to the rest of the space.

 

wheel

 

‘70

wheel

Figure 3. Drawing of two bicycles. *

Two bicycles drawn by the same girl with Williams syndrome, at age 5 years 9 months (left) and 10 years 9
months (right). Both times, the child was given a blank piece of paper and asked to draw the best bicycle
she could. The labels on the left drawing were provided spontaneously by the child.

‘From “Williams syndrome” by Mervis and Morris (2007) in Mazzocco & Ross (Eds), Neurogenetic

Developmental Disorders: Variation of Manifestation in Childhood. Copyright 2007 by the MIT press.
Reproduced with permission.

27

They concluded that persons with WS are local processors rather than global
processors. Recent articles (Bertrand, 1997; Bertrand & Mervis, 1996) now challenge the
preceding theory concerning WS’s drawing ability. Based on their four-year longitudinal
study, while persons with WS are significantly delayed in this area, they obviously follow
the same developmental path as children with typical development (TDC) (see Figure 3).

In the DAS pattern test, 88% of the subjects belonged in the first percentile of the
T-score, approximately 50% of the subjects stayed in the lowest possible score, and only
10% were in the normal range of the pattern construction (Mervis et al., 1999). With
WISC-R subtest, 6 subjects (10-17 year olds) completely failed the test. It was even
difficult for them to maintain the overall 2 X 2 arrangement of the blocks (Bellugi et al.,
1988). Mervis and Morris (Mervis & Morris, 2007) concluded that the difficulty with
block construction appeared to be due to spatial representation difficulties. Construction
ability seems to improve over time.

The difficulties in visuospatial ability hinder the performance of everyday
activities: using tools, scissors, knives, tying Shoelaces, copying simple lines or figures
(which is related to hand writing), or plugging a cord into outlet. These activities involve
processing visual information, calculating distances, and executing tasks. Sports activities
require visually tracking balls or other moving objects and estimating trajectory.
According to a Utah Survey (Semel & Rosner, 1991), parents report that 97% of children
with WS have difficulties in drawing, 89% in cutting with a knife, 77% with personal
grooming, and 77% using scissors. While fine motor skills that are required to perform
such task are receptively preserved, visual perceptual problems are responsible for

difficulties in completion.

Visuospatial concept difficulties in the WS population are suspected to also be a
factor in mathematical abilities (Ansari et al., 2003; Ansari & Karmiloff—Smith, 2002;
O'Heam & Landau, 2007). However, the exact mechanism of interference with
mathematics performance is still not fully understood. Visual aids are often suggested for
individuals with cognitive impairment, but because of their difficulties in visual

information processing, they might not be helpful for individuals with WS.

Afiinity to Music

The musical ability of those with WS is often exaggerated words such as
wonderful, amazing, extremely good, fascinating, and extraordinary (Semel & Rosner,
2003). Musicality is defined as the “sensitivity to, knowledge of, or talent for music,
and/or the quality or state of being musical” (Merriam-Webster Inc., 2003). Ansdell, a
British music therapist, expresses musicality as “a quality which involves both ‘musical
thinking’ (or ‘know-how’) and ‘musical feeling’” (1995, p. 210). The Williams Syndrome
Association (WSA) uses the term, “affinity to music” to describe the magnetic attraction
individuals with WS have to music: an attraction defined as “a natural attraction, liking,
or feeling of kinship” (J arrold, Baddeley, & Hewes, 1999; Klein & Mervis, 1999). This
study will use a description of musicality in individuals with WS, to also include the
affective bonds of music.

There is ample evidence of individuals with WS exhibiting a high degree of
musical afl'rnity. A 1991 WSA survey of parents reported that 86% of children with WS
easily memorized songs, 71 % had a musical talent, and 87% liked to sing (cited in Semel

& Rosner, 2003). In a similar study, more than 90 % of children with WS reported that

they loved music (cited in Semel & Rosner, 2003). Parents, clinicians, and educators who
work with individuals with WS continuously report the latent musical talent of many
individuals with WS (Mervis & Klein-Tasman, 2000). The most nationally known
musician with WS might be Gloria Lenhoff, born February, 11, 1955 in the book, the
Strangest Song (Sforza, 2006). She is a lyric s0prano and accordionist and has Williams
Syndrome. She has a repertoire of over 2,000 pieces in more than 20 languages, but she
cannot read music. She has performed with members of the LA. Opera, the Boston Lyric
Opera and numerous other settings around the world.

Several studies confirm that individuals with WS also have great strengths in
auditory rote memory (Sforza, 2006, p. 61). This ability is may be a strengthening
mechanism for both vocabulary acquisition and grammatical development (Sforza, 2006,
p. 61). Another characteristic of WS is a hypersensitivity to sound, sometimes expressed
as hyperacusis (Bellugi, Lichtenberger, Jones, Lai, & St. George, 2001; Levitin &
Bellugi, 2006; E. Semel & Rosner, 2003). This hypersensitivity is especially problematic
in infancy. While it can be ameliorated as children with WS mature, it does not
completely disappear. Studies of this issue generally suggest that the cause of hyperacusis
is mainly due to the hyperexcitability of cortical regions associated with auditory
processing (Levitin & Bellugi, 2006).

Auditory abilities of children with WS are also exceptional as they relate to
music. As a music therapist, I was inspired by WS children’s ability in music and how
they might be used to compensate for othe deficits. A number of examples exist
describing the extraordinary abilities of children with WS. Amada was diagnosed with

WS at age three. Sandy Miller, Amanda’s mother, said, “. . from the second she could

30

stand up, [she] would hang on the piano. She wasn’t even tall enough to see the keyboard.
She’d stick her arms up high over her head until she felt the keys. She would not pound
away indiscriminately, as so many toddlers did: instead, she picked out perfect,
harmonious notes” (Sforza, 2006, p. 18). Sally Meersman has a similar story about her
daughter, Mary, who was diagnosed at age two. “Mary would snap to attention whenever
she heard music. She’d listen with the most comically intense look on her face.
Mesmerized” (Mills et al., 2000). The musical intensity of Gloria Lendroff during her

toddler years was captured by Scheiber (2002):

“He (Howard: Gloria’s father) settled onto the sofa, turning pegs
and plucking strings to bring the guitar into tune. A room away,
Gloria snapped to attention, as if the sound were a bolt of
electricity. She dropped her favorite toy -- a jingle key chain she
would shake and shake until Howard wanted to scream — and
crawled as fast as she could to her father’s feet, hauling herself up
and standing almost on the guitar strings, watching, listening,
with almost comical intensity. Howard strummed Elizabethan
folk songs, classical pieces, sea yarns, and cowboy tunes. Gloria
stared wide-eyed at the strings the entire time, hypnotized,

mesmerized by their luscious sound.”

Individuals with WS Show a unique ability to reproduce a melody after only
hearing it once, and to remember songs and complicated rhythm patterns. Neurological
studies on individuals with WS report relatively well-maintained or slightly-spared areas
in the prefrontal cortex, the auditory association cortex, and the neocerebellum, all of

which are important in auditory and language processing (Baharloo, Johnston, Service,

31

Gitschier, & Freimer, 1998; Zatorre, 2003). Children with WS tend to eagerly participate
in music activities and give music a longer attention span in comparison to other daily
activities. In consideration of these facts, researchers recommend using simple percussion
instruments such as tarnbourines, wood blocks, and castanets during the infant and
toddler years (Miyazaki & Rakowski, 2002; Peretz & Hyde, 2003).

Absolute pitch (AP), also known as perfect pitch, refers to the ability to accurately
identify by name and produce the pitch of musical notes without benefit of a reference
note (Levitin & Rogers, 2005). The incidents of AP in WS is higher than the incidents in
typically developed individuals (Lenhoff et al., 2001; Semel & Rosner, 2003). AP is a
rare ability that is exhibited in approximately 1 in 10,000 people. Even though the
definition of, and the manner in which AP is tested is varied and complicated, research
suggests that early musical training plays an important role, but is not sufficient for the
development of AP (Baharloo et al., 1998). i

There are controversial discussions concerning the benefits and disadvantages of
AP (Dowker, 2003). Peretz and Hyde (2003) state that there is increasing evidence that
individuals with AP have an innate, music-specifically neural network, and that these
networks can be selectively compromised by congenital disposition and brain
abnormalities including brain damage, abnormal brain structures, and abnormal
perceptual processing. Levitin and Rogers (Donnai & Karmiloff-Smith, 2000) point out
that even though there is a difference between the processing of color and pitch, it is
possible that categorical perceptions for pitch can be transferred, and an equivalence of

categorical processing might be achieved.

32

Pitch processing and advantages/disadvantages of AP are still elusive. However, I
speculate that there is the possibility that AP can be used to teach abstract concepts to
children with WS. There is an anecdotal report on this speculation. During my previous
music therapy interactions with children with WS in Korea, color recognition was
observed as a problematic domain for toddlers. I tried to teach different colors to a boy
named Mike for several weeks with unsuccessful and frustrating results. With knowledge
of AP in mind, I used a color-coded instrument. I have a xylophone that has an individual
color for each note; red for C, orange for D, yellow for E, etc., forming a rainbow of
color. The same concept is supported by other instruments and objects, such as boom
whackers, bells, scarves, and color sticks. Music was composed based on the colors on
the xylophone with distinctive melodic and rhythmic structures. Mike showed remarkable
improvement in color recognition after the first trial, and I observed generalization with

other objects (scarves and color sticks) in the following weeks.

Neurobiological Factors

Each expression of WS symptoms has a complex phenotype defined by a specific
subset of symptoms and limitations in functional domains. They are caused by abnormal
neurological development believed to be the result of the deletion of the ELN gene, and
some surrounding genes in chromosome 7 (Reid & Lienemann, 2006). Research on WS
is rapidly expanding and adding detailed information, thanks to advancing technology in
brain imaging and genetic mapping. Brain imaging methods from positron emission
tomography (PET: one dimensional computer scans) to voxel-based morphometry (V BM:

three dimensional scans) allow researchers to examine the specific areas of the brain in

33

terms of volume, density, shape, and degree of activation (National Research Council
(U .S.), 2000; Ormrod, 2003). Linguistic capability is largely managed by the left brain,
and the right brain handles visual and spatial information. Prior to development of the
fluorescence in situ hybridization (FISH) test for WS in 1993, WS was suspected, but
could not be confirmed as the unifying cause of a range of symptoms that were treated
separately with mixed results. Because individuals with WS have spared language
capability and visuospatial difficulties, researchers in 1993 expected to find lesions
within the brain regions related to WS disabilities. However, once the FISH test verified
these subjects as having WS, no indications of lesions or other apparent abnormalities
were found in either the left or right hemispheres in the brains of individuals with WS.
Recent improvements in technology and newer conceptual models have helped
researchers reconsider the relationship of the areas of the brain to atypical behavioral and
cognitive profiles. Recent research has confirmed developmental abnormalities of the
brain’s volume, shape, and neural density.

Research in the neuroanatomy, neurophysiology, and neuroscience of WS is
providing a better understanding of WS and helping to establish a better comprehension
of brain function in the general human populace. As better evidence has been obtained, a
broader consensus has developed. By conducting research using individuals with WS,
scientists are now considering stronger relationships between genetic structure and
cognitive maps as a more distinctive case of human cognition and genetic expression.
Neurologists are investigating the differences in individuals with WS’ brain structure to
better understand the connections between localized brain function, cognition, and

language. They anticipate that a better understanding of the idiosyncratic characteristics

34

of WS’s brains will give them more refined insight into typical human neurological
development.

Genetic studies of WS focus on identifying the genes that are responsible, the
cause of the deletion, and gene mapping within the WS common deletion (Feinstein &
Reiss, 2006; Korenberg et al., 2001). The common WS deletion is ~1.6 Mb, including
~20 genes; the elastin gene (ELN) and LIM-kinase 1, and extends to the area from
WBSCR20 through GTF21 (Morris, 2006). The ELN encodes the protein elastin, which is
a major component of the elastic fibers found in many connective tissues such as heart,
skin, and intestines. The elastin gene is responsible for congenital heart disease
(supravalvular aortic stenosis) and other vascular stenoses. LIM-kinase 1, is speculated to
contribute to the characteristic cognitive profile, in part through the spatial deficit. A
growing number of other genes mapped in the region include three major areas of
deletion of the chromosome band (see Figure 4). Some genes are responsible for
characteristic features of WS cognition and some with the characteristic facial features
illustrated in Figure 4.

There is a great deal of research about the volume and the shape of the brain of
individuals with WS. Reiss et al. (2001) observed that the brains of individuals With WS
have smaller total volume (13%) than those of individuals with typical development.
They also found, as previous studies suggest (J emigan & Bellugi, 1990; Jemigan,
Bellugi, Sowell, Doherty, & Hesselink, 1993), that reduction in brain volume is
proportionately uneven depending on regions. While overall brain tissue volume
reduction was at 13%, cerebellar volume was reduced only 7%, and brainstem tissue

volumes were reduced 20%.

35

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37

 

Comparing right and left total cerebral tissue volume between the four lobes of
the brain, frontal-lobe, parietal-lobe, temporal-lobe, and occipital lobe tissues indicates
that the gray matter tissue volume of parietal lobe seemed to be disproportionately
increased (Overy, 2000, 2003). In contrast, overall tissue volume in occipital lobe is
decreased predominantly in the gray matter in the right hemisphere (Reiss, et al., 2001).
The superior temporal region is the only area in the brain to be preserved in size and
shape. This area is believed to be the center for perception and processing of music as
well as auditory and language processing.

As indicated by their behavioral phenotype, the frontal cortex of individuals with
WS is relatively larger than the posterior (visual) cortex. The visual cortex indicates an
abnormal clustering of cells (Feinstein & Reiss, 2006). The visuospatial-motor (VSM)
difficulties of individuals with WS may be based on this neuroanatomical difference,
specifically in the dorsal cortical stream (Meyer-Lindenberg, Mervis, & Berman, 2006).
There are two areas involved with transmission of visual information; the dorsal cortical
stream and the ventral cortical stream. The dorsal cortical stream sends the visuospatial
information (the “where” of the object) to the parietal lobe, and the ventral cortical
stream transmits visual identity information (the “what” of the object) to the temporal
lobe, reflecting a deficit of frontoparietal processing (Meyer—Lindenberg et al., 2006).

Post—mortem investigations on individuals with WS reveal reduced brain size,
Chiari I malformations, more convex shape of the corpus callosum, and altered cell-size
and density in primary visual cortex (Mervis & Morris, 2007). Functional Magnetic
Resonance Imaging (MRI) indicates isolated hypo-activation in WS in the parietal

portion of the dorsal stream when the participants performed tasks that are related to

38

 

spatial locations of drawings or block reconstruction assignments (Meyer-Lindenberg et
al., 2004; Meyer-Lindenberg et al., 2006).

,_.., ._...-. .4...

Anatomy of WS' Brain 1

Frontal Lobe

Parietal Lobe
The frontal cortex is relatively larger The gray matter tissue volume of parietal
than their posterio (visual). cortex.

lobe appears to be disproportionately
increased.

 
    
    
 

 

5' Visual Cortex
' The visual cortex indicates an
abnormal clustering of cells.

0

 

Temporal Lobe

Q
Temporal lobe is relatively preserved in
size. shape. and function .

e
e
e
‘1

Occipital Lobe
The overall tissue voulume in occipiatal

lobe is decreased predominantly in the
gray matter in right hemisphere.

Overall brain tissue volum reducation at 13%

Figure 5. Anatomy of WS ’ Brain.

Illustration by Hyang Eun Lee

The parietal portion of the dorsal stream, which is immediately adjacent and
anterior to the intraparietal sulcus region is identified by the volume of reduced gray
matter (Meyer-Lindenberg et al., 2005; Meyer-Lindenberg et al., 2004; Meyer-

Lindenberg et al., 2006; Schmitt, Eliez, Bellugi, & Reiss, 2001).

39

Other aberrant brain morphologies include relatively spared structures within the
limbic system of the temporal lobes. It is speculated by researchers that the temporal
lobes may be associated with memory and emotion. Another difference in the brains of
afllicted individuals includes a relatively large cerebellum, especially in the
neocerebellum, which may incorporate language function. One last consistency among
individuals with WS involves enlarged sections of the temporal lobe and planum
temporale, which is involved in the processing of auditory verbal and music stimuli. The
volume of the superior temporal gyrus is preserved, providing a basis for their cognitive
strengths and musicality.

Different patterns of neural organization in individuals with WS have been
discovered by functional magnetic resonance imaging (MRI) (Levitin et al., 2003).
Compared with individuals with typical development, the flVIRI of participants with WS
indicated decreased activation in the temporal lobes and more variable and diffuse
activation throughout the brain, including the right amygdala, cerebellum and brain
stemz. When the control group processed music stimulus, it showed focused activation in
particular areas of the temporal lobe. Between the responses to music listening verses
noise listening, two groups showed differences in activation patterns, while the control
group mainly activated temporal lobes for music processing. The subject group did not

significantly differentiate those two different stimuli. Levitin et a1. (2003) concluded that

 

2 The brain stem (brainstem or corticorubrospinal) is evolutionarily the most primitive part of the brain and
is responsible for sustaining the basic functions of life such as maintaining breathing and blood pressure
(Adams et al., 1998; Heimer, 1995). In terms of motor control, the brain stem provides maintenance of
axial body tone which is necessary for erect posture and continuously modifies the body posture to
maintain equilibrium. (Adams et al., 1998; Love, 1992). Voluntary movement is controlled by many
different part of brain. The cerebrum initiates selective voluntary movement by normal volition. The basal
ganglia and cerebellum help to control complex patterns of muscle movement by controlling the intensity
and direction, timing, and selection of the appropriate muscle group in order to achieve smooth movement.
The brain stem maintains the axial body tone which is necessary for erect posture and modifies the body
posture in order to maintain equilibrium.

40

significantly differentiate those two different stimuli. Levitin et a1. (2003) concluded that
these findings are congruent with behavioral patterns in individuals with WS toward
music and auditory stimulation, and their neural organization seems to be different from
typically developed individuals.

When WS was confirmed as a genetic disorder, scientists wanted to compare WS
with Down Syndrome (DS) because of the common characteristics; a genetic disorder
and cognitive impairment. Scientists believed that by comparing the two disorders they
could generate a better understanding of WS based on the existing body of literature in
DS. They found a number of similarities, but at the same time many differences. Studies
about WS and DS reported that, although the full-scale IQ scores in both groups were
similar, the pattern of cognitive strengths and weaknesses were different.

In addition, based on neurological research (Feinstein & Reiss, 2006; Reiss et al.,
2001) and high-resolution MRI studies (2003), scientists suspect that the difference in the
brain structures of those who have DS and WS, as well as the specific degrees to which
their respective syndromes are activated, demonstrate fundamental differences in the
processing of stimuli. This helps differentiate the distinctive phenotypes of DS and WS
and provides explanations for preservation of language skills in individuals with WS, and
significantly better spatial and motor abilities of individuals with DS. Therefore, even
though people with both disorders are categorized as cognitively challenged,
interventions for each syndrome should be differentiated based on its specific expression
and underlying cause. While the neuropathology of WS remains elusive, it is important to
refine the understanding of specific causes and characteristics of WS difficulties and to

use this knowledge to develop music therapy strategies that address them.

41

 

Limbic System I

Amygdala
(emotions)

 

       
      
  
 
   

 
 

Cingulate Cortes

(pain and visceral responses) .
Hippocampus

Fornix (memory acausltlon)

Corpus collosum

Thalamus

Cerebellum

Hippocampus
(memory acausition)

 
    

Amygdala
(emotions)

Basal ganglia removed

While the control group mainly activate temporal lobes for music processing,
individuals with WS adivatejmygdala, cefleflum and brainstem as well

 

 

 

 

r The subject group did not significantly differentiate music and noise process.

 

Figure 6. Limbic System in WS’ Brain.
Illustration by Hyang Eun Lee

42

 

Bellugi, George, and Galaburda (Ansari & Karmiloff-Smith, 2002; Mervis &
Becerra, 2007) state that their research with children and adolescents with WS suggests
that neural systems in WS differ in typical paths from those seen in TDC. Fisher and A15
(2004), neurologists specializing in neonatal development, state that neurological wiring
is dzflerent, not delayed. The theory of diflerent, not delayed is extremely useful because
it sheds light on the differences in genetic disorders and suggests that different

approaches for each disorder are required.

Mathematical Development

Anecdotal reports by parents and related personnel, and the results of various
cognitive tests indicate a significant deficit in numerical cognition in individuals with
WS. Mathematical development has remained relatively neglected and has not yet been
systematically investigated other than through limited references in articles by O'Heam
and Landau in 2007 (Ansari et al., 2003; Karmiloff-Smith, Ansari, Campbell, Scerif, &
Thomas, 2006; O’Shea, O’shea, & Algozzine, 1998).

Mathematical development, especially during early childhood (ages 0 to 7), is
generally overlooked because young children typically gain an understanding of basic
mathematical concepts without requiring much intervention by adults. Young children
learn by playing with toys and manipulating objects, and by interacting with parents,
siblings, and friends. In addition, there are other issues with children with WS that require
more immediate attention such as medical needs, early language issues, toilet training,
and self-help skills. However, some individuals with WS do not seem to develop

mathematical concepts in their early years. When they enter school, mathematical

43

problems such as addition and subtraction, are introduced. WS children’s lack of
understanding quickly becomes apparent to teachers and parents. The underlying problem
appears to be rooted in their lack of foundational mathematical development.

Developmental guidelines in mathematics for children ages 2 to 7 years give
insight concerning individuals with WS and the missing developmental milestones
(Clements & Sarama, 2004). Clements and his colleague established developmental
guidelines for numerical operations, algebra, geometry, measurement, data analysis, and
probability (see Appendix H). According to their guidelines within nmnerical operations,
individuals need a range of concepts and a knowledge base in order to perform simple
operations.

Dowker (2003) stated that there may be as much as a seven-year gap in the typical
ll-year old’s highest and lowest mathematical performance. However, for children with
WS, the gap widens to 10, 11, or 12 years. Research on numeral discrimination in infants
suggests that, while infants with DS showed a significant impairment, infants with WS
showed a similar response to both their chronological and mental age control group
(Paterson et al., 2006). However, as adults, individuals with WS show a more serious
deficit on a wide variety of number tasks than individuals with DS. These findings
suggest that mathematical cognition development in individuals with WS is disturbed, to
some degree, after infant- and toddler-hood and that several levels of components in the
numerical system need to be taken into account. The exact pathway and means of
mathematically cognitive development, and which level of cognition hinders
mathematical development in WS, remain elusive. However, Donnai & Karmiloff-Smith

(2000) suggest that low-level cognitive processes, such as the ability to count and to use

44

counting to determine exact quantities, are affected by deleted genes. As a result, while
general development proceeds, different levels of cognition are uniquely affected (Donnai
& Karmiloff-Smith, 2000). Mathematical difficulties are directly related to basic living
activities involving numbers, such as money, telling time, and finding pages in a book.

Elementary mathematics as logic (solving a problem by logical thinking) rather
than abstract thinking can be taught with strategic instruction, i.e., a step-by-step
approach. The literature in academic learning of WS, emphasizes the importance of using
auditory and verbal modes and the order in which to teach new concepts (Carpenter,
Fennema, F ranke, Levi, & Empson, 1999, p. xiii). A strategy of instruction was
developed and its effectiveness documented by researchers in special education
(Dehaene, Bossini, & Giraux, 1993). In practical terms, the strategy is defined as “a
series of ordered steps that will allow a student to perform a task” (Reid & Lienemann,
2006, P.18). Therefore, when teaching mathematics, it is important to analyze
mathematical tasks and know which mathematical concepts the student does not
understand. For example, the mother of a lO-year old child with WS reported that her son
could not understand the meaning of “carry over” in two-digit addition. Songs to
illustrate or teach how to carry-over could be composed and implemented providing
students with step-by—step instructions and self- guidance as a possible music therapy
intervention.

The following table gives examples of problem solving logic in mathematics.

45

Table 1. Example of task breakdown for two-digit by two-digit multiplication *

 

Steps

Skills required

 

Multiply 1’s column.

Bring down the 1’s digit part of
the answer.

Carry the 10’s digit part of
the answer.

Multiply across, the bottom l’s digit
to the top 10’s digit.

To that answer, add the number that
you carried and write that down.

Under that answer write a O in the 1’s
colurrm.

Multiply the bottom 10’s digit to the
top 1’s digit.

Bring down the 1’s digit part of the
answer put it in the 10’s column.

Carry the 10’s digit part of the
answer.

Multiply the 10’s digit colurrm.

To that answer add the number that
you carried and write that down.

Add your two answers; the number
that you get is the answer.

Write it down.

Knowledge of place value
Knowledge of multiplication facts

Knowledge of place value
Where to write answers to vertically written math problems

Knowledge of place value
How to carry numbers

Knowledge of place value
Knowledge of multiplication facts

Knowledge of place value
Where to write answers to vertically written math problems

Knowledge of place value

Knowledge of place value
Knowledge of multiplication facts

Knowledge of place value
Where to write answers to vertically written math problems

Knowledge of place value
How to carry numbers

Knowledge of place value
Knowledge of multiplication facts

Knowledge of addition facts
Where to write answers to vertically written math problems

Knowledge of addition facts

Where to write answers to vertically written math problems

 

*Note. From Strategy Instruction for Students with Learning Disabilities by R. Reid and Toni Ortiz
Lienemann, 2006, New York, NY: The Guilford Press. Copyright 2006 by the Guilford Press. Reprinted

with permission.

46

From simple acknowledgement of the symbolic presentation of Arabic numbers to
division, there are more than 50 steps to complete a mathematical computation. Table 2

illustrates the basic building blocks in mathematics.

Table 2. Major building blocks on the way to division

 

Building Blocks

Know Arabic number system.
Write down the numbers.

Add one digit up to 10
Understand carrying-over concept
Add one digit up to 18

Add two digits

Subtract one digit less than 10
Understand borrowing concept
Subtract two digit concepts
Memorize all 9 multiple tables
Multiply one digit

Multiply two digits

Divide one digit

Divide two digits

 

 

.. _- “Another consideration for this project is the mathematic developmental level of
each individual informant. This study is intended to teach addition, subtraction,
multiplication, and division, which are basic to mathematic development appropriate
levels of mathematic deve10pment. However, if an informant does not have basic
mathematic comprehension such as understanding the Arabic number symbol system, the
quantity represented by each number, or the meaning of sequencing numbers, there is a
need to focus on lower-level cognitive processes.

Mathematical development is complicated. While language acquisition can
continue despite missing pieces in language development, mathematic development is

sequential and built on prerequisite mathematical acquisitions. As with gross motor

47

development in the first year of life, infants advance from laying down, to developing the
muscle in the neck to controlling the movement of the head and trunk , learning to sit,
crawl, stand, and finally walk. Walking is not possible without head and trunk control. In
much the same fashion, mathematical development is a sequentially developmental
process that is more similar to physical development than language development.

The visuospatial aspect of mathematics is neither clearly defined nor thoroughly
studied because the process of manipulating and calculating numbers within the brain is
still elusive and controversial (Bachot, Gevers, Fias, & Roeyers, 2005; O'Heam &
Landau, 2007). The visuospatial difficulties in the WS population might manifest in the
form of an abnormal perception of the mental number line (Dehaene, Dupoux, & Mehler,
1990; Zorzi, Priftis, & Umilt, 2002), which refers to the spatial perception and
representation of a number along a left-right orientation in a continuous analogical
format, such as that of a ruler (Dehaene et al., 1993; Zorzi et al., 2002). Research on the
mental number line suggests that this concept might be an inherited psychological aspect
rather than a learned behavioral. Even those in right-left-oriented written documentation
cultures, there is evidence that they possess left-right orientation with the mental number
line (Davis, 1999; Peters, 2000). There is no specific research on the perception or

recognition of a mental number line in individuals with WS.

Music as a Therapeutic Medium
Music provides experience and stimulation in various developmental domains for
individuals with special needs (National Research Council (U .S.), 2000). A preliminary

study in Korea of three young children with WS ranging in age from 40 to Sl-months

48

reported that during 10 weeks of musical intervention, subjects showed distinguishable
improvement in word acquisition, including the names of body parts, animals, fi'uit, and
instruments. However, when comparing abstract concepts, subjects continued to show
difficulties distinguishing a large drum from a small drum (Song, 2003). The study
suggested the possibility of using various aspects of music such as rhythm, melody, pitch,
harmony, and dynamics to give toddlers with WS Opportunities to deve10p expression as
well as receptive language skills. I think it is possible that the same concept could be
extended to the development of mathematical skills. The concept of music therapy
provides a theoretical framework for the use of music as a therapeutic medium with
regard to mathematical functions. Music can be beneficial for a particular cognitive
domain, learning process, memory process, dyslexia, and as a therapeutic medium in

memory and attention.

The Learning Process

Human learning, the recognition and absorption of concepts and processes,
involves multi-level and multi-dimensional processes. The nature of childhood and the
nurturing received during childhood are interwoven and impact an individual’s learning
process. The learning process is sophisticated so any learning difficulties further
complicate an already complicated process. Learning is not the mere memorization of
simple facts, but includes knowledge, comprehension, application, analysis, synthesis,
and the ability to apply it to multiple situations (Reid & Lienemann, 2006). Therefore, if
the structure or characteristics of musical activities can aid in any of these processes, then

specific training using music can facilitate and promote the over-all learning processes.

49

Research conducted in developmental psychology, cognitive psychology, and
neuroscience advocates the findings that, “learning changes the physical structure of the
brain, these structural changes alter the functional organization of the brain; in other
words, learning organizes and reorganizes the brain, and different parts of the brain may
be ready to learn at different times” (National Research Council (U .S.), 2000, p. 115).
Music can be used to create a structure that integrates activities with learning to help the
formation of brain structures. Based on an animal learning process experiment, “learning
adds synapses; exercise does not” (National Research Council (U .S.), 2000, p. 120). In
other words, more physical movement (such as treadmill exercise) does not generate new
neuron synaptic connections, but the exercise combined with cognitive processing does
generate connections (such as occurs with acrobats). Based on those findings, learning

through musical activities can add new pathways for performing tasks.

Memory Process

Despite differing perspectives from various fields in human memory processes,
the prevailing theory is a dual-store memory model (Ormrod, 2003). The term dual-store
comes from the two distinctive memory types; short-term and long-term memory. Figure
7 depicts a simplified dual-store model (Ormrod, 2003, p. 187).

Sensory input is registered for a short time period of a few seconds at most.
Depending on sensory modality, the remaining information is changed and remembered
for a fraction of a second longer than visual information and for two to four seconds
longer than auditory information (Ormrod, 2003). In order to process input at the next

level (i.e., short-term/working memory), the receiver needs to pay attention to the stimuli.

50

Since only registered information moves into working memory, attention is the important
factor in memory. Human beings consistently need to choose the most salient stimuli
because of their limited human capacity. Ormrod (2003) identified six factors that
influence attention: size, intensity, novelty, incongruity, emotion, and personal
significance (National Research Council (U S), 2000; Ormrod, 2003). The interesting
factor in these six factors is personal significance. Aside from the factors, personal
significance can change the direction of attention, which explains why different

individuals in the same situation perceive things differently.

 

 

 

Lost Lost Lost?
Input \ Sensory Short- Long-Tenn
Input Register Tenn/Working Memory
In ut _—_’ ' '
P Memory
—>

 

 

 

 

 

 

 

 

 

Figure 7. A simplified dual-store model of memory. *

‘From “Human Learning (4th Ed)” by Ormrod, p.187. Copyright 2003 by the Pearson Merrill Prentice Hall
Reproduced with permission

When information is moved into the working memory stage, individuals start to
think or manipulate the information they receive in order to process it into the long-term
memory. The strategies used to incorporate information into the long-term memory
include organization (chunking), retrieval (scanning process of all the information), and

maintenance rehearsal (connecting new information with old. information). Storage and

51

retrieval are two main control processes in long-term memory. Storage happens when the
learner understands, organizes, and integrates new information with pre-existing
information.

The retrieval of information from long-tem memory involves a considerably more
sophisticated process than when retrieving information from working memory. Since
human beings retain huge amounts of information, it is important to focus on the correct
memories. All three steps in the storage process (understanding, organizing, and
integrating) are directly related to successful information mapping and retrieval. Fully
understood concepts that are organized and richly detailed have a better chance of being
retrieved successfully. The more connections that are made with previously memorized
information, the better the chance to retrieve it in the future. A comprehensive
understanding of the memory process is important to gain an understanding into how

music can influence the memory process for individuals with cognitive challenges.

Dyslexia and Music

Research on the relationship between language and timing skills suggests that the
difficulties rising from language-learning impairments may result from a fundamental
deficit in the ability to process rapidly changing sensory input. Tallal and colleagues
(1996) designed a training procedure using prolonged speech patterns that preserved
spectral content and the natural quality of speech. They reported that after 3 to 4 weeks of
intensive temporal processing (3 hours a day, 5 days a week at the laboratory for 1 to 2
hours a day, with 7 days a week as homework) seven participants (with a mean

chronological age of 7.3 years) dramatically improved their receptive language skills. A

52

small amount of similar research has echoed these findings. Results from several other
studies support the conclusion that there is a correlation between dyslexia and the
temporal encoding of sensory information that can be improved by temporal processing
training. The relationship between dyslexia and the temporal aspect of music (rhythm)
has been reported in two directions. Dyslexic musicians reported difficulties with rhythm,
and dyslexic children have problems keeping a steady beat (Granschow et al., 1994;
Miles et al., 2001; Oglethorpe, 1996(Overy, 2003)). Overy (2000) applied the concept of
applying temporal processing training with dyslexic individuals to dyslexia and music.
Overy (2003) and Overy, Nicolson, F awcett, and Clarke (2003) used music, which has an
intrinsic temporal organization, for remediation of temporal processing deficits and
conducted research to identify a particular area of music on which to focus in order to
irnprove timing processes. The results indicated that music activities that place a strong
emphasis on rhythm improve both phonologic and spelling skills, but not reading skills.
Overy (2003) also pointed out that the participants exhibited no sign of difficulty
with the use and recognition of pitch. This provides additional insight into the relatively
preserved language ability of individuals with WS, despite their cognitive impairment.
Individuals with WS do not show any signs of difficulties in phonological or spelling
skills, and their auditory sensitivity and musical abilities (pitch and rhythm) are part of a
well-known phenotype. For other children with cognitive challenges, using visual or
whole word teaching strategies is often suggested to teach them to read a word, whereas
the phonic system or auditory modality to break words into components is generally
suggested for individuals with WS (Mervis & Morris, 2007; Semel & Rosner, 2003).

Processing speech involves developing the ability to make fine discriminations between

53

sounds rapidly. Music therapy interventions that require the child to distinguish between
high and low pitches, loud and soft tones, and the sounds of various instruments can train
the child for auditory perception in time with clear self-evaluation tools (i.e., the students

understand the expectations of therapists and whether or not they successfully meet

them).

Mnemonic and Strategic songs

Music has been used as a mnemonic device or memory aid, and has been
particularly useful for children with memory issues including mental retardation,
cognitive impairment, and learning disabilities. Gfeller (1999) suggests using short and
simple melodies to recall information such as multiplication tables. Songs for
multiplication tables have been used by the general public. Some children in the US
learn their multiplication tables with changing lyrics of the melodies from familiar songs
such as ‘Jingle bells,” “Row Row Your Boat,” and “Yankee Doodle.”

The guidelines for composing a song memory aid, called a jingle, can be found in
the analysis of the composition of songs used in commercial advertising. Commercials
use melodies that are catchy and made to be easily remembered. Based on their empirical
experience, Zager(2003) suggests four basic guidelines to make a song memorable:

l. Melodies have to be simple: more conjunct (stepwise) motion than disjunct

(skips) motion in the melodic structure, but interesting.

2. The harmony should be simple.

3. Songs should be easy to remember after the first hearing.

54

4. The song should be easy for the average person to sing along with (Zager,
2003, pp. 94-98).

Phrasing in music through musical elements helps chunk verbal information and
is the most crucial informational organization strategy for composing music for
rrmemonic purposes. Thaut (1999) explains the role of phrasing by musical elements for
memory processes. Chunking refers to the memory organization strategy of grouping
single, small pieces of information into larger units that are easily remembered as a
whole. For example, seven digit telephone ntunbers are chunked into two units of 3 and 4
digits. Rhythm, melody, and harmonic elements are all excellent structures that enhance
the organization, sequence, and recall of verbal information. A short melodic phrase can
trigger the recall of a long series of words and sentences. One of the best known
examples might be the ABC song. While it is a considerable challenge to remember 26
separate bits of information, the ABC song is organized in 4/4 meter, and uses musical
phrasing to create four, easily manageable phrases of information (Thaut, 1999).
Therefore, songs, chants, rhymes, and such can be used efficiently by music therapists to
facilitate memory process.

Strategic instructions, such as providing strategies to solve academic problems,are
strongly encouraged for students with learning disabilities (Karmiloff-Smith, Ansari,
Campbell, Scerif, & Thomas, 2006; Semel & Rosner, 2003). New concepts can be
learned and internalized through a series of necessary steps: repetition, comprehension,
and application. Most songs for young children use simple rhythmic structures and
straightforward melodies that repeat lyrics. When children listen and interact with others,

they learn either the actions along with music, or some portion of the lyrics, usually the

55

chorus as it is the most repeated section of a song. Repetition is the important part of the
learning process and songs can be repeated many times without making children bored.
The multi-sensory aspects of music are often additional benefits to the
memorization process (Gfeller, 1999; Thaut, 2005). Using multi-sensory stimulation as
much as possible gives individuals more opportunities to retain and retrieve new concepts
(more connections) and provides more chances to apply new concepts in general
situations. Music as a multi-sensory activity gives children with learning disabilities a
chance to overcome their difficulties using a favored sensory modality (Overy, Nicolson,

F awcett, & Clarke, 2003; Peters, 2000).

Issues in Attention and Motivation

Lack of motivation and attention deficits are often reported as salient
characteristics in learning disability. Lack of motivation and failure in academic pursuits
can lead to a learned helplessness, which in turn feeds back into a negative cycle that
reinforces itself. Attention deficits do not mean that children with learning disabilities
cannot pay attention, but that they often pay attention to unnecessary or unimportant
stimuli, or can be easily distracted by insignificant stimuli (Ormrod, 2003). Aldridge
(1996) claims that four elements are required to achieve sensory integration: arousal,
attention, affection, and action. These four concepts are essential to the learning process.
In order to learn, the learner should maintain alertness (arousal), pay attention to the
subject (attention), enjoy the subject (affection), and demonstrate an understanding of the
subject (action). Music activities require synthesizing auditory stimulation (music

listening) and appropriate response (playing, singing, or acting), which can be easily

56

observed and redirected if the action is inappropriate.

Based on this theory, it can be hypothesized that, within a rhythmic structure,
variation in musical elements could create an attention-gripping moments in the music.
Attention can be focused on and manipulated using music from the perspective of the
following six factors; size, intensity, novelty, incongruity, emotion, and personal
significance (Ormrod, 2003). In other words, while the rhythmic structure of song (meter)
gives the listener familiarity, changes in pitch (which comprise melody and eventually
phrases), loudness, and timbre, will catch and maintain attention. For those who have an
aflinity to music, especially individuals with WS who also have hyperacusis and
transfixion to different sounds, music interventions can also provide significant emotional
and personal connections that increase attention. Music is processed in the brain.
Knowledge about the function and role of each element in music as it is processed in the
brain provides potential frameworks to use to promote specific goals and objectives in the

learning process.

Wish List from Parents and Music Therapists
Before this study, I conducted a preliminary pilot project for developing a
rationale for music therapy programs for individuals with WS. After reviewing the
literature about WS, interviewing a parent with a child with WS, and music therapy
colleagues, it was possible to define at least a partial wish list for those who work with
individuals with WS. Nancy, who is the mother of Jessica, a three—year old with WS
stated that she is concerned about Jessica’s cognitive function. Since Jessica has good

language ability and relatively acceptable cognitive skills for her age, and considering her

57

diagnosis, she hopes that Jessica can reach a relatively normal range of cognitive
fimctioning.

Nancy is an actively involved parent; she studies WS and seeks interventions for
her daughter’s condition. Since Nancy found the information about using songs to teach
individuals with WS, she has taught Jessica how to spell her name, her telephone number,
and her address using songs. Diane, a music therapist who has two clients with WS,
expressed that she wishes to have more information about WS. She feels that the current
information about WS is “purely neurological and medic-a1.” She thinks there is enough
information regarding difficulties, abnormalities, characteristics, or extreme anecdotes
about idiopathic examples of some individuals with WS, but there is not enough
information about what kinds of interventions are useful to compensate for cognitive
differences, and how to use their special abilities in their daily lives. For example,
because individuals with WS have good musical skills, many of them could successfully
perform in public. However, from a music therapist’s standpoint, it is equally important to
know how one can use their special music abilities to help improve non-musical skills.
There is not enough information and research about these areas.

There may be music therapists who encounter individuals with WS, but have
limited resources and time to investigate and develop individualized music therapy
interventions. Vicky, a music therapist at the WS summer music camp, suggested that
music therapists should provide appropriate and accurate information about the role of
music therapy for individuals with WS. Then, administrators, physicians, and parents
would have a better understanding of the use of music therapy with individuals with WS,

an important first step into school systems and toward being included as regular therapy

58

for individuals with WS in their individual education plan (IEP). The fact sheet about
WS, which is compiled by the American Music Therapy Association, could be the official
document presented for music therapy service. However, the fact sheet does not currently
give enough information to explain the benefits of music therapy for individuals with
WS.

Singing activities suggest the need for further investigation because of current
successes and limitations. My pilot study reported that Diane (a music therapist with two
children with WS clients) mentioned that Jessica and Tom do not like to sing during
sessions. Diane speculated that Tom might not like his singing voice. This may be due to
the deletion of the gene for elastin, the vocal chord connective tissue in persons with WS.
As a result, individuals with WS typically have “hoarse” voices. The reported incidence
of this characteristic varies dramatically from 20% to 100%, depending on the study
(Semel & Rosner, 2003). Unlike Tom, Jessica seems to be shy and also does not sing
during seSsions, but she does sing the songs from her sessions in the car with her mother
during the trip home afterwards.

Fidler, Lawson, and Hodapp’s work (2003 supports the need for further
investigation of the use of songs. They discuss parents’ desire for modifications to their
children’s current educational programming for children with three different genetic
syndromes (Down Syndrome (DS), Prader-Willi Sindrome(PWS), and WS). They
concluded that the parents of children with DS need speech therapy and reading services,
parents of children with PWS desire increases in adaptive physical education, and parents
of individuals with WS need adaptive music instruction and musical aides in the

classroom. One parent expressed that a special educator underestimated her child’s

59

musical capabilities, and another parent believed music needed to be incorporated into
regular classroom instruction. They also cite that parents wish to have available “. .. [a]
class with individualized learning strategies, such as visual learning with music” (Fidler,
Lawson, & Hodapp, 2003, p. 201). In accordance with the needs expressed by parents,
early music therapy interventions for children with WS should begin as soon as the
diagnosis is made, and should be continued for each stage of the child’s life to help them

overcome and compensate for difficulties and limitations.

Reason for Study, Initial Theory, and Research Questions

This study attempts to develop effective music therapy interventions in
mathematical leaming for individuals with WS by utilizing music therapy tools as
instructional aids based on existing theories of music therapy. Responses of three
participants to these techniques were observed and analyzed in relation to the expected
effectiveness of the interventions.

Information exists regarding strengths and weaknesses in the characteristics of
individuals with WS, as well as the use of music therapy for individuals with special
needs. However, the use of music therapy to remediate cognitive difficulties in WS and
music therapy has not been tested and evaluated to determine its role for specific
academic needs of individuals with WS. A cyclic process of research, redefined theory,
and practice are needed to maximize the effectiveness of music therapy interventions in
such a context.

Individuals with WS require alternative strategies and tools to address

mathematical development because of their unique neurological issues affecting

60

development. Traditional methods used to teach mathematics are ineffective for
individuals with WS. Speculative causes for these difficulties in mathematics include
cognitive limitations, visuospatial deficits, and attention deficits. Ifthe identification of
these causes is accurate, music therapy theory suggests that music may, to some extent,
counteract cognitive limitations and lack of attention.

Existing theories of music therapy for individuals with special needs suggest that
music can be used to promote cognitive development through the singing of songs that
contain the content of academic learning concepts such as the alphabet and numbers
(Adler, 2006). Music also functions as focus and reinforcement for attention and as a
structure that may promote mathematical development as discussed in the previous
section on music as a therapeutic medium. In addition to these therapeutic benefits,
individuals with WS’ affinity for music can be an additional asset to amplify the benefits
of music. The basic premise for the effectiveness of music therapy intervention for
individuals with WS is drawn from the individual’s known musical affinity and specific
neurological responses to music. Individuals with WS have a certain type of musical
nature: they are drawn to music and share idiosyncratic musical abilities, such as
extraordinary melody memorization, distinctive rhythmic ability, and absolute pitch.
Even in the most severe cases of WS, this characteristic affinity for music is not lost. By
combining arithmetic with musical activity, individuals with WS might be motivated to
participate in activities and their attention can be drawn and held for longer periods that
would likely enhance their ability to learn and retain the lessons.

Interventions developed for the individuals in this study are based on theories of

neurology, child development, and music therapy. Individuals with WS often have

61

difficulties memorizing some mathematical concepts, such as the value of coins or
multiplication tables. The lack of strategies to work on mathematical tasks has been
suggested as an integral part of their difficulties in mathematics (Karmiloff-Smith et al.,
2006; Semel & Rosner, 2003). Therefore, by providing them with efficient strategies and
self-instruction routines through songs, these difficulties may be reduced. Songs can be
composed to reinforce mathematical concepts, as well as provide strategic steps or
routines to complete mathematical tasks. In this way, music therapy might offer these
individuals an opportunity to use their musical strengths to overcome other difficulties.
Music addresses the emotional and psychological needs of individuals with WS and
promotes a more effective learning environment as illustrated in the following diagram of
a cognitive map supporting this study.

This study used observation, video analysis, and interpretation to deveIOp an
understanding of the underlying reasons for the mathematical difficulties of those with
WS, to improve the understanding of music as a therapeutic medium in this particular
population, and to develop music therapy strategies to support mathematical reasoning
ability. This study applied a qualitative research approach based on a collective case
study foundation, offering the opportunity to collect the information needed for this initial

examination of the issue.

Research Questions
The research questions that prompted this study are listed below:
1. What are the obstacles in the mathematical development of individuals

with WS? What can they do and what can they not do mathematically?

62

2. Based on existing music therapy theories for individuals with special
needs, as described in the section on music as a therapeutic medium, are
there music therapy interventions that can serve as strategies to support

learning in mathematics?

U.)

How do individuals with WS respond to music therapy interventions to
overcome their challenges? How are these interventions useful or
effective in meeting their needs?

Figure 8 illustrates how these three questions are interrelated to one another in

this study.

 

Figure 8. The initial cognitive map linking mathematics and music.

63

CHAPTER III

METHOD

Research methods provide a framework for the collection, analysis, and
interpretation of data. While there is a great deal of research available concerning
individuals with Williams Syndrome, including neurological perspectives on their
musicality and their abilities involving rhythm and absolute pitch, little work has been
directed toward designing music therapy interventions for them. Aldridge (1996) states
that qualitative and quantitative research methods have unique and valuable contributions
to understanding phenomena in music therapy. According to bibliographic research by
Brooks (2003), music therapy, especially within the United States, tends to favor the
quantitative research method. Meanwhile, qualitative research is often used to develop a
perspective based on the interaction between the researcher and the participants being
studied (Wheeler, 2005, pp. 13-15). For this study, qualitative inquiry offers a useful
research method because of the lack of foundational knowledge needed to determine the
relationship between WS and music therapy. The processes behind the therapeutic effect
of music on clients with WS are elusive. Therefore, research incorporating individualized
sessions with participants may provide insight into these mechanisms and the

development of music therapy for those with WS.

64

Methodological Framework

A collective case study (Stake, 1995) was chosen as the methodological
framework for this study. A qualitative case study is “an exploration of a ‘bounded
system’ of a case (or multiple cases) over time through detailed, in-depth data collection
involving multiple sources of information rich in context” (Creswell, 1998, p. 61). The
goal of a qualitative case study is to develop a thorough analysis of a single case or
multiple cases in order to generate a hypothesis based on ontological and epistemological
paradigms (Smeijsters & Aasgaard, 2005). Different qualitative case study methods, such
as intrinsic case study, instrumental case study, and collective case study, have different
benefits and limitations, especially when applied to the previously undocumented use of
music to reinforce mathematical concepts for children with WS. While an intrinsic case
study focuses on a particular case for its own sake, and an instrumental case study means
that the case is used as an instrument to understand the phenomena. The central tenet of a
collective case study is to study particular cases as instruments to gain insight and
understanding of research questions, or phenomena (Stake, 1995). The collective case
study method was selected as the best approach to serve the purpose of this study

The collective case study approach provided an overarching picture of the
phenomena in this study and aided in establishing the relationship between music therapy
interventions that attempt to improve mathematical operations for people with WS, and
the possible role of music in clarifying their mathematical difficulties. This approach
provided freedom and flexibility to explore different aspects of these phenomena.

Observations and interpretations from this study will provide a foundation for more

65

extended research into music therapy interventions for mathematical development in

individuals with WS.

Researcher 3‘ Lens

My undergraduate training in mathematics led me to work for several years as a
computer programmer. The logic and structure of algorithms and code drew me to
programming in much the same way that the structure of musical notation and sequences
underlies my love of music. Eventually I found that music blended with my tendency to
want to help others, and I began to pursue an education in the field of music therapy.

Before I learned the details associated with WS, I was already falling in love with
these individuals because of their affection for music. I was amazed by their fascination
with sounds and music and the way I could use music to communicate with them. During
the earliest sessions with my first WS client, I felt like a music teacher who had stumbled
upon a musical prodigy. This client only had to listen to a song or selection a couple of
times before he could repeat it back exactly. He could imitate complicated rhythms and
patterns, and maintain a musical dialogue with me, something of which few adult
musicians are capable—and he was only 5 years old.

Several other clients of mine with WS have demonstrated similar talents in
varying degrees. They seemed entranced by new sounds and instruments such as finger
cymbals, a gong, a rain stick, and the thumb piano. My limited piano playing ability was
enough to enthrall them. I felt like a world-famous, concert pianist engaging my
audiences. Music makes these individuals happy. In fact, when music is involved in

therapy sessions, their behavior significantly improves. When music therapists and

66

individuals with WS focus on musical experiences, and the clients are engulfed by the
music, clients seldom show their frustrations and limitations such as uncontrollable
tamper-tantrums, emotional roller-coasters, and obsessive-compulsive. Those moments
do not typically emerge while individuals are happy and content. When I became more
deeply involved with individuals with WS and spent more time with them outside
sessions, such as working at the music camp for individuals with WS, and attending their
conferences and gatherings, I witnessed the desperation, frustration, and agony that are
also part of their daily lives.

“They are smart enough to know that they are different,” said Dr. Barbara Pober,
director of the Williams Syndrome Clinic at Massachusetts General Hospital in a TV
documentary (CBS, 1997). Another statement fiom the documentary was that persons
with WS did not feel that they “fit in.” When they are with other individuals with WS,
they feel a sense of belonging, and can relate to, and understand each other. The sense of
hopelessness—‘there is nothing I can do to improve this condition,’ ‘I tried this and that,
and nothing has been working’—is a devastating feeling. If the love of music by
individuals with WS can be used to augment mathematical ability, it could be a
significant breakthrough that could make their lives, and their parents’ lives, a little
easier.

Today’s materialistic and product-oriented society does not readily understand
individuals with WS’ inabilities in certain areas. In some ways I empathize with the day-
to-day interactions and prejudices someone with WS might experience. English is not my
mother tongue and I have found over the past ten years in the United States that this can

cause a range of negative responses. I am not perceived as foreign, necessarily, but as

67

language-impaired and this sometimes translates into being treated as ignorant, or less
than competent, something a person with WS experiences intimately and often. The fact
that I am normal in my native Korean society does not help me feel better. Shifting from
my linguistic challenges to individuals with WS’ mathematic obstacles, I rrright claim that
I have experienced something similar because of the perceived lack of ability. This allows

me to relate to how WS individuals might feel in their daily lives.

Participant Selection

The participants in my study were three individuals diagnosed with WS as
confirmed by fluorescence in situ hybridization (FISH) testing. The participants’ ages
fiom 10 to 25 years of age and were selected to provide a broad spectrum of
mathematical development. Gender was not a basis for selection. Individuals who were
non-verbal and non-ambulatory were excluded because of out-of-normal distribution of
functions in Williams Syndrome3 (Bellugi et al., 2001).

In order to understand mathematical function in different age groups, three
participants were recruited; one from the 10 to 15 year age group, one from the 16 to 20
year age group, and one from the 21 to 25 year age group. I met my three participants at
WS conferences, music camps for individuals with WS, and through my work with
parents in the WSA. Detailed information about the participants will be shared in each

case study.

 

3 Bellugie, et al (2001), cited 47 studies on cognitive firnction of individuals with WS using standardized
intelligence tests, mainly the Wechsler Intelligence Scale for Children-Revised (WISC-R), and Differential
Ability Scales (DAS) (Martens et al., 2008). The mean IQ of individuals with WS was between 50 and 60
with a range of 40 t0100. Another source for the average or typical performance is the Williams Syndrome

Cognitive Profile(WSCP) (Frangiskakis et al., 1996; Mervis, et al., 2000).

68

Sessions Procedures

Music therapy sessions took place over a ten week period, once per week, for 40
to 60 minutes, and in each participant’s home environment. Session duration varied
according to the participant’s needs. The participant’s home was used instead of a clinical
setting to maximize the effectiveness of music therapy. The study encompassed ten
weeks, which may not have been enough time for the participants to fully learn and
internalize the concepts and strategies. The effectiveness and sustainability of the
interventions and their residual effects were also concerns, even after the project was
ended. Therefore, providing the sessions at home was meant to increase the residual
effects without having to be as concerned about generalization of learning outside of a
music therapy clinic. In addition, the parents and siblings had opportunities to observe
and participate in music therapy sessions and to learn songs to support the mathematical
development for future use.

The initial intervention designs were based on literature and individual stories
drawn from anecdotal experiences that included the WSA, Parents List Servers, third
person narratives, and empirical/personal experiences with various WS gatherings. These
experiences suggested likely difficulties that might be encountered during music therapy
sessions and helped formulate plans for music therapy. As the project unfolded, the
interventions evolved, and were modified and then adapted to each participant’s
firnctional level, in order to address their individual needs and difficulties. Video
recordings of each session were reviewed and transcribed prior to the next session in
order to revise preparations based on a growing awareness of each participant’s needs.

Concurrent investigations into the mathematical development of the participants were an

69

important part of this study. They shed light on the underlying difficulties in each
participant’s mathematical development and revealed specific impediments of their

ability to process mathematics.

Data Collection

Video Recording. All thirty individual sessions with the three participants in this
study were digitally recorded and converted to mp4and DVD file formats using the
Macintosh iMovie software program. Each video recording focused primarily on the face
and hands of each participant, and the materials with which they worked. Each session
was reviewed and transcribed within two days to preserve nuances and other elements
that may or may not have been directly related to the sessions.

Parent Interviews. Before beginning the study, the mother of each participant
was interviewed using open-ended questions to establish her attitudes and fi'ustrations as
the parent of a child with WS, her expectations for the study, and her desires for her
child’s future. After the sessions were completed, each parent was asked to share
observations, both positive and negative, and suggest improvements for future music
therapy interventions. These conversations provided a secondary data source for
triangulation.

Worksheets (Artifacts). Worksheets of math problems were periodically provided
to each participant. The worksheets were individualized for each participant’s specific
skill level. Each worksheet was analyzed and adapted for future sessions. A comparison
of all completed worksheets was used to identify common mistakes among participants,

and to analyze the benefits or disadvantages of particular songs used in the sessions.

70

Personal Journals and Field Notes. A personal journal of each session and field
notes taken during sessions were maintained. The journal recorded observations,
difficulties during the various processes, questions about behaviors and responses, and
speculations regarding changes in the subjects’ use of mathematics. The field notes
served as a reminder of particular events in the sessions that may, or may not, have been
captured by video tape.

Peer Debriefing. To assure trustworthiness, peer debriefing took place on a
weekly basis during the study. Peer debriefing was done with a music therapy faculty
member at Michigan State University (MSU), who reviewed the personal journals, field
notes, and video recordings. Afterwards, the tentative themes and interpretations were
reviewed and discussed with other professors and music therapy graduate students at
MSU. Music therapists at the WS music camp and an occupational therapist, who had
experience working with WS populations in a music therapy camp for individuals with

WS discussed recurring themes and possible interpretations.

Data Analysis

Data analysis began with transcription of the video recordings. These
transcriptions were reviewed before the next session and any repetitive behaviors or
patterns were given a code and added to a cumulative data set (see Appendix B for the
final list of codes and themes). Each code had either its own font color or highlight color.
Red highlights were used for mathematical difficulties, yellow highlights for
distractibility, blue font for musical responses, and interesting moments without a

corresponding code were highlighted in gray (see Appendix B). After all thirty sessions

71

were transcribed, all uncoded items (gray highlighted items) were reviewed to aggregate
any similarities and reinforce comparisons. A couple of new codes emerged from the
review and the videos were reviewed again, and the new codes were applied.

Session progress charts (see Appendix G) were used to outline the main events of
each session and provide changes across sessions in either developmental progress or
regression during sessions. The mathematical mistakes made by each participant were
gathered in a video format for each participant as well as compiled in written format (see
Appendix J). This process was used to provide easy access to the mistakes and illuminate
any themes and relationships in the foundational difficulties of the participants’ individual
mathematical development. The three compiled video files were reviewed multiple times _
by participant, intervention, and side—by-side by using two or three monitors to uncover
the similarities and differences. Each code was cumulatively reviewed by searching for

patterns, consistencies, and outliers for emergent themes and trends.

T rustworthiness

Thick descriptions, analytical memos, repeated analyses, auditing, peer
debriefing, and triangulation using different types of data (video tape, worksheet, parents’
interview, and personal journal) were employed to establish the trustworthiness of the
data. Each research step was carefully and rigorously implemented as suggested by
Smeijsters and Aasgaard (2005) and Stake (1995) with “standards of integrity for
qualitative music therapy research” in mind as suggested by Bruscia (1998). Authors of
qualitative research methods demand that qualitative researchers be fully dedicated to

their study while conducting research. They must hold to rigorous standards, not merely

72

for personal academic achievement, but also to maintain a self-motivated and intrinsic
curiosity about the relationships among a phenomenon, the client, and the music. I was

mindful of theses requirements during the research process.

73

CHAPTER IV

DESCRIPTION OF MATHEMATICAL INTERVENTIONS, SONGS, AND PROPS

In this chapter, the principal mathematical interventions, songs, and props that
were used for this study are introduced. They are introduced before the case studies are
described in order to more clearly understand the descriptions and purposes of
interventions, songs, and props. The literature review identified key themes and theories
in the cognitive development of WS as well as the principle difficulties involved with
fine motor control and visuospatial difficulty. The interventions in this study were
designed to capitalize on the musical ability of WS in a way that might help to
compensate for their difficulties in understanding mathematical concepts. There are
seminal interventions that were developed based on literature reviews of WS, music
therapy theory, and prior experiences of individuals with WS, as well as others with
cognitive challenges. Also included are adapted and modified interventions fiom seminal
interventions. In addition, I developed new interventions that were necessary as new
difficulties emerged. Interventions, new song compositions, and appropriate props to
teach mathematical concepts were also imperative to this project. The descriptions of the
songs and props, the way to design and develop them, and their purpose are included in

this chapter.

74

Mathematical Interventions
According to the Principles and Standards for School Mathematics by the

National Council of Teachers of Mathematics (NCTM), mathematics has five categories:
numbers and operations, algebra, geometry, measurement, and data analysis/ probabilities
(Clements, 2004). The area of number and operations is most directly related to tasks
such as manipulating money and the most frequently discussed topic in mathematics.
Three mathematical tasks in number and Operations were selected for this project because
of their relationship to everyday life and other difficulties as cited in the body of literature
on WS. Tasks included an intervention focused on coin use, written addition, and reading

an analog clock.

Coin Related Intervention

Coin usage is one of the most necessary mathematical skills in everyday life and
was the starting point of this project. As illustrated in the prologue, many individuals with
WS have difficulties manipulating coins. I performed informal testing related to the
concept of coins with individuals with WS at the WS music camp and the WSA
convention. In my test 1 quarter and 2 pennies were often identified by WS individuals as
‘3’ without specifying the total amount. A couple of campers and attendees said ‘75’ also
without the unit i.e., coins or cents. I concluded their answer was probably was the
memorization of the number sequence of 25’s—25, 50, 75—based on quarters. Only
about one-third of campers were able to correctly answer ‘27’ cents.

In a task analysis of coin usage, several steps have been identified: coin

discrimination, coin knowledge, coin calculation, coin combinations for commercial

75

transactions, and value subtraction for receiving change. The first step in calculating
monetary values and in manipulating coins is to understand the differences between
coins. The next step is to discriminate the value of each coin. Then individuals are ready
to process the amount and give and take reciprocally. All of these processes are necessary
to be able to fimction in daily life; therefore, interventions related to coin usage began
with coin discrimination, calculating the amount of values, and then practicing the
exchanging of coins. None of the participants in this study was able to move on to

calculating subtraction for receiving change.

Written Addition

Single digit addition is often considered the first real mathematical skill. Addition,
subtraction, multiplication, and then division is typically the order of teaching operations
in a general education setting. There are many anecdotal examples in the body of WS
literature and documentaries of the inability of individuals with WS to add two single
digit during observations in various settings. Since written addition is the most used form
in the educational system, and the ability to perform mathematical tasks is tested by
written addition, written addition was included in this project. There was a pre-test
worksheet in order to determine the current mathematical skill level of each participant.
The worksheet indicated that the participants had not yet developed a reliable
understanding of addition. I developed a couple of individualized worksheets based on
each participant’s needs. Single digit addition was used as the starting point. Initially, the
plan was for them to move on two digit addition, but this did not take place, because of

the limited ability of the participants.

76

Clock Related Interventions

Reading an analog clock is also a basic life skill. Although digital clocks are
readily available and can be used as an alternative, analog clocks are still considered
important and included in elementary school curricula. Reading a clock is a
comprehensive activity that requires accumulated knowledge and understanding and is a
challenging task for any young child. Difficulties in telling time from an analog clock
were reported in the literature on WS. There are a number of related concepts that form
the foundation for understanding the use of time; the comparison of words such as the big
and little hands, the directionality of their sweep around the clock, or clockwise, and the
different values for hours and minutes. The concept of fractions is also involved in
reading an analog clock. For example, half an hour, a quarter after 2, or 5 minutes to 10
are concepts that must be mastered in order to read a clock face. If any of these
competences are not overcome, the task can not be successfirlly accomplished.
Understanding comparison words such as big and little, visual perception difficulties, and
problems understanding changing values based on units are common for in individuals
with WS. Interventions using a clock were designed to help identify the degree and

intensity of these difficulties in the participants and to help find a way to intervene.

Lexicon of Mathematical Terms and Concepts

Many mathematical terms will be used to describe the mathematical functions -
used by each participant. The detailed description and implication of each term will be
discussed for each case study. The following table serves as a reference for further

reading.

77

Table 3. Lexicon of mathematical terms and concepts*

 

 

Math Terms and Description

Concepts

Cardinality The number of elements in a set

Carrying A digit that is moved fiom a given place value, or column, to
another column of greater place value.

Commutative The sum is not changed by the order of addends in addition

property of addition

Complements of ten

Composing and
decomposing number

Fact extensions

Flexibility of starting
Identity property of
zero

Magnitude

Mental calculation

Mental number line

Subitizin g

“Teens” concept

 

If number X and Y are added and equals 10, then X is Y’s
complement of 10 and vice versa

The operation of combining and separating the numbers. For
example, children recognize 6 (wholes) as the number 1 (part)
and 5 (part), or 2 and 4, or 3 and 3.

Calculations with larger numbers using knowledge of basic
facts. Knowing 2 + 1 = 3 can be used to solve problems 20 + 10
= 30

Counting not only from 1, but being able to start from any
number

The sum of zero and any number is that number (2 + 0 = 2)

The ability to recognize which numbers, quantities, or amounts
are larger than the other

The mathematical operation done by the human brain without
help from other tools

A number in the brain which is spatially represented, possibly
with a left-right orientation, in a continuous analogical format
like in a ruler

An action which refers to quickly glance at the small number of
objects and to be able to make judgments of number.

Knowing fourteen to nineteen is same sequence as four to nine
with the suffix “teen”

 

* This list is based on Adding It Up: Helping Children Learn Mathematics by Kilpatrick, Swafford, and
Findell (2001), Children’s Mathematics by Carpenter, Fennema, Franke, Levi, & Errrpson (1999),
Engaging young children in mathematics by Clements and Sarama (2004), and Glossary of Math Terms by

Benson (2003).

78

Songs and Props

Music functions in many capacities and influences various domains such as the
psychological, emotional, cognitive, and physical. Using an appropriate intervention to
each target area at the right moment is the key to success. Therapeutic interventions often
include purposefully composed songs for children with cognitive impairment. However,
not enough specific interventions for individuals with WS have been developed. Further,
interventions to facilitate the mathematical development of individuals with WS are not
available. Therefore, as a part of this study, songs and props were deve10ped that focused
on mathematics, and the responses of participants were carefully observed to improve the
quality of further sessions. In the next section, the most frequently used songs and props
for all three participants are explained. The particular songs and props for individual

participants will be explained in each case study.

Songs

A series of songs were used with all three participants. These included “How
Many Pennies in a Quarter?” “Nickel, Dime, and Quarter;” “Give Me a Quarter!”
“Finding the Bigger Number;” “Nobody’s Perfect,” and “You Can Do It.” Songs are from
various sources; some are pre-composed songs by third parties, some were composed by
the participants and the therapist, and others are motives from other songs. There were
also a couple of songs and interventions spontaneously improvised either by the
participants or the therapist. Sometimes, modifications of the improvised songs were
necessary for later sessions. Songs used only for specific participants are listed in their

individual case studies or in Appendix A.

79

How Many Pennies in a Quarter? The song, “How Many Pennies in a Quarter?”

(Figure 9) was composed by Big Red and used to learn the value of each coin. The lyrics

were prepared prior to the session and given to Big Red at his first session. Because of

previous experience with Big Red, I knew of his improvisational ability and

acknowledged it. I asked him to compose a song and to rhythmically chant the lyrics and

add rhythms with his drum set. Based on his melodic line, I provided an accompaniment

chord progression and transcribed the song as follows (see Figure 9).

How Many Pennies Are in a Quarter?

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Big Red
a = 82
h C Am G G7
111' n i r r i\ i ix 4] i i I: i L L i m i r J j
In \J l l I II 1 U I I I J T I I r I I I
‘ i if I I1 I I i; i i;- I l1 i. I I UL fl
How ma- my pen - nies are in a quar - ter twen - ty five
5 G7 G7 c c
J 4 r . l r K r I r 1 T r r i r i 1
L I ' I It i sure! ' a ii i as. 4.: - l
W 1 fl 5 '1. i 51 5' ._ I: n
How ma - ny pen - nies are in a dime - ten pen-nies
9
_A F Fr 1 . C . C.
7 If r 1 K I k r I I l\ J l r L I I I
J f I l ]\I I 1\ Ir Ii I } IL {\1 i. i I}
i {r ij—‘d—i- I ('1 =1 r V r" 5rd
How ma-ny pen - nies are in a nic - kel five pen-nies
13, G7 Dm c c
I 1 fl I 1 I l 1 1]
A I J i L I L I I I I\ If r J r I r I _ I]
If“ I r l l\ I 1‘ l ‘ J L I T F I I J I i L]
‘ ' I I I I I I L l ‘ Tl
- d‘ '° 51 d
How ma - my pen - mes are in a quar - ter twen - ty five—

Figure 9. How Many Pennies in a Quarter?

While the song was interesting and catchy, I noticed that it did not serve as a

mnemonic device. The structure of the song was useful for questioning and confirming

8O

acquired knowledge of coin concepts, but was not functional as a mnemonic device.
There was no immediate connection between each coin and its value, and the melodic
pattern used for each coin was too similar to the others and did not make distinct
differences between the coins. In other words, any combination of the name of coin and
value was possible; quarter — ten pennies, nickel —twenty five. While the song was
pleasant and musically successful, it was not itself useful for the purpose of facilitating
memorization.

Nickel, dime, and quarter. I composed the song, “Nickel, Dime, and Quarter” (see
Figure 10) and used it to assist in making direct connections between the name of a coin
and its value after, “How Many Pennies in a Quarter?” failed as a mnemonic device. I
tied a rain stick to the dime to reinforce the mnemonic purpose of the song by providing
additional auditory stimulation. The initial thought was to use a triangle for the nickel and
a woodblock for the quarter. However, it became too difficult to switch between three
instruments within a short period of time. Because it is concise and succinct with few
distractions, this song serves as a mnemonic device, but is a less aesthetically pleasing

piece of music than, “How Many Pennies in a Quarter?”

Nickel, Dime, and Quarter
Eunmi Emily Kwak

D Em A7 D A A7 D

 

Nic kel five Dime

 

ten Quar ter twen ty five

Figure 10. Nickel, Dime, and Quarter.

81

Give Me a Quarter! The song, “Give Me a Quarter! ”4 was based on the motive of
“The Money Song" by Robert Lopez and Jeff Marx from the musical Avenue Q (Lopez &
Marx, 2004). This song includes rest measures that provide time to process the meaning
of each request and to accomplish each task. This song was designed and used to ask
questions and prompt answers that would help with the discrimination of coins.

Finding the bigger number. The song “Finding the Bigger Number” (see Figure
11) was improvised with Big Red as a strategic song to help reinforce the process used to
solve mathematical problems. It was a truly spontaneous song. While participants worked
with the addition worksheet, their lack of understanding of the associative property
became apparent. Accordingly, the concept of the associative property was explained and
reinforced at the beginning of the tasks using single digit addition. Repetitive verbal
instructions were used in an attempt to guide them, as has been suggested for individuals
with WS (Semel & Rosner, 2003). The idea to improvise a song based on these verbal
instructions came about while working through the worksheet with Big Red. He asked to
make a song using the verbal instructions. Although he preferred to compose using duple

meter, he was asked to try using triple meter instead. He created the following song.

 

4 Due to a c0pyright issue, the modified score of "Give me a quarter" cannot be included

in this dissertation.

82

Finding the Bigger Number

 

 

 

 

 

 

 

 

Big Red
fl 1* I \ I 4 :I
u I I I r n I
L] I I I I K I f A 1 I\ j
(J I I :I Ti r II D I I JVI I L I
d j I -» d d J J
“I

Fin-ding the big - ger num-ber

 

 

 

 

 

 

 

I r I\ I I\ I s
. . h. . m Ir] 1 i l as I .1? l : 9:1
_\dz_+ I r 1 1 r r r r :1 v r v i 1
cl j ;: 3‘ i 4 4 ‘
num - ber with ydur fin - gers say-ing the big - ger number

 

that is your an - swer

Figure 11. Finding the Bigger Number.

Roxie ..

These lyrics indicate the steps that the participants needed to use in order to solve

a problem, with rest sections to provide opportunities to answer, and time for the

participants to work on each task. The participants were asked to circle the bigger

numbers when this song was introduced to them. After a couple of questions, the

participants showed that they could skip this process and complete the task without

mistakes. When the song was sung by me or together with a participant, the participants

pointed to or said the bigger number on their papers. The next part of the song is, “Make

the smaller number with your fingers.” The participants were asked to represent the

smaller number with their fingers. They were taught to count upwards from the larger

number. For example, the problem was 5 + 3. They began counting at 5 and speak each

step: “six, seven, eight.”

83

Nobodyis perfect by Hanna Montana and You can do it. Because of their cognitive
limitations, the participants already had difficulty with academic achievement, and they
showed signs of disappointment and distress when faced with obstacles. They were
expected to fail. While their individual characteristics influenced their attitude toward
experiencing failure and the emotional responses of hopelessness, individuals with WS
can also be expected to have some level of learned hopelessness. This was observed in
differing degrees with each participant. Two main songs were used for emotional support.
The first was “Nobody’s Perfect” sung by Hanna Montana. Roxie and Angela both knew
the song and were fans of the artist. The chorus section of the song is:

Nobody’s perfect. . .I gotta work it

Again and again till I get it right
Nobody’s perfect...ya live and ya learn it
And if I mess it up sometimes...hey
Nobody’s perfect

(Gerrard & Nevil, 2006)

The lyrics focus on keeping a positive attitude and emphasize that effort alone can
and should be respected. While the participants already knew the song, they seemed to
have overlooked the meaning. The message of this song and its resonance for participants
can be magnified and internalized by participants if the song is used in a suitable context.
For this reason, participants were given the lyrics on a sheet of paper to help them
classify the words. After the meaning of the song was discussed, when the participants
made mistakes and showed obvious distress during activities and interventions, the song

was played on the piano or a recording was played and the participants were encouraged

to sing along.

84

“You Can Do It” (the composer is anonymous) was also used for emotional
support. The lyrics emphasize effort, not results, and encourage one to continue to try and
not focus on the product. The lyrics “if you try (4th and 8th measure)” were changed in
various versions, such as “if you want” and “if you really work hard.” These two songs

were used to redirect the participants’ focus from the actual accomplishment to the

 

 

 

 

 

 

attempt itself.
You Can Do It!
Anonymus
A . i r 1 it . . r J l k
‘ 7‘ '. I ‘ JIAFII i? l i\1 J"; g}. iffififfl
g) 4 «d"_ if? d 4' '. J:
You can do it You cando -1t You can do it If you try

Figure 12. You Can Do It.

Props

Props for use with each intervention were carefully designed to provide more
tactile than visual information because individuals with WS are known to have
difficulties with visuospatial perception. Even with training as a therapist, it is difficult to
plan and use alternate strategies that do not incorporate visual information. Age-
appropriateness was also considered, and childish illustrations and toy-like materials
were avoided. There were props that were selected and designed before the study, and
props that were designed during the study depending on participants’ needs. The prOps

that were developed because of individual needs had more immediacy and practical use

85

than the ones that were developed without interactions and observations with the
participants.

Coins. Real coins instead of imitations (toy coins or educational coins) were used
in this project to assist in the generalization process as well as to increase age-
appropriateness. Participants had their own jar in their homes, each containing
approximately 40 quarters, 40 dimes, 40 nickels, and 50 pennies. The coins were
sanitized prior to use and left with each participant in their home to provide them with
greater opportunities to work with the coins during the week.

Coin hand props. The coin hand props were to help performance and reinforce the
comprehension of different coin denominations and values. While I worked with Big Red
using my hands and his hands to make the nickels and dimes distinct, the idea of using
drawings of hands to explain the concepts of different coin values emerged. Coin hand
props were prepared for later sessions. A nickel was represented through the illustration

of a hand with five fingers to show the equivalent of 5 cents (see Figure 13).

 

 

 

‘5

 

 

 

 

 

 

Figure 13. Nickel-hand-prop and Dime-hand—prop. *

*This is not an actual size. The actual size example is in Appendix I, Figure 1-3 & 1-4.

A dime was represented with two hands (see Figure 13), and a quarter was

represented with five hands. There were two types of quarter-hand-props (see Figure 14)

86

to put the pattern 1 and 2 together makes 50 cents presentation. For the later sessions, the

actual coin was added to hand props as shown in the figures.

Figure 14. Quarter-hand—prop Pattern 1 & 2. *

 

 

 

 

 

 

 

 

 

 

 

 

 

*This is not an actual size. The actual size example is in Appendix I, Figure l-5 & I-6.

American coins are based on multiples of 5, except pennies. Therefore, using a
method to count by increments of 5 is a useful tool in making money and using
mathematical fimctions. It also became a good tool for promoting the comprehension of
using different units, such as hands or fingers, in place of actual nickels, dimes and
pennies as a way to instill the abstract concept of value.

The coin-classification—chart. During sessions with participants, 1 observed that
they did not have strategies for adding coins and do not know how to arrive at the
designated amount. Participants attempted to calculate the aggregate values of various

coins by adding lower value coins first, then the values of larger denominations. For

87

example, when the participants had a quarter and a nickel, they began with 5 cents and
attempted to add 25 cents instead of adding 5 cents to 25 cents. Participants had a limited
ability to perform addition, but it was a far simpler process to add 5 to 25 by using the
finger counting method. Unfortunately this was impossible to perform when 25 was
added to 5; when the larger denomination is added to the smaller. Knowing the names
and values of coins is not beneficial unless it also takes the form of ftmctional knowledge.
People with WS need a strategy or procedure to efficiently calculate the sum and value of
an arbitrary selection of coins. Figure 15 shows the coin-classification-chart was utilized
to facilitate their calculations. Participants were asked to place coins in order from largest
to smallest and to add from left to right. Strategies for each coin were introduced. The
value of a dollar was used to provide a consistent scale and the number of quarters used
to reach 100 cents was memorized using songs developed for each denomination, i.e., 25,
50, 75, and 100.Participants were instructed to tap twice when saying the numbers in 5’s
for dimes (the first tap with 5, the second tap with 10); nickels were a tapping when

saying the numbers in 5’s’; and pennies were a tapping with saying the numbers in 1’s.

 

 

25 10 5 1

 

 

Figure 15. Coin-classification-chart. *

*The chart is not actual size. The actual size chart is in Appendix I, Figure L2. The circle did not represent
the size of coin but the placement for the quarters, dimes, nickels, and pennies.

88

Number flashcard. The article by O'Hearn and Landau (2007) regarding
mathematical skills in individuals with WS was newly published while this study was in
progress. The article indicates that individuals with WS appear to have difficulty with the
magnitude of numbers. Number flashcards with diagrams (dots, stars, squares, or
triangles etc.,) or objects (stamps, seals, or smiley faces etc.,), and with and without
Arabic number were used to evaluate the participants’ abilities to distinguish orders of
magnitude. Commercially manufactured number flashcards were avoided because of their
child-oriented illustrations. The range of numbers of objects used was fiom l to 10 and
20. A card with 20 dots was included to confirm their ability to differentiate 20 dots from
other cards. The objects were organized in two line formations; the top line has 5 objects

and the second line has from I to 5 objects.

 

 

 

 

 

 

 
 

 

 

 

 

 

 

Figure 16. Example of number flashcard. *

"‘ This is not the actual size of the cards. The actual size of these cards was a 4 by 6 inch index card.

89

Analog clock. An actual analog clock that included marked and numbered
increments for minutes (i.e., 5 for 1 o'clock, 10 for 2 o'clock, 15 for 3 o'clock and so on;
see Figure 17), was used in this project. An actual analog clock was chosen rather than an
educational practice clock or toy clock, to provide age-appropriate materials as much as
possible. The plastic cover over the face of the clock was removed in order to allow
participants touch the big and little hands for additional tactile stimulation. The minute
increment of this analog clock was based on units of five, echoing the fimction and value
of a nickel. The relationship of clock increment and the use of a nickel could be naturally

reinforcing and strengthen the concept of multiples of 5.

 

Figure 1 7. Example of analog clock.

90

Address numbers for address plaque. Brass address numbers typically used for
number plaques for homes were used in addition to written numbers as part of the

continuing effort to provide more kinesthetic and tactile stimulation.

12378

Figure 18. Example of address numbers.

Summary
In this chapter, the principle interventions and the major foundational materials
for the therapeutic interventions were introduced and explained. The songs and props
were developed prior to, and in the course of the project. There were some unexpected
responses to the songs and props. Detailed information about interventions, songs, and
props were provided here in order to more clearly understand the case studies and to help

music therapists and others to use them for other individuals with WS.

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CHAPTER V

THREE CASE STUDIES

The three participants in this study told their stories to me over 10 weeks. I have
attempted to convey their stories without my own analysis, but in some cases, it was not
possible to solely narrate the facts. The rich information about themselves, their
difficulties, frustrations, and enjoyment was astonishing. In this chapter, I will introduce
them as separate cases so as to understand the participants as unique and authentic
individuals. The participants’ responses to mathematical exercises are rather difficult to
understand with typical logic. Their way of thinking was different; they did not think and
process the mathematical tasks the ways that most people do. Accordingly, the
participants’ exact wording and mistakes and the contexts in which these happened are
illustrated in order to follow their patterns of thinking and thought processes. When I
believe it may be difficult to understand the participant’s rationale, answers will be

provided in the analysis and interpretation sections.

92

Big Red Clown

When Big Red was asked to select a pseudonym for this project, he
proudly announced, “My name is Big Red Clown” without a
moment’s hesitation. Besides being a drummer, being a clown is his
secondary passion. He was dignified when he handed me his
business card with a picture himself in a clown costume.

Big Red Clown was 21 years old and has Williams Syndrome. He is involved in a
local band whose members are typically developed individuals (TDI), individuals with
WS, and individuals with special needs. I first met him at the 2006 Great Lakes Region of
the American Music Therapy Association Conference. Big Red was a speaker for the
presentation along with his music therapist. He started his portion of the presentation with
an opening comment, ‘Williams Syndrome is the best thing that ever happened to
me,” and stated, “Music is my life.” His speech was impressive, touching, and moving.
He talked about his journey with music and how it helped him to overcome his
difficulties. During that extraordinary and thought-provoking presentation, the ‘dove’
incident that I described earlier occurred. Big Red could not pay attention to his
presentation and kept looking at the dove. The contrast between his outstanding speech
and his difficulty in focusing his attention were marked.

Big Red is an outstanding musician. The day after his presentation at the
conference, I noticed he was the drummer in a band that had performed music for the
music therapy conference during breaks and lunch periods. When he played with his
peers, be blended into the group; he was simply the drummer. However, knowing that he

is diagnosed with WS changed my appreciation for his talent. There are a few WS

musical celebrities, but I have never encountered them personally. This conference was

93

my first time actually observing someone with WS who had firlly developed his musical
potential and played as a professional musician.

Big Red has several mantras that he often recites: “Music is my life,” “Practice
makes perfect,” and “I need to focus.” He has picked up many sayings and verbal
mannerisms from various teachers, tutors, and other significant individuals, but there did
not appear to be much application of these messages in his behavior. “Music is my life"
was the statement that initially caught my attention, but after a while, it became
bothersome to listen him say it over and over. The first time I spoke with him and heard
repeat the phrase I was impressed, but the more I heard him say it the more irritated I
became because of the inconsistency between his quotes and behavior. His inability to
execute what he had memorized indicated that either he did not actually comprehend the
meaning, or there were great difficulties in self-regulation and executive functions.

During my sessions with Big Red, my main focus was interaction with him, but,
from time to time, I needed to take care of other things for the session, such as changing
camera angles and preparing for next intervention. While I was transcribing his sessions,
I noticed that he looked sad and dejected when I was not interacting with him. His facial
expression radically altered according to the level of my attention to him. The more I
evaluated and scrutinized the videos, the more I felt uneasy about his mood swings. Why
did he not appear content when by himself? Were his smiles real or fake? He only looked
happy when I interacted with him and gave him my full attention. While many people
have a tendency to use a different persona depending on with whom they are interacting,

the extremes Big Red demonstrated were atypical. The happiest moments were when we

94

played or sang together. He seemed to be confident in what he was doing. He knew how
to lead the musical experience and that made him happy as he could be.

Big Red was interested in my country of origin and he wanted to learn how to say
some words in my language such as hi, good-bye, bravo, and funny. He memorized these
words and used them often. Big Red is a born actor; he knows what actions go with
which song and makes appr0priate gestures and facial expressions for each song. He has
a big Muppet for his clown job and made several different voice patterns for his it
depending on the situation and fluidly used them for his act. He seemed happiest and
most fulfilled when he performed for young children as a clown.

When he knew he had performed well, or hit it, as he called it, he laughed proudly
and announced, “I am impressed with myself!” When he realized that someone was
uncomfortable, it seemed as though he could physically feel it and he wanted to ease their
pain. When he was stressed because of his inability to solve basic mathematical
questions, such as finding the sum of coins or the number on a card, he became
distraught. I often found myself asking, “Does he have to know this concept?”, or, “Is this
a necessary skill.” Big Red really made me doubt the necessity of mathematics in his life
and to reconsider the importance of the role of mathematical interventions in his therapy

setting.

Language
Talking with Big Red was an interesting experience. I had spoken with him
outside this project at the music therapy conference, as well as in other WS conference

settings and outings in the Michigan area, and at the 2007 WS Christmas party. I also

95

interacted with him at music camps in which I was a music therapist and he participated
as a camper, and as a counselor who worked as a helper for my music therapy sessions at
the camp. My initial conversations with him were polite, elegant, and pleasant. He always
greeted people with appropriate small talk or questions concerning the weather, or travel.
In my case, since he knew I drove about an hour and half to get to his home and it was
wintertime in Michigan, he was often concerned with whether or not I had eaten and what
the driving conditions had been. He is a gentle man with good manners and cares about
how others feel. He uses polite speech, such as “Can I ask. . .?”, ‘Would you mind. . .7”,
and "May |...?” When we unexpectedly met at the National AMTA conference in Kansas
City, Missouri, in 2006, he expressed joy to see me there and asked me why I was
attending. There were no awkward moments during these initial conversations, but after
these formal greetings, the conversation did not progress until we talked about music, his
favorite band, the Beach Boys, or the album Smile by Brian Wilson who was the lead
singer of the Beach Boys. Conversations then became awkward and choppy and he
always began to look for someone with whom to start a new one.

Big Red sometimes articulated well how he perceived situations. I often wished
for a pen and a paper to write down his delightful words. When I asked him to compose
the song, “How Many Pennies in a Quarter?” I let him know that I would be using the

song for other participants. To that he replied,

“Sometimes it is hard for people to learn it a different way.
Because they are so confused and they don’t know. When they
teach them the right way and they got it. So it helps me. I can
sing the song everyday to learn (Big Red 1-27208).”

96

Even without my explanation, he understood the benefit of using music for his
mathematical skills and was able to express it in his own terms. Later in the same session,
he said, “It really gives me the ability to work with math in easier ways (Big Red 1-

33:32).”

Finger Dexterity

Big Red’s fingers looked as if he had advanced arthritis; his joints were stiff and
the bones were twisted. When he picked up small objects like coins and pencils from a
flat surface, he was not well coordinated and it took him a long time to do so. When he
held up fingers or curled them down, it did not look natural; it looked like it took a great
amount of effort and appeared to create pain. The way he held up three fingers was
similar to that of a one-year old who is trying to move a single finger for the first time.
When I asked him to do complete tasks that required finger dexterity, I felt as though I
was giving him a challenging task that caused him extreme discomfort. I wanted him to
use his fingers to help with adding numbers, but difficulty in finger dexterity slowed
down the process could not be used effectively. His limited understanding about nmnbers
and operations required him to have a concrete object as a reference. Hands and fingers
as concrete props would have been useful because he carried them everywhere. However,
because of his difficulty with finger dexterity, his hands and fingers could not be used

efficiently (Big Red 8-45zl3).

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Attention

Big Red had significant difficulties in maintaining attention. His attention span

was short and his self-regulation ability was limited. He knew what he was supposed to

do, but distractions, especially auditory distractions, quickly diverted his attention.

Big Red:

Big Red:

Big Red:

Emily:

Big Red:

Big Red:

Big Red:

Big Red:

Emily:
Big Red:

(A phone is ringing and he tries to not look
away)
We ignore the phone.

(Emily does not hear the cell phone ringing
during the session during the video clip
analysis)

1, 2, 3, 4, 5, 6, 7, Oh! No! 8, 9, 10

That’s my cell phone going off.

That’s 5, right?

No

Think before you answer.

(A bird screeches in background; he looks
away)

Oh! My!!!

(A bird is singing and a phone is ringing, and
he looks away)
Uh-Oh! ...(calls out to father) Who is that?

6, (A bird screeches and he looks away)
It’s 7.

Yeathocus

It’s 7.

Big Red had a pet bird with an open-door cage. When the bird flew around and

made noise, he looked at her. He also had a big grandfather clock that chimed on the

quarter hour that he looked at each time it chimed. When his father came back from

98

work, Big Red needed to greet him immediately, and this also distracted him from tasks.
He could sometimes hear things I did not and that also distracted him. Some of these
auditory distractions included: Each time he looked away from the task, he tended to lose
his train of thought. He often had to restart tasks. Even if he could keep working on the
task after the distraction occurred, he usually did not get the correct answer.

Music, as an auditory stimulation, was also a distraction for him. I could use
songs to ask him questions, but while he worked on the task, I needed to sit still and be
quiet. At the beginning of this project, instead of providing complete silence, I provided a
short musical phrase. However, while music played, he could not focus on tasks. Instead,
he would start harmonizing with the song, finish the musical phrase, or tap his knee and
then losing track of what he was doing. Several times during these events, he would
cover his ears and say, “I have to concentrate.” It was difficult for me to sit and wait for
him to complete his tasks because, as a therapist I had a tendency to want to help, and
doing nothing seemed to be waste of session time. However, it quickly became clear that
he needed to have absolute silence to work.

All sessions were recorded using a digital video camera. Each time it was set up,
Big Red wanted, and needed, to greet the camera. In most cases, distraction caused by a
camera wears off after 3 or 4 sessions. However, Big Red needed to be reminded to
ignore the camera every session. During the second half of the project, he incorporated a
comment that reminded him to disregard the camera, but he continued to greet the camera
through the final session.

Big Red was interested in many personal things about me such as marriage status,

family, age etc. “How old are you?” “Are you married?” ‘What did you eat for

99

lunch?” (Big Red 3-41 :24) These unrelated questions were randomly asked while he
worked on his task. Later on during the project I asked him if his questions were related
to the task. He typically answered, “No, it is not related." Then I observed that he asked
unrelated questions and instructed himself saying, “Do not ask a girl’s age. That is not

appropriate," “Do not ask personal things” (Big Red 2-43:18).

Big Red: 22 dollars
(sings the song, Don‘t forget the change.
Right after he finishes song)

Big Red: How do you say bravo in Korean, when
someone...?

Emily: I cannot think about Bravo, Bravo, Bravo.
Big Red: How do you say good?
Emily: fag—act.

Big Red: That’s was so error Yeah!!!
(Big Red 8—40216)

Big Red often said “focus" with the hand gesture of touching both sides of his
temples with his hands, but he was only able to focus for a few seconds (Big Red 3-6200).
His sayings, body gestures, and behaviors suggested that he was aware of his difliculty.
He had been redirected many times by other authority figures. However, he did not

develop self-regulation skills and had limited ability to redirect himself to focus on a task.

Mathematics Exercise

Big Red was clearly aware of his inability in mathematics. He had. a math tutor
prior to this project. His mother decided it would be better to have one person assist him
in mathematics than two people working on the same subject but in different ways. Big

Red had many roadblocks that prevented him from improving his mathematical skills. I

100

needed to readjust his interventions every session because he did not understand the
prerequisite concepts required to solve the math problems.

Coin discrimination. My first session with Big Red began with identifying
different coins using the song, “Give Me a Quarter!” He did not have problem with
pennies, but the other three coins were too complicated for him. While he could
discriminate between the four different coins in his hand, he had a difficult time finding
different coins when they were disorganized in a pile. When he was asked to find each
coin, he often said “nickel normal,” “quarter big,” and “dime tiny (Big Red 2-1 1 :15, 3-
14210).” However, this appeared to not help him find each designated coin because he
lacked the understanding of the meanings of normal, big, and tiny. Occasionally he
asked, “A nickel is a nickel smaller or bigger?,” this suggests that he had not .
completely understood the connections between the names and sizes of coins. Rather, he
had worked on this task previously and somebody had explained how to find different
coins using a different method than I wanted. This also indicated that his strategy to find
different coins was to use verbal instructions to guide him.

Big Red could not process information fast enough to keep up with the tempo of
the song “Quarter, Dime, Nickel, Penny”. In addition, his fine motor skills were not
fluent and hindered his performance, causing him to take a long time retrieving a coin
from the table. I sang the type of coin in two musical beats and gave him six beats to
work, which helped him to find the coins. However, when he only had two beats to
complete the task, he became agitated and anxiously rubbed his hands together or

nervously shook his two hands in front of his chest like some individuals with autism.

lOl

He said, “Nickel normal; dime, tiny; quarter bigger” to guide him to find the
designated coin. However, because he often confused the size of the coins, the mnemonic
song, “Normal, Tiny, Big” was composed to help his performance. The simple rhythmic
pattern helped him distinguish different coins and was introduced based on his own

understanding that a nickel is the normal size, a dime is a tiny, and a quarter is big (see

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 19).
Normal, Tiny, Brg
Eunmi Emily Kwak
9 A
E {Jr 1 1 1 L I,1 ?1 r} 1 H
U . :l l-_.
Nick - el Di - ime and Quar-ter
Nor - mal Ti - ny Big

Figure 19. Normal, Tiny, Big.

Although he had no difficulty singing the mnemonic song, he did not appear to
comprehend the meaning of sizes such as normal, tiny, and bigger, because he had
difficulty understanding comparison vocabularies. He also used my facial expressions to
confirm that he had given the correct answer (Big Red 2-1 1 :00). If he felt agitated or that
the pace of the question and answer was too fast, he randomly picked the coins.
Occasionally, he did not even look at the coins, but fixated his eyes on me and picked up
the coins. He tried hard to answer correctly without knowing what else he could do and
kept asking, “How am I doing so far?" His performance in discriminating different

coins was inconsistent.

102

Coin calculation. The concept of different values for each coin was introduced in
introduced in the second session. Big Red was confused by different coin values and kept
asking about it. “Dime is worth 5 cents? Dime is worth 25 cents? 10?” His questions
demonstrated that he knew that there are three values for the three different coins, but that
he could not make a clear connection between the name and the value of the coins. I
provided the lyrics for “How Many Pennies in a Quarter?” and Big Red composed the
song. The song itself was delightful and enjoyable, but it failed to help him understand
the concepts (the details of this song were discussed in the songs and pr0ps section. p.
74). After the song “Nickel, Dime, and Quarter” was introduced, instead of asking
questions about the value, he started singing the song and made a better connection
between the coins and their values. At the end of the third session, he was able to recite
that “a nickel is 5, a dime is 10, and a quarter is 25” However, I continued to doubt that
he truly comprehended the distinction. The lyrics for the song “Five Pennies Make a
Nickel” were provided and the song was composed by Big Red to help him understand
the concept. I used a visual prop to go along with the lyrics in order to help reinforce the

concept as illustrated in Figure 20.

103

 

Penny

00000 ©0000

 

Nickel II II

 

 

Dime O

 

 

 

Figure 20. Visual prop for five pennies make a nickel.

*This is not an actual size of the prop. The circle in the actual prop is the actual size of the coins and the
participants can place different coins in each spot.

Big Red and I made errors by changing the names of coins in the song “Five
Pennies Make a Nickel.” For example, instead of singing “two dimes and a nickel make a
quarter every time,” he sang “two dimes and a guarter make a quarter every time,”
or instead of dimes, we used nickels or quarters or whatever unit came to mind. On paper,
it was a clear and easy concept to grasp, but when we sang the song, he needed to think
about the name and value of the coins with rapidity. “How Many Pennies in a Quarter?”
was not effective for learning because the melodic lines of the various phrases were too
similar to each other, making the lines interchangeable. The song could not promote the
comprehension of coin combinations unless a solid understanding of the value or sum of

each coin was already in place. Therefore, the song failed to serve its primary purpose.

104

Five Pennies Make a Nickel

Big Red

 

 

 

 

 

 

Five pen - nies makea nic - kel Two nic - kels makea dime

 

 

 

 

 

 

 

5
n l l — l 1
I I I\ I I j I k I A 1 I K j I 1
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c) . . v\——" ' d d . 0
Two d1me_ and a mc-kel make_ a quar-ter eve-ry time—

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

9
h
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II I I I t . IX 1 I I I l'\ I I j
Jm 4 I I I I l I J *I IL I I I II I I j
W I l I I ‘I I F I; I A' A I I 0 ’ I I I
I can trade a quar-ter for three mc - kels and a_ dime—
13
9 If FIT I T K I A . 41 IK I T I
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Four quar - ters makea dol - lar Make a dol-lar every_ time
17
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I J_ I I L l I 17 ]
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1 can_ tra e__ a quar-ter make a dol - lar every - time
21
A 1 J
J L I l I I I IX I II
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4155 I II I I\ I II I I I L I I I I .II 2 IX I II
‘ ‘ ‘ I I I I I I J V V II I ll
. ° V v d 4 d 4 «f' 4
Five pen - mes makea mc - kel make a dol-lar every - time

Figure 21. Five Pennies Make a Nickel.

Beginning with the second session, the relationship between the value of coins

and the numbers of fingers on hands was established to connect the idea that a nickel is a

hand, and a dime is two hands (1 0 fingers). He could add two sets of five fingers as 10

fingers when using actual fingers. He could add up to “four hands” to 20 fingers by

counting by 5 (i.e., 5, 10, 15, and 20). However, immediately after he finished this

105

addition using four hands, I changed the method by asking the same question in a slightly

different format (from my point of View); then he completely lost track of his process.

Emily: You have ten fingers and my ten makes

Big Red: 5, 15

Emily: No, No. Think about it your hands have 10
fingers. My fingers are 10. All together?

Big Red: 10, 15, 20, 25 (counts my first hand as a 10)

He appeared to be able to recite numbers in five’s, but, since I used 10 to describe
the10 fingers of a person, he thought that he needed to start from 10, which led him to the
wrong answer. In addition, while we worked on the dime hand-props and nickel hand-
props, he demonstrated that he could not add up the dime hand-props for 10 plus 10. This

indicated that he did not comprehend the operation and solution of 10 + 10 = 20.

Emily: (shows a dime-hand-prop)

Big Red: 10, right?

Emily: (nods her head “yes” and shows the second
dime-hand-prop)

Big Red: 25 (examines Emily’s facial expression)

Emily: (shows non-approval expression)

Big Red: 10, 20. There we go.

Emily: (shows the third dime-hand-prop)

Big Red: 25 (examines Emily’s facial expression)

Emily: (shows non-approval expression)

Big Red: 30, there we go.

Emily: (put down a dime-hand-prop and a nickel
hand
prop)

Big Red: 10. Just 10 (points to a dime-hand-prop).

Emily: Ah-ha, this is 10. (points a nickel-hand-prop)
and this

Big Red: 25

106

Between hands and fingers, he had a hard time understanding that, depending on
the unit, the answer needed to be changed. One hand has five fingers is the same concept
as a nickel is five cents. When given four nickels and asked to figure out the sum, he
made similar mistakes: instead of 5, 10, 15, 20, he counted 1, 2, 3, 4 and answered “4”
without specifying the unit i.e., 4 hands or 4 fingers. For Big Red, changing the value to a
different unit was an abstract concept. His difficulties in converting between the units
could be understood in light of the fact that typically developed individuals have
difficulties when converting from inches to centimeters or from pounds to kilograms.

The reason he kept saying 25 was unclear, but indicated that he did not
comprehend the pattern of using increments of 10. In order to show him the differences
between a dime and a nickel, a dime hand-prop was placed on the table as a constant, and
the other props were switched back and forth between a dime hand-prop and a nickel
hand-prop. In other words, if I held up the dime hand-prop, he needed to add 10 and the
answer would be 20, and if I held up the nickel hand-prop, he needed to add 5 and the
answer would be 15. We practiced together. However, while he could sing “Nickel, Dime,
and Quarter,” he could not figure out whether the dime hand-prop or the nickel hand-prop
was 5 or 10. The best example of his confusion was when I presented three dime hand-
props. He declared the first line of the dime hand-prop as 5, the second line as 10, 15, and

the third line as 30 (see Figure 22; Big Red 3-48:57, 3-49z51, 3-51207).

107

 

 

W Sit ’5

W W -)10,15
-)30

Wfiiew

Figure 22. Three dime-hand-props and Big Red 19 answers.

 

 

 

 

 

 

 

 

 

The majority of TDI understand that 5 plus 5 equals 10, therefore, a nickel (5) and
a nickel (5) equals a dime (10). This is not an abstract concept. TDI are able to transfer
the understanding of ‘5 + 5:10’ to ‘a nickel + a nickel = a dime’. Big Red’s facial
expression suggested that this was an abstract concept that he did not comprehend. This
was difficult to explain, because the concept was so fundamental. It was difficult for me
to think of other ways to introduce the connection or explain it. While it could have been
my language limitations or my different cultural background, even when my limitations
are considered, his capacity for understanding a basic mathematical skill seemed to be
limited.

During the fourth session, a quarter-hand-prop was introduced and he was asked
to figure out how many hands and how many fingers were there. Instead of counting all
the fingers together, he counted the fingers on the prop asl , 2, 3, 4, 5; and started from 1
again for each hand. Since we worked on dime and nickel-hand-prop in the previous

session, I expected him to count by 5 and come up with 25. However, when I asked him

108

how many fingers in the prop, he kept saying “5” and he counted the fingers on one hand.
When I covered up the other 3 hands and showed him 2 hands like a dime hand-prop, he
was able to say 10 fingers. The slight change from showing only one hand (a nickel) or
two hands (a dime) to five hands (a quarter) confused him, and he was unable to identify
the similarities between the three different props. During the first attempt, I tried to teach
him a quarter hand-prop with 10 for the first line, 20 for the second line, and 25 for the
third line. After he practiced, when he was asked to count fingers in a quarter hand-prop,
he counted 10, 20, 30, 35, 40, as shown in Figure 23. Immediately after his mistake, and

without correcting it, I asked him to do it again as seen in Figure 24.

 

(ti/l -) 10, 20
l 9&“7

(\tlll/ééttl? 930,35

 

 

 

 

 

 

 

(it; 940

Figure 23. T we dime-hand-props and one nickel-hand-prop, and Big Red’s answers.

 

109

 

gill/Z (WI/7 -) 10, 20

\
/

 

 

W32; Rm” 925, 30

9 35
iii

 

 

 

 

 

 

 

Figure 24. Two dime-hand-props and one nickel-hand—prop, and Big Red 19 answers.

During the session with the combinations of nickel, dime, and quarter hand-props,
it was clear that Big Red did not understand the relationship between 1 hand and 5
fingers. Because of his inability to understand this concept, as well as his confusion with
different coins, he was not able to keep working on coin calculation.

Written addition. Big Red was able to do some single digit addition. The
following worksheet (Figure 25) is an example of how he performed on an addition
worksheet without help. He picked the equations he knew he could handle. He appeared
to memorize the answers for the questions with two same number addition problems,
such as “5 + 5” or “2 + 2.” The interesting thing was his response to questions # 7 and #
9. While he scanned these questions, he skipped # 7, but worked on # 9 and correctly
answered it. This incident indicates lack of understanding of an associative strategy to
switch the order of the variables.

He correctly answered “3 + 6.” The process by which he arrived at his first

answer was pointing to imaginary dots on the paper. This was not successful. The second

110

attempt was by holding up 3 fingers with one hand, and making 6 with the other hand by
putting his thumb up (his way of indicating 6), and air-fingering for 6 and counting 3
fingers by speaking “7, 8, and 9,” He reached the correct answer, but when he counted 3
fingers on his hand, then pointed to his ring finger once, pointed to his index finger twice,
and skipped the middle finger. It was difficult to decide whether it was a simple mistake
or because of the camera angle that it looked like he pointed at the same finger twice and
missed another.

The song “Finding the Bigger Number” was composed by Big Red to help with
the associative strategy used for written addition problems. He had a difficult time
holding up the correct number of fingers, especially with 6, 7, 8, and 9, because of his
lack of understanding of composing and decomposing—the concepts of “parts” and
“wholes” that is 6 as 5 + l, 4 + 2, and 3 + 3—and his fine motor skills difficulty. When
those digits appeared in the questions, he often asked, “How do I do that?” or randomly
held up fingers and waited for me to confirm whether the answer was correct. While we
worked on this task, it became clear that he needed to master the skill of holding up

designated fingers before he could master it.

111

BIGRED !!!!!!!!!!!!!!!

L9+2=ii
25+5=i€i
38+2=§3
43+2=1f
5.6+5=

Figure 25. Big Red’s worksheet. *

"‘ Hand written

Magnitude. While contemplating Big Red’s inability to perform calculations, it
became clear that he was missing a much deeper foundational mathematical concept. A
clue was provided in the Development Guidelines for Number and Operations by
Clement and his colleagues (see Appendix H) and the newly published article about WS’s

mathematical abilities (O'Heam & Landau, 2007); difficulties in magnitude and

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cardinality. Magnitude refers the understanding that if there are more than two sets of
objects, which set has more objects. Cardinality means understanding that the Arabic
number represents a certain number of objects (Nickson, 2004; O'Heam & Landau,
2007)

Big Red’s ability to subitize—to glance at the object and label it with a number—
was tested and practiced with a set of flashcards numbered from 1 to 10 objects and 20
objects (see Props, Number flashcard. p. 86.) He easily mastered the cards up to 5
objects, and the card with 20 objects. However, he was not able to find cards with 6, 7, 8,
9, and 10 objects. The cards with 5, 6, 7, 8, 9, and 10 objects were randomly placed in
front of him and then the placement was not changed for each card. For the cards with 5,
6, 7, 8, 9, and 10 objects, Big Red was instructed to count from 5 and then by using an
increment of 1 instead of counting from 1. Each card was counted together to help him to
learn the flexibility of starting from any number. The objects on the cards were organized
in 2 lines as shown in Figure 16 The first line always has 5 objects so it could be
processed as one unit that has five objects. Therefore, the first line did not need to be
counted one by one. This fact was explained every time he counted the first line during
our practice.

After we counted the cards together, and practiced finding the cards without
changing the placement for about 25 minutes, he was still not able to find the cards with
6, 7, 8, 9, and 10 objects. He kept trying to count from 1; he could not start from 5. In his
logic, it did not make sense, as numbers have to start from 1. In addition, he often
counted the same objects twice or skipped an object. When it happened the first or second

time, I overlooked it. However, after the project had been completed, and after repeated

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viewings of his video clips, those incidents appeared to not be a simple mistake. It raised
doubt that he would ever develop the concept of one-to-one correlation with an increment
of 1. He frantically moved his hand over the cards one after the other. He looked so
desperate. He gazed back and forth between the cards and me with a “silent scream for
help.” He could not find the designated number of object cards. Out of 6 different cards
(from 5 objects to 10 objects) with the same placements and 25 minutes of practice on the
same task, he was not able to find the designated card.

Address numbers for address plaque. Brass address numbers typically used on
houses were used for some exercises. While the design of the font'represented the
numbers well, Big Red wanted to know, “Why this is cut off here (points number 8 in
Figure 26)?” His other concern was between number 1 and 7; while the difference
between 1 and 7 is obvious, when they are presented separately, The numbers] and 7
have similar shapes except the longer front line for 7. Big Red was confitsed by those two

and made same mistakes several times.

   

Figure 26. Address numbers.

Clock. Big Red had a difficult time distinguishing between the big and
little hands of the clock. However, he understood clockwise direction, and could correctly

apply the concept when reading an analog clock. To make it easier for him to identify the

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big and little hand, I asked him to put his left index and middle fingers at the end of the
little (hour) hand. The instructions for the reading the clock were used with much the
same wording, the same tactile cues, and the following sequence with a tactile one as
illustrated in figure 27:

i. Find the big hand

ii. Clock goes this way

iii. It passes _ but does not pass _

iv. o'clock.

v. OK, Find the little hand

vi. Clock goes which way?

vii. sing 5 number song and minutes
viii.use the number

 

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Figure 27. Clock with tactile cues.

Illustration by Hyung Bun Lee

Big Red could follow these step-by-step instructions and read a clock with my
guidance. However, he was not able to perform the process by himself. Even with the
tactile cues of the big and little hands, he often asked me to confirm his answer. To use
either of the increments of 1 or 5 depends on context—the big or the little hand—and this
confirsed him. When the minute hand pointed to the 1 1 (55 minutes), the hour hand
seemed to be closer to the next hour than the actual hour. The visual and spatial

information offered by an analog clock was too complicated and confused Big Red.

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Reading such a clock requires a number of small steps that themselves require different
strategies. After all our sessions, I came to the unfortunate conclusion that reading an

analog clock was not learnable at this point in his level of function.

Music

Big Red is an accomplished musician. He is a drummer who can play many
different rhythmic patterns on the drum and perfectly support either a single musician or
an ensemble with his meticulous playing. He also learned how to play piano with another
music therapist. He plays with both hands, the right hand for the melody line and the left
for the accompaniment. His accompaniment pattern is two notes chords, typically the
bass and 5th note of each chord instead of a complete triad. He effortlessly played and
seemed to easily find the two notes that went along with the melody line instead of
playing by reading chord symbols or other instructions. When he did not ‘like’ the
‘sounds,’ he tried different combinations of notes with the melody line and decided what
he liked most. He appeared to learn the music by ear, not by reading the notation. He
could also play the djembe (African hand drum), as well as sing and dance.

Big Red improvises well. The first time I asked him to create a song based on
some lyrics, he started to rhythmically chant, picked the rhythmic pattern, added a little
melody, and started singing, creating the song, “How Many Pennies in a Quarter?” While
giving presentations at the national AMTA conference and workshops, Big Red’s video
clips of improvising the song “How Many Pennies in a Quarter?,” which depicts the
process of his improvising a song based on the lyrics, impressed audiences with his

ability. A music therapy student approached me after one of the presentations and said

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that she was not sure she could keep up with his musical talent. His musical ability made
me feel the same way. He made me doubt my capacity to fully support his musical
potential.

Whenever I asked him to make a new song out of lyrics, he usually had no
difficulty. He simply read through the words a couple of times, picked a rhythmic pattern,
and began tapping, drumming, or singing. He preferred to improvise with his drum set.
One exception was during the Christmas season when we sang several Christmas songs
before the session began. He kept singing the lyrics with different Christmas tunes. I
asked him to compose a song in 3/4. When I pointed out that he sang it with a Christmas
tune, he started with a new melody, but he kept going back to the Christmas tune. At that
particular moment, Big Red’s pet bird was especially noisy, and he seemed to be agitated
by the bird. He touched and rubbed his forehead, which is his sign of distress, told
himself “focus,” and regained his composure. After he reorganized his thinking, he was
finally able to improvise “Finding the Bigger Number.”

His songs are catchy, repetitive, and easy to follow (see Figures 9 and 11). The
melody lines reflect the characteristics of the words such as intonation and long and short
vowels. Big Red continues to have drum lessons and other music-related tutoring
sessions. However, although he has not had formal composition lessons, he seems to
compose intuitively. He created five major songs during our 10 sessions along with other
short musical phrases. When I asked him to make a different drumming pattern to
describe the words normal, tiny, and bigger, the rhythmic patterns sounded the same. I

kept asking him make it different, but it appeared he could not. However, when I brought

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the video home and listened again, there was subtle difference, that I had not detected
earlier.

Another song he composed was “Five Pennies Make a Nickel” (see Figure 21).
Almost immediately after I gave him lyrics and asked him to rhythmically chant them, he
said that he wanted to use a Calypso rhythmic pattern, and the song was coming out of
his mouth. Then he added a drum beat accompaniment, as if he knew the song from the
bottom of his heart, and just needed an opportunity to sing it. While he improvised and
composed a stunning song with the lyrics, Big Red and I kept changing the quantities and
names of coins.

Before this study, I myself did not know which coin was labeled as a nickel, dime,
or quarter. I did grasp that the big one was the quarter, but the others I knew as 5 cents
and 10 cents. I am from another country, and my own second language difficulty was my
own obstacle. Not knowing the names of coins did not bother me living in an American
culture until this study. When I first wrote the lyrics, I had time to think about them: “OK,
a nickel is 5 cents and a penny is 1 cent; therefore, 5 pennies makes a nickel”; 2 nickels
are 10 cents and 10 cents equals a dime... etc. However, when I needed to sing along
with Big Red, I did not have time to consider the value of, the name of, and the sum of
the coins. The lyrics were too complicated and the tempo of the song was too fast. We
both made mistakes with nickel, dime, and quarter, and his mother could not stop
laughing at us. The song failed as a mnemonic device, because I had inserted too many
concepts into the song.

If the provided lyrics did not fit into a standard measure format, Big Red

spontaneously added a line or two lines to make lyrics into an 8 or 12 measure format..

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The original lyrics I prepared for “How Many Pennies in a Quarter?” were:
How many pennies in a quarter? 25
How many pennies in a dime? 10
How many pennies in a nickel? 5
However, he added pennies after 10, and 5 to fit the melodic contour, and added
one more line “how many pennies in a quarter? 25” to make a song with 8 measure
format as followed:
How many pennies in a quarter? 25
How many pennies in a dime? 10 pennies
How many pennies in a nickel? 5 pennies
How many pennies in a quarter? 25
Another similar incident happened later in the project during session 7 when we
worked on composing a song for addition. The initial lyrics were:
Finding the bigger number
Making the smaller number with your fingers
Saying the bigger number
This strategic song was composed to remind him of the process of addition and the steps
necessary to complete the task. When he started to improvise this song, he added “putting
it all together” as a forth line to finish up the song.
A third case was with “Five Pennies Make a Nickel.” Big Red added two
measures, “Five pennies make a nickel, Make a dollar every time” to make a song with
12 measures. The second part of the added lyric was not logically correct, but melodically

it matched perfectly with the song. Sometimes I did not notice his additions until I

reviewed the video, because they were so smoothly blended into the song. He did not

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stop improvising or hesitate. I was astonished to observe him create music so naturally
and effortlessly.

During music camp, about a year before I began the study, I improvised songs
based on themes from the musical Beauty and the Beast. Big Red appeared from nowhere
and began playing the drums, and we played together for about 30 minutes. That was the
first time I noticed his ability to improvise and I really enjoyed that time with him. While
we communicated musically, he seemed to be so gratified and contentment radiated in the
air. Looking back at the sessions, when Big Red is musicing5, he knows he is good and
needs no additional encouragement or external esteem boosters. He is neither arrogant
nor conceited but accepts his ability as a given. He could lead our music experiences,
articulate what he wanted musically, and clearly express his musical preference for that
moment. We were two individuals drawn to the music who shared our joy together. When
he composed or improvised the songs, he did so effortlessly, unlike his difficulty on
mathematical tasks. It was like he had a jukebox in his head, and as soon as Ipushed the
button, music came out of his brain. The more I worked with him, the more musical
moments we shared that were precious and treasured. We were no longer defined as a
therapist and client, but as two people who wanted to share our musical passion and could
understand each other without words.

The project ended just before Christmas. After we sang several Christmas songs, I

sang for him “We Wish You a Merry Christmas.” When I finished, he started singing:

 

5 The term used by D. J. Elliot (1995) means ‘music making.’

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I am gonna miss you this year.
I hope see you next year
So we all have fun again.
See you in 2008.
The wording matched with the melodic contour of the song perfectly, and the

content of the song was appropriate for that session. The song was a nice conclusion to

our 10 week journey (Big Red 10-50z53).

Emotional Rollercoaster
How am I doing so far?
Is that working?
Is that right?
This is the one, right?
15. right?
How do you like it?

Big Red knew that he had limitations and difficulties and when not musicing he
was concerned about his non-music performance. He often asked the questions above.
When he knew that he had done a good job, he commented on his performance as
follows:

So, why do I have good memory?
Why am I Ieaming so fast?
We make good progress today. Wow!!!

During the fourth session, he asked me, “How am I doing so far?” Previously,
when he had asked similar questions, he was reminded to think about his performance
and answer himself rather than asking the same questions. Immediately after he asked me

about his performance, he stated, “You (Emily) don’t have to answer that question.

Because | (Big Red) know I’m doing good.” When he did a good job, he was proud of

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himself. He laughed as though he could not believe that he had gotten the correct answer.
When he made a beautiful and interesting song for the lyrics to 10 +3, he said, “I am
impressed myself.” I

When he could not answer a question or follow the steps as instructed, he became
desperate and agitated—nervously rubbing his chin, flipping his hands in fiont of his
chest, and anxiously pressing against both of his temples. When he was frustrated and
confused, he often touched his forehead and had a desperate look on his face.
Nonetheless, he swayed between positive and negative comments like a pendulum. When
I tried to minimize the emotional effect of his mistakes by comforting him, he realized
that I was trying to cover up his mistake and felt my distress. He admitted his mistakes
and was discouraged by his inability to do what I asked. He told me that he knew he did
not do good job that day and that I did not need to lie about it. I am relatively good at not
showing how I feel and think because of my training as a therapist and my personality. As
a result, not many people notice how I really feel. However, Big Red accurately read
what I was feeling and immediately perceived my disappointment and discomfort. In
front of him, I felt like I was an open book. Since he was keenly aware that my
discomfort was caused by his inability, he was frustrated and wanted to me make me
happy. However, he often did not know what he could do. Perhaps because he had
difficulties in so many areas, he was negative toward himself. There were so many
mundane things that he could not do by himself. Big Red had reasons to get frustrated,
and he was sensitive to other people’s feelings. His ability to notice my discomfort
showed me that individuals with WS are acutely sensitive to the feelings and emotions of

others.

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In contrast, his posture, facial expressions, and speech all changed when Big Red
worked on a music-related activity. Instead of being passively involved and checking my
facial expression every minute, he expressed himself easily with music; the tempo, the
style, and the choice of instrument. He was confident and ready to share his joy of music
with others. He had two extremely different attitudes toward mathematics-related
activities and music-related activities. When math and music were combined, he was
confident and happy, until he realized that he needed to perform some math related
manipulations in order to finish the song. Because of his limited math ability, strategic
songs were often interrupted to give him more time or to help him to finish that part of
the task. When he would get fi'ustrated because of some concept or procedure, I would
suggest that he play or sing a musical phrase with me to divert his attention from his
hopelessness toward his strength. Musicing could change his mood almost
instantaneously so that he did not require verbal redirection.

In summary, Big Red is a great musician who plays drum set and loves to play
music with others. He is a gentle man who cares about others, and a clown who loves to
work with children. Despite his wonderful personality and characteristics, his daily life is
full of obstacles. It was not pleasant for me to observe his struggles and difficulties in
simple mathematical tasks. It made me wonder whether starting interventions when he

was much younger would have made it possible or easier to overcome his difficulties.

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Roxie Montana

It was early in the morning on Saturday the first time Roxie and I
met each other at her home. When she was asked what her
favorite music was, she proudly announced Hannah Montana,
brought her CD player and CD and played Nobody’s Perfect. She got
up and then started dancing in her pajamas. She moved with music
and felt the music with her body. She was in the zone with the
sunshine coming through the window and illuminating her, it was a
moment of perfection as she listened and danced to her favorite
song.

Roxie was a lively and cheerful 16 years old girl with dreams, wishes, boyfriend
issues, clothing issues, make-up issues, and rebellious teenager behaviors. Roxie is the
oldest child in the family, with two brothers, one 11 and one 10 years old. Her parents are
supportive and caring. The brothers want to help her in any way they can. Her parents
moved Roxie’s schedule around to make sure a weekly meeting with me was possible.
The family members were lively, energetic, and dynamic. The extended family,
grandparents, aunt, and uncle live within a walking distance and many family gathering
and activities were going on. Kelly Montana, her mother, described that she treats Roxie
the same as the other two children, and Roxie was not given any leeway for having WS
diagnosis. Mrs. Montana brings her to many concerts; the concert setting mesmerizes
Roxie, and Roxie is so happy to be there.

Roxie is mainstrearned in the public school system. She struggles with
handwriting and spelling. Her parents have considered purchasing a dictation software
program to help compensate for her spelling difficulties and to give her another tool to
connect with other people through letters, e-mails, and communal websites such as
Facebook. Her parents had emphasized the importance of reading a clock since she was a

child, but she had not deve10ped the ability to do so. Roxie’s mathematical development

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lagged behind her peers, and she clearly acknowledges her inability in the area of math.
She witnessed her younger brothers’ math skills surpass hers. When her brothers teased
her when she had difficulties, she said, “You know I don’t get it.” She gave up and did
not put any more effort into developing her mathematical skills.

Roxie did not enunciate every word. She seemed to lack the minute motor skills
necessary to clearly form each word, and she seemed to not be fully present. Her lack of
enunciation had the charm of childhood speech. The emotions she seemed to feel were
the extreme highs and lows of a rollercoaster ride; when she was happy, she was
extremely happy, when she was sad, she was devastated. When I told her that she was
doing great, she nodded, smiled, looked at me, and said, “I know” with a tone of pride.
When she easily accomplished a task, she proudly announced, “I’m so good at this,”
and then the very next minute, when she was disappointed, she threw her head into her
arm saying “I messed up,” and “I am not good at math.” In WS circles, “Drama
Queen” is an unofficial term used to describe these emotional ups and downs. I heard the
terms and was finally able to clearly understand and feel what is meant by the “Drama
Queen” title when I was with Roxie. Her emotions did not appear to occupy middle
ground.

Roxie was mesmerized by piano playing, saxophone, and the iPod; everything
related to music. The first time I played for her in a “show-off” piano style, she
exclaimed, “It is so cool. Bravo!!! Excellenté!!!" (in Italian pronunciation). She made
me feel very special as a musician and fed my ego. She was attracted to the iPod and

wished to have one of her own. She was amazed how many songs could be stored and

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accessed on a tiny iPod, and she was so happy to find the songs by Hanna Montana. She
especially liked the glockenspiel, keyboard, saxophone, and rain stick.

From the initial interview, Roxie was open and fiiendly. She did not hesitate to
talk with me and to ask some questions about my gadgets and instruments. She did not
display any sign of discomfort because of my presence in her home. She clearly
expressed what she liked, and what she wanted to do. It was difficult to believe that this
was our first meeting. Coin discrimination was her first task using music in the

mathematical interventions.

Finger Dexterity

Roxie appeared to have relatively good finger dexterity compared with Angela
and Big Red. While there was room for improvement in her fine motor skills, she handled
coins and other small objects almost as well as typically developed individuals. She could
easily display different hand gestures; however holding up different numbers of fingers
took longer, not due to physical limitation but due cognitive limitation. She needed to

count each finger one by one.

Attention

Roxie was easily distracted: the dog, her cousin, siblings, or father—whoever
passed by our session room which was the formal dining room of her home, distracted
her. The living room had two openings; one was toward a dinning room and the other one
was toward the foyer. The third side was the window. Whenever someone passed by, she

felt the necessity to communicate with them—saying hello, petting the dog, telling them

126

what she had done. While her distraction could be understood as a normal response to the
stimuli, the degree of distractibility was more severe than it would be with TDI. There
were two questions to which I wanted answers: How much attention is appropriate at her
age, and if the environment is changed, could her behavior be changed?

At the beginning of the project, she talked to the video camera a lot. “Hello!
Cameral,” “I love people and Lansing6. See you soon.” However her interest in the
camera diminished around the third session, and she no longer needed to greet the
camera. Her attention span changed depending on the day’s event. When she expected
Travis, her fiiend from Grand Rapids, Michigan, to come just after the session, she did
not able to pay attention to the task. The day she had a basketball practice after the
session, she was distracted and checked the time almost every 5 minutes.

Musical cues, interestingly, were more effective than verbal redirections for her
distractive behaviors. When I verbally redirected her to focus on her task, she rolled her
eyes or talked back like “I’m doing it,” or “I’m working on it.” She displayed typical
adolescent behaviors; however, when musical cues were given, she usually smiled and
gave me the signs of understanding the meaning of a musical phrase by a wink or a hand

gesture pointing at her worksheet or the materials, and focused back on her task.

Mathematical Exercises
Coin discrimination. Roxie could easily distinguish between different coins. She
already knew the names of each coin and could order them from the most valuable to the

least valuable. I used song “Give me a Quarter!” to evaluate her ability to discriminate

 

6 Roxie knows I live in Lansing, Michigan and the particular day she was supposed to go Lansing,
Michigan for a football game after the session.

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be4tween different coins. She successfully handed in the designated coins. In order to
challenge her to speed up her processing speed, I asked her to find each coin along with
the improvised song, “Quarter, Dime, Nickel, Penny.” She had difficulty keeping to this
tempo (two beats with a tempo of approximately 100 beats per minute tempo). It seemed
to be too fast a pace to process as well as too challenging for her fine motor skills.
Picking up flat objects from the table was not easy for her. Even though the task was too
difficult, she thought it was funny and threw her head in her arm laughing. After we
enjoyed the moments, the song slowed down and gave her 6 more beats to complete the
task. She effortlessly completed it. Next, I gave her two beats to name the type of coin
and two beats for processing; she easily accomplished that task as well.

Coin calculation. Roxie demonstrated that she understood the concept that
different coins has different values; a nickel is 5 cents, a dime is 10 cents, and a quarter is
25 cents. She could also recite the incremental sequence of each coin: 5, 10, 15, 20, 25
for nickels, 10, 20, 30, 40, for dimes, and 25, 50, 75, 100 for quarters, each all up to one
hundred. Nevertheless, Roxie was not able to apply these concepts to calculating the
amount of change in her hand. When I presented to her a nickel or nickel-hand-props (see
Figure 13), she answered in increments of 53, and she could do the same for the dime and
the quarter as well. However, when the coins were mixed, she had no strategy to calculate
the sum. For example, when I presented four nickel—hand-props one by one, she answered
5, 10, 15, 20. Then I separately presented a dime-hand-prop, and I expected that she
would answer either 10 (just for the prop) or 30. Instead she answered “25,” which was

the next number in the 5 sequence (see Figure 28).

128

 

 

 

 

 

 

W7 910

 

 

 

(fl/i9 -) 15
((1in -)20

 

 

 

 

 

W at e 22

Figure 28. Four nickel-hand-props and one dime-hand-props and Roxie ’s answer.

 

 

 

This suggested that she did not grasp the concept, but based her response on rote
memorization. When I pointed out a dime-hand-prop and asked her again to give me just
the amount of the dime-hand-prop, she was able to answer 10. A nickel-hand-prop was
added after a dime-hand-prop, and she correctly answered 15. Then another nickel-hand-
prop was added, and she incorrectly answered 16 (see Figure 29). After a couple more
questions, the same question in same sequence was tried the second time, and she made
the exact same mistake. The precise reason for her incorrect answers and how she
processed the information were elusive but indicated that she did not comprehend the

process.

129

 

910

 

 

 

 

 

 

 

 

 

Figure 29. One dime-hand-props and two nickel-hand—prop, and Roxie ’5 answer.

During the third session, I asked Roxie to calculate amounts by putting coins in a
coin-classification-chart (see Figure 15) and then adding the amount from right to left. To
teach adding a dime to a quarter, I initially used the concept of the identity property of
zero (the sum of zero and any number is that number i.e., 2 + O = 2) and fact extensions
(Calculations with larger numbers using knowledge of basic facts. For example, knowing
2 + l = 3 can be used to solve problems 20 + 10 = 30). In other words, a quarter (25) plus
a dime (10) equals 5 + O = 5 (identity property of zero), and then 2 + 1 =3 make 35.
However, Roxie comprehended neither the concept of identity prOperty of zero nor fact
extensions. After the initial teaching method had failed, I developed and explained the
tapping strategy to Roxie.7 The tapping strategy is that a dime is ten cents, and that a
dime is also two fives, just as two nickels equal 10 cents; therefore she can calculate the
sum of different coins by tapping twice on a dime, and say the numbers in 5’s and tapping
once on a nickel. A quarter plus a dime, tapping a dime twice and a nickel once with

saying 5’s (i.e., a quarter plus a dimes is 25, 30, 35 or two quarters plus a nickel 25, 50,

 

7 The tapping strategy was developed after all three participants had same difficulty as Roxie had. They did
not comprehend the identity property of zero and fact extensions.

130

55) method was introduced. This strategy appeared to simplify her thinking process. She
was able to use the tapping method and could figure out the amount of cents for different
combinations with the song “How Much Is It?” However, in practicing the tapping
method with various other combinations of coins, she had difficulty.

While she was able to calculate combinations with quarters and other coins by
using increments of 25 and then the tapping method for nickels and dimes, and then using
increment of 1 for pennies, she had a hard time finding how to start without a quarter.
With 2 dimes and 1 nickel, she calculated the amount of coins by saying 10, 20 for dimes;
and then she did not know what to do with the nickel. She answered 45, without
providing a reasonable rationale. In the same session, she tapped the dime twice saying
10. She ended up answering 2 dimes (10, 20, 30, 40) and l nickel (50) equals 50. These
mistakes could be explained by confusion between old knowledge and new knowledge.
Roxie knew that dimes are numbers in 10’s (the old knowledge) and then she learned that
she could count a dime with tapping (the new knowledge). However, she did not
understand the sameness in the two methods or that both methods arrived at 10. When
she needed to decide which method to use, she could not decide which method was the
correct one.

The task of giving change requires more steps than calculating the amount for
given coins. While understanding the different types and values of coins and the process
of adding the coins based on using 5’s, giving change the concept of bigger and smaller
numbers, and then choosing the best way to arrive at the same total amount. Roxie could
make different combinations of the same monetary amount; however, she tended to use

the easy combinations, such as using 3 dimes to make 30 cents, 7 pennies for 7 cents

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instead of a nickel and two pennies, 2 dimes and 7 pennies for 27 cents. Dimculties often
came when the using quarters and nickels. When Roxie was asked to give 40 cents, she
selected 4 dimes. When challenged to use quarters, she said “(pointed 2 quarters) 25, 50”
and added a nickel “55,”and then looked at me. I prompted her to figure out which
number was bigger, 40 or 50. She answered 40. After I told that two quarters are more
than 40, I asked her to use nickels or dimes. Her choice was 3 nickels, and she was able
to give me 40 cents. It was difficult for her to come up with different combinations for
the same amount. In real life situations, every coin would not be available for making
change in a transaction.

Written addition. Roxie was relatively good with tasks requiring single digit
addition. However, her pattern of mistakes revealed that she did not understand the
commutative property (also known as the order property—the sum is not changed by the
order of addends (Kilpatrick, Swafford, & Findell, 2001). As shown in the Figure 30,
Roxie correctly answered 9 + 2, but she did not correctly answer: ‘6 + 5 (question # 5),’
‘2 +7 (question # 7),’ ‘3 + 9 (question # 8),’ or ‘3+6 (question # 10).’ The common factor
in these questions is that the second addend is larger than 5. While she was able to handle
questions that had a smaller second addend, she could not correctly answer with the
larger second addend. That indicates that she did not have a strategy to switch the first
and second addend.

After the song “Finding the Bigger Number” was introduced, she was able to
switch the first and second addends and could arrive at the correct answers. She enjoyed
the song; she swung side to side along with the song and eagerly displayed her fingers for

the smaller number, and enthusiastically counted the fingers. She proudly wrote down the

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answers on the paper and looked at me to start all over again for the next question.

Figure 30. Roxie ’s worksheet. *

* Hand written

Roxy!!!!!!!!!!!!!l
L9+2=l\
zs+5=lo

3. 8+2= lo
43+2=g
56+5=10
62+2=4
72+7=7
83+9=M2
97+2=(k

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One incident particularly stood out interesting. Roxie worked alone on a single-
digit addition worksheet, while I prepared for the next intervention. She counted all of the
questions on the worksheet correctly. I praised her for doing this by herself and moved on
to the next intervention. However, when the video clip was transcribed, I noticed an
interesting detail was revealed. For the question “9 + 6,” the correct way to calculate,
according to the song, was to hold up the smaller number using her fingers (6 fingers),
and air-counting the bigger number (9), and count holding up fingers from 10. Instead she
said the smaller number 6, but held up her 7 fingers. Instead of air-counting for 9, she
said 9 with her pointing finger, and counted each finger one-by—one; she got the right
answer “15.” The two mistakes made her final answer correct, but the process of arriving
at the answer was not right. This incident should inform teachers in special education
settings or any educational setting. When one leaves a student alone to work on a task,
the answer is checked for success or failure; however, the process of the solving the
problems is not necessarily checked by this procedure. Individuals with WS illustrate that
they think and process problems differently from TDI. Unless the teachers or therapist
knows how the students approached a problem, it is not possible to intervene with their
way of thinking.

Sense of numbers. Roxie showed that she was missing a couple of key elements in
her sense of numbers. First of all, the ability to subitize, which typically develops around
6 years of age (Clements & Sarama, 2004), was difficult for her. She had difficulty when
she was asked to glance at a number of objects and identify the number beyond 4 or more
objects. Even, when I showed less than 5 objects, she felt the necessity to count them

individually. It was unclear whether she could perceive smaller units, or if she just

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preferred to count as confirmation. Roxie knew that one hand represents 5 fingers and
two hands—5 fingers plus 5 fingers—equals 10; however, when she encountered 5
pennies in a row, and added another 5 pennies in a second row, she counted from 1 and to
10. Even afier she was specifically asked to count from 5 with verbal explanations of the
sameness between 5 fingers in a hand and the hidden 5 cents in a nickel and practiced

6‘19,

counting objects from 5, she had to start from to count 10 pennies. Her facial
expressions suggested that she was wondering why she needs to use the new method
when she has a method in her hand that walked perfectly well. She made me feel as if I
had intentionally given her a difficult time over an unimportant thing.

Similar patterns were observed in her finger counting. When I held up seven
fingers, she could not identify it as the quantity of 7 without counting. The pattern was
the same as when counting objects. When I held up eight fingers or nine fingers, she
counted from 1. This indicates she did not build concepts of composing and
decomposing, which means a whole can be the sum of parts (decomposed) and the parts
can be brought together and became a whole (composed). This occurs when 5 is a partner
number that where children comprehend 6 as 5 + 1, 7 as 5 + 2, 8 as 5+3 up to 10.
Typically developed children are reported to develop this skill by the age of 6 years
(Clements & Sarama, 2004).

When we did the finger counting interventions—presenting 6, 7, 8, 9 fingers, and
she needed to figure out the number of fingers in less than two seconds—her inability to
identify the sum at a glance became more obvious. After I asked her not to count from 1,

because she already knew one hand had 5 fingers and had practiced counting from 5, she

was finally able to do the computation. However, she still needed to count fingers; that is

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“5, 6” for 6 fingers, “5, 6, 7” for 7, “5, 6, 7, 8,” for 8 fingers and so on. This indicated she
did not have a concept of 6 as 5 and 1, 7 as 5 and 2, and so on—composing and
decomposing.

When she moved on to identifying the number of fingers from 11 to 20, she
revealed she did not have the concept of “teen” (knowing fourteen to nineteen is the same
sequence as four to nine with a suffix “teen”). While she easily grasped that she did not
need to count 10 fingers from 2 hands like 5 from 1 hand, she needed to count 11, 12, 13,
14, 15, 16, 17, 18, and 19 instead of seeing 5 fingers and saying fifteen or 6 fingers and
saying sixteen. Roxie was missing two key elements: teen concepts and the ability to
subitize the number of objects. These two missing concepts prohibited her from figuring
the total number of fingers on 4 hands.

Even though she grasped the concept of counting fiom 5, or 10, her mouthing “1,
2, 3, 4, 5” was often observed. Counting from 1 seems to be the only solid technique
Roxie seems to understand absolutely, and she utilized that technique for counting the
quantity of objects. She could not count by 2’5, 3’5 or 4’3, but she could count by 5’s. She
knew a quarter is 25 cents, a dime is 10 cents, and a nickel is 5 cents, but she did not
know a quarter and a dime together equals 35 cents. Roxie grasped some the facts of
mathematics; but not the process of mathematics. Her mathematical skills make me think
of a 1000 piece puzzle; she has only a couple of pieces out of 1000 pieces, but she did not
know where the pieces fit into the big picture.

2 fingers in right and 5 fingers in left for 25. Roxie held up 2 fingers on her right
hand and five fingers on her left hand and proudly announced “25.” It was

understandable and acceptable; even charm and cute; I assumed she used the 2 fingers to

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represent two groups of ten, and the five fingers to represent 5; so I did not intervene.
However, a problem emerged when I presented 5 + 2. I used my right (her left) for 5 and
my left (her right) for 2, she yelled, “25!” It was clear that while most typically developed
children will understand the differences between the two settings, it might be not the best
way to present 25 with 2 in one hand, and 5 in another hand for individuals with WS.
Since she did not grasp the concept of abstract representation, it was a confusing abstract
concept. The best way might be to hold up 10 fingers themselves, 10 fingers from a
second person, and 5! fingers from a third person. Since individuals with WS have
difficulties with quantity, it will be better let them feel and see the large quantity of 25.
Clock. Roxie was not very comfortable reading the clock. During the pre-
interview, Kelly, her mom, expressed in five different questions in the pre—test (see
Appendix J. Roxie) that Roxie most likely figures out the clock questions, because Kelly
had really emphasized reading the clock since Roxie was in school. Unlike Kelly’s
expectation, Roxie did not get the right answer. She selected 7:15 as an answer out of
four choices for 3:30. When I asked her to read the clock during the session (it was
10:10), she said, “Let’s me guess.” That word choice indicated her insecurity with her
clock reading skills. She read 10:10 as 9:20. There was no pattern or consistency in the

mistakes. This told me that she did “guess” in her answers.

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Music
Emily: Does she enjoy listening to music?
Mom: 100%

Kelly, Roxie’s mother, answered the questions without a moment of hesitation.
After school, Roxie comes home and plays her favorite music and dances hours and
hours like she did it when we met for the first time. Roxie takes pleasure in listening to
High School Musical, Hair Spray, and Chicago (it is not a mystery to speculate from
where she selected her pseudonym). When music is played, she has a hard time not
dancing to it. At that point, her favorite singer was Hannah Montana.

Roxie has a strong memory for melody. She could remember a song after hearing
it only a few times. The week before the first session, when I had a pre-interview with her
mother present, I played The Money Song from the musical Avenue Q as background
music. Because I expected to use the song for the sessions, I let her hear it a couple of
times along with other prospective songs while I interviewed her mother. When we
settled down to begin the first session, she spontaneously sung the melody from The
Money Song, and expressed that she wanted to listen to the song. It was very impressive
that she could memorize the melody and sing it back a week later. Throughout the
project, she relatively easily learned new songs; she usually memorized the songs after a
few repetitions we sang them together. She has a tendency to sing when she counts the
numbers and reaches 100 or at the end of counting and makes rather exaggerated endings.

Roxie has an eerie musical sense. When we improvise together; she predicted the
next note and we could sing in unison; as when two individuals say an exactly same word

at the exact moment in dialog. A case in point happened 5 minutes after the very first

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session. While she worked on coin discrimination, I improvised based on the word

quarter, dime, nickel, penny with the following pattern:

PartA Part B

J
J r r I I I

PartA & B

 

 

 

Figure 31. Quarter, dime, nickel, and penny improvisation.

She jumped into the song at the 5‘11 note and sang along with me. It was not
certain whether the part B section of improvising was her idea or my idea. It appeared as
though we had practiced it previously. After the first two measures of the improvisation,
she was able to continue the melody by herself with my basic chord accompaniment
without my paying the melodic line. Since she did not have any formal music training,
this incident provided an insight that she is aware of a basic chord progression, not
theoretically, but as a second nature. Another similar episode happened when I
rhythmically chanted based on the word “done.” It seemed to remind her of a similar tune
with hand gestures. She danced with her improvised singing and beamed with joy. Her
gesture and movement reminded me of the scene Born to Hand Jive in the movie Grease.

Roxie has a piano and a drum set at home, she also said that she wanted to play
the flute; however, she does not actually know how to play any of these instruments.

Roxie was instructed to play a rain stick when we sang that the Dime part in Nickel,

139

Dime, Quarter. She smiled when she played the rain stick and was fascinated with the
sounds. She especially liked to play a rain stick with improvisations on the piano. She
enjoyed musicing and described our playing together as “very relaxing.”

Roxie likes to play the glockenspiel. Roxie played an unidentifiable melody on
the glockenspiel and claimed that she had played Mary had a Little Lamb. I asked her to
play it again, and I was able to identify the song. She played the song in D dorian mode

with an introduction as shown in Figure 32.

l
---l

 

 

 

r
‘ 1 rr 1 r I
(farm! I I_ 11 I 1 r 1 I r
:1 1‘ d i—J——fij :1 'L :1
4

 

Figure 32. Roxie 's version Mary had a little lamb.

While she sang the song correctly, she played it in a modified version. When I
attempted to determine how she ended up playing with an incorrect version, it was
noticed that if she had started the song using the C note instead of the D, she would have

had the correct version (see Figure 33).

140

 

Figure 33. Correct version in the key of C, Mary had a little lamb.

The reason for her inability to recognize the differences between the tune when
sung and when played was not clear. One possible explanation is that she may have
learned how to play the tune on piano or keyboard instrument fiom someone, but did not
accurately remember the starting note on the keyboard. This speculation comes from my
observation with other individuals with WS; once an individual with WS could audiate a
song, they were going up and down a keyboard and finding the next pitch based on their
melodic memory. Even when a person started with a wrong pitch, the person will frnd the
next pitch based on the relation with the previous note. If she learned by herself, she
might do the same thing, but she played the white keys on the keyboard by sequencing—
up, up, up, and down—as she plays in the key of C.

Another speculation is possible based on theories of child development and
musical ability. Typically developed children learn melodic contour of the music before
they can reproduce the precise interval and pitch for each note in a melody (Dowling,
1991, 1993). Contour is defined as “the sequence of ups and downs in a melody,
regardless of interval size” (Dowling, 1991; Dowling & Fujitani, 1971, p. 524).The
contour of two figures is matched at earlier stage of musical development and that gives

another way of explaining her “mistake.”

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“Nobody’s Perfect” and “You Can Do It” worked well for Roxie as an encourager.
These two songs were sung during the first session and we discussed the meaning of the
songs. During the second session, she was so fi'ustrated because of her inability to
calculate with dimes and nickels. She growled, thrust her hands downward, and
proclaimed “I always messed up.” I said, “No” and started singing “You Can Do It”
and I stopped after “If you,” she appeared to be momentarily lost, but she came up with
“try.” She laughed after she answered and she was able to get back on task. During the
project, she liked to substitute the ‘If you try” part of the song “You Can Do It” with
“nothing to it.”

During the 3rd session, when she commented about her poor performance in math:

“I am not good at math. I keep forgetting.”

I started improvising on her intonation as follows;

 

I keep for

Figure 34. Intonation of Roxie 's “I keep for getting. ”

I sang for her with that motive and Roxie started improvising the melody with my
basic chord progression accompaniment between I and V in the key of F major. The lyrics
were as follows;

I don’t want to give up on counting penny, dime,
nickel, and quarter: dime, penny, and nickel.

I don’t want to mess up on counting quarters and
dimes but if I can I will do well, so let’s do it all over.

142

She pointed out her difficulties, and expressed her frustration, and described her
next action plan. The melodic line fit well with the lyrics and was quite an impressive
piece of work.

There was an incident that illustrated the power of music in her behavior. When
Travis, her best friend who lives about two hours away was supposed to visit, she was
very distracting. She looked at the clock almost every five minutes and had a big smile on
her face. When the phone rang, she got up to answer it, because it might be Travis. When
her mother redirected her to pay attention to the task, Roxie became very angry, growled
and sat down in her chair. Since she had just finished her worksheet for the day, I wanted
to play Roxie’s congratulation song for her. When I started playing the song, her
emotions switched from angry to happy as though she had an on-ofl switch activated by
the music. Roxie immediately shouted “l , 2, 3, 4,” and clapped with the song.

Music and musicing can change Roxie behaviors and attitude toward
mathematical tasks. She has a reason to be frustrated in mathematical tasks, however
when the mathematical tasks presented in musical context, she takes it as music activity
than math activity. Even if she cannot play any instrument, her desires to learn how to

play an instrument exist, but she appears to not meet the right person to teach her yet.

Emotional Support
Roxie consistently made mistakes during the 2nd session when asked to sum the
value of the coins. She was so troubled that she exclaimed
“I always messed up”
“Messed up every time."

“I always messed up on those one with the dime and
nickel."

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I also was so fi'ustrated and worried about the whole project. However, the
expression of ‘always’ and ‘every time’ raised concerns. This particular incident, the
song “You Can Do It” did not do the trick. I took time to talk about why she needed to
keep trying, and sang “Nobody’s Perfect” and “You Can Do It” again As I mentioned in
Roxie’s music section, she improvised songs about how she want to handle situations.
She also added a line “nothing to it” for the song “You Can Do It.” Her performance in
mathematics is not very strong, so perhaps she had a reason to be agitated and fi'ustrated;
however, her responses to music for emotional support suggest that she “self-medicated”
using music to ease her distress.

In summary, Roxie is a cute and adorable 16 year old girl. She is energetic and
lively and is ready to laugh at any given moment. She wanted to teach me how to dress
and to put on make up. She was really easily distressed by her simple mistakes in
mathematical tasks and was ready to give up. She was mesmerized by music and
different sounds from various instruments. When I called her and wanted to get hergfull
attention, I chanted her name with a descending major 3rd (G to E), she would answer
with “Yes, MA’AM.” It became our ritual and she and I enjoyed our tradition. Her mother
started using the same tone, and Roxie and her mother also enjoyed these moments.
When I met their family, the year after the project at the music camp, Roxie’s mother told
about “Roxie-Yes, Mme,” that she still used that tune to get Roxie’s full attention and it
worked so well. She is an adorable teen age girl. I cannot wait to see her again as a grown

up woman.

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Angela Hill

At the beginning of 6th session, Angela told me she had just
watched a documentary of the Beach Boys that morning and loved
Surfin’ USA. I told her I had Surfin’ USA in my iPod. She begged me
to play the song. When the music started, so did Angela. She
wiggled and danced and grooved! Never did the smile leave her
face. She danced with such joy and delight, adding lip syncing to
complete the performance. She was keenly aware of her mom, me,
and the video camera. When she made very cute and adorable
movements, she looked at the camera as if she wished that the
video camera had caught her joyful moment. It did. Her mother and
I had never seen her act like that before; we looked at each other
and laughed. The gentle, calm, and quiet Angela had gone wild with
Surfin’ USA.

Angela Hill is a very calm, gentle, cute, adorable, and happy 10 year old girl. Like
other children with WS, she loves music; she has her own CD collection and brings her
CD case everywhere she goes. The music she likes ranges from Hanna Montana to the
Beach Boys to children’s Gospel music, but she does not like loud noise and wears ear
plugs in noise environments.

Angela’s communication abilities seem to be the same as a typical lO-year-old;
we had no communication difficulties during our sessions. She does have a rather
reserved personality, as WS literature often suggests, and is generally quiet. She tends to
be very calm and well-mannered and does not offer a broad range of emotions in her
interactions.

When we had spoken on the phone prior to the interview, Angela’s mother, Lee

Hill had expressed some reservations regarding my accent and pronunciation, and was

concerned about my ability to communicate with Angela. I stressed that face-to-face

145

interaction would demonstrate my ability to work with Angela and we agreed that after
we met, Angela would decide whether to participate in the study. During this initial
interview in their home, Angela was curious and inquisitive as I spoke with her mother.
When her mother asked Angela if she could understand me, she eagerly declared that she
could understand me perfectly. While it was not clear exactly how much of my actual
speech she understood, it was clear that she enjoyed meeting a new person and wanted to
please me. It was also apparent that she wanted to spend some time with my attractive
gadgets: a bag of instruments including, a keyboard, a set of sticks, a glockenspiel, an
iPod, and a file box. She was especially interested in the contents of the file box.

Mrs. Hill advised me that Angela has difficulty with reading comprehension,
handwriting, and spelling. According to her mother, she is doing well relative to the
others in her special needs class regarding her mathematical development. In general, her
attention span is quite good—as opposed to what the body of literature on WS suggests,
and in comparison with other children with WS that I have observed. Angela’s home
environment was calm, quiet, and very well-organized, which may have helped her to
maintain her focus and attention. Of all the study participants, Angela was the most
prepared for our sessions. Mrs. Hill’s interaction with her daughter is pleasant and very
gentle. She was very interested in Angela’s performance and ready to help her with her
tasks. She is also deeply involved with Angela’s schoolwork, after school activities, and
in the use of additional education programs to improve her reading and comprehension.
Mrs. Hill is very reserved and modest, and has a more formal, even old-fashioned
speaking style that is quieter and focuses on enunciation and pacing. To a great extent,

Angela imitates her mother’s speech patterns. She usually speaks very softly, and often

146

nods her head rather than vocalizing. When she wanted to do something other than what I
asked her to do, she tended to give a wide-eyed response that could be called “puppy-do g
eyes,” as a form of emotional appeal or manipulation, rather than a provocative or violent
response.

As Angela became more comfortable with me and with the process, we began to
build a rapport and a form of mutual understanding. She more clearly indicated what she
wanted to do, and also what she could and could not do in terms of mathematical tasks.
As the sessions continued she began to talk about subjects outside our exercises, and she
started sharing some of her wishes, desires, and concerns. During our sixth session, she
announced that she wanted to get her ears pierced. The way she presented this was
exaggerated and overly dramatic, a type of response I had not observed from her before.
However, this scene was very familiar; similar scenes are frequent during conversations
with many individuals with WS. For Angela, this level of exuberance also manifested
when I used musical reinforcement after she was successful with math problems during

our sessions.

Language

While Angela does not have predominant difficulties in spoken language, she has
difficulties with written language: reading, comprehension, handwriting, and spelling.
She and her mother had used the “Hooked on Phonics” academic program in an attempt
to improve her reading skills with limited results. When I asked her to select a
pseudonym for this study, I also asked her to spell it out; her facial expression in reaction

to the request was one of distress. Her tone of voice changed and took on a hint of

147

sadness. She seldom appeared as distressed during our later sessions, even when making
mistakes while working with mathematics, as she did following my initial request to spell
her name. Although she finds spelling, reading, writing, and math all equally challenging,

her spelling ability troubled her the most.

Finger Dexterity

Angela has difficulties with her finger dexterity. Her pinky is approximately one-
third shorter than it is in most people; her pinky does not reach the first joint of her ring
fingers. She is right-handed and has more difficulties with her left hand. When she was
asked to hold up 3 fingers, she worked laboriously to hold up just three fingers, especially
with her left hand. When I helped her during her attempt it took some time to succeed; I
physically banded her pinky and brought her thumb down to hold the pinky down. Her
pinky did not want to stay in position; she needed to use her other hand to hold down her
thumb. She even has a difficult time holding up just two fingers with her left hand; she
had to use her right hand to hold down the other fingers. When her fingers resisted this
positioning, she said, “You stay there! I am serious,” but in a very cheerful manner. In
order to provide opportunities to practice finger dexterity, the song “Thumb and Pinky
Meet Each Other” (see Appendix A. Figure A-3) was improvised and used during later
sessions. A Hawaiian greeting hand gesture of greeting, called the “Shaka” signs, “V” for
victory sign, and “I love you” from American Sign Language were demonstrated and
practiced to encourage her to exercise using different formations with her fingers (see

Figure 35).

 

8 This gesture requires a person to extend the thumb and pinky finger as other three middle fingers bended
toward to palm of the hand.

148

     

Shaka sign V sign I Love you sign

Figure 35. Hand gestures.

Illustration by Hyang Eun Lee

Attention
Angela is usually relatively good at focusing on her tasks, with one notable
exception. When the window of the session room was covered with sheer blinds, she

thought the windows were fogged up, and became distracted.

Emily: Dime 10 (plays rain stick together)

Angela: Wow! Wow! What happened?

Emily: What happened?

Angela: I think the windows are fogged up. Wow!
Lee noticed her distraction and asked her whether it bothered her or not. Angela asked
Lee to draw up the blind. After the blinds were adjusted, she was able to focus more on
the tasks.

Angela was also distracted by the video camera. In order to verify the visibility

and apprOpriateness of the camera angle, the display window of camera was faced toward
Angela could watch herself through the camera lens and enjoyed doing so. She became

discouraged when she was instructed to ignore the camera during the initial sessions. Her

tendency to look at the camera was expected to lessen after a couple of sessions; however

149

the video analysis revealed that she had stopped her extensive waving to the camera and
making faces, yet she would momentarily look at the camera on occasion, or quickly
raise her hands, and/or momentarily make faces when she thought I didnot observe her.
This implied that she could not resist the temptation fiom the camera and indicated some

degree of weakness in self-regulation.

Mathematical Exercises

Coin discrimination. The first step to calculate and manipulate coins and paper

money is to recognize the differences between coins and paper money. When we
began our sessions, Angela was not able to discriminate among different coins, but she
quickly learned and began to enjoy doing so. Initially she had a difficult time
distinguishing between a nickel, a dime, and a quarter. She did not confuse these other
coins with the penny. However, coins of the same color but with different size confused
her. It was also hard for her to remember the specific names and values of each of the
coins. The songs, “Give Me a Quarter!,” “How Many Pennies in a Quarter?,” and
“Nickel, Dime, and Quarter,” were used to help her learn these concepts. The tactile
differences between a quarter and a dime, with their rough edges, and a nickel, with its
smooth edge, were emphasized to enhance her ability to categorize these coins. On
several occasions she ran the edge of a coin along the back of her hand, just as she had
when the coins were first introduced to her, in order to determine their texture. When the
coin hand props (see Figure 13 & 14) were first brought into the sessions, Angela was
asked how many “fingers” were in the quarter-hand-prop. While she had been told there

are five “hands,” each with five “fingers,” in the quarter coin hand prop, her incorrect

150

answer was that there were only five “fingers” (see Appendix K. Angela Episode 5). This
illustrates her difficulty combining different units to reach a numerical value. Singing the
song “How Many Pennies in a Quarter?” and the visual presentation with a quarter with
25 fingers did not help her to understand that the concept of a quarter being the same as
25 individual items. She could not make connections between these concepts by herself
without direct instructions. It seemed that she had a hard time understanding how one
thing can represent 25 things. This symbolic representation is an abstract concept, one
that she had difficulty grasping, as well as the related concept of conversion.

Coin calculation. After Angela could relate the name and value of the each coin
relatively well, she was asked to work on calculating the total amounts for several groups
of coins. She shared that it was difficult for her to apply her knowledge (i.e., a quarter is
25 , a dime is 10, and a nickel is 5) to solve coin calculation questions without help with
her thinking process. She knew a nickel represented five cents and she counted intervals
of 5 to 100, but failed to combine those two notions to figure out the value of just two
nickels. The following conversation took place after we worked on the value of the nickel
as 5 cents and operation that 5 plus 5 is 10. She eventually came to the correct answer;

however, it was necessary to verbally redirect her to guide her thinking process.

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Emily: (put down two nickels) How much is it?

Angela: And, 5 and... 25.... (Thinks with tapping her
chin)... 29

Emily: 29! No, No, No. This is 5. OK. Nickel five,
(Sings Nickel, dime, and quarter)

Angela: (finishes the line for quarter) 25

Emily: (Sings Nickel, dime, and quarter) Nickel five.
(points the second nickel) How much is it?

Angela: 10 (reading Emily’s facial expression)

Emily: (points to a nickel to make sure she really has
the right answer) This is???

Angela: 25

Emily: No, a nickel is

Angela: 5, 10!!!!!!!

Angela also constantly made obvious errors substituting the value of different
coins (see following example; as well as Appendix K. Angela episode 1, 4, 6, 7, & 13).
Like the previous example, she could not apply the abstract lmowledge to solve the

question. She needed to have helped to accomplish this task.

Emily: (points to a quarter) How much is it
Angela: 25.

Emily: This is 25. And this is (points to a nickel)
Angela: A nickel?

Emily: Yes! Nickel; and 25 plus 5 equals.
Angela: 29!

Emily: 29!

Angela: 90... wait. I am confused

She needed to be reminded that different coins have different values and that she needed
to use different values for different coins. In addition, she needed to be assisted in adding
25 to 5. It was interesting to observe that she made the same mistakes in the first and
second examples, where she added 25 plus 5 and an'ived at 29. When she thought of two
nickels as 5 and 25, she answered 29, and the question of a quarter plus a nickel and she
answered 29. Any logical explanation was not available; however, she arrived at the same

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answer twice with the same addends, it suggested that in her logic it might make sense,
and I was wonder how she an'ived at that answer.

After working on the value of two quarters and a dime (50 plus 10) is 60 cents by
written addition by emphasizing the vertical representation of the question, she was able
to do so as shown in the figure 36. However, when the problem of adding two quarters
and a dime (50 plus 10) was verbally asked, Angela proclaimed that is “65.” It almost
seems that when she is adding two digit numbers in vertical format, she think that the 5 or
other numbers is mysteriously appeared instead of her comprehending the concept of the

identity property of zero.

ED mg

+i0 "thi

f.

s a
5
fl \.\ \l 7;,

/.

 

 

 

 

Figure 3 6. T wo-digit addition formation. *

* Hand written

Angela appeared to be not ready to mentally calculate the problems of adding a
number to 10, therefore she was taught the tapping strategy that is a dime is ten cents; and
that a dime is also two fives, just as two nickels equal 10 cents. After she added up
quarters in the coin—classification-chart by increments of 25, she was instructed to tap
twice or once dependent on a nickel and say in increments of 5. This tapping strategy

does not require mental calculation and has concrete and tactile components in the

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process. Angel’s then became more accurate with the different combination of the coins.

Angela likes to manipulate with coins, especially pennies, which are the easiest
way for her to count instead of the combinations of different values of coins. Once she
asked me to say 16 cents and she cheerfully collected all pennies from the pile. This
tendency was observed several times during the course of intervention. When Angela was
asked to pull out a certain amount from an array of coins, she completed the task in the
easiest way, for example, 4 dimes for 40 cents, three dimes for 30 cents, sometimes even
wanting to use only pennies despite of the large quantity of cents, such as in the case of
55 cents; she asked me if she could use only pennies and happily counted out 55 pennies.
Other combinations like a quarter, a dime and a nickel for 40 cents, or a quarters and a
nickel for 30 cents did not flash upon her on mind. Angela was lather challenged to use
other combinations.

Angela was asked to produce 55 cents. She previously solved this task by
selecting two quarters and a nickel, or five dimes and a nickel. This time she was stopped
initially because she could only find single quarter in the pile, similar to the real life
situation. She was then reminded to use the coin-classification-chart (see Figure 15)
which helped her too complete the task.

Emily: You can’t find any more quarter?
Angela: Um... um...
Where is it? um.... um...
Emily: Let’s use this.
(points the coin-classification-chart)
Let’s put a quarter here.
Angela: Ah!!!
Emily: What is the next thing you can put it?
Angela: (points a dime twice) 30, 35
Emily: Ah-ha!

Angela: (points a dime twice) 40, 45
(points a dime twice) 50, 55

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During the next session, she was asked to collect 40 cents, and her response was
to give four dimes. When she was challenged to use a quarter instead, she was easily able
to adapt to the task and use nickels for the balance. She previously required some level of
guidance to do so, but she was able to do this by herself for the first time.

Emily: (sings Give me a 40?)

Angela: (counts dimes but says) 1 nickel, 2 nickel

Emily: That’s a nickel or dime?

Angela: Oh! Dime (laughs) 10, 20, 30,40 (4 dimes)

Emily: Can you use a quarter?

Angela: Urn ~~~~~~~~~~~~~~ quarter quarter quarter.
Come on. Quarter 25, 30( a nickel),
30,40 I mean 35 and 40 (1 quarter and 3
nickels)

She was next asked to present 55 cents. This time when she could not find a
quarter, her response was to use dimes. She was able to find 3 dimes but no more were
available; this stopped her and she appeared to have no idea of what to do next and she
could not complete the task by herself. She was able to shift from a quarter to a dime, but
when she needed to switch from a dime to another coin, she seemed to lose her train of
thought and looked to me to help her. When she was redirected to use the coin-
classification-chart (see Figure 15); she came up with the idea of using a nickel. The coin
chart appeared to offer a way to organize the coins in order to make 55 cents. The chart
seems to give a visual and concrete aid to think the next step.

Coin versus written. At the beginning of the project, she was better with the two-
digit written format addition than she was with coin calculation; however as the project
progressed, she became stronger in calculating using coins. When Angela became more

familiar with coin calculation, she was asked to do addition in a written format. A very

interesting incident happened when she worked on the written worksheets. The way in

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which the question itself was presented, vertical or horizontal format, appeared to change
her way to complete the problem. She calculated the problem in vertical formation by
writing the “5” for the first column, and then do addition for “2+1” and

writing down “3” in the second column. She differently calculated the problem in
horizontal format.

She perceived “1. 9” as 19 and “2. 5” as 25. Then, she attempted to solve the
problem. She was able to solve 19 plus 2 by doing 19, and holding up one finger, and
saying 20, and the second finger 21. Her mistake was missed until it happened again. She
struggled with adding 25 and 5, and used the same strategy as she had to work with 19
and 2; that was saying 25, and holding up one finger, and 26, and the second finger 27

and so on.

@9+2=%ii
@;5+5=l

 

f'

D, ‘7
J

Figure 3 7. T wo-digit addition in horizontal and vertical formation. *

* Hand written

156

Previously she worked on a quarter plus a nickel, i.e., 25 + 5, and quickly and
successfully came up with the answer of 30. When the same question was presented in a
vertical written format, she sighed, tapped the table, touched her nose, said “um” 4 times;
and then she held up fingers to add up from 25. When she reached the answer, she yelled
“30.” I drew the line between “2.” and “5” and she was able to instantly answer 10. In the
example of another problem, 50 + 20, she was able to add 2 quarters (knowing 2 quarters
equal 50) and 2 dimes (20); however, she was not able to repeat this in the abstract
written form. These examples indicate that she did not comprehend the sameness in three
different formations and she perceived three different addition formats as three separate
problems, and uses three different strategies.

Written addition. The song, “Finding the Bigger Number,” was developed to help
with written addition. Before introducing the new strategic song, I wanted to confirm that
Angela had solid and effective strategies for written addition, because it had been
observed that when the new strategy is introduced, Angela, as well as other participants,
was confused between the old and the new strategies. Therefore, if she already had her
own way of doing, I wanted to reinforce and respect her own strategy. To verify whether
she had or did not have a strategy for written addition, the addition worksheet was
provided and her reaction and way of working on the worksheet was observed and
analyzed.

The analysis of the two successive sessions on the written addition suggested that
she seemed to not possess her own strategy for the calculation. It was observed that she
could add “7 + 2,” but not “2 + 7.” The way that she adds two one-digit numbers is that

she spoke the value of the first addend, and then hold up her fingers. She could easily do

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7 plus 2, by sequentially holding up another two fingers, however with 2 plus 7, when she
began with 2, and tried hold up additional fingers one at one time, from 3 to 4, to 5, she
seemed to lose track of her progression.

When Angela worked on the question on the worksheet (9 + 6), she held up 6
fingers, looked at the ceiling, tapped her lips thoughtfully, looked at the questions again,
said “urn” twice, and then tapped her lips again, and desperately looked at me. I allowed
about 30 seconds passing to see if she needed to have additional support or if she could
figure it out on her own. However, she did not seem able to solve the given question,
even though its pattern and presentation were the very similar to that of the previous
session, and we went through steps very similar to the lyrics of “Finding the Bigger
Number,” but without the melody. She was asked if she was willing to learn a song that
might help her solve the problem. She excitedly agreed to learn the song, which was
sung, and the corresponding steps were taught with the lyrics. The 10 questions were
practiced with the song “Finding the Bigger Number.” She was able to sing along after a
few practice rounds and knew what the next step was. While she worked on the
worksheet with song, she maintained eye contact with me and she did not tap her lips or
the table, or look away from the worksheet.

Another technique was also provided to make addition easier and more effective:
the concept of the teen numbers, which is fourteen to nineteen, as similar to using four to
nine and adding teen. The concept was introduced using a simple chanted rhythm while
Angela held up her 10 fingers and, I changed the fingers 1‘ held up from 4 to 9. After
working on this concept with different combination of fingers with chant, Angela had her

‘ah-ha’ moment. When she worked on 6 plus 7, she counted up to 10, and then she

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noticed that she had another 3 fingers left and she yelled “13!! Woo-hoe!!!” instead of
keeping counting her fingers.

With two-di git vertical addition practice, she indicated she did not completely
understand the concept of carrying (A digit that is moved from a given place value, or
column, to the another column of greater place value). The numbers in the questions are
the common numbers in combinations of coins. It was intentionally delivered in that
manner in order to make connections between coin calculation and written addition.
Figure 38 illustrates that she was able to add “5 + 5,” and other questions using the
identity property of zero, but made a mistake with “15 + 10” (the second question of the
first line. It indicated that she still did not completely understand the identity property of

ZCI'O .

 

 

 

g l E) l O
a)" B . - l C) t l o
2‘ /fl ’7’ 5 if“

 

 

Figure 38. Addition in the vertical formation. *

* Hand written

159

Figure 39 displays Angela’s mistake. She added “5 + 5” and wrote “0” for the first

column and “2 + 2” and wrote “4” for the second column. When I mentioned about

carrying and wrote “1” for Angela, she said “ah-ha!” and wrote 5. When the same

question was presented a second time, she answered 50.

 

First time Attempt

Second time attempt

 

Emly's handwriting to remind
her for carrying

Angela wrote 5 over 4 after
' writing "1" for her.

  

Angela initially wrote 4.

 

 

 

Figure 39. 25 + 25 in the vertical format. *

* Hand written

Figure 40 shows the difficulty of carrying for Angela. She got the answer of

“1070” for the problem of adding 75 and 25. The process seemed to be “5 + 5” and wrote

“0,”she wrote 7 from some where—might be from the question——and “1 + 7 + 2” is 10.

\

And wrote down before “70,” and arrived at “1070.”

160

._ WB .90.-.
4 >43 r >4;

(.4, 7 l.

 

.....I'

Figure 40. ”75 + 25 ” and “50+25” in the verticalformat. *

* Hand written

Number buddies song. During the initial interview, Lee reported that Angela was
working on her subtraction in school at that point. The song “Number buddies” (see
Appendix A. Figure A-2) was composed and taught in order to help her subtraction skills.
This song was developed based on the concept of complements of ten. The concept is that
if number X and Y summed equals 10, then X is Y’s complement of 10 and vice versa. In
the example “3 + 7 = 10,” 3 is 7’s complement, and 7 is also 3’s complement; they are
partners and buddies. This concept gave the participants another tool to use in solving
math problems. The entire song was sung, and then each line of lyric (9 and 1 equals 10,
8 and 2 equals 10, 7 and 3 equals 10) was separately reintroduced with many repetitions.
Each line was repeated (at least 15 to 20 times) in different registers using a keyboard, as
well as a different tone for voices (such as singing baritone or soprano) and in different
voice qualities (such as singing like a bird, or a dog) to help keep her attention. The
glockenspiel was also used to provide a different tonal quality to keep her attention.
Angela was able to sing along with my accompaniment and had no difficulty memorizing

the lyrics. There were a number of props to turn the abstract concepts of numbers and

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values into tangible and concrete images including a children’s abacus with 100 beads,
and a number chart (see Appendix I. Figure I-l). She had no problem using the abacus to
solve problems; however, it did not appear as beneficial in facilitating her mathematical
development. She easily answered questions using it, but when she was asked to solve the
same question without it, even though she had just answered that question, she was
unable to do so. It was apparent that when she uses an abacus, she did not use her mental
arithmetic skills, but counted to solve the problems.

When the worksheet that a directly applied the elements of the song Number
Buddies i.e., “10 — 9 = __,” “10 — 7 = __,” and “10 — 8 = _,” was used, she was
able to solve the problems immediately. However, when she was presented with the
application problems, “11 — =”and “11 — 8 =” and “11 — 9=,” she wrote down “3,” “2,”
and “1” as answers same as the complements of 10 (see Figure 41). She did answer the
application questions based on 7, 8, 9, and did not think of the relationship to 11. She
needed help to apply the knowledge to solve these applied problems. This indicates that
she memorized the matching numbers, but did not understand how to apply to the

question.

162

 

 

     

Figure 41. Angela 's subtraction. *

*Hand written. 4 is wrdte over 3. 3 is wrote over 2. 2 is wrote beside 1.

Sense of Numbers. Angela did not exhibit any difficulty with the magnitude of
numbers or their cardinality. She could pick the appropriate ntunber flash card when
requested. However, she did not skip numbers even though the diagrams were organized
in sets of 3’3, 2’3, or 5 ’s as illustrated in Figure 42. After she was instructed to count
objects in increments of 2, 3, and 5, she did not understand the purpose of doing so. I
explained that it is important to do so in order to speed up the process of counting objects.
We counted the object together in increments of 1 and increments of 33. She recognized
the differences in the increments and understood the reason behind the instructions. After
a couple of attempts, she was able to do so. Nevertheless, without my redirection and

reminder, she preferred to count by 1’3.

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Figure 42. Number flash card examples.

 
     

 
 

Angela could subitize up to 4 objects. In other words she did not need to
individually count groups of less than 4. She did prefer to count groups of more than 5
objects. She easily adapted the technique of counting from 5 and was able to use this tool
without my verbal directions. She did understand the concept of composing and
decomposing using 5 as a complement and was able to say ‘6 is 5 and 1’ and to use the
concept to hold up 6 fingers.

Lack of strategies. The sessions with Angela revealed that she lacked a range of
strategies to solve math problems. Lack of strategies is often mentioned in WS literature,
and her preferred way to count using her fingers provided some insights.

There are a couple of ways to solve mathematical tasks: one is to break down the task and
make the problem solving process easier, a technique that children with typical (TDC)
development develop by interaction with older individuals and manipulating objects
without specific instructions. For example, most TDC can flexibly start verbal counting
in increments of 1 from any point by age 6. If more than five fingers were held up, she

did not skip count up to five; she counted from 1. That is if 7 fingers were held up (5 in

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one hand, and 2 in another hand), she had to count from 1 to 7 instead of 5, 6, and 7.

Counting from 5 was encouraged and practiced many times. When five fingers in
my right hand and other combinations of fingers in my left hand were presented and she
was asked to answer how many fingers were up, she counted from her left side. This is a
good indication that she has a solid strategy for organized information from left to right.
However, it also indicated that she did not develop a strategy to glance at the five fingers
first, and then count from 5. She was finally able to count from 5 at the end of the project
without verbal redirections. According to Clements’ (2004) Developmental Guidelines for
Number and Operations, (see Appendix H), TDC can glance at and identify collections of
or collections of 1 to 6 objects and if the objects are organized in patterns up to 10 objects
(that is objects in groups of 25, 3s, 4s, and 55). This incident provides an indication that
she lacks this type of pattern recognition. Angela eventually learned this strategy by the
end of this study, which was evidenced by her spontaneous counting of objects using
increments of 3. When 9 objects in groups of 35, were presented, she counted “3,” “6,”
and “9” and arrived at 9.

Clock. The song “Increments of 5” (see Appendix A. Figure A—l) was introduced
to help her read the analog clock. The lyrics of this song simply utter numbers in 5’3 with
very succinct wording. Angela could easily recite multiples of 5 with the song and
pointed out the numbers along with singing. We sang the song up to 60 and 12 o'clock
was presented as 60 minutes. The problem occurred when she tried to read the clock with
an hour. While Angela was previously able to read hours, she now said “3:60” for 3
o'clock and “4:60” for 4 o'clock. After 3 sessions of working on reading a clock, it was

observed that she used the same wording and tactile cues to read the clock. While

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working with Angela, additional tactile cues were cultivated along with verbal
instructions. To help with the difficulty to distinguish the big and little hand, she was
instructed to put her left index and middle fingers between the end of a hand and the side
wall of the clock. The hand which she can put her two fingers is a big hand (a clock
hand), and the one she cannot is a little hand (a minute hand).

The next step was to put her right index finger from the 12 and go toward the
larger numbers on the clock with verbal instruction of “the clock goes this way.” When
her right index finger met her left fingers, the number on that spot indicates the hour. She
occasionally confused the process, but in general, was able to accurately read the clock

by the end of project.

Music

Angela had no particular musical ability and played no instruments, however she
liked to listen to music of many different genres and to sing along with recorded music.
Lee (her mother) stated that it is very hard to find a song she does not like. She was
fascinated with almost everything related to music including my iPod, a rainstick, a
glockenspiel, and a piano. I organized my iPod for the sessions and it includes various
genres and each participant’s favorites. She was amazed at the number and variety of
songs stored in the tiny iPod. When she had a chance to choose a song from an iPod, she
wanted to listen to Harmah Montana, the Beach Boys or Christian gospel songs especially
by Steve Green. Her choice of the Beach Boys was interesting given her young age; even

considering the Beach Boys are timeless; it felt like a little odd choice.

166

Angela’s singing quality is not notable; her voice is stretched, pushed and
pinched; but she surely enjoys singing and dancing. Although she could not play
instrument, she showed a great deal of interest in the piano and the glockenspiel. She was
taught for her to play a simple melody the glockenspiel by demonstrating and modeling
with a modified scores (see Appendix E) and she was joyful when she was able to play
by herself. She enjoyed different percussion instruments—practically anything
provided— including a rainstick, finger cymbals, and wind chime; practically anything
provided. She was able to play “Number Buddy Song (see Appendix A. Figure A-2),”
“Mary Had a Little Lamb,” and “Twinkle, Twinkle Little Star” on the glockenspiel and
could play Chopsticks on the piano.

Angela has a very interesting speech pattern. She repeats words and phrases three
times: “Ten cents, Ten cents, Ten cents,” and “a quarter, a quarter, a quarter” in a
manner of chanting. She made many different exclarnations with different intonation and
used sound effects such as “woo, wow, opps, hee-haw, and hurrah.” She seemed to
enjoy the diverse vocalized sounds. Her repetitive speech patterns sometimes were used
as motives in improvisation and became a song.

Angela’s chant evolved into melodic lines, which were more often observed after
the 6‘h session on. Her melodies were often based on descending minor-3rd and similar to
motives from the well-known songs like Chim Chim Cher-ee, as seen in the following
example. Improvised songs are sometimes identifiable like the Westminster Quarters, but
most often the familiar melodic line was not identifiable.

When she recited numbers and reached “100,” around 90% of the time when she

sang “100" it was in a big operatic voice producing—a dramatic ending. Her musical

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responses were encouraged and applauded; her rhythmic and melodic patterns usually
became the motive for improvisation and sometimes turned out to develop into a song.
Angela demonstrated her ability to make cadences. She was able to match the rhythmic
pattern and melodic contour of the beginning of the song. We enjoyed our musical
dialogs, which deepened our relationship, and out later sessions developed into
something similar to movies and musicals.

Angela did not feel comfortable during the first session because of a dental visit
and during the fourth session because of cold-like symptoms. She wanted to continue our
session even though she was not feeling well; she even retumed home early from school
because of how she felt, but she did not want to cancel the session. Her mother was
concerned and asked her if she wanted to take a break, but she said that she wanted to
continue, and she insisted that she was fine. This episode suggested that the power of
music as a motivator; she wanted to participate in the music therapy session despite of
her physical discomfort.

Music can influence her attitude, mood, and behavior. As the Angela’s dance with
Surfin ’ USA illustrated, to see this dramatic change in her behavior speaks loudly for the
power of music in her life. Normally a very shy girl with a calm demeanor, she changes
to a dancing queen when music is in the air. Angela was in awe of a flowery piano style
with fast moving passages, glissando, and ornamented riffs. She often asked me to play
“one more time” and expressed that she wants to play like that. Therefore, playing her
favorite passage worked as a very effective reinforcement for her.

In addition, Angela’s tempo of the physical movement has been changed by the

tempo of music. At the beginning of the project, I often improvised a simple motive to

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provide background music for her while she worked on her task. For example, when she
worked on the finding nickels from the pile of coins, she moves her hand too quickly to
match the tempo of my playing and cannot find any nickels. I did not notice this during
the session; however, while monitoring the video clip, it was observed. After noticing
this, I varied the tempo of music, and she changed her movements along with the tempo

of the music. It was remarkable how much impact music had on her.

Learned hopelessness and emotional support

Angela did not clearly show signs of learned hopelessness in her math skills, until
we worked on the similarities between written format and coin addition. She had two sets
of strategies, one for written format and one for coin calculation, she was unable to
recognize similarities to cross utilize concepts. I thought if she understood the sameness
in two formats, it could be a major break through in her mathematical development. After
several parallel explanations between written equations and coins, she became very
agitated and said “I’m confused,” “I cannot do” and “I don’t know” and finally stated “I
CANNOT LIKE... DO LIKE... THE HARD MATH.”

Another case in point, after struggling with distinguishing different coins and
adding up, Angela affinned, “I’m always confused.” There are few incidents that she
made negative declarations toward her mathematical ability; however those general
statements imply she has a general notion that she is not good at mathematics.

The negativity toward her performance was redirected by “Nobody’s Perfect” and
“You Can Do It.” At the beginning of the project, the songs were used to prevent any

apprehension she might feel about her abilities. That faded with the progression of the

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study. Later sessions, when she made negative comments, and I played “Nobody’s
Perfect” and “You Can Do It,” she began to sing the song and was able to regain her
composure and could focus on her project. Short musical phrases for approval,
congratulation, and disapproval worked very well conveying the meaning of what I was
communicating without verbal explanations, and she enjoyed listening to those phrases,
as evidence by her smile, and request for more playing.

In summary, Angela has such a cute and adorable smile. When she smiles, I feel I
need to do whatever she wants me to do so. She makes me feel like I am one of the best
musicians in the world and that she wants me to play more for her. A short musical
excerpt reinforced her as much as a piece of candy for young children. She was relatively
good at mathematical problem solving compared with the other two participants;
however, she mentioned that she could not do “hard math” when she worked on the
question “50 + 20.” She did not display any particular musical ability, but she
demonstrated potential ability in improvisation and was interested in learning how to play

a keyboard instrument.

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CHAPTER VI
ANALYSIS

In this chapter, stories of Big Red, Roxie, and Angela were collectively viewed as
individual chapters in a book. While the chapters share commonalities, they are also
different. After all thirty video recordings were transcribed and printed in color; the
segments of mathematical mistakes were indexed and complied into a file for each
participant using the Macintosh iMovie software program. The compiled files were
individually reviewed, and then the same interventions used for two or three participants
were simultaneously reviewed on two or three monitors. The similarities and differences
among the participants became defined and specifics emerged from this method of
observation.

The video clips were also edited and compiled dependent on the themes and/or
purposes for the various presentations during this study. The unexpected benefit of this
editing and compiling process was that while I tried to find the best clips and editing
process, I had to watch the same clips frequently giving me additional opportunities to
observe behaviors and responses. The participants’ difficulties, mistakes, every
movement, facial expressions, and gestures were imprinted in my mind, which, later,
proved most helpful in finding the related moments out of 30 session recordings, as well
as accessing the pivotal moments by theme. Sometimes, the same moment was used for
different themes. For example, a clip could be used to identify the theme of mathematical

mistakes as well as learned hopelessness. Side by side editing of three participants gave

171

the viewer a chance to know and understand the participants, but at the same time gave
me opportunities to observe them with a new lens. When I give public presentations of
the work, the comments and questions from the audience also make me look at the clips
with different perspectives.

The literature review on mathematical development, which concurrently occurred
with the sessions and analysis, sheds light on the participants’ mistakes and difficulties in
mathematics. This process of seeking more information was begun out of the desperation.
The participants’ answers for some questions did not make sense, and at times I could not
understand or even come close to speculating reasons for their responses. Having their
mistakes and incorrect answers in my hand via the video recordings and reading literature
fi'om the books and articles about typical mathematical development as well as
mathematical difficulties, provided insight into a deeper understanding of the participants
as individuals and as a group. Many of the incidents and mistakes which are mentioned in
this chapter were already described in the case studies, but here are collectively viewed in

order to understand them as a group.

Case Study Summary
In broad terms, when compared to Roxie and Big Red, as well as other children
with WS (from descriptions in the WS literature), Angela showed a higher level of
accomplishment in mathematical tasks and her understanding of mathematical concepts
was much faster than the other two participants, especially Big Red. While Angela
improved in mathematical functions and learned advanced concepts, Big Red needed to

go back to foundational mathematical concepts. It became clear that Big Red did not have

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the prerequisite concepts to work on addition. He could not figure out the quantity of the
collections of 1 to 10 items, which is supposed to be developed by the age of 5 or 6.
Roxie’s functional mathematical level was in between that of the other two participants,
and her other characteristics, such as music ability and attention span, were also between
those of the other.

While Big Red was the oldest, Angela was the youngest, and Roxie was in
between them. The level of functioning in mathematics for the three participants was in
the opposite direction: in the music area, Big Red was the star. Big Red could play a
drum like a professional drummer, and a piano with some I-IV-V-I basic chord
progressions with two fingers, the other participants did not have any particular ability for
playing an instrument. Roxie enjoyed singing along and was able to improvise a couple
of songs during the project. Though Angela liked to listen to music, she did not
particularly display her music-making ability. Angela exhibited much better attention,

whereas Big Red could be very easily distracted, especially by auditory stimulation.

Mathematical Interventions
At the beginning of the study the same exercises and interventions were intended
to be used with all three participants; however, during the course of the sessions these
were adapted and individualized interventions by modifications made to suit individual
participant’s needs. This included the changes in the tempo of songs, and specific songs
and subset of the verbal instructions. For instance, while the song “Finding the Bigger
Number” was used for Roxie and Big Red, the tempo for Big Red’s was much slower

than for Roxie’s. This difference was revealed by the video analysis with two monitors; it

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was not intentional. This adjustment occurred naturally as Big Red, for instance, required
_ more time to register information, and to accomplish the task. In another instance, when
one of participants did not understand a prerequisite concept in the song, another song
was developed to teach the needed concept. Comparisons and contrasts of Big Red,
Roxie, and Angela are discussed in four major categories of mathematical interventions—-

coins, written addition, magnitude and cardinality, and clock.

Coins

Roxie could distinguish between different coins; Angela could not identify the
different coins at the beginning of proj ect, however she mastered this skill within her first
two sessions. Meanwhile, Big Red did not master this skill until the end of the project.
The other coin related tasks were to be able to identify the sum of a combination of coins,
and to be able to give them to me. A coin—classification chart was used to help clarify this
concept and Roxie and Angela became familiar with the chart and effectively used it for
coin calculation. When Big Red counted coins or the coin hand props, he displayed his
confusion of quarters, dimes and nickels. Even though Big Red could sing the song
Nickel, Dime, and Quarter, he could not determine which coin was 5 cents and which
was 10 cents. Even though the sizes of each coin hand props was different and suggested
a different quantity, none of those cues—songs, visual hands props, and the size of
props—helped him to understand the value of coins. He seemed not to be able to grasp
that one coin can represent 5 things (a nickel), or 10 things (a dime), or 25 things (a

quarter); “1” is"‘one.” Although he did not understand the value of each coin, Big Red

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practiced with the coin-classification-chart’. However, he failed to understand when he
needed to use two taps for a coin (a dime), or only one tap (a nickel), and to count by 5 ’3.
While he struggled with these concepts, the necessity of retrogression to instill key
elements of mathematical development became obvious. In Big Red’s case it became
necessary to work on the concept of quantities—magnitude and cardinality—rather than
coin discrimination and calculation, and written addition.

On the surface, determining the sum of a group of coins and giving a certain
number of coins to another person would seem to involve the same process; however,
they actually require different skills. The identification of the sum of a group of coins can
be simple addition: “25 + 5,” “30 + 2” etc, but when we must to pay exactly 72 cents, for
instance, the task also depends on what we have on hand. If we have 3 quarters, it
requires handing over all three; if we have 4 quarters, 2 dimes, and 5 pennies, then we
can offer 2 quarters, 2 dimes, and 2 pennies, or three quarters and expect change in
return. Therefore, figuring out which coins are needed and paying the required amount
involve different steps and strategies. All three participants had a difficult time
completing this task, echoing the events at the Starbuck’s coffee shop described in the
prologue (see Prologue, p.1).

Roxie and Angela had difficulty calculating the exact change when they had more
change than required. For example, when asked to collect 40 cents, they began with 2
quarters, but did not know how to proceed. They needed and an explanation that

compared two numbers—the sum they have on hand and the required amount. If the sum

 

9 After the experimental period was finished and analysis of sessions had bee corrrpleted, it was clear that
without understanding the value of coin, the coin-classification-chart is not helping. But while I worked
with the participants, the apparent logic was overlooked. This issue is discussed in limitation and
consideration section in the conclusion.

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is the larger number, put down the last coin added; and then use smaller value coins. Then
they needed to repeat this step until they reached the required amount. Roxie and Angela
had the same tendency to go for the simplest amount, such as a dimes to reach 40, rather
than mixing coins and amounts, such as by using a quarter, a dime, and a nickel. While
Angela became accustomed with the more complicated combination of coins, Roxie had

a difficulty time to mix the coins and prefer to remain with the simplest way.

Written addition

A worksheet on single digit addition was used to work with written addition.
While Roxie and Angela could do all 10 questions with some mistakes (about 60 %
accuracy), Big Red could only answer 7 questions out of 10 questions without assistance
(see Big Red section, Figure 25). While Big Red did not even attempt to work on those he
perceived as ‘hard’ questions, Roxie and Angela did, but they also answered a few of
those ‘hard’ questions incorrectly. The quantity of the second addend seemed to be a
factor in the level of difficulty they perceived; if the second addend was larger, like 7, 8,
or 9, Big Red did not even attempt the questions, while the others often made mistakes.

The reason for this tendency can be found in the way the participants solved
problems. They appeared to add the second number either by using their fingers or by
mentally adding one to their running total, and by speaking the new total. If the second
number was too large, they became confused while using their fingers or during mental
calculations. When Angela worked on “19 + 2,”she did not hesitate; she immediately
answered 22 (even thought it was an incorrect answer), but when she attempted to add

“25 + 5,”she became agitated and counted aloud, “26, 27, 28, 29, 30” while she held up

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her fingers one by one. Angela’s attitude towards using the numbers 2 or 3, and the
numbers 5, 6, 7, 8, and 9 was obviously different and suggested she expected this to be a
‘hard’ question. Roxie added “9 + 6” by mouthing the next height number while holding
up her fingers, just like Angela did. Big Red occasionally used his fingers to guide him;
however, he seemed to basically memorize some questions and answers like double
numbers “2 + 2,” “3 + 3.” He would scan the questions and find the one he already knew
like “5 + 5,”and “9 + 2. Then he would write those answers and leave the others blank.

The song “Find the Bigger Number” was used to help compensate for these
difficulties and in an attempt to teach them commutative property (see Table 3). This song
worked very well for Roxie and Angela, but not Big Red. He knew which number was
bigger than the other; however, his difficulties with manual dexterity prevented him from
successfully accomplishing the task while singing the song. Instead of working on the
addition worksheet, Big Red practiced accurately holding up a certain number of fingers.
Roxie and Angela continually worked on the written addition worksheets and they could
finish the worksheet slightly faster than they first worked on the same worksheet before
learning the song and singing it. Also, the accuracy of the answers greatly improved and
the hesitative behaviors were not present.

It was interesting that Roxie and Angela could not recognize the similarities
between written addition and coin calculation. The two girls were able to complete the
written additions “5 + 3” by employing the finger counting strategy as a crutch. However,
when the exact same numbers were used but presented as a nickel (5) and three pennies
(3), Roxie and Angela were not able to use the same strategy. Since a nickel is a single

object, knowing that it had an abstract value of five cents was not something they

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comprehended. Roxie and Angela “connected the dots” between the fact of “a nickel is
five cents” and the process of converting a nickel to 5 cents in order to derive the sum of
a group of coins.

Like Roxie and Angela, Big Red could not transfer their strategies fiom written
addition to coin calculation, Angela could not transfer the strategy for coin calculation to
written addition. Angela could do the a quarter plus a nickel by tapping the nickel and
speaking the next higher number in increments of 4, but when she tried to add “25 + 5,”
she used counting strategy “26, 27, 28, 29, 30.” This illustrated that she did not perceive
the similarity between the coin calculation and written problems. She solved the
problems using rote memorization, not by creatively employing the mathematical
concepts.

Even after these similarities were pointed out and explained the two girls, they
continued to have a difficult time using the same strategy for different tasks. I can explain
the fact and knowledge to them, but I cannot make them understand; the understanding
occurs within them. While Angela finally came to understand that a nickel is equivalent
to 5 pennies, and she was able to use the same strategy for both version of the ‘5 + 3’

problem, Roxie was unable to learn this concept.

Magnitude and Cardinality

Magnitude itself means the amounts of each group of objects. Understanding of
' magnitude in mathematics refers to recognizing which one of two or three sets of objects
is the larger amount in the set. Cardinality is defined as the number of elements in a set.

Understanding of cardinality denotes the ability to say the word for the number

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corresponding to the object and to know the last number said is equivalent to the total
number of items in a set (O'Heam & Landau, 2007). For example, when a child ‘sees’
two die; dice A with l dot and dice B with 6 dots, a child, who understands magnitude
and cardinality, understands that dice B has more magnitude, and has 1 dot in dice A and
6 dots in dice B. If a child understands magnitude but has not yet developed an
understanding of cardinality, the child knows dice B has more dots, but does not
understand the last number he just counted means the amount of dots in the die.
Understanding of magnitude is observed as early as 18 months for two groups of objects
(Clements, 2004; Cooper, 1984). Understanding of cardinality is typically mastered by
the age of six, for collections of l to 6 objects, and if the objects are organized in patterns
of up to 10 objects (that is objects in groups of 23, 3s, 4s, and 55) (Clements, 2004).

Big Red displayed that he had not mastered cardinality, and might not be able to
master even the concept of magnitude. Furthermore, it suggests that the mistakes—he
counted his finger twice or skipped counting his fingers—might not be a mistake, but
indication of his lack of understanding the one-to-one correlation with each number and
each object (enumerating objects). He was able to distinguish the difference in magnitude
between 10 and 20 objects, but not the difference between 6, 7, 8, 9, or 10 objects. While
Big Red struggled with magnitude, Roxie and Angela both became accustomed to using
the concepts of magnitude and cardinality.

All three of the participants had not learned the “teen” concept. When this concept
was first introduced and practiced, Angela was able to learn the concept relatively
quickly, and could readily use it to solve a single digit addition questions. Roxie did learn

the concept and was able to use it; however, she needed to be reminded and redirected to

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use it at times; she preferred her old way. Because Big Red had to put great effort into
acquiring and using small scale cardinality, we were unable to work with the “teen”
concept during his sessions.

All three participants were good at reciting numbers in a sequence. Reciting
numbers in increments of 5’s and 10’s were also not difficult tasks. However, the ability
to recite numbers was not an indication of their understanding of the value of these
numeric symbols. This is similar to when typically developed individual could learn how
to count objects in other languages, such as in Spanish, German, or Chinese, they may
memorize the numbers in the other language did not mean that the person has an
understanding of the meaning of the numbers. An example might be not knowing the
meaning of cinco in Spanish. This metaphor also can explain why the participants need to
count from 1. We might not know what cinco means, but if we memorized the sequence
of numbers in Spanish, we could figure out the meaning of cinco by counting from uno
with one finger, do’ with adding the second finger, and so on. Several incidents suggested
that they appeared to be able to recite sequential numbers because they have memorized
the words themselves, rather than having learned their abstract values.

The first example of their level of understanding the meaning of numbers is
revealed by a comparison between Angela and Big Red. They showed the different levels
of function with understanding of cardinality. While Angela could remember the different
values an each card, Big Red pointed to any card in front of him as illustrated in Table 4.
The cards were organized with the first line with 5 dots with the second line variations
(see Figure 17). He seemed to distinguish cards with one line (a card with 5 objects) and

two lines (cards with 6 and more objects) and could find the card with 5 objects; however

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he failed to discriminate between 6, 7, 8, 9, or 10 objects. The placement of the card was
not changed between trials to enhance the opportunity to recognize different cards; but
even then, he could not remember the different quantities represented by each card, even
though their placement was the same. Roxie and Angela showed a similar functional
level; after they counted all the cards, and knew where the cards were, they were able to
identify them individually when asked. When the placements of the cards were changed,
they still found the designated cards and that suggested they understand the cardinality of

each card.

Table 4. The comparison between Angela and Big Red

Angela:

Angela

Emily: (sings Which one is 8?)

Angela: (her lips formed 8 and she picks a card with 8
objects)

Emily: (sings Which one is 7?)

Angela: (picks the 9-card and quietly counts and
touches her mouth; picks the 7-card)

Emily: (sings Which one i310?)

Angela: (whispers 10) This one.

Emily: Yeah!!! I think you are right.

Oh! Yes. Oh! Yes.

(sings Which one is 5?)

Angela: (pick the one; whispers 1,2,3,4,5)

Emily: (sings Which one is 10?)

Angela: (immediately points to the 10-card) This one

Emily: (sings Which one is 7?)

(immediately points to the 7-card) This one

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Table 4 cont.

Big Red
(Previously each card was counted together.)
Emily: (points to 10-card) This one?
Big Red: 5... Think before you answer.
Emily: You need to count by 5
Big Red: 5, 10
Emily: (points to the 9-card) This one is?
Big Red: 5, 10; No. 5, 6, 7, 8, 9
Emily: Which one is 8?
Big Red: We just went over it.
Emily: Yes, Can you find it?
Big Red: (sighs) OK.
(points to the 10-card) This one, right?
Emily: No. This one is?
Big Red: Oh! (points to the 7-card) Here it is.
Emily: Let’s count it.
Big Red: 5, 6, 7
Emily: How about this one.
Big Red: 1, 2, 3, 4,
Emily: Uh-uh! Don’t count from one. Count from 5
Big Red: 5, 6, 7 Oh! This one is 8.
Emily: Yeah! Yes!!! That’s great!!!
Which one is 5?
Big Red: (points to the 5-card)
Emily: Right! Which one is 7?
Big Red: (scans; points 5-card, and then 8-card) This
one right Here.
Emily: Uh-uh (no)!
Big Red: That’s 8. This is...

(points to the 10-card) This one is 7.

The second example of the understanding of the meaning of numbers is based on
an observation of the meaning of a quarter. The first time I asked Angela what a quarter
represented, she answered 52 pennies. After this mistake, I asked her the same question a
second time, she got the right answer. It could have been considered a simple mistake, but

as since 52 was not a random number, it raised a question of the reason for the mistake.

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Emily: I have four quarters and how many pennies?
Angela: (shrugs her shoulders) I don’t know.

Emily: You don’t know. One quarter equals...
Angela: 52 pennies?

Emily: One quarter?

Angela: 25

A possible explanation came from Roxie’s mistake rather than an assumption that
it was due to dyslexia. When Roxie worked on the value of a quarter, she often used 2
finger with her right hand, and 5 fingers with her left hand, while she spoke “25.” This
kind of presentation had been used habitually in various situations and I thought that it
could not hurt her development, and she was so cute when she did that. However, it
became a problem when we worked on addition using fingers as a reference tool. We
worked using fingers on both bands such as “2” using fingers of the right hand, and plus
“3” in from the left. When I asked her what five fingers rose in my right hand, and two
from my left hand meant, she proudly announced “25.” At that moment, I realized that for
individuals with WS, symbolic gestures can be confusing. When Roxie made her gesture
of “25 ,” I assumed she understood the concept of 2 fingers for two groups of 10, and 5
fingers for 5, but that was only my wishful thinking. If this is the case, Angela’s mistake
in saying that 52 cents were in a quarter, might not have been a sheer mistake; it may
have been an indication of confusion of the concepts.

Each participant displayed a different mathematical level when identifying the
value represented by a set of fingers (always the 5 fingers of one hand and another
combination on the other hand). While Angela could easily identify the value with or
without counting the actual fingers, Roxie needed to count all of the fingers on both

6‘] ,9

hands beginning at Big Red yelled out random numbers; he even offered “1 5” for

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“7” fingers. However he could answer “5” by one hand and “10” by two hands. When I
presented different combination of fingers, Big Red tried to reproduce it with his own
hands, and then to count his fingers by touching them with his other hand. Even with
touching fingers, he sometimes counted the same finger twice or skipped a finger. All of
the participants were instructed to count beginning at 5 when more than 5 fingers were
presented; however only Angela adapted to this method and used it appropriately. Roxie
understood the concept and occasionally used it, but preferred to count from 1 and she
seemed to be more confident using that as her main strategy. Big Red did not understand
the concept, he could not use it by himself; I needed to start with him. While Angela
showed composing and decomposing of numbers by saying “5 and 1” for 6, “5 and 2” for
7, neither Roxie or Big Red display the understanding of composing and decomposing of
numbers. Roxie was taught the concepts of composing and decomposing, especially in
relationship to units of 5; for example, 8 as “5 + 3,” 7 as “5 + 2.” While she did not fully
understand these concepts, she memorized the fact and for addition questions in

coordination with the song “Finding the Bigger Number.”

Clock

Reading an analog clock is a comprehensive test of accumulated knowledge and
concepts. Many smaller ideas lie hidden within this task and each concept must to be
mastered before a clock face can be read successfully. All three participants had difficulty
telling time using an analog clock, a problem elaborated at length in the WS literature.
Big Red, Roxie, and Angela revealed their similarities and differences in their skills in the

following ways:

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l. The comparison of the little hand and the big ha_n_cl

Big Red, Roxie, and Angela could not identify which hand was longer than the
other due to visuospatial deficits. Besides recognizing the different hands by size, these
conceptual values were counter-intuitive; as the “big hand” represents minutes, the
smaller unit and the “little hand,” the hour, or larger unit. The tactile information
provided by touching the end of the hand and the inside wall of the clock helps them to
determine which hand is the little or big. Because of Big Red’s problems with manual
dexterity, this was not an easy task for him. It took him too long to check with his two
fingers, and to process this information in a reasonable timeframe. Angela and Roxie
made a great improvement after they were prompted to touch the clock. This tactile
method appeared to offer a way to confirm their speculation, as well as give them the
opportunity to correct themselves.
2. Clockwise

Roxie and Angela appeared to understand “clockwise” as the direction that hands
travel around a clock face. Angela and Roxie could also understand that the clock
advances from smaller numbers to bigger numbers, however, Big Red was occasionally
confirsed by this.
3. The different systems of numbers for the little hand and the bi ,9, hand

The hours are counted by the little hand: 1, 2, 3, 4, 5 etc. through 12; and for the
big hand, the minutes were measured in increments of 5 on the clock they used, thus 5 —
55. All three participants could not grasp the concept of the two different number systems
in a “same” clock. However, they could count in units of 5 from memory. Angela was

confused when she reached 60 minutes and needed to switch back to “00.” For example,

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when the clock read 3:00, and Angela said 3:60. In response to Angela’s mistake, Roxie
and Big Red were taught to count from 5 to 55, and 0. The participants occasionally
confirsed which system to use in formulating their answers, and needed to be reminded
which system should be used for the minute hand and for the hour hand. However, Roxie
and Angela were eventually able to correctly choose the relevant system, even though Big
Red continued to make mistakes.
4. When the little hand is between two numbers and seems to be closer to the m
mtg

Different clock designs and differences in visual acuity, even for TDI, can result
in fiequent mistakes in judging the positional of the hour hand on many clocks. Big Red,
Roxie, and Angela and had a very difficult time with this function. After suggesting that
they place their two fingers from right hand at the end of the hour hand and move their
left index fingers from the small number to the big number, and when two hands meet,
the number at which the left index finger point is the number for hour, their performance
improved (see Figure 41). While Angela was able to accurately read the clock by the end
of project, Roxie was still occasionally confused, and Big Red had few opportunities to

develop this skills because of his other issues.

5. Switching between increments of 5 to single minutes to reach the actual time. such as

 

4:27
This skill is typically introduced in Grade 3 mathematical curriculums based on

the Standards and Focal Points adopted by the National Council of Teachers of

Mathematics (NCTM, 2008). None of the participants had attained this level of skill.

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Fine Motor skills

Fingers can provide a useful and versatile reference for someone who has just
started working with the concept of addition. In order to effectively use their fingers in
such a way, the manipulation of fingers must become automatic and facile. Ifit requires
physical manipulation and/or mental effort, if it takes too long, the usage of the fingers
themselves becomes a distraction and interferes with the task. While Roxie did not have
difficulty with her fine motor skills, Angela and Big Red had difficulty with their manual
dexterity.

In Angela’s case the problems were more physical, while in Big Red’s case it was
both cognitive and physical in nature. Angela’s pinky was a little shorter than that of the
average individual; her pinky did not reach the first joint of her ring finger. Because of
this structural difference and the actual stiffness of her joints (contractures), she had a
hard time holding up three fingers, especially with her left hand. Big Red’s contractures
appeared to be more advanced and especially hindered his finger movement, which was
very awkward and labored. He could not subitize nor did he understand the concepts of
composing and decomposing numbers. This meant that when he held up a certain number
of fingers, he also needed to hold up each finger individually, from one through the
ntunber. He did comprehend the units of “5”and “10,”but any other unit he needed to
count by using his fingers. In addition, because he could not flexibly start counting
numbers, he would hold his hand open to display five, and then add the fingers required
to get to a certain number. In other words, he did not have any means to expedite the
process of holding up a certain number of fingers, and his finger dexterity hindered him

from doing it efficiently and smoothly. Furthermore, his attention deficit hindered his

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focus on the task. As a result of all these factors, he did not have many chances to finish

his task holding up 7, 8, and 9 fingers.

Music

Big Red, Roxie, and Angela loved to listen to music. They liked many different
genres, from musicals, gospel, to popular music, and all shared a love of the Beach Boys.
Angela and Roxie both loved Hanna Montana. They were mesmerized with the different
and unique sounds created by different instruments including the glockenspiel, rain stick,
saxophone, and finger cymbals; they wanted to play with each.

Roxie and Big Red showed their ability to improvise. Roxie was able to express
how she was frustrated by her inability to solve some math questions by singing an
improvised song. Roxie showed that she could predict the next melodic line, and so
improvised with the therapist. Big Red composed four songs for this study, and numerous
songs and musical phrases for his own use, and exhibited a great sense of rhythm for the
lyrics and catchy melodic lines. His ability to change the lyrics of existing songs or to add
lyrics to complete an improvised song was very impressive. Angela did not show any
particular ability in improvisation, but she did have the tendency to complete a number
sequence by singing it operatically.

Their ability to improvise inspires awe, especially when considering that they
have no formal music training and improvise solely by listening and informal learning.
They create musical dialogues while not consciously understanding the formal elements
and rules of music. It is understood that for individuals with WS learning the structure

and language of music is the same as learning any language. People communicate even

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when they do not understand formal grammar, the rules of syntax, or even all the jargon
they use. It is speculated that people with WS might understand music in the same
fashion, by learning by ear. By listening, they learn where the harmony changes; when
they need to take a break or a breath, and when to emphasize a particular note or syllable.
Big Red can play the drum set, the djembe, and the piano. He could play a drum
set like a professional drummer. He knows many popular rhythms, and can successfully
accompany a band. His piano skills are less developed than his drum skills. He can play
with both hands; melody with his right hand and 2 finger accompaniment patterns with
his left. Roxie and Angela did not have any particular ability to play instruments. Roxie
did play Mary Had a Little Lamb in a minor key on keyboard and Angela was able to
learn to play a part of the song Chopsticks using a modified score on the keyboard and to
play “Number Buddy Song (see Appendix A. Figure A-2),” “Mary Had a Little Lamb,”
and “Twinkle, Twinkle Little Star” using a modified score on the glockenspiel. Roxie
showed a great interested in playing the saxophone, but she was not able to develop the
breath control necessary to use the instrument by the end of project.
All three participants were influenced by music, and when it was presented, Big Red
wanted to harmonize with the song or tap along to the beat, Roxie wanted to dance; and
Angela wanted to just listen the song. Angela occasionally matched the tempo of her
movement to the music. When Roxie did not dance with the song, she had a similar
tendency like Angela did, she moved to the tempo of a song. After this discovery, the

tempo of music and the types of music were considered to be congruent with the type and

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pace of their movement (Patterned Sensory Enhancement)’0. In addition, if they moved
too quickly, or if I tried to push them a little faster, I could use the tempo to alter their
pace within a limited range. For Big Red, complete silence was often necessary because
of his poor attention focus. When music was provided to fill the break between a question
and an answer, instead of working on the task, he focused on harmonizing or tapping on
surfaces. Individuals have a tendency to unconsciously move to the tempo of the music
with a tapping on the desk with fingers, keeping time with their feet, and/or nodding their
heads1 1; however these three participants displayed that they were more responsive to the
music. Other individuals would more likely still attend to the task and change the pace of
their movements to match the tempo of the background music within the range of
adaptable tempo; however, the participants’ responses to the tempo of the music were
instantaneous and replaced the physical movement of the task even out of tempo range
for their current tasks. They wanted to forget the task and move with the music by
dancing, tapping, etc. This phenomenon can be explained by the different degree of
activation in the amygdala, cerebellum and brain stem in the WS’ brain (Levitin et al.,
2003) (see Neurobiological factor, p. 35). The cerebellum and brain stem are directly
related physical movement of muscle group. There might be some explanation of their

immediate physical response to the beat of music.

 

’0 Patterned Sensory Enhancement (PSE) is defined as “rhythmic, melodic, harmonic and dynamic aspects
of music to provide temporal, spatial, and force cues for movements which reflect functional exercises and
activities of daily living” (Thaut, 1999).

H External rhythmic auditory stimulation (music) is mediated by internal perceptual shaping, and can
arouse and raise the excitability of spinal motor neurons mediated by auditory-motor circuitry at the
reticulo-spinal level (subcortical level). As a result, individuals are unconsciously influenced by the tempo

of music, and the groups of muscles involuntarily move to the tempo of music (Paltsev & Elner, 1967;
Rossignol & Melvill Jones, 1976; Thaut, 2005).

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CHAPTER VH

INTERPRETATION

The individual stories of Big Red, Roxie, and Angela depict the uniqueness of
each participant. The comparison and contrast of their stories shed insight into the
meaning behind the scene. In this chapter the meaning of this study, based on their stories
and various theories from many different disciplines is seen through my lens. The body of
literature regarding Williams Syndrome establishes the difficulty for peOple with the
syndrome when attempting to use math, both in understanding abstract values and in
processing or manipulating numbers. While the difficulty is well documented, its
causality seems to have been overlooked in the literature. It quickly became apparent
during the experimental phase of this study that some interventions were more effective
than others, and that specific techniques worked better with some participants than others.
One immediate example—the limited ability to perform addition—is itself a complex
combination of different cognitive and perceptual limitations, as well as conceptual
difficulties. Mathematical development theories give a basis for understanding their
difficulties. Neurobiological, behavioral, and physical factors in WS might seem
unrelated at first, but produce a Gestalt effect, one that severely restricts the capabilities
of these participants beyond what I expected. A detailed analysis of observations, session
notes, and digital video recordings were used as an instrument to reveal the bases of

mathematical difficulties and responses to music. The results helped to refine the theories

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underpinning interventions and suggested more effective strategies to compensate for

deficits and future music therapy practice.

Roadblocks to Mathematical Development

Over the course of these sessions all three participants continued to repeat the
same or similar mathematical mistakes, which provided a sort of window into their
thought processes. These observations lead to the identification of several roadblocks that
stand in the way of their mathematical development and improvement. These deficits
stem from difficulties in WS. It is not enough to ask what these roadblocks are, or how
they may inhibit the WS; we must also ask what those with WS are still able to do and
how capable they remain in the face of these obstacles. The analyses of the session data
suggest the following roadblocks: lack of metacognition, visuospatial deficits, attention
deficits of various levels of severity, slower data and problem processing, fine motor skill
deficits. As a result of those five factors, individuals with WS miss mathematical

developmental milestones, and they learn hopelessness.

Metacognition

Metacognition is typically defined as knowing how to think; it can also be defined
as the ability to understand a task and to choose appropriate strategies specific to the task
or a series of tasks, as well as the ability to organize the order of the problem solving,
such as in a flowchart (Hallahan & Kauffrnan, 2003; Overy et al., 2003). There are many
different names and terms for cognitive limitation in WS literature. Out of the many

different forms and descriptions of cognitive limitations associated with WS, the analysis

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of the participants in this study suggests that metacognition is a more serious and core
problem than is indicated in the existing literature. Metacognition is the broad enough
and specific enough to describe the difficulties in their cognition and can be the best
Option to be used. All three participants showed various levels of difficulty with
metacognition. During an individual’s early years, an infant or toddler learn some facts
and strategies along with their physical development. These facts and strategies are
informally learned early and reinforced cognitively and socially, through rewards and
recognition, as they are incorporated into more sophisticated structures. However, in the
case of WS these more complex strategies either do not develop or are slow to develop,
leaving them with only simple and basic tools and strategies.

The typically developed individual learns the basic properties and concepts of
mathematical processes at an early age, including commutative and identity properties
(see Table 3), as well as other miscellaneous concepts. For Big Red, Roxie, and Angela,
these concepts often remain beyond their grasp, although the memorization of specific
facts in mathematics and solution to specific set for the certain questions can simulate the
development of these skills, they appeared to memorize the facts and the solutions case-
by-case, rather than by understanding the big picture.

The following example is drawn from the music therapy sessions with the study
participants and reveals this limitation. Each participant was given the same worksheet of
math problems; one specific question asked for the sum of the values of 5 and 25, a task
that typically requires few steps to solve. It also takes little effort for most people to
switch the order of the variables from “5 + 25” to “25 + 5,” but those with WS are unable

to make this switch. In addition, it became apparent that some participants did not seem

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to understand the addition of actual abstract values, but had memorized sequences—the
multiples of prime numbers, the number 5, doubles (2 + 2, 4 + 4, 7 + 7) for example—in
order to simulate the use of addition or multiplication.

One potential explanation for the inability of the participants to add “25 and 5” is
the inability to apply the commutative property, to switch “5 + 25” to “25 + 5,”which is
easier for them to manipulate, because they can associate values with their fingers in a
quick hand-calculation. In Roxie’s case, she could only add “25 + 5” by using the coin
mechanism, or multiples of 5 to add the value of a quarter to that of a nickel to then reach
the sum of 30. In Angela’s case she was only able to add the specific sequence “25 + 5”
as abstract values, but not the converse, “5 + 25.” Even then, Angela only arrived at her
solution by talking herself through this process one integer at a time: “1 , 26; 2, 27; 3, 28;
4, 29; 5, 30.” When “5 + 25” was presented, she was unable to solve the problem with
any method. In a similar vein, she could solve “2 + 9,” but not “9 + 2.” Based on these
observations I speculate that participants are not able to use an associative strategy to
switch the order of the variables to sequences they can solve, a method typically taught to
elementary school students

Several incidents detailed in the case studies suggest that the participants
overused specific strategies incorrectly for unrelated problems; it appears that the
participants cultivate a single successful strategy at a time and discard earlier techniques
after a new tool produces success. The ability to differentiate between strategies and
recognize when and where they are best employed is not part of their consideration. As a
result, rather than develop repertories of strategies for different situations, the latest

strategy overrules the older approach. In Angela’s case, she had previously recognized 3

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o’clock by matching the hands of an analog clock to their positions or as she put it, when
the hour hand is on the “3” and the minute hand on “12” it is 3 AM or 3 PM. However,
after she learned the technique of reading the minutes within an hour, her description of 3
PM became “3:60”; she added the multiple of 5 minutes, 60, the base time of 3 PM, as
identified by the hand positions, and she lost the distinction between “0” and “60” as both
occupy the same visual position.

Mathematical problems and their solutions involve the manipulation of signs and
symbols, typically in a textual or written format. These signs and symbols are both
abstractions and the difficulty of WS to solve mathematical problems seems to lie in their

‘61,,

inability to understand the abstractions involved: the is as much an object as an apple,
or a penny and as an abstract principle, like the representative value of a coin, is difficult
to grasp. Numbers, sequences, the value of a specific denomination of money, for
instance, are memorized absolutes rather than being understood as representative values.
Because of their limitations in transferring knowledge and strategies from one

format (physical calculation: coins) to another (written calculation: worksheet),

participants perceived the two formats as unrelated sets of tasks (see Figure 43).

 

Coin Calculation Vertical Format Addition

 

A quarter 1

Ani'ckel @5+5:!O

Figure 43. 25 + 5 in two different formats. *

 

 

* Partially hand written.

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Angela added coins by saying 25 and then tapped a nickel while saying the next
factor in the series of 5 (30). For vertically formatted written addition problems, she did
this by talking through this process one increment at a time: “1, 26; 2, 27; 3, 28; 4, 29; 5,
30.” In Angela’s case the process of adding coins did not transfer into the written format,
a more abstract symbolic process of manipulating numbers.

Although the participants had previously acquired strategies to solve
mathematical problems, the participants responded to newly successful strategies as their
latest universal tool, and abandoned their previous strategy. In order words, instead of
accumulating concepts and strategies, and evaluating their appropriateness, they
immediately attempted to apply acquired techniques to all tasks. It was very interesting to
observe, and it raised the question of how to teach them the ability to discern the
differences between situations.

Angela, nevertheless, showed promise; she displayed spontaneous learning and
understanding. When she needed to count a group of several nickels, she spontaneously
suggested arranging them in rows of 3. Previously she had counted coins in a random
order; her suggestions came just after she had counted “smiley faces on a card in rows of
three in increments in 3. This indicated that Angela has the ability to understand the
concept in some degree—the “ah-ha” moment.

The mistakes, which the participants made, are not new or unique mistakes for
young children; these mistakes are common in the early developmental path for typically
developed children. The seriousness and significance of these mistakes are the
participant’s ages, 21, 16, and 10 years. The counting mistake (counting an object twice

or skipping counting an object) made by Big Red is a mistake made by 2 or 3 year olds.

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The questions we might encounter are “Would and/or could he pass this developmental

age?” and “Is it merely significantly delayed or a definite deficit?”

Visuospatial Perception Difliculties

Several reasons have been considered for these missing early developmental
milestones in WS. The visuospatial perception difficulties may impact the development
of children with WS in more ways than discussed in the literature. The difficulties in
visuospatial perception and tasks that rely on such perception have been reported in the
literature from a number of sources (Bellugi et al., 2001; Mervis & Klein-Tasman, 2000;
O'Heam & Landau, 2007; Semel & Rosner, 2003); yet the relation of this issue to
problems using math have not been clearly stated and may not be fully understood. On
the other hand, the visuospatial skills and their association with performance in math in
TDC and in Fragile X syndrome have been recently reported, and that gives the possible
explanation for the deficits noted (Bull, Espy, & Wiebe, 2008; Mazzocco, Singh Bhatia,
& Lesniak-Karpiak, 2006).

While computation does not particularly require visuospatial perception, recent
articles have pointed out that visuospatial perception and discrimination appear to be
important in order to understand basic math concepts (Aunio, 2008; Booth & Siegler,
2006; Bull et al., 2008; Holmes & Adams, 2006; Jarvis & Gatlrercole, 2003; Jordan,
Hanich, & Kaplan, 2003; Mazzocco et al., 2006). In comparing the test results, a strong
correlation between visuospatial ability and mathematical ability is apparent (Bull et al.,
2008; Mazzocco et al.’, 2006). Based on their longitudinal study, Bull, Espy, and Wiebe

(2008) suggest that the function of visuospatial perception on the formation of short-term

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memories during the first year of primary school (age 4-5) strongly correlate with
achievement at age seven in math. The ability to organize and process the visual stimuli
is a necessity in establishing the sense of magnitude; however, the deficits in visuospatial
processing of WS seem to hinder the ability to absorb the concept of quantity (Geary,
2004; Mazzocco et al., 2006).

There are many theories of how visuospatial difficulties interfere with
mathematical development. One is that visuospatial perception hinders the development
of the sense of magnitude further impeding an established mental number line. Also,
alignment of digits is very important to understand place-value and to borrow and carry
in arithmetical calculations (Mazzocco et al., 2006). The concepts of borrowing and
carrying are a known difficulty area for individuals with WS.

Big Red was not able to tie his shoes, to fold a paper in thirds in order to put it in
an envelope, and had difficulty plugging in a cord. He did not develop the sense of
magnitude and has difficulties with very fundamental mathematical concepts. He is not
necessarily typical of the entire WS population, but his symptoms in the visuospatial area
and his ability in mathematics are suggested in the literature. During free play or in
instructional settings, children engage in various types of everyday math-based activities,
and they intuitively learn these mathematical concepts. In contrast, because of their
visuospatial perception difficulties, children with WS may participate in the activities but

be limited in that they learn from these ordinary activities.

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Attention Deficits and Slower Processing Speed

All three participants have different measures of attention deficit disorder, as well
as different mental processing speeds. Big Red had a very difficult time focusing on
tasks, he was quiet sensitive to stimulation, especially auditory stimulation, and had
difficulties in self-regulation, Roxie showed these tendencies, though to a less severe
degree. Angela, in contrast, was fairly well disciplined and maintained focus well in
comparison to the other participants. Attention deficit alone affects performance; however
when it interacts with processing speed, the difficulties becomes much clearer.

There is no standardized definition of processing speed, and it varies depending in
discipline. In addition, the mechanisms underlying processing speed remain elusive, and
the factors involved in processing information, perceptions, thinking, and output speed
are still speculation and debate (Benner, Nelson, Allor, Mooney, & Dai, 2008; Shanahan
et al., 2006). In this study processing speed refers to how fast participants solve a task.
The comparison among three participants and their performances during the coin
discrimination tasks revealed several differences that Big Red processes much more
slowly than Roxie and Angela, perhaps partially due to his problems in fine motor
control. By comparing their use of tempo with the song “Finding the Bigger Number,”
Angela was slightly faster than Roxie. Big Red was about half of Roxie’s speed. Big Red
required more time to register questions, to cognitively solve the problems, and to
physically execute the tasks than did Roxie and Angela. Roxie missed more concepts and
facts in math than Angela: that missing pieces slow down her process speed. For
example, Angela could hold up 6 fingers by 5 and 1 concepts, while Roxie needed to hold

up fingers one-by-one.

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In addition to slow processing speeds, the attention problems in WS also make the
matter worse. As seen with my participants, they were always aware of the video camera.
I assumed that after five or six sessions, they would forget that I was videotaping, but this
was not the case; occasionally, they would stop in their task to wave and say hello to the
camera. While working on a task, if some distraction occurred, the participants lost their
focus, which necessitated starting from the beginning again. For this reason, Big Red did
not have many opportunities to go through the practice sequences and attempt to solve
the math problems. He would restart, become disturbed, stop and have to star over. While
he was the one who required the most practice, he was also the one with the least
opportunities to practice. Angela processes faster and has a relatively good attention span.
In contrast, Big Red both processes more slowly and has much more difficulty
concentrating on a specific task. If the processing speed could be enhanced there would
be less of an opportunity to become distracted. While attention deficits and processing
speed might not be the primary factors of mathematical development, they do influence

overall mathematical performance and training.

Fine Motor Skills

While fingers can be a very useful and handy gadget for individuals who learn
counting and number sense, because of difficulties in fine motor skills, Angela could not
effectively use her fingers. While working on Angela’s finger dexterity, a short musical
phrase was established. The lyrics suggest putting down the pinky first and then holding
it with the thumb. During the music camp for the young children with WS fi'om 6 to 12

years of age, there were more children who have the same type of pinky as Angela has

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and they have the same difficulties as Angela did. “Thumb and Pinky Meet Each Other”
(see Appendix A, Figure A-3) was used to help Angela as well as with the children at the
music camp. The other finger gestures which I practiced with Angela could be also used

to develop fine motor skills.

Missing Developmental Concepts

When limited ability to calculate is further scrutinized, the more obvious it seems
that this might not be due just to misunderstanding the concepts of addition and
subtraction. Because of difficulties in 5 identified areas, the three participants appear to
have missed many developmental milestones in their mathematical development. It is
relatively easy to recognize that in order to answer questions requiring division, an
individual must also understand the concepts of addition and subtraction, and know how
to handle remainders. However, it was not as easy to identify the lack of prerequisite
concepts required to handle basic addition. To perform addition, individuals must fully
graSp the number sense, which includes the meaning and use of quantity, and composing
and decomposing. In addition, understanding of the concepts of comparison between two
numbers, adding to and taking away, and grouping and place value is beneficial and can
expedite the processing speed.

Big Red was able to verbally count using integers of 1 to 100, and this seems to
be his only solid comprehension he has about number and operations area. He counted
by multiples of 10 or 5, to reach 100. (We did not attempt to count beyond 100.)

However, he appeared to not understand how to use his ability to skip numbers (that is

201

using multiples of 5) for nickel counting. In this respect, Roxie was more skillful in
mathematical operations than Big Red.

Roxie had already internalized some basic number skills including magnitude,
and cardinality. She could subitize up to 2 or 3 distinct objects but preferred to count each
object. She still struggled with groups of more than 4 objects; she needed to count each
object individually rather than using smaller sets of group to reach a total in order to
expedite the process of counting. She could neither group the objects nor compose and
decompose numbers. In contrast, Angela showed, even though she did not completely .
comprehend it, the ability to group objects, and compose and decompose numbers,
especially using units of 5 as an aid. Angela also showed that she could use of the concept
of the complement of ten, which was introduced to her during this study. In addition, she
demonstrated an understanding of the concepts of addition and subtraction. Angela was
able to advance her mathematical knowledge base on this foundation. However, Big Red
and Roxie could not advance to the next concepts because they did not possess the
foundational skills that were needed, and I had to develop an understanding of their
functional level. Big Red, especially, needed to begin with the basic concepts of
magnitude, and this lead to the realization of the length and depth of the difficulties he
faced in mathematical development.

The importance of the “key elements” of mathematical development was
especially apparent in the use of the concept of “teens.” While transcribing Roxie’s video
clip, it became apparent that while Roxie used the song “Finding the Bigger Number” to
solve the addition questions, she had also counted using her fingers. While many

typically developed individuals count 3 group of 20 objects by first grouping 10, and then

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continuing upwards using 11, 12, 13, etc: Roxie did not use this strategy, and she counted
using individual fingers to add to her running total. It seemed that she might not
understand the concept of numbers of the “teens.” To verify this suspicion each
participant was tested on the concept, which confirmed that none of them understood it. It
was quite interesting since they enjoyed finding patterns in language, and they enjoyed
different prosody. Why they did not master the language pattern of the ‘teens’ by the ages
of 21, 16, or 10? After working on this concept, Angela could use it for coin calculation
and addition. Roxie did understand the concept, but she did not use it spontaneously. She
needed to be encouraged to use it and retained her preference to count each object
individually. Unfortunately, Big Red never learned the concept.

According to Developmental guidelines for number and operations by Clements
(2004) (see Appendix H), TDC understand “teen” as 10 and more by the age of 6. The
“teens” issue indicates a lack in their development of number pattern recognition.
Clements (2004) explains this as language structure of the number system. In English the
words for numbers for the hundreds and thousands are regular; the words between 10 and
100 have irregularities. For example, 3,333 is said “3 thousand 3 hundred thirty 3” not “3
ten 3” (Fuson, 2004).

The participants demonstrated that they had missed a several mathematical
developmental milestones: counting by multiples, magnitude and cardinality, and having
the mental number line. The lack of these underlying concepts appears to be the root of
their inability to add. The participants, especially Big Red, appear to be missing several
key prerequisites of mathematic development. This suggests that while the difficulties in

mathematics became obvious during kindergarten or early elementary schools, the

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fundamental problems might have their origin as early as around the age of 2 or even
earlier. While the weak foundations of mathematical development were expected, the
severity and gravity of the issue was not accurately foreseen. The following figures
illustrate the expected foundation of their mathematical development prior to the
beginning of the study (Figure 44) and again, after the project was completed (Figure 45).
I was expected that even the participants had weak foundations, they might able to build
their mathematical development based on their weak foundation as illustrated in Figure

44.

 

Figure 44. Expected mathematical foundation with the imagination of the game Jenga.

Illustration by Hyang Eun Lee

204

Unlike my expectation, their foundations for the mathematical development was
practically non-existent especially for Big Red (see Figure 45). He memorized some facts

in math, but he did not understand the concepts behind the facts.

 

Figure 45. Perceived mathematical foundations afier the project completion.

Illustration by Hyang Eun Lee

Learned hopelessness
Big Red, Roxie, and Angela all made numerous comments about their own
perceptions of their performances.
Big Red And think about me I have trouble with math.

(Music) really help me.
It really gives me ability to work with math in

easier way.
Roxie “I always messed up"
"messed up every time."
Angela “I cannot like... do like... the hard math"

205

In addition to Big Red’s verbal statements, the different attitudes and body langue
he held towards math related tasks, and music-related tasks suggested that he was keenly
aware of his relative inability to use math. When he was requested to work on math tasks,
he kept popping up and down from his seat, or if he was standing, he moved back and
forth and his shoulders were downcast and dangling. He asked repeatedly, “How am I
doing so far?” and would nervously touch his chin or temple. “Let me guess.” He
specifically chose the word “guess” instead of “find” or “see” when he tried to figure out
the different numbers of the object cards—“guess.” While he offered answers to different
problems, he keenly searched the researcher’s facial expressions instead of concentrating
on the task, props, materials, and actually solving the problem. He used the researcher’s
facial expressions as a barometer for the accuracy of his answers. In contrast, when he
worked on his musical task, his shoulder was opened and broadened. He sat up straight,
made eye—contact, and his eyes were sparking with expectations.

Roxie was readily distressed by her confusion with mathematical concepts and
was too willing to quit and give up too quickly. While she said that she did not want to
make mistakes, the missing elements of her mathematical development inhibited her
performance and really upset her. Angela did not clearly show her distress regarding math
related task until she commented about her statement about “hard math.” She categorized
tasks as easy or hard, and she defined “50 + 20”was “hard math.” She was definitely
aware of her limitations concerning math.

The participants showed learned hopelessness to different degrees: Big Red was
tenified, Roxie agitated, and Angela concerned and aware. After years of frustration and

repeated failures gave them reason to believe that they are not able to use math,

206

something which is a fact to some degree. They were all missing some of the basic tools
necessary to solve mathematical tasks. They felt hopeless, but it is a “learned
hopelessness” which might not be the most appropriate term in their cases. This
hopelessness leads to “giving up”; they are ready to give up at any give moment. The
years of experience with math tells them ‘no matter what, I am not going to understand
the concept.’ One more trial, or one more day of practice could make the breakthrough,
but they do not even want to try. That eventually means no attempt, no opportunity to
learn, and no gain. That is the real problem in ‘hopelessness.’ When their answers were
correct and when they truly knew that they had succeeded, they were truly happy and
gained a huge boost to their self-esteem, at least for the moment. It they could achieve
more successes and victories, especially in their “war with math,” they might be able to

reverse some of this learned hopelessness.

207

Judgment: Music as a Therapeutic Medium Based

on the Results of this Study '3 Interventions

 

 

Music as a therapeutic medium

W “a-

 

[1 /

i ’l,”’”(/ 9‘.“‘\x‘ ./ ‘\\\ if i
.. ’ * Music as a . 2

~- _ Music for ‘ - ._ 1’ . Musrc as a structure
F _ cogmtwe _"i motivator and , provider l
development " encourager

,. . 2. l

J“; \\ /

 

l l .
,r' ‘.‘_ ’ l I

 

Figure 46. Music Therapy Theory Diagram.

The discussion of the use of music as a therapeutic medium for individuals with
WS will consider three primary functions: music used to advance for cognitive
development; music used to encourage and motivate; and music used to provide
structure. This discussion will also consider the benefits, limitations, and additional

considerations drawn from the responses to the interventions.

Music for Cognitive Development

Music used as a memory aid or a mnemonic device proved successful and was a
useful tool for the participants of this study. The key to this success was mainly the
structure of the songs. Participants could quickly learn and easily sing the mnemonic
songs developed and used in this study (other than the songs “How Many Pennies in a
Quarter? ” and “Five Pennies Make a Nickel.”) The participants memorized the songs in
fewer than four iterations and showed no difficulty in retrieving the melody, but the
information contained built into these songs had a little different aspect. Although the

melodies were simple and catchy, if the lyrics were too verbose and became distracting

208

and meaningless, the result was that the participants frequently used incorrect lyrics. For
the songs “How Many Pennies in a Quarter?” and “Five Pennies Make a Nickel,” the
participants randomly replaced “penny” with nickel, dime, or quarter. In later sessions,
these two songs were replaced with “Nickel 5, Dime 10, and Quarter 25” which is much
more succinct and directly connects the name of the coin to its value.

As suggested in the literature review, songs used as memory aids and mnemonic
devices should be succinct and the musical motives for each concept need to be distinct.
In addition to a distinct melody, using interesting sound effects from different instruments
(such as rain sticks, glockenspiel, and wind chimes) seemed to help hold their attention
due to their affection for different sounds. Concise lyrics and distinct musical motives
appear to be the key to success for these songs to work as mnemonic devices.

Learning involves several steps. While facts can be memorized fairly easily using
mnemonic songs, the other aspects of learning—comprehension, application, the analysis
and synthesis of the information, and the ability to apply knowledge to new situation— _
are not as easily transferred. Ann Badeau (Carpenter et al., 1999, p. xiii), a second-grade
teacher said, “It is only when you build from within that you really understand
something. If children don’t build from within and you just try to explain it to a child,
then it’s not really learned. It is only rote, and that’s not really understanding.”
Understanding of knowledge cannot be forced and music therapy intervention cannot
provide the illuminating or the “ah-ha” moment. However, music therapy interventions
could give individuals with WS more opportunities to have the “ah-ha” moment. Very

few opportunities are available for individuals with WS; because they avoid working on

209

the math related tasks as much as possible. This is because they have tried it before and
they knew they do not perform well on those tasks.

The participants had difficulties applying the information they memorized to new
situations. Strategic songs were designed to help compensate for this difficulty. While the
participants were readily able to recite the lyrics of the mnemonic songs, using those
songs as aids to solve math problems proved a different issue. The mnemonic songs were
not used unless the participants were given specific instructions to use them. Knowing the
song “Nickel 5, Dime 10, and Quarter 25” may have provided the answer to the direct
questions of “how many cents in a nickel,” but it did not immediately help them identify
the total amount of a mixed group of coins. That would be an example of applying the
knowledge gained. In order to calculate the value of a more random combination of
coins, the participants needed to classify them first by type, using a coin-classification-
chart, and then to calculate the amount for each coin: in increments of 25 for quarters,
tapping method for dimes and nickels, and in increments of l for pennies. Verbal
instructions were used to guide this process. “Finding the Bigger Number” was
developed based on these repeated verbal instructions; it worked very effectively to guide
the process using written addition. The participants were able to solve the written
addition questions slightly faster without verbal instructions.

The (mathematical tasks used in the study required the participants to follow a
series’of logical sequences, such as “Finding the Bigger Number,” the lyrics of song
guide this logic. The key element to development of this song was the analysis of the
task, which in this instance was single digit addition, a relatively simple task to base a

strategic song upon (see Figure 47).

210

 

C 3

Finding the bigger number

 

 

 

 

Holding up the smaller number

 

 

 

 

Saying the bigger number

 

 

 

 

s
Counfing

 

 

 

 

 

Figure 4 7. Flowchart for Finding the Bigger Number Song.

Other tasks required different types of decision making dependent on the
requirements of the process. For example, coin calculation requires a rather complicated
procedure. As shown in the following flowchart (see Figure 48), the song for this
procedure should direct the learner to the next concept, and move them through

progressively as indicated in the chart.

211

Figure 48. Flowchart for coin calculation.

/’ \
1 Start I
r, /

\ 4/
x
L

 

:3-
Put the coins
in the coin classification chart in order

 

 

 

 

 

Count each coin by 25's

 

 

 

Remember the last number

 

 

Say the last number

\ _>----_-— “:1 5"
~\,\\ 15 there a dime. Yes ' Tap the coin twice by saying 5's
1

 

 

 

 

Say the last number

\k

/
\

 

 

 

 

Tab each coin by 5's

 

Say the last number
Count each coin by 1's

 

 

 

 

 

 

 

/’/'
i . .
\ Finish order

The difficulty with this concept is that each condition statement of the point of

decision needs to have a distinct melody. If the melody is too similar to others or if each

212

condition statement has a much different verse, individuals with WS will mix-and-match,
and lose the process embedded in the song. In both cases, “How Many Pennies in a
quarter?” and “Five Pennies Make a Nickel,” the participants reshufiled and mixed the
lyrics. This makes composing strategic songs much more challenging than mnemonic
songs. If the task is appropriately analyzed, and the logic flow is developed considering
their skill level, individuals with WS can be taught the new strategies by strategic songs.
These songs can be beneficial and effective for supporting their particular mathematical

task.

Music as a Motivator and Encourager

The participants showed the advantage of using music as a motivator and
encourager and suggested that a music therapist can effectively use music for that
purpose for individuals with WS. Gfeller (1999) states that “Music is a valued, enjoyable
stimulus and social event for many people. As such, the opportunity to listen to music or
participate in musical activities can be used as a reinforcer (reward) in behavior
modification programs (p.267).” In addition to these, music attracts individuals with WS
as a magnet draws iron or steel objects.

At the camps, conventions, and meetings of individuals with WS, they gathered
around a performer using an instrument forming lines similar to those in a magnetic field.
At these events, if there was music, there were individuals with WS and if there was no
music, they had a tendency to make music by singing and tapping. They crowded so close
to the performers that the performers did not usually have elbowroom to play their

instruments. The staff members at these events often warned performers of the tendency

213

of WS to crowd too close to performers, and tried to set the boundaries. Even then, the
performers were usually overwhelmed by the crowd and their enthusiasm. Throughout
this research, indicators remained strong that music can be a motivator, and serve to
encourage participating math-related music interventions for individuals with WS.

Angela’s willingness to participate in music therapy session despite her physical
discomfort suggests that music has some power to alleviate this discomfort and to
motivate her to work. Big Red demonstrated the benefit of music activity for
mathematical development by his attitude and behavior toward music-related math tasks
as opposed to math tasks that did not incorporate music. As previously mentioned, his
attitude and behavior altered dramatically when he was presented with a task involving
math. He became very tense, and agitated. However, when this same task incorporated
music, in even as simple a form as asking a question in a song, he relaxed. Roxie
typically sighed and rolled her eyes when she was given a worksheet, but she swung her
forearms from side to side along to the tune of “Finding the Bigger Number” while she
worked on the worksheet. When Angela worked on her worksheet and could not
remember how to do solve a problem, she stopped, tapped her chin and looked up the
ceiling. When she was informed that she could learn a song that might help her face lit up
with a big smile. These are all instances indicating that when math tasks are synchronized
with music, the participants perceive a math task that incorporated music as a musical
task more than as a math task at least, until they begin to have difficulties.

The participants enjoyed listening to short selections of musical pieces as a
reinforcer, in addition to verbal reinforcement, or just by itself. At the beginning of

project, verbal reinforcement was provided in addition to musical reinforcement, but this

214

was no longer necessary after a couple of times of demonstration with verbal and musical
reinforcement. Angela was fond of the little excerpt from “Congratulation and
Celebration” byCliff Richard. This song used to praise her successes, and she always
wanted to hear the excerpt again. She often asked to for this excerpt to be played, but to
keep it as a reward it was only played when she successfully accomplished her tasks. The
music reward was more powerful than was verbal praise.

Roxie also enjoyed Congratulation and Celebration. However, there was a short
improvised musical phrase used to congratulate Roxie, and she preferred to listen to it
rather than Congratulation and Celebration. In Big Red’s case, giving him the
opportunity to play his drum after a success worked as encourager. At the end of a task or
at the end of a session, we played songs together; we used the songs that supported tasks
or any other music he chose. He liked sharing musical moments with others, and he took
our playing together as a reward.

Minor descending scales or minor chord progressions were used to indicate
incorrect answers rather than spoken responses. Since the participants often required
redirection, this was a better way to handle the errors than to repeat negative statements.
Because they were easily frustrated and often ready to give up, these musical phrases
conveyed the same error message but in a more constructive manner. In Big Red’ case
after I played these minor themes, Big Red would look at me, smile, and say, “Let me
start over.” Roxie usually burst into laughter and started the task again. Angela typically
said, “That’s so cool.” They needed to know that they had not gotten the correct answer,

but at the same time their disappointment should be turned into a positive and reinforcing

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experience whenever possible. It is a way to avoid self “put-downs” affecting their self-
perceptions of their math performance.

Using music to focus attention was discussed in the literature as a potential tool.
Music does hold their attention; however, the participants sometimes became too
attracted to the music itself and could not remember their task or did not want to work on
their task. Big Red would harmonize with the song or tap the table instead of finding the
designated coins. Roxie liked to dance with the songs, and more often than not, she
danced to the music. While Angela worked on finding designated coins, she moved her
hands too quickly to match the tempo of the music and she was not able to find the coins.
When I provided some background music while she completed a worksheet, she often
looked away from the worksheet and said, “I like it” or “That’s pretty.” There is a very
fine line between music as an attention holder or as a distracter. Music has a powerful
influence on the participant’s attitude, mood, and behavior. That power allows music to

work as a motivator and encourager for individuals with WS.

Music as a Structure Provider

From the beginning of music therapy history, the time aspect in music and the
physical structure (forms and patterns) in music have been pointed out as one of the
major therapeutic tools in music. Gaston (1968) stated that “Music is structured reality
(p.24)” and Sears (1968) pointed out that “The music must be carried through in its time
order (p.35).” A musical tempo helps structure the perception of time and can be used to
adjust processing speeds of individuals with WS when music accompanies a task. The

structure of music can help to organize information in order and provide opportunity to

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express their feelings and emotions in musical context. Lathom (as cited in Peters, 1987)
states that “For the child [or adult] whose world is often a confusing chaos, the order of
music is a welcome structured experience in which the child [or adult] may feel free from
confusion and safe in the ability to predict the activity and the person associated with the
music” (p. 51). Musicing and listening to music is not a ‘total’ chaotic experience; there
are expectations, unwritten rules, and certain level of predictabilities. As well the order
and structure in music provides the framework to work on the participant’s mathematical
tasks.

There is relatively well-established theory and practice in the role of rhythm and
tempo in physical movement (Thaut, 2005). Rhythmic auditory stimulation influences
kinematic movement in a person, and it is reported that when rhythmic auditory
stimulation is applied, the parameters, such as a cadence, spatial trajectory, and efficacy,
in physical movement can be modified within a 5% auditory time perception threshold
limitation12 (Thaut, 2005). When Angela worked on finding different types of coins, she
matched the cadence of her arm movement with improvised music. According to the
tempo of the music, she moved faster or slower.

The time structure (or tempo) provided by music appears to be used to control
process speeds in individuals with WS. It was possible to challenge Angela and Roxie to
complete tasks by playing a song with a faster tempo, yet still within a threshold. If the

tempo was changed too noticeably, the participants typically looked at me and wanted to

 

’2 The increase or decrease in 5 % from the current tempo of movement was suggested by a “Weber
fraction.” The “Weber fraction” is the percentage of the different thresholds obtained for different sensory
stimulus. For example, in order to perceive the difference between electric shocks, a person needs to have
1.3 % difference between them. In comparison, to taste sodium, 8.3% difference is needed. It was found
that the Weber fraction for auditory time perception is 5 % from 0.4 sec to 2.0 seconds interval auditory
stimuli. The imperceptible changes in the tempo of music were essential to make training as comfortable as
possible (Epstein, 1985; Getty, 1975; McBurney & Collings, 1977).

217

know the meaning of sudden tempo change; sometimes they appeared to become
confused, other times they took as it a game and laughed at the assumed challenge.
Although it is difficult to gauge how far they can be challenged, it is worth researching
the possibility of using the tempo of the song to alter or enhance their processing speed.

Playing a musical instrument, especially those with keyboards, requires hand-eye
coordination and visuospatial perception. When playing a song on a keyboard or an
instrument, it requires the coordination of different muscle groups to play the next note or
passage. The player needed to know the distance between the notes; calculate how much
movement is required for the next note by the visual information; and then physically
manipulate their hands to move that note. The relationship between instrument playing
skills and visuospatial ability has been mentioned in anecdotal examples. The individuals
who have better instrument playing skills, appear to have a better visuospatial ability. It is
not clear whether they are able to play an instrument because they have a better
visuospatial ability, or they have a better visuospatial ability because they practice with
an instrument. However, it is clear that playing an instrument demands eye-hand
coordination and practice with an instrument will strengthen this ability.

One unexpected benefit during the study was seen in the participants’ ability to
improvise. Big Red was particularly talented in improvisation. Roxie showed a great deal
of possibility, but her improvisation ability appeared to not be fully developed yet.
Angela’s improvisation was made up of excerpts from miscellaneous songs rather than
her own original work, though she exhibited that she has potential by embellishing given
motives and phrases from a song. Big Red and Roxie seemed to know the basic chord

progressions and were able to improvise songs. Big Red, without apparent conscious

218

deliberation, extended three line lyrics into four line lyrics on the spot, and added a word
phrase to finish the song with a tonic chord. Such a musical framework (physical
structure) allows a musical conversation to be possible between two individuals, a
conversation enjoyed by both participants and researcher.

Big Red has had drum lessons for more than a decade and has been exposed to
many different musical settings. Roxie and Angela listen to music in a great deal, but are
not involved with the formal music training. However, the participants were acquainted
with the musical style. Their understanding of music allowed the music therapists to work
with them at an advanced level.

Improvisation was used mainly for two functions: to express feelings and
emotions, and individual compositions to match their specific cognitive needs. In Big
Red’s case, he improvised a song not only for himself, but for other two participants.
When Big Red experienced difficulty counting increments of 25 for quarters, he was
encouraged to chant the numbers (25, 50, 75, 100) and he improvised a simple melodic
phrase to help him memorize the numbers, which worked very well. He improvised the
songs “How Many Pennies in a Quarter?” and “Finding the Bigger Number,” which
became the two major songs used in this research; after the songs were introduced to and
used by the other participants. When he was told that his songs had been introduced and
used for other participants, he showed a great deal of pride in his accomplishment,
especially as he knew the others through WS gatherings. He expressed his pleasure to
have had a chance to help Roxie and Angela. Roxie was able to vent her frustration with
her poor performance by her improvisation, and was able to positively redirect herself to

focus more attention in her next effort not the final product (process vs. product).

219

Summary of Interpretation

The following diagram (see Figure 49) offers a visual relationship of several of
the key elements discussed in this interpretation. Individuals with WS have many
stumbling blocks during the course of their mathematical development. They have
difficulties in metacognition, visuospatial ability, attention, processing speed, and fine
motor skills. The length and depth of these difficulties vary by individual. While many of
the effects of these difficulties overlap and are interrelated, each can require a different
form of intervention and music therapy can offer various approaches to help them cope.

Metacognition difficulties can be helped by using mnemonic and strategic songs.
While the participants did not have difficulties memorizing the mnemonic songs, they
needed help with the procedure in solving the mathematical tasks. Strategic songs must
be composed based on the steps or logical flow produced from the analysis of specific
tasks. Practicing how to play the instrument by using music as a structure provider might
be a tool to improve WS visuospatial abilities

Music used as a motivational tool and as positive reinforcement can help focus
attention. The participants are drawn to music; they enjoyed listing, playing, singing, and
improvising. Music is the powerful tool used that can gain their attention and redirect it
or lessen outside distractions. Once music had their attention, they tried to focus more on
their task; however, there is the risk that they may enjoy the music and become distracted.
The time taken to process information can be affected by music tempo. The participants
were easily influenced by music and its tempo. For fine motor skills, playing an
instrument as well as working on action songs that require finger movement might be

another tool to investigate.

220

 

 

 

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lrnprovisation in musical structure could be used to aid their emotional needs and
their cognitive needs for memorization. Their hopeless feelings were eased by singing
encouraging songs. The various functions in music allow helping with participants’
different needs. Understanding music’s foundational function in human behavior is a

highly useful to define how to skirt many of their cognitive deficits.

221

CHAPTER VIII

CONCLUSION

This study investigated the teaching of mathematical concepts to children with
Williams Syndrome (WS) through music to provide a foundation for developing music
therapy interventions in their cognitive development. The results of this study suggest a
number of possible limitations, considerations, and suggestions for the practice of music
therapy in relation to individuals with Williams Syndrome, and future avenues of

exploration.

Suggestions for Music Therapy Practice
In this section, the thoughts, ideas, and clinical applications for music therapy
practice are shared. Some are the direct observations from this study, and others are
additional suggestions based on my experiences and observation with individuals with

WS and their parents.

Think Like Individuals with WS

A key to finding a more effective way to help individuals with WS might be to
attempt to think like them. Many unforeseen impediments became variables and factors
during the course of this study. The specific example of Roxie’s confusion with 2 fingers
on one hand and 5 fingers on the other as representing “25,” or Big Red’s mistakes in

counting, or Angela’s shorter pinky. A greater understanding of neurological pathology

222

and attempting to more accurately predict reactions might have helped to avoid potential
obstacles that limited the effectiveness of the techniques used to advance cognitive
development.

Hallahan and Kauffman (2003) summarized the differences in different genetic
disorders as illustrated in table 5. Each genetic syndrome has different characteristic, ,
strengths, and weaknesses. There is an enormous amount of information behind simple
descriptions of WS, which is to be expected of any genetic disorder.

Up to this point, music therapists have overlooked these differences in cliental and
applied practice based on prior knowledge, theories, and research regarding cognitively
challenged individuals and other similar symptoms. With a vast array of genetic disorders
and conditions in existence, it is impossible for any therapist to be familiar with all of
them. There is no way to know each disorder in advance and in many cases, music
therapists are overloaded because of large caseloads and required documentation.
However, this study suggests that it is necessary to know each disorder in order to help
therapists understand WS’s cognitive development and become more familiar with

possible techniques and approaches.

223

Table 5. Links between genetic syndromes and behavioral phenotypes*

 

Behavioral Phenotype

 

 

 

 

 

Genetic syndrome Relative Weaknesses Relative Strengths
Down Verbal skills, especially grammar Visuospatial skills
Syndrome Problems interpreting facial emotions
Cognitive skills tend to worsen over
time
Early onset of Alzheimer’s
Williams Visuospatial skills Expressive language,
Syndrome Fine-motor control vocabulary
Anxieties, fears, phobias Facial recognition and
Overly friendly memory
Musical interests and
skills
Fragile X Short-term memory Verbal skills, including
syndrome Sequential processing vocabulary
Repetitive speech pattern Long-term memory for
Social anxiety and withdrawal information already
acquired
Prader-Willi Auditory processing Relatively high IQ
syndrome Feeding problems in infancy (Average about 70)
Overeating, obesity in childhood and Visual processing
adulthood Facility with jigsaw
Sleep disturbances puzzles.
Compulsive behaviors

 

* From “Exceptional Learners: Introduction to Special Education” by Hallahan and Kauffman (2003),
p.126. Copyright 2003 by Pearson Education, Inc. Reproduced with permission

It is not practical to do intensive research on every incoming client’s disorder. A
solution might be to research a particular population and share information through
publications and presentations. An individual cannot know every disorder, but music
therapists, as a collective group, could hold requisite knowledge. There are experts in
various areas, such as hearing impairment, neonatal intensive care, and physical
rehabilitation. There should be more experts in genetic disorders. Individuals who have
genetic disorders often share the same etiology, neuropathology, and characteristics
within a range. This provides a good opportunity to develop a complete set of music

therapy programs that incorporate research and develop assessment tools or interventions.

224

In order to further development of the profession of music therapy, it is necessary to
examine the effectiveness of music therapy interventions, not only as they bring about
improvement in individual clients, but also as they reflect on etiology and neurology, and

as they are relevant to of the phenotypes of each client.

What is the Most Crucial “Key Element ” in Mathematical Development?

While it is relatively easy to figure out what milestones the participants missed
and which key elements the participants did not have, it was more difficult to define
which milestones and key elements were indispensable to understanding “number

3, 66

calculation, coin calculation” or other targeted mathematical skills. After the therapist
“chooses the battle,” individuals with WS need to achieve the target “goal” with as few
new concepts as possible. Unlike typically developed individuals whose education
attempts to build up a foundation for future education by learning all five mathematical
foundation areas, individuals with WS are not always able to master even one. It may be
analogues to working on the foundation for a pentagon-shaped building with ten stories.
Individuals with WS may benefit most by building a triangular—pyramid-shaped building
with one or two stories (see Figure 50). They might not need to build the foundation for
the other part of the building. Therefore, it is crucial to focus only on the necessary
concepts and skills. At this point, it is essential to research early mathematical
development of children in order to identify the “crucial key element” in mathematical
development for each target goal. The importance of assessment has been emphasized,

but because of complexity and breadth, a standardized assessment tool has not been

developed in the music therapy field. We often have the same difficulties as other areas

225

working with other clientele. However, when music therapists can identify the crucial key
elements for each target area, the assessment tool could be developed based on those key

elements.

   

Algebra

Geometry

Number & Operations

Figure 50. A pentagon-shape building versus a triangular-pyramid—shaped building.

Based on these assessment tools and identified key elements, music therapists
could develop sets of music therapy interventions for at least three different levels of
functioning. The first group is individuals with WS, like Big Red, who do not develop the
comprehension of magnitude and cardinality. The interventions should be focused on
developing magnitude and cardinality. The second group is individuals with WS who
develop these basic concepts, but still struggle with operating numbers such as

composition and decomposition. Based on identified key elements for the readiness of

226

addition and subtraction, the individuals in this group need to work on those key
elements. The last group is individuals with WS who develop a basic understanding of
numbers and operations and are ready to move on to addition and subtraction. There are
ways to help with processing addition using songs such as “Finding the Bigger Number”
and “Number Buddy.” According to this study, the chronological age of participants was
not a precise indicator of the mathematical development age. Therefore, knowing the
mathematical level of functioning of each individual and using an appropriate set of

interventions, it may be possible to move to a higher level of mathematical development.

Music Score Reading and Ear Training

Music score reading is not an easy task for individuals with WS. Considering their
visuospatial deficits, the lines, spaces, and symbols in the score have too much
Visuospatial information to assimilate. There are anecdotal reports by parents that
children with WS gave up music lessons because they could not learn to read music. This
does not mean they cannot learn to read music; there are some individuals with WS who
can. However, since there are so many other skills for daily life that they need to master,
perhaps their ability to learn by ear should predominate. Teaching individuals with WS to
read music notation should not be a primary focus, and the inability to read music should
not be a good reason to give up music lessons. There are other music pedagogical
methods, such as the Suzuki Method, which emphasizes earning 1 “by ear” that would
give better opportunities to learn how to play an instrument.

Another tool to teach an individual with WS to read music is a modified score. In
music therapy field, score modification has been developed using numbers, letters, or

colors. Roxie and Angela used the scores modified by letters (see Appendix E).

227

Individuals with WS have a very good memory for rhythmic patterns, but sometimes
cannot figure out the melody by ear. Chromatic and accidental notes in music often
confuse them, and large intervals often pose difficulties in figuring out the next note in a
sequence. They might not need to have notation for the rhythm, but the adapted melody
notation could help a great deal when playing instruments.

Ear training appears to be necessary to enhance the ability to learn music by ear.
As mentioned earlier, the participants in this study had difficulty perceiving half-step
intervals, leaps, and accidentals. Similar tendencies were noticed during interactions at
the music camps. Many individuals with WS who are able to play keyboard instruments
have difficulty finding the notes with accidentals and chromatic notes. While they were
able to identify typical intervals and progressions, they had a hard time figuring out less
familiar melodic lines.

Besides traditional ear training, another method can be used in music therapy.
When individuals with WS listen to music and want to play, they learn the chorus part
first, and then the verses. The bridge and introduction are often the last section to be
learned, or they may never learn those parts of the song. iTunes, Windows Media Player
(digital media players for computers), or other MP3 music players with minute and
second indicators which displays how many minutes and seconds have either elapsed or
remain might be very useful tool. When they do not grasp a certain part of the song, those
players can be manipulated to play the exact spot (for example, 3 min and 22 seconds is
the starting point for the bridge). Then the individual can work on small segments of
music, instead of hearing the entire song and losing focus. Compact disc players and tape

players could work in the same way using the rewinding function, but they do not work

228

as precisely as digital media or MP3 players do. The other benefit of using an actual song
for ear training is motivation. When I used this method for the campers at the WS music
camp, I could use their favorite songs, and they were highly motivated to learn. They

appeared not to perceive this process of ear training as a tedious or unnecessary chore.

Choose Your Battles

There are five major areas in preschool and kindergarten mathematics: number
and operations, algebra, geometry, measurement, and data analysis. There are six major
sections in the number and operations areas alone: counting, comparing and ordering,
addition/subtraction, composing and decomposing, grouping and place value, and equal
partitioning. Each of these areas contains several key elements. In the case of WS,
participants did not exhibit the ability to apply analogies or similarities. They did not
transfer techniques or tools across tasks, such as from physical (coin) calculation to
written calculation, or vice versa. While written mathematical tasks are dominant in
academic settings, the usage of more practical forms to handle daily life skills is an issue
of some debate.

There are ongoing discussions on the WS online discussion server concerning
mathematical difficulties and strategies to overcome difficulties in the WS population. I
monitored the server discussions to help develop a better understanding of the needs and
wishes of parents. One discussion began with a parent asking for help for her daughter’s
mathematical difficulties. This parent focused on what the parents did to address their
children’s difficulties, as did the responses. The age range of the WS children in the

discussion appeared to be from eight to 20 years old. After a long debate about the best

229

way to cope with mathematical difficulties, the discussion split into two fundamentally
different groups of parents The first group had 3 “been there/done that,” attitude and felt
that fiirther gains were not worth the effort. They felt that mathematics was a weakness
and that the emphasis should be on their child’s strengths, not weaknesses. The other
group believed that their child should be given as many opportunities as possible to learn.
One parent said, “I don’t want to give up while my kid is in elementary school.” There
are so many difficulties to overcome and so little time in which to do so. The parents
worry about what will happen when they are no longer around Sometimes they only wish
that they may live just one day longer than their child. Some basic mathematic skills are
not academic skills, but life skills. No one but these parents can make these decisions, but
we, as therapists, can help inform their choices.

The decisions are not in the therapist’s hands. We can merely address aspects of
math and the benefits and disadvantages of working on them to guide the parents’
decisions. Besides mathematical difficulties, individuals with WS have other major issues
that affect their daily lives: medical issues, reading difficulties, emotional crises, etc.
Instead of sprinkling water over a burning house, we need to focus the fire hose on the
origin of the fire, one room at a time. The wishes and needs of individuals with WS and
their parents need to be seriously taken into account. There are many areas where music
therapists can assist individuals with WS, and teaching parents so they can make
informed decisions about what they want for their child is the most important. “Choose

your battles” is the best approach.

230

Miscellaneous Thoughts

Recorded as opposed to live music. As explained in the analysis and interpretation
chapters, the same songs for all three participants were used, but with different tempos,
especially for Big Red. This was not planned but was a response to their different
processing speeds. The songs were slowed downed or the tempo was increased. The
tempo adjustment between Roxie and Big Red points to the need to respond and
spontaneously adjust to the needs of the individual. The tempo of the songs, especially
strategic songs, must match with the individual’s processing speed. If the tempo of a song
is faster than their processing ability, or the learner does not understand one of the
concepts in the song, and the song will not be beneficial for that individual. Therefore,
when each song is introduced and practiced, live music has a considerable advantage over
recorded music. With live music, the tempo and content of the song can be adjusted and
individualized. Only after the individual masters a song should the song be recorded and
used as a tool to maintain the acquired knowledge.

Managing with coins. The name of a coin is a practical concept. However, is it
absolutely necessary? When Big Red, Roxie, and Angela worked on coin tasks, their
struggles reminded me of when I first moved to the US. I had a difficult time learning to
quickly identify different coins when I paid for merchandise. Knowing the value of each
coin is more important than knowing the proper name of the coin. And, as I admitted
earlier, I did not know the names for 5 and 10 cents until this study. “The big one is a
quarter, and the value is 25 cents.” Within that simple statement, individuals with WS

need to understand the spatial concept (big), the name of the coin (quarter) and the value

231

of the coin (25 cents). It might be better to focus on the value of the coin first and then
move on to the name of the coin.

When I moved to US, it was not easy to distinguish between different coins; why
was the smallest coin’s (dime) value more than a larger one (a nickel)? Silver shiny coins
that look similar and a small coin that has more value do not make the task easy. It was
not easy to determine the exact amount quickly, especially because of quarters. While
other coins are based on 55 and 103, a quarter falls out of logic and developing strategies
took me quite a while. Eventually I had my own “ah—ha” moment and started counting 25
first and then adding the others, just as I did in teaching the participants.

Meanwhile, instead of figuring out the exact change, I gave cashiers only dollars
or let the cashiers pick the change from a handful of coins. It was not easy to stand in line
paying with cash, trying to identify each coin, and then calculating the total needed while
others stared at me—making me uncomfortable. It is important to be able to solve money
related mathematics but the processing speed of the task is also a very important factor in
daily transactions. The participants’ slower processing Speed in general adds more weight
to their difficulties in coin calculation. Some individuals with WS might need to learn to
ignore the coins and use dollars that add up to more of the sum needed and simply
receive change. At the end of the project, Big Red learned the following song, and

practiced it with paper money. That might be the helpful for some individuals with WS.

232

One More Dollar

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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Change your voice to emphasize
the meanin of the

    

one more do] lars_ Don't for get the change_-_

Figure 51. Big Red One More Dollar.

Importance of pr0ps. Special education and music therapy practices rely heavily
on visual props rather than tactile or auditory props. Individuals with WS have adequate
vision, but their perception and ability to process visual information is different, and it is
unclear how much it deviates, and how big a problem it may create. It was interesting to
find props that did not rely on visual information. Although the greatest effort was made
to use tactile props in this study, it was almost impossible not to incorporate visual
materials. This project was a great example of the predicament therapists face when
choosing which props to use for the WS population. For example, the number cards

require visual information. I was wondering if there was any other way to present the

233

same information. The following questions exemplify some unanswerable questions I had

during the project:

1. “Is it necessary to use tactile information, like felt-tip stickers or push
pins?

2. Because individuals with WS need to improve their visuospatial function,
should only visual presentations be used?

3. Is it possible to start with tactile information coupled with visual cues and
then gradually focuses only on visual stimuli?

4. How do auditory stimuli fit into the picture?

The participant’s performance in organizing coins was greatly improved by the
coin-classification-chart. While Big Red did not have the benefit of the coin-hand-props,
Roxie and Angela understood counting by different units, i.e., hands and fingers. Big
Red’s inability to distinguish between coins is most likely due to his lack of
comprehension of the comparison words: normal, big, and tiny. Without those concrete
reference points, the words normal, big, and tiny were meaningless to him. (It might be
beneficial to have some tangible reference. For example, I used three real coins (without
a penny) as a reference point, or a card that reproduces the actual outline of each type of
coin. These props could be used as educational tools, or on a quick reference chart, like

typical consumers employ a tip chart.

234

Transcription of a session as an educational tool for music therapy students
whose first language is not English. Transcribing the sessions produced an unexpected
benefit. As a non-native speaker, a major struggle was to improve the way I could
articulate in English. After several sessions, I found that I made many grammatical
mistakes and did not have enough verbal expressions to explain the mathematical
concepts to the participants. There were certain things I could not succinctly articulate,
and I felt there was a better way to explain the facts and concepts. I looked for
professional help and found a speaking tutor to help me with these difficulties. At the end
of the sessions, I realized how much my English had improved. The process of
transcribing gave me opportunities to monitor my own speech patterns, mannerisms, and
mistakes. Working on specific expressions for particular situations gave me the chance to
learn new expressions and verbal usages. Sometimes, I did not understand the meaning of
what the participants’ said, but the video material allowed me to review the clips with
another therapist and figure out the meaning of what was said. For example, when Roxie
sang “nothing to it” for “You Can Do It,” I perceived the meaning as “there is nothing she
could do about it” instead of “it is easy.” Since the meaning did not fit into the song, I
asked a music therapist and she explained the meaning to me. Another example was
Angela’s thingamabob. At the first listening, I thought she was talking gibberish. Two
sessions later, she used it again, I was not sure about the meaning, and I again asked
another music therapist what thingamabob meant. I might have heard this before, but did
not grasp the meaning of it. Therefore, I think that transcribing the sessions and walking
through them with a supervisor or peer could be very beneficial in working on subtle

language skills, which are necessary for non-native speakers.

235

Attention and study room environment. The question about a study area
environment was raised when analyzing the data. Since the participant’s were easily
distracted by even minor incidents, especially auditory stimulation, is it better to have
fewer distractions in the environment to keep them on task, or is it better to work through
those distractions so they could improve their self-regulation skills? While this may
generate differing Opinions, the issue does need to be thoroughly examined. At this point,
I am in favor of having fewer distractions. Individuals with WS need to have an
environment in which they can truly focus on the task. Because of their distractibility,
figure-ground difficulties, and slow processing speed, new knowledge did not always go
through their short-term, and into their long-term memories. Providing an atmosphere
with as few distractions as possible might be the best way to learn new information for
the WS population. Self-regulation skills could be simultaneously treated in other settings

for generalization or by other interventions that are designed to address those skills.

Limitations and Other Considerations
Mathematical Difliculties are the lip of the Iceberg
Mathematical difficulties in individuals with WS appear to be the just the tip of
the iceberg of cognitive limitations. Visuospatial difficulties, fine motor skills, attention
difficulties, and self-regulation deficits are entangled in the iceberg. I performed a review
of the literature of mathematical development in WS prior to this project and focused on
the use of addition and subtraction. However, the actual difficulties originate in more

basic math skills that were overlooked until they became apparent in the middle of this

236

project. Big Red’s inabilities indicated that his problems were not caused by a weak
foundation, but by no foundation for mathematical development.

Ten sessions over ten weeks were not enough time to make substantial progress in
understanding the mathematical concepts addressed. The project moved too quickly for
the researcher to fully understand individual abilities, or to test ideas, speculations, and
interventions. There was not enough time to adapt and modify the strategies for each
participant. Once a lack of foundational knowledge became apparent, all three
participants required different sets of interventions: songs, props, and worksheets. After
the sessions ended, and even while analysis and interpretation were in process, the study
seemed like nothing but a series of chaotic observations. The analysis of what seemed to
be chaotic observations revealed a connection. Repetitive elements in the video
recordings and artifacts became themes reinforced by more focused surveys of the
literature. In addition, consultation with experts drawn from other fields and researchers
who have worked with WS greatly helped to define and delineate the embedded
meanings.

In retrospect, I do not understand why I worked with the participants for such a
brief period; the sessions now seem hurried. Big Red did not understand some
fundamental concepts in mathematics throughout his 21 years of life, but I expected them
to happen during my 10 weeks with him. It became obvious that each participant did not
understand prerequisite concepts to some degree, nor did they have a solid enough
foundation to move on to the next level. I, however, moved on at the next session
nonetheless. This proved especially problematic during initial sessions. Since Big Red

began the sessions first, Roxie second, and Angela third, he and I traveled further in the

237

wrong direction than the other two. I expected more from him because of his age, social
interaction skills, and musical skills. Unfortunately, those proved irrelevant in predicting
his mathematical ability. My previous impressions and experiences with him did not
indicate such a high degree of difficulty using math. I knew that he needed to have time
to process, and that it was important to verify his comprehension, but I was clouded by
my eagerness to successfully complete the project. I attempted to test too many concepts,
interventions, and techniques. The freedom and flexibility granted by using a qualitative
research method was misapplied. It is more important to teach one concept at a time and
to take the time to make certain each participant understood the concepts than to move
through each successive concept in a chain. After the analysis and interpretation were
finished, I felt as though I had only explored a continent by flying over it, when the
continent needed to be explored by walking on my feet. While this is obvious now, I
believe I missed it in the course of the study. This study is, therefore, a pilot study to

stimulate fiirther research.

Fish is Fish

The beginning of the book How Pe0ple Learn by the US. Research Council
(2000) uses quotes from Fish is F ish (Lionni, 1970) to point out the danger of
interpretation and construction of new knowledge based on existing knowledge. Fish is
Fish is a children’s book that tells the story of a fish and a tadpole. The two became very
close friends when they were young. As they got older, the tadpole moved away from the
pond and the fish remained in the pond. When the tadpole came back from his journey as

a frog, the tadpole tried to explain what he had experienced living on the land. No matter

238

what the frog described, the fish could only imagine it from a fish perspective; a bird is a
fish with wings, cows are fish with udders, etc.

While working on this project, I was frustrated by the enormous amount of
knowledge required to understand the difficulties in mathematics experienced by
individuals with WS. The participants’ simple mistakes often led me into a completely
new world where I needed to learn a new language and culture. A simple mistake was
often not simple at all, but one rooted in much deeper difficulties. There were too many
speculations on my part, and no answers forthcoming on my own. I needed to have in-
depth knowledge about child development, cognitive development, neuroscience, and
mathematical development. I did my best using my limited knowledge, but I wanted to be
able to interpret what really happened in the mathematical development of the
participants. I wanted to understand my participants’ responses to my music therapy
interventions and I kept thinking that I might miss some of the major points going on in
the process. I feared I might overlook something crucial. I might not make connections
between obviously related events. I sincerely hope that this study leads to more thinking,
more questions, and even more wonderment about cognitive function in WS. Even if my
theories and suggestions are supplanted by new ones, I will be highly gratified to be the
one who stimulated a process to help us understand more about intervention strategies for
WS. I sincerely hope my study will be a springboard to help them overcome cognitive
deficits. However, my experiences with music and individuals with WS were able to turn
a noxious task into a joyful one that instilled confidence in myself. This last observation
served to remind me what an effective tool music can be to improve quality of life. I also

shall never forget that Fish is Fish.

239

Future Research

The Mental Number Line

While working with these participants, the underlying rational of dyscalculia was
continually researched and evaluated. One of the themes that manifested during my study
was the possibility that individuals with WS have an inability to visualize a mental
number line. A mental number line refers to the way numbers are represented
conceptually or spatially within the mind, possibly with a left to right orientation and in
continuous format analogues to that of a ruler (Bachot et al., 2005; Dehaene et al., 1990;
Zorzi et al., 2002). While the process of manipulating and calculating numbers in the
brain is still elusive and controversial, the visuospatial aspect of mathematics is also not
clearly defined nor thoroughly researched (Dehaene et al., 1993). While there is no
particular research on the use of the mental number lines in individuals with WS, it has
been suggested that the visuospatial difficulties within the WS population might manifest
themselves as an aberration of the visualization of such a “line” (Ansari et al., 2003;
Bachot et al., 2005; O'Heam & Landau, 2007; Zorzi et al., 2002). Research on the mental
number line suggests that this concept might not be a learned concept, but rather an
inherited aspect of human psychology. Even people in a culture who use right to left
oriented writing indicate that their conceptions of numbers run left to right (Dehaene et
al., 1993; Zorzi et al., 2002). The area of the parietal lobe is thought to be responsible for

this inherited ability (O'Heam & Landau, 2007; Zorzi et al., 2002).

240

 

 

 

 

 

 

 

...]
8 .
7|
5 6
4|
2 ’3

 

H

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 52. Hopscotch in a vertical line and a horizontal line.

A handful of interventions were considered to compensate for this lack of ability,
such as walking or jumping, or hopscotch using a horizontal linear number-line, instead
in a vertical line (see Figure 52). Another intervention is to use keyboard instruments
with numbers on the keys in order to help organize thoughts based on auditory (pitch)
and visual information. The instrument can be marked with the Arabic numeral 1 for

middle C, 2 for the middle D, and so on, up to 10 for B one octave higher (see Figure 53).

241

Possible music therapy interventions using this concept are singing songs that relate to
number concepts with the instrument, counting a number of objects with an ascending
scale, and playing a song with a modified number score13 . The duration of the
intervention also needs to be tested; how long it will take? Ten sessions a week?
Everyday 5-10 minutes for a month? In order to conceptualize this intervention, the
frequency and duration of need to be taken into consideration. Using keyboards as

described above might have great potential for the WS population and beyond.

8910

 

 

 

 

 

 

Figure 53. A keyboard instrument with numbers.

Music Aptitude Tests and Musical Abilities

The musicality of individuals with WS has been well publicized and has drawn
great interest within the WS community and beyond because of some well-known music
geniuses who have Williams Syndrome. Unfortunately, there is little intensive research.
The music abilities of the participants in my study covered a broad range. While Big Red
has taken private music lessons and performed in various bands, Roxie and Angela did
not have such a background and had less opportunity to be exposed to different musical

environments. It was unclear whether Big Red’s advanced musical ability was due to his

 

‘3 “Twinkle Twinkle Little Start” can be played by 11 55 77 5 44 33 22 l in key of the C.

242

exposure to music, or because of his innate musical ability (nature vs. nurture). Roxie
showed her ability to improvise with her spontaneous song about her feelings. One could
wonder if she had more training and opportunities to improvise, her own skills would
advance.

The potential for music aptitude can be measured by using a music test such as
the Primary Measures of Music Audiation (PMMA; Kindergarten through Grade 3), the
Musical Aptitude Profile (Grades 5-12), the Intermediate Measures of Music Audiation
(IMMA; grades 1 through 6), and the Advanced Measures of Music Audiation (Grade 7
through adult) (Gordon, 1997). While some of these tests are standardized, they have not
yet been actively applied to the WS population. Two studies using Gordon’s aptitude test
as a measurement tool have been published (Don, Schellenberg, & Rourke, 1999;
Hopyan, Dennis, Weksberg, & Cytrynbaum, 2001). Don et al used PMMA for 8- to 13-
year—olds (n=l9), and Hopyan et al used MAP for average 12 year olds (n=14). Don et a1
concluded that the performance of the WS group showed results similar to the matched
group on receptive language skill (average 7 year old), but below the chronological age.
Hopyan et al (2001) reported that the WS group scored lower on tests of pitch, rhythm,
and interpretation, but were similar to the chronological age group on the musical
expressiveness subtest. Research on music abilities with standardized music aptitude test
will help to identify WS individual’s musical ability. Is it myth or truth? The previous
research on their music abilities suggests the results are inconclusive (Deruelle, Sch,
Rondan, & Mancini, 2005; Don et al., 1999; Hopyan et al., 2001; Lenhoff et al., 2001). In
addition, because of the disconnect between chronological and cognitive ages, difficulties

with visuospatial perception, and in format test settings, the most appropriate of these

243

tests need to be researched and identified.

“Think before you answer” (see Figure 54) is a song composed to emphasize the
use of the strategies taught in sessions. Big Red and Angela could not sing the song in
tune”. Considering their ability to memorize other melodies, it may be that this specific
melody uses half-step intervals. While Big Red and Angela have fewer difficulties with
stepwise or other intervals, the inability to identify the half step needs to be investigated.
It might be related to the discussion of potential talent and Ieaming opportunities, i.e. the
argument of nature or nurture. This would be an intriguing investigation, especially if the

ability to produce these intervals can be gained through training and practice.

Think Before You Answer

Eunmi Emily Kwak

 

Think—__ be - fore you an - swer

Figure 54. Think Before You Answer.

Attention Deficit, Hyperacusis, and Odynacusis

Distractibility, especially from auditory stimulation and poor concentration, was a
very serious issue in the learning process with the participants. Their hyperacusis and
attraction to auditory stimulation appears to be partially responsible. According to

parents’ reports on hyperacusis questionnaires, the incident of hyperacusis ranges from

 

14 . . . . . .
This song was not used for Roxre as much as others, and she dld smg the song With me but not srng the
song by herself; therefore, it was not clear she could or could not sing it herself.

244

84% (Gothelf, F arber, Raveh, Apter, & Attias, 2006), 92 % (wain, 1990), to 95%
(Nigam & Samuel, 2007). The medical definition of hyperacusis has a slightly different
meaning; ‘heightened sensitivity to sound, with aversive or pained reactions to normal
environmental sounds’ (Dirckx, 2005) and ‘an abnormal sensitivity to sound’ (Venes,
2005). Levitin, Cole, Lincoln, and Bellugi (2005) point out that while ‘sensitivity’ in
medical terms refers to “lowered hearing thresholds, that is an ability to hear soft sounds
that others cannot” (p.515). Using the term “hyperacusis” indiscriminately to describe for
different reactions to auditory input is not appropriate. Levtin and his colleagues establish
the detailed descriptive terms for four different conditions; true hyperacusis, Odynacusis,
auditory allodynia, and auditory fascinations. True hyperacusis is characterized as “an
ability to hear soft sounds that other cannot.” (p.515). Odynacusis means there is a lower
threshold for the volume of sounds. Auditory allodynia refers to fear toward particular
sounds that do not evoke any aversion responses or pain in typical individuals. Auditory
fascinations indicate fascination or unusual attachment to certain sounds. They concluded
that while there are only 4.7% of the subjects in the true hyperacusis category, 79.8%
were in Odynacusis and 90.6% in the auditory allodynia category, 9% of the participants
were identified as having auditory “fascinations.” Odynacusis and auditOry allodynia are
speculated to be due to neurobiological and structural differences and auditory nerve
dysfunction (Levitin, Cole, Lincoln, and Bellugi, 2005).

As shown in the participants of this study, Big Red was very sensitive to sound
and had a very good musical sense. Angela was very sensitive to loud noises, and wears
ear protection in noisy environments, such as at a party or the airport, but did not display

any particular music ability except being fascinated with music. Roxie did not exhibit any

245

particular sensitivity toward different sounds like Big Red or fear of loud noises like
Angela. It would be interesting to research the relationship of the above conditions of
auditory sensitivity and musical ability to what types of sensitivity play an important role
in musical ability. Big Red seemed to hear what I did not hear, but he did not display
particular difficulties with loud noises, but he was the most distracted individual. I kept
wondering if the sensitivity to music hindered the participants’ ability to focus. The

relationship between sensitivity and distractibility would be a fruitful research agenda.

Research Opportunities in the WS population

Another difficulty in WS research is the accessibility of participants. It is not easy
to find a large number of WS participants within one state, much less within a single city.
If the study requires a certain age group, and/or a particular functional level, the
availability of participants drops to almost nothing. The Williams Syndrome Association
(WSA) and the parents of individuals with WS are very willing to participate in research
in order to improve treatment options available for their children. The WSA convention, a
biannual event, is a great opportunity to propose research. The WSA has accepted
research proposals from various Universities, granted permission to conduct research on
the conference site, and allowed researchers to gather prospective participants. The
convention usually gathers between 200 - 300 individuals with WS from ages as young as
8 months, to some as old as 60.

Other opportunities to network and suggest research occur at the annual music
camps. WSA has two music camps in Grand Rapids, Michigan; one for children 6 to 12

years old and another for those 13 to 30 years of age. Depending on the research proposal

246

and goals, the association permits researchers to conduct research projects at the camp.
These conferences and camp settings have definite limitations and restrictions to research
designs, and ways of conducting the research; however, they can offer good opportunities

otherwise unavailable 15.

Final Thought

Individuals with WS are different. They are not the same as typically developed
individuals, not the same as other persons with cognitive challenges, Ieaming disabilities,
or visual impairments. Individualized and specialized music therapy interventions for WS
are necessary because of their different ways of processing auditory information and
difficulties with visual perception. Often, these differences are overlooked when they are
taught in a traditional educational setting. Even in a special education setting, such
instruction often fails.

Individuals with WS’ inability to learn using traditional means such as visual aids
and rote repetition brings to mind the fable of the Fox and Stork by Aesop. Besides the
moral of this fable, the container for the food draws my attention. The container and its
manner of presentation determine the ability to absorb the contents. Mathematical facts
and concepts in a musical “container” instead of a visual “container” may provide a better

way for WS to absorb content.

 

l5 . . . . . . . .
This statement was sent to WSA and they approved its inclusron 1n thrs dissertation.

247

 

 

 

 

 

 

The Fox and the Stork

After her long migration,
the fox sent the stork a dinner invitation.
When she came he served
thin soup in a marble dish 50 flat
that the stork could fish out nothing with her bill.
She served minced food in a narrow-mouthed um
and tortured her hungry guest
by plcklng the best bits Out till she'd had her fill.
While the fox was desperately llcklng
the neck of the urn, the blrd
remarked, so I've heard,
'He got what he deserved. Tit for tatl'

 

 

 

Figure 55. The F 0x and the Stork (Michie & Lord, 1989, p. 132)

Illustration by Hyang Eun Lee
248

Study participants liked mathematical manipulations when working on written
addition with music, engaging with coins, and other manipulations. They did not hate
“math.” What they did not like were the struggles, difficulties, and frustrations. Just as in
the fable, the presentation of numeracy to WS needs to fit with their learning style as a
musical learner for internalization to occur. The natural repetition that occurs in
melodies, and the attentiveness to music that individuals with WS all share, create a link
to concepts and strategies that help usher them fiom short-term working memory to long-
terrn memory to be used as tools that will be readily available for daily use.

When I started this journey for my dissertation, I thought I knew a great deal
about WS because of my years working at various camps, attending conferences,
gatherings, and past interviews with parents and professionals who work with individuals
with WS. None of this had actually prepared me for the most basic discoveries, and what
they revealed to me about the mind of individuals with Williams Syndrome.

The range, depth, complexity, and entanglement of the matter were much more
complicated than I had expected. None of these factors is isolated and each day brought
new theories, concepts, and idioms to my interpretation. New and updated articles were
published during my research, increasing the resources available but, at the same time,
overwhelming me. Besides being overwhelmed, I was frustrated, mesmerized, and full of
remorse. I was overwhelmed because of the amount of knowledge I needed to absorb in
order to analyze and interpret the observations. I was frustrated because I felt I should
have known more before beginning the project. I was mesmerized because of the

participants and their similarities and differences And, I was remorseful because I truly

249

felt sorry for my previous clients with WS who had worked with me before I learned
from this project. I do not feel I did my best for any of them.

The boundaries of knowledge in therapy are unclear. Obviously, training in the
mathematical development of young children is not part of my curriculum to be a music
therapist, nor part of my expertise. In my experience as a music therapist, I became aware
that bridging the gap between knowledge of a particular disease and evidence-based
music therapy strategies is important to further development in the music therapy
profession. This project confirmed my belief that without understanding the condition
being addressed, it is difficult, if not impossible to deliver the appropriate interventions.
This study brought me to another place and has strongly affected my philosophy as a
therapist. Where should the boundaries of knowledge be for the therapist? Within my
own limits, how much do I need to know to treat people? Collaboration, interdisciplinary
teams, and consulting might offer avenues to explore. The processes used to develop
music therapy interventions for those with WS may also be applicable to other genetic
cognitive impairments, such as Prader-Willi Syndrome (PWS), Angelrnan Syndrome
(AS), and DS. My hope is that this study will not only suggest elements of music therapy
programming for individuals with WS, but also provide a framework for developing
music therapy interventions for those with related genetic disorders.

“What we discover by doing research is just how complex the world is (Strauss &
Corbin, 1998, p. 55).” This statement encompasses the process of this dissertation. The
more I learned, the more this study reminded me how much I did not know. This project
raised more and more questions, but provided no concrete answers. The realization that

more research, a more exhaustive review of the literature, and practice is necessary and

250

possibly painful, but is more important than any single “answer.” I tried to describe, as
much as possible, what I observed in order to provide as much information for future
researchers and clinicians involved with individuals with WS. It is clear for me now that
individuals with WS have often untapped potential, that music therapists have effective

tools, and that it is our responsibility to find the key to these often locked doors.

251

EPILOGUE

June 2008, Grand Rapids, Michigan: Music Camp for Children with Williams Syndrome

Eight Orff instruments were marked with the Arabic numeral 1 for middle C, 2 for
middle D, and so on up to 10 for E one octave higher. Paul, Peter, Tom, Adam, Mary,
Debra, Ellis, and Dina, with ages ranging from 9 to 10, enjoyed playing up and down the
scale. They sang the scale using numbers; like fixed Solfegio using numbers instead of
the sol-fa syllables. Their voices were stretched with higher notes such as C an octave
higher. While Tom did a wonderful job playing the song Itsy Bitsy Spider with a modified
number score (e. g., 1112 33 32123 l-CCCD EE EDCDE C in the key of C), Mary had a
hard time playing her instrument. She could not move her hand fast enough to follow the
song, and she did not know where the next notes were. Debra, Ellis, and Peter were better
than Mary, but also needed assistance.

The children were next asked to play the pattern of CGEG (1 — 5 — 3 — 5) using a
mallet. Adam had a hard time moving from C to G (1 to 5). Mary, Debra, Ellis, and Peter
did not perform well either. Why was this hard for them? Was it because of a lack of
experience or was it another manifestation of visuospatial perception problems? Perhaps
it was an eye-hand coordination difficulty. A set of 6 different number cards with dots
were given to those eight campers. We counted each card in the scale; “1” (C), “2” (D),
“3” (E), “4” (F), “5” (G) for the 5-dot-card; 1 (C), 2 (D), 3(E), 4(F), 5(G), 6 (F) for the 6-
dot-card. We sang the different cards up through the lO-dot-card. When they were asked
to find the 7-dot-card by singing, “Which one is 7?” Tom and Peter were already holding

up the card flashing beautiful smiles. Mary could not identify it. She desperately looked

252

back and forth between the 6 different cards, the counselor, and me—as if she thought we
would give her the answer. “Give me any sign, please”—her eyes were pleading. Her
hands frantically moved between them. She picked the 5-dot—card and counted them one
dot at a time, and put it away. She picked the 8-dot-card and counted them one by one.
Sometimes she counted the same object twice, or skipped an object. This scene was too
familiar. Mary’s face was overlapped with Big Red in my mind’s eye, displaying the
same desperation, hopelessness, and disappointment. Mary started holding up random
cards hopelessly. The counselor gently removed three cards from Mary’s pile and Mary,
now, only had the 5-, 7-, and 8-dot-cards. The counselor quietly counted with Mary until

she finally got the 7-dot-card. I was left wondering:

Were these interventions helpful to developing a concept of counting?

Could playing 00?" instruments with numbers help to develop a mental number line?
What prevents Mary and Big Red fi‘om grasping the understanding of numbers?

Why can ’2‘ they identify those ridiculous number flash cards? Do they need to know the

differences in numbers? Is it necessary?

I still wonder .......

253

Appendix A

Song and Lyrics

Increments of 5
Eunmi Emily Kwak

Eb c m Ab 1357 Eb Bb BB7

 

5 10 15 20 25 30 35 40 45 50 55 60 Rest
5 ER Cm Ab Blfl El)

 

65 70 75 8O 85 90 95 and 100

Figure A-I. Increments of5

The lyrics of the song simply utter numbers in SS with refined wording.

Number buddies 9 - 7

Emily Kwak

 

9 and 1 makes 10 8 and 2 makes 10

 

7_ and 3 makes 10 those are num ber bud - dies

Figure A-2. Number Buddy.

This song was used for the concept complement of 10 for Angela.

254

Thumb and Pinky Meet Each Other

Eunmi Emily Kwak

V

 

 

Thumb and pin - ky meet each 0 - ther—
Thumb and ring fin - ger meet each 0 - ther—
Thumb and mi - ddle__ meet each 0 - ther—

 

how are you? _ how are you?_
how are you?_ how are you?_
9 how are you? _ how are you? __

    
  

Thumb and in dex__ meet each 0 ther_

I3

 
   

. O 9
V V
how are you?_ To day!!!

Figure A-3. Thumb and pinky meet each other.

This song was used to promote finger dexterity for Angela.

255

Appendix B

Final list of codes and themes

 

Symbol*

Theme

 

Negative comments or concerns about their. performance (H:4xl)**

 

 

Gaps inknowledge of social interéCtiOn (H 2X11)

 

Distractibility (H 1x1)

 

DistraCtion present but can maintain focus ontaSks. (H 71x11) and (F 4x4) 1:

 

Mathematical problems (H 2x 1)

 

Clock problem (H 1x38)

 

Lack of strategies (H 5x2)

 

Parent’s coMents (Red ,boX)

 

Looking for mom’s help (H 2x3)

 

Interesting moments (H 2x3)

 

Language (H 4x2) ,

 

Musical responses (F 5x3)

 

Happy moments (F 2x2)

 

Got it right (F 6x2)

 

Emotional support (F 4x2)

 

Areas of concern (F 1x5)

 

Therapist’s musical Strategies (F 7x5)

 

Fine motor (F 4x4)

 

 

23>@;§:1< tir©¥§€ elm: ~< co®m cox

 

Developing Strategies (H 5x2) and,(F 3x4)

 

256

 

* Symbols were assigned to more readily access data by a word search for analysis.

** The numbers represent a column and row of either highlight color (H 4x1) or font
color.(H 4x1). H 4x1 means 4th column and 1St row in highlight color chart.

 

Highlight Color Chart Font Color chart
{.Vofi‘fiflfilflr 5"!- V ‘ fir: W'-\“‘I"Wi’:

r z: '1

l: .

imwmasu: hummus-ix int: raw—cad

   

... =0
t, .'
git“? ‘WC’PJ

 

 

257

Appendix C

Table for Abbreviation

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

ADHD Attention Deficit and Hyper Activity Disorder
AMTA American Music Therapy Association
AP Absolute Pitch
ASD Autism Spectrum Disorder
CP Cerebral Palsy
DAS Differential Ability Scales
DS Down Syndrome
ELN Elastin Gene
FISH test Fluorescence In Situ Hybridization
fMRI Functional Magnetic Resonance Imaging
KBIT The Kaufman Brief Intelligence Test
PPVT Peabody Pieture Vocabulary Test
PWS Prader-Willi Syndrome
RAS Rhythmic Auditory Stimulation
SVAS Supravalvular Aortic Stenosis
TDI Individuals with Typical Development
TDC Children with Typical Development
WS Williams Syndrome
WSA Williams Syndrome Association.

 

258

 

Appendix D

Example of Session Transcripts with Codes

This is example of Roxy’s 5’h session. It is the beginning of the first three pages. In here
is printed in black and white, but as indicated in Appendix C, the final copy is in full
color. Different shades of gray highlight and font color indicate there are some codes in

that transcription.

Session Transcription

Special interesting moment

She was tired because it was time for dinner(around 6pm) and she had attended school
that day. When I arrived at their house, she was a little grumpy; her mom warned me
about her mood, “You will see What happens when she is t-i-r-e-d. However, she did not
show much behavior changes during the session.

 

 

 

    

... 5P "@y
n) (This IS the great moment to
explain how musical cue can get there attention) (H 2x3)

 

 

 

 

Good. If you play as writen you canlay Mary haadv Little Lamb.

0:10 (making lamb sounds)
(laughing)

(laughing)
0:29 (playing C instead of E)
Where is E?

Right here.
No, that’s C
0:32 right here?

259

That’s D

  

‘ (playing Glockenspiel rhythmically) (F 5x3)
You can see it here (it rs too small to see it as well as the lighting)

E, D, C
0:56 Oh! Wait. E, D, C (playing Mary had a Little Lamb with glockenspiel; I
was
pointing the note which need to be played)
8 1: 35 ' ____Yeah!. '! (even though there was a lot of noise and music going on, she

could concentrate and could finish the song.) (H 1x1) & (F 4x4)
You play and I will accompany you.

' G 1:40 to 2:47 . (she moved the score to her right side; @ by herself) (F 5X3)

2:51 (trying to play mouth piece)

You need to stretch your lower lip over your teeth.

2:54 (she wasn’t able to make a sound out of 7 attempts) I did it before.
3:05 Yes you did it before. You can make it.

(4 attempts; sighing)

 

3:17 (1 success out of 3)
3:25 (I attempt; sighing) I can’t do it.
3:47 Yeah

Can I have a mouthpiece and put it here? (I wanted to end the saxophone)
Wait. (1 success out of 3)
(1 success out of 3)

 

 

4:00 (1 success out of 3) Yeah! !!
(© laughing)
4:17 (2 success out of 3)

That’s the sound I am looking for. That’s the sound. Great.
(3 success out of 3)

 

Yeah! ! !!
4:39 (1 success out of 3: very nice sounds but high tone)
4:48 (2 success out of 5)

Practice it.
Practice makes perfect.
Yes. Yep, Yep, Yep

 

5:14 (1 out of3)
Good
5:20 (1 out of 3)

You need to make lower tone. You make high tone (® the piano), 1 want you to make

Low tone (® the piano)

(1 out of 3)

(1 out of 4)

Good, Thank you.

You want to play this one more time. Actually I have Nobody is Perfect here. You want to

 

 

260

play this one. (® nobody 19 perfect). Play from here. AAAAAB. ..

 

6:24 (® nobody ’s perfect: pointing #5)
7:10 Yeah!
You want to play Mary had a Little Lamb.
7:26 I’m just looking at.
7:30 Roxy! Is my name.
Yes, that’s your name.
Roxy, Roxy
Roxy is your name.
7: 38 Roxy. Roxy Roxy (language rhythmic pattern) .. ..
& 7: 51 ' _(unidentifiable melody; @Mary had a Little Lamb; made mistake)
8: 05 (sighing) Wait (playing Mary had a, Little Lamb) Yeah! ! !,
Justdoing :one more time. You’re dOing great. (F 5x3)
(@Mary had a Little Lamb)

 

9:07 Ah-ha. I love your playing.

(CPD unidentifiable melody)

9:14 I want you practice with it. I will leave the glockenspiel and I want you practice it.
OK

9:31 OK Go away. Let’s put it here. Today I want you to do. This one

9:55 Roxy
Yeah!

I want you to say Yes, Ma’ am

(laughing) Yes, Ma’ am

 

 

 

 

w (© addition song) 2 + 2
' (making 2 and making 4) (F 4x4)

 

® 10: 5- 11:3 (good example with fine motor skill problem; remark video clip Roxy

Fine motor clip 1)

Let’s2. Making2.. .12

(she is able to make 2; adding 1, 2) 4
Yes, that’s four. You need to have“ 3, 4.

You need to make 4. You need to make 4

1.. .That’ s 3. Let’s do it this way. Can you make four?
10' 47 ’ ________(holding 4)

You already have 2.

3, 4, 5,6, (H 2x1)

Can you write down 6?

 

 

11:03 Hey! Tom! !! You are on Camera. You are famous. (Does she feel good on
camera?)

Good

11:14 I will do my hair done on Saturday (that day, WSA will have a Christmas

party

and she is so exited about that)
Saturday; Ah—ha

261

(nodding)
Did you get the dress for this Sat. ?

 

 

 

 

© » I get it tomorrow. But I am exited. (F 2x2)
Good. Which color do you want to have?
Red
11:25 Ah-ha: Christmas color.
(nodding) yeah!
Tom, you will be there?
(brother) Yeah!
Yeah!!! We have a big party.
11:37 Is it gonna be huge?

I heard it will be around 70 people. It is huge. I will bring some pecan pie.
Em... pecan. I never try pecan pie. How many people are there so far?

 

70. 7 .. 0

70 people!!! Oh! My God!

It will be big party.

Roger will be coming (her WS friend from WS camp). I am exited.
12:05 (@ addition song) 2 + 5 Let’s make 5.

(holding 5 with left and adding 2 with right)

You already have 2. Don’t make it.

2, 3, 4, 5, 6, 7 7

How much is it? Doesn’t look like seven. (remark 11/29/07 worksheet)
Don’t erase it. I want you write down here another one.

(writing 7)

 

 

 

 

 

262

Appendix E

Modified orefor Angelaan

Twinkle Twinkle
Little Star

 

CC

GG

AA

 

FF

EE

DD

 

GG

FF

EE

 

GG

FF

EE

 

CC

GG

AA

 

 

EE

DD

0 C) U U 0 CD

 

FF

 

 

 

 

 

if

 

Appendix F

Handwriting Samples for Angela

Angela thinks the question numbers are the
part ofquestion, i.e., 9 + 2 as 19 + 2. She
did not get the right answer for 19 + 2.
When she worked on the second question,

I notice her mistake and drew the line
between the questions number and the
question. It is understandable, but at the
same time raises the question about the
reason of question. She did not answer
questions four and five correctly. The
reason for this mistake might be because of
her way of doing some of additions; she
doubled the number and subtracted; i.e., for
question number three, she doubled 3 and
got 6, and need to subtract 1, but she added
1. She made the same mistake for the
question number five: she doubled 6, and
got 12, and added 1, instead of subtracting

l.

264

 

265

 

 

 

 

 

 

 

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280

 

 

 

 

 

Figure 1-3. Nickel Hand Prop

 

 

 

 

 

Figure [-4. dime Hand Prop

281

 

 

 

 

 

 

 

 

Figure 1-5. Quarter Hand Prop Pattern 1

282

 

 

 

 

 

 

Figure [—6. Quarter Hand Prop 11.
To put the pattern 1 and 2 makes 50 cents presentation.

283

 

 

Appendix J

Pre-test of Angela, Roxy, and Big Red

Angela

1. What time is it?

a.4:3o b.7215 c.6t03 4%

2. Put these numbers in order, from largest to smallest: 7, 12, 3, 8K

 

11.3.7.8.12. 19

h. 12. was .

010.12.“ .3 .1. 1
Q4117. s. w

3. You find 3 nickels. 1 quarter, and 2 dimes in your pocket. How much money do you have?

a. 6 cents

13. 30 cents

C. 47 cents
()0 cents

4. How do you read the equation II = 9 + 2 out loud?
@Elex-en equals nine plus two.
h. l‘ilexren minus nine equals two.

e. lxleven equals nine minus two.
d. lileven plus nine equals two.

5. Look at the pietograph. Which person has the most coins?

 

  

 

    
    
 

....

 

55) ”'1 i “ 2:3
:‘4. a ': " "i
. .:3‘ 1"! tiff. ~_ r
Sara t- -
— t . ,I‘ i ,/ ...- ' -
. . - _ ‘ ' E. « . .
l".l'llll\‘ . a
“"—‘ , 4 .. 'n. ‘m ' "' “ _2
. #5, an. 47‘. i
. ~ ' x“ ' ' . ' .. ‘ 1
8" I“! ’ qfi‘h( 2. , "I i

' Sunn) . ~’

 

 

:1. Sara h. Fniily C'. Sunny

284

Roxy

I. What time is it?

 

3. You find 3 nickels, 1 quarter. and 2 dimes in your pocket. How much money do you have?

376 cents

h. 30 cents
c. 47 cents
d. ()0 cents

4. How do you read the equation ll = 9 + 2 out loud?
0. Eleven equals nine plus two.
[1. [Eleven minus nine equals two.

c. Eleven equals nine minus two.
(1. Eleven plus nine equals two.

5. Look at the pictograph. Which person has the most coins?

 

 
    

Sara

   

 

 
 
 

l_imil\

 

._
l
l
l
I
.

'Sunn)

 

l

l

l

t .
Le... -

 

 

at. Sam h. Emily C. Sunny @Kevin

285

Big Red

I. What time is it?

 

'4 J

0

21.4230 h. 7:|5 96:03 d. 3:

2. Put these numbers in order, from largest to smallest: 7, 12, 3, 8, l9.

0 7 2)

21.17.842.19 [q l 2 a
h. ll. 19. 3. 7. 8
10. 12.8.7.3 tol 731?-
d.l2.3.7.8.19 lfiéf” «7‘7,
3. You find 3 nickels, 1 quarter, and 2 dimes in your pocket. How much money do you have?
a. 6 cents
b. 30- cents

47-cents
d. otlieents
4. How do you read the equation 11 = 9 + 2 out loud?
at. Eleven equals nine plus two.
h. Eleven minus nine equals two.

c. Eleven equals nine minus two.
Eleven plus nine equals two.

5. Look at the pictograph. Which person has the most coins?

 

. __.- ., . ._-.. _,‘.__,,.__ -— —...Y

 

 

     

 

 
 

 

Sunny" -
- Keyin a .
:1. Sara h. Emily C. Sunny

286

 

 

Appendix K

Mathematical Mistakes and Interesting Incidents

Some of interesting and thought-provoking episodes are presented in this section. Some
of them are already mentioned in the case studies as well as analysis, and interpretation

sections.

In the conversation, Emily is identified by the dialogue without a line, and the
participants dialogue has lines.

Big Red

 

Episode

Conversation

 

Big Red 1

Give me a nickel
Normal

 

Big Red 2

Let's make 25 pennies here.
1, 2,3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19,
20, 21, 22, 23, 24, 25
(Did not use any strategies for faster process such as
counting by 2's or 5's)

 

Big Red 3

A Nickel... is a nickel smaller or bigger?
You need to think about it.
I think normal nickels are little bigger.

 

Big Red 4

Dime has the tingly feeling. (singing Give Me a Nickel)
Normal

(singing Give Me a Nickel e)
tiny so tiny

 

Big Red 5

Give me a dime.
Dime is worth 5 cents
Uh-huh
Dime is worth 25 cents.
Uh-huh
10

 

 

Big Red 6

 

(Asked him to find a quarter within a two count)
Quarter, one - two
Oh!!! (cannot finishes the task within a time frame, he was

frustrated and he touched forehead)

Dime, one - two
One and two

Nickel, one - two

One and two
One and two (cannot fine a nickel, anxiously rubs his hands together, has
agitated body posture, fine motor difficulty is presented)

 

287

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Big Red 7 (He supposed pick two quarters; he picks a quarter and a nickel)
Look at this
That’s not good. This is right. This is wrong.
(Rubs his hands together)
Big Red 8 Five pennies same as (expect him to answer a nickel)
Five nickel
Big Red 9 (sings Five Pennies Make a Nickel) Two dimes and a quarter
make a quarter every time.
Big Red 10 (sings Five Pennies Make a Nickel) Two nickels, Two nickels
(supposed to be a dime; growls; rubs his forehead)
Big Red 11 (work on finding nickel, dime, and quarter)
14:10 Nickel, dime, and quarter
A nickel... A dime... and a quarter...
There we go.
Mm-hmm
This is nickel. Because this is tiny.
No, a nickel is normal.
It's the original size.
Big Red 12 How many? (points out a nickel)
10 (he examines my facial expression for the hidden
message.)
That's 5.
(add one more nickel)
10
(add one more nickel)
15. No way! (he thinks he got the wrong answer)
15! Yeah! you are right! you are right! (add one more nickel)
OK
(point two nickels)
10
(add one more nickel)
20, No, 10, 15, 20 (It is supposed to be 5, 10, 20)
10,15, 20. No!!!
Big Red. 13 OK, How many hands are there?

 

1, 2, 3, 4, 4!
How many fingers are there?

................... 5, 10, 15, 20
Good. This is (show my one hand)

5
(show one more hand) makes?

10
Put your 5 fingers and

5 (I expect him to say 15)
And all together

10 (I expect him to say 20)
Your ten and my ten makes

 

288

 

 

 

Big Red 13
(cont)

10, 15. No!
No, No Think about it your hands have 10 fingers. My fingers are 10. All
together?

25
Think about it. Your hands have ten fingers. Like this one and this one
(point the nickel hand prop). Let's count together.

5, 10, 15, 20, there we go
OK. 10 and your ten. Makes?

10, 15, 20, 25 (count my first hand as a 10)

Ah-ha. I got you. I think I know why you think that way. OK. 5, 10, 15,
20
(show him to one hand by one hand)

5, 10, 15, 20
Let's do it.

5, 10, 15, 20

5, 10, 15, 20 (he points each hand one by one).

 

 

Big Red 14

(show a dime hand prop)

10, right?

(show the second dime hand prop)

25 (examines Emily’s facial expression)
10, 20. There we go.

(show the third dime hand prop)

25 (examines Emily’s facial expression)
30, there we go.

(put down a dime hand prop and a nickel hand prop)
10. (points a dime hand prop) Just 10.
Ah-ha, this is 10.

(points a nickel hand prop) 25

Before you tell me, think about it.

C. (a dime hand prop)

O (a nickel hand prop)

5, 10, 15

 

 

 

 

 

 

 

5, 10, 15 There we go.

 

(switch to a dime hand prop)
O. (a dime hand prop)
O. (a dime hand prop)

5, 10

 

This gotta be 5.
This is 5.
5, 10, 15, 20

 

Good, OK (switch to a nickel hand prop)

O. (a dime hand prop)

O (a nickel hand prop)

(points the top line dime hand prop) 5

 

 

 

289

 

 

 

Big Red 14
(cont)

(points the second line nickel prop)10
(examines Emily's facial expression; points the second line
nickel prop again) 15
(points the left hand from the dime hand prop) 5,
(points the right hand fi‘om the dime hand prop) 10,
(points the second line nickel prop) 20, No
(touches his forehead)
(show approval facial expression; points one hand by one hand)
5, 10, 15
Good.

 

 

 

 

 

 

Big Red 15

..5
.10. 15
.. 30

 

Big Red 16

Five pennies make a nickel.
Two pennies make a dime.

 

Big Red 17

Five pennies make a nickel.
Two dimes make a dime.
Two dimes and a quarter make a nickel.

 

 

Big Red 18

 

That's gotta be a quarter.

Can you tell me how many fingers are there?
1, 2, 3, 4, 5

That’s how many hands. How many fingers are there?
1, 2, 3, 4, 5

There are more fingers, isn't it?
1, 2, 3, 4, 5

1, 2, 3, 4, 5

1, 2, 3, 4 5
1, 2, 3, 4, 5

(laugh) Good How can you count that? We practice that one, 5.
5,10,15, 20, 25

V

Yeah!!! How many fingers are there?

5 (shows his one hand)

Here. .. how many fingers?

10, No way. 1, 2, 3, 4, 5

That’s how many hands. How many fingers are there?

1,2, 3,4,5

How many fingers are here?

1, 2, 3, 4, 5 (points one finger by one finger in one hand)

5, 10,fingers 15 fingers 20 fingers

25 fingers (a phone is ringing and looks away)
(sing How Many Pennies in a Quarter)
(sings the answer) 25.

This one is? (show a quarter hand prop)

25
How many fingers are there? (show another a quarter hand prop)

 

290

 

 

 

Big Red 18
(Cont)

 

5, 10, 15, 20, 25. 25 again, Wow

Quarter?
25

Quarter 25 (sing Nickel five, Dime 10, Quarter 25 song)

(Answers the song) 10, 25

(A phone is ringing again, and tries to focus and to not look
away. Was not successful)
We ignore the phone.

 

291

 

Roxy

 

Episode

Conversation

 

Roxy 1

(Works on 5 pennies same as a nickel, 10 pennies same as a dime, and 25
pennies same as a quarter)
It is same as a quarter (points out 5 pennies in 5 rows), and five pennies
same as a nickel.
How many pennies?
35
Why
Because there is 35 pennies.
35 pennies? This is 25 and this is 5.
25 plus 5 equals
35
Nope-Nope

 

Roxy 2

Roxy, you have 10, and take away 5. How much you have?
(manipulates her fingers, but could not do

it)

(shows her 10 fingers and take away 5)
5

 

 

Roxy 3

 

(previously we work on coin hand props; showed her nickel hand prop and
worked on counting by 5; showed her dime hand prop and worked on
counting by 10) '

(presents 4 nickel hand props)

5, 10, 15, 20
(presents dime hand prop)
25
No, just this (covers the 4 nickel hand props and points dime hand prop)
1 O
Mm-hrnm (adds nickel hand prop)
1 5
(adds one more nickel hand prop)
1 6
(shakes my head) 10
20
Yeah! (takes out one nickel hand prop). This is 10
5
10 and 5
15
(add one more nickel hand prop)
Er!!!!!!!!!!!!! (manipulates her fingers) 16?

How many fingers (points a hand by a hand)? Count fiom here.
5, 10, 15, 20. Now I got it.
I think you got it. This is one hand, but how many fingers?
5 (holds a hand in a air and adds another hand) and this is
10

 

292

 

 

(hold up my right hand) all together
. . . 1 5
Give me your 10. Give me your 10 fingers (hold up my right hand)
20,
This is
1 5
(add my left hand)
10
Yeah! I have 10 and you have
10
Together
25
10 and 10
20

 

 

Roxy 4

 

(work on mixed dimes and nickels)
(two dimes)
10, 20
(add a nickel)

5
Twenty. . .five

25
10, 20 (add a nickel)

5
Twenty-five

_hohum, 25
(present a dime)

1 0
(add one more dime)

20
(add a nickel)

25
(add a nickel)

30
(add a nickel)

35
(add a nickel)

40
(take away a nickel)

30. (looks at my facial expression; growls; thrusts her hand
downward) I always messed up.
No. (Sing You Can Do It; stop after the lyric "if you")

try (laughs and changes her mode)

 

293

 

 

 

 

Roxy 5

 

(show her a nickel)

5
(add a penny)

15
No, if I this one (put the dime), it is 15 . But this is?

6
Yes (add more pennies).

7, 8, 9, 10
(add a dime)

20
(high five; add a penny)

25
How much?

One. Messed up every time. Yai yai ya (throw herself into her
lap)
No (singing, You Can Do It song: You can do it, you can do it, you can do
it, if you. . .)

I always messed up on those one with the dime and nickel

(roles her eyes and throw herself backward toward to the couch)

 

294

 

Angela

 

Session 1

 

Episode

Conversation

 

Angela 1

How many pennies in a quarter?
52 pennies?

 

Angela 2

(Working on counting by 5)
5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 10 I mean 60.

 

Angela 3

(holds up 8 fingers: 3 fingers in right hand and 5 in left hand: In her view,
I hold up 3 in her lefi side and 5 in her right side)

1, 2, 3 (counting fingers in my left side and then) 4, 5, 6, 7, 8
[Better strategy is counting 5 and 6, 7, 8 for the faster process]

 

Angela 4

(puts down two nickels) How much is it?
And, 5 and. . .. 25.... (Thinks with tapping her chin)... 29
29! No, No, No. This is 5. OK. Nickel five, (Sings nickel, dime, quarter)
(finishes the stanza for quarter) 25
(Sings Nickel, Dime, Quarter) Nickel five. How much is it?
10 (read my Emily’s facial expression)
(points a nickel) This is???
25
No, nickel is
5. 10!!!!!!!

 

Angela 5

(points the quarter finger chart) How many fingers?
1, 2, 3, 4, 5,
No, No, No. (points a quarter in the middle of the quarter finger chart)
That’s?
25
Look at this (points out each fingers).
5, 10, 15, 20, 25 (high five)

 

Angela 6

(points a quarter) This is how much?
25.

This is 25. And this is? (points a nickel)
A nickel?

Nickel and 25 plus 5 equals.
29!

29
90... wait. I am confilsed

 

Angela 7

(presents 4 dimes and a nickel) 10
10, 20, 30, 40,50... 55
(Gives her surprised facial expression; sings Think Before You Answer)
10, 20
(together) 30, 4O
40 plus 5 equals
45
Together: (high five)

 

 

Angela 8

 

10 plus 5 equals

 

295

 

 

holds up fingers one by one) 11 is 1, 12 is 2, 13 is 3, 14 is 4, 15 is 5.

 

Angela 9

(presents in verbal format) 25 + 10 equals
30

(presents in a written format) 25 + 10 equals
(works on the 25 + 10)

Yes, you need to write down 5 there.
35!

 

Angela 10

(Sings How Much is it Bum, Bum, Bum? )
(a dime, a nickel, and a penny) 1, 11, 25 I mean, I mean...

 

19...
Let’s me share the secret with you. Put the most valuable first (points out
coin classification chart). 10 + 5 equals
............ Um
10 + 5 equals
(shouts) 10
10 + 5 equals ? (points coin hand props with a dime and a nickel)
(counts coin hand props) 5, 10, 15
This one (points a penny)
21 (in singing voice)
This is One (penny)

15, what is the next number?
20 (very careful voice)
15...
16

 

Angela 11

(Sings How Much is it Bum, Bum, Bum .7; 3 quarters )
25 ...... ah. . .. 75 (in singing voice) I mean 25 (in singing
voice). 70 (in singing voice)
25 (shows quarter hand prop pattern 1 and 2 together)
5, 10, 15, 20, 25, 30, 35! (with exited voice)
Uh-oh
75 wait... (covering her face with her hand)
OK, you can do it.
How much?
(shows one hand)
5
(shows two hands)
10
(points two hands in the quarter hand props)
10, 20, 30, 40, 50

 

Angela 12

(© How Much is it Bum, Bum, Bum ?; three nickels)
10, 20, 35?
(CS) Think before you answer) How much is it? (CS) nickel five)

 

 

Angela 13

 

(© How Much is it Bum, Bum, Bum .7; a dime & two nickels)
75 (nervously nods)... 5

 

296

 

 

Is it same size?
10, 20.... 25
(shows her “No” facial expression)
I mean (covers her face with her hands)
(© nickel five) (lets her feel the smooth edge of a nickel)
10 + 5 equals
20, 15
Plus 5 is
15 and 20

 

Angela 14

(points to 9:30) What time is it?
10
But clock goes this way, so
7...
No
6 (Angela reads the number where the big hand points)
No, Uh-uh (no). It passes 9 but does not pass 10. That’s why you need to
say -
9 o’clock
This is?
6 o’clock (it supposed to be 30 minutes)
No
Big hand goes like.
5, 10, 15, 20, 30 I mean 25, 30
Yes! 9:30.

 

 

Angela 15

 

Clock goes which way?
That way.
Pass 4, but not pass
5
Which means?
30. ..
Uh-oh !!! (sing Think Before You Answer)

 

297

 

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