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PROSPECTIVE MIDDLE SCHOOL TEACHERS’
KNOWLEDGE ABOUT DATA ANALYSIS AND ITS
APPLICATION TO TEACHING

presented by
MARIA ALEJANDRA SORTO

has been accepted towards fulfillment
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PROSPECTIVE MIDDLE SCHOOL TEACHERS’ KNOWLEDGE ABOUT DATA
ANALYSIS AND ITS APPLICATION TO TEACHING

By

Maria Alejandra Sorto

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

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ABSTRACT

PROSPECTIVE MIDDLE SCHOOL TEACHERS’ KNOWLEDGE ABOUT DATA
ANALYSIS AND ITS APPLICATION TO TEACHING

By
Maria Alejandra Sorto

The purpose of the study was to identify the important aspects of statistical
knowledge needed for teaching at the middle school level and to assess prospective
teachers’ conceptions and misconceptions of statistics related to teaching data analysis.
An analytic study of the current literature, including state and national standards, was
conducted to identify the important aspects of statistical knowledge for teaching. A
written assessment instrument was developed and administered to a sample of 42
prospective middle school teachers. The purpose of the instrument was to gather data in
order to describe teachers’ conceptions for teaching data analysis and statistics. A subset
of the sample (n = 7) was interviewed to provide deeper insight into their conceptions and
to assure reliability of the instrument.

Results show that state and national standards differ greatly on their expectations
of what students and teachers should know about data analysis and statistics. The
variation is also large for the emphasis or importance given to the content. The average
emphasis of all the documents reviewed is given to the selection and proper use of
graphical representations of data, and measures of center and spread. Important aspects
of knowledge applied to teaching are proper selection and use of teaching strategies and
inferring students’ understanding from their work and discourse.

Prospective teachers that participated in this study performed better at the level of

pure statistical knowledge than at the level of application of this knowledge to teaching.

In particular, they showed abilities on reading, interpreting, and constructing graphical
representations, and computing measures of center and spread. Difficulties were shown in

judging students’ comments and identifying students’ mistakes.

©Copyright by
MARIA ALEJANDRA SORTO
2004

To my dear husband, Alex and my adorable daughter Isabel.

ACKNOWLEDGMENTS

This dissertation has been the product of many people and institutions that in one
way or another have made great contributions to its completion. First, I would like to
thank my major advisor, Dr. Sharon Senk, for her guidance, help, and great effort to
make this a quality work. Also, I would like to thank the members of my committee:
Joan Garfield for her statistical education contributions and for serving as a great model
and inspiration; Dr. Glenda Lappan for her very thoughtful comments and for providing
the necessary vision; Dr. Joan Ferrini-Mundy for her broad prospective contributions and
motivation, and Dr. Melfi for his encouraging comments and statistical perspective.

Second, I would like to thank the faculty and staff of the Mathematics
Department; in particular I would like to thank Dr. Zhou and Dr. Ulrich for their time and
effort to make me learn advanced mathematics; and especially Barbara Miller for all the
things, big and little, she has done to make this possible. Also, I would like to thank the
faculty and staff of the Statistics Department for treating me as one of their own.

Third, I am grateful to my colleagues and friends. To Sarah Sword and Andy
Jones for their words of comfort and suggestions on draft versions of this work. To
Romelia and Irvin Widders for serving as my family during my years in East Lansing.

Finally, thanks to my family. To my husband, Alex, for all his support in many
Ways. For being a father and a mother to our daughter in my numerous absences, for his

cOrnfort when things were difficult, for the many useful discussions about making this
Work meaningful, and for his love for me and respect for what I do. To my daughter,

Isabel, for allowing me to spend time not playing with her or watching her teeth grow.

vi

TABLE OF CONTENTS

LIST OF TABLES .................................................................................. x
LIST OF FIGURES ................................................................................ xii
CHAPTER 1
INTRODUCTION .................................................................................. 1
Mathematics for Teaching ................................................................................................ 2
Knowing and Learning .................................................................... 2
Data Analysis and Statistics: New Content Aligned with Reform ............................. 3
Historical Prospective ...................................................................... 3
Rationale ........................................................................................................................... 8
Research Questions ......................................................................................................... 10
Data Sources and Assumptions ....................................................................................... ll
Significance ..................................................................................................................... 11
The Structure of this Dissertation ................................................................................... 12
CHAPTER 2
REVIEW OF LITERATURE AND THEORETICAL FRAMEWORK .................... 15
Data Analysis and Statistical Content ............................................................................. 15
Statistical Content in School Mathematics .......................................................... 16
Statistical Content in College ......................................................... 18
Conceptions and Misconceptions of Data Analysis and Statistics .................................. 20
Kindergarten - Grade 12 Students ......................................................................... 20
College Students .................................................................................................... 26
Pro-service and In-service Teachers ...................................................................... 29
Theoretical Frameworks .................................................................................................. 33
Learning Statistics ................................................................................................. 33
Teaching Statistics ................................................................................................. 35
Assessing Statistics ................................................................................................ 40
Teachers' Knowledge ............................................................................................. 41
Summary ....................................................................................................................... .50

vii

CHAPTER 3
ASPECTS OF KNOWLEDGE FOR TEACHING DATA ANALYSIS AND

STATISTICS ......................................................................................... 52
Data Analysis and Statistics in Middle Grades ............................................................... 54
Background on Measuring Content ....................................................................... 55
Content Maps: A Visual Representation ............................................................... 66
Analysis of Content Maps ..................................................................................... 69
Data Analysis and Statistics in Middle Grades Teachers ................................................ 79
Content Analysis and Maps ................................................................................... 80
Relationship Between Content for Students and Teachers .............................................. 83
Statistical Content Knowledge Applied to Teaching ...................................................... 85
Teachers' Knowledge as Suggested by Documents ............................................... 86
Classification of Documents by its Characteristics ......................... 86
General Aspects of Teacher's Knowledge .................................... 87
Mathematical Aspects of Teacher's Knowledge .............................. 90
Statistical Aspects of Teacher's Knowledge .................................. 94
Summary and Discussion .............................................................................................. 101
CHAPTER 4
METHODOLOGY FOR DEVELOPING AN INSTRUMENT TO ASSESS
STATISTICAL KNOWLEDGE FOR TEACHING .......................................... 103
Instruments .............................................................................. .........103
Statistical Knowledge for Teaching Assessment .............................................. 103
Development Procedures ............................................................ 106
Description of Items ................................................................. 109
Procedures ................................................................................ 112
Participants ........................................................................ 114
Analysis of Statistical Knowledge for Teaching Assessment ....................................... 115
Rubric ...................................................................................... 116
Follow-Up Interviews ........................................................................... 119
Purpose and Description ................................................................. 119
Analysis ..................................................................................... 120
CHAPTER 5
MEASURING STATISTICAL KNOWLEDGE FOR TEACHING ........................ 122
Item Level of Performance .................................................................. 122
Item 1 ...................................................................................... 122
Item 2 ....................................................................................... 128
Item 3 ..................................................................................... 136

viii

Item 4 .................................................................................... 144

Item 5 .................................................................................... 154
Item 6 .................................................................................... 160
Item 7 .................................................................................... 165
Item 8 .................................................................................... 171
Overall Performance ......................................................................... 178
Performance by Domain of Knowledge ............................................. 179
Performance by Cognitive Demand .................................................. 181
Percentage Success on Assessment .................................................. 183
Summary and Discussion .............................................................. 185
CHAPTER 6
CONCLUSIONS AND RECOMMENDATIONS ............................................ 186
Summary and Discussion of Main Findings ................................................................. 188
Significance and Implications ...................................................................................... 203
Recommendations for Future Research ......................................................................... 206
APPENDIX A
State and National Student Standards ........................................................... 208
APPENDIX B
Codes Assigned to Student Standards ........................................................... 220
APPENDIX C
Statistical Knowledge for Teaching Assessment .............................................. 235
APPENDIX D
Item Scores and Statistics for Assessment ..................................................... 247
APPENDIX E
Interview Protocols ............................................................................... 250
APPENDIX F
Interview Transcripts ............................................................................. 271
REFERENCES .................................................................................... 297

ix

LIST OF TABLES

Table 3.1 Language Frequently Associated with Cognitive Demands ...................... 58
Table 3.2 Data Analysis and Statistics Content Matrix ...................................... 61
Table 3.3 Number of Student Standards Analyzed and Codes Assigned .................. 69

Table 3.4 Number of Specific Statements Analyzed and Codes Assigned
for Teachers Documents ......................................................................... 80

Table 3.5 Summary of Aspects of Content Knowledge for Data Analysis
and Statistics ........................................................................................ 84

Table 3.6 Summary of Teaching Tasks for Knowledge for Teaching Data Analysis

and Statistics ....................................................................................... 100
Table 4.1 Distribution of Items by Content and Cognitive Demand ....................... 109
Table 4.2 Distribution of Subjects by Gender, by Age, and by Class ..................... 114
Table 5.1 Distribution of Scores for Item 1 (n = 42) ....................................... 123
Table 5.2 Distribution of Prospective Teachers’ Conception of “Typical” ............... 125
Table 5.3 Distribution of Scores for Item 2 (n = 42) ....................................... 128
Table 5.4 Distribution of Scores for Item 3 (n = 42) ....................................... 137
Table 5.5 Distribution of Responses to Item 3a by Recognition
of Mistake and Its Source ...................................................................... 138
Table 5.6 Distribution of Scores for Item 4 (n = 42) ....................................... 144
Table 5.7 Distribution of Scores for Item 5 (n = 42) ........................................ 154
Table 5.8 Distribution of scores for Item 6 (n = 42) ........................................ 160
Table 5.9 Distribution of scores for Item 7 (n = 42) ......................................... 166
Table 5.10 Distribution of scores for Item 8 (n = 42) ....................................... 171

Table 5.11 Distribution of Responses of Computational and Conceptual
Knowledge for the Standard Deviation ....................................................... 177

Table 5.12 Descriptive Statistics of Percentage Scores by Domain of Knowledge ...180
Table 5.13 Descriptive Statistics by Cognitive Demand ................................... 182
Table 5.14 Percent Success on Items Measuring Pure Statistical Knowledge .......... 183

Table 5.15 Percent Success on Items Measuring Knowledge Applied to Teaching. ...1 84

Table B.l Codes for Connecticut Students Standards ...................................... 221
Table 3.2 Codes for Florida Students Standards ............................................ 222
Table B.3 Codes for Georgia Students Standards ........................................... 223
Table B.4 Codes for Kentucky Students Standards .......................................... 224
Table 8.5 Codes for North Carolina Students Standards ................................... 226
Table B.6 Codes for Missouri Students Standards .......................................... 228
Table 8.7 Codes for Ohio Students Standards ............................................... 229
Table B.8 Codes for Oregon Students Standards ............................................ 230
Table B.9 Codes for Virginia Students Standards .......................................... 232
Table 3.10 Codes for West Virginia Students Standards ................................... 233
Table B.11 Codes for NCTM Students Standards ........................................... 234
Table D.1 Items Scores and Statistics ......................................................... 248

xi

LIST OF FIGURES

Images in this dissertation are presented in color

Figure 2.1. Distributions Presented to Children to Compare

“Which Group is Better” ............................................................................ 24
Figure 2.2. Item Given to Science and Mathematics Pro-service Teachers ................. 32
Figure 2.3. Teacher’s Knowledge Developing in Context .................................... 44
Figure 2.4. Interaction of Knowledge Domains ................................................. 45
Figure 2.5. Framework to Investigate Mathematical Knowledge for Teaching ............ 49

Figure 3.1. Florida’s Standard on Data Analysis and
Probability for Grades 6-8 ......................................................................... 62

Figure 3.2. Three Virginia standards on Probability and
Statistics for Grade 6 ................................................................................ 63

Figure 3.3. Excel Spread Sheet to Record Codes for Three Specific Florida

Standards ............................................................................................. 65
Figure 3.4. Frequency Matrix for Codes Observed in Florida Standards ................... 66
Figure 3.5. Content Map for Florida Standards Grades 6-8 .................................. 68
Figure 3.6. Content Maps for the Connecticut and Florida Standards ........................ 71
Figure 3.7. Content Maps for the Georgia and Kentucky Standards ......................... 72
Figure 3.8. Content Maps for the Missouri and North Carolina Standards .................. 73
Figure 3.9. Content Maps for Ohio and Oregon Standards .................................... 74
Figure 3.10. Content Maps for Virginia and West Virginia Standards ...................... 75

Figure 3.11. Content Maps for Principles and Standards for School
Mathematics (N CTM, 2000) ....................................................................... 76

Figure 3.12. Content Map for the Mathematical and Problem-Solving Goals in
Connected Mathematics Teacher’s Guide Grade 6 and 8 Textbooks ........................ 77

xii

Figure 3.13. Content Map for the ten states, the Mathematical and Problem-Solving
Goals in Connected Mathematics Teacher’s Guide Grade 6 and 8 Textbooks and
Principles and Standards for School Mathematics (N CTM, 2000) .......................... 78

Figure 3.14. Content Maps for the Topics Covered in PRAXIS 11 Middle School
Mathematics Test and The Mathematical Education of Teachers (CBMS, 2001). . . . . . ....82

Figure 3.15. Combined Content Map for The Mathematical Education of Teachers
(CBMS, 2001) and the Topics Covered in PRAXIS II Middle School Mathematics. .....83

Figure 3.16. Structure for Analyzing Aspects for Knowledge for
Teaching ............................................................................................. 101

Figure 4.1. Content Map for the ten states, the Mathematical and Problem-Solving Goals
in Connected Mathematics Teacher’s Guide Grade 6 and 8 Textbooks and Principles and
Standards for School Mathematics (N CTM, 2000), The Mathematical Education of
Teachers (CBMS, 2001) and the Topics Covered in PRAXIS 11 Middle School

Mathematics ........................................................................................ l 05
Figure 5.1. Item 1 .................................................................................. 123
Figure 5.2. Item 2 .................................................................................. 128
Figure 5.3. Item 3 .................................................................................. 136
Figure 5.4. Item 4 .................................................................................. 144
Figure 5.5. Example of a Typical Notational Mistake in Prospective Teachers .......... 146
Figure 5.6 Data Set Used in Interview to Ask Prospective Teachers to Find the Mean
without Computational Algorithm ............................................................... 147
Figure 5.7. Example of a Procedure Used by a Prospective Teacher to Find a
Distribution with Mean 3.5 ....................................................................... 151
Figure 5.8. Item 5 .................................................................................. 154
Figure 5.9. Example of a Prospective Teacher’s Strategy to Find the Median of
Categorical Data .................................................................................... 158
Figure 5.10. Item 6 ................................................................................. 160

Figure 5.11. Example of Typical Mistake when Constructing a Stem-and-leaf Plot.....162

Figure 5.12. Item 7 ................................................................................ 165

xiii

Figure 5.13.
Figure 5.14.
Figure 5.15.
Figure 5.16.
Figure 5.17.

Figure 5.18.

Item 8 ................................................................................ 17]
Distribution of Percentage Scores of 42 Prospective Teachers .............. 178
Mean Percentage Scores by Domain of Knowledge and Overall ........... 180
Distribution of Scores for Statistical Knowledge ............................ 18]

Distribution of Scores for Knowledge for Teaching .......................... 181

Mean Percentage Scores by Level of Performance for Statistical

Knowledge Domain ............................................................................... 182

xiv

CHAPTER 1

INTRODUCTION

As statistics becomes more prevalent in the K-12 school mathematics curriculum
and nearly ubiquitous in everyday discourse, the need for a statistically literate population
becomes imperative. Implementation of K-12 school mathematics curriculum aligned
with reform movements has challenged teachers and teacher educators in many ways.
New content, such as statistics, is one of these challenges. Many experts agree that
teachers’ own deep and substantial knowledge of mathematics content is a key factor to
provide quality mathematics education (Ball, Lubienski, & Mewbom (2001); NRC,
2001b). The inclusion of statistics topics across the school curriculum from kindergarten
to Grade 12, as suggested by Principles and Standards for School Mathematics (National
Council of Teachers of Mathematics [NCTM], 2000), provides the rationale to
investigate how this new content relates to teachers’ own knowledge of the subject.

The purpose of this study is to address two main questions fundamental to the
preparation of middle grades mathematics teachers. First, what are the important aspects
of content knowledge for teaching statistics? Secondly, how can we assess the extent to
which prospective teachers possess this knowledge? Since these questions are
inextricably linked to context and content, this study focuses in particular on the teaching
of statistics, often called data analysis, at the middle school level. This chapter places

this study within the context of current movements in mathematics education. It starts by

outlining the problems of the content preparation of teachers. Next, the case of statistics
education is described as an example of new content aligned with reform movements. As
new content, statistics poses a big challenge for teachers. In the next three sections, the
rationale, significance, and purpose of this study are stated. Research questions and

assumptions underlying this study follow.

Mathematics for Teaching
Knowing and Learning

Teaching mathematics is a practice that involves the knowledge of what, how and
whom to teach. The “what” is usually determined by a curriculum guide or textbooks
and teachers supposedly learn the content by taking mathematics classes. The “how” is
suggested in teachers’ editions and is presumed to be learned in methods and education
classes. Although in practice all these components are intertwined, teachers learn them in
separate academic departments. Schools of education are usually responsible for
teaching courses in methods of teaching and educational psychology, whereas the subject
matter is the responsibility of academic departments (Ball, 2000; Lagemann, 1996).

Research on teachers knowledge (Ball & Bass, 2000; Berenson, F riel, & Bright,
1993; Bright, Friel, & Berenson, 1993; Even, 1993; Even &Tirosh, 1995; Heaton, Prawat,

& Remillard, 1992; Lloyd & Wilson, 1998; Ma, 1999; Putman, et. al., 1992; Tirosh, 3

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2000; Russell, Goldsmith, Weingerg, & Mokros, 1990; Vacc & Bright, 1999) has shown ,

that many teachers have misconceptions about mathematics, have difficulty explaining in ,5
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simple terms an abstract idea and cannot correct or anticipate students’ mistakes, even

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though they have had the required classes to become teachers. It is clear that something
is missing in their preparation to become teachers.

Thus, mathematics education researchers, mathematicians, and teacher educators
are trying to answer in a more systematic manner the questions “What do mathematics
teachers need to know to teach well?” and “How can teachers develop the knowledge of

mathematics they need to teach well?”

Data Analysis and Statistics: New Content Promoted by Reform
Historical Prospective

The introduction of statistics in school mathematics started at the beginning of the
20"h century when the Mathematical Association of America (MAA) in 1916 appointed
the National Committee on Mathematical Requirements. The committee’s final report
published in 1923 (known as “The 1923 Report”) advocated a general mathematics
program for grades 7-12 which included integrated study of arithmetic, informal
geometry, elementary algebra, graphs and descriptive statistics (Jones, 1970). In
particular, the report suggested the study of measures of center as part of either tenth- or
eleventh-grade courses.

Although “The 1923 Report” suggested an integrated approach for grades 7-12,
this cuniculum was only implemented at the junior high school level. This was due
partly to the philosophy of mathematics at that level which is exploratory and partly to
materials and textbooks developed at this time. Relative to secondary school, two major
curriculum reports were published in 1940. One was, Mathematics in General Education

by the Progressive Education Association (PEA) committee, and the other was a product

of the Joint Commission of the MAA and the National Council of Teachers of
Mathematics (N CTM). The PEA committee selected broad categories of mathematical
behavior including one on Data, stating “The teacher must help students become aware of
differing kinds of data and their characteristics such as accuracy and relevancy. The
student should acquire the ability to collect and record data, understand the measurement
process, and be familiar with the construction and use of tables and graphs.” (Jones,

1970, p. 226). Unfortunately, these recommendations were not implemented. A national
war crisis intervened and people felt that “new and formerly ‘pure’ mathematics was
needed by the technicians, engineers, and scientists of an expanding wartime technology”
(Jones, 1970, p. 233).

Afier the war in 1945, the secondary school’s college preparatory program based
the objectives on The Harvard Report in which the key word to describe the role of
mathematics was “appreciation”. The report suggested a course in the senior year which
comprised a survey of elementary trigonometry, statistics, and precision of measurements
and use of graphs. At the same time the Commission on Post-War Plans of the NCTM
reports suggested statistics as one of the twenty-nine key concepts for junior high school.

Even though clear efforts were undertaken to improve the secondary program, a
survey conducted by the Educational Testing Service in 1954 indicated that there were
problems. Hence, in 1955 the Committee on Examinations of the College Entrance
Examination Board (CEEB) appointed a commission formed by college mathematicians,
high school teachers, and college teachers of mathematics education. The suggestions of
the commission published in 1959 were minor with respect to change in content but

emphasized important changes in instruction and teaching. As in the previous reports,

the commission recommended descriptive statistics in grade 9 and the study of
probability with statistical application as an optional course in grade 12. For the first
time attention was paid to the teaching of probability and statistics.
The probability and statistics course the commission proposed represented an area
that had never been studied seriously at the high school level. Consequently, the
commission felt its recommendation should be accompanied by a demonstration

of its feasibility; members of the commission prepared a textbook for this course
and taught it in experimental classes. (Jones, 1970, p. 265)

Suggestions about including the study of statistics and probability in school
mathematics in the late 505 were reflected by the inclusion of the content in curriculum
materials developed by the School Mathematics Study Group (SMSG). The SMSG
consisted of college teachers of mathematics, high school teachers and supervisors, and
representatives of corporations such as Rand and Bell Telephone Laboratories. The study
group created a book for a one semester course on probability and statistics for twelfth
grade. The materials were tried out in schools, edited, and revised in 1958 and 1959.

By the mid 19603 the secondary school program was well established with many
exemplary experimental programs, textbooks and new materials and the reform efforts
turned their attention toward the elementary school level. In the summer of 1962,
mathematicians and representatives of the NSF met in Cambridge to discuss the state of
mathematics in elementary and secondary schools. The following year they published
Goals for School Mathematics; one of the key ideas was that “some ‘feeling’ for
probability and statistics was considered important for all students” (Jones, 1970, p. 291).

The Cambridge Report also included cuniculum plans for grade K-2 and grades

3-6 and two proposals for grades 7-12. The suggestions of this report focus more on the

intellectual ability of children and coherence among grade band levels. Critique of the
report lead to debates and controversy.

In the late 703 the National Council of Supervisors of Mathematics (N CSM), an
organization composed of mathematics leaders at district, state and university levels,
published its Position Paper on Basic Mathematical Skills (1977), which defined "basic skills"
as including not only computation but also estimation, geometry, problem solving,
computer literacy, and statistics and probability. More specifically, the report suggests
that students should know how to read, interpret, and construct tables, charts, and graphs.
This report, and others, prompted NCTM to appoint a committee to develop
recommendations for school mathematics for the '803. The product, An Agenda for Action
(1980), was one of the earliest position statements from NCTM and a definitive step
toward reform. It set problem solving as the curricular focus, and recommended that the
definition of "basic skills" be broadened to include such mathematical skills as estimation
and logical reasoning, and promoted the use of calculators and computers in the
classroom at all grade levels. Another publication that awakened the general public to
crisis in the United States was A Nation at Risk (1983). Commissioned by the National
Commission for Excellence in Education (N CEE), this report lays out the critical status
of students’ performance and gives recommendations about content, expectations, time,
and teaching. In particular, the report suggests that “The teaching of mathematics in high
school should equip graduates to: (a) understand geometric and algebraic concepts;

(b) understand elementary probability and statistics; (c) apply mathematics in everyday
situations; and ((1) estimate, approximate, measure, and test the accuracy of their

calculations. In addition to the traditional sequence of studies available for college-bound

students, new, equally demanding mathematics curricula need to be developed for those
who do not plan to continue their formal education immediately” (Retrieved June 28,
2004, from http://www.ed.gov/pubs/NatAtRisk/recomm.html ).

In 1986, the Board of Directors of the NCTM established a Commission on
Standards for School Mathematics to help improve the quality of school mathematics.
The commission published in 1989 Curriculum and Evaluation Standards for School
Mathematics (N CTM, 1989) and later Principles and Standards for School Mathematics
(N CTM, 2000) and placed, for the first time, statistics and probability on an equal footing
with numeration, measurement, algebra, and geometry in their importance in
Kindergarten through Grade 12. These standards suggest that instructional programs
should enable all students to formulate questions that can be addressed with data, collect,
organize, and display relevant data; select and use appropriate statistical methods to
analyze data; develop and evaluate inferences and predictions that are based on data; and
to understand and apply basic concepts of probability.

As before, the new content and teaching philosophy suggested by the reform
movement led to the production of curriculum materials, which included data analysis
and probability. For example, all of the new National Science Funcation (NSF) funded
comprehensive middle school curricula — Connected Mathematics Project (CMP)
(Lappan, Fey, Fitzgerald, Friel, & Phillips, 2002), Mathematics in Context
(http://www.wmich.edu/cpm)), and MathScape: Seeing and Thinking Mathematically
(http://www2.edc.org[MathscapeSTM) include a statistics and probability strand. At this

point, as the new curricula are being implemented there is a need for professional

development and teacher preparation in statistics, since teachers may have not studied the
subject for teaching in grades K—12 (CBMS, 2001, p. 114).

At the end of 2003, the American Statistical Association [ASA] and the
University of Georgia held a planning conference on the statistics education of future
teachers as a response to The Mathematical Education of Teachers (CBMS, 2001) report
published in fall 2001. The participants were a group statisticians and mathematics
educators which came together to start a conversation on current models of teacher
education, research on how students learn, and exposition of current activities used for
preparing future teachers. The group, which is known by the acronym Math/Stat TEAMS
(Math/Stat Teacher Education: Assessment, Methods, and Strategies), is scheduled to
meet again in 2005 to continue planning for setting an agenda on the integration of

statistical content and pedagogy into teacher preparation.

Rationale
In statistics education, teachers’ knowledge is of special interest since the reform

and new curricula are challenging teachers not just with new teaching approaches but
with new content as well. While there is consensus about their importance as basic
statistical concepts in the school mathematics curriculum, few studies have been
conducted on learning and teaching statistical concepts. Although in the last decade,
significant advances (have been made regarding students’ conceptions of measures of

._ center; we still do not MOVIES much about how teachers learnirnportant concepts of
statistics and data analysis. This is particularly true for the concepts of variability and

-—-..._.,_ «—.-.

distribution. Despite this lack of research, new curricula emphasizing statistics have

become available for use at the K—12 levels in this country (c. g. Quantitative Literacy
Series (Landwehr & Watkins, 1986), the University of Chicago School Mathematics
Project (Senk et al., 1998), Data-Driven Mathematics Series (Burrill & Hopfensperger,
1997), and Connected Mathematics Project (Lappan, Fey, Fitzgerald, Friel, & Phillips,
2002). Also, in the United Kingdom, the Schools Council Project in Statistical Education
has published materials for secondary students (Garfield, 1988).

At this point we only know that teachers as adults may have the same difficulties

found in college students (Mevarech, 1983; Pollatsek, Lima, & Well, 1981). Even i f ‘

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though it is important to explore teachers’ conceptions of topics he or she teaches, we

need to explore them in teaching contexts. Teachers need to understand concepts not just

to know them and apply them in real world problems, but to use that knowledge in the

context of practice to help students learn (Ball, 2000; Ball et al., 2001). More research is
needed on knowledge for teaching statistics and data analysis. ’

The approach taken here attempts to measure certain aspects of prospective
teachers’ subject matter knowledge of statistics and knowledge of statistics for teaching.
Several of the measurement items are posed in the context of teaching to examine K
teachers’ decision making and knowledge of statistics. Much, if not all, the formal ; I IV
training teachers receive in statistics is in university classes geared, not to teachers of the
subject, but instead to practitioners and to consumers of statistics. Do problems which
arise in the context of teaching unveil incomplete understanding and misconceptions
different from those typically assessed in these university statistics courses? When
exposed to situations where they need to teach and help others learn, do prospective

teacher’s call upon their statistical knowledge, rely solely upon concepts and lessons

learned from methods and education classes or successfiilly combine the two sources?
Describing teachers’ knowledge and any differences requires the creation of a
measurement tool and its administration. This is a primary goal of this dissertation.

As the importance of statistics in the middle school curriculum is a fairly new
phenomenon, it is crucial to be clear about what statistical content teachers are currently
required to teach. In particular, what are the important concepts or big ideas that they
need to know? And what are the teaching problems that arise when they are teaching it?
Furthermore given constraints of time, money and manpower, which concepts and I ,, z). r
problems can be feasibly measured? So, before trying to assess teachers’ knowledge, it is l
necessary first to find out what aspects and topics can and should be assessed. . if

Hence, this study will contribute to the literature on teacher preparation and {I "i i 0
statistics education in several ways. First, it will identify the important aspects of content
knowledge for teaching statistics and data analysis at middle grades level. Second, it will
produce a reliable instrument for the assessment of statistical knowledge for teaching,
and it will measure and describe the knowledge some prospective teachers have with
respect to some of these aspects.

Research Questions

The central questions addressed by this study are:

1. What are the important aspects of content knowledge for teaching data analysis and
statistics at the middle school level? More specifically, what are the important
statistical content topics taught in middle school for which teachers need to be

prepared?; What are the cognitive demands (such as memorize, perform procedures

10

 

and solve non-routine problems) that are related to the content?; What are the
important aspects of knowledge for teaching that relate to the aspects above?

2. What are the conceptions and misconceptions prospective middle school teachers
have with respect to these important aspects of knowledge for teaching data analysis
and statistics?

Data Sources and Assumptions

Although content knowledge for teaching is a construct still under development
(Ball et al., 2001), for this study, knowledge for teaching will be understood as the
application of content to teaching contexts. The identification of the aspects of content
knowledge for teaching to be studied will be based on integrating different kinds of work:
the content suggested by state and national standards at the student and teacher level; the
statistical content in the middle school mathematics curriculum; research and theoretical
work on learning and understanding data in particular and other statistical concepts in
general; and research and theoretical work on teachers’ content knowledge as used for
teaching.

A written instrument designed to measure some of these aspects will given to
senior prospective middle grade teachers from selected universities and followed by face-
to-face interviews with selected respondents.

Significance

The vast majority of statistics presented in the media deals with the basic

statistical concepts of shape, center and spread. Furthermore, clear understanding of

these concepts is fundamental to any further study of statistics. Hence there is a need to

11

prepare teachers who are capable of knowing and explaining these concepts to their
students.

To address this need, programs (e. g. ASA-NCTM Quantitative Literacy Project),
have been developed and more are underway (e. g. Explorations in Statistics: The
Learning Math Project) to train in-service and pre-service middle and high school
teachers. Some universities (e. g. San Diego State University, Kennesaw State University
and University of Missori-Columbia) offer statistics courses that are especially targeted
to future statistics teachers. These efforts, however, are being carried out in the near
absence of research on the conceptions and misconceptions that preservice teachers and
other college students bring to the statistics classroom. Even less is known about what
knowledge and conceptions teachers transfer from their time as a student to the classroom
as teacher.

This study will contribute to the research effort on teacher preparation and
statistics education in several ways. First, it will develop a systematic method to organize
and identify the important aspects of content knowledge for teaching statistics and data
analysis at middle grades level. Second, it will produce a reliable instrument for the
assessment of statistical knowledge for teaching, and it will measure and describe the

knowledge some prospective teachers have with respect to some of these aspects.

The Structure of this Dissertation
Chapter 2 provides the theoretical basis for this study. The first section of this

chapter summarizes the recommendations about statistical content from Kindergarten to

12

College level. It describes the conceptions and misconceptions related to data analysis
and statistics for K-12 students, college students, and pre-service and in-service teachers
as reported in the literature. The last section of this chapter describes the different
frameworks that exist for learning and teaching statistics and for investigating teachers’
knowledge.

In Chapter 3 the first research question is considered. A variety of documents are
analyzed to determine the important aspects of content knowledge for teaching data
analysis and statistics at the middle school level. The first three sections review the
content standards and goals for middle school students and teachers. A specialized
instrument called a content matrix and its map are used to compare and contrast the
standards demanded by the different documents at the two levels. The following section
describes the analysis of standards for knowledge applied to teaching. The different
frameworks used in the documents reviewed are described and the most important
aspects identified. Finally the last section provides a summary of the results.

Chapter 4 describes the procedures used for the development and administration
of a written instrument and follow up interviews used to assess content knowledge for
teaching data analysis and statistics. The first section discusses the choice of items, pilot
testing and reliability of the written instrument. The second section lists the procedures
used to administer the written instrument including a description of the participants
selected. The next section explains the process used to score and analyze the results of
the written instrument. The final section gives the purpose of and method of analysis used

for the follow up interviews.

13

The principal results of the written instrument and follow up interviews are
presented in Chapter 5. The two sections report the item by item performance and overall
performance, respectively.

Finally, Chapter 6 provides a summary and recommendations, and discusses
limitations of this study. The first section lists major findings of the study along with its
limitations, the second section discusses their significance and implications and the final

section provides recommendations for future work.

14

CHAPTER 2

REVIEW OF LITERATURE AND THEORETICAL FRAMEWORK \‘

This chapter is organized around the research questions. First the
recommendations about the statistical content students and teachers should know is
summarized to set the grounds for identifying the important aspects of content
knowledge. Second, research on students’ and teachers’ conceptions and misconceptions
are described to help understand how these are connected to content knowledge and
identify aspects of statistical knowledge for teaching. The chapter ends with a section on
theoretical frameworks on learning, teaching, and assessing statistical concepts and a
framework to measure teachers’ knowledge. These help develop the instrument to

measure the desired construct of statistical knowledge for teaching.

Data Analysis and Statistical Content
Basic concepts in statistics such as measures of central tendency (mean, median,
mode) and dispersion (range, variance, standard deviation) appear everywhere. In
everyday life, newspapers, and work, statistics are used to inform (or mislead) the public
on a broad range of issues. These basic statistical concepts, traditionally called

descriptive statistics, are included in a broader concept called data analysis. Shaughnessy

15

(1996) points out that “the current meaning of data analysis emphasizes organizing,
describing, representing, and analyzing data, with a heavy reliance on visual displays
such as diagrams, graphs, charts and plots. Conceptually, data analysis looks for
patterns, centers, clusters, gaps, spreads, and variations in data” (p. 205).

Burrill (1998) also adds, “Understanding statistics and probability enables people
to reason from and make conclusions based on data, judge the quality of other people’s
conclusions, recognize the degree of uncertainty in any endeavor, and quantify that
uncertainty”.

Statistical Content in School Mathematics

Although research in teaching and learning statistical concepts is still in its
infancy, its basic concepts are now being introduced in pre~college levels guided by the
Curriculum and Evaluation Standards (N CTM, 1989) and the Principles and Standards
for School Mathematics (NCTM, 2000); the Guidelines for the Teaching of Statistics
(ASA, 1991); and the Benchmarks for Science Literacy (American Association for the
Advancement of Science [AAAS] , 1993).

The study of basic statistical concepts, as recommended by most curriculum
guides, should start in elementary grades allowing students to work with some data, but it
is in middle grades that these understandings begin to be developed in depth. The

”expectationsuof the NCTM are that at the end of high school, students should have a hill
range of data-analytic skills and be comfortable designing an appropriate study for a

a" "‘

question of interest to them, collecting data, and summarizing the results. J

The Principles and Standards for School Mathematics (N CTM, 2000)

recommends that students should begin their study of Data Analysis in preschool and

16

progressively build their skills until 9th grade. In Grades Pre-K-2, students begin by
describing parts of the data and the set of data as a whole to determine what the data
show. In Grades 3 — 5 they should continue describing the shape and important features
of a set of data and comparing data sets, but with an emphasis on how the data are
distributed; using measures of center, focusing on the median, and understanding what
each does and does not indicate about the data set; and comparing different
representations of the same data and evaluating how well each representation shows
important aspects of the data. Finally in Grades 6 — 9 students complete their study of
describing data by finding, using, and interpreting measures of center and spread;
discussing and understanding the correspondence between data sets and their graphical
representations, especially histograms, stem-and-leaf plots, box plots, and scatterplots.

These standards envision the learning of basic statistical concepts embedded in
the exploration of real world data. Students must make sense of the data before they
simply compute the algorithms. Furthermore, the learning of basic concepts goes along
with the use of appropriate graphical representation, inferences and predictions about the
data, and design of statistical experiments.

Besides the national standards mentioned here, content standards/cuniculum
frameworks/objectives exist for each state in the country. Other reports and documents
suggest what students and teachers should know and able to do in statistics. Some of
them are examined and analyzed in Chapter 3 to identify the statistical content in a more

systematic way.

17

/’,

Statistical Content in College
Statistical content at the college level has a broader scope than the content taught
at K-12 level. Colleges and universities teach statistics as a general education
requirement which usually consists of an introductory non-calculus based course; as part
of an undergraduate degree in statistics or mathematics; and as a research tool for
undergraduate and graduate programs in other disciplines. Recently, some institutions

have created a new course in statistics designed specially for future teachers as well.

The content taught in these courses varies according to their purpose and level. if: . I:

Recommendations and guidelines for the non-calculus based introductory course are
described by Cobb (1992) in the report of the Join Curriculum Committee of the ASA
and the Mathematical Association of America [MAA] entitled Heeding the Call for
Change. The report suggests three major recommendations: 1) Emphasize statistical
thinking; 2) More data and concepts: Less theory, fewer recipes; and 3) Foster active
learning. These recommendations are being revised and updated by the Guidelines for
Assessment and Instruction in Statistics Education [GAISE] Project of the ASA. The
latest draft of the GAISE College Report (J. Garfield, personal communication, June 29,
2004) builds on the previous recommendations and suggests the following: 1) Emphasize
statistical literacy and develop statistical thinking; 2) Use real data; 3) Stress conceptual
understanding rather than mere knowledge of procedures; 4) Foster active learning; 5)
Use technology to develop conceptual understanding and analyze data; and 6) Use
assessments to improve and evaluate learning. The draft also includes goals for students
regarding what it means to be statistically educated and suggestions for how to

implement them.

18

Recommendations for undergraduate statistics and mathematics majors are given by
the ASA Curriculum Guidelines for Undergraduate Programs in Statistical Science
(http://www.amstat.org/education/index.cfrn?fuseaction=Curriculum_Guidelines ) which
suggest statistical, mathematical, computational, non-mathematical, and substantive area
topics. They also recommend the approach to teaching these topics which is similar to

the guidelines mentioned previously:

. Emphasize real data and authentic applications.

. Present data in a context that is both meaningful to students and indicative of the
science behind the data.

0 Include experience with statistical computing.
. Encourage synthesis of theory, methods, and applications.

. Offer frequent opportunities to develop communication skills. a

) ~
‘I
\._'\

.
I

I . . .I»=.~
3.x. ‘ . '-

,. I_. \

(3 ’.'

As for the content recommended for futureteaChers, the report ’by the Conference
Board of the Mathematical Science (CBMSIAeIntitled The Mathematical Education of
Teachers (CBMS, 2001) suggests that future teachers, in particular middle grade
teachers, should know and be able to design simple investigations and collect data to
answer specific questions; understand and use a variety of ways to display data; explore
and interpret data by observing patterns and departures from patterns in data displays,
with particular emphasis on shape, center, and spread; draw conclusions with measures of
uncertainty; and know something about current uses of statistics in many fields.

The recommendations described in this section provide a basis for the

identification of the content in data analysis and statistics teachers need to know. A more

systematic analysis of other documents such as national and state standards at the student » »

l9

and teacher level is necessary to identify this content. Chapter 3 addresses this issue in

detail.

Conceptions and Misconceptions of Data Analysis and Statistics
Studies of student understanding of basic statistical concepts are relatively new
and most of them focus on measures of center (mean, median and mode). There is very
little research on measures of dispersion, and the few which do exist have only addressed ‘ .
the range and standard deviation. This section describes findings of research on students’ ’-
conceptions and misconceptions at the different levels: K-12 students, college students, if
and pre-service and in-service teachers. , .
Kindergarten — Grade 12 Students
Measures of center
Strauss and Bichler (198 8) investigated the understanding of properties of the
mean with 80 children of ages 8, 10, 12, and 14. The purpose of their study was to
determine the development of children’s understanding of the mean and to assess the
effect of the type of task presented to students (continuous, discrete, story, concrete and
numerical). Strauss and Bichler found that about half of 8 year-olds and almost all 10-,
12-, and 14-year-olds could solve a task involving the property of the mean being located
between the extreme values. For the property about the sum of the deviations about the
mean being zero, it was found that at all ages few children offered correct judgements.
An example of a task that measured this property follows:
Children brought cookies to a party they were having.
Some children brought many and some brought few.

The children who brought many gave some to those
who brought few until everyone had the same number

20

of cookies. Was the number of cookies given by
those who brought many the same as the number of
cookies received by those who brought few? Was it
more? Less? Why do you think so? (Strauss &
Bichler, 1988, p. 69)

The most dominant justification was “that we could not know the answer to the
question because we did not know how much material there was or how many children
received it”(p. 72). The most difficult property for children to understand was the
average as representative of the values being averaged. Given a list of data points
children could not come up with a number that would be representative of all the data.
Some young children picked the biggest number claiming that it was the biggest that best
represented the whole data set.

In summary, this study concluded that for some properties of the mean there are
development differences and that the material used or the medium had no effect.
However, a replication study by Leon and Zawojewski (1993) with fourth grade, eight
grade and college students with a paper-and-pencil test (as opposed to interviews) found
significant differences between students’ responses to items in story format and
numerical format.

Although the two studies give some understanding about children’s
conceptualization of the mean, neither of them provides insight in how children
understand and construct indicators of center, or what the average tells us about the
population it summarizes. In a qualitative study by Goodchild (1988), 13 and 14-year-
old students were interviewed to explore their understanding when they are confronted

with the word average in the context of everyday situation. When students were asked

what the sentence printed on a matchbox “Average contents 35 matches” meant to them,

21

they had different interpretations. The word average was viewed as a representative
number, as a measure of location, and as an expected value. The words “around”,
“about”, “roughly”, “not exactly”, and “close to” where indicators of measure of location
and were used by 15 out of 17 students (Goodchild, 1988). Four of them indicated the
idea of expectation with the use of words such as “normally” and “usual amount”.
Goodchild wanted to explore these perceptions about the mean more deeply and asked
the students to hypothesize the distribution of the contents of 100 boxes. The
distributions the children constructed suggested that the students see the contents of the
boxes as variable. This particulartask gave more insight into children’s perception of
variability, which will be discussed in a later section of this chapter.

" " ’ Mokros and Russell (1995) also looked at children’s own construction and
interpretation of the mean. They interviewed fourth, sixth, and eight graders using open-
ended problems. An example of one of the problems used was to put price stickers on
pictures of nine bags of potato chips so the “typical or usual or average” price of the
chips would be $1.38. Many of the fourth graders associated the average with the mode,
making all or most of the values the same as the mean. One fourth grader who used the
modal strategy argued that the typical value was 15 cents (1.38 divided by 9) because it
occurs nine times and was the result of (mis)using the mean algorithm (p.28). Older
students (mainly eight graders) used the mean as a “middle” point, placing one value at
$1.38 and creating a symmetrical distribution around that value. “Symmetry had a strong
attraction for these students” (Mokros & Russell, 1995, p.32). However, as Mokros and

Russell pointed out, students’ strategies were efficient and elegant in dealing with

symmetrical distributions but when they introduced a constraint that did not allow them

22

to make a perfectly symmetrical distribution (e. g. forbidding the use of the value $1.38),
all of them, except one, were stumped.

Mokros and Russell (1995) noted that some of the oldest group of children tried to
apply the property of the sum of the deviations about the mean is zero. The children
thought about this property informally using the idea of “balance” or that “a higher value
must be balanced by a lower one” (p.32). However, when they tried to apply this idea to
construct the data, none of them fully understood the way in which the data on either side
of the mean must balance. Instead, they used a phenomenon called by Mokros and
Russell as balancing totals, where students create data so that the sum of the data points
below the mean is equivalent to those above the mean. Mokros and Russell interpret this
misconception by saying “the procedure of balancing totals on either side of the average
is modeled on a notion of balance represented by a pan balance or by an equation:
Whatever is added to one side must be added to the other. Viewing the mean as the
fulcrum of a pan balance leads students to pay attention to the value of each data point
only, rather to the distance between the mean and the piece of data” (p.34).

Mokros and Russell also encountered students who relied almost exclusively on
the algorithm for finding the mean, and none of them used it effectively. “For them the
average means a series of steps involving addition and division; it is not a mathematical
object” (p.35).

The last study about students’ understanding of the mean reviewed in this section
is related to student visual representation of the distribution and comparison of data sets.
Comparison of data sets is the beginning of statistical inference and it is of particular

interest because in order to compare data sets students start by using the basic concepts of

23

spread, center and shape. Gal, Rothschild, and Wagner (1989) gave elementary school
children pairs of data sets represented by a line or dot plot with different conditions:
distributions in which means are very different, distributions in means are close or equal,
and distributions in which have different sizes. Children were asked to decide whether
one of the groups was “better” or the same. Their responses where matched with the
actual comparison of the mean of the groups.

Virtually all the children answered correctly the problems where the means of the
groups are very different, even the one that had an outlier. The children had more
difficulty when asked to compare distributions that overlap significantly, especially with

the distributions that had the same mean and mode but different range (see Figure 2.1’).

GroupA * GroupB *
*IIHII *
*IIHIUFIIIIII
*4"?
*IIUI‘

Figure 2.1. Distributions Presented to Children to Compare “Which Group is Better”

The results of Gal, Rothschild, and Wagner (1989) agree with those of Mokros
and Russell about young children focusing exclusively on the mode and deciding in favor
of the group that had the “tallest” column, but without consideration of the actual value of
the modal column. But the most difficult problems were the ones with different group

sizes, “only about one third of the 3th-graders, and two thirds of the 6th-grades, gave any

 

' Adapted from “Which Group is Better?: The Development of Statistical Reasoning in Elementary School
Children,” by 1. Gal, K. Rothschild, and D. Warner, 1989, paper presented at the meeting of the Society for
Research in Child Development, Kansas City, MO, p. 7.

24

indication that group-size information was taken into account in forming a decision” (p.
3).

Besides looking at the responses of the students, Gal and colleagues looked at the
strategies used. There were students who summarized the data by estimating the mean,
or looking where the “bulk of data” lay in each group. Others concentrated only on some
part of the data (e. g. the “tallest” column) or trying to “balance” high and low scores
within a group without using that knowledge to compare the groups. Watson and Moritz
(1999) extended this study to 88 students in grades 3 to 9 to explore their developmental
understanding. Their findings suggest that comparison of two equal size data sets with a
visual approach is appropriate for grade 3. For older students it is possible to use the
comparison of groups to develop higher order statistical thinking.

Less attention has been given to the other measures of location, the median and
the “1999...

computational errors in calculating the mode, the median and the mean:

0 Mode: Take the highest absolute frequency.

0 Median: Failing to order the data before calculating the median; take the
central value of the absolute frequencies ordered increasingly; take the mode
instead of the median.

0 Mean: Take the mean of the frequency values; not to take into account the
absolute frequency of each value when calculating the mean.

Cobo and Batanero (2000) also point out that the algorithm of the median is a

complex one, as it is not uniquely defined. It depends on whether the number of data

points is even or odd, and on whether the data is presented in grouped or ungrouped form.

25

Measures of spread
Goodchild (1988), in his attempt to explore deeply students’ understanding of

expectation, asked how much difference from 35 they would expect in one matchbox

N4-

4 fa

when the average across matchboxes is 35. The answers range from one or two to five
(i.e. 35 i 1, 2, 5). He asked them the same question for 100 boxes and the majority of
students thought it would be 10 or more (i.e. 350 i 10, 15). Goodchild says that “one of
the reason the pupils do not make allowances for the swarnping affect of a large number
of boxes may be because they see all the boxes having the same number of matches in
them, and ‘average contents’ being used only as a guide to that number, rather than as a
descriptor of the outcome as a stochastic process” (p. 79). Thus, to explore this
possibility further, be asked them to hypothesize a distribution for the content of 100
boxes. Students’ distributions actually showed that they see the contents as variable but
without any particular form. Goodchild mentions that older students were able to give a
symmetric bell-shaped distribution. What doesthis tell us about students’ understanding

of the variance? The answer is still pending.

College Students
Measures of center
Although college students can easily compute the mean of a group of numbers by
the algorithm "add-them-up-and-divide”, results from studies by Pollatsek, Lima, and
Well (1981) and Mevarech (1983) show that a large proportion of them do not understand
the concept of weighted mean when calculating the average of averages. Furthermore,

computing the simple mean was the only method they had available (not just the easiest

26

or most obvious) to attack the problem. One of the problems presented by Pollatsek and
his colleagues to students was the following:

There are ten people in an elevator, four women and six

men. The average weight of the women is 120 pounds,

and the average weight of the men is 180 pounds. What is

the average of the weights of the ten people in the

elevator?

Although only four students were interviewed for this particular problem, two of
them calculated the unweighted mean of 120 and 180 (150 pounds). Mavarech (1983)
extended the work of Pollatsek et a1. (1981) by investigating students’ misconceptions of
other properties of the mean. She tested 103 freshman and sophomore college students
majoring in education, all with at least one course in statistics completed. She found that
students mistakenly attribute group structure properties like associativity, identity and
closure to the operation of computing means. For example, only 40% of the students
recognized the following misconception that concerns the “identity element”:

A score of zero (0) was added to a set of 5 scores
(52, 68, 74, 86, and 90) with a mean equal to 74.
What is the mean of the new set?

Solution: The mean will not be changed because
adding zero to the sum does not change the sum.

As the college students studied by Pollatsek et al. (1981) these students thought it
possible to “average averages” (closure) by "add-them-up-and-divide” algorithm. These
results are especially troubling for teachers, because they occurred after the students had
an elementary course on descriptive statistics.

In summary, college students have difficulties solving weighted and simple

average problems and these problems can be attributed to the fact that students think of

the Operations of the average as binary operations satisfying the four laws of an additive

27

group. Furthermore, conceptual understanding did not seem to be modified by exposure
to formal instruction in statistics. These two studies were conducted with college
students and more research was needed, at that time, in identifying specific aspects of
conceptual knowledge such as properties and interpretations.

As mentioned before, the literature about the learning of measures of dispersion is
extremely scarce. Even the few studies that try to get at how students understand
variability are within other contexts and little can be learned from them.

Mevarech (1983); besideslooking at the conceptual, understanding of the mean in
terms of its non-additive group properties, looked at the concept of variance. She found
that all students could calculate and recognize the formula for variance but only a few
students possessed the conceptual knowledge to solve the problems. That is, students
thought that in order to calculate the overall variance of different groups they needed to
calculate the variance of the variances, or that adding zero to the data would not change
the value of the variance. Unfortunately, Mevarech does not provide examples of the
tasks used to measure these misconceptions nor does she provide much elaboration.
Measures of spread

Loosen, Lioen, and Lacante (1985) investigated the interpretation of the words

variatiofldispersion, and spread among 154 college freshmen who had not received any

- - .-M_.‘-.—
w... .— u”-

instruction in dispersion. They showed two different sets of blocks A and B. The length
of the blocks in set A were 10, 20, 30, 40, 50, and 60 cm. and the length of the blocks in
set B were 10, 10, 10, 60, 60, and 60 cm. When they asked the students which of the two
sets presented more variability, they got the following responses: 50% of them thought

that the set A was more variable, 36% that set B was more variable, and 14% that both

28

had the same variability. Loosen and colleagues interpret these responses as evidence

that the intuitive concept of variability is equivalent to “not similar”, that is, how do the
values vary with respect to each other, as opposed to varying with respect to a fix point.
In that sense, set A is in fact more variable, however the standard deviation is bigger for

set B. ix ~. ,_ ,4

. I
\. '1. "i

Pre-service and In-service Teachers ‘
l

M

Although limited, there exists some research on teachers’ knowledge of statistics“

and the need for more research has been expressed.“ Russell et al. (1990) point out that as

-- “which- Arno. i

researchers become more interested in how students learn basic statistical concepts, they
become curious about the teachers’ misconceptions and understandings. Furthermore,
investigating the conceptions and approaches teachers bring into the classroom,
investigators gain insight into what teachers are likely to view as important and the
complexities they face in translating their own knowledge about describing and
summarizing data.

One approach, described in Ball, Lubienski, and Mewbom (2001), focuses solely

on teachers and their knowledge of statistics. That is, researchers have interviewed in-

IMEHV-or

service and pre-service teachers about conceptual knowledge on, topics like measures of

iHrr-4

center and graph comprehension without placing them in the context of the classroom.

fl...

Russell, Goldsmith, Weinberg, and Mokros (1990), studied teachers’ knowledge

of the mean and dispersion by giving teachers the same tasks they gave the students (see

...m

summary of Mokros and Russell (1995) in the section of K-12 students of this chapter).
Eight teachers (2 elementary, 4 middle, and 2 mathematics coordinators for K-S) were

interviewed. For the Potato Chip Problem, the task was to put prices on nine bags so that

29

the “typical or usual or average” price would be $1.38. Five of the eight teachers used
approaches which involved “middle” or “midpoint” while seven of them used the
algorithm at some point (usually to check if the less algorithmic approach was “right”).
Different from the students, teachers used these two approaches in combination (Russell
et al., 1990).

Here is some of the language used by one of the teachers:

“Kind of like the mid-point of the middle or the average”. . .”I would say there had

to be a number of these above that figure and then also probably a comparable

number below the figure” (p. 4)

Teachers were successful at constructing distributions and they apparently
understood of the relationship between the data and the mean. However, when teachers
were confronted with larger data sets, which were more spread out and/or less
symmetrical, difficulties arouse. One teacher tried to choose a point of balance such as
pieces of data on one side of that point balanced by a piece of data symmetrically placed
on the other side of that point, ignoring pieces of data of value zero. Another teacher
who successfully had applied the strategy of balancing deviations in the Potato Chip
Problem tried to create a non-symmetrical distribution with large number of data points
using the same strategy used by some students, “balancing totals” (Makros & Russell,
1995). She explains her definition of balance:

Evelyn: Balance [means] that if you add something to this side of the

$1.50 you have an equal addition to the other, I mean, the equation

idea.

Interviewer: So, for instance, if you added, let’s say you added six
[tiles] here at $1.00, then how would you think about...

Evelyn: Then I’d have to put something over here [greater than the
mean given] that would equal $6.00.

30

Teachers’ solutions differed from students’ solutions in various ways and
dimensions. Teachers were more flexible in their use of two or more strategies in
combination and used the arithmetic mean almost exclusively as their definition of
average.

Berenson, Friel, and Bright (1993) assessed the understanding of thevje‘c‘enter or
middle of the data” and “typical” of 55 elementary teachers. The assessment instrument
consisted of a non-symmetrical line plot (number of raisins in a box) and a histogram
(length of cast in inches). Teachers were asked to a) determine the center or middle of
the data; b) determine what is typical; and c) predict assuming the data was a
representative sample. Their findings show that teachers tend to look at the center of the
range or the center of the horizontal scale as the center of the data. Most teachers (like
the students) picked the mode as “typical”, even though the distribution was skewed.
Few differences were found in the interpretation of these concepts between the two
representations, but the extra information in the histogram produced alternative
conceptions such as the average of the vertical scale and the size of the sample as typical.

In a comparative study among science and mathematics prf'SEIXIP-g teachers
Gfeller, Niess, and Lederman (1999) looked at the use of multiple represent‘ations—‘in

-WQ’PEII? EESILPIFPPICHIS; “Teachers were given an instrument with 10 items

(various contexts) and they were asked to solve each problem in two different ways. An

example of one item is shown in Figure 2.22.

 

2 From “Preservice Teachers’ Use of Multiple Representations In Solving Arithmetic Mean Problems,” by
M. Gfeller, M. Niess, and N. Lederman, 1999, School Science and Mathematics, 99, p. 256.

31

Below is a chart depicting a data set and its mean. The data point for week 3 is missing. Place a
dot on the graph that would correspond to week 3 so that the mean of all the data is 4 inches and
explain how you determined that.

ONQOO

Number 5

of “EMS 4 Mean = 4 inches

r-‘Nw

 

Week

Figure 2.2. Item Given to Science and Mathematics Pre-service Teachers.

The findings of this study are consistent with previous ones. The method most
often used equally by both groups was the computational algorithm, however
mathematics pre-service teachers balanced deviation more often than science teachers.

As part of the implementation of a curriculum, Friel and Bright (1998) designed a
project3 to help elementary teechers in their professional development in statistics
education. A variety of assessment instruments were developed, including a statistics
content survey and a pedagogical survey. Major findings in the content survey were that
despite instruction, “teachers continued to be quite confused about the median as a

EEEESPE of center” (p.106); also, teachers demonstrated some confusion about reading
data from a bar graph (looking for data values instead of frequencies on the y-axis). As
for the pedagogical survey, teachers’ perceptions about statistical concepts changed from
an “isolated content” prospective to a “process” perspective. For example, instead of

organizing the statistical concepts via simple but isolated activities such as graphing and

organizing data, teacher viewed the statistical concepts in terms of more integrated

32

 

2“

\‘(AI

r-g .

A
Lg

v
.I
I

I.

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processes such as interpret data. Also, teachers showed notions of data representation in

a variety of ways, extended discussions, follow-up questioning and question formulation.

Theoretical Frameworks
A variety of theoretical frameworks have been developed by researchers in the
field of learning, teaching, and assessing statistics. Some researchers have proposed the
theory as a result of their experience in the field (e. g. Garfield, 2002; Moore, 1997;
Shaughnessy, 1992) and others have accompanied these theories with empirical results
(e.g. Friel, Bright, Frierson, & Kader, 1997). Similarly, in the field of teachers’
knowledge researchers have developed and tested different frameworks for understanding
teachers’ knowledge (e. g. Lappan, 2000; F ennema & Franke, 1992; Shulman, 1986) and
frameworks to measure that knowledge (e.g. Ball, Lubienski, & Mewbom, 2001). In this
section, these empirical results and frameworks are summarized.
Learning Statistics
In this subsection two frameworks of basic statistical concept development are
described as guides to organize the research on learning statistics, as well as determine
the knowledge of students’ cognition about these concepts teachers must have.
Shaughnessy (1992) suggests a model characterizing stochastic conceptions based on
research results and his own practice. He identifies four types of conceptions: Non-
statistical, Naive-statistical, Emergent-statistical, and Pragmatic-statistical. These four

types of conceptions can be explained in terms of students’ understanding of the mean

and its relation to variability.

 

 

3 Gideon, Joan, Ed. (1997). Professional Development Manual Teach-Stat for Teachers Statistics: A Key to
Better Mathematics. Palo Alto, CA: Dale Seymour Publications.

33

 

 

Non-statistical. When people do not operate in a statistical setting and use
the mean as a representative of data with no variation. For example, the
misconceptions that the mean must be one of data values or that the mean is
represented by the mode.

Naive-statistical. When people understand that the mean represents data that
varies and it is the balancing point, however they do not understand how the
balancing occurs. For example, the misconception of “balance totals” or that
the mean is represented only by the median.

Emergent-statistical. Interpret the mean as a mathematical balance with
small and symmetrical data only. For example, misconceptions about taking
average of averages for unequal groups and difficulties with data with large
variation.

Pragmatic-statistical. In-depth understanding of the mean and its relation to

variability in any context.

These type of conceptions are not necessary linear or mutually exclusive

(Shaughnessy, 1992). A person does not need necessarily to be first a “naive statistician”

in order to become an “emergent statistician”. Besides, people can operate in several of

these conceptions depending on the settings and the nature of the task.

If we agree with Shaughnessy that most students who take beginning courses

in statistics are usually in one of the first two categories, non-statistical or naive-

statistical, the process of teaching and learning should aim for the last two categories.

However, he points out that “the latter two conceptual stages will not occur without

34

carefully guided learning experiences under the tutelage of a well-trained teacher who is
mathematically and statistically competent as well as sensitive to the types of beliefs and
misconceptions that students have about stochastics” (p. 486).

To better understand students’ conceptions and misconceptions, the Godino and
Batanero (1994) framework on understanding a mathematical object can also be applied
to measures of center.

0 Common errors on procedural skills: e. g. not ordering the data before

computing the median

o Misconceptions due to notations, displays or words used to represent the

concept: confusing the sample mean and population mean

0 Difficulties in understanding and justifying the properties: what is the effect

on the mean and median of adding a large number to the data set.

0 Difficulties using the concept in relation to others: determining the center or

spread given a histogram or box plot.

Teaching Statistics
The teaching reform movement in mathematics and science has influenced the
teaching of statistics and probability. However, Moore and Cobb (2000) point out the
major differences between mathematics and statistics, stating that the practice of statistics
is not strictly mathematical but rather is characterized by a dialogue between data and
models. They stress the importance of real data and less mathematical and probability

theory, as well as interpretation and communication. “Statistics combines computational

35

activity in a meaningful setting with the exercise of judgment in choosing methods and
interpreting results” (Moore, 1990, p.96)
David S. Moore (1997) presents the following summary of research-based reformed
diagnosis and prescription.
0 Goals: Higher-order thinking, problem solving, flexible skills applicable to
unfamiliar settings.
0 The old model: Students learn by absorbing information; a good teacher
transfers information clearly and at the right rate.
0 The new model: Students learn through their own activities; a good teacher

encourages and guides their learning.

0 What helps learning: Group work in and out the classroom; explaining and
communicating; frequent rapid feedback; work on problem formulation and
open-ended problems.

Moore (1997) elaborates on the first and second point saying that instruction
should concentrate on interpretation of graphics, strategies for effective exploration of
data, basic diagnostics as preliminaries to inference. Technology is also an important part
of helping students learn. See Shaughnessy (1996) for a review of research on computer
._ software for teaching statistics. Many agree that the appropriate use of technology ’
empowers students to do data analysis that is interactive and exploratory, using
visualization and simulations to understand statistical concepts.

Although the above claims are supported mostly by research in probability and

randomness (see Shaughnessy (19,92) for literature review) there are some studies thaf

. “L '

focus on the—teaching of basic statistical concepts. For example, Mevarech (1983)

36

studied the effects of exposing college students to the instruction of Mastery Learning
Strategies (MLS)4 in overcoming statistical misconceptions (in particular the
misconceptions about treating the set of means as an additive group). Instruction for the
experimental group (N=75) differed from the instruction of the control group (lecture-
discussion strategy, N=64) only with respect to the provision of feedback and the
implementation of corrective activities.

Results showed that students in the experimental group achieved significantly
higher scores than the control group and that 70 percent of the students in the
experimental group demonstrated mastery (grade of B or higher) compared with only 40
percent in the control group. These results do not necessarily show that students
overcome their misconceptions, but Mevarech explains that the posttest to measure
achievement required more than computational knowledge, “students working with these
formative tests have to understand the concepts, apply the rules and evaluate solutions”
(p. 426).

In another more qualitative study, George (1995) compared the nature and extent

of the procedural and conceptual understanding of the mean developed by two groups of
students exposed to different forms of instruction. Six seventh-grade students
participated in the study; two of them were from a school which employed the Visual
Mathematics curriculum5 and the remaining four received instruction on the numerical
“add-them-up and divide” algorithm. After giving each student five different tasks

George concluded that the students that learn the numerical algorithm were confident and

 

’ Mastery Learning Strategies is defined by Bloom (1976) to consist of a) unit mastery requirements, b)
provision of feedback, and c) corrective activities.

In this curriculum, students learn the visual “leveling-off” method for finding the average of a given set of
numbers.

37

successful in finding the mean when a complete set of data was given, regardless of size
of the set and number sizes. By comparison, students from the Visual Mathematics
“revealed greater flexibility in moving back and forth between the numbers in the set and
the mean” (p. 7).

Bright and F riel (1998) also did a study on the impact of an instructional unit
’ [developed specifically to highlight the connections between pairs of graphs (e. g. bar
graphs for ungrouped and group data, line plot and bar graph, stern plots and histograms).
They tested and interviewed students in grades 6, 7, and 8 before and after the unit and
found that students don’t necessarily make translations between representations easily
and quickly and that students should have the opportunity to compare multiple
representations of the same data set to encourage recognition of the similarities and
differences in what is communicated by each representation. Although this study did not
focus on the concept of mean or variance but on the overall understanding of graphical
representation, its results show how different representations of data influence how
students interpret properties of the data.

Statistical ideas have their own substance and modes of reasoning, hence statistics
has its own pedagogy. What do teachers need to know about statistical pedagogy to help
students learn? Moore (1997) suggests a “synergy” among content-pedagogy-
technology. “Content and pedagogy — our understanding of what students should learn
and of effective ways to help them learn- should drive our instruction. Technology
should serve content and pedagogy. Yet technology has change content and allows new

forms of effective pedagogy. The most effective teachers will have a substantial

knowledge of pedagogy and technology, as well as comprehensive knowledge about and

38

 

experience applying the content they present” (p.134). Moore suggests the following
framework called “Synergy in Statistical Education”:
0 Content <:>Pedagogy
Data analysis <:> Hands-on work
Statistics in practice <:> Communicate, cooperate
Concepts 42> More explanation, less proof
0 Pedagogy <:>Technology
Visualization (Multiple representations)¢:> Automate graphics
Problem solving: Automate calculations
Active learningc> Multimedia
0 Technology <:>Content
Computingc> Data analysis, diagnostics, bootstrap,. ..
Automation <:> More and larger concepts
Simulation <2> Alternative to proofs
Moore (1997) expands on each of the components but of special interest in the
pedagogy of statistics is the use of data visualization and technology. Tufte (1983) as
cited in Shaughnessy (1997) suggests that excellence in statistical graphics consists of
complex ideas communicated with clarity, precision, and efficiency. “Students can see
an enormous variety of data in the media, which vary in appropriateness and which
sometimes are intended to deceive. . . Such distortions can provide points of departure for
data-handling topics” (p. 218). Computer software has long been available for statistical
analysis, however, its role in teaching and learning is still evolving. As in teaching

mathematics, as the capabilities of technology increase it is important to consider the

39

most appropriate use of technology in facilitating students’ learning of statistics and
consider potential disadvantages of using computing in statistics courses (Rubin, 1991).
Assessing Statistics

Assessment tools and frameworks in statistics, as in many fields, have a variety of
purposes. Some measure student performance with the goal of assigning a grade, others
are designed to measure students’ knowledge and understanding with the goal of learning
their conceptions and misconceptions to improve instruction. Whatever the purpose is,
the assessment tool or instrument should focus on providing sufficient information about
student learning and should be aligned with curricular goals (Garfield, 2003).

At the K- 8 level, Friel, Bright, Frierson and Kader (1997) suggest a framework
for assessing knowledge and learning in statistics, in particular, graphical representations.
The model consists of designing tasks or items that are 1) meaningful to students at the
proper age; 2) ask different kinds of questions focusing on the process of data reduction;
and/or 3) ask different kinds of questions to reflect the different levels of graph
comprehension (read the data, read between the data, and read beyond the data). Friel et
al., (1997) conclude that their work in the area of data representation “has provided a
model of one way to build both understanding based on consideration of the development
of graph knowledge and of strategies of assessment that may be used to support and
evaluate this development” (p. 62).

b - g
_ ‘4 SJ T
. l). I

At the college level, Garfield (2003) and Lui (1998) provide a valid and reliable /i ire ,9;

/

___.

instrument to measure the construct of statistical reasoning. The instrument consist of 20

__,,-
—

- multiple-choice questions carefully designed to identify correct reasoning skills and

misconceptions in statistics and probability concepts. Items for the assessment were

40

based on the nature of reasoning skills in statistics and the identification of
misconceptions found in research of student learning.

Another framework for instruction and assessment that builds on the work of
Garfield’s is the one suggested by delMas (2002). He suggests considering three
instructional domains, basic literacy, reasoning, and thinking. Basic literacy refers to
identification or recognition, computation, construction of graphs; reasoning refers to
explaining why or how results were produced or why a conclusion is justified; and
thinking refers to the application of students’ understanding to real world problems, to

critique, evaluate, and generalize.

Teachers’ Knowledge

Scholars during past years have proposed several frameworks and models of
teachers’ knowledge. They all provide insight into defining teacher knowledge and new
ways of thinking about it. The approaches are not inconsistent but rather they build upon
each other. In order to identify useful conceptions of teacher knowledge taken into
account by the various aspects of the models proposed, a review of the consistencies,
inconsistencies and relationships follows.

Shulman (1986) proposed a framework that analyzes teachers’ knowledge by
considering different categories. These categories are: subject-matter knowledge,
pedagogical content knowledge, and curricular knowledge. Subject-matter knowledge is
the “amount and organization of the knowledge per se in the mind of the teacher” (p. 9).
Pedagogical content knowledge includes “the most useful forms of representation of

those ideas, the most powerful analogies, illustrations, examples, explanations, and

41

demonstrations — in a word, the ways of representing and formulating the subject that
make it comprehensive to others.” He also includes “an understanding of what makes the
learning of specific topics easy or difficult, the conceptions and preconceptions that
students of different ages and backgrounds bring with them to the learning of those most
frequently taught topics and lessons” (p. 9). Curricular knowledge is the “set of
characteristics that serve as both the indications and contraindications for the use of
particular curriculum or program materials in particular circumstances” (p. 10).

Peterson (1988) builds on Shulman’s framework arguing that it is not only
important for teachers to know how students think and how to facilitate growth in student
thinking but also self-awareness of her or his own thinking in mathematics. She claims
that knowledge of content will be useless in structuring the classroom so that students can
learn unless teachers understand their own cognitive processes.

Lloyd and Wilson (1998) also extend Shulman’s types of knowledge by defining
content conceptions “to encompass the range of conceptions that a teacher might hold
about a particular topic or concept that is taught” (p. 250). The word “conceptions” for
Lloyd and Wilson refer not only to knowledge but also to beliefs, understandings,
preferences, and views of a particular topic. They also assume that teachers’ instructional
decisions and actions are closely linked to their conceptions of mathematics, teaching and
the cognition of their students. Shulman, Peterson; and Floyd and Wilson describe many
of the same components of teacher knowledge, emphasizing the importance of the
content, its organization, and how it should be studied. In addition, Shulman and
Peterson make a strong case for the importance of considering knowledge of the leamer’s

cognition about subject matter.

42

In an attempt to built a cognitive model of teacher knowledge, Leinhardt, Greeno,
Putrnan, Stein and Baxter (cited in Fennema & Franke, 1992) propose that the skill of
teaching is determined by at least two fimdamental and related systems of knowledge:
subject matter (content knowledge) and lesson structure (practical knowledge). For
Leinhardt and her colleagues, knowledge of lesson structure consists of organizing the
lesson with an overall goal, some procedures or activities to achieve the goal, and some
routines that enable the class to function smoothly. Knowledge of the subject matter
includes not only mathematical knowledge but knowledge of curricular activities,
methods of presentation, and assessment procedures. Leinhardt et a1. tested the model on
expert and novice teachers and found that the expert teacher’s knowledge tended to be
more organized, used richer systems of representations and presented more detailed
conceptual and procedural knowledge. The only critique of the model, according to
Fennema and F ranke (1992), is that it lacks attention to individuals and the actual
mathematics they are being asked to learn.

Another framework that can be used to understand teacher learning is that of
situated knowledge, “the mathematics that teachers learn must be learned in a context that
is much broader than traditional in-school learning so that the teachers’ knowledge is
more similar to what we have called the nature of mathematics” (p. 160). Ball and Cohen
(1999) agree that teachers’ knowledge, in the way it has been defined, “could be used
only in complex interactions in the unpredictable situations that we call classrooms” (p.
10). Recommendations of learning experiences in context for pre- and in service teachers
are reviewed by Putnam and Borko (2000). They include the used of integrated

multimedia environments, written cases, student work and videotapes of lessons.

43

 

'

f.

In an attempt to include all the important components of teachers’ knowledge
pointed out by the above scholars, Fennema and F ranke (1992) propose a new model for
its examination and discussion, centered on teachers’ knowledge as it occurs in the
context of the classroom (see Figure 2.36). The model includes knowledge of the content
of mathematics, knowledge of pedagogy, knowledge of students’ cognition, and teachers’

beliefs. All of these components agree with the ones previously analyzed, except the

knowledge of mathematics.

 

 

Pedagogical

Knowledge of
‘__' Context H Knowledge

Mathematics

 

 

 

 

 

 

 

 

 

Knowledge of
learners’
cognition

 

 

 

Figure 2.3. Teachers’ Knowledge Developing in Context.

For F ennema and Franke, knowledge of mathematics includes knowledge of
concepts, procedures within the domain in which they teach, “it also includes knowledge
of the concepts underlying the procedures, the interrelatedness of these concepts, and
how these concepts and procedures are used in various types of problem solving” (p.
162).

What differentiates this model from previous ones is the center triangle indicating
the teachers’ knowledge and beliefs in context or as situated. “Within a given context,

teachers’ knowledge of content interacts with knowledge of pedagogy and students’

 

6 From “Teachers’ Knowledge and its Impact,” by E. Fennema and M. Franke, 1992, Handbook of
Research on Mathematics Teaching and Learning, p. 162.

44

cognition and combines with beliefs to create a unique set of knowledge that drives
classroom behavior.” (p. 162).

Although, Fennema and F ranke make it clear that the study of the components of
teacher knowledge out of context or in isolation will not reflect the dynamic nature of
teacher knowledge, part of their model have been used by some scholars (Swafford,
Jones, Thornton, Stump, & Miller, 1999) in isolation. Swafford and her colleagues
designed a professional development program that included content courses, seminars on
pedagogical practice and research seminars on student cognition as separate components.

The study of teachers’ knowledge components in isolation from each other and in
teacher education programs has caused researchers to rethink the conceptualization of
such knowledge. Lappan (2000) views the components (or domains) of teacher
knowledge in a nonlinear fashion. She proposes the interaction of knowledge domains
illustrated by the Venn diagram in Figure 2.47, and claims that “effective teaching occurs

in the intersection of these domains of knowledge” (p. 321).

Content for

Teaching Pedagogy

   

Learning Evaluation

Figure 2.4. Interaction of Knowledge Domains.

 

7 From “A Vision of Learning to Teach for the 21St Century,” by Glenda Lappan, 2000, School Science and
Mathematics, 100, p. 321.

45

The domain Content for Teaching in this model includes not just the content of
mathematics but knowledge of the mathematics taught in preK-12 programs and how the
ideas at different level relate. The Pedagogy domain includes “compelling examples of
tasks that can be used to elicit particular kinds of thinking and investigation on the part of
the learner” (p.322). Knowledge of student development as it is related to learning and
teaching mathematics is included in the Learning domain. It also includes knowledge of
and experience in listening to what students can make sense of and where they have
problems. Although not in the diagram, Lappan (2000) also mentions as an important
domain, a vertical vision of curriculum that includes the lesson, unit, school year and pre-
K-12 level. Finally, Lappan adds one more component to the teacher knowledge that is
the Evaluation domain, which includes assessment of students’ knowledge and evaluation
of programs.

As the conceptualization of teacher knowledge becomes more complex, the
challenge is how to blend all the components or domains to prepare teachers to help all
students learn. Ball (2000) proposes three problems that need to be solved in order to
bridge the gap between content and pedagogy: 1) How to determine which content
matters for teaching; 2) How subject matter must be understood to be usable in teaching;
and 3) How to create opportunities to learn to use subject matter knowledge in the
context of practice.

For problem number one, Ball (2000) suggests that scholars “ identify core
activities of teaching, such as figuring out what students know; choosing and managing
representation of ideas; appraising, selecting, and modifying textbooks; and deciding

among alternative courses of action, and analyze the subject matter knowledge and

46

insight entailed in these activities” (p.244). The second problem on how to use this
knowledge is the capacity to deconstruct or unpack one’s own knowledge into a less
polished and final form, which is related to Shulman’s definition of pedagogical content
knowledge. The third problem is related to where and how to use this knowledge, that is
in the context of practice. Ball suggests that scholars “design and explore opportunities
to learn content that are situated in the contexts in which subject matter is used, a core
activity of teaching.” (p. 246). Outlining these problems, Ball mentions some of the same
components or domains that other scholars do, however, she looks at it in at least two
different ways. One, is that knowledge for teaching is more rooted in content knowledge;
and two, she provides practical solutions for teaching and learning such knowledge which
helps better understand its conceptualization.
Measuring Content Knowledge for Teaching

It is one thing to define a construct and another to identify it or to measure it.
Ball, Lubienski, and Mewbom (2001) give a review of the literature on measuring
teacher’s knowledge using a historical prospective on teachers’ mathematical knowledge.
The authors explain that the identification of the mathematical knowledge is related to the
way it has been measured. For example, the research that focuses on the assumption that
knowledge of mathematics is essential for teaching; measures it by counting the amount
of mathematics courses or workshops taken, the degree or certification earned. The
research that focuses on the nature of teachers’ knowledge assumes that content
knowledge is intertwined with other aspects such as how students learn, and what they
find difficult. This approach is basically trying to measure, in part, the construct of

pedagogical content knowledge by interviewing teachers - or prospective teachers — to

47

probe their knowledge about a specific mathematical topic. Ball, Lubienski, and
Mewbom (2001) describe this method:

Researchers working in this approach often use methods that probe teachers’

knowledge and that situated the questions in an around questions that might arise

in teaching. Such questions respond to a student’s confusion, explain a

mathematical procedure whose meaning is buried inside rules of thumb, or

consider the connections among ideas. Those questions create conditions where
teachers have to make explicit their understanding of the mathematical ideas and

procedures behind the questions. (p. 444)

Although this approach helps understand the nature of teachers’ knowledge it
does not produce a quantitative measure and thus it can not be associated to any other
factor that may influence student learning. Hence, this approach to measuring teachers’
knowledge is more attractive to researchers than to policymakers. Another limitation on
this approach, because of its qualitative nature, is that researchers can only focus on a
small number of teachers and/or on a specific topic. Most of the studies of this type have
been conducted with preservice elementary teachers in the areas of numbers and
operations, functions, geometry and measurement, and proof (for detailed review on these
areas see Ball, Lubienski, and Mewbom (2001)).

Stylianides and Ball (2004) suggest an integrative framework for studying the
mathematical knowledge for teaching which accounts for different data sources and sites
of inquiry. The authors propose six different approaches: analyzing experts’
perspectives, teacher’s mathematics curricula, teachers’ mathematical knowledge,

students’ mathematics curricula, students’ mathematical knowledge, and school

mathematics practice (see Figure 2.58). The organization of these approaches is a

 

8 From “Studying the Mathematical Knowledge Needed for Teaching: The Case of Teachers’ Knowledge
of Reasoning and Proof”, paper presented at the 2004 Annual Meeting of the American Educational
Research Association, San Diego, CA.

48

network of dynamic relationships that bring together practice, policy, and research. The

way Stylianides and Ball envision this system is that the first five approaches are in

coordination and alignment and that the sixth approach, school mathematical practice,

would confirm the utility of these contributions. The study of knowledge for teaching

could also start from examining school practice and see how it plays out in the actual

work of teaching. The dotted segments in Figure 2.5 suggest the existence of relations

among these elements and the dotted arrows suggest that these elements are influenced

by outside factors such as cognitive psychology, the structure of mathematical discipline,

learning and pedagogical principles among others.

 

 

 

Analyzing experts’
perspective

 

 

 

What mathematical knowledge is
needed for teaching?

 

 

 

 

 

School
mathematics
practice

Analyzing teachers’
mathematical
curricula

 

 

How should teachers’
mathematics curricula develop?

A

 

A

 

 

 

 

 

 

Mathematical
knowledge
for teaching

 

Analyzing teachers’
mathematical
knowledge

 

Which aspects of teachers’
mathematical knowledge are
worth analyzing?

 

 

 

 

 

 

 

 

 

 

9

Analyzing students
mathematical
curricula

 

How should students’
mathematics curricula be
developed?

A

 

 

 

 

 

 

Analyzing students’
mathematical
knowledge

 

 

 

 

 

Which aspects of students’
mathematical knowledge are
worth analyzing?

A

 

 

 

 

< .............

Figure 2.5. Framework for Investigating Mathematical Knowledge for Teaching

Studying the development of teachers’ knowledge by looking at the different

frameworks suggested by researchers, it is clear that it has become more complex as

more is learned about the field. Investigating statistical knowledge for teaching taking

into account all of the components suggested by the latest suggested framework are
beyond the scope of this dissertation. However, the first research questions mentioned in
chapter 1 combines two of the components of this framework, what statistical knowledge
is needed for teaching? And which aspects of teachers’ statistical knowledge are worth
analyzing? The starting point for this study will be analyzing experts’ perspectives, and
analyzing mathematical curriculum for students and teachers. From here, the component
of analyzing teachers’ statistical knowledge is investigated to see what can be learned
about prospective teachers’ knowledge for teaching data analysis and statistics.
Summary

In order to identify what knowledge is necessary to teach statistics and assess
whether the teachers, in fact, possess this required knowledge, important components
discussed in this chapter; content, learning, and teaching, and their interaction in
particular contexts must be taken into account. Although some studies of teacher’s Ti i
knowledge in statistics exist, at present no studies on teachers of statistics that consideri
the interaction of the different components have been published. «J.

All of the studies mentioned above focus their investigation on teachers’
understanding of measures of center. As in the students’ case, not much has been
investigated about teachers’ knowledge of measures of dispersion or other components of
the process of statistical investigation, although some conclusions could be drawn from
teachers’ understanding of distributions. Furthermore, these studies have concentrated on
the conceptual knowledge and understanding of basic statistical concepts or on how to

teach these concepts to students, i.e. on content or pedagogy separately. No research

appears to exist on teachers’ understanding of student cognition. Are teachers, when

50

placed in a given context, able to combine their knowledge of statistics, pedagogy, and
student cognition to create an effect learning environment? In other words do they posses
the statistical knowledge for teaching? In order to investigate teacher content knowledge
about a particular topic and to see its effects in instructional practices and students’

achievement, it is necessary to develop a reliable and valid way to measure it.

51

CHAPTER 3

ASPECTS OF KNOWLEDGE FOR TEACHING DATA ANALYSIS AND

STATISTICS

This chapter addresses the first research question — what are the important aspects
of content knowledge for teaching data analysis and statistics at the middle school level?
More specifically, it seeks to identify the important aspects of content (i.e. “the big
ideas”) at the student and teacher level and to identify the important aspects of knowledge
for teaching.

Several approaches are taken to investigate this question. First, the analysis of
policy documents, (e. g. national and state standards, books, and reports) is a reflection of
experts’ perspectives giving insight on what it is that they value or consider important.
Second, the analysis of students’ mathematical curricula (e. g. mathematics textbooks and
teacher’s guide) is a reflection of what teachers are supposed to teach with an intended
curriculum. This second approach identifies what statistical knowledge teachers need to

have in order to implement these curricula in their classrooms.

Specifically, the data sources for statistical content knowledge at the student level
are content standards at the middle grades level from ten states, Principles and Standards

for School Mathematics (N CTM, 2000), and the Mathematical and Problem-Solving

Goals of the units on data analysis from the Grade 6 and 8 textbooks produced by the

52

Connected Mathematics Project (Lappan, Fey, Fitzgerald, Friel, and Phillips, 2002)
Grade 6 and 8 textbooks.

The Connected Mathematic Project (CMP) textbooks constitute one of the recent
middle grades curricula supported by the National Science Foundation. A set of two
units on Data Analysis and Probability from the Connected Mathematics Teacher’s Guide
(Lappan et a1, 2002) where examined. The analysis was conducted on a list of
mathematical and problem-solving goals published in each unit on the preliminary pages
(Data About Us, page 1a; and Samples and Populations, page 11) and the “big ideas”
published in the introduction of the textbooks. Each goal served as the unit of analysis
and the “big ideas” served as a reference to determine the intent of the goals. Although
the textbooks provide many other sources for analysis such as the “summary of the
investigations” and “mathematical highlights”; the lists of mathematical goals provided
the most closed form of the intent of the content for the textbooks and facilitated the

process of coding.

The ten states included in the analysis are the states that require middle school
certification for teachers; they are Connecticut, Florida, Georgia, Kentucky, Missouri,
North Carolina, Ohio, Oregon, Virginia, and West Virginia

(http://www.enc.org/professional/standards/state/).

For statistical content at the teacher level the data sources are: the topics covered
in the PRAXIS 11 Middle School Mathematics (ftp://ftp.ets.og[pub/tandl/0069.pdf ), and
The Mathematical Education of Teachers (CBMS, 2001).

For content knowledge applied to teaching, the data sources are Knowing and

Learning Mathematics for Teaching (NRC, 2000), Adding It Up (NRC, 2001a), Middle

53

l
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/
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.- I;

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childhood through early adolescence/Mathematics Standards (National Board for
Professional Teaching Standards, 1998 or http://www.nbpts.org/pdf/mceaimath.pdf ),
Professional Standards for the Accreditation of Schools, Colleges, and Departments of
Education (National Council for Accreditation of Teacher Education, 2002), National
Middle School Association Middle Level Teacher Preparation Standards
(http://wwwnmsaorg ), professional standards from four states, Connected Mathematics
Teacher’s Guide (Lappan, Fey, Fitzgerald, Friel, & Phillips, 2002), and The Mathematical
Education of Teachers (CBMS, 2001).

The first section of the chapter describes and summarizes the data analysis and
statistical knowledge students are expected to know. The second section describes and
summarizes the statistical knowledge teachers are expected to know. The third section
describes and summarizes the content knowledge applied to teaching evident in teaching

documents and suggested by recent research.

Data Analysis and Statistics for Students in Middle Grades

National and state standards vary in form and content. In order to identify
common content and the level of emphasis suggested by national and state standards in a

systematic way, an instrument that identifies common language was needed.

,.//-

Although there are several frameworks that organize content, including Bloom’s
taxonomy (Bloom, B., Englehart, Furst, Hill, & Krathwohl, 1956) and TIMSS assessment «if?
framework (V alverde et a1, 2002), they are not suitable for this research. Bloom’s
taxonomy is not specific to mathematics, and the TIMSS assessment framework is not
designed to compare or find commonality among different documents. The framework

that better suits the purpose here is the content matrix developed by Porter, Kirst, Osthoff,

54

Smithson, and Schneider (1993). This instrument was adapted to meet the needs of this
study.
Background on Measuring Content

The conceptual development of the content matrix started when Porter and others L
(1993) needed to measure what was taught in high school mathematics and science
classes in six states. In this study, Reform Up Close, descriptors of high school
mathematics and science were organized into three dimensions: t0pic coverage, cognitive ‘
demand, and mode of presentation. Topic coverage consisted of mathematics topics such
as ratio, volume, and relation between operations. Cognitive demand included nine
descriptors including memorize, perform routine and non-routine procedures, and
conjecture. Among the seven modes of presentation were exposition, pictorial models,
and equations/formulas. A content topic was defined as the intersection of all three
dimensions. Porter refers to this “language” as a rich and systematic way to describe
instructional content. Teachers or observers were trained to use codes for the descriptors
for daily logs and observation protocols.

Although this conceptualization gave Porter and others a language and coding
scheme to compare teachers’ reports and observation reports of a given lesson, it had its
limitations. The survey instrument could not be constructed in a simple way to include all
three dimensions and not to impose a burden to the teacher. So, Porter and Smithson
(2001) proposed a simpler conceptualization using only two dimensions — content
category and cognitive demand — displayed in a matrix format. Furthermore, they
reduced the number of categories of cognitive demand from nine to five.

Deciding which topics, cognitive demand, and labels to use is key to the internal

validity of the instrument. Porter and Smithson (2001) claim that the language used in

55

their study is reform-neutral, with the hope that “the language should be translatable into
reform language distinctions so comparison to state and other standards is possible” (p.
8).

The Topic coverage dimension was organized to establish a comprehensive list.
For mathematics, they suggested a list of topics for elementary, middle, and high school
organized by area such as numbers, operations, measurement, algebra, and data analysis
and statistics. For middle school data analysis and statistics they suggest the following
list of topics:

Bar graph, histogram

Pie charts, circle graphs

Pictographs

Line graphs

Stern and leaf plots

Scatter plots

Box plots

Mean, median, mode

Line of best fit

Quartiles, percentiles

Porter and Smithson are aware that this is not the only way to organize content,
especially if the instrument is used for other purposes. They suggest an alternative to
organize content using instead “big ideas” such as change or shape as the framework,
which is the one desired for this study. However, they claim, that in practice, content is
still organized along the traditional lines of algebra, geometry, statistics, etc. not in terms
of big ideas and that this approach can be taken when practice is reformed to better reflect
these big ideas.

The cognitive demand dimension was defined as the cognitive activities that

engage students in the content topics; therefore they are behaviorally defined. The five

cognitive demands used by Porter and Smithson (2001), and Porter (2002) are: memorize

56

facts, definitions, formulas; perform procedures/solve routine problems; communicate
understanding of concepts; solve nonroutine problems/make connections; and conjecture,
generalize, prove. For memorize facts, definitions, formulas, classroom activities focus
on recalling traditional math skills and knowledge. For example, recall the formula for
the mean. For perform procedures/solve routine problems, classroom activities focus on
demonstrating basic skills, selecting and applying various computational methods. In
statistics, this is interpreted as creating graphs, computing measures of center or spread,
and collecting data for analysis. For communicate understanding of concepts, classroom
activities focus on students sharing their mathematical understanding in both oral and
written form. In statistics these activities include choosing the appropriate graph and
measure of center and spread to answer questions about the data, as well as matching
verbal descriptions of the data to distributions and their graphs. For solve nonroutine
problems/make connections classroom activities focus on students applying mathematical
knowledge to solve unfamiliar problems or seeing relationships between topics within
mathematics and to other content areas. Examples of nonroutine problems include
analyzing a complicated data set, recognizing patterns in data, and using statistics to
explore real world problems. Finally, for conjecture, generalize, prove students focus on
making and justifying conjectures. In the context of statistics at the middle school level,
this translates mainly to informal inference and prediction from data about real world
problems. Table 3.1 gives the language associated with each category.

The content of instruction is then described at the intersection between topics and
cognitive demand, based on data gathered from teacher surveys. Porter asked the
teachers to indicate, for the past school year (a) the amount of time devoted to each topic

and then, for each topic, (b) the relative emphasis given to each student expectation.

57

Table 3.1

Language Frequently Associated with Cognitive Demands

 

 

 

A B C D E
Memorize facts, Perform Communicate Solve nonroutine Conjecture,
definitions, procedures/solve understanding of problems/make generalize, prove
formulas routine problems concgts connections
0 Recognize - Do 0 Communicate 0 Apply and 0 Complete
0 Identify computations mathematical adapt a variety proofs
0 Recall 0 Make ideas of appropriate 0 Make and
o Recite observations 0 Use strategies to investigate
. Name 0 Take representations solve mathematical
. Tell measurements to model nonroutine conjectures

0 Compare mathematical problems a lnfer from
Develop ideas ' APPI)’ data and
fluency 0 Explain mathematics in predict

findings and context OUtSidC 0 Determine the
results from of mathematics truth of a
statistical o Analyze data, mathematical
analyses recognize pattern or

0 Explain patterns proposition
reasoning 0 Explore

0 Describe 0 Judge

0 Select

 

Note. From “Measuring the Content of Instruction: Uses in Research and Practice,” by Andrew C. Porter,
2002, Educational Researcher, 31, p. 13.

Identifidng Data Analysis and Statistics Content

Instrument

In his more recent publication Porter (2002), suggests that the content matrix can
be used to go beyond describing content instruction in the classroom. He gives examples
of the use of the instrument to measure alignment of assessments with content standards,
alignment of instruction with assessments, and as a tool to create powerful graphical
displays to describe the content emphasized (and not emphasized) in state standards. It is
for the latter that the instrument will be put in this study.

Since the only area of interest for this research is the content in Data Analysis and

Statistics in Middle grades, a modified context matrix was created, based on the list

58

t
r/"ivn'
A

suggested by Porter and Smithson (2001) for Data Analysis/Statistics topics in Middle
School presented above. Porter and Smithson’s list of topics is the product of the Reform
Up Close study, a Consortium for Policy Research in Education project and follow up
studies. Since the articulation of their latest approach is still in the beginning stages of
development, a modified version of Porter’s content matrix toward the organization of
topics by “big ideas” was used. For example, Porter and Smithson (2001) list “bar graph,
histogram” as a topic. For the modified matrix, a re-organization of the graphs and plots
was made to reflect the big ideas in data analysis of graphical representation of
categorical and numerical data. Graphs and plots are organized according to the type of
data they describe. For example, graphs and plots that are used to organize categorical
data are grouped together. Similarly, the graphs and plots to summarize numerical data
and bivariate data appear as separate topics. Porter & Smithson (2001) suggest measures
of center but leave out measures of spread such as range and interquartile range.
Therefore a new topic was added to take spread into account. The big idea of describing
the shape of distributions was also added to account for other features that characterize
distributions of data such as skewness, outliers, clusters, and gaps. Finally, since some of
the standards encountered address the process and design of statistical investigation, a
separate topic focusing on the formulation of questions, collection of data and design of
studies was added. Following is the modified content matrix used in this research.

As for the list of categories of the cognitive demand dimension, the ones
suggested by Porter (2002), with the proper interpretation, can be seen as a more refined
version of a framework suggested by statistical educators (Garfield, 2002 and delMas,
2002). Garfield and delMas suggest three categories of learning outcomes: Statistical

Literacy, Statistical Reasoning, and Statistical Thinking. Literacy refers to recognition or

59

computation; reasoning refers to explaining why and how a specific process works; and
thinking refers to applying statistics to a context, critiquing, generalizing. A link can be
made between these three learning outcomes and Porter’s five categories of cognitive
demand. For example, the description of Porter’s category of Memorize and Perform
routine procedures is similar to Garfield and delMas’ Statistical Literacy category which
is mainly associated with creating graphs or plots, finding measures of center and spread.
Communicate understanding is similar to the Statistical Reasoning category associated
with the appropriate use and selection of graphs and measures. For the purpose of
achieving a finer analysis of the content of state and national standards and assessment,
all five categories from Porter’s (2000) work were used in this section. In later sections
where it was decided that coarser grid is more appropriate, however, only the three levels

suggested by Garfield and delMas are used.

60

 

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Data Analysis and Probability

Standard 1:
The student understands and uses the tools of data
analysis for managing Information. (MA.E.1.3)

collects, organizes, and displays data in a variety
of forms, including tables, line graphs, charts, bar
graphs, to determine how different ways of
presenting data can lead to different
interpretations.

understands and applies the concepts of range and
central tendency (mean, median, and mode).
analyzes real-world data by applying appropriate
formulas for measures of central tendency and
organizing data in a quality display, using
appropriate technology, including calculators and
computers.

 

 

Figure 3.1. Florida’s Standard on Data Analysis and Probability for Grades 6-8.
Content is then described at the intersection between topics and cognitive

demands, based on data gathered from content standards and assessment.

Content Analysis

The documents analyzed at the student level were content standards for middle
grades students from ten states, Mathematical and Problem-Solving Goals of the
Connected Mathematics (Lappan, Fey, Fitzgerald, Friel, & Phillips, 2002) Grade 6 and 8
textbooks and the Data Analysis standards for Grade 6-8 (N CTM, 2000). The states
considered were those that require a middle school certification, and they are:
Connecticut, Florida, Georgia, Kentucky, Missouri, North Carolina, Ohio, Oregon,

Virginia, and West Virginia. For the complete set of standards see Appendix B.

Some states present their standards in a general statement followed by a set of more
specific standards. For example, Standard 1 from the state of Florida is shown in Figure

3.1(www.firn.edu/doe/curric/prekl2/frame2.htm ). Others do not present any general

62

/__

Content standards vary in their structure and level of specificity from state to state.

statement and only list specific standards. For example, see Figure 3.2 from the state of

Virginia (www.pen.kl 2.va.us).

 

Probability and Statistics

6.18 The student, given a problem situation, will collect,
analyze, display, and interpret data in a variety of
graphical methods, including line, bar, and circle graphs
and stem-and-leaf and box-and-whisker plots. Circle
graphs will be limited to halves, fourths, and eighths.

6.19 The student will describe the mean, median, and mode
as measures of central tendency and determine their
meaning for a set of data.

6.20 The student will determine and interpret the probability
of an event occurring from a given sample space.

 

 

 

Figure 3.2. Three Virginia standards on Probability and Statistics for Grade 6.

Because standards needed to receive a code or codes, it was decided to make the

unit of analysis the specific standard. In the case that a general standard is presented, like i

in the case of Florida, it was used to help identify the intent of topics and cognitive
demands of the specific standards to follow. Each unit of analysis received a code or
codes according to the t0pics and cognitive demands observed. Codes consisted of a digit
from 1 to 7 corresponding to the seven topics (see Table 3.2) and a letter from A to E
corresponding to each cognitive demand.
Mapping the Standards to Cells on the Matrix

After becoming familiar with the instrument — the content matrix- and the units of
analysis -— specific standards, the first thing to determined was which topics on the content
matrix were the best match for the specific standard. Identifying the topics in the matrix

that were appropriate was done parsimoniously. The first principle was, “stick to the

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language of the standard”. Coders were instructed not to overgeneralize or to interpret

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63

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intended topics that were not noted in the standard. Second, if the standard applied to

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more than one topic, all of those topics were coded} or example, consider the following

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standard “select, create, and use appropriate graphical representation of data, including

 

histograms, box plots, and scatterplots”. This refers to two topics of the matrix,
“Numerical data representation” and “Bivariate data representation”, but not the topic
“Categorical data representation”.

Similarly, matching the cognitive demand(s) on the content matrix to the already
chosen topic was also done parsimoniously. Here too, the principle “stick to the .. ._
language of the standard” was applied. For example, if the standard was “describe the
shape of the data using range, outliers, and measures of center, including the mean,
median, and mode”, the correct cognitive demand would be “communicating
understanding”. The raters did not assume that in order to “describe the shape of the
data” the student would also be able to find or select measures of spread and center
(“performing routine problems”) or to formulate conjectures or infer beyond the data. In
the case where the standard did not specify whether the student should be able to solve
routine or non-routine problems and it was not possible to determine whether the problem
was routine or non-routine from the context, then both cognitive demands were selected.

For example, the Florida standard “The student collects, organizes, and displays
data in a variety of forms, including tables, line graphs, charts, bar graphs, to determine
how different ways of presenting data can lead to different interpretations. ” received all
of the following codes: 1B, 2B, 2C, 4B, and 4C. Although the standard does not mention
some of the graphical representations listed in topic 2, like pictographs and some listed in
topic 4, like scatter plots, there is at least a representation of categorical and bivariate

displays listed in topic 2 and topic 4. In terms of the cognitive demand, the “collects,

64

 

organizes, and displays data” corresponds to category B: “perform routine procedures”
and the last part about different interpretations corresponds to the category C:
“communicating understanding”.

A spreadsheet for each state was created to record the codes. Columns were
created for each specific standard. For example, Figure 3.3 shows three out of the four
specific standards for the state of Florida, which correspond to the specific standards
shown in Figure 3.1. They are labeled FLl, F L2, and F L3 and form the columns of the

spreadsheet with the corresponding codes assigned. Note how each specific standard

received more than one code.

 

 

 

 

 

 

 

 

- _ -4: : _ _ . l—EE’ agig'gtxf‘: :m- —:——— —~r—;_—.=. l-‘Lxls 5::
AJ 9 l C I D I E I F l G .I
in State: | Florida I
i Grade I Grades 6-8 I

 

 

 

 

 

Figure 3.3. Excel Spread Sheet to Record Codes for Three Specific Florida Standards.

To assure reliability of the coding process, one/rater Was trained to code the
specific standards. The trained rater and the researcher coded the specific standards
independently. The researcher adjudicated when there was disagreement between them.
A total of 67 specific student standards were coded and since each specific standard could

receive one or more codes, there were a total of 171 codes recorded.

65

 

Content Maps: A Visual Representation

The next step in the analysis was to create a visual representation of the data 2“. f
analysis from each document reviewed. Content maps were used to provide a
representation of the content in a given set of standards using a surface area chart which
results in a graphic similar to contour or topographical maps, except that these graphs
better suite the categorical nature of the data.

The graphs were created using a simple Mathematica software code. To
demonstrate how this was done consider again the example of Florida standards. First, a
frequency of the codes for all standards was recorded in a two-dimensional matrix (see
Figure 3.4), the colurrms correspond to the seven topics and the rows to the five cognitive
demands. These frequency values are what would constitute the measurement cell of the
content graph. In each cell of the matrix, the number of times a code was observed was
recorded. That is, the code Al was observed zero times, the code B1 was observed one
time, and so on. Note that the sum of the values of each cell should correspond to the
total number of codes assigned to the standards of Florida, 15 codes in this case. Also

note that the highest frequency of codes is one, which corresponds to about 6.7%.

 

 

 

 

 

 

 

 

 

feiT T? ‘ "T ‘ Tc" T T T ‘ Eifiiig—é:::f—_T—L€ —;::‘ 5 foIs 5:2: :3
A l B j c l b j I F T o l H l
1 Topic
2 Demand 1 2 3 4 5 e 7
"E: A o o o o o o o
_4_ j e 1 1 o 1 o o o
5 c o 1 o 1 o 1 1
H633 o o 1 o 1 o 1 o
.13 E 1 1 1 o o 1 1
R

 

 

Figure 3.4. Frequency Matrix for Codes Observed in Florida Standards.

66

 

 

Then the matrix was copied into a Mathematica file and defined as “fl”. A small
function defined with the specific arguments created the desired colored graph:

fl = {{0, 0, 0, 0, 0, 0, 0}, {1,], 0,1,0, 0, 0}, {0, l, 0,1, 0,1,1},{0,1, 0,1, 0,1,

0},{1,1,1, 0, 0,1,1}}

graph[ mat__] := Module[{ } ,

rects = Table[Rectangle[{i,j}, {i+1,j+1}] , {i, 0, 6}, {j, 0, 4}];
Map[Graphics, Flatten[MapThread[{Hue[colorfun[#1], brightfun[#1], l ],
#2}&, {mat, rects} , 2],1 ]]]

The grid of colored rectangles in the graph identifies the seven topics (indicated
by columns) and five cognitive demands (indicated by rows). The intersection of each
topic and category of cognitive demand represents a measurement cell and corresponds to
a particular cell of the content matrix. The image of the map is simply a computer—
generated graphic from Mathematica based on the frequency values for each cell. The
color of a cell indicates the percentage of the content observed by the raters for a given
topic and category of cognitive demand. The darker the cell the more frequently the

content (code) was observed across the standards. The resulting content map for the

Florida standards is shown in Figure 3.5.

67

 

 

Florida

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I I 7 r 7 I 7 7 r Conjecture,generalize, , 100/+
prove I: —] °
~e _._ 7 7 , * “*4 , 8-9.9°/
Solve non routine L ,i °
problems i ] 6-7.9%
.6 T ,7
a T I ? T T ”T ' 4—5.9‘r
2 Communicate M 0°
g understanding L , J 2'3-9/0
. L - L 7 E ,4 A ,
“a El 0.149%
6
EL, ,, _ g fig 7 7 Perform procedures 0%
Memorize Percentage of content
c: ‘3 1: m a; 3
1°: ‘3 8 8 E e
s 9 8 g 8 a
.— 1: E 5 Q t... I...
8 8 w m '5 ° °
'5 E a 3 .2 8 8
a e .. a v ‘5 a
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8 3 ‘= E E
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til: .E on g .2
8 Z CD
63

Figure 3.5. Content Map for Florida Standards Grades 6-8.

Content matrices and maps were created for each of the ten states, the NCTM 1
(2000) standards and the CMP Mathematical and Problem-Solving Goals. To allow for;
easy comparison, the same color scheme and scale was used for all maps. The scale,
shown in the legend of each content map, was chosen to highlight the important features
of the map for each document as well as the map summarizing all of the documents
reviewed. Note that for the majority of documents the frequency observed for each cell

observed is either 1 or 0. Hence the two colors represent the presence or absence of that

68

content in the document. Table 3.3 gives the total number of specific standards coded,
total number of codes assigned, highest frequency of codes observed and its
corresponding percentage for each set of state standards, the NCTM standards and the
CMP Mathematical and Problem-Solving Goals.

Table 3.3

Number of standards analyzed and codes assigned

 

 

 

 

Number of Number of Highest Highest
specific codes assigned frequency of percent of
standards codes content 1
State observed '
Connecticut 4 10 l 1/10=10%
Florida 4 15 1 1/15 5: 7%
Georgia 7 17 1 1/17z6% 34-;
Kentucky 9 21 2 2/21 z 10% " ' ‘
Missouri 4 l7 2 2/17 z 12%
North Carolina 6 10 1 1/10=10%
Ohio 5 l 1 1 1/1 1 z 9%
Oregon 8 15 1 1/15z7%
Virginia 7 18 1 1/18 :2 6%
West Virginia 6 l6 1 1/16 z 6%
NCTM 7 21 2 2/21 z 10%
CMP 9 36 3 3/36 z 8%
Total 76 207 12 12/207 z 6%

 

Analysis of Content Maps

Figure 3.6 thru Figure 3.10 show content maps for each set of state standards and
Figure 3.11 shows the content map for Principles and Standards for School Mathematics
(N CTM, 2000). State standards vary considerably on the topics, cognitive demands and
the emphasis given to those. For example, the states of Connecticut, Florida (see Figure
3.6), and West Virginia (see Figure 3.10) all show emphasis on the topics of measures of
center and categorical data representation, but Florida has these topics at the level of
solve non-routine problems while Connecticut and West Virginia have them at the level

of perform procedures.

69

 

Virginia (see Figure 3.10) has one of the most comprehensive content maps. It
covers all the topics but one, shapes of distributions; and it is pretty uniform on the
emphasis of level of cognitive demand. The content map for Oregon (Figure 3.9) does
not show clusters around any particular topic; instead it puts emphasis on all but
categorical representation of data. The map for Georgia (Figure 3.7) covers all topics on
data representation at pretty much all levels of cognitive demand, but leaves out the
concept of shape of distribution and includes finding and using measures of center and
spread.

Figure 3.11 shows that the Principles and Standards for School Mathematics
(NCTM, 2000) includes all topics except categorical data analysis (which is included in
the standards for Grades 3 — 5) and the explicit language of shapes of distributions; and
includes all levels of cognitive demand. Nevertheless, these standards also show higher
percent of content on the interpretation of numerical and bivariate data representation and

on the top level of cognitive demand: conjecture, generalize and prove.

The content described in the Connected Mathematics Teacher’s Guide (Lappan et
a1, 2002) is also different from eleven sets of standards for students reviewed earlier. The
most emphasis is given to the process of statistical investigation and to inference about
that data using shapes of distributions, measures of center and spread. This is a reflection
on the emphasis of studying statistics as an exploration process of investigation. These
textbooks include topics related to the design of studies (survey), random samples, and
the comparison of sampling distributions to make conclusions about the population.

There is only one commonality among all 12 sets of standards examined. None

\
%\A
I
1

covers any topic at the level of memorization.

7O

Connecticut

 

 

 

 

 

 

Conjecture,generalize, a 10%+
rove ,
Z 1 _ [L J 8-9.9%
0 ve non routine #
"5 problems E: 619%
- 4-5.9°/
E Communicate [j 2_3 90/0
E understanding w “ ' o
: - 0.1 -1.9%
(3 0
Perform procedures [:1 0A,
Memorize Percentage of content

 

 

 

Florida
L 'Ihnics >

 

    

 

 

 

 

 

Conjecture,generalize, a 10%+
prove m : 8-9.9%
Solve non routine ' V 6-7 90/
problems E ' o
E [- 4-59%
B Communicate W13 2_3 9%
5 understanding " ' '
.5 - 0.1—1.9%
'c:
O‘V
8 Perform procedures 0
Memorize Percentage of content
a g I: H 'U
is; a 8 a E 1%: i
m 8 g 8 g 0 3‘
'33 c: :1 '0 a. c...
8 8 8 g .5 o o
w 8 2 2 a a a
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§ 90 3 “ .. ‘H a a
o a :3 S .‘3 g o o
8 a a 45‘ a c» 2 2
£1 .5 7,; E 0 E
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a 5 E
0

Figure 3.6. Content Maps for the Connecticut and Florida Standards.

71

 

 

 

Conjecture,generalize,

 

 

 

prove a 10%+
Solve non routine I “I 8-9.9%
problems |:' 6-7.9%
'U
a Communicate - 4-5.9%
S understanding 23.9%
E - 0.1-1.9%
8 Perform procedures [:I 0%
Memorize Percentage of content

 

 

 

 

 

 

 

 

 

 

 

Kentucky
I; 'lbnim >
Conjecture,generalize,
prove 10%+
Solve non routine U 899%
problems E: 6-7.9%
E . - 4-59%
Communicate ‘
is understanding 2-3.9%
:§ - 0.1-1.9%
8 Perform procedures I 0%
Memorize Percentage of content
:1 =1 :1 5'5 '3
'55 _o .° IO 3 ... 2
'3 fs’ 3 3 '3 § a
'3 g :1 =1 B '-H «.1
8 q to g 8 E O O
m .0 2 u . 8 8
“SE, a E :83 é a s
8 ‘9. s S 8 ° § §
8 g g a g a 2 2
a: .a 5 g g g
I: 'n .
go 5’ .2
m
(3 2

Figure 3.7. Content Maps for the Georgia and Kentucky Standards.

72

Missouri

 

 

 

 

 

 

i Thins >
Conjecture,generalize, a 10%+
prove ‘
I II; I 8-9.9%
So ve non routine ‘ ‘
problems E3 619%
g _ 4-59%
5 Communicate [7'1 239%
understanding a—
g - 0.1-1.9%
0
8 Perform procedures E] 0%)
Memorize Percentage of content

 

 

 

 

 

 

 

 

Conjecture,generalize,
prove a 10%+
Solve non routine L] 8'9'9%
problems 1: 6-7.9%
"E _ 4—5 9%
3 Communicate '
a ‘ understanding a 2-3.9%
3% - 0.1—1.9%
8 Perform procedures [I 0%
Memorize Percentage of content

 

 

 

Process of statistical
Measures of center
Measures of spread

investigation
Categorical data representation
Numerical data representation
Bivariate data representation
Shape of distributions

Figure 3.8. Content Maps for the Missouri and North Carolina Standards.

73

 

 

(b nitivc Demand

" Demand

0)

 

 

 

 

 

 

 

'3
3
a
.‘3
J:
O
2
g
a:

E
Eu
‘5
m
2
. S

Categorical data representation

Numerical data representation

Bivariate data representation

Shape of distributions

Measures of center

 

Measures of spread

Conj ecture, generalize,
prove

Solve non routine
problems

Communicate

understanding

Perform procedures

Memorize

Conjecture,generalize,
prove

Solve non routine
problems

Communicate
understanding

Perform procedures

Memorize

Figure 3.9. Content Maps for Ohio and Oregon Standards.

74

 

10%+
8-9.9%
6—7.9%
4-5.9%
2-3.9%
0.1 -l.9%
0%

 

DIEIDEH

 

 

Percentage of content

 

rm 10%+
8-9.9%
6-7.9%
4-5.9%
28.9%
0.1 - l .9%
0%

DIEIEE

 

 

 

Percentage of content

Virginia

 

 

 

 

 

 

 

 

 

 

 

 

 

 

L 'lbn'nq >
Conjecture,generalize, a 10%+
rove ‘
p LJ 8-9.9%
Solve non routine .A .7. . 0
problems D 6‘7'94’
E - 4-59%
Communicate *,
E ' _ o
:3 understanding U 2 39A
:5 - 0.1—1.9%
0
8 Perform procedures |:] 0A’
Memorize Percentage of content
West Virginia
I_ 'Ibn'm >
Conjecture,generalize,
prove a 10%+
Solve non routine L: 899%
problems E 6—7.9%
'6 0
5 Communicate - 469/"
g understanding E I 2-3.9%
1% - 0.1 4.9%
8 Perform procedures C] 0%
Memorize Percentage of content

 

 

Measures of spread

H
g
D
U
Q-I
O
a
a
a
D
2

Process of statistical
investigation
Categorical data representation
Numerical data representation
Bivariate data representation
Shape of disu'ibutions

Figure 3.10. Content Maps for Virginia and West Virginia Standards.

75

 

‘k _~- —_

NCTM Standards

 

 

 

 

 

 

Conjecture,generalize, fl 10%+
pmve U 8-9.9%
Solve non routine E 6-7.9%
problems - 4_5 9V
3 ' 2 3-900
5 Communicate D - ' A,
‘9: understanding - 0.1-1.9%
- 0
e: [:1 0/°
:3 Perform procedures
Percentage of content
Memorize

 

 

Process of statistical
Shape of distributions
Measures of center
Measures of sprea

investigation
Numerical data representation

Bivariate data representation

8
'5
E
'm’
8
5?
S
3
3
'3
o
co
8
as
D

Figure 3.11. Content maps for Principles and Standards for School Mathematics (N CTM,
2000)

76

 

 

Conjecture, generalize,

 

 

 

 

prove a 10%+
Solve non routine I J 8-9.9%
problems E 6-7.9%
'U
2 Communicate E! 4-5.9%
5 understanding ‘ 3 23.9%
:E - 0.1-1.9%
6 Perform procedures 0%
Memonze Percentage of content

 

 

Process of statistical

investigation
Measures of center
Measures of spread

Categorical data representation
Numerical data representation
Bivariate data representation
Shape of distributions

Figure 3.12. Content Map for the Mathematical and Problem-Solving Goals in
Connected Mathematics Teacher’s Guide Grade 6 and 8 Textbooks.

When all matrices are put together the content map of Figure 3.13 is created. This
map allows one to “see” this set of 12 documents as a whole and identify the statistical

content that students in middle grades are expected to know. The map indicates that the

i
1

least covered topics are shapes of distribution and the process of statistical investigation. “ix
Instead, emphasis is placed on representation of data, particularly numerical
representation and measure of center and spread at the level of communicate

understanding. That is, 5.8% or more of the content in middle grades described in these

documents is dedicated to communicating understanding with graphical displays of

77

numerical data. Whereas between 5.1% and 5.8% of the content is dedicated to
communicating understanding with measures of center. In general, the main focus of the
standards is at this middle cognitive demand. Only for the topic of process of statistical

education is the highest level of conjecture, generalize and prove emphasized.

 

F m >

Conjecture,
generalize, prove

Solve non routine

 

 

 

 

 

 

 

problems
'0
S .
E Communicate
é understanding
.2.
e
:3 Perform Procedures
Memorize - 6% +
m 5- 5.9%
S S s: m H l I 4-4.9%
c: S ‘3 c: -
E; -8 ,9 -8 g 3 .g ,9 5;; E f, 1 3-39%
.F. v—l 0d _ ..s 0 Q.
.88.» .2 .2 Iz—m
m o o D 0 U.‘ "" U) E O o 2
g‘ .... U: U a o m w m V. l-l.9%
ofifigéérae v v ‘
§'§522953‘8 g g .o.1-o.9%
8 a; a o o 0%
a: .s g 2 2

Percentage of content

Figure 3.13. Content map for the ten states , the Mathematical and Problem-Solving
Goals in Connected Mathematics Teacher’s Guide Grade 6 and 8 Textbooks and
Principles and Standards for School Mathematics (N CTM, 2000)

78

 

Data Analysis and Statistics for Middle Grades Teachers
We now focus our attention on the content the teachers, as opposed to students,
are expected to know in the area of data analysis and statistics. For teachers, content

standards are not as developed as they are at the student level. Only four of the ten states

mama”-.. W- .- _ ,4;

(Florida, Georgia, Missouri, and North Carolinazaanalyzfleduhadprofessional standards for (I

teachimgych were available at the time this research was conducted. F urtherrnore the V

standards for these states are very broad and not content specific. For example, North

Carolina’s content pedagogy standard states in part “The teacher understands the central

concepts, tools of inquiry, and the structures of the discipline ...”. However, for teachers 7‘6)
1" .. a.

.-
d'
‘
’ I

to get accreditation to teach mathematics at the middle grades level in the ten states under

vs
,a
'0

consideration, they need to pass a standardized content test called PRAXIS 11: Middle ; ‘9’ i; «9“ ‘ r"

School Mathematics (0069). The content covered in this examination, sets the base of
what teachers need to know. The actual test was not available for analysis, but the
Educational Testing Service (ETS) publishes an on-line booklet called Tests at a Glance
(fip://fip.ets.org/pub/tand1/0069.pdf ). Tests at a Glance includes content outlines, sample
questions in each content area with a rationale for the best answers, and test-taking
strategies. The topics covered are organized by content area and the description includes
level of cognitive demand required. The content standards on the Data, Probability and
Statistical Concepts in the on-line booklet for test 0069 were analyzed for this research.
Recently, the Conference Board of the Mathematical Science (CBMS) published a
report called The Mathematical Education of Teachers (CBMS, 2001), which was "ll
designed to be a resource for those involved in the education of mathematics teachers.

The recommendations in this report are organized by elementary, middle, and high school

level and by areas of content. The second set of standards for teachers analyzed for this

79

research were the specific statements in the summary of data analysis and statistics
content (page 113) under the section “The Mathematical Content Needed by Prospective
Teachers”.
Content Analysis and Maps

Content analysis for these documents was preformed similarly to the analysis for /
students’ documents. The same content matrix (Table 3.2) and language was used even «3V
though some of the documents included more advanced topics]. Table 3.4 shows the
number of specific units of analysis examined and the number of codes assigned to each
document. The table also shows the highest frequency of codes observed and the
corresponding highest percent of content, which correspond to the highest level of
shading on the content maps.

Table 3.4

Number of specific statement analyzed and codes assigned in three sets of standards for

 

 

 

teachers.
Number of Number of Highest Highest
specific codes frequency percent of 5,
Document statements assigned of codes content "’3
observed
Topics covered in PRAXIS II 3 13 1 7.7%
Middle School Mathematics
The Mathematical Education
of Teachers (CBMS, 2001) 8 14 l 7.1%
Total 1 l 27 2 7.4%

 

Figure 3.14 shows the content maps for the PRAXIS 11 test and The Mathematical

Education of Teachers (CBMS, 2001). These two documents show two different patterns

 

' In order to facilitate comparisons between the standards for teachers and students some advanced topics
were excluded from this analysis. Advanced topics which were considered well beyond the middle grades

80

Mm
.tL.

CFOL

Ea

'M
w

~.
at

‘3.
:gfi
'|

)-
K‘s.

corresponding to their nature; one is a multiple choice assessment and the other is a report
on recommendations. The PRAXIS 11 content pattern is quite different from the content
suggested for students in Figure 3.13. The most striking difference is for the topic
numerical data representation. For the PRAXIS II the cognitive demand of perform
procedures and solve nonroutine problems are emphasized over communicate
understanding while the reverse is true for the student level content map in Figure 3.13.
In contrast, The Mathematical Education of Teachers (CBMS, 2001) shows its highest
percent of content across all topics, most of them at the middle level of cognitive demand.
Note that the emphasis for data representation is not on performing procedures (i.e.
construction of graphs) but rather on communicating understanding (interpreting graphs).
In addition to the topics described in the content matrix, the MET suggests that
prospective middle school teachers should know how to design simple investigations
including random sampling or random assignment to treatments; sampling distributions,
margin of error, confidence intervals, and expected values.

As was done with the content for students, the two content matrices were put
together to form a single content map (Figure 3.15) to identify the commonality between h’
the two documents. The map shows emphasis on categorical data representation,

bivariate data representation and measures of center and spread.

 

level and were not coded included expected values, random assignment to treatments, sampling distribution
and procedures of formal statistical inference such as confidence intervals and hypothesis testing.

81

 

 

Topics Covered in PRAXIS II

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Education of Teachers (CBMS, 2001)

 

 

 

 

 

l 7 'Ihnirs
Conjecture,generalize, , ,
prove [:1 10%+
Q ’ ’ ”TAT T Solve non routine J 899%
problems 1 J 6-7.9%
E —T 7" ' 7 FM 7 7 Communicate U 459%
3 understanding L J 2-3.9%
:E 7’ 7 7 7 CI 0.149%
:3 i i Perform procedures 0%
Memorize Percentage of content
The Mathematical

 

 

 

 

Conjecture,generalize, .
' 10%+
prove E j
Solve non routine LJ 899%
problems L J 6-7.9%
E I" 4 590/
6 Communicate D " °
g understanding [A] 2-3.9%
1: a 0.149%
:3 Perform procedures [:1 0%
Memorize Percentage of content
11
a g .5 é a ‘e‘
‘5 a 53 s '5 8 %
33 t: 5 c: B 9.. us
8 g 3 3 at”) E ° °
‘” .° 8 H B .m 3 8
“a a a g a 6 '5 a
m DO I- “ H ‘H g m
a '5 :3 8 s 3 o s
8 e a 3 a g 2 2
5: .E "‘ Te 0 s:
8 .0 ‘5 W
‘5 a '5
co >
e g a
6

Figure 3.14. Content Maps for The Mathematical Education of Teachers (CBMS, 2001)
and the Topics Covered in PMXIS 11 Middle School Mathematics Test.

82

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Conjecture,generalize, , o
prove L I 106+
_ o
7 F *4 Solve non routine L 7 J 8 99A)
problems l J 6-7.9%
'° ~—~* , 7 *7 - r m" 4-5.9°/
g r Communicate L] o
8 understanding L, J 239%
3g L—Hi , 7 . CI 0.1—1.9%
c:
8 Perform procedures 0%
Memorize Percentage of content
: L- "U
‘5 a S a ‘g 8 e
'5 c: ‘3 t: t... u—
o ‘1’ a.) r9 o o
8 q u) "’ w 5
£2 .9 £9. E E .2 8 8
3 ‘31 s? g a? 33 E ‘3‘
8 '9, :3 S s 3 8 8
8 a e i3 e e 2 E
d: .S '3 8 3 a
8 'CJ _<u
o “9 5
or) >
a g a
0‘“

Figure 3.15. Combined Content Map for The Mathematical Education of Teachers
(CBMS, 2001) and the Topics Covered in PRAXIS 11 Middle School Mathematics Test.

Relationship between Content for Students and Teachers

As we compare the two sets of documents, one at the student level and the other at #-
the teacher level, we observe some differences in coverage and emphasis. On the one
hand, Figure 3.12 shows clearly two clusters around communicating understanding of
numerical data representation and measures of center. On the other hand, the map of
content for teachers (Figure 3.15) shows clusters around communicating understanding of
categorical data representation and measures of center and spread, as well as computing
measures of center and spread and using bivariate data representations to solve nonroutine
problems. However, it is important to recall that the teacher level map in Figure 3.15 is
based only on two documents, while that student level map is based on 12 documents.

Thus caution should be taken not to overgeneralize these findings.

83

While the content matrix and its corresponding map are useful to visualize the

breadth and emphasis of coverage of the various standards analyzed, it is also desirable to

have a list of specific tasks students and teachers are expected to perform. This is

especially true if the ultimate goal is the creation of an instrument to assess if teachers in
fact possess the knowledge demanded of them. To this end, Table 3.5 includes the most

important tasks found in the documents reviewed in this study. The columns represent

levels of cognitive demand using the framework developed by Garfield (2002) and

delMas (2002).

Table 3.5

Summary of Tasks of Content Knowledge for Data Analysis and Statistics

 

 

Statistical Literacy Statistical Statistical Thinking
Reasoning
Knowledge of Data 0 Identify categorical o Formulate 0 Make decisions
Analysis and and numerical data. questions that can be on what and how to
Statistics 0 Create and read addressed through measure.
information presented data collection. 0 Extend, predict,
in data displays. 0 Understand or infer from
0 Find and compute what constitutes a information presented
mean, median and random sample. in data displays to
mode. 0 Understand answer implicit
0 Find and compute how surveys are questions.

range.

0 Identify clusters,
gaps, outliers,
symmetry, modality,
and skewness.

undertaken and how
experiments are
designed.

O Interpret and
integrate
information
presented in data
displays

0 Interpret what
measures of center
and spread tell about
the data.

0 Identify misuse
of cause-and-effect
interpretations of
correlations.

0 Use measure of
center to make
predictions and
inferences from data
about the group to
which the data
pertains.

0 Use the spread
and shape of a data
set to make
judgments about the
accuracy and
reliability of the data
and make inferences
fi'om data.

 

/
\.

/

Statistical Content Knowledge Applied to Teaching

Identifying the content knowledge that is needed to teach data analysis and
statistics in middle grades is a much more difficult task than identifying the “pure”
content knowledge for several reasons. One, authors of theoretical and empirical
literature of the past and present propose divergent elements to be considered for such
knowledge. Second, the development of a construct of the nature of teachers’ knowledge 7!
and its organization is still in the works (www.soe.umich.edu) with clear evidence that the
construct has multi-dimensionality characteristics. Third, the variability in and sometimes
lack of complete structure of documents which suggest teachers’ knowledge make
systematic comparison difficult. Finally, as the literature and researchers point out (Ball,
Lubienski, and Mewbom, 2001) this knowledge is better manifested or embedded in the
actual practice of teaching. However, a comprehensive observational study of classroom
practice or in—depth interviews with a large sample of in-service teachers is beyond the
scope of this present study.

Nevertheless, in this section, an attempt is made to take a look at written
documents related to teachers’ knowledge in order to describe their similarities and
contrasts and finally to smnmarize the important aspects. As a proxy for examining
actual (or modeling ideal) classroom practice, the Teacher’s guide of the Connected V
Mathematics (Lappan et a1, 2002) data analysis and statistics unit “Data About Us” was I}
also studied. The hope is that the discussion here will serve as an important step in the

development of content area-based teaching standards on par with those that currently

85

 

exist for content knowledge for students, and perhaps more importantly a framework in

which to base teacher preparation programs and assessment instruments.

Teachers ’ Knowledge as Suggested by Documents

For content knowledge applied to teaching, the data sources examined were
Knowing and Learning Mathematics for Teaching (NRC, 2000), Adding It Up (N CR,
2001a), Middle childhood through early adolescence/Mathematics Standards (National
Board for Professional Teaching Standards, 1998), Professional Standards for the
Accreditation of Schools, Colleges, and Departments of Education (National Council for
Accreditation of Teacher Education, 2002), National Middle School Association Middle
Level Teacher Preparation Standards (http://wwwnmsaorg), four state professional
standards (Florida, Georgia, Missouri, and North Carolina), Teacher’s Guide Grade 6
Statistics Unit: Data About Us (Lappan et a1, 2002), and The Mathematical Education of
Teachers (CBMS, 2001). An attempt to examine all ten state professional standards was

made, however only four were accessible at the time this research was conducted

Classification of Documents by its Characteristics

Documents can be classified by their structure into three different categories based

upon the level of specificity of the recommendations contained. In the first category, with

the most general recommendations, are professional teaching standards. These
documents are written to suggest knowledge for teachers in general terms without
specifying any particular subject matter or, with the exception of the National Middle
School Association Middle Level Preparation Standards (http://wwwnmsaorg), even the

grade level. Documents that fall into this category are the Professional Standards for the

86

It:

Accreditation of Schools, Colleges, and Departments of Education (National Council for
Accreditation of Teacher Education, 2002), National Middle School Association Middle

Level Teacher Preparation Standards (http://wwwnmsaorg), and the states’ professional

 

standards.

In the second category, the documents suggest teachers’ knowledge for
mathematics specifically, but do not focus on any content area within mathematics.
Documents falling into this category are Knowing and Learning Mathematics for
Teaching (NRC, 2001b), Adding It Up (NRC, 20013), and the Middle childhood through
early adolescence/Mathematics Standards (National Board for Professional Teaching
Standards, 1998).

Finally, the third category is made of two documents: The Mathematical
Education of Teachers (CBMS, 2001) which gives separate recommendations based on
specific content area and grade level; and the Connected Mathematics Teacher’s Guide
Grade 6 Statistics Unit: Data About Us (Lappan et a1, 2002) in which the application of
statistical knowledge is implicitly suggested in the development of lessons. Note that
these two documents were also analyzed for the statistical content needed for teachers in a
previous section. In this section the focus is on the content for teaching and the analysis
is more descriptive and in depth.

General Aspects of Teachers ’ Knowledge

General aspects refer to those aspects suggested for all teachers regardless of their
area of specialization and grade level. Professional teaching standards, national and state,
shows that these aspects are organized by domains of knowledge following a similar 44
framework as that suggested by Lappan (2000). They all suggest that teachers and/or

prospective teachers should have in-depth knowledge and understanding of the subject

87

_ A.“ —~ .‘.. -~-»

“W“ ~

 

"3

Ca.

(It:

'1’!

matter they plan to teach, knowledge of pedagogy, knowledge of students as learners, and?"
knowledge of assessment. Although the structure among all standards is similar, the
language used in these documents is not at all uniform. For example, some standards

prefer the language “knowledge of teaching practice” or “art of teaching” for the
pedagogical component. Others like to use better “pedagogical content knowledge” as
defined by Shulman (1986) and include the categories of content, pedagogy and

knowledge of learners. Furthermore, a few include technology and planning as separate
domains of knowledge.

The two documents related to national teacher standards: Professional Standards
for the Accreditation of Schools, Colleges, and Departments of Education (National
Council for Accreditation of Teacher Education, 2002) and the National Middle School
Association Middle Level Teacher Preparation Standards (http://www.nmsa.org), as a
whole, put the emphasis of the pedagogical or pedagogical content knowledge on the
appropriate use and selection of teaching strategies. In terms of students as learners, the
emphasis is on child developmental processes and ways of learning.

As for state standards, they all have different patterns in terms of the emphasis of
the domains of knowledge. For example, professional standards from Florida refer more
to knowledge about students’ developmental learning and explanations but have no
mention of assessment. In contrast, Georgia professional standards do not mention any
topic related to knowledge of students as learners and the emphasis is on content and
pedagogical content knowledge. Professional standards from the state of Missouri cover
all domains of knowledge but very superficially. In particular, they mention very little
about students as learners, student work or classroom discourse. Finally, North Carolina

standards follow more the pattern of national documents providing coverage of all

88

domains with emphasis on student development, accessing student thinking though
written work and classroom interaction.

These documents suggest one kind of general organization of teachers’ knowledge
by defining four domains of knowledge with specific aspects each: 1) Depth and breadth
of subject matter knowledge teachers plan to teach which includes demonstration of
understanding of central concepts; 2) knowledge of pedagogy which includes the
appropriate use and selection of teaching strategies or methods, the ability to explain and
present important concepts in different ways and contexts, and the use and selection of
materials, and technology; 3) knowledge of students as learners which includes the use of
students’ prior knowledge, design instruction appropriate for social, cognitive, and
emotional; access student thinking through classroom discourse and written work; and 4)
knowledge of assessment, which includes selection, development, and use of informal
and formal assessment strategies for the purpose of measuring achievement and to adjust
instruction.

Although a framework is provided by the national and state teachers’ standards
and some aspects of knowledge for teaching are identified, they do not provide clear
expectations in terms of cognitive demands for teachers or what this knowledge looks like
in teaching practices. Hence, the guidelines they provide are too general to be useful for
the development of assessment instruments or teacher preparation programs. A look at
documents related to teaching mathematics in particular is necessary to have a clearer

view of this knowledge.

89

Mathematical Aspects of Teacher ’s Knowledge

These aspects refer specifically to teachers of mathematics, not necessarily to a
specific grade level. The first document considered is a book based on the Proceedings of
the Mathematics Teacher Preparation Content Workshop, held on March, 1999 at the
National Academy of Sciences. The workshop was designed to set the stages for defining
and identifying the mathematical knowledge teachers need to know to teach mathematics
well. The published document is entitled Knowing and Learning Mathematics for
Teaching (NRC, 2001 b). The document addresses many different issues related to
mathematical knowledge for teaching; one of interest for this study is the organization of

knowledge for teaching around what they call “tasks of teaching practice”. The

discussion was motivated around the fact that often teaching is seen as presenting \' 1‘

material to students, but of course teaching includes many more small and large tasks -\

such as figuring out what students know, composing good questions, assessing and
revising textbook lessons, and so on. The key question was what are some of these

recurrent tasks of teaching that require the use of mathematics?

The participants of this workshop listed many recurrent tasks and discussed which lie,-

required the use of mathematics. They organized the tasks into six groups: 1) managing "

class discussion, 2) establishing a classroom culture for mathematical reasoning, 3)
designing and selecting tasks, 4) analyzing student thinking and work, 5) planning
instruction, and 6) assessing student learning.

Managing class discussion, involves selecting the language/tenninology to use to \7
explain an idea or procedure, to pose a task, or to relate to students’ explanations and
observations; anticipating misconceptions, deciding when to give feedback and what type

of feedback to give to students, deciding when to acknowledge “good” mathematical

90

thinking or an explanation and when to remain nonjudgmental, deciding how to build on
what students say, deciding which student solutions or strategies to focus on in whole-
group discussions, and assisting students by providing hints to move them along in their
thinking.

Establishing a classroom culture for mathematical reasoning involves sharing or
developing criteria with students for their work, developing definitions as a group,
examining and critiquing student ideas and work as small groups or a whole class, and
discussing expectations with students for their mathematical explanations.

Designing and selecting tasks for students involves aspects of pedagogical content
knowledge such as selecting the language to use to describe a task, making tasks
accessible to a range of learners, selecting a context for a task, evaluating mathematical
tasks through a child’s eyes to determine the “hard” parts, sequencing the use of
mathematical tasks, remodeling mathematics tasks, and selecting mathematical tasks that

will yield the best results for student learning.

Analyzing student thinking and work involves interpreting student explanations ,

l

and making sense of what they are saying, determining the mathematical validity of a
student strategy, solution, or conjecture; determining a student’s prior knowledge of a
mathematical idea; figuring out what students know and do not know, as well as what
conceptual knowledge connections are missing or are fragile; and examining student
strategies and solutions to determine which are more elegant and sophisticated requiring

that teachers have a sense of the range of potential strategies and solutions.

Planning instruction relates to deciding what mathematical topics to teach, 3;,

composing good questions, making long-range and short-range plans, assessing and

revising textbook or resource book lessons, designing lessons, selecting mathematical

91

‘ 2

'f

models and manipulative to use, and making decisions regarding the amount of time to
spend on a topic, lesson, or activity.

Finally, assessing student learning, is similar to the previously defined knowledge
of assessment seen in the first category of documents. Teachers need to design formal
and informal assessments, set criteria to make judgments about student work and analyze
and use information from assessments to guide student learning.

Authors of these lists of teaching tasks are aware that some tasks may fit into
more than one category but they used their judgment to place each tasks to the best fit.
With this type of organization one is forced to look at the work of teachers rather than just
examining their mathematical content knowledge. This is particularly useful for
designing assessment instruments; it provides the contexts to develop application
problems where teacher’s work is connected with their mathematical knowledge. One of
the pitfalls of a long list like this one is that educators may think that every t0pic in
mathematics can be connected with every task.

The other document that gives recommendations about teaching mathematics is
Adding It Up (NRC, 2001a), which addresses the question of “What does it take to teach
for mathematical proficiency?” In doing so, the authors discuss the kinds of knowledge
needed to develop proficiency in mathematics and problems involved in this task. Similar
to the framework observed on the state professional standards for teachers, this document
lists three kinds of knowledge: knowledge of mathematics, knowledge of students, and a.
knowledge of instructional practices. Knowledge of mathematics includes not only
understanding of concepts correctly and being able to perform procedures accurately, but
also understanding mathematics “in ways that allow them to explain and unpack ideas in

ways not needed in ordinary adult life” (p. 371). This definition can be viewed as the

92

“pedagogical content knowledge” idea used by Shulman and the teachers’ standards.
Knowledge of students and how they learn mathematics refers to how ideas develop in
children including common difficulties with concepts and procedures. Knowledge of
instructional practice refers to knowledge of curriculum, tasks and tool for teaching,
design and management of classroom discourse, and knowledge of classroom norms.
This kind of knowledge includes knowledge of both pedagogy and planning mentioned in
a previous section.

These kinds of knowledge, according to this framework, must be connected so
that their use has an impact in children learning. Furthermore, the authors suggest that
the connection needs to be made to classroom practice. The suggestion is then to require
several interrelated components of proficiency in the context of teaching: conceptual
understanding of the core knowledge required in the teaching practice, fluency in carrying
out basic instructional routines; strategic competence in planning effective instruction and
solving problems that arise during instruction, adaptive reasoning in justifying and
explaining one’s instructional practices and in reflecting on those practices, and a
productive disposition toward mathematics, teaching, learning, and the improvement of
practice. J ,4) 'g

If we look carefully at this framework we can relate the first four components, that H
is all except for the productive disposition, to the levels of cognitive demand used for
statistical content knowledge for students. For example, understanding of core
knowledge and fluency would correspond to solving routine problems or in the case of
statistics, statistical literacy. Adaptive reasoning would correspond to communicating

understanding or statistical reasoning, and strategic competence would correspond to

93

‘fi:
\
.lu

t

HA

4.,
mtg

l“ 9;
Li LN

solving non-routine problems or statistical thinking. The comparison is not perfect, but
one can see similarities.

Taken from this point of view, the short-comings of the present national and state
standards become clear. As opposed to standards for student learning, standards for
teacher preparation generally describe what a teacher needs to know, but do not explicitly
define the different levels of cognitive demands, nor do they relate these general domains

of knowledge to specific tasks teachers must perform in the classroom.
Statistical Aspects of Teacher 's Knowledge if}

Finally, these are the aspects of knowledge for teaching specifically statistics at

the middle grade levels. The only document that explicitly refers to knowledge needed J

l

for teaching statistics is The Mathematical Education of Teachers (CBMS, 2001) report. J
Although this document does not provide a framework or separate categories as
documents mentioned previously, one can isolate the following specific recommendations

in terms of content in the discussion (CBMS, 2001, pp. 115-117).

0 Prospective teachers must develop both skills for calculation and those for
interpretation within the context of the problem.

0 Teachers themselves need to learn to be critical consumers of data and statistical
claims.

0 A teacher’s time is better spent on learning to interpret graphs and related summary
statistics rather than undertaking tedious calculations.

0 Prospective teachers should have practice with and develop understanding of the role
of conjecturing using sample data, and they should understand that conjectures as to
why certain patterns appear in data are part of the exploratory process. They should
encourage their future students to think about data in this way.

' Examples of the misuse of statistical association to make cause-and-effect statements
can be brought into class discussions.

94

0 Teachers should understand the process of making inferences through simulated
sampling distributions (which can be done effectively in middle grades) and its
relationship with more mathematically based inference procedures taught at higher
levels.

Other recommendations are also made but these listed above are the ones that
relate to teaching the subject. This document focuses more on recommendations about
understanding the statistical content than in the other domains. Hence, a textbook was
also examined to identify more aspects for teaching statistics as an approximation of
teaching statistics in practice. The textbook examined was the Connected Mathematics
Teacher’s Guide Grade 6 Statistics Unit: Data About Us (Lappan et a1, 2002).

The content of data analysis and statistics is covered in two units, one in grade 6
called “Data About Us” and the other one in grade 8 called “Samples and Populations”.
The “Data About Us” unit focuses on formulating questions; gathering, organizing,
representing, and analyzing data; interpreting results from data. The “Samples and
Populations” unit focuses on using samples to reason about populations and make
predictions; comparing samples and sample distributions. Each unit cover subtopics in
small sections called “investigations” and each investigation consist of several problems
which address a concept or procedure.

Two investigations of the writ of “Data About Us” were selected for examination.
This unit and investigation were selected because they best represent aspects of content
identified previously in the students’ standards and The Mathematical Knowledge of
Teacher (CBMS, 2001) report. One investigation entitled “Using graphs to group data”
focuses on collecting and organizing numerical data in steam-and-leaf plots, locating
measures of center and spread, describing the shape of the data, including the location of

clusters and gaps, determining what is typical about the data, and comparing two data sets

95

using back-to-back stem-and-leaf plots and using statistics, such as median and range.
The knowledge for teaching identified in this investigation in terms of content was
knowing different graphical representation for numerical and categorical data, and
knowing that a data set can be represented with different types of graphs. Teachers also
need to judge when to use a specific type of graph and justify the choice. In terms of
knowledge of students, teachers need to know how to respond to students who want to
use an inappropriate type of graph, how to create questions that middle school students
can ask in order to collect data and that might involve using a stem-and-leaf plot, and how
to assess students’ responses making judgments on their reasoning.

As for pedagogical knowledge, teachers need to engage students in exploration of
the data given by having them suggest questions that might have originated from the data
and methods for collecting the data, leading students in the process of constructing a
stem-and-leaf plot and finally, pose questions about data organized in the stem-and-leaf
plot that guide students to “read the data”, “read between the data” by focusing on reading
the stem and on identifying intervals; and “read beyond the data” by focusing on the
mode, median, and shape. This investigation also includes reading back-to-back stem-
and-leaf plots for comparing two data sets. Here the main aspect of knowledge for
teaching identified was knowing how to conduct a discussion with students about the
comparison of two data sets. In particular, a teacher needs to know how to respond to
students that only focus on a particular statistic or students that have difficulties
comparing data sets with different number of data points.

The second investigation of this unit, entitled “What do we mean by mean?”,
focuses on the concept of the mean as the “balancing” point of the distribution, finding

and interpreting of the mean of a data set using physical models leading to the algorithm

96

and the proper use of the mean, median and mode. The investigation is split into five
different problems. In the first problem, students explore different ways to describe the
average number of people in six households and the mean is introduced through a visual
model using cubes. The model used employs towers of the cubes to represent the 6
different observed data values. Students are then asked to move the blocks around to
create 6 towers of equal height. This height represents the mean value. For this problem
teachers need to know that there are different ways to determine the average of a data set
and how to use them to solve problems. In terms of pedagogy, they need to know
advantages and limitations of physical models to introduce the concept of the mean as the
“evened out” number; how to make connections between the physical model and the line
plot; how to pose questions to students to help them see that the physical model and the
line plot display the same information. Teachers need to be aware that in the physical
model it is easier for the students to find out the sum of the data values than in the line
plot. Finally, teachers need to explain the mean as the balance point in the distribution
and relate the “evening out” model to the line plot.

The second problem in this investigation is a continuation of the first and students
are supposed to apply what they learn in the first problem. Here the teacher needs to
understand how students are thinking about the data when they use different strategies
and models to find the mean, and how to assess proper statistical reasoning for justifying
students’ strategies.

The third problem links the two previous problems and explores the idea that
different sets of data may yield the same mean. As for content, teachers need to know
how to create data sets with the same mean and different number of data points; create

data sets with different mean and same number of data points; create data sets with the

97

 

mean as part of the data set and not part of the data set. Pedagogically, teachers must
understand how to work with the physical models in the classroom. They need to know
how to create data sets with the same mean but a different distribution using physical
models, to relate the physical model with line plots, to make connections about the
number of data points, the total value and the mean, to make the transition between
having the sum of the data values as the unknown to the mean as the unknown using
models, and to lead students to the discovery of the algorithm of the mean and why it
works. Finally, in terms of knowledge of students, teachers need to know how to respond
to students who think that it is impossible to have many data sets with the same mean.

In problem four, students use a larger data set than they have used before
motivating them to develop an algorithm for computing the mean. For this problem
teachers need to know how to assess student’s explanations of their strategies for finding
the mean of a large data set and why it works. They also need to understand the
difference between the median as the physical middle of the values and the mean as the
balancing point of the distribution.

The last problem was designed to broaden student’s understanding of the mean by
introducing some extreme values. The knowledge for teaching needed for this problem is
to anticipate students’ answers or interpretation to an investigation question such as “How
many movies did you watch last month?” and be able to pose questions to students that
lead them to see the effect that outliers and/or any new data values have on a stern plot
and the mean.

The lack of a common framework and the different purposes of the documents
reviewed in this section make it difficult to summarize the different aspects of knowledge

addressed in them. However, as with the statistical content analysis, it is useful to list the

98

important tasks, in this case teaching tasks, teachers are called on to perform. The most
specific knowledge for teaching statistics was found in the examination of the Connected
Mathematics textbooks which approximate the actual practice of teaching statistics in
middle school. It is here where the identification of knowledge for teaching statistics was
found most useful in terms of developing items for assessment. Table 3.6 summarizes the
teaching tasks for knowledge for teaching. The rows represent the domains of knowledge
suggested by the state professional standards and the columns represent three of the

components of proficiency described in Adding It Up (NRC, 2001a).

99

Table 3.6

Summary of Teaching Tasks for Knowledge for Teaching Data Analysis and Statistics

 

Understand core
knowledge, fluency on
teaching routines

0 Describe and
anticipate students’
misconceptions and
limitations when
analyzing data.

0 Know the

Adaptive reasoning

0 Select language to use
and context to describe a
task.

0 Select and sequence
statistical tasks that will
yield the best results for

Strategic competence

o Create, adapt, and use
tasks for diverse purposes.

0 Making decisions about
when and how to probe for
deeper understanding, to
give feedback, to react to

 

 

a development of student learning. Students miStakCS.

g statistical ideas 0 Determine and use 0 Compose good

33 according to age, students’ prior knowledge questions.

3 abilities, and interest. in connection with new . Assist students with

,2 information. hints to move them along in

g o Interpret students’ oral their thinking.

3% and written responses in 0 Examine students’
relation to statistical strategies and solutions to
concepts. infer students’

0 Determine the understanding and plan

statistical validity of a future instruction.

student strategy, solution,

or conjecture.
0 Describe different 0 Select powerful 0 Use appropriate
teaching strategies and teaching methods teaching strategies,
give examples and according to the statistical materials and technology.
counterexamples of concept to teach. 0 Develop and adapt
statistical concepts. 0 Revise textbooks or instructional materials and

23:6 0 List concrete resource book lessons. plans.

go material and 0 Select statistical

E technology available models and manipulative

for instruction in to use.

statistics.

0 Know formal and 0 Select assessment 0 Design and use a
informal assessment strategies according to variety of formal and
strategies. educational purposes and informal assessment.

’5 students developmental o Analyze and use

4&2 level. information from

a o Align assessment assessment to guide student

a strategies with what is learning and inform

< taught and how it is taught. students, their parents, and
0 Set criteria to make school.

judgments about student
work.

100

Knowledge for Teaching

 

domains

 

 

 

 

l I
Content Applied to Teaching
[ dimensions '1 subdomains ——|
Topic X Cognitive Demand Students Assessment Pedagogy
as
Learners

Figure 3.16. Structure for Analyzing Aspects for Knowledge for Teaching.

Summary and Discussion

Figure 3.16 displays the structure used to organize the analysis of documents in $
this chapter. Knowledge for teaching was separated into two major domains: Content
Knowledge and Content Knowledge Applied to Teaching. As shown in Figure 3.16, each
domain was then further subdivided. The aspects of content knowledge are tasks that
represent a cross between topic and cognitive demand. The tasks needed for teaching in
the middle school grades are suggested by state and national standards, and The
Mathematical Education of Teachers (CBMS, 2001) and are summarized in Figures 3.13,
3.15 and Table 3.5. National standards suggest a broader list of topics and higher level of
cognition than state standards. Furthermore, state standards vary considerably in their
emphasis of statistical content as well as the level of cognitive demand.

The important aspects in data analysis and statistics identified in these documents,
as a whole, are data representation, in particular numerical data representation such as

line plots and histograms, and measures of center and spread. The emphasis is on

101

performing procedures (making the graphs and computing the measures of center and
spread), and communicating understanding (appropriate selection and interpretation).
Less emphasis was given to solving nonroutine problems (application of data analysis and
statistics to real world problems) and conjecture, generalize, prove (make predictions and
inferences from the data).

Knowledge applied to teaching is subdivided into three subdomains (Students as
Learners, Pedagogy and Assessment) suggested mainly by national and state teachers’
standards. Most of these documents are not subject matter specific and only provide
general guidelines for what is needed to teach well. Like content standards at the student
level, these documents vary in the domains of knowledge suggested for teachers and the
cognitive demand. The documents place emphasis on the pedagogy domain, in particular
teaching strategies. Second in importance is the domain of knowledge of students as
learners, in particular classroom discourse; and informal assessment. The most specific
aspects for knowledge for teaching statistics was found in the examination of the
Connected Mathematics textbooks which approximate the actual practice of teaching
statistics in middle school. The aspects, which represent teaching tasks, that were
suggested by the NRC documents and the Connected Mathematics textbooks are

summarized in Table 3.6.

102

CHAPTER 4

METHODOLOGY FOR DEVELOPING AN INSTRUMENT TO ASSESS
STATISTICAL KNOWLEDGE FOR TEACHING

The analysis reported in Chapter 3 identifies the important aspects of statistical
knowledge needed for teaching at the middle school level. A second question naturally
arises, what do prospective teachers know about the various aspects of statistical
knowledge for teaching? In particular, what do they know about the content and what is
their pedagogical content knowledge of data analysis and statistics? This chapter
describes procedures used to develop and administer instruments for assessing statistical
knowledge for teaching.

Instruments
Statistical Knowledge for Teaching Assessment

The Statistical Knowledge for Teaching Assessment instrument has been
designed to measure two major domains of knowledge. One is purely statistical
knowledge, and the other is statistical knowledge applied to teaching. The analysis in
Chapter 3 of documents, state and national standards, and several units from a set of a
curriculum materials identified many more aspects of knowledge than can be measured
by a single instrument. For example the content matrix in Table 3.2 contains 35 cells,
representing 35 potential different content aspects and Table 3.6 contains over 25

teaching tasks. To narrow the focus to a more manageable level, a selection of aspects

103

was necessary. To determine the topics and levels of cognitive demands to best assess
purely statistical knowledge the content maps in Figures 3.13 and 3.15 were combined
and modified to create Figure 4.1. The five levels of cognitive demands were reduced to
three. Given the fact that none of the 14 documents analyzed contained any content at
the level of memorization, this level was dropped. Furthermore, the two highest levels
were collapsed. With this modification the levels of cognitive demands can be matched
with the framework suggested by Garfield (2002) and delMas (2002). Perform
Procedures corresponds to Statistical Literacy, Communicate Understanding to Statistical
Reasoning and the combined levels of Solve Nonroutine Problems and Conjecture,
Generalize, Prove correspond to Statistical Thinking. The different shaded cells on the
figure correspond to the intersection of a topic with a particular cognitive demand.
Examination of Figure 4.1 reveals that 15 of the 21 content cells have a percentage of 4
or higher. Of these content cells 9 were included in final instrument. The cell with the
highest percentage, process of statistical investigation at the statistical thinking level, is
associated with designing studies for inference and is beyond the scope of a timed written
instrument. All cells with next highest percentages, representation of numerical data and
measures of center, are included in the instrument. Items on the representation of
categorical data were included in a piloted version of the instrument. However, they
were found to be too easy to be of use and were dropped from the final instrument. The
lowest level — statistical literacy - of the process of statistical investigation is associated
with formulation of questions to produce data. Although an item related to this topic was
included in the instrument its focus is on the domain of knowledge for teaching. The two

levels — statistical literacy and reasoning - of the measures of spread with percentages

104

above 4% are included, but due to space and time constraints, bivariate data
representation was not. Shape of distributions, which has the lowest percentages of
coverage (below 2.9%), was not included in the instrument. It should be noted however
that the distinctions between the different cells is not entirely clear cut. For this reason,
one item may measure more than one cell. For example, item 4a was created to asses the

interplay between measures of center and spread; so it was counted in two cells.

 

1 Jam >

 

Statistical Thinking

Statistical Reasoning

(b nitive Demand

Statistical Literacy

 

 

 

8% +

7-7.9%
6-6.9%
5-5.9%
44.9%
3-3.9%
2-2.9%
1-1 .9%
0-0.9%

EEIEII

Categorical data
representation
Numerical data

representation
Bi-variate data
representation
Measures of center
Measures of spread

E3
H
.2
..
§:
“.2
07-3
3.20
§7§
5.5

Shape of distributions

win”!
I

 

 

 

Percentage of content

Figure 4.1. Content map for the ten states standards, the Mathematical and Problem-
Solving Goals in Connected Mathematics Teacher’s Guide Grade 6 and 8 Textbooks,
Principles and Standards for School Mathematics (NCTM, 2000), The Mathematical
Education of Teachers (CBMS, 2001) and the Topics Covered in PRAXIS II Middle
School Mathematics Test.

105

As for statistical knowledge applied to teaching, the instrument focuses on the
knowledge of students as learners. Pedagogical knowledge and assessment knowledge,
the other two domains described in the previous chapter are not considered here and are
left for fiiture investigation. Twelve teaching tasks in this domain are listed in Table 3.6.
The following two were considered most conducive to this type of assessment and are
included in the instrument:

0 Interpretation of students’ oral and written responses in relation to the content.

0 Examination of students’ strategies and solutions to exercises to make

inferences about their understanding.

Development Procedures

Two pilot studies were conducted. In the first study, several items were tested
using only a written format. Items were analyzed and a second phase was conducted to
refine the selected items and perform a couple of interviews. The participants in the first
phase (Fall 2001) were prospective teachers planning to receive K-8 teaching certificates
from three different institutions: Michigan State University, American University, and
Montgomery College. A total of 42 prospective teachers participated in the first phase,
the majority (30 out of 42) of which were females in their early twenties. All the
participants had taken at least three credits of a mathematics for teachers’ course, and an
average of nine credits of college mathematics. About two thirds of the participants had
at least three credits of college level statistics.

Seventeen items were tested for construct reliability, wording, and format. The

main purpose of the first phase was to analyze these items, that is, to identify what items

106

—-—

 

were more successful measming the construct desired, the effect of the format of the
responses (close form or open ended), and the nature of responses. Furthermore, the
analysis helped to develop the initial rubric or classification of the responses into patterns
and dominant answers to help the researcher to create a rough picture of prospective
teachers’ statistical knowledge.

Items were examined in isolation first. For closed format items, responses were
tabulated accordingly to the correct answer and distracters. For open ended and short
format, direct responses were recorded creating naturally emerging categories, with the
purpose of not imposing any preconceptions on the nature of responses. When available,
these categories were compared against existent literature. Items were coded and a small
database was created.

Items covered different levels of performances, from computational and
procedural skills to higher-order thinking. The items were adapted and/or used from the
literature to address the different aspects of knowledge for teaching described earlier.
Item sources included the Collecting, Representing, and Interpreting Data Module, San
Diego State University, TIMSS 1999 Assessment, Friel, Bright, Frierson, and Kader,
(I997), PRAXIS 11: Middle School Mathematics (flp://ftp.ets.org/pub/tandl/OOé9gdf ) ,
Liu (1998), Watson (1997), Susan Jo Russell (personal communication July 16, 2001),
and Senk et a1 (1998).

The seventeen items, each with multiple parts, were separated into two forms.
Form 1 had eight items and Form 2 nine items. Items in the two forms were distributed
so that the forms covered similar content and difficulty. Pilot responses were coded for

correctness (utilizing a rubric for each item) and statistical tests of reliability were

107

conducted. Reliability of the pilot instrument was tested using a Cronbach-Alpha. The
results showed the following values: Form 1 or = .4731, Form 2 CL = .6627.

Items that did not provide extensive insight into either content knowledge or
knowledge about students were discarded. Some were re-written to improve the wording
and format, and others were added. As expected, prospective teachers were able to
perform well on the items that required extracting information from a graph, and
computing and defining measures of center and spread. Although it is necessary to keep
these types of items because they help provide a fuller picture of teachers’ knowledge,
fewer of them appear in the final version of the instrument. For example, pilot results
show that the majority of prospective teachers can extract information from and interpret
bar and pie graphs. The mistakes identified were related to proportional reasoning or
scale errors. The final instrument measures the use of graphs for numerical data instead.

In addition, the pilot instrument did not cover some of the important aspects of
statistical investigation such as the prospective teachers’ ability to formulate questions
and their knowledge of measures of spread. These two aspects were added to the final
version.

This new version was pilot tested in a second study (Spring 2002) with 16
prospective teachers from the state of Maryland. Two prospective teachers were invited
to participate in a follow-up interview. The second instrument piloted consisted of 11
items, items 1 — 5 consisted in short questions about statistics and items 6 — 11 where
short questions about statistics applied to teaching. Most of the items of the second pilot

were kept for the final version. Some were used for the follow-up interviews.

108

Description of Items

The final version of the instrument is given in the Appendix C. It consists of 8
free-response items with multiple parts intended to be answered in 50 minutes. Some of
the items cover statistical knowledge without reference to teaching while others cover
statistical knowledge for teaching in relation to students’ thinking. Table 4.1 shows the
distribution of items for the domain of statistical knowledge. Note that a single item or
part of an item may measure more than one aspect of statistical content. For example,
item 4a measures numerical data representation and measures of center because this item
consist of the presentation of a graphical display where the prospective teacher is asked to
find the mean (see instrument assessment in Appendix C).
Table 4.1

Distribution of Items by Content and Cognitive Demand

 

Cognitive Demand

 

 

Aspects of statistical content Statistical Statistical Statistical

literacy reasoning thinking
Numerical data representation 1a, 4a, 6b 1b, 6a 1c, 8b
Measures of center 4a 2, 4b, 4c 1c
Measures of spread 8a 8b

 

Items were also designed to assess statistical knowledge applied in teaching, in
particular, to applications to students’ thinking about data, data displays, and measures.
In each item the prospective teacher is confronted with either a student response or
solution to an exercise. In contrast to the items related to statistical knowledge, these

items are not characterized by cognitive demand. In some items (3a, 3b, 5a, and 5b), the

109

prospective teacher is asked to judge whether a response is correct and then to explain
what thought process the student might have used to arrive at that response. In others
(items 7a and 7b) the prospective teacher is asked to describe the method or solution used
by the student and then to make inferences about his/her understanding.

Item I, which involves a stem-and-leaf plot, has been adapted from an example of
an investigation presented in Friel et a1. (1997). It was originally designed to measure the
three levels of graph comprehension identified by Friel, Curcio, and Bright (2001): (a)
extract information, (b) find relationships, and (c) move beyond the data. These levels
closely correspond to the levels of cognitive demand of the framework used in this
research.

Item 2 measures knowledge of the appropriate use of measures of center based on
the shape of the data and the effect of outliers. This item is a variation of an item from
the Statistical Reasoning Assessment (Lui, 1998) which originally was developed to
measure the understanding of how to select an appropriate average, to identify the
misconception that averages are the most common number, and the mistake of failing to
take outliers into consideration. In addition, the context of this item suggests the view of
the average as a signal in noise (Konold, 2002). According to this view, “each
observation is an estimate of an unknown but specific value. Each observation is viewed
as deviating from the actual weight by a measurement error, which is viewed as
‘random’. The average of these scores is interpreted as a close approximation of the
actual weight” (p. 269).

Item 3 measures graph comprehension (histogram) at the level of interpretation.

Part (a) measures the ability of the teacher to recognize a common student misconception

110

and to interpret the source of misunderstanding. Part (b) measures interpretation and
judgment of a student’s oral response. While similar in nature, the two parts differ in the
degree of misunderstanding demonstrated by the student. In part (a) the student answer is
incorrect, while in part (b) the student statement is more incomplete than incorrect. The
item hopes to measure the ability of the teacher to differentiate between the two. This
item is a variation of a problem from the Collecting, Representing, and Interpreting Data
Module developed at the Center for Research in Mathematics and Science Education at
San Diego State University. The misconceptions and students’ responses are authentic,
taken from the pilot study.

The data for items 4, 5 and 6, the graphical displays and suggestions for student
responses were taken from the Connected Mathematics units (Lappan et al., 2002). Item
4 measures knowledge of calculating the mean from a line plot, which implies assessing
for knowledge of graph comprehension and the concept of the mean beyond the
algorithm. It also measures properties of the mean and the ability to create distinct
distributions with the same mean. Item 5 measures knowledge about formulating
questions to generate data and measures of center and range for categorical data. This
question measures the ability to identify errors in students’ responses. Item 6 measures
the proper selection of data representation, taking into account the shape of the data and
the ability to create a graph.

Item 7 measures the level of interpretation of a graph and the ability to examine
and make judgments about student work based on statistical reasoning. It also measures

the ability to describe students’ thinking and infer about their understanding. This item

111

was adapted from an item shared in a personal communication with Susan Jo Russell
(July 16, 2001).

Finally, item 8 measures in part (a) computational knowledge of the mean, range
and standard deviation from a line plot of two data sets. Part (b) measures the
recognition of the two data sets’ shape in relation to the measures calculated in part (a).
This item is based on an example from Functions, Statistics, and Trigonometry (Senk et
aL,1998)

Validity of the instrument was not conducted by the judgment of experts on the
field but instead by using the analyses of documents described in chapter 3 and by using
the theoretical perspective described in chapter 2.

As in the pilot study, reliability of the instrument was tested using a Cronbach-
Alpha. The results showed greater reliability with or = .80. Reliability was also tested for
the two difi‘erent domains of knowledge measured. The part of the instrument measuring

statistical knowledge (items 1, 2, 4, 6, 8) has a = .74 and the part that measures statistical

knowledge for teaching (items 3, 5, 7) has on = .53.

Procedures
The population under study is the set of prospective middle school teachers in the
last stage of their professional education in the US. The states considered for the sample
are those that both require a standardized mathematics content test1 and offer Middle
School Certification. Ten states meet these criteria, and they are Connecticut, Florida,

Georgia, Kentucky, Missouri, North Carolina, Ohio, Oregon, Virginia, and West

112

Virginia. It is believed that colleges and universities in these states are more likely than
others to have teacher preparation programs that focus on middle school content and
pedagogy.

Three major universities in these states were selected to participate in the study.
These are the University of North Carolina, Kenessaw State University and the
University of South Florida. These universities were selected for two reasons. First, each
of these institutions has a large program and strong reputation in teacher preparation.
Second, personal contacts in each were willing to cooperate with the study. In addition
students from the University of Maryland and Towson University participated in the
study. Although the state of Maryland does not have middle school certification, the
College of Education at the University of Maryland is among the top ranked programs in
the nationz. Furthermore, their geographic proximity to the researcher made the process
of interviewing convenient.

Within each university a faculty member, or some representative of the
researcher, identified prospective teachers who qualify for the study. Although in most
cases these students were clustered in a mathematics education class, some were
identified by the faculty member within their education program. Subjects were asked to
answer a few questions about the mathematics, mathematics education, and education
courses they had taken. These questions were customized for each institution according

to the official academic program found in the respective web site.

 

I PRAXIS 11: Middle School Mathematics (0069) (Retrieved June 30, 2004 from
fin://fip.ets.org/pub/tandl/OO69.pdf )

2 University of Maryland’s Curriculum and Instruction Program raked I 1th in the nation by 2004 US.
News & World Report Guide.

113

Participants

A total of 42 prospective middle school teachers participated in the study. Most of

subjects that participated in the study were female seniors in their twenties. Table 4.2 -
presents the distribution of the sample by gender, age, and class.

Table 4.2

Distribution of Subjects by Gender, by Age and by Class

 

 

Variable Frequency
Gender
Female 33
Male 9
Age
19-23 22
24-29
30-35 1
Over 35 12
Class
Junior 9
Senior 29
Graduate 4
Total 42

 

Subjects for this study have a strong mathematics background. All of them had
taken at least one semester of Calculus, and 38 of the 42 prospective teachers surveyed
had taken at least one course beyond calculus. The average number of mathematics
classes starting at calculus is 5.2. Some of these classes include Calculus II and III,
Differential Equations, Linear Algebra, Geometry, Number Theory, and History of
Mathematics among others. All but four students had taken a basic or introductory
statistics class at the college level. The performance of these four students was not

different than the rest of the participants and therefore were included in the final analysis.

..._.4

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Mathematics education classes were harder to account for; for some institutions
these belong to the education department and for others they belong in the mathematics
department. The emphasis on content or pedagogy was also hard to tell, that is, it was
unclear if the mathematics education classes were more focused on content knowledge
for teaching or pedagogical knowledge. For the purpose of this study, all of these classes
are labeled mathematics education classes. Subjects participating in this study had an
average of 2 mathematics education classes.

Finally, participants were also strong in education classes, with an average of 4.5
classes. Some of the titles of the courses taken were Foundation of Education, Education
Psychology, Human Development and Learning, Multicultural Prospective, Teaching &

Schools, and Curriculum and Instruction.

Analysis of Statistical Knowledge for Teaching Assessment

The aim of the analysis of the written instrument is to create a global picture of
prospective teachers’ statistical knowledge for teaching. The responses were
characterized not only by identifying levels of correctness but also by observing response
patterns and solution strategies. Therefore, the analysis of the written instrument was
done at two levels, item analysis and global analysis. Item analysis was conducted by
coding each item’s responses utilizing a rubric for levels of correctness. For many of the
items in the final version of the instrument, rubrics were developed using the pilot data,
but the rubrics were refined with the responses from the final version. To assure

reliability of the scoring, two graders —- the researcher and a trained grader - coded the

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responses independently. The researcher adjudicated when there was disagreement
between graders.
Rubric

Since many questions on the instrument are free-response, a rubric was needed to
score the items. For items that measure statistical knowledge, it was desirable to have a
consistent scale in terms of level of performance and conceptual understanding that can
be applied across all items. The scores can then be combined across the items that
measure statistical knowledge to obtain an overall score and give a clearer picture of the
level of performance of prospective teachers. A general rubric outlined below describes
the important criteria that were taken into consideration for all items measuring statistical
knowledge. This general rubric was then interpreted to develop a more specific rubric for
each item.

A holistic-scoring method, adapted from two sources: an analytic-scoring method
offered by Garfield (1993) for evaluating students’ solutions to practical statistical
projects and a holistic-scoring method used by Thompson & Senk (1993, 1998) to assess
problem solving and conceptual mathematical understanding, was used to score the items
in the instrument. Garfield’s analytic scoring method considers multiple dimensions such
as communication, visual representation and interpretation of results and uses criteria that
are specifically adapted to statistical content. However, each item is given multiple
scores, one for each dimension. A holistic-scoring method described by Thompson &
Senk (1993) creates a single scale and makes the process of scoring and data entry faster
and easier taking into account the assessment of procedural and conceptual

understanding. By adapting Garfield’s ideas to a single holistic scale, it is hoped to

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obtain the best of both methods. The integration of both methods was done by
associating the levels of correctness with the appropriate language used in statistics. The
criteria listed below provide a generic rubric. Prior to actual grading, the criteria were
further specified and customized to create item specific rubrics.

Successful responses

4 Solution is complete and correct. Language and notation used is correct; tables
and graphs are correctly constructed; all decisions are made correctly; data sets
are interpreted correctly using all appropriate information.

3 Solution is almost complete and correct, but some minor error is made, perhaps in
use of language, missing labels and scale on graphs, or calculation error with
valid reasoning.

Unsuccessful responses

2 Response is in the proper direction and contains some substance, such as a chain
of reasoning. But either the prospective teacher stops about halfway through the
solution or the complete solution contains some major conceptual errors. The use
of language is partly appropriate or some decisions about the selection of graphs
for representing data and summary measures seem inappropriate or the
interpretation is too brief, the prospective teacher fails to interpret some important
information or weak conclusions are made, but some attempt is made to look
beyond the data.

1 Some work is correct, but the student reaches an impasse early. The work shows
no evidence of a chain of reasoning. The prospective teacher uses statistical

words in a context that does not make sense or errors in calculation lead to

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unreasonable answer or the respondent is unable to interpret the plots and
measures or draws conclusions not substantiated by the data.
0 Work is all wrong or meaningless. No correct statistical knowledge is used for a

solution or there is not enough information to evaluate.

With numerical scores assigned to each item, percentages of prospective teachers
that reach a certain level of correctness on each item is reported. In the case of the items
that fall into the pure statistical knowledge category, 3 percent of prospective teachers
that reach each level of performance is reported as well. Since the goal is to characterize
teachers’ knowledge as a whole, individual scores are not reported.

The global analysis was done by combining scores of the items that correspond to
the two major domains of knowledge: statistical knowledge and statistical knowledge
applied to teaching. To summarize the overall performance, average scores are reported
as well as averages of each domain of knowledge. Furthermore, the percentage of
prospective teachers who demonstrate at least a given level of correctness (0 to 4 scale)
on all the items within a content category is reported. This type of summary permits an
analysis and comparison of the performance in each category. For example, the
percentage of teachers who are successful on reading graphs (at least level 3 on all items
related to reading graphs) can be compared to the percentage who demonstrates success
on measures of center.

For statistical knowledge applied to teaching, the performance was assessed from
two perspectives. First, the ability of prospective teachers’ to identify the correctness of

student responses was measured by combining the results from items testing knowledge

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of student thinking. Secondly, for the items that prospective teachers are asked to explain
what thought process the student likely used to arrive at that response. Responses were
examined to characterize their understanding of the statistical reasoning of students. The
key ingredients of this characterization was the level of agreement between the
misconceptions cited by the prospective teachers and the literature on statistical education
reviewed in chapters 2 and 3, and the depth of insight demonstrated by the prospective
teacher’s responses.
F ollow-up Interviews
Purpose and Description

The second stage of data collection consisted of face-to face interviews with a
subset of prospective teachers from the sample. The interview has two proposes: (1)
clarify individual student responses of the written instrument and to probe more deeply
into student understanding of knowledge for teaching, and (2) assure reliability on the
written instrument. The information gathered from the written instrument is sufficient for
a general description of some of the aspects of knowledge for teaching, but it is limited
and sometimes hard to interpret. By asking prospective teachers to explain their thinking,
reflect on their responses or give explanations for incomplete responses, a more accurate
and detailed picture of the subjects’ knowledge for teaching can be developed.
Furthermore, the interview responses complement the information on questions that are
too difficult to answer in another format.

Ten prospective teachers were selected to participate in the interview process. The
criteria for selection consisted of physical proximity to the researcher and representation

of low, middle, and high level of overall performance in the written instrument. Seven of

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those indentified met the appointment for the interviews. Attempts to reach the other
three were unsuccessful. After the selection and invitation to participate in the interview,
subjects’ individual responses to the written instrument were analyzed prior to the
interview and a customized interview protocol was prepared for each subject (see
Appendix E).

Customized protocol questions were guided by the nature of the written
responses. For example, for Item 1 a new stem-and-leaf plot was presented to the
interviewee according to what they have said was the typical value of the data set
presented in the written instrument. The new plots varied according to what they had
picked for the mean, median, or mode. This was done with the purpose of finding out
whether prospective teachers took into consideration the shape of the distribution when
choosing measures of center or they associated the word “typical” with any particular
measure of center.

Of the seven subjects from the North Carolina and Maryland sites interviewed,
two had reasonably high performance in the written instrument, two were about average,
and three had low performance. Interviews were conducted following the protocols;
besides the protocol questions, informal probing questions based on the specific answers
each subject gave to the prepared questions were asked to clarify ambiguous responses
and to unveil specific dimensions that seemed important.

Analysis

The interviews were audio taped. The analysis began by listening to the taped

interviews and transcribing them. The complete transcripts are in Appendix F. Then

each interview transcript was analyzed by item, and by domain of knowledge.

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By item: For those items where several interviewees are questioned, a summary table for
each item was created across subjects. Special attention was given to the method used for
answering the item, ways of thinking about a concept, ideas, ways of assessing students’
comments and responses, and other significant comments.

By domain of knowledge: As in the analysis of the written instrument, items that
measure pure statistical knowledge and knowledge for teaching were be combined within
each domain and responses were summarized. Special attention was given to the way
prospective teachers think of concepts in and out of the teaching context.

The results of the interviews were used to enrich the item analysis of the written

instrument and appear embedded when appropriate in Chapter 5.

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CHAPTER 5

MEASURING STATISTICAL KNOWLEDGE FOR TEACHING

This chapter addresses the second research question - what do prospective
teachers know about the various aspects of statistical knowledge for teaching? The first
section describes the performance of prospective teachers at the item level followed by a
section on overall performance. The last sections describe prospective teachers’
knowledge of statistics and knowledge for teaching. The final section summarizes the
findings.

Item Level Performance

The written instrument consisted of 8 free- response items with multiple parts.
Each item was graded on a scale of 0 — 4. For a general description of the rubric see
Chapter 4. Each item is followed by a description of what the item was designed to
measure and the corresponding distribution of scores. Next, the scoring rubric is
described followed by the analysis of results. Last, a summary and discussion of results
is presented.

Item I
Item Specification
Item 1, which involves a stem-and-leaf plot, was adapted from an example of an

investigation presented in Friel et al. (1997). It is designed to measure the three levels of

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graph comprehension identified by F riel, Curcio, and Bright (2001): (a) extract

information, (b) find relationships, and (c) move beyond the data.

 

The stem-and-leaf plot below shows the number of minutes it takes

students in a class to travel to their school.

Minutes to Travel to School

0 335789

1 02356689
2 013335588
3 05

4 5

 

a. How many students are in the class?

 

b. How many students took less than 15 minutes to travel to
school?

c. What is the typical time it takes for students to travel to
school?

 

Explain your answer.

 

 

 

Figure 5.1. Item 1
Table 5.1

Distribution of Scores for Item 1 (n = 42)

 

 

 

Score
Item 0 l 2 3 4
1a 4 0 0 0 38
1b 3 0 O 0 39
1c 3 8 6 9 16

 

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Description of the Rubric for Item I

For part 1a, a score of 4 was given to the correct answer, “26 students” and 0
otherwise. Similarly for part 1b, a score of 4 was given to the correct answer of “9
students” and 0 otherwise.

For part lo, a score of 4 was given to responses that calculated measures of center
correctly, that is, responses with the mean (18.46) or the median (18.5), or the mode (23).
Score of 3 was given to responses that reported measures of center of clusters followed
by appropriate statistical reasoning. For example, the following response got a score of
3, “the range of 20-28 is the largest range for minutes to travel to school, and there are 3
students who take 23 minutes. Therefore, I chose 23 minutes for typical time.” Also, a
score of 3 was given to responses with correct computation for the mean but rounded
from 18.46 minutes to 18 minutes. Score of 2 was given to responses that chose some
kind of measure of center but the explanation was vague. For example, “20 minutes.
About 1/2 between 16 and 25 min. which is about 20”. A score of 1 was given to
responses that chose some kind of measure of center but made a mistake in the

calculation or did not provide any work or explanation.

Analysis

In terms of correctness, 93% of the prospective teachers can successfully
“extract information” and “find relationships” from a steam-and-leaf plot. More
specifically, they can correctly count how many data points are represented in the plot
and how many of those data points are below a certain data value (see Item 1a in Figure

5.1 and Table 5.1). In contrast, only 59% of the prospective teachers were able to “move

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beyond the data” and provide a successful response (4 or 3 on the rubric scale) to the
question “What is the typical time it takes for students to travel to school? Explain your
answer.”

The responses for part 1c, which reflect prospective teachers’ conceptions of and
reasoning about what is typical and their reasoning behind their choice, show that
prospective teachers do not associate the word “typical” with measures of center. About
a third of them used the mean to describe a typical value, about 21% chose the mode, and
about 14% chose the median. The other third of prospective teachers chose an alternative
measurement or description. Table 5.2 summarizes the type of responses for this part of
the item.

Table 5.2

Distribution of Prospective Teachers’ Conception of “Typical”

 

 

 

“Typical” as Frequency
the mean 14

the mode 9

the median 6

a range of a cluster 5

a mode/median of a 5
cluster or stem

Other 3
Total 42

 

Here are some examples of responses in the category of “typical as a range”:
“Data clustered from 0 to 28.”
“10 to 25”

“midteens to nearly twenties, basically it seems by look to be what the average
would be”

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Explanations are more sophisticated for the category of “typical as mode or median of a
cluster or stem”. Here are some examples:

A stem of 2 has most students in that stem. Most of data centered around 20
minutes.

The larger cluster occurs at 23 minutes, there are also clusters at 16 minutes, 25
minutes and 28 minutes. So it takes students, roughly, 20 minutes to get to
school.

Most of the students were between 10-28 the majority between 20-28 with 23
occur 3 times while others 1 or 2.

Further inquiry was made about the interpretation of “typical” when six prospective
teachers were given follow up interviews. Prospective teachers were presented with a
second data set in which their previous interpretation of “typical” was no longer
appropriate. Every prospective teacher interviewed used the same method of finding the
“typical” value regardless of the distribution of the data set. The following vignette, from
Prospective Teacher A, illustrates the choice of picking the median because the data are
already organized. For this particular prospective teacher, another data set was presented

where her first choice for typical was not appropriate.

Int.: Part c of Item 1 refers to what is the typical time for students to travel to
school, and I am curious to know what each person thinks “typical” means.

A: It could mean any kind of a center, to me it could mean median, or it
means. . .you know. . .not normally I don’t look at mode, but it could mean mode.

Int.: So, you picked the median.
A: I thought it was easier, because it is already organized.

Int.: OK, so, let me ask you the same question for this other set of data.

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Minutes to travel to School

0 335567
100148
2112

3 5

4 33567

 

A: I would find the median, it is organized, so...

Int.: Would you still stick with the median?

A: Ummm. ..[pause]...it is about 18, right? To me it is easier if it is organized or

some kind numerical order.
Summary and Discussion

Prospective teachers show graph comprehension at the level of extracting
information and finding relationships. However at the level of moving beyond the data,
prospective teachers tend neither to use statistical measures nor statistical reasoning to
infer about the data. It seems that the performance at the latter level is influenced by the
undefined term of “typical”, which has several meanings.

Item 1 is a variation of an item used by Friel et al. (1997) to assess graph
comprehension for middle school students. Friel and others found similar results with
middle school students after instruction. They found that students’ alternative responses
to measures of center fit into two categories, responses that identify a cluster of time and
responses that provided a tally or range of numbers that occurred most frequently. An
attempt to use the same framework was done with prospective teachers expecting similar
reasoning. However, different patterns were found for prospective teachers. Prospective
teachers not only looked at cluster of times, but also focus on the center of those clusters

by finding the median or mode of the cluster.

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Item 2
Item specification
Item 2 measures knowledge of the apprOpriate use of measures of center based on
the shape of the data and the effect of outliers. This item is a variation of an item from
the Statistical Reasoning Assessment (SRA) developed by Lui (1998) and Garfield
(2003). In the SRA, the item has a multiple-choice format where incorrect and correct
reasoning were given as choices. In this study, the item was changed to an open-ended

format.

 

Nine students in a science class weighed a small object on the same scale
separately. The weights (in grams) recorded by each student are shown below:

6.2 6.0 6.0 15.3 6.1 6.3 6.2 6.15 6.2

The students want to determine as accurately as they can the actual weight of this
object. They may use the following methods:

I. Use the most common number, which is 6.2.

11. Use the 6.15 since it is the most accurate weighing.

III. Use the result of adding up the 9 numbers and dividing by 9.

As a teacher, what method would you prefer your students to use?
a. Method I
b. Method 11
c. Method 111
d. Other
Explain your choice.

 

 

 

 

Figure 5.2. Item 2

Table 5.3

Distribution of Scores for Item 2 (n = 42)

 

 

 

Score
Item 0 l 2 3 4
2 10 5 8 8 l 1

 

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Description of the rubric for Item 2

The level of correctness for this item was measured judging both the method
picked and its explanation. An ideal response (score 4) would involve choosing an
alternative method of throwing out the outlier and taking the mean of the remaining data
values with indication of the effect of the outlier in the explanation. For example, “I
would not want students to use method III because of the outlying data value of 15.3. If
this data value were used to find the mean weight it would raise the mean value (cause it
to be high). It would be best for students to find the mean of the data while excluding the
outlier value of 15.3”.

A nearly ideal response (score 3) would involve responses that gave indication of
the same correct reasoning as the ideal response but either did not communicate clearly
the reasoning or choose the mode/median being aware of the outlier and viewing the
measure of center as a balance point. For example, “I would use the mode because 15.3
would not give me an accurate average, it is considered to be an outlier. I would get rid
of the outlier and do the mean if I had my way” or “I usually would have the students add
up the numbers and divide by 9, but the 15.3 really throws the average off. So, 6.2 is the
most common number and all the other numbers are slightly above 6.2 or slightly below
6.2.”

A score of 2 was assigned to those responses that indicated awareness of the
outlier but chose the mode with an incorrect reasoning. For example, “the most common
is most likely the most accurate” or “measurement is not exact therefore having 6.2 as a

measurement seems practical”. A score of 1 was given to those responses that had no

129

evidence of awareness of the outlier and chose the mode or chose the mean but in their
explanation gave indication of the effect of outliers. For example “If the weight of the
object appeared the same a couple of times, then that would be the most accurate to me”
or “use the mean, this way each student’s measurement is weighted the same. Although,
if there were any outliers, I would have them throw it out.”

Finally, a score of 0 was assigned to those that chose any of the methods given
with none or limited explanation or gave incorrect reasoning. For example, “use the
mean because it is the method that can allow you to find the average” or “Taking the

average will be more accurate”.

Analysis

A few less than half of the prospective teachers (19 out of 42) provided a
successful response to this item (see Table 5.3). Although there was evidence in all of
the responses of the effect of the outlier, their explanations vary in complexity. Some
were very simple as “It will skew the data if a straight average is taken.” Others were
more sophisticated as:

I would not want students to use method I because it is not usually the case that
the most common answer is correct or close to correct. However, in this situation
because all of the data values (with exception of 15.3) are so close in value, this
method would not be a terrible method. I would not want students to use method
11 because the scale might not show zeros to the right of the decimal. If this were
the case, every other reading (i.e. 6.2) would actually have a zero on the end (i.e.
6.20). Therefore, every reading would really contain numbers to the same place
value. I would not want students to use method 111 because of the outlying data
value of 15.3. If this data value were used to find the mean weight it would raise
the mean value (cause it to be high). However, it would be best for students to
find the mean of the data while excluding the outlier value of 15.3.

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A few of the prospective teachers identified with the statement in the item that
says “As a teacher, what method would you prefer your students to use?” and used some
pedagogical language in their explanations; the following responses illustrate this point:

I chose this method because I would explain to my student that by collecting all

the data and figuring out average; that would be the most accurate way. I would

go on to tell them that if we notice most of the numbers are in a range of 6.0 to

6.3, therefore our average is around those numbers. I would tell that the reason

that we would not include 15.3 is because it is an outlier, that somewhere we

made an error because it is not consistent with all the others.

This is the method a real scientist would use. I would want my students to know

that. Also, the student who answered “15.3” needs some help with measuring

weights and balances. The students also need to know why methods I, II, and III

are not the best choices.

Furthermore, Prospective Teacher B thought that choosing a method in this case
is like choosing different strategies to solve a mathematics problem where the strategy is
a matter of preference.

Int. .' Why not method I?

B: You could, this is just another way to find the average. It depends of[sic] the

average you want, it is not wrong. As a teacher you have to teach and accept

other ways. But I would prefer the method I choose.

Unsuccessful responses also vary in their type, and many conceptions are
unveiled with this open-ended format. Prospective teachers that choose the mode or the
most accurate weighing as the most accurate method to determine the actual weight of the
object have different levels of reasoning. Their responses fit mainly three categories.
One category of responses is to choose the mode or most accurate weighing simply

because of effect the outlier, for example:

I choose method I because I felt the 15.3 would give a false average and there is
no way to tell if 6.15 is the most accurate weight.

131

I would tell the students to pick the mode instead of mean because the last weight
of 15.3 is too extreme to get an accurate measurement.

I choose method 11 because the most common doesn’t make the weighing the
most accurate and the average would be skewed by the 15.3.
Another category fit responses that chose these methods because of the conception that
accurate measurements are reflective of the ability of a person to read a scale. These
subjects do not view each observation as deviating from the actual weight and there is no
evidence that takes the outlier into account.

Because when weighing something and in any case when the majority of students
make an experiment and get the same answer, then I would go with the majority

Because if the weight of the object appeared the same a couple of times, then that
would be the most accurate to me. In this case I might have the students take the
average of 6.0 and 6.2 b/c they both appear more than once.

Small object is always the same weight; looked at the data to determine 6.2 since
it is the mode.

Finally, a third category fit those responses that do not show evidence of
statistical reasoning. Such as

The mode is the most reflective of the data set.

Might want to think about 6.15 because it is more ‘accurate’ maybe round it.

I would want my class to understand the importance of using the most accurate
weight, so I would definitely have the class use method 11.

Although choosing the mean (Method 111) in this problem was a measure of
failing to take outliers into consideration in the Statistical Reasoning Assessment (SRA)
developed by Garfield (2003), five out ten prospective teachers that chose the mean said
that they would pick it because they want to take into account the low and high values or

because they want representation of every data value.

132

The best way to find the average weight in this example would be to find the
mean. That would take into account the higher numbers, lower numbers, and
the numbers in the middle. If the mode was used, the high and low end of the
results wouldn’t even be seen.

Every student has different weights on a small object. . .. It is unfair if I use

other numbers instead of the average of their weights. Also, these numbers

may not be the actual weight of this object.

This way, each student’s measurement is weighted the same.

Method III, because I would want the students to find the average of mean of

the weights they found. That way instead of picking one of their findings, the

class would be able to see the average weight from all of their findings.
Again, only two students (out of the ten that picked the mean) viewed the average

as a signal in noise interpretation. However they failed to take the outlier into

consideration.

Sometimes or most of the times the weight is off because of human error so avg
out all the weight should come out to be more precise.

 

I would use the third method, for several reasons. There is always room for error
when measuring an object. How are students supposed to know that 6.15 is the
most accurate weighing? In using the third method, I would make sure that the
same object would be used for each group. Just because one # may be more
common than others, does not mean that is the most accurate, so I would not use
the I method. You could have a tenth group who got 6.0, what would the students
then do?

A deeper view of prospective teacher’s reasoning is illustrated with Prospective
Teacher C’s comments during the interview. Prospective Teacher C chose to use the
mean even though she recognized the effect of the outlier. Her thinking revels that she
knows for a fact that the mean is sensitive to high values but when she looks at the

distribution of the data, she is confused.

Int. : You said that you chose method III (the mean) because it is more accurate;
what do you mean by that?

133

C: because it’s clustered all around a close sort of range. I thought OK with the
mean. . .you know. . .I thought the mean and the median might be close in this
situation, but some times when you have a really, really high, like say you have a
score of 8.5, I might want to say I want to ignore the 8.5 and do median instead. I
say that the extreme score is going to weight the mean more than it would with
the median.

Int. .' and in this case...

C: In this case the mean and the median are going to be really close, because the
data are all clustered together.

Int. : around. . .cluster around. . .

C: Well, you have the one... that’s what brought me to think about it. Like I said,

well, even though I say maybe we want to take that one into account, because

sometimes you do. . .so I said, well it might be weird but maybe we should take

him into account . So I thought maybe in this situation you want to. That’s why I

said the mean, I just didn’t know. They are all clustered except for that one guy,

that’s what made me think, “Do I really want to take it into account or not?”
Summary and discussion

Prospective teachers are able to identify outliers in a data set but not necessarily to
use their effect to choose the appropriate measure of center. Some actually want to take
the outlier into account for representative reasons; others for pedagogical reasons. It is
important to note that prospective teachers may be using this reasoning because the item
asks “to determine as accurately as they can” and one of the choices is to “use the 6.15
since it is the most accurate weighing” given the indication that the task is about finding
the most accurate measurement and not the most appropriate measure of center.

Garfield (2003) describes the results of a version of this item given to college
students. Her version was multiple choice with the three methods given here plus a

fourth method: compute the mean with the outlier thrown out. She concluded that

college students who do not choose the fourth method are ignoring the outlier. Due to the

134

design of the item used in the current study, however, we can see that the attraction of the
mean (method III) appears to be stronger and more complicated than first thought.
Results here indicate that prospective teachers may recognize the outlier but still choose
the mean (method III).

Konold and Pollatsek (2002) give this item as an example of a context in which
the average is interpreted as signal in noise, where the average is a close approximation
of the actual weight and each observation is viewed as deviating from the actual weight
by a measurement error. There was no evidence that prospective teachers that chose to
use the mean after throwing out the outlier take this view. Only 2 out of the 19 mention
that the average “would help reduce error”. Others mention that the average will be the

“most accurate way”.

135

Item 3
Item specification
Item 3 measures graph comprehension (histogram) at the level of interpretation.
Part (3) measures the ability of the prospective teacher to recognize a student’s
misconception and to interpret the source of misunderstanding. Part (b) measures
interpretation and judgment of a student’s oral response. This item is a variation of a
problem from Collecting, Representing, and Interpreting Data Module developed at the

, Center for Research in Mathematics and Science Education at San Diego State

University.

 

The following graph gives information about the adult female literacy rates in
Central and South American countries.

 

3 - r__

Frequency

 

 

 

 

 

 

 

45 50 55 60 65 70 75 8’0 85 90 95 100 I

Adult Female Literacy Rate (%)
Adult Female Literacy Rates in Central and South America

a. Suppose you ask your students to tell you how many countries are
represented in the graph. One student says, “there are 7 countries represented”.

Is this student right or wrong?

 

In your opinion, what is the student’s thinking to arrive to that conclusion?

b. Suppose now you ask your students to explain what the third bar from the
right indicates. One says, “It indicates 85% to 90% literacy rate”. Comment
on the response.

 

 

 

Figure 5.3. Item 3

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Table 5.4

Distribution of Scores for Item 3 (n = 42)

 

 

 

Score
Item 0 l 2 3 4
3a 1 6 9 3 23
3b 15 4 3 5 l 5

 

Description of the rubric for Item 3
For part 3a, a score of 4 was assigned to responses that stated that the student is
wrong and identified the proper source of the mistake, counting bars. A score of 3 was
assigned to responses that answer the question with “yes” but identified correctly the
source of the mistake. A score of 2 was assigned to responses that stated that the student
is wrong, identified the mistake of counting bars, and suggested the wrong representation
of the bars. This is an example of a response with a score of 2, “The student is wrong.
The student thinks that each bar in the histogram represents a country, not the frequency
of adult female”. A score of l was assigned to responses that either stated that the
student was wrong and did not identified correctly the source of mistake or stated that the
student was neither right nor wrong because there was way to tell the number of countries
represented. Finally, a score of 0 was assigned to responses that stated that the student
was right. This is an example of a response with score of O, “The student is right,
different bars shows different rates. Every country has its own rate. So bars could be
countries.”
For part 3b, ideal responses, which were assigned a score of 4, were expected to
say that the explanation is incomplete and the student needs to add that there are three

countries with the rate mentioned. A score of 3 was assigned to those responses that only

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stated that the student needs to focus on the frequency as well, without providing
evidence of understanding of the label “Frequency”. Responses that only mentioned that
the comment of the student is incomplete or mentions in general terms, like “the student
is only look[sic] at the information on the x-axis and not how the x and y axis relate.”
received a score of 2. Less clear responses such as, “you are on the right track, but look
closely at what percent rate goes with what bar” received score of 1. Scores of zero were
given to those participants who said that the student’s comment was correct or provided

an incorrect interpretation of the bar in the histogram.

Analysis

For part 3a, the about three fourths correctly identified that the student’s statement
in part (a) was incorrect. Nine said that the graph did not provided enough information to
determine if the student was right or wrong, and only one said that the student was right
(see Table 5.4). Out the 32 prospective teachers who correctly said that the student was
wrong, 20 also identified the source of the mistake by saying that the student focused
only on the number of bars (see Table 5.5).
Table 5.5

Distribution of Responses to Item 3a by Recognition of Mistake and Its Source.

 

 

 

 

Student is Student is Can’t tell/Don’t Total
_ right wrong know
Correct source of 0 20 0 20
fistake
Incorrect source 1 12 9 22
of mistake
Total 1 32 9 42

 

138

Some prospective teachers went further in their explanation and explained what
the student is not focusing on, providing information of their own knowledge of graph
comprehension. Some of these extended explanations were correct and some were not.
Examples of correct explanations are:

Each bar actually represents a range which had at least 1 country reporting that
score.

Student did not take into account that the height of these bars showed the

frequency (or number) of countries within a specific range of % females who are

literate. Had the students correctly read the graph, he or she would have noticed

that 15 countries are actually represented.

The student is thinking that each bar represents 1 country, not that the height of

the bar tells the frequency of the countries.

Examples of incorrect explanations varied and fit different categories. This type
of responses was assigned a score of 2 or 1 for level of correctness. Some prospective

teachers thought that the student is not “reading” the x-axis correctly.

The student looked at the graph and counted the bars, but either did not read the
x-axis or doesn’t have a grasp on reading graphs.

The student is looking at the bar graph and assuming that the x-axis represents

seven different countries, when in actuality it does not. I can understand why a

student might interpret the bar graph that way, because it is misleading.

Other prospective teachers thought that the student made the mistake because he
or she did not understand what the bar represents providing an incorrect explanation for
the bar representation. These prospective students did not provide a successfisl response

for the second part of the problem, either. Examples of misconceptions about the

representation of the graph fit mainly two categories. Some prospective teachers thought

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that the bar represents the percentage of literacy rate in Central and South American
countries.
“The bars represent the % of literate adult females in the central and ...”

“The student thinks that the adult female literacy rate(%) represents the different
countries”

“They counted the bars on the graph, not realizing it is the literacy rate (%)”

Others thought that each bar represent the frequency of adult female.

The student thinks that each bar in the histogram represents a country, not the
frequency of adult female literacy rate.

Due to frequency, I think that means the number of women at the Adult Female

Literacy Rate.”

The nine prospective teachers who said that they could not tell whether the
student was right or not gave the following explanations:

The graph is not explicitly detailed enough to be able to tell, depending on the
grade level some students will say 7 countries.

We don’t know if the student is right or wrong. He is assuming that if there is
some frequency of female literacy in every country it would register.

Yes and no, yes there are 7 bars, but there may be stats that fall on a frequency of

0.

Results for part 3b are consistent with the previous part. About half of the
prospective teachers successfully recognized that the student had a partially correct
answer and correctly identify what was missing in her/his explanation (see Table 5.4).
Four of those who answered correctly showed some pedagogical knowledge.

The third column from the right does indicate a 85% - 90% female literacy rate,
but it’s that in one country, more than one country, or the entire region? I would

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then ask the student to examine and explain what this graph is representing to get
him or her to answer my previous question.

I would ask 85% to 90% literacy rate of what? I would ask them open-ended
questions to get them to see that it explains that 3 countries have a 85% to 90%

literacy rate.

I would ask the student what the graph is telling us about that rate. What does the
height of the bar indicate?

Seven responses received scores of 2 (n = 3) or 1 (n = 4) and 15 responses fall
into the category of zero score. Four of the 15 said that the student was correct or the
response was “an excellent response”. Three thought that the student’s comment was
incomplete by neglecting to say that “the frequency of 3 tell us how many women had
85% to 90% literacy rate”. The rest provided responses that gave evidence of no
knowledge for teaching. Such as

It is the interval of the literacy rate and not specifically 85% or 90% literacy rate.

This response is vague, does not define what ‘it’ is that indicates 85%-90%
literacy rate.

I would explain that since it is a histogram, the bars are connected, but in reality

that piece of data ends for 85% it is just connected to 90.

In particular, when Prospective Teacher A was asked to elaborate on her written
response about not knowing if the student was right or wrong for the first part of the item,
she indicated her struggle to understand student’s mistakes.

Int.: When you say that “we don’t know” if the student is right or wrong, do you
mean that we don’t have enough information in the graph to say how many countries
are represented?

A: We assumed that we have data for everybody, but apparently we didn’t .
Sometimes you have something missing, like I thought, OK there is quite a bit
missing here [pointing at the gap between 50% and 70%] why do we lay out our

histogram this way, and I thought, maybe there is something where there was no
literacy rate. . .some country where there is zero.

141

Int. : That’s why those gaps?

A: You never know. . .I though it was a strange histogram. Normally, you would just
said, well you have a bar for everything, but you don’t always know that, right? When
you have a gap, you don’t always know about your gap. This bothered me a little,
this gap.

Int.: Why do you think the student said 7?

A: I don’t know, because he wasn’t really looking ...well, he was only looking at one
bar or two and then saying. . .you know what I mean. I couldn’t figure. . .error with
kids always bothers me. . .in math. . .I look at the error and I go. . .it is hard to know
what they mean.

Furthermore, the prospective teachers seemed to have a difficult time correcting
the mistake or guiding the student in the right direction when they themselves have other
limitations. In this item, some prospective teachers were able to say that the student was
wrong in saying that there were 7 countries represented and that the student was in fact
counting bars. However, when asked in an interview how they would correct the mistake
or guide the student to a correct answer, prospective teachers did not seem to understand
the information given by the histogram.

Int.: You said the student is wrong because...

D: The graph said is for Central and South American countries, it does not say which
ones.

Int.: But you do understand why the student said 7 bars?
D: The student is counting bars.
Int. : What would you tell the student to lead him to the right answer?

D: I would tell him to look at the height of the graph and remember always to follow
the height of the bar to the left to see how tall it is and that is the frequency.

Int.: If you ask the student what does that mean, frequency 3, what would you want
him to tell you?

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D: The bar indicates 85% to 90% literacy rate for 3 women.

Discussion and summary

About 75% of the prospective teachers can recognize a typical student’s
misconception when reading a histogram and about 50% can also identify the reason why
the student is making the mistake. However, that does not necessarily mean that the
prospective teachers have other limitations with the histogram and are unable to correct
the mistake properly. In other words, prospective teachers correctly make the judgment
that the student is wrong but give the incorrect way to correct the mistake.

Results here show that prospective teachers misinterpret the representation of the
bar in a histogram mainly because there is only one variable labeled “Adult Female
Literacy Rate (%)” and they do not understand how to interpret the meaning

“Frequency” in the content of the problem.

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Item 4
Item specification
Item 4 measures knowledge of calculating the mean from a line plot, which
implies assessing for knowledge of graph comprehension and the concept of the mean
beyond the algorithm. It also measures properties of the mean and the ability to create
distinct distributions with the same mean. Graphical displays were taken from materials

developed by the Connected Mathematics Project (Lappan et al., 2002).

 

The following line plot shows the number of people in households in a neighborhood.

Number of People in Households

X X
X X X X

1234567

 

a. Find the mean. Show how you find it.

b. Is it possible to have other sets of data with the same mean?
Explain why or why not.

c. Is it possible to have a data set of six households with mean 3% people?

If yes, give an example.
If not, explain why.

 

 

 

Figure 5.4. Item 4

Table 5.6

Distribution of Scores for Item 4 (n = 42)

‘

 

 

Score
firm 0 1 2 3 4
4a 3 0 4 7 28
4b 5 6 9 1 5 7
_‘4c 1 0 0 4 4 24

 

144

Description of the rubric for Item 4

For part 4a, a score of 4 was assigned to responses that correctly stated that the
mean was 4 people per household and provided a valid justification. A score of 3 was
assigned to the responses that stated the correct mean but provided an incorrect equation
or made a small computational mistake. A score of 2 was assigned to responses that
either divided by 7 instead of 6 or did not give any justification. No score of 1 was
assigned, and a score of O was assigned to responses that were blank or gave incorrect
reasoning.

For part 4b, a score of 4 was assigned to responses that answered “Yes” to the
question “Is it possible to have other sets of data with the same mean?” and provided a
valid justification based on the algorithm connected to the concept of the mean as a
balanced point of a distribution. A score of 3 was assigned to responses that stated “Yes”
but only provided an example of another distribution with mean 4. A score of 2 was
assigned to responses that said “Yes” but provided an unclear justification like, “there are
many sets of data with the same mean”. A score of 1 was given to even more unclear

j ustifications, like, “Yes, it is possible” or, “Yes, as long as the average = 4”. Finally,

Scores of 0 were assigned to responses of “No” or blanks.

For part 4c, a score of 4 was assigned to responses that stated “Yes” to the
q nestion “Is it possible to have a data set data set of six households with mean 3%
people?” and provided a correct distribution as an example. A score of 3 was assigned to
responses that stated “Yes” and in the attempt to create an example made a small mistake.
A Score of 2 was assigned to responses that stated “Yes” but did not provided an example

or Stated an unclear justification like, “because the mean is the average it does not have to

145

be whole number”. A score of 0 was assigned to those responses that stated that is it not

possible to have a mean of 3 1/2 because of the fraction.

Analysis

For part 4a, which measures the ability of prospective teachers to “extract
information” from the graph, the results are surprising. About two thirds of the
prospective teachers correctly found the mean and wrote a valid mathematical equation to
justify their answer (see Table 5.5). About 16% of the prospective teachers successfully
identified the data values and the number of data points but made a notational error of
equating the value of the sum to the ratio and therefore to the mean. An example of this

mistake is shown Figure 5.5.

 

Show how you find it.
2+5+3+ mow: 3% T (v =4

 

 

 

Figure 5.5. Example of a Typical Notational Mistake from a Prospective Teachers’
Work.

Only 2 students failed to actually divide the sum of the data value by the correct
number of data points, 6. Instead these prospective teachers divided by 7. Note that there
are 7 numerals on the axis of the plot. All of the prospective teachers found the mean
using the mathematical algorithm. Only one person attempted to use the idea of center
and unfortunately he used it incorrectly, saying, “using the idea of ‘cluster’ point

associated with the mean, the mean is 3, for the data set above.”

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The most preferred method to compute the mean was to identify each data value,
add them up and divide by the number of data points, reading the graph from left to right.
For example, (2+3+3+4+6+6)/6 = 4. A quarter of the prospective teachers that
successfully found the mean used a weighted mean approach by identifying the data

2(1) + 3(2) + 4(1) + 6(2)
6

 

value and its frequency. For example = 4 . One person read the

data values in horizontal “layers”, (2+3+4+6+3+6)/6=4.

As expected, almost all of the prospective teachers could compute the mean of a
given distribution. Interviews aimed to investigate if they could find or estimate the
mean for a larger data set without reaching for the algorithm by providing either a larger
data set where computation is not practical or asking them to find the mean by another
method. The scenario used in the interviews was related to teaching. The interviewee
was first presented with a new bigger data set (see Figure 5.6) and then asked how would

a student find or estimate the mean without doing any computations.

 

X x

X x

X x x

X x x x x x

X x x x X x x

X x x x X X X X x X
26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

Number of Raisins in a Box

Figure 5.6. Data Set Used in Interview to Ask Prospective Teachers to Find the Mean
without Computational Algorithm.

Prospective Teacher D had a method to estimate the median from a line plot but

struggled to find a way to estimate the mean for a larger data set. In the intent to estimate

147

the mean, she “moves Xs around” and tried to relate the concept of balance but ended up
finding the average of the frequencies instead!

Int. .' Suppose you are teaching the mean to your students and you want them to
estimate the mean before they use the algorithm. How could they do it?

D: They could go. . .count one from each side [crossing out an X on each
side]. . .they can take the average of the two left over and the answer will not be 4
but very close to it.

Int. .° How about with a bigger data set?

D: Do the same. . .cross out one on each side, one, one, two, two,. . . [cross out one
X on each side and ends up with two X’s under the number 31], you have two
31’s, then you would not have to divide by two. The answer is 31.

This is the median.

Int. : How about the mean? How can the students see the picture and estimate the
mean?

D: It would not have to be exact.

Int. .' No, but we want to find a good estimate.

[Long pause]

Int. .' So the median would be easier to estimate for students than the mean?

D: That is what it seems to me. . .because I can’t do it off top of my head. . .Maybe
arrange the X’s ...like I would put this X down here and see how they average it
out. . .that is the only way I can see how to do it. I would send them to the board
and have them arrange the X’s and make them line them up and that will be the
average.

Move these over here, and erase these.

All the X’s are even now, that would be three. Humm. . . [pause] I guess that is 3
raisins hold on. . . [more thinking]...I am trying to figure out...

You could do adding up again and then divide. You add 28, 28, and 28 or 28
times 3.

Parts of 4a and 4c were designed to measure the ability of the prospective teacher

to explain why it is possible to have different distributions with the same mean and to

create distributions with a particular mean. About half of the prospective teachers gave a

148

successful response to Item 4b by providing a mathematical argument, or giving an
example of different distributions with equal means in the context of the problem (see
Table 5.6). However, only a third of those provided the ideal response of connecting the
algorithm with the concept of the mean as a balance point of a distribution. Following
are two examples of mathematical arguments given.

Yes, because the average of numbers can be the same depending on what # you
are dividing by and what #’s you are adding.

All it takes is to have a numerator divisible by the denominator that comes out to
be 4.

Yes, as long as the total is 4x the number of responses.
About 12% of prospective teachers gave arguments in relation to the context of the
problem — people in households — to explain the possibility of having another data set
with the same mean. Here are some examples of these types of responses.

Sure, the data could have been 20 people and 5 households, or 28 people and 7
households.

Yes, the family that has 2 children and a family that has 3 children could move
out of the neighborhood, then two new households could move in one with 1

person the other with 4 people.

Yes, a house of 4 can be replaced by two houses of 2, and two houses of 3 can be
replaced by one house of 6.

Yes, you could. If you still only asked 6 households, you could still total 24

people. Data could be 2,3,3,4,5,7 = 24/6=4 or could ask less households have

12/3=4 or 32/8 if you asked more households. 4 has many multiples.

A third category of responses is one that talks about distribution, sample size, or
variation. However, these types of responses were too vague to be accepted as valid

arguments. Following are some examples.

Yes, the distribution can be proportionately spread out.

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Yes, you can have a smaller or larger sample size that may/may not add to 24 but
still possible to have the same outcome.

Yes, other sets could be clumped together around the 4 or spread out.

Prospective Teacher E - who justified her answer for this part of the item by
relying on the algorithm - was asked about this item during the interview.

Int.: How would you convince the children that is possible to have other data set
with the same mean without reaching for the algorithm?

B: You can start with the same numbers and then switch them around , like if you
have 4,4,4 take away one from the first 4 and make it into 3 and then put one
more and make a 5. Just playing with numbers.

Int. : Can you do it with the bigger data set?

E: Let’s see. . . [pause] I guess you should move... I don’t how to do it with data, I

just know how to do it with numbers. Because you can’t move an X to 39
because this represents 3 of 38, isn’t?

As for creating a data set with a specific mean (see Item 4c), about 70% of the
prospective teachers were able to create a correct data set with the specified mean and
number of data points. About half of those chose a symmetric distribution about the
point 3.5, and the other half chose a non-symmetric distribution but with clear
indication that they created it by finding the sum of the data values first (see Figure

5.7).

150

 

 

 

 

Figure 5.7. Example of a Procedure Used by a Prospective Teacher to Find a
Distribution with Mean 3.5.

Finally, about 15% of prospective teachers argued that it was not possible to have
six households with a mean of 3.5 people because when it comes to people, it does
not make sense to have a non-integer value (see examples of responses below).

No because a household can not be represented by half point.
No! because the mean 3 '/2 people is logically incorrect.
No, because there is no such thing as 1/2 people.

You can have a mean w/ 3.5, but not when it comes to people, because you
can not have half of a person.

Prospective Teachers B and C were asked what it means to have a mean of 3.5
people per household and how they would explain it to a child. Prospective Teacher B
seems to have a good way of explaining.
Int. : What does an average of 3.5 people per household mean?

B: That is the mean, you take all the numbers and add them all up and divide by the
number you have.

Int.: Suppose you are explaining to a kid. Would the kid be bothered by the 3.5?
B: I don’t think so, if you explain it in certain way.
Int. : Like what?

B: If you say, well nobody really has 4.5 kids, but over a big data set some people
have 2 kids, some people have 3, some people have 6, some people have 8. So over a

151

big data set it could be 4.7, it could be 3.2, it just means over a big data set that is
your average. I think kids understand that.

You have to say, well, you take an average and the average means that that matters,
that every one matters, that’s why you would get 4.7 kids. It just means that people
are all over the range, some people have 1, some people have 2, 3, 4, 5, 10.

Int.: Now, I am going to play the kid. The kid would say “why doesn’t the answer
come out to be a whole number”?

B: I think you have to teach them about the median, maybe, that there is another
number that could be a whole number, but not always. Even the median could be 4.5,
but...

In contrast Prospective Teacher C was unable to explain the meaning of a mean of 3.5

people per household.

Int. .' What does and average of 3.5 people per household mean? How do we interpret
this number?

C: I guess when the average. . .many many houses average about 3.5. I don’t know

what that means; I don’t know how to explain what that means though. Because the
kids are gonna go “what is 3.5 of a person?!”

Summary and discussion

About half of prospective teachers are able to compute and estimate the mean of a
small data set represented in a line plot. When given a bigger data set and asked to
estimate the mean, they either estimate the median without observing the distribution or
in their attempt to find the mean; they find the average of the frequencies instead. The
latter seems to be a reflection of the methodology taught in elementary and middle school
of leveling off stacks of cubes.

About half of prospective teachers know that it is possible to have many sets of

data with the same mean. However, only a third of those can justify it with an argument

152

that relies on both the algorithm and the concept of the mean as a balance point. About
two thirds of prospective teachers are able to create a distribution with a specific mean
that is not a whole number.

More than half of prospective teachers interviewed and probed for knowledge for
teaching did not show sufficient evidence of this type of knowledge. Most could not
estimate the mean of a data set without reaching for the algorithm, or describe how to
convince a child that there could be several data sets with the same mean, or explain what

a average of 3.5 people means.

153

Item 5
Item specification
Item 5 measures knowledge about formulating questions to generate data and
measures of center and range for categorical data. Furthermore, this question measures
the ability of prospective teachers to identify errors in students’ responses. The data and
suggestions for student responses were taken from materials developed by the Connected

Mathematics Project (Lappan et al., 2002).

 

One middle school class generated data about their pets shown below.

 

Pet Frequency

bird 2

cat 4

cow 2

dog 7

duck l
2
l
3
3

 

fish
goat
horse
rabbit

 

 

 

a. Give a possible question the teacher could have asked
the students to generate the data.

b. Students were talking about the data and one said:
“ The mode is dogs, the median is duck, and the range is 1 to 7.”

If you think the student is right, explain why.
If you think the student is wrong, identify the mistake(s).

 

 

 

Figure 5.8. Item 5

Table 5.7

Distribution of Scores for Item 5 (n = 42)

 

 

 

Score
Item 0 1 2 3 4
5a 14 0 0 7 21
5b 1 8 21 2 1 0

 

154

Description of the rubric for Item 5

For part 5a, ideal responses that received a score of 4 were those that correctly
stated a question that could have generated the data. A score of 3 was assigned to
responses that did not state the answer in a question format but provided a procedure or
way to generate the data. No scores of 2 or 1 were assigned. A score of 0 was assigned
to responses that stated a question that can be answered about the pets, such as, “What pet
does the majority of the students have?” or, “How many pets are there in all?”

For part 5b, the ideal response was to state that the student is only right when he
or she says that the mode is “dog”. The median and range do not apply to categorical
data such as type of pets. A score of 3 was assigned to responses that came close to the
ideal responses by giving some indication that the median and range did not apply in this
case, but made some incorrect statement. A score of 2 was assigned to responses that
stated that the mode and the range were correct, but the median was not because of the
type of data. A score of 1 was assigned to responses that stated that the student was right
saying that the mode was “dog”. A score of O was assigned to responses that did not
make any correct judgment about the student’s statement.

Analysis

For part 5a, half of prospective teachers were able to create a correct question that
teacher could have asked the students to generate the data. Seven out of 42 did not write
their response in a question format, but indicated a way to create the given Table 5.. Of
the remaining 14 responses, 12 stated a question that could be answered using pets and 2

did not provide any response.

155

The typical question created by prospective teachers that could be asked to
generate the data was of the following kind.

“Everyone tell me what kind of pets you have?”

“What type of pets do you own and how many?”

As before, some prospective teachers identify with the teaching context and
provide responses with some pedagogy related. Here are some examples.

List all the different types of pets and have the students come up to the board and
put a tally next to the pets in their home.

Make the sound of your animal (that you own) and get a group by making those
sounds. Now select a leader to count the number of kids in the group! How
many of you own one of these animals. (Provide chart).

The second part of this item turned out to be the most difficult question of the
assessment instrument. No one responded correctly that the student is right when he talks
about the mode, but for categorical data the definition of measures of center and spread
do not apply in the same way as they do for numerical data. Only one person got close to
a correct response (score of 3) by saying:

The data is showing how many people have each type of pet. When we refer to

measures of central tendency, we are using numerical data. In this case dog is

the most frequent choice, but it has no numerical value. . . .We cannot order non-
numerical data. ...There are no data values to subtract, you do not use frequency

to find the range, and if you did the range would be 7-1 = 6.

Two others showed some awareness of the type of data and mentioned something
about not making sense of measures of center in this case. These two responses fall into
the category of unsuccessful with the score of 2.

It is difficult to arrange non-numerical data into the same type of

asending/desending[sic] order that numbers can be arranged into. ‘Dog’ is the

most frequent response, so I might agree that it could be considered the ‘mode’.
However, ‘duck’, although it falls in the middle of the alphabetic listing, can

156

hardly be considered the median (since 9 answers are past it in the list and 15 are
before it). Alphabetically, the range would be ‘bird to rabbit’.

The student is right on the mode & the range but wrong on the median part.
Median can’t be duck. It has to be a number. Median is the average # of animals
in each type of pet that a one middle school class should have.

The rest of the responses split into those that recognized that the student is
correct in saying that the mode is “dog”, but were incorrect about of the judgment of the
rest of the student’s statement (21 out of 42) and those that incorrectly judged the entire
statement, receiving a score of zero (18 out of 42).

The typical response for the 21 that received a score of one is that the student is
wrong about the median being “duck” because the student is just finding the middle
animal in the list and he/she needs to took into account the frequency as well.
Furthermore, prospective teachers take the frequency into account in many different
ways, coming out with at least five different strategies to find the median.

One strategy (4 responses) is to find the median by ordering the frequency values
and picking the middle value. This is illustrated by the following response:

“The mode is dogs and the range is I to 7. However, the median is
2....112®347 “

The second strategy (2 responses), is not to pick the middle value of the
frequencies but to pick the corresponding pet the goes with the middle frequency. As
shown in Figure 5.5, the prospective teacher picks “cow” as the median.

A third strategy (1 response) is similar to the previous one. However, instead of
picking a pet arbitrarily within the frequencies, the names of the pets are listed in

alphabetical order. The median would then be the pet that corresponded to the median of

157

the frequency values (see Figure 5.6). When this prospective teacher was asked in the
interview “you said that the median is ‘fish’, what does this tell you about the pets?” She

responded, “that the average student in the class have fish [sic]”.

 

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Figure 5.9. Example of a Prospective Teacher’s Strategy to Find the Median of
Categorical Data.

A fourth strategy (3 responses) is to order the frequency values, not with the
purpose of finding the median of these values, but with the purpose of making a list of the
25 pets being considered: {duck, goat, bird, bird, cow, cow,. . ., rabbit, cat, eat, eat, cat,
dog, ..., dog}. The median is the13th pet in the list, namely “rabbit”. As one student put
it “. . .the student incorrectly stated that the median is duck, because he or she failed to
place the data in order of increasing frequency first. Once the data is arrange [sic] in the
correct order, it is clear that the median is rabbit.”

Finally, the filth strategy (6 responses) is a modification of the fourth strategy.
Instead of ordering the list of pets by its frequency, the prospective teachers decided to
keep the list in alphabetical order: (bird, bird, cat, cat, cat, cat, cow, ..., dog,. . ., rabbit,
rabbit, rabbit}. Then, they picked the 13th pct, which in this case is “dog”. In their own
words “. . .median is incorrect. It should be the pet that is listed in the middle of the 25

pets, which would be dog.”

158

One of the students who used the fifth strategy argued that this is the way to order
the animals “. . .because when finding the median, you’re suppose to put it in order from
smallest to largest and when it comes to animals how to judge which is larger or smaller.
There are dogs smaller than ducks, ducks smaller than cat and vice versa.”

With respect to the statement about the range being 1 to 7, 27 out of 42
participants said that the statement was correct. The typical explanation was that the
minimum number of pets is 1 and the maximum is 7, thinking more in terms of the range
of the frequency values. Another group of participants (5 out of 42) said that the student
was wrong because he or she should say that the range is the difference between the
largest and the smallest number, in this case it would be 7 —1 = 6.

Only one prospective teacher gave evidence that the range should be consistent
with the other measurers in terms of non-numerical value and said “. . .alphabetically, the

range would be ‘bird to rabbit’”.

Summary and discussion

Almost all prospective teachers can create or formulate questions to generate data.
However, most can not recognize students’ mistakes about median or range for
categorical data. Most prospective teachers could not describe what the measures of
center of spread tell about the data when trying to find them. Instead they use a
procedural method to get an answer. In this particular case, prospective teachers were

very creative inventing their own methods to have a median for categorical data.

159

Item 6
Item specification
Item 6 measures the prospective teachers’ ability to select and create a suitable
representation for a data set, taking into account the shape of the data.. The data for this
item was taken from materials developed by the Connected Mathematics Project (Lappan

et al, 2002).

 

 

 

 

 

 

 

Student’s Time in
A middle school class was wondering how much time ”B‘s" ”'23“
it took each student to travel to school each morning DD so
and collected the following data. iii: 33
Students are asked to make a graphical display of the data to FH 25
show how much time the majority of students take to travel 3:; i;
to school. EN 20
vn 20
1w 17
a. Which one of these plots seems the most appropriate AS 15
for the data? 52 :3
AS 10
- MS 8
Cl Histogram C] Stem-and-leaf Plot RS 5
El Bar Graph Cl Other
Explain your choice.
b. Make the plot picked in part a.

 

 

 

Figure 5.10. Item 6

Table 5.8

Distribution of scores for Item 6 (n = 42)

 

 

 

Score
Item 0 1 2 3 4
6a 1 3 6 7 12 4
6b 7 0 5 2 28

 

160

Description of rubric for Item 6

Correct responses for the first part of the item, 6a, were the choices of stem-and-
leaf plot or histogram together with a correct explanation. Prospective teachers were
expected to justify their choice by mentioning that these two representations would show
the shape of the data and could answer the question in consideration. In addition, if they
chose steam-and-leaf plot, they were expected to say that this plot shows all the data
values. These responses were assigned a score of 4.

Still considered successful responses, but with a score of 3, were those responses
that chose stem-and-leaf or histogram for practical reasons related to the type of data such
as its range, frequency of 205 and teens, etc. These responses did not attend to the
question needing to be answered -— how much time the majority of students take to travel
to school?

Unsuccessful responses with scores of 2 or 1 were those that chose the
appropriate plots but provide a non-statistical reason such as — it is the quickest, it is less
confusing, it would suit it better, it would give the best visual, etc. Scores of 0 were left
for those that chose a bar graph with incorrect reasoning or provided no response.

For part 6b, a score of 4 was assigned to a correctly made graph that was
consistent with the choice in part 6a. A score of 3 was assigned to graphs with minor
mistakes, such as miscalculating the frequency of data values within an interval when
constructing a histogram. A score of 2 was assigned to graphs that contained major
mistakes, such as not showing the stem of 403 because there were no values in the 403 in

the data given. No scores of 1 were assigned and a score of 0 was assigned to blank

161

responses or responses that showed a graph that was inconsistent with the choice in part
6a.
Analysis

Results show that almost 40% of prospective teachers provided successful
responses. Approximately 30% scored 2 or 1 and another 30% scored 0.

A typical reasoning prospective teachers stated for choosing a bar graph is that
they thought that it was important to pick a graph that would show the time with the
corresponding student, claiming that this is more powerful than any other choice. One
used a different reasoning.

I would use a line graph, only because there are two pieces of info given to the

student: initials and time. If there were only one piece available to the student

then I would use a stat plot, or a histogram.

The second part of the item, 6b, had better results. Almost 75% of prospective
teachers were able to create the graph they chose in part 6a, 28 correctly and 2 with minor
mistakes. Five made a major mistake when constructing the graph. The typical mistake
when constructing the stem-and-leaf was to omit the stem corresponding to the 405 since

no values are represented in the table (see Figure 5.11).

5'8
05'5’5'7

00155
05'

0

0

quJD—O

 

Figure 5.11. Example of Typical Mistake when Constructing a Stem-and-leaf Plot.

162

Seven responses received a score of zero, four of which were inconsistent with the
choice in part 6a. Participants that chose a histogram made a bar graph instead. Two
were blank responses.

In the interview, prospective teachers were also asked why they did not pick the
alternative choices. Most of them based their choices on non-statistical reasoning as the
following vignette shows.

Int.: Tell me why did you pick the steam-and-leaf plot?

A: I thought the histogram is easier, but I always do the steam-and-leaf plot before I
do the histogram because I want to see how does [sic] the data actually clusters.
Cause there are so many in two [points to the numbers in the 203] and then I might...
I would have to split my groupings slightly so I wouldn’t have so many in just one
part of the histogram.

Int.: How about in terms of the data? Would you do the same for another set of data?
A: It depends on the students’ ability level, but you can do a histogram.

Int.: What can you tell me about the bar graph?

A: Well, time is continuous in general, it’s a measure I suppose as discrete data. I
don’t know, you could do it I suppose. I wouldn’t tell a child no you can’t ever try
this. They can try it and see how it works, right?

Int.: Can you answer the question with a histogram?

C: No, you can do it with the histogram. . .it would look good. You could actually
visualize the frequency a little bit better. But I thought this would be easier for a

student to draw on its own, because with a histogram you have to find the frequency
that you are going use between each number and there is a lot more stuff you have to

do.

Discussion and summary
Prospective teachers were able to create graphical representation of data, however

they often based their choice on practical reasons for construction. There was very little

163

evidence that prospective teacher took the question needing to be answered into account
for the choice of the graph.

The most common choice was a bar graph showing the initial of the students on
the x-axis. It seems important to them to show the correspondence between the time and
each student. This is an interesting point since statisticians understand data more as

distributions rather than as representing separate pieces of information.

164

Item 7
Item specification
Item 7 was designed to measure the level of interpretation of a non-standard data
representation and the ability to examine and make judgments about student work based
on statistical reasoning. It also measures the ability to describe students’ thinking and
infer about their understanding. This item was adapted from an item shared in a personal

communication from Susan Jo Russell (July 16, 2001).

 

Imagine that two second-grade students in the same class have created the
following representations to show the number of teeth lost by their classmates.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Student 1: Student 2:
99 ii I ‘ manning
1+ ‘ (:1 ..
f0 ,. Bafifig
6 3.21
l 7 Q“ 3 . $7
Leif ‘7“ “it“: 0J4?)
10 ®

 

 

 

 

 

a. Compare and contrast the two representations. That is, how are they alike?
How are they different?
b. What do you think each student understands about the data?

 

 

 

Figure 5.12. Item 7

165

Table 5.9

Distribution of scores for Item 7 (n = 42)

 

 

 

Score
Item 0 1 2 3
7a 7 7 23 4 1
7b 18 9 1 1 4 0

 

Description of the rubric for Item 7

For the first part of the item, 7a, where participants were asked to compare and

contrast the two representation, a response with a score of 4 or 3 was expected to

mention, or give some indication, that the representations are alike and different in the

sense that a bar graph and a line plot or histogram are different when used to represent

the same data.

An unsuccessful response, with a score of 2, was identified as a response that

compared and contrasted the representation in terms of physical construction and gave

indication of graph comprehension. An example of this type of response is

Student 1 has represent[sic] each student with a picture placed beside the number
of teeth he/she has lost, starting with the least and moving to the most. Student 2
represents each student with a vertical stack of blocks, each block representing
one tooth lost. The representations are similar in that they are very visual and did

not require a lot of explanation.

Responses that compared the representation in terms of superficial characteristics

yet with some indication of graph comprehension received a score of 1. An example of

such a response is the following:

“Both students show the same information. 19 students were polled and both
charts show that the same amount of teeth were lost by the 19 students. Student 1
did a better job of labeling and using creativity.”

166

A score of zero was left for blank responses and responses of that focused solely
on superficial characteristics.

“alike: represent same form of data (one is horizontal and the other vertical),
different: no labels and not quite clear”

For part 7b, the ideal response was expected to state, or give some indication, that
Student 1 has grouped or reduced the raw data represented by Student 2 and that student
1’s representation enables him/her to answer different questions about the number of

teeth than Student 2’s representation.

A score of 3 was assigned to responses close to the ideal response by either
mentioning that Student 1 has a picture of the distribution of number of teeth or by
mentioning that Student 1 is grouping or reducing the data. A score of 2 or 1 was
assigned to responses that stated some evidence of understanding at least what each
student is focusing on. Following are two examples of this type of responses.

Student 1 understands that for each amount of teeth lost, a picture could represent

the number of children who lost those teeth. Student 2 understands that every

child who lost teeth should get their own stack to represent them.

They both understand how to model the situation. Yet Student 1 represented # of

people that lost teeth and how many vs. where[sic] the Student 2 represented # of
teeth lost by the whole class, if you count the squares individually.

Finally, a score of 0 was assigned to blank responses or incorrect reasoning.

Analysis

For part 7a, only one prospective teacher provided the correct ideal response and

4 participants got close to it. A little more than half scored 2, which means that they were

able to at least describe the differences of the graphs in a physical way.

167

The second part of the item, 7b, was even more difficult than the first.
Prospective teachers had a hard time talking about students’ understanding about the
data. No one provided an ideal response. About half of the participants scored 2 or 1
which means that they mentioned or gave some indication of understanding of student’s
thinking.

More than 40% of the sample scored 0. Seven were blank responses. From the
eleven that answered the question, some categories of responses can be identified. About
half of them think that both students had the same understanding about the data but
represented it differently.

“Frequency is represented in both graphs, I think they both understand and show
that.”

“Same but different perspectives.”

“Both understand what to do, however, they choose to express it in different

9,

way.

A couple of other prospective teachers provided answers to questions that the
graphical representation may answer, such as
“more students have lost 8 teeth than any other group.”

“Well, there was a total of 115 teeth lost by their classmates.”

Finally, there were a couple of prospective teachers that expressed that they could
not comment on students’ understanding because of a lack of explanation:

“Depends on how it is interpreted. There is no explanation so I have no idea
about what each student is thinking.”

“how to count, student 1 realizes to separate but student 2 is going somewhere
and 1 don’t know where.”

168

When interviewed, prospective teachers were asked to think statistically, that is,
to think about how the students’ work differed in a statistically sense. Prospective
teacher A thought that Student 1’s representation was like a pictograph and Student 2’s
was like a bar graph. Although Student 1 has pictures of children, the representation does
not resemble a pictograph but a line plot. Towards the end of the conversation, she noted
that Student 1 reduced or grouped the data. This was what was expected for an ideal
response.

Int.: You described how these representations are different and understand what each

one tells you about the teeth lost. Now, I want you to think statistically. This one

resembles what type of graph? [Pointing to Student 1’s]

A: A pictograph

Int. : And this one? [Pointing to Student 2’s]

A: like a bar graph

Int. : Now think how are they different

A: They are very similar, if you put this one sideways [pointing at Student 1’s] it
is also a bar graph, the different[sic] the pictograph is that there is only one
number, but then you can have as many pictures as that many number kids are
represented and then... but in the graph bar every student is
represented. . .where. .. will be a lot more space ...this is... I personally will do
a pictograph. You can see that 3 people had 7 then for the bar graph type you

have to count each one up, where the numbers are right there for you.

Int.: How about in terms of questions each one answers? For example, what is the
total number of teeth lost?

A: I think the pictograph because you can see 2 people in the 23, so that is 2 times
2, and 2 people in the 33, so 2 times 3, and so on. While this one you have to
add up because it is not laid out for you.

Int.: What is the most common number of teeth lost?

A: The pictograph shows it a little bit better, you can see it in the bar graph, it is
very close to the seventh category it is not as visual as this one.

169

Int. : If you were assessing these two students understanding about the data, what
would you say? Remember to think statistically.

A: The pictograph because that involves more knowledge to group them and make it
more readable whereas this one they have the knowledge of how to make the

graph and all the information they need to put in it, but they do not understand
how to put it together yet and so they just laid it all up.

Differently than Prospective Teacher A, Prospective Teacher C judges students’
understanding about the data in terms of their communication and presentation.

Int.: If you were assessing these two students understanding about the data, what
would you say? Remember to think statistically.

C: I would actually say Student 1 understands better. Student 2 does not have

anything that tells me what it is. If they have told me a key, or if they have told

me what this axes represents here and what this represents I would have an easier

time to. . .but they organize it beautifully. I like this second grade work, it shows

that they did it themselves or in a little group and they really thought about it.
Summary and Discussion

Analyzing and judging students’ own data representation in terms of statistical
reasoning was extremely hard for prospective teachers. When asked to compare and
contrast their work, most focused only on the procedural and physical differences of the
graph representations. Since both students’ work are representations of the data,
prospective teachers do not perceive any difference in students’ understanding of the data
claiming that “they both show the same information in different way”. Part of the

difficulty of this item is the fact that prospective teachers are asked to infer about

students’ statistical thinking for data representation that are non-standard.

170

Item 8

Item specification

Item 8 measures in part 8a computational knowledge of the range and standard

deviation from a line plot of two data sets. Part 8b measures the recognition of the two

data sets’ shape in relation to the measures calculated in part 8a. This item is based on an

example from Functions, Statistics, and Trigonometry (Senk et al., 1998).

 

Consider the dot frequency distributions below with the heights in inches of the

 

 

a. The mean of each distribution is 75. Calculate the following for each

10 players on two basketball teams. Team B
TeamA ‘
O
O
C
C
O O
o o O o O o o o o O O o '
111111111111 IIIIJIIIIIAIII
7I0 I I I I 7I5 I I I I 8'0 I 70 75 80
Heights Heights

 

distribution.
I Team A I Team B
Mean 75 75
i)Range I
ii)Standard deviation

 

standard deviation, without calculating.

 

b. Explain how you could have determined which distribution has the largest

 

Figure 5.13. Item 8
Table 5.10

Distribution of scores for Item 8 (n = 42)

 

 

 

Score
Item 0 1 2 3 4
8ai 2 0 0 25 l 5
8aii 19 3 7 3 10
8b 21 3 3 7 8

 

171

 

Description of the rubric for Item 8

In Item 8a, computations for the range and standard deviations were scored
separately. For the range, a score of 4 was assigned to responses that computed the range
as a single measurement value of 10. A score of 3 was assigned to those that reported the
range as “70-80” or “70 to 80”. No scores of 2 or 1 were assigned for this item and a
zero score was assigned for incorrect computations.

Scoring the computational knowledge of the standard deviation, took the whole

range of the scoring rubric. Again, a score of 4 was assigned to correct calculations using

_ 2 — 2
x. — x x — x
either formula, s = Z(—'1—) or 0' = L . A score of 3 was assigned to
n — n

 

 

those responses that made a small computational mistake when dividing by n or n — l, or
the value was incorrect in the tenths or hundredths digit. A score of 2 was assigned to
those computations with a major mistake such as not dividing by n or n — 1 or computing
the mean average deviation (MAD) or not giving a numerical answer by indicating that
Team B has a smaller standard deviation than Team A. A score of 1 was assigned to
responses that made a mistake in the early stages of the procedure to compute the
standard deviation. Scores of 1 were also given to responses that indicated that all they
remember about the standard deviation was the fact that for normal distributions 68% of
the data values are within one standard deviation. Finally, responses that did not
provided any or incorrect values were given scores of 0.

Item 8b was designed to measure the understanding of the measures of center and
spread in relation to the shape or distribution of the data, that is, the application of the
measures of center and spread as a tool to compare two distributions. Scores of 4 were

those that correctly communicated the meaning of the standard deviation applied to the

172

context of the problem. An example of this type of response was “Team A has a more
spread out height distribution, whereas Team B had more players closer to the mean.”
Responses that did not mention that the spread was about the mean or did not refer to the
context of the problem received a score of 3. For example, “The standard deviation told
me that graph A is more spread out than graph B”. Score of 2 and 1 were assigned to
responses that indicated partial understanding of what the measures calculated in the first
part say about the distribution but either did not communicate clearly or part of the
response was incorrect. Responses with scores of 0 were those that did not provide any

answer or all of the statements were incorrect.

Analysis

Results for Item 8ai show that more than 90% of prospective teachers have
computational knowledge of range. Out of the 40 who were successful on this item, 15
gave the answer as a single measurement and 25 reported the range as an interval.

About a quarter of the prospective teachers correctly found the standard deviation
for both data sets (five used calculators and five did it by hand). Three of them made a
small mistake. Thus, 13 out of 42 (31%) successfully responded to this part of the item.
No patterns of major computational mistakes were identified here, and the alternative
algorithm used for the standard deviation was the mean absolute deviation (MAD).
However, the subjects made no indication of the fact that this was a different
measurement.

About 45% of the participants scored 0. Six participants left the question blank

and the typical response for those that provided an answer was an equal value of 5

173

standard deviation units for each distribution. No computational procedure was shown
by any of the participants that arrived at this answer. However, one of them indicated in
the second part of the problem that “The students’ height range in total of 10 in. from
shortest to tallest with 75 being in the middle so 5 inches difference up or down.”

During the interviews prospective teachers that successfully computed the range
and the standard deviation were asked whether their answers made sense and why. They
were also asked to say what the number they computed means in terms of the data. Their
answers reveal different levels of understanding about this measure of spread.
Prospective Teacher A

Int.: Do your answers make sense?

A: It does, this stuff is all scattered so it is a higher standard deviation. This stuff,

you only have two aberrations and the rest is all clustered right at the mean, so it

makes sense that one would be smaller than the other.

Int.: What does 2.14 means?

A: 2 units, it means that from your mean one standard deviation takes into

account 68% of your data and two standard deviations take a 95%, that’s all it

means.
Prospective Teacher C

Int. .' Do the answers make sense?

C: Yes, I think my answers are right.

Int. .' Because. . .what does the standard deviation tell you?

C: It is the distance away from the average.

Int. .' Does it make sense that this one [Team B’s] is bigger?

C: Huumm, yeah because these are more the same. . .and these come down and

out, I guess, so that it become. . .I am not. . .I want to say that make sense. But I
would actually do it in paper to see if it is correct.

174

Int. : So, you are saying that number represent how far each data is away from the
mean. So 3.57 would mean that...

C: That the average will go ???. . ..that far away.

Prospective Teacher B
Int.: Do the numbers make sense?

B: The standard deviation is from the lowest number to the highest number how
much they are changing.

Int.: So, does it make sense that Team A has a higher standard deviation?

B: Probably not to a child, Team A is more spread out but Team B is way up there

towards the middle.

Prospective Teacher D

Int. : Do your calculations make sense?

D: For team B we see that it is 2.28, that means that the standard deviation is
smaller than team A which is 3.58. That makes sense because none of the
answers are closer together for team B given a smaller deviation and
further apart make a larger deviation.

Int.: What are the units for the standard deviation? What does 2.28 means?

D: Is it inches? no. . .ah..l don’t know what that means.
How far apart, how frequently the grades occur?

More than a third of prospective teachers scored 3 or 4 on this item, giving
indication of understanding of the meaning of what the standard deviation measures.
Eight of the fifteen successful subjects mentioned that the variation is about the mean and
the rest only mentioned the variation. Nine of the fifteen successful responses for this
item also provided a successful response for the calculation of the standard calculation.

That is, nine out of 42 prospective teacher were able to compute and make sense of the

175

measurement. Six either made a major mistake on the computation or computed the
MAD but still understood the meaning of the standard deviation.

For this item, responses were more dichotomous. Very few participants (6) were
assigned scores of 2 or 1. They either knew about what the standard deviation tells us
about the data or they did not. Consequently, this is the item among all the items in the
instrument that has the highest percent of 0 scores (21 out of 42). The majority of these
(17/21) could not compute the standard deviation, either. The remaining four subjects
showed computational knowledge but incorrect statistical reasoning. Some of the
misconceptions are illustrated in the following responses:

Because the standard deviation of Team B is less than that of Team A, I can

conclude that team B represents a more normal distribution than does team A.

The less the standard deviation the more normal shaped distribution of data
because values are located closer to the mean of the data.

“Standard deviation indicates the + or — away from the mean and the percentage
of subjects in such deviations eg: i standard deviation (approximately 17%)”

“that number represent how far each individual is away form the average height”

The results are summarized on Table 5.11. Successful responses for this item
correspond to conceptual understanding about the standard deviation; unsuccessful
responses with score of 2 or 1 corresponds to partial understanding; and scores of 0

corresponds to no conceptual understanding.

176

Table 5.11

Distribution of Responses of Computational Skills and Type of Responses for the
Standard Deviation.

 

 

 

 

 

Correct Incorrect Total
computation computation

Successful responses 9 6 15
(score 3 or 4)

Unsuccessful responses 1 5 6
(score 2 or 1)

Unsuccessful responses 4 1 7 21
(score 0)

Total 14 28 42

 

Discussion and summary

Prospective teachers who have computational knowledge of the range and the
standard deviation do not necessarily have understanding of the concept. However,
results here show that the number of prospective teachers who have computational
knowledge and poor or none conceptual knowledge is lower than the number of those
who actually understand or partially understand the concept but do not have the
computational knowledge. That is, there are more prospective teachers that show
understanding without knowing how to compute than prospective teacher that can

compute without understanding.

177

Overall Performance

Prospective teachers’ overall performance on the written instrument was obtained
by scoring each question (some items had more than one question) on a scale of 0 to 4.
The total number of questions was 18, making a total of 72 possible points. Totals for
each participant were converted to a percentage. Descriptive statistics across all 42
percentage scores were computed. There were four prospective teachers that did no
report any background in statistics at the college level, however their performance in the
assessment was about or above average. Therefore, they were not excluded from the
analysis. Figure 5.14 shows the distribution of scores; the distribution is slightly skewed
left, with mean 58.86%, median 60.42%, and standard deviation of 16.21%. None of the
participants showed mastery or near mastery level (score 4 or 3 from rubric) on all the

questions of the instrument.

Frequency

liffi' ff .

 

 

 

 

2» . .

1O 20 30 40 O 60 70 80 90

0|

Percent scores

Figure 5.14. Distribution of Percentage Scores of 42 Prospective Teachers.

178

The lowest score was 19% and the highest 83%. The distribution does not show
large spread about the mean or gaps between scores. Half of the scores (22) fall between

60% and 80% and the majority fall between 50% and 80%.

Performance by Domain of Knowledge

The assessment instrument was designed to measure two main domains, statistical?
knowledge and statistical knowledge for teaching. The domain of statistical knowledge _____ 5
was measured with 12 (out of 18) questions from the assessment, making a total of 48
possible points for that part. The 12 questions are Items 1a, 1b, 1c, 2, 4a, 4b, 4c, 6a, 6b,
8ai, 8aii, 8b. The domain of knowledge for teaching was measured with 6 (out of 18)
questions, making a total of 24 possible points. The 6 questions are Items 3a, 3b, 5a, 5b,
7a, and 7b. Scores were added for each part and converted to percentages.

Prospective teachers performed better in the domain of statistical knowledge than,- 5:
the domain where they had to apply this knowledge to teaching with means 65.7% and i
45.1%, respectively (see Figure 5.15). Furthermore, scores for the domain of knowledge j
for teaching are more spread out about the mean than the scores in the statistical
knowledge domain (see Table 5.12). Again, none of the prospective teachers showed

mastery or near mastery level of correctness in either domain of knowledge considered

here (see Appendix D for complete database).

179

Table 5.12

Descriptive Statistics of Percentage Scores by Domain of Knowledge

 

Mean(%) Mediafi/o) SD(%)

 

 

 

Statistical knowledge 65.7 67.7 17.6
Statistical knowledge for 45.1 45.8 18.1
teaching

Overall 58.9 60.4 16.2

 

 

e8
88

Mean Percentag
a 3 a s

 

0

Overall Statistical Knowledge Knowledge fu’ Teaching

Figure 5.15. Mean Percentage Scores by Domain of Knowledge and Overall.

Distributions of scores for separate domains of knowledge show a better picture

‘I

of how these compare. Figure 5.16 and Figure 5.17 shows that the distribution of h it

i

statistical knowledge has more of a bell shape than the distribution of knowledge for it

teaching. The distribution for the knowledge for teaching looks pretty uniform for scores i
between 40 and 70%, while the distribution for statistical knowledge is skewed left. j
About 76% of the prospective teachers scored 60% or better on the statistical knowledge
part of the assessment, while about 76% of the participants scored 60% or less on the
knowledge applied to teaching.

Another major difference between the two distributions is the percentage of

prospective teachers that score below 30%. For statistical knowledge about 7% of the i

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prospective teachers score below 30%, while about 29% of the participants scored below 1'

30% for knowledge for teaching. ‘ J

Frequency

 

Percent scores

Figure 5.16. Distribution of Scores for Statistical Knowledge

Frequency

 

Percent scores

Figure 5.17. Distribution of Scores for Knowledge for Teaching

Performance by Cognitive Demand
The assessment instrument also measures three different levels of cognitive
demands for the domain that only involves statistical knowledge. The level of Statistical
Literacy was measured with 5 questions (Items 1a, 4a, 8ai, 8aii, and 6b); the level of
Statistical Reasoning was also measured with 5 questions (Items 1b, 2, 4b, 4c, and 6a);
and the level of Statistical Thinking was measured with 2 questions (Items 10, 8b). As

expected, prospective teachers perform better at the lowest level of performance —

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Statistical Literacy. This involves mainly extracting information from a graph and
recognition, identification, computation or basic understanding of concepts. At the
higher levels, statistical reasoning and thinking, prospective teachers do progressively
worse (see Table 5.13). Statistical reasoning refers to the way students reason with
statistical ideas when asked why or how results are produced. Statistical thinking refers
to the application of students’ understanding of real world problems; for measures and
distribution this might mean using them to make predictions and inferences about the
group to which the data pertain.

Table 5.13

Descriptive Statistics by Cognitive Demand

 

Mean(%) Median(%) SD(%)

 

 

 

 

 

Statistical 74.0 75.0 18.5
Literacy
Statistical 63. 1 65.0 20.9
Reasoning
Statistical 51.5 50.0 30.4
Thinking
80
70
60
50
3E 40
30
20
1 0
0
Statistical Literacy Statistical Statistical Thinking Overall
Reasoning

Figure 5.18. Mean Percentage Scores by Cognitive Demand

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Percentage Success on Assessment
The two different domains of knowledge assessed can be analyzed by observing
the percent of prospective teachers that successfully scored 3 or 4 on the items that fall
into each category. Furthermore, since each item was designed to measure a specific
aspect of content or content applied to teaching; the percent of success on these aspects
can be analyzed. Table 5.14 shows the percent of success for statistical knowledge.
Table 5.14

Percent Success on Items Measuring Statistical Knowledge

 

 

 

Item Percent Topic Cognitive Demand
Successful
8ai 95 Range Literacy (compute)

1b 93 Steam-and-leaf plot Reasoning(interpret)
la 90 Steam-and-leaf plot Literacy (read)
4a 83 Mean Literacy (compute)
6b 71 Graph representation Literacy (construct)
4c 67 Mean Reasoning(property)
1c 60 Steam-and-leaf plot Thinking(infer)
4b 52 Mean Reasoning(property)

2 45 Median Reasoning(proper use)
6a 38 Graph representation Reasoning(proper use)
8b 36 Standard deviation Reasoning/Thinking

8aii 31 Standard deviation Literacy (compute)
Mean % 63.42
successful

 

Note that most of the items with the highest percent of success are the items that
fall into the level of statistical literacy, except for item lb. Statistical literacy includes
mainly computational skills, constructions of graphs; and the lowest level of graph
comprehension (reading the data). These items with high percent of success do not fall
into a single category of topics, they range from graphical representation, range, and

mean. In contrast, the items with the lowest percent of success all, except for item 8aii,

183

1
fall into the statistical reasoning. Interestingly this is the level of cognitive demand that 5"

showed the highest emphasis suggested by state and national standards. Topics for the l
most difficult items also vary, however the topic of standard deviation show low percent '
of success both at the level of statistical literacy and statistical reasoning. l.

Similar analysis can be made for items that measured the domain of knowledge
applied to teaching. Table 5.15 shows the percentage success for this domain and the
corresponding teaching tasks.

Table 5.15

Percent Success on Items Measuring Knowledge Applied to Teaching

 

 

 

Item Percent Topic Teaching Task
Successful
5a 67 Formulating Select language appropriate
question for students’ level
3a 62 Histogram Identify students’ mistake
3b 48 Histogram Judge students’ comment
7a 12 Data representation Analyze students’ work
7b 9.5 Data representation Assess students’ thinking
5b 2.4 Mean, median, mode Judge students’ comment
Mean % 33.48
successful

 

For this domain the highest percentages of success is shown for teaching tasks
that involved the prospective teachers formulating questions that may have generated a
data set using appropriate language for middle grade students. Also with high percent

success among these items is identifying students’ mistakes; however, judging the

_._.-._.__ i
—»‘ 2...; '-

validity of students’ comments shows greater degree of difficulty. With a very low

‘3
I
1

percentage of success (2.4%) is item 5b which involved judging students’ comments it

about measures of center. Note that computing, recognizing properties or using measures l,

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of center had a higher percent of success for items that did not involve the application of

teaching (see Table 5.14).

Summary and Discussion

Prospective teachers of middle school level that participated in this study showed 7’ ,1]
1
better performance for items related to statistical knowledge than for items that had some /
application to teaching. These results are consistent with other studies related to other I
areas (Ball, 2001; Even, 1993; Even &Tirosh, 1995; Ma, 1999) which show that
prospective teachers may know the subject matter well but not well enough to teach it s,‘
based on children’s conceptions. In this particular study, it is shown that most
prospective teachers have the ability to formulate a question that may have generated a
data set; construct correctly a graphical display and be able to extract information from it;
compute and know how to find measures of center and spread for a set of values or small
data displayed in a line plot. i

In contrast, the majority of prospective teachers were neither able to identify a

,1

correctly the source of erroneous or incomplete conceptions, nor could they infer or judge ;

l
a child’s level of understanding based on their work. Mi

.‘

185

CHAPTER 6

CONCLUSIONS AND RECOMMENDATIONS

Two central questions were investigated in this study. First, what are the
important aspects of content knowledge for teaching data analysis and statistics at the
middle school level? More specifically, what are the important statistical content topics
taught in middle school that teachers need to be prepared for?; What are the cognitive ‘
demands (such as memorization, performing procedures and solving non-routine
problems) that are related to the content?; What are the important aspects of knowledge
for teaching that relate to the content aspect above?. Secondly, what are the conceptions "
and misconceptions prospective middle school teachers have with respect to these

important aspects of knowledge for teaching data analysis and statistics? ,

Chapter 3 describes the important aspects for teaching data analysis and statist'cs
and Chapter 5 describes prospective teachers’ knowledge in relation to these aspects.
Although these descriptions and findings may not tell the complete story, they do
motivate discussion and ask better questions for future research on teacher preparation.
As Ball, Lubienski, and Mewbom (2001) point out, the problem of what teachers need to
know remains an unsolved one. Stylianides and Ball (2004) suggest that one of the
reasons for this is “that people have been looking at it from different angles, having

available only limited research that concentrated on connecting the knowledge produced

186

by different perspectives” (p. 3). Stylianides and Ball (2004) suggest a framework that
brings together, in an iterative cycle, different perspectives such as analyzing experts’
perspectives, teachers’ mathematics curricula, teachers’ mathematical knowledge,
students’ mathematical curricula, students’ mathematical knowledge, and school
mathematics practice with the hope that the integration of all these resources may help
the study of the different domains of teachers’ knowledge for teaching.

This study partially investigates some of the approaches suggested by the
Stylianides and Ball framework. The choice of the approaches studied here are practical
and based on the availability, at the time, of resources in the field of statistics education.
The approaches taken here are the analysis of experts’ perspectives by examining
national and state standards at the student and teacher level, teachers’ mathematical
knowledge by administrating an assessment, and the examination of a students’
mathematical curriculum. Teachers’ mathematical curricula and school mathematics
practices were not examined for this study.

The chapter is divided into three parts: summary and discussion of main findings,
significance and implications, and recommendations for future research. A reminder note
for the reader, findings and conclusions of this experience are based on limited sources
and its generalizations should be done with caution. Aspects of content knowledge for
teaching were identified by looking at written documents and recent research, not by
observations of teaching practice. Prospective teachers’ knowledge was measured by a
written instrument only on a convenience sample of 42 subjects taken from five eastern

universities.

187

Summary and Discussion of Main Findings

The identification of the most important aspects of knowledge for teaching, which
addresses the first research question, was motivated by the need to create a framework
upon which to develop an instrument to measure this construct. The identification of
these aspects was done by analyzing experts’ perspectives expressed in state and national
standards, reports and a curriculum. A systematic analysis of these documents required
searching for tools that have been applied to other similar purposes, such as content map
analysis (Porter, 2003). This methodology was more successful with standards than with
the rest of the documents given the format and language in which the standards are
written. Complementing this methodology there were many other sources that help in
understanding this complex idea. One is the literature on content knowledge for teaching
in areas like numbers and operations or functions. Research here is well documented (see
Ball, Lubienski, and Mewbom (2001) for a review of literature in these areas) and
provides clear examples of the important aspects and its organization. Other resources
include the most recent documents focused on teacher preparation published by the
National Research Council (NRC, 2001) and the Conference Board of the Mathematical
Sciences (CBMS, 2001) which give clear suggestions of the mathematical and statistical
education of future teachers. Other documents are the numerous set of national and state
standards at the student and teacher level on which many of the curriculum and programs
for teacher preparation are based. Finally, in an attempt to get as close as possible to
teaching practice, units developed by the Connected Mathematics Project (Lappan et al.,

2002) on statistics and data analysis at the middle school level were examined.

188

Recently, current research on knowledge of teachers has shifted from looking at
teachers’ background to looking at teachers’ knowledge to finally looking at knowledge
in and for teaching (Ball et al. 2001). Because there is very little evidence of any
association between teachers’ credentials, such as course work in mathematics, and
teacher effectiveness or student performance; researchers have taken the approach of
studying the nature of teachers’ mathematical knowledge focusing on the different kinds
of knowledge. This last approach, although valuable, generates results that are based on
hypothetical settings and does not provide evidence that teachers have applied it in
practice. Hence, a new approach is suggested by Ball and others, mathematics
knowledge in and for teaching that focuses on the core activities of teaching. This new
approach requires watching teachers using their knowledge while they teach. “What
emerges from these observations is that being able to talk about mathematics is different
from doing it” (Ball et al., 2001, p. 450). This new approach has given new directions
and hope for better answers to the problem of teachers’ mathematical knowledge.

For this study, the approach taken was a mixture of the two latest approaches.
Observation of actual teachers was beyond the scope of this study; so, in order to
identify the aspects of knowledge for teaching statistics and data analysis a systematic
analysis of documents and a textbook was conducted. As for measuring these aspects, a
reliable instrument was constructed basing the teaching settings as close as possible to
real practice, with the understanding that this instrument can only partially tell the story
about what prospective teachers know and how they know it when confronted with

teaching the subject.

189

Aspects of Knowledge for Teaching Data Analysis and Statistics
First a summary of the content that is expected of students at the middle grades
level is given, not because this study is about students, but because the introduction of
data analysis and statistics in the middle school curriculum is relatively new, in relation
to other areas such as geometry and algebra. Minimally teachers have the responsibility
to meet these expectations and given the newness of the statistics curriculum may have
difficulty fulfilling this responsibility. This new material creates an extra challenge to
teachers and prospective teachers, they themselves need to become familiar with an area
that may never have been part of their pre-college education and must rely only on one or
two statistics courses - not necessarily geared towards teachers — at the college level.
A systematic analysis of national and state standards shows that prospective
teachers, as future teachers, should know and be able to teach the following content:
0 Process of statistical investigation (includes formulating questions, collecting
data, designing studies)
0 Categorical data representation (includes bar/pie graphs, pictographs, tables)
0 Numerical data representation (includes stem-and-leaf plots, histograms, box
plots)
0 Bivariate data representation (includes scatter plots, line graphs, regression line)
0 Shapes of Distributions (includes shapes of distributions, skweness, gaps, outliers,
clusters)
0 Measures of center (includes mean, median, and mode)

0 Measures of spread (includes range, interquartile range)

190

Prospective teachers are also responsible for engaging students in cognitive

activities related to these content topics, the cognitive demands identified on the national

and state standards are:

Perform routine procedures/Statistical literacy (includes collecting data,
reading/ creating graphs and data displays, finding and computing measures of
center and spread, identifying clusters, gaps, outliers)

Communicating understanding/Statistical reasoning (includes making
decisions on what and how to measure, justifying the use and selection of
graphical representation and measures of center and spread, interpreting
graphs, and explaining findings and results)

Solve nonroutine problems/make connections/Statistical thinking (includes
applying statistics to real work contexts, analyzing data recognizing patterns,
judging or critiquing statistical methods)
Conjecture/generalize/prove/Statistical thinking (includes making inferences,

predictions, and generalizations from data displays and measures)

The list above is a comprehensive one; results show that state and national

standards at the student level vary a great deal in terms of the inclusion of and emphasis

given to topics and activities. In particular, state standards differ on the level of cognitive

demand for each topic. When both topic and cognitive demand are accounted for, only

measure of center at the level of communicating understanding is found in the standards

of all 10 states analyzed. The national and state standards differ significantly as well. On

the whole, state standards emphasize categorical data representation and performing

procedures more than the national standards do.

191

Fewer documents exist which provide standards for middle school teachers as
opposed to middle school students. Two documents were reviewed for this study, the
PRAXIS II and The Mathematical Education of Teachers (CBMS, 2001). The latter
places emphasis on the process of statistical investigation and shapes of distributions,
while the former emphasizes categorical data representation. Both documents place
higher emphasis on bivariate data representation at the level of solve non routine
problems than do the standards for students.

Using a methodology to identify content adapted from Porter (2003) it was found
that the average emphasis of all the documents reviewed is given to: 1) graphical
representation, more numerical than categorical, at the level of creating and using them
properly; and 2) measures of center and spread at the level of computation and proper
selection. Hence, these aspects of content are included in the assessment instrument
developed for this study. A limitation on using this methodology is that it depends on the
language used in the documents and not on the intent experts have given in the
recommendations, therefore documents that are not written with clear distinction between
topics and cognitive demands were very hard to analyze (e. g. The Mathematical .
Education of Teachers (CBMS, 2001) report).

Aspects of knowledge for teaching data analysis and statistics were identified
from teachers’ standards, National Research Council (NRC) documents, and the
Connected Mathematics Project’s two units on statistics from the middle grade
curriculum. Each document suggest a different organization for looking at teachers’
knowledge. State and national teachers’ standards give general recommendations and

organize the knowledge into four kinds:

192

0 Knowledge of the subject matter
0 Knowledge of pedagogy (some include pedagogical content knowledge)
0 Knowledge of students’ learning
0 Knowledge of assessment
Adding It Up (NRC, 2001) besides focusing on these kinds of knowledge, also
adds cognitive activities that teachers engage in relation to these kinds of knowledge
more specifically tied to mathematics. The authors argue that teachers’ learning, like
students’ learning, can be conceived in terms of interwoven strands. The strands or
components that are required in teaching for mathematical proficiency are the following.
o conceptual understanding of the core knowledge required in the practice of
teaching;
0 fluency in carrying out basic instructional routines;
o strategic competence in planning effective instruction and solving problems that
arise during instruction;
0 adaptive reasoning in justifying and explaining one’s instructional practices and
in reflecting on those practices so as to improve them; and
c productive disposition toward mathematics, teaching, learning, and the
improvement of practice.
Another NRC document, Knowing and Learning Mathematics for Teaching
(NRC, 2001), focuses instead on practical activities of teaching practice that are directly
related to mathematics. Six categories of recurrent teaching tasks are identified that
require the use of mathematics: 1) managing class discussions, 2) establishing a

classroom culture for mathematical reasoning, 3) designing and selecting tasks, 4)

193

analyzing student thinking and work, 5) planning instruction, and 6) assessing student
learning.

All these different frameworks and organizations intersect in one or more points.
Each one of the categories, strands, components or kinds of knowledge for a particular
organization could be mapped into another framework. For example, we can map the
teaching task of “analyzing student thinking and work” onto “knowledge of students” in
the first framework and also to “strategic competence” in the second.

The different organizations of knowledge for teaching help create a general vision
of the different aspects that one needs to consider for assessment, which is the ultimate
goal of this study. However, these organizations are too general. It is not clear if all of
these aspects apply in all areas within mathematics, in particular in data analysis and
statistics. If they do, do they apply to all topics? The necessary intersection of the actual
content and all of these aspects of knowledge of teaching is still absent. As mentioned
before, observing teachers teach the subject would have been the ideal path. Failing that,
relying on research done in this area would suffice. However, the first was beyond the
scope of this study and to date there is no research in this area. The best approximation
of this intersection was the teachers’ guide to a unit in statistics from a middle grade
curriculum.

The book examined was the Teachers’ Guide of the Connected Mathematics
Statistics unit: Data About Us (Grade 6) (Lappan et al., 2002). Although this unit covers
several topics identified previously, only the two topics which received greater emphasis
in the standards and documents were examined, namely organizing and representing data,

and measures of center.

194

For teaching data representations the knowledge of students in relation to the

content was identified to be:

Know how to create questions that middle school students can ask in order to
collect data and might involve using a specific type of graph;
Know how to respond to a student who want to use an inappropriate type of graph

Assess students’ responses making judgments about their reasoning;

The Teachers Guide also identified aspects related to pedagogy for data representation:

Engage students in the exploration of data by having them suggest questions that
might have originated the data and methods of collecting the data;

Lead students in the process of constructing a stem-and-leaf plot, and;

Pose questions that lead to “read the data”, “read between the data” and “read

beyond the data”.

As for measures of center, in particular, the mean; the following pedagogical aspects

were identified, most of them dealing with the proper use of physical models:

Know the advantages and limitations of physical models to introduce the concept
of the mean as the “evened out” number;

Know how to make connections between the physical model and the line plot;
Know how to create data sets with the same mean but different distribution using
physical models; and

Lead students to the discovery of the algorithm of the mean and why it works.

Finally, aspects of knowledge of students in relation to the measures of center were also

identified:

195

0 Understand how students are thinking about the data when they use different
strategies and models to find the mean;
0 Assess proper statistical reasoning for justifying students’ strategies;
0 Respond to students who think that is impossible to have many data sets with the
same mean, and
o Anticipate students’ answers or interpretation to an investigation question and be
able to pose questions to students that lead them to see the effect of outliers or/and
new data values have on the distribution and the mean.
Some of these aspects were considered in the measurement of prospective teachers’
knowledge for teaching, especially the ones related to knowledge of students. Some were

addressed as items in the written instrument and some in the interviews.

Prospective Teachers ’ Statistical Knowledge

To address the second research question about measuring teachers’ knowledge an ”i
instrument was developed, piloted twice, and administered at five eastern universities. if
The instrument was divided into two major domains, statistical knowledge and statistical \'

1
knowledge applied to teaching. Number and allocation of items measuring the statistical }
knowledge were selected to represent the percentage of content identified in the content ‘
maps. The applications to teaching were restricted to students’ work and responses in
relation to statistical concepts given the constraints of time and the written format. The
reliability of the instrument is CL = .80.

Although the items were divided into content and content applied to teaching, the

items measuring the latest domain are rooted in content and therefore related to the first

196

domain. However, the division provides a way to compare how prospective teacher
know the content with and without teaching contexts.

The sample consisted of 42 senior prospective teachers mainly females in their .
early 203 with strong background in mathematics and education. All of them had taken at I
least one course in mathematics education and all but four had taken a course in statistics.
Prospective teachers were selected conveniently from five different universities which
have a program in middle school mathematics and are located in a state that requires a
middle grade certification. A subset of the sample (7) participated in follow-up
interviews.

Written responses were scored according to a 0 — 4 scale rubric. Analyses at the
item and overall level were conducted. Interview responses helped enrich the analysis of
the instrument. Following are summaries of the results organized by topics or big ideas
within each domain of knowledge.

Knowledge of Statistics

The aspects of statistical content measured in the instrument were: 1)
Formulation of questions to generate data, 2) Numerical data representation; and 3) 1
Measures of center and spread. An attempt to measure each of these aspects at all levels
cognitive demand was made, but it was not possible given the limitations of a written

instrument. However, the instrument has the following representation of aspects of

content and cognitive demand.

197

Formulation of questions

Participants in this study were given a table showing different kinds of pets and its

respectively frequency (see Item 5 of instrument). They were asked to create a possible

question that could have generated the data. Two thirds of the prospective teachers were ,
able to create a correct possible question, although some of them did not use a proper i
question format.

Numerical data representation

About 70% of the prospective teachers were able to create graphical
representation of data; however they based their choice on practical reasons for
construction. There was very little evidence that prospective teacher took the question
needing to be answered into account for the choice of the graph.

The data given was a list of students (initials) and their time to get to school (in
minutes). The most common choice was a bar graph showing the initial of the students
on the x-axis. It seems important to prospective teachers to show the correspondence
between the time and each student. This is an interesting point because statisticians

understand data more as distributions rather than as representing separate pieces of

information. Some of the prospective teachers argued that because of the time variable,

they needed to use a line graph.
Nearly all (93 %) of the prospective teachers were able to read and interpret data Q
in a stem-and-leaf plot. However, about 60% of prospective teachers used neither

statistical measures nor statistical reasoning to make inferences about the data. When

interpreting histograms, the main difficulty was to interpret what each bar represented.

198

Many prospective teachers focused on the x-axis variable and did not know how to relate
that to the “Frequency”.

Measures of center and spread

About 80% of prospective teachers were able to compute and estimate the mean
of a small data set represented in a line plot. When given a bigger data set and asked to
estimate the mean, they either chose the median as an estimate without observing the
distribution or in their attempt to find the mean, they found the average of the frequencies
instead. The latter seems to be a reflection of the methodology taught in elementary and
middle school of leveling off stacks of cubes.

About half of prospective teachers knew that it is possible to have many sets of ”I
data with the same mean. However, only a third of those could justify it with an argument
that relies on both the algorithm and the concept of the mean as a balance point. About -_
two thirds of prospective teachers were able to create a distribution with a specific mean if
that is not a whole number. Prospective teachers interviewed and probed for knowledge-J!
for teaching did not show sufficient evidence of this type of knowledge. They could
neither estimate the mean of a data set without reaching for the algorithm, nor convince a
child that there could be several data sets with the same mean, nor explain what an
average of 3.5 people means.

”HT?

About half of prospective teachers were able to identify outliers in a data set but i

,l'
/

not necessarily to use this knowledge to choose the appropriate measure of center. Some 1'

actually wanted to take the outlier into account for representative reasons, others for

pedagogical reasons. When asked to say what is the typical value for a data set given,

prospective teachers choose a measure of center or a range of values independently of the

199

shape of the distribution. Furthermore, about half of them could not make sense of the
computed measures of center and the corresponding shape of the distribution. One likely
explanation for these results is the use of the word “typical” which is not a well defined
statistical term. It could be thought as the mode, median or mean. As prospective
teachers relate this term to one of the measures of center, they tend to attach a single
measure of center to the word “typical” (e. g. the mean) and use it to describe the center of
the distribution regardless of the shape.

Prospective teachers who had computational knowledge of the range and the ’}
standard deviation did not necessarily have understanding of the concept. This result was {
somehow expected. However, results here show that the number of prospective teachers I
who have computational knowledge and poor or no conceptual knowledge is lower than
the number of those who actually understand or partially understand the concept but do

not have the computational knowledge.

Knowledge for Teaching Statistics
As for statistical knowledge applied to teaching, the instrument focuses on the

knowledge of students as learners. The aspects considered for the instrument within this M]

"M...“

domain are: 1) interpretation of students’ oral and written responses in relation to the

content, and 2) examination of students’ strategies and solutions to exercises to make

‘E ~3-~.M .w- '-

inferences about their understanding. Results in this part of the assessment were lower l
than the part on statistical content and unveiled conceptions and misconception about
statistical content that were not captured by items with no teaching contexts. One of the

reasons for lower performance might be due to the new format of items. Most

200

prospective teachers’ responses were based on their pedagogical rather than statistical
content knowledge. The expectation that the responses needed to be focus on the
statistical content was not explicitly given in wording of the item. This attempt was done
only with the interviewed subjects (see Interview Protocols in Appendix E). For
example, when analyzing student work, prospective teacher were asked how students’
own graphical representations resemble the standard statistical graphs. Some prospective
teachers were able to correctly identify the features in students’ graphs that would
resemble a bar graph or a stem-and-leaf graph. In contrast, others would focus on the
physical drawing and said that the students’ graphs would resemble a pictograph because

the student drew classmate faces.

Interpretation of students ’ written responses
About three fourths of prospective teachers could recognize a typical student’s

misconception when reading a histogram and half could also identify the reason why the

1.3..

student is making the mistake. However, the results show evidence that the prospective
teachers have other limitations with the interpretation of the histogram and are unable to
correct the mistake properly. In other words, prospective teachers correctly make the
judgment that the student is wrong but give the incorrect way to correct the mistake. i
In contrast, most could not recognize students’ mistakes about the l;
inappropriateness of using the median or range for categorical data. Most prospective i
teachers could not describe what the measures of center or spread say about the data

‘».,\

\
when trying to find them. Instead they use a procedural method to get an answer. In this 1

201

l

particular case, prospective teachers were very creative inventing their own methods to

compute a median for categorical data.

Examination of students ’ strategies and solutions

Analyzing and judging students’ data representation was extremely hard for l
prospective teachers. When asked to compare and contrast their work, they focused only
on the procedural and physical differences of the graph representations. Since the word
of both students presented represented the same data, prospective teachers generally
could not describe any difference in students’ understanding of the data claiming that
“they both show the same information in different way”.

It is not clear from this investigation the reason why prospective teachers do not ,

1‘
l

have the knowledge required to teach statistics. One potential cause, and that needs /
further investigation, may be the lack of opportunity to learn applications to teaching in f/

I
the way they were assessed in their own teacher preparation programs. Since this J}
variable was not measured, these particular prospective teachers should not be held
accountable for their lack of knowledge. Another reason could be the integration of
pedagogy issues and content in each item; prospective teachers may not have had a clear
notion what was expected from them. A successful response was expected to focus on
statistical content, but most of prospective teacher focused on pedagogical issues. Better
efforts are needed to create items to measure this construct so that prospective teachers
know what is expected of them.

Although the assessment instrument was clearly divided into two domains,

statistical content and statistical content applied to teaching, inferences on what

202

prospective teachers know about statistics for teaching can not be separated in this
fashion. As statistical content is present in both domains, claims can only be made about
the way the context of teaching changes how prospective teachers think and understand

the content.

Significance and Implications

The measure and description of prospective teachers’ knowledge, its application
in teaching, and understanding about basic statistical concepts should be of interest to
several communities: teacher educators, statistics educators, mathematics teacher
educators, prospective teachers’ curriculum developers, statistics professors, and
assessment developers. The implications of the results are theoretical, practical, and
methodological. The first two implications are related to the goals of the study — the
identification of important aspects of statistical knowledge for teaching, and the
descriptions of the knowledge prospective teachers have with respect to the above
aspects. The third implication has to do with the way the aspects were identified and the
development of a reliable instrument to measure them.

Several aspects of statistical content knowledge were identified as very important
for middle grades teachers. The choice of aspects was based on a systematic integrated
analysis of several documents such as students’ state and national standards, which in
part, are based on theoretical and empirical work on student learning. This approach of
identification of content is of particular interest to those that are trying to make hard
choices on what to include — or not to include — in curriculum guides and assessment for

preservice teachers and professional development developers for inservice teachers.

203

The aspects of statistical knowledge for teaching were not identified in the same
fashion as the content. This kind of knowledge is much more complex and has too many
dimensions to be analyzed the same way that students’ content is. However, several
pieces of work were examined and integrated: teachers’ state and national standards,
research and theoretical work on teachers’ knowledge and its role in teaching, and
teachers’ guide to a statistics textbook. These documents provided a general framework
to view teachers’ knowledge as well as specific aspects that come from the actual practice
of implementing curriculum. They are all important as we need the “big picture” to
create vision in teacher preparation programs and the “little picture” to make it happen in
the classroom and to create authentic assessment instruments. The identification of these
aspects for teaching is a starting point for a discussion of what do middle grade teachers
need to know about statistics in order to teach it well and a continuation of how to
measure this knowledge in the other areas of mathematics.

Statistics and mathematics educators are concerned about the way statistics is
taught at all levels, in particular at the elementary and middle grades levels where the
inclusion of this content is relatively new. They are also concerned about the way
teachers understand statistical concepts, as they are different from mathematical concepts
(Bright & Friel, 1998; Burrill, 1998; Cobb & Moore, 1997; Coob, 1992; Friel & Bright,
1998; Garfield, 1995; Garfield & Ahlgren, 1998; Mokros & Russell, 1995; Moore, 1997;
Shaughnessy, 1992). The measurement and description of the statistical knowledge that
prospective teachers have about central statistical topics taught in middle grades in this
study raises concerns about the opportunities future teachers are having in their

preparation to learn statistics in a way that they can teach it to young adolescents. Lack

204

of statistical reasoning to justify and judge proper use of graphical representations and
measures of center and spread, knowledge of procedures and facts without understanding
where the procedures come from or what do the facts tell you about the data, are just
some of the findings that may keep teachers from teaching statistics in a meaningful way.

Current reform efforts in teacher preparation (e. g., Wilson, Floden, & Ferrini-
Mundy, 2001; NRC, 2001a) point out that the subject matter preparation is important and
the current research results are disappointing. They also argue for the need for studies
that give insights into the nature of content and quality across areas and levels. This
study provides needed information about the limitations of middle grades teacher
preparation programs in the area of statistics and data analysis. Changes should be made
so that middle grades mathematics teachers will understand more profoundly the basic
concepts of statistics and data analysis taught in middle grades, be exposed to children’s
work and strategies to represent and reason about data, and understand the study of
statistics as a process of data investigation, not as a collection of procedures and
formulas.

Furthermore, this study contributes to the argument that the knowledge of the
subject matter is a necessary but not sufficient condition for teachers to teach it well.
Most of the subjects that participated in this study were prospective middle school
teachers, seniors with strong mathematical and education backgrounds, including two

., -' ' \
mathematics education courses, on average; and one basrc course 1n statistics. That is, ‘1

1

making a course or two a requirement in their program because they are likely to teach it,

does not guarantee that they would be able to teach it well. It seems that a required

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statistics course needs to be backed up with a course that integrates the content and its

205

application in teaching because results show that although the prospective teachers have

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taken a course in statistics their performance is low in the area of application. Statistics ,-

1

teachers at the university and college level may use this information to make arguments ,

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for developing new courses geared toward future teachers that will better serve them in !
practice.

The methodological implications of this study are concerned with the systematic
way to analyze important content and the development of a reliable instrument to
measure a construct such as knowledge for teaching. Although the written instrument is
limited to some aspects of data analysis emphasized in middle school, it has the
advantage of being able to assess a fairly large scale number of subjects and what is most
important, it provides a quantity associated to those aspects. Conscious of the limitations,

some written responses were followed-up with interviews complementing some of the

gaps and providing more detail.

Recommendations for Future Research
This study investigated statistical knowledge for teaching by studying only some
of its components. Mainly by analyzing experts’ perspectives and measuring teachers’
statistical knowledge. The answer to the problem is not given in this single study
conducted primarily by a single person and it is far from complete. For example, the

study of teachers’ statistical curricula, students’ statistical knowledge, and school

L ‘ - __..-_..H ,,_

statistical practices is needed. What it is hoped to be accomplished is a contribution to -
the solution to the problem by understanding some of its pieces. There is much more to

be learned about this process of teaching and learning statistics for teaching.

206

One of the difficulties in developing the instrument was to identify the important
aspects of statistics applied to teaching situations. There is no research that focuses on {I
l
the statistical knowledge in the context of teaching, that is, studying how the knowledge it
of statistics interacts with the actual work of teaching it. Research is needed in this area“;
to strengthen the validity of the instrument and create more authentic teaching-related
items.

The instrument developed for this study only covers a limited number of
statistical topics and aspects of knowledge for teaching. Topics like bivariate data
representation and distributions need to be added as well as other aspects of knowledge
for teaching such as how to help students learn statistical concepts, evaluation of
curriculum materials, and development of student assessment to name a few.

It would be useful, as well, to generate more items and pilot them to create a
closed version of the instrument. At the moment the instrument is short and free response
which is hard to grade, time-consuming, and limits the number of participants.

One of the beauties of measurement is that we can assign a quantity to a construct,

./”l
not necessarily for accountability, but to make association with other variables. One of

the most needed associations is between teachers’ knowledge and students’ achievement.”

The ultimate goal of this project is to be able to measure that association. As researchers,

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we would like to answer the question of whether there is a relationship between what /

teachers know for teaching and their students’ understanding of statistics. 1

207

APPENDIX A

State and National Student Standards

208

Connecticut State Department of Education

Mathematics Curriculum Framework

Division of Teaching and Learning

Retrieved June 22, 2004 from http://www.state.ct.us/sde/dtl/curriculum/Frmath.pdf

("router/it Standard 7: Statistics and Probability

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Students will use basic concepts of probability and statistics to collect, organize, display
and analyze data, simulate events and test hypotheses.

K-12 Perf mancevStandards

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Educational experiences in Grades 5-8 will assure that students:

0 make conjectures; design simulations and samplings; generate, collect,
organize and analyze data; and represent the data in tables, charts, graphs and
creative data displays;

0 make inferences and formulate and evaluate hypotheses and conclusions based
on data from tables, charts and graphs;

0 describe the shape of the data using range, outliers, and measures of central
tendency, including mean, median and mode;

0 select and construct appropriate graphical representations and measures of
central tendency for sets of data.

209

Florida Department of Education

Sunshine State Standards

Mathematics Grade 6-8

Retrieved June 22, 2004 from http://wwwfim.edu/doe/curric/prek12/frame2.htm
Data Analysis

Standard 1:

The student understands and uses the tools of data analysis for managing
information.

o collects, organizes, and displays data in a variety of forms, including tables, line
graphs, charts, bar graphs, to determine how different ways of presenting data can
lead to different interpretations.

o understands and applies the concepts of range and central tendency (mean, median,
and mode).

0 analyzes real-world data by applying appropriate formulas for measures of central
tendency and organizing data in a quality display, using appropriate technology,
including calculators and computers.

Standard 3:

The student uses statistical methods to make inferences and valid arguments about
real-world situations.

0 formulates hypotheses, designs experiments, collects and interprets data, and

evaluates hypotheses by making inferences and drawing conclusions based on
statistics (range, mean, median, and mode) and tables, graphs, and charts.

210

Georgia Department of Education
Quality Core Curriculum Standards
Retrieved June 22, 2004 from http://www.gl_c.k12.ga.us/qcc/homepg.asp

Grade 6 Mathematics Statistics

Topic: Data Collection, Data Organization, Data Display, Scale
Standard: Collects and organizes data, and determines appropriate method and
scale to display data.

Topic: Data Collection, Data Organization
Standard: Constructs tables, charts, pictographs and bar, circle, and simple line
graphs to display data.

Topic: Mean, Median, Mode, Range
Standard: Finds median, mean, mode, and range of a given set of data.

Grade 7 Mathematics Statistics

Topic: Charts, Tables, Graphs, Distributions

Standard: Collects, organizes data, determines appropriate method and scale to
display data, and constructs frequency distributions, bar graphs, line graphs, circle
graphs, tables, and charts.

Topic: Measures of Central Tendency and Spread
Standard: Uses mean, median, and mode to describe central tendencies of a data
set, and uses range to describe spread of the data.

Topic: Charts, Tables, Graphs, Distributions
Standard: Reads and interprets data in frequency distributions, diagrams, charts,
tables, and graphs; and makes predictions or conclusions based on this data.

Grade 8 Mathematics Statistics

Topic: Data Collection, Data Organization, Data Display, Scale

Standard: Collects and organizes data, determines appropriate method and scale
to display data, and constructs frequency distributions; bar, line, and circle graphs;
tables and charts; line plots, stem-and-leaf plots, box-and-whisker plots, and
scatter plots.

Topic: Mean, Median, Mode, Range
Standard: Uses mean, median, mode, and range to describe tendencies of a data
set and make predictions.

211

Kentucky Department of Education

Core Content for Assessment - Mathematics

Retrieved June 22, 2004 from
http://www.education.kv.gov/KDE/Instructional+Resources/Middle+SchoolMathematics
/Kentuckvs+Curriculum+Documents+for+Mathematics.htm

Grade 6 through 8 with Assessment in Grade 8"I

Statistics

Concepts - Students will describe properties of, define, give examples of, and/or apply to
both real-world and mathematical situations:
0 Meaning of central tendency (mean, median, mode)

0 Meaning of dispersion (range, cluster, gaps, outliers)

0 Characteristics and appropriateness of graphs (e.g., bar, line, circle), and plots
(e. g., line, stem-and-leaf, box-and-whiskers, scatter)

Skills - Students will perform the following mathematical operations and/or procedures
accurately and efficiently, and explain how they work in real-world and mathematical
situations:

0 Organize, represent, analyze, and interpret sets of data

0 Construct and interpret displays of data (e.g., table, circle graph, line plot, stem-
and-leaf plot, box-and-whiskers plot)

0 Find mean, median, mode, and range; recognize outliers, gaps, and clusters of data

Relationships - Students will show connections and how connections are made between
concepts and skills, explain why procedures work, and make generalizations about
mathematics in meaningful ways for the following relationships:
0 How different representations of data (e. g. tables, graphs, diagrams, plots) are
related

0 How data gathering, bias issues, faulty data analysis, and misleading
representations affect interpretations and conclusions about data (e. g., changing
the scale on a graph, polling only a specific group of people, using limited or
extremely small sample size)

0 How probability and statistics are used to make predictions and/or draw
conclusions

212

North Carolina Department of Education
Current Standard Course of Study and Grade Level Competencies (1998)
Mathematics Curriculum - Middle Grades 6-8

Retrieved June 22, 2004 from
http://www.dpi.state.nc.us/curriculum/mathematics/middle.html

Data, Probability, and Statistics

Competency Goal 4
Grade 6

Create and evaluate graphic representations of data.
Use measures of central tendency to compare two sets of data.

Construct convincing arguments based on analysis of data and interpretation of
graphs.

Grade 7

Interpret and construct histograms.

Compare and relate bar graphs and histograms.

Construct circle graphs using ratios, proportions, and percents.

Create, compare, contrast, and evaluate both orally and in writing, different
graphic representations of the same data.

Identify appropriate uses of different measures of central tendency.

Grade 8

Interpret and construct box plots.

Collect data involving two variables and display on a scatter plot; interpret results;
identify positive and negative relationships.

Interpret the mean, explain its sensitivity to extremes, and explain its use in
comparison with the median and the mode.

213

Missouri Department of Elementary and Secundary Education
Missouri's Frameworks for Curriculum Development
Mathematics

Retrieved June 22, 2004 from
http://dese.mo.gov/divimprove/curriculum/frameworks/math.html

By the end of grade 8, all students should be able to

develop, analyze, and explain methods utilized to collect, organize, and describe
data

make, read, and interpret multiple representations including tables, charts and
graphs of data

formulate, predict, and defend positions taken that are based on data collected
analyze information and arguments that are based on data collected

investigate the power of making decisions based on statistical methods and the
applications of probability in the real world

use computers, graphing calculators, and/or other forms of technology to enhance
understanding of numbers, data, and the resulting analysis

214

Ohio Department of Education

Joint Council of the

State Board of Education and the Ohio Board of Regents
Mathematics Academic Content Standards

Retrieved June 24, 2002 from
http://www.ode.state.oh.us/academictcontent_standards/acsmath.asp

Data Analysis Standard

Students pose questions and collect, organize, represent, interpret and analyze data to
answer those questions. Students develop and evaluate inferences, predictions and
arguments that are based on data.

By the end of the 6 — 8 program...

Select, create and use appropriate graphical representations of data including
histograms, circle graphs, box plots and scatter plots and justify the selection of
the graph types.

Evaluate different graphical representations of the same data to determine which
is the most appropriate representation for an identified purpose.

Find, use and interpret measures of center and spread, including mean and
interquartile range, and use these measures to compare two sets of data.

Interpret the mean, explain its sensitivity to extremes, and explain its use in
comparison with the median and mode.

Construct convincing arguments based on analysis of data and interpretation of
graphs.

215

Oregon Department of Education

Academic Standards

Retrieved June 24, 2004 from
http://www.ode.state.or.us/cifs/learningresource/searchstandards.aspx

Mathematics — Statistics Grade 6-8

0 Find, use, and interpret measures of center and spread, including mean and
interquartile range for given or derived data.

0 Formulate questions and design experiments or surveys to collect relevant data.

0 Represent and interpret data using frequency distribution tables, box-and whisker-
plots, stem-and-leaf plots, and single- and multiple- line graphs.

0 Determine the graphical representation of a set of data that best shows key
characteristics of the data.

0 Recognize distortions of graphic displays of sets of data and evaluate
appropriateness of alternative displays.

0 Analyze data from frequency distribution tables, box-and whisker-plots, stem-and-
leaf plots using measures of center and spread and draw conclusions.

- Predict and evaluate how adding data to a set of data affect measures of center.

0 Use observations about differences between two or more samples to make
conjectures about the populations from which the samples were taken.

216

Virginia Department of Education

Standards of Learning

Mathematics

Retrieved June 22, 2004 from http://wwwmenkl2.va.us/go/Sols/math.html

Statistics

Grade 6

The student, given a problem situation, will collect, analyze, display, and interpret
data in a variety of graphical methods, including line, bar, and circle graphs and
stem-and-leaf and box-and-whisker plots. Circle graphs will be limited to halves,
fourths, and eighths.

The student will describe the mean, median, and mode as measures of central
tendency and determine their meaning for a set of data.

Grade 7

The student will create and solve problems involving the mean, median, mode,
and range of a set of data.

The student will display data, using frequency distributions, line plots, stem-and-
leaf plots, box-and-whisker plots, and scattergrams.

The student will make inferences and predictions based on the analysis of a set of
data that the student(s) collect.

Grade 8

The student will use information displayed in line, bar, circle, and picture graphs
and histograms to make comparisons, predictions, and inferences.

217

West Virginia Department of Education
Mathematics Content Standards and Objectives for West Virginia Schools
Retrieved June 22, 2002 from http://wvde.state.wv.us/policies/p2520.2_ne.pdf

Standard 5: Data Analysis and Probability

Students will:

formulate questions that can be addressed with data and collect, organize, and
display relevant data to answer them;

select and use appropriate statistical methods to analyze data; develop and
evaluate inferences and predictions that are based on models.

Data Analysis and Probability Objectives

Grade 6

collect, organize, display, and interpret data using line graphs, circle graphs, bar
graphs, histograms, stem-and-leaf plots, tables, and charts.

Create and solve problems involving the mean, median, mode, and range of a data
set of data.

Grade 7

Read and interpret multiple line graphs;

extrapolate information from multiple-line graphs, circle graphs, bar graphs,
histograms, tables, and frequency distributions (tally charts);

collect, organize, graphically represent, and interpret data using frequency
distributions, line-plots, stem-and-leaf plots, box-and-whisker plots, and scatter
plots;

determine measures of central tendency (mean, median, mode, range) and
dispersion from data, table, and experiment.

Grade 8

draw inferences and construct convincing arguments based on data analysis.

218

Principles and Standards for School Mathematics
National Council of Teachers of Mathematics (2000)

Data Analysis Standard for Grades 6 - 8

In grades 6 - 8 all students should ——

formulate questions, design studies, and collect data about a characteristic shared
by two populations or different characteristics within one population;

select, create, and use appropriate graphical representation of data, including
histograms, box plots, and scatterplots.

Find, use, and interpret measures of center and spread, including mean and
interquartile range;

discuss and understand the correspondence between data sets and their graphical
representations, especially histograms, stem-and-leaf plots, box plots, and
scatterplots.

Use observations about differences between two or more samples to make
conjectures about the populations from which the samples were taken;

make conjectures about possible relationships between two characteristics of a
sample on the basis of scatterplots of the data and approximate lines of fit;

use conjectures to formulate new questions and plan new studies to answer them.

219

APPENDIX B

Codes Assigned to Student Standards

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9:5. mafia mean”. 953. 953. 9:5. mafia
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234

 

APPENDIX C

Statistical Knowledge for Teaching Assessment

235

MICHIGAN STATE UNIVERSITY
DEPARTMENT OF MATHEMATICS

January, 2002
Dear Prospective Teacher:

I am a doctoral student in Mathematics Education at Michigan State University. I am
conducting a study as part of my dissertation about the conceptions of prospective
teachers about data analysis and statistics as they relate to the teaching of these subjects.

Please answer the following questionnaire to the best of your knowledge without
consulting another person or any other resources. The approximate time needed to
complete the questionnaire is about 50 minutes. Responses will be confidential; that is,
your privacy will be protected to the maximum extent allowable by law. Responses will
be used only with the purpose of understanding how prospective teachers know and apply
statistical knowledge in teaching. In some cases, actual written work may be used in the
discussion of results in potential subsequent publications. If this is done, your name will
not be used.

You indicate your voluntary agreement to participate by signing your name on the next
page and by completing and returning this questionnaire. Please provide your e-mail as
well, in case we need to contact you for clarification of your responses. If you have any
questions about this study, please contact me by email: sortomar@msu.edu. If you have
questions or concerns regarding your rights as a study participant, or are dissatisfied at
any time with any aspect of this study, you may contact - anonymously, if you wish -
Ashir Kumar, M.D., Chair of the University Committee on Research Involving Human
Subjects (UCRIHS) by phone: (517) 355-2180, fax: (517)432-4503, e-mail:
ucrihs@msu.edu, or regular mail: 202 Olds Hall, East Lansing, MI 48824.

Sincerely,

Maria Alejandra Sorto
Mathematics Department
Michigan State University
East Lansing, MI 48824

236

 

MICHIGAN STATE UNIVERSITY
DEPARTMENT OF MATHEMATICS

Statistical Knowledge for Teaching Assessment

Instructions

Please write all your responses in this booklet. Some questions ask about “your students.”
When answering these questions, imagine you are actually teaching students at a grade
level you hope to teach.

You may use a calculator

237

 

 

 

Background Questions
Name: e-mail:

 

1. Male B Female [:1
2.Agerange: 19—23 B 24—29 C] 30—35 D over35 El
3. Class: Freshman [:1 Sophomore [:1 JuniorEl

Senior E] Post-grad El

4. Discipline major :
Discipline minor :

 

 

5. Please check the mathematics courses you have taken at the college level.

 

_ MATH 10 : Algebra

__ MATH 30: Trigonometry and Analytic Geometry
__ MATH 31: Calculus of One Variable I

__ MATH 32: Calculus of One Variable II

__ MATH 67: Number and Algebra

__ MATH 81: Discrete Mathematics

__ MATH 131: Eucledian and Non-Eucledian Geometry
MATH 111: Developing Math Concepts

MATH 115: History of Mathematics

STAT 11: Basic Concepts of Statistics and Data Analysis
STAT 31: Introductory Statistics

Other:

—
_—
—
_
.—

 

6. Please check the math education courses you have taken at the college level.

EDUC 86 Teaching Mathematics in the Middle Grades

Other:

 

7. Please check the education courses you have taken at the college level
__ EDUC 65: Introduction to Teaching
__ EDUC 66: Planning for Teaching in the Middle Grades
__ EDUC 69: Teaching Skills Laboratory
_ EDUC 96: Teaching Internship
__ EDUC 97: Seminar on Teaching
_ Other:

 

8. Are you using a calculator? Yes D No D

If Yes, indicate the brand and model (e.g. T183)

 

 

238

 

 

 

 

1. The stem-and-leaf plot below shows the number of minutes it takes students
in a class to travel to their school.

Minutes to Travel to School

0 335789

1 02356689
2 013335588
3 05

4 5

 

a. How many students are in the class?

 

b. How many students took less than 15 minutes to travel to
school?

c. What is the typical time it takes for students to travel to
school?

 

Explain your answer.

 

 

239

 

 

 

2. Nine students in a science class weighed a small object on the same scale
separately.The weights (in grams) recorded by each student are shown below:

6.2 6.0 6.0 15.3 6.1 6.3 6.2 6.15 6.2
The students want to determine as accurately as they can the actual weight of
this object.
They may use the following methods:

I. Use the most common number, which is 6.2.

II. Use the 6.15 since it is the most accurate weighing.

III. Use the result of adding up the 9 numbers and dividing by 9.
As a teacher, what method would you prefer your students to use?

a. Method I

b. Method II

c. Method 111

d. Other

 

Explain your choice.

 

240

 

 

 

3. The following graph gives information about the adult female literacy rates
in Central and South American countries.

 

3 - [—

D

45 50 55 60 65 7O 75 80 85 90 95 100 I

Frequency

 

 

 

 

 

 

 

 

 

Adult Female Literacy Rate (%)

Adult Female Literacy Rates in Central and South America

a. Suppose you ask your students to tell you how many countries are
represented in the graph. One student says, “there are 7 countries

represented”.

Is this student right or wrong?

 

In your opinion, what is the student’s thinking to arrive to that
conclusion?

b. Suppose now you ask your students to explain what the third bar from
the right indicates. One says, “It indicates 85% to 90% literacy rate”.

Comment on the response.

 

 

241

 

 

 

 

4. The following line plot shows the number of people in households in a
neighborhood.

Number of People in Households

X X
XXX X

1234567

a. Find the mean. Show how you find it.

b. Is it possible to have other sets of data with the same mean?
Explain why or why not.

c. Is it possible to have a data set of six households with mean 3%
people?
If yes, give an example.

If not, explain why.

 

242

 

 

 

 

5. One middle school class generated data about their pets shown below.

 

Pet Frequency

bird 2

cat 4

cow 2

dog 7

duck 1
2
1
3
3

 

fish
goat

horse
rabbit

 

 

 

a. Give a possible question the teacher could have asked
the students to generate the data.

b. Students were talking about the data and one said:
“ The mode is dogs, the median is duck, and the range is 1 to 7.”

If you think the student is right, explain why.

If you think the student is wrong, identify the mistake(s).

 

243

 

 

 

 

 

 

 

6. A middle school class was wondering how much time Student’s Time in
it took each student to travel to school each morning Initials minutes
and collected the following data. DB 60
Students are asked to make a graphical display of the DD 50
data to show how much time the majority of students SE 35
take to travel to school. AB 30
FH 25
a. Which one of these plots seems the most appropriate CL 25
for the data? DR 22
BN 20
D Histogram El Stem-and-leaf Plot VH 20
IW 17
Cl Bar Graph [:1 Other AS 15
KS 15
VC 15
Explain your choice. AS 10
MS 8
RS 5

 

 

 

b. Make the plot picked in part a.

 

244

 

 

 

 

7. Imagine that two second-grade students in the same class have created the
following representations to show the number of teeth lost by their classmates.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Student]:
21k 45 47‘
39 'Q’ | en
tr 6" '
___F—‘—
5 m 6
c) Q {9‘
7 © (53 37'
3 ,0 V“ ml a?» 4,) In
to ®
Student 2: :1

 

 

 

Illlllll

 

 

 

 

 

 

 

lillllllllLl

 

 

llllllllll

a. Compare and contrast the two representations. That is, how are they
alike? How are they different?

b. What do you think each student understands about the data?

 

245

 

 

 

 

8. Consider the dot frequency distributions below with the heights in inches
of the 10 players on two basketball teams.

 

 

o
TeamA TeamB .
o
o
o o ’
. Q 0 O O Q Q Q Q C O Q 0
éoiiirisrifitlglol joiilfijs‘fiiilslol—
Heights Heights

a. The mean of each distribution is 75. Calculate the following for each

 

distribution.
Team A Team B
Mean 75 75
Range

Standard deviation

 

 

b. Explain how you could have determine which distribution has the
largest standard deviation, without calculating.

 

246

 

 

APPENDIX D

Item Scores and Statistics for Assessment

247

Item Scores and Statistics

Table 0.1

Item Scores

Thinking_

Reasoning

Literacy

75

50

50
37.5

100

75
80

80
75
45

90

60

75
25

75
75
75
90
100

100
65

87.5

95

87.5

70

50
50
37.5

100
75

70
70
45

25
87.5

75
60

75
60

1 00
90

100

60

12.5

65

75

50
37.5

80
70
60
60

80

85

37.5

70
65

50
62.5

60

65

37.5

80

80

20
35

12.5

70
50
80

37.5

50
65

100

12.5

25

35

37.5

90
35

37.5

75

 

For
Teach

Pure

Stat

67

85

33
63

75
77

17
58
67

50
85

63

67

85

83

81

67

92

46

69

67
43

88
79

27

63

21

25

60

42
46

75

71

50
29
58
42

60
63

73

71

21

27

50
29

52

 

%

79
61

57

72
39
76

52
28
55
46
57

79
82
83
61

59
6O

60
43

43
31

78
74
25
49

53

18
35

4664

45

63

405660

57

41

51

37

49

19
47

14

33
75

25

24

1 8
49
32

68

 

8b Sum

8aii

7b 8ai

6b 7a

5b 6a

4b 4c 5a

1b 1c 2 3a 3b 4a

0

0

 

 

10

11
12
13
14
15
16
17
18
19
20
21

248

22
23
24
25

26
27
28

29

75
37.5

65

80
60

65

12.5

65

75
75

12.5

80
60

62.5

95
55
75
95

25

55
40
95

87.5

25
87.5

60

75
90
60
95

100
87.5

65

55
75

87.5

1 00
51

35

75

63

74
75
1 9

30

21

 

67

73
58

29
42

46
29
29

67

75

63

63

83

71

81

29

63

85

42
45

63

66

18

18

 

71

51

49

35

54 6O
60

39

43

60

43

43

31

32

76
65

55
47

72

52

51

37

75

40
42

59
60

16

12

 

1.6

4

4

1.7

 

3O
31

32

33

35

37

39
40
41

42

Mean

 

Median
SD

249

 

 

APPENDIX E

Interview Protocols

250

 

 

Interview Protocol for subject A

1. In question 1c you the median for typical, what would you say is the typical time for
this data?

Minutes to Travel to School

0 33556778
1 001489

2 1123

3 55

4 335678

 

2. Suppose that there are two modes, how would you reconcile that?

3. There is no way to find out how many countries are represented? What does
“frequency” means in this case?

4. Suppose you did not know the algorithm of the mean, how would you estimate
the mean from the picture? How would you explain 4b?
How would you interpret 3.5 people as the mean of the data?

Take a look at this other set of data, can you estimate the mean?

 

X x

X x

X x x

X x x x x x

X x x x x x x

X x x x X X X X x X
26 27 28 29 3o 31 32 33 34 35 36 37 38 39 40

Number of Raisins in a Box

5. Interpret what “bird 2” means for your question.

6. You would need to list students initials to do the bar graph or the histogram?
So, how much time the majority of students take to travel to school?

7. You did not answer this one, do you want to think out loud how would you answer this
questions?

Statistically speaking, how would you rank these students?

You said that the second student does not understand order, what do mean by that?

251

8. Do you calculations make sense? What do these numbers mean? How did you get
them?

Which team would you bet on? and why?

252

 

Interview Protocol for subject B

1. In question 1c you the median for typical, what would you say is the typical time for
this data?

Minutes to Travel to School

0 33556778
1 001489

2 1123

3 55

4 335678

 

2. Why would the mean be more accurate?

3. There is no way to find out how many countries are represented? What does
“frequency” means in this case?

What would be the ideal response here?

5. Suppose you did not know the algorithm of the mean, how would you estimate
the mean from the picture? How would you explain 4b?
How would you interpret 3.5 people as the mean of the data?

Take a look at this other set of data, can you estimate the mean?

 

X X

X X

X x x

X x x x x x

X x x x x x x

X x X x X X X X x X
26 27 28 29 3o 31 32 33 34 35 36 37 38 39 40

Number of Raisins in a Box

5. What would be the actual question? Interpret what “bird 2” means for your question.
So, what would be the right response for the mode, median, and range?

6. How about in terms of the data? Does the data tell you what display to use?
So, how much time the majority of students take to travel to school?

253

 

7. You did not answer this one, do you want to think out loud how would you answer this
questions?

Statistically speaking, how would you rank these students?
You said that the second student does not understand order, what do mean by that?

8. Do you calculations make sense? What do these numbers mean? How did you get
them?

Which team would you bet on? and why?

254

 

Interview Protocol for subject C

1. In question 1c you picked a range of values for typical, what would you say is the
typical time for this data?

Minutes to Travel to School

0 3 3 5 5 6 7 7 8
1 0 0 1 4 8 9

2 1 1 2 3

3 5 5

4 3 3 5 <5 7 8

 

2. Were you going to say more about method 111?, what about method I and II?

3. How would you correct this student?
It seems that you did not have enough time to finish part b, what you tell the student?

4. Suppose you did not know the algorithm of the mean, how would you estimate the
mean from the picture? How would you explain 4b?
How would you interpret 3.5 people as the mean of the data?

Take a look at this other set of data, can you estimate the mean?

 

X x

X x

.x x x

)( x. )< x .x x

)< .x )( :x :x x: :x

ix x: )( :x x .x :x x :x x
26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

Number of Raisins in a Box
5. Interpret what “bird 2” means for your question:

You wrote up here that the median is 12.5 but you say down here that the median is
horse, which one is it?

6. So, how much time the majority of students take to travel to school? What do you
mean by rightly represented?

7. You did not answer this one, do you want to think out loud how would you answer this
questions?

255

Statistically speaking, how would you rank these students?
You said that the second student does not understand order, what do mean by that?

8. Do you calculations make sense?
You mention that are positive, does this tell you something? You say that “s.d. is 7

away from the mean”, 7 what?

Which team would you bet on? and why?

 

256

 

 

Interview Protocol for subject D

1. In question 1c you picked the mean for typical, what would you say is the typical time
for this data?

Minutes to Travel to School

0 335789

1 023566889
2 01333

3 1

4

5

6 2

 

2. What about method I or method II?

3. The bars are not countries, so what do they represent?

How do you think we can correct this mistake?

What would be the ideal response you are expecting the student to answer?
You said the graph is misleading, in what way?

So, is the student’s response wrong?

4. For 4b, why is it possible?

For 4c, you say the only way possible is if some of the numbers are 21/2 and so on?
Suppose you did not know the algorithm of the mean, how would you estimate the mean
from the picture? How would you explain 4b?

How would you interpret 3.5 people as the mean of the data?

Take a look at this other set of data, can you estimate the mean?

 

X x

X x

X x x

X x x x x x

X x x x X x x

X x x x X X X X x X
26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

Number of Raisins in a Box

5. Interpret what “bird 2” means for your question.

257

You can see how students come to this conclusions, but is the student correct?

6. Is this a histogram?

7. Tell me what student I understand? Do you think questions about the number of teeth
lost can be answer by any of the two representations?

Statistically speaking, how would you rank these students?

8. Do you calculations make sense?

Which team would you bet on? and why?

258

 

Interview Protocol for subject E

1. Can you show me how you got 87 minutes for the typical time?
In question 1c you picked the average for typical, what would you say is the typical
time for this data?

Minutes to Travel to School

0 335789

1 023566889
2 01333

3 1

4

5

6 2

 

2. Why not the most common or the most accurate?

3. The bars are not countries, so what do they represent?
How do you think we can correct this mistake?

4. Suppose you did not know the algorithm of the mean, how would you estimate the
mean from the picture? How would you explain 4b?

How would you interpret 3.5 people as the mean of the data?

Take a look at this other set of data, can you estimate the mean?

 

X x

X x

X x x

X x x x x x

X x x x X x x

x x x x x x x x x x
26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

Number of Raisins in a Box

5. Interpret what “bird 2” means for your question.
What would be a correct way to order the data? What would the median be?

6. What do you mean by “group into range”? Why not the others?

7. Explain to me again how are they alike and different?

259

Do you think questions about the number of teeth lost can be answer by any of the two
representations?
Statistically speaking, how would you rank these students?

8. You said “that number” which one? The SD? How did you find these numbers? Do
they make sense?

Which team would you bet on? and why?

260

Interview Protocol for subject F

1. In question lc you picked the mode for typical, what would you say is the typical time
for this data?

Minutes to Travel to School

0 335559
1 023566889

2 01334

3 1

4 2

 

2. Why is it ‘unfair’ to use the other methods?
What do you mean by “these numbers may not be the actual weight of this object”?

3. The bars are not countries, so what do they represent?
How do you think we can correct this mistake?

4. Suppose you did not know the algorithm of the mean, how would you estimate the
mean from the picture? How would you explain 4b?
How would you interpret 3.5 people as the mean of the data?

Take a look at this other set of data, can you estimate the mean?

 

X x

X x

x x x

X x x x x x

X x x x x x x

x x x x x x x x x x
26 27 28 29 3o 31 32 33 34 35 36 37 38 39 40

Number of Raisins in a Box
5. Interpret what “bird 2” means for your question.

6. What do you mean by “continuing”? Explain how is this a histogram. Why not the
others? Can you tell from here how much time the majority of students take to travel?

7. Do you think questions about the number of teeth lost can be answer by any of the two
representations?

261

 

 

Statistically speaking, how would you rank these students?
8. How did you find these numbers? Do they make sense?

Which team would you bet on? and why?

262

Interview Protocol for subject G

1. In question 1c you picked the average for typical, what would you say is the typical
time for this data?

Minutes to Travel to School

0 335789

1 023566889
2 01333

3 1

4

5

6 2

 

2. What do you mean by “many of the data are varied’?
When you say that it is a mathematical way, you mean...

3. You say that the bars represent a range of countries between certain literacy rates, so
how many countries are represented here?

How do you think we can correct this mistake?

4. Why a ‘clump’ data about 4 would have the same mean, what about spread out?
How would you interpret 3.5 people as the mean of the data?

Take a look at this other set of data, can you estimate the mean?

 

X x

X x

X x x

X x x x x x

X x x x x x x

X x x x X X X X x X
26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

Number of Raisins in a Box
5. Interpret what “bird 2” means for your question.
What would be a correct way to order the data? What would the median be?

7. Do you think questions about the number of teeth lost can be answer by any of the two
representations?

Statistically speaking, how would you rank these students?

263

8. Which team would you bet on? and why?

264

Interview Protocol for subject H

1. In question 1c you the mode for typical, what would you say is the typical time for this
data?

Minutes to Travel to School

0 335559

1 023566889

2 0 1 3 3 4
3 1
4 2

 

2. So, you want all the data to be represented? Take a closer look at the data.

3. There is no way to find out how many countries are represented? What does
“frequency” means in this case?

4. Suppose you did not know the algorithm of the mean, how would you estimate the
mean from the picture? How would you explain 4b?
How would you interpret 3.5 people as the mean of the data?

Take a look at this other set of data, can you estimate the mean?

 

X x

X x

X x x

>< .x )( :x .x :x

X x x x x x x

:x )( 1x )( .x )( )( :x :x .x
26 27 28 29 3o 31 32 33 34 35 36 37 38 39 40

Number of Raisins in a Box

5. Interpret what “bird 2” means for your question.
Explain more how to find the median here.

6. So, how much time the majority of students take to travel to school?

265

7. You did not answer this one, do you want to think out loud how would you answer this
questions?

Statistically speaking, how would you rank these students?

You said that the second student does not understand order, what do mean by that?

8. Do you calculations make sense? What do these numbers mean?
You mention that are positive, does this tell you something? You say that “s.d. is 7

away from the mean”, 7 what?

Which team would you bet on? and why?

266

Interview Protocol for subject I

1. In question 1c you picked the median for typical, what would you say is the typical
time for this data?

Minutes to Travel to School

0 33556778
I 001489

2 1123

3 55

4 335678

 

2. In this case, is it important to take all the number into account? What about method
II?

3. The bars are not countries, so what do they represent?
How do you think we can correct this mistake?
What would be the ideal response you are expecting the student to answer?

4. Suppose you did not know the algorithm of the mean, how would you estimate the
mean from the picture? How would you explain 4b?
How would you interpret 3.5 people as the mean of the data?

Take a look at this other set of data, can you estimate the mean?

 

X x

X x

X x x

X x x x x x

X x x x x x x

X x x x X X X X x X
26 27 28 29 3o 31 32 33 34 35 36 37 38 39 40

Number of Raisins in a Box

5. Interpret what “bird 2” means for your question.
Is the mode and range correct?
What is the meaning of the fish as median?

6. What about a histogram?

267

7. Do you think questions about the number of teeth lost can be answer by any of the two
representations?
Statistically speaking, how would you rank these students?

8. Do you calculations make sense?
You said “these facts also are true because the mean and range are the same”, does that

mean that if the mean and range weren’t the same...

Which team would you bet on? and why?

268

Interview Protocol for subject J

1. In question 1c you picked the mean for typical, what would you say is the typical time

Minutes to Travel to School

for this data?
0
1
2
3
4
5
6

2.

 

335789

023566889

01333

1

3. When you say “yes and no” you mean that the student is right and wrong? Explain.
So, he/she is not taking the correct range?

5. For 4b, you gave an example, how would you show that there are many data sets?
For 4c, for this context it is not possible, right?

Suppose you did not know the algorithm of the mean, how would you estimate the mean

from the picture? How would you explain 4b?

How would you interpret 3.5 people as the mean of the data?

Take a look at this other set of data, can you estimate the mean?

 

X x

X x

X x x

X x x x x x

X x x x x x x

X x x x X x x x x x
26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

Number of Raisins in a Box

5. Interpret what “bird 2” means for your question.

What about the median?

269

6. So, how much time the majority of students take to travel to school with your line
graph?

7. Do you think questions about the number of teeth lost can be answer by any of the two
representations?

Statistically speaking, how would you rank these students?
You said that the second student does not understand order, what do mean by that?

8. Do you calculations make sense?
What about Team B?

Which team would you bet on? and why?

270

APPENDIX F

Interview Transcripts

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Question 1

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Question 2

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Subject B

Part c of question 1 refers to what is the typical time for students to travel
to school, and I am curious to know what you think “typical” means.

It could mean any kind of a center, to me it could mean median, or it
means. . .you know. . .not normally I don’t look at mode, but it could mean
mode.

So, you picked the median.

I thought it was easier, because it is already organized.

OK, so, let me ask you the same question for this other set of data.

I would find the median, it is organized, so...

Would you still stick with the median?

Ummm. . . [thinking]...it is about 18, right. To me it is easier if it is
organized or some kind numerical order.

You said that you chose method 111 (the mean) because it is more accurate,
what do you mean by that?

because it’s clustered all around a close sort of range. I though OK with
the mean. . .you know. . .I though the mean and the median might be close
in this situation, but some times when you have a really, really high, like
say you have a score of 8.5, I might want to say I want to ignore the 8.5
and do median instead. I say that the extreme score is going to weight the
mean more than it would with the median.

and in this case. ..

In this case the mean and the median are going to be really close, because
the data are all cluster together.

around. . .cluster around...
Well, you have the one... that’s what brought me to think about it. Like I
said, well, even though I say maybe we want to take that one into account,

because sometimes you do. . .so I said, well it might be wired but maybe
we should take him into account . So I thought maybe in this situation you

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Question 3

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want to. That’s why I said the mean, I just didn’t know. They are all
cluster except for that one guy, that’s what made me think, do I really want
to take it into account or not.

When you say that “we don’t know” if the student is right or wrong, do
you mean that we don’t have enough information in the graph to say how
many countries are represented?

We assume that we have data for everybody, but apparently we didn’t .
Sometimes you have something missing, like I though, OK there is quite a
bit missing here [pointing at the gap between 50% and 70%] why do we
lay out our histogram this way, and I though, maybe there is something
where there was no literacy rate. . .some country where there is zero.

That’s why those gaps

You never know. . .I though it was a strange histogram. Normally, you
would just said, well you have a bar for every thing, but you don’t always
know that, right? When you have a gap, you don’t always know about
your gap. This bother me a little, this gap.

Why do you think the student say 7?
I don’t know, because he wasn’t really looking ...well, he was only
looking at one bar or two and then saying. . .you know what I mean. I

couldn’t figure. . .error with kids always bother me. . .in math. . .I look at the
error and I go. . .it is hard to know what they mean.

How do you know there are more data sets with the same mean?

If you want to have another data set with the same mean 4, all you do, you
can move one up and one down or two up and two down. You have to
keep it equal, the same number.

Estimate the average with this data set? What would the kid would say?

Their best guess would be 32 or 33 because half are here and half are over
here.

What is the meaning of 3.5?

That is the mean, you take all the numbers and add them all up and divide
by the number you have.

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Question 5

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Suppose you are explaining to a kid. Would the kid he bother by the 3.5?
I don’t think so, if you explain it in certain way.
Like what?

If you say, well nobody really have 4.5 kids, but over a big data set some
people have 2 kids, some people have 3, some people have 6, some people
have 8. So over a big data set it could be 4.7, it could be 3.2, it just means
over a big data set that is your average. I think kids understand that.

You have to say, well, you take an average and the average means that that
matters, that every one matters, that’s why you would get 4.7 kids. It just
means that people are all over the range, some people have 1, some people
have 2, 3, 4, 5, 10.

Now, I am going to play the kid, the kid would say, why does the answer
come out to be a whole number?

I think you have to teach them about the median, maybe, that there is

another number that could be a whole number, but not always. Even the
median could be 4,5, but...

What is the question the teacher asked?
How many people have. . .?
What does “bird 2” means”

That would mean that two people have one bird OR one people have two
birds.

For question b, is the student completely wrong?

He is analyzing the data as continuous and it is a discrete data. For pets,
you can’t really have a median or mode, it is discrete data. . .I don’t see
that, because five people could have count for all the pets and does not
mean anything.

What do you think about the range?
We don’t even know that, we don’t even know what the range is of the
number of pets. That doesn’t tell us, we don’t know how many pets does

the average student has, so we don’t know what the range is. Is the range
0 to 10 pets? We don’t know that, it does not tell us that.

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Question 6

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Question 7

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Tell me why did you pick the steam-and-leaf plot?

I thought the histogram is easier, but I always do the steam-and-leaf plot
before I do the histogram because I want to see how does the data actually
clusters. Cause there are so many in two and them I might I would have to
split my groupings slightly so I wouldn’t have so many in just one part of
the histogram.

How about in terms of the data? Would you do the same for another set of
data?

It depends on the students’ ability level, but you can do a histogram.
What can you tell me about the bar graph?

Well, time is continuous in general, it’s a measure as suppose as discrete
data. I don’t know, you could do it I suppose. I wouldn’t tell a child no
you can’t ever try this. They can try it and see how it works, right?

How much time the majority of students take to travel to school?

There in their 203

What type of graphs?
There are bar graphs.
What is the total number. . .?

Student 1 has an easier time counting them up, because is a pictograph and
in second grade they are more into pictograph, they are more into seeing it
visually. So he or she can just count.

Assessing?

I would actually say student 1. Student 2 does not have anything that tells
me what it is. If they have told me a key, or if they have told me what this
axes represents here and what this represents I would have an easier time
to. . .but they organize it beautifully. I like this second grade work, it
shows that they did it themselves or in a little group and they really though
about it.

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Question 8

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You want students to understand each other?

You would have to talk to each other and give advice to each other.

Do your answers make sense?

It does, this stuff is all scatter so it is a higher standard deviation. This
stuff, you only have two aberration and the rest is all cluster right at the
mean, so it makes sense that one would be smaller than the other.

What does 2.14 means?

2 untis, it means that from your mean one standard deviation takes into
account 68% of your data and two standard deviations take a 95%, that’s
all it means.

If you were a coach...

Either one, you know what? I know this is going to sound really strange
but I would probability would like to coach Team A. Because I wouldn’t
mind coaching all different ability levels, because I am just different.
Whereas these two extremes, I would feel so bad for this kid that is way up
here and so bad for this kid that is way down here. Sure it would be easier
to coach these kids (Team B) it would be horrible, this kid would be
always be playing and this kid would never be playing.

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Question 1

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Question 2

Subject C

In question 1c you picked a range of values for typical, why?
I think, I use the mean, try to find the mean.
What is typical for this other data set?

I’ll use the median, because we have two groups with the same number, so
I can’t just pick one.

[Time was limited for this subject and no questions were asked here]

Question 3

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Question 4

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Question 5

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How do you tell the student that he/she is wrong?

The graphs only shows the literacy rate, from the graph there is no way to
find out how many countries.

What does the third bar represents?

The third bar represents 85% to 90% literacy rate for 3...

Suppose you are introducing the concept of the mean, how would you
estimate the mean without reaching to the algorithm? Show raisin plot.

Some of them would say to equate the stacks to have at the same level, and
then I want them to say that the mean will be each of the 3 boxes, the high
ofthe boxes 3,3,3...

Is it possible to have a data set of 6 households with a mean 3.5 people?

I can’t have half of person, it depends on

You wrote up here that the median is 12.5 but you say down here that the
median is horse, which one is it?

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Student: I try to bring the number in order, sum the number which is 25 to find how
many pets, then divided by 2 because is odd, and counted 12 and 12 from
each end and pick the middle

Question 6

Researcher: Which one is easier to understand?....

[Time run out and interview had to be suspended]

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Question I

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Question 4

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Subject E

How did you get 87?

I added the times3, 3, ..., and then divided by 26.
Does that make sense to you?

No, it does not. I think I did something wrong...

I am going to show you another data set. What will be the typical time
here?

Add them up, 3 plus 3, plus 3,..., and divided by 21. The answer is 17.85

Why not method I?
You could, this is just another way to find the average. It depends of the

average you want, it is not wrong. As a teacher you have to teach and
accept other ways. But I would prefer the method I choose.

What do you want the student to say the bars represent if not countries?
The percent of Adult. ..

How would you correct the student?

The student would not able to actually say a number, because you don’t

know how many countries are there in Central and South America on top
in your head.

Can you arrive to the answer without knowing the algorithm?

They could go. . .count one from each side [crossing out an X on each
side]. . .they can take the average of the two left over and the answer will
not be 4 but very close to it.

How about with a bigger set?

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Do the same. . .cross out one on each side, one, one, two, two,. . . [cross out
one X on each side and ends up with two X’s under the number 31], you
have two 31’s, then you would not have to divide by two. The answer is
31.

This is the median.

How about the mean? How can the students see the picture and estimate
the mean?

It would not have to be exact?

No, but we want to find a good estimate.

[struggle here]

So the median would be easier to estimate for students than the mean?

That is what it seems to me. . .because I can’t do it off top of my

head. . .Maybe arrange the X’s ...like I would put this X down here and see
how they average it out. . .that is the only way I can see how to do it. I
would send them to the board and have them arrange the X’s and make
them line them up and that will be the average.

[show work]

Move these over here, and erase these.

All the X’s are even now, that would be three. Humm. . . [trying to make
sense] I guess that is 3 raisins hold on. ..[more thinking]. . .I am trying
to figure out...

You could do adding up again and then divide. You add 28, 28, and 28 or
28 times 3.

It is possible to have other set with the same mean?

You have one house with 2 people, two house with 3 people, you got one
house with 4 people and two houses with 6 [see picture]. Then you got,
count the X’s, 24 people and let them write circles or something else to
even it out, then you get the answer.

How did you come up with this numbers?

I figured up the total by multiplying 3.5 by 6, that is 21. Then divide that
into six numbers.

What does that mean?

That would say that you have some range of number of household, a lower
range and an upper range and 3.5 is in the middle.

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Question 6
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Question 7
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What does “bird 2” means?
One kid has 2 birds or two kids have one bird.
What would be the right order to find the median?

We have one dog, one goat, two fish,
....(dgffcocobbhhhmcacacacadodododododo. . .) and then pick the middle.

We have the mode and the median to be a pet and the range a number.

Right.

What do you mean by “group into range”?

I though it would be easier to see how many students fall into groups of O
to 10 and 11 to 20, and so on.

Why not the others?

I though with the histogram would be easier to show the minutes. . .it is just
a way.

How are they different?

This student is at the level where he doesn’t have to represent each tooth,
he can just see in his head and say, ok I am going to draw a picture and he
is going to be one person and that’s gonna represent 2 and this one gonna
be another person and it is going to represent 2 teeth that he lost. But this
one, he may be in a lower grade. . .he does not have to be behind. . .and he
just goes one two, one two, that is one person and another person that lost
two tooth... the same information. . .it is just organized in different way.

Would both graph answer the question what is the total number of tooth
lost?

This would have to count 2 and 2, 3 and 3, 4, 5. But this one have to
actually add.

What is the most number of teeth lost?

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This one would have to look this one. . .but this one would have to look
how many people this way.

I would prefer this student to get at this because is more advance thinking.

Do the answers make sense?

Yes, I think my answers are right.

Because. . .what does the standard deviation tell you?

It is the distance away from the average.

Does it make sense that this one is bigger?

Huumm, yeah because these are more the same. . .and these come down
and out, I guess, so that it become. . .I am not. . .I want to say that make

sense. But I would actually do it in paper to see if it is correct.

So, you are saying that number represent how far each data is away from
the mean. So 3.57 would mean that...

That the average will go ???. ...that far away.
Which team would you bet on?

I don’t want to bet in team 13 just because they are taller. Team A is more
like average, every one is more like in the middle.

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Subject F

For the first question, if I give you another set of data and ask you the
same question.

5 because more people took 5 minutes to travel to school. Typical means
the most common number.

You mention that you take the mean because “it is unfair” to use the other
method. Why do you say that?

because if you take any of the number of weights, it is unfair.
Why? What do you mean?

Maybe the object is not that weight.

Look careful to the numbers, do you notice something?

The 15.3

What about it?

I think that is not the real weight of the object.

Would you still consider the average (method III)?

No. Maybe if not that one, the most common number.

Why?

....can‘t hear!!!

If we want to correct this mistake, what would you say to the student?

I would tell them to look at the frequency to see how many countries.
There are 15 countries.

What does each bar represent if they do not represent countries?

The percent of adult female literacy.

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So, this person that said that the third bar from the right indicates 85% to
90% literacy rate, is he right?

It needs to say that three countries have

 

Can you find the mean in another way? Look at this data set, can you
estimate the mean?

I would pick the middle one, it looks like 32 for this data.

Is it possible to have other sets of data with the same mean? (Do not use
the algorithm)

You can have five numbers and all those numbers are 4, then the mean is
4. Or you can change the set with 3, 4, 5 and still have mean 4.

What does 3.5 people tell you?

The average of number of people.

Suppose someone does not know what “average” means, what would you
tell that person? In other words, what does the statement “the average

number of people per household is 3.5” mean?

It means that the total number of people in each household divided by the
number of households is 3.5.

You are reaching for the way you got the number, for the algorithm. If
someone does not know the algorithm, it would need another way to
explain what “average means”. How would you explain?

I would tell them to look at the dictionary and find the definition.

Ha, ha, What would you think the dictionary would say?

That the average is the way to describe the mean.

: What question was asked to generate the data?

How many pets do middle school student have? (corrected from How
many pets are there?)

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Question 8

What does “bird 2” means in the chart?

There are two people with one bird each.

You picked a histogram because “their datas are continuing”, what do you
mean by that?

The data follow a continuing pattern
As suppose to what? What would be a data that is not continuing?
[silent]

One of the characteristics of histogram is continuing, and time is
continuous data.

Is there is any other reason why you pick histogram?
[silent]

How much time the majority of student. . .?

35

Show me how you did it?

How would you make one student understand the others representation?

Which student understand better?
Student 1
Why?

It pictures the number of teeth lost with each student, student 2 is difficult
to understand.

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Tell me how you got the numbers?
Calculator

Do they make sense? Why?

Yes, the shape of the graphs.

If you were going to bet on one, which one?

Team B.

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What does typical mean to you?
I PICKED THE MODE
What about for this data?

5, that is the mode for this data.

You picked the average (mean) for this one, were you aware of the outlier
and picked the mean to make every data value to count?

Yes.

You said the student is neither right nor wrong because...

The graph said is for Central and South American countries, it does not
said which ones.

But you do understand why the student said 7 bars?

The student is counting bars.

What would you tell the student to lead him to the right answer?

I would tell him to look at the height of the graph and remember always to
follow the height of the bar to the left to see how tall it is and that is the

frequency.

If you ask the student what does that mean, frequency 3, what would you
want him to tell you?

The bar indicates 85% to 90% literacy rate for 3 women.

Think about how would you answer these questions if you were teaching
average for the first time to your students and they do not know the
algorithm.

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Maybe they would look at the middle, like 4 or 3 and 6 because they both
have two Xs.

What do you think they would say for a bigger set?

They would say 32 o 33 because they are looking for the middle or some
type of balance.

How would you convince the children that is possible to have other data
set with the same mean?

You can start with the same numbers and then switch them around , like if
you have 4,4,4 take away one from the first 4 and make it into 3 and then
put one more and make a 5. Just playing with numbers.

Can you do it with the bigger data set?

Let’s see. . . [pause] I guess you should move... I don’t how to do it with
data, I just know how to do it with numbers. Because you can’t move an

X to 39 because this represents 3 of 38, isn’t?

What does and average of 3.5 people per household mean? How do we
interpret this number?

I guess when the average. . .many many houses average about 3.5. I don’t

know what that means, I don’t know how to explain what that means
though. Because the kids are gonna go “what is 3.5 of a person?”

You answer that the data could be generated by asking the question “What
type of pets do you have in your home?” So, what does bird 2 means?

It could mean two different things. It could mean one child has two birds
in the house or two different children have one bird.

To find the median you say that the student “would have to count through
all the numbers of pets to find the middle” What do you mean by that?

You need to order 1 1 2 2 2 ...and pick the middle.

I know this is completely wrong.

Is it?

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I think so, because this is really wide and I think is kind of confusing. But
at first I look at every one and though of histogram but then I thought that
with a graph you can see it better. Then I try to do it and it turns out some
wide.

Can you tell me how much time the majority of students take to travel?

I would say this, but like I said things are so confusing because this is so
wide and that is skinny but that’s taller. I guess I would say this.

More were in between 15 and 20.

How are the students thinking statistically?

The second student is doing like a bar graph and the first student is doing
more like a stem-and-leaf.

Do you think they can answer questions about the teeth? Like the total

number of teeth lost? Do you think that one of the representations would
answer this question easier than the other?

Ithink both ways.

How about if we want to find out how many teeth were lost the most?
Either one.

How about the rank?

Student 2.

How would you make one student understand the other?

Do the numbers make sense?

The standard deviation is from the lowest number to the highest number
how much they are changing.

So, do it make sense that Team A has a higher standard deviation?

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there towards the middle.

Researcher: If you were a coach, which team would you pick?

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Subject I

For number 1, just tell me if I am right here, ummm, you say you found,
OK you counted 26 then you said half of them are 13, then you counted 13
this way and then this way and then divided by two this remains me of
the procedure of. ...not the mean but...

the median

I want you to answer the same question here with typical, for this data set
1,2,. . ., 26, so also the 13th position, 1, 2, 13, so between 8 and 9 again,
taraaal. It will be 18 and a half

So the same typical time will take for this set that that set

Yeah

Cause of you don’t take all of them. . .in other words if you don’t take the
mean, ummm you are leaving out, you know, the extremes the high
extreme, the low extreme. You don’t really,. .. you just know what’s in the
middle, which is good sometimes, but you also know the range.

So, they are not represented if you don’t take all of them into account

Right

I will explain that the bars represent just central and south america, not
specific country and that when you go up the graph that’s the frequency, so
I don’t know what we understand here, it doesn’t say one thousand, one
million, but that will be the frequency of how many people.

So when he says that the third bar here indicates 85% to 90%, he is
missing. . .he should said also that...

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He left out the frequency, of what that means. 85 to 90 percent what? you
know or something like that. ‘case it is not literacy rate because the
percent is literacy rate but you wanna know how many, so

That will be how many. . .is this case is 3.. .3 what?

units, whatever that is

How did you figure out the 21?

All I did ummm I made everything,... I put all the numbers to 31/2 and
then I changed them. I did 31/2, 31/2, , 31/2.[makes a list of six 31/28]
Then I went. ...and I just gave a little from here [points at the first 31/2]
and put it here [the last 31/2], so in other words, I put, I took maybe 11/2
from here made this a 2 and gave 11/2 here so plus 11/2 to make this 5 and
then I just distributive it, and I kept the total of this numbers to 21 so when
I divided it by 6 will be 31/2 still.

In this case why can’t just pick this as an example?

Because I know that if I have all 31/28 the mean it’s gonna be 31/2, is that

' what you asking me?

Aha, would that be a good example too?

Sure,.. well, possible. . .no, you should not have a half a person per
household, but if you are working with a different unit.

Does the answer make sense?

It does because some people may have 3 people in the household and some
may have 4 and when you average them out...

In the Raisins problem how would you estimate the mean?

I would first told them to take into account ALL of the numbers, you
can’t just look at one column to find uh ask them to take an educated guess
of how many on average they had and uhmm

What would you think they would say?

They’d probably said is somewhere something like 35 because is closer to

the middle and it is a tall one, but I don’t think that will be the correct
answer. uhumm and after they give the answer I would start out by

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Researcher:

Student:

Question 5
Student:

Student:

Researcher:

Student:

Question 6

Researcher:

Student:

Researcher:

Student:

explaining to them how to do a very simple way without using any
algorithm, just by show them to find the median. . .I mean, to find the
average we can find the median. . .so we start by crossing one out from the
beginning cross one out from the end, cross another one from the
beginning, cross another from the end... and go all the way down until you
find the middle, whatever that is.

What about part b of the question for this data.

I can give them like rods or something to work with and ahh I can give
them a scenario say. I tell them to put ...maybe 16 blocks in one stack,
maybe 32 in another row and maybe 64 in another row and then I have

them level out the blocks and then they find that 32 blocks will be in each
column.

a. That 2 students in my class have bird.

Mode would be correct because the mode is the number that occurs the
most times , so you don’t have to order for the mode and the range you can
see what is the lowest number and the highest number is, you have to order
it, I mean, unless you have a large number of data then it would be easier
to order it, but is not necessary, it is only necessary for the median.

You find that the median is “fish”, what does that tell you about the pets
when you said that fish is the median?

That the average student in the class has fish.

Tell me can we answer the question?

You just looking at it, do the crossing out of the number and find the
median, that is the average.

We couldn’t do it with the histogram?

No, you can do it with the histogram. . .it would look good. You could
actually visualize the frequency a little bit better. But I thought this would
be easier for a student to draw on its own, because with a histogram you
have to find the frequency that you are going use between each number
and there is a lot more stuff you have to do.

293

 

Question 7

Researcher:

Student:

Researcher:

Student:

Researcher:

Student:

Researcher:

Student:

Researcher:

Student:

Researcher:

Student:

Question 8

Researcher:

Student:

Researcher:

a. Think statistically
This one resembles what type of graph?

A pictograph

and this one?

like a bar graph

Now think how are they different

They are very similar, if you put this one sideways it is also a bar graph,
the different the pictograph is that there is only one number, but then you
can have as many pictures as that many number kids are represented and
then... but the graph bar every student is represented. . .where. .. will be a
lot more space ...this is... I personally will do a pictograph. You can see
that 3 people had 7 then for the bar graph type you have to count each one
up where the numbers right there for you.

How about in terms of questions each one answers? What is the total
number of teeth lost?

I think the pictograph because you can see 2 people in the 28, so that is 2
times 2, and 2 people in the 3s, so 2 times 3, and so on. while this one you
have to add up because it is not lay out for you.

What is the most common number of teeth lost?

The pictograph shows it a little bit better, you can see it in the bar graph, it
is very close to the seventh category it is not as visual as this one.

How can you tell this student to do the other?

The pictograph because that involves more knowledge to group them and make it more readable
whereas this one they have the knowledge of how to make the graph and all the information they
need to put in it, but they do not understand how to put it together yet and so they just laid it all up.

Do you calculation make sense?

For team B we see that it is 2.28, that means that the standard deviation is
smaller than team a which is 3.58. That makes sense because none of the
answers are closer together for team B given a smaller deviation and
further apart make a larger deviation.

What are the units for the sd?

294

 

Student: is it inches? no. . .ah..I don’t know what that means.
How far apart, how frequently the grades occur?

Researcher: Which team would you bet on?
Student: Team B, because they look like a stronger team, they all fall within the
same range whereas here some of them they fall into the lower range and

some fall on the higher to get that mean of 75. While most of them in
team B are at 75.

295

Question 1

Researcher:

Student:

Question 2

Researcher:

Student:

Question 3

Researcher:

Student:

Researcher:

Subject J

What would be the typical time for this data?

Still take the mean, because. . .like. . . just. . .I don’t know. If you take the
mode, what would you do if you have two most common times.

You picked the average because...

I would use the third method, for several reasons. There is always room
for error when measuring an object. How are students supposed to know
that 6.15 is the most accurate weighing? In using the third method, I
would make sure that the same small object would be used for each group.
Just because one number may be more common than others, does not
mean that is the most accurate, so I would not use the first method [the
mode]. You could have a tenth group who got 6.0, what would the
students then do?

When you say “yes and no”, does that mean that the student is right in
some sense?

Well, by just looking at the graph he sees 7 bars, so that is right. But there
could be some countries unaccounted for with some literacy rate that fall
on the gap with zero frequency, so we don’t know how many countries are
represented.

What would be the ideal response?

296

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