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4thesis entitled

A STUDY OF THE EFFECTS OF EXPLICIT

READING INSTRUCTION ON READING

PERFORMANQi’esented by

IN MATHEMATICS AND ON PROBLEM SOLVING
Virginia Sue Mmski ABILITIES

OF SIXTH
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Date ay 12, 1.978

 

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@ Copyright by
VIRGINIA SUE MURASKI

1978

A STUDY OF‘THE EFFECTS OF EXPLICIT
READING INSTRUCTION ON READING PERFORMANCE
IN MATHEMATICS AND ON PROBLEM SOLVING

ABILITIES OF‘SIXTH GRADERS

BY

Virginia Sue Muraski

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

College of Education

1978

ABSTRACT
A STUDY OF‘THE EFFECTS OF EXPLICIT
READING INSTRUCTION ON READING PERFORMANCE

IN MATHEMATICS AND ON PROBLEM SOLVING
ABILITIES OF SIXTH GRADERS

BY

Virginia Sue Muraski

A. Population
The population for this study was a sixth grade class at an
elementary school located near Grand Rapids, Michigan. The
population was randomly assigned to the experimental and

control groups, with thirteen students in each group.

B. The Problem
This study explored the effects of a five—week (450-minute)
program in the reading of mathematics on the problem
solving abilities of thirteen sixth graders. The reading
program consisted of three 30-minute lessons for each of
five selected reading subskills of word recognition and
comprehension. The five reading subskills selected for
this study were:
1. Instant Recognition of Special Symbols of Mathematics

and Their Voiced Equivalents (WOrd Equivalents)

2. Structural Analysis

3. Contextual Prediction

4.

5.

Inferential Reasoning

Evaluative Thinking

The fifteen reading lessons were designed and taught by the

researcher.

The objectives of this experiment were:
Describe the behaviors of sixth graders who were being
taught how to use the five selected reading subskills
of word recognition and comprehension in reading
mathematics
Determine the impact of the teaching of the five
selected reading subskills on the abilities of these
sixth graders to use these skills in
a. Interpretation of mathematical symbols
b. Mediation of an unrecognized mathematical word
through its structural units
c. Obtainment of factual comprehension of a
mathematical message by contextual prediction
d. Exploration of understanding by inferring relations
among mathematical concepts
e. Formulation of mathematical problems from given
problem situations through evaluative thinking
Determine statistically the impact of teaching these
five selected reading subskills on the problem solving

abilities of sixth graders

C. Data Collected

Statistical data were collected from a pretest and a

posttest to measure problem solving abilities. The Null

and Alternate Hypotheses tested in this study were:

HO: There will be no difference between the mean scores of
the two sets of differences obtained by subtracting,
for each group, each subject's score on the pretest
from his score on the posttest.

H : The mean score for the experimental group will be
greater than the mean score for the control group.

A t-test was used on the two mean scores to determine if

the difference between them was significant.

Other measures consisted of a retest for retention,
classroom teacher's assessment of the subjects' classroom
behaviors, a daily log kept by the researcher and

interviews for problem solving analyses conducted by the

researcher.

D. Results and Conclusions
According to the test statistic, the gains on problem
solving by the experimental group were significant at the
0.005 level. It became apparent through observation and
interviews by the researcher that as the sixth grade
subjects decoded a written mathematical passage into
language and tried to resurrect the intended message, each
of the selected reading subskills played important roles in
the performance of these subskills in mathematics and

problem solving.

Dedicated to:
My husband, Edward
and

My children, Edward, Jr. and Michél

ii

ACKNOWLEDGMENTS

I wish to express my gratitude to the members of my
doctoral committee--Professors William Fitzgerald, Chairman,
Gerald Duffy, Glenda Lappan, William Cole and Bruce
Mitchell--for their generous contributions of time and

talent in assisting me in the writing of this dissertation.

iii

TABLE OF‘CONTENTS

LIST OF TABLES O O C O O O 0 O O O O O O O O 0

LIST OF‘FTGURES . . . . . . . . . . . . . . .

Chapter

I.

II.

III.

IV.

INTRODUCTION..............'

Introduction . . . . . . . . . . . .
Statement of Problem . . . . . . . . .
Purpose of the Study . . . . . . . . .
Assumption . . . . . . . . . . . . . .
Areas for This Research . . . . . . .
Importance of the Study . . . . . . .
Outline of Procedure . . . . . . . . .
Limitations of the Study . . . . . . .

REVIEW'OF‘RELATED LITERATURE . . . . . .

Language and Mathematics . . . . . . .
Verbalization and Mathematics . . . .
Logic and Mathematics . . . . . . . .
Special Symbols and Mathematics . . .
VOcabulary and Mathematics . . . . . .
Reading and Mathematics . . . . . . .
Problem Solving and Mathematics . . .
Related Areas . . . . . . . . . . . .

DESCRIPTION OF‘THE INVESTIGATION . . . .

The Problem . . . . . . . . . . . . .
Procedure . . . . . . . . . . . . . .
Summary . . . . . . . . . . . . . . .

RESIJLTS O O O O O O O O O O O O O O O 0

Introduction . . . . . . . . . . . . .
Analysis of Data . . . . . . . . . . .
Observations . . . . . . . . . . . . .
Problem Analyses . . . . . . .

Problem Posing on Pretest and Posttest
Classroom Teacher' 3 Observations . . .
Retest for Retention . . . . . . . . .
Summary . . . . . . . . . . . . . . .

iv

23

23
27
29
31
33
35
42
44

49

49
50
69

71

71
72
75
93
100
106
107
109

V. SUMMARY, CONCLUSIONS

Summary .
Conclusions

Recommendations

RECOMMENDATIONS

APPENDIX A Materials for Reading Instruction

APPENDIX B Evaluation Instruments . . . . . .

SELECTED REFERENCES

LIST 0 F TABLES

Table l.--Data for Experimental Group on Pre- and
Posttests O O O O O O O O O O O O O O O O 73

Table 2.--Data for Control Group on Pre- and
Posttests . . . . . . . . . . . . . . . . 74

Table 3.--Data on Test for Retention . . . . . . . . . 107

vi

LIST OF‘FTGURES

Figure 1 THE CHILD AND THE READING ENVIRONMENT . . . 10

FigureZAREASFORRESEARCH............. 12

vii

CHAPTER I
INTRODUCTION

Introduction
Rudolf Arnheim(l73) has stated that language has

been termed "a better vehicle for thought" than anything
else. The more extensive a person's linguistic development
the better this person would be able to articulate his
thoughts clearly and succinctly to himself as well as to
others. While detached, theoretical thinking--which is
sometimes the case in mathematics--can possibly function
without words, there seems to be no question that language
helps thinking. Piercey(161) writes:

Many ingredients make up successful learning and

successful performing. But perhaps the major, the

pivotal, ingredient is the ability to manage the

specialized language peculiar to a special field.

. . . Ability to manage the language of a subject area

can be developed, and that development is a function of

teaching. It is estimated that 80 percent of the

learning students are asked to do involves the printed
word.

Earle(73) maintains that any content objective in
mathematics can be mastered by psycholinguistic activity at
one or more of the following levels of reading:

1. Perceiving Symbols

2. Attaching Literal Meaning

3. Analyzing Relationships
4. Solving word PrOblems

If our humanness is gauged in part by our ability
to use written language, then the quality of reading and
writing produced in mathematics classes ought to be a
central social issue. Pauk(l55) asserts, "Learn the
language of mathematics, and learn to use the words
precisely. Only then can you fully and accurately
communicate with others, orally or in writing, in the realm
of mathematics." The language of mathematics is composed
of words, word phrases, numerical symbols, diagrams,
graphic and tabular symbols, as well as other special
mathematical symbols. This language is the language found
in mathematics textbooks and articles in journals,
newspapers, magazines and other material related to
mathematics. Learning to read the language of mathematics
would entail being able to understand messages encoded by
using each of these symbol systems.

Research regarding what in the reading act
prohibits or enhances the understanding of written messages
in mathematics is difficult and has really just begun.
Aiken(6) believes that "a significant degree of overlap
between verbal and mathematical abilities remains I
unexplained" and that the pivotal variable may not be
general intelligence but rather reading or linguistic
ability. Research(6,7) has shown that reading is an

important factor in the acquisition of mathematical

3
knowledge — the how's and the why's have not been clearly
defined or described. Many educational researchers(72,73)
agree that the mathematical development of children
correlates highly with their ability to read and that the
more meaningful the approach taken in learning mathematics
the more important reading ability becomes.

Thomas Barretts's(l6) description of reading is:

Reading involves the visual perception of written
symbols and the transformation of the symbols to their
explicit or implicit oral counterparts. The oral
responses then act as stimuli for a thoughtful reaction
on the part of the reader. The type or level of
thought induced by the stimuli is determined, in part,
by the intent and the background of the reader and the
nature of the materials. In addition, the effort
expended in the perceptual act and the intellectual
impact of the written materials on the reader are
influenced by his interest in the specific selection
and by his attitude toward reading in general.

To read mathematics, one must know the language,
both words and the way the words form.thought patterns, as
well as how the special symbols of mathematics fit into
these patterns. Many factors probably influence the
attitudes(70) or motivation of students in mathematics, but
it is doubtful that mathematics can be meaningfully studied
‘without reading as one essential means of subject mastery.
An instructor who develops reading skills for prdblem
solving would prObably help prevent discouragement and
frustration and could possibly sustain student interest.

A student who learns how to cope with the language of a

subject area will have the means for continued learning for

a life time. As in the proverb:

4

If you give a man a fish, he will have a single meal;
if you teach him how to fish, he will eat all his life.

In the literature of Duffy and Sherman(68), it is
maintained that reading is the most crucial of the
functional skills and consists of thousands of different
subskills that extend from the simple to the complex but
which fall in to four categories: word identification,
comprehension, efficient study and literary understanding.
Identifying an author's scheme for stringing words together,
thinking it through and then arriving at some conclusions
are apparently behaviors necessary for a reader to interact
with the author. Piercey(161) claims that all students
need help with these behaviors some of the time and:

. . . teachers do not give this help is both tragic and
deplorable. It is tragic from the standpoint that some
learners spend six or seven periods a day, five days a
week spurning the printed word instead of learning with
it: it is deplorable from the standpoint that without
needed help during their school years odds are that
they will continue to ignore or misuse the printed word
as a resource for data-gathering and pleasure in adult
years.

Duffy and Sherman(68) claim that a person, who is
unable to read or who has difficulty reading, lacks one or
more of the basic skills needed to be a successful reader;
consequently, instruction in the basic skills of reading
must be one of the major components of a good reading
program. It seems that very few humans learn the basic
concepts of mathematics with little or no direct
instruction. Many humans find the understanding of

mathematics a very difficult task--a mysterious process

with no apparent logic; a foreign language with no order or

5
understanding possible. Do such human beings, especially
school students, need a systematic program emphasizing both
reading skills instruction and how these skills can be
applied specifically to reading mathematics?

According to Duffy and Sherman(68):

. . . a primary cause of reading failure is that
teachers have not been adequately trained in the skills
of reading. Because teachers do not know the skills,
they are unable to teach them, and even when a teacher
does know the skills, instruction is too frequently
haphazard and inefficient because the instructor lacks
a system for determining what skill each learner needs
at any given moment and also lacks an instructional
strategy for teaching that skill.

Since it is likely that there are reading related
cues which have an impact on learning mathematics, it seems
important to identify them. Before specific instructional
strategies can be devised for teaching teachers to teach
reading in mathematics, attempts ought to be made to
identify what in the reading act inhibits or enhances
students' learning mathematics. What are the reading-

mathematics related questions for which answers are

desired?

Statement of Problem
Although reading is a complex activity in which
many skills are apparently used, this study examined only a
few of the beginning stages of reading that are frequently
encountered in the elementary schools and explored their
possible impact on the learning of mathematics.
This study was designed to investigate the effects

of the explicit teaching of five selected reading subskills

6

of word recognition and comprehension on the performance of
these skills in mathematics and on the problem solving
abilities of a group of sixth graders. The reading
subskills of word recognition and comprehension selected
for this study were:
1.. Instant Recognition of Special Symbols of Mathematics

and Their Voiced Equivalents (WOrd Equivalents)
2. Structural Analysis
3. Contextual Prediction
4. Inferential Reasoning
5. Evaluative Thinking
For the purpose of this study, Duffy and Sherman's(68)
description of the reading process and of these five
reading subskills were used. This description is discussed
on pages 8 through 18.

This investigation attempted to answer the
following question: Does the teaching of specific and
explicit reading lessons in these five reading subskills
effect performance of these skills in mathematics or

problem solving abilities of sixth graders?

Purpose of the Study

 

Mathematics means different things to different
people. Whorf(211) maintains that mathematics is a special
language, and Bloomfield(24) claims that mathematics is a
verbal activity and that logic is a study of verbal
activities. Language, logic and much of mathematics

apparently are inseparable. Linguists(93,136) maintain

7
that for someone Who really wants to learn a language, the
best advice is that he go to an area where that language is
used. Falk(82) writes, ". . . if the teacher himself does
not speak the language fluently, his students will never
learn it unless they are able to find some additional
source from.which to learn." 4

These observations have implications for the
teaching of mathematics as well as the teaching of the
reading of mathematics. In order to uncover one of man's
most valuable creations, in an independent way, one must
learn to read mathematical language. Streby(l9l) maintains
that "the language of mathematics under-girds the decisions
and facilitates the reports that citizens in all walks of
life have to make."

One of the major goals of instruction in
mathematics is to develop students' abilities to solve word
prdblems and to confront prdblem situations successfully
and independently. Research(6,73) supports the claim that
most good prdblem solvers are also good readers.

Kane(123) suggests that reading is the perception
and comprehension of a message transmitted by written
language and maintains that in order for a student to
become a competent reader in mathematics, special skills
are required and the acquisition of such skills
necessitates special instructional techniques in the

reading of mathematical language.

8
The purposes of this study were:
1. Describe the behaviors of sixth graders who were being
taught how to use the five selected reading subskills
of word recognition and comprehension in reading
mathematics
2. Determine the impact of the teaching of the five
selected reading subskills on the abilities of these
sixth graders to use these skills in
a. Interpretation of mathematical symbols
b. Mediation of an unrecognized mathematical word
through its structural units

c. Obtainment of factual comprehension of a
mathematical message by contextual prediction

d. Exploration of understanding by inferring relations
among mathematical concepts

e. Formulation of mathematical problems from given
problem situations through evaluative thinking

3. Determine statistically the impact of teaching these
five selected reading subskills on the problem solving

abilities of sixth graders

Assumption
In teaching the basic reading skills, word
recognition and comprehension are not to be isolated from
each other and the interactions that exist between them

should not be obscured(68). These interactions are crucial

9

and can be more readily seen by defining the reading

process as a "psycholinguistic guessing game"--actually

three games played simultaneously by the reader, with each
game consisting of a set of signals(68):

First--graphemic signals move the words from the page into
the reader's head; alphabetic letters represent
speech sounds.

Second--syntactic signals automatically group words into
phrases, sentences or paragraphs according to the
constraints of the grammar of the English language.

Third--semantic signals contribute the meaning of the
printed message, meaning to the syntactic structure
of the printed words.

In the words of Duffy and Sherman(68):

. . . the interrelationship of word identification,
syntactic constraints, and semantic prediction is
apparent. The total process contains elements of both
word recognition and comprehension, but these elements
are never mutually exclusive in the good reader. . . .
The good reader guesses or predicts his or her way
through the message, using graphemic, syntactic and
semantic signals.
Duffy and Sherman's description of the reading
process is schematically explained in Figure l, and details

of this process may be found in Systematic Reading

Instruction(68).

10

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Figure 1 shows that the skills of comprehension
reflect the syntactic and semantics signals and that the
mastery of the graphemic signals of the English language
gives the reader the skills of word recognition. An arrow
goes from.the mediated stream to the instant stream to
reflect the fact that after an unknown word has been
mediated several times, it should become a sight word. The
third comprehension stream focuses on helping the reader
make judgements about what they read and can only be valid
if it follows both information and manipulative thinking.
Duffy and Sherman(68) write, "It is absurd to ask children
to do such an activity without the prerequisites of
comprehension found in the preceding factual and
manipulative levels." Fbr this reason, arrows in the
diagram go from the informational stream to the manipulative
stream and from the manipulative stream to the evaluative
stream to show a hierarchy of dependence. A double-headed
arrow is found between the word recognition streams and the
comprehension streams to reflect the interconnectedness and
interrelatedness of the graphemic, syntactic and semantic

signals.

Areas for This Research
Figure 2 schematically depicts those areas

selected for exploration in this study.

12

WORD RECOGNITION

Strand Specific Subskill

Instant Recognition Recognition of Special
Symbols of Mathematics

Structural Analysis

 

Mediated Identification
COMPREHENSION
Strand Specific Subskill

Information Gathering Contextual Prediction

 

Manipulative Thinking Inferential Reasoning

 

 

Evaluative Thinking Problem Posing

Figure 2 AREAS FOR RESEARCH

13
WOrd Recognition

Experts in the field of reading(68,l6l) insist
that it is necessary for all teachers to recognize the
importance of a child's developing effective word
recognition skills and to support this commitment with a
sophisticated understanding of the learning tasks and
instructional techniques that influence their acquisition.
Most of the skills in the area of word recognition reflect

the graphemic code.

Instant recognition

With the mastery of a large number of sight words,
decoding becomes an automatic process, hence, freeing the
reader to concentrate on syntactic and semantic
considerations. Call and Wiggin(48) write, "WOrds are
linguistic signs which call up or reactivate a memory
trace; they are symbols of concepts. Wbrds derive their
meanings from the context in which they are found."

Reading specialists(68,186) point out that instant
word recognition is not a natural phenomenon for most
children and requires instruction and that children learn
to pair what is seen with what is said. Readers make the
connection between the discriminated stimulus (a written
word) and the spoken word.

In Robinson's(l7l) opinion, reading appears to be
easier when one symbol system at a time is used. In
mathematics, most often it is impossible to use only one

symbol systeme-words, numerals, letters, graphics and

14
special signs are intertwined-~and the interrelationships
among the symbol systems must be understood and utilized
by the student in order to solve problems.

According to Wickelgren(212), problems often rely
heavily on verbal language, and the first and most basic
step in problem solving is to represent the verbal
infbrmation concerning givens, actions and goals in
special symbolic or diagrammatic form. What impact does
instant recognition of special mathematical symbols and
their voiced equivalents have on the learning of
mathematics? It is likely that instant recognition of
these special symbols and their voiced equivalents would be
less of a natural phenomenon than instant word recognition
and would require special attention and instruction. When
should this experience with special mathematical symbols be

started?

Mediated recognition

Mediating strategies serve as a bridge between not
knowing words at all and recognizing them instantly. From
the list of subskills for mediating recognition,
structural analysis was selected for exploration in this
study. What part does the understanding of the structures
produced by the prefixes and stems used in mediating an
unrecognized word play in the learning of mathematics?
Does the history of a word or of its parts stimulate or
enhance learning? Sawyer(l74) feels that if one could

trace the way in and progress by which mathematical terms

15
were developed from every-day words, then one would
understand and appreciate what mathematics really is.
Could the time spent in developing the structural analysis

reading subskill reap worthwhile benefits in mathematics?

Comprehension

Even though learning to read does demand the
identifying of words, it is essentially a communication
process in which obtaining meaning is of primary
importance. The reader decodes the written page into
language and tries to resurrect the intended message. Once
the words are identified then they must be comprehended as
they are used syntactically and semantically. In the
words of Adler(2), "We express our thoughts in words. For
this reason, the structure of thought is parallel to the

structure of language."

Comprehension as an informationggatheringpprocess

At this level of reading, the reader is trying to
learn the factual content of a message. What does the
message say literally? According to Duffy and Sherman(68),
"As children listen to a communication, a portion of their
understanding comes from their sentence sense that allows
them simultaneously to know the idea of the message (its
content) and to predict from both this knowledge and their
grasp of grammatical relationships what word will follow

what word and what idea will fOIIOW’What idea."

16
The contextual clue is a strategy that is often
employed by the efficient adult reader to obtain factual
comprehension of a message(28). Since contextual prediction
is a powerful tool in the comprehension of a written
message, it has been selected as the subskill in this
stream for exploration. What impact does it have on the

learning of mathematics?

Comprehension as a manipulative thinking process

 

Once the informational process is secure, readers
must think about this information and explore its
implications and inferences. The reader moves from a
literal to an inferential understanding.

Inferential reasoning is an important comprehension
skill because not all information worthy of Obtainment is
always stated literally. Children must be taught to think
beyond the information level of a written communication by
"reading between the lines" and to manipulate factual
information in order to discover both implications and
inferences(68).

Do I multiply or divide? Many human beings can
perform basic arithmetic operations with success but
cannot solve word problems or cope with problem situations.
Friebel(88) gives a sixth grader's system for solving word

problems:

17
If there are lots of numbers, I add. If there are only
two numbers with lots of parts, I subtract. But if
there are only two numbers and one is littler than the
other, it is a hard problem. I divide to see if they
come out even. If they don't, then I multiply to get
the answer.

A common difficulty of students in problem solving
is the failure apparently to generate a map or a series of
possible logical processes or inferences. What impact does
the reading skill of inferential reasoning have on the

learning of mathematics to solve problems?

Comprehension as an evaluative thinking process

According to Duffy and Sherman(68), comprehension
in the informational and manipulative areas is passive in
the sense that readers interpret what the author meant, but
at the evaluative level, readers assert themselves and
strike out on their own, making a judgement that may or may
not be what the author intended. Are there comparable
skills at this level of comprehension which would have an
impact on the reading for successful prOblem solving?

Bagford(l4) maintains that one of the biggest
influences on comprehension growth in children is the
nature of the questions asked by the teacher during reading
class and during other parts of the day. Suppose the
students were trained in asking the questions--recall or
recognition type questions, as well as questions requiring
evaluative thinking? What impact would this skill have on

reading to solve problems in mathematics?

18

In this age of accountability, one of the major
goals of mathematics instruction for many teachers is to
prepare students to cope with the mathematical problems
and prdblem situations which they will encounter during
their lives. In a realistic problem situation, there may
not be a clear statement of what prOblem is to be solved
and the students will be forced to formulate a clear
problem from the problem situation before formulating a
solution procedure. In order for children to read
successfully on the evaluative level, they must strike out
on their own, making judgements and decision. Is it the
case that prdblem posing activities could be effective in
stimulating areas of creative thinking and decision-making
skills necessary for problem solving and confronting

prdblem situations?

Importance of the Study

To this investigator, the needs of today's children
and young people do not differ radically from those of
yesterday's--learning to cope with change and being equipped
to deal with the future. Analytic reading--the reading
used in methematics--probably ought to be an integral part
of a curriculum designed to develop students' abilities to
acquire and use knowledge. Acquiring and using knowledge
involve bringing order out of information chaos by finding
a theme or pattern that can be communicated to others--
behaviors used in prOblem solving, coping with change and

dealing with the future.

19

In the words of Taschow(l98), ". . . the language
in these mathematical fields with their subdivisions is
concise, and the reading is precise, particular, and
critical . . . without mastering this complex reading
process, the student is vexed even if he retreats to
memorization." According to Samuels(l73), ". . . any
student, regardless of his experience, cultural background,
or past performance can acquire the basic language skills,
naturally and promptly, if he is properly motivated." Is
this the case with the language of mathematics?

The implications are:

1. Much of mathematics and language are inseparable(24,25,
211)

2. Complex thought and language are inseparable(46,207)

3. Reading is a complex process made up of many skills
(68,137)

4. Oral verbalization is an important behavior in
successful reading and learning(l78,186)

5. Effective handling of the printed word--critical
reading-—in any area is an important consideration for
successful coping with a changing social and physical
environment(161)

6. Curricula do not reflect this need for the teaching of
reading in the mathematics area(10,80)

7. In teacher-training courses, very little is done in the

area of the teaching of reading in mathematics(10,77)

20
8. The relationship between reading and problem solving
in mathematics has not been clearly explicated(158)
The attempt to shed additional light on the
relation of the learning of mathematics for problem solving
to the learning to read mathematical language could be

important.

Outline of Procedure

The main purpose of the study was to determine the
effects of the explicit teaching of five selected reading
subskills on the performance of these skills in reading
mathematics and on the problem solving abilities of
thirteen sixth graders.

The researcher planned three 30-minute lessons for
each of the five selected reading subskills for a total of
fifteen 30-minute lessons in reading mathematics. Each
reading subskill was covered during one week with one 30-
minute lesson being held on alternate mornings. The
researcher did the instructing for five weeks in these five
reading subskills as applied to mathematics.

A sixth grade population of twenty-six students
was randomly divided into the experimental group and the
control group. The experimental and control groups were
pretested for problem solving ability and, after five weeks,
were posttested for problem solving ability. Both
statistical analysis and observations were used to describe

the results of the program in reading and the effects that

21
this reading program had on the prdblem solving abilities

of the subjects participating in this study.

Limitations of the Study

Time was a limitation of this study. Any activity
used for the reading instruction had to be within the
confines of the 30-minute period assigned for this project.
There was no extra time for remediation for those students
Who may have needed it. Three 30-minute lessons for each
of the reading subskills seemingly would be an
unreasonably short time to spend on apparently very
powerful reading skills for problem solving. There were
no regular or systematic evaluation nor corrective
instruction. Individual deficiencies were not identified
so that students Who failed to reach mastery level of a
subskill could be given additional assistance and time to
master the objective. No attempt was made to learn what
reading skills individual students already possessed.

Diagnostic instruction was not used because of
time limitation, the absence of any established well-known
reading cues related to the learning of mathematics, and
the lack of knowledge in this area on the part of this
researcher.

Another limitation of this study was the necessity
of teaching the reading of mathematics in isolation from a
quality mathematics curriculum involving comprehensive

prOblem solving. In the words of Earle(73), ". . .

22
successful teaching of mathematical reading is most likely
to occur in classes where both the content of mathematics
and the process of reading are emphasized equally and
constantly."
Other possible limitations of the study could be
the presence of the researcher and the extra time spent on

mathematical activities by the experimental group.

CHAPTER II

REVIEW OF RELATED LITERATURE

Language and Mathematics

Hogben(110) has pointed out that some of the
difficulties with mathematics arise because of the language
difficulties. According to Kolata(129), "Translating
mathematics into English was described by one mathematician
as being more difficult than translating Chinese Poetry."

In the opinion of Bruner(42), "There is one part of
the picture in the building of mathematical curriculum now
in progress where I see a virtual blank. It has to do with
the investigation of the language and concepts that
children of various ages use in attempting intuitively to
grasp different concepts and sequences in mathematics."
Madden(l4l) believes that mathematics is a language as well
as a way of thinking with this language, and he feels that
teachers must elicit from their children expressions of
their language patterns because, according to him, there is
great need to study differences between children's everyday
language patterns and the more formal mathematical sentence
models. Madden(l4l) maintains that "mathematics is

internalized as language."

23

24

Davis(62) alludes to Bruner's experiment which
claims that language provides the means of getting free of
immediate appearance as the sole basis of judgement in
dealing with mathematics tasks relating to things in the
real world. Language makes possible the acquiring, recall,
and growth of experience; it permeates intelligence; and it
is a requisite vehicle in all methods of teaching and
study, not excluding mathematics.

Chomsky(52) claims that there is reason to believe
that the language-acquisition system may be fully
functional only during a critical period of mental
development. Lenneberg(136) writes:

. . . language acquisition becomes more difficult when
the physical maturation of the brain is complete. . . .
There is evidence that children at this age [preteen
years] are capable of developing language in the same
natural way as do very young children [preschoolers].

. . . healthy children have a quite different propensity
for acquiring foreign languages before the early teens
than after the late teens. . . . Neurological material
strongly suggests that something happens in the brain
during the early teens that changes the propensity for
language acquisition . . .

In the opinion of Cooper(56), children enter first
grade with several years of oral language experience--the
basic tool for learning to read, that is, to understand the
meaning of something written or printed. Cooper(56) also
believes that children's oral mathematics language is very
limited. If the line of analysis of the reading process as

described by reading specialists(68) and psycholinguists

(178) is correct, before a student, in general, can begin

25
reading mathematical language intelligently, he must learn
to speak it.

Stauffer(186) feels that in the early childhood
period activity and language need close association and
that language development, as a part of maturation or all-
around mental capacity, influences much of a child's
progress from thought that is predominantly perceptual and
intuitive to thought that is conceptual and logical.
Brownell and Hendrickson(39) write:

. . . the process of conceptualization has been related
very closely to development in the ability to use
language. . . . Not only does language enable us to
escape the limitations of space and time in our search
for meanings and relationships but it also enables us
to think with understanding about things that do not
exist and cannot exist, as, for example, the
abstractions of mathematics. It is not without reason
that language is our chief vehicle of instruction.

According to Bruner(42), language profoundly
affects behavior and "Pupils progress when ideas of size,
degree, and relationship shine clearly to them through
mathematical language." Braunfeld(31) maintains that
because the language of mathematics is more precise than
that of ordinary English, students must have a good deal of
experience in using this precise language so that an
appreciation for the benefits, derived in mathematics from
this very precise formulations of prdblems and results that
mathematics demands, can be developed. It appears that, if
a reading curriculum for mathematics is to be developed,

the Observations of these language specialists and

researchers should be considered in light of the claim that:

26
1. Mathematics itself is a language
2. Oral language is the basic tool for learning to read
that language in print

Rose and Rose(l72) claim that childhood training in
language is crucial for performing well in mathematics;
actually suggest a sociopsychological variable of language
style as one of the intervening explanatory variables in
arithmetic performance. In the opinion of Aiken(6), this
accounts partially for the relatively greater mathematical
abilities of children from upper socio-cultural backgrounds.
In Peters'(lS7) study of kindergarten children of low socio-
economic backgrounds, he found verbal training to be
significantly more effective than noncued, visual-cued, or
no training When the criterion was immediate learning, but
if delayed retention was the criterion, then both verbal
training and visual-cued training were more effective than
the other two procedures.

Sollee(182) concludes from her study of first and
second graders that verbal competence is related to the
achievement of stable and generalized levels of
conservation and that this relationship remained
significant when intelligence, measured nonverbally, was
held constant. weil's(210) study supports Piaget's claim
that language is not a sufficient condition to advance
operational thinking but found that the relationship
between the language measures and the concept measures were

significantly correlated. Pelfrey(156) concludes from.his

27
study that the lack of language in deaf children has
significantly affected their cognitive development.
Pullman(l66) maintains that both verbal conditionality and
cognitive structure could be used in explaining part of the
variance in mathematical ability not explained by
intelligence.

In learning algebraic symbols and syntax, Ausubel
and RObinson(12) claim that the prdblems are the same as in
learning a second language. The literature supports
Aiken's(6) claim that many researchers maintain that
mathematics is a language and should be taught as such(12,

33,56,69,110,133,141,l49).

Verbalization and Mathematics
Irish(ll7) reports positive effects on problem
solving by modifying instruction in mathematics to include
emphasis upon children's verbalization. Palzere(153) and
Stern(187) found no significant difference in requiring
verbalization in mathematics; whereas, Greider(95) claims
that verbalization does affect learning in a positive way.
Hendrix(102) claims:
. . . correct verbalization of a discovery does emerge
immediately after the dawning of awareness, but these
cases are rare. They are confined to learners with
unusual powers of correctness and precision in the use
of a mother tongue, and they are confined to situations
in which the learner happens to possess all the
vocabulary and rules of sentence structure needed to
formulate the new discovery.

Hendrix(102) also maintains that dialogue between teacher

and students, as well as between the students, plays a very

28
important role in the initial stages of mathematics
learning. Aiken(6) feels that one of the important
influences of verbalization is permanency in retaining
learned concepts.

Piaget(l60) claims that the needs a child tends to
satisfy when he talks are neither linguistic nor logical
but psychological. Piaget(l60) feels that, "The mere act
of telling ones thought—-of telling it to others or telling
it only to oneself—-must be of enormous importance to the
fundamental structure and functioning of thought in general
and of child logic in particular."

Davis(62) writes:

Anyone who watches children between the ages of three
and about eight years old can hardly fail to be
impressed by the child's tendency to verbalize about
nearly everything. While he does something, he
produces a gratuitous verbal description, aloud,
concerning what he is doing. If our present line of
analysis is at all correct, this talking-about-what-
you-are—doing-while-you-are-doing-it behavior is one of
the most basic learning activities of all.

Chase(51) maintains that children want to talk but that they,

by 4th grade, have been turned off and trained not to talk

about their academic subjects.

vygotsky(207) claims that a child's intellectual
growth is contingent on his mastering the social means of
thought, which is language, and that verbal intercourse
with adults becomes a powerful factor in the development of
the child's concepts. Ausubel(ll) writes:

Verbalization, I submit, does more than verbally gild
the lily of subverbal insight; it does more than just

attach a symbolic handle to an idea so that one can
record, verify, classify, and communicate it more

29
readily. It constitutes, rather, an integral part of
the very process of abstraction itself. When an
individual uses language to express an idea, he is not
merely encoding subverbal insight into words. On the
contrary, he is engaged in a process of generating a
higher level of insight that transcends by, in clarity,
precision, generality and inclusiveness--the previously
achieved stage of subverbal awareness.

Irish(ll7) reports to have obtained evidence that
growth was Obtained When instruction in computation was
modified to include emphasis upon children's growth in
ability to formulate verbal generalizations appropriate to
the topic under study. Horn(ll4) maintains that
investigations have shown that at higher grade levels
students who cannot understand what they read usually
cannot understand what they hear and that there seems to be
close correspondence between the adequacy of a student's

verbal statement of an idea and the clearness with which he

grasps the idea.

Loqic and Mathematics

Elder(81) reports that explicit instruction in
logic improved the abilities of college students to
verbalize generalizations discovered while working in a
programmed algebra text. According to Rimoldi(168), there
is an interaction between language and logical structure;
that is, logical structure and language are not
experimentally independent variable. Suppes(193) claims
that although a class of fifth grade students proceeded
much more slowly than the usual undergraduate class in

college mathematical logic, the results indicate that

30
children of this age were able to achieve a level of
accomplishment comparable to college students in the skills
of deduction essential to mathematical reasoning when
instruction was geared to their level of experience.
Paris(154) concludes from.his experiment that
neither adults nor children interpret linguistic connectives
according to formal propositional relationships prior to
explicit instruction. Inhelder and Piaget(ll6) maintain
that formal aspects of logic begin about age eleven.
Hyram's(llS) work suggests that direct teaching of the
logical rules may bring about the improvement in children's
performances in dealing with valid inferences. The
recognition of valid inferences is one of the main skills
in reading comprehension of mathematical materials.
According to Hill(109), children of six, seven, and eight
years old are able, to a marked degree, to recognize the
validity of logical inferences and these same children can
also demonstrate their understanding in reasoning from
hypothetical premises.
Hyram(115) writes:
. . . the fact that thinking does not appear to be
inherently logical is of little significance with

regard to the possibility that it can be developed into
characteristically logical patterns through experience
and habit formation. . . . Children in the upper grades
of the elementary schools are, in general, mentally
capable of acquiring the necessary understanding of
logic and a proficiency in the use of its rules. . . .
the most effective way of helping children to acquire
the necessary working knowledge of the principles of
logic is through direct instruction.

'31

According to Suppes(l95), "We must provide for and
take advantage of the opportunity to pay more than
incidental attention to the development of the student's
reasoning powers. . . . The encouragement of critical
thinking should be acknowledged as one of the legitimate
Objectives of our schools . . . critical thought requires
the ability to make logically correct inferences, to
recognize fallacies, and to identify inconsistencies among
statements." These traits for critical thinking are also
embedded in the skills required to read. Muscio(150)
claims that the close relationship between language and
thinking must be recognized as we develop appropriate

learning experiences for children.

Special Symbols and Mathematics

According to Suppes(l94), his empirical findings
indicate that mathematical concepts generally can be
understood by primary grade children provided that these
concepts are presented precisely with the help of a
consistent notation. Skemp(l76) believes that the
distinction between a symbol and its associated concept,
being two different things, is not trivial in mathematics.
He continues by claiming that making an idea conscious is
closely related to associating it with a symbol and "Once
the association has been formed, the symbol seems to act as

a combined label and handle, whereby we can select (from

32
our memory store) and manipulate our concepts at will. It
is largely by the use of symbols that we achieve voluntary
control over our thoughts."

Suppes'(l94) experimental work also has shown that
primary grade youngsters do not in general have any
difficulty in grasping new notation. Earle(72) feels that
the more times that students are exposed to the sight and
sound of important mathematical symbols, the more likely
they are to recognize those symbols accurately and rapidly.
Taschow(l98) maintains that sheer recognition of
mathematical words and symbols without the ability to
verbalize or pronounce them may restrain comprehension and
later interfere with interpretation and evaluation. He
suggests that writing the word or symbol will provide a
third sensory attack on new or possibly forgotten terms and
symbols. Skemp(l76) maintains that "The power of mathematics
is immense, and at all stages, symbols make a major
contribution to this power. But without the ability of
mathematicians to invest them with meaning, they are
useless."

Phillips and Uprichard(158) found that translation
from ordinary English to the special language of
mathematics correlated consistently with Ninth Grade
students' ability to solve verbal problems. Dahmus(59)

claims that teachers must develop students' abilities to

33
translate verbal situations into proper mathematical
symbols. Polya(l63) suggests that mathematical notation is
a language and that:
When we face a new problem, we must choose certain
symbols. . . . Choosing a suitable notation may
contribute essentially to understanding the prOblem.
We need experience to choose the more suitable notation
as we need experience to choose more suitable words.
Vocabulary and Mathematics

Georges(90) maintains that reading difficulties
caused by mathematical terminology are predominant, with
difficulties caused by symbols and notation being next.
Suppes(l94) claims that the use of explicit notation
permits a clarity of definition of concepts that would
otherwise be impossible.

Buckingham(46) writes, "Since reading and
vocabulary are so intimately related, it follows that the
setting for the learning process in a single prOblem may be
so changed by an unknown word as to make the total problem
incomprehensible to the student."- This agrees with the
findings of Bogolyubov(25), which indicate that if words
are not understood or if they are misunderstood, the process
of prOblem solving becomes extremely complicated and
possibly impossible for some students. Bogolyubov writes,
"Teachers work very little with children on words in the
solution of problems, and in particular, do not teach the
significance of certain words in everyday life. The latter
fact frequently creates great difficulties in the study of

arithmetic."

34

According to Bruner, Olver and Greenfield(43), the
development of clear and adequate terminology is essential
to cognitive growth. Johnson(lZO) found that, by
development of a meaningful understanding of vocabulary
beyond that which was provided by the textbook, an
experimental group achieved significantly greater gains
than did the control group in both vocabulary and prOblem
solving. Treacy(203) reports a great need for stressing
the meaning of terms, general and mathematical, as an
approach to improving prOblem solving.

Vander Linde(205,206), from the results of his
study, maintains that the study of the technical vocabulary
of arithmetic cannot be left to chance but must be an
essential part of the arithmetic program. Langer(l34)
maintains that the development of vocabulary is an
inextricable part of concept development and difficulty in
reading comprehension often stems from difficulty in the
understanding of words and the concepts they represent.
Drake(65) reports that students demonstrated superior
achievement in algebra when the vocabulary of the subject
was stressed. Johnson(121) found that in considering
prOblem solving in relationship to general intelligence
through the factors of vocabulary and reasoning that
vocabulary rated highest in both prOblem solving and

general intelligence. Johnson(121) writes:

35
Perhaps it can be explained by knowing that words are
our vehicle of thought and the better the vehicle the

better the thinking. And the better the thinking one
does the more intelligent he is and perhaps the better

problem solver.
Dresher(67), Edwards(79), as well as Lyda and Duncan(l40),
found that their results show a considerable influence of
vocabulary upon achievement in mathematics.

Bruner(42) maintains that teachers need clear and
correct mathematical words to describe problem situations,
words to question students' unreasoned statements in
mathematics, words to encourage further student research
and reading in mathematics. Bruner(42) continues by
saying, ". . . good mathematical language challenges--
relights the flame of curiosity. . . . Men use words to
solve most of their perplexities, if not all of them. But
it is not easy to use words properly in solving prOblems.
. . . In helping students to think reflectively . . . the
teacher should help them to understand the use of words
. . ." Taschow(l96) writes:

Studies emphasize that without an understanding of
words, comprehension is impossible . . . a large
vocabulary is related to high comprehension. . . .
Vocabulary should not be taught in isolation but should
always be related to meaningful content . . . in order
for the reader to achieve greatest possible
comprehension he must become meaning-conscious. This
skill is taught by means of word structure analysis and
context clues.
Reading and Mathematics
Georges(90) suggests that accuracy of statement in

mathematics, as well as accuracy of computation, is the

only criterion of acceptable work and that this is

36
impossible without accuracy of reading in mathematics.
Call and Wiggin(48) report the results of an experiment in
which second year algebra high school students were taught
by an English teacher with some training in the teaching of
reading but no experience in teaching mathematics and in
which the control group was taught by an experienced
mathematics teacher. The findings show that the
experimental group did better on the criterion test in
mathematics than the control group. The initial
differences in reading and mathematics test scores were
statistically controlled. [Call and Wiggin write, "If‘by
teaching reading, instead of mathematics, we got better
results, it seems reasonable to infer that the competent
mathematics teacher might get considerably better results
if he were trained to teach reading of the kind encountered
in mathematics."

Kress(l30) found that nonreaders, when compared to
readers, lacked (a) versatility, flexibility, and
originality in devising suitable hypotheses for testing,

(b) initiative and persistence in verifying solutions,

(c) ability to draw inferences from relevant clues, (d)
ability to analyze factors or to devise adequate labels,
and (e) ability to form adequate concepts for dealing with
language. Muscio(150) reports that arithmetic achievement
depends on both general intelligence and verbal ability and
that achievement on the measure of quantitative

understanding is closely related to mathematical vocabulary

37
and general reading ability. Gilmary's(9l) study indicates
a definite relationship between reading and arithmetic
procedures, and the effects of the reading instruction had
a significant transfer value for the arithmetic classes.

Cottrell(57) found that the strong relationship
between reading, psycholinguistics and arithmetic are due
to general linguistic ability. Skillman(l77) concludes
that it is possible to obtain improved achievement in
mathematics by emphasizing reading techniques.
Ziesemer's(218) study shows that the fifth graders, with a
pattern of high reading and low arithmetic achievement,
scored significantly higher than the students with high
arithmetic and low reading achievement on a series of
standardized tests measuring intelligence, mental abilities,
picture vocabulary abilities, conceptualization, symbolic
mediation, language association and application of verbal
skills to new situations.

Buckingham(46) writes, "It cannot be assumed that
an understanding of the problems as they occur in the text
will materialize regardless of vocabulary or reading
ability. . . . Mathematics has its own settings, words and
meanings Which must be taught in parallel with prOblem
solving." Pribnow(165) maintains that the root of the
inability of students to solve word problems is concerned
with reading mathematics as opposed to the "Johnny can't
read lament" because without the identification and

comprehension of the given information or of What must be

38
found, further analysis of the prOblem is impossible.
There have been other studies dealing with the relation of
reading, and language in general, to problem solving in
mathematics(8,60,76,97,l75).

Taschow(l98) recommends that the reading of
mathematics must be set in the curriculum in a systematic,
direct and planned fashion and must not be left to
incidental learning. Streby(l9l) claims:

. . . reading is very complex, involving many skills

and understanding. . . . Reading verbal problems in

mathematics texts does require a different technique

than reading descriptive material or fiction. The

inability to grasp the full meaning of a statement is

more serious in mathematics than in any other subject.
Georges(90) claims that the types of reading required in
technical subjects, such as mathematics, are quite
different from those required in courses in English.
Georges(90) also found in his work that “any answer to a
prOblem satisfies the careless reader whether it is the
desired quantity or not."

Eagle(7l) writes:

The function of selecting and organizing the essential
data is important in life situations to a greater
extent than in most textbook problems. . . . More
emphasis on this aspect of mathematics would relate the
work of the schools more closely to practical
applications, thereby making the mathematics courses
more functional and probably better motivated. In such
courses reading abilities . . . would be of even more
importance in relation to achievement in mathematics,
and they would be therefore developed to a greater
extent than is the case in the usual mathematics
courses.

Horn(ll4) asserts that low achievement in reading, in

listening, and in oral or written composition is PRIMA

39
EACIE evidence of poor learning in other curriculum fields.
Horn(ll4) also maintains that investigations have shown
that at higher grade levels students who cannot understand
what they read usually cannot understand what they hear.
Communication of ideas depend upon the ability to associate
written and oral symbols with the ideas.

Chase(51) writes:

Every normal child uses his natural language quite
fluently at a very early age, yet millions of normal
ten- and twelve-year-olds are categorized as nonreaders.
They find it difficult to transfer sounds into

symbols. . . . homo sapiens are talking animals Whose
brain centers for writing and reading have only
recently been opened up.

Stauffer(186) claims that critical reading and
concept development can be taught from the first grade on,
through a reading program which encompasses language and
experience, reading and thinking, phonics, syntax, and
semantics in a unified course of instruction and that word
attack skills and concept development skills must be so
taught that the reader can proceed in a self-reliant way
regardless of the content. In the opinion of Skemp(l76),
the greatest and final contribution of an excellent
mathematics teacher is to reduce the learner's dependence
on him and this would entail the learner's being able to
read the language of mathematics independently. According
to Bruner(42), far too many students will not even attempt
to read mathematics or to speak or to write about

mathematics, and those who do attempt it, most often use it

"muddyly, inconcisely, incorrectly and incogently."

40

Feeman(83) asserts that teachers of mathematics
have long realized that many difficulties encountered.by
students in learning mathematics are reading related
difficulties rather than intrinsic mathematical
difficulties and that these reading difficulties are
related to the reading of mathematics specifically. Feeman
also claims that, in general, mathematics teachers do not
see themselves as reading teachers, but strictly as
mathematics teachers and that reading teachers per se
seldom.have much training in mathematics. Clark(53)
writes, "Mathematical reading is highly specialized. The
person best equipped to guide the reader is the mathematics
teacher, who conceives of the teaching of reading of
mathematical ideas as an integral part of the learning in
his field."

The Harvard Survey(10) found that social studies
and science were basically the only content areas in which
reading skills were "sometimes" developed. The lack of
time and teacher "know'how" were cited as the chief
barriers for this absence of helping students master the
printed word in subject matter areas. According to Earp(77),
there is very limited evidence that reading skills are
being taught in the content areas, especially in the area
of mathematics, and on examination of professional reading
texts and basal reading series, he fOund reading
recommendations are slanted toward the reading of

informational materials more representative of social

41

studies rather than mathematics. Austin and Morrison(10)
claim that the graduating student in elementary education
is not always adequately trained to undertake the
responsibility of helping children master the printed word
even in a general way much less for mathematics reading.

Research(6) which relates reading and mathematics
seems to indicate that poor readers will probably be poor
prOblem solvers and that the overall mathematical
development of children correlates highly with their
ability to read mathematics. Aiken(6) suggests that we do
have enough evidence to conclude that instruction in
reading in general and in reading in mathematics in
particular improves performance in mathematics. He
continues by writing, "It seems reasonable that attempting
to cultivate the skill of reading carefully and
analytically in order to note details and understand
meanings, thinking about what one is reading, and
translating What is read into special symbols would improve
performance on many types of mathematics prOblems."

According to Bagford(l4), Bormuth(27) and
Bornick(28), a deeper understanding of the message of
printed material can be fostered among students by use of
the cloze procedure, since the students are required to
read carefully, to use contextual clues and to become
actually involved with what they are reading in order to
predict correctly the words or the special symbols which

have been deleted from.the reading material. The cloze was

42
developed by Wilson Taylor(l99) in 1953 and is a
contextually interrelated series of blanks to be completed
by the students, with the word or special symbol that makes
the most sense, given the context. Reading specialists
(Duffy and Sherman(68), Earle(72), Stauffer(186),
Taschow(l97)), for example, agree that the contextual clue
is a powerful strategy most Often employed by efficient
readers. Research (Kane(124), Hater(99), Culhane(58),
Rankin(l67)) has shown the cloze to be a very valid and
reliable measure of reading comprehension as well as a
means for teaching reading. According to Culhane(58),
"Rarely does a method of instruction emerge that is as
easily learned or as easy to put to use in the classroom as

the cloze method."

Problem Solving and Mathematics

Treacy(203) suggests that students may be troubled
in problem solving by the inability to retain significant
facts as well as by the inability to read. Treacy's(203)
results support the value of the pencil-paper reading skill
in mathematics. According to Dahmus(59), texts and
teachers fail to teach the vocabulary the students need and
to develop the abilities of the students to translate a
prOblem situation into proper mathematical symbols: hence,
the students are unable to read or write mathematics.
Dahmus(59) also claims that in mathematics classrooms, the
answer is the "little god" and that the students' habit of

trying to work prOblems mentally also makes teaching

43
difficult. He suggests that the mental arithmetic
exercises are partially responsible for creating this
unfortunate situation. Dahmus'(59) work supports the need
for children to be able to write the language of
mathematics as well as to speak it, understand it spoken,
and to read it, if successful prOblem solving is to be
accomplished.

Henney(lO4,lOS) found that special instruction in
reading verbal problems and supervised study in verbal
prOblems both improved children's abilities to solve verbal
prOblems. Linville(l38) concludes from.his study that
syntactic structure and vocabulary level are important
factors in solving verbal problems, with the vocabulary
level perhaps the more crucial. He also found that
children of high reading ability scored significantly
higher on a verbal arithmetic prOblem test than did the
children with low reading achievement. Pribnow(165) claims
that the key to the inability of students to solve word
prOblems appears to be organizing and analyzing the prOblem
which apparently are skills related to reading for
comprehension. Phillips and Uprichard(158) write,
"Obviously, success in solving word prOblems depends upon
skills in reading and computation, however, the relative
contribution of these skills has not been clearly
explicated." In their review of the literature, Phillips

and Uprichard(158) conclude that prOblem solving has not

44
been systematically investigated by mathematics educators
and that studies of prOblem solving yield conflicting

results.

Related Areas
Intelligence and Mathematics

According to Aiken(6), in 1958, Wrigley concluded
from.his study that high overall intelligence is the first
requirement for success in mathematics, but Aiken feels
that "a significant degree of overlap between verbal and
mathematical abilities remains unexplained.“ Aiken goes on
to demonstrate that the correlation between general
intelligence and mathematical ability is appreciably
reduced when reading ability or scores on other linguistic
tests are partialed out. Aiken(6) concludes that the
pivotal variable may not be general intelligence but rather

reading or linguistic ability.

Verbalization in Mathematics Classrooms

Cooney(54), Fey(85), Gregory(94), Kysilka(l3l), and
Lamanna(132) have investigated the verbal interactions of
teachers and students in the mathematics classrooms.
According to Aiken(6), these studies have "increased our
knowledge of the kind, frequency and effects of verbal
exchange between students and teachers in mathematics
classrooms." According to Bagford(l4), one of the biggest
influences on comprehension growth in children is the nature

of the questions asked by the teacher during reading class

45
and during other parts of the day. Bagford(l4) writes, "If
teachers ask only questions which call for one word,
factual answers, pupils form the habit of looking for and
remembering only facts as they read. Consequently, the
pupils do not develop the capacity to perceive and
comprehend the larger ideas expressed in the reading
material." Guzak(96) found that 70 per cent of the
questions asked by teachers were either recall or

recognition type questions.

Language Arts and Mathematics
Using
1. Similarities between teaching mathematics and the
language arts(49,108,l42)
2. Children's literature(159,l90)
3. Anecdotal style of content presentation(151)
have been investigated and found to be effective
motivational and instructional media for the teaching of

mathematics.

Textbooks and Mathematics
Browning(40) maintains that there is a great
disparity in terminology usuage from one mathematics
textbook series to another and that the frequency of the
usuage of most terms is very inadequate in all series
examined. According to Willmon(214), the number of new
mathematics words introduced in the textbooks for primary

grades is too great for the textbooks to be the sole

46
teaching vehicles. From Stevenson's(188) investigation of
third grade mathematics textbooks and first and second
grade readers, he concludes that the teachers must teach
the new vocabulary in the mathematics books as no
dependence upon the basic reading program is warranted.
Stevenson also advises that teachers must devote time to
teaching the reading of mathematics. Stauffer(lBS) reports
that, in his investigation of primary textbooks, he found
that even if a child in first grade mastered all the
different words presented in all of the seven reading
series, he would still need to learn to read at least
one-half of the words presented in the arithmetic series in
the arithmetic class. This means that he would need to be
prepared to deal with these words semantically (meaning) and
phonetically-structurally (speaking) in order to grasp and
deal with arithmetic prOblems and discussions. Stauffer
found a total of 1331 words, appearing in the arithmetic
list made for grades 1, 2 and 3, which did not appear
anyWhere in the reading list made for the same grades.
According to Stauffer, "It is apparent that every teacher

is truly a teacher of comprehension or of reading."

Readability and Mathematics Materials
Hiatt(107) reports that a creative problem solving
test measured certain mathematical thinking abilities not
measured by standardized measures of achievement and

intelligence. Kane(123,124,125) maintains that reading

47
formulas for ordinary English are not appropriate for use
in mathematics; they are valid only when applied to
materials written in ordinary English. Hater(99,100) found
the Cloze Procedure to be highly reliable and to be a valid
predictor of the comprehensibility of mathematical
language. Kane(124) claims that cloze scores exhibit high
positive correlation with Shannon letter redundancy scores
and that letter redundancy is a powerful determinant of the
learning of verbal materials. According to Kane(123),
"Extensive research shows that mean cloze scores are valid
measures of reading difficulty for ordinary English. . . .
cloze tests adapted for use with mathematical passages are
measures of reading difficulty in the language of
mathematics."

Instruments to measure reading in mathematics have
been constructed by LeDuc(l35) and Brunner(44). Brunner
found her instrument to be more than an intelligence test
or a general reading test. LeDuc's instrument would only
measure some abilities that are important in the reading of
mathematics as he left out of his study two very important
aspects-—technical vocabulary and symbolism. Byrne's(47)
study reports the development of a measure of the
familiarity of mathematical terms and of mathematical
symbols for 7th and 8th grade students.

Loehrlein(139) suggests that readability formulas
for ordinary English are inappropriate for rating the

difficulty of mathematics materials. Her study confirms

48

the theory that the language of mathematics possesses a
number of unique characteristics which influence its
comprehensibility and supports the hypothesis that teacher
judgement is a highly reliable measure of the readability
of materials written in the language of mathematics.

Bankston(15) and Thompson(202) both found that the
readability of mathematics materials has a significant
effect of achievement in mathematics. Smith(181) and
Heddens(101) conclude that readability of mathematics
materials is extremely important. In the words of
Smith(181), "If the child cannot read the statement, then

his abilities to think, analyze, and compute are hampered."

CHAPTER III

DESCRIPTION OF‘THE INVESTIGATION

The Problem

 

This study identified some questions which are
related to the two reading skills of word recognition and
comprehension, and what role these skills have in the
acquisition of mathematical concepts from written
descriptions of these concepts. Specifically, this study
was designed to investigate the effects of explicitly
teaching certain selected reading skills on the performance
of these skills in mathematics and on the solving of verbal
mathematical problems.

The five reading subskills, as described by Duffy
and Sherman(68), selected for this study were:

1. Instant Recognition of Special Symbols of Mathematics
and Their Voiced Equivalents (Voiced WOrd-Equivalents)

2. Structural Analysis

3. Contextual Prediction

4. Inferential Reasoning

5. Evaluative Thinking

49

50

The objectives of this experiment were:

1. Describe the behaviors of sixth graders who were being

taught how to use these five selected reading subskills

in reading mathematics

2. Determine the impact of the teaching of these five

selected reading subskills on the abilities of these

sixth graders to use these skills in:

a.

b.

Interpretation of mathematical symbols

Mediation of an unrecognized mathematical word
through its structural units

Obtainment of factual comprehension of a
mathematical message by contextual prediction
Exploration of understanding by inferring relations
among mathematical concepts

Fbrmulation of mathematical problems from given

problem situations

3. Determine statistically the impact of teaching these

five selected reading subskills on the prOblem solving

abilities of sixth graders

Procedure
Selection of Sample

The sample for this study was drawn from a sixth

grade class in a suburban, K-6 elementary school near Grand

Rapids, Michigan. The twenty-six students in the sixth

grade class selected were randomly divided into two groups

of thirteen for the control and experimental groups.

51

Development of Instructional Materials

The instructional materials for the explicit
teaching of selected reading skills were designed by the
researcher. There was a total of fifteen lessons—-three
thirty-minute lessons for each of the five selected reading
subskills. Copies of the material covered in the reading
lessons may be found in Appendix A. A description of the
objectives and materials used for each of the five reading

subskills selected for this study follows.

Subskill I: Instant Recognition of Special Symbols of
Mathematics and Their Voiced Equivalents
(Voiced WOrd-Equivalents)

Performance Objectives: Given a set of special symbols,
provide word equivalents and voice
them.

Given a word phrase or word
sentence, provide special symbolic
representation.

Most of the special symbols chosen for this
subskill were those symbols judged to be essential for
reading the sixth grade textbook being used by the subjects
in this study. Some symbols, such as\/F_: were chosen
because of their importance to hand-held calculators. A
special-symbol list, a special-symbol card game,
construction paper models, and a list of word phrases and
sentences for translation were used to accomplish the

objectives for this subskill.

52
Subskill II: Structural Analysis
Performance Objective: Given an unknown content word
which has structural units,
identify and give meaning.

The mathematical words for this subskill were
selected because of their structural units and their
relation to the special symbols chosen for Subskill I. The
material for this reading subskill consisted of word cards,
pictures or diagrams demonstrating meaning of structural
units of the content words, and a list of prefixes and

their meanings.

Subskill III: Contextual Prediction
Performance Objective: Given a set of sentences (one or
more) in which unknown words or
special mathematical symbols are
present, supply each with meaning.
For this subskill, the materials used were of three
types:
1. A set of sentences using "mathematical" words in
nonmathematical context
2. A set of mathematical sentences with certain key
mathematical words underlined which were to be defined
given that context
3. Cloze procedures constructed over mathematical
passages. In these cloze procedures, every fifth word

was deleted from the passages. The student was

53
required to complete each blank with the word or symbol

that made the most sense in the given context

subskill IV: Inferential Reasoning

Performance Objective: Given a story problem, determine
the mathematical processes or
Operations to be used.

EOr the reading subskill of inferential reasoning,
each of the three lessons consisted of a set of verbal
prOblems in which the children, after reading the problem,
inferred the types of mathematical processes or operations

required to solve these problems.

subskill V: Evaluative Thinking

Performance Objective: Given a situation, pose a prOblem.
To develop this fifth reading subskill, each of the

three lessons contained a set of prOblem situations. From

each of these problem situations, the children were to pose

a prOblem.

Teaching Procedure

Preliminary arrangements

As hand-held calculators were to be used in some of
the reading lessons, the researcher designed and presented,
one week prior to the implementation of the reading
lessons, five one-hour lessons on the hand-held calculator.
The entire sixth grade class, participating in this study,
received a daily one-hour lesson on the calculator for a

total of five one-hour lessons.

54

Each student was provided a four-function
calculator for these calculator lessons, the reading
lessons, as well as for the pretest and the posttest. The
calculator lessons were primarily given to this sixth grade
class to eliminate any computational advantages of
calculator use that the experimental group could possibly
gleam from the use of the calculator in the reading
lessons.

All twenty-six subjects participating in this study
were instructed in certain aspects of a calculator to
promote basic calculator literacy and were given activities
in which to practice using the calculator to find out some

of the things it can do and some things it can not do.

Teaching,schedule

This study was carried out during the Fall Term,
1977. The researcher instructed the subjects in the
experimental group in the five selected reading subskills
while the control group was doing independent work such as
individual reading of selected stories, homework or working
on special projects in art. The explicit reading
instruction in mathematics was given to the experimental
group in addition to their regular mathematics instruction.
The control group and the experimental group received the
customary instruction in mathematics together as a class,
instructed by the regular classroom teacher.

The explicit reading lessons were scheduled for

three days per week: Monday, wednesday and Friday--for

55
thirty minutes each—-9:00 A.M. to 9:30 A.M.-~for five
consecutive weeks. This schedule permitted three thirty-
minute lessons for each of the five selected reading

subskills to be investigated.

Presentation of instructional materials

 

Each reading subskill was described at the
beginning of the first lesson for each subskill. Why the
skill is important to reading mathematics was also
discussed, as well as how the students would be asked to
demonstrate their learning of the skill.

The teaching of the selected reading skills
included for each subskill a series of three 30—minute
lessons in a sequence designed to lead the student to
develop the subskill as applied to reading mathematics.
Each student had the opportunity to practice the skill
being taught in completing the designed activities
individually as well as going over them as a group. The
students were instructed to collaborate on some of the
tasks and to consult with each other after completing other
exercises individually. Students were given the opportunity
and encouraged to share as well as to defend their
understanding. The exercises were designed to provide
opportunities for experience in verbalizing orally in the
language of mathematics as well as opportunities for
interacting with the printed messages of mathematics. A

description of how each subskill was developed follows.

56
Subskill I: Instant Recognition of Special Symbols of
Mathematics and Their Voiced Equivalents
(Voiced WOrd-Equivalents)

The lessons pertaining to this particular subskill
illustrated that words and special symbols are used in
mathematics as a way of encoding ideas and as a means of
talking about mathematical situations. The stage for the
idea of a special symbol representing a mathematical idea
or an unknown was set via a short exercise, having students
think about things which they could not see, hear, taste,
smell or touch--mystery animals, mystery candy, or mystery
persons, having certain characteristics or properties--the
"guess-who" game or the "guess-which" game.

Example: (Clues given for dogfish)

I am a small animal.

I can be spiny or smooth.

I come from the NOrth Atlantic,
California coasts or southern European
waters.

My "namesake" is a carnivorous
domesticated mammal.

Part of my name is also used by humans to
mean a worthless, wretched person.

My name also denotes a device for holding
or fastening as well as an andiron.

My name also means to hunt or track.

I am a small shark.

At the beginning of the second lesson, dealing with
unknowns was continued via having the students explore, in
their minds, the future--the "what—if" game--what if it

rained coke tomorrow? The future is not explicitly known

and because they were dealing with an unknown, as

57

frequently required in mathematics, the subject was

necessarily quite abstract. But, the students could

explore in their minds!

The children were then reminded that people often
in everyday life refer to an object in a way which
indicates that they are speaking of an arbitrary element
belonging to a certain set, for examples:

A car is an element of the set of automobiles.
A redbird is an element of the set of birds.

The unknowns and the arbitrary elements were used to

introduce the idea of a variable, such as letting x

represent a number.

Calculators, straightedges, construction paper
models and cards were used to present the special symbols of
mathematics and their corresponding word codes to the
children. The children were:

1. Asked to respond orally to special symbols on a card or
to a construction paper-model by providing
corresponding verbalizations

2. Given verbal commands, spoken and written, and asked to
respond by using the correct special symbols on a
hand-held calculator or by drawings or sketches on
paper

3. Given words and asked to provide written symbolic

representation

58
For Lesson 1, the following activities were used:
A. Each student was given a list of symbols and, in turn,
provided oral verbalizations and examples
Examples: S; x S 10; The number x is less than or equal to
ten; 7 S 10.

H; AE; Ray determined by points A and B;

: A \
- /

A B
B. A deck of cards of the special symbols and of

corresponding words were dealt

A student "read" orally a card from.his hand; the card
was placed on the table in the center of the group; all
"related cards" were "read" and discarded. Students
"decided" on the correctness of responses with

supervision by the researcher.
F———_ ‘

 

 

 

The sum of Five is
Example: 5 + 7 five and added to 7 + 5
seven. seven.

 

 

 

 

 

 

 

The The terms

       
 

233:: :3 addition of a sum
. of five are 5 and
five.
and seven 7-

 

 

 

C. Construction paper models were identified orally by

students and special symbol-labels written on paper

Example: Model: Student Response: triangle

 

On Paper: AABC
D. Verbal commands were given. Students produced special

symbolization with pencil and paper and on a calculator

59
Example: Command: The sum of six and eight.
Pencil/Paper: 6 + 8
Calculator Sequence: 6, +, 8
E. Verbal commands were given. Students produced the

special symbolization diagrammatically with labels
Example: Command: Circle C Diagram:

Label : 0C
Fbr Lesson 2, the students were presented a list of
symbols and were asked to "read" them by giving voiced word
equivalents. Verbal commands were given and students
produced special symbolization with pencil/paper and
calculator when applicable.
Examples: Command: Find the sum of five and nine.
Pencil/Paper: 5 + 9 = 14.
Calculator Sequence: 5, +, 9, =

Command: Draw an angle whose measure is less

 

than 90°.
Pencil/Paper:
l//}a m(LA) < 90°.
A ~%

Fbr Lesson 3, the children were given a list of
word phrases and sentences and were asked to translate into
special symbols. These translations were then discussed

orally.

60
Example: The difference of eight and the second power of a
number.
A possible translation: 8 - x2
The students were also given a list of verbal commands to
"read" and respond by using the correct symbols on a
calculator and recording results on paper.
Example: The square of 19.
Calculator Sequence: 19, X, 19, =

Pencil/Paper: 192 = 19 X 19 = 361.

Subskill II: Structural Analysis

Structural clues rarely define a word completely.
Pointing out structural clues to students was used to make
them more sensitive to frequently used prefixes, suffixes
and word roots in mathematics. This could also encourage
them to draw on their own knowledge of word parts and to
try making a relationship between what they know and a word
which might be unknown or hazy to them or which might have
multiple meanings.

The selected content words for developing this
particular subskill were presented to the students on word
cards, accompanied by the following oral questions:

Can you pronounce this word?
Do you know what it means? or

Does any part of the word look familiar to you?

 

Example: bisect

 

 

61
The students were also provided with "pictures" or
diagrams in some cases to help demonstrate the meaning of
the content words as gleamed from prefixes or word parts.
The students were asked to "read" these diagrams to promote
the verbalization of words with structural units as well as
to promote the relation of special symbols to words and the

interrelatedness of reading skills.

 

 

Example: bisect midpoint

 

 

Diagram: -

AC a CB
Reading: C is the midpoint of AB.
C bisects A3.
Line segment AC is congruent to line segment CB.
On the third day (See Appendix A.), the students
were given:
1. A list of some prefixes and their meanings
2. A list of expressions depicting meanings for
mathematical words with these prefixes
3. A list of these mathematical words
The students were to match each word with the expression
depicting meaning for the content word.
The children were also given a list of commands to

"read" and to respond with pencil and paper sketches to

demonstrate understanding.

62

Example: Draw an angle with counterclockwise rotation.

Possible response:

For some of the content words (See Appendix A.),
completely unknown to any of the children or with
interesting parts or origins, clues were given to develop
the subskill of structural analysis. These new technical
terms were taken apart to define their structural units.
The students were asked to examine the examples carefully
and to try to figure out a possible definition of the
content word under consideration. It was conjectured that,
if students had some experience with the history of the
parts of a content word, they would possibly recall an
interesting origin upon seeing the visual stimulus and then

would be reminded of the meaning of the word.

Subskill III: Contextual Prediction

To illustrate the importance of syntactical
structure and content in semantics, the first part of the
first lesson for this subskill presented a set of
"mathematical" words, each being used in a nonmathematical
context. The children were asked to provide a sentence
using each of these words in a mathematical context.
Example: Bring the clothes ligg into the house.
Response: A ligg is a geometric figure stretching

endlessly in both directions.

63

In the second part of the first lesson for this
subskill, the children were given a set of mathematical
sentences with certain key items underlined. The chihdren
"read" the sentences and were then asked to explain the
meaning of each of the underlined vocabulary items.
Example: Draw a polygonal region with a triangular

boundary.

Response: The union of a polygon and its interior is called
a closed polygonal region. If the polygon of
this union has three sides (triangular), then it
is said to be the boundary of the polygonal
region, and is called a triangular boundary.

The other two lessons for this subskill used cloze
procedures (See Appendix A.). The children were asked to
read silently the entire cloze first before attempting to
fill in the blanks. It was felt by the researcher that
this would help the children make maximum use of redundant
information and contextual clues throughout the passage.

If the children were going to make use of context, it seems

that it would be important that they read up to the unknown

word and beyond it for additional clues. Students
suggested words or special symbols which could semantically
and syntactically fit into the blanks of the cloze and were
then asked to offer reasons for their choices.

Example of a sentence taken from a cloze procedure:

64
The perimeter of a is defined as the

of the lengths of sides.

 

Response: polygon, sum, its

Subskill IV: Inferential Reasoning

Fbr the reading subskill of inferential reasoning,
the lessons were designed to develop the children's
abilities to infer the mathematical processes or operations
necessary to solve verbal problems.

Example: An airplane flew 3600 miles in 16 hours. What
was its average speed per hour?
Oral response: To find the average speed, divide 3600 by
16.
Written response: Let S represent average speed. Then
S = 3600 + 16.

Some of the exercises were completed orally with
each student being given the opportunity to respond. Others
were completed as a group, with a student recording
suggestions by the group on a small portable blackboard.
Other exercises were written.

Not wishing to make the students feel locked into
an inflexible progression of steps nor have them conclude
that there is one and only one acceptable way to solve a
problem, no rigid instructions were given by the researcher
for the reading subskill of inferential reasoning. Fbr the

written exercises, the investigator circulated through the

65
group to assess progress and to provide suggestions for
those subjects who wanted to "get started" or to get
"unstuck."

The aim of any offered suggestion was to get the
students to read carefully and then to analyze the problem
systematically, that is, to read critically and
analytically. The suggestions given were:

1. Read the first time to obtain a general view of the
situation

2. Read a second time for comprehension in full-—read with
a questioning mind and with a pencil and paper--
translate the words into special symbols in order to
see relationships more clearly

3. Read a third time to regain a comprehensive, full
picture of the problem

Some of the students were encouraged to make a diagram or a

flow chart so that they could "see” the problem situation

visually as well as "see" in the mind--making it possibly

easier to detect relationships so that a valid inference

could be made.

Subskill V: Evaluative Thinking

The rationale for this reading subskill was
motivated by presenting to the students the following
questions:
DO you think it important:
1. For you to speculate about possible consequences of

events

66
2. To develop abilities to imagine trying to survive as
adults in the future

3. For you to develop your forecasting talents

To develop this fifth reading subskill, the lessons
provided opportunities for the children to strike out on
their own by creating or posing mathematical prOblems from
given situations after evaluating the given situations.
From.the list of given situations, the children posed
problems, both orally and in writing.
Example: On a 20-word spelling test, Denise missed 3 words.

Response: What per cent of words were missed?

Limitations of presentation

The entire experimental group received instruction
in the subskills selected for this study. Individual
deficiencies were not identified so that students who failed
to reach mastery level of a subskill could be given
additional assistance and time to master the objectives.
NO extra remedial exercises or activities were provided fOr
those who may have needed them at the point where they were
having difficulty. Even though each unit did not
necessarily depend upon mastering a previous unit, there
was a cumulative payoff in mastering sequential lessons.

The program in reading had to be usuable within the
confines of the assigned 30-minute period, since it would
have been unrealistic to assume that students requiring

extra work would do it outside of the 30—minute session.

67
All sessions lasted 30 minutes, except the first lesson for
the first reading subskill. This lesson lasted for

approximately 50 minutes.

Testing and Evaluation Procedures
Testing instruments and schedule

Two equivalent instruments were designed by the
researcher as a pretest and a posttest to measure the
problem solving abilities of the sixth graders involved in
this study and to determine the effects of the explicit
teaching of certain selected reading subskills on the
problem solving abilities of these sixth graders. The
posttest items were Obtained from the pretest by
interchanging the order in which the problems were presented
and, in some cases, by making other minor changes, such as
substituting different numbers within a problem. The
pretest was given 4 days before the start of the reading
lessons and the posttest was given 2 days after the last
reading lesson.

Previous to determining the final version of the
items on the pretest, the instrument was given to a group
of thirty-five prospective elementary teachers, enrolled in
a preservice college mathematics course. This was done to
obtain information and feedback in regards to the possible
reading subskills required for solving the problems on the
test. Copies of the pre- and posttests can be found in

Appendix B.

68

Neither the pretest nor the posttest was timed but
neither exceeded sixty minutes. A hand-held calculator was
furnished each student for both tests.

Test validity and reliability were based upon the
scholarly analyses of test items by mathematicians and
mathematics educators and researchers in conjunction with a
reading specialist at Michigan State University.

The tests were corrected by the investigator, using
a predetermined score-analysis sheet. Credit was given in
the following manner:

1. For a solution without error, the maximum point score
of five was given

2. Fbr a partially correct solution, a score of one to four
was given

3. In cases of a totally incorrect solution or no solution,

a score of zero was given

Statistical evaluation

Each subject's score on the pretest (yi) and on the
posttest (xi); 1 = l, 2, 3, 4, 5, 6, ..., 13; were paired
for the same person to find the set of differences
(xi — yi) for each of the two groups. The mean score of
these differences for each group was determined. A t-test
was used on these two mean scores to determine if the

difference between them was significant.

69
The Null and Alternate Hypotheses tested in this
study were:

HO: There will be no difference between the mean scores of
the two sets of differences obtained by subtracting,
for each group, each subject's score on the pretest
from.his score on the posttest.

H : The mean score for the experimental group will be

greater than the mean score for the control group.

Other measures

A set of test items from the posttest was selected
for further detailed study. This study took the form of
interviews with a randomly selected subset of seven subjects
from the experimental group. The results of these
interviews were used to help describe the effects of the
reading instruction in mathematics.

A daily log, kept by the researcher, was used to
help determine the impact of the reading instruction.

The classroom teacher's periodically written
comments regarding the behaviors of the subjects in the
experimental group in the regular classroom were read to
Obtain his observations of the effects of the reading
lessons in mathematics on the subjects' classroom
behaviors.

A retest for retention was given to the
experimental group five weeks after the posttest. A copy

of this retest for retention can be found in Appendix B.

70
Summary

This study explored the effects of a five—week
(450-minute) program in the reading of mathematics on the
problem solving abilities of thirteen sixth graders. The
reading program consisted of three 30-minute lessons for
each of five reading subskills as identified and described
by Duffy and Sherman(68). The reading subskills selected
for this investigation were:

1. Instant Recognition of Special Symbols of Mathematics
and Their Voiced Equivalents (Voiced WOrd Equivalents)

2. Structural Analysis of Mathematical Words

3. Contextual Prediction in Mathematical Writings

4. Inferential Reasoning in Mathematical Writings

5. Evaluative Thinking in Mathematical Situations

The fifteen reading lessons were designed by the
researcher. Researcher-designed pre- and posttests were
used to measure performance of these skills in mathematics
and problem solving abilities of the twenty-six subjects
who participated in this study. The population in the study
was divided randomly into control and experimental groups.
A t-test was used to measure any significant difference in
the gains of the two groups.

A daily log kept by the researcher, written
comments of the regular classroom teacher, interviews with
seven randomly selected subjects from the experimental
group for certain problem-analyses and results of a retest
for retention five weeks after the posttest were used to

help describe the effects of the experiment.

CHAPTER IV

RESULTS

Introduction

 

Most reading specialists feel that the skills for
reading can be categorized as skills of decoding the
writing system to its associated language and skills in the
use of this code for comprehension. According to these
reading specialists, the thousands of different subskills
for reading extend from the simple to the complex-—from
visual discrimination of symbols to judging content. The
subskills described by Duffy and Sherman(68) and selected
for investigation in this study were:

1. Instant Recognition of Special Symbols of Mathematics
and Their Voiced Equivalents (Voiced WOrd Equivalents)

2. Structural Analysis

3. Contextual Prediction

4. Inferential Reasoning

5. Evaluative Thinking

The purposes of this study were:

1. Describe the behaviors of sixth graders who were being
taught how to use the five selected reading subskills
of word recognition and comprehension in reading

mathematics

71

72
2. Determine the impact of the teaching of the five
selected reading subskills on the abilities of these
sixth graders to use these skills in
a. Interpretation of mathematical symbols
b. Mediation of an unrecognized mathematical word
through its structural units
c. Obtainment of factual comprehension of a
mathematical message by contextual prediction
d. Exploration of understanding by inferring relations
among mathematical concepts
e. Fbrmulation of mathematical problems from given
problem situations through evaluative thinking
3. Determine statistically the impact of teaching these
five selected reading subskills on the problem solving

abilities of sixth graders

Analysis of Data
Two random samples, the experimental group and the
control group, were selected from a population of twenty-
six sixth graders. For each group, each subject's score on
the pretest (yi) and on the posttest (Xi) were paired to
find the set of differences (xi - yi). The mean score of

these differences for each group was determined. A t-test

73

was used on these two mean scores to determine if the

difference between the two mean scores was significant.

Table l.--Data for Experimental Group on Pre- and Posttests

 

 

 

 

 

Student Pretest Posttest xi - yi Deviations di
Yi Xi d1
N 1 21 20 "12.923 167.004
0 3 45 42 9.077 82.392
p 0 31 31 "1.923 3.698
Q 11 55 44 11.077 122.700
R 12 75 63 30.077 904.626
8 2 44 42 9.077 82.392
T 10 24 14 '18.923 358.080
U 6 29 23 "9.923 98.466
v 0 15 15 ‘17.923 321.234
w 8 66 58 25.077 628.856
x 8 31 23 ’9.923 98.466
Y 13 44 31 "1.923 3.698
2 7 29 22 "10.923 119.312
13
From Table 1, 2 (xi - yi) = 428.

M1 =

i=1

428 divided by 13 =

Experimental Group.

32.923—-Mean Score for the

74

Table 2.--Data for Control Group on Pre- and Posttests

 

 

 

 

 

Student Pretest Posttest xi - yi Deviations d3
Y1 Xi d2
A 2 2 0 ”1.615 2.608
B 4 11 7 5.385 28.998
c 7 2 '5 ‘6.615 43.758
0 1 3 2 0.385 0.148
E 2 2 0 ’1.615 2.608
F 9 8 '1 ‘2.615 6.838
G 5 8 3 1.385 1.918
H 5 10 5 3.385 11.458
I 9 11 2 0.385 0.148
J 7 7 0 ‘1.615 2.608
K 6 5 ‘1 ‘2.615 6.838
L 8 16 8 6.385 40.768
M 7 8 1 '0.615 0.378
13
From Table 2, Z (Xi - yi) = 21.
i=1
M = 21 divided by 13 = 1.615--Mean Score for the Control

2

Group.

75

32.923 - 1.615

 

2990.924 + 149.074 _1_+_1__
13 + 13 - 2 13 13

6.979 df = 24

 

t

A t = 6.979 with 24 degrees of freedom is significant at
the 0.005 level.

Based on the t-test for making comparisons among
means, it can be concluded that the Nu11 Hypothesis,

H M - M = 0, must be rejected in favor of the

o‘ 1 2

alternate, HA: Ml - M2 > 0, at the 0.005 level of

significance.

Observations

 

As a prelude to the discussion establishing the
objective of the reading program, the following questions
were asked:

1. Do you like mathematics

2. What do you think mathematics is

3. Do you like to read

In summary, the essense of the responses to these questions

was:

Question 1: All the girls said that they did not like
mathematics and that it was boring. The boys
were less emphatic in their negative attitudes
toward mathematics but most felt it was boring.
One boy said that he really liked solving
problems with the calculator. The other
children then said, "Oh: But that was

different!"

Question 2: The only comment received was that mathematics
is, "Finding the answer with numbers."

76
Question 3: The girls enjoyed reading some things. On the
other hand, the boys in general said that they
did not like to read.
A daily log kept by the researcher was used to help

describe the impact of the reading instruction for each of

the five reading subskills.

Subskill I: Instant Recognition of Special Symbols of
Mathematics and their Voiced Equivalents
(WOrd Equivalents)

The students were not at all familiar with most of
the special nondiagrammatical symbols of geometry or voiced
equivalents of special symbols of geometry. Fbr example,
no student responded orally to the question as to the
difference between the meaning of the term "circle" and the
term "circular region." After being shown special
diagrammatic symbols for these two mathematical concepts,
all thirteen students demonstrated, in writing, a
respectable intuitive understanding of the difference
between these two items. In general, the responses were:

circle: "Just the line."
circular region: "What's inside the circle."

The students recognized the symbols such as +, -,

X and + as representing operations but were not familiar
with "-" as in -3, nor were they knowledgeable about the
verbalizations of these symbols as operations. EOr

example, they did not know the meanings for sum, product,

quotient or difference. To all the students:

77

+ means add,

- means subtract,

x means times,

+ means divide,

and all meant, "Find the answer."
These students were also totally unfamiliar with most of
the other algebraic symbols such as: 3(2 + 7), 2b, 9, <,
S, >, 2, 8/2 (meaning divide), and 32. Two of the students,
after some contemplation, remembered the meaning of the
square root sign and of the exponent 2 from their lessons
with the calculator. Once the calculator was mentioned, a
discussion ensued between the children, whereby, all were
able to demonstrate that they had remembered the meanings
of these two symbols from the calculator lessons.

In giving examples of numbers, not one student
volunteered an example of a nonpositive nor noninteger
number. Fbr example: Fbr 3 < x < 9, what are some
possible values for the number x? The students agreed that
the gn y values were 4, 5, 6, 7 or 8. The students were
not aware that 3.85 represented the same number as
3.85000000, nor were they knowledgeable of the ordering
relation of numbers such as 3.4563 and 3.4573. In fact,
the students were very unfamiliar with decimal fractions
and negative numbers.

The students seemed more relaxed in working with and
talking about the geometric symbols and the ideas

represented by these symbols. They seemed uncomfortable,

78
being asked to talk about +, -, x, and + or about most of
the algebraic symbols or about numbers in general. There
were signs of frustration or embarrassment--tense body,
high voice, twitching fingers and probably damp palms as
some would wipe hands on their sleeves.

Generally the students seemed to grasp the ideas
rather rapidly if paying attention when symbols and
associated ideas were explained. Even though most of the
students seemed easily distracted, it was amazing how much
they remembered from one lesson to the next, especially
concerning geometric items. There did not seem to be as
much quality retention for the algebraic symbols.

In providing verbalizations for the special symbols
in the second lesson, the students performed rather well.
In many of the items, very little coaching was needed for
most students to verbalize as well as to provide examples
to demonstrate some comprehension.

Example: éx--"One-half of a number x."--"If x equals six,
fix is three."
Even though the students could verbalize expressions such
as 4x - 3 < 5 or 4x - 3 = 5, as well as paraphrase (You
multiply 4 times the number x, then you subtract 3 and that
should give you a number less than 5 or you multiply 4
times some number x, then you subtract 3 and you'll get 5.),
they did not know how to find specific values of x to
produce true statements. No student suggested trial and

error--guessing and checking. NO time was spent on

79
explaining how this was done specifically as this was
classified as strictly mathematical concepts, according to
this investigator. The suggestion of trial and error was
offered to the students but not any other algebraic methods
for solutions of these open sentences.

Some of the students had trouble distinguishing
between the decoding of < (less than) and > (greater than).
Some of the students verbalized L (angle) as "less than"
not using contextual clues to predict meaning.~ The other
students quickly made the corrections.

The students in general did not perform well in
translating word symbols to special symbols. Most all the
students worked carelessly, made rather quick and rash
decisions. In general, the girls did better than the boys.
If this investigator worked individually, coaching by
asking questions, each student could successfully translate
most of the required expressions. Most of the students had
trouble working individually, even the brighter students.
They seemed to need constant assurance and frequent
feedback. They were excited and enthusiastic if working
with this investigator but seemed incapable or unwilling to
do the required translations alone.

There appeared to be some perceptual errors such as
in Exercise 13 of Lesson 3, as three students drew "two"
lines instead of "three," but in Exercise 12, only two lines
were called for. Upon being asked to reread Exercise 13,

all three students "caught" their error.

80

Fbr the activity of reading a list of verbal
commands and responding by using the correct symbols on a
calculator, one student finished long before the rest of
the group, a feat which the student was Obviously proud of.
A quick check of his work showed several errors. The
investigator asked if this student had "checked" his work.
The response was, "No, I haven't."

"Why not?"

"I don't have the answer booki"

"Is that the only way to check your work?"

"Why yes! It is!"

"Do your results make sense? Can't you check this

without an answer book?"

The student did not know What "making sense" meant
in these calculator problems. As it turned out, none of
the students really understood What "making sense" meant,
so a short discussion about estimation was held--then the
calculator lessons were remembered. Results making sense
and estimation had been covered in the calculator lessons.

From.the informal Observations made by this
investigator, these sixth graders are capable of dealing
with the special symbols in the reading lessons for this
study. They did not seem.awed or afraid of these symbols.
They did seem surprised to be asked to 3315 about the ideas
represented by these symbols and to 351;; about mathematics
in 393d symbols. They appeared to be totally inexperienced

in verbalizing in mathematics-~that is, in expressing

81
meaning and demonstrating understanding of mathematical

concepts represented by the special symbols.

Subskill II: Structural Analysis

The students seemed totally unfamiliar with the
reading subskill of structural analysis--not just in the
area of mathematics. In answer to the question, "How does
one obtain meaning of a word?", the response given was,
"The teacher tells you what it means." Not one student
even suggested the use of a dictionary. Yet, in spite of
themselves or unconsciously, a few of the students would
use contextual clues to give words a bit of meaning in some
cases. Below is an approximation of the type of dialogue
which evolved during this set of reading lessons.
Investigator: What does ascending mean?
Student 1: Don't know--never heard of the word.
Investigator: Does any part of the word suggest some

meaning of the word to you?
Student 1: (After some thought) No-—never heard of the
word.
Investigator: What does, "I'm ascending the stairs," mean?
(The whole group is quiet. Student 1 smiles rather timidly
but does not answer. Finally Student 2 volunteers.)
Student 2: Either you're "going up" or "coming down" the
stairs.

Investigator: Why do you think that?

Student 2: That's what you do to the stairs!

82
Investigator: Don't we paint or carpet or sweep the stairs
too?
Student 2: Yeah . . . I guess we do? (All were very
puzzled now.)

After some discussion among themselves, it was
unanimously decided by the students that "ascending" means
"going up" but no student could or would explain why they
felt this way. After the discussion of the etymology of
the word, the students seemed to feel very confident and
very proud of themselves for having made the correct
decision.

The students acknowledged that they had studied
prefixes and suffixes but could not define any of the
prefixes presented to them even though all of the students
knew where the prefix and the suffix belonged in the
composition of a word.

After the explanation of this subskill and after
the discussion of several examples of words with structural
units, each student was given the chance to demonstrate
understanding of this reading skill. Fbr example:
Investigator: If "quad" means "four" and "lateral" means

"side," in the context of mathematics, what
do you suppose "quadrilateral" means?
Student: A four-sided figure.

When asked the meaning of "quadrant," the students

responded that it had "something to do with four things."

Upon being shown the diagrammatic symbol of a plane being

83
separated into four quadrants, the students very quickly
identified correctly a quadrant as applied to a plane.
After the discussion of "bisect," the students were able to
describe "trisect" quite accurately when reminded to think
also about the meaning of "triangle."

The origins and historical notes seemed to be quite
interesting to the students, and they clearly enjoyed
trying to "guess“ the meanings of words from the clues
given. The students worked enthusiastically with the
exercises of matching words containing structural units
with the expressions depicting meaning for these content
words.

The activity generating the most enthusiasm
involved a list of commands to be "read" and to be
responded to with pencil/paper symbols to demonstrate
comprehension. The children enjoyed these exercises. Some
of the comments overheard and recorded by this investigator
as the students completed the activity were:

"Oh boy, this is fun!"

"Hey! Look at mine!"

"Boy, this is neat!"

"Wow! This is fun!"

"Once you get the hang of this, it is real easy!"

"I wish we did this stuff in math class!"

For this subskill of structural analysis, the
students in general had very few decoding errors; that is,

phonetic and syllabication difficulties were minimal. But

84
the perceptual error of omission was rather strongly
demonstrated in this subskill as in the first. This
problem seemed to stem from careless reading and
nonrereading. Once the students were instructed to reread
the verbal command more slowly and to check against their
response, the majority of students were able to discover
most of their errors of omission.

In this section, processing errors were rather
numerous but the students in general were able to perform
successfully upon being encouraged to read and reread and
then paraphrase or to talk over the command with the
investigator or with each other. When an unknown word was
related to something which the students were already
familiar with--something in their everyday, ongoing lives--
almost all the students could give successful responses.
Students seemed to remember the unknown word if they
remembered their own participation and contributions in

group discussions.

Subskill III: Contextual Prediction

The children "read" the sentences in which
"mathematical" words were used in a nonmathematical context
with aplomb. That is, each student "called out" the words
of these sentences perfectly. Most of the students had
trouble providing, without coaching, a meaningful sentence,
using the mathematical words in the context of mathematics.
Most of the students had trouble explaining many of the

meanings as used in the everyday sense. They were not very

85
meaning-conscious readers. Reading to them appeared to be
"the pronouncing of words"--a type of oral performance.
The students seemed surprised, if not a bit agitated, to be
asked to explain terms and discuss meanings, especially as
applied to mathematics.

Even though the students in general seemed to be
perturbed in attempting to explain meanings or to provide
mathematical sentences, after being presented with clues,
most of the students were able to provide mathematical
sentences of ordinary or mediocre quality. Two of the
students were able to produce sentences of a rather
moderate to high excellence.

Examples: Here is an anglg of 45°.
The answer to a problem is the solution to the
prOblem.
This ggy goes off in that direction without
ending.
Two is a 231mg number.
A rectangular region whose length is 2 inches and
width is 5 inches has an gggg of 10 square inches.

In the second part of the first lesson fOr this
subskill, the children were given a set of mathematical
sentences with certain key items underlined and asked to
explain the meaning of each of the underlined vocabulary
items. Their success with this activity was again
accomplished with coaching in the form of cue or clue

questions and sometimes with diagrammatic symbols where

86
appropriate. Some of the students could demonstrate some
intuitive comprehension by drawings or sketches, but did
not possess the necessary mathematical vocabulary to give
their explanations in words.

Fbr their first cloze procedure, the children were
very much at a loss and had to be not only coached
frequently but also coaxed and Wheedled. They were ready
to quit almost immediately and again this group of
nonperformers included, at the beginning, the brighter ones.

"This is too hard!" "I can't do this!"

"This seems stupid!"

They were very familiar with "fill in the blanks"
nonmathematical tests with prejudged words being left out,
in which, in the opinion of this investigator, it is hardly
ever necessary to read critically, analytically or
carefully. Also, in the ordinary "fill in the blanks"
tests, knowledge of the syntactical structure of language
is not, in the opinion of this investigator, as important
as in the cloze procedure; nor is the use of the reading
subskill of contextual analysis as prevalent. To supply
missing words or special symbols in a cloze procedure, a
person must "read" critically--must be acutely cognizant of
the graphemic, syntactic and semantic signals as indicated
explicitly or implicitly.

Even though the students admitted to having studied
verbs, nouns and other parts of speech, they in general

found it difficult to predict what part of speech was

87
needed to complete a cloze. The students also found the
situation of "more than one correct answer" uncomfortable
if not actually intolerable.

"How can there be two right answers?"

After the completion and discussion of the first
cloze procedures, the children commented that they could
now understand the importance of the knowledge of parts of
speech and how this knowledge could help them understand
better the printed word, but "What has all this to do with
mathematics?" During a discussion of the importance of
reading in mathematics, all the children not only agreed
that reading was important but could offer reasons why
reading is important in mathematics and problem solving.
Yet, it was demonstrated time and time again during this
experiment that these children were not psychologically
tuned to the realization of how important language and
reading in particular were to mathematics.

Even though the situation of "more than one correct
answer" still was not palatable to most of the children, by
the third lesson, they did become rather self-assured and
competitive. Before they would change their choice for a
blank to agree with someone else's, they had to be "proven"
incorrect. If the investigator agreed that two different
responses were correct, then "Mine is the best," could be

heard in many cases.

88

In the opinion of this investigator, the reading
subskill of contextual analysis as developed through cloze
procedures could have a positive effect on the reading
abilities in mathematics of elementary children at the
sixth grade level as well as at other elementary grade
levels. This investigator feels that it is rather
unreasonable to expect much progress within the short time
spent on the cloze with these thirteen sixth graders, but
through the observations made, these thirteen sixth graders
demonstrated that progress can be made even in a very short

time.

Subskill IV: Inferential Reasoning

With many of the students, their approach to the
reading of the problems to infer the necessary mathematical
processes for solution was a source of difficulty. Most of
them, from.their comments, had developed a deep-seated
dread of "story" problems. After an inadequate first
reading, most were inclined to panic. Some were impatient--
dismissing a prOblem as "too hard for me" if they could not
see as well as organize the relationships instantly and
mentally. They hesitated to manipulate the definitions or
other ideas and concepts to explore the relations so that
an inference could be made.

NOne of the students used the pencil/paper reading
at the onset—-did not note important ideas and relations,
or make sketches nor translate to special symbols. Some of

the students seemed to slight preliminary reading, striking

89
out blindly as if expecting to comprehend without reading
all the way through. If they reached parts that they did
not understand, some of the students simply stopped. They
felt compelled to have meaning before going forward
possibly to pick up contextual clues--until they were
reminded of the previous reading subskill.

Few students thought that more than one reading
would be necessary-—they sat for the most part almost if
waiting for a "bolt of lightning" to show them What had to
be done. Most of the students were prone to take rather
superficial approaches to the task of inferring the
relations required. A popular belief seemed to be that
"the last sentence in the prOblem always tells what is to
be done." The students looked for signal words or symbols
to tell them what processes to use.

The students seemed totally inexperienced in
describing or in writing down sentences explaining the
mathematical processes used in solving a word problem. If
an inference was made, most of the students were prone to
try to carry out any computations mentally without denoting
which calculations were to be used. As the investigator
called attention to this situation, most Of the students
were inclined to defend their actions with, "Well, I got
the right answer, didn't I?" or "Well, that's the answer

and that's the most important thing!"

90

Obviously these students were not experienced in
using special symbols to describe “how to" in a problem.

If they were reminded of their first reading lesson, some
of the students would then try to translate the words into
special symbols, whereby some success in ascertaining the
correct procedures was accomplished. Other students had to
have more explicit clues given to them, such as, "Suppose
we let P represent the unknown product. NOw, describe fOr
me in special symbols how to find P."

Inferring "what to do" or "how to" seemed to have
been an unusual experience to these children. All of the
students experienced difficulties with this reading skill,
partly because of their lack of knowledge and understanding
of the concepts of mathematics and partly because of their
lack of knowledge of the language of mathematics and its
uses.

By the end of their third lesson in this reading
subskill, a little progress had been made by most students.
They apparently though could not proceed without the
investigator's presence with coaching questions and
immediate feedback as to the correctness of their
inferences and their descriptions of these inferences. It
was gratifying to notice that many of the students were
discussing with each other the possible applications of the
previous reading subskills. Some students had also
developed a rudimentary mathematical language which gave

them enough courage to debate timidly the possible

91
mathematical processes needed for the exercise prOblems and
to ask themselves and each other questions which led them
into seeing the relations required for making the necessary

inferences.

subskill V: Evaluative Thinking (Posing PrOblems)

It did not seem difficult to convince the children
that in a realistic prOblem situation, the students could
be forced to formulate a precise prOblem from a prOblem
situation and that posing problems could be helpful in
developing forecasting talents. But the general response
was, "What has this got to do with mathematics?" The
students seemed unusually perturbed and rather uneasy
trying to relate the objective of these lessons to what
they had in mind as being classroom mathematics.

By the third lesson for this subskill, in general,
the students exhibited very positive attitudes in trying to
accomplish the task of posing problems. This task of
posing prOblems seemed to be even more difficult than
inferential reasoning. Some of the students would lose
track of what was to be done and instead of trying to pose
a problem they would be trying to solve a problem.which was
not even identified.

The difficulties experienced by the children in
"striking out on their own" in posing prOblems from given
prOblem situations probably were due to their lack of
facility with mathematical language as well as to their

lack of knowledge and understanding of mathematical

92

concepts and their relations. Their own mental attitude
toward mathematics apparently played an important part in
this activity. The cases in which students managed to pose
a meaningful prOblem demonstrated a tendency for the
students to ask questions which called for a single
computational process. Most of the students were impotent
in posing meaningful problems from situations containing no
numbers--"HoW can you make a prOblem if there're no
numbers?"

Below are some examples of problem situations and
the corresponding problems posed by some of the students.

Situation: Jane's father finds that his car travels 16.75
miles per gallon of gasoline.

Problems: How many miles can he drive a day?
How many miles can you get out of a gallon?

How many miles would Jane's father travel with 3
gallons?

How many gallons of gas would you have to use
for 83.75 miles?

Situation: Douglas bought 4 pairs of socks for $4.76.
PrOblems: How much change would he get?

and he had 10 dollars, how much change would he
get?

How much would one pair cost?
How much would a dozen pairs cost?

Situation: Mr. Cartwright bought a city lot 75 feet wide
and 99 feet deep.

Problems: How many feet all together?

How wide could the store or the thing that he
wants be?

93
What would the area of two be?

If Mr. Cartwright divided his lot in half, how
wide would each half be?

Situation: The bottoms of some of the Great Lakes are
below the level of the ocean.

Problems: HOW come?
Why is this so?
How does it do that?

Situation: When a boy tried to fly his kite, he found that
it would not go up in the air.

Problems: How come the kite wouldn't fly?
Why doesn't it fly?

Why wouldn't the kite fly in the air? Circle
one of these answers:

It wasn't a good kite.
There wasn't enough wind.
PrOblem Analyses
Problems 16, 17 and 18 from the posttest were
selected for further analyses as to the reading cues which
may have hindered the students in the solutions of these
prOblems. These prOblems were chosen because the students
did not give acceptable solutions to these problems. Seven
randomly selected subjects from the experimental group were
interviewed individually by this investigator for
approximately thirty minutes. Each of these students were
instructed:
Read the prOblem orally.
Verbalize your thinking as you solve the problem.

Describe what in the problem gave you trouble.

Problem 16:

94

Consider the figure below With the given data.
Find m(Lb) and the measure of the exterior
angle at La. '

 

 

 

m(La) = 70° m(Lb) =
m(Lc) = 50° m(exterior angle at La) =
a b
6 \
‘—7

PrOblem 17:

PrOblem 18:

Student R

Problem 16:

PrOblem 17:

The measures of the sides of a triangle are 4,
6, and 10 feet respectively. Find the
perimeter of a similar triangle whose longest
side is 60 feet.

Note: In constructing the posttest from the
pretest, the decision to change
measures of the sides of the triangle
for this prOblem resulted in a triangle
of area zero. This was an oversight.
Even though the children had been made
aware of the relation between the
measures of the sides of a triangle,
they overlooked, or were not bothered
by this degenerate triangle.

One side of a triangular flower bed must exceed
a second side by 6 feet, while the third side
must be twice the second side. If the
perimeter of the flower bed is to be at least
18 feet, find the smallest possible measure of
the shortest side.

Decoded the problem correctly and verbalized a
correct solution. Did not know Why he did not
manage a correct solution on the test.

Decoded the problem correctly and verbalized a
correct solution. Again, he did not know why

he did not solve the problem correctly on the

test.

Problem 18 :

Student W

Problem 16:

PrOblem l7:

PrOblem 18:

Student Q
Problem.l6:

95

Decoded the problem correctly. Sketched a
triangle and verbalized the correct relation of
sides but could not symbolize this relation.
The student was coached with the question,
"What if we let the second side have a

measure of x feet?" The student set up the
following procedure but did not know, "How to
find x."

x + 6 x P = x + 6 + x + 2x = 18

 

2x

Decoded the problem correctly and verbalized
a correct solution. Did not know why he did
not solve the problem correctly on the test.

Decoded the problem correctly and verbalized
a correct solution. Guessed that if he had
sketched a diagram as was done in the
interview, he would have been able to solve
the prOblem.

Decoded the prOblem correctly. Could not
verbalize the correct relation of sides even
though a sketch of a triangle was made. The
student guessed that he just didn't "read

good enough." With several coaching questions,
the student finally was able to verbalize and
symbolize the relation of sides. The
following procedure was then set up by the
student but the student did not know, "HOW to
find x."

x + 6 x
P = x + 6 + x + 2x = 18

 

2x

was not able to decode problem correctly
without coaching. Read the symbol for "the
measure of" simply as an "m" and decoded the
symbol for angle as "less than." After the
first reading, the student said, "That doesn't
make sense!" He was reminded of the reading
subskills, especially of contextual analysis.
The student then decided that "L" should be
decoded as "angle" but still hesitated with

Problem 17:

Problem 18:

96

the "m". After being asked to reread the
given data, the student was able to decode the
"m" as "the measure of." Still could not
verbalize a correct solution. The student
could only offer "I don't know how to do the
problem" as the reason for not being able to
accomplish a solution. The interviewer
conjectured that the student could not infer
the relation of the measures a, b and c. After
prompting, student guessed that the sum of
these measures was 100°, therefore,

m(Lb) = 100 — 120. "No, no! That can't be
right!"--exclaimed the student. After being
given the correct relation, a correct measure
for angle b was obtained by the student.
Student could not then infer the relation of
angle a to its exterior angle. After being
coached about a straight angle, the student
obtained a correct solution to this problem.
was not able to offer any reasons for
difficulty, except, "I just didn't know it."

Decoded the prOblem correctly. Did not
remember what perimeter or similar meant.

Given clues concerning the parts of the word
"perimeter," the student verbalized a correct
definition. After several coaching questions,
a correct diagram depicting the relation of
"similar" was made by the student and the
student then verbalized a correct solution.

The student said that the meanings of perimeter
and similar were the trouble spots.

Decoded the problem correctly. Even with
coaching, could not infer the correct relation
of sides. After a diagram depicting this
relation was described by the interviewer, the
student inferred that "NOw the sides would be
added together to find the perimeter" but
could not verbalize precisely what this sum
would look like. The interviewer verbalized
the required inequality but the student did
not know what to do with it. Again, the
student responded to the question concerning
the trouble spots with, "I just can't do the
problem." It seemed that the big trouble was
not being able to translate words to special
symbols in order to see relations more clearly.
Inferential reasoning was also an important
reading cue.

W

PrOblem 16:

Problem 17:

97

Decoded the symbols like "m(Lb)" as "m times
angle b." After a period of silence, the
student decided that he could not do the
problem, but didn't know why, except that he
"just didn't know What to do." Even after
being encouraged to reconsider the decoding of
"m," the student still maintained the "m
times" meaning because "parentheses always
means multiply!" He was then asked, "Does
this make sense, in this context?" After a
seemingly frustrating adjustment of his
thinking and after being encouraged to use
contextual analysis, the correct decoding was
accomplished. The student then remarked that
the measure of angle b could be found if he
could just "remember the sum of the angles of
a triangle." A correct measure for angle b
was obtained by the student after being given
the relation of the measures of angles a, b
and c. The student then guessed that the
other required measure was 110'. An
explanation was asked for and the student
described what he thought was the relation
between angle a and its exterior angle. This
turned out to be the correct relation.

Decoded the problem correctly. Described

correctly the meaning of perimeter but had
some difficulty with "similar." Drew two

triangles like so:

4A6

10

 

60

The student remarked that "60 is similar to 10."
In attempting to explain this statement, he saw
the light and quickly labeled the measures of
the other sides of the second triangle as
below:

24 36

 

60

PrOblem 18:

Student Y

Problem 16:

Problem 17:

Problem 18:

98

This diagram was very proudly presented to
the interviewer. After a pause and a puzzled
look, the student asked, "Isn't that right?"
He was then instructed to reread the problem,
was able to discover that the problem was not
finished and then effect a solution.

Decoded the prOblem correctly. Immediately
drew a triangle and labeled as follows:

 

2 x x + 6

Remarked, "Now you add them all up!" After it
was suggested that the problem be reread, the
student claimed, "Oh, after you add them all
up, you get 18." Student could not symbolize
this sum. After several coaching questions,
the student relabeled the triangle, showing
the correct relation of sides, and after much
encouragement, the student recorded

x + 6 + x + 2x = 18, but could not find x.

Decoded problem correctly. Verbalized a
correct solution. Did not know why he did not
solve prOblem correctly on test.

Decoded problem correctly. Verbalized a
correct solution. Did not know why he did not
solve problem correctly on test.

Decoded prOblem correctly. Verbalized the
correct relation of sides but could not
symbolize this relation. After being coached
into introducing the variable x, the student
symbolized the following procedure but could
not effect a solution:

x + 6 x

2x + x + x + 6 = 18

 

2x

Student S

PrOblem l6:

Problem.l7:

Problem 18:

99

Decoded "the measure of" as simply an "m."
After several rereadings, the student
maintained that he did not know how to work
the problem.but could not pinpoint or

describe the trouble spots. Even after being
told of the meaning of "m," the student made a
guess which could not be explained in any way
by the student. It was "just a number which
came to my head." After being reminded of the
relation of the measures of angles a, b and c,
the student was able to infer the correct
procedure and effect a solution. With several
coaching questions, the student was able to
accomplish a correct solution for the measure
of the exterior angle at angle a.

Decoded the problem correctly. The student
could not decide the meaning of perimeter as
opposed to area. Given clues concerning the
parts of the word "perimeter," the student
described perimeter correctly. The student
described similar triangles as being "alike
but one is bigger than the other." Even after
the student sketched two triangles with the
60-foot side corresponding to the lO-foot side,
he could not decide on the correct arithmetic
to use. He tried to use an additive term of
50 instead of a multiplicative factor of 6.

To the interviewer, this indicates a
misunderstanding of the meaning of "similar
triangles." After being given the correct
arithmetic to use, the student was able to
find the correct solution.

Decoded the problem correctly. Could not infer
the correct relation of sides even Mdth
coaching. After a diagram was described by

the interviewer, the student remarked that
"Now, you add up all the sides." After

several rereadings and coaching, the student
still could not symbolize the required sum.

The interviewer asked, "Do you know what this
sum would be?" After another reading, "Oh
yeah! You get 18!" The interviewer verbalized
the correct inequality but the student did not
know what to do with it. The student's
explanation of his difficulty with this problem
was, "I never could do story problems."

"Why not?"

"I just can't!"

Student P

PrOblem l6:

PrOblem l7:

PrOblem.18:

100

Decoded the problem correctly except "the
measure of" was simply read as an "m."

Before any suggestions or coaching could be
given, the "m" was decoded as "the measure of."
The student inferred the correct process to be
used for finding the measure of angle b but
could not remember the sum of the measures of
angles a, b and c. After obtaining a correct
measure for angle b, the student had to be
reminded that the problem was not totally
solved. After reading again, the student
verbalized a correct measure for the exterior
angle at angle a. He was not able to shed any
light on why this omission was made, except,
"I was just careless."

Decoded the prOblem correctly. Even though
the student was able to describe the meanings
of perimeter and similar, no solution was
verbalized. It was suggested that the student
make a diagram, and after some hesitation and
erasing, the student sketched a correct
diagram and then verbalized the correct
solution.

Decoded the prOblem correctly. Could not
verbalize the relation of sides. "Picked up"
some numbers, combining them in a way that did
not correspond with the correct relation
between what they represented. With several
coaching questions, the student managed to
label two sides x and 2x but could not
translate "exceed" into "+" even though he
paraphrased "exceed" to "more than." His
translation was 6x. After the appropriate
translation was made for the student, the
student verbalized, "Now, we add up all the
sides to get 18." The student had to be
coaxed into translating this statement into
special symbols, but even then, with the
special symbolization at hand, the student
could not find x.

PrOblem Posing on Pretest and Posttest

The description of the problems posed by the

children on the pretest as compared to the ones posed on

the posttest will be reported as they appeared on the

101
children's papers for Problems 19 and 20. Even though in
solving most problems the skill of problem posing may be
employed, Problems 19 and 20 for both the pretest and
posttest would demonstrate most effectively the students'
abilities for using this subskill and would also demonstrate
any change that may have occurred during the period in
which the reading lessons were given. Besides inferential
reasoning, reading for problem posing could be one of the
most important reading skills to be used in a meaningful

approach to studying mathematics.

Pretest Posttest
19. A faucet is leaking. 19. How many salmon go up
How much water is the river during
being wasted? mating season at the
dam in downtown Grand
Rapids?
20. How many leaves on a 20. How many deer hunting
tree? licenses would you

issue each year in
Michigan if you were
in charge of issuing
such permits?

CONTROL GROUP

 

 

Student B

19. Left Blank l9. thousands
20. Left Blank 20. 1

Student E

19. Left Blank 19. Left Blank

20. Left Blank 20. Left Blank

102
Student G
19. Left Blank

20. Depends on how young
or Old the tree is.

Student I

19. Left Blank
20. Left Blank
Student L

19. Lots

20. 100
§tudent J

19. The amount of water
that is leaking.

20. 36

§tudent C

19. 100 000 000 gallons
20. l 000 000 000 000 000
Student A

19. Enough

20. All of them

§tudent F

19. As much water we
usually drink.

20. If the tree is a
healthy tree it has
more than a sick tree
would.

19.
20.

19.

20.

19.

20.

19.

20.

19.

20.

19.

20.

19.

20.

Left Blank

Left Blank

1000

50

Lots

Put electric eye beam
on every river. They
can count them for
you.

Left Blank

A lot

A lot

A lot

A lot

More than 1000 fish

About 500

W
19. Left Blank
20. 11 000 000
Student D
19. Left Blank
20. Left Blank

Stuggnt M

103

19. It depends on how long

the faucet was
leaking.

20. It depends on how'big
or small the tree is
or if it is dead or

alive.

Student K

19. It depends on how long

it leaks.

20. There's a different
number of leaves on

each tree.
knows.

NObody

EXPERIMENTAL GROUP

Student R

19. [(12 x 60) x 48] x 335

= 11 577 600 drops a

year.

20. Depends on how old the

tree is.

19.

20.

19.

20.

19.

20.

19.

20.

2000 a day

Left Blank

250

800

Depends on how many
fish there are.

150

Left Blank

Left Blank

20 in an hour.
24 x 20 = 580 in a day.

Find the average of
deer in about 15
places.

104
Studentgg

19. I don't know.

20. Depends on the size--
small, medium, large

Student S

19. NOt very much.

20. 1000

Student U

19. Left Blank

20. Depends on hOW'big the
tree is.

Student X

19. Depends how long it
was leaking.

20. Could be 1 or 1
trillion.

Student Z

19. Left Blank

20. Left Blank

19.

20.

19.

20.

19.

20.

19.

20.

19.

20.

First catch some fish
and put tags on them.
If you catch the same
fish throw him back in.
Then move down stream
and do the same thing.
Say you used 100 tags.
You moved in 2 spots.
You divide 2 into 100
and get 50. 50 would
be the average.

You would tag each
deer. If you caught
20 deer, you would
hand out 10 licenses.

It depends on the
season and how clean
the water is.

Depends on how many
deer there is.

It depends what time of
year and how clean the
water is.

It depends how many

deer are in the woods
and the weather.

Depends what time of
year and how cold.
Depends what season.

Depends if it was deer
season.

Left Blank

Left Blank

 

 

 

 

 

 

Student 0

19. Gallons
20. Left Blank
Student N

19. Enough

20. All of them
Student P

19. Left Blank
20. Left Blank
Student W

19. Left Blank
20. Left Blank
Student T

19. Left Blank
20. 100
Student V

19. Left Blank

20.

Left

Blank

105

19.

20.

19.

20.

19.

20.

19.

20.

19.

20.

19.

20.

Depends on the time of
year and how the
weather is.

Depends on how many
deer there is.

If 40 fish go up 24
hours, 120 go up in 3
days.

Same as 19 but with
hunting licenses.

First you would have
to see how pure the
water is. And you
would have to tag a
lot of fish.

Just keep track of how

many you issue. Also
you would tag deer.

Depends on the weather
and how many fish are
there now.

It depends on how many
deer there are.

1000

700

Left Blank

Left Blank

106

 

Student Y

19. Plug it, measure it 19. Left Blank
often.

20. Left Blank 20. Left Blank

Classroom Teacher's Observations

The cooperating classroom teacher in this study was
asked at the onset of the project to be prepared to provide
an assessment of the program through classroom Observations
of the subjects in the experimental group. This evaluation
will be reported by listing the written descriptive comments
made by the classroom teacher periodically during the
experiment.
Oct. 3-—Very positive response. Using calculators fun!
Oct. 5--It's going well because students who came back

enjoyed it. Others who were not chosen wanted to

go also.
Oct. 7--Students very enthusiastic!
Oct. 10——Very positive-—other students wanted to go again!

Oct. l4--Liked learning about angles--congruent figures!

Oct. l7--Enthusiastic! No real improvement in attitude
toward mathematics.

Oct. l9--Liked learning mathematics! Girls liked it! Boys
said it was fun!

Oct. 26--Other students wanted to go also!

Oct. 28--Students enjoy going. Have seen no other
enthusiasm for math in class.

Oct. 31--Student enjoy going!
Nov. l—-Excited!

NOV. 4--This has been a worthwhile experience for the
students.

107
In summary, the teacher's comments indicate very
positive affective results of the reading program in
mathematics, which can somewhat be corroborated by the
remarks written by the children on their papers, such as:
I'm glad I got to go with you.
Hi there! (A smiling face)

Thank you!

Retest for Retention
A ten-item test (See Appendix B.) was given to the
subjects in the experimental group five weeks after the
posttest. The data for this test are recorded in Table 3.
After the tests were corrected, each student was
interviewed for approximately ten minutes in regards to

their performance on this retention test.

Table 3.--Data on Test for Retention

 

StudentNOPQRSTUVWXYZ

 

Score % 85 70 85 70 100 75 75 85 55 100 80 85 60

 

 

The mean percentage score for the retention test
.was 78.8%.and the range was 55%P-100%. The median score
was 80%.

In the interviews, the students were asked to read
orally each of the prOblems, and then the problems missed

were discussed. The decoding errors were minimal. Some

108
of the students decoded aloud "the measure of" simply as
an "m? but it was apparent that in their minds they had
comprehended "m" as "the measure of."

Some of the students, possibly feeling forced to
come up with an answer, made vague leaps into an answer to
some of the prOblems'by performing meaningless arithmetic
operations or by making an unrelated meaningless guess.

Many of the prOblems missed on the retention test
were solved correctly by the students as they talked over
the problems with the interviewer.

The results of interviews in regards to three
problems presenting difficulty for many of the students are
summarized‘below.

Problem 4: Give an example of 3 consecutive whole numbers
if the first one is x.

Incorrect Responses: x, y, x
l, 2, 3
x + l + 2 + 3
Left Blank

Interviews: Most of the students had retained the idea of
"consecutive" but admitted that the "x"
disturbed them "left as an answer." They felt
that they had to give one and only one specific
answer with specific numbers. One student
remembered the word "consecutive" but had
"forgot what it means." After being coached,
he gave the response of "x, y, z."

PrOblem 6: Give an example of a nonnegative number.

Incorrect Responses: Left Blank

Interviews:

Problem 8:

109

"NOnnegative" was the trouble spot. All the
students had retained the idea of ”nonnegative"
as being "not negative" but were not
knowledgeable of or comfortable with negative
numbers to be able to give an example of a
number that did not possess this quality. All
admitted to having remembered discussions of
these numbers in the reading lessons. Most of
the students were able to give examples of
negative as well as nonnegative numbers to the
interviewer. "Why not on the test?"--insecure
"giggles" were the popular responses to this
question.

Given the figure below with the noted data.
Find the m(exterior angle at La).

m(La) = 65° a

 

Incorrect Responses: 20°, 65°, 180°, 30°, 70°

Interviews:

All these students defined "interior" as
"inside the triangle" and "exterior" as
"outside the triangle" and explained their
results on the retention test as, "I just
guessed at it." Two of the students could not
describe precisely which angle should be
considered for measuring even though they
"pointed" to the exterior region immediately
adjacent to angle a. The children who had
responded with "180°" on their tests added
that they had remembered "it has something to
do with a straight angle." After "talking it
over" with the interviewer, two students were
able to effect a correct solution. The six
other students were able to verbalize a
correct solution upon being coached to
consider the problem's relation to a straight
angle.

Summary

Previous instruction in the selected reading

subskills for this study apparently had not produced

mastery learning in these students in their reading classes

as the children seemed rather unfamiliar with them in

110
general at the beginning of the study. Perceptual errors
of omissions appeared more often than decoding errors.
Phonetic and syllabication difficulties seemed minimal for
these children in comparison to the processing errors for
comprehension.

In inferring what mathematical processes are
required in solving a problem, many of the students at the
onset seemed to give no thought that more than one reading
would be necessary--unaware that complete stops might be
called for-—thought time in addition to decoding time.

This situation with the children's reading became rather
apparent during the interviews.

The first set of prOblems for inferential reasoning
did not require a substantial amount of abstract insights—-
just systematic sequential thought as to what the words
were saying. In general, most of the children could not at
the onset infer clearly without coaching the sequential
steps as indicated by the message of the problem. They
were not skilled in organizing information from.the printed
words of mathematics and skipped unsystematically back and
forth through the entire problem.

The results of the posing of prOblems on the pretest
compared to the results of the posing of problems on the
posttest show improvement in the quality of the problems
posed by the children in the experimental group. There
seems to be an improvement in their willingness to attempt

to pose a prOblem from a situation.

111

During the week of the reading lessons for the
subskill of recognizing and verbalizing the special symbols
of mathematics, the students demonstrated their abilities
to learn this skill but were unable with the minimum
training received to use this skill effectively in reading
to solve problems. This skill appears to be important in
solving prOblems as the students were easily thwarted by
these symbols if not recognized immediately. Contextual
analysis was not used to give some meaning to these special
symbols until the students were coached to do so.

In analyzing some of the prOblems not solved on the
posttest but which were solved by some of the students
during the interviews, it was demonstrated that vocalizing
thoughts as relationships are analyzed could be effective
in teaching students to read fOr problem solving.

The statistical evidence is decidedly positive and
encouraging--especially considering the relatively short
period of time between pretest and posttest. The
calculated value of the test statistic does fall in the
rejection region: therefore, the Null Hypothesis of no
difference between the change in the prOblem solving
abilities of the subjects in the control group and the
subjects in the experimental group is rejected. The data
present sufficient evidence to indicate that the subjects in

the experimental group did make significant gains.

CHAPTER V

SUMMARY, CONCLUSIONS AND RECOMMENDATIONS

Summary
This study investigated the effects of explicit

reading instruction in selected reading subskills of word
recognition and comprehensions as described by Duffy and
Sherman(68) on the performance of these subskills in
mathematics and on verbal problem solving abilities of
sixth graders.

The selected reading subskills were:
1. Instant Recognition of Special Symbols of Mathematics

and Their Voiced Equivalents (WOrd Equivalents)

2. Structural Analysis
3. Contextual Prediction
4. Inferential Reasoning
5. Evaluative Thinking

Statistical analysis was used to determine the
effects of the reading instruction on the problem solving
abilities of sixth graders. Observations and interviews by
the researcher were used to describe the impact of the
instruction in these selected reading subskills on the
abilities of sixth graders to use these skills in

mathematics.

112

113
Statistical Results

All thirteen students in the experimental group
made gains in their posttest scores on problem solving.
The range of these gains was from 140% to 525%. The
maximum gain for the control group was 175%.and there were
cases in which students regressed. A t-test indicated that
the gains by the experimental group were significant at the

0.005 level.

Other Results
This study identified and obtained partial answers
to some mathematics-reading-related-questions. These
questions were:

1. What impact does the ability to recognize and voice
special symbols of mathematics have on performance in
mathematics

2. What role does the understanding of structures produced
by word parts play in the learning of mathematics

3. What effect does the subskill of contextual prediction
have on literal comprehension in mathematics

4. What impact does the training to infer logical processes
have on manipulative thinking in mathematics

5. What impact does problem posing from evaluating problem
situations have on evaluative thinking in mathematics

It became apparent through observations and
interviews by the researcher that as the sixth grade
subjects decoded a written mathematical passage into

language and tried to resurrect the intended message, the

114
selected reading subskills played important roles in the
performance of these subskills in_mathematics and problem

solving.

Conclusions

 

Some of the students in the experimental group of
this study claimed that they could read quite effectively
but that they could not read mathematics. With the results
of the observations and interviews, this investigator has
concluded that these students could "read" orally with
apparent facility even in mathematics but, at the onset of
the study, comprehend only a small part of the material
which they read. They seemed to "read" by pronouncing
empty words and apparently felt little responsibility for
knowing what they were reading, apart from some of the
literal facts. As the study progressed, the students
receiving explicit reading instructions in mathematics
began to take on, faintly in some cases, some of the
behaviors of critical readers. This conclusion is
supported by the statistical evidence and the interviews
for problem analysis. By the end of the five weeks of
reading lessons, the children in the experimental group
seemed more honestly aware of their abilities to cope with
complex problems, to delve into a variety of mathematical
concepts and to construct mental images of "things," not
physically seen, through the manipulation of the language

of mathematics--printed and spoken.

115

Interpretation of Mathematical Symbols

Verbal labels seemed to play an important role in
enabling the children to discriminate between concepts.
This investigator has concluded that not being able to
distinguish clearly between the symbol and the concept
represented by the symbol caused some difficulty initially
for comprehension and for problem solving. As students
became more comfortable with the concept, the symbol became
more a part of the concept. Mbst of the students appeared
to remember and.be more comfortable with the geometric
items or with the algebraic items integrated with geometric
ideas as opposed to simply algebraic items. This could
stem partially from the fact that they could'become more
"physically" involved by drawing special diagrammatic
symbols representing these geometry-related ideas.

The special symbols were apparently used easily by
the children if recognized with meaning. If a special
symbol was not recognized at all or recognized without
meaning or without the associated word equivalents, then
comprehension difficulties were encountered. Translations
from.words to special symbols presented more difficulty for
all the children than recognizing the special symbols and

voicing them.

WOrd Mediation through Structural Units
Structural analysis seemed to be effective as well
as affective in a very positive way. The children appeared

to become less fearful of their inability to mediate

116
meanings of mathematical words. The meanings of structural
units as derived from their origins produced a clearly
observable increased student participation and interest.
Developing meaning of a word from its parts possibly lends
a humanizing quality to mathematics and its language, hence

enhancing interest and possibly comprehension.

Contextual Prediction

Contrasting mathematical meanings with
nonmathematical meanings of words made the students more
aware of meaning being obtained from context, as they
seemed to demonstrate more comprehension of mathematical
concepts denoted by these words. At the end of the
contextual prediction lessons, the children in general
seemed to be beginning at least to realize the tasks
involved in "reading" mathematics for meaning. Taking the
sentences apart by defining certain items seemed to be
helpful in developing this realization. The cloze
procedures were apparently effective in promoting not only
critical reading but self-confidence in their own work and
observations and in their abilities to read analytically

and critically.

Inferential Reasoning
The results of this study show that in general
sixth graders can be taught to read mathematics critically
and to make logical inferences based on information

contained in the readings. The inferential reasoning

117
subskill of reading apparently encouraged the children to
organize systematically the relations explicitly stated in

a problem so as to be able to infer the implied relations.

Evaluative Thinking

This study supports the idea that sixth graders in
general can learn to make critical judgements about
statements Which they read in mathematics. The subjects in
the experimental group demonstrated that they could in some
cases formulate mathematical problems from given prOblem
situations. The lessons on posing problems seemed to
encourage the children to pose questions in prOblem solving.
This could partially explain Why the children became more
adept in seeing relations important in making inferences to
solve prOblems. This observation indicates the dependency

and interrelatedness of the reading subskills.

Problem.Solving

Solving prOblems is a task which requires making
sense of and bringing meaning to all kinds of signs and
symbols. Before children can understand what they read,
they must be able to understand the direct meaning of
words, special symbols, phrases and sentences. They must
recognize important details and understand the sequence of
events. The statistical results of this study indicate
that for these sixth graders the particular reading
subskills selected fOr the explicit reading instruction

were important in the reading related to prOblem solving.

118

It is concluded by this researcher that the oral-
verbalizing component of the reading lessons and interviews
was an important, effective and affective component. Some
of the youngsters were initially shy about verbalizing
mathematical thoughts. As the study progressed, all the
subjects improved in their abilities to use mathematical
language. The children clearly became more conscious of
the control which they could have over their own thinking
activities. In many cases, students exhibited enough of
their ideas for the researcher to follow thought paths.

The effectiveness of vocalized problem solving showed up in
the interviews for the retest for retention. It is
concluded by this researcher that the vocalized prOblem
solving technique was effective partly because the children
were able to read mathematics and think more clearly about
the meaning Of the printed words. vague ideas could
apparently be expanded and relations possibly seen more
clearly. The human contact with the researcher in the
vocalized activities could also have been an affective
component.

Besides the vocalizations making an idea more clear
and multiple ideas jell, vocalizing also seemed to provide
an encouragement or an excuse to gesture. Both the
vocalizing and the gesturing apparently aided in
establishing the relations or processes needed to
understand the message of the printed words of a prOblem.

Once the verbalizations triggered the gesturing,

119
communication appeared to be enhanced, and often times, the
children were able to convey their basic intentions to the
interviewer without words. But then, with encouragement,
they were often able to describe these intentions with
words.

The teacher-student dialogue seemed more effective
than the student—student dialogue or the student-whole
group dialogue. In the student-whole group dialogues, the
children did not appear actually to listen to each other.
If one student were reading aloud, it appeared difficult
for the other children to listen courteously or
inquiringly. It is concluded that verbal intercourse with
adults does become an important factor in the development
of a child's concept of mathematical ideas. When this
verbal intercourse was one—to-one, adult-student, the
result was clearly more positive than the teacher-whole
group dialogue.

Being able "to think" in complete sentences as
demonstrated by the students' oral verbalizations of their
thoughts seemed to play an important role in their
comprehension and problem solving. It is concluded from
this study that the sentence is prObably the primary unit
for the expression of thought in mathematics. If the
children talked in single words or phrases, not in
sentences, it was not only difficult for the researcher to
find out what they were trying to say but, in most cases,

the students themselves demonstrated that they were having

120
difficulty with comprehension. The reading lessons
apparently promoted the idea of complete sentences in the
printed word and in the spoken word of mathematics. The
students began to recognize the sentence structure even in

the special-symbolic writings of mathematics.

Recommendations

Since mathematics has (or is) a language--a precise
and efficient language for expressing ideas compactly--a
language to describe succinctly the world around us and how
this world might possibly be in the future—-it seems
important to this writer that children be given the
opportunities and encouragements at all grade levels to
develop their talents and abilities for handling this
language and its uses.

The United States Government is two hundred years
old; the English language is seven hundred years old; and
mathematics is over 4000 years old. The study of
mathematics and its language can help the human being learn
something about thinking itself: how to state prOblems
clearly, sort out the relevant from.the irrelevant, argue
coherently and abstract common properties from many
individual situations. The heart of mathematics is problem
solving--finding order, building models and creating
abstractions. Mathematics with its language seems to be
the ideal arena in which to develop skills in the areas of

information organization, problem analysis and argument

121
presentation. These skills are described by the reading
specialists as being the necessary skills for critical
reading.

In light of the results of this study, it is
recommended that time and effOrt be spent in developing
well-trained teachers Who will be qualified in mathematics
and in the teaching of the reading of mathematics so that
"nonreaders" will be taught how to read if they are
academically and mentally capable. Throughout their
education, language--written and spoken--probably will be
one of the most important ways in which our children will
represent their mathematics, as well as having it
represented for them.

The ability to translate physical imagery into
significant mathematical symbols clearly could be one of
the rarest gifts to be developed by our schools. The
results of this study show that, in general, sixth graders
can successfully deal with mathematical symbolism and can
be taught to read nontrivial mathematics in order to solve
nontrivial problems.

The types of reading required in mathematics are
prObably different enough from those required in courses in
ordinary English to warrant the explicit teaching of skills
necessary to obtain reading objectives in mathematics.
Even the children in the experimental group of this study
began to see that words often passed over quickly in

informational writing or general prose had to be defined

122

precisely and carefully in mathematics. The complex

sentence structures, often ignored, had to be clearly

examined in order to decide on the appropriate logical
inferences.

Reading as an end product may be a generalized
ability but it is suggested from the results of this study
that reading should be taught as a composite of specific
skills related to a specific discipline. This research
suggests that the five subskills selected for this study
ought to be included in a set of reading skills for
mathematics. These five reading subskills were selected as
a representative subset of the many reading subskills
associated with word recognition and comprehension. There
are many other reading subskills which could also play an
. important role in the learning of mathematics for problem
solving and confronting prOblem situations.

In addition, it is recommended from the results of
this study that:

1. This research be replicated with possible improvements
in procedure by enlarging the sample and extending the
study to include numerous and varied populations of
elementary children

2. The contribution to prOblem solving abilities of each
of these reading subskills be investigated by

psycholinguists

123

3. The effects of explicit instruction in other reading
subskills of word recognition and comprehension on the
performance of these skills in mathematics and on
problem solving abilities be studied, using similar
and appropriate procedures on populations of various
grade levels

4. The impact of vocalizing on problem solving abilities
be investigated by psycholinguists and educational
philosophers

5. The language of children of various ages, which is used
in everyday language patterns and the more fOrmal
mathematical sentence models in attempting to grasp
concepts of mathematics be compared

6. The contribution of cue questions on problem solving
abilities be examined

7. Long range studies be done to investigate the effects
of continuous and systematic reading instruction on the
mathematical abilities of children through the elementary
grades

8. Strategies for classroom implementation of explicit
reading instruction in mathematics be investigated by
educationalists possibly in conjunction with classroom
teachers at various grade levels

Being able to communicate in the language of
mathematics--printed and spoken--could be the most important
mathematical activity, second only to individual thinking.

As in the words of Henkin(103):

124

Whatever philosophical view one adopts about the

relation between mathematics and language, it is

evident that the teaching of mathematics involves to

some extent the teaching of new linguistic patterns.

The researcher readily admits that the growth by

the experimental group as shown by the test statistic could
be partially the result of the presence of the researcher
or of the extra time on task required of the subjects in
the experimental group. However, since this study involved
the scientific method, it can be justifiably assumed that

the results of this study are significant and the

recommendations made are appropriate.

APPENDIX A

APPENDIX A

MATERIALS FOR READING INSTRUCTION

Subskill I: Instant Recognition of Special Symbols of

Mathematics and Their voiced Equivalents

Performance Objectives: Given a set of special symbols,

provide word equivalents and voice
them.

Given a word, word phrase or word
sentence, provide special symbolic
representation.

Introduce some of the special symbols of
mathematics, giving their voiced equivalents.
These symbols will simply be used to talk about
mathematical objects and their relationships.
The children will be asked to respond orally to
these special symbols placed on cards and to
construction paper-models by providing
corresponding verbalizations. They will be asked
to produce uses of some of these symbols by
displaying examples on a hand-held calculator or
by diagrams on paper.

in +3 or +3
in 7 + 3
in 9 — 4

in -3 or -3

ab as in a x b (4t)

Lesson 1
l. + as
2. + as
3. - as
4. .- as
5.
6.()
7. / as
8. 2 as

as in 4(2 + 5)
in 2/4

in 32 or 3 as in 43

125

9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.

34.

126

\/F—as in \/f;—'

as in 4 = 8/2

#'as in 1/3 # 0.3
< as in 5 < 6

K as in 6 fl 5

> as in 6 > 5

i as in 5 f 6
SasinxSlO
2 as in x Z 10

S as in 3 S x S 9

< as in 3 < x < 9

% as in 60%

m/s as in metres per second
F as in C = nD

° as in 90° angle

as in 1,2,3,4,...

L as in LA (angle A)

-as in AB

m as in m(AB)

6-9
é-Pias in AB

_|_.. 1. .3123

H as in AB H CD

A as in AABC

circle

35.

36.

37.
38.

39.

40.

 

127

circular region

 

 

a as in

A32

triangle

triangular region

CD

~ as in AABC ~ ADEF

 

4P
3” , (4.3)
2»
l“
25 - . ., . . . . .\.
V ‘7
. 1 2 3 4
W Rectangular
I Coordinates

 

Lesson 2 The children will provide word equivalents for
the symbols below and voice them and will give
examples for some of them by using a calculator.

[3+5

'3 + x
x/2
A”

8.
9.
10.
11.
12.
13.

14.

E'x
4x2

22 + 33
(4:02

5 - (6 + 7)
3E} 4A

29 + 72 = 101

128

 

 

 

15. A >0 27. 60%of 75=x
16. 6<n+3<7 28.E]+2km/h=88km/h
17. [3 s7
18. y < 0 and y > 10 29.
7 4 cm
19. y<00ry>10 .
3 cm

4A-... e
21. 4x-3=5 31. Qca 00'
22. -4Sy <5 32. 111(5): 3 cm
23. 2A- 3 = 15 33. m(LA) = 30°

2 0") 0'9
24. A = nr 34. AB H CD
25. c=n0 35. °XY°_L§?

D 36. O,3,6,9,12,...
26- 3A-T=4

Lesson 3 The children will translate the following word
phrases or sentences into special symbols of
mathematics. They will also be given verbal
commands and asked to respond by using the
correct special symbols on a calculator.

1. The product of seven and a number.

2. The difference of eight and the second power of a
number.

3. Add eight to the quotient of a number and two.
4. A number increased by the square root of nine.

5. The measure of a side of a square is equal to the
square root of the area of the region bounded by the
square.

Eight diminished by the square of a number.
The sum of the nonnegative multiples of three.

Pi is a number between three and four.

melm

A rectangle is twice as long as it is wide.
10. The product of two numbers.

11. The coordinate of the point on a number line halfway
between zero and seven is three and one-half.

12.
13.
14.

15.

16.

17.

18.

129

Two lines intersect to form four congruent angles.
Three coplanar lines do not intersect.

The perimeter of a polygon with sides measuring a, b
and c units respectively.

The area of a rectangular region is six units and its
length is one third of its width.

If a number is decreased by seven per cent, the result
is twenty.

The speed of the car was eighty-five kilometres per
hour.

The sum of the measures of two sides of a triangle is
greater than the measure of the third side.

The children will respond to the following commands by
using the correct symbols on a calculator.

o

N Rare N h-rd P‘P‘ H bald F‘t‘ H
u said :310 u>~q 01L” 9 to N r4 o
O O O O O O O O O O O O I I

seventeen decreased by 9
the product of 52 and 37
the sum of 82 and 49

the fifth power of 2

39 increased by 42

the difference of 98 and 79
the quotient of 39 and 13
the square root of 196

the square root of 225

the fourth power of 3

the product of 22 and 73
the quotient of 94 and 47
89 increased by 102

117 decreased by 83

the difference of 712 and 526
the third power of seven
the square root of 256
seven times as large as 56
the sum.of 412 and 713

the quotient of 2187 and 243
8 diminished by 23

the product of 92 and 24
the quotient of 7434 and 18

130

24. the square of 19

25. the square root of 361
26. the sum of 1832 and 1478
27. half as large as 728

28. the double of 273

29. the triple of 413

30. 173 diminished by 34

Subskill II: Structural Analysis

Performance Objective: Given an unknown content word which
has structural units, identify and
give meaning.

The following three sets of content words will be used to
develop abilities for using this particular subskill for
successful reading in mathematics. Each of the selected
content words will be presented to the children on word
cards, accompanied by the following oral questions:

Can you pronounce this word?
Do you know what it means? or
Does any part of the word look familiar to you?

Fbr some items totally unknown to any of the children and
with interesting units or origins, clues will be given.

Lesson 1
l. ascending 12. diagonal
2. associative 13. diameter
3. binary 14. displacement
4. binomial 15. distributive
5. bisect l6. concentric
6. boundary 1?. concurrent
7. circumscribe 18. congruent
8. circumference l9. coplanar
9. clockwise 20. counterclockwise
10. collinear 21. counterexample
ll. commutative 22. descending

\OCDxlO‘U'IwaI—i

N N to N H H H H H H H H H H
00 N H O \D (D Q 01 U1 9 u N p—o o
O O O I O I O O I O O O I O

24.

131

Lesson 2

decade
decagon
decimal
decimetre
equality
equiangular
equidistant
exterior
factorial
factorization
horizontal
identity
inequality
inscribe
interior
intersecting
midpoint
nonadjacent
nonmetric
nonnegative
nonparallel
nonpositive
nonterminating

nonzero

\OQQO‘UI
I

10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.

Lesson 3

octagon
octahedron
polygon
polygonal
quadrant
quadrilateral
reference
reflection
rotation
semicircle
subscript
subset
substitute
summation
terminating
transformation
translating
transversal
trichotomy
trinomial
trisection

variable

The following content words and their corresponding
origins(9) will be given to the children to promote the use
of structural analysis in reading mathematics.

geometry -

angle —
diagonal -

diameter -

g3 meaning earth

metron meaning measure

angulus meaning a corner

dia meaning through

gonia meaning angle
It passes through the angles.

Greek -
Latin -

Greek —

Greek -

dia meaning through

metron meaning measure
It is a measure through the circle.

132

polygon - Greek - poluz meaning many
gonia meaning angle

isosceles - Greek - isos meaning equal
skelos meaning leg

perimeter — Greek - peri meaning around
metron meaning measure
Measure around a polygon.

circumference - Latin - circum meaning around
ferre meaning to carry
The measure is carried around
the circle.

dimension - Latin - dimensio meaning apart
metiori meaning measure
A rectangle is taken apart into two
measures - length and width.
11 A rectangle is a two-dimensional

figure.
plane — Latin - planus meaning flat or level
circle — Latin - circulus meaning a little ring
fraction - Latin - fractus meaning broken
quadrilateral - Latin - guattuor meaning four

latus meaning side

parallel - Greek - para meaning beside
allelon meaning of one another

rectangle - Latin - rectus meaning right

The children will match each word with the expression below
best depicting meaning of the word. The children will also
be given a list of meanings suggested by certain prefixes
or structural units.

_____1. binomial ____13. diagonal
____.2. bisect ____i4. diameter
.____ 3. circumscribe ____15. equiangular
____ 4. circumference ____16. equilateral
_____5. collinear ____i7. exterior
____ 6. concentric ____18. factor

____ 7. concurrent _____l9. factorization
_____8. congruent ____20. inequality
_____ 9. coplanar _____21. inscribe
_____10. decade ____22. interior
____il. decagon ____23. invariant
_____12. decimal .____24. nonadjacent

133

____25. nonnegative ____34. substitute
____26. nonparallel ____35. quadrant
_____27. nonpositive _____36. quadrilateral
.____28. nonterminating ____}7. translation
____29. nonzero ____S8. transversal
_____30. polygon _____39. trichotomy
_____31. polygonal .____40. trinomial
_____32. subscript _____4l. trisect
_____33. subset 42. variable

IHWNH
no.

10.
11.

12.
13.
14.
15.
16.
17.
18.

19.

20.

 

Division into three parts or classes.
To divide into three congruent parts.
Consisting of three mathematical terms.

A straight line segment joining two nonadjacent
vertices of a polygon.

A Ohord passing through the center of a circle.

To put a word, symbol or number value in place of
another word, symbol or number value.

A distinguishing symbol or letter written immediately
below and to the right of another character.

A mathematical set each of Whose elements is an
element of an inclusive set.

Outside, as the angle between a side of a polygon and
an adjacent side extended.

A quantity that may assume any one of a set of values.

A closed plane figure made with three or more straight
line segments as sides.

A closed plane figure bounded by a polygon.
To divide into two congruent parts.
Consisting of two mathematical terms.

A polygon of 10 sides and 10 angles.

A period of ten years.

A number expressible in powers of ten.

Something that actively contributes to the production
of a result such as any of the numbers that when
multiplied together form a product.

Process or result of finding numbers that when
multiplied together form a product.

Any of the four parts into Which a plane is divided by
perpendicular reference lines used for graphing.

134

21. A polygon with four sides and four angles.
22. The distance around a circle.

23. To encircle.

24. The inside, as the inside region of a polygon.
25. Not equal.

26. Invariable quantity, unchanging.

27. To draw one figure within another.

28. Having all angles congruent.

29. Having all sides congruent.

30. Meeting in the same point.

31. Lying in the same plane.

32. Harmony or correspondence between things, like
geometric figures having the same shape and the same
size, so that they can be made to coincide.

33. Having a common center.

34. Lying in the same straight line.

35. To change over from one medium to another.

36. Lying across or intersecting a system of lines.
37. Lines which intersect on a plane.

38. Net next to.

39. Numbers x such that x S 0.

40. Numbers x such that x 2:0.

41. Numbers x such that x <:0 or x > 0.

42. Not ending.

Meanings suggested by certain prefixes or structural units.

bi- ---- two parts

circum- ---- around

co-, con-, col- ---- with, together, jointly
deca-, dec- --—- ten

di- ---- twofold

dia— ———- through or across

equi- --—- equal

ex- ---- outside, outer

fact -——- a thing done

in- ---- not, into, within

135

lateral --—— side

linear —--- relating to a line
non- -—-- not, an absence of
poly- —--- many

quad- ---- four

quadri- -—-- four

sub- ---- under, below

trans- —--- across, beyond, through
tri- -—-- three

vari- ---- diverse, changeable

The children will read the following commands and respond
with the correct symbols to demonstrate comprehension.

l. Inscribe a triangle in a circle.
. Draw a 90°-angle. Bisect it.

. Draw a line segment. Denote its midpoint.

photo

Draw a plane region with a quadrilateral as a
boundary. Shade in the interior region of this
figure.

Circumscribe a circle about a triangle.

Draw a circle and its diameter.

Draw a quadrilateral and its diagonals.
Sketch an ggpilateral triangle.

\DmQO‘U'I
0

Sketch an pguianqular triangle.

10. Sketch an angle rotated cggpterclockwise.

11. Draw three concurrent lines.

12. Sketch a nonsguare rectangle.

13. Sketch two congruent squares.

l4. Sketch five collinear points.

15. Draw 2 nonparallel lines cut by l transversal.

Subskill III: Contextual Analysis

Performance Objective: Given a set of sentences (one or
more) in which unknown words or
special mathematical symbols are
present, supply each with meaning.

136

Lesson 1

Part

Part

N
o

(DQO‘UI-bw
o

1: The children will make a sentence, using the
underlined word in each sentence, in a
mathematical context.

Bring the clothes line into the house.

The Democrats are now in ppwer in the federal
government.

The intersection of Baldwin Avenue and 20th Street is
not far from here.

 

The sun's rays are warm in July.
These chemicals form an explosive solution.
My father likes prime rib for dinner.

In washington, D.C., the avenues form transversals for
the streets.

The Capital building in washington, D.C. is the center
from which the avenues radiate.

What is your angle?
The area of knowledge in medicine is growing.

2: The children will explain what the underlined
vocabulary items mean in the following sentences.

The difference between the length and width of a
rectangle is 10 inches and a diagonal of this
rectangle is 12 inches. Find the dimensions of the
rectangle.

Find the area of an egpilateral triangle Whose side is
one unit.

Multiply the binomials (10 + 3)(10 - 3).

Is the ordered pair (2,3) a solution to 3x + 4y = 18?
One-third can be written as a nonterminatinggdecimal.
Draw a polygonal region with a triangular boundapy.
Draw three different concentric circles.

0-9 '9
Consider the horizontal ray AB below. Draw ACo
representing a counterclockwise rotation of 90 .

‘7. i3 9
Three hundred forty and sixty three hundredths written

in decimal notation is 340.63, which means

3(102) + 4(101) + 0(100) + 6(10’1) + 3(10'2). This
notation shows groups of tens in descending powers.

 

 

10.

137

Consider the set of four ordered pairs of numbers
below as determining the consecutive vertices of a
quadrilateral. Locate these four points on a graph
and sketch the quadrilateral. Does this quadrilateral
appear to be a parallelogram? Why?

(2.1). (6.2). (4.3) and (3.4)

Lesson 2 The children will complete the following

paragraphs by using the Cloze Procedure. They
will be instructed to read silently the entire
cloze first.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Carla looked at the on a shelf. She
, "There are six boxes
each row. There are rows. How many
boxes there in all?"
Jay :
3 rows of 6 is equal to 1
boxes.

3 x 6 l8
fa:E;r fa:I:r /T\
The perimeter of a is defined as the

of the lengths of
sides. Since a circle not ordinarily
called a , the distance around it

 

its circumference. Perimeter is

 

by the number of units

 

it contains. The of any polygon can
found by adding the of

 

the sides. If knows that a figure
a polygon and that has

 

all congruent sides, the perimeter can

be by multiplying instead of

 

138

Lesson 3 The children will complete the following
paragraphs by using the Cloze Procedure.

1. A "dip" in the is the opposite of
"rise." A temperature "below

 

" is the opposite of

 

 

temperature "above zero." A " " is the
opposite of "surplus." "Below the sea"
the opposite of "above

 

 

 

 

 

 

 

 

 

 

 

 

 

 

sea." A "loss" is opposite of a
"gain." " . C." is the opposite
"A. D." Can you of
other opposites to to this list?
2. The object of the of Geometry is to
about the properties of
lines, planes, triangles, squares, ,
circles and other figures up of points.
Just the aim of algebra
to learn about the of sets of numbers,

 

aim in studying geometry

 

 

to learn about the of sets of points.
numbers, geometric points and

 

 

 

 

 

 

of points such as ,
planes and circles, are concepts, not
concrete objects. helps his thinking
by points by dots, lines

 

strokes, and planes by surfaces.

Subskill IV: Inferential Reasoning

Performance Objective: Given a story problem, determine
the mathematical processes or
operations to be used. (This one
type of inferential reasoning is
not synonymous in all aspects to
the inferential reasoning
associated with classical
mathematical logic.)

139

In each of the three sets of exercises, the children will
explain how they would solve the following problems.

Lesson 1

1.
2.
3.

4.

10.

11.

12.

Find the sum of 142 and 48.
What is the product of 14 and 21?

An airplane flew 3600 miles in 16 hours. What was its
average speed per hour?

At the start of a vacation Mr. Watson's car
speedometer registered 39 624 miles. When he returned
home it registered 42 383 miles. How far had he
driven?

Mr. Moore rents an apartment for $225 per month. How
much rent does he pay in a year?

John wants to buy a bicycle for $95.75. He has $28.98
in savings and he received $5.00 for his birthday.
How much more money does he need?

One Saturday Bob caught six fish. They weighed 2 1b.,
3i 1b., 1% 1b., 2—1— 1b., 1 lb. and 3%; lb. What was
their average weight?

What is the perimeter of a rectangular flower bed that
is 25 feet long and 12 feet wide?

Approximately how far would a jet plane fly in 12
hours if its average speed was 550 miles per hour?

If the middle of three consecutive Whole numbers is y,
the first is and the third one is

 

Each triangle in this figure is equilateral. Name as
many pairs of congruent triangles as you can.

A
AVA

A quadrilateral ABCD is inscribed in a circle.

m(LA) = 8x, m(LB) = 7x, m(LC) = 2x and m(LD) = 7x. If
theasum of the interior angles of a quadrilateral is
360 , find the measures of the four angles of this
quadrilateral. Sketch a diagram.

140

Lesson 2

1.

10.

11.

12.

Kate Johnson bought a $44.78 plane ticket for herself
and one for her niece at half price. How much did she
have to pay altogether for the tickets?

In a certain city normal rainfall for July is 4.01

inches. One year 3.17 inches of rain fell in July.
How much below normal was the average rainfall that
year?

Ruth's uncle went by plane from New York to Honolulu,
5051 miles, then to Juneau, Alaska, 2825 miles, and
back to New York, 2874 miles. How far did he travel
in all?

How many square yards are there in the area of a
circular flower bed that is 50 feet in diameter?

The closest that the moon comes to the earth is about
221 000 miles. About how long would a rocket traveling
26 000 miles an hour take to reach the moon at this
distance?

Sound travels about 1100 feet per second. Suppose you
hear the thunder 5 seconds after you see a flash of
lightning. How far away is the lightning?

One aspirin tablet weighs about 325 milligrams. How
many aspirin tablets weigh 4875 milligrams?

One barrel of crude oil produces about 72 litres of
gasoline. How many barrels are needed to produce
11 160 litres of gasoline?

A golfer had a total score of 288 in a 72 hole
tournament. What was the golfer's average score per
hole?

Theoexterior angle at vertex C of AABC has a measure of
105 and m(LA) = 60 . Find m(LB), if the sum of the
measures of the interior angles of a plane triangle is
180°.

Suppose AABC has two angles with equal measures and
the third angle is twice as large as either of the two
congruent angles. Find the measures of the angles of
this triangle.

Consider the case in which each side of an equilateral
triangle is doubled. Describe what happens to its
perimeter by identifying the factor by which it is
multiplied.

141

Lesson 3

1.

11.

12.

If the sun is 93 million miles away from the earth and
if light travels about 186 000 miles per second, how
many minutes does the light from the sun take to reach
the earth?

The sun's diameter is about 864 400 miles, which is
about 100 times greater than the diameter of the earth.
What is the diameter of the earth?

Uranus Was discovered in 1781. How many complete
revolutions has it made since then?

What is the speed in land miles per hour of a ship
going 20 knots?

An empty ship weighs 10 000 tons. How many tons of
water does it displace? The ship is loaded with 3000
tons of fuel and cargo. What is the total weight of
the water it displaces?

How many years would it take an airplane flying 500
miles per hour to reach the sun?

How many Saturday nights will there be by the time you
are 100 years old?

If you drove around the equator of the earth, how long
would a round trip take?

How much does the air in your classroom weigh?

If the perimeter of a triangle is 18 decimetres and
one side is twice another, is it the case that the
only possible whole number lengths of the sides are 4,
6 and 8 decimetres? Explain.

How high is a flagpole which casts a fifty-foot shadow
when a man six feet tall casts a 7i~foot shadow?

Describe how you would mathematically find the largest.
possible diameter of a circular table top which can be
carried through a doorway 7 feet by 4 feet. Between
what two whole numbers do you think this diameter
would lie?

Subskill V: Evaluative Thinking

Performance Objective: Given a situation, pose a prOblem.

The following three sets of situations will be given to the
children. The children will pose a problem from each
situation.

142

Lesson 1

Mike spelled 40 out of 50 words correctly in a spelling
test. Jim took the same test and made 5 mistakes.

 

 

2. Jane's father finds that his car travels 16.75 miles
per gallon of gasoline.

3. Mr. Cartwright bought a city lot 75 feet wide and 99
feet deep.

4. Douglas bought 4 pairs of socks for $4.76.

5. Don gained 5.25 pounds in 6 months.

6. On a 20-word spelling test Jane missed 3 words.

7. A gallon of water weighs 8.33 pounds.

Lesson 2

1. Since pioneer days, wild plants and animals have been
disappearing in our country.

2. The first United States manned suborbital space flight
reached a height of 601 920 feet.

3. The total cost to a club of 160 members for a charter
flight to Europe was $62 880.

4. A school has 3538 students and 122 teachers.

5. Most of our rain and snow comes from water that was
once in the oceans.

6. Turning off electric lights that are not needed can
help conserve fuel.

7. During a thunderstorm, the lightning is always seen
before the thunder is heard.

Lesson 3

l. The average depth of the soil is from 3 to 6 feet:
200 000 acres of good soil are washed into rivers
every year and 500 to 1000 years are needed to form a
layer of good soil 1 inch deep.

2. Much of the soil along the lower Mississippi River
comes from higher land far to the north.

3. The bottoms of some of the Great Lakes are below the
level of the ocean.

4. You have a block of oak wood and a block of pine wood.

5. Good eggs sink to the bottom of a jar of water, but

spoiled eggs float.

143

6.. The water in the ocean is salty.

7. When a boy tried to fly his kite, he found that it
would not go up in the air.

APPENDIX B

7.

APPENDIX B

EVALUATION INSTRUMENTS

Pretest

Name

 

Draw a quadrilateral with nonparallel opposite sides.

Give an example of three numbers x, y and 2 Which
satisfy the given data: x > y and y = 2.

Write the next five terms of this sequence:
0, 1, 0, O, l, 0, 0, 0, l, ...

If the quotient of two numbers is 1, what would be
their difference?

If m/3 represents a whole number, then describe the
relation of m to 3.

Consider the figure below. What would be true if CD
were an angle bisector?
C

 

 

A B
D

In the figure below, if m(LBXA) = 40° and m(LCXB) = 503
find the number of degrees in LCXA. Draw XE such that
m(LAXD) = 180‘.

 

x g 9

144

10.

11.

12.

13.

145

Draw a diagram of three collinear pointstA, B ppd C.
Whpp seems to be the relation between m(AB), m(AC) and
m(BC)?

Examine the diagram below. What are the specific
numbers represented by the symbols x and y if the
point labeled (x,y) is the midpoint of AB?

 

 

y-numbers

5’? B
4 .
3 . (my)

2 .

1 h A

<: c - e - ‘ :> x-numbers
l 2 3 4 5
17

Consider the figure bGIOW'With the given data. Find
m(Lb) and the measure of the exterior angle at La.

m(La) = 80°

nflLc) 50°

 

Draw a figure showing a circle circumscribed about a
square. If the radius of the circle is 4 inches, find
the length of a diagonal of the square.

A certain polygon has a total of five diagonals. How
many sides does this polygon have? Sketch a diagram
of this polygon.

Mr. Jones buys stones to cover a rectangular patio of
160 square feet. If the patio may be between 8 and 10
feet wide, what limitations are there on its length,
assuming that Mr. Jones uses all the patio stones?

14.

15.

16.

17.

18.

19.

20.

146

a. The number twenty-four (24) has eight positive
whole number factors. Find these eight factors of
24.

b. There are four different-shaped rectangular
regions of area 24 and with whole number lengths
and widths. Identify these four rectangular
regions by describing the dimensions of the
boundary of each region.

The measures of the sides of a triangle are 8, 10 and
12 feet respectively. Find the perimeter of a similar
triangle whose longest side is 84 feet.

One side of a triangular flower bed must exceed twice
a second side by 5 feet, while the third side must be
twice the second side. If the perimeter of the flower
bed is to be at least 15 feet, find the smallest
possible measure of the longest side.

Draw at least four different quadrilaterals, each
having all sides congruent and at least two of them
being equilateral and equiangular. Draw each figure's
diagonals. What seems to be the common feature of
each set of diagonals?

 

If y = x E 1 and if x increases from 2 on, what

happens to y? (Remember x and y represent numbers.)
Does it increase or decrease? Does it appear to be
approaching a specific number as a limit?

A faucet is leaking. How much water is being wasted?

How many leaves on a tree?

we
Name

Give an example of three consecutive whole numbers if
the first one is x.

Write the next six terms of this sequence:
2: 0: 20 or 00 2: O: 0: OI 2: or or 0: OI 2: 00-

If the quotient of two numbers is 1, what would be
their difference?

If y/5 represents a whole number, then describe the
relation of y to 5.

147

5. Sketch a quadrilateral with nonparallel sides.

6. Consider the diagram.below; What conclusion can be

drawn if CD were an angle bisector?

 

w

7. Draw a diagram of three collinear points, 5‘ Y ang_Z.
What appears to be the relation between m(XY), m(YZ)
and m(XZ)?

8. Examine the figure below. What are the specific
numbers represented by the symbols x and , if the
point labeled (x,y) is the midpoint of AB?

x =
y = 4\ y-numbers
5 db
4 «P
B
3 " (4.3)
21' (my)
1.. A
(2.1)

 

 

x-numbers <F-—A— : : ; : :
\I/ 59

9.

10.

11.

12.

13.

14.

15.

148

In the diagram below, if m(LAXB) = 30° and
m(LBXC) = 60°, find the number of degrees in LAXC.
Draw XL such that m(LAXD) = 180°.

4\.

cl)

 

 

x : 9

Draw at least four different quadrilaterals, making at
least two of the quadrilaterals equilateral and at
least two of them equiangular. Draw each
quadrilateral's diagonals. What seems to be the
common feature of each set of diagonals?

Circumscribe a circle about a square. If the length
of a diagonal of the square is 8 inches, find the
length of the radius of the circle.

 

If y = 1 and if x increases from 2 on, does y
x - 1

increase or decrease? Does y appear to be approaching
a specific number as a limit?

a. The number twenty—four (24) has eight positive
whole number factors. Find these eight factors of
24.

b. The number of different-shaped rectangular
regions of area 24 and with whole number lengths
and widths is 4. Identify these four rectangular
regions by describing the dimensions of each.

Mr. Smith buys stones to cover a rectangular patio of
160 square feet. If the patio may be between 8 and 10
feet wide, what limitations are there on its length,
assuming that Mr. Smith uses all the patio stones?

A certain polygon has a total Of five diagonals. How
many sides does this polygon have? Sketch a diagram
of this polygon and its diagonals.

16.

17.

18.

19.

20.

149

Consider the figure below with the given data. Find
m(Lb) and the measure of the exterior angle at La.

 

 

m(La) = 70° m(Lb) =

m(Lc) = 50° m(exterior angle at La) =
c

l a b

.\ -

 

The measures of the sides of a triangle are 4, 6 and 10
feet respectively. Find the perimeter of a similar
triangle whose longest side is 60 feet.

One side of a triangular flower bed must exceed a
second side by 6 feet, While the third side must be
twice the second side. If the perimeter of the flower
bed is to be at least 18 feet, find the smallest
possible measure of the shortest side.

How many salmon go up the river during mating season
at the dam in downtown Grand Rapids?

How many deer hunting licenses would you issue each

year in Michigan if you were in charge of issuing such
permits?

Test for Retention

Name

 

Sketch a quadrilateral.
Draw two nonparallel line segments.
Draw an angle whose sides are perpendicular.

Give an example of 3 consecutive whole numbers if the
first one is x.

10.

150

Consider the diagram below. If m(LAXB) = 40° and
m(LBXC) = 30°, find m(LAXC).

(I B

X e \

Give an example of a nonnegative number.

Draw 2 lines which are parallel and then draw a third
line which is a transversal for the first two lines.

Given the figure belOW’With the noted data. Find
m(exterior angle at La).

0

m(La) = 65 .

If_§b'bisects the following lipe segment and if
m(AB) = 6 units, then find m(AC).

TD

A C B

Sketch a figure to denote that a circle is
circumscribed about a triangle.

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