SELECTED PIAGETIAN TASKS AND THE ACQUISITION
OF THE FRACTION CONCEPT IN REMEDIAL STUDENTS
BY
ROBERTA LEA DEES
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1980
Copyright 1980
by
Roberta Lea Dees
ACKNOWLEDGMENTS
I wish to acknowledge the inspiration of Jean Piaget,
whose productive life ended in September, 1980.
I wish to thank my committee members: Dr. Mary Grace
Kantowski, who understood what I wanted to do; Dr. John K.
Bengston, without whose assistance and support I could
never have done this; Dr. Donald H. Bernard; Dr. John F.
Gregory; and Dr. ArthurJ. Lewis.
I am grateful to the other Dees women, Jennifer,
Sarah, and Suzanna, who always had faith in me.
I also wish to thank my friend, Sidney Bertisch, for
his help and encouragement.
TABLE OF CONTENTS
PAGE
ACKNOWLEDGMENTS iii
LIST OF TABLES IN TEXT vii
LIST OF FIGURES viii
ABSTRACT ix
CHAPTER
ONE INTRODUCTION 1
TWO RESEARCH REVIEW AND RATIONALE 7
The Content: Fractions 8
Fraction Hierarchies 8
Interpretations of Rational Numbers 15
The Learner 24
Piaget's Theory 25
Disadvantaged Students 41
The Interaction of the Learner and the
Content 42
Piaget's Fractions 43
Assessment of Students' Knowledge
of Fractions 51
Diagnostic and Prescriptive Teaching 56
Clinical Study 64
Related Piagetian Research 65
Concrete versus Abstract Modes of
Presentation 85
Rationale 93
The Student's Notion of Fraction 94
Concrete or Manipulable versus Pictorial
or Written Presentations 102
Clinical Methodology Used 103
Question 1 104
Question 2 105
THREE PILOT STUDY 106
Subjects 106
Instruments 106
Piagettype Tasks 108
Fractions Tests 110
Procedure 112
Findings and Discussion
Piagettype Tasks
Concrete Fractions Test
Written Fractions Test
General Observations
Resulting Modifications
Tasks Instrument
Fractions Tests
FOUR MAIN STUDY
Subjects
Instruments
Tasks
Fractions Tests
Procedure
Testing
Scoring
Planned Data Examination
FIVE FINDINGS OF MAIN STUDY AND DISCUSSION
Findings
Overall Data
Possible Relationships, Questions ]
and 2
Qualitative Data from Student Protc
Discussion
Overall Data
Possible Relationships, Questions I
and 2
Qualitative Data from Student Protc
Limitations of the Study
SIX IMPLICATIONS FOR RESEARCH AND FOR TEACHING
Implications for Future Research
Question 1
Question 2
Other Issues
Suggested Research
Implications for Teaching
D TASKS (REVISED)
E CONCRETE FRACTIONS TEST (REVISED)
F WRITTEN FRACTIONS TEST (REVISED)
G TASK RESULTS
H CONCRETE FRACTIONS TEST RESULTS
I WRITTEN FRACTIONS TEST RESULTS
REFERENCE NOTES
REFERENCE LIST
BIOGRAPHICAL SKETCH
TABLES IN TEXT
Page
1. Sample of Results of Lankford Study 58
2. Success in Area Subtasks 124
3. Percentage of Students Successful on Tasks 143
4. Success on Subtasks of the Tasks 145
5. Success on Concrete and Pictorial Forms of Tasks 146
6. Percentage of Students Successful on Sections
of Fractions Tests 147
7. Success on Items of the Concrete Fractions Test 148
8. Success on Items of the Written Fractions Test 150
9. Percentage of Students Successful on Sections
of Fractions tests by Test Sequence,
ConcreteWritten and WrittenConcrete 151
10. Average Percentage of Students Correct Per
Item by Model of Fraction 165
FIGURES
1. Walbesser Contingency Table
2. Task I and Fractions Section A
3. Task V and Fractions Section B
4. Task IV and Fractions Sections A, B, and C
5. Task IV and Fractions Section A
6. Conservation of Number, Concrete and Pictorial
7. Seriation, Concrete and Pictorial
8. Classification, Concrete and Pictorial
9. Class Inclusion, Concrete and Pictorial
10. Conservation of Distance, Concrete and Pictorial
11. Conservation of Area, Concrete and Pictorial
12. Concept of Fraction (Discrete Model), Concrete
and Written Forms
13. Concept of Fraction (Area Model), Concrete and
Written
14. Equivalent Fractions, Concrete and Written
15. Frequency Distribution of Scores on Classifica
tion, Subtask A and Subtask B
viii
Page
152
154
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155
156
157
157
157
158
158
158
159
160
160
179
Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
SELECTED PIAGETIAN TASKS AND THE ACQUISITION
OF THE FRACTION CONCEPT IN REMEDIAL STUDENTS
By
Roberta Lea Dees
December, 1980
Chairperson: Mary Grace Kantowski
Major Department: Curriculum and Instruction
This clinical study was designed to answer two ques
tions: 1, is there a relationship between the acquisition of
cognitive structures, as exemplified in Piagettype tasks,
and the acquisition of the fraction concept; and 2, is
there any difference between the concrete or manipulable
and the pictorial or written modes of presentation in as
sessing students' knowledge?
Three instruments were developed. The first was a set
of tasks, similar to those used by Piaget to test for the
cognitive structures thought to be related to the concept
of fraction: conservation of number, seriation, classifica
tion, class inclusion, conservation of distance, and conser
vation of area. Tasks were prepared in concrete or mani
pulable and pictorial forms.
The other two instruments were fractions tests, one
concrete or manipulable, and one written, containing
parallel sections on the concept of fraction (discrete, num
ber line, and area models) and equivalence and comparison of
fractions. The written test also included addition and sub
traction of fractions with like denominators.
A pilot study was conducted with four students at
Gainesville High School, Gainesville, Florida, in summer,
1979. The main study was done in spring, 1980, with 10 girls
and 15 boys in the tenth, eleventh, and twelfth grades (me
dian age 16 years), who were enrolled in the compensatory
mathematics classes in Eastside High School, Gainesville,
Florida.
Tests were administered individually; interviews were
recorded. The tasks were administered first. The two frac
tions tests were given on the next available day, 12 students
taking the concrete form first and 13 taking the written
test first.
In general, students scored very low. No students
were successful on conservation of area tasks; 8% were
successful on classification tasks. The best scores were
56%, conservation of distance; 44%, seriation; and 36%,
conservation of number.
No student passed all sections of either fractions
test. Three students passed both forms on concept of
fractions, discrete model. On the concrete form, scores
were better on the discrete and area models of the concept
of fraction (39% and 56%, respectively, being the average
percentage of students correct per item in those sections)
than on the number line model (average of 14% correct per
item). Performance was poor on the equivalent fractions
section (average of 19% correct per item); no student could
do the comparison of fractions task.
On the written test, results were similar except on
equivalent fractions: 2 students (8%) passed the section,
and 7 other students (28%) answered 3 out of 4 items
correctly, apparently using a reducing algorithm. Six stu
dents could add and subtract fractions, but were incorrect
on many items related to the concept of fraction.
To answer the two main questions, data were examined
using Walbesser contingency tables. No strong trends were
evident, but there were some patterns. Students who could
conserve number performed more satisfactorily on the dis
crete model of fraction. Students performed better on the
concrete form of the class inclusion task than on the pic
torial (56 to 8%).
Students taking the concrete form of the fractions
test first were more successful on the written test than
those who took the written test first, indicating that
learning may have occurred during the administration of
the concrete test.
CHAPTER ONE
INTRODUCTION
Students who enter secondary school have all received
some instruction in basic mathematical concepts and skills.
In spite of having received instruction, there exists a
group of secondary school students who apparently have
not learned these concepts and skills.
The majority of students are able to master operations
with whole numbers, except possibly long division. The
introduction of fractions signals the beginning of a
dramatic separation of students into those who succeed in
mathematics and those who do not. Many frustrated educators
have seen the advent of the handheld calculator as salva
tion for those students who cannot operate with fractions;
they say that students will ho longer be blocked from
further progress in mathematics because they can not remem
ber when to "invert," etc. Others feel that real progress
will still be missing unless students have a basic under
standing of the concept of fraction, on which the concepts
of decimals and percentages are logically based.
Those responsible for educating the masses realize
that there is a lower end to every distribution; a normed
test will always have Stanines 1 and 2. But in recent years,
there has been an attempt in many states to agree upon
specific mathematics competencies for all students. These
competencies would become the minimum achievement required
1
for graduation from high school. This idea represents
a considerable advance over the old "grading on the curve"
method, in which the person who did the "worst" automati
cally failed. Competency testing also has its imperfec
tions. Nevertheless, only after competencies were chosen
and tests administered did educators become aware of the
large number of students who had not mastered these mini
mum skills. The finding was consistent with the results
of the 197273 mathematics assessment by the National
Assessment of Educational Progress (NAEP).
In many secondary and adult schools, compensatory
or remedial instruction is scheduled for students who have
not mastered basic skills. Often smaller classes are ar
ranged; teaching aides or assistants are sometimes provided
to lower the studentteacher ratio even more. However,
the instruction appears to consist of essentially the same
strategies that did not produce competence in previous
years of schooling. Prejudicing chances for success,
some schools assign their beginning teachers to these
compensatory classes, the more experienced teachers being
given the brighter students and the college preparatory
subjects. Compounding the dilemma still further is the
3
current shortage of mathematics teachers; teachers teaching
"out of field" are often assigned to compensatory classes.
It seems evident that it is not a simple task to
compensate for the learning not yet achieved in eight,
nine, or ten years of school; it will require experience
and knowledge both of the subject matter to be taught and
of how students learn. It will probably not be accomplished
by repetitious drill or by more of the same instruction
that was not successful in the past.
Fifteen years of teaching in various school settings,
producing some success with these challenging students,
have given this investigator a basic belief, which is as
follows: To remediate or compensate mathematical deficien
cies, first, one must start where the student is; secondly,
the student usually needs a handson or manipulative
approach to discover concepts for himself or herself. This
study provides an opportunity to begin to see whether these
ideas can be substantiated.
One purpose of this exploratory study is to investigate
why certain secondary school students have not mastered a
specific portion of mathematics content, the fraction
concept. What can be learned about students' understanding
of fractions could eventually lead to more successful in
structional strategies; first, it is necessary to identify
what they know. Knowledge of where the individual student is
4
with respect to a mathematical concept has not been gained
from standardized tests. More sensitive, individual
testing, such as that used in clinical studies, is needed
to reveal the student's thinking about mathematics.
The secondary school student has through the years
accumulated some information and misinformation about
operating with fractions. Some mathematics educators
attempt to isolate and identify the errors, and then
remediate them. But such a method may not be successful
if the student does not have a firm understanding of the
concept of fraction on which to base the operations. What
is the necessary foundation?
Two things are needed: better diagnosis of what a
student knows now about mathematics, and knowledge of what
basic structures are required for the student to be able to
learn mathematics. In trying to meet the first need,
better diagnosis, one is led to the methods of Jean Piaget,
who pioneered in the use of the clinical interview to try
to understand children's thinking. His results in turn
lead to possible insights into the second need. For in
Piaget's work with young children, there are described
behaviors remarkably like those observed in secondary
school students who were having trouble with fractions.
For some reason the cognitive development of certain
students has been delayed, so that they do not have the
cognitive structures often assumed to be present in all
secondary school students. The question arises: is this a
coincidence, or are the two deficiencies related? Could
their tardy cognitive development be the cause of some
students' difficulties in learning mathematics? It is not
surprising to find that students who are behind in one
academic area are slow in something else as well. But if
specific Piagetian concepts are found to be related to the
specific mathematical concepts to be taught, the finding
could be very helpful to those who diagnose student defi
ciencies and plan instruction. In this study an attempt is
made to identify basic cognitive structures necessary for
an understanding of the fraction concept.
Another interest of some mathematics educators,
especially those involved in elementary school mathematics,
is the laboratory method of teaching mathematics. The
method is not new; the resurgent interest can probably be
traced to the late sixties and the Nuffield Project in
England. The use of concrete or manipulable objects in the
learning of mathematics is fairly well established in
elementary schools, but is rarely seen in secondary schools.
A second purpose of this study is to consider whether there
may be any justification for the use of these materials
in secondary schools.
6
In a specific portion of mathematics content, the
concept of fraction, this study will explore the following
two questions:
Question 1. Is a particular level of cognitive
development, as indicated by performance on certain
Piagettype tasks, prerequisite to the student's acquisi
tion of the fraction concept?
Question 2. Does the mode of presentation of mathema
tics content, concrete or symbolic, make a difference in
students' performances? Or are students who are successful
at a task in one format always able to perform the same
task when it is presented in the other manner?
CHAPTER TWO
RESEARCH REVIEW AND RATIONALE
In designing instruction on fractions, an educator
must consider the goal, or what the student is to learn
about fractions, and the present status, or what the student
already knows about fractions. The teacher's instructional
plan is the strategy that is expected to effect movement
from the present status toward the goal.
Mathematics educators have studied the instructional
process from both ends. In this section, the goal, the
mathematics content to be taught, will be considered first.
Pertinent research about the student's acquisition of the
fraction concept will be reviewed. Next, attention will be
given to the learner's competence with respect to the goal.
The congitive development theory of Jean Piaget will be
presented as the theoretical framework for describing the
development and knowledge of the learner with respect to
fractions.
The third section will consider the interaction between
the learner and the mathematics content. Related aspects
to be discussed include Piaget's study of how children
understand fractions and educators' attempts to assess the
learners' present knowledge of fractions. Current research
trends to be reviewed include diagnostic and prescriptive
studies; studies of students disadvantaged in mathematics;
studies employing a clinical method; and studies which
attempt to apply Piagetian theory to education. The ways
in which students respond to the concrete and symbolic modes
of presentation of mathematics content will also be
examined.
The Content: Fractions
In considering mathematics content, not everyone agrees
that the understanding of, and computation with, fractions
are worthy goals. In an article on the metric system and
mathematics curriculum, for example, Sawada and Sigurdson
(1976) suggest that common fractions should be studied only
at the conceptual level, and that decimal numeration should
receive major attention.
Fraction Hierarchies
Those researchers who do select fractions as mathematics
content to be taught often perform task analyses after the
work of Gagne (Gagne, Mayor, Garstens, and Paradise, 1962).
A specific concept or skill is analyzed by its subconcepts
or subskills. A learning hierarchy, or a network of par
tially ordered subconcepts or subskills, is developed on
the basis of logical relationships (Gagne, Note 1). The
assumption is that if the hierarchy is valid, it gives a
sequence, perhaps optimal, for teaching the component parts
of the concept of skill. However, it appears that these
"expert" generated learning hierarchies are not equivalent
to student generated learning hierarchies with the same
terminal behavior (Walbesser and Eisenberg, 1972). A task
analysis technology, described by Resnick (1976) and
Walbesser and Eisenberg (1972), has been developed to test
the validity of a hierarchy with students, to find out
whether the various hypothesized dependencies of the hierar
chy are supported.
Greeno (1976) is concerned with showing how psychologi
cal theories might be used in formulating instructional
objectives. He has attempted to identify the "cognitive
objectives" needed to produce the desired outcome behaviors.
Presenting his work as a serious proposal about what the
goals of instruction are, he says,
It may be that when we see what kinds of
cognitive structures are needed to perform
criterion tasks, we will conclude that
something important is missing; but if that
is the case, it also will be important to
identify a more adequate set of criterion
tasks in order to ensure that instruction
is promoting the structures we think are
important. (p. 124)
Adding fractions was Greeno's first example. He
constructed a "procedural representation," or a flow chart,
for adding fractions. Recognizing that finding equivalent
fractions is necessary both before and after the actual
addition, he looked at three different models, or procedures,
for finding equivalent fractions. The first is based on
"spatial processing" of a region (or an area model). The
second is a "settheoretic" (or discrete) model. And the
third is simply an algorithm, "operating directly on numeri
cal representations" (p. 133). Had he extended his reason
ing one step further, he might have wondered what
10
understanding of area a child would need to be able to use
the spatial processing model, or what concept of number, to
use the discrete or algorithmic model.
Uprichard and Phillips sought to generate, then vali
date, a hypothesized hierarchy for adding (1977) and sub
tracting fractions (Note 2). The authors intended to give
consideration "to both psychological and content (discipline)
factors" in identifying and hierarchically ordering tasks.
The procedures for both studies were essentially the same.
Fraction addition and subtraction problems were divided into
two levels, those with like and unlike denominators. Within
each level, classes were identified by the nature of the
denominators, prime or composite, and the nature of the
relationship between the two denominators. Further, there
were sum or difference categories, depending on various
renamings required. Both studies were done with students in
grades four through eight; the majority were in the fifth
and sixth grades. Items were compared by two methods, the
Walbesser (Walbesser & Eisenberg, 1972) contingency table,
with ratio levels of acceptability as determined by Phillips
(1972); and pattern analysis, after Rimoldi and Grib (1960).
The end results were two lists of problems, in order of
ascending difficulty. Conclusions were that problems of
certain types should be taught before problems of other
types, based on the assumption that those missed most often,
and therefore, by definition, the hardest must depnd on
the easier problems as prerequisites.
Examination of these dependencies yields examples of
concepts which could, logically, seem prerequisite to
others, but which have a reversed order of difficulty for
students. The following example is taken from the sub
traction study: It was found that tasks involving whole
number sums greater than one ( such as 5 1/6) were more
difficult than those involving mixed numbers (such as
1 1/6 2/6) (Uprichard and Phillips, Note 2, p. 10). Per
haps students were performing a rote algorithm on 1 1/6
(denominator times whole number plus numerator), rather
than realizing that 1 can be renamed as 6/6 and 5 as 30/6.
The authors say in summary that the results
support the notion that both epistemological
and psychological factors be considered when
developing teaching sequences in mathematics.
Some of the implications above would not
necessarily be derived from logical analysis
alone. Also, in interpreting the results of
this study one must be conscientious of the
limitations of indirect validation procedures.
For example, confounding variables such as
prior educational experience of subjects and
errors of measurement must be considered. (p. 11)
Students older than their subjects will have had even
more experiences in school, and the partial learning they
bring to a taskmay function as a confounding variable. The
rote application of a poorly understood, or poorly remem
bered, algorithm is an example of this. In discussing the
Uprichard and Phillips work, Underhill (Note 3) remarked
that a hierarchy may be valid for original learning, but
not necessarily for remediation when instruction has
already been given. There may be some "subskill retention
hierarchies" that could be omitted in remediation.
As pointed out by Kieren in his review (1979) of the
addition study, Uprichard and Phillips's analysis treated
fractions as symbols to be manipulated according to formal
algorithms (a very limited view). Kieren further suggested
that such studies needed to have a sound epistemological
basis from which to work, and that clinical evidence needed
to be given to support the statistical analyses. It is
knownwhich problems were missed, but it is not know why
they were missed.
Novillis (1976) studied more basic subconcepts of the
fraction concept, with subjects in grades four, five and
six. Each subconcept was depicted by a model that had as
its unit either a geometric region (the partwhole model);
a set (the partgroup model); or a unit segment of a number
line (which the author considered a specific form of the
partwhole model). The investigator constructed a hierarchy
of dependent subconcepts of the fraction concept and designed
a fraction concept paperandpencil test of 16 subtests,
one for each of the subconcepts. Most of the subtests
contained one item each of the following types:
a) given a fraction, the student was asked to choose
the correct model.
b) given a model, the student was asked to choose
the correct fraction.
c) given a model, the student was asked to select
another model for the same fraction.
d) given four models, the student was asked to choose
the one that did not depict the same fraction as the others.
To validate the hierarchy, Novillis (1976) used a
category system equivalent to Walbesser's contingency table,
and analyzed the individual dependency relationships using
ratios developed by Gagne et al. (1962) and Walbesser
(Note 4). Support was found for 18 of the 23 dependencies
in the hierarchy.
The author concluded that certain subconcepts were
prerequisites to others. The main dependencies are given
below, with the subconcept on the left being prerequisite
to the subconcept on the right.
Lower order subconcept Higher order subconcept
associating fractions with associating a fraction
partwhole and partgroup with a point on a number
models line
associating a fraction using a fraction in a com
with a partwhole model prison situation invol
or with a partgroup ving the respective model
model
associating a fraction associating a fraction with
with a partwhole model the respective model where
or with a partgroup the number of parts was a
model multiple of the denominator
and the parts were arranged
in an array that suggested
the denominator
associating a fraction associating a fraction
with a partwhole model with the respective
or with a partgroup model having noncongruent
model having congruent parts, where (in the case
parts of partwhole models),
the parts.were equal in
area. (Novillis, 1976,
p. 143)
The author noted that the study was exploratory but
inferred that elementary school students were not exposed
to a sufficient variety of instances of the fraction concept
or negative instances (cases where it is not valid) to per
mit generalization of the concept.
Because they are relevant to the present study, two
of Novillis's examples are given here:
Many students can associate the fraction 1/5
with a set of five objects, one of which is
shaded, but most cannot associate the fraction
1/5 with a set of ten objects, two of which
are''shaded, even when the objects are arranged
in an array that clearly indicates that one out
ofevery five is shaded. . .
If two rectangular regions have been separated
into five parts such that in one case the parts
are congruent and in the other case the parts
are neither congruent nor equal in areas, and
in each case one of the parts is shaded, then
many students associate the fraction 1/5 with
each of these regions and indicate that 1/5 of
each region is shaded. (p. 143)
Since the instrument for the validation was a written
test, the intriguing question of why they missed the items
cannot be answered. In the case of the second example, was
the difficulty due to their concept of area? A clinical
study, in which individual students could have been observed
and interviewed, might have yielded further information.
In a discussion of directions for research, Lesh
(1975a) suggested that mathematics educators should apply
Piagetian techniques and theory to rational numbers, and
referred to Kieren's (1975) paper as "a first step in the
direction of a Piagetian analysis of the concept of rational
numbers" (Lesh, 1975a, p. 15). The work by Kieren seems to
be motivated by curriculum development more than by the
theory of Piaget. However, of available published work,
it is closest in focus to the present study. Therefore it
will he:discussed at length.
Interpretations of Rational Numbers
Kieren was concerned about the different possible
interpretations of fractions, particularly the "algebraic"
aspects of operations on rational numbers, which are usually
not presented when fractions are introduced, and which
sometimes get lost. He attempted to show the connection
between the mathematical, cognitive and instructional
foundations of rational numbers in the following way: He
named seven different interpretations of rational numbers.
For each of these interpretations, he stated the mathematical
structures emphasized. Then he listed a set of related
cognitive structures and a set of instructional structures
(or sequences of necessary experiences). It is not always
clear whether he was summarizing existing educational
practices or whether he was making recommendations for
instructional sequences.
Kieren suggested these seven interpretations of
rational numbers:
1. Rational numbers as fractions. This is his label
for the most common interpretation of rational numbers, the
symbols used in computation. In this interpretation, the
associated mathematical structure is a set of procedures
(or algorithms) for manipulating the symbols. Kieren gave
very little attention to the other two kinds of structure
for this interpretation:
The corresponding cognitive structure is a
set of skills. It is not necessary under
this interpretation to assume any other
structures underlying the skills. The pre
requisites for these skills would be skills
incomputation with whole numbers and not
developed concepts of partwhole relation
ships or proportionality.
The major instructional strategy is diagnosis
and remediation both based on elaborate task
analysis. (Kieren, 1975, p. 107)
It seems doubtful that the student could in fact
organize and memorize these skills (the 160 different addi
tion types he mentioned, for example) if there not some
other cognitive structures on which to anchor the skills.
Concerning the instructional strategy, even though
Kieren did say the interpretations were not independent
(p. 103), this passage might still lead one to believe that
this narrow, symbolic interpretation is to be readily
found in classrooms. Actually it is highly
unlikely that a teacher would present the algorithms
for the first time without some attempt to give meaning
to the processes by appealing to one or more of the other
interpretations, or to some concrete device.
2. Rational numbers as equivalence classes of fractions.
A rational number is defined as a set of ordered pairs of
integers. In mathematical structure, the rational numbers,
together with the operations of addition and multiplication,
constitute an ordered field.
The principle underlying concept needed, according
to Kieren, is that of an ordered pair of numbers. He sees
three phases: perceiving a real situation and its coordinate
parts in order, being able to represent these coordinates
symbolically, and associating the symbols again with a
coordinate reality. Kieren continues:
With rational numbers the child must learn
to identify partwhole situations, learn
verbal and numerical codes for these, and
learn to correctly identify a code (fraction)
with a partwhole setting. As a cognitive
capstone of this ordered pair concept set,
the child must realize that a partwhole
setting can be seen in a set of equivalent
ways, and that the various fractions which
represent the elements of this set are
equivalent. (Kieren, 1975, p. 109)
But logically, it is not necessary to understand
anything about partwhole situations to use the equivalence
class concept of rational numbers. For example, a rational
number (a, b) can be said to be equivalent to (1, 2) if
(a 1) / (b 2) = 1/2. Alternately, one could give
a geometric meaning of equivalence. Referring to
Kieren's (1975) graph (p. 108), partially reproduced to
(,,1) left, one could say that the
rational number (a, b) is
equivalent to (1, 2) if (a,b)
<0," lies on a line through (0, 0)
and (1, 2).
Kieren suggests that the proper instructional strategy
for this conceptual development is exposure to a wide
variety of partwhole settings. He mentions four settings:
statestate (static comparison between a set and one of
its subsets), stateoperator (divide 3 cookies among
5 persons), operatorstate (use 5 of a dozen eggs) and
operatoroperator (cut a pie in eighths, serve 5). The
student must also understand that these ordered pairs are
numbers. This understanding must include the relationship
of this new set of numbers to whole numbers, and a "notion
of operations consistent with the fractional and equivalence
notions" (p. 110). This second notion, Kieren feels, de
pends on the ability to partition both discrete and continu
ous quantities. Examples he gives are: dividing 15 plants
among 5 pots, dividing a rope into 5 equal pieces, and
dividing some crackers among 4 people.
In concluding this section, he says that in this
interpretation,
the child must be able to assign a pair of
numbers to a partwhole situation. This, of
course, entails the ability to logically
handle the partwhole relationship in both
the discrete and continuous cases. The ability
to handle class inclusion may be very important
in the former case, while partitioning plays
a role in the latter. (Kieren, 1975, p. 110)
3. Rational numbers as ratio numbers. An example of
a ratio number is the number x, where x is to 1 as 1 is to 8.
This interpretation leans heavily on the previous one, as it
depends on ordered pairs and operations proceeding from
equivalence classes. This interpretation is a sophisticated
one and Kieren does not expect the child to be able to
deal with it until the proportionality schema is developed,
probably not until later adolescence.
4. Rational numbers as operators or mappings. In this
interpretation, 2/3 is an operator which maps 3 onto 2,
yielding a line segment 2/3 as long as the original. A
finite analog would be giving 2 boxes of crayons to every
3 children. (Thus 6 children would need 4 boxes, etc.;
equivalence can be seen in operators in this way.) Of the
operations, multiplication and division can each be thought
of as one operator following another, and are easier than
addition and subtraction. Kieren says that three cognitive
structures are critical to this interpretation. One is the
notion of proportion. However, he says,
the rational number notions in this interpre
tation can be developed as concrete generali
zations about a large number of concrete
situations. Thus, these notions from the
point of view of the child can be considered
preproportional. It should also be noted
that the fraction notion in this interpreta
tion is based on the quantitative comparison
of two sets or two objects; hence, part
whole or class inclusion notions are not cen
tral to the interpretation. (Kieren, 1975, p. 115)
The child must also have a structure of composition
(one operator followed by another) and be able to replace
these transformations by their product.
The third structure is that of properties, parti
cularly those of inverse and identity, and the underlying
reversibility notion.
Instructional strategies would include work with simi
lar figures, which Kieren calls "preproportional," and ex
change games with finite sets.
5. Rational numbers as elements of a quotient field.
The rational number x is a solution to an equation of the
form ax = b, where a and b are integers. Field axioms are
assumed. This interpretation relates rationals to abstract
algebraic systems, and "is not closely related to the
natural thought of the child" (Kieren, 1975, p. 121). Be
cause it requires formal reasoning, this interpretation will
not be detailed further.
However, Kieren says that the more primitive cognitive
structure underlying the quotient concept is partitioning:
if there are 6 pizzas for 5 children, what is an
equal share for each? His simpler illustration is this:
Here are 20 letters to be divided evenly in 5 mailboxes.
This problem can be solved by distribution of the letters
one at a time into the mailboxes, like dealing out cards
(Kieren, 1975, p. 121).
6. Rational numbers as measures. Rational numbers
are points on the number line. Addition is the simple
laying of two vectors endtoend and reading the result.
This interpretation gives an intuitive notion of order.
Kieren gives the cognitive structures that seem
particularly important:
The first is the notion of a unit and its
arbitrary division. The child must realize
that one can partition the unit into any
number of congruent parts. Second, the
child must be able to conceptualize part
whole relationships in this context and
recognize equivalent settings arising from
partitioning of the unit (1/2 = 3/6).
Third, the child must develop the concept
of an order relation. This involves both
the ability to order physical reality and
the ability to use correctly the symbolic
order statements. Underlying these
structures are more general structures,
conservation of length and substance, and
a general notion of ordinal number. (p. 125)
Instructional activities are suggested by both forms
of division, measurement and partitioning. Equivalences
can be shown with rods or paper strips of different colors.
7. Rational numbers as decimal fractions. In this
interpretation, rational numbers are those which can be
expressed as either terminating or repeating decimals.
The operations are extended from those for whole numbers,
making computation simple. In division, a remainder is not
needed. Teaching from this viewpoint would not provide
preexperience for the rational expressions of algebra.
The cognitive structures necessary are similar to
those for measurement. However, the child must be able
"to generalize in the symbolic domain" (Kieren, 1975,
p. 126). Also, one out of six parts, or 1/6, is a natural
extension of counting; saying "about .16" is not. There
fore measuring and estimating are critical. Estimating,
he says, involves a general notion of unit and the ability
to think hypothetically.
Instructional activities would include any work with
the numeration system, and operations with whole numbers.
Metric system measurement and money also provide natural
models for decimal fractions. And estimating length is
a predecimal fraction activity. Kieren says that "the
processes of seriating and comparing are of paramount
importance as is the whole notion of order" (p. 127).
Having described these interpretations of fractions,
Kieren makes the point that all should be considered in
the curriculum. Given these interpretations, he says, a
curriculum developerinstructional designer "can then
ascertain the necessary cognitive structures for meeting
the objectives and develop sequences of instructional
activities which contribute to the growth of these
structures" (Kieren, 1975, p. 128). He says further that
a researcher who asks, "How does the child
know rational numbers?" must go through a
similar process. He can study selected
interpretations in more detail and identify
what he believes to be the most important
cognitive structures. Settings can then
be developed or used which allow one to see
the extent to which a child has such struc
tures. The growth of such structures can
then be studied developmentally. Alterna
tively, the importance of such structures can
be tested. Here, one would test the effect
of having or not having some structure on
attaining some rational number objectives. (p. 128)
As mentioned elsewhere, students' learning does not always
proceed logically, or according to researchers' expectations.
Therefore, to ascertain these necessary cognitive struc
tures as Kieren suggests may require indepth study of
students and their learning.
Kieren then summarizes the "conglomerate picture of
rationals," including some work that has been done in
developing hierarchies of skills, and suggests curriculum
research. He further suggests clinical research such as
that of Inhelder and Piaget (1969) on the growth of logical
thinking, saying,
Some aspects and behaviors of rational number
will be impossible to study in their "natural
state." They will undoubtedly be colored by
instructional experience. (p. 140)
As already observed, secondary students will have had many
such instructional experiences, which may confound the study
of their concept of fraction.
Kieren also says that "it would seem that conservation
of area and length might be related to continuous partitive
division" (Kieren, 1975, p. 141). This idea will be dis
cussed further in following sections.
The Learner
Two aspects of the learner will be considered: first,
what is known about the learner's cognitive structure, as
described in the cognitive development theory of Piaget;
and secondly, what is known about the remedial student.
Not all scholars agree with everything Piaget says.
In fact, according to Flavell (1963),
the system has an extraordinary penchant for
eliciting critical reactions in whoever reads
it. Piaget has done and said so much in a
busy lifetime that foci for possible contention
and disagreement abound. More than that, he
has consistently done and said things that run
so counter to accepted practice as to make for
an immediate critical reaction in his reader,
almost as though he had deliberately set out
to provoke it. (p. 405)
Flavell also disagrees with certain parts of the theory,
but concludes that Piaget's work "is of considerable value
and importance, with a very great deal to contribute to
present understanding and future study in the are of human
development" (p. 405).
Piaget's theory of cognitive development is not a
theory of education, but of knowing, which may or may not
be related to the knowledge purveyed in schools. Piaget
has left educational implications and applications to
others (Sigel, 1978, p. xvii). However, the
following discussion of his theory will indicate that the
cognitive structures he describes are of importance to
school learning.
Piaget's Theory
Piaget is an epistemologist. He studies the nature of
knowledge; he is concerned with finding out how the ability
to know develops. He looks for commonalities in children's
knowing that do not depend on what school they attend, their
emotional state or other factors (important though they may
be to the overall condition of the child).
During decades of study on hundreds of children,
Piaget concluded that there were definite levels of cognitive
development which were invariant in the sequence in which
they emerged. Unfortunately, this idea gives rise to the
first of many common misinterpretations of Piaget's ideas.
An example is the following:
The research of Piaget, et al. suggest that
all students by about the age of 12, should
be able to correctly use an external frame of
reference to properly predict water level,
pendulum position, etc. (Dockweiler, 1980, p. 214)
A review of the work cited (Piaget and Inhelder, 1956) fails
to turn up the suggestion by Piaget and Inhelder that any
student should do anything at a particular age. Piaget does
not view cognitive development level as age dependent.
Flavell emphasizes Piaget's position on the stage
age question:
Piaget readily admits that all manner of
variables may affect the chronological age
at which a given stage of functioning is
dominant in a given child: intelligence,
previous experience, the culture in which
the child lives, etc. For this reason, he
cautions against an overliteral identifi
cation of stage with age and asserts that
his own finTings give rough estimates at
best of the mean ages at which various
stages are achieved in the cultural milieu
from which his subjects are drawn. . Of
course not all individuals need achieve the
final states of development. . Piaget
has also for a long time freely conceded
that not all "normal" adults, even within
one culture, end up at a common genetic
level; adults show adult thought only in
those content areas in which they have been
socialized. (Flavell, 1963, p. 20)
The present study is not focused on the chronological
age at which a student has reached a stage; all of the
subjects are "behind" Piaget's children. Attention is
given instead to whether the student has reached a stage,
and whether having reached it has anything to do with
enabling the acquisition of mathematics concepts.
In this discussion, the major developmental stages
themselves will be called "periods," in accordance with
Piaget's stated preferences (Flavell, 1963, p. 85); the
word "stage" will refer to subdivisions with the period
(except where reference is made to authors who use the
former nomenclature).
The first period, called the sensorimotor period,
lasts from birth to about two years of age. The last
one, the formal operational period, in which an individual
becomes able to think about thoughts and reason about
reasoning, has been found by Piaget to be completed at about
age 15. These periods at the two ends of the developmental
scale were not exhibited by the students in this study.
Therefore, attention will be paid only to the middle period,
called by Flavell "the period of preparation for and
organization of concrete operations" (Flavell, 1963, p. 86).
The first of two major subperiods is that of preopera
tional representations, and the second is that of concrete
operations.
In the preoperational subperiod, found by Piaget to
last roughly from 2 to 7 years of age, the child is learning
to use language as representation of thought. His under
standing of space increases to include such concepts as
more and less, larger and smaller, before and after. He
learns to discriminate differences in objects, colors, etc.
Yet, in the preoperational child, perception is a stronger
influence than reason.
During the concrete operational subperiod, about 7 to
11 years in Piaget's findings, the child acquires a conser
vation schema. She can classify objects on the basis of a
common characteristic. She learns to seriate, or put things
in order from smallest to largest or vice versa.
The following are some examples of Piagettype tasks
and how children react in each of the two subperiods.
Conservation of number. In Piaget's theoretical
analysis, the concept of number is derived from "a synthe
sis of class inclusion and seriation" (Sinclair, 1971,
p. 152). Piaget's own volume, The Child's Conception of
Number (1965), includes conservation of quantity, oneto
one correspondence, logical classification and order
relations, each of which were given at least one chapter.
In spite of its complexity, conservation of number has
been chosen to present first, because it can make a vivid
illustration of what Piaget means by conservation.
What will be presented here is a simplistic version, with
emphasis on the tasks themselves, based on work by Copeland
(1979), Formanek and Gurian (1976), and Lesh (1975b).
A child is shown two rows of beads displayed as follows:
0 0 0 0 0
0 0 0 0 0
The child agrees that there are the same number of beads in
each row.
If one row is now spread out, like this,
0 0 0 0 0
0 0 0 0 0
the child of 5 or 6 years may think that there are more
beads in the bottom set, because the row is longer. Even
if he counts each row, he may still be influenced by what
29
he sees, the length of the rows, in making judgment about
which set contains more beads. He would be said to have the
ability to conserve number if, in this case, he could
realize that the number of beads remained constant even
when the beads were rearranged.
There are three stages into which children's responses
can be divided:
I: Says that the second row contains more beads.
Pressed for a reason, says, "Because I can tell by looking,"
etc.
II. Is transitional. Appears to conserve, but is not
sure; counts to see whether the rows are equal in number.
III. Realizes that rearranging does not change the
number. Asked for a reason, replies, "Because you didn't
put any more or take any away."
These three stages are typical of the sequences Piaget
finds in other conservation tasks (quantity, length, area,
volume, mass). The stages are summarized by Flavell (1963):
I, no conservation; II, conflict between
conservation and nonconservation, with
perception and logic alternately getting
the upper hand; and III, a stable and
logically certain conservation. (pp. 312313)
Flavell also says that, in this task, "a genuine
concept of cardinal number is by no means guaranteed by the
ability to mouth appropriate numerical terminology [or count]
in the presence of objects" (p. 313).
30
Seriation. Seriation is the act of putting things in
order. Its beginnings are in the child's broad discrimi
nations between big and little.
In one version of the seriation task, a child is given
about 10 sticks of different lengths and is asked to put
them in order. In stage I, a child can order two, or maybe
three, sticks at a time, but there is no overall scheme.
In stage III, the child has a plan, and methodically selects
the longest (or shortest), then the next longest (or next
shortest), etc., and completes the series efficiently. If
some sticks are introduced as having been "forgotten," the
stage III child can insert them with no problem (Copeland,
1979, p. 96).
The stage II child can usually form the series by
"trial and look." as Copeland calls it (p. 96). But a plan
or system is noticeably absent. In fact, the child may end
up with two or three unconnected subseries, as Lesh
(1975b, p. 97) shows:
Even if such a child can complete the series, he may still
be unable to insert a "forgotten" stick, Lesh says (p. 97).
Copeland says this child "considers the series already
built to be complete and feels no need to insert the addi
tional sticks" (Copeland, 1979, p. 94).
In Piaget's language, the preoperatory levels, stages
I and II, "lack coordination in that subjects can put two
or three elements in order at a time but cannot put all
the elements in order. The operator level sees a general
(reversible and transitive) coordination linking these
specific actions into a whole" (Piaget, 1976, pp. 300301).
One source of difficulty in the seriation task lies
in the tendency of some stage I and II children to make
pairs. It may not be simply that they can only attend to
two at a time (that is, they can consider a < b, but not
a < b and b < c simultaneously). Another factor may be,
Piaget says, that
the conceptualization on which the cognizance
is based, which starts from the results of
the act, is not only incomplete but often
incorrect as well, because the child's pre
conceived ideas influence his reading of the
situationthat is, he sees what he thinks he
ought to see. (1976, p. 300)
In this context, once a stage I child picks up two sticks
and orders them, she may continue making pairs in that
fashion, disregarding the original instructions, because
she hears the directions she thinks she ought to hear.
Classification. In a simple classification task, a
child is given a collection of objects or pictures and
asked to put "the ones that are alike" together. Flavell's
(1963) discussion of children's responses will be abbre
viated. In stage I,
the child tends to organize classifiable material,
not into a hierarchy of classes and subclasses
founded on similarities and differences among
objects, but into what the authors [Piaget
and Inhelder] term "figural collections"
[like pictures]. . It is a relatively
planless, stepbystep affair in which the
sorting criterion is constantly shifting
as new objects accrue to the collection.
. Partly in consequence of this inch
byinch procedure bereft of a general plan,
the collection finally achieved is not a
logical class at all but a complex figure
(hence figural collection). The figure
may be a meaningful object, e.g., the
child decides (often post hoc) that this
aggregation of objects is "a house." Or
instead, it may simply be a more or less
meaningless configuration. . Frequently,
at least part of the child's collection is
founded on a similarityofattribures basis.
What often happens is that the child begins
by putting similar objects together, as
though a genuine classification were in
progress, and then "spoils" it by incor
porating his "class" into a nonclass, con
figurational whole. (Flavell, 1963, pp. 304305)
Flavell says that the stage I child may also begin
by putting squares together, but fails to include all the
squares or contaminates his collection with nonsquares.
This is an illustration of his inability to differentiate,
and hence coordinate "class comprehension (the sum of
qualities which define membership in a logical class) and
class extension (the sum total of objects which possess
these criterial qualities)" (p. 305). He explains:
In a genuine classification, these two pro
perties must always be in strict correspon
dence: the definition of the classification
basis determines precisely which objects
must constitute its extension, and the nature
of the objects in a given collection places
tight constraints on the definition of the
class they together form. But for the young
child, there seems to be no such strict
correspondence. (p. 305)
It is noted by Flavell that these gaps in the child's
understanding may be hidden. "The child's ability to bandy
about classificationrelevant phrases (e.g., 'dogs are
animals,' "some of these are red,' etc.) either under
ordinary questioning or in spontaneous discourse, is likely
to be a most unreliable guide" (Flavell, 1963, p. 306).
A stage II child can form nonfigural collections on
the basis of similarity of attributes. He can generally
assign every object in the display to one or another group.
Still troublesome are groups or collections with only one
member, or, worse yet, no members. Copeland reports
Inhelder and Piaget's (1969) findings: "The concept of the
singular class is not operational until eight or nine years
of age, and the empty or null class is not operational until
ten to eleven years of age" (Copeland, 1979, p. 69).
Stage III does not occur, for Piaget, until the child
has mastered class inclusion. This will be treated below
as a separate task.
Class inclusion. The important aspects of class inclu
sion can be exemplified by these two tasks, taken from
Flavell (1963, pp. 307309). In one, the child is to be
tested on the notion of "some" and "all" (a reflection of
the understanding of class comprehension and extension,
discussed above). A series of objects is shown, such as
the following collection:
Red Blue
Blue Blue
The questions asked take two forms:
a) Are all the blue ones circles? or
Are all the squares red? etc.
b) Are all the circles blue? or
Are all the red ones squares? etc.
Being able to answer questions like those in a does not
guarantee that the child can answer questions like those
in b.
In the second experiment, the child is shown a set of
flowers with a large subclass of primroses and a few other
(various) flowers. It is first established that the child
understands that the primroses are flowers. Then questions
are asked on the "quantification of inclusion" (Flavell,
1963. p. 308):
1) If I took away all the primroses, would there
still be flowers left?
2) If I took away all the flowers, would there still
be primroses left?
3) Are there more primroses or more flowers?
Strangely enough, some children can answer questions
1 and 2 correctly and still "fail" question 3. In Piaget's
interpretation, if B is the set of flowers and A is the
subset of primroses,
The child can recognize that A and A'
comprise B when he focuses attention on
the whole B (thus, he can perform B = + A'),
btE "losesT B (and the fact that A = B A')
when he isolates A as a comparison term.
With B momentarily inaccessible as an object
of thought, the child cannot do other than
compare A with its complement A'. (Flavell, 1963,
p. 309) CFlavell's emphasis]
Conservation of distance. Distance and length are not
the same thing. Length is the measure of something which
takes up space (onedimensional) and distance is space
(onedimensional) which can be filled up with something.
If movement is involved, the situation is complicated
further, according to Piaget, Inhelder and Szeminska (1960):
Questions about the strips of paper . .
may be asked in terms of "static" length or
in terms of distances travelled. The answer
is not always the same in both cases and the
two languages should not be confused. (p. 106)
The conservation task to be discussed below, adapted
from Formanek and Gurian (1976, pp. 3234) concerns the
linear space between two points.
Two small toys, such as cowboys or soldiers, are
placed about 50 cm apart. The child is asked if the toys
seem to be "close together" or "far apart." (Either one
is satisfactory; this establishes a frame of reference.)
Then a low screen, or barrier, is placed about
midway between the toys, as if it were a fence
separating them. The child is then asked whether they
are still as close together or as far apart, depending
on the child's first reply. The screen is then replaced,
first with a larger screen, high enough to hide the two
toys from each other, then with an obviously three
dimensional object, like a block of wood. Each time the
child is asked to make a judgment about whether the
distance has changed and why.
In stage I, children are thrown off by the partition
and no longer seem to be able to consider the total
distance between the two toys; they will only look at
the distance each toy is from the screen.
A stage II child can consider the total distance, but
the distance seems less, because the obstruction is
taking up space. For them, distance is empty space.
Children in stage III realize that the obstruction
is irrelevant to the distance between the two toys; they
state confidently that they are just as far apart because
they haven't moved.
Conservation of area. Logically, adding one more
dimension would tend to complicate matters. The "farm"
task, adapted from the version given by Piaget, Inhelder
and Szeminska (1960, pp. 262273), will illustrate some
of the complexities involved in considering area.
The child is shown two rectangular sheets of green
cardboard and told that they represent fields of grass.
It is established that they are the same size, by putting
one on top of the other, if necessary. Then a tiny model
of a cow is placed on each field and the child is asked
whether both cows have the same amount of grass to eat.
Thus, the frame of reference is established. Then the
investigator begins to change things. Two identical "barns"
are added, one to each field, and again the child is asked
whether the cows have the same amount of grass. According
to the authors, every child says that they have (Piaget,
Inhelder and Szeminska, 1960, p. 263).
A second barn is then introduced into each field, but
in a different arrangement: in one, the barn is juxtaposed
to the previous one; in the other, the second barn is
placed elsewhere in the field, not near the first barn.
The child is asked the same question; if it is answered
correctly, a third barn is added (in a row in one field,
spread out in the other), then a fourth barn, and so on.
The authors found results analogous to the previous
ones:
During stage I we find it difficult to pursue
the enquiry, but at stage IIA children are ob
viously interested, yet they refuse to admit
that the remaining areas are equal, often at
the very first pair of houses. Here there is
no trace of operational composition, and
judgment is based entirely on perceptual
appearances. At level IIB we find a complete
range of intermediate responses: up to a
certain number of houses the subject admits
the remaining meadowlands are equal; beyond
that number the perceptual configurations
are too different. Here there is intuitive
articulation in varying degrees, but not
operational composition. At stage III,
however, . children recognize that the
remainders are always equal, relying on an
operational handling of the problem which
convinces them of the necessity of their
reasoning. (Piaget, Inhelder and Szeminska, 1960,
pp. 263264)
Variations in the above procedure produced some
surprises. In the discussion above, it is not mentioned
where each of the first two barns was placed on each field.
In the experiment, they were placed identically. Yet in
other experiments, it turned out that if one of the barns
was placed in a corner, and the other in the center of the
other field, the remaining space did not seem equal to
all children (p. 263).
The authors used rectangular "bricks" to represent
the barns. They relate one example in which the bricks were
first placed in identical positions, and the child identified
as GAR agreed that the amount of green was the same. Then;
investigators And like this (one in the
centre of Bi with the length of the brick
parallel with the length of the meadow,
another at one end of B2 and laid breadth
ways)? [GAR] No, there's more green left
here (B2). Investigator] Why? LGAA Because
there's all this left (free space). (p. 264)
Bi C Bz 2
39
This situation is reminiscent of some optical illusions
in which the orientation changes the appearance of a
length.
After carrying out other experiments, the authors
noted further that the conservation of "space remaining"
did not necessarily occur simultaneously with conservation
of "space taken up" (Piaget, Inhelder and Szeminska, 1960,
p. 286), and that once an area (or plane surface) had been
cut, its area might not seem the same to some children,
even when it was put back together (p. 295).
Other features of the theory. There are many other
experiments Piaget has done which would be illustrative of
his theory and his method. These six tasks were chosen
because of their relationship to the present study. Some
of the other relevant features of Piaget's theory, taken
primarily from Flavell (1963) and Travers (1977), are
discussed below.
The child develops a "schema," an organization of
ideas or behaviors, a structure in the intellect which
enables the child to understand. New information that is
found is "assimilated," or added into the existing schema.
The process of assimilation entails adding knowledge or
behavior consistent with actions already organized within
the schema. Later, as the child acts on the environment,
the child changes the schema or builds a new one to
accommodate new behaviors in response to new situations.
"Accommodation" is this building of new schemata or modifying
of old schemata to adapt to new situations. A child
adapts to the environment by an interplay of assimilation
and accommodation.
There are definite stages of cognitive development,
invariant in sequence. Each stage is the foundation for
the next stage. To go from one stage to the next, the
child needs to mature chronologically, and also needs
experience with the environment. Further, there has to
be a problem that the child wants to solve. The child is
not satisfied with the solution produced by the present
stage of development; Piaget calls this a state of
"disequilibrium." When the child finds a new solution
to the problem at a higher cognitive level, equilibrium
is restored. This process is called equilibrationn."
Thus, cognitive development results from the child's
interests and drives in interaction with the environment.
Conjectures could be made about what might happen
when some of these requirements for development are not met.
For example, a child might have matured chronologically
without having had the experiences which induce development.
The particular environment may not have presented problems
that the child wanted to solve. Or the child's interests
may have been in art or some other endeavor which did not
induce the conflict, or disequilibrium, necessary for
cognitive growth. In these cases, the expected structures
may not have developed.
Such a situation does not preclude further growth.
Piaget's theory does not put a ceiling on development at
any age. Therefore, the theory is compatible with the
possibility that children of intellectually deprived
environments may not yet have achieved the cognitive
development of which they are capable. The next section
will focus on these disadvantaged students.
Disadvantaged Students
Pikaart and Wilson (1972) "examined the research on
the slow learner in mathematics and found it lacking"
(p. 41). The meager research that is available, they say,
parallels the development of the idea that intelligence
is quantifiable. IQ scores are of little use, they say.
A more fruitful approach . is to consider
specific learning aptitudes of slow learners
and to adapt instruction to take account of
these individual differences. (p. 42)
In Suydam's (1971) summary of research on teaching
mathematics to disadvantaged pupils, she notes that the
summary does not contain many studies done with students
in the secondary school. One of the reasons she gives for
this is that
there are not as many slow learners or low
achievers or otherwise disadvantaged students
still enrolled in mathematics courses in the
secondary school. The process of selection or
tracking precludes most students in any of
the subsets of the disadvantaged from going
beyond a general mathematics course. (p. 3)
To "enrolled in mathematics course," she might have added
"enrolled in school." With compulsory attendance over at
around age 16, many who have not been successful by then
drop out.
The studies concerning disadvantaged students that
are listed by Suydam (1971) usually focus on comparing
different teaching methods, and will be mentioned later.
Compensatory and remedial programs have proliferated;
still mathematics education researchers interested in
secondary school mathematics have devoted the bulk of
their resources to studying the students who are in the
college bound track, taking courses in algebra and
geometry. It is hoped that this study of disadvantaged
secondary students will be a start in the direction
suggested by Pikaart and Wilson (1972).
The child's development is left now for a consideration
of the student's school learning, as the learner interacts
with mathematics content.
Interaction of the Learner and the Content
The discussion of what happens when the learner
interacts with mathematics content must be limited for this
review. The topics chosen can be explained by first
summarizing the previous two sections.
First, efforts to study the content, the fraction
concept and operations with fractions, were discussed.
Included were calls for research to find the underlying
cognitive structures of fractions. Secondly, in looking
at the learner, relevant aspects of cognitive development
theory were described. The lack of research on students
who have difficulty learning mathematics was mentioned.
This section, on the interaction of the learner with
the fractions content, will relate the preceding sections.
For example, in spite of his disavowal of educational
objectives in general, Piaget did consider what are almost
prefraction concepts in some detail. This work on fractions
will be reviewed. Next will be a description of assess
ment efforts aimed at finding out what students in general
know about fractions, and then of diagnostic and prescrip
tive studies, concerned with why the individual student
has not learned fractions and what might be done about it.
Attention will also be given to clinical studies, which
often include detailed observation of interactions between
learner and mathematics content. The neoPiagetian
research will be included. Last will be a discussion of the
concreteversussymbolic modes of presentation of mathe
matics content in attempts to assess students' knowledge.
Piaget's Fractions
Much of Piaget's work has been done with small children,
so he has not given much attention to fractions. He has,
however, considered "Subdivision of Areas and the Concept
of Fractions" as Chapter 12 of The Child's Conception of
Geometry (Piaget, Inhelder and Szeminska, 1960). He
describes work with children whose ages range from 4 to
around 7 years. He is not studying "fractions" as they
are normally taught in school, however. For example, a
child is asked to cut a cake, to "divide it up so that the
man and the woman will both have the same amount of cake
to eat" (Piaget, Inhelder and Szeminska, 1960, p. 304).
The child does not have to know either the notation "1/2"
or the words "one half" to be able to perform the task.
When Piaget writes about "their idea of a fraction" (p. 310),
he seems to be talking about the children's idea of
partitioning, a basic component of, or perhaps even a pre
concept to, the idea of a fraction. (The fact that the
work is a translation may add to the confusion.)
The procedure was as follows:
The child was expected to use a wooden knife to divide
a circular cake made of modelling clay equally between two
dolls. After the division was performed, the child was
asked whether, if the pieces were put back together, it
would be equal to the original whole. Those children who
could divide the cake into halves were then asked to
divide the cake between three dolls, and so on, up to six
dolls.
The youngest children often cut two pieces of arbitrary
sizes for the two dolls, leaving the remainder of the cake
(either ignoring it or pushing it aside). When pressed by
the interviewer as to what was to be done with the remainder,
a child might refuse to discuss it (p. 305) or even try to
hide the leftover part (Piaget, Inhelder, and Szeminska,
1960, p. 306). At this stage the child was concerned
neither with equality of shares, nor with exhausting the
whole. Some children also seemed to think that two
pieces required two cuts. More advanced children could
correct their mistake, having made two cuts, by subdividing
the remainder and parceling out more cake to the dolls, so
that the cake was exhausted, at least, whether equally
subdivided or not.
In trying to comprehend the children's behavior,
Piaget suggests that the halftowhole, and generally,
parttowhole, relationship can be understood by the child
perceptually. But, he says,
it is a far cry from such perceptual or
sensorimotor partwhole relations to
operational subdivision. There are syste
matic difficulties in understanding part
whole relations on the plane of verbal
thinking. . When we used phrases like
"a part of my bunch of flowers is yellow,"
or "half of this bunch is yellow," etc.,
we found that even children of nine or
ten thought of the whole bunch as yellow
because they thought of the part (or half)
as something absolute rather than as being
necessarily relative both to the other
part (or half) and to the whole. Typical
replies were these: "What's a half?
Something you've cut off.What about the
other half?The other is gone." Obviously
the half that is cut off and thought of
as a thing apart without reference either
to the whole or to the other half echoes
the little pieces which are cut off in
actual fact by children of two to four.
S. Quite early on children elaborate
means of dealing with reality at the level
of action, and even at the level of con
crete operations, but these solutions still
need to be reworked at the level of verbal
thinking by means of formal schemata.
(Piaget, Inhelder, and Szeminska, 1960, p. 308)
Thus, when a half is cut off, it may become to the
child an entity on its own, with no further reference
to the whole of which it was a part. Piaget refers to
his earlier work on the partwhole relation, when he was
studying the child's conception of number:
Thus, when shown a large number of brown
beads alongside two white beads, all these
beads being made of wood, the child under
seven could not understand that there were
more wooden beads than brown beads for he
persisted in forgetting about the collection
as a whole when concentrating on the brown
beads and therefore came to the conclusion:
"There are more brown beads than wooden
beads because there are only two white
beads." (Piaget, Inhelder, and Szeminska, 1960, p. 308)
In the partitioning task, again, subdivision must be
reconstructed in thought, but with reference to a concrete
situation. It cannot be assumed that a child who is able
to physically partition an object can verbalize the actions.
In analyzing the actions of the smallest children,
mentioned earlier, Piaget says that the most striking
thing is the presence of a partpart, rather than a
partwhole, relationship. For them, "the relation
between parts is one of juxtaposition and not of a nesting
series" (p. 309). The child ignores the quantitative
aspect, that two halves are equal, for example, and
also the relation of the part to the whole, "from which
it may be parted in fact but to which it still relates
in thought" (Piaget, Inhelder, and Szeminska, 1960,
p. 309).
Piaget's analysis of the fraction concept continues:
The notion of a fraction depends on two
fundamental relations: the relation of part
to whole (which is intensive and logical)
and the relation of part to part, where
the sizes of all other parts of a single
whole are compared to that of the first
part (a relation which is extensive or
metric). (p. 309)
He describes the necessary components of the notion of a
fraction as follows:
1. The child must see the whole as composed of
separable elements, i.e., divisible. Very young children,
he says, see the whole as an inviolable object and refuse
to cut it. Later, the children are prepared to cut it, but
then the act of cutting it may make the object lose its
wholeness.
2. A fraction implies a determinate number of parts.
Children who do not realize that the number of shares
should correspond to the number of recipients begin by
randomly breaking off pieces.
3. The subdivision must be exhaustive, i.e., there
must be no remainder. There has been mention already of
children who
refuse to share out the remainder, apparently
satisfied that when they have made up the two
parts they were asked for, anything left over
is neither part nor whole and has nothing to
do with the two real parts: these alone are
real because these alone go to make up their
idea of a fraction. (Piaget, Inhelder, &
Szeminska, 1960, p. 310)
4. There is a fixed relationship between the number
of subdivisions and the number of intersections, or cuts
to be made.
5. The individual parts must be equal.
6. The parts themselves have a dual character: they
are parts, but they can also be wholes, and thus are sub
ject to being subdivided further. This is the understand
ing necessary for finding fourths by halving halves.
7. The sum of the parts equals the original whole.
Somehow, cutting the cake changes it for some children.
A subject identified as SOM thinks there is more in two
halfcakes than there is in one whole (p. 327). Subject
GIS says that they are the same, but: asked.to choose between
a whole cake and two halves, chooses the whole, saying,
"I get more to eat this way" (p. 320). In a measured
understatement, Piaget says, "We see how paradoxical are
these replies" (p. 329).
Some of the conditions for understanding the fraction
concept may seem obvious, but Piaget has discovered their
necessity by seeing their absence in the thinking of
children. He further says that these seven conditions must
still be part of a general structure to be operational.
There must be an anticipatory schema: children must be able
to anticipate the solution before they can solve the prob
lem. That is, they must plan ahead where all the cuts will
be before they make the first cut. In the absence of such
a plan, successive fragmentation of the cake is made.
Piaget emphasizes the complexity of the task:
The subdivision of an area . is fraught
with considerable difficulty for young
children and its complications compare in
every respect with those pertaining to
logical subdivision of the nesting of
partial classes within an inclusive class.
(Piaget, Inhelder and Szeminska, 1960, p. 333)
Piaget defines the substages in the subdivision of
areas according to whether the seven conditions are met by
the children, and whether they can halve, trisect, quarter,
etc. (trisecting being more difficult than quartering,
since quartering can be done with two successive dichotomies).
He concludes the chapter with the following summary:
The facts studied in this chapter show not
merely that there is a clear parallel between
the subdivision of continuous areas and that
of logical classes, but also that notions of
fractions and even of halves depend on a
qualitative or intensive substructure. Before
parts can be equated in conformity with the
extensive characteristics of fractions, they
must first be constructed as integral parts
of a whole which can be decomposed and also
reassembled. Once that notion of part has
been constructed it is comparatively easy to
equate the several parts. Therefore, while
the elaboration of operations of subdivision
is a lengthy process, the concept of a fraction
follows closely on that of a part. For parts
which are subordinated to the whole can also
be related to one another, and when this has
been achieved, the notion of a fraction is
complete. (Piaget, Inhelder and Szeminska,
1960, pp. 334335)
Although Piaget says that "the notion of a fraction
is complete," it must be noted that his discussion has
dealt basically with one interpretation of fraction, that
of subdividing continuous substances. He studied primarily
one medium, clay, which is threedimensional, though he
referred to the task as "subdividing areas." Piaget did
try some different shapes and some plane figures, finding
it easier, for example, for the children to trisect a
rectangle than a circle, and the longer the rectangle, the
easier. Some of the children were given a "sausage" of
modeling clay and it was found to be the easiest solid to
trisect, presumably because it was like an elongated
rectangle (p. 319). Piaget's work has given valuable insight
into some of the components necessary to a child's concept
of fraction. But he did not treat a linear model or a
discrete model. He considered only unit fractions, and then
with small denominators. He did not look at equivalence,
or comparisons between different fractions. And certainly
it was not his purpose to study how children learn about
51
fractions as ratios or quotients or how they come to per
form mathematical operations with fractions. These other
aspects need to be given indepth study also.
The next topic is students' general knowledge of frac
tions as taught in school, and as evidenced by assessment
tools.
Assessment of Students' Knowledge of Fractions
Students have all received instruction in fractions by
the end of the sixth grade, so it is appropriate to ask
what understandings and skills they carry with them into
junior high and high school.
In view of most elementary school mathematics pro
grams today, Carpenter, Coburn, Reys, and Wilson (1978)
say, "13yearolds should be thoroughly operational with
fractions" (p. 34). However, in their summary of the NAEP
mathematics assessments, they say that overall results on
fraction concept tasks were low. Of all three groups, 13
yearolds, 17yearolds, and adults, no more than about
two thirds responded correctly to an exercise that dealt
with fraction concepts (p. 34).
Some of the NAEP results were reviewed earlier (Carpen
ter, Coburn, Reys, & Wilson, 1976), when only two exercises
were released. The first was:
1/2 + 1/3 = (p. 137)
Only 42% of 13yearolds and 66% of 17yearolds were
successful in solving this exercise. Of various incorrect
responses, the most common was obtained by adding both the
numerators and the denominators (30% and 16%, for 13 and
17yearolds, respectively). In speculating about this and
other errors, the authors say the results suggest "that
students are not viewing the fractions as representing quan
tities but see them as four separate whole numbers to be
combined in some fashion or other" (Carpenter, Coburn,
Reys, & Wilson, 1976, p. 138).
The multiplication exercise was:
1/2 x 1/4 = (p. 137)
The students performed better on this exercise, getting
62% and 74% correct answers. The incorrect responses did
not show a pattern.
The authors noted that these results were consistent
with data from various state assessments and other research
(p. 139).
The later, more complete report (Carpenter, Coburn,
Reys, & Wilson, 1978) describes the testing of the concept
of fraction:
Asked what fractional part of a small set of marbles
was blue, 65% of 13yearolds answered correctly (p. 37).
The three older groups were given this problem:
There are 13 boys and 15 girls in a group.
What fractional part of the group is boys? (p. 38)
Described as "very disappointing," the results were 20%,
36%, and 25% correct answers, for 13yearolds, 17yearolds,
53
and adults (Carpenter, Coburn, Reys, & Wilson, 1978, p. 38).
The authors commented that
the cause of the errors cannot be determined
from the data. Perhaps there is a problem
with the language "fractional part" that
would contribute to the"I don't know" re
sponses. But the committed errors must be
due to a lack of mastery of fraction con
cepts and their application to problem con
texts. (p. 38)
In a multiple choice exercise, two common fractions
less than 1 were given; respondents were asked to select
another fraction between them. Correct answers were given
by 56% and 83% of 13 and 17yearolds (p. 39). However,
when given six very common fractions less than 1 and asked
to write them in order from smallest to largest, "no age
group could perform this task adequately" (p. 39).
Asked which fraction was the greatest of 2/3, 3/4, 4/5,
and 5/8, 26% of 13yearolds and 49% of 17yearolds an
swered correctly. The authors say,
The strongest distractor for both the
13yearolds and the 17yearolds was 2/3.
The exercise clearly shows that 13yearolds
are not yet operational with fractions. (p. 40)
In a survey intended to find out whether deficiencies
in fractions skills were due to current instructional pro
grams, Ginther, Ng, and Begle (1976) went to "the most ad
vantaged schools" in their area and tested about 1,500
eighth graders. The students were in intact classes iden
tified as average by their teachers. A battery of fractions
tests was designed, to include the cognitive levels of
54
computation, comprehension, and application. In the compu
tation section, the second easiest problem (the second high
est percentage correct) is given for each of the operations,
with the percentage of students correct:
1/6 3/4 5/8 x 32 7/8 5/16
+ 5/8 2/5
63% 58% 65% 40%
correct
(Ginther, Ng, & Begle, 1976, pp. 34)
The authors were apparently not concerned by these low
percentages, commenting in the conclusion that the students
had a reasonable understanding of the fraction concept (p.
9). They did, however, decry the students' lack of under
standing of structure.
In the comprehension section, items were intended to
be answered very easily by students who understood the
structure of the rational number system. The following are
examples, presumably still the second easiest:
2 x[= 1 .
42% correct
(p. 4)
I I 1 1 I I I I I
0 A 1
A is
1/2 1/3 1/4 3/8 5/8
62% correct
(p. 5)
The fifth easiest diagram question was as follows;
There is a drawing on the left. Part of the
drawing is shaded. The drawing suggests a
fractional number. You are to choose the
fraction on the right which names the same
fractional number as the shaded part of the
drawing. Circle the letter in front of your
answer choice.
1/6 3/6 5/6 7/6 None of these
88% correct (Ginther, Ng, & Begle, 1976,
p. 6)
The following example from the applications section
was the fourth easiest in its subsection:
A girl weighs 64 1/2 pounds. Her brother
weighs 1/2 as much as she weighs. How many
pounds does he weigh?
54% correct (p. 7)
These results were included here to illustrate that
even in the "most advantaged" schools, many students do not
have a thorough understanding of fractions. The authors
concluded that the poor understanding of the structure of
the rational number system was due to poor instructional
programs, and that until elementary and junior high school
teachers could teach fractions in a more meaningful way,
much of the work on fractions should be postponed to secon
dary school (p. 9). An alternative explanation is that the
students may not have reached the level of cognitive deveop
ment necessary to profit from the instruction.
Efforts to help the individual student will be
discussed next.
Diagnostic and Prescriptive Teaching
Diagnostic and prescriptive teaching is not new, but
is emerging as an important area in mathematics education.
The State of Florida has recently passed a law requiring
early childhood teachers to use diagnostic and prescriptive
techniques when teaching the basic communication skills.
In the teaching of mathematics at all levels, the techniques
seem especially appropriate.
The pioneers in using diagnosis and prescription in
the teaching of mathematics were Brownell, Brueckner and
Grossnickle, who did extensive work in the field beginning
in the twenties and working through the forties. Interest
in that effort waned during and after the war, but the
preoccupation in the late sixties with disadvantaged
students and the current emphasis on ensuring that minimal
competencies are mastered has caused a rebirth of interest
in the field.
The term "diagnosis" refers to knowing not just that
the student missed the problem, but why (in the sense of
"what type of error was made?"). "Prescription" means the
assignment of instruction specifically designed to correct
that type of error. This method of teaching has been des
cribed as shooting with a rifle, rather than with a shotgun
(Glennon and Wilson, 1972, p. 283).
There have been some attempts to find the causes of
"discalculia," or mathematical disability (FarnhamDiggory,
1978), including studies of brain damage (Luriya, 1968) and
of the hemispheres of the brain (Davidson, Note 5). Concen
trating more on psychology than biology, Scandura (1970)
reviewed research in "psychomathematics." He concluded
that
there are a large number of unspecified, but
crucial, "ideal competencies which underlie
mathematical behavior. These need to be
identified. . There is also the urgent
need to consider how the inherent capacities
of learners and their previously acquired
knowledge interact with new input to produce
mathematical learning and performance. (p. 95)
These urgent needs might best be met through indepth obser
vations of individual students and their learning, as is
done in clinical studies.
In the meantime, many diagnosticians have taken the
pragmatic viewpoint: they would like to know how students
learn mathematics, but meanwhile, they try to find out
specifically what students are doing wrong and to correct or
remediate those errors.
Glennon and Wilson (1972) wrote a stateoftheart
paper for the 35th National Council of Teachers of Mathematics
(NCTM) Yearbook, The Slow Learner. They defined diagnostic
prescriptive teaching as "a careful effort to reteach success
fully what was not well taught or not well learned during
the initial teaching" (p. 283). They suggested the interview
technique perfected by Brownell (Brownell & Chazal, 1935)
for finding out what students were doing wrong.
Lankford has used individual diagnostic interviews to
survey the computational errors of seventh graders as they
worked problems involving whole numbers and fractions. He
tested 176 students in six intact seventh grade classes.
In the interviews he directed students to "say out loud"
their thinking as they computed (1974, p. 26). The percen
tages correct on the fraction exercises are not surprising,
in view of the national assessment data; in general "the
performance was much below that with whole numbers" (Lank
ford, 1972, p. 30). A sampling taken from that article
(pp. 2030) follows:
Table 1
Sample of Results of Lankford Study
Percentage of
Exercise Attempted Exercises Correct
3/4 + 5/2 47
3/4 1/2 58
2/3 x 3/5 63
9/10 3/10 41
Which is larger,
2/3 x 5 or 1 x 5? 61
It should be noted that in the last exercise cited above,
there were two choices; students could have been correct
50% of the time by chance. In fact the interviews showed,
Lankford said, that sometimes students gave the correct
answer for the wrong reason.
59
The main findings, of course, were students' thinking
patterns. In addition of fractions, for example, out of
97 incorrect answers, 62 were found by adding the numerators
and also adding the denominators; 10, by adding the numera
tors and taking the larger denominator; and 6, by adding
the numerators but multiplying the denominators (p. 30).
These errors might have been predicted by experienced
teachers, but Lankford says, "relatively large whole numbers
were a 'surprise' as when 3/4 + 5/2 = 86 . and 3/4 1/2
= 22" (p. 31). Students were adding or subtracting the
numerators and denominators; the surprise lies in the manner
in which the results were written. Apparently either a
fraction did not have meaning as a small number for these
students, or the students did not connect the meaning of a
fraction with computations done on paper.
Another error demonstrates the lack of understanding
of the meaning of a fraction:
3/8 + 7/8 = 11/15 (p. 31)
The answer was derived from 3 + 8 = 11 and 7 + 8 = 15; the
procedure may have been a persevering pattern from the
column addition of whole numbers.
And to change 3/4 to an equivalent fraction, one
student reasoned, "4 times 1 equals 4 and 1 + 3 is 4, so
4/4" (p. 31). The conclusion that 3/4 = 4/4 again indicates
that the student did not understand the concept of fraction,
60
or did not connect the concept with the computation. Some
students even stated that "2/3 is greater than 1" (Lankford,
1972, p. 34).
In concluding, Lankford gives pointers in the use of
the diagnostic interview, suggesting that teachers can
learn how well instruction has been imparted by using this
method with their own students.
Glennon and Wilson (1972) also recommend Brownell's
models of ideographicallyy oriented procedures,"which they
feel are effective techniques for both diagnostic and pre
scriptive teaching. They give credit to both Brownell and
Piaget for their contributions to the development and use
of idiosyncratic procedures in mathematics education, but
cite as the more easily understood and readily used the
work of Brownell (p. 308).
Even with Brownell's and Piaget's clinical procedures,
especially frustrating and challenging are those students
called "disadvantaged" or "slow learners" or "low achievers."
Many teachers feel that if a formula could be found to
enable their learning, all students would benefit from
the formula. The only logical way this idea could be in
61
error is for slower students to actually learn in a quali
tatively different manner from the more successful students'
manner. What has been discovered, if anything, about this
possibility?
In Suydam's (1971) summary of research on teaching
mathematics to disadvantaged students, cited earlier, she
lists the following as one of the statements that can be
implied from the research:
The mathematical characteristics which distinguish
disadvantaged from advantaged pupils appear to
exist in degree rather than kind. That is,
disadvantaged and advantaged pupils have
similar abilities and skills, but differ in
depth or level of attainment. (p. 13)
It is an assumption of this study that the above statement
is true, and that what is learned about the learning of
disadvantaged students will help the advantaged students
as well.
Suydam also found that "active physical involvement
with manipulative materials, which is believed to be
important for all children, may be even more so for the
disadvantaged" (p. 13). However, as she noted earlier,
"little research has been done on this specific topic with
specific sets of disadvantaged pupils" (p. 5). She
concludes,
Groups of disadvantaged pupils are not all
disadvantaged in the same way. There is
as much need to individualize instruction
for disadvantaged students as for other
groups of students. (p. 13)
Currently many compensatory and remedial instructional
programs aimed at teaching basic skills do not take these
individual differences into account.
There is an older study which was designed to address
the problems of these students in a substantive way. Al
though the students were not in secondary school, but in
the upper elementary grades, the spirit and method of the
study and the questions asked make its review appropriate.
The purpose of the study, reported by Small, Avila, Holtan
and Kidd (1966), was to "explore factors related to low
achievement and underachievement in mathematics education
and to determine if there are individual levels of abilities
in abstractive thought with respect to mathematics concepts"
(p. 4).
This pilot study was an attempt to identify charac
teristics of low achievers and underachievers in mathematics
in grades 4, 5 and 6, in hopes of finding new approaches
to remediation, thereby making it possible to intervene in
the processes which often lead to failures and dropouts.
Low achievers were defined as students of average IQ
whose average percentile scores on all sections of a
standardized achievement test were at least two deciles
below their present grade placement level. Underachievers
were students of average IQ whose nonmathematics scores
were equal to or above their grade placement, but whose
63
mathematics computation and concepts scores were two or more
deciles below their nonmathematics percentile averages..
Small et al. (1966) used a case study approach with
12 underachievers and 11 low achievers. Each student was
tested individually on two concpets, place value and linear
measurement. There were three levels of questions on each
subtest: concrete (the test material was a physical model
which could be manipulated by the subject); semiconcrete
or pictorial (materials used were photographs of real ob
jects); and abstract (questions were asked verbally or sym
bolically). The report included affective results.
First, there was no consistent pattern on levels of
abstraction; the ability to operate on the different levels
is an individual problem and must be identified for each
student.
Secondly, both the low achievers and the underachievers
seemed to experience more emotional adjustment problems than
did the typical student population. The underachieving stu
dent was most often a child with a large amount of anxiety
and a relatively unharmonious home in which high achievement
was considered importatn. The low achieving student probably
needed a comprehensive compensatory program at school.
Several recommendations were made for underachievers,
basically aimed at reducing their anxiety. The authors
recommended a diagnosis and remediation plan involving
levels of abstraction, for testing by other researchers.
The Small et al. (1966) study serves both to introduce
the general field of clinical studies and to focus attention
on the concreteversusabstract question.
Clinical Study
In the study discussed above, a "case study" approach
was used with 23 subjects. Tests were administered indivi
dually. (The testing instruments are given, but the report
is brief and details of the diagnostic interviews are not
available.) No hypotheses were being tested; rather, the
researchers were searching for factors which might be used
to form hypotheses concerning low achievers and under
achievers.
In many clinical studies the interview, as developed
by Piaget, is used as the basic technique to gain informa
tion about children's thinking. This approach may seem
unscientific to some researchers trained in standardized
testing, for, as Flavell (1963) says, no two children will
ever receive exactly the same experimental treatment. Even
though the initial questions may be uniform,
in the course of this rapid sequence, the
experimenter uses all the insight and
ability at his command to understand what
the child says or does and to adapt his own
behavior in terms of this understanding. (p. 28)
The same individual attention used in diagnosis needs
to be used in studying the interaction of the learner with
instruction, as in "teaching experiments." According to
Steffe, teaching experiments share these characteristics:
65
They are usually long term interventions, with a small
number of students. Researchers study how children learn,
or the "dynamic passage from lack of knowledge to knowledge
present" (Steffe, Note 6).
This microscopic attention to individual students is
expected to yield much information, in contrast to tradi
tional paperandpencil standardized testing, where "there
is no way of knowing exactly what respondents were thinking"
(Carpenter, Coburn, Reys, and Wilson, 1976, p. 137).
In mathematics education research, according to
Kilpatrick, we not only want to know that certain people do
better at certain things; we also want to know their
characteristics, and what interaction is occurring. These
things, he says, can not be learned from statistical analyses.
Neither can anything be learned without sensitivity. A
suggested approach is,"Let me look very intensively at a
small number of people and see what is happening" (Note 7).
This is the approach of a clinical study.
The next topic will be a cursory look at how others
have interpreted Piaget's works and the resulting impact
on mathematics education.
Related Piagetian Research
The many efforts to make sense of, and subsequently,
to make use of Piaget's voluminous output can be roughly
categorized as follows:
1) validation (or invalidation) studies, where
attempts are made to replicate his experiments;
2) "training," or learning, studies, where experi
menters test to see whether children can be taught the
cognitive structures Piaget has described (classification,
seriation, etc.);
3) applications or extensions of his theory and/or
his methods to other situations, to education in particular.
(In a sense, of course, group 2 is a subset of group 3.)
There will be no attempt here to give a comprehensive
review of this work. The earlier works can be located in
Flavell's (1963) definitive book on Piaget's work, and
Lovell (1971a) has reviewed "twentyfive years of Piaget
research in intellectual growth as it pertains to the
learning of mathematics" (p. 2).
Some general comments will be made, including mention
of a few relevant studies, with the primary attention given
to the third group.
1) The validation studies generally support Piaget's
theory, although variations are reported. Lovell (1971a)
summarizes a group of these:
By and large the stages in the development
of the structures, proposed by Piaget, are
found but there are differences. The age range
for the elaboration of a particular structure is
considerable even in children of comparable
background and ability as judged by teachers
or by test results. (p. 5)
67
Lovell states further that the situation, the actual
apparatus used, and the previous experiences of the children
are all variables affecting their behaviors. "It is now
clear that the tasks are subtle, that the relevant ideas
have to be carefully devised and that analysis has to be
thoughtfully considered" (Lovell, 1971a, p.6).
2) Flavell (1963) reviews 20 training or learning
studies (pp. 370378) which pertain to the teaching of the
various cognitive structures. Results were mixed; only
a few reported significant differences between the trained
groups and the control groups. Flavell comments,
Probably the most certain conclusion is that
it can be a surprisingly difficult undertaking
to manufacture Piagetian concepts in the
laboratory. Almost all the training methods
reported impress one as sound and reasonable
and wellsuited to the educative job at hand.
And yet most of them have had remarkably little
success in producing cognitive change. It is
not easy to convey the sense of disbelief that
creeps over one in reading these experiments.
(p. 377)
Just as they are difficult to induce, the conservation
concepts are difficult to extinguish when actually once
acquired, he says. The one study he reported in which
the training group clearly outperformed the control group
was one by Smedslund (1961), in which the keynote of the
training procedure was the induction of cognitive conflict
and the absence of external reinforcement.
Piaget's response to these efforts is usually amuse
ment. In the first place, he does not understand why edu
cators want to accelerate what he considers the child's
natural development. Even assuming that such acceleration
is a worthwhile goal, he is skeptical. Whenever he is told
that someone has succeeded in teaching operational struc
tures, there are three questions he asks. First, is the
learning lasting, two weeks, a month later? "If a structure
develops spontaneously, once it has reached a state of
equilibrium, it is lasting; it will continue throughout the
child's entire life" (Piaget, 1964, p. 184). And when the
learning is achieved by external reinforcement, he asks,
what are the conditions necessary for it to be lasting?
Secondly, how much generalization is possible? "You
can always ask whether this is an isolated piece in the
midst of the child's mental life, or if it is really a
dynamic structure which can lead to generalizations"
(p. 184).
The third question is, "What was the operational
level of the subject before the experience and what more
complex structures has this learning succeeded in achieving"
(p. 184)? We must see, he says, which spontaneous operations
were present at the outset and what operational level has
now been achieved after the learning experience.
69
Recent training studies by mathematics educators have
included those by Coxford (1970), Johnson (1975), Kurtz and
Karplus (1979), Lesh (1975b), and Silver (1976). Some re
port successful training and some do not.
3) The unsettled questions just mentioned bear on the
present section. In his article on psychology and mathema
tics education, Shulman (1970) says that Piaget's charac
terizations of numberrelated concepts have helped shape
our ideas of what children of different ages might learn
meaningfully. This has thus influenced some current concep
tions of readiness:
To determine whether a child is ready to learn
a particular concept of principle, one analyzes
the structure of that to be taught and compares
it with what is already known about the cogni
tive structure of the child of that age. If
the two structures are consonant, the new con
cept or principle can be taught; if they are
dissonant, it cannot. One must then, if the
dissonance is substantial, wait for further ma
turation to take place. (p. 42)
If the degree of dissonance is small, Shulman says,
Piaget's theory does not recommend, but neither precludes,
training procedures aimed at achieving the desired state of
readiness.
Brainerd (1978) disagrees entirely with Piaget's
model of learning. Since he assaults major theses, not
trivial details, his arguments will be mentioned. He first
takes issue with the notion that concepts will arise na
turally and need not be trained. Brainerd's is a typical
oversimplification of Piaget's "notion," which actually
includes as requirements for this "natural" development
not only chronological maturation, but also an appro
priate set of experiences, providing for disequilibrium
and subsequent, higherlevel equilibration (Copeland, 1979;
Flavell, 1963).
Further, Brainerd says that those Piagetians who do
training experiments insist that the training be as natural
as possible and include opportunities for selfdiscovery.
Brainerd maintains that there is not a continuum from
artificial to natural, and that there is no evidence that
natural is better (Brainerd, 1978, pp. 8384). The same
original sources, in this case Piaget's theory as stated
by his coworkers, can yield different interpretations.
Another person, reading the same quotations Brainerd has
selected (pp. 6978), might summarize them using the phrase
"relevant to the child," for example, instead of the word
"natural." (This interpretation would render irrelevant
Brainerd's admitted digression on Rousseau (pp. 7984),
subtitled "Is Mother Nature Always Right?") If Brainerd's
recommended methods of teaching, or training, are worthwhile,
whether natural or not, then of course they should be used.
For example, he mentions "correction training," in which
verbal feedback from the experimenter is accompanied by
"a tangible reward (e.g., candy or a token) following
correct responses" (p. 86). This method of teaching
71
is not recommended by some psychologists. Not only may the
reinforced behavior be extinguished when the reinforcements
are removed, but also, extrinsic rewards may actually de
crease the intrinsic value of the learning activity for the
subject, thus doing more harm than good (Levine & Fasnacht,
1974, p. 820).
Other types of training Brainerd mentions as success
ful are "rule learning" and "conformity training." In rule
learning, as the name implies, the students are taught a
rule or rules "which may subsequently be used to generate
correct responses on a concept test" (Brainerd, 1978, p. 87).
In conformity training, children who missed the concept
questions on protests are grouped with children who answered
the pretest questions correctly. Asked to arrive at "con
sensual answers," the conservers apparently convinced the
nonconservers. Brainerd says that "79% of the pretest
nonconservers learned all five concepts. . All improve
ments were stable across a 1week interval" (p. 88). One
must accept the statement that the 79% gave correct res
ponses; Piaget would want to wait more than a week to see
whether the children had "learned all five concepts."
In further critique, Brainerd selects three predictions
he says the Piagetian theory makes. First, learning inter
acts with children's knowledge of tobetrained concepts.
But, Brainerd says, few learning theories would not say
that.
Secondly, preoperational children cannot learn con
crete operations concepts. Brainerd says that this has
been disproved (Brainerd, 1978, p. 105).
And thirdly, concepts belonging to different stages
must be learned in a certain order. Brainerd says that
this is a trivial outcome; the way the stages are set up,
each stage includes the concepts of the previous stage
(pp. 100101). He concludes that "although we may need a
readiness perspective on concept learning, Piaget's
approach does not seem to be it" (p. 105).
The basic thrust of this study concerns the possibili
ty of improving our knowledge of how students learn, or fail
to learn, mathematics, the fraction concept in particular.
The value of Piaget's theory in this effort, if any, will
not be that it is correct and aesthetically satisfying in
every detail, but that it adds to our knowledge of how
students learn or fail to learn, that it enriches our diag
noses of students; difficulties, and possibly, that it,
eventually, inspires more successful teaching techniques.
Consequently, the first and third predictions, which
Brainerd finds insignificant, are not weak points in this
context; the ideas might prove to be valuable to an educa
tor attempting to sequence instruction for the student's
maximum success.
73
In trying to refute the second prediction ascribed
to Piagetian theory, Brainerd again violates Piaget's
assumptions. In setting the stage for the studies that he
says prove that preoperational children can learn concrete
operational concepts, he states, "preschoolers should be
almost completely untrainable. . in a sample of 3 to
4yearolds . it should be safe to assume that concrete
operational mental structures are not present" (p. 96).
As mentioned previously, Piaget's theory does not say what
should be, but describes what has been observed. The mental
structures of a child are developed individually and may not
be congruent with those of his age group. It seems evident
from Piaget's experiments that it is not "safe to assume"
anything about a child's thinking. Brainerd cites a train
ing study on number and length conservation with 4yearolds,
saying that
there was clear evidence of transfer. The
same subjects passed roughly 41% of the
items on the mass and liquid quantity post
tests. (Brainerd, 1978, p. 100)
With the item format not available, it is not convincing
that 41% correct answers represents clear evidence. In the
other experiments mentioned, retention was again tested only
one week after training.
If Brainerd's objection is correct, however, and
Piagetian concepts can be trained, and if certain Piagetian
concepts are found to be related to mathematical concepts,
then the path is obvious: students should be trained in
Piagetian concepts before, or in conjunction with, their
mathematical instruction. Certainly many mathematics
educators have seemed to heed Piaget's (1973) invitation:
If mathematics teachers would only take the
trouble to learn about the "natural" psycho
genetic development of the logicomathematical
operations, they would see that there exists a
much greater similarity than one would expect
between the principal operations spontaneously
employed by the child and the notions they
attempt to instill into him abstractly. (p. 18)
Piaget optimistically says that
one can anticipate a great future for coopera
tion between psychologists and mathematicians
in working out a truly modern method for
teaching the new mathematics. This would con
sist in speaking to the child in his own lan
guage before imposing on him another readymade
and overabstract one, and, above all, in
inducing him to rediscover as much as he can
rather than simply making him listen and repeat.(p. 19)
Lovell has called for studies which give other than
pass or fail responses, and suggests that more emphasis
should be placed on careful observation of the schemes
which lead to correct solutions. He says that
such studies are likely to throw light on the
nature of the schemes (in respect of mathema
tical ideas) available to normal as compared
with dull and disadvantaged pupils. . The
classical Piagetian structural model must be
supplemented. (Lovell, 1975, p. 187)
Carpenter expressed the research need as follows:
What is essential is the construction of good
measures of children's thinking and the iden
tification of specific relationships between
performance on those measures and the learning
of particular mathematical concepts. (p. 76)
Several studies have used Piaget's cognitive structures
as measures of children's thinking and have attempted to
relate them to mathematics learning. Those most pertinent
to the study of fractions will be discussed.
Hiebert and Tonnessen (1978) wanted to extend Piaget's
analysis of fractions in continuous situations to other
physical interpretations. They decided to replicate the
experiments with continuous models and to investigate
whether Piaget's analysis applied equally well to a discrete
model of fractions. Nine children, 5 to 8 years old, were
given three tasks in videotaped interviews. They were
asked to divide a quantity of material equally among a
number of stuffed animals so that the material was used up.
In the area task, a circular "pie" of clay was used; in
the length task, a piece of licorice, and in the set/subset
task, penny candy (four times as many candies as animals).
Two children had tasks dealing with halves, three with
thirds, three with fourths, and one with fifths.
Six of the nine children succeeded in discrete (set/
subset) tasks; only two succeeded in both length and area
tasks. The explanations offered by the authors are that
discrete quantity tasks do not require welldeveloped
anticipatory schemes, while continuous quantity tasks do.
Discrete tasks were solvable by number strategies (e.g.,
counting); length and area tasks first required a sub
division into equal pieces.
Concerning the developmental sequence, the authors
said that in area representation, some children were
successful with halves and fourths, but not with thirds.
In the length task, the level of difficulty corresponded
with the number of parts. And in the set/subset task,
no orderofdifficulty sequence was observed. The pre
dominant onetoone partitioning strategy was used with
equal success for all fractional numbers.
Hiebert and Tonnessen (1978) conclude that the Piagetian
conceptual analysis of fraction is adequate to describe the
children's strategies in the length and area tasks, but not
in the set/subset task. Nothing inherent in the task
forces the child to use the partwhole approach, since the
task can be solved by simpler strategies (counting and one
to one partitioning).
Further, they say that meaningful comparison of the
discrete and continuous interpretations of fractions was
not possible. They did not find generalizable identifying
criteria that define a complete partwhole fraction concept
across all physical interpretations. "It appears that
further theoretical work involving a conceptual analysis
of fraction must include psychological, as well as
logical, analyses if this comparison is to be meaningful"
(Hiebert and Tonnessen, 1978, p. 378).
In upper elementary school, the ratio interpretation
of fractions is important. But if, as Piaget has suggested
(Lovell, 1971a, p. 8) and Lovell and others have confirmed
(Lovell and Butterworth, 1966), proportional reasoning is
not available to children until they reach the period of
formal operations, then how can they understand fractions
as ratios and solve proportions? Steffe and Parr (1968)
investigated the success with fractions of fourth, fifth
and sixth graders who had been exposed to two curricula,
one using fractions as ratios, the other, as quotients, or
fractional numbers. Among the authors' conclusions were
the following statements:
Children solve many proportionalities
presented to them in the form of pictorial
data by visual inspection both in the case
of ratio and fractional situations.
Whenever the pictorial data, which display
the proportionalities, are not conducive
to solution by visual inspection, the
proportionalities become exceedingly diffi
cult for fourth, fifth and sixth grade
children to solve, except for the high
ability sixth graders. (p. 26)
The authors raise this question, in implications for
further research:
Is it possible to construct a "readiness
test" for the study of ratio and fractions
in the elementary school? Such a test
may have its foundation in the psychological
theory of Piaget. (Steffe & Parr, 1968, p. 26)
Efforts are being made to use Piagettype tasks in
classroom diagnosis. Johnson (1980) suggests that ele
mentary teachers can use such tasks in diagnostic inter
views. Information thus gathered, along with that obtained
through traditional means, "allow the teacher to develop a
program based on the diagnosed strengths and weaknesses of
the child" (p. 146). A set of 18 tasks are described.
The two tasks that are relevant to this study are Task 17,
"Meaning of a fraction," and Task 18, "Concept of a
fraction."
Task 18 is an abbreviation of the cakecuttings of
Piaget, discussed in Chapter 2. However, Johnson's di
rections do not seem to be complete enough for a teacher's
guide. The teacher may not know how to interpret it when
a child cuts off two small slices for the two dolls,
leaving a large portion of cake (perhaps trying to get rid
of it under the table). Some sample expected answers could
be provided, along with some criteria for deciding which
answers exhibit what sort of understanding.
79
Task 17 purports to test the student's understanding
of the meaning of a fraction. The materials are two 4inch
by 8inch rectangular regions. Here are the directions:
Take the two regions and ask the child if they
are the same size. The child should be allowed
to place one on top of the other to verify.
Now mark region A and region B as in the diagram
below.
A B
Ask, "What is a fraction name for each part of
region A?" "What is a fraction name for each
part of region B?" Now point to a part of
region A and ask if that part is the same size
as one part of region B (pointing to a part of
B). (Johnson, 1980, p. 164)
There may be confounding factors in the above example,
relevant to the tasks used in the present study. First, an
optical illusion may be operating; the horizontal length
of region B may appear to be greater than that of region A,
when in fact they are the same. There is also the example
in Piaget's study of conservation of area, reported earlier
in this paper, where the different orientation of two iden
tical bricks changed a child's perception of the area re
maining in the field. While this situation is not exactly
analogous, it casts doubt as to whether the child will see
the horizontal parts of region B as equivalent to the
vertical parts of region A.
It seems to be assumed by Johnson that the child can
conserve area. Piaget has reported protocols in which
children have started with two rectangles exactly alike;
having cut one into two or more parts, the experimenter
asked whether there was as much room in each, the cut
rectangle and the uncut rectangle. Several children
maintained that there was more room in the rectangle which
had not been cut (even when the experimenter put the cut
pieces back together, right on top of the uncut rectangle)
(Piaget, Inhelder and Szeminska, 1960, pp. 275277). Could
the markings on Johnson's rectangles function in the same
way, to make the child think the area had changed? If the
vertical marks changed region A, did they change A in the
same way that the crossed marks changed B, if they changed
B?
Care also needs to be taken in the use of vocabulary.
What is the interviewer's definition of "the same size?"
And does it happen to be the same definition the child is
using? A tall skinny man and a short fat man might have
the same mass or perhaps the same volume (or possibly even
both?), but one might not say that they are the same size.
These considerations echo the comment of Lovell (1971a)
He was, in turn, quoting Mayer (1961), who said that future
teachers needed a "course which attempts to explore the
profound aspects of the deceptively simple material they
are going to teach" (Lovell, 1971a, p. 12). Certainly Task
17 was more complicated than it appeared on the surface.
81
Of all the models of fraction, the area model seems to
be appealed to most often in schools. Therefore Taloumis's
(1975) area study may bear on the teaching of the fraction
concept. Taloumis wanted to standardize the reporting of
abilities of primary school children in area conservation
and area measurement. Also to be studied was the effect of
test sequence on performance. Of the 168 children in
grades 1 through 3, half did the area measurement tasks
first, the other half, the area conservation tasks first.
Tests were administered individually.
There were three conservation tasks. In the first one,
two rectangles (index cards) were shown. As the child
watched, one of the index cards was cut on the diagonal.
The halves were separated, rotated and rearranged into an
isosceles triangle. The child was asked whether the two
shapes (rectangle and new triangle) had the same amount of
space.
The second conservation task was the farm problem
discussed earlier in this paper. In the third task, the
congruence of two green "gardens" and the congruence of
two brown "plots of ground" for flowers were established.
The brown plots were placed in the gardens, and one of the
plots, which was sectioned, was changed into successively
longer rectangles. The child was asked whether each garden
had the same amount of ground for flowers, or, if not, which
one had more.
82
In the area measurement tasks, the child was to use as
measuring devices 1unit squares, 2unit rectangles, and
halfunit squares to compare two noncongruent shapes (the
unions of rectangles). In the second task a triangle was
to be compared with a polygonal shape.
Taloumis found that the sequence of presentation did
affect the performance on the second group of area tasks.
The conclusion includes the following:
If area conservation tasks are administered
first, the scores on area measurement tasks
are increased, and vice versa. The impli
cations for future researchers are: l)train
ing in area measurement may improve a child's
performance in area conservation; 2) learning
takes place across Piagetian tasks given in
sequence. (Taloumis, 1975, p. 241)
She concludes that Piaget's theory that the ability to
measure quantities is dependent on acquired concepts of
conservation does not appear to be completely tenable.
Piaget's stand may not be be completely tenable. On
the other hand, there might be a simple explanation for
Taloumis's results: the two tasks are not all that different.
In the first conservation task (Ci), for example, two plane
figures are being compared. In the first measurement task
(MI), two plane figures are also being compared, but with
the assistance of some smaller increments of area (unit
squares, etc.).
Consider Piaget's work on area. In a conservation of
area task, a child is being asked to compare the area of
a rectangle with a second one which has been transformed
into a pyramid. After asking the usual question about the
amount of room in each shape, the interviewer says, "What
if I covered it with cubes" (Piaget, Inhelder, & Szeminska,
1960, p. 281)? The child is then led to cover first one
area, then the other, with the cubes, which serve exactly
the same function as Taloumis's unit squares do in task M,.
For Piaget, the tasks C, and MI are both conservation
tasks. Therefore it is not at all surprising that they
were found to be interdependent.
When Piaget studies the measurement of area, the task
is slightly different. He again asks the child to compare
the areas of two polygonal regions, but using two separate
techniques. With the first method, there are enough or
nearly enough measuring cards to cover the area being
measured. He wants to discover the age at which children
will use the smaller cutouts as a middle term, or common
measure. In the second method, the subject is presented a
limited number of square unit cards which he must then move
from one part of the surface being measured to another.
The point then being observed is not simply that the child
answers "equal" or "not equal," but whether the child
realizes the transitivity of a common measuring term, a
basic component of measurement (Piaget, Inhelder, & Szemin
ska, 1960, pp. 292293).
In explaining the dependence of measurement of area
on conservation of area, Piaget mentions the "harder prob
lem," the conservation of completmentary areas, where the
child must not only understand the space of "sites" which
are occupied and those which are vacant, but also the re
ciprocal relation between the area within a perimeter and
the area outside it (Piaget, Inhelder, & Szeminska, 1960,
p. 291). A child may be able to comprehend the area of a
thing which takes up space before the area of the "site,"
or space taken up. The analogous difference in one di
mension was mentioned in the discussion of conservation
of distance.
In addition to realizing the transitivity of a common
measuring term, in order to measure area, the child must
"understand composed congruence (i.e., that a number of
sections taken together equal the whole which they cover)"
(p. 294).
Taloumis (1975) says further that the scores showed
that significant learning took place during the assessment,
and that there seemed to be transfer of learning in both
directions (p. 241). This result is not incompatible with
Piagetian theory. For children who were transitional, the
testing situation may have provided the necessary cognitive
conflict, or disequilibrium, to enable equilibration at a
higher level with regard to the conservation of area. In
85
fact, the "keynotes" in Smedslund's (1961) training study,
conflict with no feedback, were apparently present in
Taloumis's assessment procedure.
The explorations with concrete manipulatives may have
also been helpful to the children in Taloumis's study.
There is considerable interest in the use of manipulatives
in instruction and, more recently, in diagnosis.
Concrete Versus Abstract Modes of Presentation
The mathematics education literature has for years in
cluded recommendations that concrete, manipulable materials
be used in instruction (Lovell, 1971a; NCTM, 1954; Suydam,
1970; & Swart, 1974). Shulman (1970) says that "Piaget's
emphasis upon action as a prerequisite to the internaliza
tion of cognitive operations has stimulated the focus upon
direct manipulation of mathematically relevant materials in
the early grades" (p. 42). Of course, as Piaget uses
"action," internal cognitive operations are actions. In
Piaget's concept, actions performed by the subject are the
raw materials of all intellectual and perceptual adaptation
(Flavell, 1963, p. 82). The infant performs overt, sensori
motor actions; with development, the intelligent actions
become more internalized and covert.
As internalization proceeds, cognitive actions
become more and more schematic and abstract,
broader in range, more what Piaget calls re
versible, and organized into systems whicare
structurally isomorphic to logicoalgebraic
systems. (Flavell, 1963, p. 82) [Flavell's emphasis]
Flavell insists that despite the enormous differences
between them, the abstract operations of mature, logical
thought are as truly actions as are the sensorimotor
adjustments of the infant (Flavell, 1963, p. 82). Piaget's
notion of development, then, is active, interactive; think
ing and knowing are actions that one performs.
Flavell also interprets certain of Piaget's beliefs
about education: In teaching a child some general principle,
one should parallel the developmental process if possible.
The child should first work with the principle in a concrete
and actionoriented context. Then the principle should be
come more internalized, with decreasing dependence on per
ceptual and motor supports (moving from objects to symbols
of objects, from motor action to speech, etc.)(p. 82).
It must be remembered that Piaget was not himself an
educator; he provided a theoretical rationale for certain
recommendations, but no practical instructions for teaching.
Some mathematics educators have tried to apply strategies
which would provide for active learning in the spirit of
Piaget. They reason that children should be provided both
concrete or manipulable objects and diagrams which could
illustrate the mathematical concepts being taught symboli
cally. It is not clear that teachers or students always
know what to do with these learning aids.
Payne (1975) reviewed research on fractions done
primarily at the University of Michigan. Most of the
studies that compare different instructional sequences
are not germane to this study, but some do relate to the
question of mode of presentation. For example, Payne says
that "where meaningful approaches to operations on frac
tions have been compared to mechanical or rule approaches,
there appears to have been some advantage for the ones
that were meaningful" (p. 149). Further, he says, "when
there was an advantage favoring meaningful approaches, it
was usually most evident on retention tests" (p. 149).
"Meaningful" and "mechanical" were not always clearly de
fined; however, Green's (1970) study, according to Payne,
had a logical development but relied heavily on physical
representations (Payne, 1975, p. 150).
Green investigated the effects of concrete materials
(one inch paper squares) versus diagrams and an area model
versus a "fractional part" model on fifth graders' learning
the algorithm for multiplication of fractions. Since the
study is not available in its entirety, excerpts of Green's
summary, as quoted by Payne, will be given. Basically the
approach using area was more effective, and the diagrams
and manipulative materials were equally effective. Of further
interest is Green's note:
The failure in finding a fractional part
of a set definitely points to the need to
find a more effective way to teach this im
portant concept. Particular attention should
be given to overcoming the difficulty chil
dren have with the "unit" idea, relating the
model and the procedure for finding a frac
tional part of a set, and delaying the rule
until there is understanding of the concept.
(Payne, 1975, p. 153)
Perhaps the difficulty alluded to is caused by the need
to have logical class inclusion firmly in place for the
understanding of a partwhole relationship (Kieren, 1975;
Piaget, Inhelder, & Szeminska, 1960).
Payne says that Green's results were better than those
of similar studies. Green's approaches all involved visual
models: either concrete materials that children manipulated
or diagrams of regions. Since all her retention scores
were almost 90% of posttest scores, Payne concludes that
the use of visual materials in developing algo
rithms has a more important effect on retention
than does a purely logical mathematical develop
ment. (Payne, 1975, p. 155)
However, the use of manipulative materials did not seem to
have the expected advantage in achievement. Payne says
that evidently it is not a simple thing to relate a child's
thought to his use of concrete materials or diagrams (p. 156)
Kurtz and Karplus (1979) undertook a training study to
see whether ninth and tenth grade prealgebra students
could be taught to become proficient in proportional reason
ing. Manipulative materials were hypothesized to be
more effective and to engender more favorable attitudes
than paper and pencil activities alone. The authors' con
clusions were that proportional reasoning was taught suc
cessfully, that manipulative materials and paper and pencil
activities provided equal cognitive gains, but that the
manipulative version was considerably more popular than the
paper and pencil version (Kurtz & Karplus, 1979, p. 397).
Except for studies such as the above, the use of mani
pulatives in instruction has been primarily restricted to the
elementary schools. An interesting result came from a study
(Barnett & Eastman, 1978) of ways to train prospective ele
mentary teachers in the use of manipulatives in the class
room. Subjects either received demonstrations only (control
group) or both demonstrations and "hands on" experience with
the manipulatives (experimental group). On the test on the
uses of manipulative materials, the authors found no signifi
cant difference between the groups. However, the experimen
tal group did better on the mathematics concept posttest.
The authors suggest that
a plausible explanation for this result may
be that although subjects do not learn to
"teach better" by actually using manipulatives,
they may better learn the mathematics concepts
involved. The results of several studies have
suggested that many preservice elementary
teachers have not reached the level of abstract
operations, and hence they might need manipu
lative aids themselves in order to learn the
mathematical concepts that they are expected to
teach. (pp. 100101)
