Selected Piagetian tasks and the acquisition of the fraction concept in remedial students

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Selected Piagetian tasks and the acquisition of the fraction concept in remedial students
Dees, Roberta Lea, 1938- ( Dissertant )
Kantowski, Mary Grace ( Reviewer )
Bengston, John K. ( Reviewer )
Bernard, Donald H. ( Reviewer )
Lewis, Arthur J. ( Reviewer )
Place of Publication:
Gainesville, Fla.
University of Florida
Publication Date:
Copyright Date:
Physical Description:
xi, 276 leaves : ill. ; 28 cm.


Subjects / Keywords:
Circles ( jstor )
Educational research ( jstor )
Fractions ( jstor )
High school students ( jstor )
Learning ( jstor )
Mathematics ( jstor )
Mathematics education ( jstor )
Rectangles ( jstor )
Schools ( jstor )
Students ( jstor )
Arithmetic -- Study and teaching (Primary) ( lcsh )
Cognition in children ( lcsh )
Curriculum and Instruction thesis Ph. D
Dissertations, Academic -- Curriculum and Instruction -- UF
Fractions ( lcsh )
bibliography ( marcgt )
non-fiction ( marcgt )
Spatial Coverage:
United States -- Florida -- Gainesville


This clinical study was designed to answer two questions: 1, is there a relationship between the acquisition of cognitive structures, as exemplified in Piaget-type' 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 assessing 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, classification, class inclusion, conservation of distance, and conservation of area. Tasks were prepared in concrete or manipulable 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, number line, and area models) and equivalence and comparison of fractions. The written test also included addition and subtraction 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 (median 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 fractions 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 students 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 discrete model of fraction. Students performed better on the concrete form of the class inclusion task than on the pictorial (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.
Thesis (Ph. D.)--University of Florida, 1980.
Includes bibliographic references (leaves 268-275).
General Note:
General Note:
Statement of Responsibility:
by Roberta Lea Dees.

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University of Florida
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Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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Copyright 1980


Roberta Lea Dees


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.










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


Subjects 106
Instruments 106
Piaget-type Tasks 108
Fractions Tests 110
Procedure 112

Findings and Discussion
Piaget-type Tasks
Concrete Fractions Test
Written Fractions Test
General Observations
Resulting Modifications
Tasks Instrument
Fractions Tests


Fractions Tests
Planned Data Examination


Overall Data
Possible Relationships, Questions ]
and 2
Qualitative Data from Student Protc
Overall Data
Possible Relationships, Questions I
and 2
Qualitative Data from Student Protc
Limitations of the Study


Implications for Future Research
Question 1
Question 2
Other Issues
Suggested Research
Implications for Teaching










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,

Concrete-Written and Written-Concrete 151

10. Average Percentage of Students Correct Per

Item by Model of Fraction 165


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


14. Equivalent Fractions, Concrete and Written

15. Frequency Distribution of Scores on Classifica-

tion, Subtask A and Subtask B

















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



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 Piaget-type 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,


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.


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 hand-held 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

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 1972-73 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 student-teacher 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


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 hands-on 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


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.


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

Piaget-type 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?


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


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


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 "set-theoretic" (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


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 part-whole model);

a set (the part-group model); or a unit segment of a number

line (which the author considered a specific form of the

part-whole model). The investigator constructed a hierarchy

of dependent subconcepts of the fraction concept and designed

a fraction concept paper-and-pencil 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
part-whole and part-group with a point on a number
models line

associating a fraction using a fraction in a com-
with a part-whole model prison situation invol-
or with a part-group ving the respective model

associating a fraction associating a fraction with
with a part-whole model the respective model where
or with a part-group 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 part-whole model with the respective
or with a part-group model having noncongruent
model having congruent parts, where (in the case
parts of part-whole 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
in-computation with whole numbers and not
developed concepts of part-whole 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 part-whole situations, learn
verbal and numerical codes for these, and
learn to correctly identify a code (fraction)
with a part-whole setting. As a cognitive
capstone of this ordered pair concept set,
the child must realize that a part-whole
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 part-whole 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 part-whole settings. He mentions four settings:

state-state (static comparison between a set and one of

its subsets), state-operator (divide 3 cookies among

5 persons), operator-state (use 5 of a dozen eggs) and

operator-operator (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


the child must be able to assign a pair of
numbers to a part-whole situation. This, of
course, entails the ability to logically
handle the part-whole 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 end-to-end 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

pre-experience 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 pre-decimal 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 developer-instructional 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 in-depth 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


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 Piaget-type 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, one-to-

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


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,"


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. 312-313)

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).


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. 300-301).

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
situation--that 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, step-by-step affair in which the
sorting criterion is constantly shifting
as new objects accrue to the collection.
. Partly in consequence of this inch-
by-inch 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 similarity-of-attribures 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. 304-305)

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 classification-relevant 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. 307-309). 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 (one-dimensional) and distance is space

(one-dimensional) 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. 32-34) 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. 262-273), 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


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. 263-264)

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


This situation is reminiscent of some optical illusions

in which the orientation changes the appearance of a


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

pre-fraction 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 neo-Piagetian

research will be included. Last will be a discussion of the

concrete-versus-symbolic 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


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 half-to-whole, and generally,

part-to-whole, relationship can be understood by the child

perceptually. But, he says,

it is a far cry from such perceptual or
sensori-motor part-whole 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 re-worked 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 part-whole 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 part-part, rather than a

part-whole, 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


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

half-cakes than there is in one whole (p. 327). Subject

GIS says that they are the same, but: 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 de-composed and also
re-assembled. 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. 334-335)

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 three-dimensional, 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

fractions as ratios or quotients or how they come to per-

form mathematical operations with fractions. These other

aspects need to be given in-depth study also.

The next topic is students' general knowledge of frac-

tions as taught in school, and as evidenced by assessment


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, "13-year-olds 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-

year-olds, 17-year-olds, 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 13-year-olds and 66% of 17-year-olds 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

17-year-olds, 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 13-year-olds 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 13-year-olds, 17-year-olds,


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 17-year-olds (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 13-year-olds and 49% of 17-year-olds an-

swered correctly. The authors say,

The strongest distractor for both the
13-year-olds and the 17-year-olds was 2/3.
The exercise clearly shows that 13-year-olds
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

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%
(Ginther, Ng, & Begle, 1976, pp. 3-4)

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 (Farnham-Diggory,

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


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 in-depth 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 state-of-the-art

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. 20-30) 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.

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,


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

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


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


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

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); semi-concrete

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


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 concrete-versus-abstract 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-


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:


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 paper-and-pencil 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 "twenty-five 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)


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. 370-378) 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 well-suited 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.


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 number-related 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


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, higher-level 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 self-discovery.

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. 83-84). The same

original sources, in this case Piaget's theory as stated

by his co-workers, can yield different interpretations.

Another person, reading the same quotations Brainerd has

selected (pp. 69-78), 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. 79-84),

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


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 1-week 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 to-be-trained concepts.

But, Brainerd says, few learning theories would not say


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. 100-101). 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.

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

4-year-olds . 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 4-year-olds,

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 logico-mathematical
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 ready-made
and over-abstract 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 well-developed

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 order-of-difficulty sequence was observed. The pre-

dominant one-to-one 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 part-whole 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 part-whole 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 Piaget-type 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


Task 18 is an abbreviation of the cake-cuttings 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.


Task 17 purports to test the student's understanding

of the meaning of a fraction. The materials are two 4-inch

by 8-inch 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

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. 275-277). 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


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.


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


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.


In the area measurement tasks, the child was to use as

measuring devices 1-unit squares, 2-unit rectangles, and

half-unit 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. 292-293).

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


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 whic--are
structurally isomorphic to logico-algebraic
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 action-oriented 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 part-whole 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. 100-101)