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STUDENTS' CONCEPTIONS OF VARIABLES
AND THEIR USES FOR GENERALIZATION
OF MATHEMATICAL PATTERNS

presented by

Donna Elaine Bird Ericksen

has been accepted towards fulfillment
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STUDENTS’ CONCEPTIONS 0F VARIABLES
AND THEIR USES FOR GENERALIZATION
OF HATHEHATICAL PATTERNS
By

Donna Elaine Bird Ericksen

A DISSERTATION

Sub-itted to
Hichigan State University
in partial fulfillment of the require-ents
for the degree of

DOCTOR OF PHILOSOPHY

Department of Teacher Education

1988

ABSTRACT
STUDENTS’ CONCEPTIONS 0F VARIABLES
AND THElR USES FOR GENERALIZATION
OF HATHEHATICAL PATTERNS
By

Donna Elaine Bird Erickson

Statement 21 £25 Problem
This study investigated students’ conceptions of
variables as tools for generalizing patterns. The specific
research questions addressed were:

I. Uhat types of aatheaatical patterns can students
recognize?

ll. Uhat types of patterns can students generalize
by using variables?

lll. Uhat aeaning do generalizations using variables
have for students?
flgihgig
The research questions were addressed by in-depth
interviews with seventh graders and high school algebra
students (n 8 13 for each group). The interviews consisted

of five tasks.

Findings

One conclusion of the research was that the saaller
the number of surface features built into the groups of
probleas the aore successful students were at recognizing
the deep patterns (concepts) which existed within the
problea. One result of this fact was that patterns in

tables were easier for students to recognize than patterns

in groups of expressions, sentences and sequences.
Patterns which were built on one operation (i.e., x + 5 and
3x) were aore easily recognized than patterns involving
two operations (i.e., 5x + 3 and 5x - 4).

Once algebra students and seventh graders recognized
a pattern they were usually able to generalize it using
variables. It was harder for students to recognize
patterns than it was for thee to generalize those same
patterns. lncorrect generalizations aade by the algebra
students usually contained too aany variables. The errors
aade by the seventh graders indicate that ease of thee
aight be confused by the difference between a constant and
a variable. i

working backwards free a generalization containing
variables to a aatheaatical situation that fits the
generalization was not easy for students. One of the
techniques that students used to try to aake sense of the
generalizations was the use of the variable to represent
the beginning initial of the object the problea was about.
Another technique used by students was to assign the
variable a constant value and to talk about it as if it
were constant. A technique which was only used by seventh
graders was to take expressions and change then into

sentences by setting thea equal to acne value.

DEDICATION

This dissertation is dedicated
to ay husband
Craig Stuart Ericksen
and to ay grand-other
Harie Tiffany Donner.

ACKNOHLEDGHENTS

Many people have supported and encouraged ae
throughout ay tiae at Michigan State University. I aust
first express ay appreciation for the work that Charles
(Andy) Anderson has put in during the last two years
helping as to finish this dissertation.

Andy: 1 night not have always enjoyed the process,
or our interactions, but I have always appreciated the aany
hours you have put in helping ae to coaplete this project.
I could not even begin to count the aany drafts you have
read or the coaaents you have written. I have learned aany
things free you. i an looking forward to continuing our
work together.

l further want to acknowledge the other aeabers of ay
coaaittee: Glenda Lappan, Perry Lanier, Linda Anderson and
Elizabeth Phillips.

Glenda: l reaeaber the first tiae that I heard you
speak at a conference. I thought to ayself that here was a

lady that I would not aind getting to know better. You
represent to ae the type of aatheaatics educator that

soaeday I would like to be. You were then, and you
continue to be, a role aodel for no. The only difference
now is that you are also a friend. Thank you for all of

your help with this dissertation and throughout ay prograa.
I look forward to continued personal and professional
interactions with you.

Perry: I could not have chosen a better person to
chair ay coaaittee. Your help and advice throughout ay
prograa was auch appreciated.

Linda: A few years ago you helped ae when I needed a
listening ear. Your support will never be forgotten.

Betty: Froa you I have learned how iaportant it is
to work with others. to work hard, and above all, to have
fun with your work. 1 aa looking forward to a continued

personal and professional relationship with you.

ii

I thank again all five of ay coaaittee aeabers. Hy
gratitude goes beyond what I have written here.

I further want to thank the three teachers and two
schools who allowed their students to take part in the
study. I also thank all of the students who willingly
volunteered to participate in the study. There would not
be a study without your support.

I thank Sigrid Uagner for her help in supplying as
with copies of articles which she had written about
students’ conceptions of variables.

I thank Hr. Cecala for setting an early exaaple for

ae of what it aeans to be a good aatheaatician. You helped

to inspire ae to chose a career in aatheaatics. i also
thank Hrs. Ruckaann. You gave ae the writing tools
necessary to carry ae through all of ay studies. Both of

you will always stand as exaaples of what excellent
teaching is.

I wish to acknowledge the support of ay faaily, both
the Birds and Ericksens. l proaise you that you do not
have to,hear about this any acre.

1 also need to thank all of the people who have been
special friends at different points during ay ties at
Michigan State University: Helene Anthony, Debbie Ball,
Theron Blakeslee, Stephene Gregg, Okee Lee, Lois
Niedzwiedski, Karen Nordhoff, Chet Pierce, Kathy Both, and
Pan Schraa. I hope that l have naaed all who have helped

ae throughout the process, but if I have left off your naae

iii

I apologize. All of ay friends have aeant a lot to as and
I thank you for your support and friendship.

I also thank ay neighbors and friends the Godwins.
Hike and Debbie you have shared your boss, children and
aany good tiaes with as. In aany indirect ways you helped
as with this work.

I thank Central Hichigan University for eaploying so
before I had finished this project. l appreciate not ever
being bothered concerning when I would finish. I also
appreciate the teaching schedules which you have given ae
which have allowed ae to work both as a faculty aeaber and
on this dissertation.

Finally, I need to acknowledge all of the support,
encourageaent and love shown to as by ay husband. Craig, I
can not even begin to properly say in how aany ways you
helped as with this project. I think that the writing of
this dissertation was half ay effort and half your
encourage-ant.

To everyone aentioned in these acknowledgaents and to

all who I have unintentionally forgotten, thanks!

iv

TABLE OF CONTENTS

LIST OF TABLES . . . . . . . . . . . . .

LIST OF FIGURES . . . . . . . . . . . . .

Chapter

II.

INTRODUCTION . . . . . . . . . . .

Background . . . . . . . . .

The Difference Between Arithmetic and Algebra

The Historical Development of Variables .
The Nature of the Difficulty Uith Variables

The Historical Versus the Hodern Presentation

of Variables . . . . . . . . .

The Research Problem . . . . . . . .
Research Questions . . . . . . . . .
Overview of the Research Design . . . .
Assumptions . . . . . . . . . . .
Limitations . . . . . . . . . . .

Overview of the Dissertation . . . . . .
REVIEW OF THE LITERATURE . . . . . . .

Introduction . . . . . . . . . . .
Articles Concerning Students’ Conceptions of
Symbols in General and Variables in
Particular . . . . . . . .
Research on, And Articles About, Teaching
Hodels Designed to Enhance Students’
Understanding of Variables . . . .
Summary . . . . . . . . . . . . .

"ETHODS O O O O O O O O O O O O 0

Overview . . . . . . . . . . . .
Subjects . . . . . . . . . . . .
Data . . . . . . . . . . . . .

Development of the Interview . . . . .
The Interview . . . . . . . . .
The Card Sort and Generalization of Card Sort
Cards . . . . . . . . . . .

V

Page

viii

p

(”#00

11
12
13
13
14

16

16
18
34
42
46
46
47
48
48
49

49

Chapter

IV.

Recognition and Generalization of Patterns in
Tables . . . . . . . . . . .

Interpretation of Expression, Sentence and
Sequence Generalizations by Development
of Word Problems . . . . . . . .

Data Collection Procedure . . . . . .
Initial Meeting Between Participating
Teachers and Principals . . . . .

Gathering Information on Student Ability .
The Interview Schedule . . . . . . .
Pilot Interviews Uith The Students . . .
The Data Collection Interviews . . . .
Data Analysis Procedures . . . . . . .
Transcription of the Audio Tapes . . . .
Analysis of the Transcripts . . . . .
Summary . . . . . . . . . . . . .

RESULTS AND DISCUSSION . . . . . . . .

Introduction . . . . . . . . . . .
Results . . . . . . . . . . . . .
Research Question One: Uhat Types of Deep
Patterns Can Students Recognize? . .
Issue One: What Deep Patterns Did
Students Recognize? . . . . . .
Issue Two: Uhat Makes a Pattern
Difficult of Easy For Students to
Recognize? . . . . . . . .
Types of Numbers Used . . . . . .
Properties of numbers . . . . . .
Common Features of more than one
Expression or Sentence . . . .
Hays of Representing Expressions,
Sentences and Sequences . . . .
Mathematical Symbols and Notations
Used . . . . . . . . . .
Issue Three: Uhat Types of Mistake
do Different Patterns Lead
Students to Make? . . . . . .
Conclusion . . . . . . . . . .
Research Question Two: Uhat Types of
Patterns Can Students Generalize by
Using Variables? . . . . . . . .
Issue One: What Patterns Can Students
Generalize and Uhat is the
Connection Between the Ability to
Generalize Patterns and the Ability
to Recognize Patterns . . . .
Issue Two: Uhat Types of Errors Do
Students Make Uhen Generalizing
Expressions and Sentences? . . . .
Conclusion . . . . . . . . . .

vi

Page
55
57
58
58
58
59
59
60
60
60
61
64
65

65
68

68
69
73
75
75
80
80
80

87
88

91

91

97
99

Chapter

Research Question Three: Uhat

Meaning

Do Generalizations Using Variables Have

For Students? . . .

Issue One: Uhat Strategies Do Students

Use To Obtain Meaning
Generalizations? .
Conclusion . . . . .
Summary . . . . . . . .

V. SUMMARY AND IMPLICATIONS . .

Overview . . . . . . .
Summary . . . . . . .
Subjects . . . . . . .
Data . . . . . . . .
Data Analysis . . . . .
Results . . . . . . .
Conclusion . . . .
Implications for Teaching . .
Implications for Future Research
Chapter Summary . . . . .

APPENDICES . . . . . . . . .

Appendix A: The Interview Tasks .
Appendix B: A Copy of the Permission

REFERENCES 0 O O O O O O O 0

vii

From

Slip . .

Page

101

102
106
108

109

109
109
111
112
112
113
116
117
121
123

124
127

129

LIST OF TABLES

Table Page

3.1 Example of a Set of Interview Cards . . . . 51

4.1 Relationships Among Research Questions, Analysis
Issues and Interview Tasks . . . . . . 67

4.2 Students’ Ability to Recognize Patterns as
Demonstrated by the Card Sort. . . . . . 69

4.3 Results From Recognition of Patterns in Tables
TESR O O O O O O O O C I O O O O 72

4.4 Uhat Surface Features Students Paid Attention to
Uhen Sorting the Cards Representing the Deep
Patterns of Multiplicative Identity and
Commutative Property of Multiplication . . 76

4.5 Uhat Surface Features Students Paid Attention to
Uhen Sorting the Cards Representing the Deep
Patterns of an Arithmetic Sequence with the
common Difference of Two and a Geometric
Sequence Uith a Common Ratio of Two . . . 77

4.6 Uhat Surface Features Students Paid Attention to
Uhen Sorting the Cards Representing the Deep
Patterns of a Area and Perimeter . . . . 78

4.7 Types of Surface Features . . . . . . . 79

4.8 Uhen Sorting the Cards Representing the Deep
Patterns of Multiplicative Identity and
Commutative Property of Multiplication Those
Cards Uhich Students Placed Together in More
Than Fifty Percent of Their Total Sorts . . 82

4.9 Uhen Sorting the Cards Representing the Deep
Patterns Arithmetic Sequences Uith a
Common Difference of Two and Geometric
Sequences with a Common Ratio Of Two Those
Cards Uhich Students Placed Together in More
Than Fifty Percent of Their Total Sorts . . 83

4.10 Uhen Sorting the Cards Representing the Deep
Patterns of Area and Perimeter Those Cards
Uhich Students Placed Together in More
Than Fifty Percent of Their Total Sorts . . 84

viii

Ability to Generalize Patterns Found on Card
Sort Cards Seventh Graders . . . . .

Results of Recognition and Generalization of
Patterns in Tables Tasks . . . . . .

Strategies of Students Uho Incorrectly
Generalized Patterns Found on Card Sort
Cards . . . . . . . . . . . .

Ability to Generate Uord Problems From
Generalized Statements . . . . . . .

Using the Variables as the Initial of the
Subject that the Problem is About . . .

ix

Page

92

96

97

103

105

LIST OF FIGURES

Figure 9389

3.1 Flowchart of the Research Interview . . . . 50

4.1 Flowchart of the Process Involved in Sorting
Interview Cards . . . . . . . . . . 88

CHAPTER ONE

INTRODUCTION

The mastery of algebraic concepts is a prerequisite
for further study for many academic majors. High school
graduation requirements across the United States are
changing from one to as many as three years of mathematics
for students. In some cases passing Algebra I is part of
the requirement. At the International congress of
Mathematics Educators (I.C.M.E. 5) held in Adelaide, South
Australia, in August 1984, members of the Algebra Theme
'Group expressed their common concerns for upgrading the
teaching of algebra. They also agreed that the most
difficult concept for high school algebra students is the
concept of variable (Carss, 1986).

Variables are the key for opening the door between
arithmetic and algebra. They give students the power to
talk not about numbers but about properties of numbers and
number systems. Historically variables were used as tools
for generalizing patterns. Today a limited picture of
variable, as place holder for solutions to equations, is
taught to students in classrooms. By painting such a
narrow view of variables it is questionable if students
recognize the large frame within which variables exist.

Research studies (Kuchemann, 1978: Clement, 1982: Hagner,

1978) suggest that students are unable to use variables for
the many mathematical applications for which they are
needed.

The link between arithmetic and algebraic
understanding and the role which variables play in making
this link is important in terms of students’ comprehension
of the field of mathematics as a whole. Hiebert and
Lefevre (1986) write about the need for investigating the
relationship between conceptual and procedural knowledge.

Students can solve algebraic problems manipulatively

without realizing that these problems represent
generalizations from arithmetic. Hhat is frequently
lacking is the underlying conceptual knowledge. These

students may come to understand the formal language and
rules of algebra, but they lack the conceptual knowledge
which gives these symbols and procedures meaning. For
these students variables become symbols to be manipulated
rather than a means of generalizing patterns from
arithmetic situations. Uithin this context algebra is not
an extension of previously learned mathematics, but a
totally new and separate branch of mathematical study.
Conceptual knowledge of variables implies a depth of
understanding which is not characterized by superficial
manipulation of symbols. This type of understanding is
defined by Lesh, Post and Behr (1987) who describe

understanding in part as the ability to do three things:

3

1. be able to recognize a concept in a variety of
representational systems,

2. manipulate flexibly the idea within a given
representational system and

3. to correctly translate the idea from one system to
another.

Lesh’s et al. (1987) definition of understanding centers
around recognition of concepts. Their definition also
involves the ability to express the concept within
different representational systems. The idea of
representational systems will be expanded later in this

chapter.

w

The Difference Between Agithmetic egg Alggggg

Pettitto (1979) describes algebra as the subject
which takes the arithmetic that students have previously
learned and organizes it into a formal system. In algebra
classes students are not expected to manipulate numbers as
they were in arithmetic classes. Instead the goal of
algebraic instruction is to teach students to manipulate
formal statements involving arithmetic operations. The
numerical content of such manipulations is irrelevant.

.Bertrand Russell (1938) also sums up the nature of
the difference between arithmetic and algebra when he
writes

"Elementary Arithmetic, as taught to children, is

characterized by the fact that the numbers occurring

in it are constants: the answer to any schoolboy’s

sum is obtainable without propositions concerning any

number. But the fact that this is the case can only
be proved by the help of propositions about any

4

number, and thus we are led from schoolboy’s

Arithmetic to the Arithmetic which uses letters for

numbers and proves general theorems."
The arithmetic which Russell refers to which uses letters
is algebra. The letters which are used to prove the
general theorems are called variables. It is the
introduction of variables and their use which extends
arithmetic into algebra.
In; flistggicel Development g1 Vegiaglgg

The historical development of algebra is considered
to have gone through three stages of development:
rhetorical algebra, syncopated algebra and symbolic algebra
(Eves, 1976). Rhetorical algebra represents the period of
algebra when a problem was written completely without
symbols, but rather as prose. A problem during this period
might be written out like this, ”what three successive
numbers added together to produce a new number equal to
24?" The next stage of development, syncopated algebra,
saw the abbreviation of frequently occurring quantities and
operations. During this stage the above problem might have
been written with the words ”added together" and equals
replaced, with symbols to reflect these commonly occurring
words and phrases. At this time symbols were used simply
as a shorthand method of writing problems. The final stage
of development, bringing algebra to modern times, was the
symbolic stage of algebra. This is present day algebra

where the above problem would be written

(y - 1) + (y) + (y + i) = 24 and accompanying directions

5
would be to ”solve for y”. In this stage the variable "y"
is standing for the middle of the three consecutive
numbers in the problem.

The introduction of the use of variables is
attributed to Diophantas (Breslich, 1939). About two
thousand years ago in his book A;i;hge;ig;_ Diophantas
introduced variables as place holders for unknowns.
Variables were easier to work with than prose. As Lamon
(1972) writes, we could say "The cube of a sum of two
numbers is the sum of the cubes of these numbers added to
the triple of the product of the sum of the numbers
multiplied by their product" but it is far more convenient
to write (a + b)3 8 a3 + b3 + 3ab(a + b).

Variables provide the bridge between arithmetic and
higher levels of mathematics because they allow
mathematicians to talk about whole classes of numbers
without specific reference to numbers at all.
Unfortunately, these tools of convenience for mathematician
often do not function as bridges, but rather barriers for
students as they move from arithmetic to algebra.

15; Meters QL Ln; Difficulty Qith_yarigbles

Bertrand Russell (1960) writes,

”When we come to algebra, and have to operate with x

and y, there is a natural desire to know what x and y

really are. That at least, was my feeling: I always

thought the teacher knew what they really were, but
would not tell me”.

Russell is summing up a problem which still exists with

algebra students. Students are confused by variables.

6
This is not surprising considering the abstract nature of
variables and the many uses of letters in mathematics.
Hagner (1983, 1980) talks about some of the roles
which letters can play in mathematics. There are several
different letters which are used in mathematics as
variables, others which are always constants, still others
which, for example in formulas, stand for abbreviations of
words. If the variable x is used in the equation
x + 2 I 2 + 3x the variable represents an unknown. However,
if it is used in the equation x + 2 I 2 + x, the variable
represents a generalized number. Depending on the context
in which variables are used mathematicians are able to
determine which usage of a letter is intended. Students
are not always as flexible in their thinking.
Skemp (1982) writes
"those who understand mathematics - who can attach
correct mathematical meaning to its symbols - pay
little attention to the symbols themselves as they
pass beyond them to the associated mathematical
ideas. But those who do not understand mathematics
do not get beyond its symbols, which rightly or
wrongly they regard as one of their main sources of
difficulty”.
Thus, there are not only the problems associated with the
abstract nature of variables, students further must deal
with the many uses of letters in mathematics, not all of
which are letters as variables. If students do not see the
mathematical picture in which variables are embedded they

will become frustrated with mathematics and in particular

the variables themselves.

7

MEMMLu-mmod new“ to 91.1w bl-
North (1965) writes:

Algebra is a generalization of arithmetic. Hhen
the operations of arithmetic- addition, subtraction,
multiplication and division - are performed upon
numbers, the numbers combine to form new numbers,
according to some particular scheme or pattern. This
pattern is an abstraction, being independent of the
particular numbers. The same set of operations could
be performed on a different set of numbers, but the
pattern of the calculations would be the same in each
case. The study and analysis of such patterns is one
of the objects of algebra. Hhen the pattern of a
calculation has been analyzed, it is often possible
to generalize it so that the original calculation can
be applied in new ways. To analyze the pattern and
to see its structure and symmetry it is necessary to
eliminate the numbers, so that the pattern behind

them is made apparent. Algebra does this by
replacing individual numbers by letters, each letter
standing for a number in the arithmetical
calculation.

Looking closely at North’s quote two definitions implicitly
lappear, that of pattern and that of variable.

Pattern - recognizable set of numerical expressions
or equations which remain invariant regardless of numbers
used.

Variable - mathematical device for generalizing
patterns and arithmetic statements. (Traditionally
thought of as the historical definition of variable.)

These two definitions describe patterns and variables in a
way which is consistent with North’s quote. They also
describe the concept of variable and patterns in view of
the historical development of algebra and variable, algebra
as generalized arithmetic. Taken in terms of Lesh’s et al.
(1987) three components of understanding, a pattern is an
instance of what they describe as a concept. Variables are

one representational mode used for expressing concepts.

Based on the presentation of variable in current

algebra textbooks variables are introduced to students as
place holders for unknowns. In textbooks instruction about
the historical development of variables has been bypassed.
Students are not taught to see variables in the historical
sense, as a powerful symbolic tool to represent
generalizations of patterns.

The clash between introducing students to the concept
of variable in a historical development sense (based on a
transition from arithmetic to algebra) and the end product
of such development is expressed by Hheeler.

Uheeler (1986) writes:

The school curriculum, by insisting that algebra
is about numbers or quantities, while bypassing the
stages of long historical development of the
algebraic language, virtually forces pedagogy to
oscillate inconsistently between presenting algebra
as a universal arithmetic egg a formal linguistic
system with arbitrary rules.

The formal linguistic system with arbitrary rules
which Hheeler refers to is the concept of variable as
Diophantas and textbooks today present it. It is a system
which, a review of literature shows, leaves students with a
weak concept of the meaning of variable. This system asks
students to solve equations using what they have come to
see as an arbitrary set of rules.

This definition of variable is different from the
definition of variable which we saw from North. In this
latter case variables are not seen as tools for

generalizing patterns. They are seen as place holders for

solutions to equations. The power of algebra is limited if

9

students only see variables in this way.

The process of generalization from arithmetic to
algebra has two aspects, on the recognition of arithmetic
patterns which can be generalized and the ability to use
variables and operations to generalize patterns. Letters
have many uses in mathematics. Letters are used as
constants (i.e.,‘H ), as variables, as abbreviations (i.e.,
ft. for feet) and as symbols of operations (i.e., x meaning
multiplication). Students need to be knowledgeable about
the different uses of letters if they are to successfully
generalize patterns. They also have to be flexible enough
in their thinking about mathematics and variable to be able
to correctly interpret expressions containing letters. Each
expression, sentence or sequence that contains variables is
part of a much larger group of expressions, sentences and
sequences. For example, the expression x r 5 could
represent a fraction or a decimal or an integer increased
by five. In other words x + 5 represents any quantity
which is to be increased by five. Students’ conceptions of
variables are limited if they are unable to interpret
generalizations containing variables. Students should be
able to see the larger group which the generalization
represents. If students only know variables as place
holders for solutions to equations, it is questionable
whether they will be able to (1) recognize mathematical
patterns, (2) use variables to generalize these patterns

and (3) be able to draw meaning from generalized

10

expressions, sentences and sequences.
In; Researgh Pgoblg!

Research studies have investigated students’
interpretation of variables as place holders for solutions
to equations. No study has attempted to address the larger

question of students’ conceptions of variables as tools
for generalizing patterns. Uhat has been previously
studied is students’ ability to manipulate variable.
Students understanding of variables is still open for
investigation.

As already stated Lesh et al. (1987) describe
understanding in part as the ability to do three things:
(i) be able to recognize a concept in a variety of
representational systems, (2) manipulate flexibly the idea
within a given representational system and (3) to
correctly translate the idea from one system to another.

Lesh et al. (1987) describe five representational
systems with-in the mathematical system:

1. knowledge organized around real world events,

2. manipulative models,

3. pictures or diagrams,

4. spoken language and

5. written symbols.

Uithin each of the five representational systems are
different modes of representation. For example, within the
written symbol representational system there exist (among
other modes) numerical symbols, variable symbols and

operational symbols. Manipulating an idea flexibly within

a given representational systems requires in part that

11

students can translate between several modes of
representation within one operational system.
Traditionally research on variables has focused on
students’ ability to manipulate literal symbols. By
focusing on one mode of representation (that of variables)
within one representational system such research does not
consider the flexibility one needs to operate within a
representational system. Such research also does not
investigate students’ ability to recognize the concept in a
variety of representational systems or to correctly
translate the idea from one system to another.

The research proposed here is designed to overcome
some of the gaps in existing research on students’
conceptions of variables. The focus of the research is on
how well students handle tasks which require them to
demonstrate skills which correlate with Lesh’s et al.
(1987) three aspects of understanding. In particular the
ability to recognize patterns (concepts) in a variety of
representational systems and the ability to translate ideas

from one system to another is investigated.

Reseagch Quegtions

This study investigates students’ conceptions of
variables as tools for generalizing patterns. It is
designed to focus on students at the point in their
mathematical experience just prior to and just after the

formal transition between arithmetic and algebra. The

12

specific research questions to be addressed are:
I. Hhat types of mathematical patterns can students
recognize?

ll. Uhat types of patterns can students generalize
by using variables?

III. Uhat meaning do generalizations using variables
have for students?

By looking at the broad questions of students’ conceptions
of variables as tools for generalizing patterns these three
questions, along with the tasks designed to investigate
them, focus on all three aspects of Lesh’s et al. (1987)
definition of what it means to understand. Using the
terminology of Hiebert and Lefevre (1986) these questions
are designed to investigate students’ conceptual rather

,than their procedural knowledge of variables.

9__r_i_o_uvov ummm

The research questions were addressed by in-depth
interviews with seventh graders and High school algebra
students (n-13 for each group). The interviews were
designed to investigate how students thought about and
answered questions involving variables used to generalize
mathematical patterns. The interviews consisted of five
tasks which investigated student conceptions of variables.
Each interview lasted approximately 45 minutes. Not all
students encountered all tasks: however, certain tasks were

common to all interviews.

13

W

This study is based on the assumption that it is
important for students to make the transition from thinking
about mathematics as dealing with specific numbers to
thinking about mathematics as the study of numbers in
general. It further assumes that this transition may not
occur in a formal way until a student begins the study of
algebra. Another assumption of the study is that viewing
variables only as place holders for solutions of equations
limits the ability for students to make this transition.

It is also assumed that this study will have
implications concerning the current practice of introducing
variables as place holders for solutions to equations.
Some researchers have suggested that the introduction of
variables needs to draw more upon the prior experiences
students have had with arithmetic (Adda, 1982: Lampert,
1985: Herscovics and Kieran, 1980). It is assumed that
this research will speak to that issue.

A third assumption is that in volunteering to take
part in this research students approached all questions and
tasks with the intention of answering to the best of their
ability.

Limitetiggg

The purpose of this study was to be begin to
understand students’ conceptions of variables as tools for
generalizing patterns. Students at the seventh grade and

algebra levels were targeted as subjects for the study.

14

Although data obtained from the two groups was to be
discussed independently attempts were made to try to match
the two groups in ability. There were certain problems in
obtaining this result. None of the same standardized test
scores were available for both sets of students. The
teachers were asked to assess the ability of the students
based on a three point scale of below average, average and
above average. The algebra teacher expanded the scale by
writing in the descriptions, average to below average and
average to above average. Both of the seventh grade
teachers had the students for mathematics, English and
history. They felt that their ratings were based not only
on mathematics but on their reflection of the students
across all three subjects. So although the groups of
students used are similar, as rated by the teacher, there
are problems with these ratings. This limitation is
minimal, however, given that the intent of the study was
to develop qualitative descriptions of the students’

thinking rather than to make quantitative comparisons among

them.
Overview 2; the Qigsertatign
This dissertation consists of five chapters. The

first chapter provides the background for the study. It
also contains the research problems and research questions,
an overview of the research design as well as assumptions

and limitations of the study.

15

Chapter Two includes a review of literature. The
literature is divided into four parts. The first part talks
about mathematical knowledge and what it means to
understand a concept. The second part describes recent
essays concerning viewpoints about symbolism in general.
The third part focuses on specific research concerning
variable. The last part reviews literature about the role
of patterns in mathematics.

Chapter Three presents the research methods used. The
subjects and sites are described. A detailed explanation
of the clinical interview is given. The data recording and
analysis processes are also described.

Chapter Four describes the connection between the
research questions, analysis issues and interview tasks.
For each research question, the corresponding analysis

issues, interview tasks and results of the research are

presented. Immediately after the data for each question a
conclusion which ties this information together is
presented.

Chapter Five contains an overall summary statement
relating all of the findings of the research. A discussion
of these results follows. Implications for teaching and

implications for research are presented.

CHAPTER THO

REVIEU OF THE LITERATURE

lntrodgction

One concern of many mathematics teachers and
researchers is how well students perform on tasks such as
solving problems and taking tests. Recently, however,
mathematics educators at all levels have begun to realize
that how well students perform on mathematics problems is
not always an indication of how well they understand
mathematics. Sometimes a gap exists between the process of
solving problems and the understanding of these problems.
Hiebert (1984) addresses this issue in an article which
discusses the link between ”form" and ”understanding”.

Form as Hiebert describes it is the syntax of the
mathematical system. Form indicates the symbols and
numerals of mathematics. Form is also the rules which are
used to join the symbols and numerals of mathematics.
Understanding, on the other hand, is the intuitions and
thoughts which mathematicians of all levels have about how
mathematics work.

In order to discuss mathematical knowledge about any
concept a clear understanding of what is meant by

mathematical knowledge is needed. Hiebert defined

16

17

mathematical knowledge as the link between form and
understanding or the relationship between conceptual and
procedural knowledge. These definitions still leave
unanswered the questions of what is conceptual knowledge
and what does it mean to understand.

As already described in Chapter one Lesh, Post and
Behr (1987) describe understanding as the ability to: (a)
recognize a concept in a variety of representational
systems, (b) manipulate flexibly the idea within a given
representational system and to (c)correctly translate the
idea from one system to another. They also describe five
representational systems with-in the mathematical system:

1. knowledge organizedaround real world events,

2. manipulative models,

3. pictures or diagrams,

4. spoken language and

5. written symbols.

Similar systems are described by Dufour-Janvier, Bednarz
and Belanger (1987) when they discuss internal and external
representations. Janvier (1987) also describes similar
ideas which he calls multiple embodiments.

The research proposed in this study is designed to
investigate students’ understanding of variables.
Understanding is defined as outlined in Lesh and Post’s
definition which is given above.

The research questions for this study are:

l. Uhat types of mathematical patterns can students
recognize?

ll. Uhat types of patterns can students generalize by
using variables? ‘

18

lll.Uhat meaning do generalizations using variables
have for students?

These questions are concerned with all three skills which
Lesh et al. state are necessary indicators of
understanding.

The subsequent review of literature will start with a
look at articles and research studies concerning students’
conception of symbols in general and variables in
particular. This section will be primarily directed at
highlighting what questions have been and what questions
have not been answered about students’ understanding of
variables.

Following this section will be a review of research
_on, and articles about, teaching models directed primarily
toward enhancing students’ understanding of variable.
Throughout this review of literature particular attention
will be paid to what the articles have to say concerning
how well students understand the concept of variable as a
tool for generalizing patterns. Lesh’s et al. (1987)
definition of understanding will help to serve as the
backbone for comparison. In the final section a summary

will be given relating all of the works.

Artigles ancerning Stgdents’ Conceptigns 21 Symbols
mmanQ—AV-rib snag-mo

Bernard (1984) expressed a need for research to
address the issue of the gee of symbolism. He reports that
researchers have considered when to introduce symbolism.

He stated, however, that clinical studies and teaching

19

experiments need to take place in order to help to develop
guidelines for using symbolism. One area that he
specifically targeted for investigation was the ways in
which students construct meaning and understanding from
symbols.

This area is one which has not been sufficiently
addressed by current research about variables. Little
research concerning variable goes beyond students’ ability
to solve equations. A focus on how students conceive of
variables and how they construct meaning from variables as
tools for generalizing mathematics is needed. Such a study
would need to start with a basic definition, such as Lesh’s
et al. (1987), of what is meant by understanding.

Only one article could be found which dealt with
research concerning how students conceived of symbols.
lwuoha (1985) wrote a doctoral dissertation concerning
eighth grade students’ conceptions of algebraic symbols.
Outlined were four levels of understanding of algebraic
symbols: (1) Fully, (2) Mathematically, (3) Pragmatic
Mechanistic and (4) Functionally. The difference between
understanding fully and mathematically was that students in
both categories could explain how a symbol worked, but only
when they fell into the first category could they explain
why a symbol worked. Students who operated under the
category labeled pragmatic mechanistic used symbols
mechanically. They felt that some higher authority had

designated the use of the symbols. Those students who

2O

operated functionally viewed symbols with little or no
level of understanding. The term symbol encompasses many
mathematical symbols besides variables (i.e., operational
signs, symbols of comparisons). Although lwuoha’s work
presents nice neat categories for classifying students’
conceptions of symbols in general, his work does not
necessarily represent students’ conceptions of variable in
particular. lwuoha’s work was hard to analyze in light of
Lesh’s et al. (1987) definition of understanding as only
the dissertation abstract was easily accessible for review.

Before discussing the specific research concerning
students’ conceptions about variables it is necessary to
emphasize briefly one distinction between variables as they
appear in algebra and what some people misconceive of as
variables in arithmetic. Both Hoodrow (1982) and
Herscovics and Kieran (1980) point out the difference
between expressions such as 3 + I] 8 S and 3 + x I 5. Uhen
students encounter the first type of equation in arithmetic
they are usually asked to fill in the box. This simply
means that the students must remember an arithmetic fact
that will make the statement true. The second type of
equation is usually accompanied with directions to ”solve
for x'. Thus, the second type of equation calls for the
student to consider the x as a variable and to act on it
accordingly. Referring to Lesh’s et al. (1987) definition
of understanding the first type of equation requires

students to regard an expression written only in a

21

numerical mode of representation.

It is the second type of equation which was under
consideration in the majority of the subsequently discussed
research. This second type of equation requires only that
students manipulate written statements involving numerical
and variable representations. The goal of such
manipulations is to arrive at a new statement which
involves a numerical expression being equated with a
variable statement. Students are not challenged to
translate or transfer between or within different
representational systems, such as would be required for
understanding as defined by Lesh et al. (1987). It is a
third use of variable, variable as used to generalize
patterns and as part of generalized statements, which
incorporates all skills necessary for understanding as
defined by Lesh and Post, which research is not addressing.
For this reason research which currently exists focuses
primarily on manipulation of variable statements not
understanding of the concept of variable.

Building partly on the work of Collis (ref. Bell,
Costello and Kuchemann, 1983) the study "Concepts in
Secondary School Mathematics and Science (CSMS)" at Chelsea
College, London was developed. One topic incorporated into
this study was students’ (ages 13 - 15+) conceptions of
Algebra. Three thousand students, drawn from 15 different
schools, took a half - hour paper and pencil test.

The steps taken in writing the test as well as the

22

results of the test have been published in several
different reports (Booth, 1984a: Kuchemann, 1978, 1981 a
1984). The research revealed that students conceive of
variables in six ways: (1) letter evaluated, (2) letter not
used, (3) letter as object, (4) letter as specific unknown,
(5) letter as generalized number and (6) letter as
variable. It needs to be noted, however, that when the
authors arrived at these categories of conception they were
focusing on variables as placeholders for solutions of
equations. Such a narrow definition of variable focuses on
procedural knowledge of one use of variables. This
definition does not consider conceptions as used in
generalizing arithmetic. I

The first category is when a student in looking at an
algebraic expression or equation assigns the variable a
numerical value from the onset. This category also refers
to problems where children are asked to find a specific
value for an unknown (for example, ”Hhat can you say about
k if k + 7 I 9?"). The second category occurs when a
student just ignores the variable (for example, if e + g I
8, e I f + g I ?, students would incorrectly ignore the
variables and say 12 because, 4 + 4 + 4 I 12). The third
category refers to a student regarding the variable as
shorthand for an object, for example, 5m means 5 meters,
not 5 times the quantity m). A student who treats
variables as in category four is taking variables to

represent one specific unknown (for example, ”what can you

23

say about c if c + d I 10?") An incorrect response from a
child which would fall into this category is that c I 4."
Category five refers to the belief and carrying out of
calculations based on the idea that the variable is able to
take several values rather than just one (for example, a
child answering the problem 'Hhat can you say about c if c
+ d I 10" and saying c I 1,2,3,4 would be giving a
response that would be classified as category five). In
the sixth category a student treats the variable as
representing a range of unspecified values, and a
systematic relationship is seen to exist between two such
sets of values. This final category is the level of
understanding desirable for students to achieve. Students
who respond in ways consistent with category four (taking
letters to represent one specific unknown) understand
variables as placeholders for solutions. Again it needs to
be stressed that understanding as it is referred to here is
only a procedural not a conceptual understanding.
Responses which fall into category four would be expected
of students who are thinking about variables as they are
traditionally presented in textbooks.

These categories not only represent ways of
classifying students’ responses to problem situations, the
categories also represent types of problems (for example, a
problem of category four would be a problem that requires
students to use a variable as a specific unknown in order

to solve the problem.) Thus, these categOries can be used

24

to classify types of problems as well as correct and
incorrect responses to problems.

Kuchemann considered these six categories as a
hierarchy of understanding. His work revealed that as items
on a test become more difficult for students, forcing them
to look at variables in one of the ways represented by the
later categories, students would often chose an
inappropriate technique (for example, ignoring the
variable) which lead to an incorrect response.

As a result of this study Kuchemann thought that
students operated at four levels of generalized arithmetic
understanding. It is important to note that what Kuchemann
considered as levels of generalized arithmetic is not
Aunderstanding of generalized arithmetic as presented in
Chapter One. Chapter One described the ability to
generalize arithmetic statements as involving conceptual
knowledge of variables. Kuchemann focused on procedural
knowledge.

At levels one and two children could only solve
problems which do not require them to operate on variables
as unknowns. The main distinction between the two levels
is in the degree of difficulty of the problems (for
example, a child who is able to solve problems at level one
would be able to correctly solve the problem, 'Uhat does x
equal if x + S I 8, but would not be able to solve the
level two problem, ”Uhat can you say about a if m I 3n + i

and n I 4?"). Children operating at levels 3 and 4 are

25

able to solve problems that required them to treat
variables as specific unknowns or as generalized numbers or
variables. Again, the main distinction between the two
levels relies on problem difficulty. Of 15 year olds, 5%
operated below level 1, 30% operated at level i, 23%
operated at level 2, 31! operated at level 3 and only 9%
were able to operate at level 4. The majority of students
were operating at level one or two which required no
understanding of algebra.

Hhile Kuchemann’s work supports the contention that
current efforts to teach students about variable are not
successful, it would be hard to agree that his six levels
provide a complete framework for describing students’
conceptions of variables. The definition of variable which
he bases his work on is the narrow view of variable as
place holder for solutions to equations. In regard to
Lesh’s et al. (1987) five representational systems it is
clear that Kuchemann’s work is only focusing in on one mode
of representation within the representational system of
written symbols. Kuchemann’s study does not focuses on how
well students can transform variables within this system.
He also does not question how well students can recognize
the variables in other representational systems. Students’
ability to translate from one system to another is also not
investigated. A more accurate picture of students’
conceptions of variables would have to take into account

students’ conceptions of the role of variable as used to

26

generalize patterns and as part of generalized statements.
These two areas of investigation incorporate all three
skills which constitute understanding. Kuchemann’s work
investigates variables procedurally as textbooks present
them. His research needs to be expanded with further
research which focuses on understanding of variable, in
particular variables as they were historically developed.
Other research findings support Kuchemanns’s claims
that children have difficulty with the notion of variable.
Uheeler (1986) writes “That students do not find ~the
arithmetic-algebra connection transparent emerges in our
research (and is well known to practitioners anyway)". He
further writes ”As soon as letters are written for numbers,
mathematical language changes. In spite of the apparent
parallelism between, say, writing 17 + 32 and a I b, where
a and b are undetermined numbers, the two expressions
belong to two different orders of reference. Students
experience (or may experience) puzzlement about writing the
"sum" of two numbers which cannot be summed because their
identity is not known". The ability to recognize this
parallelism requires that students are able to transfer
knowledge between numerical systems and variable systems.
Research has not yet formally addressed the question as to
how flexibly can students transfer variable conceptions from
one representational mode to another within a given
representational system, namely as in these observations the

written symbol system.

27

Following along with this last idea was a finding by
Adda (1982). She found that in problems where the
directions stated ”let n be a number” a typical reaction of
students was to reply "n is not a number, it is a letter."
This exclamation by students might explain why some
students, as in Kuchemann’s level two simply ignored
variables in problems. If variables are regarded as other
than representing numbers it is no wonder that they get
ignored when students solve problems.

A study by Tonnessen (1980) also discussed the level
of attainment of students’ conception of variable. Eight
criteria were formulated to assess attainment of the
concept. Four of the criteria dealt with concept
acquisition. The other four dealt with concept use. The
study involved 202 algebra, 178 trig. and 186 calculus
students enrolled at the University of Hisconsin.
Tonnessen’s work revealed that the levels of attainment of
the concept of variable were low. A further conclusion was
that the level of acquisition of the concept of variable
could not be used as a predictor of the level of use of the
variable. Unfortunately, only a dissertation abstract for
Tonnessen’s work was easily available so it is impossible
to fully explore exactly to what extent his work further’s
understanding about variable, with understanding modeled by
Lesh’s et al. (1987) definition.

Tonnessen’s work focused only on variable as place

holder for solutions to equations. Research is needed

28

which assess the attainment level of variable as tool for
generalizing patterns. In other words, as already noted,
research is needed which focuses on conceptual rather than
procedural knowledge of variables.

A problem about which much has been written (Rosnick
and Clement, 1980; Rosnick, 1981: Clement, 1982: Clement,
Lockhead a Monk, 1981: Uollman, 1983: Sims-Knight and
Kaput, i983: Kaput, 1986 and Lockhead, 1980) and which
illustrates Kuchemann’s ”letter as object category" is the
student-professor problem. The problem is as follows:

There are six times as many students as
professors. Using P to represent professors and S to
represent students write an algebraic expression to
represent this situation.

Hhere Kuchemann’s focus was on procedural knowledge this
problem begins to look, using one example, at Lesh’s et al.
(1987) third requirement for understanding. This problem
requires that students translate a problem from the spoken
language system to the written symbol system. The majority
of students given this problem write the incorrect equation
65 I P instead of the correct solution 6P I S. Clement,
Lockhead & Monk (1981) listed two strategies students used
when solving the problem. The first involved ”word order
matching” or direct mapping of the words in the problem to
the equation. The second was called the ”static -
comparison" method. Students who used this method could

draw pictures to illustrate that they Understood the

relationship correctly, but they still represented it

29
incorrectly. Their conclusion was that students were

taking the letters as a direct label meaning students
rather than as a variable meaning number of students. In
other words, for this example, students were unable to
translate the idea from one representational system to
another.

Booth (1984) conducted a research project designed to
provide further insights into Kuchemann’s six levels. She
interviewed students to try to find the cause of their
errors. Fifty students, with approximately 16 from each of
the second, third and fourth years of high school, were
interviewed. These students made the types of errors
described by Kuchemann, based on the CSMS tests.

The results of these interviews revealed that errors
might result from three things: (i) the meaning children
attach to variables, (2) the process of operating with
variables and (3) difficulties with notation and
convention. One of the errors that students made which was
classified as an error caused by the meaning students
attach to variables was a confusion with variable
representing number as opposed to letter representing an
object. This is the same type of misconception which
results in the student - professor problem mix-up. A
second error which students make is in assuming that
different variables represent different numbers. Other
mistakes included assuming that variables represent only

whole numbers and in ignoring the meaning of the variable.

30

Students who ignored the meaning of a variable used three
different strategies in simplify problems of the type (2x +
By + 3x). One approach was to add up all of the numbers,
then put down each variable that occurs (once only). That
approach would lead to the incorrect simplification 13xy.
A second approach was when the student would add up all the
numbers, then put the variable for every time it occurs in
the expression. This approach would lead to the incorrect
solution 13xyx. The last faulty approach involved adding
up all of the numbers, than putting down the variable that
occurred most often. An answer of 13x was the result of
using this method. Students who made errors of this type

had trouble separating the meaning of variable
.representations versus numerical representations.

In other research Assab (1978) found that students
had what was labeled a "single letter fixation". His
research revealed that students were unable to accept
algebraic expressions containing more than a single
variable as representing real numbers. Such a finding
might reveal why students over simplified expressions such
as (2x + 8y + 3x) and particularly why 13x would be a
common simplification.

The final types of errors that Booth labeled as
errors involving the meaning children attached to numbers
were errors when the child assumed a ”pattern" between a
variable and the number it was suppose to represent. For

example assuming that x,y,z meant 2,3,4 or 10.20.30.

31
Another error of this type was saying that y is "higher"

than p.

Sigrid Uagner (i977, 1980, 1981, 1983) has written
several papers about this same idea of y being "higher"
than p. She referred to this as the childs’ inability to
conserve letter. She found that when children were
presented with equations such as 7 x w + 22 I 109 and 7 x n
I 22 I 109 and ask which was larger they would respond with
such answers as ”w because it is further in the alphabet.”

These students had trouble working within the variable

representational mode within the written symbols
representational system. The tasks she presented to
students involved looking at equations and functions. She

labeled students who gave the above response as

nonconserving. Students who gave the response "you can’t
tell without solving the equation" were said to be
transitional. Those students who gave the correct answers

were said to be conservers of equation and function under
transformations of variables. She found that the majority
of students were either nonconserving or transitional in
their responses.

Booth’s second category of types of errors were
caused by what she called difficulty concerning the process
of Operating with variables. She found that students quite
often needed to see a problem worked with numbers before
they were able to consider it without numbers. She felt

that often if students were unable to work with an

32

arithmetic process then they were unable to do a similar
problem involving variables. Another common response that
falls into this category would be responses such as "I
can’t do it, I don’t know what the letters are.”

The third cause of errors that Booth reported were
errors that dealt with difficulty involving notation.
Errors that fell into this category were saying that a I m
was the same as am: k I 2 I k2: 4m I m,m,m,m (four m’s not
4 times m): 2 lots of x I 2x: 2 lots of 7 I 27 and xy, if x
I 3 and y I 2, equalled 32. She also found confusion with
the idea of exponents. She further found that students
felt that brackets were unnecessary since you should
perform operations in the order in which they occur in the
equation. Several of these same results were found by Matz
(1980). She felt that these errors could be attributed to
misuse of existing rules or the students making up their
own rules. These three types of errors resulted when
students look procedurally at using variables as place
holders for solutions to equations. It would be
interesting to note whether similar errors result when
students are questioned on tasks which require them to
regard variables conceptually.

Booth conducted a second set of interviews to further
test her hypothesis involving the three causes of students
errors. This second set of interviews involved 17 students
from the first interview phase and 7 new students. Again

her results were quite similar.

33
As a by product of research that Nelson (1986) was

doing concerning children and Logo he found that students
were often confused when a diagram was kept the same, but
given different variable dimensions. An example of this
problem would be to let a square of total given area of 25
cm. be labeled with sides 'b' cm. If students solved for
“b” they’d find that "b” represents 5. Teachers often save
themselves some work by using the same diagram but changing
the total given area, for example letting it be 64 cm. In
this later problem the "b" would represent an 8. He said
that it was erroneous to assume that all students
understand this demonstration since students see dimensions
as fixed. In other words the students could not operate
flexibly using variables to represent the dimensions.
Harper (1980) interviewed 12 pupils in years one
through five of High School. He drew two lines a long red
line which he labeled "b" cm and a shorter green line which
he labeled "s” cm. He asked students questions such as,
”which line is longer? Could they ever be the same in
length? Hhen would they be the same in length?” He found
two types of responses. Some children focused only on the
physical length of the line. Others talked about the
variables a and b. Of the year one students 11 times as
many were the first type. In years, two, three and four
there were about 3 times as many of the first type of
students than the second. By year five there were only

twice as many students who focused on the physical length

34
as opposed to the variable measurements. These two

research studies reveal two “buggy algorithms” which might
affect students’ overall conceptions of variables. Detailed
research which takes into account Lesh’s et al. (1987)
overall model for understanding needs to be undertaken in
order to see just how these misconceptions fit into a

larger picture of understanding.

Resegrch on, And Articles AboutI Teaching Model; Qegigned
£9 Enhange sgudents’ Understanding of Varigbles

Research and subsequent instructional interventions
which approach variable as a tool for generalizing patterns
are lacking. Research needs to be done which assess
students’ understanding of variable as part of a system of
generalized arithmetic. Such research and subsequent
teaching would incorporate all three skills necessary for
understanding as described by Lesh et al. (1987). The
majority of articles described in this section of the paper
focus on student instruction of the procedural use of
variables. One exception is an article by James and Mason
(1982) which looks at introducing variables conceptually as
tools for generalizing patterns.

James and Mason (1982), Skemp (i982) and Freudenthal
(1973) discuss the need to be careful when introducing
symbols to children. Freudenthal argues that textbooks
used to be very careful when introducing symbolism to
children. Hith the introduction of New Mathematics he
feels that the emphasis was so heavy on introducing the

symbols that little attention was paid to how they were

35

introduced. In the language of Lesh et al. (1987)
Freudenthal is stating that the emphasis was placed too
heavily on variable as a representational mode rather than
variable as a concept which can express ideas from several
representational systems.

Both Skemp (1982) and James and Mason (1962) talk
about the need to draw from the experience of a child when
introducing symbols. Skemp offers four steps for helping
students to integrate symbols in mathematics. These steps
are to: (1) give children as many physical embodiments as
possible (2) assimilate new concepts - not allowing
students to memorize steps of symbolic manipulation (3)
stay longer with verbal mathematics and (4) allow students
to use their own symbols in order to experience advantages
and disadvantages. Skemp stated that children needed to be
pushed to use many physical embodiments. Skemp is arguing
for the type of symbol education which is consistent with
Lesh’s et al. (1967) skills necessary for understanding.

Based on difficulty concerning the student-professor
problem Rosnick and Clement (1980) attempted to tutor
students on the problem. The tutoring methods they tried
included: (1) telling the students that the reversal is
incorrect, (2) telling the students that the variable
should be thought of as "number of students," not
"students", (3) pointing out with pictures that the student
group is bigger and that the need to multiply by 6 to get

equal sets, (4) having students test equations by plugging

36

in numbers (5) having students draw graphs and tables (6)
showing students how to set up a proportion to solve the
problem and (7) demonstrating a correct solution using an
analogous problem. Despite their efforts seven out of the
nine students they tutored persisted with their
misconceptions.

Booth (1964 a.) as part of the final stage of her
research also included a teaching phase where students were
instructed with worksheets which were designed to reveal
their mistakes concerning variables to them. For example
they might be asked to find the area of a square after
which they would discuss the difference between a x a and a
x b. These worksheets were aimed directly at instructing
students in using variables as expressed by Kuchemann’s
level six within the context of variables as placeholders
for solutions to equations. The conclusion after the
teaching was that the program had been effective in
improving students’ conceptions about variable, but the
gains that occurred were not overly impressive.

It is possible that the lack of success with teaching
variables based on the emphasis of variable as a tool for
solving equations is a result of the fact that this
instructional approach is based on an incomplete conception
of children’s problems in understanding and using
variables. This approach does not focus on students’ prior
knowledge about mathematics. Using this approach children

are only required to operate within one representational

37
system. In order to achieve conceptual understanding
children should also be required to recognize variables in
other representational systems. They should also be
encouraged to translate among different representational
systems.

Herscovics and Kieran (1980) attempted to teach
students about variables in a way that would improve their
conceptions about variable use. Their approach was to
start first with the childrens’ ideas about arithmetic and
from there to move into algebra. They did this by first
having students look at equations without variables. The
students used their fingers to cover numbers in the
identities. They then had the students replace numbers
with letters. They worked with having the students hide
more than one number and replacing their fingers with a
variety of letters. From this point the students built
different identities by replacing the letters with numbers.
No statistical information was given to suggest whether or
not this instructional sequence led to improved conceptions
about variables, but the authors felt that they were seeing
positive results. The approach suggested by Herscovics and
Kieran is consistent with Lesh’s et al. (1987) skills of
recognizing a concept in a 'variety of representational
systems and correctly translating from one representational
system to another.

James and Mason talk about three activities designed

to help students draw from their wealth of experience when

38

learning about symbols. One of the activities is
particularly interesting because it deals with the
introduction of variable by using patterns. Students first
built frames around various size pictures. They learned
that a 2 x 2 picture’s frame used 2 blocks on each side
with one block in each corner. The 3 x 3 block used three
blocks on each side and one block in each corner in order
to build its frame. Students continued building frames and
noticing the pattern. Eventually the students wrote out
the following pattern which they observed:

1 in the corner becomes 1

size of the picture becomes [1
add one onto the size and

.Then add all of these together: ([1 I 1) x 3 I I] I 1
(Note: Hhere I have used a i] the children actually used a
cloud. They chose the cloud to represent the thinking cloud
often used in childrens’ comic books.)

The students were eventually led by the teacher to
substitute an n for the I) (cloud). The final expression
became 3(n I 1) I n I 1 equals the number of cubes needed
to surround a square picture of size n. The article did
not address the question of why the students were led to
this equation as opposed the more simplified 4([] I 1).
The initial importance of such a teaching model is not what
generalization students eventually arrive at but rather,
the fact that this example is one instance where students
were beginning to investigate variables conceptually as

opposed to procedurally. This article is a good example of

a demonstration of a task which requires a student to

39

translate a pattern using pictures to a parallel statement
using written symbols. Research articles are missing
which discuss students’ success at tasks which require them
to recognize and generalize patterns with variables. Such
research would fit directly with Lesh’s et al. (1987)
model of understanding.

Niegenann and Parr (1986) conducted a study which
involved presenting students with thirty three word
problems of which there were two or three problems each of
various types (i.e., mixture, transportation, saving money
etc...). The problems were selected from ninth grade
algebra and physic books. The students were told not to
solve the problems, but to sort the problems into those
piles that could be solved in the same way.

Niegenann and Paar found that students sorted the
problems according to five classes of criteria: structural,
surface, difficulty, physical principles and a final group
labeled other. They further found that the number of
surface categories increased from 21% for experts, to 38%
for college students to a high of 53! for high school
students.

This study provides information concerning the types
of features which students rely on when viewing patterns in
story problems. It does not provide any information
concerning how variables would be used by the students to
generalize patterns they had identified.

One article which does investigate students’ ability

40

to recognize patterns is an article by Lopez-Real (1984).
She described a situation where students were using
multilink cubes to build frames with different size edges.

The students built frames such as the following:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

They then filled in the following table:

Edge of Square 3 4 5 6

 

Number of Cubes 8 12 16 20

 

The two girls working on this problem were pushed by the
teacher to describe a pattern. They realized that the
number of cubes needed to build the squares were multiples
of four. The teacher realizing. that recognition of
patterns is a first step toward'proof pushed the students
to explain why the number of cubes was always a multiple of

four. Uhat the teacher did not do was to push the students

41

to generalize the pattern with variables and to use this
generalization to predict how many cubes were needed to
build different size squares. In fact, no research was
found that reported students’ success and failure at
generalizing patterns with variables. In other words,
students were not questioned concerning how skillful they
were at recognizing patterns in various representational
systems. They also were not questioned concerning their
ability to generalize variables using patterns.

One last article which discussed the need for
students to be able to recognize and articulate
symbolically patterns was a report by Burton (1984).
Burton defined mathematical thinking as making sense out of
mathematical situations. He said that the study of
relationships is central for doing mathematics. He further
stated that mathematical thinking involved three steps (1)
manipulating (2) getting a sense of pattern and (3)
articulating that pattern symbolically. He considered
articulation of a pattern to be either verbal, diagrammatic
or symbolic. Burton’s definition of manipulations is much
different than the definition used in most of the research
and teaching methods presented in this chapter. His
definition reflects a conceptual look at manipulation which
is consistent with Lesh’s et al. (1987) definition of
understanding, not a procedural approach. Despite their
importance, however, these areas do not appear to be

explored by existing research. Few existing teaching

42

methods focus on helping students to be successful with
these three skills. It is not enough to say that students
need to be able to successfully handle all three tasks.
Research needs to address the quality of students
performance on tasks of this nature.

Summary

The three research questions outlined at the
beginning of this chapter are all built around the overall
question of how students conceive of variables as tools for
generalizing mathematical patterns. The ability to
generalize patterns involves all three skills which Lesh et
al. (1987) state are necessary for conceptual
understanding.

A review of the literature reveals the difficulty
students have with mathematical symbolism and in particular
with variables. Several misconceptions and erroneous uses
of variable were revealed through numerous research
projects. These errors led Kuchemann to identity six
levels at which students can operate with variables. Booth
furthered Kuchemann’s work by identifying three causes for
why students made the type of errors which Kuchemann had
identified.

Although Kuchemann and Booth, as well as other
researchers, talk about levels of attainment of generalized
arithmetic in students, their definition of generalized
arithmetic is narrow. Of Lesh’s et al (1987) five

representational systems only the written symbol system is

43

investigated extensively. Students’ abilities to transfer
within and translate between systems as well as their
abilities to recognize variables in various
representational systems are only considered slightly in
existing literature. As explained earlier Uoodrow,
Herscovics and Kieran point out a difference between having
students solve the equations 3 I [J I 5 and 3 I x I 5. A
similar distinction between generalized arithmetic as
described in this review of literature and generalized
arithmetic as described in Chapter One needs to be made.

Levels of generalized arithmetic as discussed in this
literature section means having the students move from
solving equations without variables to solving equations
containing variables. This definition provides a limited
view of what it means to have students use variables to
generalize arithmetic. North’s definition and the
definition given in Chapter One is much broader.

Students who can use variables to generalize
arithmetic as described in Chapter One, and in particular
as described by North, understand generalized arithmetic in
a much broader sense. They not only can recognize and
flexibly use variables within one representational system
they are also able to recognize variables in other systems.
They further are able to translate variables to these other
systems. This view of generalized arithmetic involves (1)
recognizing that patterns exist in groups of equations,

expressions and sequences and (2) using variables to

44

generalize those patterns. Very little of the research
reported investigated the process of children generalizing
arithmetic using variables in this broader sense.

Research concerning students’ ability to recognize
patterns as relationships between sentences, equations and
formulas is almost non-existent. Niegenann and Parr’s
study is one exception. Their research investigated the
types of features which students pay attention to when
looking for patterns. Their study did not, however,
address the issues of what patterns students could
recognize, nor was the question of whether or not students
could use variables to generalize the patterns which they
recognized addressed.

Lopez-Real and James a Mason illustrated the
importance of using patterns to motivate the concepts of
variables and proof. Burton defined mathematical thinking
as the process of making sense out of mathematical
situations. He identified getting a sense of the pattern
and articulating the pattern symbolically as being key to
mathematical thinking. Being able to recognize and

generalize patterns is the essence of doing mathematics.

Research studies which only focus on procedural
understanding of a concept are only investigating
superficial understanding. How well students understand a

concept can not be measured by how well students perform on
traditional tasks. Students’ conceptions need to be

thoroughly investigated by focusing on tasks which require

45

them to move beyond specific problems to the
generalizations of those problems. Such problems involve

all three aspects of Lesh’s et al (1987) definition of

understanding.

CHAPTER THREE

METHODS

Qverview
The purpose of the study is to investigate students’
conceptions of variables as tools for generalizing
patterns. The transitional step from arithmetic to

generalized arithmetic formally occurs when students begin
to study algebra. In order to understand students’
conceptions both before and after this point subjects at
the seventh grade and algebra levels were interviewed.

Students were interviewed for thirty to sixty minutes
using a clinical interview. The interview was designed
around the following three tasks: (1) sorting cards (2)
looking for and generalizing patterns in tables and (3)
interpreting generalized expressions, sentences and
sequences. These tasks were designed to villuminate
students’ understanding of variables as tools for
generalizing patterns. Understanding is defined according
to Lesh, Post and Behr’s (1987) definition which is given
in Chapters One and Two.

The data was analyzed using seven analysis issues.
Tables were generated to organize the findings. Instances
of student responses support the arguments. The subjects,

research design, development of data collection instruments

46

47

and analysis procedures are explained in further detail
within this chapter.
Subject;

The subjects for this study were drawn from four
seventh grade and two algebra classes. The seventh graders
attended a middle school in a small rural town. The
algebra students came from the neighboring high school in
the same community. The four seventh grades were taught by
two different teachers. Each teacher taught two classes.
The two algebra classes were both taught by the same
teacher. The teachers and classrooms involved in the study
were selected because of the teachers’ willingness to
participate in the study. The students were selected on a
volunteer basis.

From the students who volunteered to participate in
the study an attempt was initially made to select fifteen
students at the seventh grade and fifteen students at the
algebra level. These fifteen at each level were to be
divided into three groups: five below average, five average
and five above average in mathematics.

Problems arose with trying to obtain standardized
test scores for the two groups. No set of the same
standardized test scores were available for both the
seventh graders and algebra students. It was decided to ask
the teachers to assess the students’ mathematical
achievement based on a three point scale of below average,

average and above average. The algebra teacher expanded

48

the scale by writing in the descriptions, average to below
average and average to above average. Both of the seventh
grade teachers had the students for mathematics, English
and history. They felt that their ratings were based not
only on mathematics but on their reflection of the students
across all three subjects. The two group of students were
matched as closely as possible based on the teachers’
ratings. The final number of students used from both
levels was reduced to thirteen per group because of time
constraints. These thirteen students at each level fell
into the groups of 4 below average, 4 average and 5 above
average in achievement. Eight students at the seventh
grade level and seven algebra students also were
interviewed during the pilot stage of the research.
ata

Developmeng g;_gg;_lnterview

The interview tasks were developed to reflect the
proposed research concerns. At no point during the
interview were students asked to evaluate expressions or
sentences containing variables. The role of variable as
tool for solving equations was not under consideration.
Instead tasks were developed that required the students to
use variables as tools for generalizing patterns and
interpreting generalizations. During pilot interviews the
data collected was analyzed and used to develop and polish

the final interview.

49

15; Interview

The interview consisted of five basic tasks: sorting cards,
generalizing the card sort, looking for patterns in tables,
generalizing patterns in tables and interpretations of
expression, sentence and sequence generalizing by creating
a story to match the situation. A flowchart of the
interview structure is given in Figure 3.1.

In; 95;; 52;; eng Generalization 2; gegg,§g£;,§;;g;

Seven sets of cards were developed for the card sort.

An example set of cards is given in Table 3.1. The
complete set of cards is shown in appendix A. Six of the
sets were used for data collection purposes. One set was

developed to use in explaining the type of task to
students. The explanatory set of cards does not model the
structure of the actual card sorts. Each of the other
sets of cards in the card sort contained twelve to fourteen
expressions, sentences or sequences printed individually on
cards. The cards were organized around three surface
features on which students might focus when sorting the
cards. A surface feature is defined as a physical
characteristic of an expression, sentence or sequence which
is noticeable by looking at one card in isolation from the
other cards. For example the sentence: 8.2 I 3 I 11.2 has
the following surface features, it is an addition problem
of a decimal plus a whole number.

Two surface features which were present in all the

sets were four expressions containing whole numbers and

50

 

 

 

 

 

 

 

 

AIOTIEISETOFCAIDS llBHlOEIMII
cnvessmanmnn
$10ch ’

 

 

 

 

 

 

 

I! mam PM!”
Ill DID mm

mm! m swan nu Mm

mu: ’ fl GIVE CAD

1
Elm: 3.1: Flowchart of n- lumcl Interview

 

 

 

 

 

 

 

 

 

51

four or five expressions containing variables. These two
surface features represented two different representational
modes within the written symbols representational system.
The third surface feature changed from card set to card
set. The third surface feature might present an 'idea
within the written symbols representational system or it
might represent the idea within a different
representational system. For example, the orange geo cards

which are pictured below illustrate the concepts of area

and perimeter within both the written symbols
representational system and the picture and diagram
representational system. The whole number cards and

variable/formula cards are examples of cards which
illustrate the numerical and variable representational

modes within the written symbols representational system.

'rablo 3.1
Err-pl: of r Set of Interviev Cards

Orange Geo (Used In Every lnterviev)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Deep Pattern: Surface Features
Uncle lulbvrv Diagras: Variables/Fornulas
Perimeter 4 Mt I l6it p I 2i) I 2h
'l
(2 All) o (2 amazon “*“NI‘W’ p . 21: o 2|
F l
3 J. 3 .-
“ Drawn“
2 .
Area Sit 5ft I 25ft 3‘ i I lv
3 5-25 ,
9in 3in I 27 in , l... i I NI
=45

 

 

 

 

 

 

 

AI:

52
Also built into the set of cards were two deep
patterns. Deep patterns are the patterns which require
looking at a whole set of expressions, sentences or
sequences.

For example the set of cards:

5 I O I 5
8.2 I O I 8.2
6 I O I 8
3.9 I O I 3.9

all have the deep pattern that can be expressed by the
generalization u I O I u. A summary of all of the deep and
surface features of all of the sets of cards is given in
Appendix A.

During the card sort the students were given the
twelve or fourteen cards which go together as illustrated
in Appendix A. Each of the cards was laid out separately
by the researcher. All of the cards were placed on the
table so that no cards overlapped. The students were
asked to place the cards into piles according to what they
felt went best together. They were then asked to explain
why they sorted the cards in the way that they had. If
they correctly sorted and explained the cards by the deep
patterns during the first sort a new set of cards was given
to them. The directions and sorting process were repeated
with this new set of cards. If students were unsuccessful
at sorting the cards according to the deep patterns they
were asked "if I were to mix up the cards and ask you to

sort them again is there another way that they can be put

53
together?" This process continued until one of three

things happened (i) the student successfully sorted and
explained the cards according to the deep patterns (2) the
student could no longer see any more sorts or (3) the
researcher felt that the student was not able to see the
deep pattern or was confused by the cards. After one of
these events occurred the student was either given a new
set of cards to be sorted or the next task. Consequently
students have from one to seven attempts at sorting the
cards recorded for them.

The order in which the card sets were given to the
students and the actual sets that each student was given to
sort depended in part on their success at the previous
sort. For example, if a student had trouble with a set of
cards they would have been given another similar set later
on in the interview. The only two exceptions were if it was
apparent that the student was too frustrated by the task to
continue seriously thinking about the sorts or if there
were no more available sets of cards. Certain sets of
cards were given to all of the students to sort, but not
all sets were given to every student.

The next task was to have the students sort a set of
cards from the card sets in Appendix A with the cards
containing variables removed. If students successfully
sorted these cards by the deep patterns they were then
asked to generalize the patterns they saw using a given

variable. For example the researcher said, “can you

54
describe the pattern that you see on the cards in this pile

by using a 't"?” If students were unsuccessful at sorting
the cards they would either be given another set of cards
to sort or the cards would be sorted by the researcher.
The researcher only sorted the cards for the students if
one of two things happened (1) the students were so
frustrated by the task that they were becoming discouraged
and reluctant to sort the cards themselves (2) the students
had already failed to sort one set of cards without
variable generalizations and could see no more sorts in a
second set of cards. After the researcher sorted the
cards, students were then asked if they saw the pattern.
If they could explain why the researcher put the cards in
the specific piles they then were asked to generalize the
pattern. If the student could not see the deep pattern the
task ended.

All of the cards which made up the deep patterns used
in this task contained either expression or sentences
written using only the numerical representational mode
within the written symbols representational system.
Successful completion of this part of the card sort task
involved two skills: (1) the recognition of deep patterns
which were written within the numerical representational
mode of the written symbols representational system and (2)
the translation from the numerical representational mode to
the variable representational mode within the written

symbols representational system.

55

The next task was to show the students the tables,
which are given in Appendix A, one at a time. Each time
that the students saw a table they were first asked, ”what
would go after a ten in this table?" This question was
designed to see if the students recognized the pattern. If
the students gave a correct reply they would then be asked
why they gave the answer that they did. If the students
stated how they found the answer in a way that illustrated
that they recognized the pattern they were then asked to
generalize the pattern using a certain variable. This was
done by asking the students what would be on the other side
of the table from a "b” (for example). At this point the
next card was shown to the students. If the students could
not say what would go after the ten the researcher would
ask the students how they found their answers. The
researcher then moved on to the next card.

If the students did not successfully see any of the
patterns in the first three tables that they were shown
(tables 1, 2 and 3 in appendix A) the task was ended. If
the students saw some of the patterns but not all of them
then the researcher might elect to tell the student the
pattern. The decision to tell a student a pattern was
based on three factors (1) the level of frustration of the
student with the task (2) curiosity by students after
unsuccessfully looking for any patterns themselves or (3)

as a way of helping students, who after several minutes of

56
looking for patterns were unwilling to end the task until

they had seen a pattern. This usually happened when the
students were stumped by tables four and five (as numbered
in Appendix A). The pattern was revealed to the students
in a way which did not give away the variable

generalization. For example, looking at the table

 

1 8
2 13
3 18
4 23
the researcher said to the students, ”one of the other
students told me that they could find the number that went
on the other side of the table by multiplying by five and
adding three. Do you think that that student was right?”
The students then would check the rule and in all cases
idiscovered that it was true. After the students
successfully showed that they recognized the pattern by
saying what went after several numbers they than were asked
to generalize the pattern by telling what would go on the
other side of the table after a specific variable. This
task involved seeing if students could recognize patterns
which were written within the numerical representational
mode of the written symbols representational system, but
which were presented within the tables, pictures and
diagrams representational system. After investigating
students’ ability to recognize these patterns students were
questioned to see if they could translate the patterns to

the variable representational mode of the written symbols

representational system.

57

lnteeeretation e1 Exeeeesion, Sentence and Seguence
Gener l ation ey,Qeveloemeet e; Verd Eeoelees

Next the researcher moved to the expressions,
sentences and sequences that are given in Appendix A. Each
of these were printed separately on cards and the students
were shown them one at a time. The task was introduced to
the students by the researcher showing the students one of
the cards and saying, ”you know how you have story problems
in your math books that you have to write equations for?
Here are some equations, but the story problem is missing.
Can you make up a story problem that would fit the equation
on the card?" All of the students were shown all of the
expressions, sentences and sequences printed on the cards
despite their success or failure on any given card. This
was done since the cards differed significantly in their
structure and success or failure on one card would not
necessarily predict success or failure on the next card.

This task differs considerably from the last two
tasks. Both of the last two tasks were designed partly to
see if students could translate deep patterns to the
variable representational mode within the written symbols
representational system. The focus of this task was an
opposite form of translation. For this task students were
given generalized expressions, sentences and sequences
which were written using the variable representational mode
of the written symbols representational system. The

students were asked to translate these generalizations into

58
stories within the spoken language representational system

which fit the generalizations. The stories the students
created were taken as reflections of their knowledge of how
the generalized statements represent real world events.
This task was designed to illustrate an overlapping of how
the students saw the generalizations of concepts within the

spoken language and real world representational systems.

Qeta Collecgion Proceduees

The Initial Meeting Between fierticieeting Teachers and
Prineieele '

The first step in data collection was to set up a
meeting with the teacher(s) and principal in the schools
where the study was to take place. During the initial
meeting with the teacher(s) and principals the expected
commitment of the teacher(s) and the principal was
explained. The teachers provided the researcher with a
list of students in the class. They also collected
permission slips from the students. The teachers released
one student per period to be interviewed. The principals
were asked to provide a room where the interview could take
place. In one case a conference room was arranged. In the
other the vice principal’s office was used.
WEI—MEMO I 0 QMM

Immediately after the permission slips were turned in
the researcher attempted to record standardized test scores
for the students. The problem of not having the same

standardized scores for both the seventh grade and algebra

59
students was encountered. The researcher decided instead

to have the teachers rate the students on a scale of below
average, average and above average. The problems with this
process have already been described earlier in this chapter
and in Chapter One.
The lnteeview Schegele

The teachers at the seventh grade level met weekly to
discuss their curriculum with each other. They also used
the same textbook and covered the same material at the same
time. For this reason it was felt that no distinction
would be made between the students in one teacher’s
classroom versus the students in the other teacher’s
classroom. For the pilot and actual interviews students
were drawn from either classroom depending on which teacher
would be least interrupted by having the students leaving
and entering the room. Hith the algebra students, the
interviews took place on days that tests weren’t being
given.
£112; Interviews 2112.122 §§ueen§s

The first seven interviews with the algebra students
and the first eight interviews with seventh graders were
considered pilot interviews. Audio tapes of each of these
interviews were quickly transcribed by the researcher.
They were then analyzed to provide the researcher
information concerning how well the students were
understanding the tasks, what kind of questions were

providing useful information and which task needed to be

60
changed. Review of these transcripts helped to fine tune

the pilot interview until it was ready to be used for
actual data collection. This phase of data collection took
approximately three weeks.
IQe geeebgollection Interviews

After the pilot interviewing was completed and the

interview protocol was revised the interviews for the study

began. During the interview students were provided with
scrap paper and pencil. Students were asked to think out
loud as they worked on a task. The tasks which students

were given were in part determined by their success on

various parts of the interview. Each interview had core
tasks which were common to all of the interviews. These
are designated as such in Appendix A. Uhile the student

was being interviewed the researcher kept notes on a
separate sheet of paper. These notes along with the audio
tapes combined to provide the transcripts of the
interviews. In all cases the interviews lasted from thirty

minutes to an hour, averaging forty five minutes.

Data Analysis Peocedures

Transcription e; the Audio Taees

 

All verbal comments made by the students were
transcribed as close as possible to verbatim. If a part of
the tape was hard to hear the researcher’s notes were used
to supplement the audio tape. The researcher’s comments
were only transcribed if they differed in any way from the

standard interview directions and explanations. For

61

example, if a student asked the researcher a question
during the interview the reply, along with the question,
was transcribed. Also, any time that a student had trouble
with a task and the researcher provided additional
information beyond the initial directions the researcher’s
comments were transcribed.

Analysis _1 £22 Transcr s

The transcripts were analyzed in much the same way as
they were recorded. The first level of analysis was to
categorize responses as a success or failure on a certain
task. At this point questions such as "how many students
successfully sorted the blue sets of cards?” were answered.
The data for the seventh graders and algebra students were
always separated.

Next the transcript for each individual student was
analyzed internally as well as being compared with other
students. This was done in order to determine if there
were certain groups of students both on the interview as a
whole and on separate parts of the interview which fell
into similar groups. For example, was there a group of
students who were never successful on the tasks, a group
who were part of the time successful and a group who were
always successful._ It was hoped that profiles of say a
successful versus an unsuccessful student could be
developed. Unfortunately, no such groups emerged. No one
task predicted success or failure on another task.

Similarly a student might fail to recognize the patterns

62

that the majority of students recognized yet would
recognize a pattern that no other student recognized. No
student profiles could be developed without producing
individual charts on each student along with results on
every tasks. The patterns that existed seemed to be
patterns of the group of students as a whole or of the
seventh graders versus the algebra students.

At this point tables were developed to illustrate the
students’ responses to the various tasks. Up to this point
success and failures on tasks or generalizations of
patterns were considered without specific reference to the
transcribed reasons eey the students completed tasks in the
way that they did. The next step in the analysis was to
,analyze the explanations students gave along with their
answers to a specific task. Each transcript was coded two
separate times.

For the card sort all of the explanations that
students gave for why they sorted cards the way they did
were compiled. These responses were then assigned a
number. The transcripts then were read through on two
separate occasions. Each time a student sorted or explained
that they had sorted according to a certain feature the
code for that feature was marked down on a sheet. For
example, sorting all of the sentences containing fractions
into one pile would be coded with a ten if the student
explained that the presence of fractions is why they sorted

the cards in the way that they did. The coding of the data

63

in the tables was done by assigning of colors to indicate
specific correct or incorrect error patterns. The story
problems that the students developed as well as the
comments they made while doing this task were coded in the
same way that the card sorts were coded.

After examining the explanations students gave along
with their performance on specific tasks analysis issues
began to emerge. These analysis issues served as a link
between the research questions and the interview tasks.
For example, the sorting of cards and the identification of
patterns in tables were designed to provide information for
the first research question (Uhat types of patterns can
students recognize?). The analysis issues functioned as a
way that data from these two tasks could be organized in
order to address the first research question. Table 4.1
illustrates the relationship among research questions,
analysis issues and interview tasks.

Once the analysis issues were identified all of the
data in the transcripts were reexamined. As patterns
emerged more tables were developed to explain the findings.
These findings were highlighted by student comments which
illustrate students reactions to tasks. Not only was the
question of nee did the students perform on the tasks
examined, but also examined was the question eey_did they
answer in the way that they did. The final step in the
data analysis was to relate this how and why in summary

tables which explained the relation between the two. In

64
answer to both questions student responses were cited in

order to illustrate the summaries presented in the tables.
Sam
This chapter has described the research methods and
analysis procedures used in this study. A detailed
description of the clinical interview was given. This
chapter also described the population involved in the
study. The data collection and analysis procedures were

also examined in detail.

CHAPTER FOUR

RESULTS AND DISCUSSION

Introduction

The purpose of this study is to address the question
of students’ understanding of variables as used to
generalize patterns. The following three research
questions served as the structure for the research:

I. Uhat types of patterns can students recognize?

ll. Hhat types of patterns can students generalize by
using variables?

Ill. Uhat meaning does generalizations using variables
have for students?

These questions were addressed through clinical interviews.
The clinical interview was designed so that students’
conceptions about variables used to generalize patterns was
explored. Students’ conceptions of variables as part of
generalized expressions, sentences and sequences were also
explored. Analysis of the interview involved reviewing
student responses to tasks while keeping Lesh, Post and
Behr’s (1987) definition of understanding in mind. Results
of the interview will be presented in light of this
definition.

Hithin the framework of the three research questions
several analysis issues emerged. In the same way that
different interview tasks were designed to address specific

65

66

research questions individual issues which emerged were
also most closely identified with a specific research
question. This relationship among the research questions,
analysis issues and interview tasks is summarized in Table
4.1.

Research question one is elaborated through the
following questions:

1. Uhat patterns from the interview were students
able to recognize?

2. Hhat makes a deep pattern difficult or
easy for students to recognize?

3. Uhat types of mistakes do different patterns
lead students to make in the process of

recognition of those patterns?
These specific issues, as well as the general research
question, were addressed through the card sort and the
identification of patterns from tables task.

The second research question is ”what types of
patterns can students generalize by using variables?”.
This question was addressed through the following issues:

1. Uhat types of patterns can students

generalize and what is the connection between
the ability to generalize patterns and the
ability to recognize patterns?

2. Uhat types of errors do students make when
generalizing expressions and sentences?

These specific issues along with the general research
question were investigated through two tasks. These tasks
included the generalization of grouped cards in the card

sort and the generalization of patterns in tables.

Research Questions

67

Table 4.l

Relationships Aaong Research Questions,
Analysis Issues and Interviev Tasks

Analysis Issues

lnterviev Tasks

-Type of Skills

Associated Vith
Understanding Task
Investigated

 

i. let Types of
Patterns Can Students
Recognize?

2. flat Types of
Patterns Can Studen
Generalize Ry Using
Variables?

3. Vhat lleanlng Do
Gmnralizations Using
Variables lava For
Students?

 

l. Vhat Patterns Free the
Interviews Vera Students Able
To Recognize?

2. Vhat Ilates a Pattern
Difficult or Easy for
Students to Recognize?

3. Vhat Types of llistahes Do
Different Patterns Lead Students
To llake Vhen They Try to
Recognize Patterns?

1. Vhat Types of Patterns Can
Students Generalize and hat is
the Connection Retveen the
Ability to Generalize Patterns
and The Ability to Recognize
Patterns?

2. Vhat Types of Errors do
Students Ilake Vhan
Generalizing Expressions and
Sentences?

l. Vhat Strategies do
Students Use to Obtain
lleanlng Pros Expressions,
Sentences and Sentences
Containing Variables?

 

Card Sort,
Identification of
Patterns Pres Tabla?

Generalization of
Groups of Cards
Vithin a Sort,
Generalised
Statenants of
Patterns in
Tables

linking up of Story
Probleas to Fit
Espressions and
Sentences

 

Recognition of Deep
Patterns within the
Imerlcal
Representational llode
of the Vritten Syebols
Representational
Systes, Recognition of
Patterns Vritten Vithin
The Numerical
Representational llode
of the Vritten Syebols
Representational Systes
Vhlch Vere Presented
Vithin The Pictures and
Diagraa
Representational
Systea.

Translation Pros The
limerical
Representatlnnl lloda
to the Variable
Representational

llode Vithin the
Vritten Synbols
Representational
Systes, Translation
0f Patterns to the
Variable
Representational iloda
ill the Vritten Syebols
Representational
Systea.

Translation ef
Statesaats Vritten
Using The Variable
Representational lode
Of The Vritten Syfiol
Representational Systes
Based on Knovladge of
lbs The Generalizations
Reflect Real Vorld
Events.

68
The final research question is "What aeaning do

generalizations using variables have for students?" This
question was investigated through analysis of the following
issue:

1. Uhat strategies do students use to obtain
aeaning froa expressions, sentences and
sequences containing variables?

This issue along with the third specified research question
were investigated through the interview task which asked
students to generate story probleas to fit given

expressions, sentences and sequences.

Results.

Research Question One: What Types 2; Deep Patterns
Can §tudents fiecognige?

During the card sort all of the students were given

 

'several different sets of cards to sort. Three sets of
cards were coaaon to all interviews. These three sets
contained the following pairs of deep patterns:

1. The aultiplicative identity and the coaautative
property of aultiplication.

2. An arithaetic sequence with the rule of each tera
being two greater than the previous tera and
a geoaetric sequence with the rule that each tera
was twice the previous tera.

3. Cards illustrating through forauias, diagraas and
nuaeric sentences the concepts of area and
periaeter.

All of these deep patterns are representative of what Lesh
et al. (1987) label as concepts. As described in detail in
Chapter Three each set of cards was also organized around

three surface features. Surface features are physical

characteristics of the expressions, sentences or sequences.

69

Issue One: Hhat Qeep fiatterns Did Students Recognize?

Table 4.2 susaarizes the success of seventh
graders and algebra students respectively on each of the
sorts listed above. Overall both the seventh graders and
algebra students were aostly unsuccessful at these tasks.
One exception was the algebra students’ ability to
recognize the multiplicative identity and to a lesser
degree the coaautative property of aultiplication. The

students were unsuccessful at recognizing the deep patterns

 

 

 

 

 

in the cards containing sequences as well as in the area
and perimeter cards.
Thble 4.2
Students' Ability To Recognize Patterns as Deeonstrated By the Card Sort
neia
ltipIicative Connutative Sequence: Sequence: Perineter Area
dentlty Property of s,r02.srt, r.2:.is,8r
MWUNHMUmIrNLHLM. ML..

th alg. 7th alg. 7th alg. 1th alg. 7th alg. 7th alg.

rd. sts. grd. sts. grd. sts. grd. sts. grd. sts. grd. sts.
Correctly J
WMMHnS D i D D D I D D D D D
‘hymwulh
Deep Pattern
Correctly
Snudmr
Later Sort Dy i 2 3 D D i 2 i I D 2 2
Deep Pattern
amDup
Pattern Vith
Iluuslhp D D i i 2 D i D D D D D
has»
Variable Rep.
lMlfinu
lased on R 2 D A it I2 D 12 12 13 ii ii
Surface
Features

 

 

 

 

 

 

 

 

 

 

 

 

 

Exaaples of two

correct, one incorrect,

aultiplicative identity

70

different students’ sorts, one
for the set of cards containing the

and the coaautative property of

aultiplication are given here.

Frank (7th grader):

Cards Sorted Together

3/4 - 1 8 3/4
18/19' 1 8 18/19

74 o 4 8 4 ' 75
754 e 1 8 754
83 e 1 8 83

11 P 3 8 3"11

7/20 ° 1/4 8 1/4 0 7/20
2/3 S 5/7 8 5/7 - 2/3

‘UU-U'D

.. a .

rel or:

I'll

'DIOI
0

Frank saw no other ways

Opal
Cards sorted Together

3/4 ‘ 1 8 3/4
18/19 e 1 8 18/19
754 s 1 = 754

83 f 1 8 83

75 s 4 8 4 e 75

11 n 3 8 3 ‘ 11

7/20 - 1/4 8 1/4 8 7/20
2/3 t 5/7 8 5/7 0 2/3

5 e I ' I O J

b s c 8 c s 1)

Reason "by Frank Placed the Cards
Together

They look pretty such the saae.
The way they’re set up. They’re
fractions and then they have
equals another fraction.

They all have a nuaber and then
an equal sign and then another
nuaber.

They have a fraction and then a
aultiplication sign and then
another fraction and then an equal
sign. Then the saae fraction just
set up in a different order.

They all have a letter that you
multiply by. The first thing is a
letter.

that the cards could be sorted.

(algebra student)

Reason Opal Placed The Cards
Together

Because anything tiaes one is
itself and so, all of those are
Just tiaes one. So it’s always
itself.

Because they both have the saae
answers. c . b is b e c. So all
of then are like that b e c 8 c - b
You always get the saae answers in
both ones so it is equal.

71
The above exaaples illustrate both a correct and incorrect

sort. As indicated in Table 4.2, the aajority of sorts were
incorrect.

Given the students’ lack of success on these tasks a
question raised is whether or not the presence of variables
influenced students inability to recognize east of the deep
patterns. This question can be answered on two levels.
On one level the answer is yes. Students were affected by
the presence of cards containing variables. This effect
will be discussed further under the issue of what aakes a
pattern difficult or easy to recognize. On the other hand,
one way that the students light have chosen to sort the
cards was to first sort all of the cards containing
variables together. They then would have been able to sort
the reaaining cards without paying attention to the cards
containing variables. Table 4.2 indicates that few
students on each task successfully eaployed this strategy.
The cards containing variables did not provide clues
helping students to recognize patterns. These cards also
did not stand directly in the way of deep pattern
recognition.

The second task which investigated students’ ability
to recognize patterns was the identification of patterns in
tables task. Tables 4.3 suaaarizes the results of this
task. Both groups were sore successful at identifying
these patterns than they were at identifying the patterns

in the card sorts.

72
“Meta

Results Free Recognition of Patterns In Tables Test

 

2

s s 0 5 3a 5: 0 8 Sr - A
7th alg. 7th alg. , 7th alg. 7th alg. 7th alg.
grd. sts. grd. sts. grd. sts. grd. sts. grd. sts.

new new n'13 at” n-u aria neSntll n-2n-D

 

Pattern
Recognized 13 13 D 11 3 12 2 l D D

 

 

 

 

 

 

 

 

 

 

 

If we look at Just the results of having students
recognize the patterns x + 5 and 3x the algebra students
were aore successful at recognizing these patterns than
were the seventh graders. Alaost all of the algebra
students recognized these patterns. All of the seventh
graders and algebra students successfully recognized the
pattern :2. Only eight out of thirteen seventh graders
identified the pattern x + S. For the pattern described by
the expression 3x the success rate for seventh graders was
only three out of twelve.

The table representing the pattern 5x + 3 was only
shown to five seventh graders as opposed to eleven algebra
students. The decision to not continue the task with the
aajority was based on their lack of success with the x + 5
and 3x patterns. Pilot interviews had revealed that if
students were unsuccessful at seeing the patterns with
these cards they would also fail at seeing the 5x + 3 and
5x - 4 patterns. Of the five seventh graders who saw the

pattern 5x + 3 two successfully identified it. Of the

73

eleven algebra students who saw the pattern only one was
able to identify it.

Issue Two: what flakgs ; Pattern Difficult 2; Easy Eor
Students ;_ Recognize?

The next question of what sakes a pattern difficult

 

 

or easy to recognize follows logically froa a look at
which patterns students did or did not identify. In
addressing this issue it was necessary to go beyond the
success or failure of students on a task to the analysis of
what reasons students gave for sorting cards in the ways
that they did. Free the responses students gave for why
they sorted cards into the groups that they did, different
features of the expressions, sentences, sequences, and
_diagraas which affected how students sorted cards eaerged.

As described in Chapter Three each of the sets of
cards in the card sort were organized around three surface
features which students eight focus on when sorting the
cards. The students often saw aany aore. Responses on the
three sets of cards which were coaaon to all of the
interviews will illustrate this point.

The cards which were to be sorted into two piles
representing aultiplicative identity and coaautative
property of aultiplication had the built in surface
features of whole numbers, fractions and variables. Table
4.4 illustrates the nuaber of students who sorted at one
point using these surface features. Each student light

have sorted the set of cards one or aore tiaes. If a

74
student sorted the cards five different tiaes and in three

of the sorts aade reference to cards being together because
they were fractions this student was only credited once
with having been influenced by the surface feature of
fractions. In this way a nuaber was obtained to reflect
how aany students were affected by each surface feature.

Looking at Table 4.4 it is apparent that students
found the presence of whole nuabers versus fractions versus
variables to be an iaportant distinction. A couple of
students were also influenced by the fact that soae of the
sentences contained the sale nuabers. Another influencing
feature for a couple of the seventh graders and algebra
students was the physical‘ appearance of .the sentence
itself. Physical structure here would be state-ants such
as the following:

"Cause it’s got one nuaber tiaes another nuaber
equals the sane nuaber on the other side.” (Xavier,
algebra student)

”These are nuabers tiaes nuabers.”(Stan, algebra
student)

“Cause they’re both like, they have a nuaber and then
a dot (a aultiplication sign) and then a number." (Grace,
7th grader)
The eaphasis is not on the nuabers the-selves, but on the
way in which they are operationally Joined.

Tables 4.5 and 4.6 illustrate other surface features
which students paid attention to which were or were not

intentionally built into the respective card sorts. By

looking at these two tables and the transcripts froa other

75
card sorts that the students did, a list of types of

surface features to which students pay attention eserged.
Students paid attention to surface features which
fell into the following five groups:
1. Types of nusbers used.
2. Properties of nusbers.
3. Cosson features of sore than one expression or
sentence.
4. Hays of representing expressions, sentences and
sequences.
5. Hathesatical sysbols and notation used.
These five broad categories along with types of surface
features which fell under each heading are sussarized in
table 4.7.
Sose exasples fros the transcripts which illustrate

each of the five types are as follows:

Types 2; Numbers Used

 

For the card sort containing the deep patterns of
sultiplication by 5 and sultiplication by 1 Frank, a 7th
grader, put the following cards together: 4/8 ' 5, 4/6 . i,
5 ' 8/9, 1 - 6/8. His reason for grouping these cards as
such was "They all have a fraction that you sultipiy by a
whole nusber.“
ngpertigs of Numbers

In sorting the sase set of cards which Frank (above)
sorted, Used, an algebra student, placed the cards 1 ° 53,
S ‘0 87 and 317” 5 together. His reason for sorting the
cards in this fashion was :Because it’s a prise nusber

tises another prise nusber.”

76

Table 4.:

Uhet Surface Features Students Paid Attention to Vhen Sorting The
Cards Representing The Deep Patterns of lIuItiplicetiee Identity
and Conutative Property of Ilultiplication

 

 

 

 

 

(RIue Cards)

Surface Features luilt in fiber of Seventh Graders Uho lie: of Algebra Students Veo

by The Researcher Sorted Paying Attention to the Sorted Paying Attention to the

Particular Surface Feature Particular Surface Feature
a 8 13 n 8 13

hole fibers 6 A
Fractions 0 5
Variables 11 S
Dther Surface Features
The Students Identified
Because they contained
a Certain Her 2 D
The Physical Structure of
the Problen 3 2

 

 

 

77

Table A.5

Hat Surface Features Students Paid Attention to Vhan Sorting The
Cards Representing The Deep Patterns of an Aritheetic Sequence uith the canon
Differem of Ten and a Geoeetrlc Sequence Vith a Canon Ratio of Ten

(Tellou Sequence Cards)

 

Surface Features built in

Mar of Seventh Graders Vho

Me: of Algebra Students Ibo

 

 

 

 

Iy The Researcher Sorted Paying Attention to the Sorted Paying Attention to the
Particular Surface Feature Particular Surface Feature
n 8 13 a 8 i3

Uhole fishers 5 5
Practices 10 9
Variables 6 D
Other Surface Features

The Students Identified

Certain “er in Co-on D 1
IIiaed Fractions A A
Parenthesis 5 2
Sane Deaosinators 1 3
Sue Itsarators i 2
Odd fibers 2 3
Even “are 3 A
Size of Mrs 1 2

 

 

 

78

Table A.S

that Surface Features Students Paid Attention to Vhen Sorting Tb
Cards Representing The Deep Patterns of Area and Periaeter

(Grange Geo. Cards)

 

Surface Features built in

Mar of Seventh Graders to

“er of Algebra Students he

 

 

 

 

 

Ry The Researcher Sorted Paying Attention to the Sorted Paying Attention to the
articqu Surface Feature Particular Surface Feature
a 8 13 n I 13
Vhole fibers 2 1
Diagraas 5 6
Variables/Pursues! DI‘I AIS
AStudents discussed Variaqu and Foraulns Differently
Dther Surface Features
The Students Identified
Epls P l A
Equals A 2 A
Squre “era 1 2
Presence of S 2 1
Feet 1 Inches (neasuruents) 2 A
Feet 2 3
Inches 1 1
Odd Isbers D 3
Even More 0 3
IIuItiplicatioa Problees 3 7
Addition Probleas 2 3
Problens vith Length and Vidth i 2
Problaus uith base and lbigbt 1 2
Certain lusber in Canon 1 1
Dee Letter Eqsl to The Others 3 3

 

 

 

79

Table A.7

Types of Surface Features

 

TVPES fl IRIERS PRDPERTII I use: REPRESMATIN w “I”.
m AID m
Fractions Prise Mars Pictorial Representations
Ilired Fractions Ddd Ruders Iserical Representations
Deni-Is Even Rushers Variable Generalizatiens
Variables Size of Rushers Used
Perfect Squares

her of digits in ausbers

Presence of certain ausbers
Use of certain letters
in foraulas a’s for area, p’s for perlaeter
lengths such as b - base, h I height
Ilaasureaent Units Used
Operations Used (9, -, a)
Physical Structure of Problena
Percelve Diffienlty of Problens if They Vera to be Solved

HATIDIATICAL sums AD WINS m

Parenthesls
3i versus3r J versus3°j

 

80
Common Featurgs g; Hogs than one Expressign g; §entgnce

Hhen sorting the cards containing the deep patterns
of cossutative property of sultiplication and
sultiplicative identity Hary, a seventh grader, placed
75 ' 4 8 4 e 75, 754 ' 1 8 754, 7/20 8 1/4 8 1/4 8 7/20,
2/3 u- 5/7 - 5/7 - 2/3 together. Her reason for placing
these four cards together was ”Cause they both have
sosething that has like 7 in it."
25y; g; Begregentigg Expressions, figntences ;QQ_Seguences

Uhen Yoyo, an algebra student, sorted the set of
cards containing the deep patterns of arithmetic sequence
with a cosson difference of two and geosetric sequences
with a cosson ratio of two he placed the cards containing
the sequences: n,2n,4n,8n,16n....

r, (r + 2), (r + 4), (r + 6), (r + 8) ...
1/5 a, 2/5 a, 4/5 a, 8/5 a, 16/5 a ... and
(s - 4), (s - 2), s, (s + 2), (s + 4) ...
together. His reason was "These ones have variables.” The
other pile he sade were all of the cards which did not have
variables on them.
Hathematical Symbols gag Notations gggg

Zoe, an algebra student, when shown the cards
containing the ease deep pattern as the set Yoyo sorted
placed: r, (r + 2), (r + 4), (r + 6), (r + 8)... and
(s - 4), (s - 2), s, (s + 2), (s + 4) ... together. Her
reason for putting these two cards together was "because
they were all in parenthesis.”

Each of these previous vignettes are just one of

sany exasples which could be used to illustrate the five

81

groups of surface features which students paid attention
to. By paying attention to surface features, of the
sequences, expressions and sentences, students often failed
to see the deep underlying patterns that existed. Thus the
surface features became noise or static which prevented the
students from receiving the picture of the deep patterns.
Evidence of the interference caused by surface
features is obvious when looking at the cards which
students placed together when sorting the various groups of
cards. Those cards that were grouped together in sore than
fifty percent of the total nusber of sorts sade by the
students is sussarized in Tables. 4.8, 4.9 and 4.10.
Students often sorted a particular set of cards sore
than one tise. The resulting total number of sorts is
higher than 26. Once a student successfully sorted a group
of cards the sort ended. These tables include all features
that were salient for students previous to and including

successful sorts.

82

Table A.3

hen Sorting The Cards Representing The Deep Patterns Df Ihultiplicative
Identity and Cc-utatlve Property of hitlplication Those Cards Vhich Students Placed

Together In Ilore Than Fifty Percent Of Tlueir Total Sorts

 

 

(blue Cards)
n 8 A2
Percent of Sorts Cards Correct Dr lncorrect htch
Placed Together based on Deep Pattern
71.A p i 8 p correct
a i 8
73.3 33 . 1 8 fl correct
15A 8 1 8 75A
”.5 15-A8A875A correct
11 8 3 8 3 8 11
90.5 213 . 517 8 517 . 213 correct
7120 8 11A 8 11A 8 7120
92.9 31A 8 1 8 31A correct
13119 8 l 8 13119
95.2 Jon-no} correct

 

 

 

83

Table A.R

Iuen Sorting The Cards Representing The Deep Patterns Aritheetic Sequences
Uith a Co-oo Difference of Ten and Geooetrlc Sequences uith a Canon Ratio 01 Too
Those Cards Vhich Students Placed Together In here Then Fifty Percent of Their Total Sorts

(Tollov Sequence Cards)

 

 

a 8 AD
Percent of Sorts Cards Correct Dr lncorrect Itch
Placed Together Rasod on Deep Pattern

51.0 mu.:oo.zos.zou,zoe. . . lncorrect
m,m,uoomoo, 1900. . .
2...“,0950oeo
3,6,12,24,45. e e

55.1 3,6,12,2A,AR... incorrect
201,203,205,2DT,2M...

55.1 10D,2DD,ADD,ND,1900... lncorrect
2,A,S,D,lD...

63.3 201,203,205,2DT,2W... correct
2,A,S,R,lD...

85.3 2, A,S,3, 10. . . incorrect
3g.,12g2‘g2‘ooe

67.3 201.203.205.207, zoo. . . Incorrect
IDD,2DD,ADD,&)D,1600...

67e3 impmgmgm. ‘meee £032.01
3,6,12,24,48...

75.5 ' r,(r02),(r0Al,(r93),(r83)... correct
‘."” ('-2),.g ‘.,2’. ('7‘)e oe

81.6 20 31A, 22 31A, 2A 31A, 3 31A, fl 31A... correct
5 213, 7 213, D 213, 11 213, 13 213...

no.0 315,615,1215,2A15,ARIS,5615... correct
11A, 21A,AIA,91A,191A. . .

 

 

 

84

Table A.1D

Ion Sorting The Cards Representing The Deep Patterns of Area and Porlneter
Ihooe Cards Vhich Students Placed Together In here Then Fifty Percent of Their Total Sorts
(Grange Geo. Cards)

 

 

a 8 53
Pete-n: of Sorts Cards Correct Dr Incorrect htch
P lee-r Together lasod on Deep Pattern
50.9 p 8 As correct
p 8 2b 8 2h
p 8 2n 8 2l
52.6 (2 8 Alt) 0 (2 8 bit) 8 20" correct
A 8 Aft 8 16ft
52.8 p 8 As \ correct
p 8 2b 0 2h
2
66.0 Din . 3in 8 271a incorrect
A 8 Aft 8 16ft
2
5ft . Sft 8 2Sft
2
71.7 A 8 a correct
A 8 Iv
A 8 bh
61.1 Poriaater Diagraas correct
63.0 p 8 Zn 8 21 correct
p 8 2b 9 2h
35.9 A 8 Iv correct
A 8 bh
“.6 Area Diagraas correct
90.6 A 8 Aft 8 l6ft incorrect
2
lift 8 Sit 8 22m

 

 

 

85

Table 4.8 shows that when sorting the cards
containing the sultiplicative identity or cossutative
property of sultiplication the following cards were placed

together sore than fifty percent of the tise:

1. j 8 s 8 s . j and b s c 8 c e b

2. 3/4 . 1 8 3/4 and 18119 8 1 8 18119

3. 7S 8 4 8 4 8 75 and 11 8 3 8 3 8 11

4. 2/3 . 5/7 8 5/7 . 2/3 and 7/20 8 1/4 8 1/4 8 7/20
5. 83 8 1 8 83 and 754 8 1 8 754

8. p . 1 8 p and a e 1 I a

uh 11s it is encouraging that all of these cards are
cOl-rectly placed together it is discouraging that students
“\e a difference between sets one, three and four, as
‘"|\4sbered above, in sore than half of the sorts. It is
‘qually discouraging that sore than half of the students
tailed to recognize that sets two, five and six contained
‘the sase deep pattern.

Sisilar results are seen in Tables 4.9 and 4.10 which
contain the inforsation for the sorts of the cards
representing the arithsetic and geosetric sequences and the
area and periseter cards respectively. Here however, not
only weren’t cards that should have gone together placed
together, but cards were incorrectly placed together as
well. An exasple of two cards placed incorrectly together
would be the following two cards: 4 8 4ft 8 16ft and
5ft 8 5ft 8 25ft2. The first card illustrates the
finding of the periseter of a square while the second card
illustrates the finding of the area of a square. A typical

reason that would accospany a sort like this was ”because

86
tLRseey’re all perfect squares" (Robin, algebra student) or

" 1 t is a nusber tises itself.” (Andy, seventh grader) Only
1.1) one instance did any student (a seventh grader) sake any
co-sent that would indicate that they were aware of the
d :l. fference between feet and feet squared.

It appears that the sore potential noise, or surface
thtures, a group of probless contains affects how
61 tficult or easy it will be for students to recognize the
dQQp patterns within these sets of cards. This say explain
why students were sore successful at recognizing the
p‘tterns in tables rather in groups of cards. The very way

1“ which a table is set up alsost elisinates any surface
1r'eatures. Thus, not having to deal with surface features
‘80hen looking at the tables students were free to
(concentrate on finding the deep patterns. This explanation
also sheds light on why students were sost successful at
identifying the sultiplicative identity. The
sultiplication by 1 could easily have been recognized in
all equations such as a surface feature is identified.
It was ispossible to separate those students who recognized
that all of the equations contained a one fros those
students who realized the deeper property of one, when one
of two factors is 1 the product always equals the other

factor.

87

I -sue Three: Uhat Types 9_f_ Histakes .d_q Different Patterns
L-‘d Students _t_c_u Make?

The third issue that was identified under research
question one was the types of sistakes which different
pas tterns lead students to sake. This question is tied very
c l cseiy with the last question. If a group of equations
cot-itain several potential surface features students often
‘81 ll focus only on these features. Any deep patterns will
be cospletely ignored. Thus the students fall in a trap of
rCiczusing only on the types of issues outlined in the
previously given Table 4.7.

Figure 4.1 illustrates a flowchart of the process

1 nvolved in sorting cards. ~ Many students are unable to
recognize the path that leads to the left. These
students get caught in the process of sorting cards by
surface features. They are unable to look beyond the
surface features to the deep patterns. If this flowchart
were three disensional the deep pattern path would have to
be illustrated somewhere hidden behind the surface feature

path out of view for sost students.

88

Stmvon

(n. Group of
Cards

 

 

 

 

   

 

Sort Deep /Vhat Features \ Surface Sort
Successful 4 0f Cards To Pay 9. Unsuccessful
Patterns \ Attention To? / Features
Scae Surface Student
Features! <<::::Given Cards
Partial Deep 0 Rose t
u

Sort Partially
Successful

       

 

 
 

 

   

 
  

  

Tee Does Student

See Another
Sort

h»
This card
Sort Ends ‘

Figure A.1r Flcvchart of the Process Involved In Sorting lnterviev Cards

   

 

 

Conclusion

The overall research question which was addressed in
this section was the question of what types of deep
patterns will students recognize? Two tasks, the card sort
and identification of patterns fros tables task, were

designed to answer this research question. In terss of

89

Leah’s et al. (1987) definition of understanding the card
sort task investigated students’ ability to identify
concepts (deep patterns) which were expressed using at
I eesast two different representational sodes within the
":8 itten sysbols representational system. Two constant
il’lelpresentational sodes of expression were the nuserical and
Variable representational sodes. One set of cards, nasely
.tho cards organized around the deep patterns of area and
periseter, also expressed the concepts using the pictures
§‘I'ned diagrass representation systes.
The identification of patterns fros the table task
‘Iias structured so as to see if students could recognize
thatterns which were written within the nuserical
representational sode of the written sysbols
representational systes but which were presented within the
pictures and diagrass representational systes. One
conclusion of the research was that the ssaller the nusber
of surface features built into the groups of probless the
sore successful students were at recognizing the deep
patterns (concepts) which existed within the probles. One
result of this fact was that patterns in tables were easier
for students to recognize than patterns in groups of
expressions, sentences and sequences because they contain
few or no surface features. Vithin the tables the nature
of the pattern to be generalized also affected the ease of
recognizing certain patterns. For exasple, the pattern x2

was easily recognized by students. This is not surprising

90

(statesidering that the sequence represented by this pattern
seosed fasiliar to sost students. Patterns which were
but It on one operation (i.e., x + 5 and art) were also sore
Oasily recognized than patterns involving two operations
( 1-e., 5x 8 Sand 5x - 4).

when groups of expressions, sentences and sequences

cfi'bt'ntain sany potential surface features seventh graders and
"-1 ‘gebra students were equally and highly unsuccessful at
chognizing these patterns. Hhen potential surface
tQaatures were sinisized both algebra students and seventh
graders were sore successful at recognizing deep patterns.
1"“ this later case, however, the algebra students were sore
‘8tuccessful than the seventh graders. The presence of the
ageneralized fors of a pattern did not help the students by

providing clues to the structure of the deep pattern.

These results indicated that students had quite a bit
of difficulty recognizing deep patterns. Students not only
had trouble recognizing patterns which were expressed
within different representational systess, such as was the
case with the area and periseter cards, but they also had
trouble recognizing patterns which were expressed with one
representational systes, as was the case with all but the
area and periseter cards. 'The fact that concepts were
expressed in all card sorts using both the nuserical soda
and variable node within the written sysbols
representational systess did not sees to either hinder or

help the students to recognize deep patterns. In terss of

91

.l.¢essh’s et al. (1987) three criteria for understanding students
1.x1 this study were usually unable to recognize deep
patterns which were expressed within one or sore

:8 e presentational systess.

Fleasearch Question Two: Uhat Types 21, Patterns Can
Students Generalize 5y Qsing Variables?

 

_!_!Lé;23 One: Uhat Patterns Can Students Generalize and What
JELEL, the Connection Between the Ability pp Generalize
\Pa tterns and the Ability Q Recognize Patterns?

 

One of the tasks which took place during the card

sSbrt phase of the interview was to have the students sort a
group of cards where the variable generalizations were not
IFbresent. If the students successfully sorted the cards
"Lhey were then asked to give a generalized statesent, using
8a given variable for the pattern they saw. Since each set
(of cards contained two deep patterns it was possible for a
student to correctly sort out one deep pattern, but not the
other. In this case the student was asked only to
generalize with a variable the pattern they correctly saw.
If a student was unsuccessful at sorting the cards they
were then sorted correctly for his/her by the researcher.
If the student could explain why the researcher sorted the
cards the way she did then the student was asked to
generalize the cards. If the student could not
successfully explain why the cards were sorted in the way
that they were, the task ended. The results of this task

are given in Table 4.11.

92

Table 4.11

Ability To Generalize Patterns Found on Card Sort Cards
Seventh Graders

 

 

Cooautatlvo Property Additive Identity hultlpllcaticn Of A
01 Addition Single Digit Ruaber Ry
a Variable
n83 n8l2 n85
Nuaber of Students
Vbo Vere 6 5 ' 2

Successful

 

Algebra Students

“'9 I'la 3'8

 

Ruaber of Students
that” ' D 3 2
Swuuhl

 

 

 

 

Table 4.11 indicates that not all students were given
the same set of cards to generalize. It depended on the
students success at various points in the interview as to
what set of cards they were asked to generalize. This table
also shows that the seventh graders and algebra students
were about equally successful at this task. At this phase
of the interview each student was only asked to generalize
one set of cards.

Table 4.12 is an expanded version of Table 4.3. This
table indicates which of the patterns in the tables
students were able to generalize. The numbers of students
who could produce generalizations with variables is in some

cases larger than the number who recognized those patterns.

93
That is because, using the sethod described in chapter
three, sose students who did not recognize the pattern were
told how the pattern worked (without any reference to
variables) and then asked if they could generate an
appropriate generalization with a variable.

Table 4.12 indicates that sore algebra students were
successful at recognizing the patterns in the tables. This
fact has already been discussed in greater detail earlier
in the chapter. As Table 4.12 indicates sany of the
seventh graders who did not recognize the patterns for
x 8 5 and 3x were not told what the patterns were and then
asked to generalize thes. This is because only the
patterns for 5x + 3 and 5x - 4 were usually told to
students unless the student expressed interest on their own
in knowing the patterns for x 8 5 and 3x. Table 4.12 shows
that the sajority of algebra students who recognized
patterns could also generalize thes. It also shows that if
the algebra students did not recognize the patterns they
still were able, in sost cases, to generalize the pattern
with variables if they were told what the pattern was. The
algebra students’ ability to generalize with variables
equations seess to exceed their ability to recognize
patterns.

An exasple of an interaction which illustrates this
point is the following:

Zoe, an algebra student, correctly identified and

generalized the patterns in the table which were

94

2
generalized by the equations s 8 5, 3k and J . Uhen

presented with the table which looked like this:

 

8
13
18
23

concr-

 

Zoe said ” Oh, I know, it also alternate, 8 and 3 and 8 and
3.” By following this alteration and also noting that each
nusber in the right colusn was increased by 5, Zoe was able
to correctly say what would go opposite a 10 in the table.
I pointed out to her that if we wanted to figure out what
went after 100 in the table, using her sethod would be
tise consusing. I asked her if she could think up a
forsula, or shortcut way, like she had used in the previous
three tables, to find out what would go opposite 100. She
was unable to do this so i eventually said to her, ”One
other student told se that they thought that if you
sultiplied the nusber in the left colusn by 5 and then
added 3 you could find the nusber that went in the right
hand colusn. Do you think that that student was correct?"
Zoe tried a few exasples and decided that the student had
given a correct rule. _ I next asked Zoe what would go
opposite a p in the table. She wrote down ” p(5 + 3).”
She then said "wouldn’t those two together it would be
8 '8 p. No it wouldn’t work. Add those two together it
would be 8 8 p. It’s 5p 8 3." So by hearing the
fictitious other student’s rule and using a bit of testing

of the first written generalization Zoe was able to

95

generalize with a variable a pattern which she was unable
to recognize on her own.

Only patterns for 5x 8 3 and 5x - 4 were revealed to
the students. For the sajority of seventh graders the task
had ended before this point so it is hard to sake a sisilar
statesent about them. Looking at the data for the seventh
grade students who saw the patterns indicated by the
expressions 5x + 3 and 5x - 4 it is clear that with these
patterns seventh graders’ ability to generalize patterns
with variables exceeded their ability to recognize the
pattern. Therefore, for the seventh graders the data tends
to indicate that their ability to generalize patterns
exceeds their ability to recognize patterns, but it is
ispossible to state this as a fact based on the available
data. Further research needs to be done before a definite
statesent about the relationship between the ability of
seventh graders to recognize patterns and their ability of
generalize these patterns with variables can be sade.

Table 4.12 indicates that the patterns that students
can generalize are dependent on the patterns that they
recognize. Students’ ability to generalize patterns is not

dependent on their ability to recognize those patterns on

their own. Recognition of a pattern is a prerequisite for
generalization of that pattern. The data indicates,
however, that if students were sore successful at

recognizing patterns they would be equally sore successful

at generalizing those sase patterns.

hsults of Recognition and Generalization of Pattern In Tables Tasks

Table A.12

 

 

 

 

 

 

 

 

 

 

 

 

Seventh Graders
Pattern I2 I 8 5 3s 5r 8 3 Sr - A
n813 n813 n812 n85 n82
Pattern
Recognized 13 D 3 2 0
Pattern
Correctly 5 5 A 5 2
Generalized
(1 student (1 student (3 students (2 students
told the told the told the told the
pattern) ‘ pattern) pattern) pattern)
Algebra Students
2
Pattern s :85 3a 5183 Sr-A
n813 n813 n813 a811 n86
Pattern
Recognized 13 11 12 l D
Pattern
Correctly 12 12 12 10 S
Generalized
(1 student (ID students (5 students
told the told the told the
pattern) pattern) pattern)

 

 

 

 

 

 

97

issue Two: hat Trees 2; Errors 9; Students Hake Uhen
Generalizing Expressions and Sentences? ‘

C

 

 

 

 

Table 4.13 looks at the types of errors students sade
when trying to generalize the cards in the card sort. This
table shows that for seventh graders the lost con-on type
of error was to generalize by making the variable parts of

the patterns constants and the constant parts of the

equations variables. For example, given the group of
equations: 17 + O 8 17
8.2 + O 8 8.2
3 + O 8 3
99.3 + O 8 99.3
Table 4.13

Strategies oi Students Ubo incorrectly Generalized Patterns
Found on Card Sort Cards

 

lncorrect Strategy 01 lncorrect Strategy 01
flaking The Constant Use oi Too lany
The Variable Variables

Seventh Graders

ntio

 

lueber oi Students Ubo
Eeployed The Strategy 6 0

 

Algebra Students

n=1+
Nunber oi Students Uho
Enployed The Strategy 3 S

 

 

 

 

98

The variable parts of these sentences are the nuaerals, 17,
8.2, 3 and 99.3. A correct generalization of these
sentences indicates this variation along sentences by use
of a variable. A correct generalization would be, b + O 8
b. The zero is constant in all of the sentences. A saaple
of the incorrect generalizations given by ease of the

seventh graders were:

8.2 + p 8 8.2 (Andy)
3 + x 8 3 (Grace)
99.3 + b 8 99.3 (Hank)

Fewer algebra students than seventh graders aake this error
when generalizing equations. As Table 4.13 shows algebra
students had a tendency to err when generalizing by using
too aany variables or by aaking the generalization overly
coaplex by adding in unnecessary nuaerals. Sons of the
incorrect generalizations they gave for the previously

given set of sentences were:

a + b 8 a (Robin)
a + b 8 c (Toe)
3b + 0 8 3b (Zoe)

Hhen generalizing the patterns in the tables the
algebra students were quite successful with their
generalizations. One error pattern consistent with
Uagner’s (1981) finding that students had difficulty
conserving variables eaerged within the seventh grade
group. Four students who atteapted to generalize the
pattern indicated by the expression t + 5 erred in their
generalizations. Of the four, three used the incorrect

technique of thinking of the variable as having a nuserical

99

equivalence. Uhen asked what would go on the other side of
a table opposite the variable "a” one student (Andy,
seventh grader) replied "f cause it’s 5 down froa it."
This student was thinking a, b, c, d, e, f. "F" is the
fifth letter froa ”a”. Siailarly, one student (Karen,
seventh grader) said that opposite a ”k" would be a ”p”.
Again thinking, k, l, a, n, o, p. The fifth letter after
”k” is "p”. Finally, the last student who applied this
sale error gave a slightly different reply. Hhen asked
what went of the other side of the table froa a ”k" this
student (Mary, seventh grader) replied "16”. ’ Hhen asked
why 16 she wrote A,B,C,D,E,F,G,H,l,J,K 11 + 5 8 16. ”K”
is the eleventh letter in the alphabet. Even when it was
pointed out to the students that the variable represented a
nusber in the table they still gave these saae replies.
Conclusion

in this section the research question ”what types of
patterns will students be able to ”generalize?" was
addressed. The generalization of groups of cards within a
sort and generalized state-ants of patterns in tables task
were designed to investigate this question. The
generalization of groups of cards within a sort task
required that students be able to do two things (1)
recognize deep patterns within the nuserical
representational node of the written syabols
representational systes and (2) to translate concepts froa

the nuaericai representational node to the variable

100

representational aode within the written sysbols
representational systea. The second task required students
to translate patterns in tables which were written within
the nuserical representational node of the written syabols
representational systea to the variable representational
lode also within the written syabols representational
system.

The results of these tasks indicated that once
algebra students realized what the pattern was their
generalizations of that pattern was usually correct. One
exception to this last statesent occurred in the card sort
when incorrect generalizations usually contained too aany
variables. One reason that this error probably did not
show up in the tables task is that the patterns which were
being generalized resulted in expressions as answers rather
than sentences.

The errors aade by the seventh graders when
generalizing patterns indicate that sons of the. eight be
confused by the difference between a constant and a
variable. when students incorrectly generalized the
additive identity as, for.exaaple, 5 + a 8 5 rather than
a + O 8 a they were exhibiting confusion between constants
and variables. The assigning of a specific numerical value
associated with position in the alphabet to variables also
indicated the confusion. This type of error showed that
these seventh graders had trouble distinguishing between

the written sysbols and variable representational nodes of

101

the written syabols representational systea.

it can not be denied that a prerequisite step to
generalizing a pattern is recognizing the pattern. Data
indicates, however, that it is harder for students to
recognize patterns than it is for thee to generalize those
sale patterns. The tasks in this section indicated that
students have aore trouble recognizing concepts within a
representational systea than they did translating concepts

froa one node of representation to another.

gesearch guegtign Three; What Hganing 29 anegaligations
Uging yariablgs flave E2; studentg?

Each expression, sentence or sequence that contains
variables is a generalization of an infinite group of
expressions, sentences and sequences. The last task of the
interview was designed to see whether or not students could
work backwards froa a generalization and provide one real
life exaaple froa the infinite group of exaaples which
could fit the generalization.

To work backwards froa generalizations to situations
which fit those generalizations was hard for students. For
exasple, no student saw w, w + 2, w + 4, w + 6... as a
series of consecutive nuabers differing by 2. This
sequence was also not viewed by any student as infinite. A
few students interpret the pattern by focusing on w,
w + 2, w + A and w + 6 as independent nuabers or as

descriptions of some nuaerical event over tile. Soae

102

exasples froa the transcripts are:

'Uell like you have w nuaber of bugs in your house
and then you get two aore to cone along and then you
get 4 aore to cone along and then 6 aore cone along.”
(irene, 7th grader)

“This rich guy, he has w houses and he goes out and
gets bored and he takes a trip around the world so he
goes and, he already has w houses so he goes and buys
2 aore houses, then he gets bored so he buys 4 aore
houses. Now he has a lot of houses and he’s still
bored and he marries a lady and he needs aore houses
so he goes out and buys six aore houses." (Penny,
algebra student)

"Jenny had w amount of stickers and her friend Becky
had the saee aaount plus 2 and Jody had the same
amount as Jenny had plus 4 and their neighbor had 6
acre than Jenny had.” (Zoe, algebra student)
Siailarly, no student saw the sentence
b + (b + 3) + (b + 6) 8 735 as the sun of three nuabers
.differing by a value of three which equal 735. After
reviewing the aany ways in which students viewed the
expressions, sentences and sequences the interesting
question of how did students try to lake sense of this task

emerged.

lssue One: What Strategies _2 Students Use 19 Obtain
Meaning Fgom Generalizations?

Table 4.14 illustrates one of the techniques which

 

 

 

students used to obtain aeaning froa the generalizations.
Hhat soae students did to lake sense of the generalizations
was to assign and use nuaeric values in their probless.
Soae exasples, taken froe the transcripts for the
generalization c + O 8 c, which illustrates the use of this

technique are:

103

TAIL! A. 1‘

Ability to Generate Vord Problens Pros Generalized Statsnests

 

 

 

 

 

 

 

 

 

Seventh Graders
Generalized c9081: tsl s,v02,est,n-9 brad” Sig-TOSS-h87
Statseent e96... (how-75
n813 n813 n813 n813 n813 n812 n811
Generated
Vord Problens
ly Assigning
Constant Values 5 2 i 3 0 i 1
To The
Variables
Algebra Students
Generalized coo-s tOS e,v02,e0A,n-O some» Gigi-70 55-1-7
Statsnents e06. . . (N087!)
n813 n813 n813 I813 I813 n813 n811
Generated
liord Problens
Iy Assigning
Constant Values 2 2 l) i 1 2 0

To The
Variables

 

 

 

 

 

 

 

104

"There are six cats and you added zero which would
equal 6 cats again.” (Cathy, 7th grader)

'lf Joe has 13 aarbles and he doesn’t have the aoney
to buy any more he will still have 13 aarbles.”
(Frank, 7th grader)

"One cucuaber plus zero cucuabers equals one
cucuaber." (Vladiair, algebra student)

"You have ten cookies in a jar and you add zero to
it. How aany cookies do you have?" (Stan, algebra
student)
These students found it easier to lake sense of the
generalizations if they thought of the variable as if it
were a constant. Or in other words if they looked at a
specific instance of the variable.

Another technique which students used to aake sense
of the generalizations was to think of the variable as
being the first initial of the object the generalization
was about. The nuaber of students who used this technique
are suaaarized in Table 4.15. Soae exasples fro. the
transcripts for the generalization w, w + 2, w + 4, w + 6
... are:

”You go to a car wash and you wash your window and
when you wash 2 and then you wash 4 and then 8.”

(Andy, 7th grader)

'Ua, you have like, you have ten gallons of water
there and someone brings you two lore gallons and
you’re adding 4 aore, 6 lore.” (Karen, 7th grader)

“Uh, let’s see. if Bill had uh, w nuaber of wheels

and he added an even consecutive nuaber, he’d have,
sosething like that.” (Yoyo, algebra student)

105

‘hfletdb

Ublng The Variables As The initial of The Subject That The Problen Is About

 

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During the interview aany students paused when they had to

come with an object that the problea was about. it

UP
soaetiaes took then a little while to aatch the variable to
a word starting with the correct letter or to give up and

generate the problea about an "initially” unrelated object.

For some students the need to aatch the variable to an
object that the problea was about was one of the goals of
the task. Cathy, a seventh grader, aade this explicit when

she said " i’a using the first letter for words."

One technique used by four seventh grade students on
one expression each was to change the expression to an
equation. By doing so they forced the variable into a
situation where it represented a specific unknown. For

106

exaaple for the generalization t + 8 the following word
probleas were given:

“if you had 12 cats, how aany would you have to add
to B to get 12?” (Cathy, seventh grader)

“There’s a certain nuaber of suckers and you add 8
and you have to equal 13." (Mary, seventh grader)

For the generalization a - 9 the following were given:

"Like Mary has so aany apples and she gave away nine
and she has nine left.” (Debbie, seventh grader)

"if Joe has blank Iarbles and he lost 9 now he has
6.” (Frank, seventh grader)

This technique was not used on both expressions by any of
the four seventh graders. For the expression a - 9 Cathy
replied that you couldn’t do it ”because there’s no
answer." Hary caae up with a correct situation for a - 9
without setting the problea equal to anything. For the
expression t + 8 both Debbie and Frank said, ”i can't think
of one for that." None of the algebra students used this
technique for aaking sense of the generalizations.
Conclusion

The third research question addressed by the study
was ”what leaning does generalizations using variables have
for students?” The aaking up of story probleas to fit
expressions and sentences task was designed to investigate
this question. The task required students to translate
stateaents written using the variable representational node
of the written syabol representational systea to the spoken
language representational systes based on their knowledge

of how the generalizations reflect real world events.

107

The meaning generalizations have for students is
measured in part by the ability of the students to
interpret those generalizations. One type of
interpretation is the ability of students to fit the
generalizations into their schema of mathematical
situations. It is clear from the data presented in this
section that working backwards from a generalization to a
mathematical situation that fits the generalization was not
easy for students.

One of the techniques that students used to try to
make sense of the generalizations was the use of the
variable to represent the beginning initial of the object
the problem was about. Another technique used by students
was to assign the variable a constant value and to talk
about it as if it were constant. A technique which was
only used by seventh graders was to take expressions and
change them into sentences by setting them equal to
something. No other recognized techniques were used to
help the students interpret the generalizations.

A test of the meaning that students give to
generalizations is their ability to use the generalizations
to create mathematical situations. it is obvious that few
students are seeing this connection. The link from
generalizations to real life mathematical situations to fit
the generalizations is weak for most students. in other
words, students were unable to correctly translate

statements from one system (the written symbol system) to

108

another system (knowledge organized around real- world
events.)
Summary

The results of this study indicate that both the
seventh graders and the algebra students had trouble
identifying patterns among groups of expressions, sentences
and sequences. Patterns in tables were more easily
identified by both groups, although the algebra students
had more success with these tasks than did the seventh
graders. The ability of students to generalize patterns
seems to exceed their ability to recognize patterns. Both
groups of students had quite a bit of trouble generating

word problems from generalizations with variables.

109

CHAPTER FIVE

SUHHARY AND lHPLICATlONS

Dismiss!

This chapter has three sections. The first section
provides a summary of the dissertation along with a summary
of the results of the research. Secondly, is a section
discussing implications for teaching. Lastly implications

for future research are presented.

Summary

The purpose of this research was to address the
question concerning students’ difficulties in using
variables. Students’ conceptions of variables used to
generalize patterns and as part of generalizations were
investigated. This question arose out of consideration for
the historical development of variables. Historically
variables were developed as a part of a shorthand system
for generalizing patterns. Today this historical
development is bypassed in schools. Students are taught to
think about variables as place holders for solutions to
equations.

By focusing instruction on variables as static place
holders for solutions to equations students are not being

presented with a picture of variables which would lead to

110

conceptual understanding of the uses of variables to
describe dynamic situations. Lesh, Post and Behr (1967)
describe three skills which are necessary for conceptual
understanding of a concept. These are:

(1) be able to recognize a concept in a variety of
representational systems.

(2) manipulate flexibly the idea within a given
representational system and

(3) to correctly translate the idea from one system
to another.

There are five representational systems which Lesh et .al.
(1978) identify:

(1) knowledge organized around real world events,

(2) manipulative models,

(3) pictures or diagrams,

(4) spoken language and

(5) written symbols.
in contrast to presenting variables as place holders for
solutions to equations the historical development of
variables involved all three of Lesh’s et al. (1967)
abilities associated with understanding.

Existing research reveals that students have
difficulty understanding and using variables. Kuchemann
(1976, 1961 & 1964) concluded from his research that
students fell into six levels of understanding concerning
variables. These levels are discussed fully in Chapter
Two. Booth (1964a) extended Kuchemann’s work with further
research. He classified the types of errors which students

make when working with variables. Other researchers also

revealed various misconceptions and difficulties which

111
students have with the concept of variable.

in order to research and discuss students’
conceptions of variables it was important to focus on a
clear definition of what it means to understand a concept.
Lesh’s et al. (1967) definition of understanding provided
the theoretical backbone for the analysis. Results of this
study concerning students’ conceptions of variables were
regarded in terms of this definition of understanding.

A further consideration of this study was the way in
which variables were defined. Past research has focused on
variable as a placeholder for solutions to equations. A
broader definition of variable has to consider variable as
a tool for generalizing patterns and as part of
generalizations. This broad definition formed the
structure for this research.

Subjects

Because the move from arithmetic to generalized
arithmetic is supposed to take place with the introduction
of algebra it was decided to interview seventh graders and
algebra students for this study. Seventh graders have had
the highest level of instruction possible without yet being
introduced to formal algebra. Such would not be the case
with eighth graders since prealgebra is often taught at the
eighth grade level. Algebra students on the other hand
will have had a formal introduction to algebra. Twelve

students at each level were interviewed.

112

 

The data for the research were collected through the
use of clinical interview. The clinical interview was
designed to answer the following three research questions:

(1) what types of deep patterns can students
recognize

(2) what types of patterns will students be able to
generalize and

(3) what meaning does a generalization have for
students?

After conducting pilot work with students at the
seventh grade and algebra levels the research interview was
put into finalized form. This finalized interview
consisted of five tasks: sorting cards, generalizing cards
in the card sort, looking for patterns in tables,
generalizing patterns in tables and interpreting
expressions, sentences and sequences by creating story
settings. The card sorts were organized around three
surface features and two deep patterns. The whole
interview was audio taped and later transcribed.

25;; Analygis

The transcribed interviews were viewed qualitatively
with the three research questions in mind. Data were
analyzed around six issues which elaborated the three
research questions. A table relating the research
questions, analysis issues, types of understanding
investigated and interview tasks can be found in Chapter
Four (Table 4.1). The analysis issues arose from the

analysis of data with the research questions in mind.

113

Tables and supporting vignettes were presented to
summarize the research results.
Results

Hhile Leah’s et al. (1987) three characteristics of
mathematical understanding provided the structure of the
study, the three research questions reflecting these skills
structured the interviews. The research questions and
interview tasks based on the questions were designed to
probe students’ conceptions of variables.

The analysis of the research revealed several results

concerning students’ conceptions of variables. Both the
seventh graders and algebra students were weak at
identifying patterns in the card sort. Uhen students were

presented with patterns in tables, both groups identified
these patterns more easily. On this task, however, the
algebra students were more successful than the seventh
graders. Hhile data were inconclusive for the seventh
graders, the algebra students were more successful at using
variables to generalize patterns than they were at
recognizing patterns. Both groups found the task of
providing real life situations for existing generalizations
difficult.

Hhile the above set of findings revealed how well
students performed on tasks, other findings helped to
explain why they were successful or unsuccessful on these
tasks. One factor which was quite influential in

distracting students from seeing deep patterns was the

114

presence of surface features in the problems. Although
only three surface features were deliberately built into
problems, students often identified many more. These
surface features fell into five groups: (1) types of
numbers used (2) properties of numbers (3) common features
of more than one expression or sentence (4) ways of
representing expressions, sentences and sequences and (S)
mathematical symbols and notation used. These surface
features directly affected the way in which students sorted
the cards and the ways in which they viewed the patterns.
Another factor which influenced the students’ ability
to see deep patterns was the structure of the patterns
themselves. For example, patterns requiring only one
operation (i.e., x2, 3x, x + 5) were easier for students to
recognize than patterns involving two operations (i.e.,
5x + 3, 5x - 4). Numerical patterns, such as those
represented in tables, were also more easily recognized
than patterns in expressions, sentences and sequences.
Students did not use the cards containing variable
generalizations to obtain clues concerning the deep
patterns. This result ties in with the fact that students
were unable to generate real life situations from
expressions or equations containing variables. it is clear
that students have limited knowledge of what a
generalization means and limited ability to translate among
representational systems. i

in trying to obtain meaning from generalizations

115

students used different techniques. Both the seventh
graders and algebra students sometimes assigned constant
values to the variables. They then generated their
problems around this constant value. Assigning constant
values to the variables helped the students to make more
sense of the generalizations. Students often thought of
the variable as designating the first initial of the object
that their problem should be about. A few of the seventh
graders were uncomfortable with the expressions. The need
to make situations that gave a precise answer seemed to
influence students to change the expressions into sentences
in order to make sense of them. No algebra students used
this technique. This change is significant since variables
in equalities only have a limited domain of possibilities
which they can assume in order to represent an equality
which is true. Variables in expressions can assume an
infinite number of values.

Another place where algebra students and seventh
graders answered differently involved their
generalizations. Uhen generalizing with variables the
cards in the card sort the seventh graders were most likely
to err by taking the constant parts of the equations and
identifying them with variables. Similarly they took the
variable parts of the equations and gave them constant
values. A few algebra students used this technique. The
majority of algebra students who erred were more likely to

make mistakes which involved using more variables then were

116

needed or making their generalizations too complex by
adding numbers which were unnecessary. The seventh
graders’ confusion between variables and constants also
showed up when they were asked to generalize the patterns
in the tables. A few of the seventh graders on this task
associated the variables with their position in the
alphabet. Thus, "a” had a numeric value of 1, "b" had a
numeric value of 2 with the pattern continuing for all of
the letters of the alphabet. Again, this error was not
seen in the algebra students. The algebra students had a
clearer conception of the difference between a variable and
constant than did the seventh graders.
Conclusion 0

Lesh’s et al. (1987) definition of understanding
requires that students recognize a concept in a variety of
representational systems, manipulate flexibly the idea
within a given representational system and correctly
translate the idea from one system to another. Students in
this study were largely unsuccessful with the first and
third aspects of understanding of variables as tools for
generalizing patterns. The second skill associated with
understanding was not investigated.

implications Lg; Teaching

in order for teachers to help students to become
knowledgeable about variables so that they can perform
successfully on skills associated with all three aspects of

Lesh’s et al. (1967) criteria for understanding they need

117

to have an idea of the present extent and structure of
knowledge. The results of this study cannot be generalized
to all groups of seventh graders and algebra students.
This study, however, begins to shed some light on students’
conceptions of variables as tools for generalizing
patterns. Emphasizing the results of the study will help
teachers to begin to get a feel of how poorly students
currently understand variables, of the naive conceptions
students have, and of the lack of flexibility in
interpretation of situations that characterizes students
knowledge. The results further speak for the need of an
alternative approach toward the teaching of variables, such

as variables as used to generalize patterns, which would

incorporate all of Lesh’s et al. (1967) aspects of
understanding. The process of using variables to
generalize real world situations and using those

generalizations to solve problems requires that students:
(1) recognize a pattern (2) generalize that pattern (3)
correctly interpret what the generalization means and (4)
substitute numbers into the generalization to solve
problems related to specific situations. These four skills
are somewhat parallel to Lesh’s et al. (1967) three aspects
of understanding.

The results of the card sorts and tables task
revealed that students were largely unsuccessful at
recognizing patterns. One exception to this statement was

the ability of seventh graders and algebra students to

118

recognize patterns consisting of one arithmetic operation
presented using the numerical mode of representation within
the pictures and diagrams representational system.
Teachers need to start by expanding students’ knowledge
about table patterns. Since students were successful at
recognizing patterns using one arithmetic operation a
logical next branch of expansion would be to give students
increased experience with patterns in tables consisting of
two operations. Uhen students begin to become successful
with the recognition of these types of patterns it is time
to present students with different types of patterns. One
example would be series of equations all illustrating the
same concepts such as those which were used in the card
sort. Opportunities to explore a wide variety of pattern
situations many help students develop strategies for
recognizing patterns. They will begin to realize that
patterns are an integral part of mathematics.

From kindergarten on up students need to be guided
toward seeing similarities and differences between
expressions, sentences and sequences. They need to be led
from interpreting mathematical statements in isolation to
thinking about the total mathematical system within which
such statements exist. This study revealed that students
are unsuccessful at making connections between expressions,
sentences and sequences which share common structural
patterns. As one example, of many, most students were

unable to recognize that 2.4.6.8,10... and

119

201,203,205,207,209... are both sequences with a common
difference of two. Teachers need to make such links
explicit to students. Teachers also need to provide
students with opportunities which lead them to experience
the power of identifying patterns to help them to make
predictions about events in their day to day life.

Once teachers have helped students to recognize
patterns they need to guide them in generalizing these
patterns. The results of this study indicate that teachers
should experience success in teaching students to
generalize a recognized pattern. it was much harder for
students to recognize patterns than it was for them to
generalize those patterns with variables.

Although students can be expected to learn to
generalizing patterns with variables, teachers need to pay
careful attention to how they organize and set tasks for
students that help them develop a deep understanding of
what instructing students as to what generalizations with
variables mean. This study revealed that students had
trouble with the third aspect of using variables to
generalize patterns: working backwards from generalizations
written with variables to generate situations which would
fit those generalizations. Teachers should help students
to be able to look at an expression such as ”k + 5" and be
successful at the following tasks:

- writing other examples with and without variables

that fit the expression,
- illustrating the expression pictorially,

120
- deriving the expression from situations,

- writing a real life situation which fits the
expression.
This last task was very difficult for students in this
study.

This research revealed that compounded with students
conceptions of variables is their conceptions of other
mathematical ideas. if student view fractions as something
other than numbers than they are going to have a hard time
knowing that fractions can be substituted into expressions
just as easily as whole numbers. Similarly if students
don’t understand certain mathematical conventions, such as
writing two variables side by side to indicate
multiplication, then they will become bogged down trying to
make sense of them. An example from this research
illustrating this problem was that when sorting the cards
with deep patterns of area and perimeter 90.6 percent of
the sorts involved having the cards 4 ' 4ft 8 16ft
and 5ft 8 5ft 8 25ft2 together. The fact that the numbers
were squared was more salient than the deep patterns.
Students did not recognize that 5ft 0 5ft 8 25ft2 had to be
an example of finding an area whereas 4 . 4ft 8 16ft
illustrated a length of 4ft occurring 4 times. The reason
that students sorted so often by surface features is in
part because they still have trouble fully understanding
the language of expressions, sentences and sequences. it

is not enough to focus instruction on improving students’

conceptions of variables. Teachers must be tuned in to all

121
aspects of how students make sense of mathematics.

implications 12;,Future Research

Lesh et al. (1987) describe aspects of understanding
as well as five representational systems. This study
focused on students’ conceptions of variables as
demonstrated in three areas: (1) students’ ability to
recognize patterns in two representational systems, (2)
students’ ability to translate patterns from the numerical
representational mode within the written symbols
representational system to the variable representational
mode within the written symbols representational system and
(3) students’ ability to translate variable representations
within the written symbols representational system to
spoken language representations based on knowledge
organized around real world events. Future research needs
to focus on all three aspects of understanding and the
interactions between them as well as all five
representational systems.

This study focused mainly on students’ ability to
recognize and represent patterns using variables. Future
research needs to focus on students’ ability to recognize
and represent patterns with manipulatives, pictures and
diagrams, spoken language and numerical representations.
One such task would be to present students with variable
expressions and let them generate numerical examples to fit
the expression. Another task might be to present students

with tables already generalized and let them generate

122

entries for the tables. These tasks are extensions of
tasks which were used in this study. Showing students
manipulatives illustrating a pattern and having them
verbalize the pattern could be another investigation task.
in general, more tasks which focus on all five
representational systems need to be incorporated into
future studies.

The flexibility of students in manipulating variable
expressions within one representational system was not
investigated by this study. Finding out what types of
problems students can solve with given generalizations will
give input as to what power generalizations hold for
students.

Finally, this study was based on the belief that the
ability to recognize patterns and generalize them using
variables were prerequisite skills for student
understanding the power behind variable expressions. An
alternative belief could be that by having students solve
several different problems using variables as place holders
for solutions to equations they are gaining a feel for the
power variable expressions have. Teaching experiments need
to be conducted. Subsequent research should begin to look
at the strengths of using the two different approaches
based on students” understanding variables, using Leah’s
et al. (1967) definition of understanding. Only after
students’ conceptions of variables are more fully

understood can curriculum materials be developed to address

123
misconceptions. Such developed curriculum would require

classroom studies and further research designed at

continued study of students’ conceptions of variables.

Mimi-sax

This chapter provided an overall summary of the
dissertation along with results of the research.
Educational implications based on Leah’s et al. (1967)
definition of understanding were presented. Lastly,
implications for future research, also based on Lesh’s et

al. (1967) definition for understanding, were presented.

APPENDICES

APPENDIX A
THE lNTERV I EU TASKS

124

Card Sort Cards

Blue Set (Used in Every lnterviev)

 

Deep Patterns Surface Features

Vhole lllmbers Fractions Variables
CautativeProperty 11.383811 2/385/785/782/3 b .c8cob
oi ilultiplication 75 . A 8 A . 75 7120 8 1/A8l/A 87/20 1 . s 8 n c i
ilultiplicative Identity 83 . 1 8 83 18119 8 1 8 18119 p o 1 8 p
' 754-1875A 3/A81831A n.18a

Vellov Sequece Set (Used in Every interviev)

 

Deep Patterns Suriace Features

Vhole )hmbers Fractions Variables
Aritlmetic Sequence 201,203,35,207,209... 53.715.915.118JSZ... r,(r92),(rrA),(r86),(r88)...
‘. 2) 2.0.6.8,10... m‘.a%,2‘%”%flu (.-‘)g (.-2)... ‘..2).(..‘)eee
Geceetric Sequence 3,6,12,2A,A8. .. SIS,6I5,12/5,2A/5,A8I5. ..g m; 1,}, a, $3 a...
(E 2) 1wgmgm’mp!Meee ‘I‘.2"g‘l‘g8l"1el‘een I,2I,.I..I.16I.u

Orange Geo (Used in Every lnterviev)

Deep Patterns Surface Featrxes

 

Vhole llvmbers Diagraas Variables/Forenlu
Perleeter A - Alt 8 lGit p 8 2b 8 2h

 
 
 

u
stamens-lg P , 2' , 2|

5‘
(2 8411) t (2‘6“)82011

3‘ #3

‘i ' «+3840: H

‘1

 

 

 

 

 

2 1.
Area Sit . Sit 8 251t -q A 8 iv

Sin 8 Sin 8 27 in2 gfiEg 815 A 8 bh
i 3: A 2

A8s

 

 

 

 

 

 

 

 

 

 

 

125

 

 

 

Teliov Set
Deep Patterns Surface Futures
Vhoie lumber: Fractions Variables
ilnltiplicetion 317 . S Alb . 5 5 . s
By!» 5807 588/9 v.5
ilnltlplicetion SS ' 1 AIS 8 1 i . c
Byl 1 . 53 1 8 SIS 1 . h
Pit Set
Deep Patterns Surface Features
mole Ilmbers Decinais Variables
CmtativeProperty 3818183 11.583.283.2811.5h8s8s8h
DfAddltion 128282812 CS.A88.988.9866.An8h8h8n
Additive Identity 17 8 D 8 17 09.3 8 t) 8 99.3 y 8 o 8 y
38083 8.280881 a808a
Green Set
Deep Pattern Surface Features
Vhole llmbers large Vhole lhmbers Variables
llultlplication A . 3 8 12 5312312 - 3 8 1593936 3 .. v 8 3v
By3 18383 2A73-387A19 r-383r
iluitipilcetion 78281A M82849“ zo2821
By2 5°2810 C323A21828166A60A22-g82g

 

 

 

 

CM. 3 C___) ("3.3 C ...... 3
CM“) CM“)

o) G)

 

 

 

 

 

 

 

126

Tables
Orange 11 Tellov 12 Green A3 Ilue AA Pint AS
1 6 1 i 1 3 1 8 1 1
2 7 2 A 2 G 2 l3 2 G
3 8 3 9 3 3 3 13 3 11
A 9 A 16 A 12 A 23 A

Generalized Statenents

c808c

t88

n-S

b8(b83)8(b86)8(b89)8735
38g870

55-k87

APPENDIX B
A COPY OF THE PERMISSION SLIP

127

Donna E. Bird

252 Erickson

Hichigan State University
East Lansing, Hichigan
March, 1967

Dear Parent, .

As part of my doctoral research at Michigan State University
i‘am interested in investigating students’ conceptions of
mathematical patterns and variables. Your childs’ teacher and
principal recognize the potential benefits of this research for
mathematics instruction. They have agreed to allow me to
interview students during class time.

Your child is a vrluable component of my research. in order
to investigate the ways in which students think about
mathematical concepts it is necessary to talk with them one
on one in a relaxed atmosphere.

The research is to consist of an interview during which the
students will be asked to think about mathematical situations.
The students responses to these situations will be tape
recorded. After the study is completed the tapes will be
destroyed. The interview will last approximately the time of
one class period.

in order to conduct these interviews with students it is
necessary that parents give their consent. At any time during
the interview either the student or the parent can withdraw
their consent.

All results of the research will be treated with strict
confidence. Subjects will remain anonymous. within these
restrictions results will be made available to subjects if
requested.

This research is an important step toward answering the
question of ”how do students think about mathematics?" if you
have any questions before giving your consent please feel free
to contact me at the below phone number.

Thank you for you cooperation.

£30an E 6(18/

Donna E. Bird
349 - 6286

128

Student Consent To Participate in the Study of Students’
Conceptions of Hathematics

By“
Donna E. Bird

i agree to allow my child to participate in the doctoral
research study being conducted by Donna E. Bird. This project
is part of an approved research program at Michigan State
University. A description of the rationale and designof the
study has been provided separately.

i understand that.as a participant in the study. my child
will be expected to:

1. Participate in an interview taking approximately one
mathematics period.

i understand that the following precautions will be taken to
protect against abuse of my childs’ confidence or the data from
this study:

1. All data collected during this study will be kept
confidential and the study will be reported without the
identification of individual students, their teachers. or
schools.

2. i may request data on my child (and a group) and review
it with the research.

3. fly child may withdraw from the study at any time without
recrimination.

Signature

 

Printed name

 

Date'

Printed name of child;

REFERENCES

129

References

Abbot, J. a Hells, D. (1965). Mathematicp Today. Florida:
Harcourt brace Jovanovich inc. Grades 6-6.

Adda, J. (1962). Difficulties with mathematical symbolism:
synonymy and homonym.V1sibie Language. 1653), 205-214.

Assad, S.l. (1976). Perspectives in Mathematics Learning.
State University at Buffalo. DAl 40A: 722 -723: August
1979. [7916745]

Bell, A.H., Costello, J., & Kuchemann. (1963). A Review 9;
Research ;p_Mathematics Education. Part A: Research on

Learning and Teaching. Uindson: Nfer-Nelson.

Bernard, J. (1964, April). Ipp_teaghing ppg_learning p;
algebra: Developing perspectives. Paper presented at
the 1964 Annual Meeting of the National Council of
Teachers of Mathematics, San Francisco, California.

Booth, L. R. (1964 a.). Algebra: Children’s strategiesa and

error: in secondpry mathematics project. Uindso r:
Nfer - Nelson.

Booth, L.R. (1964 b.) Misconceptions Leading to Error in
Elementary Algebra. Journal 2; Structural Learning.
Vol. 6, 125 -136.

Bresiich, E.R. (1939). Algebra ;,Ap interesting Language.
New York: Newson & Company.

Brown, G. U. (1964). Searching for Patterns of
Division.Arithme§ic Teacher. 32(4), 32-34.

: The Struggle
n Mathematics

Burton, Leone. (1964). Mathematical Thinking
for Meaning. Journal for Research 1
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