florida state university librariesdiginole.lib.fsu.edu/islandora/object/fsu:254168/... · the...

Florida State University Libraries

Electronic Theses, Treatises and Dissertations The Graduate School

2006

How Graphing Calculators and VisualImagery Contribute to College AlgebraStudents' Understanding the Concept ofFunctionRebekah M. Lane

Follow this and additional works at the FSU Digital Library. For more information, please contact [email protected]

http://fsu.digital.flvc.org/

mailto:[email protected]

THE FLORIDA STATE UNIVERSITY

COLLEGE OF EDUCATION

HOW GRAPHING CALCULATORS AND VISUAL IMAGERY CONTRIBUTE TO

COLLEGE ALGEBRA STUDENTS’ UNDERSTANDING THE CONCEPT OF

FUNCTION

By

REBEKAH M. LANE

A Dissertation submitted to the

Department of Middle and Secondary Education

in partial fulfillment of the

requirements for the degree of

Doctor of Philosophy

Degree Awarded:

Summer Semester, 2006

The members of the Committee approve the dissertation of Rebekah M. Lane defended

on June 5, 2006.

_________________________

Leslie Aspinwall

Professor Directing Dissertation

_________________________

Diana Rice

Outside Committee Member

_________________________

Maria L. Fernández

Committee Member

_________________________

Elizabeth Jakubowski

Committee Member

Approved:

Pamela S. Carroll, Chair, Department of Middle and Secondary Education

The Office of Graduate Studies has verified and approved the above named committee

members.

ii

TABLE OF CONTENTS

List of Tables ................................................................................................iv

List of Figures ................................................................................................v

Abstract ......................................................................................................vii

1. INTRODUCTION ..........................................................................................1

2. LITERATURE REVIEW ...............................................................................6

3. METHODOLOGY .........................................................................................36

4. RESULTS OF VISUALIZER ........................................................................44

5. RESULTS OF NONVISUALIZER................................................................86

6. CONCLUSIONS.............................................................................................138

APPENDICES ................................................................................................150

A MATHEMATICAL PROCESSING INSTRUMENT..........................150

B MATHEMATICAL PROCESSING QUESTIONNAIRE ...................151

C APPENDIX C .......................................................................................164

D LETTER OF CONSENT FOR ADULTS.............................................165

E SECOND LETTER OF CONSENT FOR ADULTS............................166

REFERENCES ................................................................................................167

BIOGRAPHICAL SKETCH ..............................................................................173

iii

LIST OF TABLES

Table 3.1: College Algebra Weeks 1-8................................................................39

Table 3.2: College Algebra Weeks 9-16..............................................................40

Table 3.3: Web Homework Scores ......................................................................41

Table 4.1: VL’s Emerging Categories .................................................................85

Table 5.1: NVL’s Emerging Categories ..............................................................137

Table 6.1: VL’s Emerging Categories .................................................................139

Table 6.2: NVL’s Emerging Categories ..............................................................140

Table 6.3: Presence of O’Callaghan’s (1998) Translating Component...............145

Table 6.4: Role of Graphing Calculators .............................................................145

Table 6.5: Visual Imagery....................................................................................147

iv

LIST OF FIGURES

Figure 4.1: Mathematical Task # 1 .....................................................................47

Figure 4.2: Visualizer’s table of numerical values for task one .........................48

Figure 4.3: Visualizer’s Mathematical Task # 1 Graph ......................................49

Figure 4.4: Mathematical Task # 2 Graph ..........................................................50

Figure 4.5: Mathematical Task # 3 Graph ..........................................................53

Figure 4.6: Visualizer’s tables of numerical values and graphs .........................54

Figure 4.7: Visualizer’s completed Mathematical Task # 3 ...............................56


Figure 4.9: Visualizer’s tables of numerical values and graphs .........................60

Figure 4.10: Visualizer’s second page of a table of numerical values ...............61

Figure 4.11: Mathematical Task # 5 ...................................................................65


Figure 4.13: Visualizer’s algebraic calculations of Mathematical Task # 6 .......69


Figure 4.15: Visualizer’s completed Mathematical Task # 7 .............................73


Figure 4.17: Visualizer’s completed Mathematical Task # 8 .............................76


Figure 4.19: Mathematical Task # 10 .................................................................82

Figure 4.20: Visualizer’s completed Mathematical Task # 10 ...........................83


v

Figure 5.2: Nonvisualizer’s tables of numerical values ......................................90

Figure 5.3: Nonvisualizer’s y = x line ................................................................91

Figure 5.4: Nonvisualizer’s y = x and y = x + 5 lines .........................................92


Figure 5.6: Nonvisualizer’s y = 2x table of numerical values ............................96

Figure 5.7: Nonvisualizer’s y = 3x table of numerical values ............................96

Figure 5.8: Nonvisualizer’s y = x + 3 table of numerical values ........................98

Figure 5.9: Nonvisualizer’s y = 2x + 3 table of numerical values ......................99

Figure 5.10: Nonvisulizer’s y = 2x + 6 table of numerical values ......................101

Figure 5.11: NVL’s Mathematical Task # 2 Graph ............................................102








Figure 5.19: Mathematical Task # 10 .................................................................134

vi

ABSTRACT

The purpose of this study was to answer the following research questions:

• What is the role of graphing calculators in understanding functions?

• How does visual imagery contribute to visual and non-visual College Algebra

students’ understanding of functions?

Interviews and document reviews were the data sets used in this study. The data were

analyzed by using two theoretical frameworks: O’Callaghan’s (1998) translating

component for understanding functions and Ruthven’s (1990) role of graphing calculator

approaches. The investigation utilized the qualitative case study method. The findings of

one of the case studies of the two College Algebra students were reported in chapter 4.

The findings of the second case study were reported in this chapter.

The two participants in this study were presented with mathematical tasks to

complete over the course of a semester. Each task was given to the students individually.

In order to thoroughly understand the students’ responses, task-based interviews were

conducted and videotaped. In addition, each participant was interviewed based on his or

her response to the mathematical tasks. The tasks captured different types of

mathematical functions. These included linear, quadratic, cubic, absolute value, and

exponential functions. Furthermore, prior to receiving the tasks, the students’ preference

for processing mathematical information visually or non-visually were determined using

Presmeg’s (1985) Mathematical Processing Instrument and Questionnaire. These tools

were chosen because they measured how a student preferred to process mathematical

information, i.e., visually or non-visually.

In this investigation, O’Callaghan’s (1998) translating component was present

during the completion of linear, quadratic, cubic, absolute value, and exponential

functions. One of the participants used the graphing calculator during the completion of

all of the mathematical tasks and exhibited Ruthven’s (1990) approaches. The other

participant used the graphing calculator during the completion of five of the tasks and

exhibited Ruthven’s (1990) approaches. In addition, one participant relied on visual

imagery during the completion of five of the mathematical tasks. The second participant

relied on visual imagery during the completion of three of the tasks.

vii

CHAPTER 1

INTRODUCTION

“Algebra, whether at middle school level, high school level, or college level often

strikes fear in the hearts of students. Generation after generation have passed down the

opinion that algebra is not only difficult, but perhaps also boring” (Stephens &

Konvalina, 1999, p. 483). In fact after teaching mathematics on the university level, I

have found the previous quote to be true for many of my students. As an educator, this

truly disturbs me. It saddens me to look at my students at the beginning of each semester

and see fear in their eyes. How can anyone learn College Algebra or any other type of

mathematics when you start out afraid of it? This is one of the many obstacles that we

(my students and I) attempt to overcome each semester. One way to try to overcome the

fear of algebra is by establishing a learning environment in your classroom where the

students feel free to ask questions and share their opinions about mathematics. I like to

think of it as an algebra friendly learning community. This type of community of

mathematics learners develops and grows throughout the semester.

What is algebra? First of all, Christy (1993) states that “mathematics is a basic

tool in analyzing concepts in every field of human endeavor” (p. 3). Bishop (1989)

expresses, “mathematics is a subject which is concerned with objectivizing and

representing abstractions from reality, and many of those representations appear to be

visual, (i.e., they have their roots in visually-sensed experiences)” (p. 8). Algebra is a

type of mathematics. The dictionary provides a technical definition of this term. Algebra

is defined as “a generalization of arithmetic in which symbols, usually letters of the

alphabet, represent numbers or members of a specified set of numbers and are related by

operations that hold for all numbers in the set” (The American Heritage, 1991, p.93).

Angel and Porter (2001), Christy (1993), and Slavit (1999) also define algebra as a way

of generalizing arithmetic. Arithmetic involves performing addition, subtraction,

multiplication, and division with numbers (Christy, 1993). According to Kieran (1990),

“the teaching of equations, functions, and the manipulation of literal expressions and

equations traditionally signals the start of algebra” (p. 96). In addition, Angel and Porter

(2001) explain the historical origin of algebra:

The root word al-jabr, which the Muslims (Moors) brought to Spain along with

some concepts of algebra, suggests the restoring of broken parts. The parts might

be bones, or they might be mathematical expressions that are broken into separate

parts and the parts moved from one side of an equation to the other and reunited

in such a way as to make a solution more obvious. (p. 257)

Kieran (1990) summarizes three evolutionary stages of algebra:

The rhetorical stage, which belongs to the period before Diophantus (ca. A.D.

250), was characterized by the use of ordinary language descriptions for solving

particular types of problems and lacked the use of symbols or special signs to

represent “unknowns”. The second stage, syncopated algebra, extended from

Diophantus, who introduced the use of abbreviations for unknown quantities, to

the end of the 16th

century….the concern of algebraists during these centuries was

exclusively that of discovering the identity of the letter or letters, as opposed to an

attempt to express the general. The third stage, symbolic algebra, was initiated by

Vieta’s use of letters to stand for given quantities. At this point it became

1

possible to express general solutions and, in fact, to use algebra as a tool for

proving rules governing numerical relations. (p. 97)

Arcavi (1994) posits that algebra is “…a tool for understanding, expressing, and

communicating generalizations, for revealing structure, and for establishing connections

and formulating mathematical arguments (proofs)” (p. 24). Similarly, Cates (2002)

believes “algebra is a way of thinking, communicating, and reasoning” (p. 33).

For example, you use a coordinate system when you consult your car map to find

directions to a new destination. You solve simple equations when you change a

recipe to increase or decrease the number of servings. To evaluate how much

interest you will earn on a savings account or to figure out how long it will take

you to travel a given distance, you use common formulas that are algebraic

equations. (Angel & Porter, 2001, p.256)

One of the main topics that is discussed in algebra is the concept function.

According to Malik (1980), “the concept of function originated when Galileo (1564-

1642) proposed a programme [sic] for the study of motion” (p.490). Leonhard Euler

(1707-1783), Johann Bernoulli (1667-1748), Nicolaus Bernoulli (1695-1726), Daniel

Bernoulli (1700-1782), and other mathematicians constructed one of the first definitions

of function as “…an analytic expression representing the relation between two variables

with its graph having no corners” (Malik, 1980, p. 490). The definition presented above

is considered as Euler’s definition and was created during the development of

“…Calculus to deal with physical problems” (p.490). In 1829, Peter Dirichlet (1805-

1859) expressed that “…y is a function of x if for any value of x there is a rule which

gives a unique value of y corresponding to x” (Malik, 1980, p. 491). In 1939, Nicholas

Bourbaki (Fraleigh, 1994) defined function as “…a rule of correspondence between two

sets…” (Malik, 1980, p. 491). As a result, “by the end of the first half of this century, the

Dirichlet-Bourbaki definition of function had become established as textbook

terminology” (p.491). Some of these definitions are mentioned below.

For example, Vinner (1983) states “…a function is any correspondence between

two sets (a domain and a range) such that every element in the domain has exactly one

element in the range that corresponds to it” (p. 298). Vinner and Dreyfus (1989)

similarly define function as “…a correspondence between two nonempty sets that assigns

to every element in the first set (the domain) exactly one element in the second set (the

codomain)” (p. 357). Kieran (1990) provides an operational definition of function as

“…an algorithm for computing one magnitude by means of another” and a structural

definition as “a correspondence between two sets” (p. 109). In addition, Angel and

Porter (2001) express “a function is a special type of relation where each value of the

independent variable corresponds to a unique value of the dependent variable” (p. 325).

Similarly, Lial, Hornsby, and Schneider (2001) purport that a function is “a relation in

which for each element in the domain there corresponds exactly one element in the

range” (p. 180).

In general, students are introduced to algebra in middle school and high school.

“Some of the mathematical objects that are met for the first time in algebra are

expressions, equations with unknowns, functions and variables, and monomials and

polynomials” (Kieran, 1990, p. 99). Subsequently, they are expected to master the

techniques of the course in college at a faster rate. Algebra I and Algebra II that are

taken for two years in middle and/or high school become College Algebra at a higher

2

education institution. College Algebra is taken for one semester. An instrument that may

help students to master algebraic concepts is the graphing calculator.

Identification of the Problem

Algebra is seen as an abstract subject to most of the students that I have taught.

According to Fey (1992), “many students do not become proficient in the skills of

algebra…[and] very few students acquire the understanding of algebraic ideas and

methods that is required to reason effectively with symbolic expressions” (p. 1).

Furthermore, students learn at various rates and in different ways. For instance, some

people are visual learners. Others possess a kinesthetic learning style. Some students are

tactile learners. Many students are auditory learners.

Presmeg (1986a) defines visualizers as being “…individuals who prefer to use

visual methods when attempting mathematical problems which may be solved by both

visual and nonvisual [sic] methods” (p. 298). According to Price (1996), visual

preferences include support using “… pictures, filmstrips, computers, films, graphs,

books, and magazines” (p. 10). Each of these methods may aid the understanding of

algebraic concepts for the visual learner.

Presmeg (1986a) also defines non-visualizers as being “…individuals who prefer

not to use visual methods when attempting...[mathematical problems which may be

solved by both visual and non-visual methods]” (p. 298). Price (1996) explains that

auditory preferences involve the inclusion of “…tapes, videotapes, records, radio,

television, and precise oral directions when giving assignments, setting tasks, reviewing

progress, using resources or for any aspect of the task requiring understanding,

performance, progress, or evaluation” (p. 10). Tactile preferences include favoring the

“…use [of] manipulative and three dimensional materials; [in addition] resources should

be touchable and movable as well as readable” (p. 10). Furthermore, Price (1996) states

that kinesthetic learners prefer “…opportunities for real and active experiences for

planning and carrying out objectives; site visits, seeing projects in action and becoming

physically involved” (p. 10).

When algebraic concepts do not appear lucid, despite how a person learns,

understanding is not achieved. A tool that may help is the graphing calculator. The

graphing calculator offers visual learners the graphs of equations, inequalities, and

functions. They can see a picture or graphical mathematical representation of these

topics on the screen of the graphing calculator. In addition, the graphing calculator

provides kinesthetic and tactile learners with the opportunity to create graphs by

constructing equations and inputting the data into the calculator. This tool also gives

auditory learners a way to examine their recollection of algebra by using its features to

review algebraic concepts that have been covered and discussed in class.

I conducted a small study regarding graphing calculators and algebraic concepts

(Lane & Williams, 1998). The subjects (n =12) for this quantitative study were tenth and

eleventh grade high school students. This study had a Pre Test / Post Test / Control

Group design. The results of the study showed that the subjects in the study using

graphing calculators actually graphed equations better than the students who did not.

This study left many unanswered questions for me to examine. Some of these questions

described below will be addressed in chapters 4 and 5.

First of all, I would like to find out how graphing calculators impact a student’s

understanding of first degree, second degree, and higher order functions. How do

3

students use graphing calculators? Do students use the table and trace keys to explore

possible boundaries and properties of functions? How does the availability of technology

encourage algebra students to construct their own functions or functions related to the

tasks they will be given to complete? Do graphing calculators benefit visual and non-

visual College Algebra students? The answers to these questions may depict graphing

calculators as an essential tool for both visual and non-visual College Algebra students.

Secondly, in addition to finding out the learning impact of graphing calculators, I

would also like to try to determine what understanding the algebraic concept of function

means to the visual College Algebra student. Do the students rely on mental images in

order to gain understanding? If so, how do the students connect these mental images with

understanding the concept of function? Can the students draw the images on paper? Will

the students translate the mental images on a graphing calculator?

This study will attempt to discover what understanding the algebraic concept of

function means to the non-visual College Algebra student. Do the students rely upon

symbolic forms of equations in order to gain understanding? If so, how do the students

connect these equations with understanding the concept of function?

O’Callaghan (1998) developed a developed a cognitive model for understanding

functions. This framework included four components: “…modeling, interpreting,

translating, and reifying” (p. 24). Translating was defined as “the ability to move from

one representation of a function to another…” (p. 25). In addition, he explained that “the

three most frequently used representations for functions are equations, tables, and

graphs” (p. 25). In the current study, the presence or absence of the translating

component (O’Callaghan, 1998) will be used to measure the visual and non-visual

mathematical learners understanding of functions. As a result, the following research

questions were developed.

Research Questions




Significance of the Study

One of the major goals in mathematics education is to ensure the success of all

students in mathematics. A way of accomplishing this goal is by incorporating different

kinds of learning experiences for the variety of learners in the College Algebra

classroom. These learning experiences should include the use of technology such as

graphing calculators and mathematical computer software (National Council of Teachers

of Mathematics [NCTM], 2000; Writing Team and Task Force of the Standards for

Introductory College Mathematics Project, 1995).

Electronic technologies – calculators and computers – are essential tools for

teaching, learning, and doing mathematics. They furnish visual images of

mathematical ideas, they facilitate organizing and analyzing data, and they

compute efficiently and accurately. They can support investigation by students in

every area of mathematics, including geometry, statistics, algebra, measurement,

and number. When technological tools are available, students can focus on

decision making, reflection, reasoning, and problem solving. (NCTM, 2000,

p. 24).

4

The variety of learners that I have encountered in my College Algebra classroom

consisted of visual and non-visual learners. Providing one method of representing a

mathematical concept did not reach all of the students. For instance, a very detailed

verbal explanation may have been enough for the students who have auditory learning

preferences. On the other hand, a graphical or pictorial form of a solution to a

mathematical problem could aid the understanding of a visual learner. In addition, a

tactile or kinesthetic learner may still not understand a mathematical concept presented

by an instructor using the previous two methods. Tactile and kinesthetic learners might

benefit from actively manipulating the keys on a graphing calculator by using the zoom,

table, and trace features as well as being involved in a hands-on activity.

If this study shows that the graphing calculator aids in the understanding of

mathematical topics, such as functions, the graphing calculator may be an essential tool

for visual College Algebra learners. In fact, one of the basic principles states, “the use of

technology is an essential part of an up-to-date curriculum” (Writing Team and Task

Force of the Standards for Introductory College Mathematics Project, 1995, p.3). The

type of technology this principle discusses includes graphing calculators. Furthermore,

“faculty should take advantage of software and graphing calculators that are designed

specifically as teaching and learning tools. The technology must have graphics, computer

algebra, spreadsheet, interactive geometry, and statistical capabilities” (p. 3).

The final basic principle purports that “increased participation by all students in

mathematics and in careers using mathematics is a critical goal in our heterogeneous

society” (Writing Team and Task Force of the Standards for Introductory College

Mathematics Project, 1995, p. 3). It is vital that every male and female College Algebra

learner see mathematics as meaningful and relevant. Subsequently, “mathematics

instruction must reach out to all students: women, minorities, and others who

have…differing learning styles….faculty must provide a supportive learning environment

and promote appreciation of mathematics” (p. 3).

5

CHAPTER 2

LITERATURE REVIEW

This chapter examined research articles and studies that pertained to the concepts

of functions, visualization, and graphing calculators. First of all, literature on the concept

of function was presented. Next, visualization studies were introduced. The third portion

of this chapter dealt with graphing calculator research. At the end of chapter 2, a

summary of the functions, visualization, and graphing calculator literature was included.

Functions

Vinner (1983) analyzed the learning processes of tenth and eleventh graders of the

algebraic concept of function. The author developed a cognitive model to help explain

how the students learn. “For each concept, assume the existence of two different cells in

the cognitive structure….One cell is for the definition(s) of the concept and the second is

for the concept image” (p. 294). The author also defines concept image as “…a set of

properties associated with the concept …together with the mental picture…” (p. 293).

The sample of this study included 65 tenth graders and 81 eleventh graders

(Vinner, 1983). They were given a questionnaire that had five function-related tasks to

complete. This author classified the students’ responses into the following four

categories:

The textbook definition sometimes mixed with elements from the concept image

cell; the function is a rule of correspondence; the function is an algebraic term; a formula,

an equation, an arithmetical manipulation, etc; and some elements in the mental picture

are taken as a definition for the concepts. (pp. 299-300)

Vinner (1983) found that 57 students fell into the first category. Fourteen tenth

and eleventh graders were in categories two and three. Only seven gave a response

similar to category four. Eight provided no answer.

In another study, Vinner and Dreyfus (1989) analyzed the images and definitions

for the concept of function. The authors defined concept image as “…the set of all the

mental pictures associated in the student’s mind with the concept name, together with all

the properties characterizing them” (p. 356). In the final analysis, the sample included

271 college freshmen and 36 junior high school mathematics teachers.

Vinner and Dreyfus (1989) “…categorized the students’ definitions of a function

into six categories…” (p. 359). The categories were correspondence, dependence

relation, rule, operation, formula, and representation. Fifty seven students and 25

teachers provided category one definitions. Three teachers and 78 students defined the

term function as a dependence relation. Three teachers and 29 students responded that a

function is a rule. One teacher and 13 students gave category four definitions. Thirty

students define function as a formula. One teacher and 24 students provided category six

definitions. Whereas, three teachers and 40 students gave other definitions of the concept

function that were not described by one of the categories.

The concept images of the students were classified in four categories. These

categories included, “various aspects of the function concept as conceived by the

students…” (Vinner & Dreyfus, 1989, p. 361). They included one-valuedness,

discontinuity, split domain, and exceptional point.

Thompson (1994) explored six topics in order “…to discuss research on students’

understanding of functions and its importance for the undergraduate curriculum” (p.21).

First of all, the author examined the concept image and the concept definition of

6

functions. These terms were discussed previously (Vinner, 1983; Vinner & Dreyfus,

1989). Similarly, Thompson (1994) stated, “…a concept definition is a customary or

conventional linguistic formulation that demarcates the boundaries of a word’s or

phrase’s application” (p. 24). Thus, the concept definition of a specific function

describes the attributes of that function using words. “On the other hand, a concept

image comprises the visual representations, mental pictures, experiences and impressions

evoked by the concept name” (Thompson, 1994, p. 24). The author believed that

students could be more successful in their understanding of functions when the concept

image and concept definition are balanced.

Secondly, Thompson (1994) investigated function as an action, process, and

object. An action conception of function occurred when students “…think of an

expression as producing a result of calculating…” (p. 26). The students only see the

mathematical operations involved in the solving process. In fact, “students holding an

action concept of function imagine that the recipe remains the same across numbers, but

that they must actually apply it to some number before the recipe will produce anything”

(p. 26).

According to Thompson (1994), “from the perspective of students with a process

conception of function, an expression stands for what you would get by evaluating it” (p.

26). In this stage, the students also saw the mathematical operations. However, their

viewpoint does not stop at the operations. The students compute using the mathematical

operations in order to attain a solution. As a result, for students with a process

conception of function, the solution was equivalent to the original expression they started

with.

“To reason formally about functions seems to entail a scheme of conceptual

operations which grow from a great deal of reflection on functional processes”

(Thompson, 1994, p. 27). According to the author, the previous statement is the

beginning of how a student could develop an object conception of function. Once this

reflection takes place, students may think about how different mathematical operations

affect many types of functions. For instance, “one hallmark of a student’s object

conception of functions is her ability to reason about operations on sets of functions” (p.

27).

The third topic Thompson (1994) explored was function as a correspondence and

co-variation. Function as co-variation involved investigating the rate of change among

quantities and /or data sets. Function as a correspondence of ordered pairs referred to the

Dirichlet-Bourbaki definition of function.

Next, Thompson (1994) discussed understanding of phenomena and representing

phenomena. In this section the author wondered “…what role conceptions of functions

might play in supporting or inhibiting students’ conceptualizations of situations” (p. 30)?

How would a student with a process conception of function view a situation in Algebra,

Calculus, or Physics? The discussion appeared to focus on math students moving beyond

the symbolization of concepts by becoming equipped to represent data in other ways.

The last two topics included mathematical operations on numbers and functions

as well as emergent issues (Thompson, 1994). He revisited the process conception of

functions. The author also presented three issues in the form of questions for further

research that may impact the learning and teaching of functions.

7

Sfard (1991) presented a conceptual framework that described how students

perceived functions. First of all, the author explained the two different conceptions of

functions. The structural conception referred to seeing functions as “…abstract

objects…” (p. 4). On the other hand, an operational conception included viewing

functions as “…processes, algorithms and actions...” (p. 4).

According to Sfard (1991), “seeing a mathematical entity as an object means

being capable of referring to it as if it was a real thing—a static structure, existing

somewhere in space and time” (p. 4). A student with a structural conception did not

require acting on the object in order to gain understanding. “In contrast, interpreting a

notion as a process implies regarding it as a potential rather than actual entity, which

comes into existence upon request in a sequence of actions” (p. 4).

Sfard (1991) mentioned three ways to move from the operational conception to

the structural conception. The first stage was interiorization. “At the stage of

interiorization a learner gets acquainted with the processes which will eventually give rise

to a new concept…” (p. 18). For example, “in the case of function, it is when the idea of

variable is learned and the ability of using a formula to find values of the ‘dependent’

variable is acquired” (p. 19). The second phase was condensation. “The phase of

condensation is a period of ‘squeezing’ lengthy sequences of operations into more

manageable units.” Thus, the student would still use processes, however, the concept

should become more concrete. For instance, “…the learner can investigate functions,

draw their graphs, combine couples of functions (e.g. by composition), even to find the

inverse of a given function” (p. 19). Thirdly, reification was the last stage. This third

phase involved “…conceiving the notion as a fully-fledged object…” (p. 19). For

example, “in the case of function, reification may be evidenced by proficiency in solving

equations in which ‘unknowns’ are functions (differential and functional equations,

equations with parameters)…” (p. 20).

O’Callaghan (1998) developed a cognitive model for understanding functions.

This framework included four components: “…modeling, interpreting, translating, and

reifying” (p. 24). In this study, the author examined the influence of these aspects by

comparing traditional algebra (TA) students with computer-intensive algebra (CIA)

students. (College algebra was considered the traditional algebra course.) “CIA is a

function-oriented curriculum that is characterized by (a) a problem-solving approach

based on the modeling of realistic situations, (b) an emphasis on conceptual knowledge,

and (c) the extensive use of technology” (p. 21).

First of all, modeling was referred to as “the ability to represent a problem

situation using functions…” (O’Callaghan, 1998, p. 25). According to the author,

interpreting was considered “the reverse procedure…” (p. 25) of the first component.

“Problems could require students to make different types of interpretations or to focus on

different aspects of a graph, for example, individual points versus more global features”

(p. 25). Translating was defined as “the ability to move from one representation of a

function to another…” (p. 25). Furthermore, he explained that “the three most frequently

used representations for functions are equations, tables, and graphs” (p. 25). Thus,

translating could refer to moving from graphs to equations or vice versa. “The final

component of the model for functions is reification, defined as the creation of a mental

object from what was initially perceived as a process or procedure” (p. 25).

8

College students were the subjects of this study (O’Callaghan, 1998). Two TA

courses and one CIA course were also included in the study. The author’s data sets

consisted of questionnaires, tests, and interviews.

The questionnaires used in the study were the Revised Math Attitude Scale and

the Mathematics Attitude Inventory (O’Callaghan, 1998). Both of the inventories were

given to the subjects at the beginning and at the end of the semester.

Pre-tests and post-tests were administered to the students (O’Callaghan, 1998).

Each question on these tests was designed to refer to one of the four aspects of

understanding functions. In addition, “the posttest was essentially an alternative version

of the pretest” (p. 29).

The interviews consisted of inquiring how and why students solved tasks that

were similar to the pre-test and post-test questions (O’Callaghan, 1998). Each interview

problem also corresponded with “…(a) modeling a real-world situation using a function,

(b) interpreting a function in terms of a realistic situation, (c) translating among different

representations of functions, or (d) reifying functions” (p. 29). In addition, the interviews

included analyzing and probing the subjects’ responses to the attitude scales.

O’Callaghan (1998) concluded the following results. In the beginning of the

semester, “the only significant initial difference found on the two attitude scales was that

the students had a better perception of me as an instructor than they did of the instructor

for the other traditional section…” (p. 32). When the attitude inventories were given to

the students at the end of the semester, responses to the questionnaires revealed that the

CIA students found the course more interesting than did their TA counterparts…” (p. 32).

According to O’Callaghan (1998), the interviews showed that the traditional

algebra students and the computer-intensive algebra students thought about functions

differently. “The traditional students demonstrated very limited knowledge of functions,

which they generally defined as equation, and very little recognition of their uses” (p.

35). On the other hand, one of the CIA students provided the following definition: “It is

an operation used to analyze a situation or solve a problem, when a certain pattern is

involved” (p. 35). Furthermore, the author explained that “all of the CIA students

described functions as dependency relationships involving input and output variables” (p.

35).

In the beginning of the semester on the first component, more mathematical

problem situations were mentioned by the CIA students (O’Callaghan, 1998). However,

the TA students were better at symbolic manipulations of functions. “The translation

tasks involving graphing uncovered some severe deficiencies in the abilities of both

groups, particularly in relation to scaling or examining meaningful ranges of values; these

problems led to difficulties in interpreting as well” (p. 35). The final component also

challenged both groups.

At the end of the semester, “…the CIA students had a better knowledge of the

individual components of modeling, interpreting, and translating as well as a better

overall understanding of the function concept” (O’Callaghan, 1998, p. 36). However,

even by the end of the semester reifying still challenged both groups.

Moschkovich (1999) examined how 18 ninth and tenth grade algebra students

understood linear functions. The authors also presented two case studies which described

four of the participants as they worked in pairs. Specifically, this study focused on

“…students’ use of the x-intercept in equations of the form y = mx + b” (p. 169).

9

The participants in the study “…completed written assessments and participated

in discussion sessions with a peer where they explored linear functions with graphing

software” (Moschkovich, 1999, p. 169). The students were also given a Pre-test and a

Post-test. The graphing software used during the study was Superplot. This software

was “…a graphing utility which allows students to graph equations and problems…” (p.

174). By using Superplot, they investigated the slopes, x-intercepts, and y-intercepts of

linear functions.

According to Moschkovich (1999), the students used Superplot during their

discussion sessions. In addition, “students were asked to discuss several predictions

before graphing, choose one, and record their answer and explanation on paper. Next,

they were asked to graph an equation on the computer to test their prediction” (p. 175).

During the discussion sessions, the author also asked the pairs of participants to be in

agreement with each other regarding the equations they chose to graph. A sample

discussion problem would be, “If you start with the equation y = x then change it to the

equation y = x + 5, what would that do to the graph” (p. 176)? A two-dimensional blank

graph was provided for the participants to use with the x and y axes labeled on it. The

author also included probing questions such as, Would the new graph “A. Make the line

steeper, B. Move the line up on the y-axis, [ and] C. Make the line both steeper and move

up on the y-axis” (p. 176).

The written assessment contained 31 graphing related questions (Moschkovich,

1999). An example of a written assessment question would be, “Graph the equation

y = x + 3 on a coordinate grid. A student said that the line y = x + 3 would go through

the x-axis at (3, 0) because in the equation you add 3 to x. Do you think this student was

right” (p. 177)? The author also asked the participants to write down the reasoning

behind their answers. She constructed three categories which described how the students

used x-intercepts. In the first category, “x-intercept graphical, a student described a line

as moving ‘left to right’ or ‘on the x-axis’ as a result of a change in the b in an equation”

(p. 177). So, in this category the students did not appear to consider b as the y-intercept

in the equation y = mx + b contributing to the upward or downward movement of a

graph. Secondly, under the “x-intercept for b [category], a student used the x-intercept of

a line for b in an equation or responded that the number b in an equation corresponded to

the x-intercept of a line” (p. 177). In the second category, the participants did not make a

correspondence between b in y = mx + b and the y-intercept. In the third category, “x-

intercept for m, a student used the x-intercept for m in an equation or responded that the

number m in an equation corresponded to the x-intercept of a line” (p. 177). In this

category, the students did not appear to consider m in y = mx + b as the slope of the

equation.

Moschkovich (1999) found six students from the Pre-test and two from the Post-

test to be in the x-intercept graphical category. Secondly, seventeen participants after

taking the Pre-test and 18 after taking the Post-test used the x-intercept for b. Thirdly,

seven students from the Pre-test results and one from the Post-test results used the x-

intercept for m.

According to Moschkovich (1999), the participants’ responses were not

misconceptions, but examples of transitional conceptions. This conception had the

following four characteristics. It “…a) arose as students made sense of this domain, b),

10

reflects an important aspect of the conceptual structure of this domain, c) can be

productive depending on the context, and d) has the potential to be refined” (p. 169).

Moschkovich (1999) also presented two case studies. The first case study

depicted a pair of high achieving students who displayed the previously mentioned

characteristic part c) of a transitional conception. For example, when the slope was equal

to one in the equation y = mx + b, the participants calculated and accurately graphed the

x-intercept. However, “even these two students, whose answers on the pre-test seemed to

reflect an understanding of the x-intercept for [the y-intercept] and [the slope] during

problems where the slope was not 1” (p. 183).

The second case study examined another pair of students who also displayed

characteristic part c) of a transitional conception (Moschkovich, 1999). These two

students improved their Pre-test score of 31

1 and

31

1 to Post-test scores of

31

29 and

31

23.

The student who made 31

29 on the Post-test “…used the x-intercept in three instances for

[the y-intercept], in one instance for [the slope], and in three instances to describe line

movement [on the Pre-test], there was not evidence of any use of the x-intercept in his

post-test responses” (p. 186). Throughout the study, the pair explained how their use of

the x-intercept fell into the previously mentioned second category. In other words,

“…they expected [the algebraic y-intercept] to correspond to the x-coordinate of [the

graphical x-intercept]” (p. 186). Next, they “…concluded that [the algebraic y-intercept]

corresponds to the opposite of the x-coordinate of [the graphical x-intercept], which is

correct for lines of slope 1” (p. 187). However, “they went on to discover that when

m = 2 [the algebraic y-intercept] does not correspond to the opposite of the x-coordinate

of [the graphical x-intercept] but to the y-coordinate of [the graphical y-intercept]” (p.

189). In addition, the pair explained how the x-intercept fell into the third category of the

x-intercept use. They used “…the x-coordinate of [the graphical x-intercept] for [the

algebraic slope]…” (p. 190). Even though the participants found instances where the

third category did not coincide graphically, by the end of the study the pair had not

reconciled an accurate relationship between the graphical x-intercept and the algebraic

slope.

In this study, Dreyfus and Eisenberg (1983), investigated 84 college students’

understanding of the concept of function. Specifically, “its purpose was to take the

aspects of linearity, smoothness (differentiability), and periodicity and determine the

extent to which these have been internalized by college level students” (p. 126). As a

result, the students were given a questionnaire.

The questionnaire contained 34 questions which pertained to the three aspects

mentioned above that the authors were trying to capture (Dreyfus & Eisenberg, 1983).

There were different kinds of problems included in the questionnaire. One type of task

consisted of sketches of pieces of the graph of a function. Some of the portions of the

graph were missing. Then the authors asked the students to “continue the graphs in such

a manner that the resulting curve is the graph of a function” (p. 127). Thus, the students

had to decide on the shape and continuity of the graphs they constructed. In addition, the

authors included algebraic tasks on the questionnaire. For instance, “…Give an example

of a function for which f (-1) = 6.5…” (p. 129).

11

Dreyfus and Eisenberg (1983) determined the following results. Most of the

students constructed continuous graphs of functions for the first type of problem. In fact,

“…28 % of the responses were (piecewise) linear continuations while 62 % were smooth

(differentiable) continuations. The remaining 10 % were neither piecewise linear nor

smooth” (p. 129). For the algebraic related task, “…79 % of the responses were linear

relationships, while only 14 % were smooth but non-linear” (p. 129).

Breidenbach, Dubinsky, Hawks, and Nichols (1992) investigated the manner in

which 59 college students developed a process conception of functions. Specifically,

“the students in the present study were mainly sophomore and junior math majors

preparing to be high school, middle school or elementary school math teachers” (p. 249).

In addition, the authors described the students’ mathematical ability as “mediocre” (p.

249).

Breidenbach, Dubinsky, Hawks, and Nichols (1992) mentioned three ways that

students construct their understanding of functions. The first phase was a pre-function

conception. In this stage, the student did not have a very clear idea about functions. In

fact, “whatever the term means to such a subject, this meaning is not very useful in

performing tasks that are called for in mathematical activities related to functions” (p.

251). Secondly, “an action [conception] is a repeatable mental or physical manipulation

of objects” (p. 251). The third stage was a process conception. “A process conception of

function involves a dynamic transformation of objects according to some repeatable

means that, given the same original object, will always produce the same transformed

object” (p. 251). According to the authors, this third conception represented that students

understood the concept of function.

In order to depict the process conception of function, Breidenbach, Dubinsky,

Hawks, and Nichols (1992) initially posed a couple of questions regarding functions and

provided a computer environment in an elementary discrete mathematics course. The

students were asked to write down their own definition of function and to provide

examples of functions. These two inquiries were given to the students at the beginning

and towards the end of the semester. The computer environment involved using ISETL.

“ISETL is an interactive, interpreted programming language that implements a number of

mathematical constructs in a syntax which is very similar to standard mathematical

notation” (p. 255). The students solved mathematical problems using this software in a

small group setting. In addition, the subjects were given 24 functions in situations

questions. On this instrument, the students were asked to examine numerical, symbolic,

or graphical situations in order to determine how many functions each represented.

Therefore, the instructional treatment “…was about learning the syntax of ISETL,

working with elementary properties of number systems, propositional calculus and sets”

(p. 255).

After the students completed the previously mentioned questionnaires, the authors

provided three post-treatment instruments (Breidenbach, Dubinksy, Hawks, & Nichols,

1992). The first instrument included two non-standard items. “There is no formula given

nor might one expect to find any algebraic relationship although a vague kind of

dependence is hinted at in the story that is told” (p. 265) in question one. The next

problem with four parts involved inferring information about functions using ISETL.

Secondly, the authors interviewed 19 of the subjects regarding their responses to the 24

functions in situations questions. The third post-treatment instrument was the class final

12

exam. According to the authors, “about 75 % of the questions of the Final Exam for the

course are appropriate for this study” (p. 266). The exam items dealt with the domain,

range, inverse, transformation, and various operations of functions.

Breidenbach, Dubinsky, Hawks, and Nichols (1992) concluded the following

results regarding the two function questions posed at the beginning of the semester.

Forty percent of the subjects’ responses had a pre-function conception. Twenty-four of

the responses expressed an action conception, 14 were process, and 21 were unknown.

Towards the end of the semester, 17 % of the students’ responses had a pre-function

conception. Thirty-six of the responses expressed an action conception, 36 were process,

and 11 were unknown.

Breidenbach, Dubinsky, Hawks, and Nichols (1992) also found out “the

percentage of correct answers for 56 students on…” (p. 268) the two non-standard

function questions. On the first non-standard item, 85.8 % of the students attained the

correct answer. “Looking at individual questions, we note that nearly everybody (48 out

of 56 students) succeeded in constructing a function and using its process to organize

information about the frequency distribution in a text (Item 1)” (p. 268). Recall that the

second question had four parts. Seventy-three point two percent got the correct answer

on part one, 65.7 % on part two, 75.3 % on part three, and 69.3 % on part four. “Looking

at the remaining items, it seems that only a few students…were unable to construct a very

complicated combination of processes and then reverse them selectively” (p. 268).

The authors conducted interviews with the students regarding their responses to

the functions in situations questions (Breidenbach, Dubinsky, Hawks, and Nichols,

1992). They determined that the responses expressed an action conception, a transition

from an action conception to a process conception, or a process conception. However,

the authors did not report on how many responses fell into each category.

Breidenbach, Dubinsky, Hawks, and Nichols (1992) reported the final exam

results as percentage scores. On the first question, 83.9 % of the subjects answered it

correctly. This problem dealt with determining the range of a function that was a

composition of two other functions. Eighty-three point five percent attained the accurate

answer to item two. The second and third problems asked the students to find the domain

and range of a function. On the third question, only 40 % of the subjects turned in the

correct answer. The next item dealt with finding the product of two given piece-wise

functions, which 87.2 % provided the correct response. The fifth problem involved

constructing a proof or counterexample of: “If f , g are both 1-1 then it follows that f + g

is 1-1” (p. 283). Fifty-four point five percent responded correctly. The next question

also involved constructing a proof or counterexample of: “If f , g are both onto then it

follows that f + g is onto” (p. 283). The results showed that 47.5% of the subjects

provided the correct answer. Seventy-five point eight percent responded accurately to the

seventh item. This problem involved having the students determine if two transforming

operations, which acted on a function, were functions. The next question was regarding

the composition of functions, which 73.7 % turned in correct answers. On the last item,

only 30.9 % responded accurately. This final question involved determining the inverse

of a function.

In conclusion, Breidenbach, Dubinsky, Hawks, and Nichols (1992) summarized

the overall progress of the subjects. The authors stated “…that of 59 students, 7 appeared

to start the course with strong process conceptions, 24 showed clear progress throughout

13

the semester…” (p. 274). They also concluded that the remaining 28 subjects “…made

only a small amount of progress or their performances appeared to oscillate” (p. 274).

In this study, Schwarz and Dreyfus (1995) investigated how ninth grade students

understood the function concept after receiving instruction in a computer software

environment. “The environment we used to achieve this goal, the Triple Representation

Model (TRM) is a computer microworld that eliminates the ‘technical load’ from tasks

on functions and stresses the use of concurrent dynamic settings” (p. 260). TRM had a

graphical, algebraic, and tabular mode.

The authors presented the students with two kinds of tasks and a questionnaire in

order to determine which skills had been attained (Schwarz & Dreyfus, 1995). First of

all, the box problem stated: “…an open box is constructed by removing a square from

each corner of a 20 by 20 cm square sheet of tin and folding up the sides. Find the

largest possible volume of such a box...” (p. 276). The second question was given in

graphical form. “In the Rectangles task, an undisclosed continuous function is chosen by

TRM; the student is presented with a rectangle and asked to find out whether the graph of

the function passes through the rectangle” (p. 277). The subjects were provided two

problems of this type. In the first one, a linear vertical straight line function passed

through the rectangle. In the second task, a cubic function was drawn closely along one

side of the rectangle. However, the cubic function did not intersect the rectangle.

Thirdly, “the questionnaire was designed to study most of the skills fostered by TRM,

and to compare experimental and control students” (p. 278).

Schwarz and Dreyfus (1995) reported the following results. First of all, “…more

than half of the TRM students who solved the Box problem fully coordinated and/or

integrated the information gleaned from various representatives they created” (p. 287).

(By representatives, the authors were referring to vertical and horizontal translations of

functions as well as increasing or decreasing the values on the x and y axes.) Secondly,

18 of the subjects from the experimental classes thought the linear function would pass

through the first rectangle task. However, nine TRM students were uncertain. On the

other hand, 30 of the subjects from the experimental classes reported that the cubic

function would not intersect the second rectangle task. Thirteen TRM students were not

sure. Thirdly, on the questionnaire, “the experimental results show that the experimental

students have a better command of transformation skills than the control students but the

difference between the groups is somewhat less striking than for the other two classes

skills” (p. 288).

The other two skills were categorized as partiality skills and link skills (Schwarz

& Dreyfus, 1995). “In summary, the scores and the distribution of answers to the

Questionnaire strongly suggest that experimental students handled skills from the

‘partiality’ class well” (p. 280). Partiality skills included interpolation between points,

recognition of representatives, and integration of various partial graphs of a function into

one graph. The authors also concluded “…that the experimental students achieved

substantially better results than the control students on the questions which related to

links between settings…” (p. 283). The link skills referred to making connections among

representatives of the same and different settings. In fact, “…the Box problem solutions

showed that in a problem solving situation, most TRM students were able to link between

representatives of the same and of different settings…” (p. 287).

14

Sajka (2003) reported on how an average high school mathematics student

understood the concept of function using Case Study research. The author presented the

student with a series of mathematical tasks to complete. These tasks included problems

on functions and inequalities. In fact, the author “…chose a set of non-standard tasks, in

which the pupil should show both familiarity with different representations of the idea of

function and show his or her understanding” (p. 232). However, in this article, the author

chose to focus on one particular mathematical task that dealt with functions.

Sajka (2003) found that the student encountered many difficulties in

understanding functions. The author believed the difficulties came from two major

sources “…(a) Kasia’s misinterpretation of the symbols used in the functional notation

and (b) her very limited procept [sic] of function” (p. 246).

First of all, “…while the function symbol f is identified correctly, it does not carry

any content, it is perceived as a label, or an abbreviation of the word function” (Sajka,

2003, p. 247). The student does not see the entire expression as one function. In

addition, she thinks the symbol f denotes “the beginning of a thought or new task” (p.

247).

Secondly, “Kasia is not able to interpret the symbol f (3) correctly” (Sajka, 2003,

p. 247). The student believed that f(3) equals zero. Even though she did not appear to

understand what the function represented, Kasia thought that when the function was

applied to three the result would always be zero. “This is probably because during

mathematics lessons she has only come across this notation when testing whether a given

argument is the zero of a function” (p. 247). Consequently, in Poland, an argument is

considered an element of the domain (Sajka, 2003).

The third way the student’s misinterpretation of functional notation involved how

she viewed f(x) and the expression it equaled (Sajka, 2003). “Kasia interprets the symbol

f(x) and the algebraic expression determining the function as the formula of a function”

(p. 247). In other words, the student saw the entire mathematical sentence as a formula.

In fact, “she does not see in that symbol the value of the function for the argument x” (p.

247).

Next, “f(x), f(y), f(x+ y) are seen as three different functions” (Sajka, 2003, p. 248)

by the student. Initially, the author posed the following problem: “Give an example of a

function f such that for any real numbers x , y in the domain of f the following equation

holds: f(x+ y) = f(x) + f(y)” (p. 233). Kasia did not appear to see any connection among

the variables or symbols in this equation. The author believed “Kasia’s interpretation is

probably influenced by the fact that in her mathematics classes, the student was rarely

faced with the situation when the argument of the function was marked by a letter

different than x” (p. 248).

Another misinterpretation of functional notation was that “the symbol y in the

notation f(y) is treated as the same as f(x) or as the ordinate of some hypothetical point”

(Sajka, 2003, p. 248). The student’s explanation, which was void of differences existing

between the symbols f(x) and f(y), may cause even more difficulty interpreting functions

numerically, algebraically, and graphically.

In addition, “the notation f(x+ y) = f(x) + f(y) was not initially seen as an equation

by Kasia…” (Sajka, 2003, p. 249). According to the author, the student’s view of an

equation would be variable expressions containing only the symbol x. “Since she had

15

never come across a notation in which the letter f appeared on both sides of the equation

mark, she does not treat that notation as an equation” (p. 249).

The seventh way Kasia’s misinterpretation of functional notation lead to

difficulties in understanding was misusing the distributive law (Sajka, 2003). For

instance, “the student sees an analogy between the notation of the distributive law in

numbers f(x+ y)=f(x)+f(y)” (p.249). She did not appear to be thinking of the definition of

a function or the purpose of the distributive law. Instead, the student seemed to look for

symbol similarities.

Her final misinterpretation also dealt with the initial problem of f(x+ y) = f(x) +

f (y) which was provided by the author (Sajka, 2003). “In the conversation summarizing

our dialogue, Kasia stated: The main thing here [f (x+ y) = fx + fy] is the equation, not

the function” (p. 249). The student appeared to look for expressions and equations which

were familiar and changed the problem into those.

Let us examine the six ways Kasia’s difficulty in understanding functions

originated from her limited procept of function (Sajka, 2003). Procept is a term

developed by Gray and Tall (1994). “It consists of three components: a process that

produces a mathematical object (or concept) and a symbol that represents either the

process or the object” (Sajka, 2003, p. 231).

First of all, “the student avoids the concept of function and even talking about it

or saying the word function” (Sajka, 2003, p. 250). She appeared to act as if the concept

did not exist. As a matter of fact, “Kasia’s reasoning is on the level of numerical

equations (identities) and she does not want to move to the level of thinking about a

function” (p. 250).

Secondly, the student saw a “function as the beginning of a new thought or new

task” (Sajka, 2003, p. 250). As the author explained previously, Kasia associated

function with the symbol f. Therefore, when the student saw the symbol f, she thought it

denoted “…the beginning of the equation…” (p. 250).

Thirdly, “for Kasia, the concept of function is often indistinguishable from the

concept of the formula of a function” (Sajka, 2003, p. 251). Thus, she did not appear to

discriminate between the specific function and its formula.

The fourth way Kasia’s difficulty in understanding functions originated from her

limited procept of function was by perceiving “function as that which determines all the

rest in the formula” (Sajka, 2003, p. 251). In other words, the student perceived that the

symbol f or f(x) determined the equation or formula of the function. At this point, she felt

that the variable x could change depending on its value, however, the specific equation

that x is substituted in would not change.

Next, the student viewed “function as a computational process” (Sajka, 2003, p.

251). She appeared to focus on the mathematical procedures of addition, subtraction,

multiplication, and/or division that were involved in solving an algebraic expression. For

instance, “about a function defined by the formula f (x) = 2x + 1, the student states: the

function f multiplies it by 2 and adds 1…” (p. 251).

The final way Kasia’s difficulty in understanding functions originated from her

limited procept of function was by looking at a “function as a kind of formula which

leads to drawing a graph” (Sajka, 2003, p. 251). Apparently, the student believed the

formula of a function should produce a graph of that function. In fact, she told the

16

author, “[function] is a kind of formula that leads to drawing a parabola or something like

that” (p. 251).

In this study, the authors’ purpose “…was to help students think of functions in a

visual way, and to help us understand the obstacles they must overcome in doing so”

(Eisenberg & Dreyfus, 1994, p. 47). Therefore, 16 high school boys were investigated

through pre-tests, post-tests, computer games, and interviews.

The same pre-test and post-test were given in the form of a questionnaire

(Eisenberg & Dreyfus, 1994). This questionnaire included a standard portion and a non-

standard portion. “The standard part contained 17 questions involving substitutions,

solving equations, graphs of parabolas, etc. The non-standard part contained 34

questions which related to the role parameters play in graphs of functions” (p. 49). These

parameters dealt with horizontal and vertical translations of functions. The non-standard

portion also consisted of questions pertaining to the stretching and shrinking of the

graphs of functions.

The Green Globs computer game is an educational software program used by the

authors to help the students to graph functions (Eisenberg & Dreyfus, 1994).

Specifically, “the program presents a number of small circles (globs) in a coordinate

system on the screen and the task is for the student to find functions whose graphs pass

through as many globs as possible” (p. 49). The students used the software for

approximately five weeks. They worked in groups of four with the authors to complete

six lessons. “Part of each lesson was devoted to guided discussion. During these

sessions, attention was drawn to certain classes of functions and transformations to which

the functions could be subjected” (p. 53).

Eisenberg and Dreyfus (1994) concluded the following results. By examining the

pre-test and post-test results, the authors found that the post-test “scores increased on all

non-standard questions…” (p. 55). Recall, that the non-standard questions referred to

vertical and horizontal translations of functions as well as the stretching and shrinking of

the graphs of functions.

Eisenberg and Dreyfus (1994) also reported on their interview results. “The

results of Question 1 are inconclusive; recognizing the transformation did not imply that

one would give a correct result and vice versa, giving a correct result did not imply that

one had recognized the transformation” (p. 56). This first question had four components

for the students to complete. On the second part of Question 1, all of the subjects

perceived it “…as a transformation [of the first part of Question 1] but only 4 of them

gave a correct answer” (p. 55). On the final component of this question, all of the

students provided the correct answer, according to the authors, however, “…only two

recognized the transformation” (p. 56).

“Most of the students needed help with the formulation of Question 2, and even

then only three were able to solve the problem to any acceptable depth” (Eisenberg &

Dreyfus, 1994, p. 56). For instance, two of the subjects in the study solved this problem

graphically using translations of the given function. Two other students solved the

question analytically by substituting the numerical value of -2 into the expression that

g(x) was defined as. Unfortunately, two of the subjects in the study did not know how to

solve the problem at all. In addition, “…one gave a correct answer based on his intuition,

but was unable to give us any indications of how he had reached this conclusion” (p. 56).

17

According to Eisenberg and Dreyfus (1994), “this [last] question was beyond the

students in the study” (p.56). In Question 3, the authors supplied the students with the

graph of a parabola that shifted below the origin by two units. The graph had two x-

intercepts. One was located at the point (2, 0). The other was (-2, 0). Then they were

asked to draw the inverse of this function. The subjects attempted to draw rough sketches

of the function however, the authors did not see how the students’ responses were related

to the original graph of Question 3.

Monk and Nemirovsky (1994) presented a case study of a twelfth-grade high

school student who used visual characteristics of graphs and the Air Flow Device to

understand functions. The study described the authors’ interviews with Dan. The

mathematical task that the student was presented with was the graph of a straight line.

The x-axis was labeled t for time. The y-axis was labeled as flow rate. Then the authors

asked Dan “…to predict the graph of Volume vs. time that would be associated with [the

flow rate versus time graph]” (p. 142) using the Air Flow Device.

The interviews were depicted in three episodes (Monk & Nemirovsky, 1994). In

episode one, the visual characteristic the student appeared most concerned with was the

steepness of a line. He consistently drew the volume versus time graphs to be steeper

than the flow rate versus time graphs. Dan also experimented with the Air Flow Device

to produce graphs. When the student saw the computer’s image of a volume graph on the

screen, he was somewhat puzzled by the way the graph appeared. Dan stated that, “I

thought it was going to be more spread apart” (p. 145).

During the second episode, the student was introduced to a new flow rate versus

time graph (Monk & Nemirovsky, 1994). This graph went up from the origin and then

came back down to the x-axis. However, it did go below the x-axis. This task presented

Dan with the opportunity to think about how the increasing and decreasing characteristics

of a graph affect the relationship between flow rate and volume. Therefore, the authors

asked the participant to produce a flow rate versus time graph with the device that was

similar to the one they presented to him. As a result, Dan experimented with the Air

Flow Device several times. Finally, on the ninth attempt, he constructed a graph similar

to the authors. After accomplishing this task, the authors asked the student to produce a

volume versus time graph that corresponded with the previous flow rate versus time

graph. Dan believed the corresponding volume graph was joined by two segments and

increased over the interval. So, the participant drew that type of graph. On the contrary,

the computer displayed a different volume graph. The computer’s graph was more of a

curved shape that increased in the beginning and then leveled off. This left the student

perplexed.

In the final episode, the authors wanted to help Dan reconcile why the computer’s

volume graph from episode two looked different from his (Monk & Nemirovsky, 1994).

Therefore, the student was provided with three new graphs. Graph one was a slightly

slanted and increasing straight line. Graph two was a curve that began increasing and

then appeared to become constant. Graph three was another curve-shaped graph that

increased over its entire interval. Dan was asked to choose which graph represented the

volume graph. First, the participant chose the straight line. Then the authors referred the

student back to the previous graph of flow rate versus time. One of the authors stated,

“Now, when flow rate touches zero….Then what happens with volume” (p. 164)? After

18

thinking about the task some more, Dan chose graph two because he felt this graph

reflected how the volume would “go up and then sort of goes down” (p. 164).

Yerushalmy (2000) investigated the problem solving strategies of two middle

school students in a function approach algebra course. “A function approach to algebra is

an approach that assumes the function to be a central concept around which school

algebra curriculum can be meaningfully organized” (p. 125). This longitudinal study

took place over a three year period. When the study began the participants were in the

seventh grade.

The students were introduced to functions using co-variation (Yerushalmy, 2000).

Their curriculum, which was divided into three parts, provided the students with a series

of mathematical tasks to complete. “The three phases are (1) emergence of the concept

of function throughout modeling, (2) manipulating function expressions and function

comparisons (equations and inequalities), and (3) exploring families of functions and

specifically linear and quadratic functions” (p. 126). The author interviewed the

participants on three similar mathematical tasks towards the completion of each part of

the curriculum. The tasks were considered “…linear break-even…” (p. 128) problems.

The participants in the study were interviewed together (Yerushalmy, 2000). “We

chose to interview in pairs, since students were working in pairs in class” (p. 127). Even

though the students appeared to work very hard, “this pair was considered to be among

the low achievers in their algebra class” (pp. 127-128).

Yerushalmy (2000) concluded the following results. On each task, the students

were asked to decide which payment method was the most economical. In addition, each

task could have been solved using a numerical, algebraic, or graphical approach. The

first task dealt with a shopping club. The pair used “…numbers as the only means of

modeling in the first interview…” (p. 142). They wrote out a long list of numerical

values as the method of solution. The second task involved parking lots. This time the

students’ method of solution included “…massive work with graphs and tables as linked

parallel representations of a situation in the second interview…” (p. 142). Finally, the

third task was a salary bonus problem. The pair used “…more symbolic representations,

i.e., expressions and sketches, in the last interview” (p. 142).

Slavit (1997) analyzed how understanding is attained through the property-

oriented view of function. “The theory of a property-oriented view of function suggests

that students come to understand the concept of function by transforming their

experientially-based perceptual patterns of functional growth behaviors into well-formed

understandings of specific functional attributes” (p. 261). The author believed that this

viewpoint was developed by two situations. “First, the property-oriented view involves

an ability to realize the equivalence of procedures that are performed in different

notational systems” (p. 267). For example, numerically solving a linear function by

plugging in zero for the x-value and graphically determining the y-intercept, would be

considered as equivalent procedures for finding where the graph of a line crosses the y-

axis. “Second, students develop the ability to generalize procedures across different

classes and types of functions” (p. 267). For instance, linear, quadratic, cubic, and many

other functions have x-intercepts.

Slavit (1997) conducted three case studies and mentioned one in detail in this

article. “Over the course of the year, three case study participants were given 44 sets of

four or five cards bearing graphs, equations, and tables of functions…” (p. 272). Some of

19

these cards included translations of linear functions. The cards also contained different

types of functions. “The participants were asked to discuss any similarities or differences

that they noticed, although they almost always focused on similarities” (p. 272). In the

beginning of the study, one of the students examined functions using numerical values to

plug into the unknown variables in the equations. In fact, she would evaluate a function

by verifying that the points on the graph corresponded with the numerical values of the

equation. The participant also looked for the same correspondence between the tables

and the equations. However, as the study progressed, the student analyzed linear

functions in terms of slope and y-intercept. Thus, the author concluded that this

participant attained a property-oriented view of function because she only described the

function in terms of its slope and y-intercept.

In this qualitative study, Karsenty (2002) investigated 24 adults’ long term

cognitive abilities of linear functions. The participants were introduced to the concept in

high school. They were between the ages of 30 to 45 and examined through Case Study

research in Israel. Half of the participants were men and the other half were women. “In

Israel mathematics is a compulsory subject throughout high school, and can be studied at

three levels, here referred to as high level, intermediate level and low level” (p. 142). In

fact, out of the 24 adults, “…12 took mathematics in the low-level track and 12 in the

intermediate or high level tracks…” (p. 121).

Karsenty (2002) concluded the following results. Based upon the adults’

responses, the author constructed seven categories. The first category was called,

“sketching a correct graph by marking two or three points in a coordinate system and

connecting them with a straight line” (p. 123). Five high levels, one intermediate level,

and one low level participant were placed in this category. Secondly, “sketching a

straight line that reflects a misinterpretation of the relationship between x and y” (p. 123)

was the next category. One high level and four low level participants were in the second

category. The third category was “sketching an incorrect graph based on a holistic

estimation of the behavior of the function” (p. 124). Only, two intermediate level

participants were in this category. The next category was “marking only one point in a

coordinate system” (p. 125). One intermediate level and two low level participants were

in the fourth category. The fifth category was labeled, “drawing a graph by allocating

segments on the x-axis and y-axis and connecting the two endpoints” (p. 125). Three low

level students were in this category. The next category was “describing the function

through equality between shapes or line segments” (p. 126). These items were not drawn

on a two-dimensional Coordinate System. One high level, one intermediate level, and

one low level participant were in the sixth category. The final category was labeled “no

response” (p. 127). Only one low level participant was placed in this category.

Karsenty (2002) also concluded that the adults in categories one through three

“…can be characterized as expressing some degree of acquaintance with representations

of linear functions in a Cartesian system…” (p. 128). On the other hand, the author

observed that “…the basic notion of linear graphing is replaced with personal on-the-spot

constructing of ideas” (p. 128) for the participants in categories four through seven.

Visualization

Krutetskii (1976) examined research concerning 34 visual learners. Among these

visual learners, the author wanted to determine how the students relied on visual images

in a mathematical problem-solving context. Specifically, “…whether he strives to

20

visualize mathematical relationships, whether he has a need for a visual interpretation of

even the most abstract mathematical systems…” (p. 314). Secondly, the author studied

the development of their spatial abilities regarding geometric shapes and figures. He

attempted to capture each student’s “…ability to visualize (to “see” mentally) the position

of a solid in space and the mutual position of its parts, the interrelationship of solids,

figures, planes, and lines (his geometric imagination)” (p. 314).

Krutetskii (1976) reported the following two perspectives based on the results of

the study. First of all, the absence or presence of showing a preference for the

visualization of abstract mathematical concepts and having a strong development of

spatial abilities “…does not determine its type” (p. 314). The second perspective was

that the presence of the previously mentioned components “…showed a very high

intercorrelation in our experiments” (p. 314).

Based on these perspectives, Krutetskii (1976) constructed a framework of three

types of mathematical learners. They were analytic, geometric, and harmonic. The

analytic mathematical learner was “…characterized by an obvious predominance of a

very well developed verbal-logical component over a weal visual-pictorial one” (p. 317).

For example, when given a choice between using equations and graphs, this kind of

learner generally would solve mathematical problems using equations. The geometric

mathematical learner was “…characterized by a very well developed visual-pictorial

component, and we can tentatively speak of its predominance over a well developed

verbal-logical component” (p. 321). For instance, when given a choice between

expressing a mathematical relationship using equations or graphs, this student would

have chosen graphs or diagrams. The harmonic mathematical learner was “…depicted by

a relative equilibrium of well developed verbal-logical and visual-pictorial

components…” (p. 326). The author mentioned two forms of harmonic learners.

Modification A of a harmonic learner depicted a “well developed verbal-logical and

visual-pictorial components in equilibrium, an inclination for mental operations without

the use of visual-pictorial means…” (p. 327). On the other hand, modification B of a

harmonic learner depicted “…an inclination for mental operations with the use of visual-

pictorial schemes…” (p. 327).

Aspinwall and Shaw (2002) used Krutetskii’s (1976) framework to analyze the

cognitive abilities of two beginning Calculus students. The authors presented each

student with four mathematical tasks to complete. These tasks included “…graphic

representations for calculus derivative functions” (Aspinwall & Shaw, 2002, p. 434).

Aspinwall and Shaw found the male student, Al, to be a geometric type of learner. In

fact,

Al’s drawing of the derivative as a curved line from quadrant III to quadrant I

illustrates his preference for using graphic representational scheme, typical of

individuals with ability in and a preference for geometric processing. He made no

attempt to describe his work analytically, for example, by inferring the function

y = x , computing y ' = 2x, and determining ordered pairs to complete the graph 2

of the derivative. (p. 436)

On the other hand, the female student, Betty, was perceived as an analytic type of learner

by the authors.

Betty’s preference for analytic representations in the form of equations became

evident in task 1, shown in figure 1. Betty began by inferring that the graph in

21

figure 1 was y = x . She then computed y ' = 2x as its derivative. Using 2

analytic representations in the form of equations, she calculated ordered pairs

[(1,2) and (2,4)] for the function y ' = 2x and produced the drawing shown in

figure 6. Betty’s procedures suggest that her cognitive structures for the

derivative function are predominantly analytic. (Aspinwall & Shaw, 2002, p.

435)

In another study, Aspinwall, Shaw, and Presmeg (1997) examined the cognitive

abilities of a Calculus III student using Krutetskii’s (1976) framework. This case study

described a male Calculus III student named Tim. The authors presented Tim with 20

mathematical tasks to complete.

Only the first four of these tasks were developed prior to the study. The

remaining tasks were developed based on analyses of interviews from the

previous tasks and were used to probe Tim’s thinking on particular points and to

test our theories of how Tim was making sense of the tasks. (p. 306)

The authors found the student to be a geometric type of learner. In addition, Aspinwall,

Shaw, and Presmeg (1997) discovered that some of Tim’s visual images hindered his

understanding of mathematical concepts.

Tim has just changed his mind about the graph of a parabola having an asymptote

with the slope of tangent lines eventually becoming vertical and undefined. But

as the interview continued, Tim had trouble with his image of the parabola having

tangent lines that eventually approach a vertical slope for increasing or decreasing

values of x. This image was uncontrollable and greatly perturbed Tim as he

thought about the behavior of parabolas. (Aspinwall, Shaw, & Presmeg, 1997, p.

310)

This ability to comprehend the nature of mathematics varies from person to

person because students learn in different ways. People also learn at different rates.

Therefore, the question becomes, what kind of learners encompass the algebra

classroom?

Zazkis, Dubinsky, and Dautermann (1996) examined aspects of visual and

analytic learners who were studying Abstract Algebra. Specifically, 32 Abstract Algebra

students were interviewed regarding how they chose to list the elements of the dihedral

group of order four. The students were also asked to calculate the products, according to

the group operation, of pairs of elements. The authors wanted to see if the Abstract

Algebra students would solve these problems using a visual learning approach or an

analytic learning approach. A visual approach entailed rotating the corners of a square.

An analytic approach involved multiplying permutations. The authors defined

visualization in the following paragraph.

Visualization is an act in which an individual establishes a strong connection

between an internal construct and something to which access is gained through

the senses. Such a connection can be made in either of two directions. An act of

visualization may consist of any mental construction of objects or processes that

an individual associates with objects or events perceived by her or him as the

external. Alternatively, an act of visualization may consist of the construction, on

some external medium such as paper, chalkboard or computer screen, of objects

or events that the individual identifies with object(s) or process(es) in her or his

mind. (Zazkis et al., 1996, p. 441)

22

Zazkis et al. (1996) also defined analytic thinking. “An act of analysis or analytic

thinking (we will use the two terms interchangeably) is any mental manipulation of

objects or processes with or without the aid of symbols” (p. 442). The authors

distinguished how the term symbols may apply to a visual or analytic learner. “When the

symbols are taken to be markers for mental objects and manipulated entirely in terms of

their meaning or according to syntax rules, then we take the act to be one of analysis” (p.

442). Furthermore, the authors explained that “when…the nature of the symbols

themselves or their configurations is used then we would consider it an act of

visualization” (p. 442).

Zazkis et al. (1996) found that most of the Abstract Algebra students combined

the visual and analytic approaches to solve the problems. In fact, the authors stated, “our

observations reveal that students who can mix, harmonize, and synthesize the strategies

usually have a more mature understanding of the problem” (p. 444). As a result, the

authors developed the Visualization/Analysis (VA) model. This model proposed that

both approaches to learning are dependent upon each other to produce successful

mathematical problem solving. The VA model consisted of five levels of visual thinking

and five levels of analytic thinking. A student began with the first level of visual

thinking and then moved to the first level of analytic thinking. This process was repeated

until all five levels were achieved.

Presmeg (1986a) examined possible internal and external factors that contributed

to the under-representation of high mathematics achievers who had a visual learning

preference in high school. This author also discussed the effects of different teaching

styles on visual learners.

Internal factors involved a time factor and non-visual learning preferences

(Presmeg, 1986a). External factors included the possibility that mathematics, school

mathematics curriculum, and the teaching in school classrooms favor the non-visual

learner. The author of this study also found the following:

…the thirteen teachers in this study fell neatly into three groups with respect to

the visuality of their teaching, i.e., a non-visual group (N = 4), an intermediate

group (N = 4), and a visual group ( N = 5). Teachers in the nonvisual group

dispensed with visual presentations whenever possible; teachers in the middle

group used visual presentations but devalued them, while teachers in the visual

group used and encouraged visual methods. (p. 308)

In another study, Presmeg (1986b) attempted to identify the “…strengths and

limitations of visual processing in high school mathematics…” (p. 42). The author also

analyzed “…some effects of different teaching styles on the learning of high school

mathematics by visualisers [sic]…” (p. 42). The methodology conducted in this study

was classroom participant observation, student interviews, and teacher interviews.

Presmeg (1986b) found five images that the visual participants used in this study.

She classified the images in the following categories: concrete, pictorial imagery, pattern

imagery, memory images of formulae, kinesthetic imagery, and dynamic imagery.

It was also found that visual methods are often more time-consuming that [sic]

nonvisual methods. Another aspect which was found to be characteristic of the

problem solving of many visualisers [sic] in the task-based interviews was a

difficulty in communicating the concepts of mathematics. Visualisers [sic]

stumbled over terminology, could not remember key terms. In these straits they

23

typically resorted to gestures or drew diagrams. ( Presmeg, 1986b, p. 45)

In addition, Presmeg (1986b) reported on different teaching styles. “It was found

that teachers in the nonvisual group were more inclined to adopt a lecturing style…and to

teach formally, logically, rigorously, in a manner which could be called convergent” (p.

46). On the other hand, “the visual teachers made connections between the mathematics

curriculum and many other areas of pupils’ experience, including other subjects, other

parts of the syllabus, mathematics learned in past years and, above all, the real world” (p.

46).

Arcavi (2003) defined and analyzed the concept of visualization. The author of

this article also discussed possible outcomes and difficulties associated with visual

representations in the learning of mathematics.

First of all, Arcavi (2003) defined visualization by merging the definitions of

Zimmermann & Cunningham, 1991, p. 3 and Hershkowitz, Ben-Chaim, Hoyles, Lappan,

Mitchelmore, & Vinner, 1989, p.75. Arcavi (2003) stated:

Visualization is the ability, the process and the product of creation, interpretation,

use of and reflection upon pictures, images, diagrams, in our minds, on paper or

with technological tools, with the purpose of depicting and communicating

information, thinking about and developing previously unknown ideas and

advancing understandings. (p. 217)

Throughout this article, Arcavi (2003) examined three possible outcomes of

incorporating visualization in the teaching and learning of mathematics. First of all, the

role of visual representations was perceived as a “…support and illustration of essentially

symbolic results…” (p. 223). Secondly, visualization was seen as “…a possible way of

resolving conflict between (correct) symbolic solutions and (incorrect) intuitions…” (p.

223). The third possible outcome depicted visual representations “…as a way to help us

re-engage with and recover conceptual underpinnings which may be easily bypassed by

formal solutions…” (pp. 223-224).

Arcavi (2003) also explained the cultural, cognitive, and sociological difficulties

of visualization. “A cultural difficulty refers to the beliefs and values held about what

mathematics and doing mathematics would mean, what is legitimate or acceptable, and

what is not” (p. 235). “Cognitive difficulties …arise from the need to attain flexible and

competent translation back and forth between visual and analytic representations of the

same situation…” (p. 235). According to the author cognitive difficulties also centered

around the question: “…is visual easier or more difficult” (p. 235). Sociological

difficulties included pedagogical differences (Arcavi, 2003). For example, “…many

teachers may feel that analytic representations, which are sequential in nature, seem to be

more pedagogically appropriate and efficient” (p. 236). Another challenge that Arcavi

(2003) placed in this category was the “…tendency of schools in general, and

mathematics classrooms in particular, to consist of students from various cultural

backgrounds. Some students may come from visually rich cultures, and therefore for

them visualization may counteract possible deficits” (p. 236).

In another study, Presmeg (1989) discussed the individual learning preferences of

two tenth grade Native American students and examines ways in which visualization may

enhance the understanding of students in multicultural mathematics classrooms.

First of all, Darren was classified by the researcher as a visual student (Presmeg,

1989). After he was presented with a mathematical word problem to complete, the

24

student responded by drawing a diagram of his answer. When the researcher questioned

the meaning of the diagram, Darren clearly explained how the answer was constructed

using the picture. According to the researcher in the study, Darren’s mathematical

achievement was average.

On the other hand, Ashwin was described as a “very structured, logical thinker”

(Presmeg, 1989, p. 18). After being given the same problem as Darren, Ashwin chose to

solve it analytically by using equations. He did not sketch any diagrams of his answer.

According to the researcher in the study, “…Ashwin is above average in his

mathematical achievement at school” (p. 18).

In addition, Presmeg (1989) explained two major ways that visualization may

enhance students’ understanding. The first way involved using diagrams in the teaching

and learning of mathematics. “Visual imagery which is meaningful in the pupil’s frame

of reference may lead to enhanced understanding of mathematical concepts at primary

and secondary levels” (p. 21). The second way included developing awareness and

appreciation of other cultures. According to the author, “…if the pupils know that the

turn of their particular cultural group will come, then steps towards mutual understanding

of cultures may be fostered in addition to the mathematical benefits of the activities” (p.

23).

According to Presmeg (1992), “a visual image is defined here simply as a mental

construct depicting visual or spatial information” (p. 596). The author continues by

discussing the challenges of generalization and categorization that 54 visual high school

students experienced in mathematics. The study involved “…188 transcribed interviews

over a period of eight months” (p. 597). Throughout the article, Presmeg (1992)

examined how prototypical images, metaphors, and metonymies “…may be essential to

reasoning in solving mathematical problems at a high school level” (p. 597).

Lean and Clements (1981) defined spatial ability as “…the ability to formulate

mental images and to manipulate these images in the mind…” (p. 267). Imagery was

defined as “…the occurrence of mental activity corresponding to the perception of an

object, but when the object is not presented to the sense organ” (pp. 267-268). The

authors explained that visual imagery is “…imagery which occurs as a picture in the

mind’s eye” (p. 268). Lean and Clements (1981) presented 116 college students with a

series of mathematical and spatial tests to find out how their learning preferences affected

their mathematical processing abilities. All of the subjects were in their first year of

engineering school at the University of Technology in Lae, Papua New Guinea.

The authors of this study used Suwarsono’s mathematical processing instrument

(Lean & Clements, 1981). It consisted of two parts. The instrument contained word

problems “…suitable for junior secondary pupils in Australian schools” (p. 280). Out of

the 116 college students, “…ten students were interviewed [by Clements] in order to

determine their preferred methods of solving the problems in the mathematics test. The

results obtained by interview were then compared with those obtained by the

questionnaire” (p. 280).

Out of the ten students, the authors classified five of the students as analytic

learners (Krutetskii, 1976; Lean & Clements, 1981). Four of the college students were

placed in the category of harmonic learners. One student was considered a visual learner.

Each student was placed in these preferential learning categories “…according to the

25

amount of ikonic visual imagery he seemed to use, or the number of pictorial

representations he made when explaining his solutions” (p. 285).

Bishop (1989) reviewed literature regarding visualization in mathematics

education by examining three aspects of visualization. First of all, the author looked at

defining visual images. Secondly, the author explained the complex process of

visualization. The third aspect that he explored included, “…the teaching procedures, the

role of the material and social environment and how the individual interacts with that

environment” (pp. 7-8).

In another study, Kirshner and Awtry (2004) explored the visual aspects of

learning Algebra. The authors attempted to “investigate the role of visual salience in the

initial learning of algebra” (p. 237). The authors defined visual salience as “…a visual

coherence that makes the left- and right- hand sides of the equations appear naturally

related to one another” (p. 229).

Kirshner and Awtry (2004) conducted a “…two-treatment teaching experiment”

(p. 232). The subjects of the study included 114 seventh grade students. Before this

study was conducted, Algebra had not been introduced to these students. Algebra lessons

included introducing visually salient rules, non-visually salient rules, and tree notation

(Kirshner & Awtry, 2004). “Tree notation expresses the hierarchy of operations in an

expression through the vertical arrangement of nodes…. in tree notation letters are used

to represent operations that may be indicated only tacitly by positioning of symbols in

ordinary notation” (pp. 232-233).

Kirshner and Awtry (2004) found that “…the treatment group using ordinary

notation performed significantly better…on the visually salient rules, but significantly

worse…on the non-visually-salient rules than the tree notation treatment group” (p. 238).

Graphing Calculators

“According to [the National Council of Teachers of Mathematics (NCTM)]

recommendations, the stage of beginning algebra should include an understanding of

such algebraic concepts as variable, expression, function, and equation, as well as the

ability to construct and analyze multiple representations of number patterns and

situations” (Friedlander & Hershkowitz, 1997, p.442). Using graphing calculators may

help students comprehend these algebraic concepts.

According to Milou (1999), research on the use of hand-held graphing calculators

in the classroom seems to indicate that, to date, calculators have had little

negative effect on the learning of mathematics at any level. From algebra I to pre-

calculus, at both the high school and collegiate levels, the majority of studies have

compared common test scores on various assessment instruments of a treatment

group in which students received instruction with graphing calculators to those of

a control group in which students received traditional instructions without the aid

of a graphing calculator. The results of most studies suggest that the use of the

graphing calculator in teaching and learning is beneficial in terms of

students’ level of understanding and achievement in algebra and

pre-calculus (p. 133).

In another study Clutter (1999) stated, “that there are numerous advantages to using

graphing calculators, including the teaching of higher-level thinking skills and allowing

students to draw conclusions about what they are learning” (p. 10). [Graphing

calculators] offer many advantages over manual graph plotting; most importantly, they

26

encourage a variety of linked approaches (numeric, algebraic, graphical) to the same

problem through permitting dynamic transformations of graphs connected to changes in

other representations” (Hennessy, Fung, & Scanlon, 2001, p. 268). Barrett and Goebel

(1990) believed that “with a classroom set of calculators that graph functions and ordered

pairs of data, solve equations, and graph and compute the parameters for regression lines,

students will be able to investigate and explore mathematical concepts with keystrokes”

(p. 205). Barrett and Goebel (1990) further stated:

When available to every student, these calculators will change the way we teach

many of the topics in the traditional secondary curriculum and enable us to focus

more attention on introducing new topics and real-world applications into those

courses. (p. 206)

Demana and Waits (1990) referred to graphing calculators as interactive

technology. “The greatest benefits seem to come from interactive technology that (1) is

under student and teacher control, (2) promotes student exploration, and (3) enables

generalization” (p. 212). They provide an example of how using graphing calculators can

change the teaching and learning of mathematics. Demana and Waits (1990) state:

The study of domain, range, inverses, geometric transformations, solutions to

equations, inequalities, systems of equations and inequalities, and applications can

be accomplished more effectively with a technological approach. Powerful

geometric representations of problem situations can be added to the usual

algebraic representations. Thus, the power of visualization can be used to study

mathematical concepts and ideas. (p. 212)

Sarmiento (1997) also agreed that graphics calculators help students understand algebraic

concepts.

Milou (1999), further stated that “…research into graphing calculators is in its

infancy….Clearly, more research will be needed before a complete understanding of the

interaction between graphing calculators and the classroom environment is completely

understood” (p. 133). As a result, the author conducted a research study with 146 high

school and middle/junior high school algebra teachers to investigate the use of graphics

calculators.

Of the 146 teachers, 108 (74.0%) responded that their school was currently using

graphing calculators (Item 1). Of the 108 teachers who responded yes to Item 1,

88 were currently using graphing calculators in their classroom (Item 2). In their

response to Item 3, 52 (59.8%) of these 88 teachers indicated that they were using

graphing calculators several times a week, 21 (24.1%) were using them once a

week, and only 14 (16.1%) were using graphing calculators hardly at all (Milou,

1999, p. 134).

Lane (1998) stated, “The most widely used machine is Texas Instruments’ TI-

82/83, which most students buy at the store and use for upper-level math courses such as

calculus or algebra” (p. 1). In fact, Lane and Williams (1998) used the TI-82 Graphics

Calculator in pilot study dealing with high school students. “The purpose of this study

was to determine if Texas Instruments (TI-82) Graphics Calculators increased students’

knowledge of algebra” (p. 21). The results of the study showed that “one of the

dependent variables – graphing equations – was statistically significant” (p. 22).

Therefore, it appeared that the subjects in this study “…who used the TI-82 graphed

equations better than the students who used only paper and pencil” (p. 22).

27

Stick (1997) also conducted a study using the TI-82 Graphics Calculator with

Calculus I and Calculus II college students. He taught two Calculus I and II classes. Two

of the classes were taught using the TI-82: Calculus I and Calculus II. The other two

classes were taught without using the graphing calculator. This author “…focused on five

specific areas [ in Calculus I]: inequalities, asymptotes, first derivatives and local

max/min, concavity, and integration” (p. 356). In Calculus II, Stick (1997) focused on

“…more integration, area and volume applications, transcendental functions, and series”

(p. 357). He concluded, “the main point that many students in the class [using the TI-82

Graphics Calculator] shared with me was that they could see what was happening and

that the learning became fun” (p. 360). In another study, Hollar and Norwood (1999)

used the TI-82 Graphics Calculator to “investigate the effects of a graphing-approach

curriculum” (p. 220). Hollar and Norwood (1999) concluded that “students in the

graphing-approach classes demonstrated significantly better understanding of

functions….” (p. 226). Beckmann, Senk, and Thompson (1999) stated, “Graphing

calculators allow investigation of functions through tables, graphs, and equations in ways

that were not possible before their proliferation” (1999, p. 451). In addition, the NCTM

Curriculum and Evaluation Standards describe the benefit of using a graphics calculator

as “the emergence of a new classroom dynamic in which teachers and students become

natural partners in developing mathematical ideas and solving mathematical problems”

(p. 128).

In a study by Ruthven (1990), the author investigated how high school students

used graphing calculators. Forty-seven of the students in the study came from classes

that participated in the Graphic Calculators in Mathematics project. “Essentially, the

project has enabled each of the teachers to work with at least one class of students having

permanent access to graphic calculators for the duration of their two-year advanced-level

(academic upper secondary) mathematics course” (p. 431). The remaining subjects in the

study came from classes which did not participate in the project. More specifically,

“…40 were in non-project classes where with the exception of 7 students who had

purchased their own graphic calculators, students had no access to graphic calculators (or

computer graphing)” (p. 433).

The author used a questionnaire with two sections to gather data about the

subjects (Ruthven, 1990). The first section asked about personal information. “The

second section was a 40-minute test containing 12 graphic items” (p. 433). During the

test, the students were permitted to use graphing calculators. In addition, the author

asked the students to write down “…any use of a graphic facility on their scripts. They

were also asked to indicate, where possible, any reasoning which led to their answers, as

well as the answers themselves” (p. 433). The items on the test pertained to the concept

of function. The two main types of questions could be classified as algebraic descriptions

and interpretations of graphs. For example, in the first type of question, the students were

given graphs of absolute value, quadratic, cubic, and trigonometric functions. Then they

were given the following directions: “you have to find an expression for y in terms of x

which describes the graph” (p. 434).

Ruthven (1990) chose to examine how students answered the algebraic

descriptions of graphs. He found that the students’ responses consisted of using an

analytic-construction approach, a graphic-trial approach, and a numeric-trial approach.

In the analytic-construction approach, “…the student attempts to exploit mathematical

28

knowledge, particularly of links between graphic and symbolic forms, to construct a

precise symbolisation [sic] from the information available in the given graph” (p. 439).

According to the author, most of the students in the study used analytic-construction

approaches in their responses. The second approach “…uses the graphing facility of a

calculator to repeatedly modify a symbolic expression in the light of information gained

by comparing successive expression graphs with the given graph” (p. 441). The author

found that a smaller number of the subjects fell into this category. In fact, “only about

one quarter of the graphic calculator owners report any use of graphic-trial approaches”

(p. 441). In the numeric-trial approach, “…a symbolic conjecture is formulated…and

modified in the light of information gained by comparing calculated values of the

expression with corresponding values of the given graph” (p. 443). The author found that

“…around one third of those students without access to graphic calculators…” (p. 444)

used the third approach in their responses.

Ruthven (1990) also concluded that the students in the project group

outperformed the students in the non-project group on the algebraic description of graphs

problems. He believed this was “…genuinely attributable to the use of graphic

calculators in the project classes” (p. 447).

In this dissertation, Browning (1988) conducted a quantitative study which

focused on “…the student’s understanding of functions and graphs within a precalculus

curriculum” (p. 1). Pre and post tests were given to eleventh and twelfth graders. Over

200 students from four Ohio high schools provided the sample for this study. “A 25 item

instrument was designed titled ‘Graphing Levels Test’ and given to the precalculus

students at the two city and two suburban high schools in Central Ohio” (p. 32). Student

interviews were also conducted by the author. Most of the students in the sample

participated in the Calculator and Computer Precalculus (C^2 PC) Project at Ohio State.

One class in the sample did not participate in the C^2 PC project.

“The intent of the instrument and the study is to determine and characterize levels

of graphical understanding based on the students’ response to the individual items on the

instrument and student interviews” (Browning, 1988, p. 1). The interviews occurred after

the pre test and post test. Twenty-seven students were interviewed.

The interview consisted of the students taking the “Graphing Levels Test” again

but this time they needed to talk their way through the items, i.e. explain what

they were doing and why. No indication was given to the student whether he/she

was right or wrong but just general encouragement to provide all steps in their

solution process. (p. 40)

Browning (1988) found that the “…learning of functions and their graphs occurs

in levels” (p. 79). Four levels were determined by analyzing randomly clustered pre test

items. “Results also imply the use of technology in the classroom improves student

understanding of functions and their graphs by providing an increased example base” (p.

79). In addition, the results showed the value of student interviews. “The student

interviews also made evident the importance of the explication phase in the process of

learning. Students need an opportunity to make conjectures and discuss their findings.

Technology provides the means to investigation and immediate feedback” (p. 79).

Quesada and Maxwell (1994) investigated how college Pre-Calculus students who

used graphing calculators performed with those who did not. The study was conducted

over three academic semesters. There were a total of 710 subjects. The experimental and

29

control groups were randomly selected. “During the first semester, students in the

experimental used the graphing calculator Casio G-7000…” (p. 207). During the second

and third semesters, “the experimental group used the TI-81…” (p. 207). Each semester

the control group “…was required to use a scientific calculator” (p. 207). In addition, “a

total of three different instructors taught the experimental group while seven instructors

taught the control group” (p. 208) over the period of the study. Both groups were

evaluated by their instructors similarly. “It consisted of four tests, the comprehensive

final exam and one or two weekly quizzes” (p. 208).

Quesada and Maxwell (1994) reported the following comprehensive final exam

results. During the first semester, the control group had an average score of 60.55 on the

final exam. The experimental group earned 78.49 for their average. During the second

semester, the control group had an average score of 61 on the final exam. The

experimental group earned 83 for their average. Finally, during the third semester, the

control group had an average score of 70.34. The experimental group earned 79.80 for

their average.

“Several questions of the final exam were selected and grouped in five different

categories: properties of functions, graphs, word problems, multiple choice questions and

equations” (Quesada & Maxwell, 1994, p. 211). The authors reported on the subjects’

responses from the first semester. For category one, the control group attained an

average of 16.07. The experimental group earned as an average 21.11. In the graphs

category, the control group attained an average of 5.51. The experimental group earned

8.85 as an average. For category three, the control group attained an average of 9.20.

The experimental group earned 15.32 as an average. In the multiple choice questions

category, the control group attained an average of 17.77. The experimental group earned

19.89 as an average. For the fifth category, the control group attained an average of 4.68.

The experimental group earned 6.89 as an average.

Quesada and Maxwell (1994) also concluded the following regarding the

subjects’ responses to graphing calculator questions. “The majority of the students’

answers to the open-ended questions indicated three main positive aspects on using the

graphing calculators: (i) facilitates understanding, (ii) provides the ability to check

answers, and (iii) saves time on tedious calculations” (p. 212).

Shoaf-Grubbs (1994) conducted a quantitative study involving two elementary

College Algebra classes from an all-women’s liberal arts college. Thirty-seven students

form the population of this study. Nineteen of the students were in the experimental

group. (They used graphing calculators.) The other 18 students were in the control

group. (They did not use graphing calculators.) The author gave the students Pre and

Post tests to determine the amount of growth in the following three areas: spatial

visualization, level-of-understanding in algebra & graphing concepts, and spatial

visualization & level-of-understanding in three main topics taught during the semester.

These three main topics that were taught included linear equations, systems of equations

& inequalities involving linear functions with two variables, and vertical parabolas.

“The instructor, lesson plans, and number of graphs examined during the class

period were the same” (Shoaf-Grubbs, 1994, p. 174). In addition, the same textbook was

used in both the control and experimental groups. In order to test the level of

understanding measure in Algebra and Graphs, the author used the Chelsea Diagnostic

Mathematics Tests for Algebra and Graphs. The Card Rotation Test and the Paper

30

Folding Test were used to measure broad spatial visualization skills. Two other tests

were used to measure spatial visualization and level of understanding specific to those

topics presented during the course of study. The author used the Spatial Visualization

(SV) test and the Level of Understanding (LOU) test.

Shoaf-Grubbs (1994) found the following results. “The significance of the gains

in spatial visualization ability made by the Calculator Group supports prior research

concluding that these skills can be taught” (p. 189). In fact, the Chelsea Graphs total

showed that the Calculator (Experimental) Group strongly understood the relationship

between a graph and its’ algebraic expression. The LOU Post test showed that linear

equations was the only significant gain between the two groups. The SV tests or data

depicted “significance was in favor of the Calculator Group for SV Linear Equations, SV

Parabolas, and SV Total” (p. 190). Furthermore, “the results of this study conclude that

the graphing calculator does enhance females’ spatial visualization skills and level-of

understanding in elementary graphing and algebra concepts” (p. 192).

In a case study conducted by Doerr and Zangor (2000), two high school

mathematics were observed. In these Pre-Calculus classes, each student used a TI-82 or a

TI-83 Graphics Calculator. In order to depict the keystrokes the students used, they were

put into groups. “This format generated discussion within small groups where students

explained to each other what they did on their calculator, or one student would show

another how to do a particular task…” (p. 148). However, the authors found out that

discussions with the entire class involving the graphing calculator were “…much easier

to observe, since the teacher and the students regularly used the overhead projection view

screen and explained what they were doing as they were doing it” (p. 148).

Stephens and Konvalina (1999) examined two algebra courses at the collegiate or

university level. Two groups of students were studied from an Intermediate Algebra

class and two groups of students were studied from a College Algebra class. One group

of students from each class used the computer algebra software called MAPLE. The

other group of students from each course did not use the software. The former group was

known as the experimental group and the latter as the control group. All of the students

used the same textbook. The Intermediate and College Algebra students were also given

the same final examination.

The results of the study showed that, “the mean scores on the final examination

were higher in the experimental group than the control group for both intermediate and

college algebra” (Stephens & Konvalina, 1999, p. 488). In addition, “the students in the

experimental group for both intermediate and college algebra were positive and

supportive of the use of MAPLE in the course” (p. 488).

In this dissertation, Paschal (1994) analyzed the effects of incorporating graphing

calculators and videotapes in the teaching and learning of College Algebra. First of all,

the author believed that “mathematical visualization is the student’s ability to draw an

appropriate diagram, whether it is with paper and pencil, or the computer, to represent a

mathematical concept or problem, and to use the diagram to achieve understanding” (p.

38). Secondly, the author believed “for students of mathematics, the constructivist theory

implies individual knowledge and beliefs about mathematics will be modified as

mathematical problems arise. Learning mathematics is constructing mathematics.

Mathematical activity is basically this type of building process” (p. 45). Her definition of

visualization and beliefs about constructivism supported the basis of this study.

31

Five sections of College Algebra were included in this study (Paschal, 1994).

Three of the sections were the technology treatment portion. The other two were non-

treatment sections. “There were three levels of treatment, each of which brought varying

degrees of visualization to the classroom” (p. 46). The first level of treatment involved

teaching College Algebra using graphing calculators. The second level of treatment also

used graphing calculators “…with the additional requirement that they [students] view

the series of content video tapes in the math lab, outside the classroom” (pp. 46-47). The

students viewed 15 of these tapes (Paschal, 1994). In addition to using graphing

calculators and watching College Algebra videotapes, the third level of treatment

included students viewing five graphing calculator training tapes. The two non-treatment

sections used the same syllabus, teaching guide, and textbook as the treatment sections.

Note that since the calculator training and content video tapes are part of the

supplementary materials provided by the publisher for all College Algebra

students, non-treatment students were free to watch content or calculator tapes.

However, the non-treatment students were not required to view any tapes.

(Paschal, 1994, p. 48)

Paschal (1994) determined the following results based on the eight hypotheses

from her study. First of all, hypotheses one was the only hypothesis that was not

rejected. Secondly, the author found “…that there was a difference among groups in

students’ problem solving abilities and their abilities to make sense of new problem

situations in the world around them” (p. 113). In addition, “hypothesis 7 showed that

there was a difference among groups in students’ views of Mathematics and technology,

and that positive shifts occurred in treatment students’ views of Mathematics and

technology” (p. 114). The qualitative data used in this study included observations,

student interviews, essays, and journal entries. Finally, this study “…indicated that

students observed in the visualization-enhanced mathematics class were more actively

and constructively involved in the learning process than students observed in the

traditional mathematics class” (p. 114).

In another study, Elliot, Oty, McArthur, and Clark (2001) examined eight sections

of two college mathematics courses. Four sections of Algebra for the Sciences, an

interdisciplinary class, and four sections of College Algebra were used. In the

interdisciplinary course, “…science topics lead to corresponding mathematics topics and

modelling [sic] is frequently used” (p. 812). The authors wanted to find out the “…effect

that this course has on students’ critical thinking skills, problem-solving skills, and

attitudes towards mathematics” (p. 811). Elliot et al. (2001) obtained the following

results from this study:

…the students in the Algebra for the Sciences course thought their course was

more interesting (p < 0.005) and practical (p < 0.005) than did students in the

College Algebra course. They also had better attitudes towards math (p < 0.05) at

the end of the semester than students in the traditional course. Although

statistically nonsignificant, a greater proportion of students in the Algebra for the

Sciences thought that math was important in life. (p. 815)

Nemirovsky & Noble (1997) used a case study to describe the manner in which a

high school student constructs graphs using a mathematical instrument. Karen, an

eleventh grader, “…used a computer-based tool to create graphs of height vs. distance

and slope vs. distance for a flat board that she positioned with different slants and

32

orientations” (p. 107). This computer-based tool, known as “the Contour Analyzer was

designed to study contours of objects, or paths, each in a given plane, along a three-

dimensional object” (p. 108). The authors found that by using the Contour Analyzer,

Karen became more aware about her understanding of mathematical concepts and how it

related to her other classes. For example, by examining part of the interview transcript

this point becomes evident.

Ricardo: …Let’s say that I tell you that, if there is something that has a slope of

this type [drawing Figure 8], that is horizontal but not zero, can you imagine what

it was, what that could be?

Karen: Well one –I just thought of something now –one of the reasons it might be

below the zero is ‘cause its’ going down [gestures a line slanting downward from

left to right] and it might be a negative slope. Where if it was going from the

bottom left to the upper right [gestures these locations in space], it might be

above the zero. Because come to think of it now when we did our velocity graphs

[in a physics class at school], which was very unusual for us because we never

worked with velocity before, when you’re going towards the [motion] sensor it

was negative. But when you went away it was positive. Even if you were still

increasing speed it [the velocity graph] would go up, and it gives you the illusion

of it getting bigger really when it isn’t… So that might be the only reason that it’s

below the zeros, because it’s a negative slope. But I thought it would be zero

because it’s a straight line, the slope doesn’t change. (Nemirovsky & Noble,

1997, pp. 115-116)

Summary. Many of the studies regarding the concept of functions examined the

role of different cognitive models in the understanding of functions (Vinner, 1983;

Vinner & Dreyfus, 1989; Thompson, 1994; Sfard, 1991; O’Callaghan, 1998;

Breidenbach, Dubinsky, Hawks, & Nichols, 1992). For example, the concept image and

definition of a function was presented as a cognitive model to help explain how students

learn (Vinner, 1983; Vinner & Dreyfus, 1989; Thompson, 1994). In fact, Thompson

(1994) believed that when the concept image and concept definition were balanced, then

understanding was achieved. In addition, Thompson (1994) examined the understanding

of functions in terms of developing an action conception, a process conception, and an

object conception. Thompson (1994) also explored the definition of function in terms of

the correspondence of variables and the co-variation of quantities.

Sfard (1991) presented a conceptual framework pertaining to functions. This

model included an operational conception and a structural conception. She described

three ways to move from an operational conception of function to a structural conception.

O’Callaghan (1998) developed a cognitive model for understanding functions.

This framework included four components. The components were modeling,

interpreting, translating, and reifying. The third component specifically referred to

“…translating among different representations of functions…” (p. 29).

Breidenbach, Dubinsky, Hawks, and Nichols (1992) investigated how 59 math

majors developed a process conception of functions. The authors presented the following

three phases: pre-function conception, action conception, and process conception.

According to the authors, attaining the third phase represented understanding functions.

Some of the function studies investigated college students (Vinner & Dreyfus,

1989; Thompson, 1994; O’Callaghan, 1998; Breidenbach, Dubinsky, Hawks, & Nichols,

33

1992; Dreyfus & Eisenberg, 1983). For instance, Dreyfus and Eisenberg (1983) analyzed

84 college students’ understanding of the concept of function. Dreyfus and Eisenberg

(1983) specifically examined three functional characteristics. They were linearity,

smoothness (differentiability), and periodicity.

Other studies examined how high school students understood functions

(Moschkovich, 1999; Schwarz & Dreyfus, 1995; Sajka, 2003; Monk & Nemirovsky,

1994). Moschkovich (1999) presented two case studies depicting how ninth and tenth

grade algebra students understood the concept of function as they worked in pairs. This

study focused on high school “…students’ use of the x-intercept in equations of the form

y = mx + b” (Moschkovich, 1991, p. 169). Schwarz and Dreyfus (1995) examined how

ninth grade students understood functions after receiving instruction in a computer

software environment. Sajka (2003) reported on how an average high school

mathematics student understood functions using case study research. Sajka (2003)

referred to a students’ understanding in terms of developing a procept [sic] of function.

Monk and Nemirovsky (1994) presented a case study of a twelfth grader who used visual

characteristics of graphs and a technology component, called the Air Flow Device, to

understand functions. Monk and Nemirovsky (1994) found the student to be concerned

with the steepness of a straight line.

The remaining function studies included Eisenberg and Dreyfus (1994),

Yerushalmy (2000), Slavit (1997), and Karsenty (2002). The purpose of the first study

“…was to help students think of functions in a visual way, and to help us understand the

obstacles they must overcome in doing so” (Eisenberg & Dreyfus, 1994, p. 47).

Yerushalmy (2000) investigated the problem solving strategies of two middle school

students in a function approach algebra course. Slavit (1997) analyzed how

understanding was attained through the property-oriented view of function. In the final

function-related study, Karsenty (2002) examined 24 adult’s long term cognitive abilities

on linear functions.

The gap in the literature appeared to pertain to College Algebra visual and non-

visual learners. What does understanding the concept of function mean to these visual

and non-visual mathematical students? Do visual and non-visual algebra learners

translate from one representation of a function to another (O’Callaghan, 1998)?

Most of the studies on visualization encompassed Abstract Algebra, beginning

Calculus, Calculus III, Engineering, middle school, and high school students (Zazkis,

Dubinsky, & Dautermann, 1996; Aspinwall & Shaw, 2002; Aspinwall, Shaw, &

Presmeg, 1997; Lean & Clements, 1981; Kirshner & Awtry, 2004; Presmeg, 1986a;

Presmeg, 1986b; Presmeg, 1989; Presmeg, 1992; Vinner, 1983). In fact, Presmeg (1989)

states, “visual imagery which is meaningful in the pupil’s frame of reference may lead to

enhanced understanding of mathematical concepts at primary and secondary levels” (p.

21). What happens on the collegiate level? How does visual imagery impact College

Algebra learners? Apparently, the gap in the research was how visualization affects

College Algebra visual and non-visual students.

Many of the graphing calculator studies that pertained to college students were

quantitative studies (Shoaf-Grubbs, 1994; Stick, 1997; Stephens & Konvalina, 1999;

Elliot, Oty, McArthur, & Clark, 2001; Paschal, 1994; Browning, 1988). They did not

explain in detail how graphing calculators aid the student’s understanding of the algebraic

concept of function. Instead, these studies usually reported that a group or class using

34

graphing calculators did better than the class not using graphing calculators. What does

this mean to the visual and non-visual College Algebra learner? How was understanding

impacted through the use of technology?

Other graphing calculator studies dealt with middle school and high school

students and/or teachers (Milou, 1999; Doerr & Zanger, 2000; Friedlander &

Hershkowtiz, 1997). For instance, Milou (1999) conducted a study involving middle

school and high school Algebra teachers using survey research. Whereas, Doerr &

Zangor (2000) examined a high school Pre-Calculus class using qualitative research

methods.

Most of the research articles had positive things to say about using technology in

the classroom. Graphing calculators were seen as a benefit in helping students’

understanding of algebraic concepts (Hollar & Norwood, 1999; Milou, 1999; Sarmiento,

1997). It is believed that this instrument helped students develop higher-level thinking

(critical thinking) skills (Clutter, 1999; Elliot, Oty, McArthur, & Clark, 2001; Shoaf-

Grubbs, 1988). Hennessy, Fung, & Scanlon (2001) felt that graphing calculators

encouraged representing functions numerically, algebraically, and graphically. In

addition, using graphing calculators encouraged the investigation and exploration of

concepts in mathematics (Barrett & Goebel, 1990; Beckmann, Senk, & Thompson 1999;

Browning, 1988; Shoaf-Grubbs, 1994). Furthermore, Demana and Waits (1990) referred

to this tool as being interactive technology. Shoaf-Grubbs (1994) also supported this

view. More specifically, the Texas Instrument (TI-82) and (TI-83) were the models used

by most students who use graphing calculators (Lane & Williams, 1998; Stick, 1997;

Hollar & Norwood 1999; Doerr & Zangor, 2000).

35

CHAPTER 3

METHODOLOGY

“Qualitative research tries to establish an empathetic understanding for the reader,

through description, sometimes thick description, conveying to the reader what

experience itself would convey” (Stake, 1995, p. 39). This description attempted to

capture who the participants in a study are and what the world means to them. According

to Bogdan and Biklen (1998), “the data collected take the form of words or pictures

rather than numbers” (p. 5). Qualitative research could also be naturalistic, inductive,

concerned with process, and interested in the search for meaning (Bogdan & Biklen,

1998; Lincoln & Guba, 1985). First of all, naturalistic referred to understanding the

content of a participant’s specific setting. Secondly, inductive pertained to how data

were analyzed. In qualitative research, one moved “...from specific, raw units of

information to subsuming categories of information in order to define local working

hypotheses or questions that can be followed up” (Lincoln & Guba, 1985, p. 203).

Thirdly, there was a concern with the process rather than only a concern with the final

product(s) of a study. Finally, “meaning is of essential concern to the qualitative

approach. Researchers who use this approach are interested in how different people

make sense of their lives” (Bogdan & Biklen, 1998, p. 7).

In order to study the College Algebra participants’ use of graphing calculators and

visual imagery in understanding functions, I used the qualitative case study method.

Lincoln and Guba (1985) explained that “the case study is primarily an interpretative

instrument for an idiographic construal of what was found there” (p. 200). More

specifically, two case studies of College Algebra students were investigated in this study.

Bogdan and Biklen (1998) stated, “when researchers study two or more subjects, settings,

or depositories of data they are usually doing what we call multi-case studies” (p. 62).

Participants

The population in the study came from two College Algebra courses in the fall of

2005. Together, the two sections were comprised of 71 students. The researcher taught

both sections. The students attended a four year historically black university that has

existed since 1887. It is located in the south-eastern portion of the United States.

Approximately 13,000 students attend the university.

Two participants were purposefully selected from the population using Presmeg’s

(1985) Mathematical Processing Instrument. This testing device measured a student’s

preference for visual thinking in mathematics. Therefore, one visual mathematical

learner and one non-visual mathematical learner were chosen.

According to Bogdan and Biklen (1998), in purposeful sampling, “you choose

particular subjects to include because they are believed to facilitate the expansion of the

developing theory” (p. 65). In addition, Lincoln and Guba (1985), posited that sampling

critical cases allows “…maximum application of information to other cases because, if

the information is valid for critical cases, it is also likely to be true of all other cases” (p.

200). With the previous quotes in mind, one of the main reasons behind the selection of

the visual and non-visual cases in the present study was to produce and explicate theory.

In order for this to happen, the selection of these cases should generate the following

conditions. First of all, a case should produce as many categories and characteristics of

those categories as possible. Secondly, the categories should relate to each other.

36

Instruments

This investigation used the Mathematical Processing Instrument and the

Mathematical Processing Questionnaire by Presmeg (1985). These tools were chosen

because they measure how a student prefers to process mathematical information, i.e.,

visually or non-visually.

First of all, Presmeg’s (1985) Mathematical Processing Instrument included three

sections (A-C) of mathematics problems for students to solve. The author recommended

section B only or sections B and C for college-level students. All 71 students were

provided with section B of the instrument. Section B had 12 mathematical word

problems to solve. Each question could be solved numerically, algebraically, and

graphically. Graphical solutions or drawing diagrams were considered as visual

solutions. Numerical and algebraic solutions were considered as non-visual solutions.

The test was scored by adding the total of two for every visual solution, one if the

problem is not attempted, and zero for every non-visual solution. The highest score

possible was 24/24 (24 out of 24). The lowest score possible was 0/24 (0 out of 24). If

the student’s visualization score was 12/24 or higher, then he or she was considered as

having a preference for visual thinking in mathematics. On the other hand, if the

participant’s visualization score was 10/24 or lower, then he or she was considered as

having a preference for non-visual thinking in mathematics. The students were required

to show their work for the solutions, however, they were not required to use a specific

method of solution over another. The participants were also asked to choose their own

method of solution and turn in their papers. (See Appendix A for a copy of this

instrument.)

Secondly, each student was supplied with a Mathematical Processing

Questionnaire (Presmeg, 1985). The questionnaire was a follow-up to the participants’

responses to the Mathematical Processing Instrument. This questionnaire provided three

or more solutions for the students to choose the one that was most similar to their

response. After the participants completed the questionnaire, they were asked to turn in

their responses. (See Appendix B for a copy of this questionnaire.)

After completing Presmeg’s (1985) Mathematical Processing Instrument, 52

College Algebra students scored from 0/24 to 10/24. As a result, these 52 students were

considered to have a preference for non-visual thinking in mathematics. In addition, after

completing the instrument, 19 College Algebra students scored from 12/24 to 20/24. As

a result, these 19 students were considered to have a preference for visual thinking in

mathematics.

The Visualizer (VL) for the present study was purposefully selected from the 19

students. The participant scored 16/24 on Presmeg’s (1985) Mathematical Processing

Instrument. In addition, she was extremely detailed regarding her answers on the

instrument. VL’s case was reported in Chapter 4.

The Nonvisualizer (NVL) for the current study was purposefully selected from

the 52 students. The participant scored 4/24 on Presmeg’s (1985) Mathematical

Processing Instrument. In addition, he was very detailed with his answers on the

instrument. NVL’s case was reported in Chapter 5.

Data Collection

“Qualitative data consist of quotations, observations, and excerpts from

documents” (Patton, 2002, p. 47). Therefore, the data sets for the present study were

37

interviews and document reviews. The interviews were based on the participants’

responses to mathematical tasks. Initially, there were three or four graphing calculator

tasks and function tasks. Subsequent tasks were selected and/or constructed based on the

visual and non-visual mathematical learners’ responses to previous tasks. For example,

task number one was given to the participants to complete. Then, the interview was

based on how the student completed task number one with an explanation. Next, task

number two was given. After that, the interview was based on how the student

completed task number two with an explanation. This process continued through task

number five. Task number six was selected based on the student’s responses to tasks one

through five. Again, the interview was based on how the participant completed task

number six with an explanation. In addition, the interviews were video-taped. The

interviews consisted of depicting how the participants use graphing calculators and

mental pictures to complete the tasks they are given. Using mental pictures included

verbally explaining and/or drawing the images on paper. As a result, the videotape was

used to capture sketches and calculator use. Bogdan and Biklen (1998) recommended

using tape recording devices “when a study involves extensive interviewing or when

interviewing is the major technique in the study…” (p. 130). Lincoln and Guba (1985)

explained that audio-and video-taping “…provide an accurate and unimpeachable

record…” (p. 272) for collecting data. In addition, the tapes in the current study were

transcribed after each interview session. According to Bogdan and Biklen (1998), the

“transcripts are the main data of many interview studies” (p. 130).

The functions examined in this study included first degree, second degree, and

higher order functions. For example, first degree or linear functions can be written in

symbolic form as f(x) = ax + b where a and b are real numbers. The graph or pictorial

form of a linear function is a straight line. Second degree functions can be expressed in

symbolic form as f(x) = ax + b where a and b are real numbers. The pictorial form of

second degree functions is called a parabola. A basic parabola can look like a bowl that

is shaped like the letter “u”. In addition, higher order functions can be written in

symbolic form as f (x) = ax + b where a and b are real numbers. Higher order functions

have an exponent of three or higher in their equations. Odd numbered higher order

functions can be expressed in pictorial form in three parts that are connected. One

portion of the graph is curved downward or decreasing. The second part of the graph

remains constant. The last portion of the graph is curved upward or increasing (Lial,

Hornsby, & Schneider, 2001).

2

3

The documents in this investigation were College Algebra Writing Journals, tests,

College Algebra Web Homework, and a researcher journal. Bogdan and Biklen (1998)

posited that documents “…can be used as supplemental information as part of case study

whose main data source is participant observation or interviewing” (p. 57). The College

Algebra Writing Journals included the students’ feelings, beliefs, and interpretations

about mathematics in their own words. They also recorded their struggles and concerns

regarding College Algebra over the semester. Specifically, the struggles and concerns

pertained to class assignments and/or algebraic concepts that are introduced in class.

Since the course was held three days a week, the students were expected to complete a

minimum of three entries per week. Each entry was at least one-half of 821 inches by 11

inches page. In addition, the two visual and non-visual learners wrote an additional entry

after every interview session. The interview session entries included the participants’

38

feelings, beliefs, and interpretations about the mathematical tasks. They also recorded

any struggles encountered in completing the tasks. The entry was at least one-half of 821

inches by 11 inches page.

The participants’ College Algebra tests were a second document data source.

Four chapter tests and one final examination were given. The concept of function

permeates the last three tests and the final examination. The students were asked to

represent functions numerically, algebraically, and/or graphically on these exams.

A researcher’s journal was the third document data source. I made entries in the

journal that pertain to all of the data collection activities. Specifically, the journal entries

began in the fall of 2005 after the students complete Presmeg’s (1985) visualization

instrument. It included the time, place, and length of each interview session. The entries

also included my feelings, beliefs, and interpretations regarding the mathematical task

interview sessions. Lincoln and Guba (1985) supported this idea by stating that “each

investigator should keep a personal journal in which his or her own methodological

decisions are recorded and made available for public scrutiny” (p. 210).

In table 3.1 on pages 39 and 40, the course content for the first eight weeks was

listed. Presmeg’s (1985) Mathematical Processing Instrument was distributed to the

students during week four. One Web Homework for section 2.1 was assigned during the

fifth week. Task #1 was given during the eight week.

In table 3.2, the course content for the rest of the semester was listed. Three Web

Homework assignments were given during the rest of the semester. The section 4.2 Web

Homework was assigned during week 11. The section 4.6 Web Homework was assigned

during week 15. The section 5.2 Web Homework was assigned as the final Web

Homework during week 16.

Table 3.1: College Algebra Weeks 1-8

WEEK COURSE

CONTENT

WEB HOMEWORK MATHEMATICAL

TASK

1 The Real Number

System and Absolute

Value

2 Polynomials and

Factoring

3 Radical Expressions

and Review

4 Visualization

Instrument was

given

Functions, Domain

& Range of a

Function

5 Graphical

Representation of a

Function, Distance

and Slope,

Operations on

Functions

Section 2.1

39

Table 3.1: Continued

WEEK COURSE

CONTENT

WEB HOMEWORK MATHEMATICAL

TASK

6 Review

7 Polynomial,

Rational, and

Absolute Value

Equations

8 Absolute Value and

Radical Equations,

Inequalities, &

Domain of a

Function

1

In table 3.2, the weeks when the mathematical tasks were given was also

included. Task # 2 was given during the tenth week. Tasks three and four were given

during week 12. The fifth and sixth tasks were given during week 13. Task # 7 was

given during week 15. Tasks eight through ten were given during week 16.

Table 3.2: College Algebra Weeks 9-16

WEEK COURSE

CONTENT

WEB

HOMEWORK

MATHEMATICAL

TASKS

9 Review

10 Equations of Lines

and Techniques of

Graphing

2

11 Graphs of

Quadratics and

Polynomial

Functions

Section 4.2

12 Graphs of Rational

Functions and

Review

3 and 4

13 Inverse Functions 5 and 6

14 Exponential

Functions

15 Logarithmic

Functions & Review

Section 4.6 7

16 Review for Final Section 5.2 8-10

40

In table 3.3, the Web Homework assignments that each participant completed

were described in terms of the scores they earned. Section 2.1 included 10 questions

about determining/graphing the union and intersection of sets. Both participants

completed this assignment. Section 4.2 included 10 questions pertaining to graphing

quadratic, cubic, square root, and absolute value functions with vertical and horizontal

translations. Both participants completed this assignment. Section 4.6 included seven

questions about finding the vertical and horizontal asymptotes of rational functions. VL

completed this assignment, NVL did not. Section 5.2 included graphing 13 exponential

functions. VL completed this assignment. NVL did not.

Table 3.3: Web Homework Scores

SECTION VL’S SCORE NVL’S SCORE

2.1 80 80

4.2 80 90

4.6 100 0

5.2 77 0

Analysis of data. This investigation examined two case studies of College

Algebra students. One of the participants was a visual mathematical learner. The other

participant was a non-visual mathematical learner. One of the goals of these case studies

was to answer the following research questions by conducting interviews and document

reviews.




According to Lincoln and Guba (1985), “data analysis must begin with the very

first data collection, in order to facilitate the emergent design, grounding of theory, and

emergent structure of later data collection phases” (p. 242). Therefore, the analysis of

data in the current study began by examining each participant’s initial interview session.

The taped interview sessions were transcribed. Each transcript was analyzed before the

next interview session occurs in order to look for any possible emerging patterns or

themes. If any patterns are found, they will be investigated in the next interview session.

“The four terms ‘credibility’, ‘transferability’, ‘dependability’, and

‘confirmability’ are, then the naturalist’s equivalents for the conventional terms ‘intended

validity’, ‘external validity’, ‘reliability’, and ‘objectivity’” (Lincoln & Guba, 1985, p.

300). In the present study, credibility will be established by using triangulation and

member checking. “The technique of triangulation is the third mode of improving the

probability that findings and interpretations will be found credible” (Lincoln & Guba,

1985, p. 305). In the present study, no assertion was considered valid unless it could be

supported by two or more pieces of data. First of all, the interviews were triangulated

among the different participants. Secondly, the interview sessions and documents were

triangulated. This inquiry will use two of the four modes of triangulation that have been

41

suggested by Denzin, 1978 cited in Lincoln and Guba, 1985. The two methods include

“…the use of multiple and different sources [and] methods…” (p. 305). In the current

study, the first mode referred to verifying information from more than one interview

participant. One visual learner and one non-visual learner was interviewed. The second

mode referred to using more than one data collection procedure. In the present study,

both interviews and document reviews were utilized in order to collect data.

“The member check, whereby data, analytic categories, interpretations, and

conclusions are tested with members of those stakeholding groups from whom the data

were originally collected, is the most crucial technique for establishing credibility”

(Lincoln & Guba, 1985, p. 314). The member checking technique was applied to the

current study by allowing the participants to read the results section (chapters 4 and 5) of

the dissertation. Each participant was asked to assess and correct any errors he/she finds.

Specifically, the students were asked to pay special attention to the overall written

interpretations of their responses to the various mathematical tasks, which helped to build

each case study. After that, the participants explicated if they were in agreement with

how the cases were written or in disagreement and provided reasons behind either

statement. In addition, each student was encouraged to indicate any information he/she

feels was left out of the case study that may be pertinent. All of the member check

responses were reported. The biggest difference between these two forms of establishing

credibility is that “member checking is directed at a judgment of overall credibility, while

triangulation is directed at a judgment of the accuracy of specific data items” (p. 316).

According to Lincoln and Guba (1985), the researcher had the “…responsibility

to provide the data base that makes transferability judgments possible on the part of

potential appliers” (p. 316). In the current study, the data base included descriptions of

the time and context of the case studies. In fact, each case study was written with thick

description in order to make transferability possible.

Lincoln and Guba (1985) posited that the dependability of a study takes place

when “…the naturalist seeks means for taking into account both factors of instability and

factors of phenomenological or design induced change” (p. 299). In order to help

establish dependability in the present study, the overlap methods of triangulation were

used (Lincoln & Guba, 1985). This technique was chosen based on the authors’

following claim which emphasized, “since there can be no validity without reliability

(and thus no credibility without dependability), a demonstration of the former is

sufficient to establish the latter” (p. 316). Thus, triangulation, which was discussed

earlier, was used to help establish credibility and in turn as an overlap method to establish

dependability. As an overlap method, the focus was on triangulation of multiple and

different methods.

In order to help establish confirmability, an audit trail was maintained throughout

the study. The audit trail will include five of the categories that have been suggested by

Halpern, 1983 cited in Lincoln and Guba, 1985. They are raw data; data reduction and

analysis products; data reconstruction and synthesis products; process notes; and

materials relating to intensions and dispositions. First of all, the raw data in the present

study were the results from Presmeg’s (1985) visualization instruments, audio and

videotaped interview sessions, and the College Algebra Writing Journals and tests.

Secondly, the data reconstruction was utilized in order to condense information and

identify any common patterns or relationships. Thirdly, data reconstruction and

42

synthesizing products occurred in the current study by identifying and organizing

common categories and/or themes, examining the participants’ interpretations and my

interpretations, reporting findings, and identifying connections to existing literature

and/or theory. Next, the process notes, which included methodological decisions, were

recorded in the researcher’s journal. The fifth category of materials relating to intentions

and dispositions of the study were also recorded in the researcher’s journal.

Specifically, the data were analyzed by using O’Callaghan’s (1998) translating

component for understanding functions. The data were also analyzed by using Ruthven’s

(1990) role of graphing calculator approaches. O’Callaghan (1998) and Ruthven (1990)

comprised the theoretical framework for the present study.

In the current study, the visual and non-visual mathematical learners’

understanding of functions was measured by the presence or absence of the translating

component (O’Callaghan, 1998) for understanding functions. In the author’s cognitive

model translating was defined as the ability to move from one representation of a

function to another…” (p.25). The representations of functions used in the current study

were tables (numeric form), equations (symbolic form), and graphs (graphic form).

In addition, the role of graphing calculators was interpreted based on the

Analytic-Construction Approach, Graphic-Trial Approach, and Numeric-Trial Approach

(Ruthven, 1990). In the Analytic-Construction Approach, “…the student attempts to

exploit mathematical knowledge, particularly of links between graphic and symbolic

forms, to construct a precise symbolisation [sic] from the information available in the

given graph” (p. 439). The Graphic-Trial Approach “… uses the graphing facility of a


by comparing successive expression graphs with the given graph” (p. 441). In the

Numeric-Trial Approach, “ a symbolic conjecture is formulated… and modified in the

light of information gained by comparing calculated values of the expression with

corresponding values of the given graph” (p. 443).

43

CHAPTER 4

RESULTS









one of the case studies of the two College Algebra students were reported in this chapter.

The findings of the second case study were reported in chapter 5.







exponential functions. Furthermore, prior to receiving the tasks, the students’ preferences





First, Presmeg’s (1985) Mathematical Processing Instrument included three


section B only or sections B and C for college-level students. The participants in this

study were provided with section B of the instrument. Section B had 12 mathematical

word problems to solve. Each question could be solved numerically, algebraically, and




problem was not attempted, and zero for every non-visual solution. The highest score


the student’s visualization score was 12/24 or higher, then he or she would be considered

as having a preference for visual thinking in mathematics and called a Visualizer (VL).

On the other hand, if the participant’s visualization score was 10/24 or lower, then he or

she would be considered as having a preference for non-visual thinking in mathematics

and called a Nonvisualizer (NVL). The students were required to show their work for the

solutions, however, they were not required to use a specific method of solution over

another. The participants were also asked to choose their own method of solution and

turn in their papers. (See Appendix A for a copy of this instrument.)

Secondly, each student was supplied with a Mathematical Processing Questionnaire

(Presmeg, 1985). The questionnaire was a follow-up to the participants’ responses to the

Mathematical Processing Instrument. This questionnaire provided three or more

solutions for the students to choose the one that is most similar to their response. After

44

the participants completed the questionnaire, they were asked to turn in their responses.

(See Appendix B for a copy of this questionnaire.)

In the present study, the visual and non-visual mathematical learners’

















In the report of Case 1, as patterns emerged from the data the researcher called

them categories. In this chapter, the categories were labeled using Roman numerals.

CASE 1

The Visualizer (VL) in the present study was a 20 year old African-American

female student. She was born on May 5, 1985 in Trinidad and Tobago. The participant

lived there for 12 years. Trinidad and Tobago is a territory located near Venezuela in

South America. Both islands comprise a country in the West Indies. VL lived in this

country until she was in the seventh grade. During this time, the participant and her

family moved to Pembrook Pines, which is a community near Ft. Lauderdale, Florida.

VL attended an historically black university because she explained, “I wanted to have a

different experience by going to a prominently black school”.

VL completed Pre-Algebra in the seventh grade. The participant earned a “B” in

this course. The next mathematics class she took was Algebra during the eighth grade.

VL earned an “A” in this course. During the tenth grade, the participant completed

Geometry. The participant earned an “A” in this course. In eleventh grade, VL took Pre-

Calculus and earned a “B”. The last mathematics class that she completed in high school

was Calculus I. The participant earned a “B” in this course.

When the researcher asked VL how she felt about math, she responded, “I

generally like math. But I don’t believe that is an easy subject of study. I specifically

liked Geometry in high school.” As a follow-up question to the participant’s response,

the researcher asked why she liked Geometry. VL explained, “I understood it very

easily. I think it’s because I’m a visual learner and therefore it was easy for me because

most problems are drawn out, such as the angles”, In addition, the participant shared

with the researcher that, “Mathematics was always fun to me, it was my favorite class

growing up”.

College Algebra was the only mathematics course that VL completed at the

university where the research took place. The participant earned an “A” in this course.

45

She had previously taken Calculus I at a community college two summers prior to the

research study. The participant shared her experience regarding Calculus as, “I never had

trouble with math throughout my school life except for Calculus”. When the researcher

asked VL what Algebra meant, she stated, “Algebra means thinking backwards to me.

That’s the way I see it, you have one piece of the puzzle and you have to try to figure out

the other”. She attended class regularly. The student also participated in class

discussions by volunteering to answer questions. In addition, VL completed in-class and

out-of-class assignments in a timely manner.

The participant was first introduced to graphing calculators in the tenth grade. VL

reported, “Our teacher supplied the class with them during the class sessions”. The

participant did not have her own graphing calculator. In addition, she had little

experience using the calculator between tenth grade and the fall of 2005.

Using the graphing calculator during the College Algebra course was optional for

the students. The instructor allowed the students to use graphing calculators in class,

however, it was not mandatory for the students to purchase them. The required

technology component was an Internet software program called EDUCO. EDUCO

offered the students on-line tutorials, quizzes, and Web homework (Sharma et al., 2002).

VL majored in Business Administration and minored in Psychology. The

participant was a junior during the time of the study. After graduation she explained,

“my future plans are to open up my own psychology business”. In addition, her hobbies

included swimming and reading poetry.

The participant scored 16/24 on Presmeg’s (1985) visualization instrument. In

addition, VL wrote in the College Algebra Journal, “after taking the extra credit

[Presmeg’s (1985) visualization instrument] I believe that I am a visual person because I

needed to draw a diagram for every problem to see exactly what I was doing”.

Task # 1

Mathematical task #1 was a linear function (Figure 4.1) by Moschkovich (1998).

The directions were for the participant to complete the task and explain her reasoning.

Task one was chosen in order to see how the student would solve a task involving linear

functions. Using the graphing calculator was optional during the completion of the task.

It was optional because the researcher wanted to see if the participant would choose to

use the graphing calculator. She did not.

In response to mathematical task #1, VL stated:

So, first I would try to find y = x + 5 and I would put in the x-values to see if it

would go up, and get steeper [writes y = x + 5 and constructs a table of values].

I’ll now plug the numbers [x-values] into the equation so I can draw, actually

graph the y = x + 5. Okay, so x + 5 [says and writes on the white paper] y = 0 +

5, y =5. y = 1 + 5 = 6, y = -1+ 5 = 4, y = -2 + 5 = 3. So, it’s going to be (pauses

and looks at unfinished table and fills in the following values) 5, 6, 4, 3.

46

Figure 4.1: Mathematical Task # 1

47

In this first example, the student looked at the blank coordinate grid in task one and

substituted zero, one, negative one, and negative two as x-values into the equation

y = x + 5. Then VL solved the equation in order to find the corresponding y – values. As

the participant substituted these specific x – values into y = x + 5 and calculated the

corresponding y – values, she constructed a table of numerical values (Figure 4.2).

Figure 4.2: Visualizer’s table of numerical values for task one

It appeared that the participant substituted specific values for the variables x and y into

the equations as part of the completion of task #1. During this segment of the interview,

the participant translated from the given symbolic form of a function to its numeric form.

In addition, in mathematical task #1, VL responded: “When x is zero, y is five.

One, two, three, four, five [counts up the y-axis and plots the point (0, 5) on the Cartesian

coordinate system provided on the mathematical task # 1 sheet]. When x is one, y is six

[plots (1 ,6)]. When x is negative one, y is four [plots (-1 , 4)]. When x is negative two,

y is three [counts up the y-axis one, two, three, plots (-2 , 3), and connect the points with

a straight line]”. In this instance, the participant took the calculated x and y values, from

the table of numerical values for y = x + 5, discussed previously and plotted the points on

the coordinate grid provided in mathematical task # 1 (Figure 4.3). Then she drew the

graph of the line y = x + 5 by connecting the points and extending the line past the

specific points. During this portion of the interview session, the student was translating

from the numeric form of a function to its graphic form. It also appeared that VL plotted

specific points of a function on a graph as part of the completion of task one.

48

Figure 4.3: Visualizer’s Mathematical Task # 1 Graph

In response to mathematical task # 1, VL stated:

And I would say make the line steeper [reads and refers to Part A]? I would say

no. It doesn’t make line steeper. It just, um, [points to the lines y = x and y = x +

5] positive –positive. Cause, it just carries the line up on the graph. But I don’t

think it makes the graph steeper because the slope is probably the same between

them.

In part A of the first mathematical task, the problem asked if you start with the equation

y = x and change it to y = x + 5, would the graph of the line y = x + 5 be steeper than the

graph of the line y = x. The student expressed that the line of y = x + 5 appeared higher

than the line y = x on the graph. Later on she explained that the line y = x + 5 was higher

than the line y = x “because the y – value increased from zero to five on the y – axis,

therefore moving the line up”. However, the participant did not report that starting with y

= x and changing to y = x + 5 would make y = x + 5 steeper than y = x if the two lines

had the same slope. It appeared that the student detected a relationship between the

concepts slope and steepness as part of the completion of task one.

Overall, O’Callaghan’s (1998) translating component was present during the

completion of this task. Specifically, the student translated the given symbolic form of

y = x and y = x + 5 to their numeric and graphic forms. The participant did not use the

graphing calculator during the completion of task one.

Thus, VL used the following three categories during the completion of

mathematical task # 1. Category I was substituting specific values for the variables x and

y into equations. Category II was plotting specific points of a function on a graph.

Category III was detecting a relationship between the concepts slope and steepness.

During the completion of mathematical task # 1, the participant used visual

imagery and non-visual methods to solve the problem. The student relied on visual

imagery regarding the linear function y = x. The participant drew her image and used it

to complete the task. The student substituted specific values for the variable x and y into

equations. VL also plotted specific points of a function on a graph. In addition, the

participant shared her personal viewpoint about this task in the College Algebra Journal,

“I understand the problem it was not difficult to me in any way”.

49


50

Task # 2

Mathematical task # 2 was another linear function (Figure 4.4) by Moschkovich

(1998). The directions were for the participant to complete the task and explain her

reasoning. Task two was chosen because the researcher wanted to see how the student

would solve a second linear function task. The researcher also wanted to know if VL

would use the same categories from task one in the completion of task two. Using the

graphing calculator was optional during the completion of the task. It was optional

because the researcher wanted to see if the participant would choose to use the graphing

calculator. She did not.

In response to mathematical task # 2, VL stated: “Therefore, [says and writes on

mathematical task # 2 sheet] I believe that multiplying x by three (pauses) would move it

[the line y = 2x] by six not just by three”. Part A of the second mathematical task asked

the participant if multiplying x by three in the equation y = 2x would produce the

equation of the lighter line (Figure 4.4). The participant appeared to believe that

multiplying the equation y = 2x by a positive three would move the graph of the function

by six units to the right along the x-axis. VL misinterpreted what multiplying x by three

in y = 2x would do to the graph of the line. The student calculated y = 2x times 3 = 6x

mentally without using pencil and paper. However, she saw 6x as representing

movement of the line along the positive side of the x–axis because six is a positive

number and x represented the x-axis. According to VL, a positive number indicated

movement of the line on the positive side of the x-axis. Specifically, VL reported the

original line (y = 2x) shifted six units to the right on the x-axis. In actuality, the x-values

for both functions remained the same. The difference occurred between the y-values.

The y-values of the two functions differed by a multiple of three. As a result, if the

original line y = 2x was multiplied by three, then the graph of y = 6x would be steeper

than the graph of y = 2x.

Next, during the completion of this task, VL responded:

Okay, if you add three to x [looks at y = 2x graph] I believe that it would be yes

cause if you add three to x, you would move from zero to three on the positive

side [points and counts on the x-axis from zero to three and draws a line through

(3, 0)]. So, actually I’m going to change my answer to no [erases] because it

would have to be a negative three for this actual graph. So, I believe that it would

be no because you would have to add negative three to x.

Part B of mathematical task # 2 asked the participant if adding three to x in the equation

y = 2x would produce the equation of the lighter line. In example two, the student

misinterpreted what adding three to the equation would do to the graph. Adding three to

x in the original function, y = 2x, would move the line up on the y-axis by three units

(Figure 4.4). Instead the participant reported the line would move three units to the right

along the x-axis because, to VL, three was a positive number and x represented the x-

axis. The student also explained that in order to change the darker line in task two to the

lighter line, add negative three to x in the equation y = 2x because negative three was a

negative number and x represented the x-axis. According to VL, a negative number

indicated movement of the graph of the line on the negative side of the x-axis.

The student completed task two by misinterpreting the graphical representation of

a function after multiplying and adding specific values to the symbolic form of a

function. Two examples of VL misinterpreting the graphical representation of a function

51

after multiplying and adding specific values to the symbolic form of a function from task

two were explained above.

Overall, O’Callaghan’s (1998) translating component was absent during the

completion of this task. The student did not translate the given graphic form of the

function to its symbolic form. In addition, the participant did not use the graphing

calculator during the completion of task two. In addition, the participant shared her

personal viewpoint of task # 2, “The task given to me was fairly easy. I believe I got the

correct answers after working out the problem. This task was not really challenging to

me, just a little time consuming”.

Since the researcher had some unanswered questions regarding the student’s

perspective on concepts such as steepness and slope, a follow-up interview to linear

functions was conducted. During this follow-up interview, the researcher asked the

participant “what does steepness mean to you”? The participant answered:

To me, steepness means that one is not parallel to the two, both lines are not

parallel to each other. Therefore, showing me that the lines are changing the

actual, I want to say slope is changing between the lines, but as examples of these

to me steepness isn’t really considered [referring to linear functions] because both

of them uh, show parallel, parallel lines. I guess.

The student related the terms steepness, parallel lines, and slope in the previous

definition. VL defined steepness in terms of lines that were not parallel to each other.

The participant reported that she saw slope as a measure of steepness. Her experience in

College Algebra introduced the student to the fact that parallel lines have the same slope.

Therefore, VL reportedly concluded two statements: (1) if y = x and y = x + 5 were

parallel lines, they would have the same slope and (2) if y = x and y = x + 5 had the same

slope, then the line of y = x + 5 would not be steeper than the line of y = x. As a result,

the student verified that if two functions have the same slope, one would not be steeper

than the other.

Task # 3

Mathematical task # 3 was a quadratic function (Figure 4.5) by Eisenberg and

Dreyfus (1994). The directions were for the participant to complete the task and explain

her reasoning. Task three was chosen because the researcher wanted to find out how the

student would solve a task involving quadratic functions. Using the graphing calculator

was optional during the completion of the task. It was optional because the researcher

wanted to see if the participant would choose to use the graphing calculator. She did.

In response to mathematical task # 3, VL stated:

I’m gonna [sic] get my x and my y’s [draws a table of values and labels it as 1

with a circle around it] and then plug them into the formula to get a few points to

see where the uh, where they are located on the graph and then I can probably

figure out, probably from cancellation which one is close to the function.

52


After sketching each of the four parabolas separately from the third mathematical

task, the participant reported that she constructed a numeric table of values for two

reasons. The first reason was to calculate specific points for each graph. The second

reason was to match each graph with its corresponding symbolic form. She constructed

the table mentioned above using x - 2x + 1. At this point, the student did not express

the terms translation or transformation of graphs. Instead, the participant continued to

create numerical tables for 1 – x and x 2 - 2x (Figure 4.6).

2

2

53

Figure 4.6: Visualizer’s tables of numerical values and graphs for Mathematical Task # 3

54

Similarly, for 1 - x VL explained: 2

I’m gonna get my x and my y’s [draws a table of values and labels it as 1 with a

circle around it] and then plug them into the formula to get a few points to see

where the uh, where they are located on the graph and then I can probably figure

out, probably from cancellation which one is close to the function. So, the first

one first number, I’m gonna use is zero, one, negative one, and two. I plug that in

for zero, that’s gonna be one. For one that’s gonna be, I’m gonna use the

calculator.

The participant substituted the values zero, one, negative one, and two for the variable x

into 1 - x . Then she calculated the corresponding y values. After that, the student

constructed a table of numerical values that corresponded to the function 1 - x .

2

2

For x 2 - 2x, VL stated:

For my next function, it states x squared minus 2x. I’m gonna write that down

[writes on additional piece of paper x – 2x and circles a 3 beside it] I’m gonna

do the same thing as the last two and use my x and y values. I’m gonna use the

same numbers I used before zero, one, negative one, and two. Then plug those

into the formula (constructs a table of values) into the uh, function. The first one

is zero, if x, zero, x squared, zero squared is zero minus 2x is zero. I’m gonna use

my one and plug that in one squared minus two times one is one minus two is

negative one [writes 1 – 2 = -1]. So that’ll give me negative one. When I plug in

negative one into my function, I’m gonna get positive one minus, minus positive

one minus two times negative one [writes 1 – 2 = -1]. That gives me one plus

two minus three.

2

This time the student substituted the values zero, one, negative one, and two for the

variable x into x - 2x. Then the participant calculated the corresponding y values. After

that, she constructed a table of numerical values that corresponded to the function x - 2x.

2

2

For the three functions: x - 2x + 1, 1 - x , and x - 2x, it appeared that the

participant substituted specific values for the variables x and y into the equations as part

of the completion of task # 3. Furthermore, the student translated from one

representation of a function to another representation. Specifically, VL translated from

equations (symbolic form) to tables of numerical values (numeric form) using x - 2x + 1,

1 - x 2 , and x - 2x. As a result, O’Callaghan’s (1998) translating component for

understanding functions seemed to be depicted.

2 2 2

2

2

The student did not create a table of numerical values for the function x + 1.

Instead, the participant chose the corresponding graph of x + 1 by process of

elimination. “And from process of elimination my last graph is gonna be [Roman

numeral] I [writes Roman numeral I on task # 3 sheet] and that function is correspondent

with x + 1” (Figure 4.7).

2

2

2

55

Figure 4.7: Visualizer’s completed Mathematical Task # 3

At this point, she had already matched the graphs labeled as II, III, and IV with their

corresponding symbolic forms. Graph I was the only graph left to choose from and

x 2 + 1 was the only equation left to choose from.

In the third mathematical task, after sketching four separate graphs labeled as

Roman numerals I, II, III, and IV, VL explained: “Now, I’m going back to the function

[referring to symbolic forms on task # 3 sheet]. I’m gonna plug in different numbers so I

can see a correspondence to the functions with the graphs – what the graphs are suppose

to look like”. In the beginning of the third mathematical task, the participant examined

the given graphs of the four functions and sketched the four graphs separately. VL

reported the reason she drew the graphs of the four functions individually as being

“because I can’t see the graphs on here [referring to Figure 4.5] but, I need to see the

graph by themselves to make sure I know what the points actually are”. After that, the

student referred back to the symbolic forms of each function. For instance, the student

reported, “ So, I’m gonna put as my answer for 1 - x , I’m gonna put graph four [writes

IV on task # 3 sheet] as my corresponding graph”. In addition, the participant explained

regarding the symbolic form of x - 2x, “So, I’m gonna match my third function with

graph II [writes II on task #3 sheet]”. By substituting specific numerical values into the

symbolic forms and plotting points on graphs, it appeared VL looked for a relationship

2

2

56

between the four symbolic and graphic forms of the functions as part of the completion of

task three.

During the completion of mathematical task # 3, VL responded:

[picks up the graphing calculator] For one, that’s gonna be one squared [presses

the x key] minus two times one plus one. And that’s zero. For negative one, I

plug it into the calculator. Negative one squared [presses x key] times negative

one [uses parentheses] plus one that gives me two and the last number I plug in is

two. Two squared [presses x key] minus two times two plus one is one. So,

then I would look at the graphs and see which one most corresponds to the points

that I just found.

2

2

2

In this excerpt on the interview session, the participant substituted the values one,

negative one, and two for the variable x into the equation x - 2x + 1 using the TI-83 and

calculated the corresponding y-values. Specifically, the student used the x , subtraction,

addition, multiplication, negative number sign, left parenthesis, right parenthesis, enter,

and numerical buttons on the graphing calculator. After calculating the numerical values,

VL looked for a correspondence between the x and y values and the graphs in task three.

For x - 2x, the student explained:

2

2

2

When I plug in negative one into my function, I’m gonna get positive one minus,

minus positive one minus two times negative one [writes 1 – 2 = -1]. That gives

me one plus two minus three. Let’s check that on the calculator [using the TI-83]

one [presses x key] minus two times negative one is a positive three. That’s

correct and two [presses x key] minus two times two is zero. So, as I said

before, we’re gonna check we know already used graph three and graph four. So,

the only ones left are one and two. So, I’m gonna look at the points and look at

my graph that I already have [referring to graphs I and II].

2

2

In this excerpt of the interview session, VL substituted the value negative one for the

variable x into the equation x - 2x and calculated the corresponding y-value. At this

point, she had already calculated the corresponding y-values of the function when x

equaled zero and one. The participant performed the calculations by writing on paper

and using the graphing calculator. Specifically, she used the x , subtraction,

multiplication, negative number sign, enter, and numerical buttons on the graphing

calculator. After calculating the numerical values, VL looked for a correspondence

between the x and y values and graphs I and II in task three.

2

2

In the two previous excerpts, it appeared the participant used the graphing

calculator for arithmetical operations as part of the completion of mathematical task #3.

Specifically, she calculated the numerical values of algebraic expressions. The student

also compared the corresponding graphical values in order to complete the third

mathematical task. This process of how VL used the graphing calculator seemed to

depict Ruthven’s (1990) Numeric-Trial Approach.

Overall, O’Callaghan’s (1998) translating component for understanding functions

was present during the completion of this task. The student translated the four given

symbolic forms of functions to their given graphic forms using the graphing calculator.

Therefore, the participant used the following three categories during the

completion of mathematical task # 3. Category I was substituting specific values for the

variables x and y into equations. Category II was looking for a relationship between the

57

symbolic form and graphic form of a function. Category III was using the TI-83

graphing calculator for arithmetical operations.

During the completion of task three, the participant used non-visual methods to

solve the task. The student substituted specific values for x and y into equations. She

also used the TI-83 graphing calculator to translate the functions.

In addition, the student shared her personal viewpoint about this task in the

College Algebra Journal. “I didn’t have any questions nor did I think it was difficult in

any way. I believe that I understood the concept of this problem and therefore

understood what it was asking and how to answer it”.

Task # 4

Mathematical task # 4 was a quadratic function (Figure 4.8) by Ruthven (1990).


The directions were for the participant to complete the task using the graphing calculator

and explain her reasoning. Using the graphing calculator was a requirement during the

completion of this task because the researcher wanted to see how VL would use it. Task

four was chosen to see how the student would solve a second task involving quadratic

functions. In addition, the researcher wanted to know if VL would use the same

categories from task three in the completion of task four.

In the fourth mathematical task, the graphic form of a quadratic function was

given and the participant was asked to construct its symbolic form. During the

completion of this task, VL explained:

First, I’m gonna [sic] get my x and y points [constructs a table of values] and I’m

gonna use points that are actually on the graph specifically the x and y values that

are labeled [examines the graph on the task # 4 sheet]. For my x values I’m

gonna use negative one. I’m gonna use one and I’m gonna use negative two and

positive two. I’m gonna know if this, this equation, this function [referring to y =

x 2 + 1 ] is actually true when I plug in these x’s into this equation and I actually

58

get the corresponding y answers from this graph [points to graph on task # 4

sheet].

After the student interpreted the given graph in mathematical task four as a parabola

earlier in the interview, she constructed a table of numerical values (Figure 4.9). VL

reported that she created this table because the student wanted to know if the equation

y = x + 1 corresponded to the graph provided in task # 4. The participant chose

negative two, negative one, zero, one and two as x-values for the table from the

numerical values located on the x-axis from the task four graph. Then she calculated the

y-values by substituting the x-values into the equation y = x + 1 and recorded the

corresponding y-values in the table.

2

2

In response to task # 4, VL also reported:

So, if we try and change the function x squared to negative x squared plus one,

hopefully, that will give us the right graph. So, I’m gonna try and implement that

into the third graph (writes y = -x + 1 on paper). I’m gonna use the same points

again (makes a table of values for y = -x +1) and I’m gonna plug them in.

2

2

At this point, the student was attempting to construct the symbolic form of a function that

matched the graphic form provided in task four. VL tried -x + 1 as a possible function.

The participant substituted the values negative two, negative one, zero, one, and two for

the variable x in the equation y = -x + 1. Then, the student calculated the

corresponding y-values and recorded the values in the table (Figure 4.10). VL continued

to substitute numerical values into y = x - 1 and y = -x - 1 (Figure 4.9).

2

2

2 2

For the expressions y = x + 1, y = -x + 1, y = x - 1, and y = -x - 1, it

appeared that the participant substituted specific values for the variable x and y into the

equations as part of the completion of task # 4. Furthermore, the student translated from

one representation of a function to another representation. Specifically, VL translated

from equations (symbolic form) to tables of numerical values (numeric form). As a

result, O’Callaghan’s (1998) translating component appeared to be present.

2 2 2 2

In the beginning of the interview regarding task four and after being given the

graphic form of a function, the participant explained: “It’s a parabola [writes parabola on

paper] and because, I would state that parabolas are usually the function is usually x

(squared) [writes parabola = x on paper] and because the answer y = in this case it

would be [pause] I want to say x squared [pauses] plus 1 [writes y = x + 1 on paper].

2

2

2

VL examined the given graph in task # 4 and stated: “so I believe by just looking at the

graph, I can see that it is a parabola”. Then the student explicated that the general

symbolic form of parabolas was y = x . Her experience in College Algebra introduced

the participant to the numeric, symbolic, and graphic forms of quadratic functions. First,

VL made a connection between y = x (symbolic form) and a parabola (graphic form).

Next, the student attempted to relate the parabola (graphic form) provided in task four

with y = x 2 + 1 (symbolic form).

2

2

59

Figure 4.9: Visualizer’s table of numerical values and graphs for Mathematical Task # 4

60

Figure 4.10: Visualizer’s second page of a table of numerical values and graphs for

Mathematical Task # 4

61

To construct the graph of y - x + 1 by hand, VL explained: 2

And graph the points so, [plots the point] negative one, one [plots the point]

negative two, two on y-axis [constructing graph by numbering the x and y axes]

one, two, three, four, five, negative one, negative two, negative three, negative

four, and negative five. And looking at my equation and my points, I don’t think,

I think I have the wrong y-intercept [circles +1 from x 2 + 1].

After creating a table of numerical values, the participant plotted the points (-1, 2), (1, 2),

(-2, 5), (2, 5), and (0, 1). The participant verified the calculations that the y-intercept of

y = x + 1 equaled one. Then she connected the points with a curve in the shape of a

parabola that opened upward (Figure 4.9). After that, the student compared her

constructed graph with the given graph in task four. Then VL concluded, “and as the

graph shows, it’s the opposite way, so I’m gonna change this formula”. Furthermore, the

student observed “the vertex [of y = x + 1] is correct but the graph [of y = x 2 + 1] is

going in the wrong direction [compared to graph given in task four]. It’s suppose to be

going down”. The participant also constructed the symbolic and graphic forms of

y = x - 1 (Figure 4.9), y = -x 2 , and y = -x + 1 (Figure 4.10) during the interview.

2

2

2 2

For the expressions y = x , y = x + 1, y = x - 1, y = -x , and y = -x + 1, it

appeared that VL looked for a relationship between the symbolic form and the graphic

form of a function as part of the completion of task # 4. Using each of these expressions,

the participant was translating from one representation of a function to another

representation. Specifically, the student translated from tables of numerical values

(numeric form) to graphs (graphic form) for y = x 2 + 1, y = x - 1, and y = -x 2 + 1. The

participant translated from equations (symbolic form) to graphs (graphic form) for y = x 2

and y = -x 2 . As a result, O’Callaghan’s (1998) translating component for understanding

functions appeared to be depicted.

2 2 2 2 2

2

During the completion of mathematical task # 4 regarding the numeric form of

y = x + 1, VL expressed: 2

So, the first one is gonna be negative one squared plus one [writes -1 + 1 =],

typing that into the calculator is negative one, I’m gonna put parentheses just in

case [inputs (-1) 2 in graphing calculator] negative one squared plus one equals

two. So, I place that there [referring to table of values under -1] I’m gonna plug

in one [using the graphing calculator] squared plus one. I know that of values

under -1] I’m gonna plug in one [using the graphing calculator] squared plus one.

I know that that’s two, but just in case so go ahead [enters 1 + 2 x + 1]. Also,

gonna plug in negative two squared plus one equals parentheses negative two

squared plus one equals five, positive five. And I’m gonna do the same thing to

two times, two squared plus one is [enters 2 + 1 in the graphing calculator and

writes on paper] is five.

2

2

2

In this excerpt of the interview session, the student substituted the values negative one,

one, negative two, and two for the variable x into the equation y = x 2 + 1 and calculated

the corresponding y – values. She also substituted zero for the variable x into y = x + 1

and calculated the corresponding y – values later in the interview. The participant

performed the calculations by writing on paper and using the graphing calculator.

Specifically, the student used the x , addition, negative number sign, left parenthesis,

2

2

62

right parenthesis, enter, and numerical keys on the graphing calculator. After calculating

the numerical values, VL sketched the graph of y = x 2 + 1. Then the participant

compared the graph of y = x 2 + 1 with the graph given in task four. The student also

substituted specific numerical values for the variable x into y = x - 1 and y = - x - 1

and calculated the corresponding y – values using the graphing calculator. After

calculating the numerical values, VL looked for a correspondence between the x and y

values and the graph provided in task four. She continued in this trial and error process

using symbolic expressions and reported the numeric form of y = - x + 1:

2 2

2

I’m going to plug some points in and see exactly where our points fall [constructs

a table of values] negative one, one, zero, negative two and two for calculations

plug those in negative one plus one equals [writes – (-1)squared + 1 =] [using the

graphing calculator enters -1 + 1] equals zero that’s negative one plus one.

That’s that point right there (plots -1, 0) on graph -1 + 1 = [using the graphing

calculator enters (-1) 2 + 1] equals two. Negative one, zero, -0 + 1 = 1, -

2 2 actually that’s -4 + 1 = -3 [writes corresponding numbers in table of values].

2

2

2

In this excerpt of the interview session, the student substituted the values negative one,

one, zero, negative two, and two for the variable x into the equation y = - x + 1 and

calculated the corresponding y – values. The participant performed the calculations by

writing on paper and using the graphing calculator. Specifically, VL used the x ,

addition, negative number sign, left parenthesis, right parenthesis, enter, and numerical

keys on the graphing calculator. After calculating the numerical values, the student

sketched the graph of y = - x + 1. Then she compared the graph of y = - x + 1 with

the graph given in task four.

2

2

2 2

For y = x + 1, y = x - 1, y = - x 2 - 1, and y = - x + 1, it appeared the

participant used the graphing calculator for arithmetical operations as part of the

completion of mathematical task # 4. Specifically, the student calculated the numerical

values of algebraic expressions. She also compared the corresponding graphical values

in order to complete the fourth mathematical task. This process of how VL used the

graphing calculator seems to depict Ruthven’s (1990) Numeric-Trial Approach.

2 2 2

During the completion of mathematical task # 4, the participant reported:

I’m gonna [sic] put in [presses y = button and x 2 button] x squared first and I’m

gonna graph that. So we get, with x squared [looks at the graphing calculator

screen and sketches on paper a parabola that opens upward on paper]. That the

parabola, it hits zero, zero (0, 0) that’s the vertex. So, let’s try graphing [using the

graphing calculator ] let’s try graphing negative x squared. Okay, that’s what we

need. So, negative x squared. This is x squared [writes y = x beside previously

sketched parabola] y = x squared [looks at the graphing calculator screen and

sketches on paper a parabola that opens downward] [says and writes] y = -x is

right here. The vertex is, equals zero, zero, but we need the vertex to equal

positive 1 [writes a 1 on the y-axis of the graph y = -x squared] and, how do we

do that? Let’s go back and graph again.

2

2

At this point, VL tried to match the equations y = x 2 + 1, y = x - 1, y = - x - 1, and y

= - x + 1 with the graph provided in task four. After making arithmetic errors for y = -

x 2 - 1 and y = - x + 1, the student concluded, “We still haven’t gotten what we need.

2 2

2

2

63

Okay, I’m gonna try graphing it on the calculator instead [referring to the graphing

calculator]”. First, the participant used the graphing calculator to examine and verify the

correspondence of y = x 2 (symbolic form) with the graph of y = x (graphic form).

Recall from the beginning of task four VL said, “parabolas are usually, the function is

usually x ”. Thus, the student used the calculator to verify what she thought about the

symbolic and graphic forms of y = x 2 . Next, the participant used the graphing calculator

to examine the graphic form of y = - x . Specifically, the student used the Y = ,

ALPHA, STO, x , negative number sign, left arrow, DEL, GRAPH, and TRACE buttons

on the graphing calculator. When VL pressed the TRACE key for y = x , the cursor

flashed on the point (0, 0) on the vertex and displayed at the top of the screen Y1 = x .

The graphing calculator showed X = 0 and Y = 0 on the bottom of the screen.

Similarly, when the participant pressed the TRACE key for Y = - x , the cursor flashed

on the point (0, 0) on the vertex and displayed at the top of the screen Y1 = - x . The TI-

83 showed X = 0 and Y = 0 on the bottom of the screen.

2

2

2

2

2

2

2

2

Next, VL used the graphing calculator for y = x + 1 and y = - x + 1. 2 2

Okay, so it let’s me graph x + 1 [referring to the graphing calculator]. It let’s me

graph x, y = x 2 + 1 [sketches on paper a parabola that opens upwards] and that

gives me the vertex is when x = 0, y = 1. So, this is where we need to be

[sketches a parabola that opens downward with vertex at (0, 1) and says and

writes on paper] y = - x [pauses] + 1 [using the graphing calculator]. Let’s

check that on the graph again. I’m gonna clear these [presses the DELETE and

arrow keys to delete x + 1 from the y = menu, – x + 1, I’m gonna try to graph

that [uses negative sign key this time – x + 1] and that’s our answer based on the

calculator.

2

2

2 2

2

This time the student used the graphing calculator to examine the graphic forms of

y = x + 1 and y = - x + 1. Specifically, she used the Y =, ALPHA, STO, x , negative

number sign, left arrow, DEL, addition, numerical, GRAPH, and TRACE buttons on the

graphing calculator. When the participant pressed the TRACE key for y = x + 1, the

cursor flashed on the point (0, 1) on the vertex and displayed on the top of the screen Y1

= x + 1. The graphing calculator showed X = 0 and Y = 1 on the bottom of the screen.

Similarly, when the student pressed the TRACE key for y = - x 2 + 1, the cursor flashed

on the point (0, 1) on the vertex and displayed at the top of the screen Y1 = - x + 1. The

graphing calculator showed X = 0 and Y = 1 on the bottom of the screen.

2 2 2

2

2

2

For y = x , y = - x , y = x + 1, and y = - x + 1 it appeared the participant used

the graphing calculator to construct a relationship between the symbolic form and graphic

form of a function as part of the completion of task # 4. The participant also used the

connections between the symbolic and graphic forms to construct the exact symbolic

form of a function to complete task four. The Analytic-Construction approach (Ruthven,

1990) seemed to be shown through how VL used the graphing calculator.

2 2 2 2

Therefore, the student used the following four categories during the completion of


y into equations. Category II was looking for a relationship between the symbolic form

and graphic form of a function. Category III was using the graphing calculator for

64

arithmetical operations. Category IV was using the graphing calculator to construct a

relationship between the symbolic form and graphic form of a function.



imagery regarding the quadratic function y = x . VL’s use of the graphing calculator

confirmed her image and the participant continued to complete the task. In addition, the

participant shared her personal viewpoint of task four.

2

I eventually figured out how to complete this task but it was not as easy as the

task that I had presently completed. For some reason, I just could not think of

how to solve it at first but after thinking about it and calculating the problem that

was given to me, I remembered what I was suppose to be doing.

Task #5

Mathematical task # 5 was a cubic function (Figure 4.11) by Ruthven (1990).




completion of this task because the researcher wanted to see how the student would use

it. Task five was chosen because the researcher wanted to know how VL would solve a

task involving cubic functions.

In response to mathematical task # 5, the participant explained:

So, I’m gonna [sic] take the graphing calculator. And I’m gonna, basically, uh,

look at my graph and try and figure out what expression that will express this

graph. First of all, I know it’s, uh, it’s x to the third [writes on task sheet x ]

because it’s not a parabola, whereas a parabola would be x squared. It’s gonna

3

65

be, let’s see [inputting x ^3 on the graphing calculator y = screen] x to the third,

positive x to the third. Let’s see what that graphs. [examining graph of x on the

graphing calculator] Okay, so we know it’s positive x to the third because of the

graph.

3

VL’s idea was confirmed after she saw the graph of y = x 3 on the graphing calculator

graphing screen. Specifically, the participant used the Y =, ALPHA, STO, ^, numerical,

and GRAPH buttons on the calculator.

Next, the student stated:

If it was negative x to the third it would give you something else. And let me see

[repeats] let me see what that’s gonna give us [entering - x in the graphing

calculator] negative x to the third. Let’s graph that. It would give us the other

way around because –x to the third would’ve been this way. So, because of

looking at the graph [referring to task # 5] we know it’s a positive x to the third.

3

After viewing the graph of y = - x on the graphing calculator, the participant confirmed

(1) the graphs of y = - x and y = x were different and (2) what the graph of y = - x 3

looked like. Specifically, the student used the Y =, negative number sign, ALPA, STO,

^, numerical, and GRAPH buttons on the calculator.

3

3 3

For y = x and y = - x 3 , it appeared that the student looked for a relationship

between the symbolic form of a function and the graphic form as part of the completion

of task five. For y = x and y = - x 3 it also appeared that the participant used the

graphing calculator to construct a relationship between the symbolic form of a function

and the graphic form as part of the completion of task five. O’Callaghan’s (1998)

translating component for understanding functions was depicted because the student

translated from one representation of a function (symbolic form) to another

representation (graphic form). In addition, Ruthven’s (1990) Graphic-Trial Approach

was displayed by how VL used the graphing calculator for y = x and y = -x . First, the

student examined the graphs of y = x and y = - x on the graphing calculator. Then

she compared the graphs of the two equations with the given graph in task five and made

conclusions.

3

3

3 3

3 3

In response to mathematical task # 5, the participant reported: “Um, let’s see

negative two, uh, plus one [writes y = (x 3 - 2) + 1 on task # 5 sheet]. Now, let’s graph

that [inputs y = x ^ 3 – 2) + 1 in TI-83]. Basically, I’m just guessing by looking actually

looking at the graph, then, uh, looking at my points [referring to x – intercepts] to see if it

looks similar to the given graph [in task five]”. At this point, VL was trying to determine

the exact symbolic form of the graph provided in task # 5. Earlier during the interview

the student located the x - intercepts from the graph given in task five and stated, “and

then we can find, we know this is (-2, 0), we know this is (0, 0), and we know this is (1,

0)”. So, according to the participant, she constructed y = (x - 2) + 1 from thinking the

graph in task five was a cubic function, using the x – intercept values, and guessing.

Then VL used the graphing calculator to display the graph of y = (x 3 - 2) + 1. After that,

she compared the graph of y = (x 3 - 2) + 1 with the graph provided in task five.

Specifically, the student used the Y =, left parenthesis, right parenthesis, ALPHA, STO,

^, numerical, subtraction, addition, and GRAPH keys. In addition, for the first time VL

mentioned using the TRACE feature when x equaled negative two to compare the graphs.

“Let’s trace and see how close we are when x is negative two, uh, with this equation

3

66

[referring to y = (x 3 - 2) + 1], y is negative ten. So, we’re way off”. When the student

pressed TRACE, the cursor flashed on the point (0, -1). At the top of the graphing

calculator screen Y1 = (X ^ 3 – 2) + 1 appeared and X = 0 and Y = -1 displayed on the

bottom of the screen. Then she used the left arrow key to move the cursor along the

curve of the graph until X = -2.12766 and Y = -10.63178 were shown on the bottom of

the graphing calculator screen.

After trying y = (x - 2) + 1, the participant suggested, “Let’s try making this

positive two and negative one [erases and writes on task sheet (x 3 + 2) – 1] plug that in, x

cubed plus two minus one [enters in the graphing calculator y = (x ^ 3 + 2) – 1]. Let’s

graph that and hit TRACE and let’s see what negative two equals”. At this point, the

participant was continuing to search for a relationship between the given graph in task #5

and a precise equation that would match the graph. The student explained the reason for

choosing the specific cubic equation y = (x + 2) -1 as, “basically, I’m just guessing by

looking actually looking at the graph” (referring to graph in task #5). After graphing the

equation on the graphing calculator VL used the TRACE feature. When the student

pressed TRACE, the cursor flashed on the point (0, 1). At the top of the graphing

calculator screen Y1 = (X ^ 3 + 2) - 1 appeared and X = 0 and Y = 1 displayed on the

bottom of the screen. Then she used the left arrow button to move the cursor along the

curve of the graph until X = -2.12766 and Y = -8.63177 were shown on the bottom of the

screen. Specifically, the participant used the Y =, DEL, addition, subtraction, GRAPH,

and TRACE buttons on the graphing calculator.

3

3

For y = (x -2) + 1 and y = (x 3 + 2) – 1, it appeared that VL used various features

of the graphing calculator as part of the completion of task five. For example, the student

analyzed the graphs of y = (x 3 - 2) + 1 and y = (x + 2) – 1 using the TRACE feature and

arrow keys.

3

3

Overall, the student did not construct an accurate equation to match the graph of

the function that was provided in mathematical task #5. This showed the absence of

O’Callaghan’s (1998) translating component in the final solution of task five because the

participant did not translate the given graphic form of the function in task five to its

symbolic form. VL attempted to obtain the symbolic form of the function in task five by

examining the graphs of y = x 3 , y = -x , y = (x 3 - 2) + 1, y = (x + 2) – 1, and y = x - 2

using the graphing calculator. She did not construct numerical tables. Then the

participant chose y = (x + 2) – 1 as the final solution. VL explained why she chose this

solution.

3 3 3

3

I’m not positive if that’s right, but that’s what I’m gonna stick with, I guess. X

cubed plus two in brackets minus one, just by looking at the graph even though

the graph and calculator doesn’t graph that but that’s what I’m gonna go with for

my answer.

Therefore, the student used the following three categories during the completion

of mathematical task # 5. Category I was looking for a relationship between the symbolic

form of a function and the graphic form. Category II was using the graphing calculator to

construct a relationship between the symbolic form of a function and the graphic form.

Category III was using various features of the graphing calculator.

In addition, the participant shared her personal viewpoint about this task in the College

Algebra Journal.

67

I simply took the problem and separated it to different parts as to see it more

clearly….In problems like these I have to see it by itself in able to work it out or

else I become confused. After completing this task, I truly believe that – I’m a

visual learner.

Task # 6

Mathematical task # 6 was a cubic function by Eisenberg and Dreyfus (1994)

(Figure 4.12).


The directions were to complete the task using the graphing calculator and explain your

reasoning. Using the graphing calculator was a requirement during the completion of this

task because the researcher wanted to see how the student would use it. Task six was

chosen because the researcher wanted to know how VL would solve a second task

involving cubic functions.

In response to mathematical task # 6, the student reported:

So first of all, I’m gonna plug in g of x for my f of x + 3. So, it says, this is the

original function [writes on blank paper (f (x) = x – 3x in Figure 4.13] and my 3 2

68

g of x is equaled to f times x + 3 [writes g (x) = f (x + 3) in Figure 4.13]. So as

this indicates I’m gonna replace all my x’s with x + 3.

The participant substituted the expression (x + 3) for x into the equation f (x) = x 3 - 3x

(Figure 4.13) because the task defined g(x) = f (x + 3). After making the substitution, the

student attempted to simplify the expression f (x + 3) (Figure 4.13).

2

Figure 4.13: Visualizer’s algebraic calculations of Mathematical Task # 6

69

VL did not multiply (x +3)(x + 3)(x + 3) accurately. She wrote (x + 3)(x + 3)(x + 3) =

x 2 + x 2 + 3x + 9 + 3x + 9 = 2 x 2 + 6x + 18. From that error, the participant stated,

“Our final answer when plugging in g of x = f times (x + 3) is negative x + 24x + 45.

And when we plug in g of negative 2 into the original equation, we get negative 20”.

Then the student substituted negative two into the function f(x) = x - 3x .

2

3 2

During the completion of task six, VL used the graphing calculator. “We add our

18x + 6x, 18 + 6 [inputs 18 + 6 in the graphing calculator] equals positive 24x. And then

we would add our 27 plus our 18 [inputs 27 + 18 in the graphing calculator] to equal 45”.

In this excerpt of the interview, the student used the calculator to add numerical values

when she simplified 2 x + 6x + 18 - 3 x + 18x + 27 (Figure 4.13). 2 2

Overall, the participant did not appear to make the connection that since task six

began with a cubic function, f(x) = x 3 - 3 x , a horizontal translation would shift the

graph of f(x) = x 3 - 3 x by three units. In addition, O’Callaghan’s (1998) translating

component for understanding functions was not depicted in task #6 because the student

did not successfully translate one representation of a function to another representation.

Specifically, VL did not accurately translate the symbolic form of g(x) = f (x + 3) to its

graphic form. This translation would have helped the student to calculate g (-2) because

-2 was the x – value and she would only need to find the corresponding y – value. In

addition, none of Ruthven’s (1990) role of graphing calculator approaches was depicted

in this task because the participant used the graphing calculator only to perform

arithmetical calculations that were unrelated to values on the corresponding graph.

2

2

The participant shared her personal viewpoint of this task:

This problem seemed to be a little difficult. I kept getting the wrong answer. I

did not understand why my answer would not correspond to the information given

in the problem. I kept trying different equations to show which one was the

correct graph. Eventually, I stuck to what I had gotten at the beginning of the

calculations even though I believe that it was actually wrong.

Task # 7

Mathematical task # 7 was another cubic function (Figure 4.14). The directions

were for the participant to complete the task using the graphing calculator and explain her


task because the researcher wanted to see how VL would use it. Task seven was chosen

because the researcher wanted to know how the student would solve a third task


At the beginning of task # 7, the student described how she would determine a

method of solution. “As I look at this problem the first thing I would do, or the easiest

thing I would do because I have access to a calculator is just graph it on the calculator.”

Then, the participant proceeded by inputting the given symbolic form of the function,

f(x) = -3x (x – 1) (x -2), into the graphing calculator.

70


So I go to my x my y plots and I'm gonna type exactly what I see on the paper

[referring to task # 7 sheet, note: VL called parentheses brackets]. I'm gonna put

negative 3x brackets x minus 1 bracket open bracket x minus 2 bracket. I'm

gonna press graph. I'm gonna observe the graph and from this graph [pauses]

from this graph I can tell by looking at the calculator I press trace and you can tell

exactly where the different distinct points are.

71

VL took the provided symbolic form of the function, f(x) = -3x (x – 1) (x – 2), and used

the graphing calculator to translate it into the graphic form which showed the presence of

O’Callaghan’s (1998) translating component for understanding functions. As part of the

translation process, the student used the following features of the graphing calculator: the

Y =, negative number sign, numerical, ALPHA, STO, left parenthesis, right parenthesis,

subtraction, GRAPH, and TRACE keys. She inputted the equation (symbolic form) of

the function using the Y = button. The participant saw the graphic form of the function

using the GRAPH key. In addition, VL analyzed specific features of the graph using the

TRACE button. In this excerpt of the interview, it appeared that the student looked for a

relationship between the symbolic form of a function and the graphic form as part of the

completion of the seventh mathematical task. It also appeared that the participant used

the graphing calculator to construct a relationship between the symbolic form of a

function and the graphic form as part of the completion of this task. In addition, from

this excerpt, it appeared that VL used various features of the graphing calculator as part

of the completion of task seven.

In response to mathematical task # 7, the student explained:

So the graph , the calculator actually gives me the midpoint which is zero, zero

[plots the point (0, 0) on the grid on task # 7 sheet] and then I can actually trace

with my calculator and find other distinctive points that would be relevant to me.

And by looking I can tell that when x = 1, y = 0. So, I'm gonna put that point also

[plots the point (1, 0) on the grid of the task # 7 sheet]

In this excerpt of the interview, the student used the TRACE key on the graphing

calculator to locate the point (0,0). When she pressed TRACE, the equation Y 1 = -3X (X

– 1) (X – 2) appeared above the graph of the cubic function and X = 0, Y = 0 appeared

below the graph on the bottom of the screen. After that, the participant used the right

arrow button to move along the curve of the graph to locate another x-intercept, the point

(1, 0). She referred to the movement of the points along the graph as “trace with my

calculator.” As the student kept pressing the right arrow key, the equation Y 1 = -3x (x –

1) (x – 2) remained above the graph and the X and Y values displayed at the bottom

changed. The X and Y values corresponded to the points on the graph of the function.

After using the TRACE feature, the participant plotted the points (0, 0) and (1, 0) on the

grid provided in task seven (Figure 4.15). She continued using the TRACE and arrow

keys to locate and plot the points (0.5, - 1), (2.5, - 6.5), and (1.7, 1) on the graph (Figure

4.15). It appeared that VL used various features of the graphing calculator as part of the

completion of task seven. It also appeared that the student plotted specific points of a

function on a graph as part of the completion of this task.

In response to the seventh mathematical task, VL also stated: “So it [referring to

the curved shape of the graph] goes up, then it goes down [connecting the points on the

graph]. And now I have to do my negative x-axis. So I'm going to continue tracing and

I'm gonna use the point at when x is - 1, y is 20”. The participant used the TRACE key to

locate the point (-1, 20), which was in the second quadrant. This time VL pushed the left

arrow button, while the TRACE feature was still on, and located the following

corresponding values of X = -1.06383 and Y = 20.180499.

72


The student rounded these values and plotted (-1, 20) on the coordinate grid provided in

task # 7 (Figure 4.15). She also connected this point with the other points in order to

complete the graph in the seventh mathematical task. VL explicated the reason for

choosing points as being, “after graphing it [referring to the function f (x) = -3x (x – 1) (x

– 2)] on the calculator I went back and traced the graph found main points and plotted it

on my graph to get the general look of the answer.” O’Callaghan’s (1998) translating

component seemed to be depicted because the participant translated from the symbolic

form of a function to its numeric form. In addition, it appeared that the student used

various features of the graphing calculator as part of the completion of task seven. It also

appeared that the participant plotted specific points of a function on a graph as part of the

completion of this task.

Therefore, VL used the following four categories during the completion of

mathematical task # 7. Category I was looking for a relationship between the symbolic



Category III was using various features of the graphing calculator. Category IV was

plotting specific points of a function on a graph.

73

Overall, the presence of O’Callaghan’s (1998) translating component for

understanding functions was shown by how the participant translated the given symbolic

form of f(x) = -3x (x-1) (x-2) to its graphic form using the graphing calculator. None of

Ruthven’s (1990) approaches were depicted by the student’s use of the graphing

calculator in task # 7. She was given the symbolic form of the function in the task. Then

the participant analyzed the graphic and numeric forms of the function using the graphing

calculator.

Task # 8

Mathematical task # 8 was an absolute value function (Figure 4.16) by Ruthven

(1990).





eight was chosen because the researcher wanted to know how the student would solve a

task involving absolute value functions.

In response to the eighth mathematical task, the participant explained: “By

looking at the graph [provided in task # 8] I can tell that this is an absolute function,

therefore, that’s the first thing I’m do with my y = [provided on task # 8 sheet], put my

absolute symbols [draws two vertical lines apart from each other as absolute value bars]

and I know its gonna be x ” [writes x in between the absolute value bars]. First, the

student examined the graph of a function given in task eight. Then, VL concluded that

the task eight graph was an absolute value function. (She specifically referred to it as an

74

“absolute function”). Her experience in College Algebra introduced the participant to the

numeric, symbolic and graphic forms of absolute value functions. College Algebra also

exposed the participant to translations or transformations of graphs. After that, the

student attempted to construct the symbolic form of an absolute value function as

y = x . From this excerpt of the interview, it appeared that VL was looking for a


completion of task eight.

In response to mathematical task # 8, the student reported: “By looking at the

graph I can tell that this is an absolute function therefore, that’s the first thing I’m gonna

do with my y equals put my absolute symbols [draws two straight lines apart from each

other as absolute value bars]”. In this excerpt of the interview, the participant plotted the

point (-1, -2) on the graph and referred to it as the vertex of the function in task eight.

Even though the graph of y = x was not sketched on the task eight sheet, VL reported

that y = x as the original graph of an absolute value function. Thus, in order to get the

picture of the graph in task # 8, the student suggested that from the origin of y = x “I

know I have to move to the left once”. The participant just described a horizontal

translation of the graph of y = x to the left of the origin. Even though the student did

not use the words translation or transformation, she expressed the meaning of these words

in terms the visual movement of a graph. In addition, it appeared that VL was focusing

on specific visual features of the graph of a function as part of the completion of task

eight.

In response to mathematical task # 8, the participant continued after the previous

excerpt:

And, so therefore I’m gonna put positive one and then I need to move down

negative two. So I’m gonna put negative two [beside y = on task #8 sheet, VL

writes x + 1 - 2]. And my reasoning for this is that [writes and says] the original

graph starts the original graph vertex is at zero, zero and the given graph is, has

moved to the left one and down two. And that’s my answer.

At this point, the student explained she would write the final symbolic form of the graph

in task eight as y = x + 1 - 2 (Figure 4.17). The participant constructed y = x + 1 - 2

because she explained that y = x + 1 corresponded to the graph of y = x that “…has

moved to the left one…” or a horizontal translation to the left by one unit. VL also stated

that y = x + 1 - 2 corresponded to the graph of y = x that “…has moved to the left one

and down two” or a horizontal translation and a vertical translation downward by two

units. From this excerpt of the interview, it appeared that the student was looking for a


completion of task # 8. It also appeared that VL was focusing on specific visual features

of the graph of a function as part of the completion of task eight.

75


The participant constructed y = x + 1 - 2 because she explained that y = x + 1

corresponded to the graph of y = x that “…has moved to the left one…” or a horizontal

translation to the left by one unit. VL also stated that y = x + 1 - 2 corresponded to the

graph of y = x that “…has moved to the left one and down two” or a horizontal

translation and a vertical translation downward by two units. From this excerpt of the

interview, it appeared that the student was looking for a relationship between the

symbolic form of a function and the graphic form as part of the completion of task # 8. It

also appeared that VL was focusing on specific visual features of the graph of a function

as part of the completion of task eight.

Overall, O’Callaghan’s (1998) translating component seemed to be depicted in

the eighth mathematical task because the student translated the graphic form of the

function to its symbolic form. The component also appeared to be shown earlier during

the interview when the participant discussed translating the graphic form of y = x to the

symbolic form of y = x . In addition, VL used two categories during the completion of


form of a function and the graphic form. Category II was focusing on specific visual

features of the graph of a function. The student did not use the graphing calculator

during the completion of this task because she was not sure where the absolute value key

was located on the calculator.


imagery to solve the problem. VL relied on visual imagery of absolute value functions

y = x , y = x + 1 , and y = x + 1 - 2. The participant also shared her personal viewpoint

of the task. “It was simple, not at all difficult. I observed the shifts in the original graph

[y = x ] to the present graph [provided in task # 8] and wrote the equation [y = x + 1 - 2]

that I believe the graph illustrates”.

76

Task # 9

Mathematical task # 9 was an exponential function (Figure 4.18) created by the

researcher.





nine was chosen because the researcher wanted to know how the student would solve a

task involving exponential functions.

In the beginning of the interview regarding mathematical task # 9, the participant

stated, “by just looking at the graph given [in task # 9], I can tell that y equals the

standard you know the standard, uh, function is two to the x [writes y = 2 ]”. From the

graphic form of the function provided in task nine, the student attempted to construct the

symbolic form of y = 2 and referred to it as the “standard function”. Her experience in

College Algebra exposed VL to exponential functions. Specifically, the participant was

introduced to the symbolic, numeric, and graphic forms of y = 2 . She reported recalling

the graphic form of y = 2 . In addition, prior to completing this task the, the student

completed a College Algebra (on-line) web homework assignment that dealt with

exponential functions. The computer assignment included the following functions:

x

x

x

x

f(x) = 2 , f(x) = 2 , f(x) = 2 , f(x) = e + 6, f(x) = e + 1, f(x) = e + 3, x−1 x−2 x−3 x x x

77

f(x) = e + 4, f(x) = e + 2, f(x) = e + 6, f(x) = e − + 2, f(x) = e + 5, f(x) = e + 3,

and f(x) = e + 1. The directions for the first problem on the homework were to “start

with one of the basic graphs and sketch the graph of f(x) = 2 ” (Sharma, 2005). The

directions were the same for the other functions. The basic graphs included f(x) = 2 ,

f(x) = e , and f(x) = e which were the primary functions in the problems before the

inclusion of the horizontal and vertical translations. It appeared, from this first excerpt of

the interview, the participant was looking for a relationship between the symbolic form of

a function and the graphic form as part of the completion of task nine.

x x − x x − x − x

− x

x−1

x

x − x

In response to mathematical task # 9, VL explained:

So, but it [the graph of y = 2 ] moves it shifts on the x-axis negative five so I

believe that I would have to [pauses] hum, look at the graph [on TI-83] since it

shifts on the x-axis you would put, hum, it would, okay so I put 2 to the x

negative five [writes y = 2 - 5 on task #9 sheet]. I’m gonna try and graph that

and see what that gives me.

x

x

After writing the symbolic form y = 2 , the student examined the graph provided in task

nine. Then, VL reported a horizontal translation of the graphic form of y = 2 by five

units to the left while she looked at the graph of the function provided in task nine. After

that, the participant suggested y = 2 - 5 would match the given graph of the function in

task nine. In addition, the student planned to verify this assumption by graphing

x

x

x

y = 2 - 5 on the TI-83 graphing calculator. It appeared that VL was focusing on specific

visual features of a graph as part of the completion of task # 9. It also appeared that the

participant was looking for a relationship between the symbolic form of a function and

the graphic form as part of the completion of task nine.

x

The student continued with mathematical task # 9 and stated:

[enters y = 2 ^ x – 5 on the graphing calculator)] That actually moves me down

on the y-axis. I need to move on the x-axis. So, let’s see, let’s change it from that

to two to the x uh, two to the [enters y = 2^ x - 5 on the graphing calculator where

- is the negative sign]. No, that’s not it. Two to the x plus five [inputs y = 2 ^ x +

5 on the graphing calculator] graph that. That still moves us up on the y-axis and

we need to move on the x-axis. Um, okay let me try this: two to the x plus one

plus five [writes y = (2 + 1) + 5 on task #9 sheet]. I’m gonna graph that: two to

the x plus one plus five [inputs y = (2 ^ x +1) +5 on the graphing calculator].

Okay I’m gonna switch it to this, should be five and this should be one [inputs y =

(2 ^ x + 5) + 1 on the graphing calculator] and graph. Um, okay so, by looking at

the graph I can tell that it moves on the x-axis negative five so I’m gonna put 5x

plus hum plus one [writes -5 + 1]. I’m gonna graph that um, plus one [enters y

= -5 ^ x + 1 on the graphing calculator]. That keeps, okay [writes y = (-5 + 1)

on task #18 sheet].

x

x

x

VL inputted the symbolic form of y = 2 - 5 into the graphing calculator. Then, the

participant viewed the graphic form of y = 2 - 5 on the graphing calculator. After

looking at the graph of y = 2 - 5, she explained that subtracting five from the exponential

function y = 2 corresponded to a downward vertical translation of the graph of y = 2 .

VL referred to the downward vertical translation as “… moves me down on the y – axis”.

Then, the participant reported that the graph in task # 9 was a horizontal translation of y =

x

x

x

x x

78

2 and described the horizontal translation as “I need to move on the x – axis”. Next, the

student modified the symbolic form of y = 2 - 5 to y = 2 + 5. VL also inputted the

symbolic form of y = 2 + 5 into the graphing calculator. Then, the participant viewed

the graphic form of y = 2 + 5 on the graphing calculator. This time, after looking at the

graph of y = 2 + 5, VL explained adding five to the exponential function y = 2

corresponded to an upward vertical translation of y = 2 . The student referred to the

upward vertical translation as “That still moves us up on the y – axis…”. After that, the

participant reported that the graph in task #9 was a horizontal translation of y = 2 and

stated “…we need to move on the x – axis”. So, the student changed the symbolic form

of y = 2 + 5 to y = (2 + 1) + 5. VL inputted the symbolic form of y = (2 + 1) + 5

into the graphing calculator. Then the participant viewed the graphic form of y = 2 +1)

+ 5 on the graphing calculator. After examining the graph, the student modified the

symbolic form of y = (2 + 1) + 5 to y = (2 + 5) + 1. VL inputted the symbolic form

of y = 2 + 5) + 1 into the graphing calculator. After that, the participant viewed the

graphic form of y = (2 + 5) + 1 on the graphing calculator. After looking at the graph

of y = (2 + 5) + 1, VL examined the graph of the exponential function provided in task

nine. Then she stated, “Um, okay so, by looking at the graph [in task # 9] I can tell that it

moves on the x – axis negative five so I’m gonna put 5x plus, hum, plus one…” . Again,

the student described a horizontal translation five units to the left of the graph of y = 2 .

The participant also suggested changing the symbolic form of y = (2 + 5) + 1 to y = (-

5 + 1). VL inputted the symbolic form of y = (-5 + 1) into the graphing calculator.

Then she viewed the graphic form of y = (-5 + 1) on the graphing calculator.

x

x x

x

x

x x

x

x

x x x

x

x x

x

x

x

x

x

x x

x

For the equations y = 2 - 5, y = 2 + 5, y = (2 + 1) + 5, y = (2 + 5) + 1, and y

= -5 + 1, it appeared that VL looked for a relationship between the symbolic form of a

function and the graphic form as part of the completion of task # 9. Using each of these

equations, the participant was translating from one representation of a function to another

representation. Specifically, the student translated from equations (symbolic form) to

graphs (graphic form) for y = 2 - 5, y = 2 + 5, y = (2 + 1) + 5, y = (2 + 5) + 1, and

y = (-5 + 1). As a result, O’Callaghan’s (1998) translating component for

understanding functions appeared to be depicted.

x x x x

x

x x x x

x

For y = 2 - 5, y = 2 + 5, y = (2 + 1) + 5, y = (2 + 5) + 1, and y = -5 + 1),

VL translated each function from its symbolic form to its graphic form using the graphing

calculator. Specifically, the student used the Y =, ALPHA, STO, negative number sign,

DEL, left arrow, right arrow, ^, right parenthesis, left parenthesis, addition, subtraction,

and GRAPH keys. For y = 2 - 5, y = 2 + 5, y = (2 + 1) + 5,

x x x x x

x x x

y = (2 + 5) + 1, and y = (-5 + 1) it appeared the participant used the graphing

calculator to construct a relationship between the symbolic form and graphic form of a

function as part of the completion of task # 9. It also appeared that the student focused

on specific visual features of the graph of a function as part of the completion of task

nine. The specific visual features included the horizontal and vertical translations

mentioned during the interview.

x x

In response to the ninth mathematical task, VL reported:

79

[after graphing] Oh no. Put outside [deletes parentheses and enters y = -5 ^ x + 1

on the graphing calculator] and graph. Okay, so we know it’s not negative it has

to be a positive [referring to the given graph in task #9] okay so I’m gonna put 2x

plus actually 2x minus five [writes y = (2 – 5) on task #9 sheet] and I believe

that’s the right answer. [enters y = (2 ^ x – 5) ]. Two x trace [pressed the trace

key and the left and right arrow buttons on the TI-83] okay the graph isn’t given

me the right thing, but I believe that it’s two x to the, two to the x minus five just

by looking at the graph and knowing that you move uh, five points to the left on

the x-axis but it hasn’t moved up so it only moved on the x-axis not the y-axis.

x

At this point, the student examined the graphic form of y = (-5 + 1) on the graphing

calculator. Then she deleted the parentheses of y = (-5 + 1) and viewed the graphic

form of y = -5 + 1 on the graphing calculator. After looking at the graph of y = -5 + 1

and the given graph in task nine, the student concluded, “okay, so we know it’s not

negative it has to be a positive [referring to the given graph in task # 9] okay so I’m

gonna put 2x plus, actually 2x minus five [writes y = 2 - 5 on task # 9 sheet]…” . After

examining the graphs, the participant modified the symbolic form of y = -5 + 1 to y =

2 - 5. VL inputted the symbolic form of y = 2 - 5 into the graphing calculator. Then,

VL viewed the graphic form of y = 2 - 5 on the graphing calculator. She pressed the

TRACE key and again looked at the graph provided in task nine. Even though the graph

of y = 2 - 5 and the given graph in task # 9 were not the same, the student chose y = 2 -

5 as her final answer. During the third excerpt of the interview, for y = -5 + 1 and 2 -

5, it appeared that the participant looked for a relationship between the symbolic form of

a function and the graphic form as part of the completion of task nine. Using both of

these equations, the student was translating from one representation of a function to

another representation. Specifically, VL translated from equations (symbolic form) to

graphs (graphic form) for y = -5 + 1 and y = 2 - 5. As a result, O’Callaghan’s (1998)

translating component for understanding functions appeared to be depicted. For

x

x

x x

x

x

x x

x

x x

x x

x x

y = -5 + 1 and y = 2 - 5, it also appeared that the student used the graphing calculator

to construct a relationship between the symbolic form of a function and the graphic form.

x x

Overall, the student did not construct an accurate equation to match the graph of

the function that was provided in mathematical task # 9. This showed the absence of

O’Callaghan’s (1998) translating component in the final solution of task nine because the

participant did not translate the given graphic form of the function in task nine to its

symbolic form. However, the student translated y = 2 - 5, y = 2 + 5, y = (2 + 1) + 5,

y = (2 + 5) + 1, y = (-5 + 1) and y = -5 + 1 using the graphing calculator. She

examined each of these graphs and compared them with the given graph in task nine. As

a result, Ruthven’s (1990) Graphic-Trial Approach appeared to be depicted in task nine

because VL used the graphing calculator to modify the symbolic forms of functions

based on comparing graphic forms of functions with the graphic form of the given

function in task # 9.

x x x

x x x



form of a function and the graphic form. Category II was focusing on specific visual

features of the graph of a function. Category III was using the graphing calculator to


80


imagery to solve the problem. VL relied on visual imagery of the exponential function

y = 2 . In addition, the student shared her personal viewpoint about this task in the

College Algebra Journal.

x

This task was a bit challenging to me. It reminded me of the previous task but I

could not remember how I shift the graph on the x – axis instead of the y – axis.

In the end I just figured that I was putting it into the calculator wrong. After

writing my journal for the last task I did remember that I put the function into the

calculator wrong because I failed to put brackets around the exponential part of

the equation when I entered it into the calculator. This took me a while to figure

out and I did not really figure it out until now.

Task # 10

Mathematical task # 10 was another exponential function (Figure 4.19) created by

the researcher. The directions were for the participant to complete the task using the

graphing calculator and explain her reasoning. Using the graphing calculator was a

requirement during the completion of this task because the researcher wanted to see how

VL would use it. Task # 10 was chosen to see how the student would solve a second task

involving exponential functions. In addition, the researcher wanted to know if the

participant would use the same categories from task nine in the completion of the tenth

mathematical task.

In response to mathematical task # 10, VL stated: “I’m gonna plug, plug that into

the calculator. It says e to the x minus one [inputs y = e ^ (x – 1) into the graphing

calculator]. Press GRAPH and TRACE and when x equals zero, y equals 3.6”. During

this excerpt of the interview, the student inputted the symbolic form of f (x) = e

provided in task # 10 as y = e ^ (x – 1) into the graphing calculator. Next, the student

pressed the GRAPH button and examined the graphic form of y = e ^ (x - 1) on the

calculator. After that, the participant used the TRACE feature. When VL pressed

TRACE, the TI-83 displayed Y1 = e ^ (X – 1) in the upper left hand corner of the screen.

The graphing calculator also highlighted the y – intercept and showed its coordinates X =

0, Y = .36787944 on the bottom of the screen. From this excerpt, it appeared that the

student looked for a relationship between the symbolic form of a function and the graphic

form as part of the completion of this task. It also appeared that VL used the graphing

calculator to construct a relationship between the symbolic form of a function and the

graphic form as part of the completion of this task.

x−1

In response to the tenth mathematical task, the participant reported:

So I’m just gonna estimate. That’s one. This is about point five so under that is

gonna be around point three six [counting on the y-axis]. Then I’m gonna go

along on my x-axis, find another point. When x is one, y is one. I’m gonna plot

that point [plots (1, 1) on the graph on the task #10 sheet], continue along my x-

axis. When x is two, y is three [plots (2, 3) on the graph on the task #10 sheet].

When x is three, nine, y is nine. So this is one, two, three, four, five, six, seven,

eight, nine. Three and nine [plots (3, 9) on the graph on the task #10 sheet] and I

connect my points and that’s the graph.

81


After plotting the point (0, 0.36) on the graph grid provided in task # 10, VL used the

graphing calculator to locate another point. Specifically, under the TRACE feature, she

used the right arrow key to move along the curve of the graph to locate x = 1.0638298

and y = 1.065911. The student estimated these values as “when x is one, y is one”.

Then VL plotted (1, 1) on the graph grid provided in the task. Next, under the TRACE

feature, she used the right arrow key to move along the curve of the graph to locate

82

x = 2.1276596 and y = 3.0884198. The student estimated these values as “when x is two,

y is three”. After that, under the TRACE feature, the participant used the right arrow key

to move along the curve of the graph to locate x = 3.1914894 and y = 8.9485308. The

participant estimated these values as “when x is three, y is nine”. After looking at the

graph of y = e on the graphing calculator, VL connected the points she plotted with a

curve (Figure 4.20).

x−1


83

From this excerpt of the interview, it appeared that the student plotted specific points on a

graph as part of the completion of task # 10. It also appeared that VL used the TRACE

feature of the graphing calculator as part of the completion of task # 10.

Overall, O’Callaghan’s (1998) translating component seemed to be depicted in

the tenth mathematical task because the student translated the symbolic form of the

function to its graphic form. In addition, the participant used the following four

categories during the completion of mathematical task # 10. Category I was looking for a

relationship between the symbolic form of a function and the graphic form. Category II

was using the graphing calculator to construct a relationship between the symbolic form

of a function and the graphic form. Category III was plotting specific points of a function

on a graph. Category IV was using various features of the graphing calculator.


imagery to solve the problem. VL relied on visual imagery of the exponential function

y = e . Furthermore, VL shared her personal viewpoint of this task. “This task allowed

me to use a calculator so I just put in the function into the calculator and observed the

graph. I traced along the graph to find points that looked important and plotted them on

my graph”.

x

In conclusion, the findings of one of the case studies of the two College Algebra

students were reported in this chapter. Several categories emerged from the data. They

were labeled in Chapter 4 using Roman numerals. To summarize, all of the emerging

categories were listed using alphabetical letters with the corresponding mathematical task

or tasks in Table 4.1.

• Category A: substituting specific values for the variables x and y into equations

• Category B: plotting specific points of a function on a graph

• Category C: detecting a relationship between the concepts slope and steepness

• Category D: misinterpreting the graphical representation of a function after

multiplying and adding specific values to the symbolic form of a function

• Category E: looking for a relationship between the symbolic form of a function

and the graphic form

• Category F: using the TI-83 graphing calculator for arithmetical operations

• Category G: using the graphing calculator to construct a relationship between the

symbolic form of a function and the graphic form

• Category H: using various features of the graphing calculator

• Category I: focusing on specific visual features of the graph of a function

84

Table 4.1: VL’s Emerging Categories

CATEGORIES MATHEMATICAL TASKS

A 1, 3, 4

B 1, 7, 10

C 1

D 2

E 3, 4, 5, 7, 8, 9, 10

F 3, 4, 6

G 4, 5, 7, 9, 10

H 5, 7, 10

I 8, 9

85

CHAPTER 5

RESULTS










The findings of the second case study were reported in this chapter.







exponential functions. Furthermore, prior to receiving the tasks, the students’ preference





First, Presmeg’s (1985) Mathematical Processing Instrument included three


section B only or sections B and C for college-level students. The participants in this

study were provided with section B of the instrument. Section B had 12 mathematical

word problems to solve. Each question could be solved numerically, algebraically, and




problem was not attempted, and zero for every non-visual solution. The highest score


the student’s visualization score was 12/24 or higher, then he or she would be considered

as having a preference for visual thinking in mathematics and called a Visualizer (VL).

On the other hand, if the participant’s visualization score was 10/24 or lower, then he or

she would be considered as having a preference for non-visual thinking in mathematics

and called a Nonvisualizer (NVL). The students were required to show their work for the

solutions, however, they were not required to use a specific method of solution over

another. The participants were also asked to choose their own method of solution and

turn in their papers. (See Appendix A for a copy of this instrument.)

Secondly, each student was supplied with a Mathematical Processing Questionnaire

(Presmeg, 1985). The questionnaire was a follow-up to the participants’ responses to the

Mathematical Processing Instrument. This questionnaire provided three or more

solutions for the students to choose the one that is most similar to their response. After

86

the participants completed the questionnaire, they were asked to turn in their responses.

(See Appendix B for a copy of this questionnaire.)

In the present study, the visual and non-visual mathematical learners’

















In the report of Case 2, as patterns emerged from the data the researcher called

them categories. In this chapter, the categories were labeled using Roman numerals.

CASE 2

The Nonvisualizer (NVL) in the current study was 21 year old African-American

male student. He was born on February 23, 1984 in Miami, Florida. The participant was

raised in Miami and has lived there all of his life. It was mainly a Hispanic and

Caucasian community. NVL’s elementary, middle, and high school environment had the

same cultural background since he took advanced and honors classes. The participant

attended a historically black university because he wanted a different cultural experience.

He explained, “I wanted to go where African-Americans were, not just where they were

but where they were busy getting an education”,

NVL completed basic mathematics classes in elementary school. In sixth grade,

he took Pre-Algebra. The participant earned an “A” in this course. During the seventh

grade, NVL completed Algebra I advanced. He earned a “B” in this course. In eighth

grade, the participant took Geometry. NVL earned a “C” in this course. During the ninth

grade, he completed Algebra II honors. The participant earned a “C” in this course. In

tenth grade, NVL took Analysis of functions. He earned a “B” in this course. During the

eleventh grade, the participant completed Calculus I honors, which was his last high

school mathematics class. He earned a “B” in this course.

When the researcher asked NVL how he felt about math, the participant reported

that he enjoyed mathematics in middle school and high school. As a middle school

student, after NVL completed the class assignment, his teacher would let him “play math

games and crosswords on the computer”. In addition, the participant shared his view of

mathematics. “It’s ordered, structured, and consistent. It doesn’t change meaning the

basic principles and rules. It’s logic behind it, not matter what happens in history, it’s

always gonna be there no matter what”. NVL explained that he was referring to the basic

principles and rules that apply to using mathematical operations.

87

During the fall of 2005, the participant took College Algebra for the second time

at the university where the research took place. He had enrolled in the course previously

and failed. After that, NVL took some time off from school and returned in the fall. He

earned an “A” in this course.

When the researcher asked NVL what Algebra meant, he referred to it as having

“actual steps [being] a process that never changes”. The participant called an equation

the “basic building block” of Algebra. In addition, NVL stated that Algebra was his

“favorite portion of math because I felt a great comprehension of it”.

The participant was first introduced to graphing calculator in the eighth grade.

His math teacher provided the calculators for the students to use. When NVL was in the

ninth grade, he purchased a graphing calculator. The participant used the calculator in

high school and college.

Using the graphing calculator during the College Algebra course was optional for

the students. The instructor allowed the students to use graphing calculators in class,

however, it was not mandatory for the students to purchase them. The required

technology component was an Internet software program called EDUCO. EDUCO

offered the students on-line tutorials, quizzes, and Web homework (Sharma et al., 2002).

The participant scored 4/24 on Presmeg’s (1985) visualization instrument. NVL

was a senior Business Administration major, however, toward the end of the semester,

the student changed his major to Accounting. He explained “Accounting has always

been my passion, I like working with numbers”. After graduation, the student planned to

become a Certified Public Accountant (CPA). In addition, his hobbies included

exercising, weight lifting, running, reading about finance and investing, and attending

church. The participant attended class regularly. The student also participated in class

discussions by volunteering to answer questions. In addition, NVL completed most of

the in-class and out-of-class assignments in a timely manner.

Task # 1

Mathematical task # 1 was a linear function (Figure 5.1) by Moschkovich (1998).

The directions were for the participant to complete the task and explain his reasoning.

Task one was chosen in order to see how the student would solve a task involving linear

functions. Using the graphing calculator was optional during the completion of

the task. It was optional because the researcher wanted to see if the participant would

choose to use the graphing calculator. He did.

In response to mathematical task # 1, NVL stated,

Oh, I was just saying. I plugged in the x’s negative one, zero, one, and two, and

so I just plugged it into the equation and so I got –1 + 5 and gives you 4, 0 + 5

gives you 5, 1 + 5 gives you 6, 2 + 5 will give you 7. Um, I guess I did this

because I was trying to look at this visually. I see x you know, on the graph

[referring to the graphical representation of the equation y = x] in my mind. But x

+ 5, I couldn’t really put it up there. So, I used this [pointing to the y = x + 5

table] to give me the coordinates for it. Then I was trying to see if I could look at

these coordinates and see if I could figure out a difference. But then I get -1, 4; 0,

5; 1, 6; and 2, 7. [He wrote the previous ordered pairs as (-1,4) (0,5) (1,6) (2,7)].

Up here the y = x, I get -1, 1, 0, 0, 1, 1 and 2, 2.

88


89

The student substituted the values negative one, zero, one, and two into the equation

y = x. Then, NVL solved the equation in order to find the corresponding y – values. As

the participant substituted these specific x – values into y = x and calculated the

corresponding y – values, he constructed a table of numerical values (Figure 5.2). He

also substituted the values negative one, zero, one, and two into the equation y = x + 5.

After that, the student solved the equation in order to find the corresponding y – values.

As the participant substituted these specific x – values into y = x + 5 and calculated the

corresponding y – values, he constructed a table of numerical values (Figure 5.2).

Figure 5.2: Nonvisualizer’s tables of numerical values

From this excerpt of the interview, it appeared that the participant substituted specific

values for the variables x and y into the equations as part of the completion of task one.

From this excerpt of the interview, it also appeared that O’Callaghan’s (1998) translating

component for understanding functions was present because NVL translated from one

representation of a function to another. Specifically, the student translated from

equations (symbolic form) to tables of numerical values (numeric form).

In response to mathematical task # 1, NVL explained,

So, I was just trying to figure out how these would look [referring to the graphs of

y = x and y = x + 5]. I’m trying to picture in my head. This isn’t looking too

well. So, I was uh, the only thing I can think of is trying to find the slope. Then,

I’m trying to remember the slope formula.

At this point, the student had just constructed the tables of numerical values for the

equations y = x and y = x + 5. From this excerpt, the participant reported that he was

trying to visualize the graphic forms of y = x and y = x + 5 without actually graphing the

equations. The student refrained from graphing y = x and y = x + 5 because initially the

problem posed questions comparing the graphic forms of y = x and y = x + 5 before he

90

was required to graph them. After saying, “this isn’t looking too well,” NVL tried to

recall the formula for the slope of a line.

After the participant admitted that, “I really can’t remember this [slope] formula”,

he continued trying to answer the first question in task one: If you start with the equation

y = x then change it to the equation y = x + 5, what would that do to the graph?”

I guess in this one [pointing to y = x + 5] x always increases by five so you have

your regular line y = x [draws a straight slightly slanted line] and then you have x

always higher by five I guess uh [pauses] I can’t graph it so I don’t know how it

will look. I want to say this [draws a line that is higher and parallel to the one

drawn previously with a little arrow sketched between the two lines]. So, I guess

that means it shifts upward. So, for my first answer is that the graph shifts

upward.

First, the student expressed from the equation, y = x + 5 that “… x always increases by

five …”. Then the participant drew a line (Figure 5.3) and referred to it as y = x.

Figure 5.3: Nonvisualizer’s y = x line

For y = x + 5, NVL reported “… I can’t graph it so I don’t know how it will look.” Then

he drew a second line above the line referred to as y = x (Figure 5.4).

From the two previous interview excerpts, it appeared that NVL reported

experiencing difficulty visualizing the graphic form of y = x before actually graphing

y = x as part of the completion of mathematical task # 1. In addition, from the two

previous excerpts, it also appeared that the student reported experiencing difficulty

visualizing the graphic form of y =x + 5 before actually graphing y = x + 5 as part of the

completion of task one, however, his images in Figures 5.3 and 5.4 captured the

relationship between the linear functions.

91

Figure 5.4: Nonviusalizer’s y = x and y = x + 5 lines

After the participant answered the original question, part A, part B, and part C of

mathematical task # 1, he asked, “So, can I graph it now?” The researcher responded,

“Yes”. Then the student immediately reached for the graphing calculator. During the

interview, the participant inputted the equations (symbolic form) y = x and y = x + 5 into

the graphing calculator. The student also changed the original graph setting to make the

y = x + 5 line darker than the y = x line. Next, NVL used the GRAPH feature which

displayed the graphic forms of y = x and y = x + 5 on the graphing calculator screen.

(Specifically, the participant used the Y =, ALPHA, STO, addition, arrow, ENTER, and

GRAPH buttons on the graphing calculator.) After using the graphing calculator to

display the symbolic and graphic forms of

y = x and y = x + 5, NVL responded, “Now we can see the difference [referring to why

he made the line y = x + 5 darker than the line y = x]”.

From this segment of the interview, it appeared that the participant looked for a


completion of task one. From this segment of the interview, it also appeared that the

student used the graphing calculator to construct a relationship between the symbolic

form of a function and the graphic form as part of the completion of mathematical task #

1. In addition, it appeared that the participant used various features of the graphing

calculator as part of the completion of mathematical task # 1.

O’Callaghan’s (1998) translating component for understanding functions seemed

to be depicted because the participant because the participant translated from one

representation of a function to another. Specifically, the student translated from

equations (symbolic form) to graphs (graphic form).

During the completion of part B (after graphing) of mathematical task # 1, NVL

reported:

Does it move the line up on the y-axis [referring to part B of the mathematical

task after graphing]? I guess. What does it exactly mean –move the line up on

the y-axis? Well, yeah, I see now [looking at the screen on the graphing

calculator]. Yes, the line is higher. Instead of just being right here at zero

[referring to the y = x line] it’s up here at five at zero [referring to the y = x + 5

line]. So, now the line is higher.

92

First, the student read the question in part B of the task. Then he asked, “What does it

exactly mean – move the line up on the y – axis”? After examining the graphs of y = x

and y = x + 5 on the graphing calculator, the participant determined that the graph of y =

x + 5 is higher on the y – axis than the graph of y = x. Specifically, NVL informed the

researcher that the graph of y = x crossed the y – axis at (0, 0) and the graph of

y = x + 5 crossed the y – axis at (0, 5).

Overall, O’Callaghan’s (1998) translating component for understanding functions

was present during the completion of this task. Specifically, the student translated the

given symbolic forms of y = x and y = x + 5 to their graphic forms using visual imagery

and the graphing calculator. The student also translated the symbolic forms of y = x and

y = x + 5 to their numeric forms. None of Ruthven’s (1990) role of graphing calculator

approaches was shown by the participant’s use of the graphing calculator in mathematical

task # 1. The role of the graphing calculator in task one was to display the graphic forms

of y = x and y = x + 5. He was given the symbolic forms of y = x and y = x + 5. Then

NVL analyzed the graphic forms of y = x and y = x + 5 using the graphing calculator.



y into the equations. Category II was looking for a relationship between the symbolic

form of a function and the graphic form. Category III was using the graphing calculator


Category IV was using various features of the graphing calculator.



imagery regarding the linear functions y = x and y = x + 5. NVL’s use of the graphing

calculator confirmed his image and the participant continued to complete the task. In

addition, the participant shared his personal viewpoint of the task in the College Algebra

Journal:

It was simple by plugging in numbers to the two various functions: x and x + 5.

That was probably the easiest part, plugging in numbers to find points. When

looking at the two tables, it seems easier to see the relationships between the

various functions. When I compared the two tables, I was able to see that the y –

values all rose by five, which helped me to determine the answers.

Task # 2

Mathematical task # 2 was another linear function (Figure 5.5) by Moschkovich

(1998). The directions were for the participant to complete the task and explain his

reasoning. Task two was chosen because the researcher wanted to see how the student

would solve a second linear function task. The researcher also wanted to know if NVL

would use the same categories from task one in the completion of task two. Using the

graphing calculator was optional during the completion of the task. It was optional

because the researcher wanted to see if the participant would choose to use the graphing

calculator. He did.

93


94

In response to part A of mathematical task # 2, NVL stated:

And let’s see, well, I don’t think you would multiply because I guess having what

we learned in class today which was good, you talking about um, parallel lines

and that for the lines to be parallel they have to have the same slope. If you

multiply by x then you change the slope. That’s what I’m thinking. I’m thinking

we’ll at least find out. Since this is y = 2x, if I took the point -1, plugging it in y

equals two times negative one which equals negative two. So, my x would be

negative one my y would be negative two and if I plugged in the point one my x

will be one, my y will be two. So, taking that I get, um, y – y 1 , which is two

minus two or two minus negative two which is two plus two over x 2 - x 1 , which

is one minus negative one which will give you, one plus one which equals four

over two which equals a slope of two. But if I went ahead and multiplied it by

three um, that would give me y equals let’s see x by three, hum, why or why not?

I’m getting confused, let me see, say y equals three times 2x. No, No, No, y

equals three times x and so if I do that, I’ll end up altering the slope cause if I plug

in the same numbers again negative one and one my y would become negative

three and three which would be one plus one over three plus three which equals

two over nine which is a different slope then two so that means that if they have

different slopes they are not parallel. That means A [referring to part A] you

would not multiply by three or multiply x by three because it changes the slope

and a different slope means they’re not parallel.

2

First, the student reported that he recalled during one of the College Algebra classes that

his instructor discussed a relationship between the concepts of parallel lines and slopes of

lines. Then, the participant explained how he completed part A of this task. NVL

substituted the values of negative one and positive one for x into the equation y = 2x.

Next, the participant solved the equation in order to find the corresponding y - values. As

the student substituted these specific x – values into y = 2x and calculated the

corresponding y – values, he constructed a table of numerical values (Figure 5.6). After

that, NVL substituted the value positive two for y , negative two for y 1 , positive one for

x 2 , and negative one for x 1 into the slope formula

2

y y

x x

2

2 1

1−−

and calculated the slope of

the line y = 2x. Next, the student substituted the values of negative one and positive one

for x in the equation y = 3x. The participant then solved the equation in order to find the

corresponding y – values. As NVL substituted these specific x – values into y = 3x and

calculated the corresponding y – values, he constructed a table of numerical values

(Figure 5.7).

95

Figure 5.6: Nonvisualizer’s y = 2x table of numerical values

Figure 5.7: Nonvisualizer’s y = 3x table of numerical values

96

After that, he substituted the value positive one for y , negative one for y , positive three

for x 2 , and negative three for x 1 into the slope formula

2 1

y y

x x

2

2 1

1−−

and miscalculated the

slope of the line y = 3x. (The student miscalculated the slope by substituting the original

x – values negative one and positive one as y – values into the slope formula. He also

substituted he original y – values of negative three and positive three as x – values into

the slope formula). After that, the participant reported that since he calculated different

slopes for y = 2x and y = 3x, “that means A [referring to Part A] of task two you would

not multiply by three or multiply x by three because it changes the slope and a different

slope means they’re not parallel”. It appeared that the student substituted specific values

for the variables x and y into the equations as part of the completion of task # 2.

From this first excerpt of the interview, the participant translated from one

representation of a function to another representation. Specifically, NVL translated from

equations (symbolic form) to tables of numerical values (numeric form) using y = 2x and

y = 3x. As a result, O’Callaghan’s translating component for understanding functions

seemed to be depicted.

In response to part B of mathematical task # 2, the student explained:

Um, let’s see [pauses] I think you would add three, well, let me see. I’d rather say

I would lean, towards yeah, three because it looks like, well, cause from the dark

to the light [referring to the graphs of y = 2x and y = 2x + 6] it shifts over one,

two, three [counting on x-axis to the left of the origin] um, I think you would

cause it shifts over by three and because since the lines are going in the same

direction, that means the slope hasn’t changed. So, they still have the same slope

still be parallel going in the same direction but just three apart. So, I guess if I do

it again. My y = 2x my slope is two. And so if I were to add three to x, let’s see

what I get, um, add three to x, y = x + 3 uh, I think that’s right. That’ll give me

for x and y negative one and one [constructing table of values] negative one plus

three oh no that’s wrong that’s wrong I’m sorry that’ll be um negative one, let’s

see. Hold up. Yeah, okay my y would be = -1 + 3 which would give me 2 and if

it was 1, y would equal one plus three which would give four and that goes to y -

y 1 over x - x 1 which would give me two over two which equals one. Um,

gives me one, is not the same slope as this unless it should be y equals 2x + 3

[writes equation on paper] which probably makes more sense. And so that means

when x is negative one and that one would go to y equals two times negative one

plus three which equals negative two plus three which would equals one and then

for y =1, two times one plus three which would equal two plus three which would

equal five. So, now I would take my slope which is five minus one over one plus

one which equals six over two which equals three. Now I’m confused because

they should have the same slope. They should have the exact same slope. But,

I’m somehow getting three instead of two [referring back to y = 2x]. Okay, I

think I might have messed up somewhere. Let me think.

2

2

First, by examining the graph provided in task two, the participant reported that the light

line shifted three units to the left on the x – axis from the dark line. The participant also

informed the researcher that he thought the dark line and the light line were parallel lines

and that their slopes were the same. Then, NVL stated, “My y = 2x, my slope is two”.

97

After that, the participant substituted the values of negative one and positive one for x

into the equation y = x + 3. The student solved the equation in order to find the

corresponding y – values. As the participant substituted these specific x – values into

y = x + 3 and calculated the corresponding y – values, he constructed a table of numerical

values (Figure 5.8).

Figure 5.8: Nonvisualizer’s y = x + 3 table of numerical values

Next, NVL substituted the value four for y , two for y , positive one for x 2 , and

negative one for x into the slope formula

2 1

1

y y

x x

2

2 1

1−−

and calculated the slope of the line y

= x + 3. After finding a slope of one for y = x + 3, the student chose to examine y = 2x +

3 because he mentioned earlier during the completion of the task that the slope of the two

lines given in task two would be equal. Therefore, the participant substituted the values

of negative one and positive one for x into the equation y = 2x + 3. Then, he solved he

equation in order to find the corresponding y – values. As the student substituted these

specific x – values into y = 2x + 3 and calculated the corresponding

y – values, he constructed a table of numerical values (Figure 5.9).

98

Figure 5.9: Nonvisualizer’s y = 2x +3 table of numerical values

Next, the participant substituted the value five for y , one for y 1 , positive one and x ,

and negative one for x 1 into the slope formula

2 2

y y

x x

2

2 1

1−−

and miscalculated the slope of the

line y = 2x + 3. (The student miscalculated the slope by adding the y – values instead of

subtracting them).

From the interview excerpt, it appeared that the participant substituted specific

values for the variables x and y into the equations as part of the completion of task # 2. It

also appeared that the student translated from one representation of a function to another

representation. Specifically, NVL translated from equations (symbolic form) to tables of

numerical values (numeric form) using y = x + 3 and y = 2x + 3. As a result,

O’Callaghan’s (1998) translating component for understanding functions seemed to be

depicted.

After the participant inaccurately calculated the slope of the line y = 2x + 3, the

researcher suggested to “try looking at your points again”. So, NVL substituted the

values negative one and positive one for x into the equation y = 2x again. The student

also solved the equation a second time in order to find the corresponding y – values. He

also verified the slope of y = 2x by using the slope formula y y

x x

2

2 1

1−−

again. Next, the

student substituted the values negative one and positive one for x into the equation

y = 2x + 3 again. The student also solved the equation a second time after substituting

negative one in for x. After calculating the same corresponding y – value, he proceeded

to substitute positive one in for x. NVL reported:

99

If you plug in one you get y equals two times one plus three which equals two

plus three which equals five. So, then I’d say y equals which is 5 – y 1 , which is

one gives you, WOW, that gives you four. Alright, there we go and then that

gives you two. And then that gives you a slope of two which is the exactly same

slope for your y = 2x meaning that they are they have the same slope meaning that

they are parallel and meaning that adding three just shifts, like this graph shows

[referring to graph on task # 2 sheet], the line y = 2x over by three points over by

three I guess to the left by three.

During this portion of the interview, the student corrected his arithmetical error by

explaining that five minus one equaled four instead of six. Then, the participant

continued calculating the slope of y = 2x + 3 using the slope formula. He found the slope

of y = 2x + 3 to be two. After that, NVL compared y = 2x and y = 2x + 3 by stating “…

they have the same slope meaning that they are parallel…”. The student also explained

what he thought the graph of y = 2x + 3 looked like compared to he graph of y = 2x

before actually graphing the function. The participant stated, “… adding three just shifts,

like this graph shows [referring to the graph provided in task # 2], the line y = 2x over by

three points over by three, I guess to the left by three”.

It appeared that the student misinterpreted the graphical representation of a

function after adding specific values to the symbolic form of a function as part of the

completion of mathematical task # 2. Specifically, NVL misinterpreted what adding

three to 2x in the equation y = 2x + 3 would do to the graph. Adding thee to 2x in the

equation y = 2x + 3 would move the line of the original function, y = 2x, up on the

y - axis by three units. Instead, the participant reported the line would move three units

to the left along the x – axis.

For part C of mathematical task # 2, NVL responded:

And then I would say no because if you look at the graph [referring to the graph

of the light line in task # 2], it only shifts over three so that means if you add six,

it would shift over six and with the old problem [referring to y = 2x] you would

still have the same slope but it wouldn’t be the same line. So, let’s see if that

makes sense. Let’s do it, [writes on paper] y = 2x + 6 um, yeah okay, well, I

guess the best way to see if that would actually be true is to see the points I

actually got actually fall on the line, um this second line over here y = 2x + 3

because if I use 2x + 6, they still have the same slope, but they have different

points. So, if it was 2x + 6, the points would be shifted over by six instead of by

three, I guess.

During this portion of the interview, the participant compared the graph of the light line,

y = 2x, and y = 2x + 6. In addition, the student explained what he thought the graph of

y = 2x + 6 looked like compared to the graph of y = 2x before actually graphing the

function. The participant stated, “… it would shift over six and with the old problem

[referring to y = 2x] you would still have the same slope but it wouldn’t be the same

line”. After that, in order to test this assertion, NVL suggested calculating points for

y = 2x + 6.

It appeared that the student misinterpreted the graphical representation of a

function after adding specific values to the symbolic form of a function as part of the

completion of mathematical task # 2. Specifically, the participant misinterpreted what

adding six to 2x in the equation y = 2x + 6 would do to the graph. Adding six to 2x in the

100

equation y = 2x + 6 would move the line of the original function, y = 2x, up on the y –

axis by six units. Instead, the student reported the line would move six units to the left

along the x – axis.

For part C of mathematical task # 2 , NVL also responded:

Okay, um, so I guess this is just to show you they have the same slope but I’ll get

different points and the points to the graph, the points will be shifted over by six.

So that would say I’ll use my regular points of x and y, that over here um y equals

negative one, one equals two times negative one plus six and y equals two times

one plus six that gives negative two plus six, which equals four and over here that

gives you two plus six which equals eight [constructed a table of values]. So, I

have my four and my eight and my slope would be my y = 8 – my y 1 = 4 , over

my x , 1 – my x = - 1 and would be positive which goes four over two which

gives you the same slope of two.

2

2 1

During this portion of the interview, the participant substituted the values of negative one

and positive one for x into the equation y = 2x + 6. Next, the student solved the equation

in order to find the corresponding y – values. As the participant substituted these specific

x – values into y = 2x + 6 and calculated the corresponding y – values, he constructed a

table of numerical values (Figure 5.10).

Figure 5.10: Nonvisualizer’s y = 2x + 6 table of numerical values

101

After that, NVL substituted the value eight for y , four for y , positive one for x , and

negative one for y into the slope formula

2 1 2

1

y y

x x

2

2 1

1−−

and calculated the slope of the line y

= 2x + 6.

From the interview excerpt, it appeared that the participant substituted specific

values fro the variables x and y into he equations as part of the completion of task # 2. It

also appeared that the student translated from one representation of a function to another

representation. Specifically, NVL translated from an equation (symbolic form) to a table

of numerical values (numeric form) using y = 2x + 6. As a result, O’Callaghan’s (1998)

translating component for understanding functions seemed to be depicted.

In response to part C of mathematical task # 2, the student stated:

Okay, um, if I were to actually plot these points um, -1, 4, [repeats] -1, 4

[counting] one, two, three, four, let’s see you get here [plots the point (-1, 4)] and

then you get 1, 8 [counting] one, two, three, four, five, six, seven, and eight okay

[plots the point (1, 8)]. That’s odd, they actually fall on the same line and if I did

y = 2x + 3 it’ll be -1, 1 [plots the point (-1, 1)] and then 1, 5 [counting] one, two,

three, four, five [plots the point (1, 5)] so, um that kind of proves my theory

wrong.

During this portion of the interview, the participant took the calculated x and y values

from the table of numerical values for y = 2x + 6 and plotted (-1, 4) and (1, 8) on the

coordinate grid provided in task two (Figure 5.11).

Figure 5.11: NVL’s Mathematical Task # 2 Graph

These two points fell on the light line provided on the graph in task two, which the

student thought was “odd”. Next, NVL took the calculated x and y values from the table

102

of numerical values for y = 2x + 3 and plotted (-1, 1) and (1, 5) on the coordinate grid

provided in task two (Figure 5.11). These two points fell between the light line and the

dark line on the graph in task two.

It appeared that NVL plotted specific points of a function on a graph as part of the

completion of mathematical task # 2. Specifically, he plotted the points (-1, 4) and (1, 8)

for y = 2x + 6. The student also plotted the points (-1, 1) and (1, 5) for y = 2x + 3. In

addition, it appeared that the participant translated from one representation of a function

to another representation. In this instance, NVL translated from tables of numerical

values (numeric form) to graphs (graphic form) using y = 2x + 6 and y = 2x + 3. As a

result, O’Callaghan’s (1998) translating component for understanding functions seemed

to be depicted.

For part D of mathematical task # 2, NVL responded: “Well, I guess I’ll say no

because we’ve already shown that if you add six, you get that line right there [referring

and pointing to the light line provided on the graph in task two]”. The student reported

that he did not think multiplying 2x in the original function, y = 2x, by six would produce

the equation of the light line provided in task two. After that, the participant proceeded to

complete the “AFTER GRAPHING” parts A through D in mathematical task # 2.

In response to the second task, NVL explained:

Okay, here we go [reaches for the graphing calculator]. Y = um, I’ll graph the

original equation 2x and then um, the next equation is 2x times 3 and then it’ll be

2x + 3, 2x + 6, and 2x times 6. So, when I graph it there’s the first line, second

line, third line, fourth line, fifth line. [coughs] Excuse me,um, just to help myself

out, I’m only gonna graph the 2x, the 2x + 3, the 2x + 6 and I’m gonna see how

they actually look. Here’s our 2x, 2x yep, then our 2x + 6 is actually closer to 3

shifting 3 over then the 2x + 3. So you have this and this. This is the correct one

[places a box around C: y = 2x + 6]. You have to add 6 to get the line over here

[lighter line on mathematical task sheet].

During this excerpt of the interview, the student used the graphing calculator to examine

the graphic forms of y = 2x, y = 2x × 3 = 2x + 3, y = 2x + 6 and y = 2x 6.

(Specifically, the participant used the Y =, ALPHA, STO, ENTER, numerical, addition,

multiplication, GRAPH, and arrow buttons.) At first, the NVL entered all five functions

together using the Y = key. For example, he inputted Y 1 = 2x, Y = 2x 3,

×

2 ×Y = 2x + 3, Y = 2x + 6, and Y 5 = 2x 3 4 × 6 using the Y = key on the graphing

calculator. After viewing the five lines on the same coordinate grid, the student said,

“…just to help myself out, I’m only gonna graph the 2x, the 2x + 3, the 2x + 6 and I’m

gonna see how they actually look”. After comparing the graphic forms of y = 2x, y = 2x

+ 3, and

y = 2x + 6 displayed on the graphing calculator to the graphs provided in task two, NVL

chose the symbolic form of y = 2x + 6 to match the graphic form of the light line.

For y = 2x, y = 2x 3, y = 2x + 3, y = 2x × × 6, and y = 2x + 6, it appeared that

the participant looked for a relationship between the symbolic form of a function and the

graphic form as part of the completion of task two. It also appeared that the student used

the graphing calculator to construct a relationship between the symbolic form of a

function and the graphic form as part of the completion of mathematical task # 2. In

addition, it appeared that NVL used various features of the graphing calculator as part of

the completion of task two.

103

O’Callaghan’s (1998) translating component for understanding functions seemed

to be present because the participant translated from one representation of a function to

another representation. Specifically, the student translated from equations (symbolic

form) to graphs (graphic form).

Ruthven’s (1990) Analytic-Construction Approach for the role of graphing

calculators seemed to be present because NVL used the graphing calculator to make

connections between the symbolic and graphic forms of functions to construct the exact

symbolic form of a function in order to complete task two. Specifically, the participant

used the graphing calculator to make connections between the symbolic and graphic

forms of y = 2x, y = 2x 3, y = 2x + 3, y = 2x × × 6, and y = 2x + 6. After that, the

student chose the symbolic form of y = 2x + 6 to match the graph of the light line

provided in mathematical task # 2.

Therefore, the student used the following six categories during the completion of

mathematical task # 2. Category one was substituting specific values for the variables x

and y into the equations. Category two was misinterpreting the graphical representation

of a function after adding specific values to the symbolic form of a function. Category

three was plotting specific points of a function on a graph. Category four was looking for

a relationship between the symbolic form of a function and the graphic form. Category

five was using the graphing calculator to construct a relationship between the symbolic

form of a function and the graphic form. Category six was using various features of the

graphing calculator. In addition, the participant shared is personal viewpoint of task two.

Another problem that I realized is that I tend to try and rush through the problem

without truly looking at all of my options. In this task, the first thing I should

have noticed was that these are parallel lines, meaning that they all the same

slope. So that automatically means that multiplying x by any number would

result in a different slope, which would result in a different line. That little

principle alone would have cancelled out choices A and D, which asks to multiply

x by 3 and 6 respectively. B and C would have worked out perfectly, since they

would have allowed for both equations to share the same slope.

Task # 3

Mathematical task # 3 was a quadratic function (Figure 5.12) by Eisenberg and

Dreyfus (1994). The directions were for the participant to complete the task and explain

his reasoning. Task three was chosen because the researcher wanted to find out how the

student would solve a task involving quadratic functions. Using the graphing calculator

was optional during the completion of the task. It was optional because the researcher

wanted to see if he participant would choose to use the graphing calculator. He did.

In response to mathematical task # 3, NVL stated:

I guess I’ll start with the first function which is uh, x - 2x + 1. Well, I know

x 2 means it’s gonna be a parabola so that’s the first thing. Um, the easier thing

would be just to graph it. I guess what I could do is just plug in the formula

[reaches for the graphing calculator] and then maybe look at the coordinates. So,

that’s what I’ll do.

2

104


The participant reported that he planned to examine the first function, x – 2x + 1, which

was listed in symbolic form in task three. The student also planned to use the graphing

calculator to examine the graphic form of the function. It appeared that the participant

was looking for a relationship between the symbolic form of a function and the graphic

form as part of the completion of task three.

2

In response to mathematical task # 3, NVL explained:

Okay, I’m just gonna plug in [using the graphing calculator] I’m going into the

graphing section and for my, Y =, I’m going to plug in the formula uh, for the

first one which is x to the second – 2x + 1 and uh look at the coordinates.

The student inputted y = x – 2x + 1 as y = x ^ 2 – 2x + 1 into the graphing calculator.

Next, he looked at the graph of y = x – 2x + 1 on the graphing calculator.

(Specifically, the participant used the Y =, ALPHA, STO, ^, subtraction, addition,

numerical and GRAPH keys). From this segment of the interview, it appeared that the

participant looked for a relationship between the symbolic form of a function and the

graphic form as part of the completion of task three. It also appeared that the student

used the graphing calculator to construct a relationship between the symbolic form of a

function and the graphic form as part of the completion of mathematical task # 3. In

addition, it appeared NVL used various features of the graphing calculator as part of the

completion of the third task.

2

2

105

Furthermore, it appeared that the participant translated from one representation of

a function to another representation. In this instance, NVL translated from an equation

(symbolic form) to a graph (graphic form) using y = x – 2x + 1. As a result,


depicted.

2

During the completion of task three regarding y = x – 2x + 1, NVL responded: 2

NVL: So, I guess can get, let’s see two and one so, the first coordinate would be

(2, 1) which is x = 2, y = 1. It goes is out too much uh, [pauses] I don’t know I

feel like I’m cheating if I do that, if I plug and push the formula and push graph

and just see it.

I: Any way that you want to solve it is okay. I just want to know what you’re

thinking and how you choose to complete it.

NVL: Okay, um I guess from just plugging in the formula it just gives you the

graph. So, that’s the easiest way to do it because I could the long way, you know,

plug in x and y values, get points, and compare it like that. But the easiest way is

just to plug the formula in. Then I can just see this graph and by looking at the

paper I automatically know its roman numeral III.

Using the TABLE feature on the graphing calculator, the student selected x = 2,

y = 1, and examined (2, 1) on the graph of y = x – 2x + 1. Then he compared the

graphic form of y = x 2 – 2x + 1 on the graphing calculator with the four parabolas

provided on the coordinate grid in task three. The participant admitted that he felt that

using the technology was cheating. However, the researcher informed the student that,

“Any way that you want to solve it [the mathematical task] is okay. I just want to know

what you’re thinking and how you choose to complete it”. After that, the participant

explained that using the technology was “the easiest way” to complete the task. The

student also reported an alternative to using the graphing calculator would be to substitute

specific numerical values for x in the equation y = x 2 – 2x + 1 and solve the equation to

find corresponding y – values. Next, NVL suggested plotting the specific calculated

points of y = x 2 – 2x + 1 and comparing the graph to the four parabolas provided on the

coordinate grid in task three. On the other hand, the student viewed the graph of

2

y = x – 2x + 1 on the graphing calculator and chose the Roman numeral III parabola

from task three to match the function.

2

During the completion of the third mathematical task, NVL stated:

And then taking this same one going back to the original y = page [referring to TI

– 83], and plug in the second one 1 – x to the second and I graph that one. That

gives me uh, the same graph a parabola just I guess flipped downward. I’m trying

to remember the word we used in class. Uh, it’s almost the same I guess you

could say almost the same reflected but different. So, it’d be the bottom one

which is Roman numeral IV or the second function.

The participant inputted y = 1 – x 2 as y = 1 – x ^ 2 into the graphing calculator.

Next, he looked at the graph of y = 1 – x on the graphing calculator. (Specifically, the

student used the Y =, ALPHA, STO, ^, subtraction, numerical, arrow, and GRAPH

buttons. Then, the student compared the graphic form of y = 1 – x on the calculator

with the four parabolas provided on the coordinate grid in task # 3. After that, NVL

2

2

106

chose the Roman numeral IV parabola from task three to match the symbolic form of 1 –

x 2 .

From this excerpt of the interview, it appeared that the participant looked for a


completion of task # 3. It also appeared that the student used the graphing calculator to

construct a relationship between the symbolic form of a function and the graphic form as

part of the completion of the third task. In addition, it appeared that NVL used various

features of the graphing calculator as part of the completion of task three. Furthermore, it

appeared that the participant translated from one representation of a function to another

representation. In this instance, NVL translated from an equation (symbolic form) to a

graph (graphic form) using y = 1 – x . As a result, O’Callaghan’s (1998) translating

component for understanding functions seemed to be depicted.

2

In response to mathematical task # 3, NVL reported:

And then the third one going back to our Y = screen is x to the second minus 2x.

I’ll graph that one and the same old the top to bottom parabola but it goes down to

the negative y values so that would make it graph II.

The student inputted y = x – 2x as y = x ^ 2 – 2x into the graphing calculator. Next, he

looked at the graph of y = x 2 – 2x on the graphing calculator. (Specifically, the

participant used the Y =, ALPHA, STO, ^, subtraction, numerical, arrow, and GRAPH

keys.) Then, the student compared the graphic form of y = x – 2x on the calculator

with the four parabolas provided on the coordinate grid in the third task. After that, NVL

chose the Roman numeral II parabola from task three to match the symbolic form of x –

2x.

2

2

2

From this segment of the interview it appeared that the participant looked for a




part of the completion of the third mathematical task. In addition, it appeared NVL used

various features of the graphing calculator as part of the completion of task # 3.

From this segment of the interview, it appeared that the participant translated

from one representation of a function to another representation. In this example, NVL

translated from an equation (symbolic form) to a graph (graphic form) using y = x 2 – 2x.

As a result, O’Callaghan’s (1998) translating component for understanding functions


In response to mathematical task # 3, NVL reported:

And now going back to the very last one, back to the Y = screen. We get x to the

second + 1 back to our graph screen. That one shows that the graph shifts upward

compared to the other ones. So, that would give us Roman numeral I.

The student inputted y = x + 1 as y = x ^ 2 + 1 into the graphing calculator. Next, he

looked at the graph of y = x 2 + 1 on the TI-83. (Specifically, the participant used the Y

=, ALPHA, STO, ^, addition, numerical, arrow, and GRAPH buttons.) Then, the student

compared the graphic form of y = x 2 + 1 on the calculator with the four parabolas

provided on the coordinate grid in the third task. After that, NVL chose the Roman

numeral I parabola from task three to match the symbolic form of x + 1.

2

2

107



completion of the third mathematical task. It also appeared that the student used the


and the graphic form as part of the completion of task # 3. In addition, it appeared NVL

used various features of the graphing calculator as part of the completion of the third

mathematical task.

From this excerpt of the interview, it appeared that the participant translated from

one representation of a function to another representation. In this example, the student

translated from an equation (symbolic form) to a graph (graphic form) using y = x 2 + 1.

As a result, O’Callaghan’s (1998) translating component for understanding functions



completion of this task because NVL translated the four symbolic forms of the functions

to the four graphic forms of the functions using the graphing calculator. None of

Ruthven’s (1990) role of graphing calculator approaches were shown by the student’s use

of the technology in task # 3. He was given four functions in symbolic and graphic

forms. Then, the participant inputted the symbolic form of the four functions into the

graphing calculator. Next, the student examined the graphic form of the four functions

using the calculator.





Category III was using various features of the graphing calculator. In addition, the

participant shared his personal viewpoint of this task in the College Algebra Journal.

Truthfully, the easiest and most efficient way of doing this problem was utilizing

the graphing calculator. Otherwise, I might have went back to plugging in points.

The only one I had any idea about was # IV. That is because it has a negative

(-x 2 ) whereas all the others are positive, so that helped out a little.

Task # 4

Mathematical task # 4 was a quadratic function (Figure 5.13) by Ruthven (1990).


and explain his reasoning. Using the graphing calculator was a requirement during the

completion of this task because the researcher wanted to see how NVL would use it.

Task four was chosen to see how the student would solve a second task involving

quadratic functions. In addition, the researcher wanted to know if the participant would

use the same categories from task three in the completion of task four.

108


In response to the fourth mathematical task, NVL stated:

Okay, um, alright well looking from the graph [on mathematical task sheet] I can

see because it’s a parabola that first thing, I’m gonna have I know I’m going to

have x (squared) um, but since it’s inverted since its not the regular you know

parabola from top to bottom, its inverted so I also know that its going to be

negative. Um, so I’ll (writes -x on paper) try that (reaching for the graphing

calculator) just to look at to see real quick how negative x looks.

2

2

2

The participant examined the graph provided in task four. Next, the student called the

graph of the quadratic function a “parabola”. After that, NVL informed the researcher

that the symbolic form of the graph provided in the fourth task included a -x . The

participant also explained that he planned to use the graphing calculator in order to view

the graphic form of y = -x . During this portion of the interview, it appeared that the

student looked for a relationship between the symbolic form of a function and the graphic

form.

2

2

During the completion of mathematical task # 4, NVL responded:

Um, so I’ll [writes -x on paper] try that [reaching for the graphing calculator] just to

look at to see real quick how negative x looks. [after graphing y = -x on TI-83] Yep.

Okay, so now we got that. Now, so far its -x um, I see that its up 1 [referring to y-axis]

so, I’m thinking if I, if I add 1 to this there’s a possibility because I’m trying to think.

I’m trying to figure out why it makes sense to add 1 and not subtract 1. Okay if you

would subtract 1 from this I believe you’ll get more negative values. So, that would

make the parabola shift lower. But if you add 1 un, if you add 1 to it because it’s

squared, you can actually get a positive value so, I’m going to try -x + 1 [ writes -

x + 1 on paper] and that’s again because from looking at the graph um, wow the y-

intercept is 1 so that would make sense. I’m assuming that would make sense. The y-

intercept is where um, where y = 0, the line touches the y-axis, yeah so that y = 0 I’m

confusing myself okay let’s see,

2

2 2

2

2

2

109

The participant inputted y = -x 2 into the graphing calculator. Next, he looked at the

graph of y = -x 2 on the graphing calculator. (Specifically, the student used the Y =,

ALPHA, STO, negative sign, x , and GRAPH keys.) Then the participant compared the

graphic form of y = -x 2 on the calculator with the graphic form of the function provided

on the coordinate grid in task # 4. After that, the student speculated what he believed

adding one to

2

-x 2 would do to the graph of y = -x instead of subtracting one from -x . The participant

also made the assertion that from the graph of y = -x - 1… “ you’ll get more negative

values. So, that would make the parabola [of y = -x ] shift lower”. The student just

described a vertical translation of the graph of y = -x downward by one unit. Even

though the participant did not use the words translation or transformation, he expressed

the meaning of these words in terms of the visual movement of a graph. The student also

tried to explain how the y-intercept affected the graph of the function provided in

mathematical task # 4.

2 2

2

2

2



completion of the fourth mathematical task. It also appeared that the student used the


and the graphic form as part of the completion of task # 4. In addition, it appeared that

NVL used various features of the graphing calculator as part of the completion of the

fourth mathematical task. It also appeared that the student focused on specific visual

features of the graph of a function as part of the completion of task four.

From this excerpt of the interview, it appeared that the participant translated from

one representation of a function to another representation. In this instance, the student

translated from an equation (symbolic form) to a graph (graphic form) using y = -x 2 . As

a result, O’Callaghan’s (1998) translating component for understanding functions seemed

to be depicted.

In response to mathematical task # 4, NVL expressed:

[says and pushes buttons in the graphing calculator] -x + 1. We’ll try that and

they look about the same, so yeah I’m gonna say -x + 1 due to the fact that okay

x is a parabola. It’s negative so that way it’ll be um from the bottom to the top

and then plus 1 because 1 over here [referring to graph on math task] is looks like

the y-intercept where the line touches the y-axis.

2

2

2

The student inputted y = -x + 1 into the graphing calculator. Next, he looked at the

graph of y = -x 2 + 1 on the graphing calculator. (Specifically, the participant used the Y

=, ALPHA, STO, negative sign, x 2 , addition, numerical, and GRAPH buttons.) Then,

the student compared the graphic form of y = -x + 1 on the calculator with the graphic

form of the function provided on the coordinate grid in task # 4. After that, NVL

concluded that both of the graphs “… look about the same…”. At the end of the task,

the student confirmed “my final answer is negative x to the second plus one”.

2

2

During this segment of the interview, it appeared that the participant looked for a


completion of the fourth mathematical task. It also appeared that the student used the

110

graphing calculator to construct a relationship between the symbolic form of a function,

and the graphic form as part of the completion of task # 4. In addition, it appeared NVL

used various features of the graphing calculator as part of the completion of the fourth

mathematical task.

During this segment of the interview, it appeared that the participant translated

from one representation of a function to another representation. In this instance, the

student translated from an equation (symbolic form) to a graph (graphic form) using

y = -x + 1. As a result, O’Callaghan’s (1998) translating component for understanding

functions seemed to be depicted.

2


completion of this task because NVL translated y = -x and y = -x 2 + 1 from symbolic

form to graphic form using the graphing calculator. Ruthven’s (1990) role of graphing

calculators’ Analytic-Construction Approach was also present because the student used

the connections between the symbolic and graphic forms of functions to construct the

exact symbolic form of a function to complete task four. -x and y = -x + 1 to construct

the exact symbolic form of y = -x 2 + 1.

2

2 2



imagery regarding the quadratic function y = x . NVL’s use of the graphing calculator

confirmed his image and the participant continued to complete the task.

2



form of a function and the graphic form. Category II was using the TI-83 graphing


graphic form. Category III was using various features of the graphing calculator.

Category IV was focusing on specific visual features of the graph of a function. In

addition, the participant shared his personal viewpoint of this task.

The first thing I knew was that x was negative. This is because the graph is up

side down. Whenever x is positive, it reaches to positive infinity. However,

when it is negative [-x , not (-x) ], it reaches to negative infinity. That rule

alone helped me to determine the beginning portion of the equation.

2

2

2 2

Task # 5

Mathematical task # 5 was a cubic function (Figure 5.14) by Ruthven (1990). The

directions were for the participant to complete the task using the graphing calculator and

explain his reasoning. Using the graphing calculator was a requirement during the

completion of this task because the researcher wanted to see how the student would use

it.

111


Task five was chosen because the researcher wanted to know how NVL would solve a

task involving cubic functions.

In response to the fifth mathematical task, NVL stated,

Okay, so, let’s go. Uh, well I guess first off looking at the equation [referring to

y=] I think it’s gonna be I know it looks like x to the third because of the way the

curvature of the line. So, I’m gonna look and see that first, make sure that’s right,

something like that. Um, hum it might be [pushing buttons on the graphing

calculator] I doubt if it’s x to the fourth [entered y = x ] uh, no, that’s what I

thought. 1

4

First, the student looked at the graphic form of the function provided in task five. Next,

he suggested that the symbolic form of the function was y = x . The participant also

reported that the symbolic form “…might be…” y = x . Then, the student inputted

3

4

y = x as y = x ^ 4 into the graphing calculator. After that, he looked at the graph of y =

x 4 on the graphing calculator. (Specifically, the participant used the Y =, ALPHA, STO,

^, numerical and GRAPH keys) The participant compared the graphic form of y = x

with the graphic form of the function provided on the coordinate grid in task five. Before

comparing the two graphs NVL said, “… I doubt if it’s x to the fourth, …”. After

comparing the two graphs, he student continued with “uh, no, that’s what I thought”.

4

4

During this excerpt of the interview, it appeared that the participant looked for a


completion of the fifth mathematical task. It also appeared that the student used the


and the graphic form as part of the completion of task # 5.

During this excerpt of the interview, it appeared that the student translated from

one representation of a function to another representation. In this example, the student

translated from an equation (symbolic form) to a graph (graphic form) using y = x 4 . As

a result, O’Callaghan’s (1998) translating component for understanding functions seemed

to be shown.

112

In response to mathematical task #5, NVL expressed:

Um, [enters x ^ 3 on the TI-83] Okay, alright, well the first thing I did was to see

if x to the third actually fit the, um, curvature of the line and it does. So, just

having that I see, um, that same little swoop that this has right here [referring to

graph in task #5] So now I see that the line actually hits um, points hits some of x

well, I’m assuming this [referring to graph in task #5] would be the x-intercept is,

let me see I want to say the x-intercept is or the y-intercept [pointing to where the

graph hits the x-axis] no, it’s the x-intercept, the x-intercept looks like -2 [says

and writes on paper]. The y-intercept is at zero, comma zero and your other x-

intercept is one. I’m not sure what I can do with that now, but I guess it’s good to

know.

The student inputted y = x as y = x ^ 3 into the graphing calculator. Next, he looked at

the graph of y = x on the graphing calculator. (Specifically, the participant used the Y =,

ALPHA, STO, ^, numerical, and GRAPH buttons.) Then, the student compared the

graphic form of y = x 3 on the calculator with the graphic form of the function provided

on the coordinate grid in task # 5. The participant referred to the shape of a cubic

function as the “…curvature of the line…”. He also referred to a turning point on the

graphs as a “…little swoop…”. The participant also reported both of the graphs seemed

to have similar curvature and swoops. After that, the student located the x and y

intercepts from the graph provided on the coordinate grid in the fifth mathematical task.

3

3

During this portion of the interview, it appeared that the participant looked for a




part of the completion of the fifth mathematical task.

During this portion of the interview, it appeared that the participant translated

from one representation of a function to another representation. In this example, the


y = x . As a result, O’Callaghan’s (1998) translating component for understanding

function seemed to be depicted.

3

In response to the fifth mathematical task, NVL stated:

Um, so find an expression for y in terms of x which describes the graph for. So, I

know I have x to the third and the points are y equals. I have a “x” up here and a

y [writing on blank paper]. My x is equal to -2 my y is equaled to zero. When

my x is equaled to zero, my y is equaled to zero. When my x equals one, my y

again is equaled to zero. So, I have these three points, um, now I have to figure

out a formula that will make them fit into a not a quadratic. I’m not sure about

that terminology for um, when you have x to the third power, I know it I just

don’t remember it at the moment.

First, the participant read the task question again. Next he reported specific x and y

values from the graph provided on the coordinate grid in task five. As the student

reported the x and y values, he constructed a table of numerical values. At this point,

NVL had not constructed the exact symbolic form to match the graphic form of the

function given in the fifth mathematical task. However, the participant explained earlier

that the graph of the function in task # 5 looked similar to the graph of y = x i.e. the

graph of a cubic function.

3

113

During this portion of the interview, it appeared that NVL was looking for a

relationship between the graphic form of a function and the numeric form as part of the

completion of task five. Specifically, he used the graph of the cubic function provided in

task five and (-2, 0), (0, 0), and (1, 0) as the numeric form. It also appeared that the

participant translated from one representation of a function to another representation. In

this example, the student translated from a graph (graphic form) to a table of numerical

values (numeric form). As a result, O’Callaghan’s (1998) translating component seemed

to be shown.

During the completion of mathematical task # 5, NVL explained:

So, I have to find an equation, I guess a cubic equation for um this, um, y equals

with these three points given and, um let me think (referring to task sheet). Let

me think. So, -2, 0, 1 (referring to x-values on the table he made).

At this point, the student was trying to find the precise symbolic form that

matched the graphic form of the function given in task five. So, the participant used the

numerical values from the table he constructed in order to complete the task. Next, the

student substituted the numerical values of negative two, zero, and one for x in the

equation

y = x . When the researcher asked NVL to explain why he substituted negative two,

zero, and one into y = x , the student explained that this was “my first guess”.

3

3

In response to mathematical task # 5 regarding substituting numerical values for x

into y = x , NVL continued: 3

And if I just plugged in -2, I get y = -2 to the third which would equal, uh, two

and two is 4, -8. So an easy way to say this would be I maybe put plus 8 but if I

get 0 it would be the same thing. If I did 1, hum no it won’t work I couldn’t get

plus 8.

Since the student had already constructed both the x and y values for his numerical table

from the graph in task five, NVL tried to obtain zero as a y-value after substituting

negative two for x in y = x . However, when negative two is cubed or multiplied by

itself three times, the result is negative eight. So, the participant reported that he added

eight to negative eight in order to get zero. He wanted to get zero because the ordered

pair from the graph that NVL recorded in his table was (-2, 0) i.e, when x = -2, y = 0.

Next, the student substituted zero fro x in the equation y = x . Then the participant

calculated the corresponding y – value as zero. After that, NVL substituted one for x in

the equation y = x . After calculating the corresponding y – value as one, he reported,

“If I did one, hum, no, it won’t work I couldn’t get a plus eight”.

3

3

3

From the two excerpts of the interviews, it appeared that NVL was looking for a

relationship between the numeric form of a function and the symbolic form as part of the

completion of task five. Specifically, he used (-2, 0), (0, 0), and (1, 0) as the numeric

form of a function and y = x as the symbolic form. In addition, it appeared that the

participant was substituting specific values for x and y into the equation as part of the

completion of the fifth mathematical task.

3

In response to mathematical task # 5, NVL continued to search for the exact

symbolic form to match the graphic form of the function given in the task.

Um, so I have my first part y = x to the third. Now, to find the points I need there

has to be a ay to figure it out. Bu my first real guess is just to do a trial and error.

So I would say y equals x to the third, hum, let’s see, plus maybe 2x.

114

At this point, the student constructed the symbolic form of y = x + 2x as a possible

match. The participant explained that he came up with this equation through “…trial and

error”. Next, the student proceeded to input y = x + 2x as y = x ^ 3 + 2x into the

3

3

graphing calculator using the Y = feature. Then the participant pressed the GRAPH key.

As NVL examined the graph of y = x + 2x on the graphing calculator, he stated, “…no

different”. At this point, the researcher wanted some clarification from the participant

regarding why he chose y = x 3 + 2x and what he meant by “no different”. The student

responded: “I guess what was going through my mind was that okay, I know that the

base, um, the base graph shift is x to the third because x to the third has a kind of

curvature to it. So, again NVL explained how he believed the symbolic form of y = x

was related to the graphic form of y = x . In addition, his experience in College Algebra

had exposed the student to various cubic functions. Specifically, the participant was

introduced to the symbolic, numeric, and graphic forms of y = x as the basic cubic

function to explain why he constructed the equation y = x + 2x.

3

3

3

3

3

Uh, when I was doing it with just x to the third plus 2x, there wasn’t really any

difference in the expansion of the line [compared to y = x ]. It was just either

curving you know more like this [makes a half “u” shape in the air with his

pencil, beginning at the bottom, then slowly rotating in a curved shape to the right

and upwards] or more like this [makes a half “n” shape in the air with his pencil,

beginning at the top, then slowly rotating in a curved shape to the left and

downwards], but it never included expanding out to include the negative two or

positive one.

3

Initially, the student constructed y = x + 2x by guessing. He recognized that the graph

of the function given in task five was some type of cubic function. After adding 2x to x

in the equation, the participant used the graphing calculator to compare the graphic form

of y = x + 2x with the graphic forms of y = x 3 and function provided in the fifth

mathematical task. When the three graphs did not match, NVL described what he

thought about the graph of y = x 3 + 2x as it related to the graph of y = x . The student

stated: “ …there wasn’t really any difference in the expansion of the line”. Next, he

reported the upward and downward turning points of the function. Thirdly, the

participant explained how the graph of y = x + 2x did not intersect two of the

3

3

3

3

3

x –intercepts (-2, 0) and (1, 0) of the function provided in task five.

During the completion of the fifth mathematical task as the student searched for

the symbolic form of the function that matched the graphic form of the function provided

in task five, NVL stated: “So, x to the third plus let me see x to the second [looks at

graph of y = x 3 + x on the graphing calculator]. Hum, minus 2x [looks at graph of y =

x 3 + x - 2x on the graphing calculator]. Now, we’re getting somewhere. Okay,

actually it looks like it fits. First, the student translated y = x + x from its symbolic

form to its graphic form using the graphing calculator. Specifically, NVL inputted y = x

+ x as y = x ^ 3 + x ^ 2 into the graphing calculator using the Y = feature. After

entering the equation, he pressed the GRAPH key. Secondly, he compared the graph of y

= x + x to the graph of the function provided in task five. Next, the participant

translated y = x + x - 2x from its symbolic form to its graphic form using the graphing

calculator. Specifically, the student inputted y = x 3 + x - 2x as y = x ^ 3 + x ^ 2 – 2x

2

2

3 2

3

2

3 2

3 2

2

115

into the graphing calculator using the Y = feature. After entering the equation, he pressed

the GRAPH key. After that, the student compared the graph of y = x + x - 2x to the

graph of the function provided in task # 5. At this point, the researcher wanted some

clarification from the participant regarding why he chose y = x + x and x + x - 2x

as equations.

3 2

3 2 3 2

First, NVL explained why he chose y = x + x + x as an equation. “Um, so I

just added an x squared to see how it would change the shape of the graph. And I saw

[using the graphing calculator] when I put x squared in, it caused it [referring to the graph

of the cubic function] to expand more”. So, beginning with the graph y = x , the student

wanted to see if the graph of y = x + x was more spread out than the graph of y = x .

The participant reported that he did not know what the graph of y = x + x looked

3 2 2

3

3 2 3

3 2

Like, therefore, he used the graphing calculator. Secondly, NVL explained why he chose

y = x + x - 2x as an equation. 3 2

Um, I know okay, now it’s expanding [referring to the graph of y = x 3 + x ] so

that means I’m on the right track. Well, let me not just change it to x squared

right now, let me see if I can go a little bit further. So, after x squared, going in

descending order comes x. Now for whatever reason I just guessed minus 2x.

2

The student used the graphing calculator to see visual features of the function y = x +

x . Based on the participant’s conjecture, he used the graphing calculator again to depict

the visual features of the function y = x + x - 2x.

3

2

3 2

Next, NVL explained how he came up with the final solution by comparing the

visual features of the graphic form of x + x 2 - 2x with the visual features of the graphic

form of the function provided in task five. Next, the student used the TABLE feature on

the graphing calculator to verify the x – intercepts of y = x + x - 2x were the same as

the x – intercepts of the function provided in the fifth mathematical task. After the

participant verified the x – intercepts were the same for both functions, he again

explained how and why he chose y = x + x - 2x as the final solution for task # 5.

3

3 2

3 2

But I knew that the reason well not necessarily this, but the reason why I lean I

guess toward -2x in an educated guess was that I figured that if I wanted to get it

out further toward the negatives I have to put a minus in there somewhere cause

everything else was (+) even though I would put (-) numbers in, but that -2x

would expand it more along the I guess (-) x-axis not the (-) on the second

quadrant going further out I guess you could say. So I just put -2x in there

(inputting in TI-83) and Voila!

So, again, the student explained that he constructed y = x 3 + x - 2x as an “educated

guess”. The participant reported that this educated guess involved finding the symbolic

form of a cubic function “… that would expand [the graph of y = x ] more along the, I

guess, negative x – axis…” in order to match the graph provided in mathematical

2

3

task # 5.


completion of this task because the student translated the graphic form of

y = x + x - 2x given in task five to its symbolic form using the graphing calculator.

Ruthven’s (1990) role of graphing calculators’ Analytic Construction Approach was also

present because the participant used the connections between the symbolic and graphic

3 2

116

forms of functions to construct the exact symbolic form of a function to complete the

fifth mathematical task. Specifically, NVL used y = x , y = x , y = x + 2x, y = x 3 +

x 2 , and y = x + x - 2x to construct y = x 3 + x - 2x.

4 3 3

3 2 2


the fifth mathematical task. Category I was looking for a relationship between the

symbolic form of a function and the graphic form. Category II was using the graphing


graphic form. Category III was looking for a relationship between the graphic form of a

function and the numeric form. Category IV was looking for a relationship between the

numeric form of a function and the symbolic form. Category V was using various

features of the graphing calculator. Category VI was focusing on specific visual features

of the graph of a function.

During the completion of mathematical task # 5, the student used non-visual

methods to solve the task. The participant located the x – intercepts of the cubic function

by using the graph provided in task five as (-2, 0), (0, 0), and (1, 0). Then NVL tried to

find an equation of a cubic function that corresponded with the x – intercepts. In fact, the

student shared his personal viewpoint of task five.

The key to this problem was to find the zeros of the function. Just from its shape

alone I knew it was a cubic polynomial. The zeros were at -2, 0, and 1. From that

you can get (x + 2) (x – 1) (x). That would give you the resulting equation.

Even though the student did not rely on visual imagery as defined by Lean and

Clements (1981), his use of visualization as defined by Arcavi (2003) appeared to be

depicted during the completion of this task. Arcavi (2003) defined visualization as “…

the ability, the process and the product of creation, interpretation, use of and reflection

upon pictures, images, diagrams in our minds on paper or with technological tools…”

(p. 217). This definition is broad enough to include the creation or production of visual

images using technological tools. In this case, the technological tool was the graphing

calculator. The participant used the calculator to display the graphs of several cubic

functions because he did not know what the graphs looked like. After the pictures are

created Arcavi (2003) explained as part of the definition of visualization that their

purpose included “…depicting and communicating information, thinking about and

developing previously unknown ideas and advancing understandings” (p. 217). During

the completion of the fifth mathematical task, using the graphing calculator, the student

repeatedly compared the graphs of cubic functions with the graph provided in the task

and made conclusions based on his understanding of functions. In addition, Ruthven’s

(1990) role of graphing calculators’ Analytic-Construction Approach also showed how

NVL used the calculator. Perhaps, the student might not have successfully completed

task five without using the graphing calculator. However, more evidence is needed

before the researcher reports it as an essential tool in NVL’s understanding of functions.

Task # 6

Mathematical task # 6 was a cubic function (Figure 5.15) by Eisenberg and

Dreyfus (1994). The directions were for the participant to complete the task using the

graphing calculator and explain his reasoning. Using the graphing calculator was a


NVL would use it. Task six was chosen to see how the student would solve a second task

117

involving cubic functions. In addition, the researcher wanted to know if the participant

would use the same categories from task five in the completion of task six.

In the beginning of the sixth mathematical task, NVL asked the research: “Um, I

guess okay, um I guess my question is, why, why is the graph here [referring to the graph

provided in task six]? This is just to help me out? The researcher answered, “That’s the

graph of f of x equals x - 3x ”. After that, the participant explained, “Well, I guess I’ll

start off, uh, with writing down my f of x equals x to the third minus 3x squared and I’m

also given g of x equals f of x plus three”.

3 2


Thus, NVL wrote the two functions that were given and continued to describe in detail

how he planned to solve this task.

First, NVL stated, “Saying f of x + 3 [written f (x + 3)] just means whatever this

is [referring to what is inside the parentheses] goes right here [drew a line from f (x + 3)

to f (x)], (Figure 5.18). So, that would change this to x + 3 cubed minus three times x + 3

squared”. The student shared his understanding of how f (x) and g (x) were related to one

another. Since g (x) = f (x + 3), he recommended substituting he algebraic expression x +

3 for the variable x in f (x) = x - 3x 2 . Next, the participant substituted x + 3 for x in 3

f (x) = x - 3x . 3 2

118

Secondly, during the completion of mathematical task # 6, NVL reported:

So, in order to change this around this goes to um, we have a formula for this too.

Alright x + 3 I think about it the easy way x + 3 times x + 3. I know this formula.

I just square this and it will become x 2 + 6x + 9 still times x + 3 minus 3 times

uh same thing x + 6x + 9 cause it’s squared over here, so we get the same thing

(writes ( x + 6x + 9 ) (x + 3) minus 3 (x + 6x + 9).

2

2 2

The student expanded (x + 3) 3 - 3 (x + 3) to (x + 3) (x + 3) (x + 3) – 3 (x + 3) 2 . Next,

he wrote (x + 3) as a perfect square trinomial, i.e., x + 6x + 9. The participant later

explained that he did not remember the perfect square trinomial formula. Therefore,

NVL multiplied (x + 3) (x + 3) together. After that, the student multiplied

2

2 2

(x 2 + 6x + 9) (x + 3) and simplified the expression by combining the like terms. The

participant said, “before I go any further I’m gonna kind of check my answer a little bit to

make sure I have it right”. At this point, he wanted to verify that (x + 3) 3 - 3 (x + 3) 2

= x + 9x + 27x + 27 – (3 x + 18x + 27). 3 2 2

In response to mathematical task # 6, NVL expressed:

using the graphing calculator. Um, I’m going to put in the original which is, uh

parentheses x + 3 to the third minus uh parentheses 3 parentheses x + 3

parentheses uh to the second . I believe that’s right [referring to the equation he

entered into the graphing calculator] y 1 = (x + 3) ^ 3 – 3(x+3)^2). Yep and then

we graph it (pressed graph button) and we should get a different kind of graph.

x+ 3.

The student inputted y = (x + 3) 3 - 3 (x + 3) as y = (x + 3) ^ 3 – 3 (x +3) ^ 2 into the 2

1

graphing calculator. Next, he looked at the graph of y = (x + 3) - 3 (x + 3) on the

calculator. Specifically, he used the Y =, left parenthesis, ALPHA, STO, addition,

numerical, right parenthesis, ^, subtraction, and GRAPH buttons on the graphing

calculator. When the researcher asked the participant what he meant by the phrase,

“…we should get a different kind of graph”, the student responded that he wasn’t sure,

but he was thinking it would be different because of the algebraic expression x + 3. At

this point, the participant did not report that the graph of g(x) was a horizontal translation

of the graph of f(x) by three units to the left.

3 2

During this excerpt of the interview, it appeared that the student looked for a

relationship between the symbolic form of a function and the graphic form. It also

appeared that the participant used the graphing calculator to construct a relationship

between the symbolic form of a function and the graphic form. In addition,

O’Callaghan’s (1998) translating component for understanding functions appeared to be

depicted. Specifically, NVL translated y = (x + 3) - 3 (x + 3) from its symbolic form

to its graphic form using the graphing calculator.

3 2

In response to the sixth mathematical task, participant stated:

This is the graph of g (x) = f (x + 3) [on the graphing calculator]. I just put the x

+ 3 into the um, x to the third and 3x . And so now I’m gonna check where I am

right now to make sure that it’s um, uh I guess it should still be the same. It

should still be the same. X to the third plus 9x to the 2

2

nd + 27x + 27 minus

parentheses 3x to the 2nd

+ 18x + 27 close parentheses graph (inputted for y = x 2

119

^ 3 + 9x ^ 2 +27x + 27 – ( 3x ^ 2 + 18x + 27) in graphing calculator). Uh-uh,

Yep, they’re the same.

The student examined the graph of g(x) on the graphing calculator. After that, the

participant inputted y = x + 9x + 27x + 27 – (3x + 18x + 27) as [NVL’s expanded

form of g(x)] y = x ^ 3 + 9x ^ 2 + 27x + 27 – (3x ^ 2 + 18x + 27) into the graphing

calculator. Next, he looked at the graph of y on the calculator. Then, NVL compared

the graph of y with the graph of y . The student reported that he graphed

3 2 2

2

2

1 2

y = (x + 3) - 3 (x + 3) and y = x + 9x + 27x + 27 – (3x + 18x + 27) in order to see

if the graphs of both equations were the same and to verify his algebra.

3 2 3 2 2

During this portion of the interview, it appeared that the participant looked for a


appeared that the student used the graphing calculator to construct a relationship between

the symbolic form of a function and the graphic form. In addition, O’Callaghan’s (1998)

translating component for understanding functions appeared to be depicted. Specifically,

NVL translated y = x 3 + 9x + 27x + 27 – (3x + 18x + 27) from it symbolic form to its

graphic form using the graphing calculator.

2 2

After viewing the graphs displayed on the graphing calculator, the student

concluded that the graphs of both cubic functions were the same. Then the participant

explained what the graphs of the cubic functions being the same meant to him. “So, that

let’s me know that so far I haven’t messed anything up or haven’t, uh, added any terms

incorrectly. So, so far, my equation is still the same. I just basically expanded

everything”.

During the completion of mathematical task # 6, NVL stated “So now I can go

ahead and apply this negative here to the rest of this right here [referring to the minus

sign in front of (3x + 18x + 27)]”. The student distributed the negative sign to 3x 2 +

18x + 27. Then, the participant simplified the algebraic expression by combining the like

terms. He continued this simplification process until he obtained the expression x +

6x 2 + 9x. The student factored x + 6x + 9x as

2

3

3 2

x(x 2 + 6x + 9). Next, he factored x(x + 6x + 9) as x(x + 3) . All three of these forms

were equaled to g(x).

2 2

In response to the sixth mathematical task, NVL admitted:

Now, I don’t know if it would be the same if I graphed it. I’m not sure. I have a

doubt in my head that it would be. So x times x + 3 just to be sure, and put in the

parentheses to the second (inputs for y = (x(x + 3 ^ 2)). I doubt it’ll be the same.

What? Yeah, it’s the same (NVL examines the graph on the graphing calculator

screen)! So, I guess I’ll check my well to make sure let me turn these two off.

3

The participant shared with the researcher that he did not think the graph of y = x(x +3)

would be the same as the graphs of y = (x + 3) - 3(x + 3) and y = x + 9x 2 + 27x + 27

- (3x 2 + 18x + 27). Therefore, he inputted y = x(x + 3) as y = x(x + 3) ^ 2) into the

graphing calculator to find out. After examining the graph of y = x (x + 3) on the

calculator, the student reported that the graph appeared the same as the other two

equations.

2

3 2 3

2

3

2

120



appeared that the student used the graphing calculator to construct a relationship between

the symbolic form of a function and the graphic form. In addition, O’Callaghan’s (1998)

translating component for understanding functions appeared to be shown. Specifically,

NVL translated y = x (x + 3) from its symbolic form to its graphic form using the

graphing calculator.

During the completion of task six, the student explained: “To make sure, I pressed the

ENTER key to un-highlight Y 1 = and Y 2 =. Yep, it is exactly the same. So, that’s the

check. All three equations in all three different forms are the same exact thing”. Thus,

the participant used the graphing calculator again to verify his final algebraic or symbolic

answer. Recall, that the sixth mathematical task stated, “Find g(-2)”. The participant

responded: “Um, g (-2) and in my original equation g(x) = f(x + 3). So, I just get f (-2 +

3) which gives you, hum, one [draws a box around 1]”. The student substituted the value

negative two for the variable x in the equation g(x) = f(x + 3) in order to calculate a value

for g (-2). He reported that the value for g (-2) was one. (Actually, he participant

miscalculated because g (-2) = f (1) = -2.) NVL made a symbolic error by reporting that

his final result was one instead of f (1). However, it appeared that during this excerpt of

the interview, the student substituted a specific value for x as part of the completion of

task six.

2

NVL also used the graphing calculator to find a value for g (-2).

I guess um, if I use the graphing calculator um, I put in close this out, um, x + 3

for g of x. This is just a straight linear equation. And then I can go to my g

equals (-) 2. I guess this is almost like saying I wanna say uh just x = (-) 2 then

the equation should equal 1. So, I can pull up my table, yep, if I go down to (-) 2

the equation equals 1.

The participant inputted g (x) = f (x + 3) as y = x + 3 into the graphing calculator.

Again, the student made a symbolic error by treating g (x) = f (x + 3) as g (x) = x + 3.

Next, the participant pressed the GRAPH button and looked at the graph of y = x + 3.

After that, the student used the TABLE feature and reported that when x equaled negative

two, y equaled one. NVL explained that the calculator verified g (-2) = 1.

4

4


completion of this task. Specifically, the participant translated y = (x + 3) 3 - 3 (x + 3) ,

y = x + 9x + 27x + 27 – (3x + 18x + 27), y = x (x + 3) , and y = x + 3 from their

symbolic forms to their graphic forms using the graphing calculator. None of Ruthven’s

(1990) role of graphing calculators’ approaches was present by the way the student used

the graphing calculator. The role of the graphing calculator in task six was to verify

NVL’s algebraic manipulations. In other words, the participant used the graphing

calculator to verify his algebra by comparing the graphs of various symbolic forms of

equations that he constructed.

2

3 2 2 2


the sixth mathematical task. Category I was looking for relationship between the



121


Category IV was substituting specific values for x and y.

During the completion of mathematical task # 6, the participant used non-visual

methods to solve the task. The student computed g (x) in an entirely algebraic or

symbolic manner. Then, NVL calculated g (-2) by substituting negative two into an

equation in order to find the value. In fact, the participant shared his personal viewpoint

of this task in the College Algebra Journal.

This task proved to be more challenging than the others, only because it involved

a lot more work to get the answer. It was nothing more than plugging in values

and then multiplying, simple enough, but quite tedious. The other portion, the G

of (-2), was simpler in steps, but still the same process.

Task # 7

Mathematical task # 7 was another cubic function (Figure 5.16). The directions

were for the participant to complete the task using the graphing calculator and explain his


task because the researcher wanted to see how NVL would use it. Task seven was

chosen because the researcher wanted to know how the student would solve a third task


In response to the seventh mathematical task, NVL stated: “Using the graphing

calculator, to me the easiest way to do it would just be type, go to our Y =, type in the

function negative 3x times x – 1 times x – 2 , alright, then GRAPH”. The student

inputted the equation y = -3x (x – 1) (x – 2) into the graphing calculator. Next, he looked

at he graph of y = -3x (x – 1) (x – 2) on the graphing calculator. While examining the

graph the participant explained : , “ And since its gonna be, well, you can tell since you

have 3x’s multiplied [referring to the given function, f (x) = -3x (x – 1) (x – 2), in task

#7] automatically you know it’s gonna be a cubic. So, it’s gonna have that, um, kind of

vertical ‘S’ shape”. During this portion of the interview, it appeared that the participant

looked for a relationship between the symbolic form of a function and the graphic form

as part of the completion of task # 7. It also appeared that the student used the graphing


graphic form as part of the completion of the seventh mathematical task.



student translated from on equation (symbolic form) to a graph (graphic form) using

y = -3x (x – 1) (x – 2). As a result, O’Callaghan’s (1998) translating component for

understanding functions seemed to be shown.

In response to mathematical task # 7, NVL explained how he located the x –

intercepts of the function. “So, then we can just go to our TABLE [referring to the

TABLE key on the graphing calculator] and we see that our zeros right here are zero and

zero at the origin, one and zero, two and zero”. At this point, the participant already

examined the graph of y = -3x (x – 1) (x – 2) on the graphing calculator. Now, he

wanted to locate the x – intercepts or zeros of the function.

122


Thus, the student used the TABLE feature on the graphing calculator and located

the values. The participant reported that he used this feature because, “ the easiest way to

get the points [of a function] is to use the TABLE [referring to the TABLE key on the

graphing calculator]”. First, the student pressed the 2nd

key. Secondly, he pushed the

GRAPH button. After following these two steps, the TABLE feature was enabled. The

graphing calculator displayed a two by three table. The first column was labeled X and

had x – values from y = -3x (x – 1) (x – 2) shown below it. The second column was

labeled Y 1 and had the corresponding y – values from y = -3x (x – 1) (x – 2) shown

below it. The three x and y values that the student focused on, from the TABLE on the

TI-83 were (0, 0), (1, 0), and (2, 0).

During this segment of the interview, it appeared that the participant looked for a


completion of task seven. It also appeared that the student translated from one

representation of a function to another representation. In this instance, the participant

123

translated from a graph (graphic form) to a table of numerical values (numeric form)

using the graphing calculator. As a result, O’Callaghan’s (1998) translating component

seemed to be shown.

After locating these x and y values using the TABLE feature, NVL explained,

“So, we can already go to this little sheet right here [referring to the task # 7 sheet] and

then just plot our zeros. Zero and zero at the origin; one and zero; and two and zero. So,

we have three consecutive points”. During this excerpt of the interview, the student

explained how he translated the numeric form of y = -3x (x – 1) (x – 2) to its graphic

form by plotting points. The participant plotted (0, 0), (1, 0), and (2, 0) on the coordinate

grid provided in the seventh mathematical task. It appeared that the student plotted

specific points of a function on a graph as part of the completion of this task.

In response to mathematical task # 7, regarding sketching the graph of

y = -3x (x – 1) (x – 2), NVL explained:

And then if we go back to the graph we can see it looks something around like

this [referring to graph of f (x) on the graphing calculator screen] right here, goes

here, down a little then it goes through right here. So, it goes through the origin.

Then going through 1 comma 0, then right back down to 2 comma 0, and then

straight down after that. So, there we have our function.

The participant used the graph of the cubic function displayed on the calculator to sketch

the curve-shaped graph of y = -3x (x – 1) (x – 2) on the coordinate grid provided in task

seven. Specifically, the student reported where the graph increased and decreased

between and beyond the x – intercepts of the function.


completion of this task because the participant successfully translated the symbolic form

of f (x) = -3x (x – 1) (x – 2) given in task seven to its graphic form using the graphing

calculator. None of Ruthven’s (1990) role of graphing calculators’ approaches were

present by the way the student used the graphing calculator. The role of the graphing

calculator in task seven was to display the graphic and numeric forms of the function that

was given in symbolic form in the task. In addition, the participant analyzed both the

graphic and numeric forms of y = -3x (x- 1) (x – 2) using the calculator.

Therefore, the student used the following five categories during the completion of




Category III was using various features of the graphing calculator. Category IV was

looking for a relationship between the graphic form of a function and the numeric form.

Category V was plotting specific points of a function on a graph.

During the completion of mathematical task # 7, the participant used non-visual

methods to solve the task. First, the student translated the symbolic form of

y = -3x (x – 1) (x – 2) to its graphic form using the graphing calculator. Secondly, the

participant translated the graphic form of y = -3x (x – 1) (x – 2) to its numeric form using

the TABLE feature on the graphing calculator. Thirdly, NVL translated the numeric

form of y = -3x (x – 1) (x – 2) to its graphic form by plotting specific points on the

coordinate grid in task seven. In fact, the student shared his personal viewpoint of this

task.

124

The first thing I realized I had to do in this task is to find the zeros. Plugging it

into the calculator took some of the work out of it. All I had to do was go to the

table and pull up the zeros. Once I plugged the zeros in, all I had to do was look

at the graph the calculator drew and replicate it. The longer way would have

been to draw a number line, do test point, shade in regions, etc., which would

have been much more time consuming.

Task # 8

Mathematical task # 8 was an absolute value function (Figure 5.17) by Ruthven.


and explain his reasoning. Using the graphing calculator was a requirement during the

completion of this task because the researcher wanted to see how NVL would use it.

Task eight was chosen because the researcher wanted to know how the student would

solve a task involving absolute value functions.


In response to the eighth mathematical task, NVL stated: “Um, the first thing I

notice about task number eight is, well, I guess the way the shape is, that it’s an absolute

value function. So, I know that’s the first thing”.

The participant examined the graph of a function given in the task. Then, the student

concluded that the task eight graph was an absolute value function because of its shape.

His experience in College Algebra exposed the participant to the symbolic and graphic

forms of absolute value functions. College Algebra also exposed the student to

translations or transformation of the graphs of various types of functions.

During the completion of mathematical task # 8, NVL responded: “Um,

secondly, I’ll take a guess here, it looks like the function could read the absolute value,

which I can type in here [after selecting the Y = button on the graphing calculator] of x –

125

2 maybe, possibly. Let’s just see”. The student already reported that the graph provided

in the task was an absolute value function. At this point, the participant was trying to

determine the precise symbolic form of the function in task eight. The first equation tried

he tried was y = x − 2 . The student explained that he chose this equation by guessing.

The participant inputted y = x − 2 into the graphing calculator as y = abs (x – 2 ). (On

the calculator abs was abbreviated for the absolute value function.) After entering the

equation, the student used the GRAPH feature in order to view the graphic form of y =

x − 2 and to compare it with the graph given in task eight.

During this excerpt of the interview, it appeared that the participant was looking

for a relationship between the symbolic form of a function and the graphic form as part of

the completion of task eight. It also appeared that NVL used the NVL used the graphing


graphic form as part of the completion of mathematical task # 8.

During this excerpt of the interview, it appeared that the participant translated



Y = x − 2 . As a result, O’Callaghan’s (1998) translating component for understanding


While the student examined the graph of y = x − 2 on the graphing calculator, he

reported: “No, Okay, so I saw that if you just did the absolute value of x minus two it

shifted your graph [from the origin] over two [referring to the graph of

y = x ]”. First, the participant admitted that he first guess of y = x − 2 did not match

the graph provided in task eight. Secondly, NVL described the graph of y = x − 2 as the

horizontal translation of the graph of y = x two units to the right of the origin. Even

though the student did not use the words translation or transformation at this point during

the interview, he expressed the meaning of these words in terms of visual shifting of a

graph. In addition, it appeared that NVL focused on specific visual features of the graph

of a function as part of the completion of this task.

In response to mathematical task # 8, NVL stated: “Or, it could be abs [sic]

[pressed the abs button on the graphing calculator], right now I’m just doing basic

guessing, um, cause I know there’s a two involved in here somewhere. So I did plus two

[referring to the equation entered into the calculator], probably shift your graph

somewhere here [pointing to the graph on the task sheet] when x equals negative two”.

The second equation he tried to match with the graph provided in the task was y = x − 2 .

Again, the student explained that he chose this equation by guessing. The participant

inputted y = x − 2 as y = abs (x + 2) into the graphing calculator. NVL also described a

horizontal translation of y = x two units to the left of the origin using the graph on the

task sheet. After entering the equation, the student used the GRAPH feature to display

the graphic form of y = x + 2 .

During this portion of the interview, it appeared that the participant was looking


the completion of mathematical task # 8. It also appeared that the student used the


and the graphic form as part of the completion of the eighth task.

126


from one representation of a function to another representation. In this instance, the


y = x + 2 . As a result, O’Callaghan’s (1998) translating component for understanding


After examining the graph of y = x + 2 on the graphing calculator, the

participant expressed, “So that makes me think that it’s the absolute value of x plus a

number”. NVL looked at the graph of y = x + 2 . The graphic form of the equation

confirmed his prediction regarding a horizontal translation to the left of the origin by two

units. At this point, the student made a prediction about the general form of the solution

to mathematical task # 8.

During the completion of the eighth mathematical task, NVL reported:

So, that’s the absolute value of x minus 2 (enters on y = abs (x) – 2 on the

graphing calculator). It did that at the same time. Let’s check our zoom

(referring to zoom feature on the graphing calculator) um, regular (presses under

zoom feature). We have it at (-) 2 now but then at the same time it actually hits

over here at um (-) 1 (referring to task #11 graph) which is kind of tricky.

The participant inputted y = x - 2 as y = abs (x) – 2 into the graphing calculator. Next,

the student pressed the GRAPH button. After that, the participant examined the graph of

y = x - 2 on the graphing calculator and compared it to the graph given in task eight.

After looking at both graphs, he said, “It did that at the same time”. When the researcher

asked the student for clarification of this statement, he explained that the graph in task

eight had both a horizontal and vertical translation. Then, the participant pressed the

ZOOM key on the calculator and reported that the vertex of y = x - 2 was (0, -2). The

student also reported that the vertex of the function provided on the task sheet was (-1, -

2). NVL described the graph of y = x - 2 having a vertical translation and the graph of

the function given in the task having both a horizontal and vertical translation as “…kind

of tricky”.

During this segment of the interview, it appeared that the participant was looking


the completion of task eight. It also appeared that the student used the graphing


graphic form as part of the completion of the eighth mathematical task. In addition, it

appeared that NVL focused on specific visual features of the graph of a function as part

of the completion of this task.

During this segment of the interview, it appeared that the participant translated

from one representation to another representation. Specifically, the student translated

from an equation (symbolic form) to a graph (graphic form) using y = x - 2. As a result,


shown.

In response to mathematical task # 8, NVL stated:

So now we have, hum, x we have absolute value x right here (on the graphing

calculator) minus two which would give you here (task #11 graph) (-) 1 so that

makes me wonder , not wonder but if we go on the first try I did absolute value of

x minus 2 shifted over two to the right so that would say the absolute value of x,

127

just to do the opposite, plus one, let’s see if that works (entered y = abs (x + 1) in

the graphing calculator). Ta Da! It does.

The student again restated what he saw from the graphs of the absolute value functions

using the calculator. First, he explained that the graph of y = x - 2 was a vertical

translation of y = x downward by two units from the origin using the graphing

calculator and the task eight sheet. Secondly, the participant reported how the graph of y

= x − 2 was a horizontal of y = x two units to the right of the origin. From this

information, the student tried the equation y = x + 1 in order to see if its graph would

include the horizontal translation shown in the eighth task. Thus, the participant inputted

y = abs (x + 1) into the graphing calculator. Next, the student pushed the GRAPH button

so that he could see the graphic form of y = x + 1 and if he graph included the horizontal

translation shown in task # 8.

During this excerpt of the interview, it appeared that the participant was looking


the completion of mathematical task # 8. It also appeared that the student used the


and the graphic form as part of the completion of task eight. In addition, it appeared that

NVL focused on specific visual features of the graph of a function as part of the

completion of this task.

During this excerpt of the interview, it appeared that the participant translated

from one representation of a function to another representation. Specifically, the student

translated from an equation (symbolic form) to a graph (graphic form) using y = x + 1 .

As a result, O’Callaghan’s 1998) translating component for understanding functions

seemed to be shown.

During the completion of the eighth mathematical task, NVL responded:

I’ll change the window settings to match x is -3 [for x minimum], 3 [for x

maximum], y is -2 [for y minimum], and 2 [for y maximum] actually match this

graph over here (task #11 graph) so we can get a better look at it. And there we

go. You have (-) 1, (-2) 2 (when x = -1, y = -2) giving us our function as the

absolute value of x plus one, minus two (entered y = abs (x + 1) – 2 on TI.-83). I

believe this is called translations and that’s it.

First, the student pressed the WINDOW button on the graphing calculator. Secondly, he

changed the x minimum, x maximum, y minimum, and y maximum settings to match the

values on the x and y axes in task eight. The participant expressed his reason for doing

this as being, “…so we can get a better look at it”. Next, the student inputted y = x + 1 -

2 as y = abs (x + 1) – 2 into the TI-83 graphing calculator. After that, NVL pushed the

GRAPH key to display the graphic form of y = x + 1 - 2. Then, he compared he graph

of y = x + 1 - 2 with the graph of the function given in the eighth task. Next, the

participant used the TABLE feature and verified (-1, -2) was included on the graph.

During this portion of the interview, it appeared that the participant was looking


the completion of mathematical task # 8. It also appeared that the student used the TI-83


and the graphic form as part of the completion of task eight. In addition, NVL used

various features of the graphing calculator as part of the completion of this task.

128

Overall, O’Callaghan’s translating component for understanding functions was

present during the completion of this task because the student translated the graphic form

of y = x + 1 - 2 given in task eight to its symbolic form using the graphing calculator.

Ruthven’s (1990) role of graphing calculators’ Graphic-Trial Approach was also present

because the participant used the graphing feature of the graphing calculator to

continuously change the symbolic form of a function after comparing various graphs with

the given graph to complete the eighth mathematical task. Specifically, NVL compared

y = x − 2 , y =, y = x + 2 , y = x - 2, y = x + 1 , y = x + 1 - 2.


the eighth mathematical task. Category I was looking for a relationship between the




Category IV was focusing on specific visual features of the graph of a function.

During the completion of mathematical task # 8, the student used visual methods

to solve the task. The participant recognized the given graph of the function in the task as

an absolute value function. Then, NVL used the graphing calculator to compare various

graphs with the given graph in order to modify the symbolic form of a function and

complete the task. In fact, the student shared his personal viewpoint of this task in the

College Algebra Journal.

The first thing I noticed about this graph is that it is an absolute value function. I

was able to tell by the shape. I had to do a little trial and error when it came to the

translations though. I guess this is where my weakness in visual learning pops up.

But trial, error, and the calculator proved to be more than enough to cripple my

visual learning impairment.

Even though the student did not rely visual imagery as defined by Lean and

Clements (1981), his use of visualization as defined by Arcavi (2003) appeared to be

depicted during the completion of task eight. Arcavi (2003) defined visualization as

“…the ability, the process and the product of creation, interpretation, use of and

reflection upon pictures, images, diagrams in our minds on paper or with technological

tools…” (p. 217). This definition is broad enough to include the creation or production

of visual images using technological tools. In this case, the technological tool was the

graphing calculator. The participant used the calculator to display the graphs of several

absolute value functions because he did not know what the graphs looked like. After the

pictures are created Arcavi (2003) explained as part of the definition of visualization that

their purpose included “…depicting and communicating information, thinking about and

developing previously unknown ideas and advancing understandings” (p. 217). During

the completion of the eighth mathematical task, using the graphing calculator, the student

repeatedly compared the graphs of the absolute value functions with the graph provided

in the task and made conclusions based on his understanding of functions. In addition,

Ruthven’s (1990) role of graphing calculators’ Graphic-Trial Approach also showed how

NVL used the calculator. Perhaps, the student might not have successfully completed

task eight without using the graphing calculator.

Task # 9

Mathematical task # 9 was an exponential function (Figure 5.18) created by the

researcher. The directions were for the participant to complete the task using the

129



NVL would use it. Task nine was chosen because the researcher wanted to know how

the student would solve a task involving exponential functions.

In the beginning of the interview regarding mathematical task # 9, the participant

stated, “Um, the first thing I notice [from looking at the graph], it’s be something to the

x”. When the student was asked for clarification regarding this statement, NVL

explained that he was looking for the symbolic form of an exponential function. The

participant’s experience in College Algebra exposed him to exponential functions.

Specifically, the student was introduced to the symbolic, numeric, and graphic forms of

y = 2 . From this first excerpt, it appeared that NVL was looking for a relationship


of task nine.

x


In response to mathematical task # 9, the participant reported:

So, let’s see, how would I figure this out? Hum, well I see at around (-) five

equals one, (-) five so I have (-) five comma one for my first point so that means

that if x equals (-) five whatever this is equals one [writes 1 = ? ]. So I have one

equals something to the (-) five.

−5

First, the student examined the graph of the function provided on the task nine sheet.

Secondly, NVL explained how he chose the point (-5, 1) from the graph. When x

equaled negative five, y equaled one. Using this ordered pair, the participant constructed

a possible equation to describe the given graph. During this segment of the interview, it

130

appeared that NVL substituted specific values for the variables x and y as part of the

completion of the ninth mathematical task. It also appeared that the student looked for

relationships between the graphic form of a function and the numeric form as part of the

completion of this task. In addition, it appeared that the participant looked for a

relationship between the numeric form of a function and the symbolic form as part of the

completion of task nine.

During the completion of the ninth mathematical task, NVL said, “Let’s see, so

what [number] to the negative five would equal one, hum, well, maybe if I did this plus

five [writes 1 = ? -5 + 5]. That means this could basically be any number and still equal

one. So, I’m gonna guess two”. At first, the student tried to think of a number that when

raised to the negative fifth power produced a result of one. Next, the participant

constructed a different equation by adding the negative fifth power to a positive fifth

power. Then, NVL shared his conclusion that the question mark’s value “…could

basically be any number and still equal one”. When the researcher asked the student for

clarification, NVL explained that he learned in College Algebra a number raised to the

zero power produced a result of one. Since the student believed that the question mark in

his constructed equation could be any numerical value, he chose two by guessing. Later

on, the participant reported to the researcher that if using two in the equation did not

match the given graph, NVL would try three and keep going in ascending order until he

constructed an equation that matched the given graph.

In response to mathematical task # 9, the student continued:

Let’s see one equals two to the (-) five plus five which would give you zero which

would equal one. So that equation would actually equal, y equals two to the (-)

five, (-) two to the x plus five [writes y = 2 on task sheet]. So, I’m going to

graph this and see if it actually holds true.

x+5

During this portion of the interview, the participant substituted two for the question mark

in the constructed equation. Then, the student solved the equation. Next, NVL replaced

(-5, 1) in the equation with the variables x and y. After that, the participant inputted y =

2 as y = 2 ^ (x + 5) on the graphing calculator. The student pressed the GRAPH

button in order to display the graphic form of y = 2 and to compare it to the graph

provided in task nine.

x+5

x+5

During this portion of the interview, it appeared that the participant substituted a

specific value for an unknown quantity as part of the completion of the ninth

mathematical task. It also appeared that the student looked for a relationship between the

numeric form of a function and the symbolic form as part of the completion of this task.

In addition, it appeared that NVL looked for a relationship between the symbolic form of

a function and the graphic form as part of the completion of task nine. Furthermore, it

appeared that the participant used the graphing calculator to construct a relationship


of this task.

In response to the ninth mathematical task NVL explained:

[using the graphing calculator] And let’s see if my points match up. Go to my

table (referring to table key located on the graphing calculator) and I’m going to

go x equals (-) five and I get one and I’ll go back and read the graph. It looks like

the same distance between the points. I have a space here [shading with his pencil

the area between the (+) y –axis and the graph of the function on the task sheet]

131

and I have space in there [referring to the space between the (+) y-axis and the

graph of the y = 2 ^ (x + 5) on the graphing calculator] and um, I’ll go with one:

y equals to the x plus five. 1

After the student inputted the equation (symbolic form) y = 2 as y = 2 ^ (x + 5) and

displayed the graphic form of the function on the graphing calculator, he used the

TABLE feature. The participant reported his reason for using the TABLE feature as

being to “…see if my points match up”. Specifically, NVL chose to use the TABLE key

to find the corresponding y – value when x equaled negative five. According to the

TABLE, when x equaled negative five, y equaled one. Next, the student examined the

graphic form of y = 2 on the graphing calculator. Then, the participant compared the

graph of y = 2 to the given graph in task nine and made conclusions. The first

conclusion that the student reported was, “it looks like the same distance between the

points” along the curves of both graphs. The second conclusion that NVL reported

pertained to the area between the graph of the function and the y – axis. The participant

explained, “I have a space here [shading with his pencil the area between the y – axis and

the given graph on the task sheet] and I have a space in there [pointing with his pencil to

the area between the y – axis and the graph of y = 2 on the graphing calculator]”.

After the student shared these conclusions, he informed the researcher that y = 2 was

his final solution to the problem.

x+5

x+5

x+5

x+5

x+5



completion of the ninth mathematical task. It also appeared that NVL looked for a


completion of this task. In addition, it appeared that the student used the graphing


graphic form as part of the completion of task nine. Furthermore, it appeared that the

participant used various features of the graphing calculator as part of the completion of

this task.


completion of this task because the student translated the graphic form of y = 2 given

in task nine to its symbolic form using the graphing calculator. None of Ruthven’s

(1990) role of graphing calculators’ approaches was present by the way the student used

the graphing calculator. The role of the graphing calculator in task nine was to display

the graphic and numeric forms of the function that was given in graphic form in the task.

In addition, the participant analyzed both the graphic and numeric forms of y = 2

using the calculator.

x+5

x+5

Therefore, the student used the following seven categories during the completion


form of a function and the graphic form. Category II was substituting specific values for

the variables x and y. Category III was looking for a relationship between the graphic

form of a function and the numeric form. Category IV was looking for a relationship

between the numeric form of a function and the symbolic. Category V was substituting a

specific value for an unknown quantity. Category VI was using the graphing calculator


Category VII was using various features of the graphing calculator.

132

During the completion of the ninth mathematical task, the participant used non-

visual methods to solve the task. First, the student chose a point from the given graph in

the task. Secondly, using the ordered pair, NVL constructed a possible equation using a

question mark as a symbol to designate an unknown quantity to describe the given graph.

Thirdly, he modified the equation. Next, the student substituted two for the question

mark and replaced the numerical values from the ordered pair with the variables z and y

in the constructed equation of y = 2 . After that, NVL analyzed the graphic and

numeric forms of y = 2 on the graphing calculator. In fact, the participant shared his

personal viewpoint of this task.

x+5

x+5

In this task, the first thing to do was to pick solid coordinates. Once I got them

(-5, 1). I plugged them into a basic formula y = a . That changed into 1 = a .

So I got to thinking that no whole number to the -5 would equal 1. The next

logical thing would be to make the exponents equal zero, so that, regardless of the

base, it is one. I chose 2 as a base because zero to a power is zero, and one to any

power is one, so either would fit the graph. Two was the next step up.

x −5

Task # 10

Mathematical task # 10 was another exponential function (Figure 5.19) created by

the researcher. The directions were for the participant to complete the task using the



NVL would use it. The tenth task was chosen to see how the student would solve a

second problem involving exponential functions. In addition, he researcher wanted to

know if the participant would use the same categories from task nine in the completion of

the tenth mathematical task.

In response to mathematical task # 10, NVL stated, “Okay, so let’s see what e to

the x minus one, well there’s a little function here for it [referring to the graphing

calculator]. You can do second, [pressed the 2nd

key] e to the, [pressed the LN button]

and plug in your x minus one”. First, the student looked at the buttons and features on

the graphing calculator. Secondly, he explained how to enter an exponential function

into the calculator. As the participant explained the process, he inputted

f(x) = e as y = e ^ (x – 1) into the graphing calculator. After entering the equation

(symbolic form) f(x) = e , the student described what he thought the graph (graphic

form) would look like. “It’s gonna be real shoo [sic] [moves hand by making a half “U”

shape in the air, beginning at the bottom, then slowly rotating in a curved shape to the

right and upwards]

x−1

x−1

I’m thinking”. The participant tried to show the researcher the visual image he had

regarding the behavior of exponential functions by demonstrating the movement with his

hand.

133


After viewing the graph of f(x) = e on the graphing calculator, NVL x−1

pushed the GRAPH button and explained what he saw. First, the participant reported that

the graph of f(x) = e on the calculator corroborated his visual image of the function.

Secondly, NVL used the FORMAT feature and turned off the x and y axis. Next, the

student described the shape of the function as he looked at it on the graphing calculator.



completion of the tenth mathematical task. It also appeared that the student used the

x−1

134


and the graphic form as part of the completion of this task. In addition, it appeared that

NVL focused on specific visual features of the graph of a function as part of the

completion of mathematical task # 10.

In response to the tenth mathematical task, the participant suggested, “So, we can

go to our table [referring to TABLE feature on the graphing calculator] and see that here

it’s these tiny little decimals barely at one. Then it jumps from point three to one just like

that and then you start going to more decimals”. During this portion of the interview, the

student used the TABLE feature and analyzed the displayed numerical values. It

appeared that NVL looked for a relationship between the graphic form of a function and

the numeric form as part of the completion of this task.

In response to mathematical task # 10, the student explained, “So, we have one

comma one here [reading from TABLE of numerical values for e ], so one and one

[plots (1, 1) on the graph]. We have a jump. All these are pretty much close to zero”.

During this segment of the interview, NVL selected the point when x equaled one, y

equaled one from the TABLE. Next, he labeled the x – axis with the value of one and the

y – axis with the value of one. After that, the student plotted (1, 1) on the coordinate grid

provided in task # 10. The student also reported that when the variable x represented a

negative number for the given function, the variable y represented numbers that were

“…pretty much close to zero”. The participant explained further that as the x value

increased from negative numbers to one “…a quick jump…” occurred in the graph

because the y values increased from decimals less than zero to one. Then, he drew a

horizontal slightly straight line along the x – axis and connected it to the point (1, 1) to

demonstrate his explanation.

x−1

During this segment of the interview, it appeared that the student plotted a

specific point of a function on a graph as part of the completion of the tenth task. It also

appeared that the participant looked for a relationship between the graphic form of a

function and the numeric form as part of the completion of this task.

In response to the tenth mathematical task, NVL stated:

You get, none of these are actually whole numbers, you jump from 2 to 7 to 20 to

54, 148, 403. Um, because this is exponential so you really pretty much gonna go

something just like this [draws a smooth curve going upward from the point (1,

1)] which is what the graph shows.

At this point, the student reported the increasing numerical values that represented y in

the equation y = e from the TABLE on the graphing calculator. (He rounded the

decimal values to whole numbers.) The participant examined the numerical values from

two to 403. After that, the student drew a smooth curve going upwards from the point (1,

1) with an arrow at the end in order to complete the task.

x−1


completion of this task. Specifically, the student translated the given symbolic form of

f(x) = e to graphic form using the graphing calculator. The participant also translated

the graphic form of f(x) = e to its numeric form using the TABLE feature on the

graphing calculator. None of Ruthven’s (1990) role of graphing calculators approaches

were present by the way NVL used the graphing calculator. The role of the graphing

calculator in task # 10 was to display the graphic and numeric forms of the function that

x−1

x−1

135

was given in symbolic form. In addition, the student analyzed both the graphic and

numeric forms of f(x) = e using the calculator. x−1


the tenth mathematical task. Category I was looking for a relationship between the

symbolic form of a function and the graphic form. Category II was using the TI-83


and the graphic form. Category III was focusing on specific visual features of the graph

of a function. Category IV was looking for a relationship between the graphic form of a

function and the numeric form. Category V was plotting specific points of a function on

a graph. Category VI was using various features of the graphing calculator.



imagery regarding the general behavior of positive exponential functions during the

beginning of the task. NVL’s use of the graphing calculator confirmed his image and the

participant continued to complete the task. From this point, he used the TABLE feature,

plotted a specific point, and drew the graph of the function, f(x) = e , using the

graphing calculator. In fact, the participant shared his personal viewpoint of this task in

the College Algebra Journal. NVL wrote, “All this task truly involved was plugging the

function into the calculator. Once I did that, I used the table to find my points. The only

ones that were needed were 1, 1 as everything else was decimals”.

x−1

In conclusion, the findings of one of the case studies of the two College Algebra

students were reported in this chapter. Several categories emerged from the data. They

were labeled in Chapter 5 using Roman numerals. To summarize, all of the emerging

categories were listed using alphabetical letters with the corresponding mathematical task

or tasks in Table 5.1.



• Category C: looking for a relationship between the graphic form of a function and

the numeric form





• Category F: using the graphing calculator for arithmetical operations





• Category J: substituting a specific value for an unknown quantity

136

Table 5.1: NVL’s Emerging Categories


A 1, 2, 6, 9

B 2, 7, 10

C 5, 7, 9, 10

D 2

E 1, 2, 3, 4, 5, 6, 7, 8, 9, 10

F 5, 9

G 1, 2, 3, 4, 5, 6, 7, 9, 10

H 1, 2, 3, 4, 5, 6, 7, 8, 9, 10

I 4, 5, 8, 10

J 9

137

CHAPTER 6

CONCLUSIONS










The findings of the second case study were reported in chapter 5.

In the current study, no assertion was considered unless it could be supported by

two or more pieces of data. First, the interviews were triangulated among the different

participants. Secondly, the interview sessions and documents were triangulated.

Specifically, the linear, quadratic, cubic, absolute value, and exponential functions were

triangulated among the College Algebra Writing Journals, Web homework, tests, and a

researcher’s journal.

Member checking was applied to the present study. The visualizer was allowed to

read Chapter 4. She was asked to assess and correct any errors found. In addition, the

nonvisualizer was allowed to read Chapter 5. He was asked to assess and correct any

errors found. All of the member check responses were recorded.







exponential functions. Furthermore, prior to receiving the tasks, the students’ preferences





In the reports of Cases 1 and 2, as patterns emerged from the data the researcher

called them categories. To summarize, all of the Visualizer’s (VL) emerging categories

were listed using alphabetical letters with the corresponding mathematical task or tasks in

Table 6.1.



• Category C: detecting a relationship between the concepts slope and steepness






138





Table 6.1: VL’s Emerging Categories


A 1, 3, 4

B 1, 7, 10

C 1

D 2

E 3, 4, 5, 7, 8, 9, 10

F 3, 4, 6

G 4, 5, 7, 9, 10

H 5, 7, 10

I 8, 9

Similarly, all of the Nonvisualizer’s (NVL) emerging categories were listed using

alphabetical letters with the corresponding mathematical task or tasks in Table 6.2.



• Category C: looking for a relationship between the graphic form of a function and

the numeric form










• Category J: substituting a specific value for an unknown quantity

139

Table 6.2: NVL’s Emerging Categories


A 1, 2, 6, 9

B 2, 7, 10

C 5, 7, 9, 10

D 2

E 1, 2, 3, 4, 5, 6, 7, 8, 9, 10

F 5, 9

G 1, 2, 3, 4, 5, 6, 7, 9, 10

H 1, 2, 3, 4, 5, 6, 7, 8, 9, 10

I 4, 5, 8, 10

J 9

Summary

The seven emerging categories that were used by both of the participants were A,

B, D, E, G, H, and I. During the completion of mathematical task #1, both of the students

used Category A. During the completion of the second mathematical task, both of the

participants used Category D. During the completion of mathematical task #3, VL and

NVL used Category E. During the completion of the fourth mathematical task, both of

the students used Categories E and G. During the completion of mathematical task # 5,

both of the participants used Categories E, G, and H. During the completion of the sixth

mathematical task, VL and NVL did not use the same categories. Both of the students

used Categories B, E, G, and H during the completion of mathematical task #7. Both of

the participants used Categories E and I during the completion of the eighth mathematical

task. During the completion of mathematical task # 9, VL and NVL used Categories E

and G. Both of the students used Categories B, E, G, and H during the completion of the

tenth mathematical task.

Mathematical task #1 was a linear function by Moschkovich (1998).


Both of the participants used this category during the completion of task one. In fact,

Category A was the first one displayed by the students. VL explained that she substituted

specific values for the variables x and y into equations to “actually graph the y = x + 5”.

The student referred to the substitution procedure as her way of trying “to find y = x + 5”.

NVL explained his reason for demonstrating Category A.

Um, I guess I did this because I was trying to look at this visually. I see x, you

know, on the graph [referring to the graphic form of y = x] in my mind. So, I

used this [pointing to the y = x + 5 table] to give me the coordinates for it.

Thus, both students translated the given symbolic form of a function to its numeric form

as an intermediate step. The next step was translating from the numeric form of a

function to its graphic form. Furthermore, O’Callaghan’s (1998) translating component

for understanding functions was present for the function y = x + 5.

Mathematical task # 2 was another linear function by Moschkovich (1998).

140


adding and/or multiplying specific values to the symbolic form of a function

Both of the students used this category during the completion of the second mathematical

task. One of the examples of how VL exhibited misinterpreting the graphical

representation of a function after adding and/or multiplying specific values to the

symbolic form of a function was provided by her explanation to Part B of the problem.

“Okay, if you add three to x [looks at y = 2x graph] I believe that it would be yes, cause if

you add three to x, you would move from zero to three on the positive side [points and

counts on the x – axis from zero to three and draws a line through (3, 0)]”. NVL also

provided an example of Category D. “I think you would add three, well, let me see. I’d

rather say I would lean, towards yeah, three because it looks like, well, cause from the

dark to the light [referring to the graphs of y = 2x and y = 2x + 6] it shifts over one, two,

three [counting on the x – axis to the left of the origin]”. Both of the students had a

misconception regarding the graphical transformation of the function y = 2x given the

graphs of y = 2x and y = 2x + 6 and the wording of question “ADD 3 TO X”.

Mathematical task # 3 was a quadratic function by Eisenberg and Dreyfus (1994).



During the completion of mathematical task # 3, both of the participants used this

category. This problem asked the students to “…write the corresponding [given] graph

(I, II, III, or IV)” beside the given symbolic form of the function. One instance of VL

reporting a correspondence regarding the symbolic form of x - 2x was, “So, I’m gonna

match my third function with graph II [writes II on the task # 3 sheet]”. Another instance

of NVL reporting a correspondence was, “So, x - 2x + 1 would be Roman numeral III”.

Both students translated among different representations of the four quadratic functions

during the completion of this task. As a result, O’Callaghan’s (1998) translating

component for understanding functions was present for y = x 2 - 2x + 1, y = 1 - x ,

2

2

2

y = x - 2x, and y = x 2 + 1. 2

Mathematical task # 4 was a quadratic function by Ruthven (1990).





During the completion of mathematical task # 4, both of the participants used these

categories. An example of VL using Category E was by stating, “It’s a parabola [writes

parabola on paper] and because, I would state that parabolas are usually, the function is

usually x squared”. NVL also looked for a relationship between the symbolic form of a

function and the graphic form in the following excerpt.

Okay, um, alright well looking from the graph [on mathematical task sheet] I can

see because it’s a parabola, that’s the first thing, I’m gonna have I know I’m

going to have x , um, but since it’s inverted, since it ‘s not the regular you know

parabola from top to bottom, it’s inverted, so I also know that’s it’s going to be

negative.

2

VL also used the graphing calculator to construct a relationship between the symbolic

form of a function and the graphic form. For instance, the student suggested, “So, let’s

try graphing [using the graphing calculator] let’s try graphing negative x squared”. She

141

inputted y = -x 2 into the calculator and said,“Okay, that’s what we need” while looking

at the graph on the calculator. In addition, NVL used Category G during the task. He

stated, “Um, so I’ll [writes -x 2 on paper] try that [reaching for the graphing calculator]

just to look at to see real quick how negative x squared looks”. The participant inputted y

= -x into the graphing calculator. Furthermore, both students translated the given

graphic form of

2

y = -x + 1 to its symbolic form using the graphing calculator. As a result,

O’Callaghan’s (1998) translating component for understanding functions was present.

2

Mathematical task # 5 was a cubic function by Ruthven (1990).






Both of the participants used these three categories during the completion of task five.

VL explained, “I’m gonna, basically, uh, look at my graph and try and figure out what

expression that will express this graph. First of all, I know it’s, uh, it’s x to the third…”.

During this segment of the interview, the student looked for a relationship between the

symbolic form of a function and the graphic form. NVL also demonstrated using

Category E. For example, “Uh, well I guess first off looking at he equation [referring to

y = ], I think it’s gonna be, I know it looks like x to the third because of the way the

curvature of the line”. VL also explained, “It’s gonna be, let’s see [enters y = x ^ 3 into

the graphing calculator] x to the third, positive x to the third. Let’s see what that graphs”.

Now, the student was using the TI-83 graphing calculator to construct a relationship

between the symbolic form of a function and the graphic form. In addition, NVL

exhibited Category G during the completion of task five. He stated, “Um, [entered y = x

^ 3 into the graphing calculator] okay, alright, well, the first thing I did was to see if x to

the third actually fit the, um, curvature of the line and it does. VL used various features

of the graphing calculator by inputting the symbolic form of a function, graphing the

function, and using the TRACE feature to analyze the graph. For example, after

inputting y = x 3 – 1 as y = (x ^ 3 + 2) – 1, the participant looked at the graph of the

function on the calculator. Then she suggested after the graph was shown, “and hit

TRACE [referring to TRACE key on the graphing calculator] and let’s see what two

equals”. NVL also used Category H during the completion of this task. For y = x +

x 2 – 2x , he inputted the function as y = x ^ 3 + x ^ 2 – 2x and examined the graph.

After that, the student stated, “ I’m actually going to check the TABLE and see [referring

to TABLE key on TI-83]. Um, yep for one, I get zero, zero I get zero, and negative one

uh, and negative two”. Both of the students translated the cubic function y = x using the

graphing calculator from its symbolic form to its graphic form. As a result,

O’Callaghan’s (1998) translating component for understanding functions was present.

3

3

Mathematical task # 7 was another cubic function.




142



Both of the participants used these three categories during the completion of task seven.

For instance, from the following excerpt of he interview, VL used Categories E and G.

[using the graphing calculator] I’m gonna put negative 3x brackets minus one

bracket open bracket minus two bracket. I’m gonna press GRAPH. I’m gonna

observe the graph and from this graph [pauses] from this graph I can tell by

looking at he calculator I press TRACE and you can tell exactly where the

different, distinct points are.

××

The participant inputted y = -3x(x – 1) (x – 2) into the graphing calculator. She examined

the graph and used the TRACE button to locate specific points that fell along the curve of

the graph of the cubic function. After that, the student plotted (0, 0), (1, 0), (0.5, -1), (2,5,

-6.5), (1.7, 1), and (-1, 20) on the coordinate grid provided on the task seven sheet, which

demonstrated Category B.

In another example, NVL used Categories E and G. He reported, “Using the

graphing calculator, to me the easiest way to do it would just be type, go to our Y =, type

in the function negative 3x times x minus one times x minus two, alright, then GRAPH”.

The inputted y = -3x(x – 1) (x – 2) into the calculator. Then, he examined the graph and

used the TABLE feature to locate specific numerical values that represent x and y in the

function. The participant plotted (0, 0), (1, 0), and (2, 0) on the coordinate grid provided

on the task # 7 sheet, which demonstrated Category B.

VL and NVL translated f(x) = -3x (x – 1) (x – 2) from its given symbolic form to

its graphic form. As a result, O’Callaghan’s (1998) translating component for

understanding functions was present during the completion of this task.

Mathematical task # 8 was an absolute value function by Ruthven (1990).




During the completion of the eighth mathematical task, both of the participants used these

two categories. For instance, VL stated, “By looking at the graph I can tell that this is an

absolute function, therefore, that’s the first thing I’m gonna do with my y equals, put my

absolute symbols [draws two straight lines apart from each other as absolute value bars]”.

During this segment of the interview, the student was looking for a relationship between

the symbolic form of a function and the graphic form. The next example showed how

NVL used Category E. “Um, the first thing I notice about task # 8 is well I guess the way

the shape is that it’s an absolute value function”, Next he examined the graph of

Y = x − 2 on the graphing calculator.

VL wrote y = x + 1 – 2. Then she explained, “And my reasoning for this is that

the original graph starts, the original graph vertex is at (0, 0) and the given graph is, has

moved to the left one and down two”. By reporting the horizontal and vertical

translations of the function, the student displayed Category I.

Regarding y = x − 2 , NVL reported, “So I saw that if you just did the absolute

value of x minus two, it shifted your graph over two”. The explanation of the horizontal

translation was an example of focusing on specific visual features of the graph of a

function.

In addition, both of the participants translated the given graphic form of

143

y = x + 1 – 2 to its symbolic form. As a result, O’Callaghan’s (1998) translating

component was present.

Mathematical task # 9 was an exponential function created by the researcher.





During the completion of the ninth mathematical task, both of the students used these

categories. For instance, VL reported:

…[the graph of y = 2 ] shifts on the x – axis negative five so I believe that I

would have to [pauses] hum, it would, okay so I put two to the x negative five

[writes y = 2 -5 on task sheet].

x

x

She inputted y = 2 ^ x -5 into the graphing calculator and looked at the graph. This

segment of the interview showed Categories E and G. In another example, NVL

explained, “Um, the first thing I notice it’s gonna be something to x”, which showed

Category E. Category G was depicted when he inputted y = 2 as y = 2 ^ (x + 5) into

the graphing calculator and looked at the graph.

x+5

Mathematical task # 10 was another exponential function created by the

researcher.







Both of the students used these four categories during the completion of the tenth

mathematical task. For instance, VL explained, “It says e to the x minus one [inputs

y = e ^ (x – 1) on the graphing calculator]. Press GRAPH and TRACE and when x = 0, y

= 0.36”. This excerpt showed Categories E, G, and H. The student also plotted the

points (0, 0.36), (1, 1), (2, 3), and 3, 9), which demonstrated Category B.

During task # 10, NVL reported: “Okay so let’s see what e to the x minus one,

well there’s a little function here for it you can do second [referring to the 2nd

key on the

graphing calculator] e to the [referring to the e ^ key on the graphing calculator] and plug

in your x minus one”. The student inputted f(x) = e as y = e ^ (x – 1) into the graphing

calculator, pressed GRAPH and used the TABLE feature. In this excerpt, NVL used

Categories E, G, and H. He also plotted the point (1,1) on the graph which showed

Category B.

x−1

In addition, both students translated f(x) = e among different representations of

a function. As a result, O’Callaghan’s (1998) translating component for understanding

functions was present during the completion of this task.

x−1

Discussion

In the current study, the visual and non-visual mathematical learners’


component (O’Callaghan, 1998) for understanding functions. In this investigation,

O’Callaghan’s (1998) translating component was present during the completion of linear,

144

quadratic, cubic, absolute value, and exponential functions. Specifically, both of the

participants translated the following functions:

• Linear: y = x and y = x + 5

• Quadratic: y = x – 2x + 1, y = 1 – x , y = x – 2x, y = x + 1, y = -x 2 , and 2 2 2 2

y = -x + 1 2

• Cubic: y = x and f(x) = -3x (x – 1) (x – 2) 3

• Absolute Value: y = x and y = x + 1 – 2

• Exponential: f(x) = e x−1

Table 6.3: Presence of O’Callaghan’s (1998) Translating Component

Linear Two Functions

Quadratic Six Functions

Cubic Two Functions

Absolute Value Two Functions

Exponential One Function

This means that the Visualizer and Nonvisualizer translated the linear, quadratic, cubic,

and absolute value functions listed above. It also means, using O’Callaghan’s (1998)

translating component, both students understood the linear, quadratic, cubic, absolute

value, and exponential functions listed above that were given in the current study (Table

6.3).

In the present study, the role of graphing calculators was interpreted based on


(Ruthven, 1990). NVL used the graphing calculator during the completion of all of the

mathematical tasks. VL used the graphing calculator during the completion of

mathematical tasks four, five, seven, nine, and ten.

Table 6.4: Role of Graphing Calculators

TASKS VL NVL

Four (Quadratic) Analytic-Construction Analytic-Construction

Five (Cubic) Graphic-Trial Analytic-Construction

Seven (Cubic) Display/Analyze graphic &

numeric forms

Display/Analyze graphic &

numeric forms

Nine (Exponential) Graphic-Trial Display/Analyze graphic &

numeric forms

Ten (Exponential) Display/Analyze graphic &

numeric forms

Display/Analyze graphic &

numeric forms

145

For mathematical task # 4, the role of the graphing calculator for both of the participants

was the Analytic-Construction Approach because the students used connections between

the symbolic and graphic forms of functions to construct the exact symbolic form of a

function (Table 6.4). For the fifth mathematical task, VL’s use of the graphing calculator

demonstrated the Graphic-Trial Approach, while NVL’s use of the calculator

demonstrated the Analytic-Construction Approach. VL depicted the Graphic-Trial

Approach because the participant used the calculator to compare different graphs to the

given graph in order to change a constructed symbolic expression based on information

interpreted from the graphs. NVL exhibited the Analytic-Construction Approach by

using connections between the symbolic and graphic forms of functions to construct the

exact symbolic form of a function. In mathematical task # 7, none of Ruthven’s (1990)

role of graphing calculators’ approaches was present by the way the students used the

calculator. Both of the participants used the graphing calculator to display the graphic

and numeric forms of the function that was given in symbolic form in the task. In

addition, the participants analyzed both the graphic and numeric forms of y = -3x (x – 1 )

(x – 2 )using the calculator. VL analyzed the numeric form of the function by using the

TRACE button on the graphing calculator. NVL analyzed the numeric form of the

function by using the TABLE feature. In the ninth mathematical task, the role of the

graphing calculator was the Graphic-Trial Approach for the Visualizer. She exhibited the

Graphic-Trial Approach by using the calculator to compare different graphs to the given

graph in order to change a constructed symbolic expression based on information

interpreted from the graphs. None of Ruthven’s (1990) role of graphing calculator

approaches’ was present by the way the Nonvisualizer used the graphing calculator. He

used the calculator to display the graphic and numeric forms of the function that was

given in graphic form in the task. In addition, the student analyzed both the graphic and

numeric forms of y = 2 using the calculator. Again, he used the TABLE feature to

analyze the numeric form of the function. For the tenth mathematical task, none of

Ruthven’s (1990) role of graphing calculators’ approaches was present by the way the

participants used the graphing calculator. The role of the graphing calculator in this task

was to display the graphic and numeric forms of the function that was given in symbolic

form. In addition, the student analyzed both the graphic and numeric forms of f(x) = e

using the calculator in order to complete the task. VL analyzed the numeric form of the

function by using the TRACE button on the graphing calculator. NVL analyzed the

numeric form of the function by using the TABLE feature.

x+5

x−1

Therefore, the Analytic-Construction Approach (Ruthven, 1990) contributed to

the students’ demonstrations of understanding functions because they used the

connections between the symbolic and graphic forms of functions to construct the exact

symbolic form of a function to complete task four. For VL, the Graphic-Trial Approach

(Ruthven, 1990) did not contribute to the student’s demonstration of understanding

functions because she did not construct an accurate equation (symbolic form) to match

the graph (graphic form) of the function that was provided in mathematical task # 5. On

the other hand, the Analytic-Construction Approach contributed to NVL’s demonstration

of understanding functions because he used the connections between the symbolic and

graphic forms of functions to construct the exact symbolic form of a function to complete

146

task five. Even though Ruthven’s (1990) approaches were not depicted, the role of the

graphing calculator contributed to the students’ demonstrations of understanding

functions in mathematical task # 7 because they used the calculator to construct the graph

of a function. For VL, the Graphic-Trial Approach did not contribute to the student’s

demonstration of understanding functions because she did not construct an accurate

equation (symbolic form) to match the graph (graphic form) of the function that was

provided in mathematical task # 9. Even though Ruthven’s (1990) approaches were not

shown, the role of the graphing calculator contributed to NVL’s demonstration of

understanding functions in task nine. Similarly, even though Ruthven’s (1990)

approaches were not depicted, the role of the graphing calculator contributed to the

participant’s demonstrations of understanding functions in the tenth task because he used

the calculator to construct the graph of a function.

The role of graphing calculators in the present study was similar to their role in

Hollar and Norwood (1999), Milou (1999), and Sarmiento (1997) because graphing

calculators were seen as a benefit in helping students’ understanding of algebraic

concepts. In addition, how the participants used graphing calculators in the current study

correlated with Hennessy, Fung, and Scanlon (2001). These three authors felt that

graphing calculators encouraged representing functions in numeric, symbolic, and

graphic manner. Similarly, in the present study, the participants used the graphing

calculator to represent functions numerically, symbolically, and graphically.

In the current study, visual imagery was used by the Visualizer in four of the

mathematical tasks and by the Nonvisualizer in three of the tasks.

Table 6.5: Visual Imagery

PARTICIPANTS MATHEMATICAL TASKS

Visualizer (VL) One, Four, Eight, Nine, and Ten

Nonvisualizer (NVL) One, Four, and Ten

VL relied on visual imagery during the completion of tasks one, four, eight, and nine.

NVL relied on visual imagery during the completion of tasks one, four, and ten.

Visual imagery contributed to both students’ demonstrations of understanding

functions because they relied on images of linear functions during the completion of task

one. Similarly, visual imagery contributed to both participants’ demonstrations of

understanding functions because they relied on images of quadratic functions during the

completion of task four. For VL, visual imagery contributed to her demonstration of

understanding functions because she relied on images of absolute value functions during

the completion of task # 8. Visual imagery did not contribute to VL’s demonstration of

understanding functions during the completion of task # 9 because she did not construct

the exact symbolic form of a function that was given in graphic form. Even though the

student relied on the visual image of an exponential function, she did not construct an

accurate equation (symbolic form) to match the given graph (graphic form) of the

function in task nine. In addition, visual imagery contributed to both participants’

147

demonstration of understanding functions because they relied on images regarding

exponential functions during the completion of the tenth task.

The participants’ reliance on visual imagery correlated with Presmeg (1989). She

stated, “visual imagery which is meaningful in the pupil’s frame of reference may lead to

enhanced understanding of mathematical concepts at primary and secondary levels” (p.

21). In the current study, how the College Algebra participants used visual imagery

contributed to their understanding of functions.

This study showed that the role of the graphing calculator for the visual and non-

visual mathematical learners aided their demonstration of understanding on many of the

functions used. These included linear, quadratic, cubic, absolute value, and exponential

functions. In fact, the graphing calculator appeared to be an essential tool by the way

both of the students used it.

In addition, this study showed that visual imagery contributed to the students’

demonstrations of understanding some of the functions. When visual imagery and

graphing calculators were used, the graphing calculator confirmed or disconfirmed the

visual images of the functions held by the student(s). Their use of the calculator made it

appear to be an essential tool for these students regarding their understanding of

functions. If the participant had a visual image (internal representation) of a function

before using the graphing calculator and he/she was not sure if it was an accurate

representation of a function then, after using the graphing calculator, he/she confirmed or

disconfirmed the picture he/she saw and continued to solve the mathematical problem.

One of the major goals in mathematics education is to ensure the success of all

students in mathematics. A way of accomplishing this goal is by incorporating different

kinds of learning experiences for the variety of learners in the College Algebra

classroom. These learning experiences should include the use of technology such as

graphing calculators and mathematical computer software (National Council of Teachers

of Mathematics [NCTM], 2000; Writing Team and Task Force of the Standards for

Introductory College Mathematics Project, 1995). Furthermore, “faculty should take

advantage of software and graphing calculators that are designed specifically as teaching

and learning tools. The technology must have graphics, computer algebra, spreadsheet,

interactive geometry, and statistical capabilities” (Writing Team and Task Force of the

Standards for Introductory College Mathematics Project, 1995, p. 3). It is important that

faculty members be open to learn about how to use technology in the classroom at all

levels. It is also important to look for ways to make the technological tools more

accessible to students. For instance, one of the participants in his study did not have a

graphing calculator. In addition, she had not used the technological tool very much. By

about the middle of the study, the student told the researcher that she was becoming more

comfortable with the graphing calculator. By the end of the study, the participant would

immediately reach for the calculator. This participant represents many of the students in

this College Algebra class. It is vital that every male and female College Algebra learner

see mathematics as meaningful and relevant. Subsequently, “mathematics instruction

must reach out to all students: women, minorities, and others who have …differing

learning styles…faculty must provide a supportive learning environment and promote

appreciation of mathematics” (p. 3).

The results of this study encouraged the integration of using technology in the

College Algebra classroom. Educators and textbook authors should be aware that some

148

students have little or no experience using the graphing calculator, so it would be helpful

to provide a step by step guide regarding how to use the calculator to explore among the

numeric, symbolic, and graphic forms of functions. For example, Lial, Hornsby, and

Schneider (2001), included a thorough plan of how to integrate the use of graphing

calculators in the teaching and learning of College Algebra, Trigonometry, and Pre-

Calculus. They explain the features of the graphing calculator. The textbook also

supplied examples in each chapter of how to complete a mathematical problem with the

calculator and without the calculator. Future teachers and textbook authors should also

integrate using the graphing calculator in College Algebra.

Limitations

There were three limitations of the current study. First, the researcher was also

the instructor. Before the study began, the participants knew the researcher as their

instructor. Therefore, the student-teacher relationship might have affected how the

participants responded to the researcher.

Secondly, the researcher brought her views of learning to the study. She believed

that learning occurs as an individual makes sense of mathematical and/or other

experiences over time. These experiences could take place inside and/or outside of a

classroom. In addition, these could be individualized experiences or shared experiences.

Thirdly, in the present study understanding of functions was measured by one

dimension of O’Callaghan’s (1998) framework – the translating component. The other

three components which were modeling, interpreting, and reifying were not investigated

during this study.

Future research. First, a future study could include a class of visualizers and a

class of nonvisualizers. The current study investigated one visualizer and one

nonvisualizer. Secondly, a future study could include only using cubic functions in the

mathematical tasks. The present study included linear, quadratic, cubic, absolute value,

and exponential functions. In addition, a future study could include only using

exponential functions in the mathematical tasks. Generally, the College Algebra course

spends less time on cubic and exponential functions and more time on linear and

quadratic functions. Another future study could include examining other types of

mathematical software such a MyMathLab and EDUCO. The current study examined the

role of graphing calculators. Furthermore, a future study could investigate Trigonometry

and/or Pre-Calculus students. The present study investigated College Algebra students.

149

APPENDIX A

MATHEMATICAL PROCESSING INSTRUMENT

Important:

(a) Do not write on this problem sheet. Write your solutions on the solution sheet

provided.

(b) For each problem, show your working as much as you can.

(c) You are required to attempt all problems.

SECTION B:

B-1. A track for an athletics race is divided into three unequal sections. The length of the

whole track is 450 meters. The length of the first and second sections combined is

350 meters. The length of the second and third sections combined is 250 meters.

What is the length of each section?

B-2. A balloon first rose 200 meters from the ground, then moved 100 meters to the east,

then dropped 100 meters. It then traveled 50 meters to the east, and finally dropped

straight to the ground. How far was the balloon from its starting point?

B-3. A mother is seven times as old as her daughter. The difference between their ages is

24 years. How old are they?

B-4. In an athletics race John is 10 meters ahead of Peter. Tom is 4 meters ahead of Jim

and Jim is 3 meters ahead of Peter. How many meters is John ahead of Tom?

B-5. At first, the price of one kg of sugar was three times as much as the price of one kg

of salt. Then the price of one kg of salt was increased by half its previous price, while

the price of sugar was not changed. If the price of salt is now 30 cents per kilogram,

what is the price of sugar per kilogram?

B-6. Some sparrows are sitting in two trees, with each tree having the same number of

sparrows. Two sparrows then fly from the first tree to the second tree. How many

more sparrows does the second tree then have than the first tree?

B-7. A saw in a sawmill saws long logs, each 16 meters long, into short logs, each 2

meters long. If each cut takes two minutes, how long will it take for the saw to

produce eight short logs from one long log?

B-8. A jar of kerosene weighs 8 kilograms. Half the kerosene is poured out of it, after

which the jar and contents weigh 421 kg. Determine the weight of the jar.

B-9. A passenger who had traveled half his journey fell asleep. When he awoke, he still

had to travel half the distance that he had traveled while sleeping. For what part of the

entire journey had he been asleep?

B-10. If you place a large, entire cheese on a pan of a scale and three quarters of a cheese

and a 43 kg weight on the other pan, the pans balance. How much does an entire

cheese weigh?

B-11. There was twice as much milk in one can as in another. When 20 liters of milk had

been poured from both cans, then there was three times as much milk in the first can

as in the second. How much milk was there originally in each can?

B-12. Ten plums weigh as much as three apricots and one mango. Six plums and one

apricot are equal in weight to a mango. How many plums balance the scales against one

mango?

150

APPENDIX B

MATHEMATICAL PROCESSING QUESTIONNAIRE

IMPORTANT:

On this questionnaire you are asked to consider how you did the mathematical

processing problems that you were recently asked to do. Every problem has three or more

possible solutions.

SOLUTIONS

SECTION B:

B-1. Solution 1: I solved this problem by imagining the track for the race and then

working out the length of each section.

Length of third section = 450-350 = 100 metres

Length of first section = 450-250 = 200 metres

Thus length of second section = 150 metres.

B-1. Solution 2: I drew a diagram that represents the track and then worked out the

length of each section.

250m

200m 150m 100m

151

350m

The length of the first section is 200 meters, the section section is 150 meters, and

the third section is 100 meters.

B-1. Solution 3: To solve this problem I drew conclusions from the information

given, and did not imagine or draw any picture at all:

Length of whole track is 450 m x+y+z

= 450

First and second sections combined is 350m x+y = 350

Conclusion: Length of third section =450-350=100m z

= 100

Second and third sections combined is 250m y+z = 250

Conclusion: Length of first section =450-250=200m x = 200

Thus length of second section = 450-200-100=150m y

= 150

---------------------------------------------------------------------------------------------------

B-2. Solution 1: I imagined the path taken by the balloon, and then worked out the

distance between the starting and finishing places. I found the distance to be 100 + 50

= 150 meters.

152

B-2. Solution 2: I drew a diagram representing the path taken by the balloon, and then

worked out the distance between the starting and finishing places.

100m

50m Distance = 100 + 50 = 150m.

B-2. Solution 3: In order to solve this problem, I noticed only the information which

was important for the solution (without imagining the path of the balloon). Then the

distance between the starting and the finishing places was 100m + 50m = 150m.

B-3. Solution 1: I solved this problem by trial and error:

Daughter’s age: Mother’s age:

2 years 26 years No

3 years 27 years No

4 years 28 years Yes.

Thus the daughter’s age is 4 years and the mother’s 28 years.

B-3. Solution 2: I solved this problem by using symbols and equations, e.g.,

Let daughter’s age be x years.

Then mother’s age is 7x years.

153

Difference between their ages is 6x years.

Therefore 6x = 24. Thus x = 4.

Thus the daughter’s age is 4 years and the mother’s age is 28 years.

B-3. Solution 3: I drew a diagram representing their ages:

Daughter’s

age

From the diagram, difference

between their ages is 6 equal

parts, totalling 24 years.

Mother’s Difference between

age their ages

Thus each part represents 4

years.

The daughter’s age is 4 years

and

the mother’s age is 28 years.

154

B-3. Solution 4: I imagined the diagram as in solution 3, and then reasoned that 6

parts represents 24 years, so one part represents 4 years (with or without using

symbols). Thus the daughter’s age is 4 years and the mother’s 28 years.

---------------------------------------------------------------------------------------------------

B-4. Solution 1: I imagined the four people and then worked out the distance between

John and Tom. John is 3 meters ahead of Tom.

B-4. Solution 2: I drew a diagram representing the four people, and then worked out

the distance between John and Tom.

l 3m l 4m l 3m

l

John Tom Jim

Peter

John is 3 meters ahead of Tom.

B-4. Solution 3: I solved this problem merely by drawing conclusions from the

sentences in the problem:

Tom is 4m ahead of Jim and Jim is 3m ahead of Peter.

Conclusion: Tom is 7m ahead of Peter.

John is 10 meters ahead of Peter.

Conclusion: John is 3 meters ahead of Tom.

155

B-5. Solution 1: I solved this problem by drawing a diagram which represented the

prices of the sugar and the salt.

Price of 1 kg sugar

Present price

Previous price of 1 kg of salt (30c).

In the diagram it can be seen that after the price of salt was increased, the price of

1 kg of sugar ws twice the price of 1 kg of salt (now 30 cents).

Thus the price of ikg of sugar is 60 cents.

B-5. Solution 2: I used the same method as for solution 1, but I drew the diagram “in

my mind” (and not on paper).

B-5. Solution 3: I solved the problem by reasoning. The price of 1 kg of salt is now

30 cents. This is 121 times the previous price; thus the previous price was 20 cents per

kg. Thus the price of sugar is 3x20 cents, that is, 60 cents.

B-5. Solution 4: I solved the problem using symbols and equations, e.g.,

Suppose the previous price of salt was x cents per kg.

Then the price of sugar was 3x cents per kg.

After the increase, price of salt is 121 x cents per kg.

156

Thus the price of 1 kg sugar is twice the present price of salt, that is, 2x30=60c.

---------------------------------------------------------------------------------------------------

B-6. Solution 1: I solved the problem by reasoning. After two sparrows flew from the

first to the second tree, the first tree had two less than before, while the second tree

had two sparrows more. Thus the second tree had four more than the first.

B-6. Solution 2: I drew a diagram.

Number in first tree after 2 birds fly Number in second

tree after 2 birds fly.

l

Number of sparrows in the first Number of sparrows

in the second

tree at first. tree at first.

The second tree has four more sparrows than the first.

B-6. Solution 3: Same method as for solution 2, but I drew the diagram “in my mind”

(and not on paper).

B-6. Solution 4: I solved this problem by using an example, e.g., suppose at first there

are 8 sparrows in each tree. After 2 sparrows fly from the first to the second, the first

tree has 6 sparrows and the second 10. Thus the second tree has 4 more sparrows than

the first.

B-6. Solution 5: I solved this problem using symbols and equations, e.g.,

157

Let the number of sparrows in each tree at first be x.

After two sparrows fly from the first tree to the second, the first tree has x-2 and

the second tree has x+2 sparrows. The difference in the number of sparrows is (x+2)

– (x-2) = 4.

B-7. Solution 1: To solve this problem I drew a diagram representing the long log

being cut into small logs.

16m

From the diagram, 7 cuts are needed to produce 8 short logs. Thus time required

is 7x2 = 14 minutes.

B-7. Solution 2: As in solution 1, but I “saw” the diagram in my mind.

B-7. Solution 3: I solved the problem by reasoning. If the long log were more than 16

meters long, one would need 8 cuts to produce 8 short logs. But the last cut is not

needed, so 7 cuts are required. Time taken is 7x2 = 14 minutes.

---------------------------------------------------------------------------------------------------


Let the weight of the jar be x kg.

Then the weight of kerosene is (8-x) kg.

158

So the weight of half the kerosene is 21 (8-x) kg.

Then x + 21 (8-x) = 4

21 . Thus x = 1.

Thus the weight of the jar is 1 kg.

B-8. Solution 2: I drew a diagram representing the respective weights.

8 kg

421 kg

Weight of kerosene Weight of half the kerosene

Weight of jar Weight of jar

From the diagram, weight of half the kerosene is 8 - 421 = 3

21 kg.

Thus weight of kerosene is 7 kg, and weight of jar is 1 kg.

(Or directly: Weight of jar is 421 - 3

21 = 1 kg.)


B-8. Solution 4: As in solution 2, but without any diagram or image at all.

---------------------------------------------------------------------------------------------------

B-9. Solution 1: I drew a diagram representing the distance traveled.

l l l l l l l

159

Half his journey Distance he slept Half

distance he traveled while

sleeping

From the diagram, if the whole journey is 6 parts, he slept for 2 parts, that is, one

third of the entire journey.


B-9. Solution 3: I solved this problem using symbols and equations, e.g.

Let the distance for which he slept be x units.

When he awoke, the remaining distance was 21 x units.

Then (x + 21 x) constitutes half the journey.

So the whole journey was 2(x + 21 x) = 3x units.

Thus he slept for one third of the journey.

---------------------------------------------------------------------------------------------------

B-10. Solution 1: I solved this problem by drawing a diagram representing the

objects.

= ¾ kg

Removing three quarters of a cheese from both scale pans, one quarter of a cheese

balances a 43 kg weight. Thus a whole cheese weighs 4x

43 , i.e., 3 kg.


160


Let the weight of a cheese be x kg.

Then x = 43 x +

43 . Therefore x = 3

Thus the weight of a cheese is 3 kg.

B-10. Solution 4: I reasoned without using a diagram or image:

One quarter of a cheese weighs 43 kg. Thus a cheese weighs 3 kg.

---------------------------------------------------------------------------------------------------


Let original amounts of milk be x liters and 2x liters.

Amounts after pouring out are (x-20) and (2x-20) liters.

Then 3(x-20) = 2x-20.

x = 40.

Thus the original amounts of milk were 40 liters and 80 liters.

B-11. Solution 2: I drew a diagram representing the amounts of milk.

20 liters (same amount poured from

both cans).

161

From the diagram, for the first can to contain three times as much as the second

after pouring, amount remaining in second can must be 20 liters. Thus original

amounts were 40 liters and 80 liters.


---------------------------------------------------------------------------------------------------

B-12. Solution 1: I used symbols and equations, e.g.

Let weight of plum be x units and weight of apricot be y units.

Then weight of a mango is (6x+y) units.

Thus 10x = 3y + (6x+y)

So x = y

Then weight of a mango is 6x+x, i.e., 7x units.

Thus one mango balances the scales against 7 plums.

B-12. Solution 2: I solved this problem by drawing diagrams representing the

weights.

balanced

( 10 plums ) (3 apricots, 1 mango)

162

balanced

( 10 plums ) (3+1 apricots, 6 plums)

From each scale pan remove 6 plums. Then 4 plums will balance 4 apricots. Thus

1 plum will balance 1 apricot. One mango balances 6 plums and one apricot, which is

thus equivalent in weight to 7 plums.


B-12. Solution 4: I solved this problem by reasoning (without imagining any picture).

One mango balances 6 plums and 1 apricot.

Thus 10 plums balance 3 apricots, 6 plums, and 1 apricot.

Thus 4 plums balance 4 apricots.

Thus (from first line) one mango balances 7 plums.

163

APPENDIX C

164

APPENDIX D

165

APPENDIX E

166

REFERENCES

Angel, A. R., & Porter, S. R. (2001). A survey of mathematics with applications

(Expanded 6th

ed.). Boston, MA: Addison Wesley Longman.

Arcavi, A. (2003). The role of visual representations in the learning of mathematics.

Educational Studies in Mathematics, 52, 215-241.

Aspinwall, L., & Shaw, K. L. (2002). Representations in calculus: Two contrasting cases.

Mathematics Teacher, 95(6), 434-439.

Aspinwall, L., Shaw, K. L., & Presmeg, N. C. (1997). Uncontrollable mental imagery:

Graphical connections between a function and its derivative. Educational Studies

in Mathematics, 33, 301-317.

Barrett, G., & Goebel, J. (1990). The impact on graphing calculators on the teaching and

learning of mathematics. In T. J. Cooney & C. R. Hirsch (Eds.), Teaching and

learning mathematics in the 1990’s (pp. 205-211). Reston, VA: National Council

of Teachers of Mathematics.

Beckmann, C. E., Senk, S. L., & Thompson, D. R. (1999, December). Assessing

students’ understanding of functions in a graphing calculator environment. School

Science and Mathematics, 99(8), 451-456.

Bishop, A. J. (1989). Review of research on visualization in mathematics education.

Focus on Learning Problems in Mathematics, 11(1), 7-16.

Bogdan, R. C., & Biklen, S. K. (1998). Qualitative research for education: An

introduction to theory and methods (3rd

ed.). Boston, MA: Allyn and Bacon.

Breidenbach, D., Dubinsky, E., Hawks, J., & Nichols, D. (1992). Development of the

process conception of function. Educational Studies in Mathematics, 23, 247-285.

Browning, C. A. (1988). Characterizing levels of understanding of functions and their

graphs. Dissertation Abstracts International, 49(10), 2957A. (UMI No. 8824468)

Burton, D. M. (1998). Elementary number theory (4th

ed.). New York: McGraw-Hill.

Cates, J. (2002). Understanding algebra through graphing calculators. Learning and

Leading with Technology, 30(2), 33-35.

Christy, D. T. (1993). Essentials of Precalculus Algebra and Trigonometry (5th

ed.).

Dubuque, IA: Wm. C. Brown.

Clutter, M. (1999, Spring). Graphing calculators: The newest revolution in mathematics.

Inquiry, 4, 10-12.

167

Demana, F., & Waits, B.K. (1990). Enhancing mathematics teaching and learning

through technology. In T. J. Cooney & C. R. Hirsch (Eds.), Teaching and

learning mathematics in the 1990’s (pp. 212-222). Reston, VA: National Council

of Teachers of Mathematics.

Doerr, H. M., & Zangor, R. (2000). Creating meaning for and with the graphing

calculator. Educational Studies in Mathematics, 41,143-163.

Dreyfus, T., & Eisenberg, T. (1983). The function concept in college students: Linearity,

smoothness and periodicity. Focus on Learning Problems in Mathematics, 5(3-4),

119-132.

Dunn, R. (2000). Capitalizing on college students’ learning styles: Theory, practice, and

research. In R. Dunn & S. A. Griggs (Eds.), Practical approaches to using

learning styles in higher education (pp. 1-18). Westport, CT: Bergin & Garvey.

Dunn, R., Dunn, K., & Price, G. (2003). Productivity Environmental Preference Survey

(PEPS). Lawrence, KS: Price Systems.

Dunn, R., & Griggs, S. A. (1995). Multiculturalism and learning style: Teaching and

counseling adolescents. Westport, CT: Praeger.

Eisenberg, T., & Dreyfus, T. (1994). On understanding how students learn to visualize

function transformations. In E. Dubinsky, A. H. Shoenfeld, & J. Kaput (Eds.),

Conference Board of the Mathematical Sciences issues in mathematics education:

Vol. 4 Research in collegiate mathematics education. I (pp. 45-68). Providence,

RI: American Mathematical Society.

Elliot, B., Oty, K., McArthur, J., & Clark, B. (2001). The effect of an interdisciplinary

algebra/science course on students’ problem solving skills, critical thinking skills

and attitudes towards mathematics. International Journal of Mathematical

Education in Science and Technology, 32(6), 811-816.

Fey, J. T. (1992). Compute-intensive algebra: An overview of fundamental mathematical

and instructional themes. Unpublished manuscript.

Fraleigh, J. B. (1994). A first course in Abstract Algebra ( 5th

ed.). Reading, MA:

Addison-Wesley Publishing Company.

Friedlander, A., & Hershkowitz, R. (1997). Reasoning with algebra. Mathematics

Teacher, 90(6), 442-447.

Gray, E. M., & Tall, D. O. (1994). Duality, ambiguity, and flexibility: A proceptual view

of simple arithmetic. Journal for Research in Mathematics Education, 25(2), 116-

140.

168

Haar, J., Hall, G., Schoepp, P., & Smith, D. H. (2002). How teachers teach to students

with different learning styles. The Clearing House, 75(3), 142-145.

Hennessy, S., Fung, P., & Scanlon, E. (2001). The role of the graphic calculator in

mediating graphing activity. International Journal of Mathematical Education in

Science and Technology, 32(2), 267-290.

Hershkowitz, R., Ben-Chaim, D., Hoyles, C., Lappan, G., Mitchelmore, M., & Vinner, S.

(1989). Psychological aspects of learning geometry. In P. Nesher & J. Kilpatrick

(Eds.), Mathematics and Cognition (pp. 70-95). Cambridge: University Press.

Hollar, J. C., & Norwood K. (1999). The effects of a graphing-approach intermediate

algebra curriculum on students’ understanding of function. Journal for Research

in Mathematics Education, 30(2), 220-226.

Holsti, O. R. (1969). Content analysis for the social sciences and humanities. Reading,

MA: Addison-Wesley.

Karsenty, R. (2002). What do adults remember from their high school mathematics? The

case of linear functions. Educational Studies in Mathematics, 51, 117-144.

Kieran, C. (1990). Cognitive processes involved in learning school algebra. In P. Nesher

& J. Kilpatrick (Eds.), Mathematics and Cognition: A research synthesis by the

International Group for the Psychology of Mathematics Education (pp. 96-112).

Cambridge: Cambridge University Press.

Kirshner, D., & Awtry, T. (2004). Visual salience of algebraic transformations. Journal

for Research in Mathematics Education, 35(4), 224-257.

Krutetskii, V. A. (1976). The psychology of mathematical abilities in schoolchildren.

Chicago, IL: University of Chicago Press.

Lane, J. (1998). Survey of technology use in mathematics at New Jersey’s community

colleges. Eric Microfiche.

Lane, R. M., & Williams, R. (1998). Texas instruments (TI-82) graphics calculators and

high school students’ knowledge of algebra. Journal of the Florida A & M

University McNair Program, 9, 21-23.

Lean, G., & Clements, M. A. (1981). Spatial ability, visual imagery, and mathematical

performance. Educational Studies in Mathematics, 12, 267-299.

Lial, M., Hornsby, J., & Schneider, D. I. (2001). College algebra and trigonometry and

pre-calculus (2nd

ed.). Boston, MA: Addison-Wesley.

Lincoln, Y. S., & Guba, E. G. (1985). Naturalistic Inquiry. Newbury Park, CA: Sage.

169

Malik, M. A. (1980). Historical and pedagogical aspects of the definition of function.

International Journal of Mathematical Education in Science and Technology,

11(4), 489-492.

Milou, E. (1999). The graphing calculator: A survey of classroom usage. School

Science and Mathematics, 99(3), 133-138.

Monk, S., & Nemirovsky, R. (1994). The case of Dan: Student construction of a

functional situation through visual attributes. In E. Dubinsky, A. H. Schoenfeld,

& J. Kaput (Eds.), Conference Board of the Mathematical Sciences issues in

mathematics education: Vol. 4 Research in collegiate mathematics education. I

(pp. 139-168). Providence, RI: American Mathematical Society.

Moschkovich, J. N. (1999). Students’ use of the x-intercept as an instance of a

transitional conception. Educational Studies in Mathematics, 37, 169-197.

National Council of Teachers of Mathematics. (1989). Curriculum and evaluation

standards for school mathematics. Reston, VA: The Council.

National Council of Teachers of Mathematics. (2000). Principles and Standards for

School Mathematics. Reston, VA: NCTM.

Nemirovsky, R., & Noble, T. (1997). On mathematical visualization and the place where

we live. Educational Studies in Mathematics, 33, 99-131.

O’Callaghan, B. R. (1998). Computer-intensive algebra and students’ conceptual

knowledge of functions. Journal for Research in Mathematics Education, 29(1),

21-40.

Paschal, S. G. (1994). Effects of a visualization-enhanced course in college algebra using

graphing calculators and videotapes. Dissertation Abstracts International, 55(09),

2754A. (UMI No. 9502845)

Patton, M. Q. (2002). Qualitative Research & Evaluation Methods. Thousand Oaks, CA:

Sage.

Presmeg, N.C. (1992). Prototypes, metaphors, metonymies, and imaginative rationality in

high school mathematics. Educational Studies in Mathematics, 23, 595-610.

Presmeg, N. C. (1985). The role of visually mediated processes in high school

mathematics: A classroom investigation. Unpublished Ph.D. dissertation,

University of Cambridge, England.

Presmeg, N. C. (1986a). Visualisation and mathematical giftedness. Educational Studies

in Mathematics, 17, 297-311.

170

Presmeg, N. C. (1986b). Visualisation in high school mathematics. For the Learning of

Mathematics, 6(3), 42-46.

Presmeg, N. C. ( 1989). Visualization in multicultural mathematics classrooms. Focus on

Learning Problems in Mathematics, 11(1), 17-24.

Price, G. (1996). Productivity Environmental Preference Survey: An inventory for the

identification of individual adult learning style preferences in a working or

learning environment (PEPS Manual). Lawrence, KS: Price Systems.

Quesada, A. R., & Maxwell, M. E. (1994). The effects of using graphing calculators to

enhance college students’ performance in precalculus. Educational Studies in

Mathematics, 27, 205-215.

Ruthven, K. (1990). The influence of graphic calculator use on translation from graphic

to symbolic forms. Educational Studies in Mathematics, 21, 431-450.

Sajka, M. (2003). A secondary school student’s understanding of the concept of function

- A case study. Educational Studies in Mathematics, 53, 229-254.

Sarmiento, J. (1997, June). New technologies in mathematics. Eric microfiche.

Schwarz, B., & Dreyfus, T. (1995). New actions upon old objects: A new ontological

perspective on functions. Educational Studies in Mathematics, 29, 259-291.

Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on

processes and objects as different sides of the same coin. Educational Studies in


Sharma, M. M., Treadway, R., Kumar, R., & Kapoor, J. (2002). College algebra (3rd

ed.).

McDonough, GA: EDUCO International.

Shoaf-Grubbs, M. M. (1994). The effect of the graphing calculator on female students’

spatial visualization skills and level-of-understanding in elementary graphing and

algebra concept. In E. Dubinsky, A. H. Schoenfeld, & J. Kaput (Eds.), Conference

Board of the Mathematical Sciences issues in mathematics education: Vol. 4

Research in collegiate mathematics education. I (pp. 169-194). Providence, RI:

American Mathematical Society.

Slavit, D. (1997). An alternate route to the reification of function. Educational Studies in


Stake, R. E. (1995). The art of case study research. Thousand Oaks, CA: Sage.

Stephens, L. J., & Konvalina, J. (1999). The use of computer algebra software in teaching

intermediate and college algebra. International Journal of Mathematical

171

Education in Science and Technology, 30(4), 483-488.

Stick, M.E. (1997). Calculus reform and graphing calculators: A university view.

Mathematics Teacher, 90(5), 356-360.

Texas Instruments (TI-82) Graphics Calculator Guidebook. (1993).Texas Instruments

Incorporated.

The American Heritage Dictionary. (2nd

ed.). (1991). Boston, MA: Houghton Mifflin.

Thompson, P. W. (1994). Students, functions, and the undergraduate curriculum. In E.

Dubinsky, A. H. Schoenfeld, & J. Kaput (Eds.), Conference Board of the

Mathematical Sciences issues in mathematics education: Vol. 4 Research in

collegiate mathematics education. I (pp. 21-44). Providence, RI: American

Mathematical Society.

Vinner, S. (1983). Concept definition, concept image and the notion of function.

International Journal of Mathematical Education in Science and Technology,

14(3), 293-305.

Vinner, S., & Dreyfus, T. (1989). Images and definitions for the concept of function.

Journal for Research in Mathematics Education, 20(4), 356-366.

Werdelin, I. (1958). The mathematical ability: Experimental and factorial studies. Lund:

Gleerups.

Writing Team and Task Force of the Standards for Introductory College Mathematics

Project. (1995). Crossroads in Mathematics: Standards for Introductory College

Mathematics Before Calculus. Retrieved from

http://www.imacc.org/standards/introduction.html.

Wiersma, W. (2000). Research methods in education: An introduction (7th

ed.). Needham

Heights, MA: Allyn and Bacon.

Yerushalmy, M. (2000). Problem solving strategies and mathematical resources: A

longitudinal view on problem solving in a function based approach to algebra.

Educational Studies in Mathematics, 43, 125-147.

Zazkis, R., Dubinsky, E., & Dautermann, J. (1996). Coordinating visual and analytic

strategies: A study of students’ understanding of the group D4. Journal for

Research in Mathematics Education, 27(4), 435-457.

Zimmermann, W., & Cunningham, S. (1991). Editor’s introduction: What is

mathematical visualization. In W. Zimmermann & S. Cunningham (Eds.),

Visualization in Teaching and Learning Mathematics (pp. 1-8). Washington, DC:

Mathematical Association of America.

172

http://www.imacc.org/standards/introduction.html

BIOGRAPHICAL SKETCH

Rebekah M. Lane was born in Wichita, Kansas. She attended Florida A & M

University (FAMU). Dr. Lane graduated Summa Cum Laude with a Bachelor of Science

(B.S.) degree in Mathematics. After that, she graduated with a Master of Education

(M.Ed.) in Mathematics Education from FAMU. Next, Dr. Lane obtained a Doctor of

Philosophy (Ph.D.) degree from Florida State University.

Dr. Lane also has experience teaching undergraduate mathematics courses. She

has used technology such as graphing calculators and software packages such as EDUCO

and MyMathLab with her students. Dr. Lane was also a McKnight Doctoral Fellow

during her Ph.D. studies. In addition, she published an article in the Journal of the

Florida A & M University McNair Program.

173

florida state university librariesdiginole.lib.fsu.edu/islandora/object/fsu:254168/... · the...

Documents