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THE IMPACT OF USING GRAPHING CALCULATORS
AS AN AID FOR THE TEACHING AND LEARNING OF
PRECALCULUS IN A UNIVERSITY SETTING.

By

Carl Wallace Norris

A DISSERTATION
Submitted to
Michigan State University

in partial fulfilment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Educational Administration

1994

ABSTRACT
THE IMPACT OF USING GRAPHING CALCULATORS
AS AN AID FOR THE TEACHING AND LEARNING OF
PRECALCULUS IN A UNIVERSITY SETTING.
By

Carl W. Norris

This was a quantitative and qualitative investigation involving a treatment and
control group of subjects from Central Michigan University. The treatment group (125
students) was comprised of three precalculus Classes engaged in a one semester
precalculus course which required graphing calculators. The control group (179 students
from four classes) took a very similar course which did not require graphing calculators.

Simple t-tests were employed at the 0.05 level to do a comparative investigation
of the differences between the two groups in three areas:

(i) algebraic skill proficiency,
(ii) function concept knowledge,
(iii) attitudes towards mathematics.
A qualitative form of inquiry, using questionnaires, was used to investigate
student/faculty reaction as per the effectiveness and potential of the graphing calculator.

QUantitative analysis of the research data revealed the following:

Carl W. Norris

1. There was no significant difference in the performances of both the treatment
and control groups on either a pretest or posttest of algebraic skills (p-values
= 0.082 and 0.15 respectively). Furthermore, the algebraic skills of the
treatment group appear not to be adversely affected by the treatment in the
sense that the mean change In achievement based on pre and posttest score
differences was not significantly different (p-value = 0.77) for both groups.

2. The performance of, the treatment group on a posttest of basic function
concepts and graphing was significantly better than that of the control group (p-
value = 0.000), even though pretest results show that the two groups were very
comparable in this area. Furthermore, the mean improvement in the
performance of the treatment group in this area when compared with the control
group was significantly better (p-value = 0.0003).

3. Results of an Attitude Pretest and Attitude Posttest revealed that there was no
significant differences (p-values = 0.73 and 0.23 respectively) in the attitude
towards mathematics of both the treatment and control groups. Likewise, there
was no significant difference in the mean Change in attitude over the semester
among the two groups (p-value = 0.40).

Qualitative data gathered through student and faculty questionnaires revealed

strong support for the graphing calculator as a visual aid for the teaching and learning

of precalculus.

Copyright by
Carl Wallace Norris
1 994

ACKNOWLEDGEMENTS

There are many people to whom I am indebted throughout my doctoral studies.
To all of them I would like to express my sincere thanks and appreciation. It is impossible
to acknowledge everyone. However, there are several in particular, to whom I owe a

special debt of gratitude.

First of all I wish to express my Sincere appreciation to the members of my
doctoral committee:

Dr. Louis Hekhuis, Chair, for his very valuable advice and kind and accessible
manner;

Dr. Eldon Nonnamaker, for his encouragement and constant reassurance.

Dr. VWlliam Cole, whose initial contact was instrumental in my decision to attend
Michigan State University.

Dr. William Fitzgerald, dissertation supervisor, for his constructive and helpful
suggestions.

Special thanks go to Dr. Edgar Goodaire, and the Mathematics Department,
Memorial University, who kindly supported me with a two year leave of absence in order
to further pursue my studies; and to Dr. Gerry Ludden and the Mathematics Department
at Michigan State University, who so graciously accommodated me with a teaching
assistantship during my stay at Michigan State.

In addition, I am especially grateful to Dr. Richard Fleming (Chair) and the
Mathematics Department at Central Michigan University for their kindness and hospitality.
WIthout their support this project would not have materialized. I am especially indebted
to Dr. William Miller who coordinated the project at Central Michigan University, and
arranged to mail data over 2000 miles.

Thanks should also go to Dr. Roy Bartlett and Dr. Melvin Lewis of the Department
of Mathematics at Memorial University who were always available to answer questions
whenever I needed statistical advice.

Last, but by no means least, I would like to say a special thank you to my sons,
Ian, Dwayne and Warren, for their patience and understanding and especially to my wife,
Joan, who spent long hours typing the manuscript and continuously provided moral
suppon.

vi

TABLE OF CONTENTS

Page

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

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

CHAPTER

I. INTRODUCTION TO THE INVESTIGATION .................... 1

Low Achievement in Precalculus ............................ 2

Using Modern Technology to Teach Precalculus ................ 3

Purpose, Scope, and Significance of Proposed Study ........... 14

ll. LITERATURE REVIEW .................................. 18
Educational Theories Related to the Use of

Graphing Technology ................................... 19

Computer Assisted Instruction in Precalculus .................. 20

The Use of Graphing Calculators in the Classroom ............. 32

The 02PC Project at OHIO State ........................... 40

Ill. METHODS AND PROCEDURES ........................... 43

Restatement of the Problem .............................. 43

Participants and Setting ................................. 47

Instructional Materials .................................... 48

Research Design and Research Instruments ................. 51

Method of Analysis ...................................... 58

Limitations of the Study .................................. 59

N. DATA ANALYSIS ....................................... 62

Analysis of Algebraic Skills Test Results ...................... 63

Analysis of Function Concepts Test Results .................... 70

Attitude Inventory Data Analysis ............................ 76

Results of Student Questionnaire ........................... 85

Responses to Instructor Questionnaire ....................... 93

vii

V. SUMMARY AND CONCLUSIONS ........................... 98

Summary of the Analysis of Achievement

and Attitude Tests ...................................... 98

Examination of Hypotheses ............................... 102

Summary of Results of Student and

Teacher Questionnaires ................................. 105

Main Conclusions ...................................... 108

Discussion ........................................... 1 1 1

Implications and Directions for Further Study .................. 113

APPENDICES

A. STUDENT ATTITUDE TOWARDS MATHEMATICS INVENTORY

PRETEST ............................................ 118
B. STUDENT ATTITUDE TOWARDS MATHEMATICS INVENTORY

POSTTEST ........................................... 120
C. PRECALCULUS MATHEMATICS ALGEBRAIC SKILLS

PRETEST ............................................ 123
D. PRECALCULUS MATHEMATICS ALGEBRAIC SKILLS

POSTTEST ........................................... 126
E. PRECALCULUS MATHEMATICS FUNCTION CONCEPTS

PRETEST ............................................ 130
F. PRECALCULUS MATHEMATICS FUNCTION CONCEPTS

POSTTEST ........................................... 134
G. STUDENT QUESTIONNAIRE .............................. 138
H. INSTRUCTOR QUESTIONNAIRE ........................... 141
l. MATHEMATICS 130 (CALCULATOR) COURSE OUTLINE ........ 144
J. MATHEMATICS 130 (REGULAR) COURSE OUTLINE ............ 146
K. ANALYSES OF SUPPLEMENTARY ATTITUDE AND ACHIEVEMENT

PRETEST SCORES .................................... 148

BIBLIOGRAPHY ................................................. 151

viii

Table

10.

11.

12.

 

LIST OF TABLES

Page
Schedule of Assessments ..................................... 57
Results of Two Sample t-test for Treatment and Control Groups
on Algebraic Skills Pretest ..................................... 64
Results of Two Sample Unpaired t-test for Treatment and Control
Groups on Algebraic Skills Posttest ............................... 66
Comparison of Responses of Treatment and Control Groups to Each
Question on Algebraic Skills Posttest ............................. 66
Comparison of Means of Differences in Scores on Pre and Post
Algebraic Skills Test for Treatment and Control Groups
Using Unpaired Two Sample t-test ............................... 69
Results of Two Sample t-test for Treatment and Control Groups on
Function Concepts Pretest ..................................... 72
Results of Two Sample t-test for Treatment and Control Groups on
Function Concepts Posttest .................................... 73
Comparison of Responses of Treatment and Control Groups to Each
Question on Function Concepts Posttest ........................... 73
Comparison of Mean of Differences in Scores on Pre and Posttest
of Function Concepts ......................................... 76
Results of Two Sample Unpaired t-test for Treatment and Control
Groups on Attitude Pretest ..................................... 78
Results of Two Sample Unpaired t-test for Treatment and Control
Groups on Attitude Posttest .................................... 80
Comparison of Means of Score Differences on Pre and Post Attitude
Test for Treatment and Control Groups by Unpaired Two Sample
t-test ..................................................... 82

13.

14.

15.

16.

17.

18.

19.

20.

21.

23.

24.

25.

 

 

Attitude of Students of Treatment Group Towards the Graphing

Calculator: A Description of Attitude Test Results .................... 82
Item by Item Analysis of Test of Attitude Towards Graphing
Calculator for Treatment Group .................................. 83

Attitude Towards Graphing Calculator: Level of Agreement on
Attitude Test Items ........................................... 84

Summary of Responses to Questions 1-10 and 12 on
Student Questionnaire ........................................ 86

Helpfulness of Graphing Calculator in the Study of Precalculus:
Description of Responses on Student Questionnaire .................. 90

Level of Helpfulness of Graphing Calculator: Percentage of
Students Who Responded Moderately Helpful or Very Helpful ........... 91

Results of t-test on Attitude Pretest Scores: Students Who
Wrote Pre and Posttest Versus Those Who Wrote Just Pretest
in Treatment Group ......................................... 148

Results of t-test on Attitude Pretest Scores: Students Who
Wrote Pre and Posttest Versus Those Who Wrote Just Pretest
in Control Group ........................................... 148

Results of t-test on Algebraic Skills Pretest Scores:
Students Who Wrote Pre and Posttest Versus Those Who Wrote
Just Pretest in Treatment Group ................................ 148

Results of Meet on Algebraic Skills Pretest Scores:
Students Who Wrote Pre and Posttest Versus Those Who Wrote
Just Pretest in Control Group .................................. 149

Results of t-test on Function Concepts Pretest Scores:
Students Who Wrote Pre and Posttest Versus Those Who Wrote
Just Pretest in Treatment Group ................................ 149

Results of t-test on Function Concepts Pretest Scores:
Students Who Wrote Pre and Posttest Versus Those Who Wrote
Just Pretest in Control Group .................................. 149

Results of t-test on Function Concepts Pretest Scores for
Students Who Wrote Pre and Posttest: Treatment Versus
Control Group ............................................. 150

Figure

10.

11.

12.
13.

14.

 

LIST OF FIGURES

Page
Casio Simulation of the Path of the Ball ............................ 6
Three Views of the Graph of V(x) = x(25 - 2x )2 ..................... 8
Graph of y = (x - 3)(2x -1)(x + 4) ............................... 9
Graphofy=5x3-12x2-63<+8 ................................ 12
Zoom-in on Roots ........................................... 13
Algebraic Skills Pretest Scores:
Frequency Distribution ........................................ 64
Algebraic Skills Posttest Scores:
Frequency Distribution ........................................ 65
Algebraic Skills Pre and Posttest Score Differences:
Frequency Distribution ........................................ 69
Function Concepts Pretest Scores:
Frequency Distribution ........................................ 71
Function Concepts Posttest Scores:
Frequency Distribution ........................................ 72
Function Concepts Pre and Posttest Score Differences:
Frequency Distribution ........................................ 75
Attitude Pretest Scores: Frequency Distribution ..................... 78
Attitude Posttest Scores: Frequency Distribution ..................... 79
Attitude Pre & Posttest Score Differences:
Frequency Distribution ........................................ 81

xi

 

 

CHAPTER I

INTRODUCTION TO THE INVESTIGATION

A fundamental understanding of the concepts of functions, graphing and
related concepts such as solving equations and inequalities (frequently referred to
as precalculus) is essential for student success in advanced mathematics and
related applied areas. This is evidenced by the fact that the study of functions
(both algebraic and trigonometric functions) and their graphs have in recent years
formed the basis of most beginning college algebra courses.

The College Entrance Examination Board (1985) states that:

the concept of functlon Is central to mathematics and students

entering college not only need to understand what functions are

In general but also to be familiar with examples ..... They should

be able to use computers and other tools to represent functions

graphically. (p. 29)

The function concept, as it is currently being taught at most secondary and
post-secondary institutions, using “traditional" teaching methods, has not always
been well understood. Several researchers in the field of mathematics education
such as Rosnick (1981) and Lochhead (1983) have argued that this
misunderstanding frequently results in low achievement in precalculus. This, In
turn, creates obvious difficulties in subsequent college mathematics courses, such

as calculus, which require a sound understanding of elementary functions and their

graphs.

2

Demana and Waits (1988), in an article entitled ls Three Years Enough?,
reported that data collected at Ohio State University revealed that less than one
in six high school students entering that university with four or more years of
college preparatory mathematics were ready to begin the study of calculus in their
first semester. It goes on to suggest that about 86 percent of the students with
less than three years of college preparatory mathematics will have remedial
placement in mathematics at the college entrance level. This presents obvious
difficulties for students who are pursuing science related programs, many of which

require early calculus.

Low Achievement In Precalculus

Dissatisfaction with the performance of our secondary school students in
the area of precalculus has been expressed nationally. A study entitled The
Second International Mathematics Study (SIMS) was designed to provide country-
by-country information about the content of the mathematics curriculum, how
mathematics is taught, and how much mathematics Students learn. In a report on
this study, McKnight, Travers and Dossey (1985) and McKnight et al (1987)
asserted that the achievement level of students in the United States in the area of
precalculus, particularly in the area of functions, graphing and problem solving,
was substantially below the international average. In fact, it was the lowest of the
advanced industrialized countries and in some cases ranked with the lower one

quarter of all countries in the survey.

3

The secondary school results for the Fourth NAEP Assessment uncovered
similar problems. Brown et al (1988) in reporting on this study state:

secondary students demonstrated some Intuitive knowledge of

functions but they were less successful In situations that used
functional notation. Their performances, however, on items
covering a broad range of functional concepts indicated a limited
understanding of the topic ....... In another item students were
presented with four graphs and asked to choose the graph that

did not represent a function. About two thirds of the students

with two years of Algebra and half the Algebra I students chose

the graph of a circle (which Is correct). However, when asked to

select the graph of a function and its inverse, less than 20

percent of the students responded correctly. (pp. 339-340)

These studies provide evidence of United States students as having a relatively low
level of understanding of the function concept.

It is the view of many mathematicians who teach college freshman
mathematics courses that the difficulties students are experiencing in the learning
of functions and graphing at the secondary school level are quite often reflected
in the poor performance of students in college algebra or precalculus at the post-

secondary level.

Using Modern Technology to Teach Precalculus
In the secondary schools, and perhaps to a lesser extent in colleges and
universities, pencil-and-paper graphing methods are often employed to graph
functions by plotting a small sample of points whose co-ordinates satisfy the
defining equation of the function. Such methods are both time-consuming and

frequently inaccurate. Furthermore, when such inexact methods are used, the

4

graph often becomes the main focal point and the global characteristics of the
function being studied gets lost in finding and plotting points. Waits and Demana
(1988), Heid (1988), Foley (1990), Leinhard, Zaslavsky, and Stein (1990) and
several other mathematicians have all addressed this concern. It has been
asserted by Demana and Waits (1987), Held (1990), and Fey (1989) that these
typical pencil-and-paper methods are sometimes inadequate to solve many of the
more interesting and practical problems which can be modeled with elementary
functions.

Given the undesirable situation that currently exists both at the secondary
and post-secondary level, some mathematicians and mathematics educators are
turning their attention to the potential use of modern technology in the teaching
and learning of functions and their graphs and other related precalculus concepts.
Two pieces of technology that have been used as teaching aids for this purpose
with some degree of success are microcomputer based graphics software and,
most recently, advanced programmable graphing calculators. These devices are
sometimes referred to as graphing utilities. The hand-held graphing calculators
have the advantage over the microcomputer of portability and being relatively
inexpensive. Furthermore, advanced programmable pocket calculators, currently
available at low prices, have most of the capabilities of a microcomputer. These
microcomputers and graphing calculators can generate quickly and flexibly the
graphs of a wide variety of functions, some of which are much too complicated to

graph by "traditional" methods. The technology can then manipulate these graphs

5

in various ways such as change of scale, rotation of axes, and translations.
Vonder Embse (1988) claims that "a student can graph more functions in a three-
hour homework session using this technology that would normally be done in an
entire year of mathematics instruction". By seeing a much wider variety of
functions and their transformations machine produced, students are better able to
investigate and explore patterns and make generalizations. Since the technology
is more efficient than a pencil-and-paper approach, it frees valuable time for study
of other important aspects of the study of functions other than the actual graphing
such as asymptotic and end-point behaviour and graphical solutions to new and
interesting problems.

Demana and Waits (1989), in an article entitled Ami the Sun in a
Graphing Calculator, attempt to promote the graphing calculator as a vehicle which
can be used to "investigate, simulate, and solve important real-world problems."
The authors go on to say that:

pocket computer technology has the potential to bring great

mathematical power to our students ...... Many Important topics

In science and mathematics can be given more than lip service

In a technology rich curriculum. Problem solving can truly

become the focus of a mathematics classroom. (p. 500)

A specific example is given in this article to show how the Casio fx - 7000G
graphics calculator can be programmed to give the following simulation of the
trajectory of a ball hit with an initial velocity of 63 ft/sec. and at an angle of 37° with

the ground where the path of the ball is appropriately described by a pair of

parametric equations x(T), y(T), T seconds after it is hit. (see Figure 1 )

 

 

 

[0.1201va0.701
rm-o. Tm'Z
increment-0.05

 

Figure 1 Casio Simulation of the Path of the Ball

This vision of computer or graphing calculator technology as a vehicle for
problem-solving was evident in an article on the role of computer technology by
Parish, Partner, and Whitaker (1987) wherein they assert that "one of the most
important roles of technical education is developing within students the ability to
analyze problems and develop procedures for their solutions, while at the same
time paying special attention to the underlying mathematics."

The graphics calculator has been recommended by Dion (1990) as a "tool
for critical thinking." In this regard, she conjectures that

the successful use of calculators requires a higher level of

understanding than that required for rote computation or

template problem solving. Precalculus students benefit from an
intuitive understanding of functions gained through the use of
graphics calculators, and calculus students gain a deeper
understanding of functions and their graphs by Interactively
using the graphics and algebraic capabilities of calculators. In

addition, the need to Interpret answers encourages critical
thinking for all students. (p 564)

7
In an article entitled Pitfalls in Graphical Computationfipr why a Single Gram
IS not Enough, Demana and Waits (1988) endorse Dion’s philosophy when they

conclude that:

computer graphing permits students to solve quickly and
effectively complex problems whose solutions are not accessible
by traditional methods. By analyzing numerous graphs of
functions in a short period of time, students are able to build
strong intuition and understanding about functions. Computer
graphing does not remove the need for students to think about
mathematics. Quite the contrary - it can serve to motivate and
enable students to think about more mathematics. (p. 183)

An example of a traditional calculus exercise that is accessible to
precalculus students with the aid of graphing calculator technology is illustrated
in the following optimization problem taken from page 164 of the text Precalculus
Mathematics: A Graphing Approach by Demana, Waits, and Clemens.

Problem: Squares are cut from the corners of a 20 inch by 25 inch piece of
cardboard, and a box is made by folding up the flaps. Determine
the graph of the problem situation and find the dimensions of the
squares so that the resulting box has the maximum possible volume.

Solution: Let x = the width of the square and the height of the resulting box
in inches. Then 20 - 2x = the width of the base of the box in inches
and 25 - 2x = the length of the base of the box in inches.

Now V(x) = x(20 - 2x)(25 - 2x) = x(25 - 2x)2 is the volume of the
resulting box.

Find a complete graph of V(x) = x(25 - 2x)2 (see Figure 2(a)). In this
problem situation 0< x < 10. The complete graph of the problem
situation is shown in Figure 2(b).

Use zoom-in to find the coordinates of the local maximum (Figure
2(c)). Read the coordinates of the high point as (3.68, 820.5282) with
an error of at most 0.01. Thus the maximum volume is 820.5282
cubic inches and this volume occurs when x = 3.68 inches.

—+—+—-+—-+—+—+—

 

. -
a
- u
.- .
. A L A A '
v v ,
.
u
- o
. u
o o
. r e . . .

[-5. 20) by [-IOOO. 1000] [0. l0] by [-IOOO. 1000) [3.63. 3.73] by [820.52. 820.53)
la) (b) (c)

Wfif _

Figure 2 Three Views of the Graph of V(x) = x(25 - 2:02

Hector (1992) in an article written for the 1992 NCTM Yearbook, indicates
strong support by the NCTM for the use of the graphics calculator as a valuable
tool for exploring the global characteristics of a function. In particular, she uses
mathematics examples to argue for the use of the graphics calculator as an
effective device to:

(i) Effectively and efficieme graph families of curves and help students
discover the connections between the algebraic and graphical
representations of a function.

(ii) Clarify algebraic methods used in teaching, as for example, in solving
inequalities and equations and distinguishing between trigonometric
identities and trigonometric equations.

To illustrate the point made in (i) above, Hector suggests that students can

use the graphing calculator to gain greater familiarity with quadratic functions, such

9

as the parabolic function, by quickly viewing the graphs of several parabolas such
as y = (x - 2)2, y = (x + 2)2, y = (x - 0)2, y = (x + 5)2, on one viewing screen and
then observe the common shapes and the shifts left and right. Similar explorations
can be done by varying the coefficient of x2 as in y = 0.5 x2, y = x"’, y = 3x2 and
observing the effect on the Slope of the parabola.
In regards to point (ii) above, Hector uses the cubic inequality

(x - 3) (2r - 1) (x + 4) <0 as an example of how the graphing calculator can be
used to solve geometrically an higher degree inequality and hence compliment the
algebraic solution to such an inequality. Students can find a graphical solution to
this inequality by quickly graphing the polynomial function y = (x - 3) 2x - 1)

(x + 4) and then estimate the solution by observing from the graph that values of
y are negative when x < - 4 and 0.5 < x < 3. The graph for this investigation is

given below.

 

Figure 3 Graph of y = (x - 3)(2x -1)(x + 4)

In concluding her article, Hector (1992) asserts that:
Graphing calculators are available and Inexpensive enough to
allow students to have access to them for learning about

10

elementary functions. Their use enables students to construct
many more graphs for observation and generalization than they
would usually do by hand. Instruction that connects graphical,
algebraic, and tabular representations of functions helps
students develop richer insight Into the nature of functions. (p.

137)

Support for Technology In Classrooms by Professional Organizations

As early as 1975 the National Advisory Committee on Mathematics
Education urged that calculators be used in the mathematics classroom
(NACOME1975, 40-43). Five years later in 1980, the National Council of Teachers
of Mathematics (NCTM) published a document called Agenda for Action (NCTM,
1980) which proposed a direction for mathematics instruction in the 1980’s. One
of the important recommendations of this document was at that all grade levels
mathematics programs should fully utilize the potential of calculators and
computers.

Corbett (1985), in reporting on the proceedings of an NCTM conference on
the Impact of Computing Technology on School Mathematics, made the following
statement:

when used for the analysis of graphic or symbolic data,

calculators and computers offer powerful new approaches to

familiar problems and access to entirely new branches of
mathematics. Applications of these same capabilities to
instruction are bringing major changes to mathematics

classrooms and the roles of mathematics teachers. (p. 244)

Further support for the above ideas was given in a position statement of the

NCTM entitled Calculators in the Mathematics Classroom (NCTM. 1986). This

11

report recommended that students use calculators to :

concentrate on the problem-solving process rather than on the calculations
associated with problems;

gain access to mathematics beyond the student’s computational skills;
explore, develop, and reinforce concepts including estimation, computation,
approximation, and properties;

experiment with mathematical ideas and discover patterns; and

perform those tedious computations that arise when working with real data
in problem-solving situations.

The following statement appeared in The use orf Computers in the Learning

and Teaching of Mathematics: A Position Statement of the NCTM. September.

1_9_8L

Computer technology is changing the ways we use mathematics.
Consequently, the content of mathematics programs and the
methods by which mathematics is taught are changing.
Students must continue to study appropriate mathematics
content, and they also must be able to recognize when and how
to use computers effectively when doing mathematics....
mathematics teachers should be able to appropriately use a
variety of computer tools, such as programming languages and
spreadsheets, in the mathematics classroom. For example,
teachers should be able to identify topics for which expressing
an algorithm as a computer program will deepen student insight,
and they should be able to develop or modify programs to fit the
needs of classes or individuals. Keeping pace with advances in
technology will enable mathematic teachers to use the most
efficient and effective tools available (p. 5)

The Mathematical Association of America also recommends that computers

and calculators be made available for use in concept development (Lichtenberg,

12
1988).

More recently the NCTM has stressed the use of computers and calculators
as tools in the mathematics classroom in their Curriculum and Evaluation
Standards for School Mathematics (NQTM, 19$). This document proclaims that

all students in grades 9 - 12 will have assess to graphing calculators and a
demonstration computer and that all students will have access to computers for
individual and group work. The Standards contain various examples of how
computers and graphing calculators can be effectively used in regular Classroom
instruction. One such example was given in the Algebra Standard and was also
recently alluded to by Rich (1990). It has to do with

finding the roots of the polynomial equation, y = 5x3 - 12r2 - 16x

+ 8. This problem is discussed at five levels of difficulty and the first

four levels use technology. Level one has students find the graph

of the equation, shown in Figure 4, by either table building or use of

a graphing utility. From the graph, students can isolate roots
between consecutive integers.

 

 

 

 

 

 

 

Figure4 Graphofy=5x3-12xz-16x+8

At the next level of exploration, students use built-in root-finding routines on

13

their calculator or computer to find the roots (or solution of the corresponding
equation 5::.3 - 12:2 - 16x + 8 = 0 ). The third level uses the graphing utility to
"zoom-in" to approximate the roots of the function as illustrated in Figure 5. Finally,
groups of students design an algorithm to find the roots. Once they have

developed an algorithm, students can Check its accuracy by writing a computer

program.

 

 

 

 

 

‘-123

 

 

 

 

 

 

 

 

 

 

Figure 5 Zoom-in on Roots

The National Research Council in Evegbogy Counts (1%) supports the
use of calculators and computers in mathematics education. It concludes, among
other things, that, with the availability of this technology, weakness in algebraic
skills need no longer prevent students from understanding ideas in more
advanced mathematics; learning can become more active and dynamic; and time
invested in mathematics study can build long-lasting intuition and insight, not just
short-lived Strategies for calculation. The council attempts to refute the arguments
of critics who claim, among other things that "students who use calculators do not

Ieam basic skills and will not learn to think." It goes on to report that:

14

students who use calculators learn traditional arithmetic as well

as those who do not use calculators and emerge from

elementary school with better problem solving skills and much

better attitudes about mathematics.... Although calculators and
computers will not necessarily cause students to think for
themselves, they can provide an environment In which student
generated mathematical ideas can thrive Innovative

Instruction based on a new symbiosis of machine calculation

and human thinking can shift the balance of learning toward

understanding, Insight, and mathematical Intuition. (1989, 48, 62-

63)

The forgoing discussion gives a clear indication of strong support by
mathematics educators for the appropriate use of calculators and computers in,
mathematics Classrooms.

Certainly the technology has progressed to the point that the function
concept is very easy to represent graphically on hand-held calculators and
microcomputers. These tools are accurate, can easily be zoomed-in and zoomed-
out to show special aspects of a graph, and can quickly plot far more points than

students can ever dream of doing by pencil-and-paper.

Purpose, Scope, and Significance of Proposed Study
In the Fall of 1990 the Mathematics Department at Michigan State University
integrated graphing calculators and related technology into the teaching of its
freshman precalculus or college algebra course using curriculum materials
developed by Frank Demana and Bert Waits of Ohio State University. Several
other college and universities in the United States have followed the lead of the

Ohio State and Michigan State groups and have taken similar action. Other

15

institutions such as Central Michigan University appear to be divided on the issue
of using graphing calculators and are conducting precalculus classes with and
without the aid of graphing calculator technology .

The main purpose of this study is to investigate the effectiveness of using
graphing calculator technology as an aid to the teaching and learning of
elementary functions, graphing, and related precalculus concepts in a university
setting. In particular, the investigation will attempt to determine the impact of this
technology on

(i) student performance in the area of basic algebraic skills,

(ii) student attitudes towards mathematics,

(iii) student performance in the area of functions, graphing and related
concepts.

The investigation will endeavour to respond to the following global questions:

(1) Does the use of graphing calculators in a university precalculus
course affect student performance on a test of traditional algebraic
skills?

(2) Does the use of graphing calculators in a university precalculus
course affect student performance on a test of basic concepts
related to elementary functions and graphing?

(3) Does the use of graphing calculator technology in a university
precalculus course affect student attitudes towards mathematics and

the use of the calculator technology?

16

The study will also endeavour to qualitatively investigate some of the
problems and difficulties, advantages and disadvantages, of integrating graphing
calculator technology into a one semester university precalculus course, as
perceived by students and instructors who are currently using the technology as
an aid to the teaching and learning of precalculus.

A quick survey of recent research studies would reveal that current research
findings on the effectiveness of graphing calculator technology as a teaching aid
are not very conclusive and, in fact, are somewhat contradictory. This might
logically explain why some mathematicians, in spite of the urgings of professional
mathematics educators, have been reluctant to unhesitatingly jump on the
"technology bandwagon." This study will be a serious attempt to respond to many
of these sceptics who are urging more research in this area.

Current literature reveals that not only has limited research been done on
the effectiveness of advanced graphing calculators in the teaching of mathematics,
but that most of the research has been primarily focused in secondary school
settings. In this regard, this study may be especially significant, in that it is
conducted in a university setting over one complete semester involving seven
Classes of precalculus students who are studying the full range of typical
precalculus concepts.

The proposed study should, of course, be of significant interest to Central
Michigan University where the data for the "control" and "treatment" groups will be

collected. This is especially so since this institution is currently conducting some

17

classes which are using a graphing calculator approach and others which are
using a traditional (non-graphing calculator) approach. Notwithstanding, a number
of colleges and universities in the United States have become frustrated with the
lack of success experienced in the teaching and learning of precalculus typically
taught by "traditional" methods. Many of these schools are constantly looking for
teaching innovations that will improve their performance in this area. Since the
proposed study will be conducted in a university setting, the data gathered in the
study should definitely be of interest to those schools who are seriously
contemplating following the lead of a few of their sister institutions who have fully
integrated graphing calculator technology into the teaching of precalculus.
Some critics of the "graphing technology approach" have expressed serious
concern that students who use this technology to study precalculus may
experience a deterioration in their basic algebraic skills which are widely accepted
as being very essential for success in subsequent mathematics courses at the
college level. This study, unlike most of the more recent studies done in the same
area, will endeavour, among other things, to seriously address this concern.
The next Chapter is devoted to a quick review of the existing literature
dealing with the use and effectiveness of computers and calculators in the
mathematics classroom. Chapter three describes in detail the methodology used
in this investigation. Chapter four includes results and analysis of the data
gathered in the study. Finally, Chapter five gives a summary of the conclusions,

limitations, and implications of this investigation.

CHAPTER II

LITERATURE REVIEW

An increasing number of articles that relate to the integration of technology
in the mathematics Classroom have appeared in various journals in mathematics
education over the past decade. The sheer number and diversity of such articles,
particularly those on computer assisted instruction, certainly reflects increased and
renewed interests in this subject. Many of these articles are based on informed
testimonials arising out of classroom experiences and consequently do not have
a strong research base. Frequently, they form conclusions and make
recommendations for Change without a sound theoretical and empirical rationale.
Notwithstanding, the results and conclusions of studies which have been done are
often inconclusive and in some cases contradictory.

Earlier research on calculators focused mainly on whether or not these
instruments have an adverse effect on pencil-and-paper basic skills or arithmetical
skills rather than on their impact on student achievement (Hembree and Dessart,
1986). Modern programmable graphing calculators have only recently been
introduced to education. Consequently, research on their usage is very limited.
Furthermore, most of the limited studies that have been done on these graphing

utilities have been directed at the secondary school level.

18

19

This chapter attempts to present a brief overview of some of the research
on computer or calculator assisted instruction that is relevant to the study

undertaken in this dissertation.

Educational Theories Related to the Use of Graphing Technology.

A rationale for the use of computers and calculators can be found in
Montessori’s and Piaget’s learning theories which proclaim that students learn
mathematics by becoming actively involved. These theories emphasize the use
of investigation and discovery where students discover mathematical structures
and verify and relate ideas both inductively as well as deductively. Computer and
calculator technology, if interactively used, provides an excellent opportunity for
students to investigate and relate mathematical concepts through a hands-on
approach and consequently make generalizations which can be subsequently
formally verified. In the opinion of Gale (1987), this is really the essence of
effective teaching practice alluded to by Piaget in a speech at Cornell University
in 1964 in which he states:

Teaching means creating situations where structures can be

discovered; It does not mean transmitting structures which may

be assimilated at nothing other than a verbal level ..... The

teacher must provide the instruments which the students can

use to decide things themselves. Students themselves must

verify, experimentally in physics, deductively In mathematics.

(Duckworth, 1964)

One could certainly claim that calculator and computer technology are likely

candidates for such instruments of discovery.

20

Two theoretical models that compliment each other and appear to lend
strong support for the integration of technology in the teaching of functions and
graphing are those of Hierbert (1986) and Kaput (1987a, 1987b, 1989). Hierbert’s
(1986) model is a dual model which focuses on both conceptual and procedural
types of mathematical knowledge. In this context, conceptual knowledge can be
thought of as being rich in relationships - a network of relationships. Procedural
knowledge consists of both the symbolic language of mathematics and the various
algorithms that relate to particular mathematics concepts. He argues that failure
to link conceptual and procedural knowledge accounts for many of the problems
in mathematics education. The essence of Kaput’s (1989) model is that
mathematical representations are composed of a variety of notational systems.
Proper understanding of these mathematical representations is enhanced if we
highlight the relationship between these notational systems (Kaput 1987a, 1987b,
1989). Kaput suggests that technology may be the key in constructing the links

between the various notational systems.

Computer Assisted Instruction In Precalculus
Robert Taylor (1980) in making a case for computers in the classroom,
proclaims that computers can serve the following functions:
(i) A tutor that instructs the students and monitors their practice.
(ii) A tool used by the teacher to augment the scope or effectiveness of the
instruction process.

(iii) A tutee that students can teach through programming.

21

The increased use of computer technology in the mathematics classroom
in recent years has certainly inspired many different approaches to the teaching
of mathematics, and consequently, has Changed the nature of the traditional
mathematics classroom. The computer, because of its capability to handle
massive computations quickly, has enhanced the ability of instructors and
students to conduct mathematical experimentation in the classroom. It allows the
"dynamic aspects" of mathematics to be visually demonstrated (Howson & Kahane
et al 1986). The end result of all this is that the roles of students and teachers
have changed (Shoenfeld, 1988). In particular, teachers have become less the
sole authority and at the same time students have more opportunity to become
more actively involved in the teaching and learning experience (Fey, 1984).

Support by mathematics educators and professional organizers for the use
of computers in mathematics in the Classroom has grown from the late sixties to
the stage today where many educators firmly believe that computers and
computer literacy should be an integral part of both the secondary and post-
secondary mathematics Classroom. A litany of recommendations, some of which
were mentioned in the introductory Chapter of this study, appear in the literature
in support of computer/calculator technologies in the mathematics classroom
(Conference Board of Mathematical Sciences, 1983; National Association for
Educational Progress, 1985; National Council of Teachers of Mathematics, 1980;
National Research Council, 1984). lnvariably, these recommendations have been
followed by warnings from the sceptics to proceed cautiously with the

implementation of new technologies in the classroom.

22

Computer assisted instruction, commonly abbreviated CAI, as a substitute
for or a supplement to the traditional methods of teaching mathematics, was
actually pioneered by Patrick Suppes (1968) in the Standard CAI Project in
elementary and secondary school mathematics. Since this project, a multitude of
studies have been done on the effectiveness of computer assisted instruction in
mathematics. This section will, for the most part, be restricted to those studies that
focus on college algebra or precalculus - more especially studies that specifically
relate to the teaching of functions, graphing, and related concepts.

In a two year study of eleventh grade algebra students, Ifieren (1968)
compared the achievements of two groups of students: a treatment group which
used computer programming and a control group which was taught in the
traditional manner. He reported that the control group scored significantly better
on two measures of achievement at the end of the treatment period. However, the
treatment group did significantly better in a trigonometry test. No other measures
showed any significant difference in performance. It was also found that the
impact of the computer was relatively stronger for average achievers than for high
achievers, and that the treatment contributed most to the organization of data, and
the learning of complex and infinite processes.

Katz (1971) also studied three groups of second year algebra students —
two experimental groups and a control group. The first experimental group wrote
algebra programs in conjunction with regular classroom presentation; the second
experimental group wrote and ran their own programs. Traditional instruction was

given to the control group. Katz reported that the second experimental group

23

scored significantly lower than the other two groups on standardized achievement
test. However, on a mid-term and final examination there was no significant
difference between the three groups.

Another study of computers in second year high school algebra was
conducted by Hoffman (1971). This study focused on the effects of computer
extended instruction on generalization skills and achievement. The control group
was taught by traditional methods and the treatment group received computer
applications in their instruction. Hoffman reported no significant differences in the
mean scores of both groups in performance on post tests. A follow-up item
analysis of the achievement test revealed that the computer did enhance specific
generalization skills.

An algebra-trigonometry high school group of students was studied by
Ronan (1971). This group used the computer as a computational and
experimental tool. When compared to a control group taught in the conventional
way, this treatment group, on the average did not perform significantly different on
tests of algebra review, trigonometric functions , complex numbers, and inverse
functions. There was also no significant improvement in the problem solving skills
of the treatment group. The treatment group did, however, score significantly
higher than the control group on tests involving exponential and logarithmic
functions. They also showed significant improvements in conceptual
understanding and skill development. A test of trigonometric identities and

formulae revealed better scores for the control group.

24

In a study done by Schoen (1972), a comparison was made of four
instructional treatments in a computer assisted instructional unit used to teach the
function concept. The treatments all differed in the degree of individualization and
personalization. Two units were developed with the second unit adapted for four
types of feedback: two levels of personalization and two levels of individualization
to incorrect responses. Three achievement tests were given during the
instructional period which lasted an average of 171 minutes. One test was given
after the first unit ( a dummy run intended to minimize the Hawthorne effect) and
one after each subunit of the experimental unit. An attitude test was administered
at the end of the course. Analysis of variance testing revealed that the effects of
the treatment were insignificant. However, students seemed pleased with the
personalization aspect of the course.

In another project Robitaille, Sherrill and Kaufman (1977) investigated the
effects of computer assisted instruction on the mathematics achievement and
attitude of ninth grade algebra students. Three groups of students were selected
from each of two schools with four months of the study spent in one school and
nine months in the other school. One group fully used the computer; another
group made partial use of the computer, and the third group (the control group)
did not use the computer. Results of the study did not generally support the
favourable claims made by proponents of computer - augmented mathematics
instruction. In the case of achievement, a significant difference was found in each
school but not in favour of the computer classes. The computer group showed

the most positive attitude towards mathematics in the shorter term evaluation.

25

However, in the longer-range study, no significant difference in attitude was
observed.

An eight month achievement and attitude study on computer enhanced
learning was reported by Saunders and Bell (1980). This study obtained data from
101 students enroled in four sections of a second year high school algebra
course. Two sections were designated control sections. Results of this study
showed that the use of computer-enhanced materials had no effect on algebra
achievement. Attitude scores showed differences in attitude favouring the control
group. In short, the findings of the study indicated that computer-enhanced
resource assignments need not infringe upon the time spent learning traditional
second year algebra and at the same time have no harmful effects on achievement
in algebra and attitude towards mathematics. The study indicated that in order to
overcome short-term negative attitudes it is important to introduce computer-
enhanced resource materials over a long time period.

Three sets of CAI Programs were used with college freshmen in a study by
Caputo (1981). Each set was divided into eight modules depending on areas of
difficulty. The programs in the first set used a verbal presentation mode. The
second set of programs used a verbal mode enhanced with graphics and the third
set consisted of a "placebo" exercise. Of the three presentations the graphics was
found to be the most favourable when dealing with deficiencies. Furthermore, the
graphics program was perceived by the students to be both effective and

enjoyable. Average course grades revealed no differences in the various groups.

26

Evans (1981) describes applications of microcomputers to the teaching of
precalculus. He actually studied the effectiveness of the computer to help students
visualize the conceptual relationship of the function so that they could explore the
algebraic concept of linearity. His study showed that students can more
accurately predict the graph of a given equation after they are repeatedly exposed
to visualization of the concept with the microcomputer.

Informed testimonials, proclaiming increased student motivation due to
computer assisted instruction, have been given by Dugdale (1982) and Swift
(1984). Dugdale also reported that using a graphing utility improved students
understanding of parameter variation of a family of functions.

An interactive software package aimed at improving the graphing
competency of algebra students was designed and implemented by Morris (1982).
Morris Claimed that because the student can see the graph drawn quickly and
accurately by a computer, concepts in the classroom are reinforced by the
supplementary computer enhanced instruction. He argues that these
supplementary instructions gives students a measure of control over events
occurring in the lessons.

Reed (1985) introduced a non-interactive computer graphics program to
study the effectiveness of computer graphics in improving estimates for story
problems in algebra. This program consisted of computer simulations observed
by students. As a result of this study, Reed concluded that "learning-by doing"

might be more effective than "learning-by-viewing". He suggests more research

27

is needed on the use of the microcomputer as a teaching aid to teach graphing
in college algebra.

When using the microcomputer as a demonstration tool in an intermediate
algebra course, Ganguli (1986) found that it had positive effects on both
achievement and attitude with medium students benefiting the most from the
demonstrations.

Payton (1987) investigated the use of graphing software in the instruction
of "college level basic mathematics". She claimed that use of the software helped
improve the performance of remedial students in functions, relations and graphs.
Such students, however, had the luxury of having a computer lab being made
available to them at their convenience. Positive but not significant differences in
attitudes towards mathematics and computers were also shown by the students
in this experimental group.

A comparison of the effectiveness of various technologically assisted
instructional methods was made by Mitchell (1987). In this study the researcher
tested the effectiveness of various instructional methods on students of differing
ability and course levels. She found that the most effective instructional method
for students at the precalculus level was a combination of computer assisted
instruction, audio tutorials and tutoring. At the higher ability levels students
benefited the most from a combination of computer assisted instruction and
laboratory sessions.

Payne (1988) studied the instructional effects of computer graphics in first

year college algebra. When computer graphics are utilized in solving equations

28

she found no significant difference between experimental and control subjects on
computations and comprehension measures. In the study, students with high
general reasoning or computational learning scores achieved more on higher level
learning behaviours from a treatment which incorporated techniques for solving
equations while review of prior training was given less Instructional time. Students
with low general reasoning or computational skills gained more on higher level
learning behaviour after receiving traditional instruction on algebraic techniques
based on his data,

No significant effects of implementing graphical software incorporated as
a lecture tool were discovered in eleventh and twelfth grade mathematics courses
by Geshel-Green (1987). However, anecdotes revealed that improvements made
by lower ability students were probably important enough to warrant further study.
Frederick (1989) similarly found no Significant effects when graphing software was
used with college students studying functions, relations, and graphs.

Some studies have identified difficulties in students interfacing with the
computer technology thus causing "perceptual distractions". In this regard,
Goldenberg (1988) Claims that scale Changes arising in technologically produced
graphics caused misconceptions among students when they tried to identify the
graph of a given equation.

In a study of the effectiveness of graphing with the microcomputer in college
precalculus mathematics, Coles (1989) determined that computers were especially
useful for comparing two or more graphs. They saved time and created more

interest in graphing when compared with traditional methods. They also motivated

29

students to do an analysis of graphics not usually done or perhaps not even
possible when graphing by hand.

Walfe (1990), used an interactive graphics tool to teach the function concept
to a mathematics class of inservice high school teachers. She reported that the
software had helped them to understand the function concept more fully and that
they were better prepared after their experience to teach the concept to their high
school students.

A study designed to develop computer graphics problem solving activities
to aid in the teaching of functions and transformational geometry was conducted
by Thomas (1990). In this study a determination was made of the impact of the
computer graphic activities on achievement and attitudes of secondary
mathematics students. No significant gain was reported in this study between
pretest and posttest scores for function concepts and student attitudes. However,
it was found that computer graphic problem solving activities did improve student
attitudes and achievement in the area of functions. However, these graphic
activities produced no significant gain in achievement on transformational
geometry concepts.

Cox (1990) studied the effects of curriculum specific computer aided
instruction on student achievement in a college algebra course. The main
conclusions reached in his study were:

(1) the use of specific computerized drill and practice can significantly increase
the mathematics achievement of students in college algebra who are

receiving a traditional lecture,

3O

(2) there is a significant relationship between a student’s pretest scores and
their level of success when given curriculum specific microcomputer aided
instruction.

An investigation of the use of computers for independent exploration in
precalculus was carried out by Alkalay (1991). The main purpose of this
investigation was to develop a laboratory manual that could be used in conjunction
with existing software for independent exploration in precalculus. Results of this
investigation indicated that there was no Change in mathematics attitude or student
attitude towards computers after students used the computer and laboratory
manual for an independent exploration. However, results of posttests did indicate
that students can learn precalculus concepts by independent exploration with the
aid of a computer. It was reported in this study that student evaluations of the
instructional materials used in this teaching approach indicated a strong positive
student reaction towards the computer in a laboratory setting when doing
precalculus. The instructor’s log indicated that the laboratory sitting appeared to
encourage good behaviour and good work habits within the students.

Funkhouser (1991) studied the effects of problem solving computer software

on the problem solving ability of secondary mathematics students. His
investigation involved 41 treatment and 31 control subjects who were doing
geometry and second year algebra courses. Overall, no significant difference was
reported in the study in the improvement of problem solving skills except in the
area of spatial ability (p =.13). It was reported in this investigation that positive

gains in attitude towards mathematics and self and mathematics as a discipline

31

were made by the treatment group. A significantly greater mathematics
performance score ( p = .004) was found for students of the treatment group who
were involved in computer-related mathematical activities.

An attempt was made by Easterling (1992) to study the impact of a
computer based mandatory laboratory on the performance level of college algebra
students. Two hundred and forty college algebra students in two sections of
college algebra were the subjects of this study. The study concluded that there
was no significant difference in the performance level of students who studied the
fundamentals of college algebra using a computer assisted laboratory and those
who studied the fundamentals using a teacher-aided laboratory. Furthermore
there were no significant gender differences. There was, however, a significant
difference between the performance of traditional-aged and nontraditional-aged
students, with the nontraditional-aged students scoring higher on the posttest of
the instructional materials. In short, it was found that students did just as well In
teacher-aided as a computer-aided laboratory course in fundamentals of college
algebra.

The computer is a power tool which undoubtedly has the capability to

improve and enrich the curriculum. It not only affects what students learn; it also
effects how students learn and how teachers teach. Nevertheless, the foregoing
Studies on the effectiveness of computer assisted instruction on the mathematics
achievement levels of students and their overall attitude towards mathematics are

both inconclusive and somewhat contradictory.

32

The Use of Graphing calculators In the Classroom

The literature shows that extensive amounts of research have been done
on computer assisted instruction in mathematics. However, since the graphing
calculator is a relatively new phenomenon, research on the use of this technology
in the teaching of mathematics is very recent and very limited.

Before the event of the more modern graphing calculators the great bulk
of research on the use of ordinary calculators focused on whether or not the use
of these calculators had an adverse affect on the normal paper-and-pencil skills.
Most of this research was conducted between the late 1960’s and the early 1980’s.
Probably the best survey of all studies done during this time period was
conducted by Hembree and Dessart (1986) in the form of a meta-analysis. The
study determined calculator effects in grades K - 12 by statistically averaging
subsets of the 524 effects derived from 79 studies. Based on their findings the
authors reached the following conclusions:

1. In Grades K-12 (except Grade 4), students who use
calculators In concert with traditional Instruction maintain
their paper-and-pencll skills without apparent harm. Indeed,

a use of calculators can improve the average students’s
basic skills with paper and pencil, both in basic operations
and In problem solving.

2. Sustained calculator use by average students In Grade 4 appears
counterproductive with regard to basic skills.

3. The use of calculators In testing produces much higher
achievement scores than paper-and-pencil efforts, both In: basic
operations and in problem solving.; This statement applies across
all grades and ability levels. In particular, it applied for low-and
high-ability students In problem solving. The overall better
performance in problem solving appears to be a result of Improved
computation and process selection.

4. Students using calculators possess a better attitude toward
mathematics and an especially better self-concept In mathematics

33

than non-calculator students. This statement applies across all
grades and ability levels. (p.96)

The authors assert that it is not a question of whether calculators should
be used along with basic skill instruction but how they should be used. In keeping
with this statement they made the following recommendations for Classroom
usage:

1. Calculators should be used In all mathematics classes of

Grades K-12.

2. Because of the apparent negative effects of calculators In Grade 4,
calculator functions In that grade should be approached with
caution.

3. Students in Grade 5 and above should be permitted to use
calculators In all problem-solving activities, including testing
situations. This recommendation ls based on these two
observations:

a. Calculators greatly benefit student achievement In problem
solving, especially for low-and high-ability students.
b. Positive attitudes related to the use of calculators may help to
relieve students’ traditional dislike of word problems. (p.97)
Throughout the 1980’s calculator research went into a decline and was replaced
by research in computer assisted instruction.

The first hand held graphing calculator was introduced by Casio in 1986.
Since that time strong support from influential mathematics educators for the

integration of graphing utilities in the mathematics Classroom, and

recommendations from national organizations to increase the use of computer and
Calculator technology, has recently given new impetus to research on more
SOphisticated graphing calculators.

A quick review of the literature will reveal that most of the research studies

0 n the graphing calculator, as was the case with the computer studies discussed

34

earlier, have been focused on the effect of this technology on

(i) Student attitudes towards mathematics (usually measured by a Likert type
instrument).

(ii) Student achievement on researcher-designed pretests and posttests on the
mathematics concepts being studied.

Other studies have focused on the effect of graphing calculators on problem

solving skills, spatial visualization skills, as well as gender differences in

mathematical achievement.

Rich (1990), taught pre-calculus to a group of honours students in two high
school mathematics Classes with the aid of a graphing calculator, and compared
their performance with the performance of students in three classes of similar
ability that were taught the same basic materials by traditional methods at another
school. Results of this experiment revealed no significant difference in the overall
achievement of the two groups but there was a positive impact of the graphing
utility on the learning of function concepts. In particular, the graphing calculator
gave students a better appreciation of graphical solutions to algebraic problems;
it also gave them a better understanding of the connection between an algebraic

equation and its graph. She also reported that the instructor who used the
graphing utility made better use of an exploratory approach to teaching and asked
more higher level questions.

It was revealed by Giamati (1990), who also used a graphing calculator to

reach variations in a family of equations and the transformations of their graphs,

"’7! at students who initially had solidly formed conceptual links between equations

35

and their graphs benefitted from a graphing calculator. On the other hand
students with poorly and partially formed links were cognitively distracted by also
having to learn to use the graphing utility. She concluded that understanding of
stretches, shrinks and transformations, a basic goal of the teaching of functions,
was not aided by graphing calculators. Her Classroom observations and student
interviews suggested that unfamiliarity with certain characteristics of the calculator
may have diminished its effectiveness as an instructional tool.

Shoaf-Grubbs (1991) studied the impact of using graphing calculators in a
college algebra course on spatial visualization. It was reported in this study that
students who used graphing calculators in a college algebra course made greater
gains in spatial visualization skills than students in a control class. Scores were
significantly higher for the treatment group on the Card Rotation and Paper
Folding Tests and on two of three spatial visualizations tests for linear equations
and parabolas designed by the researcher. Vazquez (1991) reported similar
results in spatial visualization for students who used graphing calculators.

The graphing calculator was used by Ruthven (1990) to study gender
differences in mathematical achievement among mathematics students as they
translated from graphic to symbolic form. This study showed marked

improvements in this area for females using graphing calculators. In the treatment
group women outperformed men on symbolization items but the reverse was true
f(Dr students in the control group. Ruthven conjectured that graphing calculators
help reduce anxiety and uncertainty and at the same time gave students more

e><r:aosure to graphical images. Consequently women using graphical technology

36

established confidence in their graphing ability and hence increased their
achievement level. Similar results were reported by Dunham (1990).

Gender differences in performance in a precalculus course which fully
integrated graphing calculators were also studied by Dunham (1990). She
reported no significant gender differences in graphing and in pretest or posttest
performance on visual items. She examined the relationship between confidence
and performance among males and females. Her findings suggest that males
exhibited greater confidence on visual items on a pretest. It was also reported that
women had significant positive correlation between problem specific confidence
and performance more often than men. However, a surprising number of high
confidence females exhibited algebraic guilt, a feeling that they relied too much
on "easy" calculator solutions and would benefit more from learning more algebraic
techniques. She recommends further research on algebraic guilt and confidence
in the technology.

Studies conducted by both Farrell (1990) and Rich (1990) suggested that
graphing calculator technology promoted problem solving activities. These

authors report that students were more willing to engage in problem solving with
graphing calculators. In particular, they were able to investigate and solve non
r Outine problems, normally not accessible with the usual algebraic strategies.
Positive changes in attitude towards mathematics for students using
graphing technology have been reported by Coles (1990), Davis (1990) and
rhomas (1990). Vazquez (1991) reported no differences in attitudes towards

"7| athematics between treatment groups of precalculus students who were taught

37

with the aid of graphing calculators. However, students using graphing calculators
were significantly more positive towards calculator use. One of the most positive
studies on attitudes towards the graphing calculator was done with 400 minority
students in a college algebra course by Whitkanack (1990). Almost all the
students in this survey reported it was important to purchase a calculator with only
one percent of the students indicating they could not afford one.

A study which focused particularly on implementation problems when
graphing calculators are integrated with regular instruction in a mathematics class
was conducted by Novak (1990). Results of her research project revealed that
teacher usage levels of calculators had no obvious predictable effect on students
learning experiences. As students struggled for an understanding of the materials
in the course, it became clear that even though students liked the calculator, they,
nevertheless, "preferred" Clearly presented moderately paced lectures by teachers
who spoke articulately, wrote legibly, and demonstrated attention to student
weaknesses.

A study of the effect of the calculator on basic skills maintenance and
algebra achievement was conducted by Whisehant (1990). The subjects for this
Study were comprised of mathematics algebra I students from a large suburban

School district in Texas. Control and treatment groups consisted of one eight
grade class, two ninth grade classes, one tenth and one eleventh grade class.
Significant differences were reported in this study in the adjusted mean scores on
tests in both of the areas of basic mathematical skills and algebra achievement.

lr~. particular, the mean scores for the control group exceeded those of the

38

treatment group on both counts with the greatest difference occurring with the
tenth and eleventh grade algebra l students.

Army (1992) studied the effectiveness of the graphing calculator as an aid
to the teaching of a college course in trigonometry. The dual purpose of this
study was to introduce new technology into the mathematics classroom and
thereby link mathematics content to real world applications in a college
trigonometry course. This study reported no evidence of any achievement gains
but some evidence of positive effects of the technology on student attitudes
towards the usefulness of mathematics and the usefulness of the graphics
calculator Furthermore, the study suggested that students who use graphing
calculators Choose to solve graphically rather than algebraically problems that
involve algebraic equations. The researcher also reported a shift of emphasis by
these same students away from algebraic manipulations and proof to graphical
investigation and the interfacing of algebra and geometry when students were
confronted with problem situations. Furthermore, it was found in this study that
graphing calculators encourage student interaction and enable instructors to better
illustrate the basic properties of trigonometry functions.

More recently Tuska (1993) conducted a study of student errors made in
graphing calculator-based precalculus classes. The main focus of this study was

to identify the most common mistakes precalculus students make in a graphing
calculator-based instructional environment. As a result of analyzing the results of
the multiple Choice mid-term exams of 1000 students, the author determined that

eight misconceptions were commonly held by precalculus students:

l5)

st.c
CO:

The

39
(1) The domain of a function cannot skip intervals.
(2) The domain is a subset of the range.
(3) To "solve" an inequality is to "find its zeros".
(4) To "solve" an inequality is to "find the intersections or cutting points".

(5) In order to solve an inequality, functions should be compared shape-wise

instead of x-wise.
(6) Every number is rational.
(7) The graph of a function on a "large viewing rectangle" is always enough to
determine the end behaviour of the function.
(8) If a function is not defined at a point then it has a vertical asymptote at that
point.
As in the case of the computer studies survey in the previous section, early
studies on the effectiveness of graphing calculator technology aren’t really
conclusive insofar as they affect student attitudes and mathematics performance.
The results are also contradictory in the sense that some studies show positive
effects on attitude and achievement for the treatment groups of the studies. Other
studies show no Significant effects. This, of course, has prompted a plea from
some educators for more research in this area. One of these dissenting voices is
that of Crosswhite (1986). In responding to NCTM recommendations concerning
the use of technology in the classrooms, Crosswhite issues a precautionary note.
He argues that adjustments obviously have to be made in the mathematics
Curriculum to accommodate calculators and computers. However, if adjustments

have to be made quickly, he cautions us that we may I"throw out the baby with the

4o

bathwater." He quickly reminds us that NCTM has recommended that school
programs take full advantage of calculators and computers not the other way
around. He then goes on to suggest that the way we respond to computering or
calculator technology in the future will largely determine the next cycle in school

mathematics. Indeed, this could be extended to all mathematics.

The CzPC Project at Ohio State

A nontraditional approach to teaching precalculus mathematics, making use
of graphing calculators and computers to accurately graph functions and to solve
a wide variety of interesting and more Challenging problems, is currently being
class tested at Ohio State University. This large project involving over 80 high
schools and 40 colleges is called the calculator and computer precalculus project
(abbreviated as CZPC). The goals of this project together with a brief description
of the project are outlined by Demana and Waits (1988) as follows:

1. To create instructional materials that make effective use of
computer based graphing and strengthen student problem
solving skills.

2. To improve student understanding of functions and related
graphical concepts, domains of mathematical deficiency in
the current college preparatory curriculum.

3. To significantly increase the number of students adequately
prepared to pursue higher education In mathematics,
science, and technical fields.

The new precalculus course Is Intended for both high school
and college students who have successfully completed Algebra
I, Geometry, Algebra II and Includes most of the traditional
topics covered in a post Algebra II functions, trigonometry and
analytic geometry course. Problem solving is emphasized and
real world applications are integrated throughout the materials.
Important algebraic skills and techniques are Included. Besides

41

providing a new approach, computer graphing also provides a

geometric Interpretation and a check for solutions found

algebraically. (p.47)

Dunham (1992) reported that some data generated by this study remains
to be analyzed. However, early results from the field test data are very
encouraging and certainly indicate that mathematics instruction with graphing
technology can have a positive impact on student achievement in precalculus,
mathematical understanding of functions and graphing, attitudes, and Classroom
roles.

Several authors who are associated with the CZPC project appear to confirm
Dunham’s findings. Rich (1990) found no significant differences on overall
precalculus performance but identified level differences that favour the use of
graphing calculators for teaching functions and graphing concepts. Davis (1990)
reported that college students who used the CZPC curriculum materials scored
significantly better on a precalculus test than did a control group. Browning
(1989a, 1989b) found that students taught precalculus CZPC materials with the aid
of graphing utilities attain higher levels of understanding than those in non-
technology classrooms. Browning’s studies also showed that there is a hierarchy
in the learning of graphing concepts. Taylor (1990) also field tested C2PC
materials in four high school Classes and reported a positive correlation between
the use of graphing calculators and the highest levels of graphical understanding.

The CZPC findings appear to suggest that students who use graphing
Utilities in precalculus receive better preparation for the study of calculus. This is

substantiated by the results of the data analyses from the C2PC project. In fact,

42

almost twice as many students in the treatment groups achieved calculus
placement scores as those precalculus students from control classes in the CZPC
project. In this respect, early results from the very comprehensive CZPC project
are more encouraging than some of the findings reported earlier.

In spite of the mixed reviews of some of the smaller studies that have been
done on graphing technology as it relates to student attitudes and achievement,
reasonable evidence exists to suggest that this technology, as with computer
graphics, has the power to enhance student understanding of basic function
concepts, as well as the ability to seriously impact on student self concepts and
student/teacher roles. Furthermore, graphing utilities appear to have the potential
to develop within students strong geometric intuition as it interfaces geometry and
algebra to find solutions to a variety of non traditional and more interesting
problems.

The next Chapter outlines the methodology used in this study as well as the
limitations of the study. The last two Chapters describe the results of the study
and provide a discussion of the findings and some of the implications of this

investigation.

CHAPTER III

METHODS AND PROCEDURES

This chapter describes the methodology used to investigate the research
questions examined in this study. It begins with a restatement of the main
problems under investigation. This is followed by a description of the participants
and the instructional materials used by the treatment and control groups. An
outline is then given of the design of the experiment and the assessment
instruments used in the study. This is followed by a discussion of the type of
analysis to be done on the results of the study. The chapter concludes with a

discussion of the limitations of the study.

Restatement of the Problem

Dissatisfaction with the performance of students who study precalculus
taught by traditional methods at the secondary school level has been expressed
nationally and alluded to in Chapter 1 of this study.

Recommendations to improve the teaching of functions, relations and
graphs, together with appeals for the implementation of computers and calculators
in the secondary classroom as powerful aids to the teaching of precalculus, have
been made by national bodies involved in mathematics education. Some of these

recommendations have already been discussed in the previous two chapters.

43

44

In response to these recommendations schools, colleges and universities
in the state of Michigan have already introduced graphing calculators as an aid to
the teaching of precalculus, using texts and instructional materials especially
designed for this purpose. Other post secondary institutions in the United States
are evidently contemplating a similar move but appear to be proceeding with
caution for reasons discussed earlier in this study; not the least of which is
concern for the erosion of basic algebraic skills, and the necessary curriculum
Changes and modification which will have to be made to accommodate the
technology.

The primary purpose of this study is to investigate the effectiveness of using
graphing calculator technology as an aid to the teaching and learning of
elementary functions, graphing, and related precalculus concepts in a university
setting. In particular, the investigation will attempt to determine the impact of this
technology on:

(i) student performance in the area of basic algebraic skills,

(ii) student performance on a test of core concepts in the area of

functions, graphing and related concepts,

(iii) student attitudes towards mathematics.

The investigation will actually endeavour to respond to the following global
questions:

(a) Does the use of graphing calculators in a university precalculus course

affect student performance on a test of traditional algebraic skills?

(b) Does the use of graphing calculators in a university precalculus course

45

affect student performance on a test of basic concepts related to
elementary functions and graphing?

(c) Does the use of graphing calculators in a university precalculus course
affect student attitudes towards mathematics?

In the light of these questions, the following hypotheses will be examined:

Hypothesis 1: After a one semester precalculus course at university, there is no
significant difference in the achievement level, in the area of basic
algebraic skills, between those students who study precalculus
with the aid of a graphics calculator and those who do not.

Hypothesis 2: After a one semester precalculus course at university, there is no
significant difference in the achievement level, in the area of basic
functions concepts and graphing, between those students who
study precalculus with the aid of a graphics calculator and those
who do not.

Hypothesis 3: After a one semester precalculus course at university, there is no
difference in the attitude towards mathematics of students who
study precalculus with the aid of a graphics calculator and those
who do not.

Questions (a), (b), and (0) above were also to be investigated by studying the

average change (loss or gain) in achievement in the areas of algebraic skill
proficiency and function concepts and also the average change in attitude
towards mathematics over a complete semester of precalculus. This average

Change in attitude and achievement in this case, is to be determined relative to

46

student performance on pre and posttests of algebraic skills, function concepts,

and overall attitude towards mathematics which are to be administered at the

beginning and end of term. With regards to this part of the investigation, the
following hypotheses will be tested:

Hypothesis 4: Over a one semester precalculus course there is no significant
difference in the average change (loss or gain) in achievement, in
the area of basic algebraic skills, between students who take
precalculus with the aid of a graphics calculator and those who do
not:

Hypothesis 5: Over a one semester precalculus course, there is no significant
difference in the average Change (loss or gain) in achievement, in
the area of basic function concepts and graphing, between
students who take precalculus with the aid of a graphics calculator
and those who do not.

Hypothesis 6: Over a one semester precalculus course, there is no significant
difference in the average change (loss or gain) in attitude towards
mathematics between students who take precalculus with the aid
of a graphics calculator and those who do not.

The secondary focus in this study is to qualitatively investigate student and
faculty reaction to the use of the graphing calculator and some of the problems
and difficulties, advantages and disadvantages, of integrating graphing calculators
into a one semester university precalculus course, as perceived by students and

instructors.

 

 

47

Participants and Setting

This study made use of both a treatment and control group of students.
The treatment group was comprised of three Classes of precalculus students (125
students) at Central Michigan University who were taking a one semester
precalculus course with the aid of a graphing calculator. The control group of
students was comprised of four classes of precalculus students (179 students) at
Central Michigan University who were taught a one semester precalculus course
in the traditional (non-graphing calculator) university style. Both these groups of
students were typical university freshmen pursuing a variety of arts & science
programs. These students were of average mathematical ability in the sense that
the top 25-30 percent of all students who take mathematics at the entry level at
Central Michigan University are assigned to a basic calculus course.

Central Michigan University had a total of seven classes of precalculus
during the Fall semester, 1992. All seven Classes and their instructors initially
agreed to participate in this study. Selection to the treatment or calculator classes
was actually self selection. These Classes were advertised in the university
calendar and students opted to take them out of self interest and because they
fitted their time tables.

Both the treatment and control groups were taught by full time faculty. The
four faculty members who taught the control group had teaching experience
ranging from 3 - 35 years. The three faculty members who taught the treatment
group were more senior faculty whose teaching experience, at the secondary and

post-secondary level, was in the 30 - 35 year range.

43

Central Michigan University, located in the town of Mount Pleasant, Michigan
is a regional state college of medium size with an enrolment of approximately
16,000 students. It is primarily an undergraduate school, with a relatively small
graduate program, offering the usual range of arts, science, and teacher education
programs. The choice of this particular school for this study was a natural one in
the sense that it was one of the few schools available to this project who offered
precalculus in two formats- with graphing calculators and without graphing

calculators.

Instructional Materials

Both treatment and control groups met four times per week during a fifteen
week semester precalculus course. The groups were taught in a semi lecture
style.

Students from the treatment group used instructional materials from a
precalculus textbook entitled Precalculus: Functions and Graphs. Second Edition
by Demana, Waits, and Clemens. This text is appropriately designed to be used
with the Cassio fx-7000G, fx-7500G, TI81 and other equivalent graphing
calculators.The book takes full advantage of the power of these graphing utilities
to apply a graphing approach to the teaching of precalculus. This technology
allows the focus of the precalculus course to be on problem solving and
exploration of mathematical ideas and at the same time hopefully develop a
deeper understanding of algebraic manipulations. A three step problem solving

process is used throughout the text and consists of:

49

(i) finding an algebraic representation of a problem to be solved.
(ii) finding a complete graph of the algebraic representation.
(iii) finding a complete graph of the problem solving situation.
This process naturally enables the algebraic and geometric representation of a
problem to interface and thus lead to a more meaningful solution to the problem.
Using graphing utilities in conjunction with the text, students are able to
produce graphs quickly and accurately and use them to study the properties of
functions and transformations and explore intuitively new mathematical ideas.
Furthermore, students using this text were given the opportunity to use the
graphing technology to intuitively investigate some of the basic concepts of
calculus such as continuity, end-point and symptotic behaviour of functions,
increasing and decreasing functions, maxima and minima. Students are thus able
to lay an intuitive foundation for the study of calculus.
Instructors and most of the students in the group used the Tl-81 calculator.
All students had regular and frequent access to these machines for homework and
normal Class activities.

Topics studied by this group included the following:

. Relations, functions and graphs

. Polynomials and solving equations and inequalities
. Polynomials, rational functions and radical functions
. Trigonometric functions

. Analytical trigonometry

. Polar co-ordinates and parametric equations

Ila:

SiJ

te

on

Io

EXE

5O

. Sequences and series
A more specific syllabus is given in Appendix I.

The control group of precalculus students were taught precalculus in a
traditional university lecture style using a standard precalculus text entitled
Preca_lculus: Functions and graphsjixth Edition by Earl Swokowski. The
students using this text used the normal pencil-and-paper graphing techniques in
the study of functions graphing, and related precalculus concepts.

One of the main goals of the Swokowski text is to make discussions of
concepts relatively easy to understand without sacrificing mathematical soundness.
To achieve this goal, each section of the text has carefully chosen examples to
help students understand and assimilate new concepts. The book has a rich
supply of applied problems from many different fields. The practical exercises are
designed in such a way that students can obtain practice on all topics by working
the odd number exercises. Labelled illustrations in the text give brief
demonstrations of the use of definitions, properties, and theorems. Most of the
exercises in the text are workable without a calculator. However, a calculator
would be useful in order to reduce the numerical computation involved in some
exercises.

Although the syllabus for the control group was taken from a different
textbook, nevertheless, the topics covered were remarkably similar. Typical
precalculus topics covered in the text include the following:

. Real numbers and graphs.

. Functions

Ar

lea

Ullll

51

. Polynomial and rational functions

. Exponential and logarithmic functions

. The trigonometric functions

. Analytical trigonometry

. Sequences and series

. Analytical geometry

. Polar co-ordinates and parametric equations.

A more detailed description of this syllabus is given in Appendix J.

Research Design and Research Instruments

The basic design of this experiment was a treatment group versus a control
group in a comparative study. The subjects that made up the control and
treatment groups belonged to seven precalculus Classes at Central Michigan
University and were described earlier in this Chapter. Participation of both groups
on the various instruments used In the study was strictly voluntary.

This investigation is both qualitative and quantitative in nature using a variety
of instruments which are described in detail in the appendices of the study. In
particular, the actual research plan included the following instruments:

(i) Student Attitude Inventory. This inventory was administered in the form
of a pretest and a posttest during the first and last weeks of the Fall
semester, 1992 to both the control and treatment groups. The pretest
contained Likert-type statements with five possible student responses to

each statement. The statements were all designed to assess the students

(ii)

52

feelings about mathematics and were placed in random order on the
inventory. Specific areas covered by the statements were mathematics
anxiety, value of mathematics to society, enjoyment of mathematics, self
concept in mathematics, motivation to study mathematics, and student
expectations. All of the statements on the pretest were also contained in
the posttest. However, the posttest contained an additional 10 statements
which were designed by the researcher to test attitudes towards the
graphing calculator at the end of the precalculus courses. These were
responded to by the treatment or calculator group only. The main
purpose of this attitude inventory was to compare the attitudes towards
mathematics of both the treatment and control groups and to study the
mean improvements in attitude that might conceivably occur during the
semester. A copy of this inventory is given in Appendices A and B.

Algebraic Skills Test This was a test of basic algebraic skills based on
material that the students in both the control and treatment groups were
exposed to earlier in their high school mathematics courses. The test was
administered in the form of a pretest and a posttest during the first and last
weeks of the Fall semester, 1992. Topics covered on both of these tests
included solving systems of equations, factorization and expansions,
manipulatives, simplifying expressions, exponents and radicals, algebraic
fractions, real number properties, absolute value, inequalities, solving
polynomials, equations, and functional notation. The aim of this test was to

compare the performance of both the treatment and control groups on

(iii)

53

basic algebraic skills both at the beginning and end of the treatment period.
This was important to the investigation of whether or not the treatment
group experienced any adverse affects on their basic skills in comparison
with the control group. In particular, a comparison of the difference in pre
and posttest scores on both groups made possible a comparison of the
mean change (loss or gain) in achievement of both groups in the area of
algebraic skills. A copy of the Algebraic Skills Pre and Posttest is given in
Appendices C and D respectively.

Function Concepts Test. This test was administered in the form of a
pretest in the first week of Fall semester, 1992 and also in the form of a
posttest in the last week of Fall semester, 1992 to both the treatment and
control groups. Furthermore, a comparison of the pre and posttest score
difference for both groups was used to compare the mean Change (loss or
gain) in achievement of both treatment and control groups on function
concept proficiency. These tests were intended to make comparisons of
the performance of both the treatment and control groups on fundamental
concepts normally associated with the notion of functions, graphing and
other related precalculus concepts both at the beginning and end of the
semester. Specific topics covered on both these tests include: functional
notation, evaluation of functions, identifying graphs of a given relation or
equation, domain and range, quadratic functions and conics, distance
formula, rational functions, intercepts, and word problems involving

functional relationships, inequalities, stretches and transformations,

(iV)

54
composition of functions, symmetry in graphing, and finding roots by
graphing. Copies of these tests are given in Appendices E and F.
Student Questionnaire. This questionnaire was administered to the
treatment group during the last week of Classes. The questionnaire
contained both open and closed-ended questions. It attempted to gauge
student reactions as per the effectiveness of using graphing calculator
technology as a teaching and learning aid in the study of functions and
graphing. The open-ended question portion of the instrument contained
ten items which allowed students to comment on the usefulness of the
calculator as a tool, problems with its usage, its effects on workload,
possible adverse affects on algebraic skills, and the extent to which
students had become "over dependent" on the technology (algebraic guilt).
Item eleven was close-ended and contained 11 parts with each part being
a statement of how the calculator could specifically impact on the teaching
and learning of precalculus. Students gave one of five responses to each
item ranging from (1) "not at all helpful" to (5) "Very helpful". Topics
covered in these items pertained to the effectiveness of the graphing
calculator in:

1. Increasing understanding of the function concept

2. Developing basic algebraic skills and manipulatives

3. Solving new and interesting problems

4. Estimation

5. Exploring and discovering mathematics ideas

(V)

55

6. Graphing new and interesting functions

7. Solving equations

8. Understanding links between an algebraic equation and its graph

9. Providing an intuitive understanding of "asymptotic" and end point
behaviour of a function

10. Bypassing the use of tables when graphing

11. Preparation for calculus.

A copy of the questionnaire is given in complete detail in Appendix G.

Instructor Questionnaire. This questionnaire was administered to all

instructors of the treatment group during the last week of Fall semester,

1992. It attempted to ascertain advantages, disadvantages, problems and

difficulties associated with the use of graphing calculators as an aid to the

teaching of precalculus. The questionnaire was strictly open-ended. It

allowed instructors to share their initial and final perceptions of the

usefulness of graphing calculator technology in teaching precalculus and

comment on any problems experienced in integrating the technology into

their mathematics Classrooms. Instructors were asked to make reference

to the impact of the technology on teaching style and workload.

Opportunity was also provided in the questionnaire for instructors to make

specific remarks on the advantages and disadvantages of using graphic

calculators visa vi their effect on student attitudes, student understanding

of basic precalculus concepts, solving new and interesting problems,

preparation for calculus. Finally, instructors were asked to state whether or

56

not they preferred to teach precalculus with graphing calculators or in the

traditional way. The specific questions asked on this questionnaire are

given in Appendix H.

In summary, the research plan contained the following evaluative

instruments:

(a) an attitude pretest and posttest,

(b) an algebraic skills pretest and posttest,

(C) a function concepts pretest and posttest,

(d) a student questionnaire,

(e) a faculty questionnaire.

Students took the attitude tests during regular Class time and the other tests
outside of Class in the evening at the University Test Center. The testing at the
University Test Center was done under the direction of the director of the Center,
with the aid of student assistants who worked at the Center. Actually, the Attitude
Inventory (pretest) was completed early in the first week of the semester during
regular Class time and the Attitude Inventory (posttest) was also completed in the
regular lecture time during the last week of the semester. Given the inordinate
amount of testing to be done in the project and the amount of lecture material that
instructors had to cover in their syllabus in a limited time, instructors found it
necessary to have students take the Algebraic Skills Test and the Function
Concepts Test at the University Test Center outside of regular class time. The
Algebraic Skills Pretest and the Function Concepts Pretest were taken together

and took approximately one hour. Similarly, the Algebraic Skills Posttest and the

57

Function Concepts Posttest were also done together within an hour. Instructors
offered incentives to students to take part in the testing at the test center. In
particular, each instructor was asked to give participating students a "free" quiz in
some manner. Instructors were also given the opportunity to use posttest
questions from the Algebraic Skills Test and/or Function Concepts Test on their

final exams. A specific schedule of all the testing is given below in Table 1.

Table 1 Schedule of Assessments

 

 

II Type of Test Testing Times ll
Attitude Inventory (pretest) First week of Fall semester, 1992
Algebraic Skills (pretest) Wednesday, Sept. 2, 7-9pm, 1992 or
Function Concepts (pretest) Thursday, Sept. 3, 7-9 pm., 1992
Attitude Inventory (posttest) Last week of Fall semester, 1992
Algebraic Skills (posttest) Monday, Dec. 7 through Thursday,
Function Concepts (posttest) December 10, 10 am. to 5 pm., 1992

 

 

 

 

 

 

In order to be able to compare performances for each of the participating
students on the pretest and the posttest for all assessments (attitude, algebraic
skills, and function concepts) students were asked to place their student
identification numbers on their answer sheets for each assessment. This also
made possible computation of the difference between posttest and pretest scores
for each student and hence proved very useful in analyzing the mean change in
the performance levels of the treatment and the control groups in all areas -

attitude, algebraic skills, and function concepts proficiency.

58
Method of Analysis

Quantitative analysis of the data gathered by the research instruments of
this study was done using a simple Minitab Software Package. Use was made of
a two sample unpaired t-test to make comparisons between the mean
performance levels of both the control and treatment groups on the pretest and
posttest assessments in the area of algebraic skills and also in the area of core
function concepts. T-tests were also used to test for differences in the mean
scores of both the treatment and control groups on an attitude towards
mathematics survey which was administered in the form of a pretest and a
posttest. A two sample unpaired t-test was also employed to assess the
differences in the mean change (loss or gain) in performance levels of the
treatment and control groups, as determined by the differences in the pre and
posttest scores of each student in the two groups, in three areas:

. Attitude towards mathematics

. Algebraic skills proficiency

. Knowledge of basic functions concepts.

In short, a two sample t-test was used to test the various hypotheses stated at the
beginning of this Chapter.

Data gathered from responses to open-ended questions on the student
and faculty questionnaires administered to the treatment group was summarized
and analyzed qualitatively. Simple descriptive statistics were used to summarize
the student responses to each close-ended statement contained in the

questionnaires.

59
IJmItatlons of the Study

Three basic limitations should be considered when interpreting the results
of this research project, and particularly when interpreting the findings of the study
to other populations or to other teaching situations. The first of these is sample
selection. All seven precalculus instructors, three of which taught the "calculator"
sections which made up the treatment group and four of which taught the "non-
calculator" sections which made up the control group, graciously agreed to
participate in the project. However, due to the inordinate amount of testing that
had to be done during a busy single semester it was decided to administer the
attitude pre and posttests during regular Class time and the algebraic skills (pre
and posttest) through the University Test Center outside of regular Class hours.
Given the demands and constraints on students time some students in spite of
incentives offered by instructors, found it next to impossible to participate in all four
tests outside of regular class hours. Consequently, some students who completed
the Algebraic Skills Pretest and the Function Concepts Pretest did not or could not
complete the corresponding posttests. A similar situation existed, but to a lesser
degree with the Attitude Tests. In light of this situation, it was decided to make
comparisons of the performance levels on the pretests, in all three areas
examined, of those subjects in both the treatment and control groups who wrote
the pretests and not the posttests versus those who erte both the pretests an the
posts. Results of unpaired two sample t-test administered for this purpose
revealed that on the Attitude and Algebraic Skills Pretests there was really no

significant difference in performance between the students from both the treatment

60

and control groups who wrote the pre and posttests and those who wrote the
pretest and not the posttest. There was, however, a mildly significant difference
in performance for these same groups of students on the Function Concepts
Pretest. This could, conceivably, have affected the Function Concepts Posttest
results for the treatment and control groups. It should be noted that students from
both the treatment and control who wrote both the pre and post Function
Concepts Tests compare very favourably. Consequently, the comparisons of the
mean improvements of both treatment and control groups given earlier should be
unaffected by this limitation. More details on this situation are given in Appendix
K.

The second limitation of this study is teacher variability. Although the
instructors that taught the classes that made up both the treatment and control
group were seasoned and experienced teachers, nevertheless, the instructors for
the control group were not the same as those who taught the treatment group.
Ideally, it would have been desirable to have the same instructors teaching both
groups.

The third limitation of the study is that the control group used a different text
than the treatment group. Even though the core topics that made up the syllabus
for both groups were the same, nevertheless, the text used for the treatment group
conceivably led to different points of emphasis and possibly different approaches
to problem solving. In view of this, it was difficult to design the function concepts
achievement tests to reflect these different points of emphasis. As a compromise,

it was decided to try and design the tests so that they tested a core of function

61

concepts common to both groups, and which, in the view of the researcher, after
many years of successful teaching experience, were important to the study of
calculus.

The research findings of this study, based on the data analysis done on all
data gathered by the different tests and research instruments, within the limitations

described above, are all reported in the next chapter.

CHAPTER IV

DATA ANALYSIS

This chapter contains analyses of the quantitative data described in Chapter
III and used to investigate the hypothesis of the study. The first part of the
chapter, and the main focus of this investigation, gives an analysis of the results
of the achievement tests (Algebraic Skills and Function Concepts Tests) and the
Attitude Inventory administered to both the treatment and control groups involved
in the study. An analysis of the data gathered in the student and faculty
questionnaires, designed to examine student-faculty reaction to the use of graphic
calculators, is reported later in the Chapter. Actually, this Chapter is divided into
five sections. The first section gives a comparative analysis of the results of the
pretest and posttest of algebraic skills for both the treatment and control groups.
The second section gives a statistical comparison of the pre and posttest scores
for both treatment and control groups on tests of basic function concepts. Section
three contains the analysis of the pre and post attitude survey for both treatment
and control groups. Finally, the last two sections of the chapter report the results
of a student questionnaire and a faculty questionnaire used to ascertain the
feelings of faculty and students on the effectiveness of the graphics calculator as

a teaching and learning aid.

62

63

Analysis of Algebraic Skills Test Results

An algebraic skills test was administered in both a pretest and posttest
format. The pretest was intended to determine if there was any significant
difference between the treatment and control groups in the area of basic algebraic
skills, before the treatment was administered. The posttest results were used to
compare the performance of both groups at the end of the treatment period.
Furthermore the difference in performance on these tests was used to measure
the average change in achievement (loss or gain) in algebraic skills proficiency for
both the treatment and control group. The pretest and posttest formats were very
similar. Both tests were comprised of 25 items.

The Algebraic Skills Pretest, a copy of which is given in Appendix C, was
administered during the first week of Fall Semester on Wednesday and Thursday
evening. Ninety-five students from the treatment (calculator) group and 102
students from the control (traditional) group participated in the pretesting of basic
algebraic skills which was given and monitored at the University Test Center.

The maximum possible score on the 25 item pretest was 25. A frequency
distribution showing the distribution of pretest scores for both the treatment and

control groups is given in Figure 6.

Proporlion ol subjecls

.9
w

.‘3
N

P
—.

greater than 20

Wm 20

    
  
    

64

11 to 15
Scores

Gtolll

, I treatment ‘

I 4 control

 

OtoS

Figure 6 Algebraic Skills Pretest Scores: Frequency Distribution

The mean score for the treatment (calculator) group on the Algebraic Skills

Pretest was slightly higher than that for the traditional group. However, results of

a two sample unpaired t-test, which are presented in Table 2, show that the

differences in the performances of the two groups is not statistically significantly

(p-value = .082)

Table 2 Results of Two Sample t-test for Treatment and Control Groups
on Algebraic Skills Pretest.

 

F:ll

 

II Group N Mean SD SE Mean t
Treatment 95 13.42 4.00 0.41
Control 102 12.41 4.10 0.41 1 .75 0. 082

 

 

 

 

 

 

 

The posttest of algebraic skills was conducted at the University Test Center

during the last week of the semester, Monday through Thursday evenings. Sixty-

Proportion of subjecls

65
six students from the traditional (control) group and 73 students from the
calculator (treatment) group participated in this test, a copy of which is given in
Appendix D.
As with the pretest, the maximum score on the 25 item posttest was 25. A
comparison of the distribution of the scores for both the treatment and control

groups is provided by the frequency distribution given in Figure 7.

 

    

I treatment

T

l
04 I ‘: control '
0.3 +
0.2 i

oratuthsnzo lBtoZO IItoIS Stall) OtoS

Scores

Figure 7 Algebraic Skills Posttest Scores: Frequency Distribution

The mean scores on the posttest for both the calculator and traditional
groups compare very favourably. As was the case with the pretest, the mean for
the calculator group was slightly higher than that for the traditional group.

However, a two sample unpaired t-test performed on the posttest data suggests

66

that this difference in performance is not significant (P = 0.15). The results of this

test are given in Table 3.

Table 3 Results of Two Sample Unpaired t-test for Treatment and Control
Groups on Algebraic Skills Posttest

 

 

 

 

 

 

 

 

 

 

Group N Mean SD SE Mean t P
Treatment 73 13.21 3.82 0.45 1.46 0.15
Control 66 12.23 4.04 0.50

 

In addition to a statistical comparison of the means, a simple comparison
was made of the responses of both the treatment and control groups for each
question on the Algebraic Skills Posttest. This was done in order to try and
identify some of the strengths and weaknesses of each group in the specific areas
covered on the posttest of proficiency in algebraic skills. A summary of the
responses for both groups is given in Table 4.

Table 4 Comparison of Responses of Treatment and Control Groups to
Each Question on Algebraic Skills Posttest.

 

 

 

 

 

 

 

 

 

 

 

Question # Concept Tested % Responding Correctly
Treatment Control
1 Solving a linear equation 61.6 75.6
2 Multiplying two binomials 100.0 92.4
3 Factorization of a trinomial 83.6 83.3
4 Multiplying exponents 76.7 83.3
5 Radical/exponent connection 39.7 62.1
6 Simplifying a radical 68.5 51.5
7 Solving a system of liner equations 80.0 72.7
s (a+b)2 = a2 + b2? 43.8 34.8
9 Solving a simple quadratic equation 80.8 71.2

 

 

 

 

 

67

Table 4 cont’d.

 

 

    

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

% Responding Correctly
Treatment Control
6.8 12.1
11 Solving an equation involving fractions 82.2 75.6
12 Factoring the difference of two cubes 61.6 63.6
13 Solving an 'absolute value' inequity 10.09 13.6
14 Adding two algebraic fractions 53.4 48.5
15 Meaning of square root 39.7 19.7
16 Reciprocal and negative exponent 17.8 18.2
connection
17 Adding fractions using negative 17.8 16.7
exponents
18 Find ab given a + b2 = o 43.8 40.9
19 Factoring via real numbers or integers 78.0 69.7
20 Solving an equation involving reciprocals 46.5 33.3
21 Solving an exponential equation 63.1 37.9
2 Solving an 'absolute value' equation 72.6 45.4
23 Finding the roots of a polynomial 4.1 9.1
24 Definition of a non-negative number 65.7 48.4
n 25 Functional notation and algebraic 53.4 47.0
manipulation

 

 

The table reveals that the treatment group outperformed the control group
on 17 out of the 25 items tested on the Algebra Skills Posttest. This is consistent
with the fact that the treatment group had a higher mean score than the control
group on the Algebraic Skills Posttest. This may be good news for the supporters
of graphing calculators who may have some doubts as to the effects of the

technology on basic skills.

68

A fundamental component of this study was to determine whether the
algebraic skills of students who used the graphing calculator to study precalculus
would be adversely affected. To investigate this question a comparison was made
of the pre and posttest scores for each student in both the treatment and control
groups who wrote both the Algebraic Skills Pretest and Posttest. There were 69
such students from the treatment (calculator) group and 53 such students from the
control (traditional) group. For each of these students the mean of the pre and
posttest score differentials was computed and used to determine the mean loss
or gain in achievement for both the treatment and control groups. Frequency
distributions showing the pre and posttest score differentials for both the treatment

and control groups is given in Figure 8.

Proporlion ol Subjecls

69

0.5
0.45 «
0.4 ‘
035 +
0.3 ’
0.25 J,»
0.2
0.15
0.1
0.05

  
 
   

l
I treatment I

I C control

lit012 4to7 0t03: «Ito-I -8 to .5 -12t0-9
Seere Differences

Figure 8 Algebraic Skills Pre & Posttest Score Differences:
Frequency Distribution
A two sample unpaired t-test was applied to test for any significant
differences between the mean loss or gain in achievement on algebraic skill
proficiency for the treatment and control groups. Results of this test are given in
Table 5.
Table 5 Comparison of Means of Differences in Scores on Pre and Post

Algebraic Skills Tests for Treatment and Control Groups Using
Unpaired Two Sample t-test.

 

 

II Group N Mean SD SE Mean t P II
Treatment 69 064 2.99 0.36 -0.29 0.77
Control 53 -0.43 4.46 0.61

 

 

 

 

 

 

 

 

 

70

An examination of Table 5 surprisingly reveals that both groups appear to
suffer a loss in achievement in the area of basic algebraic skills with the mean loss
in achievement for the control group marginally less than that for the treatment
group. The size of the p-value (p = 0.77) certainly shows no significant difference
in losses which in turn suggests that students who used the graphing calculator
(treatment group) did not fair significantly worse in the area of algebraic skills than

their counterparts who were taught precalculus in the traditional manner.

Analysis of Function Concepts Test Results

A test of basic function concepts was administered in a pretest and posttest
format during the same time slots as the algebraic skills tests and for the reasons
already explained in the beginning section of this chapter. Approximately one hour
was used for the pretesting and one for the posttesting. Copies of the Pre and
Post Function Concepts Tests are given in Appendices E and F respectively.

The Function Concepts Pretest was administered at the University Test
Center during the first week of the semester on Wednesday and Thursday
evenings. One hundred and one students from the control (traditional) group and
95 students from the treatment (calculator) group participated in the pretest.

As was the case with the Algebraic Skills Pretest, the maximum score on the
25 item Function Concepts Pretest was 25. A frequency distribution showing the
distribution of pretest scores for both the control and treatment groups is given in

Figure 9.

Proporlion of subjects

P
m

0.45

P
:-

0.35

c: o
LaF'qu’
mnmw

0.1

71

    
    

 

 

7 I
i, I I Treatment ;
J. T Control !
l ,

T

t

t

|

T

orestlthanI'I 16t020 ilto 16 atoll) 0t05
Scores

Figure 9 Function Concepts Pretest Scores: Frequency Distribution

The mean scores on the Function Concepts Pretest for both treatment and
control groups is remarkably similar with the treatment group showing a slightly
higher mean. It is therefore not surprising that a two sample t-test performed on
the pretest scores reveals that there is a nonsignificant difference between the
performances of both groups at the beginning of the semester on a test of
proficiency in basic function concepts. This is evidenced by the results of the t—

test given in Table 6.

Proportion of subjects

72

Table 6 Results of Two Sample t-test for Treatment and Control Group on
Function Concepts Pretest.

 

 

 

 

 

 

E a
Group N Mean SD SE Mean t P
Treatment 95 9.71 3.57 0.37 0.45 0.65
Control 101 9.49 3.22 0.32

 

 

 

 

The posttest of function concepts was held at the University Test Center
Monday through Thursday evenings during the last week of the semester.
Seventy-four students from the treatment group and 57 students from the control
group participated in the test. The distributions of the scores for both groups on

the 25 item test, based on a maximum score of 25, is described in Figure 10.

 

 

 

 

 

 

 

 

 

 

 

 

0.. "r
I treatment
0.5
C] control
yestertIIenZO 16t020 lltols 6toIII OtoS
Scores

Figure 10 Function Concepts Posttest Scores: Frequency Distribution

73

Table 7 gives the results of a two sample unpaired t-test used to compare
the mean pretest scores of both the treatment and control groups. The very low
p-value (p = 0.000) gives strong evidence that the treatment group did significantly
better than the control group on a test of knowledge of basic function concepts. '

Table 7 Results of Two Sample t-test for Treatment and Control Groups
on Function Concepts Posttest.

 

 

Group N Mean SD SE Mean t P
Treatment 74 16.38 3.91 0.45
Control 57 13.16 4.00 0.53 4.61 0.000

 

 

 

 

 

 

 

 

 

As was the case with the Algebraic Skills Posttest, a question by question
comparison was made of the responses of both the treatment and control groups
for each question on the Function Concepts Posttest. This was done in order to
compare the relative strengths and weaknesses of each group on the specific

areas of proficiency covered by the test. A summary of the responses is given in

 

 

 

 

 

 

Table 8.
Table 8 Comparison of Responses of Treatment and Control Groups
to Each Question on Function Concepts Posttest
Question # Concept Tested % Responding Correctly

Treatment Control

1 Finding functional values 93.2 80.7

2 Finding co-ordinates of a point on a 21.6 17.5

graph

3 Identifying the graph of a function 95.9 87.7

4 Finding the domain of a function 83.8 63.2

5 Perpendicular bisector of a line segment 43.2 31.6

 

 

 

 

 

 

n“

l I

 

//

Sam)

tlealr

74

Table 8 cont’d.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Question # Concept Tested 96 Responding Correctly
Treatment Control
6 Vertex of a parabola 85.1 45.6
7 Equation of a circle (radius and center) 89.2 78.9 1|
8 Distance formula 52.7 54.4
ll 9 Finding y intercepts 48.6 66.6
10 Obtaining the point of intersection of 83.8 49.1
two graphs
11 Identifying the conics 41.9 61.4
12 Functional notation 58.1 49.1
13 Range of trigonometric functions 70.3 24.6
14 Meaning of'complete' graph 90.5 61.4
15 Finding x - intercepts 85.1 54.3 ll
16 Expressing functional relationships 31.1 21.1
(word problems)
17 Complete graph of a discrete function 73.0 46.0
18 Range of a "square root" function 48.6 46.0
19 Number of real roots of an equation 51.4 21.1
20 Solving an 'absolute value' inequity 67.6 57.9
21 Stretches and shrinks (transformation) 64.9 42.1
22 Finding a point on the graph of an 55.4 36.8
equation given specific information
about the equation
23 Obtaining vertical and horizontal 58.1 28.1 ll
asymptotes
24 Composite functions 63.5 47.3
25 Symmetry and graphing 47.3 40.4

 

The results displayed in Table 8 certainly compliment the results of the two

sample unpaired t-test given in Table 7 in that they also show very Clearly that the

treatment group outperformed the control groups on the posttest of function

Proporlion of Subjects

75

concepts. More specifically the treatment group showed higher achievement than
the control group in all but three of the 25 function concepts tested.

The differences in the pre and posttest scores for each student of both the
treatment and control groups who wrote both pre and posttests were examined.
The means of these differences were than used to construct the mean change
(loss or gain) in achievement for both groups in the area of basic function
concepts proficiency. Figure 11 gives, in the form of a frequency distribution, a

comparison of the pre and posttest score differences for both groups.

0.45 T
0.4 «»
0.35 ..
0.3 4
025 4

 

I Treatment

Ci Control

 

 

 

P

d P

0'1 N
l
I

 

P
‘

 

0.05 4r

 

 

 

 

   

 

 

 

 

 

 

a l .
Into 14 St“ MM -5 to -I ~IIIto-6
Scere Differeeees

Figure 11 Function Concepts Pre & Posttest Score Differences:
Frequency Distribution

To test for poSsible significant differences between the mean change (loss
or gain) in achievement for both treatment and control groups on function concept
proficiency, a two sample unpaired t-test was applied to the pre and posttest

score differences for each group. Table 9 gives the results of this test.

76

Table 9 Comparison of Means of Differences in Scores on Pre and
Posttest of Function Concepts

 

 

Group N Mean SD SE Mean t P
Difference
Treatment 71 6.13 4.27 0.51 3.75 0.0003
Control 49 2.96 4.72 0.67

 

 

 

 

 

 

 

 

 

The size of the p-value (p = 0.0003) given in Table 9 is strong evidence of
a significant difference in improvement between the treatment and control group.
The treatment group appear to have made significantly greater improvements in

their knowledge of basic function than the control group.

Attitude Inventory Data Analysis

Students in both the treatment and control groups were given a Likert-type
attitude inventory in the form of a pretest and posttest. These tests were
administered in Class during the first and last weeks of the semester respectively.
The tests were designed by the researcher. Their main purpose was to compare
the attitudes of both groups of students towards mathematics at the beginning and
end of the semester and thus determine any resulting change (positively or
negatively) in attitude for each group over the course of the semester. The
Attitude Pretest (see Appendix A) consisted of 22 statements each of which
expressed an attitude towards mathematics. The posttest contained the same 22
items, but in addition contained ten items which were especially designed to
examine student attitudes towards the graphics calculator. These ten items were

responded to by the treatment (calculator) group only. The Attitude Inventory

77

contained both positive and negative statements. For the positive statements a
value of five was assigned to each "strongly agree" response, four to each "agree"
response, three to each "undecided" response, two to each "disagree" response
and one to each "strongly disagree" response. The scoring scheme was reversed
for the negative statements. A copy of the Attitude Posttest is given in Appendix
B.

One hundred and seventy students from the control group and 118
students from the treatment group wrote the Attitude Pretest. Since the test was
done during regular class hours the participation rate was very high. The mean
scores for both groups was remarkably similar with the mean score for the
traditional group being 73.7 and that of the treatment group being 73.1, based on
a total score of 110. A distribution of the scores for the treatment and control

groups is furnished by the frequency distribution given in Figure 12.

Proporlion oI Subjecls

100
to
110

90
to
99

Figure 12 Attitude Pretest Scores: Frequency Distribution

78

 
 

70 60 50

to to to

79 59
Scene

, I Treatment

, : Control

  

30 less
to than
39 30

A two sample unpaired t-test applied to the test data for both groups shows

no significant differences in the initial attitude towards mathematics of both

treatment and control groups (p = 0.73). The results of this test are given in Table

 

 

10.
Table 10 Results of Two Sample Unpaired t-test for Treatment and Control
Groups on Attitude Pretest
II Group N Mean SD SE Mean t P ll
Control 170 73.7 14.7 1.1 0.34 0.73
Treatment 118 73.1 14.5 1.3

 

 

 

 

 

 

 

 

 

Ninety-eight students from the treatment group and 115 students from the

control group wrote an attitude posttest during the last week of semester. As was

79
the case in the pretest, the results of the attitude posttest showed both groups as
having very similar attitudes towards mathematics. The mean scores for the
treatment and control groups, again based on a total score of 110, were 72.1 and
74.7 respectively. Frequency distributions, giving the distribution of scores for both

groups are contained in Figure 13.

0.35 7

 
 
     

I Treatment

3 Control I

 

Proportion of subjects

 

I It] 99 99 79 69 59 49 39 30

Figure 13 Attitude PMest Scores: Frequency Distribution

Statistical support for the similarity of the attitudes of both treatment and
control groups is given by the results of the two sample unpaired t-test given in

Table 11.

Tn

gro
diffs

tea

In t

Don

and

the:

give

80

Table 11 Results of Two Sample Unpaired t-test for Treatment and Control
Groups on Attitude Posttest

 

 

 

 

 

 

 

 

 

 

Group N Mean SD SE Mean t P
Treatment 98 72.1 16.0 1 .6 -1 .20 0.23
Control 1 1 5 74.7 1 5.3 1 .4

 

The mean scores on the Attitude Posttest of the treatment and control
groups and the size of the p-value, as given in Table 11, suggests no significant
difference at the end of the semester in the attitude towards mathematics of the
treatment and control groups.

In an attempt to measure attitude change towards mathematics (positive or
negative Change) over the semester, 8 comparison was made of the differences
in the Pre and Post Attitude Test scores for students from both the treatment and
control groups who wrote both attitude tests. The score differentials were then
constructed and the means of these differentials were computed for both groups
and used as a measure of the mean change (loss or gain) in attitude for each of
the treatment and control groups.

Distributions of the pre and post test score differences for both groups are

given in Figure 14.

tli-
'- m-

054

title
and
cont
the 2
Over
"”90
deter
Deng,
dia‘SIe

°ll the

Proportion of Subjects

81

   

j—ET
‘ I Treatment j
1 ,
j E Control

    

4 i
21m 30 1110 20 11010 -9 tell -19to-10 -29 to ~20
Score Differences

Figure 14 Attitude Pre & Posttest Score Differences: Frequency Distribution

An examination of the means of the score differences revealed a mean

difference of -1.0 for the 72 students from the treatment group who did the Pre

and Post Attitude Test and a mean difference of 0.10 for the 100 students from the

control group who likewise participated in both tests. These statistics show that

the attitude towards mathematics of both groups underwent a very small change

over the course of the semester. The attitude of the control groups appear to

improve very marginally, whereas, those of the treatment group appear to

deteriorate very marginally. Predictably, the results of a two sample unpaired t-test

performs on the score differences for the two groups show no significant

difference in the change of attitude for both groups (p-value = 0.40). More details

on the results of this test can be seen from Table 12.

ll

 

"0:11

we
low
Whll
pos
ten

In TI

til/en),

82

Table 12 Comparison of Means of Scores Differences on Pre and Post
Attitude Test for Treatment and Control Groups by Unpaired Two

 

 

Sample t-test.
Group N Mean of SD SE Mean t p
Score
Difference
Treatment 72 -1 .0 10.3 1 .2 -0.69 0.49
Control 100 0.1 1 0.3 1 .0

 

 

 

 

 

 

 

 

 

As already indicated the last ten of the 32 items on the Attitude Posttest
were designed to determine the attitude of the treatment (calculator) groups
towards the graphics calculator. The first of these items (item 23) was a statement
which reflected a negative attitude. The last nine items were all statements of a
positive attitude towards the graphics calculator. The total possible score for all
items was 50. A detailed description of the overall results for all the group is given
in Table 13.

Table 13 Attitudes of Students of Treatment Group Towards the Graphing
Calculator: A Description of Attitude Test Results

——7

SE Mean Min Max 01 j 03 H
0.729 15 so 34 42

 

 

 

 

 

 

 

 

 

An item by item analysis was done for each of the 10 items. A description
of this analysis showing the mean and median scores for all of the 96 students in
the treatment group on each item is given in Table 14. The scoring scheme used
to construct Table 14 was exactly in reverse order to that described for the Attitude

Inventory on page 77.

83
Table 14 Item by Item Analysis of Test of Attitude Towards Graphing

Calculator for Treatment Group

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

m =1
Item # Mean Median SD SE Mean Min Max 01 01
#23 3.719 4.000 1.176 0.120 1.000 5.000 3.000 5.000
#24 2.198 2.000 0.969 0.099 1 .000 5.000 2.000 3.000
#25 1.979 2.000 0.995 0.102 1 .000 5.000 1.000 2.000
#26 2.146 2.000 0.962 0.098 1 .000 5.000 1 .000 3.000
#27 2.406 2.000 0.947 0.097 1 .000 5.000 2.000 3.000
#28 2.406 2.000 2.251 0.230 1 .000 5.000 2.000 3.000
#29 2.115 2.000 0.972 0.099 1.000 5.000 1.000 3.000
#30 2.625 3.000 0.976 0.100 1 .000 5.000 2.000 3.000 1|
#31 2.500 3.000 0.821 0.084 1 .000 5.000 2.000 3.000 ll
#32 1.927 2.000 0.921 0.094 1 .000 5.000 2.000 2.000 I

 

 

Table 14 shows a mean score of 3.719 and a median score of 4.000 for
item #23. This suggests that a substantial number of students either disagree or
strongly disagree with the statement that the graphics calculator hampers student
ability to perform "algebraic manipulations" - a concern often expressed by critics
of the use of graphing calculators in the classroom. The relatively low scores on
items 24, 25, 26, 27, 28, 29 and 32 appear to confirm a generally positive attitude
towards the graphics calculator in the limited directions tested by these items. A
score of 3 on item #30 is an indication that students are divided in their opinions
as to whether or not they developed a more positive attitude towards mathematics
over the semester. Similarly, a score of 2.5 on item #31 suggests the students
were undecided as to whether or not the graphics calculator enabled them to

solve more realistic and interesting problems.

84

A simple statistical account of the level of agreement by all students in the

group on each of the 10 items tested is provided by Table 15.

Table 15 Attitude Towards Graphics Calculator: Level of Agreement on
Attitude Test Items

 

 

Item 96 Agreeing or Strongly
Agreeing
23. Using a graphing calculator hampers my 20.8

ability to perform algebraic manipulations.

 

24. Use of graphing calculator has increased 71.9
mathematical confidence.

 

25. Use of graphing calculator helps visualization 83.3
of concepts and thus increases level of
mathematical understanding.

 

 

 

26. Graphical method is a fun way to solve 66.7
mathematical problems.

27. Since using graphing calculator students feel 61.4
more at ease when performing mathematical
tasks.

28. Graphing mathematical relationships 72.9

expressed in a problem leads to a better
understanding of the problem.

 

29. Graphing calculators are useful for future 70.8
work in mathematics and related disciplines.

 

30. A more positive attitude towards 43.8
mathematics has developed since using the
graphing calculator.

 

31. Can solve more realistic problems which are 50.0
neither contrived nor boring by using
graphics calculator.

 

32. Graphics calculator frees time for more ' 79.2
interesting mathematical tasks.

 

 

 

 

leve

93

Palm
(Cale

efidec

:0 Cor

85

Table 15 shows, in simple statistical form, that there was a relatively high
level of agreement among the students in the calculators group on the fact that the
graphics calculator has the potential to:

(i) increase mathematical confidence,

(ii) help visualization and thus increase the level of mathematical
understanding,

(iii) provide graphical methods for problem solving and hence lead to a better
understanding of a problem and a fun way to solve it,

(iv) free time normally spent on tedious tasks that can now be devoted to more
interesting mathematical tasks,

(v) be of use in other areas of college work that relate to mathematics.

Of particular interest is the fact that a relatively small percentage of the
respondents (20.8%) feel that using graphing calculators hamper their ability to
perform algebraic manipulations. This result is consistent with an earlier account

of the performance of the treatment group on the Algebraic Skills Tests.

Results of Student Questionnaire
In order to gauge student reaction, as per the effectiveness of the graphics
calculator, as an aid to the study of precalculus, students in the treatment
(calculator) group were given a questionnaire containing both closed and open-
ended questions. A copy of the questionnaire is contained in Appendix G.
The first ten questions on the questionnaire gave students an opportunity

to comment on the perceived advantages and disadvantageous of using the

86

graphics calculator, some of the problems with its usage, and how they were

overcome.

Question #12 was intended to determine whether the students

preferred the graphics calculator or non-graphics calculator (traditional) approach

to the study of precalculus. A simple summary of the responses to questions 1-10

and 12 as given by the 77 students who participated is given below in Table 16.

Table 16 Summary of Responses to Questions 1-10 and 12 on Student

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Questionnaire
Question # Question 96 Responding
Yes No Undecided
1. Mbted feelings initially about 30 70 0
using the graphics calculator as
study/learning aid?
2. Change in initial attitude? 31.2 66.2 2.6
3. Used graphing calculator in 44.2 55.8 0
school?
4. Use of graphing calculator in 77.9 11.7 10.4
school would have been helpful
in the precalculus course.
5. Were there problems with usage? 11.7 87.0 1.3
6. Does use of calculator 16.9 81.8 1.3
technology increase workload?
7. Was graphics calculator useful in 32.5 67.5 0.0
other university courses?
8. Has the use of graphics 84.4 11.7 3.9
calculator helped improve
performance in precalculus
courses.
9. Has the use of the graphics 18.2 81.8 0.0
calculator had a detrimental
affect on basic algebraic skills?

 

 

 

 

 

 

87
Table 16 cont’d.

 

 

 

Question # Question % Responding
Yes No Undecided
10. Have you become "over 40.3 0.0 0.0
dependent" on the calculator?
12(a) Prefers calculator method? 87 13 0.0
12(b) Prefers non-calculator 10 87 0.0
method?

 

 

 

 

 

 

 

An examination of these statistics reveals that an overwhelming majority of
students (84.4%) agree that the graphics calculator has helped improve their
performances in precalculus. At the same time the graphics technology. has
appeared to have not increased student workload nor has it had, for most
students, any detrimental effects on algebraic skills. The vast majority of the
students (87%) have not experienced problems with the usage of the graphics
technology, and by far most of the students (87%) prefer to learn precalculus with
the aid of the graphing calculator rather than the traditional (non-calculator) way.
A large proportion of the students (77.9%) felt, however, that introducing graphics
calculators in the school would have been very helpful to them in their university
precalculus course.

Samples of types of negative and positive comments given by students on
questionnaire in response to questions 1-10 are given below:

. More useful than I thought.

- Gives better understanding of how equations relate to their graphs.

88

No serious problems encountered with usage; however, there were some
things I didn’t get to understand; more teacher training on the calculator would
have eliminated some problems.

By using the calculator I got to understand the domain and range of a function
better.

Can use it to Check on my algebraic computation.

One problem was that l relied on the calculator rather than my own knowledge.
It has decreased my workload; time saved by not having to plot graphs by
hand.

I feel that in high school we should learn how to graph by plotting points and
know the graphs before being given the opportunity to Cheat by using the
calculator.

If I had used the graphics calculator in high school, I would have been more
prepared for this class.

The only thing I was apprehensive about was the price; I was not sure it was
worth it. Now I think it was worth the money and it was very helpful.

I had never used a graphing calculator before this course and now I really
enjoy using it and would recommend its use.

Sometimes if I am really frustrated and tired I use the calculator to "Cheat". You
really have to understand the basic concepts in order to properly use the
calculator.

I still feel we need to know how to do things by hand.

I feel the graphing utility made this course more complicated.

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. It has enhanced the class by eliminating repetition of graph drawings and by
allowing quick visualization of graphs.

. Familiarity with the calculator in high school would have allowed the Class to
move more smoothly and expeditiously. This would have allowed for more
emphasis of the teaching of the material and not on the use of the calculator.

. It has made my work simpler but not to the point where my performance has
increased.

. Yes, my initial negative attitude towards the calculator has since Changed. It’s
a great chance to see if your answer is right. But your brain would go to
waste if you just used the calculator without thinking about the answer first.

. Seeing what I was doing helped me to understand better.

. Yes, I think the machine helps but its the lazy way out.

. It brought a better understanding of the types taught and enabled us to learn
at a faster pace.

. My initial attitude has changed for the better.

. If I couldn’t get a function or equation to work out correctly by hand I could do
it on the calculator, see it and understand it better.

Question #1 1 on the questionnaire gave students an opportunity to express
their views on the extent to which the graphics calculator was helpful in their study
of various topics in precalculus. Students were asked to express the extent to

which they felt the calculator was helpful in eleven basic areas of their precalculus

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studies by responding to each of the eleven statements covering these areas in
accordance with the following code:

1. Not at all helpful 2. A little helpful 3. Undecided 4. Moderately helpful
5. Very helpful

A summary of the responses for the 77 participants based on the above scoring
system is given in Table 17.

Table 17 Helpfulness of Graphics Calculator in The Study of Precalculus:
Description of Responses on Student Questionnaire

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Item # Mean Median SD Semean Q1 Q1
11 (i) 3.805 4.000 1.113 0.127 3.000 5.000
- (ii) 2.351 2.000 1.121 0.128 1.000 3.000
- (110 2.506 3.000 1.242 0.142 1.000 3.000
- (iv) 3.584 4.000 1.331 0.152 3.000 5.000
' (v) 3.597 4.000 1.091 0.124 3.000 4.000
' (vi) 4.5714 5.000 0.7852 0.0895 4.000 5.000
- (vio 4.273 5.000 0.941 0.107 4.000 5.000
- (viiD 4.2597 4.000 0.8492 0.0968 4.000 5.000
' (ix) 4.169 4.000 1.056 0.120 4.000 5.000
' (x) 4.455 4.000 2.118 0.241 4.000 5.000
' (xi) 3.532 3.000 1.142 0.130“ 3.000 4.000

 

 

The above scores attest to the fact that the graphics calculator proved to
be, for the most part, either moderately or very helpful in all areas of precalculus
examined. To get a Closer look at the level of helpfulness provided by the
graphics calculator in each area, a simple statistical analysis was made of each

item in Question 11 by determining the percentage of respondents that responded

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either 4 (moderately helpful) or 5 (very helpful) on each item. This simple analysis

is given in Table 18.

Table 18 Level of Helpfulness of the Graphics Calculator:
Percentage of Students Who Responded Moderately Helpful or

 

 

 

 

 

 

 

 

 

 

 

 

Very Helpful.
% Responding

Item # Description of Item Moderately Helpful Very Helpful

11 (0 Increasing level of 75.3 74.0
understanding of function
concepts

' (i0 Developing basic algebraic 14.0 18.2
skills and manipulatives

' (iii) Solving new and interesting 17.0 22.1
problems

' (iv) Estimating/thinking about 48.0 62.3
whether answers make sense

' (v) Exploring, conjecturing and 44.0 57.1
discovering mathematical
facts

" (VI) Graphing new and interesting 70.0 90.9
functions

' (vifl Solving equations 64.0 83.1

" (viio Understanding the link 66.0 85.7
between equations and their
graphs

" (ix) Providing intuitive 63.0 81.8
understanding of asymptotic
and end-point behaviour of
functions

' (x) Bypassing the construction of 62.0 80.05
tables of values when
graphing

" (x1) Preparation for subsequent 38.0 49.4
courses in calculus

 

 

 

 

 

 

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The results given in Table 18 suggests that a substantial number of the
students in the calculator group (70% or more) feel that the graphics calculator
has been either moderately or very helpful in the following areas:

. increasing their level of understanding of function concepts,

. graphing new and interesting functions,

. solving or finding roots of equations,

. helping them understand the link between equations an their graphs,

. providing intuitive understanding of properties of functions such as asymptotic
and end-point behavior,

. bypassing the use of tables and tabular information when graphing.

Strong support for the effectiveness of the graphics calculator in the above areas

is also suggested by the analysis of the responses given in Table 22.

The group appears to be somewhat divided in its opinion of the
effectiveness of the graphics calculator in exploring conjecturing and discovery of
mathematical ideas as well as its effect on their preparation for calculus.

A relatively small percentage of the group, as given in Table 18, felt that the
graphics calculator was either moderately or very helpful in developing basic skills
and manipulatives or solving new and interesting problems. This fact is consistent
with the results of Table 17.

It is interesting to note the relatively large percentage of the group (74%)
who felt that the graphics calculator was either moderately or very helpful in
increasing their level of understanding of the function concept - a major goal of

any precalculus course. This fact is consistent with the performance of the

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treatment (calculator) group on the posttest of basic function concepts, wherein

they out performed their non-calculator counterparts.

Responses to Instructor Questionnaire

This component of the study tried to identify, through a questionnaire, the
reaction of faculty to the use of graphing calculators in their classrooms. In
particular, the questionnaire attempted to determine some of the advantages and
disadvantages of graphing calculators as perceived by faculty who taught the
treatment (calculator) groups. In addition, the questionnaire sought to identify any
problems that had to be solved and adjustments that had to be made by faculty
in order to accommodate the technology into their classrooms.

The faculty questionnaire, a copy of which is contained in Appendix H, was
open-ended and consisted of six questions addressing the issues described
above. A question by question summary of the responses to these questions, as
given by the three faculty members who taught the calculator classes that made
up the treatment group, is given in this section of the study.

Question #1. Initially did you have mixed feelings about teaching this course
using graphing calculators?

All instructors were anxious to gain facility with the calculator and study its

potential. Initially some concern was shown about the negative effects that the

calculator might have on traditional skill development. Would the students become

too dependent on the calculator and hence use it as a crutch thus interfering with

the learning of fundamental mathematical principles? Some concern was

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expressed about achieving the proper balance between calculator related activities
and those that traditionally related to skill development in Algebra and

Trigonometry.

Question #2. Has your Inltlal attitude towards this technology since
changed? If so, how?

After using the graphing calculator for one full semester, all instructors showed a

positive attitude toward the technology. Any change in attitude was certainly for

the better. In some cases instructors saw even greater application for the

technology than was initially envisioned. It was generally felt that graphing

calculators gave students a better conception of what a function looked like and

how it behaved.

Question #3. Were there adjustment problems for both you and your
students when you first attempted to use this technology to
teach precalculus? If so, in your opinion, how and to what
extent were these addressed and overcome?

There appeared to be no major adjustment to the technology for both students

and faculty. However, extra time and in some cases extra instructional sessions

(voluntarily attended by students) were required to give the students sufficient

familiarity with the technology. It also took extra time to identify calculator "tricks"

that were effective and those that were not. An overhead grapher proved to be
useful in this regard. Many students experienced problems adjusting and setting

the range and viewing rectangle of the calculator in order to view certain graphs.

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Some students who were using a Cassio instead of a TI81 calculator were
organized as a group to assist each other with the differences between the two

gadgets.

Question #4. To what extent, If any, has the use of this technology changed
your teaching style and your workload In this course?
Any increases in workload appeared to be mainly related to the extent to which
instructors needed to become familiar with the calculator and to the amount of
time required to instruct students on the use of the technology. Several points
were raised in regard to teaching style. Among these was the necessity of
representing a problem both algebraically and geometrically. In some cases this
was addressed by integrating the technology in such a way that the instructor
could easily move back and forth from the chaulkboard to the calculator. It was
mentioned that the choice of worked examples for the calculator assisted classes
was different from the traditional class. In fact, instructors using the graphing
calculator could do more interesting and more non routine problems - some of

which were virtually impossible to tackle by the traditional lecture method.

Question #5. (Advantages)

Some of the advantages of the calculator as seen by instructors include the
following:

- Wider and more interesting range of problems can be solved - less restriction
on the Choice of examples that when teaching precalculus by the traditional

lecture approach.

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. The calculator provided more opportunity to explore and discover new and
interesting mathematical relationships.

. The types of problems being solved and the variety of concepts being
developed give students a better preparation for calculus.

. With the use of the visual aid, students developed better understandings of
functions, graphs and geometric transformations.

. Time saved on graphing and regular computations could now be better spent

developing other concepts.

Question #5. (Disadvantages)

Some of the disadvantages of the graphing calculator approach as perceived by

instructors are:

. Less time and emphasis on the development of traditional algebraic and
trigonometric concepts and skills could result in an erosion of these skills and
consequently students may be disadvantaged in subsequent mathematics
courses.

. Students sometimes become too "dependent" on the calculator.

. The calculator seems to work better for the better students. The weaker
students may learn to graph and explore functions via the calculator but may
not really understand the underlying mathematical concepts.

. Learning about the calculator and how to use it sometimes presents an added

burden especially at the beginning of the term.

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Question #6

It was unanimously agreed that teaching precalculus via a graphing
calculator was preferable over the traditional (non-calculator) approach and that
the advantages outweighed the disadvantages. Furthermore, it appeared that
students also liked this approach and that the approach adds to student interest.
One instructor went so far as to suggest that any loss in algebraic skills practice
in the calculator sections was offset by the gain in the knowledge and
understanding of functions.

A summary of the research findings resulting from the quantitative analysis
of the data gathered in this study as well as a summary of the qualitative data
collected through student/faculty questionnaires is given in the next and final

chapter.

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CHAPTER V
SUMMARY AND CONCLUSIONS

This chapter begins with a summary of the analyses of the results of the
achievement and attitude tests used to investigate the hypotheses of the study.
This is followed by an examination of the hypotheses of the study outlined in
Chapter III in the light of these analyses. The third section of the chapter gives a
summary of the results of both the student and faculty questionnaires. The fourth
section of the chapter outlines the main conclusions of the study. A short
discussion of the main conclusions is given in section five of the chapter. The
sixth and final section of the chapter discusses implications of the study and

directions for future research.

Summary of The Analyses of Achievement and Attitude Tests

Results of unpaired two sample t-tests performed on pre and posttest
scores showed no significant difference in the performance of both the treatment
(calculator) and control (non-calculator) groups on algebraic skills. The p-value
for the pretest was .082 and the p-value for the posttest was 0.15 at the 0.05 level.
Notwithstanding, the mean pretest and posttest scores for the treatment group
was slightly higher than that for the control group; 13.42 versus 12.41 on the
pretest and 13.21 versus 12.23 on the posttest, based on a total score of 25.

The differences in the means of the posttest scores is consistent with the

fact that the treatment group out-performed the control group on 17 out of the 25

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items tested on the posttest. Based on a comparison of the differences in the pre
and posttest scores, both groups showed minor loss in achievement on basic
algebraic skills over the time span of the semester. However, results of a two
sample unpaired t-test applied to differences in pre and posttest scores revealed
that the treatment group did not fare significantly worse (p-value = 0.77) than the
control group in this area. These last results may be comforting to those who fear
that the use of graphing calculators may result in an erosion of basic skills
proficiency.

A comparison of the results of the function concepts pretest revealed that
both treatment and control groups were very comparable at the beginning of the
term with the mean scores for the treatment and control groups being 9.75 and
9.49 respectively (based on a total score of 25). It was not surprising that results
of a two sample unpaired t-test (p-value = 0.650 at the 0.05 level) showed no
significant difference in achievement in this area based on pretest results.

Unlike the pretest results, results of the posttest showed the treatment
group as performing at a significantly higher level than the control group. In
particular, the mean for the treatment group on the posttest was 16.38 as
compared with 13.16 for the control group. A two sample unpaired t-test applied
to posttest data showed a p-value of 0.0000 at the 0.05 level and hence provided
strong evidence of the superiority of the treatment group. A close examination of
individual items on the posttest revealed that the treatment group outperformed
the control group on all but three of the 25 items tested. Upon examining the

differences in the pre and posttest scores for each student in the treatment and

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control groups, it was discovered that the treatment group made significantly
greater improvements over the term in their initial performance in the area of basic
function concepts. Evidence of this was provided by the results of a two sample
unpaired t-test performed on the pre and posttest score differences for both
groups which showed a p-value of 0.0003 at the .05 level.

Analysis of the attitude pretest data showed both control and treatment
groups as remarkably similar in their initial attitude towards mathematics. The
mean attitude score for treatment and control groups was respectively 73.1 and
73.7 out of a total possible score of 110. Results of a two-tailed unpaired t-test
performed on the posttest data yielded a p-value of 0.73 at the 0.05 level thus
suggesting no significant difference in the two groups.

Analysis of the attitude posttest data produced results similar to those
obtained for the pretest. The mean scores on the test were again very similar
(72.1 for the treatment group and 74.7 for the control groups). A two sample
unpaired t-test applied to the posttest data again revealed no significant difference
in the attitude towards mathematics of the groups (p-value = 0.23). The standard
deviations for both post and pretest scores were remarkably similar.

As was the case with the algebraic skills and the functions concepts, a
comparison was made of the differences in the pre and posttest scores for the
attitude tests in order to determine possible change in attitude towards
mathematics over the semester. An examination of the score differences revealed
that the attitude of the treatment group showed minor deterioration over the term

with the mean of the pre and posttest score differences given by -1.0. On the

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other hand, the control group showed a minor improvement in their attitude in the

sense that the mean of the score differences for the group was 1.0. Nevertheless,

there was really no significant difference in the change of attitude for both groups

as evidenced by the results of unpaired two sample t-test performed on the pre

and posttest score differences for both groups (p—value = 0.40 at the 0.05 level).
The attitude survey conducted in this study actually involved two

components:

(i) attitude towards mathematics (both groups) and

(ii) attitude towards the graphics calculator (treatment group only).

Ten of the 32 items on the attitude posttest were especially designed to survey (ii).

All 10 items reflected an attitude towards the graphics calculator. Statistical

analysis of the responses to these items revealed that the students in the treatment

(calculator) group generally showed a positive attitude towards the graphics

calculator in the limited directions suggested by the survey items. Based on the

percentage of students who responded agree or strongly agree to the appropriate

survey items reflecting a positive attitude towards the calculator, a relatively high

level of agreement existed among the students on the fact that the graphics

calculator has the potential to:

(i) increase mathematical confidence (71.9% in agreement).

(ii) help visualization and thus increase the level of mathematical understanding

(83.3% in agreement).
(iii) provide graphical methods for problem solving and hence lead to a better

understanding of a problem (72.9% in agreement).

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(iv) free time normally spent on tedious tasks that can now be devoted to more
interesting mathematical tasks (79.2% in agreement).
(v) be of use in other areas of college work that relates to mathematics (70.8% in
agreement).
it was surprising to find that students appear undecided as to whether the
graphics calculator enabled them to solve more realistic and more interesting
problems. Of particular interest was the fact that a relatively low percentage of
the students surveyed (20.8%) felt that the graphics calculator hampered their
ability to perform basic mathematical tasks - a result consistent with the

performance of the treatment group on algebraic skills analyzed earlier.

Examination of Hypotheses
Six hypotheses were listed in the theoretical framework of chapter III of this
study. These hypotheses were examined in the light of the research findings of

the study mentioned earlier.

Hypothesis 1. After a one semester precalculus course at university, there is
no slgniflcant difference in the achievement level, in the area
of basic algebraic skills, between those students who study
precalculus with the aid of a graphics calculator and those

who do not.

Based on results of a posttest of algebraic skills this hypothesis should be

accepted. The result of a two sample unpaired t-test performed on posttest

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scores (p-value = 0.15 at the 0.05 level) suggest no significant difference in the

performance level of both treatment and control groups.

Hypothesis 2. After a one semester precalculus course at university, there is
no significant difference in the achievement level, in the area
of basic function concepts and graphing, between those
students who study precalculus with the aid of a graphics

calculator and those who do not.

Results of a t-test performed on the posttest scores of both groups in the area of
function concepts showed that the treatment group performed at a significantly
higher level than the control group (p-value = 0.0000 at the 0.05 level). Hence

Hypothesis 2 is rejected.

Hypothesis 3. After one semester precalculus course at university, there Is
no significant difference in the attitude towards mathematics
of students who study precalculus with the aid of a graphics

calculator and those who do not.

Analysis of the attitude posttest results for both groups reveal no significant
differences in the attitude towards mathematics of the treatment and control
groups (p-value = 0.23 at the 0.05 level). On this basis, Hypothesis 3 should be

accepted.

Hypothesis 4. Over a one semester precalculus course there is no significant

difference in the average change (loss or gain) in -

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achievement, in the area of basic algebraic skills between
students who take precalculus with the aid of a graphics

calculator and those who do not.

A two sample unpaired t-test performed on the differences in the pre and posttest
scores of both the treatment and control groups revealed that although both
groups experienced a loss in achievement, nevertheless, the loss for the treatment
group was not significantly worse than that for the control group (p-value = 0.77

at the 0.05 level). This is a reasonable basis for the acceptance of Hypothesis 4.

Hypothesis 5. Over a one semester precalculus course there is no significant
difference in the average change (loss or gain) in achievement,
in the area of basic function concepts, and graphing, between
students who take precalculus with the aid of a graphics

calculator and those who do not.

This Hypothesis should be rejected. Evidence for this is supplied by the results
of a two sample unpaired t-test performed on the differences in the pre and post
function concept test scores for both groups wherein it was found that the t-test

p-value was 0.0003 at the 0.05 level.

Hypothesis 6. Over a one semester precalculus course, there is no
significant difference in the average change (loss or gain) in

attitude towards mathematics between students who take

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precalculus with the aid of a graphics calculator and those

who do not.

Earlier analysis of the differences in pre and posttest scores of both groups,
revealed that the treatment (calculator) group made significantly greater
improvement over the span of the semester, in the area of function concepts and
graphing than did the control (non-calculator) group. Evidence for this is provided
by the results of a two sample unpaired t-test performed on the pre and posttest
score differences for both groups (p-value = 0.0003 at the 0.05 level). On this

basis Hypothesis 6 should be rejected.

Summary of Results of Student and Teacher Questionnaires

Results of the student questionnaire administered to the calculator (treatment
group) revealed that student reaction to the use of a graphics calculator as an aid
to the study of precalculus was, for the most part, favourable. A large majority of
the students (84.4%) agreed that the graphics calculator had really helped them
improve their performance in precalculus. In particular, a substantial number of
the students (70% or more) felt that the graphics calculator had either been
moderately helpful or very helpful in the following directions:
. increasing their level of understanding of the function concept,
. graphing new and interesting functions,
. providing intuitive understanding of the properties of functions such as

asymptotic and end-point behaviour,

. helping them understand the link between equations and their graphs,

106

. solving or finding roots of equations,
. bypassing the use of tables and tabular information when graphing.

it was the feeling of most of the students surveyed (81.8%) that the graphic
calculator technology had no serious adverse affects on their algebraic skills nor
did it adversely affect their student workload. However, a large proportion of the
students (77.9%) felt that the introduction of the graphics calculator in high school
would have been very helpful to them in their university precalculus work.

The vast majority of the students (87%) appeared to experience no
problems with the usage of the graphics technology and the same percentage
preferred the calculator approach to the traditional approach in their study of
precalculus. However, 40% of the students surveyed felt they had become "over-
dependent" on the calculator.

One of the arguments frequently presented in favour of the graphics
calculator by supporters of the technology is that it is a vehicle for solving new and
interesting problems and for exploring and discovering mathematic ideas.
Contrary to this, only 17% of the respondents surveyed considered the calculator
moderately helpful in solving new and interesting problems. By the same token,
only 44% considered the calculator moderately helpful in exploring, conjecturing,
and discovering mathematic ideas.

As was the case with the students of the treatment (calculator) group,
reaction among faculty who taught the treatment group based on their responses
to an "open-ended" questionnaire, was generally very favourable. Although some

instructors were initially somewhat apprehensive about achieving the proper

107

balance between calculator related activities and traditional skill development,
nevertheless, after one semester of calculator usage, most instructors showed a
positive attitude toward the technology and some saw even greater applications
for the technology than they had initially envisioned.

Faculty, as well as students, experienced no major adjustment problems in
integrating the technology into their classrooms. Any increase in workload was
mainly due to extra instructional time on calculator usage and often depended on
the extent to which instructors needed to become familiar with the operation of the
calculator.

In regard to teaching style adjustment, some instructors found it necessary
to represent a problem both algebraically and geometrically. Their choice of
worked examples was also different than what was normally done in a traditional
mathematics class. In particular, instructors felt that the availability of the graphics
calculator enables them to do more non-routine and more interesting problems -
thus giving them a better preparation for calculus.

Generally instructors felt that the use of the visual aid enabled students to
develop a better understanding of function concepts, graphing and geometrical
transformations. Furthermore, time saved on graphing and regular computation

could now be spent developing other concepts.

108

Main Conclusions

The following is a summary of the main conclusions drawn from this study

based on the qualitative and quantitative analysis of the data gathered in the

investigation:

1.

There was no significant difference in the performances of both the treatment
and control groups on either a pretest or posttest of algebraic skills.
Furthermore, the algebraic skills of the treatment group appear not to be
adversely affected by the treatment in the sense that the mean change in
achievement based on pre and posttest score differences was not significantly
different for both groups.

The performance of the treatment group on a posttest of basic function
concepts and graphing was significantly better than that of the control group
even though pretest results show that the two groups were very comparable
in this area. Furthermore, the treatment group made significantly greater
improvement in their performance, in the area of basic function concepts, than
did the control group over the span of one semester, when the means of the
differences in pre and posttest scores is compared for both groups.

Results of an Attitude Pretest and Attitude Posttest revealed that there was no
significant differences in the attitude towards mathematics of both the treatment
and control groups. Likewise, there was no significant difference in the change
in attitude over the semester among the two groups in the sense that the mean

change in attitude of both groups was not significamly different.

109

4. Data gathered through a student questionnaire very definitely revealed that
students in the treatment group exhibited a positive attitude towards the
graphics calculator in the light of its effectiveness and potential as a teaching
and learning aid. In particular, there was unanimous argument among the

students that the graphics calculator had the potential to:

(i) increase mathematical confidence and help relieve mathematical
anxiety,
(ii) help visualization and thus increase the level of mathematical

understanding of a problem.
(iii) provide graphical methods for problem solving and hence lead to a
“fun" way of problem solving,
(iv) free time normally spent on tedious tasks which can now be devoted
to more interesting mathematical tasks,
(v) improve student performance in precalculus,
(vi) be of use in other areas of college work that relate to mathematics.
5. Consistent with the results given in 4 above, the results of the student
questionnaire also showed that a large majority of the students surveyed (70%
or more) expressed the view that the graphics calculator was either moderately
or very helpful in:
(i) increasing the students’ level of understanding of the function concept.
(ii) graphing new and interesting functions,

(iii) finding the roots of a polynomial,

10.

11.

110

(iv) providing intuitive understanding of some of the basic properties of
functions which are regarded as being useful to the study of calculus,

(v) understanding the link between equations and their graphs,

(vi) bypassing the construction of tables of values when graphing.

Only 18.2% of the students surveyed in the treatment group felt that using

the graphing calculator hampers their ability to perform algebraic

manipulations.

An overwhelming majority of the students surveyed in the treatment group

(84.4%) agreed that the graphics calculator had helped improve their

performance in precalculus.

Approximately 87% of the students surveyed in the treatment group did not

experience problems with the usage of the graphics technology and the

same percentage of the students actually preferred to learn precalculus with

the aid of a graphics calculator rather than by the "non-calculator" approach.

81.8% of the students surveyed in the treatment group agreed that the

graphics calculator did not increase their workload.

Approximately 40% of the students surveyed in the treatment group

expressed the view that they had become "over dependent" on the graphics

calculator.

Results of an open-ended faculty questionnaire completed by instructors of

the treatment group suggests that faculty, as well as students, reacted very

favourably to the effectiveness of the graphics calculator. As was the case

with the students, faculty did not experience major adjustments to the

re

111

calculator technology. However, extra time was required in order to acquaint
students with the basic operation of the calculator. Some minor adjustments
in teaching style were needed to accommodate the graphics technology.
Concern was expressed by faculty that students may become over
dependent on the technology and also that the proper balance may not be
achieved between calculator related activities and traditional skill
development. Notwithstanding, faculty generally felt that the “visual aid"
provide by the calculator led to a better student understanding of function

concepts and graphing and hence led to a better preparation for calculus.

Discussion

When one reflects on the main conclusions of this study, probably one of
the biggest surprises, and certainly one of the most significant results of the
quantitative analysis of the study, was that the use of the graphing calculator
appeared not to adversely affect the algebraic skills of most students over one
semester of precalculus.

Evidently, this is good news for the advocators of the use of graphing
calculators in the mathematics classroom. Another positive finding of the study
was that the performance of students who were required to use graphing
calculators in precalculus was significantly better than that of students who were
taught without the aid of graphing calculators.

The qualitative data gathered through the study is consistent with the above

results and certainly suggests that graphing calculators were a positive influence

112

in the mathematics classroom primarily because they:

. helped students solve more interesting (non-routine) problems by graphical
methods, thus giving them a better preparation for calculus.

. helped students to a better understanding of basic function concepts, graphing,
and geometric transformations.

. did not hamper their ability to perform algebraic manipulations.

In addition to the positive findings of this study, much has already been
written by mathematics educators in support of graphing calculators and
technologically assisted instruction. Indeed, this is part of the so called
precalculus and calculus reform that is invading the mathematics classroom in
North America. Notwithstanding, it is the view of the researcher, after many years
of successful teaching of post secondary mathematics at the junior undergraduate
level, that there is no substitute for the experiences that students gain by actually
doing mathematics themselves. The danger of giving a mathematics problem to
a calculator or computer which outputs a final solution is that many important
informative steps in the solution get omitted along the way. Admittedly, as long
as graphing calculator or computer technology remains a supplementary
component to the learning of mathematics, and not the main focus of it, it does
certainly have tremendous potential to open doors for students, insofar, that it can
provide them with solutions to many motivating and exciting real world problems.
However, if the technology is not properly integrated in a mathematics program,
it may obscure the actual mathematics that is occurring. Consequently, instead

of being enhanced by the technology, the students’ understanding of the subject

113

matter may indeed be jeopardized.

Whereas most of the scientific community have moved on to take advantage
of new and existing technology, mathematicians have been slow to adapt to
technological ideas. The availability of technology is still causing new questions
to be raised among mathematicians regarding technologically assisted instruction.
The mathematical community is currently undergoing a period of transition as it
experiments with a variety of technological changes for both precalculus and
calculus. In spite of the scepticism that currently exists, technology in the
classroom is here to stay. It is no longer a question of whether we use technology
but how we use it. To respond promptly and appropriately to this question is

certainly one of the biggest challenges facing mathematics educators today.

Implications and Directions for Further Study

Instructor reaction to questions raised in the faculty questionnaire completed
in this study suggests that teaching precalculus with the graphics calculator as a
visual aid will help students develop a better understanding of the function
concept. Results of the comparative analysis done on posttest scores for the
treatment and control groups involved in this study appear to confirm this finding.
Furthermore, it was the general feeling of instructors for the treatment group that
the graphics calculator gave both instructors and students an opportunity to solve
a greater variety of non-traditional problems. Consequently, in the opinion of the
instructors, students were given a better preparation for the later study of calculus.

This suggests a need for future research to do follow-up studies on students who

114

take precalculus at university with the aid of a graphics calculator in order to
determine how well these students perform in a typical calculus course as
compared with those who do precalculus in the traditional way. Many universities
are now offering calculus two ways:

(i) by the traditional lecture approach,

(ii) with the aid of computer software packages.

It might be interesting in a follow-up study to compare the performances in
calculus courses of type (i) and (ii) of students who took precalculus using a
graphing calculator with those students who took precalculus without using a
graphing calculator.

Results of student/faculty questionnaires completed in this study indicated
that neither faculty nor students, who were involved with the "calculator" approach
to precalculus, encountered any major adjustment problems. However, the vast
majority of students surveyed felt that regular use of the graphics calculator in their
secondary school mathematics classes would have saved valuable time and
certainly would have been an asset to them in their university precalculus course.
This is a strong argument in support of usage of the graphics calculator in
secondary school mathematics programs. Teachers involved in the "calculator"
approach in this study appear to have spent extra instructional time acquainting
students with the calculator operations. Clearly, extra time needs to be set aside
from the regular syllabus for teachers to address any student problems in this
area. Indeed special training sessions, set up before the course begins and

earmarked for this purpose, would be highly desirable.

115

A substantial number of the students surveyed in this study (40.3%) in the
treatment (calculator) group felt that they had become "overdependent" on their
calculator. Further studies on the graphics calculator or other related technology
might do well to focus more seriously on this problem of "overdependence" or as
some sceptics in the field say "algebraic guilt". This "overdependence" on the
calculator may indeed cost students opportunities to also sharpen their basic skills
and develop their mathematical presentations. It could also result in a deemphasis
of the importance of good mathematical style and sound presentation which could
be detrimental to students in subsequent mathematics courses.

Mathematics educators who are rather "lukewarm" towards complete
integration of graphics calculators and other technology in their mathematics
classes frequently express concerns about how the mathematics curriculum will
have to change to accommodate modern technology. In particular, they often ask:
"what parts of the curriculum will have to be replaced and what will replace it?"
One could continue this line of thinking and further inquire as to what the impact
of cutting some of the traditional topics from a typical college freshman
mathematics course to accommodate technology will have on the future
performance of these students in subsequent mathematics courses which are
delivered in the traditional manner. This is an important consideration and one that
will have to be addressed as we seek to develop a core curriculum at colleges and
universities in the late 90’s.

Extensive use of graphics calculators and related technology in the

mathematics classroom will undoubtedly affect not only what is being taught but

116

also how it is being taught and how it is being evaluated. In this vein, further
studies on the effectiveness of this technology might examine, in much more detail
than was done in this study, how the integration of the technology affects teaching
strategies, teaching style, and the student evaluation system. In particular, one
could ask: "what teaching strategies, learning experiences, and system of
evaluation is most desirable in the technologically orientated mathematics
classroom?"

This study was, for the most part, an attitude and achievement study,
involving college freshmen of generally average mathematical ability. There are
those in the field of mathematics education who will argue that technologically
assisted mathematical instruction is best suited to students of high academic
ability. To respond to this concern, future researchers might be well advised to
compare the relative achievement levels of students of low, middle, and high
academic ability who are taught mathematics with the aid of graphing calculators
or modern computers. in particular, it might be interesting in this regard, to
investigate correlations between student mathematics achievement in both
treatment and control groups of a study and their performances on SAT and ACT
Tests.

Comparisons of the achievement levels of treatment and control groups
based on gender, age, and academic programs could also be a subject for future
studies of this sort.

Given the increasing number of college freshmen who either drop or fail a

basic precalculus course on their first attempt at North American colleges, it would

117

be highly desirable to determine whether or not a technologically assisted
precalculus course would be beneficial to these so called "repeaters" who have to
repeat their freshman mathematics studies and hence tie up scare human and
physical resources in these times of financial restraint. In particular, it might be
useful to compile and examine data which compares both the pass rate and also
the dropout rate in precalculus for such students who are taught this subject with
the aid of graphing calculators with those who are taught it in the traditional way.

One measure of the effectiveness of the graphics calculator and other
computer technology in the teaching of college freshmen mathematics courses is
the participation and success rate of students taught in this manner in subsequent
mathematics courses. Data comparing the participation and success rate in
subsequent mathematics courses of students who took precalculus with the aid
of graphics calculators and those who did not might be very revealing.

Results of this study do lend support to some of the arguments advanced
by advocators of wider use of technology in the mathematics classroom.
Nonetheless, in view of the concerns addressed earlier in this section and the
corresponding directions proposed for further study, further investigation will be
needed in these directions to respond to the genuine and sincere concerns of
those sceptics in the field who at present are hesitant to fully integrate technology

in their mathematics classroom.

APPENDICES

APPENDIX A
STUDENT ATTITUDE TOWARDS MATHEMATICS INVENTORY

PRETEST

123

PRECALCULUS MATHEMATICS
ALGEBRAIC SKILLS
PRETEST

P RP AN DIR I

This skills test is part of a research effort to assess the impact of using graphing

calculators on the attitude and performance of students in precalculus.

Participation in the test is voluntary and deeply appreciated. If you choose to
participate you indicate your voluntary agreement to do so by completing and
returning the answer sheet provided. It should not take more than 30 minutes to

complete.

All results of this research will be treated with strict confidence and you will
remain anonymous in any reports of research findings. Any inquiries regarding this

study should be addressed to:

Dr. Richard Fleming or Dr. Gerry Ludden

Chairman Associate Chairman
Department of Mathematics Department of Mathematics
Central Michigan University Michigan State University
Mount Pleasant, Ml 48859 East Lansing, MI 48824
Instructions:

1. Do not write or mark on this test.

2. Do all computations on scratch paper.

Record al answers to the test questions on the answer sheet provided by blackening the
appropriate circle for each answer on the answer sheet. Return the test questions together
with the answer sheet to the proctor.

Your score will consist of the number of questions answered correctly.

5. You have 30 minutes to complete the 25 items on this test.

121

IF YOU ARE TAKING A PRECALCULUS COURSE USING A GRAPHING CALCULATOR, PLEASE
RESPOND TO ALL STATEMENTS 1-32 BELOW, BY MARKING ONE RESPONSE TO EACH
STATEMENT ON THE ANSWER SHEET PROVIDED. IF YOU ARE ENROLLED IN A PRECALCULUS
COURSE CLASS WHICH IS NOT USING A GRAPHING CALCULATOR. PLEASE RESPOND TO
STATEMENTS 1-22 ONLY.

1. I'm taking Mathematics only because it is necessary for my program of studies.
2. I feel very much at case when doing Mathematics tasks.

3. I really enjoy studying Mathematics.

4. I occasionally like to study Mathematics concepts outside of the regular syllabus.
5. I feel a sense of insecurity when performing Mathematical tasks.

6. Mathematics make me feel uncomfortable, restless. irritable and impatient.

7. Mathematics helps us better understand todays technological world.

8. Mathematics is one of my favorite subjects.

9. l have good confidence in my ability to solve Mathematical problems.

10. I'm always under a terrible strain when writing Mathematics tests or exams.
11. I can get along perfectly will in my future career without Mathematics.

12. Many computational tasks in Mathematics are both tedious and boring.
13. It really scares me to have to take a Mathematics course.

14. When I hear the word Mathematics l have a feeling of dislike.

15. I am happier in a Mathematics class than in most other classes.

16. Most of the problems we solve in Mathematics have little practical value.
17. I enjoy talking to students and teachers about Mathematics.

18. I could do much better in Mathematics if I worked harder.

19. I do not expect to do very well in Mathematics at university.

20. I enjoy exploring new ways to solve Mathematical problems.

21. It's difficult for me to get in the mood to do Mathematics assignments.

22. I often think 'I can't do it' when a Mathematics test or assignment seems difficult.

23. Using a graphing calculator hampers my ability to perform algebraic manipulations.

24. The use of the graphing calculator for homework and tests has increased my confidence in
my Mathematical ability.

25. The graphing calculator helps me to visualize Mathematical concepts and thus increases my

level of understanding of the concepts.
26. The graphical method is a fun way to solve Mathematical problems.

27. Since using the graphing calculator, I feel more at ease when performing Mathematical
tasks.

28.

29.

30.

31.

32.

122

Graphing the Mathematical relationships expressed in a problem enables me to understand
the problem better.

Graphing calculators should be helpful for my future work in college in Mathematics and
other related disciplines.

I have developed a more positive attitude towards Mathematics since using the graphing
calculator.

I can solve more realistic problems which are neither contrived nor boring by using the
graphing calculator.

The graphing calculator frees valuable time normally spent on tedious computations that
can now bedevmed to more interesting Mathematical tasks.

APPENDIX C
PRECALCULUS MATHEMATICS
ALGEBRAIC SKILLS

PRETEST

126

PRECALCULUS MATHEMATICS
ALGEBRAIC SKILLS
POSTTEST

PURPOSE AND DIRECTIONS

This skills test is part of a research effort to assess the impact of using graphing

calculators on the attitude and performance of students in precalculus.

Participation in the test is voluntary and deeply appreciated. If you choose to
participate you indicate your voluntary agreement to do so by completing and
returning the answer sheet provided. It should not take more than 30 minutes to

complete.

All results of this research will be treated with strict confidence and you will
remain anonymous in any reports of research findings. Any inquiries regarding this

study should be addressed to:

Dr. Richard Fleming or Dr. Gerry Ludden

Chairman ' Associate Chairman
Department of Mathematics Department of Mathematics
Central Michigan University Michigan State University
Mount Pleasant, Ml 48859 East Lansing, MI 48824
Instructions:

1. Do not write or mark on this test.

Do all computations on scratch paper.

Record all answers to the test questions on the answer sheet provided by blackening the
appropriate circle for each answer on the answer sheet. Return the test questions together
with the answer sheet to the proctor.

4. Your score will consist of the number of questions answered correctly.

You have 30 minutes to complete the 25 items on this test.

127

RECORD THE ANSWER TO EACH OF THE FOLLOWING
QUESTIONS ON YOUR ANSWER SHEET

. If6z+3(z +1): z—2(z—3), then 2: equals

(1) 119- (2) g (3) 13—0 (4) 2 (5) none of the above

. The product (y - 3)(2y + 5) equals
(1) 3y” +10y + 2 (2) y’ + 3 (3) 3y“ — 3 (4)3312 - 8y - 3 (5) none of the above

. A complete factorization of 62:2 - 52: — 6 is

(1) (6:: + 3)(z - 2) (2) (2.1: — 3)(32 + 2) (3) (62: — 3)(2: — 2) (4) (22: + 3)(39: — 2)
(5) none of the above

. The expresison (zy)2(-332y) equals

(1) 3:1:‘3/3 (2) «33:33;3 (3) 92:63,:4 (4) —3:z:‘y3 (5) none of the above
' s
. When % is simplified, the result is
a:
(1) of (2) 2‘? (3) e? (4) xi (5) none or the above

. \7 128 - 73/1—6 may be simplified to
(1) t/’ 112 (2) 26/5 (3) —10\’/1_6 (4) t/fi (5) none of the above

. The system of linear equations 2m - n —1 has solution

m — 3n 2
(1)m=2 (2)m=§ (3)n=—2 (4)n=1
n=-3 n=—3 m=3 -m=—1

(5) none of the above
. When 2a2 + 3b’ is divided by 2a + 3b, the result is

(l) a + 3b (2) a + b (3) 20 + 3b (4) a - b (5) none of the above
. If 52:2 - 45 = 0, then 1: equals

(1) 3 (2) —3 (3) :l:3 (4) 134/:51 (5) none of the above

5

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

128

The quantity (—8)§ + (7)’§ equals

32 -32 .

(1) ‘3— (2) T (3)0 (4)1
If-—l-——3' lvedf th It’
21) - 3 — 11 IS SO 01' p, e resu 18

4 -17 17

(1) 1—7‘ (2) 7 (3) T _ (4) T

A complete factorization os 22:3 — 16 is

(1) 2(23 — s) (2) 2(1: - 4)(:r: + 4) (3) 21(32 — 8)
(5) 2(2 — 2)(n2 + 22 + 4)

If I31: - 2| > 3, then

5 —5
(1)z>§or:r<—l (2)—{<z<§ (3)»; (4):r:<—1

 

The expression equals

_ 3
22—1 224-1

1 2z+1 2z+7
(1) 42:2 — 1 (2) 2’ + 7 (3) 4t:2 -1 (4) 4x2 -1

 

 

 

2

 

 

 

(5) none of the above

(5) none of the above

(4) 2(1: — 2)(a:2 + 4)

(5) none of the above

(5) none of the above

If a: is a real number, the expression «1% is real valued if
(1) a: > 1 (2) z 2 1 (3) :r: < 1 (4) :c 5 1 (5) none of the above
W may be Simplified to give
(1) l + 1 (2) a + y (3) 3’ + z (4) z” (5) none of the above
a: 3] my y + a:
The expression (rf’2 + I14)"l equals
b3 + a“ a’b’ l 1

2 2 __ __ _ _
(1) a + b (2) a’b’ (3) a3 + b3 (4) a3 + b3 (5) none of the above
If a + b = 0 then ab equals
(1) 1 (2) 0 (3) -b (4) 4” (5) 5’

Which one of the following can be factored using real numbers but cannot be factored

using only integers?

(1) 1:2 - 4 (2) a." - 9 (3) 4a“ - 16 (4)21:2 - 2 (5) 2:2 - 3

20.

21.

22.

23.

24.

25.

129

 

 

 

 

If 1 + l = -1-, then 2 equals
a: y z
I! 1? 1' ll 1'3!
3 4

()z-l-y ()z-l-y () 2y ()z-l-y (5)noneoftheabove
If 2’ = 4‘", then :1: equals

1 2 3
(1) 2 (2) 2 (3) 5 (4) 2 (5) none of the above
If I31: — 2| = 3, then 1: equals

5 —l 5 —1 -5
(1) 5 (2) T (3) 5 or T (4) 3 (5) none of the above

If 2 = —1 is a root of the equation 23:3 + K22 - 172: + 30 = 0, then K equals
(1) 0 (2) —45 (3) 45 (4) -49 (5) none of the above

Which of the following is non-negative for any value of 1:?

(1) 22—1 (2): (3)1—n2 (4)1—2: (5) (1-t't)2
If f(1:) = 22 — z + 1, then f(:r — 1) equals

(1)22-z+1 (2)32-3z+1 (3)32—3—1 (4)(:r:—1)'2 (5)22-3z-j-3

APPENDIX E
PRECALCULUS MATHEMATICS
FUNCTION CONCEPTS
PRETEST

130

PRECALCULUS MATHEMATICS
FUNCTION CONCEPTS

PRETEST
P RP E ND DI I N

This skills test is part of a research effort to assess the impact of using graphing

calculators on the attitude and performance of students in precalculus.

Participation in the test is voluntary and deeply appreciated. If you choose to
participate you indicate your voluntary agreement to do so by completing and
returning the answer sheet provided. It should not take more than 30 minutes to

complete.

All results of this research will be treated with strict confidence and you will
remain anonymous in any reports of research findings. Any inquiries regarding this

study should be addressed to:

Dr. Richard Fleming or Dr. Gerry Ludden

Chairman Associate Chairman
Department of Mathematics Department of Mathematics
Central Michigan University Michigan State University
Mount Pleasant, Ml 48859 East Lansing, MI 48824
Instructions:

Do not write or mark on this test.
2. Do all computations on scratch paper.

Record all answers to the test questions on the answer sheet provided by blackening the
appropriate circle for each answer on the answer sheet. Return the test questions together
with the answer sheet to the proctor.

Your score will consist of the number of questions answered correctly.

5. You have 30 minutes to complete the 25 items on this test.

131

RECORD THE ANSWER TO EACH OF THE FOLLOWING
QUESTIONS ON YOUR ANSWER SHEET

1. If [(2) = Ti-ifl' then the value of f(-l) is

(1) undefined (2)2 (3) 3 (4) 0 (5) —4
2. If the point (-3,a) is on the graph of y = t/l - 1: then the value ofa is
(1) a 2 (2) 2 (3) j (4) 73 (5) -2

3. Which of the graphs below is the graph of a function?

Marital— 3K

4. What is the domain of the function f(:)— - r?

-8

(1) {3\2 21} (2) (1.00) (3) (-00.1) (4) [-1.1] (5) (~00, 1]

5. An equation of the straight line which is the perpendicular bisector of the line segment joining
(-l,5), (5,-1) is _ ,

(l)y=z+4 (2)y=z-4 (3)y=-J:+4 (4)2y—1:=3 (5)y=z
6. The vertex of the parabola defined by y = 4x - 2:2 is

(1) (2.4) (2) (-2»4) (3) (2. -4) (4) (-2.-4) (5) (4. -2)
7. The equation of the circle with center (-1, -3) and radius J3 is

(1)(1:+1)2+(y+3)2=3 (2)(z+1)7+(y-3)’=3

(3)(z—1)2+(y-3)2=3 (4)(z+1)2+(y+3)2=3

(5) 1:2 + y2 = 3
8. The distance between the points P(—l, 3) and Q(-4, 7) is

(1) m (2) 5 (3) 3 (4) 6.4 (5) none of the these
9. Which of the following gives the y intercept(s) of the graph of 41:2 + 9y2 = 36

(1) 2 (2) :l: 3 (3) :i: 2 (4) 3 (5) —2

10.

ll.

l2.

13.

The point of intersection of the two graphs defined by y = z and --22 + y = —l is

(1) (-1. 1) (2) (1,-1) (3) (-1.-1) (4) (1.1)
The graph of the equation 312 — 4:2 = 16 is

(l) a parabola (2) an hyperbala (3) an ellipse

(4) a circle (5) none of these

If f(:r:) = 2:: + l, which of the following defines f(f(:r)?
(1)(2z-l-1)’ (2) 4£+3 (3) 412+? (4) 4z+3

Which of the following intervals gives the range of f(1:) = S in z?

(5) (1.2)

(5)41:+1

l4.

l6.

17.

18.

19.

20.

21.

22.

(1) [-1.1]

(2) (-oo,oo)

132

(3) (-1.1)

(4) [01] (5) [-35.13]

Which of the following is the complete graph of y = 1:2 — 1?

 

 

1:, which of the following defines 13(2)?
(1) P(.1:)= 2: + 1—5

(4)P=:"§"11:-1:2

Which of the following is the complete graph of f(2:) :: {

“I IS)
‘2
‘ .%‘ LIA“

"I

3 >1.

(2)

 

 

(2) Pm = 2(1: + $)

(nP=z+g

The range of the function f(1:) = 1 - fl is

(1) {ny $1}

(2) Mil < 13}

(3) [1.00)

I2) (31 .(SI
. a
. The z-intercept(s) of the graph of f(.r) = |2z - 1| — l is (are)
(1) l (2) ~71 (3)2 (4) 3.31 (5)0 and 1
The area of a rectangle is 25 sq. inches. If its perimeter P is written as a function of its width

(3)P=:+?
1&1, xaéO?
0, 2:0

(4) (40.00) (5) [0, 0°)

The number of real solutions to the equation 1:3 — 651: + 10 = 0 is

(n2

(2) 1

If I21: - 5| < 4 then

(l)z<§orz>%

(4)z>%

(3) 3

A
H
A
who ”"0

(4) none

(5) 4

(3):<§

Which of the following equations represents the graph produced as a result of applying a
vertical stretch of 3 units followed by a vertical shift of 1 unit downward to the graph of

 

yzzz?

 

(1) y: 31.-2+1
(4) y: 3(.1.’+ I)?

 

(2)1! = 3(2 -1)2

(5) none of these

(3)y=3z2-1

Assume than the point (—3,6) is on the graph of y = f(1:). If the point (—5, K) is on the
graph of y = f(: + 2), then the value of It' is

(1)2

(2) -4

(3) 6

(4)5

(5) —2

133

23. The equations of the vertical and horizontal asymptotes of the graph of y = $5 - 1 are
respectively
(l)z=—3,y=0 (2)::3,y=0 (3).t=—-3,y=2
(4)z=-3,y=-1 (5)z=—3,y=0

24. If f(.r) = 1:", 9(2) = \/1:_-_1 then (fog) (5) is
(1) -? (2) 50 (3) 2 (4) 1/5 (5) 4

25. If the graph ofa relation R is symmetric with respect to the origin and (—3, a) is on the graph
of R which of the following points must also lie on the graph of R?

(1) (-3. -a) (2) (13.-0) (3) (3.0) (4) (0.3) (5) (a. -3)

APPENDIX F
PRECALCULUS MATHEMATICS
FUNCTION CONCEPTS

POSTTEST

134

PRECALCULUS MATHEMATICS
FUNCTION CONCEPTS
POSTTEST

PURP E ND DIR T

This skills test is part of a research effort to assess the impact of using graphing

calculators on the attitude and performance of students in precalculus.

Participation in the test is voluntary and deeply appreciated. If you choose to
participate you indicate your voluntary agreement to do so by completing and
returning the answer sheet provided. It should not take more than 30 minutes to

complete.

All results of this research will be treated with strict confidence and you will
remain anonymous in any reports of research findings. Any inquiries regarding this

study should be addressed to:

Dr. Richard Flaming or Dr. Gerry Ludden

Chairman Associate Chairman
Department of Mathematics Department of Mathematics
Central Michigan University Michigan State University
Mount Pleasant, Ml 48859 East Lansing, MI 48824
Instructions:

Do not write or mark on this test.
2. Do all computations on scratch paper.

Record all answers to the test questions on the answer sheet provided by blackening the
appropriate circle for each answer on the answer sheet. Return the test questions together
with the answer sheet to the proctor.

Your score will consist of the number of questions answered correctly.

You have 30 minutes to complete the 25 items on this test.

10.

11.

12.

135

RECORD THE ANSWER TO EACH OF THE FOLLOWING
QUESTIONS ON YOUR ANSWER SHEET

. If f(£) = fifi?‘ then the value of f(—l) is

(1) undefined (2) g (3)3 (4)0 (5) -

If the point (-3, —a) is on the graph of y = m then the value of a is
(1) i 2 (2)2 13) l (4) «5 (5) —

Which of the graphs below is the graph of a function?

IZI

161% al— 1’—

. What ts the domain of the function f(z)_ - 771'?

(1) {IV 2 1} (2) (1.00) (3) (-00. 1) (4) [-1.1] (5) (-°°. ll

An equation of the straight line which is the perpendicular bisector of the line segment joining
(11 -5): (-511) IS

(l)y=z+4 (2)y=z-4 (3)y=—z+4 (4)2y—z=3 (5)y=z

. The vertex of the parabola defined by y = 1:2 - 4:1: is

(1) (2.4) (2) (4.4) (3) (2, -4) (4) (-2. -4) (5) (4,-2)

. The equation of the circle with center (—1, 3) and radius J3 is

(1)(z+1)’+(y+3)2=3 (2)(z+1)’+(y—3)’=3
(3)(2-1)’+(y—3)’=3 (4)(z+1)’+(y+3)’=3
(5) 22+y3=3

. The distance between the points P(l, -3) and Q(4, -7) is

(1) m (2) 5 (3) 3 (4) 6.4 (5) none of the these

. Which of the following gives the y intercept(s) of the graph of 1:2 + ~11]: = 36 ?

(1)2 (2) :i: 3 (3) :I: 2 (4)3 (5) -2
The point of intersection of the two graphs defined by y = -z and 2.1: + y = -1 is
(l) (-111) (2)01-1) (3)("'11-1) (4) (111) (5) (112)

. 2 a .
The graph of the equation 1|; - ‘7 = 1 IS
(1) a parabola (2) an hyperbala (3) an ellipse
(4) a circle ' (5) none of these

If f(3) = 22 - 1, which of the following defines f(f(z)?
(1) (21:+l)'2 (2)4z—l (3)4z+2 (4)4z-3 (5)4z-l-l

13.

14.

I5.

16.

17.

18.

19.

20.

21.

22.

136

Which of the following intervals gives the range of f(2:) = 2 cos .2"?

(l) [-1.1]

-22]

(4) [0.1]

Which of the following is the complete graph of y = 1 - a." 2

 

(2)

 

 

The z-intercept(s) of the graph of f(.r) = I22: + 1| — l is (are)

(1)%

(2) ’51

The perimeter of a rectangle is 25 inches. If its area A is written as a function of its width 2:,

(3) 2

which of the following defines A(z)?
(1) A(z) = 2:: + 2,-5
(4) A(:r) = 3.}: — :2

(4) *2-1

(2) 21(2) = 2(2 + 333)

(5)/1(2): 2+ 5%

(5) ['T'. 1}]

(5)0 and —l

(3)/1(2): 2+ ’35-

. . . 3L4, 2: ¢ 0
Which of the following is the complete graph of f(1:) = 6 ?
1: _

if!)

27%|»:-

(2)

 

The range of the function f(2:) = 1 + (E is

(1) {ny >1}

(2) {ny < 13}

(3) I1~00)

(4)
a. —%a

 

(4) (40.00)

(5) [0.00)

The number of real solutions to the equation x3 - 31:2 + 31: - l = 0 is

(1)2

(2) 1

If I21: — 5| > 4 then

(l)z<
(4):)

Which of the following equations represents the graph produced as a result of applying a
vertical stretch of 3 units followed by a vertical shift of 1 unit upward to the graph of y = :2?

A
2
.1.

2

9
Of2>2

(1) y: 322+1
(4) y: 3(J:+ 1)2

Assume than the point (~3,6) is on the graph of y = f(2:). If the point (-5, K) is on the

(3) 3

(4) none

 

(2)11 = 3(1: -1)2

(5) none of these

graph of y = f(.c - 2), then the value of K is

(1) 2

(2) —4

(3)6

(4) 5

(5) 4

(3).r<-2-

(3)y=3z2-1

(5) —2

23.

24.

25.

137

The equations of the vertical and horizontal asymptotes of the graph of y = :3—5 + l are
respectively

(1)z=-3,y=0 (2)::3,y=0 (3)z=-3,y=2
(4)z=—3,y=-1 (5)::3,y=1

If f(1:) = :7, 9(1) = t/le then (gof) (5) is

(1) -2 (2) 50 (3) 2 (4) 2J6 (5) 4

If the graph of a relation R is symmetric with respect to the origin and (3, -a) is on the graph
of R which of the following points must also lie on the graph of R?

(1) (-3. -a) (2) (-3.0) (3) (3.0) (4) (0,3) (5) (a. -3)

 

 

APPENDIX G

STUDENT QUESTIONNAIRE

138

STUDENI QUES I IQNNAIRE

This questionnaire is an attempt to assess student reaction to the use of graphing calculators as an aid to
the learning of Pre-calculus. It is part of a larger research effort to assess the impact of using graphing
calculators on the attitude and performance of students who are studying Pre-calculus at the college level.
Participation in the questionnaire is voluntary and deeply appreciated. If you choose to participate Mat;
you; volugm smut by guiding and mm' g the questionng'm. The questionnaire should not take
more than 20 minutes of your time. All results of this research will be treated with strict confidence and you
will remain anonymous in any reports of research findings. Any inquiries regarding this questionnaire should
be addressed to: Dr. Gerald Ludden, Associate Chairman, Department of Mathemtics, Michigan State
University.

 

1. Initially, did you have mixed feelings about taking this course with the aid of a
graphing calculator? If so why?

2. Has your initial attitude towards this technology since changed? If so, how?

3. Did you use graphing calculators in your high school Mathematics
courses? yes _
no

4. Do you feel that introducing graphing calculators in your high school Mathematics
program would have been helpful in this course? If so, how?

5 . Were there serious problems with the usage of the technology in this course? If so,
in your opinion how and to what extent were these overcome?

5- Do you feel that learning Pre-calculus with the aid of graphing technology has
increased your workload in this course? yes _
no _

 

10.

11.

139

Was the graphing calculator useful to you in university courses other than
Mathematics? yes _

no

Do you honestly believe that the graphing calculator technology has helped you
improve your performance on the various topics of Pre-calculus being taught in this
course? yes_ , no _, please explain.

In your opinion, have your basic ”algebraic skills" "suffered" as a result of using this
technology? yes _
no _

Have you become" over dependent" on the graphing calculator to solve problems and
work exercises in this course? yes __
no

Consider each of the items (1) to (xi) below and indicate to what extent you feel the
calculator has been helpful to you in the area described in each item by circling one
of the five numbers below each item according to the following code:

1. not at all helpful 2. a little helpful 3. undecided
4. moderately helpful 5. very helpful

(1) increasing the level of understanding of the function concept.
1 2 3 4 5

(ii) developing basic algebraic skills and manipulatives.
1 2 3 4 5

(iii) solving new and interesting ”word" (story) problems.

1 2 3 4 5

12.

(iv)

(V)

(Vi)

(Vii)

(viii)

(11!)

(X)

(xi)

14o

estimating/thinking about whether answers make sense.
1 2 3 4 5
exploring, conjecturing & discovering mathematical facts.
1 2 3 4 5
graphing new and interesting functions.
1 2 3 4 5
solving (or finding the roots) of an equation.
1 2 3 4 5
understanding the link between an algebraic equation and its graph.
1 2 3 4 5

providing an intuitive understanding of the "asymptotic” and ”end-point"
behavior of a function.

1 2 3 4 5

bypassing the construction of a table of values when graphing.
ll 2 3 4 5

preparation for subsequent courses in calculus.

1 2 3 4 5

Indicate by placing a tick ( ) in the space provided which of the following approaches
to the teaching and learning of Pre—calculus you would prefer.

(a)
(b)

the graphing calculator approach
the non-graphing calculator or more traditional approach

APPENDIX H

INSTRUCTOR QUESTIONNAIRE

141

PRECALCULUS
INSTRUCTOR QUESTIONNAIRE

This questionnaire is an attempt to assess instructor reaction to the use of graphing calculators in
the teaching of precalculus. It is part of a larger research effort to assess the impact of using
graphing calculators on the attitude and performance of students in precalculus. All information in
the questionnaire will be held strictly confidential. Your cooperation in responding to the questions
being raised is deeply appreciated. If you choose to participate you indicate your voluntary
agreement to do so by completing and ramming the questionnaire to: Dr. William Miller, Dept. of
Mathematics, Central Michigan University, Mt. Pleasant, Ml 48859.

 

1. Initially. did you have mixed feelings about teaching this course using graphing calculators?

2. Has your initial attitude towards this technology since changed? If so, how?

142

3. Were there adjustment problems for both you and your students when you first attempted
to use this technology to teach precalculus? lf so, in your opinion, how and to what extent
were these addressed and overcome?

4. To what extent if any. has the use of this technology changed your teaching style and your
workload in this course?

143

5. What, in your Opinion, are some of the Manes, and disadvantages of this technology?
(In your answer to this question make particular reference to student attitudes, student
level of understanding of mathematics concepts, the learning of algebraic skills, solving new
and more interesting problems. preparation for calculus, etc.).

6. Would you prefer and recommend teaching precalculus with the aid of this technology over
the traditional lecture approach? Why or why not?

APPENDIX I
MATHEMATICS 130 (CALCULATOR)

COURSE OUTLINE

144

Mathematics 130 (Calculator)
Course Outline

Text: Demana. Waits and ClanenS: Wellness: WM
1 I“.

 

1 .1 Cartesian Coordinate System and Complete Graphs

1 .2 Functions and Graphing Utilities

1 .3 Graphs and Symmetry

1 .4 More on Functions

1 .5 Linear Functions and Linear Equalities

1 .6 Quadratic Functions and Geometric Transformations

1 .7 More on Quadratic Functions and Geometric Transformations
1.8 Operations on Functions

 

2. 1 Solving Equations

2.2 Solving Systems of Equations

2.3 Solving Inequalities

2.4 Inequalities Involving Absolute Value

2.5 Solving Higher Order Inequalities Algebraically and Graphically
2.6 Maximum and Minimum Values

2.7 Increasing and Decreasing Functions

 

3.1 Continuity and End Behavior
3.2 Real Zeros of Polynomials
3.3 More on Real Zeros

3.4 Complex Numbers as Zeros
3.5 Rational Functions, Part 1
3.6 Rational Functions, Part 2

 

4.1 Inverse Relations

4.2 Exponential Functions

4.3 Economic Applications

4.4 Logarithmic Functions

4.5 More on Logarithms

4.6 Equations, Inequalities, and Extreme-Value Problems

Q t SCI. |.E I.

5.1 Angle

5.2 Right Triangle Trigonometry

5.3 Trigonometric Functions of Any Angle (Sec. 5.4)

5.4 Radian Measure and Graphs of sin 1: and cos x (Sec. 5.5)
5.5 Graphs of tan 1:, cot 1:, sec 1:, and csc 1: (Sec. 5.6)

5.6 Inverse Trigonometric Function (Sec. 6.3)

5.7 Solving equations and inequalities (Sec. 6.4)

 

6.1 Polar Coordinates and Graphs (Sec. 7.3)
6.2 Parametric Equations (Sec. 7.5)
6.3 Motion Problems and Parametric Equations (Sec. 7.6)

W
7.1 Parabolas (Sec. 8.1)
7.2 Ellipse

7.3 Hyperbolus
7.4 Quadratic Forms and Conics (Sec. 8.4)
7.5 Nonlinear Systems of Equations and Inequalities (Sec. 8.5)

 

8.1 Sequences and Mathematical Inductions (Sec. 9.1)
8.2 Series and the Binomial Theorem (Sec. 9.2)

 

APPENDIX J
MATHEMATICS 130 (REGULAR)
COURSE OUTLINE

146

Mathematics 130 (Regular)
Course Outline
Ten: Earl Swokowski. W
W
1 .1 Real Numbers
1 .2 Algebraic Expressions
1 .3 Equations and Inequalities
1 .4 Rectangular Coordinate Systems
1 .5 Lines
W
2.1 Definition of Function
2.2 Graphs of Function
2.3 Quadratic Functions
2.4 Operations on Functions
2.5 Inverse Functions

 

4.1
4.2
4.3
4.4
4.5
4.6

Graphs of Polynomial Functions

Division of Polynomials

Complex Numbers

Zeros of Polynomials

Complex and Rational Zeros of Polynomials
Rational Functions

ential 'thmi uncti

Exponential Functions

The Natural Exponential Functions
Logarithmic Functions

Graphs of Logarithmic Functions
Common and Natural Logarithms
Exponential and Logarithmic Equations

147

ome '

Angles

Trigonometric Functions of Angles
Trigonometric Functions of Real Numbers
Values of the Trigonometric Functions
Trigonometric Graphs

Additional Trigonometric Graphs

Inverse Trigonometric Functions (Sec. 6.6)

 

Mathematical Induction (Sec. 9.1)

The Binomial Theorem (Sec. 9.2)

Infinite Sequences and Summation Notation (Sec. 9.3)
Arithmetic Sequences (Sec. 9.4)

Geometric Sequences (Sec. 9.5)

 

Conic Sections (Sec. 10.1)

Parabolus (Sec. 10.2)

Ellipses (Sec. 10.3)

Hyperbolas (Sec. 10.4)

Rotation of Axes (Sec. 10.5)

Polar Coordinates (Sec. 10.6)

Polar Equations of Conics (Sec. 10.7)

Plane Curves and Parametric Equations (Sec. 10.8)

APPENDIX K
ANALYSES
OF

SUPPLEMENTARY ATTITUDE AND ACHIEVEMENT PRET EST SCORES

148

ANALYSES

OF

SUPPLEMENTARY ATTITUDE AND ACHIEVEMENT PRETEST SCORES

Table 19 Results of t-test on Attitude Pretest Scores: Students Who Wrote Pre
and Posttest Versus Those Who Wrote Just Pretest in Treatment

 

 

Group.
Fi
Group N Mean SD t P
Treatment (wrote both) 72 74.3 14.8 1.39 0.17
Treatment (wrote pre 46 70.5 14.0
not post)

 

 

 

 

 

 

 

 

Table 20 Results of Host on Attitude Pretest Scores: Students Who Wrote
Pre and Posttest Versus Those Who Wrote Just Pretest in
Control Group.

 

Group

Control (wrote both)

Control (wrote pre not
post)

 

 

I—L=

N

100

75

 

 

 

Mean SD
74.6 1 4.9
72.5 1 4.2

 

 

 

0.91

 

II

 

0.36

Table 21 Results of t-test on Algebraic Skills Pretest Scores: Students
Who Wrote Pre and Posttest Versus Those who Wrote Just
Pretest in Treatment Group.

 

 

I===
Group N Mean SD t P
Treatment (wrote both) 67 13.76 4.20 1.40 0.17
Treatment (wrote pre 28 12.61 3.41
not post)
=== =

 

 

 

 

 

 

 

 

 

149

Table 22 Results of t-test on Algebraic Skills Pretest Scores: Students
who Wrote Pre and Posttest Versus Those Who Wrote Just
Pretest in Control Group.

 

 

" Group N Mean SD t P
Control (wrote both) 53 13.09 3.95 1.79 0.076
Control (wrote pre 49 11.65 4.15

not post)

 

 

 

 

 

 

 

 

Table 23 Results of t-test on Function Concepts Pretest Scores: Students
Who Wrote Pre and Posttest Versus Those Who Wrote Just
Pretest in Treatment Group.

 

 

 

Group N Mean SD t P
Treatment (wrote both) 67 10.27 3.40 2.36 0.022
Treatment (wrote pre 28 8.36 3.67

not post)

 

 

 

 

 

 

 

 

Table 24 Results of t-test on Function Concepts Pretest Scores: Students
Who Wrote Pre and Posttest Versus Those Who Wrote Just
Pretest in Control Group.

 

 

Group N Mean SD t P
Control (wrote both) 48 10.19 2.82 2.11 0.37
Control (wrote pre not 53 8.77 3.45

Post)

 

 

 

 

 

 

 

 

150

Table 25 Results of t-test on Function Concepts Pretest Scores for
Students Who Wrote Pre and Posttest: Treatment Versus
Control Group.

 

 

Group N Mean SD t P
Treatment 67 1 0.27 3.40 0.1 4 0.89

 

 

 

 

 

 

 

 

i Control 48 10.19 2.82

 

 

 

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