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sections of PIECa
unit included def;

TWO SECtiong

learning Strategy

ABSTRACT

LIMITS: A MASTERY LEARNING APPROACH TO A
UNIT ON LIMITS OF SEQUENCES AND FUNCTIONS IN A
PRECALCULUS COURSE AND ACHIEVEMENT IN
FIRST SEMESTER CALCULUS
by

Douglas William Nance

The teaching of limits to calculus students is a task faced by most college
instructors. This study was an attempt to find a method of presenting limits
to students in such a way that they would be prepared to work with a formal
definition of limit of a function in a first semester calculus course. Specifi-
cally, a unit on limits of sequences and limits of functions was taught in four
sections of Precalculus Mathematics at Central Michigan University. This
unit included definitions of both limit of a sequence and limit of a function.

Two sections that received the unit on limits were taught using a mastery
learning strategy. This consisted of (1) stating the instructional objectives,
(2) presenting the unit of instruction, (3) administering a formative evaluation,
(4) utilizing instructional alternatives, and (5) administering a summative
evaluation. Advocates of this approach maintain that generally three fourths
of the students can achieve at the same high level that only one fourth of the

students usually achieve. This "achievement level" is referred to as the

mastery level or c
eighty-five percent
receive a grade of
attain this level at

A unique aspet
mastery level on I]
These Students we:
calcm118 the follow
0f the mastery leai

TWO sections i
received iDStrucu'c
sections Were used
limits. Performan
Calculus I COUrse t
to PrecalculuS Stud

TICatmem €fo
two ways, One Wa

etalculus sectiox

Douglas William Nance
mastery level or criterion. Ordinarily, this level is between eighty and
eighty-five percent or equivalently a criterion that would allow a student to
receive a grade of A or B in a regular classroom situation. Students who
attain this level are said to have "mastered" the material.

A unique aspect of this study was to identify students who attained the
mastery level on the summative evaluation but not on the formative evaluation.
These students were given the label "delayed mastery" and their progress in
calculus the following semester was analyzed to determine carry-over effects
of the mastery learning strategy.

Two sections of Precalculus students studied the same unit on limits but
received instruction via an expository presentation. Five other Precalculus
sections were used as a control group and received no instruction regarding
limits. Performance of students from all nine sections was analyzed in the
Calculus I course the semester following the presentation of the unit on limits
to Precalculus students.

Treatment effects on the affective domain elements were determined in
two ways. One was by interviewing randomly selected students from the
Precalculus sections. The other was by analyzing responses to a question-
naire completed by Treatment and Control students enrolled in Calculus I. In
addition to these analyses, enrollment figures in Precalculus and Calculus I
for the years 1971, 1972, and 1973 were examined to see if the Treatment

affected the percentage of Precalculus students enrolling in Calculus I.

The effect
at two stages c
first. Student
limits and (2) t
dent provided :1
Findings 01
1. The mt

the stu

 

t0 thirt

expositt
2- There v
behveen
here u
between
Control
There w
between

COrltr'ol

Douglas William Nance

The effect of the treatment on achievement in Calculus I was analyzed

at two stages of the course. Data from the test covering limits was used

first. Student performance was measured on (1) the questions involving

limits and (2) the total test scores. The semester percentage for each stu-

dent provided the second set of data.

Findings of the study include the following:

1.

The mastery learning strategy resulted in sixty-five percent of
the students attaining the mastery level of eighty percent compared
to thirty-four percent of the students attaining that level in the
expository treatment group.

There was no significant difference in achievement in Calculus 1
between delayed mastery students and other mastery students.
There was no significant difference in enrollment in Calculus 1
between students in the Treatment sections and students in the
Control sections.

There was no significant difference in achievement in Calculus I
between students in the Treatment sections and students in the

Control sections .

LU

UMTC

 

LIMITS: A MASTERY LEARNING APPROACH TO A
UNIT ON LIMITS OF SEQUENCES AND FUNCTIONS IN A
PRECALCULUS COURSE AND ACHIEVEMENT IN

FIRST SEMESTER CALCULUS

by

Douglas William Nance

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Secondary Education

1974

 

Iwould like ‘
dissertation poss

Professor Jo
adtice and gmdai

Protessors ]
doctoral commit

Professor W
Project.

Last in 0rde
tmist, ”Glen M,

have enabled me

ACKNOWLEDGMENTS

I would like to thank the following individuals who helped make this
dissertation possible:

Professor John Wagner, advisor and Committee Chairman, for his
advice and guidance during the last five years.

Professors Joseph Adney, Julian Brandon, and William Fitzgerald,
doctoral committee members, for their advice and encouragement.

Professor William E. Lakey for his continued support of the research
project.

Last in order but first in thought, I would like to thank my wife and
typist, Helen M. Nance. Her patience, encouragement, and perseverance

have enabled me to complete this project.

ii

 

LIST OF TABLE

LIST OF FIGL'R
CHAPlER

I. IbeII()I

bCeex

hias‘

Dela

I-eve

Disc

State

TABLE OF CONTENTS

LIST OF TABLES .......................
LIST OF FIGURES . . . . ......... . .........
CHAPTER
I. INTRODUCTION ............... . . . . .
Need for Study ...................
Mastery Learning ..................
DelayedMastery .......
LevelofMaterial . . . . . . . . . .........
Discussion . . ...... . . ...........

Statementonypotheses . . . . . . . . . . . . . . .

Organization of the Dissertation . . ...... . . .
II. REVIEW OF RESEARCH . . . . . ...........
Limits . . . . ......... . . ........

MasteryLearning..................
Summary....................
Classification of Objectives . . . . . . . . . . . . .
III. PROCEDURES
PopulationandExperimental Design . . . . . . . . . .
PersonsAssistingintheStudy............

iii

Page

xi

10
10
14
20
21
26
26

27

Mas

For

Con

Inst

Ev:

 

Ins
Su
De
W- Fth:

Pt

Page
Mastery Learning Strategy. . . . . . . . . . . . . . 27

Formulation of Instructional Objectives: Treatments
A and B O I O O O O O O I I O O O O O O O O O O O 28

Construction of Unit of Instruction: Treatments A and B . 30
Instruction: Treatments Aand B . . . . . . . . . . . 31
Quizzes............ ....... .. 32
Evaluation: TreatmentsAandB . . . . . . . . . . . 33
Wilson'sModel..................33
GradingtheTests ..... . . . . . . . . . . . 33
Diagnostic Procedures for TreatmentA . . . . . . . . 36
Instructional Alternatives: Treatment A. . . . . . . . 38
SummativeEvaluation................ 39
Delayed-Mastery Students . . ..... . . . . . . . 39
AnalysisofData. . . . . . ......... . . . . 39
IV. FINDINGS.......................44
PostHocTest...................44
MasteryLearning..................44
Summary....................50
Enrollment.....................S4
Summary....................S7.
Withdrawals from Calculusl. . . . . . . . . . . . 57
AffectiveDomain..................58
Interviews....................58
Questionnaire.................. 61

iv

Page
Cognitive Domain . . ......... . ...... 62
TestonLimits..................62
Semester Average in Calculusl . . . . . . . . . . 70
Summary............ ........ 73
V. CONCLUSIONSANDIMPLICATIONS . . . . . . . . . . . 76
Summary of Procedures . . ..... . . . . . . . . 76
Results......................77
MasteryLearning................78
Enrollment ...... ...........78
AffectiveDomain.................79
CognitiveDomain................80
Delayed-Mastery..... ..... 82
Conclusions....................83
Implications for Education . . . . . . . . . . . . . . 84
InstructionalUnit................. 84
MasteryLearning.................84
SourcesofVariation 85
OtherComments................. 86
Suggestions for Further Research. . . . . . . . . . . 87
APPENDIX
I.InstructionalUnit...................9O
11. Daily Classroom Record for TreatmentA . . . . . . . . lll
III.Quizzes........................114

V

IV. Summative Evaluations - Treatment B . . . . . . . . . .
V. Formative Evaluation- TreatmentA. . . . . . . . . . .
VI. Summative Evaluation - Treatment A. . . . . . . . . . .
VII. Instructional Alternative Worksheets. . . . . . . . . . .
VIII. Attendance Summary for Alternative Instruction Sessions. .
IX. CalculusTests.. ...... ...........

X. Description of Random Selection Procedure for
Interviewees

XI. Intervieruestions
XII. Questionnaire Given to Calculus Students . . . . . . . . .

XIII. Interview Responses Tabulated for Various Groupings of
CellsoftheExperimentalModel. . . . . . . . . . . . .

XIV. Cell Components of Chi Square Values for Questionnaire

Itms I I I I I I I I I I I I I I I I I I I I I I I I I

XV. Cell Components of Chi Square Values for Enrollment
my8is I I I I I I I I I I I I I I I I I I I I I I I

XVI. Calculus I Test Results and Final Percentages . . . . . .

XVII. Contingency Tables for Chi Square Test Statistics Computed
for the Sum of Items Two, Three, and Four of the Calculus

I Test I I I I I I I I I I I I I I I I I I I I I I I I

HBLIOGRAH-IY I I I I I I I I I I I I I I I I I I I I I I I I

Page
115
117
119
120
124

125

130
131

132

133

134

138

139

143

145

 

4.1

4.2

4.3

4.4
4.5
4.6

4.7

4.8

4.9
4.10
4.11
4.12
4.13
4,14
4,15
4.16

4.17

Analysis

Formatj‘
Test For

Form atj
Test F0

Format
SUmmat
SUmmaI

Summ at
Test Ft

Stimma
Test F<

Stimma
Enronr
Enronr

Intel-vi,

4.1

4.2

4.3

4.4

4.5

4.6

4.7

4.8

LIST OF TABLES

Analysis of A.C.T. Scores: Post Hoc Test . . . . .

Formative Evaluation Item Results -- Treatment A --

Test Form C

Formative Evaluation Item Results -- Treatment A --

Test Form C

Formative Evaluation Scores:

Treatment A

Summative Evaluation Item Results -- Treatment A .

Summative Evaluation Scores:

Summative Evaluation Item Results -- Treatment B --

Test Form A

Summative Evaluation Item Results -- Treatment B --

Test Form B

Summative Evaluation Scores:

Treatment A

Treatment B

Enrollment Summaries . . . . . .

Enrollment Analysis Results .

Interview Responses . . . . .

Questionnaire Responses . . .

Questionnaire Data . . . . . .

Calculus 1 Test Item 1 -- Forms, A, B, and C

Calculus I Test Item 2 -- Forms A, B,C,D,E

Responses by Cells to Limit Items 2, 3, and 4 of the

Calculus I Test

Vii

Page

45

46

47
48
49

50

51

52
53
55
56
59
63
64
66

67

68

 

4.19

4.20

4.21

4. 22

4.23

4. 24

4.1
14.2
4.3
4.4
4.5

Ah

Analysis <
Calculus I

Group Me
Scores .

One-W ay

Scheffe i
Calculus

Semeste
EXPerin

Analysi

SCheffe
Toni P

Attenda
Inter-Vi
Chi Sqi
Chi sq
Galen]

Chis:

4.18

4.19

4. 20

4.21

4. 22

4. 23

4. 24

A.l
A.2
A.3
A.4
A.5

A.6

Analysis of Student Performance on Limit Items in
calcmus I TeSt I I I I I I I I I I I I I I IIIIII

Group Means and Standard Deviations for Calculus I Test
scores I I I I I I I I I I I I I I I I I I I I I I I I

One-Way Analysis of Variance for Calculus I Test Scores. .

Scheffe Multiple Comparison Test for Equality of Means
calcmus I TeSt I I I I I I I I I I I I I I I I I I I I

Semester Averages by Groupings of Cells of the
Experimerltal MOdel I I I I I I I I I I I I I I I I I I

Analysis of Variance Table for Semester Averages by Cell .

Scheffe Multiple Comparison Test for Equality of Means
Total Percentage Grade for Calculus I . . . . . . . . . .

Attendance Summary for Alternative Instruction Sessions. .
InterviewResponses..................
Chi Square Values for Questionnaire Items . . . . . . . .
Chi Square Values for Enrollment Analysis . . . . . . . .
Calculus 1 Test Results and Final Percentages . . . . . .

ChiSquare Values for CalculusITest . . . . . . . . . .

viii

Page

68

69

70

72

72

73
124
133
134
138
139

143

 

3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
4.1
4.2
4.3

5.1

P0pulatic
Instructi
1. W. W11
Classific
Cujdeline
Diagnosu-
Room Ar

Delayed I

3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
4.1
4.2
4.3

5.1

LIST OF FIGURES

Population and Experimental Design . . . . .
Instructional Strategies for Treatment Sections
J.W. Wilson's Taxonomy . . . . . . . . . .
Classification of Test Items. . . . . . . . .
Guidelines for Assigning Partial Credit , . . ,
Diagnostic Forms for TreatmentA . . . . .
Room Arrangement for Small Group Sessions .

Delayed Mastery Students. . . . . . . . . .

Number of Students Selected for Interview by Group

Summary of Interview Responses . . . . . . . .

Analysis of Anxieties and Attitudes Toward Calculus I

Summary of Questionnaire Results . . . . . . . . .

Profile of Delayed Mastery Students . . . . . . . .

Page
26
29
34
35
36
37
38
40
41
6O
61
65

82

CHAPTER I

INTRODUCTION

 

This study investigated a method of presenting the topic of limits to
students. Specifically, a unit on limits of sequences and limits of functions
was taught in four sections of Precalculus Mathematics at Central Michigan
University (CMU). This differs from the current practice at CMU of teaching
limits“ only in the calculus course. This approach is consistent with G.
Rising'sl belief in presenting limits prior to calculus. It also gave a longer
introduction to the definition of limit of a function and its subsequent use than

is currently done at CMU.

Need for Study

 

The concept of "limit" is an essential part of a beginning calculus course.
C.B. Allendoerfer said, "The essential idea in calculus is that of 112133 . . . "2
Support for the importance of limits is given by the fact that several basic
calculus topics are defined in terms of them. In particular, continuity of
functions, differentiation of functions, and integration of functions all use
limits in their definitions.

The central role of this concept makes it apparent that mathematicians

 

"‘ The word 1imit(s) is imprecisely used and refers to the more specific topics

of "limits of sequences" and "limits of functions". Since these are closely re-
lated but nOt always taught in the same course, the general term "limit(s)" will
be used to mean either or both, whichever is more appropriate for the context.

1

2

and mathematics educators should be concerned about finding the most effec-
tive methods of presenting this topic to the beginning and prospective calculus
student. The need for this concern was emphasized by R.G. Long in 1968
when he included in his "Report of the Cornell Conference", under topics for
research, the following item: "In freshman calculus, what is the mechanism
involved in acquisition of the limit concept ?"3

There are differing opinions as to when, where, and how limits should
be taught. E.W. Ferguson, 4 D.W. Hight, 5 M. F. Hubley, 6 and G.R. Rising7
have supported the teaching of calculus (which includes limits) in the secon-

8 A. Beninati, 9 and

dary schools. On the other hand, C. B. Allendoerfer,
J. H. Neelley10 are opposed to introduction at that level. Another difference
of opinion involves the amount of formal logical procedures used in present-
ing the concept of limits. One approach uses a minimum of formal logic and
avoids a precise definition, but attempts to give students an intuitive idea of
the limit concept. Another approach utilizes a precise definition of limit
and subsequent formal logical development leading to the proof of theorems
involving limits. Since the definition of limits frequently employs the Greek
symbols "¢ " (epsilon) and "5 " (delta), this is referred to as the (6 , J ) or
rigorous approach.

The intuitive and rigorous approaches exist as opposite ends of a spec-
trum and most calculus instructors would place themselves somewhere

between these extremes. J.R. Anderson, 11 for example, believes in an

intuitive approach with an (6 .5 ) treatment postponed until advanced calculus.

3

Others such as C.B. Allendoerfer12

and R. P. Boas, Jr.l3 believe in starting ‘
with an intuitive approach and leading up to an (6 , J ) treatment in a begin-

ning course.

Mastery Learning

 

Another feature of the study was that a mastery learning strategy was
employed to teach two of the sections involved in the experiment. Histori-
cally speaking, major mastery learning attempts date to the early 1900's.
J.H. Block says, "As early as the 1920's there were at least two major

attempts to produce mastery in students' learning. "14

Mastery learning
consists of stating objectives, presenting the unit of instruction, adminis—
tering a "formative” evaluation, providing feedback from this evaluation,
utilizing alternative instructional techniques, and then administering a
"summative” evaluation. The effectiveness of this strategy is indicated
by J.H. Block in the following statement, "In general, three fourths of the
students learning under mastery conditions have achieved to the same high
standards as the top one-fourth learning under conventional, group-based

instructional conditions. "15 A description of the aspects of mastery learning

is given in Chapter II.

 

* These terms are defined on pages 19-20 of this document.

 

Delayed Master

Some stude
of this study. '
attain the mast:
reach this level
StUdents is a lit
group 0i studen

learning approe
Level of ‘ .
w

Limit pro]
learning techm
Problems (Cha
the material t:
Applicatmn, 0
most for Sch.
‘er leirrung

1191113. A dis

Delayed Mastery

 

Some students will be identified as "delayed-mastery" for the purposes
of this study. These will be the students in the mastery treatment group that
attain the mastery level (80 percent) on the summative evaluation but do not
reach this level on the formative evaluation. The identification of these
students is a unique aspect of this study. Subsequent achievement of this
group of students will be analyzed to determine what effect the mastery

learning approach in Precalculus had on their performance in Calculus 1.

Level of Material

 

Limit problems can be of many levels of difficulty. Since mastery
learning techniques have proven to be effective for lower level (Computation)
problems (Chapter II), an additional aspect of this study was to categorize
the material to be mastered by the treatment groups at the Comprehension,
Application, or Analysis levels in the cognitive domain of J.W. Wilson's
model for school learning. Part of the analyses will then be to see if mas-
tery learning can be effectively employed in the learning of higher level

items. A discussion of the components of this model are given in Chapter 11.

Discussion

 

Nine sections of precalculus students at Central Michigan University

constituted the population for this experiment. Each section contained

5
approximately thirty students. Five of these sections were used for control.
Of the remaining four, two were administered a section on limits of se-
quences and limits of functions via a classroom lecture method. The remain-
ing two sections were administered the same unit of material via a mastery
learning strategy.

Several reasons were considered when selecting the groups for treatment
or control. Both mastery learning sections were taught by the experimenter.
The other two treatment sections were taught by Dr. Douglas D. Smith who
was the only other staff member at CMU teaching two sections of Precalculus
Mathematics during the semester the experiment was conducted. The other
five sections were used as a control group. Since these were taught by five
different individuals, it was decided these would be the ones most appropriate
for control since any other assignment of treatment groups would have meant
at least three instructors working with treatment groups.

The results of this study involved both the affective (interest, attitude,
etc.) and cognitive (achievement) domains. The affective domain results
were checked by interviews, questionnaires, enrollments, and retention
figures. The cognitive domain results were obtained from the calculus
course given the semester following the teaching of the treatment units.
Results of the test covering the section on limits were analyzed and achieve-

ment in the entire course was examined.

Statement of Hypotheses

The hypotheses to be studied are:

l.

A mastery learning strategy, employed on a unit of limits of
sequences and limits of functions, will result in no significant
difference between the proportion of students achieving the mastery
level in a mastery learning situation and the proportion of students
achieving the mastery level in an expository classroom situation.
A unit on limits in Precalculus will not affect attitudes

(a) prior to taking Calculus I

(b) during Calculus I.

The analysis of this hypothesis will be qualitative and will consist
of reporting results of interviews and questionnaires completed

by students involved in the study.

There will be no significant difference between the percentage of
students enrolling and remaining in Calculus I after having had a
unit on limits and the percentage of students from the Control
group enrolling and remaining in Calculus I.

There will be no significant difference in achievement in Calculus I
between students in the treatment groups that achieve mastery and
those that did not achieve mastery.

There will be no significant difference in achievement in Calculus I
between delayed mastery students and other mastery students in

the treatment groups.

7
6. An introduction to limits prior to Calculus I will result in no
significant difference in achievement
(a) on the work with limits in Calculus I
(b) in the first semester of Calculus I.
This hypothesis was tested by examining test scores and semester
averages of students involved in the study who enrolled in Calculus I.

Hypotheses l, 3, 4, 5, and 6 were tested at the significance level of

oL=.05.

Orggnization of the Dissertation

The remainder of the dissertation is divided into four chapters. Chapter
11 contains a review of relevant literature. Material involving when, where,
and how limits are taught is examined. Explanations of and references to
mastery learning and learning models are included. Chapter 111 describes
the procedures used during the experiment. Included therein is the experi-
mental model. Chapter IV contains the findings of the study and Chapter V

gives the conclusions and implications.

FOOTNOTES -- CHAPTER I

lRising, Gerald R. "Some Comments on Teaching of the Calculus in
Secondary Schools", The American Mathematical Monthly, March, 1961,
68:287-290.

2Allendoerfer, C. B. "The Case Against Calculus", The Mathematics
Teacher, Nov. 1963, 56:482-485, p. 484.

3Long, R.G. "Report of the Cornell Conference on Mathematics
Education", 1968.

4Ferguson, Eugene W. "Calculus in the High School", The Mathematics
Teacher, Oct. 1960, 53:451-453.

 

5Hight, Donald W. "The Limit Concept in the Education of Teachers",
American Mathematical Monthly, Vol. 70, 1963, pp. 203-205.

 

6Hubley, Martin F. and Charles W. Maclay, "An Experiment in High
School Calculus", The Mathematics Teacher, Nov. 1970, pp. 609-612.

 

7Rising, Gerald R. "Some Comments on Teaching of the Calculus in
Secondary Schools", The American Mathematical Monthly, March 1961,
68:287-290.

 

8Allendoerfer, C.B. "The Case Against Calculus", The Mathematics

Teacher, Nov. 1963, 56:482-485.

9Beninati, Albert, "It's time to take a closer look at high school
calculus", The Mathematics Teacher, Jan. 1966, pp. 29-30.

loNeelley, J.H. "A Generation of High School Calculus", The American
Mathematical Monthly, Dec. 1961, 68:1004-1005.

11Anderson, John Robert, "A Comparison of Student Performance in a
One Year Freshman College Calculus Course Resulting from Two Different
Methods of Instruction", Unpublished Ph.D. Dissertation, Purdue University,
1970.

 

 

 

 

12Allendoerfer, C.B. "The Case Against Calculus", The Mathematics
Teacher, Nov. 1963, 56:482-485.

 

l3Boas, R.P. , Jr. "Calculus as an Experimental Science", American
Mathematical Monthly, Vol. 78, June-July, 1971, pp. 664-667.

8

9

14Block, James H. (ed.) Mastery Learning: Theory and Practice, Holt,
Rinehart and Winston, Inc., New York, 1971, p. 3.

 

ls'Ibid.

CHAPTER H

REVIEW OF RESEARCH

Limits

 

A college level course in beginning calculus is concerned with the
concepts of limits of functions, continuity of functions, differentiation of
functions, and integration of functions. Limits of functions is perhaps the
most important of these since the others are defined in terms of them.
R.V. Lynch referred to limit as . .the central concept of calculus. . . "1
These thoughts were supported by D. Kleppner and N. Ramsey when they
said, "The idea of limit. . .is at the heart of calculus. . . "2 and "Once you
have a real feeling for what is meant by a limit you will be able to grasp
the ideas of differential calculus quite readily. "3 Further support for the
importance of limits was given by D. W. Hight when he said, "It is the
introduction of limits that allows one to proceed from elementary mathema-
tics to a study of calculus and all its ramifications. "4

There are several approaches to presenting the concept of limits to
students of calculus. These range from a brief discussion of the concept
with some examples (frequently called the "intuitionist" approach) to a
development based on precise definition of limit and subsequent logically
developed proofs and applications of theorems. The question of how much

rigor should be applied in developing the concept of limits has been discussed

by several people. C. B. Allendoerfer has said, "All too often we begin the

10

11

course with an off-hand reference to limits as something too hard for the
students to really understand. . ."5 He continues with, "There are those,
however, who begin the course with a brief, but full dress, treatment of
limits using the epsilon-delta technique. This almost universally is wasted
on the class, for they are confronted with a diffith new idea without an
intuitive preparation. "6

A study by J.R. Anderson7 compared two approaches to presenting
limits; one used the (E , 5 ) definition and subsequent development while the
other used an intuitive development with the ( é , S ) definition coming in
advanced calculus. The second method was found to be Significantly better
with respect to achievement but there was no change in attitude as a result
of method. Included in the concluding remarks of this study was, "Evidently
neither method could make the (G ,5 ) notation and proof any more palatable
for the students. "8 T. H. McGannon, in a more general study,9 examined a
rigorous versus intuitive approach to calculus. Among the results of the
analyses were the following: there is no difference between the rigorous
method and the intuitive method of teaching calculus with respect to achieve-
ment of (a) a proficiency in manipulative skills, (b) an understanding of the
fundamental concepts, and (c) an ability to make practical applications of
the principles learned.

The fundamental nature of the limit concept in calculus and the lack of
agreement regarding its presentation have resulted in attempts to teach

limits prior to calculus. Gerald Rising supports this idea and recommends

12

having such a unit taught in a high school course. He says, "Students can
spend a substantial amount of time on two topics which most college teachers
will agree deserve more time than they are able to allot to them. These two
topics are limits and the definition of the derivative. Rather than gloss over
these topics superficially in a lesson or two, as is so often done in college
courses, the secondary program offers the possibility of spending two or
three weeks on each topic. "10 He continues by saying, "By solid teaching of
these topics [limit, derivative] the secondary teacher can help remove what
I believe to be two of the toughest hurdles to mathematics students in the
college program. ,,11

Rising is supported by D. W. Hight who says, "The teaching of the con-
cept of a limit should begin in the secondary schools. . . "12 and "The greatest
gap between the secondary school mathematics prOgram and that of the col-
leges seems to be due to the present treatment of topics that involve limits. "13
Hight's belief that limits should be taught in secondary schools is a result of
a stronger belief that limits should be taught prior to calculus. He says,
. .it is important that students have a good understanding of the concepts
of limit prior to entering calculus. . ."14; . . the students need a gradual
introduction to the limit concept in order to grow in mathematical maturity
and gain a readiness for calculus. "15; and, "If he is unacquainted with the
limit concept, he can hardly be prepared for calculus when the definitions of

both the derivative and the integral involves limits. "16

A concern for acquisition of the limit concept and success in calculus

l3

prompted the author to initiate this study which investigated some of the
aforementioned variables. Currently at Central Michigan University, limits
are taught only in the calculus course and are developed using a rigorous
(6 , S ) approach. Many of the calculus students did not have a section on
limits in their high school mathematics classes; consequently they were
receiving instruction regarding limits that, according to Allendoerfer, is
". . .almost universally wasted on the class. . ."17

The ideas of Rising, Hight, and others were implemented in this study
by including a unit on limits of sequences and limits of functions in four sec-
tions of CMU's Precalculus course. This unit included more than an intuitive
presentation of the concept of limit. It contained precise definitions (involving
[5 , S ] notation) and subsequent development of several theorems regarding
limits. This approach is supported by the work of D. T. Coon18 who, in a
study involving the intuitive concept of limit possessed by precalculus students,
found there was no significant correlation between intuitive knowledge of limit
and achievement in calculus.

One aspect of this study was the use of a special instructional strategy in
two of the treatment sections. This strategy consisted of teaching the unit on

limits using a "mastery learning" approach. Various aspects of this approach

are analyzed in the next section.

 

Mastery Learni:

Mastery lee
students can lea
variables are id
StudentS, perha
them, and it is
students to mas
mine what is m«
materials Whicl
master-3:519

The Variab

11) aptitude fOr

(3) 31311er to Un‘

l4

Mastery Learning

 

Mastery learning is an educational strategy based on the premise that
students can learn almost all of the material presented to them if certain
variables are identified and controlled. Benjamin S. Bloom said, "Most
students, perhaps over 90 percent, can master what teachers have to teach
them, and it is the task of instruction to find the means which will enable
students to master the subject under consideration. A basic task is to deter-
mine what is meant by mastery of the subject and to search for methods and
materials which will enable the largest proportion of students to attain such

..19
mastery.

The variables affecting learning were identified by John B. Carroll20 as
(l) aptitude for particular kinds of learning, (2) quality of instruction,
(3) ability to understand instruction, (4) perseverance, and (5) time allowed
for learning. Carroll believes that, . .if the students are normally dis-
tributed with respect to aptitude, but the kind and quality of instruction and
the amount of time available for learning are made appropriate to the charac-
teristics and needs of each student, the majority of students may be expected
to achieve mastery of the subject. "

Several mastery learning strategies have been devised which attempt to
control the variables described by Carroll. Bloom, when discussing what

his group at Chicago had been doing, said, "Our approach has been to supple-

ment regular group instruction by using diagnostic procedures and alternative

15
instructional methods and materials in such a way as to bring a large propor-
tion of the students to a predetermined standard of achievement. In this
approach, we have tried to bring most of the students to mastery levels of
achievement within the regular term, semester, or period of calendar time
in which the course is usually taught. "22 Samuel T. Mayo23 outlined a more
descriptive procedure when he summarized a national meeting that dealt with
the topic of mastery learning. He indicated a mastery learning strategy
should include the following.

(1) Inform students about course expectations, even lesson expectations
or unit expectations, so that they view learning as a cooperative
rather than as a competitive enterprise.

(2) Set standards of mastery in advance; use prevailing standards or
set new ones and assign grades in terms of performance rather
than relative ranking.

(3) Use short diagnostic progress tests for each unit of instruction.

(4) Prescribe additional learning for those who do not demonstrate
initial mastery.

(5) Attempt to provide additional time for learning for those persons
who seem to need it.

The success of mastery learning strategies has been documented in

several studies. Peter W. Airasian used a mastery strategy in a graduate-

level course in test theory. According to his summary of the study, the

method used produced successful results. He said, "The main result was

l6

striking. Whereas in the previous year only 30 percent of the students
received an A grade, 80 percent of the sample achieved at or above the pre-
vious year's A grade score on a parallel exam and thus received A's. "24

Kenneth M. Collins employed a mastery strategy in the teaching of four
college mathematics courses for freshmen. Concerning the results of his
study, he said, "In the modern algebra courses, 75 percent of the mastery
compared to only 30 percent of the non-mastery students achieved the mastery
criterion of an A or B grade. The calculus classes' results were similar:
65 percent of the mastery compared to 40 percent of the nonmastery students
achieved the criterion. "25

The results of other studies using mastery strategies are summarized
in the following: (1) Fred S. Keller26 had 65 percent to 70 percent of the
class receive A's or B's; (2) Mildred E. Kersh27 had 75 percent of a class
achieve mastery as compared to 19 percent that achieved mastery in a con-
trol class; (3) Hogwan Kim28 had results that indicated 74 percent of the
experimental group compared to only 40 percent of the control group attained
the mastery level; and (4) Samuel T. Mayo, Ruth C. Hunt, and Fred
Tremmel29 found when working with students in an introductory statistics
course that 65 percent of the mastery learning students received a grade of
A in contrast with 5 percent of the comparison group.

Mastery learning strategies accomplish more than having a high per-

centage of the students achieve at levels formerly attained by only a small

percentage of the students. There are several affective consequences of such

strateg.‘
a stude:

others

17

strategies. For example, J.H. Block says, "In a system with few rewards,
a student may not be rewarded no matter how well he learns so long as
others learn better. If this situation occurs repeatedly, he is likely to
eventually stop trying to learn well. In a system of unlimited rewards, by
contrast, a student who learns well can be rewarded even though others may
learn still better. Successful and rewarding learning experiences are likely,
in turn, to kindle a desire to continued learning excellence."30 Benjamin S.
Bloom, 31 when discussing mastery learning, lists the following affective
consequences: (1) students develop more interest for the courses they master,
(2) students develop a more positive self concept, (3) mastery learning can
develop a lifelong interest in learning. In addition to these consequences,
Bloom says, "Mastery learning can be one of the more powerful sources of
mental health."32

The success of mastery learning strategies induced the author to use
such a strategy in one of the treatment sections of this experiment. It was
based on the outline presented by S. T. Mayo. In accordance with item 1 of
his outline, terminal objectives (unit expectations) were selected prior to
starting the experiment. The difficulty of such a selection is documented by
K. Collins when he says, "Specification of objectives is perhaps the most
difficult variable to properly prepare. "33

The next step in preparing a mastery strategy is to select standards of

mastery. With regard to this, Bloom says, "While absolute standards care—

fully worked out for a subject are recommended, they are often difficult to

set. One metl
in a particular
grading standa-
used."34 Alo-
Objective rule
"A. . . standar.
for the subjec
Under master
ms‘ery teacl
med studex
be useful ma:
"The empiric
of the Skills :
COgru'u'Ve and
This Work a1
or nearly all
expectation i
may 11an “la

IOWal'd the 1E

18

set. One method might be to use standards derived from previous experience
in a particular course. For example, grades for one year might be based on
grading standards arrived at the previous year if parallel examinations are

used."34 Along these same lines, Block says, "There are no hard and fast

objective rules for setting mastery grading standards. "35 He further says,
"A. . .standard setting method is to transfer existent grading standards set
for the subject under non-mastery learning conditions to the courses taught
under mastery conditions. Standards from previous or concurrent non-
mastery teachings of the subject can be used. Generally, scores which
earned students learning under non-mastery conditions A's and B's seem to

be useful mastery grading standards. "36

Block supports Bloom by saying,
"The empirical work to date suggests that if students learn 80 to 85 percent
of the skills in each unit, then they are likely to exhibit maximal positive
cognitive and affective development as measured at the subject's completion.
This work also suggests that encouraging or requiring students to learn all
or nearly all (90 to 95 percent) of each unit, besides being an unrealistic
expectation in terms of student and teacher time and effort (Bormuth, 1969),
may have marked negative consequences for student interest in and attitudes
toward the learning (Block, 1970; Sherman, 1967). "37

The classroom instructional techniques for a mastery learning situation
do not have to differ from what a teacher would ordinarily employ. With

regard to this, Bloom says, "In the work we have done, we have attempted

to have the teacher teach the course in much the same way as previously.

19

That is, the particular materials and methods of instruction in the current
year should be about the same as in previous years. "38 It is the belief of
Bloom and others of his group that this is the only way mastery learning can
be effectively spread. He feels that if extensive retraining of teachers 18
required, then the necessary effort will not be expended by the teachers.

The next step in formulating a mastery learning strategy is the creation
of diagnostic progress tests. These have been labeled "formative evalua-
tions" by Michael Scriven. 39 According to Airasian, "Formative evalua-
tions seek to identify learning weaknesses prior to the completion of instruc-
tion on a course segment--a unit, a chapter, or a lesson. The aim is to
foster learning mastery by providing data which can direct subsequent or
corrective teaching and learning. "40 Another description of the purpose of
formative evaluations is given by Bloom when he says, "For those students
who have thoroughly mastered the unit, the formative tests should reinforce
the learning and assure the student that his present mode of learning and
approach to study is adequate. . . For students who lack mastery of a particu-
lar unit, the formative tests should reveal the particular points of difficulty--
the specific questions they answer incorrectly and the particular ideas, skills,
and processes they still need to work on. "41

The formative evaluations should be followed by diagnostic reports for
each student. Learning difficulties indicated on these reports are to be cor-

rected by instructional alternatives. A variety of alternatives will provide

for some of the learning variables given by Carroll (page 14). Block lists the

follovin

Blo
by his I
STOUps .
and to h
and "
likely tc

effectht

20
following "learning correctives" : 42
(1) Reteaching
(2) Small group problem sessions
( 3) Individual tutoring
(4) Alternative learning materials
(a) Alternative textbooks
(b) Workbook and programmed instruction
(c) Audio-visual materials
((1) Academic games and puzzles
Bloom strongly advocates the small group problem sessions as evidenced
by his remarks, "The best procedure we have found thus far is to have small
groups of students. . .review the results of their formative evaluation tests
and to help each other overcome the difficulties identified on these tests. "43
and, "Where learning can be turned into a cooperative process with everyone
likely to gain from the process, small group learning procedures can be very
effective. " 44
The last phase of a mastery learning strategy is administration of the
"summative evaluation". This is the phrase used by Scriven45 for the test
given to determine how well the students have "mastered" the objectives
specified earlier in the unit of instruction.
Summary: The mastery learning strategy used in one of the treatment

groups of this experiment is based on common aspects of the various strate-

gies previously mentioned and incorporated suggestions made concerning the

compo

Classi

tional
this st
n0t su
Classi:
Benjan
may fj
bel‘I'ng

mat, I

21

components of such strategies. A schematic diagram is given in Figure 3. 2.

Classification of Objectives

 

A successful mastery learning strategy requires specification of educa-
tional objectives for the material being taught. Since one of the purposes of
this study was to foster better "understanding" of the limit concept, it was
not sufficient to specify the educational objectives; it was also necessary to
classify them according to the type of learning involved. According to

'9

Benjamin 8. Bloom, . . .a teacher, in classifying the goals of a teaching unit,
may find that they all fall within the taxonomy category of recalling or remem-
bering knowledge. Looking at the taxonomy categories may suggest to him
that, for example, he could include some goals dealing with the application of
this knowledge and with the analysis of the situations in which the knowledge
is used. "46

The taxonomy used for classifying objectives in this study is James W.
Wilson's model. 47 The basic categories in the cognitive domain are:
(A) Computation, (B) Comprehension, (C) Application, and (D) Analysis. A
result of the use of Wilson's taxonomy was the inclusion of objectives at the

Comprehension, Application, and Analysis levels of behavior. A c0py of

this taxonomy is contained on page 34.

 

FOOTNOTES -- CHAPTER II

lLynch, Ransom V. A First Course i_n’ Calculus, Ginn and Company,
New York, 1964, p. 157.

 

2Kleppner, Daniel and Norman Ramsey, (Eick Calculus, John Wiley
and Sons, Inc., New York, 1965, p. 53.

3Ibid.

 

4I-Iight, Donald W. "The Limit Concept in the Education of Teachers",
American Mathematical Monthly, Vol. 70, p. 203.

5Allendoerfer, C.B. "The Case Against Calculus", The Mathematics
Teacher, Nov. 1963, 56:482-485, p. 484.

 

 

6mm.

7Anderson, John Robert, "A Comparison of Student Performance in a
One Year Freshman College Calculus Course Resulting From Two Different
Methods of Instruction", Unpublished Ph.D. Dissertation, Purdue University,
1970.

8Anderson, John Robert, "A Comparison of Student Performance in a
One Year Freshman College Calculus Course Resulting From TWO Different
Methods of Instruction", Dissertation Abstracts, Vol. 31, p. 3980A.

9McGannon, Thomas Herbert, "A Comparison of Two Methods of
Teaching Calculus With Special Inquiry Into Creativity", Unpublished Ph. D.
Dissertation, Northwestern University, 1970.

10Rising, Gerald R. "Some Comments on Teaching of the Calculus in
Secondary Schools", The American Mathematical Monthly, March, 1961.
68:287-290, p. 289.

 

nlbid.

12Hight, Donald W. "The Limit Concept in the Education of Teachers",
American Mathematical Monthly, Vol. 70, 1963, pp. 203-205, p. 204.

131m.

141mm, p. 203.

22

IS
Americ
161:

I7

 

Teache:

18

by Pre-e
Achieve

HI.

K

mags
. 1968.

23

15Hight, Donald W. "The Limit Concept in the Education of Teachers",
American Mathematical Montlgy, Vol. 70, 1963, pp. 203-205, p. 204.

“mm.

l7Allendoerfer, C.B. "The Case Against Calculus", The Mathematics
Teacher, Nov. 1963, 56:482-485, p. 484.

18Coon, Dorothy Trautman, "The Intuitive Concept of Limit Possessed
by Pre-Calculus College Students and Its Relationship With Their Later
Achievement in Calculus", Dissertation Abstracts, Vol. 33, p. 1537A.

19Bloom, Benjamin S. "Learning for Mastery", UCLA-CSEIP Evaluation
Comment, 1, No. 2, 1968, p. 1.

20Carroll, John B. "A Model of School Learning", Teachers College
Record, 64, 1963, pp. 723-733.

 

 

 

 

 

ZlBloom, Benjamin S. "Learning for Mastery", UCLA-CSEIP Evaluation
Comment, 1, No. 2, 1968, p. 4.

 

271de p. 7.

23Mayo, Samuel T. Measurement 32. Education: Mastery Learning and
Mastegy Testigg, National Council on Measurement in Eudcation, East
Lansing, Michigan, 1970.

24Block, James H. , (ed.) Mastery Learning: Theory ;an_d Practice,
Holt, Rinehart and Winston, Inc., New York, 1971, p. 98.

 

 

 

251mm, p. 111.

26Keller, Fred S. "Goodbye, Teacher. . .", Journal 2f Applied Behavior
Analysis, _1_, 1968, pp. 79-89.

271(ersh, Mildred E. "A Strategy for Mastery Learning In Fifth-Grade
Arithmstic", Unpublished Ph.D. Dissertation, University of Chicago, 1970,

Reported in Mastegy Learning: Theogy___ and Practice by J.H. Block.

28Kim, Hogwan, et. al. The Mastery Learmn rnigg Project i_n t_h__e Middle
Schools, Seoul: Korean Institute for Research in the Behavioral Sciences,
1970. Reported in Mastery LearniggtTh eo_1_'y _a_nd Practice by]. H. Block.

29Mayo, Samuel T. , et.al. "A Mastery Approach to the Evaluation of
Learning Statistics", National Council on Measurement in Education, Chicago,
Ill. , 1968.

 

 

30I

Master
Rinehm

31E
Comme

321

33(

Master}
Annual

 

34B

COmm e]
\

35B

Masten
ms;

361i

‘

3711

388:
c .
m

39S

24

30Block, James H. "Operating Procedures for Mastery Learning",
Mastggy Learning. Theory and Practice, Edited by J. H. Block, Holt,
Rinehart and Winston, Inc., Chicago, 1971, p. 65.

31Bloom, Benjamin S. "Learning for Mastery", UCLA-CSEIP Evaluation
Comment, 1, No. 2, 1968, pp. 10-11.

 

321mm, p. 11.

33Collins, Kenneth M. "An Investigation of the Variables of Bloom's
Mastery Learning Model for Teaching Mathematics", Paper presented at the
Annual Meeting of the Amer. Ed. Res. Asso. , Chicago, 1972. (ERIC 065 596)

34Bloom, Benjamin S. "Learning for Mastery", UCLA-CSEIP Evaluation
Comment, 1, No. 2, 1968, p. 8.

35Block, James H. "Operating Procedures for Mastery Learning",
Masteg Learning: Theory and Practice, Edited by J.H. Block, Holt,
Rinehart and Winston, Inc., Chicago, 1971, p. 68.

361.2411.-

371mm, p. 70.

 

 

38Bloom, Benjamin S. "Learning for Mastery", UCLA-CSEIP Evaluation
Comment, 1, No. 2, 1968, pp. 8-9.

39Scriven, Michael. "The MethodOIOgy of Evaluation", Perspectives 9i
Curriculum Evaluation, Edited by Robert Stake, Chicago: Rand McNally and
C0,, 1967, p. 7.

 

4OBlock, James H. (ed.) Mastegy Learnigg: Theogy and Practice,
Holt, Rinehart and Winston, Inc., New York, 1971, p. 79.

41Bloom, Benjamin S. "Learning for Mastery", UCLA-CSEIP Evaluation
Comment, 1, No. 2, 1968, p. 9.

 

 

42Block, James H. (ed.) Mastery Learning: Theory 329 Practice.
Holt, Rinehart and Winston, Inc., New York, 1971, pp. 71-73.

43Bloom, Benjamin S. "Learning for Mastery", UCLA-CSEIP Evaluation
Comment, 1, No. 2, 1968, p. 10.

 

4‘I‘Ibid. , p. 5.

5 .
SCI'IVI
and C0,, 19¢

46810011
1956, p. 2.

25

45Scriven, Michael, "The Methodology of Evaluation", Perspectives 3f
Curriculum Evaluation, Edited by Robert Stake, Chicago: Rand McNally
and Co., 1967, p. 7.

46Bloom, Benjamin S. , et.al. (eds.) Taxonomy 3f Educational Ogjec-
tives, Handbook 5 Cognitive Domain, New York: David McKay Co. , Inc. ,
1956, p. 2.

47Bloom, Benjamin S. , et.a1. Handbook 93 Formative and Summative
Evaluation at: Student Learning, McGraw-Hill Book Company, New York,
1971, p. 646.

 

 

 

 

CHAPTER III

PROCEDURES

Population and experimental design

The population for this study consisted of all students in nine sections of

Precalculus Mathematics at Central Michigan University during the Fall

Semester of the 1973-74 Academic Year. The study involved two treatment

groups and a control group. Treatment A consisted of teaching a unit on

limits of sequences and limits of functions to two of these sections using a

mastery learning strategy as described on page 29. Treatment B consisted

of teaching the same unit to two other sections but without employing a mas-

tery learning approach. The remaining five sections were used as a control

group and received no instruction regarding limits. The experimental design

is given in Figure 3.1.

 

 

 

 

 

 

 

 

 

 

Treatment A Treatment B Control
n = 60 n = 61 n = 152
Maste
n = 39ry Non- Non-
mastery Mastery mastery
(NNM) (SM) (SNM) (C)
Early Delayed n = 21 n _._ 21 n = 40
mastery mastery
(NEM) (NDM)
n = 5 n = 34
FIGURE 3.1

Population and Experimental Design

26

 

Persons Assi:

Three pe
E. Lakey, wh
all materials
iors for Trea
Douglas D. Sr
reviewed the 3
formative eva
Student, who ,
between the fc

ment A.

W

SpeCts fmm t1

27

Persons Assisting in the Study

Three persons assisted in some aspect of this study. Professor William
E. Lakey, who currently teaches the Calculus I classes at CMU, reviewed
all materials used in Treatments A and B, helped select the terminal behav-
iors for Treatment A (page 28), and conducted some interviews. Professor
Douglas D. Smith taught the two sections that received Treatment B. He also
reviewed the instructional materials and assisted in the construction of the
formative evaluations. The third person was Dave Rajala, an undergraduate
student, who was used as a resource person during the small group sessions
between the formative and summative evaluations for the students in Treat-

ment A.

Mastery Learning Strategy

 

The mastery learning strategy used in Treatment A differed in some re-
spects from the strategydescribed by S. T. Mayo in Chapter 11. One differ-
ence was that the strategy was employed on a unit of instruction rather than
the entire course. Consequently, only one formative evaluation was admin-
istered. Secondly, students were not informed of all terminal objectives
prior to the start of the unit because of the terminOIOgy involved. They were,
however, informed of the objectives as soon as the required notation and
terminOIOgy were introduced. For example, objective 4 (page 28) was stated

after the definition for limit of a sequence was given. A third variation was

 

that the iten
Application,

Figure .
used in the t
had to recor.

to achieve m
I“‘Ol'mulation
\

The obje
mm by Pro.
Unit on limits

(1) Give

Part

28

that the items to be mastered would be classified in the Comprehension,

Application, or Analysis levels of James W. Wilson's learning model.1

Figure 3. 2 is a schematic representation of the instructional strategies

used in the treatment sections of this study. Students in Treatments a and B

had to record a score of at least 80 percent, as suggested by J.H. Block, 2

to achieve mastery of the unit.

Formulation of Instructional Objectives: Treatments A and B

 

The objectives for the unit of study in Treatments A and B were estab-

lished by Professor Lakey and the author. At the conclusion of studying the

unit on limits, the students were expected to be able to do the following:

(1)

(2)

(3)

(4)

(5)

(6)
(7)

Given a sequence (function) that does not have a limit, indicate what
part of the definition it fails to satisfy.

Give an example of a sequence (function) that does not have a limit
and include explanation.

Given a real number, be able to produce a nonconstant sequence
(function) whose limit is that real number and include explanation.
When working with sequences, find the associated N 5 for a
particular e .

When working with functions, find the associated 6" for a
particular 6 .

Determine limits of sequences using the results of theorems.

Determine limits of functions using the results of theorems.

29

 

Treatments A and B

4L

Construction of unit of instruction:
Treatments A and B

Formulation of instructional objectivesJ

 

 

 

 

Presentation of
tructional objectives:
Treatment A

 

 

 

 

 

Instruction:
Treatments A and B

.L

Evaluation: -
Treatment B - Summative flEnd of Treatment B J
Treatment A - Formative '

3L

[Diagnostic procedures for Treatment A

J.

Instructional alternatives: Treatment A
1. Supplementary instruction ‘
2. Small-group sessions
3. Individual tutoring
4. Supplementary reading

L

[ Summative evaluation: Treatment Aj

_ L _
[ End of Treatment A J

*—

 

 

 

 

 

 

 

 

 

 

 

 

 

FIGURE 3. 2

Instructional Strategies for Treatment Sections

(8) Prove a
for limf
(9) Prove a
indepei
Objectives
Wilson's mode]
0f Previously s
For example,
9

hm BX“

“75

These Can 110‘

30
(8) Prove a given sequence has a particular limit using the definition
for limit of a sequence.
(9) Prove a given function has a limit at a particular value of the
independent variable using the definition for limit of a function.
Objectives six and seven are classified in the application level of
Wilson's model. Satisfactory performance would require appropriate use
of previously stated theorems to determine limits of sequences (functions).

For example,

 

. 2
2 11m (3x + 2)
3x + 2 x-—)O . . . .
l = - 11m t of uotient of functions
x—DO 3-1 £30034) 1 ‘1

lim 3x2 + lit-:10 2

£4 1 x .

limo xz _ lim (4) limit of sums of functions
1"" x—tO

These can now be simplified by previous work and Other limit theorems to

obtain

Construction of Unit of Instruction: Treatments A and B

 

The unit of instruction for Treatments A and B was compiled by the
author. Ideas, order of presentation, and problems were obtained from the

following: (1) Limits and Continuity by W.I(. Smith;3 (2) Limits: A Transi—
5

 

tiongCalculus by O.L. Buchanan, Jr. ;4 (3) Sguences by K.E. O'Brien;

(4) Calculus f_o_i
Mme

the instruction

Instruction: '1
\—

The expe:
ninth week of
sider several
They then an
were refined

given as f011<

a p<

Not

31

6
(4) Calculus for College Students by M.H. Protter and C.B. Morrey, jr. ;

 

and (5) The Advanced Calculus 9_r One Variable by D.R. Lick.7 A copy of

 

 

the instructional tmit is located in Appendix 1.
Instruction: Treatments A and B

The experimental unit for Treatments A and B was started during the
ninth week of a fifteen-week semester. It began by having the students con-
sider several examples of sequences, some of which did not have limits.
They then attempted to define when a sequence has a limit. These attempts
were refined by the use of counterexamples. Eventually the definition was

given as follows:

Definition: A sequence {any has a limit L if for each 6 > 0, there is
a positive integer N such that if n) N, then [an - LI ( 6 .

Notation: lim a = L
n-toe H

To better understand the 6 - N relationship, students were asked to find an
N that would "work" for a particular value of 6 and a particular sequence.
The smallest value of N that satisfied the definition for a given 6 was re-
ferred to as the "N associated with the given a " and was denoted N6 .

After finding several N6 '5, the students examined a proof of the statement

lim l
119’ n+1

 

= 0 that utilized the definition given above.

The students next considered more complicated sequences. For example,
2

+
they were asked to evaluate lim flit—n . The difficulty of finding limits
nw

32

for these sequences made them realize the necessity for theorems that would
allow them to evaluate such limits. Six theorems concerning limits of se-
quences were then stated, two of which were proven by using the definition.
(See pages 95 and 96 of Appendix I ).

Limits of functions were studied next. Examples were used to illustrate
notation and the intuitive concept. Students were then asked to formulate a
definition for the limit of a function at a particular value of the independent

variable. Eventually the definition was given by the instructor as follows:

Definition: The function f(x) has limit L as x approaches a if for each
é } OthereisaJ } Osuchthatif0( lx-aIL J , then
[f(x) - L, ( é .

Notation: lim f(x) = L
H a

After several examples and problems were studied, the need for theorems
was again noted by considering limits of more complicated functions. Six
theorems were then stated, three of which were proven. (See pages 105 to
107 of Appendix 1) Students completed study of the experimental unit by
evaluating the limits of functions at indicated values of the independent
variable and attempting proofs of some results. A daily classroom record
for Treatment A is available in Appendix II.

Quizzes: A study by Merrill, M8 has shown that specific review
following incorrect responses makes students' learning increasingly efficient.

This result was implemented in Treatment A by administering six

33
instructional quizzes during the teaching of the unit. The specific review was
then accomplished by working the problems immediately after the class had
completed them and by providing written comments on the quiz papers that
were returned to the students. These quizzes were not used as part of the
mastery evaluation or grading procedure. They consisted of one problem
each and the students were allowed at most ten minutes to work on them. If
students were absent on the day a quiz was given, they were asked to work
it outside of class and hand it in so appropriate comments could be made.

These quizzes are listed in Appendix III.

Evaluation: Treatments A and B

 

The unit of instruction for Treatments A and B required nine instructional
days. An evaluation was given on the tenth day. These were summative
evaluations in Treatment B and formative evaluations in Treatment A. A
seven-item test was used for the fifty-minute testing period. Parallel tests
were randomly assigned to the treatment sections. Copies are available in
Appendices IV and V.

Wilson's Model: Part of this study was to examine the mastery of items

 

at specified levels of a learning model. James W. Wilson's model, given in
Figure 3. 3, was used and the items were tested at the Comprehension, Appli-
cation, and Analysis levels. Figure 3. 4 indicates the classification of items
in the tests.

Grading the tests: The tests in Treatments A and B were graded by the

 

 

m>p~ZOOU

 

m>pUm~h~h~V

m

34

 

 

 

 

 

 

 

 

 

 

BEHAVIOR
g A.l Knowledge of specific facts
o 3 A. 2 Knowledge of terminology
<3 8, A. 3 Ability to carry out algorithms
g
, o
8.1 Knowledge of concepts
8 B. 2 Knowledge of principles, rules,
'53 and generalizations
o E B. 3 Knowledge of mathematical structure
a5 1% B. 4 Ability to transform problem elements
g E from one mode to another
i: 8 B. 5 Ability to follow a line of reasoning
E B. 6 Ability to read and interpret a problem
g g 0.1 Ability to solve routine problems
o g C. 2 Ability to make comparisons
U :3 C. 3 Ability to analyze data
a C. 4 Ability to recognize patterns,
<1 isomorphisms, and symmetries
D.l Ability to solve nonroutine problems
a, D. 2 Ability to discover relationships
:3 51; D. 3 Ability to construct proofs
d 0.4 Ability to criticize proofs
g D. 5 Ability to formulate and validate
generalizations
g m 3.1 Attitude
o 3 g E. 2 Interest
. 3 B E. 3 Motivation
g m 8 g E. 4 Anxiety
5 E E . 5 Self- concept
3‘. e
m a
< o g F.l Extrinsic
n: g F. 2 Intrinsic
8: F. 3 Operational
<:
FIGURE. 3. 3

J. W. Wilson's Taxonomy

 

 

auth

C0118

35

 

 

 

LEVEL TYPE OF ITEM

8.1 Given a sequence (function) that does not have a limit,
indicate what part of the definition it fails to satisfy.

B.l Give an example of a sequence (function) that does not
have a limit and include explanation.

8.1 Given a real number, be able to produce a nonconstant
sequence (function) whose limit is that real number and
include explanation.

C.l When working with sequences, find the associated N
for a particular 6 .

C.1 When working with functions, find the associated 5'
for a particular & .

C.l Determine limits of sequences using the results of
theorems.

C.l Determine limits of functions using the results of
theorems.

D. 3 Prove a given sequence has a particular limit using
the definition for limit of a sequence.

D. 3 Prove a given function has a limit at a particular value

 

of the independent variable using the definition for
limit of a function.

 

FIGURE 3. 4

Classification of Test Items

author and Dave Rajala. One type of item on all four tests was scored before

considering another type of item. Guidelines for assigning partial credit are

given in Figure 3.5.

 

36

 

 

Type of Item Partial Credit Assignment
(see page 28)
1 Right idea, insufficient explanation - 9/11

 

a. 6/12 for correct example but insufficient

 

 

 

 

reasoning
2, 3
b. No credit if functions and sequences are
interchanged
4,5 5/15 for writing‘An - LI 4 e
a. Improper use of the quotient theorem: -3
6, 7
b. Max. of 8/15 credit if all reasons are not
stated
8,9 8/20 for considering [f(x) - L] ( G or

[An-Ll44

 

 

FIGURE 3. 5

Guidelines for Assigning Partial Credit

Diagnostic Procedures for Treatment A

The tests given on the tenth day in Treatment A were used as formative
evaluations. After they were graded, a diagnostic form was completed and
returned to each student. This provided specific reference regarding the
areas in which students needed additional work. A copy of this form is

given in Figure 3.6.

 

37

 

_r

According to the results of your test, you need more work in the following

areas:

 

a)
b)
C)
d)
e)
f)
8)
h)

i)

Sequences that do not have limits

Functions that do not have limits

How to produce a sequence (function) with a given limit
Finding an N ‘ for a given E

Finding a S 6 for a given 6

Evaluating limits of sequences using results of theorems
Evaluating limits of functions using results of theorems
Proving a given sequence has a limit using the definition

Proving a given function has a limit using the definition

 

 

Test results:

 

Problem

1

2

Score

 

t

 

 

 

FIGURE 3. 6

Diagnostic Forms for Treatment A

Instructional I

Alternati‘
ment A after '

was group lec

Students . 1}]
formative ev:

A seconc

 

group sessio

38

Instructional Alternatives: Treatment A

 

Alternative learning strategies were made available to students in Treat-
ment A after the formative evaluation was administered. The first of these
was group lecture sessions covering material missed by the majority of
students. These sessions were held on the first class day following the
formative evaluations.

A second instructional alternative consisted of three two-hour small
group sessions. For these sessions, the chairs in the classroom were

separated into four areas as depicted in Figure 3.7.

 

 

 

 

xx xx
xx xx xx xx
xx D xx xx C xx
xx xx xx xx
xx xx
xx xx
xx xx xx xx
xx A xx xx B xx
xx xx xx xx
xx xx
FIGURE 3. 7

Room Arrangement for Small Group Sessions

Area A contained review material for items of type 1, 2, and 3 (see page 27);
area B material for items of type 4 and 5; area C material for items of type 6

and 7; and area D material for items of type 8 and 9. Worksheets (Appendix

39
VII) were provided for each area and students could work in whichever area
they chose. The author, Dave Rajala, and students who achieved at the
mastery level on the formative evaluation were used as resource persons for
these sessions. A record of attendance at all extra sessions in located in

Appendix VIII.

Summative Evaluation

 

Students in Treatment A received a summative evaluation on the thir-
teenth class day following the introduction of the unit on limits. The scores
on this evaluation were used as part of the course grade. A copy is contained

in Appendix VI .

Delayed-Mastery Students

 

One aspect of this study was to identify "delayed-mastery" students in
Treatment A. These were students who attained the mastery level (80%) on
the second evaluation but who did not attain it on the first one. Figure 3. 8

indicates the number so designated.

Analysis of Data

 

The data used to test the cognitive domain hypotheses (see page 6 ) was
obtained from tests given in the Calculus I course at CMU during the second
semester of the 1973-74 academic year. The first set of data came from the

test given during the fifth week of the semester. Subsequent analyses

eXE
lirr
thi:
ter
Cell

met

4O

 

 

 

 

 

 

 

Test Section Mastery Non-mastery

1 8:00 2 24
1 9:00 3 32
2 8:00 17 9
2 9:00 22 13
Delayed mastery: 8:00 15
9:00 19

Total 34

FIGURE 3. 8

Delayed Mastery Students

examined both the performance on items referring specifically to the study of
limits and the performance on the total test. Copies of parallel forms of
this test are available in Appendix IV. The second set of data was the semes-
ter averages for the course. A one-way analysis of variance using the six
cells indicated in the experimental model of Figure 3.1 was utilized. The
method of multiple comparisons was then used to identify differences.

Both qualitative and quantitative analyses were conducted in an attempt
to determine the effect of the treatments on the affective domain. One quali-
tative analysis consisted of interviewing randomly selected (see Appendix X )
students during the last tWO weeks of the Precalculus course. The interviews

were conducted by the author and Professor Lakey. Figure 3. 9 indicates the

various subgr

for intervi ewi

l K

Gro‘

Nance
Mastery

 

41

various subgroups, their respective sizes, and numbers of students selected

for interviewing.

 

 

 

 

 

 

 

 

 

 

 

 

 

Number
Group Size Selected Number
For Interviewed
Interview
Early 5 2 2
Nance
Mastery Delayed 34 5 5
Nance
Non-mastery 21 5 5
Smith
Mastery 21 5 3
Smith
Non-mastery 40 5 5
Control 152 15 13
FIGURE 3. 9

Number of Students Selected for Interview by Group

The fifteen selected from the control group were chosen by selecting three

from each of the five sections. A copy of the questions asked during the

interviews is available in Appendix XI .

A second qualitative analysis was conducted using the results of a

questionnaire completed by students in Calculus I at the conclusion of study-

ing the section on limits of functions. The responses to this questionnaire

were analyzed using a chi square test statistic with several groupings of the

cells in the experimental model. A copy of the questionnaire is available in

 

5pr

the 1
lus j
A cl.
of 5

Cal

AC:
of ‘

an:

42
Appendix XII.

The quantitative analyses for the affective domain consisted of examining
the percentage of Fall Semester Precalculus students that enrolled in Calcu-
lus I the following semester and the percentage that completed Calculus I.

A chi square test statistic was used with four cells and compared the number
of students that enrolled in Calculus I to the number that did not enroll in
Calculus I. The cells consisted of (1) Treatment A, (2) Treatment B,

(3) control group, (4) Precalculus students from the Fall Semester of the
Academic Year 1972-73, and (5) Precalculus students from the Fall Semester
of the Academic Year 1971-72. A chi square test statistic was also used to

analyze enrollment in Precalculus for the cells of the experimental model.

CHAPTER III -- FOOTNOTES

1

Bloom, Benjamin S. , et. a1. Handbook on Formative and Summative
Evaluation 9_f Student Learning, McGraw-Hill Book Company, New York,
1971, pp. 646—647.

2Block, James H. "Operating Procedures for Mastery Learning",
Mastery Learning: Theory and Practice, Edited by J. H. Block, Holt,
Rinehart and Winston Inc. , Chicago, 1971, p. 70.

3Smith, William K. Limits and Continuity, The Macmillan Co. ,
New York, 1974.

4Buchanan, Lexton O. , Jr. Limits: A Transition to Calculus,
Houghton Mifflin Company, Boston, 1970.

f5O'Brien, Katharine E. Sequences, Houghton Mifflin Company,
Boston, 1966.

 

 

 

 

 

 

6Protter, M.H. and C.B. Morrey, Jr. Calculus for College Students,
Addison-Wesley Pub. Co., Massachusetts, 1973.

7Lick, Don R. The Advanced Calculus o_f One Variable, App1eton-
Century-Crofts, New York, 1971.

 

 

8Merrill, M. David, Keith Barton, and Larry E. Wood, "Specific
Review in Learning a Hierarchial Imaginary Science", Journal 2f Educational
Psychology, 61, 102-109, 1970.

 

43

11

 

This chap
ing to the toll.
and equality c

(4) affective (

CHAPTER IV

FINDINGS

This chapter contains findings of the study. These are reported accord-
ing to the following sections: (1) post hoc test for homogeneity of variance
and equality of mean, (2) mastery learning, (3) enrollment summaries,

(4) affective domain elements, and (S) cognitive domain elements.

Post 'Hoc Test

 

A.C.T. scores for students in the nine sections of Precalculus involved
in this study were checked for homogeneity of variance and equality of means
to see if there were initial differences between sections. The Bartlettl test
for homogeneity of variance fell within the acceptable range at the at '-= . 05
level. Subsequent analysis of variance for equality of means produced an F
ratio that implied the acceptance of the hypothesis of equality of means
between sections. Data used to produce these statistics are contained in

Table 4.1.

Mastery Learning

 

Treatment A consisted of using a mastery learning approach to teach a
section on limits to two of the four treatment sections. An outline is given
on page 29 of Chapter III. The effectiveness of mastery learning was deter-
mined by an analysis of the formative and summative evaluation results for

44

45

 

 

Section Sample Size Mean Standard Deviation
Treatment A 12 25. 5 l. 98
Treatment A 14 25. 5 2 . 88
Treatment B 13 25.1 4. 86
Treatment B 15 27.4 3. 29
Control 12 24.1 3. 09
Control 18 25. 6 3. 68
Control 11 26. 5 4. 70
Control 12 27. 4 2. 97
Control 12 26. 9 2. 50

 

Bartlett test statistif = 13.9 with 8 degrees of freedom

Rejection value: OS(8) = 15. 5

 

 

Analysis of Variance Table

 

 

Source Sum of Squares DF Mean Square F Ratio
Between means 131. 0261 8 16. 3777 1. 3784
Within 1306. 9448 110 11. 8813

Total 1437. 9664 118

 

Rejection value: F OS(8,100) = 2.97

 

 

TABLE 4.1

Analysis of A.C.T. Scores: Post Hoc Test

the four treatment sections of Treatment A and Treatment B. Treatment A
formative evaluation item responses are given in Tables 4. 2 and 4. 3.
Analysis of these responses determined the material retaught during the
group lecture part of the alternative instructional methods. Both sections
in this treatment group produced poor results on items of type two, four,

and seven. Consequently, these items were discussed in class the following

 

46

 

 

 

 

 

 

 

 

 

 

 

 

Item"
Type 1 2 3 4 6 7 9
Score 11 pts 12 pts 12 pts 15 pts 15 pts 15 pts 20 pts
0 12 8 l3 5 l7 3
l
2 2 l
3 4 l
4 4
5 l 1
6 5 6 l
7 1
8 2 l
9 1
10 l 2 l l 1 1
ll 18 2
12 7 9 2 7 l
13 4 l 1
l4 2
15 l 6 l 1
l6 1
17 4
18 1
l9 6
20 3
Average
performance 87% 38% 55% 31% 64% 17% 66%
on item

 

*This refers to the instructional objectives on page 28 of Chapter 111.
TABLE 4. 2
Formative Evaluation Item Results

Treatment A -- Test Form C

 

henl

ii

Score

 

47

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Formative Evaluation Item Results

Treatment A -- Test Form D

Item
Type 1 2 3 4 6 7 9
Score 11 pts 12 pts 12 pts 15 pts 15 pts 15 pts 20 pt—sA
0 22 7 14 ll 25 3
l
2 1 l 2
3 1 1 4
4 l 1
5 l 3 1 l l
6 l 3 8 2
7 l
8 4 l 1
9 3 2 l
10 4 l l
11 22
12 7 17 l 7 2
13 5 6 l
14 6 3
15 l 4 3 3
l6 4
17 l
18 l
19 9
20 8
Average
performance 91% 30% 65% 41% 55% 15% 74%
on item
TABLE 4. 3

day. Table 4.4 contains scores obtained by students on the formative evalua-

tion in Treatment A.

 

 

 

 

 

 

Early Mastery Nonmastery
Scores Scores
87 78 72 66 62 59 55 50 43 38 27 24
83 77 71 64 60 58 54 49 41 37 27 23
82 77 71 64 60 57 54 48 40 32 26 11
81 75 71 64 60 57 51 46 40 31 26 11
80 73 70 63 60 56 50 45 39 28 25 4
Mean = 52.2
Standard Deviation = 20.1361
TABLE 4.4

Formative Evaluation Scores: Treatment A

These scores were recorded for the purpose of identifying early-mastery
students.

The formative evaluation for students in Treatment A was followed by
alternative learning procedures as indicated in Chapter III. A summative
evaluation was administered at the conclusion of these procedures. Item
responses and scores for this evaluation are contained in Tables 4.5 and 4. 6.
Thirty-nine students attained the mastery level, five of whom had been

identified as early mastery by the formative evaluation.

 

49

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Item
Type 1 2 3 5 6 7 8
Score 11 pts 12 pts 12 pts 15 pts 15 pts 15 pts 20 pts
0 4 ll 1 1 4 1
1
2
3 6 l l 2
4 l
5 1 6 l
6 4 3 2
7 1
8 6 3 2 9
9 13 l 3
10 l 3 l 10 10
ll 27 1 2 l
12 49 38 2 8 7 2
13 13 7 2
14 1 l
15 39 37 18 11
16
17
18 6
l9 2
20 26
Average
performance 81% 89% 77% 91% 90% 68% 81%
on item
TABLE 4. 5

Summative Evaluation Item Results

Treatment A

 

50

 

 

 

 

 

 

Mastery Scores Nonmastery Scores
99 98 94 92 88 87 83 81 79 75 70 65
98 97 93 90 88 86 83 81 77 75 70 65
98 96 93 89 88 86 83 80 76 74 68 61
98 95 92 89 88 86 82 80 76 73 65 55
98 95 92 88 88 84 82 76 72 65 44
Mean = 82.5
Standard Deviation = 11.8937

TABLE 4.6

Summative Evaluation Scores: Treatment A

The summative evaluation for students in Treatment B was given the
same day the formative evaluation was given in Treatment A. Item re-
sponses and scores on this evaluation are given in Tables 4. 7, 4. 8 and
4. 9.

Summary: Thirty nine of the sixty students (65%) in Treatment A
attained the mastery level while twenty one of sixty-one students (34%) in
Treatment B attained that level. The average score on the summative
evaluation in Treatment A was 82. 5 compared to an average score of 63. 9

in Treatment B.

 

51

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Summative Evaluation Item Results

Treatment B -- Test Form A

Item
9
Type 1 2 3 4 6 7
Score 11 pts 12 pts 12 pts 15 pts 15 pts 15 pts 20 pts
0 2 5 7 8 9 23 3
l
2
3 3 4
4
5 l l 2
6 4 4 1
7 l
8 3
9 4 2 l l
10 2 l 2 l
11 22 l
12 19 22 3 8 4
13 2
14 l
15 11 ll 5
16
17
18
19
20 20 g
Average
performance 86% 76% 75% 59% 58% 23% 69%
on item
TABLE 4.7

 

52

 

Item

Type

 

Score

11 pts 12 pts 12 pts 15 pts

15 pts

15 pts 20 pts

 

\OCDNO‘Uer-OONt—Io

10
ll
12
13
14
15
l6
17
18
19
20

t—INh-IN
00

O\

14 l

 

5

widths—lid

18 2

26

 

 

Average
performance
on item

77% 78% 65% 58%

68%

30% 93%

 

TABLE 4. 8

Summative Evaluation Item Results

Treatment B -- Test Form B

 

53

 

 

 

 

 

 

Mastery Scores Nonmastery Scores
100 98 85 84 79 70 67 60 54 45 40 34
100 94 85 83 78 70 67 59 48 44 40 28
100 92 85 80 77 70 65 55 47 44 39 27
100 88 85 80 76 69 64 55 47 43 38 22
100 85 85 80 74 68 61 54 46 40 38 19
80
Mean = 65.4
Standard Deviation = 22.1505
TABLE 4.9

Summative Evaluation Scores: Treatment B

I'he analyses in the remainder of this report involved grouping cells of
the experimental model. Groupings used include the following".

(1) Treatment -- Nance Early Mastery (NEM), Nance Delayed Mastery
(NDM), Nance Nonmastery (NNM), Smith Mastery
(SM), Smith Nonmastery (SNM)

(2) Mastery -- NEM, NDM, and SM

(3) Nonmastery -- NNM and SNM

(4) Treatment A -- NEM, NDM, and NNM

(5) Treatment B -- SM and SNM

(6) Delayed mastery -- NDM

(7) Other mastery -- NEM and SM

 

54
Test statistics and other results were then analyzed for the following
comparisons:
(1) Treatment versus Control
(2) Mastery versus Nonmastery versus Control
(3) Treatment A versus Treatment B versus Control
(4) Delayed mastery versus Other mastery versus Nonmastery
versus Control

(5) All six cells of the experimental model

Enrollment

 

Enrollment figures were examined to see if the treatment had an effect
on the number of students taking Calculus 1. Analysis of these figures was
accomplished by considering Precalculus and Calculus I enrollments for
three years. All fall semester Precalculus and winter semester Calculus 1
sections were used for the Academic Years 1971-72 and 1972-73. Enrollment
figures obtained from these years were compared with figures obtained from
the nine Precalculus sections involved in the experiment. Table 4.10
contains this information.

Data from Table 4.10 were analyzed by using the chi square test sta-
tistic and a test for proportion. Six groupings were considered. The chi
square statistic was used to test the null hypothesis of independence between
classifications at thefi = .05 level of significance for five of these six

groupings. The other grouping, Treatment versus Control, was tested with

55

 

 

 

 

 

Number Number

of students of students Percent
Semester Group completing enrolling in enrolled in

Precalculus Calculus I Calculus I
Fall 1971 All sections 438 173 39
Fall 1972 All sections 363 169 47
Fall 1973 NEM 5 5 100
Fall 1973 NDM 34 13 38
Fall 1973 NNM 21 8 38
Fall 1973 SM 21 13 62
Fall 1973 SNM 40 15 38
Fall 1973 Control 152 85 56
Fall 1973 Treatment A 60 26 43
Fall 1973 Treatment B 61 28 46
Fall 1973 Mastery 60 31 52
Fall 1973 Nonmastery 62 23 38
Fall 1973 Other mastery 26 I8 69
Fall 1973 Treatment 121 54 45

TABLE 4.10

Enrollment Summaries

a 2 test statistic for equality of proportion with the alternative hypothesis

of proportion of Treatment students enrolled in Calculus I is not equal to

the proportion of Control students enrolled in Calculus I. A summary of

these tests and conclusions is contained in Table 4.11. Three of the group-

ings produced chi square values that implied the rejection of the hypothesis

of independence between grouping and enrollment. Cell components of the

value of the test statistic that produced the rejections are located in

Appendix XV .

 

 

56

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03: 03.?

833080 eonooflom sunbeam game. 33:80

 

 

 

 

 

Summary:
|
Control group

students from i
on enrollment \
lhis analysis Ill
e111'011edin Calt
mastery and No

The 2 test

 

 

 

between the Tre
was not Sufficiel
two'sided test a
“Quite an * 1e
last was used in
ment effect Won
tation of the ins
however, that e
Therefore: rail
test was used.

if a one‘tailed I

%

study withdrew

these were in t1

57

Summary: The statistics of Appendix XV indicate that the Fall 1973
Control group had significantly more students enroll in Calculus I than did
students from the previous two years. Consequently, the treatment effect
on enrollment was determined by considering groupings within the year 1973.
This analysis indicated that more Other mastery students than expected
enrolled in Calculus I while enrollment was less than expected for Delayed
mastery and Nonmastery students.

The Z test statistic was used to test equality of enrollment proportion
between the Treatment group and the Control group. The Z value of 1. 86
was not sufficient to reject the hypothesis of equality of proportion for a
two-sided test at the A = .05 level. For the produced Z value, it would
require an at level of . 0628 to reject the null hypothesis. The two-tailed
test was used in this case because the original hypothesis was that the treat-
ment effect would increase enrollment. Student comments during the presen-
tation of the instructional unit in the Treatment group made it apparent,
however, that enrollment might decrease as a result of the treatment.
Therefore, rather than reverse the inequality of the hypothesis, a two-tailed
test was used. A statistically significant difference would have been produced
if a one-tailed test had been used with the proportions interchanged from the
original conjecture.

Withdrawals from Calculus 1: Nine of the 139 students involved in this

 

study withdrew from Calculus 1 before the end of the semester. Two of

these were in the Nance nomnastery group, one in the Smith nonmastery

 

group, and s:

ment did not
Affective Dc

determined
the Precalc
the intervi.
appronma.
invesu‘ gate
of alum'ltBtie
toward the
given in T
Figure 4.:

The i
of Each in

<1)E

m

(2) \

58
group, and six in the Control group. These results indicate that the treat-

ment did not have an effect of the number of withdrawals from Calculus I.

Affective Domain

 

Interviews: Part of the treatment effect on the affective domain was
determined by analyzing responses obtained from interviewing students in
the Precalculus sections. A discussion of the selection of these students and
the interview procedures is given in Chapter III. The interviews required
approximately ten minutes apiece to conduct and were primarily used to
investigate two elements in the affective domain: (1) the existence or lack
of anxieties regarding the first semester calculus course, and (2) attitude
toward the first semester calculus course. Results of the interviews are
given in Table 4.12 and a summary of these responses are contained in
Figure 4.1.

The interviewers responded to the following two items at the completion
of each interview:

(1) Briefly discuss your conception of student anxieties or lack of

anxieties regarding Calculus 1.
(2) What is your opinion of the student's attitude toward Calculus I?

Summaries of responses to these items are contained in Figure 4. 2.

59

momeamom BoEouE

 

 

 

 

 

N3. mafia.
e N a a e e o a a o N N tunes 3:5
a N a N a. N N N N o N N Naccscdacz
e a N a N a a a N e c N E382
a N_ a N c a a c S o N 2 successes
N N N a. N N a a. N o N N N accesses...
N N. N N a N N N N e c N < accessed.
N N c a o 0 addresses 8e e N N chase
N N a N N N a N _ o a N 22m
o N a N e e e a N e a N 5
N a e a N a N N a o a o 222
o a N o N a a e a e N N 292
e N o N e o o o N o N N 52
< <4 m r :2 z :z a :2 32 4... 3385
3 Na N32: N N sue

 

 

 

 

 

 

 

L

 

 

 

Item

I

Seven

Eight

Nine

 

60

 

Item

Summary of Responses

 

 

Six

Seven

Ei ght

Nine

Ten, Eleven,
and Twelve

Thirteen

Fourteen

 

One of thirteen Control group students thought he was
doing below average while six of twenty Treatment
group students thought they were doing below average.

Ten of twelve students in Treatment A indicated a lack
of positive attitudes regarding the Precalculus course,
while twelve of twenty-one students in Treatment B
and the Control group had positive attitudes.

Eight of thirteen students in the Control group thought
Precalculus was harder than they expected compared
to seven of twenty in the Treatment group.

Eight of ten students who achieved mastery in either
treatment group exhibited positive attitudes towards
the unit on limits, while only two out of ten students
in the nonmastery group favored the unit.

Four students in the treatment sections changed their
minds about taking Calculus 1. Of these four changes,
one decided not to take calculus because of a curricu-
lum change, one decided to take calculus because of
an advisor's recommendation, and two decided against
calculus because of their unsatisfactory performance
in Precalculus. None of the thirteen interviewed in the
Control group changed their minds regarding calculus.

Thirteen out of seventeen students from the treatment
sections thought Calculus I would be harder than other
math courses they had taken, while only one out of
ten in the Control group had the same reaction.

All Delayed-mastery students interviewed from Treat-
ment A expect to achieve at an above average (A or B)
level in calculus.

 

FIGURE 4.1

Summary of Interview Responses

 

Group
NEM

NDM

MN

SM

sun

Comm

 

61

 

NDM

NNM

SM

SNM

Control

 

w»

, forward to it. They did not Object to the section on limits,

 

These students thought Calculus I would be more difficult
than Precalculus, but they were not overly concerned. They
recognized the sequence as the natural order and had healthy,
positive attitudes. Students interviewed in this category ex-
pected to receive A's or B's in calculus and did not exhibit
anxiety.

These students expressed appreciation for the mastery learn-
ing approach. Positive attitudes and lack Of anxieties con-
cerning Calculus I characterized this group. They expected
calculus to be more difficult than previous mathematics
courses, but they thought they could do at least B level work.

The students interviewed thought Calculus I would be very
difficult. TWO Of the five said they were "afraid" Of calculus.
There were negative attitudes toward Calculus I and the
section on limits in Precalculus.

These students had attitudes and reactions similar to those
Of the Nance early-mastery group. Their attitudes were
positive and they did not exhibit anxiety toward Calculus I.

These students exhibited a negative attitude concerning
Calculus I. They think it will be hard and are not looking

however, because they thought it would help them.
The students interviewed in this group had positive attitudes

and no anxieties about taking Calculus 1. Nine Of the ten
responses indicated that it would not be a very difficult course.

 

 

FIGURE 4. 2-

Analysis of Anxieties and Attitudes Toward Calculus I

Questionnaire: Treatment effect on the affective domain was partially

 

analyzed by examining the results Of a questionnaire completed by Calculus I

students during the Winter Semester 1974 (Appendix XII). All but eight Of the

62
one-hundred thirty-nine students involved responded tO the questionnaire.
Table 4.13 contains tables formed by considering the responses Of each cell
in the experimental model to the five questions. The cell sizes Of these
tables were insufficient to produce statistically valid results when six groups
were crossed with five responses per question. Consequently, responses
and cells Of the experimental model were grouped to produce larger cell
sizes. The most refined grouping which produced this was a three-level
response scale rather than a five-level scale. This was accomplished by
combining responses "a" and "b" and responses "d" and "e". The groupings
of the cells Of the experimental model were those indicated On page 53.
Table 4.14 contains results produced by these groupings. A chi square test
was used to analyze the results and significance was tested at the at = . 05
level. The chi square values on twelve Of the twenty questions reported in
Table 4.14 indicate a rejection Of the hypothesis Of independence between
groups and responses. An analysis Of the values contributing to these

rejection figures is reported in Appendix XIV. A summary Of these results

is contained in Figure 4. 3.

ngnitive Domain

 

Test on limits: A test covering limits was administered to the Calculus I

 

students during the fifth week Of the Winter Semester 1974. Five forms
(A, B, C, D, E) of the test were used. Items 2, 3, and 4 were parallel on all

forms while Item 1 was parallel on forms A, B, and C. The test forms

 

 

1

00000

0.1000

10
0

10

00000

11

24.477

629 33

0

38268

30 37

Group
NEM
NDM
SNM
Control
Group
NEM
NDM
NNM
SM
SNM
Control

SM

 

 

 

 

 

 

 

 

63

 

 

 

4
4

ll 34 25
13

a
0
3
5
a
10 51

Group
NEM
NDM
SM
SNM
Control
Group
NEM
NDM
SM
SNM
Control
Group
NEM
NDM

 

 

 

 

 

04.

31

05

NNM
SM

 

 

 

 

 

 

TABLE 4.13

6
Questionnaire Responses

5

245 21

Control

SNM

 

64

 

 

 

 

 

Degrees Chi Square Chi Square
Grouping of Critical Value
Freedom Value by Question

TreatmentA Q1 -- 6.02

Q 2 -- 15.44 X
TreatmentB 4 9.4 Q3 -- 4.30

Q4 -- 7.44
Control Q5 -- 15.92 X

Q1 -- 5.78
Treatment 2 5.99 Q2 -- 13.87 X

Q3 -- 3.25
Control Q4 -- 6.97 X

Q5 -- 10.28 X
Mastery Q1 -- 10.57 X

Q 2 -- 16.61“ X
Nonmastery 4 9.4 Q3 -- 4.43

Q4 -- 8.81*
Control Q5 -- 11.58 X
Delayed mastery

Ql -- 17.57* X
Other mastery 6 12.59 Q2 -- 21.72“ X

Q3 -- 12.62* X
Nonmastery Q 4 -- 8.82*

Q5 -- 18.53 X
Control

 

 

*Fewer than 80% of the cells had five or more responses.
X -- Reject the null hypothesis of independence.

 

TABLE 4.14

Questionnaire Data

 

65

3353 0505500 3 555m

 

 

 

 

 

 

m .0 0330a
.350Bm 3.555502
05 03 55 555553 53 335 3 >005 05 0503 3503a F5532 m 35:00
.3525 3.555502 05 03 55 535 53030 0503 550:5 F5552 n 3555502
.5555 55 505: 53 5303003 03.505 35030 3555502 _ F5552
.355 53003 5:3 55555.3 000505 3550553 5035 55555 0.5. m
.3550 503 505 05
55 500: 0.58 .5 535m 030:5 350:5 5555.5. .533 3050955 v
.Fnowuumo
05m 05 5 0.55 3550553 05 3503a 3550 05:3 F5550 3550
:505: 05 5 55055 .533 3550353 05 35005 505 5555.3. N 5555.5.
.Faowouao :55
5553 53.. 05 5 55055 533 05 350:5 m 5555.3. 333
F5550 :555553: 05 5 55055 533 05 35005 < 5555.5. m 3.350
_ 530095 55 m “5553.
505: 53 m 53030 3505 350Bm 4 5555.5. 533 355058 N < “555.5.
.5355 .550 55 .555 35005 F5555
-52 0» 5:53 0.53 503 m5 :0 55055 .3595 F5555 00530 n
.3525 >553: .550 05
05 55 530050 0.55 335: .3 >035 05 053 35005 F5555 00530 m
630005 55 .535 53030 053 35005 F5555 550 3.23 35000
0305 5 530030 5 305 an 00 53030 053 3500.3 F5055 00530 N 3555502
.0355 55 .505: 5303005 F5555 550
05.3 350:5 F5555 00530 .3595 5558 550 3 005950 a 5555 0053a
3353 53500 53:80

 

 

 

 

66
differed only in that A, B, and C had a proof about limit of a function, whereas
forms D and E had a proof about derivative of a function. Other items on the
tests did not involve limits (see Appendix IX). Item and test scores are
given in Appendix XVI.

Scores on Item I (twenty points possible) of forms A, B, and C were such
that analysis of variance was not possible on the group means because the
Bartlett test statistic for this item had a value of 28.1 while the critical value
at the“ = .05 level of significance was 9.5. Responses were subsequently
categorized into two levels: (1) those with scores less than eighteen, and
(2) those with scores of eighteen, nineteen, and twenty. The number of
responses in these categories by groupings of cells of the experimental

model are given in Table 4.15.

 

 

 

 

 

 

Grouping l Score Mean
I 0-17 1 18-20 J
NEM 0 I 20. 0
NDM l 8 18. 8
NNM . 2 3 15. 0
SM 0 6 l9. 7
SNM 0 6 l9. 7
Control 5 38 18. 9
Treatment A 3 12 17.6
Treatment B 0 12 19. 7
Mastery I 15 19. 2
Nonmastery 2 9 17.6
Other mastery 0 7 l9. 7
Treatment 3 24 18. 5
TABLE 4.15

Calculus I Test Item I -- Forms A, B, and C

67
Scores on Item 2 for all five forms of the calculus test also produced
group means that did not permit analysis of variance. The Bartlett test
statistic for this item was 14. 2 while the critical value was 11.07. As before,
the responses were categorized and reported by various groupings. This

information is contained in Table 4.16.

 

 

Grouping Score Mean
-17 I -

 

 

NEM 1 4 l9. 0
NDM 3 10 18. 2
NNM 5 2 15. 6
SM 2 11 19. 3
SNM 4 10 18. 0
Control 28 52 17. 2
Treatment A 9 l6 l7. 6
Treatment B 6 21 18.6
Mastery 6 26 18. 8
Nonmastery 9 12 17. 2
Other mastery 3 15 19. 2
Treatment 15 37 18. 5

A

 

TABLE 4.16

Calculus I Test Item 2 -- Forms A, B,C,D,E

Each of the five test forms had three parallel items involving limits.
Question two involved finding the limit of a quotient of functions with reasons
given for each step. Questions three and four required students to produce
examples of functions which met certain conditions and to include reasoning.
The total possible points for these three items was thirty three. Analysis of
these items was achieved by dividing the total scores into two categories and

using a chi square test statistic. The categories were (1) scores from zero

 

68
through twenty nine, and (2) scores from thirty through thirty three.
Table 4.17 contains the responses by category and Table 4.18 contains

results obtained from analyzing these scores.

 

 

 

 

 

 

Grouping L Score I Mean
| 0-29 T 30-33 1
NEM 2 3 30.0
NDM 5 8 28. 5
NNM 6 1 24.9
SM 3 10 31.0
SNM 6 8 28.6
Control 48 32 27. 2
Treatment A 13 12 27. 8
Treatment B 9 18 29. 8
Mastery 10 21 29. 8
Nonmastery 12 9 27. 4
Other mastery 5 13 30.7
Treatment 22 30 28. 8
TABLE 4.17

Responses by Cells to Limit Items 2, 3, and 4 of the Calculus I Test

 

Degrees of Chi square Chi square

 

 

 

 

 

 

Grouping freedom test statistic critical value
OR = .05

NEM, NDM, NNM, SM, SNM, C 5 11.30 11.07
Treatment-Control 1 3. 96 3. 84
Mastery- Nonmastery-Control 2 0 5. 99
Delayed mastery-Other mastery- 3 7 3 7 8

Nonmastery- Control ° '
Treatment A-Treatment B- 2 7. 2 5. 99

Control

TABLE 4.18

Analysis of Student Performance on Limit Items in Calculus I Test

 

 

69
Chi square contingency tables for grouping of cells of the experimental
model are contained in Appendix XVII.

The data of Table 4.18 indicate that the Treatment group performed
significantly better than the control group on the limit items. Further exa-
mination of this data and the material in Appendix XVII indicate that most of
the achievement difference was due to students in Treatment B. Treatment A
students performed as expected while fewer Treatment B students than
expected scored low (below thirty) and more than expected scored high
(thirty and above).

The last analysis made from data obtained from the Calculus I test was
an examination of total scores. Means for groupings of cells of the experi-

mental model are given in Table 4.19.

 

 

 

 

Grouping Size l Mean 1 Standard
Deviation
NEM 5 91. 4 5. 9
NDM 13 83. 4 14.1
NNM 7 67. 4 13.1
SM 13 91. 8 6. 3
SNM 14 81. 4 l4. 2
Control 80 83. 0 14. 5
Treatment A 25 80. 5 14. 3
Treatment B 27 86. 4 ll. 9
Mastery 31 88. 2 10. 6
Nonmastery 21 76. 7 14. 7
Other mastery 18 91.7 5.9
Treatment 52 83. 6 l3. 7
TABLE 4.19

Group Means and Standard Deviations for Calculus I Test Scores

 

 

70
The Bartlett test statistic for homogeneity of variance of this data is 12. 7
while 1205(5) = 11. 07. However, the F ratio for an analysis of variance
of these means is 3. 33 and €056,120) = 2. 29. Consequently, one can be
reasonably sure that the analysis of variance result is statistically signifi-
cant. The analysis of variance table in Table 4. 20 indicates a rejection of

the hypothesis of equality of means.

 

 

 

Source Sum of Degrees of Mean F
Squares Freedom Square Ratio
Between means 3081. 8 S 616. 4 3. 3
Within 23324. 4 126 185.1
Total 26406. 2 131
TABLE 4. 20

One-Way Analysis of Variance for Calculus 1 Test Scores

Due to unequal sample sizes, a Scheffe2 multiple comparison test was used
to identify differences between means. Table 4. 21 contains the results of
these comparisons. The null hypothesis of equality of means was rejected
only for the comparisons of Nonmastery versus Mastery and Nonmastery
versus Other mastery. All other comparisons, in particular Treatment
versus Control, do n_ot reject the hypothesis of equality of means.

Semester average in Calculus I: The second analysis of the treatment
effect on the c0gnitive domain was conducted by examining students' semester
averages in Calculus I (Appendix XVI). The average for each student was

obtained by equally weighted percentage scores from the following three

 

71

A

 

 

Comparison Confidence Interval Conclusion"
Treatment A - Treatment B - 19. 2 —_ 7. 5 Do not reject
Treatment A - Control - 13. 5 __ 8. 9 Do not reject
Treatment B - Control - 6.7 — 13. 8 Do not reject
Mastery - Nonmastery .4 —— 28.5 Reject
Mastery - Control - 4.7 —- 16.3 Do not reject
Nonmastery - Control —20.5 —- 3. 2 Do not reject
Delayed mastery - Other mastery -25. 8 —— 9. 4 Do not reject
Delayed mastery - Nonmastery - 7.6 —- 25.6 Do not reject
Delayed mastery - Control - 13.4 —— 14.1 Do not reject
Other mastery - Nonmastery 1.0 — 33.3 Reject
Other mastery - Control - 4.6 -— 21.7 Do not reject
Treatment - Control - 8.6 —— 8.6 Do not reject

 

 

*Rejection of the null hypothesis of equality of means occurs when
the confidence interval does not include zero.

 

TABLE 4. 21

Scheffe Multiple Comparison Test for Equality of Means
Calculus I Test

sources: (1) ten quizzes, (2) three one-hour examinations, and (3) a final
examination. Averages for groupings of cells of the experimental model
are given in Table 4. 22. Means were analyzed by a one-way analysis of
variance. The Bartlett test statistic was 11.15 while the rejection value was
X2055) = 11.07. However, the F ratio for analysis of variance was 5. 57
compared to E05(5,120) = 2. 29. As before, this allows us to reject the

null hypothesis of equality of means between cells. The analysis of variance
table is given in Table 4. 23. Results of the Scheffe multiple comparison
test are contained in Table 4. 24. The only comparisons that produced a

statistically significant difference of means were those that involved the

 

72

 

 

 

 

Grouping I Size l Mean I Standard
Deviation
NEM 5 85. 0 ll. 0
NDM 13 71. 9 l7. 3
NNM 6 58. 0 17. 4
SM 13 80. 0 12. 4
SNM 14 74.1 9. 6
Control 79 79. 9 9. 9
Treatment A 24 71. 2 l7. 8
Treatment B 27 76.9 11.0
Mastery 31 77 . 4 l4. 7
Nonmastery 20 69. 3 13. 8
Other mastery 18 81. 4 ll. 6
Treatment 51 74. 0 l4. 9
TABLE 4.22

Semester Averages by Groupings of Cells of the Experimental Model

 

 

 

 

Sum of Degrees of Mean F
Source Squares Freedom Square Ratio
Between means 3652 5 730. 4 5. 57
Within 16268 124 131. 2
Total 19920 129
TABLE 4. 23

Analysis of Variance Table for Semester Averages by Cell

 

 

 

73

 

 

Comparison Confidence Interval Conclusion“
Treatment A - Treatment B -16. 8 6. 0 Do not reject
Treatment A - Control - 17. 9 1. 4 Do not reject
Treatment B - Control - 11. 5 5. 8 Do not reject
Mastery - Nonmastery .7 25.1 Reject
Mastery - Control - 9. 8 7.9 Do not reject
Nonmastery - Control - 24. 2 3. 4 Reject
Delayed mastery - Other mastery —25. 4 4. 2 Do not reject
Delayed mastery - Nonmastery - 8. 5 20. 2 Do not reject
Delayed mastery - Control - 19. 6 3. 6 Do not reject
Other mastery - Nonmastery 2. 5 30. 3 Reject
Other mastery - Control - 8. 5 l3. 7 Do not reject
Treatment - Control - 13. 5 l. 3 Do not reject

 

*Rejection of the null hypothesis of equality of means occurs when
the confidence interval does not include zero.

 

TABLE 4. 24

Scheffe Multiple Comparison Test for Equality of Means
Total Percentage Grade for Calculus I

Nonmastery students.

Summary: Two implications arise from the data of Table 4. 22 and
4. 24. The first is that the average score for Delayed mastery students (71. 9)
is much closer to that for Nonmastery students (69. 3) than it is to that for
Other mastery students (81. 4). Although the difference between Delayed
mastery and Other mastery is not statistically significant, it is an indication
that Delayed mastery students tend to perform more like Nonmastery students
than Other mastery.

The second implication is that the Treatment students scored 5. 9

percentage points lower than the Control students. This is not statistically

 

74
significant, but it suggests further investigation into the feasibility of taking

time from the Precalculus course to introduce special topics.

FOOTNOTES -- CHAPTER IV

1
Dixon, Wilfrid]. and Frank J. Massey, Jr. Introduction to Statis—
tical Analysis. McGraw Hill, New York, 1969, p. 308.

 

 

71mm, p. 167.

75

CHAPTER V

CONCLUSIONS AND IMPLICATIONS

Included in this chapter are (l) a summary of the procedures of the
study, (2) findings of the study, (3) conclusions based upon analysis of the
data, (4) a discussion of selected components of the study, and (5) implica-

tions for further research.

SummarLof Procedures

 

This study investigated the effects of teaching a unit on limits to pre-
' calculus students at Central Michigan University. Nine sections of approxi-
mately thirty students each constituted the population. Two sections received
Treatment A, two received Treatment B, and five were used for a Control
group. Treatment A consisted of using a mastery learning strategy to teach
the unit; Treatment B consisted of presenting the same unit in a regular“
classroom situation; and sections in the Control group received no instruc-
tion regarding limits.

The mastery learning strategy used in Treatment A consisted of (l) for-
mulating instructional objectives, (2) presenting the unit of instruction,
(3) administering a formative evaluation, (4) providing each student with

cfiagnostic results, (5) using alternative instructional procedures, and

 

*"Regular" means no special strategy or technique was used. The instructor
taught the way he normally would present any other material. This consisted
mainly of lecturing and answering questions.

76

77
(6) administering a summative evaluation. Students who achieved at or above
the mastery level (80%) on the summative evaluation but not on the formative
evaluation were identified as "delayed-mastery" while those who attained
that level on the formative evaluation were identified as "early-mastery .

Treatment effects on the affective domain elements were determined in
two ways. One was by interviewing randomly selected students from the
Precalculus sections. The other was by analyzing responses to a question-
naire completed by Treatment and Control students enrolled in Calculus I.

In addition to these analyses, enrollment figures in Precalculus and Calculus I
for the years 1971, 1972, and 1973 were examined to see if the Treatment
affected the percentage of Precalculus students enrolling in Calculus I.

The effect of the treatment on achievement in Calculus I was analyzed at
two stages of the course. Data from the test covering limits was used first.
Student performance was measured on (1) the questions involving limits and
(2) the total test scores. The semester percentage for each student provided
the second set of data. For the purpose of these analyses, students in Treat-
ment A were identified as Nance early mastery, Nance delayed mastery, or
Nance nonmastery; and students in Treatment B were identified as Smith

mastery or Smith nonmastery.
Results

The results of this study will be reported according to the organization

implied in Chapters III and IV. First the mastery learning results will be

78
given and then the treatment effects regarding Calculus I will be divided
into enrollment, affective domain, and cognitive domain results.

Mastery Learning: The null hypothesis of H0: the proportion of students

 

achieving mastery in Treatment A (P A) equals the proportion of students
achieving mastery in Treatment B (P8) was tested using a Z test statistic
with a significance level of o( = .05. The alternative hypothesis was

Ha: P A) PB. Thirty nine of sixty Treatment A students achieved mastery
compared to twenty one of sixty one in Treatment B. The Z test statistic
for these proportions is Z = 3. 33 where the null hypothesis is rejected for
values of Z > 1.645 whendk = . 05. Therefore, the alternative hypothesis
of P A) PB was accepted.

Enrollment: (A) The null hypothesis of H : the proportion of Treatment

 

0
group students (PT) enrolling in Calculus I equals the proportion of Control

group students (PC) enrolling in Calculus I was tested using a Z test statistic
with a significance level of UK= . 05. The alternative hypothesis was
H a: PT ¥ PC

Sixty students in Treatment A, sixty one in Treatment B, and 152 in the
Control group completed Precalculus. Of these students, twenty six, twenty
eight, and eighty five respectively enrolled in Calculus I. This produced
percentage enrollment figures of fifty-six percent for the Control group and
forty-five percent for the Treatment group. The Z value was 1. 86 where the

rejection value at the ds = . 05 level for a two-sided test was I. 96. Therefore,

the null hypothesis of equality of proportions between Treatment and Control

79
groups was accepted. (Comment: The hypothesis concerning enrollments
was that enrollment percentage would be increased as a result of the treat-
ment. The computed enrollments, however, indicated that eleven percent
EVE. students enrolled in Calculus I from the Treatment group. Conse-
quently, a two-sided test was used with the alternative hypothesis Ha: PT 7‘ P C
rather than a one-sided test with the alternative hypothesis Ha: PT) PC .
If the hypothesis had been that the treatment would decrease enrollment, then
a one-sided test with alternative hypothesis Ha: PT< PC would have pro-
duced a statistically significant difference in enrollment.)

(B) The percentage of students remaining in Calculus I was also not
affected by the treatment. Fifty one of fifty four (94%) Treatment group
students completed Calculus I compared to seventy nine of eighty five (93%)
from the Control group. The Z test statistic for equality of proportions
between these groups is Z = -. 25 where the rejection value for 0k = . 05
is Z ) 1.645.

Affective Domain: Treatment effect on the affective domain was analyzed

 

by considering student responses to interviews and a questionnaire. The
hypothesis as stated in Chapter I is: a unit on limits in Precalculus will
improve attitudes (1) prior to taking Calculus I and (2) during Calculus I. The
findings with respect to this hypothesis are reported below.

(A) Interviews conducted at the end of the Precalculus course indicated
the following results:

1. Treatment group students thought they were doing poorly in

80
Precalculus.

2. Treatment group students thought Calculus I would be very diffi-
cult as opposed to Control group students who thought it would
not be harder than other courses.

3. The treatment effect caused negative attitudes and anxieties toward
Calculus I for the Nonmastery students. Otherwise, Mastery and
Control students had positive attitudes and no apparent anxieties
toward calculus.

(B) Questionnaire responses obtained during the third week of the
Calculus 1 course are given in Chapter IV. These responses were analyzed
using a chi square test statistic for each question on the questionnaire.
Significance was tested at the ok= .05 level. The following two results of
this analysis deserve special mention.

1. Significantly more than expected Control group students thought
Calculus I was "harder than expected" compared to significantly
fewer than expected Treatment group students who responded to
the same item.

2. Treatment group students experienced significantly less frustra-
tion when studying limits than did the Control students.

gignitive Domain: Treatment effect on achievement was examined by

 

considering the Calculus I test covering limits and the final percentage score
for Calculus 1. Analysis of scores on the calculus test covering limits was

accomplished by using a null hypothesis of H : there is no significant

0

81
difference in achievement for cells of the experimental model. This was
tested by using a one-way analysis of variance. When the items covering
limits were tested, the Bartlett test statistic of 20.7 was too high to permit
use of analysis of variance. Consequently, scores were categorized and a
chi square test statistic was used. There was a significant difference at the
$= . 05 level between the achievement of Treatment and Control students.
Subsequent analysis of the chi square partial values indicated that the
Treatment group students performed better than expected.

Analysis of the total test scores also used the null hypothesis of H0: there
is no significant difference in achievement for cells of the experimental model.
In this case, the Bartlett test statistic was within the acceptable range and a
one-way analysis of variance produced an F test statistic of F = 3. 3 where
the rejection value was F05(5,120) = 2. 29. Consequently, the null hypothesis
of equality of means was rejected. Subsequent multiple comparisons were
made using a Scheffe test and indicated that the only statistically significant
differences were between (1) the Nonmastery and Mastery groups and (2) the
Nonmastery and Other mastery groups.

Semester averages in Calculus I were also analyzed based on the null
hypothesis H0: there is no significant difference in achievement for cells of
the experimental model. A one-way analysis of variance“ produced an F test

statistic of F = 5. 57 where the rejection value was F05(5,120) = 2. 29.

 

*The Bartlett test statistic for homogeneity of variance was 11.15 while the
rejection value was 2 (5) = 11.07. However, the F ratio was large

0
enough to reject the null gypothesis and perform multiple comparisons.

82
Multiple comparisons then produced statistically significant differences be-
tween the following. (1) Mastery and Nonmastery, ( 2) Nomnastery and Control,
and (3) Nonmastery and Other mastery. All other comparisons produced no
significant difference.
Delayed Mastery: A unique aspect of this study was the identification of

delayed mastery students in Treatment A. An analysis of their subsequent
behavior was obtained by constructing a profile of this group compared to

other cells of the experimental model. This information is contained in

 

 

Figure 5.1.
Item Group Most Closely
Resembled

Interview responses Other mastery
Questionnaire responses Smith nonmastery
Enrollment in Calculus I Nonmastery
First calculus test:

(1) limit items (1) Smith nonmastery

(2) total score (2) Smith nonmastery
Semester percentage in

Calculus 1 Smith nomnastery

 

 

 

FIGURE 5.1

Profile of Delayed Mastery Students

The purpose of this identification and subsequent profile was to examine
what carry-over effect the alternative instructional procedures and additional
time spent studying limits would have on these students. Figure 5.1 indicates
that the Delayed mastery group responses to all evaluation items used during

the Calculus I course were similar to those of Nonmastery students in either

83

Treatment A or Treatment B. Consequently, the alternative instruction and

extra time seem to have had little carry-over effect on the Delayed mastery

students .

Conclusions

 

Reject the hypothesis that a mastery learning strategy, employed
on a unit of limits of sequences and limits of functions, will result
in no significant difference between the proportion of students
achieving the mastery level in a mastery learning situation and the
proportion of students achieving the mastery level in an expository
classroom situation.

Do not reject the hypothesis that a unit on limits in Precalculus
will not affect attitudes

(a) prior to taking Calculus I

(h) during Calculus I.

Do not reject the hypothesis that there will be no significant differ-
ence between the percentage of students enrolling and remaining

in Calculus I after having had a unit on limits and the percentage
of students from the Control group enrolling and remaining in
Calculus I.

Do not reject the hypothesis that there will be no significant differ-
ence in achievement in Calculus 1 between students in the treatment

groups that achieve mastery and those that did not achieve mastery.

84

5. Do not reject the hypothesis that there will be no significant differ-
ence in achievement in Calculus 1 between delayed mastery students
and other mastery students in the treatment groups.

6. Reject the hypothesis that an introduction to limits prior to calculus
will result in no significant difference in achievement on the work
with limits in calculus.

7. Do not reject the hypothesis that an introduction to limits prior to
calculus will result in no significant difference in achievement in

the first semester of calculus.

Implications for Education

 

Instructional Unit: Approximately one-half of the time spent studying

 

limits was used for the study of limits of sequences. Since the Calculus I
course at Central Michigan University does not cover this topic, the time
devoted to this may have been more effectively used in additional study of
limits of functions. A second point is that notation was a problem for some
of the students. The mathematical phrase n ) N tended to be confusing.
Perhaps a change to n ) M would correct this in future work.

Mastery Learning: The mastery learning strategy of Treatment A was

 

effective as indicated in the conclusions on page 83. However, several
problems occurred which should be considered for future work. First, the
time spent for alternative instructional techniques may have produced the

same results in an expository situation. Related to the alternative instruction

85
problem is the question of what to do with the early mastery students between
the formative and summative evaluations. In this study, they were used as
resource people during the small- group alternative instruction sessions.
They might not enjoy doing this if an entire course were to be taught via a
mastery strategy.

A second problem is that the difference between early mastery and
delayed mastery students may not be an accurate indication of their abilities.
Students in this study were informed that the formative evaluation would not
count on their grade and that there would be a summative evaluation several
days later. These reasons may have resulted in students being identified
as delayed mastery when they could have achieved at the mastery level if
only one evaluation were given. Closely related to this problem is that of
' how many evaluations should be given. Would those students who scored
between sixty and eighty on one summative evaluation socre above eighty on
a subsequent evaluation? How many examinations are required to maximize
class progress and still account for individual differences?

A third area of concern is that both Treatment A and Treatment B
resulted in less time being spent on the topics ordinarily studied in Pre-
calculus. Since the semester average in Calculus I of Treatment students
was 5.9 percentage points lower than the Control group, this may mean that
prior knowledge of limits has less effect on success in Calculus I than the
knowledge of other topics in Precalculus.

Sources of Variation: Sources of variation affecting the results of this

 

86

study include the following:

1.

Treatment A students spent three more days studying limits than
did Treatment B students.

Both Treatment A and Treatment B students spent less time studying
the other Precalculus topics than did the Control group students.
Five different instructors taught the Control sections.

Treatment A and Treatment B sections were taught by different
instructors.

Treatment A sections met at 8:00 a.m. and 9:00 a.m. while
Treatment B sections met at noon and 1:00 p.m.

Calculus I was taught at two different times: 10:00 a.m. and 2:00
p. m.

Students in Calculus I had nine different recitation instructors.
Students were not randomly selected for this study. However, a
post hoc test of A. C.T. scores indicated that the sections were

evenly matched for homogeneity of variance and equality of means .'

Other comments: Statistical analyses of affective domain elements are

 

difficult to obtain. In this study, it was not possible to show that the Treat-

ment effect changed attitudes toward Calculus I. It was possible, however,

to show that a statistically significant percentage of students from the Treat-

ment group felt they experienced less frustration when studying limits in

Calculus I than did the students from the Control group.

The next comment concerns the Calculus I test that produced data for

87
part of the analysis in the cognitive domain. The limit items on these tests
had such high averages that analysis of variance was not possible. If these
items had been better able to distinguish between ability levels, Treatment
effects could have been more thoroughly examined.

The last comment has social and economic implications. Although the
Calculus I enrollments for Treatment and Control students were not signifi-
cantly different ( d~ = . 05), it is true that enrollment of Treatment students
was 11% less than that of Control students. If some of the Treatment students
did not take Calculus I because the unit on limits made them realize that
Calculus was not what they expected, they might have enrolled in Calculus I
and then dropped the course had it not been for the unit on limits. These
students have thus saved some of their tuition and some time that would have
been spent in Calculus I. On the other hand, enrollment figures are of pri-
mary importance in these days of careful auditing of student credit hours.
Conceivably a mathematics department could suffer economic consequences
if the Treatment of this study were extended and the enrollment percentages
remained the same. An examination of this problem is suggested as a topic

for further research.

Suggestions for Further Research

 

Following are listed related questions that deserve further consideration:
1. Do the carry-over effects of a mastery learning strategy differ from

the carry-over effects of an expository strategy? This study has

88
shown there was no significant difference in achievement in Calculus I
between Delayed mastery and Other mastery students. However, the
profile of Figure 5.1 indicates that the Delayed mastery students
most closely resemble Nonmastery students when one considers
the analyses of this study. Perhaps subsequent research could more
carefully define and control parameters relating to Delayed mastery
students.
Would a greater variety of alternative instructional techniques pro-
duce even better results for mastery learning? (Audio-visual tapes,
more effective use of the reading list, tutoring, etc.)
Would mastery learning be effective if all instructional objectives
were tested at the Analysis level of Wilson's taxonomy? In this
study, mastery of most items was tested at lower levels of difficulty.
Results of the mastery strategy indicate it might be as effective if
all items were tested at the highest level of the taxonomy.
What effect would a different Treatment (no sequences, no ( e , J )
notation, longer intuitive approach, etc.) concerning limits have on
achievement in calculus '?
How does a student's prior conception of the difficulty of calculus
affect his achievement in calculus ?
Does a unit on limits in Precalculus cause a decrease in enrollment
in calculus ? One result of this study was that Treatment group

students had an eleven percent lower enrollment rate in Calculus I

89
than did the Control group. This was not statistically significant
at the ck = . 05 level, but it was substantial enough to warrant a
replication of the study with particular emphasis paid to enrollment.
7. Does a unit on limits in Precalculus result in more; achievement
in calculus ? The Treatment group students of this study had an
achievement mean 5.9 percent lower than the Control group. This
was not statistically significant at the 0k = . 05 level, but it is a
justification for replication of the study.
The comments of this section suggest a replication of the study. Atten-
tion should be directed toward detecing a decrease in enrollment and achieve-

ment as a result of Treatment.

APPENDICES

90

APPENDIX 1: Instructional Unit

 

LIMITS OF SEQUENCES

Examine several sequences and plot them on a number line.
Examples: 1, 1/2, 1/3, l/4,...
2, 4, 8,16,...

l, - U3, U9, -l/27,...
Others

Find the 7th, 8th, or 9th terms of each of the above sequences.

Find the nth term of each of the above sequences.

NOTATION: a,a,a,...,a,... 0r a
l 2 3 n n n=1

1
Example: 82:11:] - 1/2, 1/4, 1/6,..., l/2n,...

Type of sequences:

DEFINITION: An alternating sequence is a sequence whose terms are alter-
nately positive and negative.

 

-1
Example: 1, -l/3, 1/9, -1/27,... = '('fi=)r—

=1

DEFINITION: A constant sequence is a sequence {an} n=l such that an = K
for some constant K.

Example: 3, 3, 3.... = (3}nzl

91
Use the above examples to examine "getting close to" (plot on the number
line).
1, 1/2, 1/3, 1/4,... —) 0
2, 4, 8,16,... ——-) ?
1, -1/3, 1/9, -l/27,... —-) ?
2, -1, 3/2, -1/2, 4/3, -1/3, 5/4, -l/4,... ——9 ?

3, 3, 3,...—.) ?

ASSIGNMENT #1

 

1. On the number line, plot several terms of the following sequences with
given nth terms:

-L

a) an—2n
2

b) a =1

n+1

n
1'1
C) {('2)}n=1
d) {2“}114

e) {42%) n=1

1
f) {n2+2 n=l

g) 1+

 

(-l) n n=1

2. What number, if any, do the above sequences "get close to"?

92
3. Find sequences (nonconstant) that " get close to" the following:
a) l
b) -2

c) "a", where a is any number

If the terms of a sequence "get close to" some number L, we say that the

limit o_f the sequence (an) is L. Since "getting close to" is not precise

 

enough to convey the concept of limit of a sequence, we use the following

definition:

DEFINITION: A sequence {a\ has limit L if for each C > 0, there is a
positive integer N such that if n ) N, then‘a n-Ll ( E.

NOTATION: lim a = L
my! 11

To illustrate this definition, consider the following examples.

Example 1:

1
Let{an}= {F1 andL= o.

If G = .01, then an N is 99 (actually, any integer grea er than
99 would also serve as N). Then for n > 99, lEl-T 0./( 01.

 

Ifé = .00001, then N = 99,999.

. 1 1 _
Note here that 1f 11 > 99, 999, then n+1) 100, 000 and n+1 100’ 000 -
But-“:1- ='—-O), hence‘an-L'(é forn)Nandthe

definition is satisfied.
Since the choice of N depends on the choice of Q , we usually write

NE rather than N as above.

6.

93

Example 2:

{a} _ where a =
n n-l n

For C = .10, find N.10 such that for n) N.10 I an - 0' < E .

-1)“
3n

1 L
Try N.10 = 3, then an = 511 and form) 3, 311 < .10.

Do the same for 6 = .001.

Using the definition of limit of a sequence, it is possible to prove that some
sequences have limits.

1
Exam 1e: Prove lim -— = 0-
p n4, n+1

Pr00f: Let C ) 0 be given. We then need to find N‘ such that

 

1
'n+l - 0’ < €whenn)N

But 34:1 ((#511—(6931-(m14enr‘l- -1.

 

 

' 1 1
Hence for a given 6) 0, let N‘) z- - 1 then for 11) N‘? 2- - 1,

l
we have n+l)'él-¢P EII<E % Ian - 01(5.

‘ lim a = 0
o o 11.,“ I)
To further illustrate this proof, let 6 = . 001.

1

Then the associated N‘ could be -0'0-1- " 1 1000 - 1 = 999.

1
Hence for n) 999, F1- < .001.

 

1’1)“
Example: Prove lim (1+ 11 ) = l
09’
Lete)0be 'en Then a -1=1+('1)n-1--1—'
g“ . In I n n.

 

 

1
Hence find N5 such that ‘3' < E for n) N‘ .

 

94
1
This can be accomplished by letting N‘>/ z

Thenn)N‘=}'é‘(n'-'-? ‘nl"<5.

ASSIGNMENT #2

 

l
1. Prove lim -—- = 0
nwmz

2. Explain why the following do not have limits:

a) 2, -1, 3/2, -1/2, 4/3, -l/3, 5/4, -l/4,...

.2.
n+l’
b)a=

n n-2
—, neven, n32
11

(Zn)...

3. Let S 1={—} n-1 and G = .001. Find the first (smallest) N 001 such

nodd

n

thatln (..001

l
4. Do the same as in problem number three for {?}n=1

1
What do you conclude about lim ‘2 ?
Il-Qod 11

Although the limits of some sequences are attainable by the definition, con-

sider a problem such as

n2+5n+6 5n+2

xii-’3‘.- (n+2)(n+3)7 °r iii—‘3.- n

 

 

If we apply the definition here, the work becomes tedious.

95
To facilitate the finding of such limits, we will now prove some basic results

which can then be applied to simplify some of the more difficult problems.

1. The limit of a constant sequence is that constant.

i.e. 11111 K = K
n-hc

Proof: Let G > 0 be given and consider the expression Ian - L' < e .
lnthiscase,a =KandL=K,hencela -Ll=lK-Kl =0.
11 n
We now have the problem of selecting N6 such that for n ) N ‘ ,
'an - K’ < e . However, we have already shown that 'an - Kl = 0,
hence any N6 we choose will have the desired property, In parti-
cular, let N. equal, say, 100. Then for n > 100, Ian - K, ( E

and the definition is satisfied. Therefore, lim K = K.
n-pv

2. "111.52% = L1 and 33.1%: = L2, then [113.5% + bn ) = Ll + L2.

Proof: Let S > 0 be given. Then there exists Nl such that for n ) N1,

6 e _ .
an - LII ( 2 . (note that —2- > 0 and III-Ionaan - Ll by hypothes1s).

 

Similarly, there exists N such that lbn - L2|( 5:- for n ) N2.

2
Now consider l(an + bn) - (L1 + L 2)l . By the triangle in equality,

(an - Ll)+ (1:)n - L2)|< Ian - Lll +

6 € _
lbn-L2|( —2—+ —2- -€ forn) max {N1’ N2} .

 

(an+bn)-L1+L2l =

 

To illustrate how these theorems can be used, consider the following

problems:

5&2 0 511+ 2
a) Evaluate 11111.9” 11 . n—jco n

2 _ . 2: =
111%(5+E)’1113u5+1159a3 5+0 5.

 

5n 2 _
-r11!21,.$n+n) -

3n +- n 3n n l
alua 'm ——2-- = m —-2' + ‘1 = im + "‘ :-
b) EV te #11990 n [Iii—ya n n 111_§¢(3 n )

1
I1131M lim‘fi' = 3+0 = 3.

There are several other theorems concerning limits of sequences that will

now be stated and illustrated but not proven.

3.

If lim a = L , then lim Ka = KL where K is a constant.
n—w n l n—)eo n 1
l = 5 .

 

 

Example: 11.131 0 = 0.

511m
oan+l n—pon+l

If {1131‘ an = L1 and 11131. bn = L2, then A151. anbn = Lle.

(n+2) 1i 1 n+2 =
_,,,m+1)(n+2) =n.‘£‘.n+1 n+2

 

 

 

Example: lim

1 1 .1 n+2 0
£~n+l n$n+2 niljnun+l 113'." '

 

 

If Almaan = L and bn = an for all n, then [Ill—Tuan = ill—Tubn' At first
glance it may appear that this theorem doesn't really say anything.
However, it is very important in that it allows us to "reduce" expres-
sions. Actually, we have used this theorem intuitively in several of the

previous examples .

2n (n + 2)
E -————- = lim = ,
xample: a) 111—?” n (n + 2) n-)ae2 2

 

2
n +5n+6 1 1
1’) i§¢2(n+2)(n+3) ‘111'13902 '2'
an L
If $31.08!!le and 111.31.915sz 9‘ 0, then 1113‘?!" = '13:.

 

 

. n2+n+l . n2+n+l n2
ample. [Ba—T‘sn + 2n - 53.33:; a? -
2 2 n2 + 11 +1
+g+l . 2
lim n = 11m 711 =
n-)0‘ n 3n +211 n—M'3n +2n
112
1 + -l- + -1-2 - l 1
lim n n _ 1111-30-0 + n+ n2) _1_
“'7” 3+2 lim (3+ 3) 3
11 mayo. 11
ASSIGNMENT #3

 

1.

step.
6n +13

a) { l3n }n=1

 

 

 

2. Prove theorem three.

Evaluate the limits of the following sequences. Give a reason for each

 

n2 + n
f) n2 - 2
n=1
2
n
8) n3 + 4 n:

98

a) Let a beasequence such that i a = 0. Prove that
n n=l an
lim a = 0.
[Hm n
b) Prove the converse of part (a). (i.e., if 1i ”an = 0, then
lim a = 0)
n—jv n
Prove that if lim a = L, then lim (-a ) = -L.
n—non n-)eo 11

Do this both by using the definition of limit of a sequence and by using

the results of some of the theorems.

LIMITS OF FUNCTIONS

We now turn our attention to examining limits of functions. Let us first

consider several examples .

1.

Let f(x) = x + 2 and suppose x approaches 2 (x—) 2).
What is f(x) approaching? (f(x)—9 ?)
It is easily seen here that as x "gets close to" two, f(x) "gets close to"

four .

Let f(x) = x2 and suppose x-—) -1.

Then f(x)—? ?

99

3. As a slightly different example, consider

_ x+l x50
f(x)-(x+2 x)0

When graphed, this would look like the following:

 

 

Let us now consider the same questions as on the previous examples.

Let x—-) 0. Then f(x)—§ ?

 

After some thought, we realize that there is a need to be more specific

about what we mean by x—) 0 and f(x)—9 . It is this need for
precision that will eventually lead to the definition for limit of a function.

In the meantime, example three illustrates the importance of knowing 1323!

x approaches the number we are considering (0 in this case). For example,
let x assume the values 1/2, 1/4, 1/6, 1/8, . . . Then the corresponding values
of f(x) would be 2 +1/2, 2 +1/4, 2 + 1/6, 2 +1/8, . . . From this it can be
seen that f(x)——) 2.

On the other hand, let x assume the values -l/2, -1/4, -1/6, -l/8, . . . The
corresponding values for f(x) are 1 - 1/2, 1 - 1/4, 1 - 1/6, 1 - 1/8, . ..

Accordingly, we would say that f(x)—9 l.

100

This function serves as an example that does not have a limit as x-—) 0.
That is because there are two possible values that f(x) is approaching rather
than one. We indicate this by saying that as x approaches 0 from the right
(x—.) 0+), f(x)—9 2; and as x approaches 0 from the left (x—) 0-),
f(x)-—) 1.
In examples 1 and 2, it made no difference how x approached the stated num-
ber (we usually say "x approached a"). When this is the case, we write
112’ a f(x) = L where L is the number that f(x) is approaching. Hence, in
example 1 we would write ’11:? 2 (x + 2) = 4 and in example 2 we would write
113 -l x2 = 1.
An adaptation of this notation could also be used for example 3. We would
have 11111; 0- f(x) =1 and #3 0+ f(x) = 2.
As you might expect, the idea of limit of a function is closely related to the
idea of limit of a sequence. The main difference is that in finding the limit
of a function it is necessary to know what value the dependent variable is
approaching. For example, ’11:; 2 (x + 2) = 4 but 3113} 3 (x + 2) = 5.
As before, the idea of "getting close to" is not sufficient for a definition.
Therefore, the formal definition for limit of a function as x approaches a is:
The function f(x) has limit L as x approaches a if for each 6 ) 0 there
isa 5‘) Osuchthatif0( lx-al L 5‘ , then|f(x)-L| ( é .
lntuitively, this says that if an interval is placed around L on the f(x) axis,
then you can find an interval around "a" on the x axis such that f(x) is in the

interval around L for all x 7! a in the interval about a.

101
Example: Let f(x) = 3x - 2 and x—, 2. ThenL = 4.
To see how this satisfies the definition, let 6 = 1/2 and con-

sider the interval (4 - 1/2, 4 +1/2) about 4 on the y axis.

 

 

/

We now find an interval about 2 on the x axis that satisfies

. the definition.

   

VIII: 1‘»

for G =1/2, we see that;

 

1/2 = 1/6 and a su1table mterval

would be (2 - 1/6, 2 + 1/6).

Foré = 1/10, find a 56 that satisfies the definition of limit
of a function.
We now show that the interval about "a" depends on the interval about L for

thefunctionf(x)=3x- 2witha=2andL=4.

102
(3x-2)-4|<6 é)
'3x-6|(£{=§ [3(x-2)|<e $9
3|x-2|(£(=) 'x-2|( e/3(ifi

2- é/3<x(2+ é/s

If|f(x) - LI ( e, then

 

Hence, for any given 5. ) 0 and the associated interval (L - é , L + 6 )
about L on the y axis, we can use the interval (2 - 5/3, 2 + 6/3) about two

on the x axis to satisfy the definition. We say let: = 5/3.

5/3 :9|f(x)-4' ( é .

Then0( lx-2l(J

This illustrates the method of proving that a particular function has a parti-

cular limit as x approaches some value.

ASSIGNMENT #4

 

1. In the following problems, let x assume several values "close to" the
specified number a, and examine the corresponding values for f(x). In

each problem, make a conjecture about 111.3 a f(x).

a) f(x) = x+3, x-—)l
b) f(x) = x2+1, x——)2
c) f(x) = -4x+5, x——)0
d) f(x) = x2-x+l,x—)l

2. In each of the following problems, for the indicated intervals about the
respective values of L, find an interval about a on the x axis such that

if x is in the interval about a, then f(x) is in the interval about L.

f)

103
f(x)=x+3, L=4, a=1, (4-1/2, 4+1/2)(é=l/2)
f(x)=x+3, L=4, a=1, (4-1/100, 4+1/100)(é =.01)
f(x)=x2+1, L=5, a=2, (5-1/3, 5+1/3)(é =1/3)
f(x)=-4x+5, L=5, a=0, (5-1/3, 5+1/3)(é =1/3)
f(x)=x+3, L=4, a=l, (4- 6, 4+ e)

f(x)=3x+l, L=7, a=2, (7- 6, 7+ E)

3. Using the definition of limit of a ftmction, prove the following:

a)
b)

C)

lim x=a

x-)a
)1132(2x+1)=5

lim (mx+b)=ma+b, m750
x—’a

There are several important parts of the definition of limit of a function that

should be noted. First, we see that the definition implies that x must ap-

proach "a" from both the right and the left. This is because it says

Ix-al($ whichisequivalentto-$( x-a (SQ; a-5(x(a+5 .

Hence one must consider values of x on either side of a.

Example: Explain why

x+1, x) 0
f(X) =
x , x$0

does got have a limit as x-) 0.

Another consequence of the definition of limit of a function is that we do not

consider what happens when x = a. This is excluded by the part that says

0 < Ix - a' . The reason for this is illustrated by the following example.

104

Consider

f(x) =

 

As x—-) a, what happens to f(x)?
After trying a few values, is it clear that f(x)—) 2? Since this is consistent
with our earlier ideas of limit of a function, we say

lim f(x) = 2

x—) 5
even if in this case f(5) 7! 2. (Note: a concept to be studied later, continuity
of functions, will help explain functions of this sort.)
Notice that if we do not insist that x 7! a in our definition, then for an interval

about L = 2 where 6= 1/2, it would not be possible to find a corresponding

interval about a = 5 that satisfies the definition because we could then use

105

x = 5 and f(x) = 6 would not be in the interval about 2.

ASSIGNMENT #5

 

Tell whether or not the following functions have limits at the indicated
values of x. For those that do not, indicate what part of the definition they

fail to satisfy. (Sketch)

 

1, x50
1. f(x) = ;a =
2, x)0
2
x +1 x41
2. = ’ ; 1
f(x) x+l,x)l
x2-4 #2
- ’ x
3. f(x) = x 2 , a = 2
10 , x=2
+ £3
4. f(x) = XI ' x ; a = 3
-2x+l , x>3

Limit Theorems for Functions:

 

As we did with sequences, we now state and prove several theorems that
enable us to evaluate the limits of various functions.
Theorem 1. Let f(x) = K be a constant function. Then 111.11;. a f(x) = K for
any real number a.
Proof: Let G- > 0 be given and consider f(x) - K. Since f(x) = K,

f(x) - K =0=0, we have that f(x)- L ( E no matter what

106
value of 5 is used. Therefore, 0 ( Ix - a, (5 (say =1)

implies [f(x) - LI < E and the theorem is proven.

 

1
£13 a 30:) = L2. Then £13 a (for) + 800) = £31, a f(X) + 1113 a g(x)

Theorem 2. Let f(x) and g(x) be functions such that )liu—n’a f(x) = L and

=L1+L2.

Proof: Let G, ) 0 be given and considerl (f(x) + g(x)) - (Ll + L2)|. As
before, we have by the triangle inequality |(f(x) + g(x)) - (Ll + L 2)l
=|(f(x) - Ll) + (g(x) - 1.2)]; [f(x) - LII + [g(x) - LZI‘ But since

1’ there is a S I such that 04—.lx - al ( ‘1 implies

|f(x) - Lll ( 6/2 and since 41.3.13“): L2, there is a ,5 2

lim f(x) = L
x-)a

such that 04 [x - al 4 52 implies' g(x) - Lzl ( ‘/2.

Hence rot-J = min {5"} 11,0 ( Ix - al ( J implies both

|f(x) - Lll L 5/2 and |g(x) - Lzl L 5/2. Therefore

'60:) + 200) - (Ll + 1.2)] 6 lfix) - LII + |g(x) - LZI 4 ‘/2 +
€/2 = 6 .

This then satisfies the definition and we have x113 a (f(x) + g(x)) =

L+L.

1 2
Example: Let f(x) = 3x2 and g(x) = 5. Then £3; 2f(x) =12 and
2 2
li =5 dli 3x+5=lim3x+ 5=
x323“) an x32< ) x—n :52
12+5=l7.
Theorem 3. Let f(x)beafunction such that 1121, af(x)=Landletheanon-

zero constant. Then lim Kf(x) = K ° L.
x—aa

107

Proof: Let G ) 0 be given and consider Kf(x) - KLl . By properties of

 

absolute value, we have |Kf(x) - KLl = IK” f(x) - Ll . Then
le(x)- KL|( £9 “(Him-1.1459 [f(x)-q ( 5m .
But since lim f(x) = L, we know there is a S ) 0 such that
x—-)a
0 (Ix - a| (S =} |f(x) - Ll ( G/|l(| . Therefore for é) 0,
thereisa S > Osuchthat0( Ix-al 45:} lf(x)-L| <
6/|1<|=.9 |1<| |f(x)- L] (6 =) le(x)- KLI (e . This
then satisfies the definition and we have lim Kf(x) = K ' L.
x—)a
Example: Let f(x) = 3x2 + 5 and suppose lim

x_’2

2 2
= ' + ' = ' = ° :7.
J11m23x £325 3,1132x +£1m25 3 4+5 1

2 2
= . +
x 4 Then x113 2 (3x 5)

There are several other theorems involving limits of functions that will now

be stated but not proven.

 

Theorem 4. Let f(x) and g(x) be functions such that )liin’ a f(x) = L1 and

Jlei-n; a g(x) = L Then 123 a f(x) g(x) =

2. Lle.

Theorem 5. Let f(x) and g(x) he functions such that 1113 a f(x) = L and

1
f(x) __ L]
lim ag=(x) L2 #0. Then 13a“? L2.

 

Theorem 6. Let f(x) be a function such that lim af(x)= L 0. Then

lim a\n/ f(x) =\n /lima f(x) =VL—.

 

108

Theorem 7. Let f(x) and g(x) be functions such that f(x) = g(x) for all x except

 

= . Th 1' = 11 'd .
x a en x12) a f(x) x 23a g(x) prov1 ed these limits exist

. x - 9 .
z = = .
Example (Theorem 7) J11m 3 x _ 3 1115 3 (x + 3) 6

 

The following examples illustrate how the previous theorems can be used to
simplify and evaluate the limits of several functions.

2 . 2
x 123 x
I . = x 3
Example. ’12 3 _ 2 (limit of quotient)

11133 (3x - 2)

 

 

9‘13 3 x) (Ly-n“ x) (limit of product)
lim (3x - 2)
x—) 3

= _____5 (previous problems)

I
\llxo

Example: (Indicate the reason(s) for each step)

+ -
11m Lil : lim (x+3) (1;+2)
x—,+2V “‘2 x—)+2 x'

(x+3)(x- 2)
VEBH x- 2

V 11m (x + 3)
x—-)+2
V 5

 

 

 

 

109

ASSIGNMENT #6

 

Find the limits of the following functions. Give reason(s) for each of
your steps.

a) lim (x2-3x+5)
x—-)3

b) lim ‘3'“;

 

c) lim

 

d) lim

e) lim £2“.-é
x—,2 x -4

V3+x -y3
0 x

 

f) lim
x-’
Letf(x)=l. Provethat l' -l- =-1 by
x x3 2 x 2

a) theorems from this section.

b) using the definition of limit of a functiOn.

Let f(x) be a function. Prove that £13 a f(x) = 0 iff £13; a f(x)| = 0.

 

Let f(x) be a function such that lim f(x) = L.
x—)2

Prove that 12.32 |f(x)| =l 14 .

110

5. Give an example of a function f(x) that does fl have a limit as x—-) 3.

6. Give an example of a function f(x) that does 93$ have a limit as x——) 3

but is such that lim f(x) = l.
x—)3

111

APPENDIX 11: Daily Classroom Record for Treatment A

 

Wednesday, October 17

About thirty minutes of the fifty minute period was spent on the new unit.
Students were made aware of the following groups involved in the study:
(a) mastery treatment, (b) regular treatment, and (c) control. The stu-
dents were also informed of impending interviews at the end of the semes-
ter. During the general discussion, students were informed that they
would still cover the core of material in the regular Precalculus course.

The remaining instructional time permitted a discussion of notation,
definitions, and examples preceding Assignment One. Assignment One
was then given for Thursday.

Thursday, October 18

Problems associated with the assignment were discussed. Quiz number
one was given.

The definition of page three is difficult to comprehend. lntuitively, the
students seem to have no difficulty with the concept of limit of a sequence.
There is some confusion between "n" and "N". To alleviate this, a proof
should be carefully checked tomorrow.

Friday, October 19

Assignment T\vo was reviewed. Quiz 2a was given and 2b was assigned
to be turned in on the following Tuesday.

Two theorems were proven. They were:

(1) lim K = K, where K is a constant
n on

(2) If {83114 and {bnknfl are sequences such that m’an = L1

= + = +
and I1113mm! L 2, then r111$".(an bn) Ll L 2
There really was not enough time today. The second theorem was
finished right at the end of the period. Consequently, there was no time
for illustration and application of the results.

Absence was a real problem today. Sixteen of the total of sixty were not
in class. This will make working with the definition diffith for those
not in attendance since this was our first good look at the proof of a
theorem using the definition.

112

Tuesday, October 23

A brief review of previously presented limit material was given. Limit
theorems three through six on page six were presented and illustrated.
Proofs of these were omitted. Several examples illustrating the appro-

priate use of these theorems were given. Assignment Three was made
for Wednesday.

Wednesday, October 24

Questions concerning Assignment Three were answered. Because of the
difficulty encountered with problems three and four of this assignment,
problem 3a was worked in class and then students were asked to turn in
either 3b or 4 on Thursday. Quiz number three was given in class.

The last fifteen to twenty minutes of class time was spent discussing the
intuitive concept of limit of a function.

Thursday, October 25

The first section on limits of flmctions was finished. The definition and
appropriate notation were covered in this section. Assignment Four
was given for Friday.

Friday, October 26

Questions from Assignment Four were answered. Students have less
trouble working with the definition this time than they did before.

Quiz number four was given. Absence is still a problem; eighteen
students were missing today. Section number two on limits of functions
was finished and Assignment Five was given for Thesday.

Thesday, October 30

Questions were answered about the assignment. The last section was
completed today. As before, only two theorems were proven in class.
They were:

(1) Let f(x) = K be a constant function. Then ‘11:; a f(x) = K for any
real number a.

(2) Let f(x) and g(x) be real-valued functions such thatILin; a f(x) = L1

and Jar-n, a g(x) = L2. Then #2; a(f(x) + g(x)) = Ba f(x) +

l . =
x.135 a g(x) Ll + L2.

 

113

The rest were stated and illustrated. Assignment Six was given for
Wednesday and quiz number five was assigned to be turned in Wednesday.

Wednesday, October 31

Questions about the assignment were answered. Quiz number six was
given. The rest of the period was used as a general review for Thursday's
test. The students are aware that this test is a formative evaluation.

Instructions to the class about testing were reviewed as follows:

(1) You may take the retest regardless of your score.
(2) The tests will be returned with diagnostic sheets on Friday,
the first alternative instruction session.
(3) Other alternative instruction sessions will be held on Monday
at 8:00 a.m. , 9:00 a.m. , and 7:00 p.m. to 9:00 p.m. Tuesday's
regular class period will also be used for alternative instruction.
(4) The summative evaluation will be given on Wednesday, Nov. 7.

Frustration seemed to be a big factor today.
Thursday, Novemberl

A formative evaluation was administered to both sections.

114

APPENDIX III: Quizzes

 

Write the fifth term of the sequence
{ -l)n
+
n 1 n=1

1

1
(a) For the sequence {72;} and G = 36 , find N such that

n=1
1 l
I-Z‘EI < '56 for n > N.
l
(b) Prove xii-$.53 = 0 using the definition for limit of a flmction.

Evaluate the limit of the following sequence. State a reason for each step.

2n2+3n- 5
3n2-n+4

 

n=1

1 1
Letflx)=2x+landlet(7- 15. 7+ 1‘6)beanintervalabout7onthe

f(x) axis. Find the associated interval (3 - 5 , 3 + S ) about 3 on the

1 l
xaxis suchthatifxisin(3- 5, 3+ 5), thenf(x)isin(7- IT)” 7+ i6).

Prove gilt-1’3 (2x + l) = 7 using the definition of limit of a function.

, x
Find ’11:; 2 x __ 2 . State a reason for each step.

 

115

APPENDIX IV: Summative Evaluations - Treatment B

MATH 120 Name

 

LIMITS - A

(11 pts.) Explain why the following sequence does not have a limit.
(Hint: Plot the terms on the number line).

1
{(4)n + ‘11-} n=

(12 pts.) Give an example of a nonconstant sequence whose limit is -2.
Include reasoning.

(12 pts.) Give an example of a function that does not have a limit as x-—)1.
Include reasoning.

 

(15 pts.) For the following sequence, find the smallest N 1 such that
1 Io_o
[an - 3' < I‘D-6 forn) NJ...’ Show your work.
100
3n + 2
:1 n=1

2

(15 pts.) Evaluate lim 2 - ' . Give the reasons for each step.
nfi~3n + 5n + l

2
x + 5x
(15 pts.) Evaluate £3 0 ‘7_x _ 2x . Give the reasons for each step.

(20 pts.) Prove, using the definition of limit of a function, that

f34(3x-2) =10.

116

MATH 120 Name

 

LIMITS-B

l. (11 pts.) Explain why the following sequence does not have a limit.
(Hint: Plot the terms on the number line.)

H)“ - é
n=1

2. (12 pts.) Give an example of a nonconstant sequence whose limit is -1.
Include reasoning.

 

3. (12 pts.) Give an example of a function that does not have a limit as x—) 2.
Include reasoning.

4. (15 pts.) For the following sequence, find the smallest N 1 such that

 

 

1 '1'0-0
I; -3l(i-0-5forn> N1 . Showyourwork.
n —
100
6n+1
2n -
n.—
5 Eal li 38-23” Gi th to h
. (lSpts.) v uaten'_11r_1”.‘m:5:n_5 . ve ereasons reac step.

2
-x + 6x
6. (15 pts.) Evaluate )l‘im O T + x . Give the reasons for each step.

7. (20 pts.) Prove, using the definition of limit of a function, that

iii-133(k-l)=ll.

117

APPENDIX V: Formative Evaluation - Treatment A

 

MATH 120 Name

 

LIMITS-C

1. (11 pts.) Explain why the following sequence does not have a limit.
(Hint: Plot the terms on the number line.)

1
{9 2)n + 1:} n=1

2. (12 pts.) Give an example of a nonconstant sequence whose limit is -3.
Include reasoning.

3. (12 pts.) Give an example of a function that does not have a limit as x—9 3.
Include reasoning.

 

 

 

4. (15 pts.) For the following sequence, find the smallest Nl such that
1 1‘66
an- 3| (I55 forn) NJ...’ Showyourwork.
1
{3n - 3
n
n:

2

5 s Eal 11 "in ”+16 Gi th f h

. (1 pts.) v uate 113092112 - 5 . ve ereasons or eac step.

3+ 3::2 - 2x

 

6. (15 pts.) Evaluate Jlim . Give the reasons for each step.

—)0 2x3-5x

7. (20 pts.) Prove, using the definition of limit of a function, that

121—“)1 (7x- 5)=2.

118

MATH 120 Name

 

LIMITS - D

(11 pts.) Explain why the following sequence does not have a limit.
(Hint: Plot the terms on the number line.)

.1
n
n=1

(-2)“ -

(12 pts.) Give an example of a nonconstant sequence whose limit is -5.
Include reasoning.

(12 pts.) Give an example of a function that does not have a limit as x—) 4.
Include reasoning.

(15 pts.) For the following sequence, find the smallest N I such that

 

l 100
Ian - 3| (IO-0’ forn) N_l_. Showyour work.
100
{gm
3n
n=l
6n3 +1011; 1

 

(15 pts.) Evaluate I1111‘.

A; .7113 + 4nl _ 2n° Give the reasons for each step.

 

3x2+2x

(15 pts.) Evaluate 111m 2 + 5x

-) 0 -2n . Give the reasons for each step.

(20 pts.) Prove, using the definition of limit of a function, that

Jl‘i‘_r_x;6(4x+5) = 29.

119

APPENDIX VI: Summative Evaluation - Treatment A

MATH 120 -- Limits Name

 

(a) (12 pts.) Give an example of a nonconstant function f(x) such that
lim f(x) = 5. Include explanation.
x—’ 2

(b) (12 pts.) Give an example of a sequence of positive terms that does
n_o_t_ have a limit. Include explanation.

(c) (11 pts.) Explain why the following function does not have a limit

asx—)-1. f - x+l,x£-l
0‘)— 3,x)-l

(15 pts.) For the following function, find 5 1 such that
1000
l
|f(x)-l3l (165.5 when0( lx-6l ( 51 .
1000
f(x)=2x+l, a=6, L =13.

 

 

2
-2 + - 3
(15 pts.) Evaluate lim n T n . Indicate reasons for each step.
n—no 4n - 5
3
x - 4x
T

(15 pts.) Evaluate £13 2 x3 + x _ 6x . Indicate reasons for each step.

(20 pts.) Prove, using the definition of limit of a sequence, that

11 n-3n2 _ 3
1133ng " °

120

APPENDIX VII: Instructional Alternative Worksheets

 

WORKSHEET l

 

Explain why the following do not have limits:

[I
n
2. {(-2 )} n=l
11
3° {3 } n=1

1
4. {411+ 3}
n=1
4
5. f(x)= {ght-S: a=5

x+l,x£--3. _
2x,x)-3' “"3

0‘
C
:2
X
V
II

I
x
1
”N
>1
‘0.
N
9)
II
N

7. f(x) -

121

WORKSHEET 2

 

In the following problems, for the given 6 , find the corresponding N g or J‘ .

l 2n+l
lo 6 i6! { n }n= L=2

l -3n+4
2- €= 17.66{T} L“
n=1

1
3. e = 1'55. f(x) = 3x+1,a=2,L=7

 

NIOD

1
4. e: 35, f(x) = -2x,a=0,L=O

1
5. 6: 555,:(x)=-3x+1,a=2,L=-5

122

WORKSHEET 3

 

Evaluate the limits of the following:

 

 

 

 

 

 

 

l 3n3-2n2
' -2nr+n-l =1
x2-9
2. f(x) = x-3’ a=3
3 f 4x4+3x3-x2 -0
e (x) _ 2114-3x2 a a"
21124-1
4. 3
-4n +2n-S n=l
+
5. {11111}
n=1
6 f x2+6x+9 _ 3
° (x) - x7+5x+6 ’ a—
x2+6x+9
7. f(x) = , a=0

x2+5x+6

123

WORKSHEET 4

 

Using the definition of limit, prove the following.

 

 

 

+
6.11m2nl=2
n-pon
7.li=0

124

APPENDIX VIII: Attendance Summary for Alternative Instruction Sessions

 

 

 

 

 

 

 

 

Session Date Time Number
Attending
Friday 8:00 a.m. 22
Group Lecture
Friday 9:00 a.m. 31
Monday 8:00 a.m. 10
Monday 9:00 a.m. 23
Small group
Monday 7:00 p.m. 29
sessions
Tuesday 8:00 a.m. 24
Tuesday 9:00 a.m. 27
TABLE A.1

Attendance Summary for Alternative Instruction Sessions

 

125

APPENDIX IX: Calculus Tests

 

MATH 202: Test 1A

1. Prove: lim (ax+b) = ac+b
He

x2
- 5 + 6
2. Find lim x . Use only one step each time a limit concept is

x-) 3 x2 -4x + 3 used and give the reason for that step.

 

For problems 3, 4, and 5, give an example of a function f such that:

3. fhasarighthandlimatz, fhasalefthandlimatZ, but hm;
does not exist.

4. £111, 1; exists and is positive for all real numbers, a.

5. J1131,31“ exists and = f' (a). (Give both f and a)

6. (a) Give the definition of : "f is continuous at a if. .

(b) Tell where f is continuous and where not continuous if

 

2-
x 3x+2 ifx 7, 2
f(X) =.- x-2
3 ifx = 2
2
7. (a) (3x3-2x2+x-4- i)‘ =

(1» «x2_ 3x) - (x3+s»' =

x3+4x

(C) 2x+7 =

 

(d) ((x2 - 3x + 010), =

126

MATH 202: Test 1B
1. Prove: lim (ax+b) = ac+b
x_.) c

2. Give one reason for each step using a limit concept and evaluate

xg- 5x+6

m3xz-4x-i'3

 

For problems 3, 4, and 5, give an example of a function f such that:

3. lim f exists and is positive for all real numbers, a.

Ha

4. lim+f exists, lim -f exists, but lim f does not exist
x-—-)2 x—)2 x--)2

5. lim f exists and = f'(a). (Give both f and a)
x-—)a

6. (a) Define: "f is continuous at a if. .

(b) Tell where f is continuous and where not continuous and give
reasons for your conclusions if. . .

 

2
x --3it+2 ifx #1
f(x) = x
2 ifx =1
3
7. (a) (4x3-x2-21t+5- E" =

2
(b) «x +2x> - (x3-7»' =

. x3-3x ,
(C) 4x+2

 

(d) «x2 - 5x +1>‘2)' =

127

MTH 202: Test 1C
1. Prove: lim (ax+b) = ac+b
x—) c

2. Give one reason for each step using the limit concept and evaluate:
1. x2 - 5x + 6
1m 2
x—’3 x - 4x + 3

 

For problems 3, 4, and 5, give an example of a function f such that:

3. lim + fexists, lim - fexists, but lim a does not exist
x...’ 3 x.) 3 x—’

4. 11m" f exists and is positive for all real numbers, a
x a

5. lim f exists and = f' (a) (Give both f and a)
x—9 a

6.. (a) Define: "f is continuous at a if. .

(b) Tell where f is and is not continuous and give reasons for your
conclusions if

 

2
xx-3lx+2 ifxifl
f(x) =
5 ifx =1

7. (a) (5X3-3x2+x-1+-:-)’ =
(b) <<x2+9x) - (x3-4n’ =
(XS'6X)"
(c) W =

(d) ((xz-sx-8)15)' =

 

128

MTH 202: Test 1D

1. Prove: If n is a negative interger, then

(xn)I = M“-1

, x2-7x+12
2. 11m 2 =

 

Evaluate using one step each time a limit concept is used. Give that
concept.
For problems 3, 4, and 5, give an example of a function f such that:

3. lim {4, exists, lim f_ exists, 11m f does not exist, and f(x) 9‘ Gl(x)
x—p x—’ l X—’ l

4. lim f exists and is negative for all real 3
x—p a

5. lim f exists and = f ' (a) (Give both f and a)
x...) a

6. (3) Define: the derivative of a function, f, at x.

(b) Use the definition of derivative to find f ' (x) if f(x) = Vx

1/5 1/5

(x+t) -x
t

 

(c) Evaluate lim
t-}0

7. (a) (3x2+2x+5+%)' =

3

(b) <(x2-3x>- (3x +5”! z

2
x +2
(C) (ans), "'

(d) ((4x2-3x+7)10)’ =

 

129

MTH 202'. Test IE

1. Prove: If n is a negative integer, then

(inn), = nx

2. Evaluate using one step each time a limit concept is used. Give the
reason to find

11m x3-7xth
H5 xZ-6x+5

 

For problems 3, 4, and 5, give an example of a function f such that:

3. lim f+ exists, lim f_ exists, lim f does not exist, and f(x) is not
x-’ 4 x—-) 4 x—) 4
the GI function

4. lim f exists and is negative for all real a

x-ga

5. limf exists and = f'(a) (Give bothfand a)
a

6. (a) Define the derivative of a function, f, at x

(b) Use this definition to find i ' (x) if f(x) =

lb— 3
(c) Evaluate lim x w 35.

w—}O w

x1...

2 l
7. (a) (4x -3x+4+;)l =
(b) ((x2+5x) . (2x3+7))' =
2
x -7
(c) (9x+4 I:

(d) ((7x2-8x+5)12)’ =

 

130

APPENDIX X: Description of Random Selection Procedure for Interviewees

 

After deciding upon the number of students to be interviewed in each group,

the following procedure was utilized:

(1)

(2)

(3)

(4)

the students in each group were numbered according to their
alphabetical listing.

a page and a column in the random number tables was selected by
a random number.

from the randomly selected starting place, the column was read
down until the desired number of digits were located (for example,
if twenty-one students were in the list and five were needed for
interviews , one would continue down the column until locating

five digits (numbers) less than or equal to twenty one).

using these numbers, the corresponding names in the list were

selected for interviews.

131

APPENDIX X1: Interview Questions

 

 

 

 

 

 

 

 

NAME DATE
GROUP
1. Status at C.M.U.
2. High School attended
Size A B C D Other
3. Class rank
4. Mathematics courses taken prior to Precalculus
5. Why did you take Precalculus ?
6. How are you doing in Precalculus ?
7. How do you like Precalculus ?
8. Is Precalculus harder or easier than you expected? In what ways ?
9. (Where applicable) What is your reaction to the section on limits ?
10. Do you presently plan on taking Calculus 1? Why or why not?
11. Did you previously plan on taking Calculus I?
12. If not included in number 10 above, indicate reasons for differences if
they exist in number 10 and number ll.
13. How difficult do you think Calculus I will (would) be compared to other
mathematics courses you have taken?
14. How do you think you will do in Calculus I? (Where applicable)

Interviewers Reaction

 

Briefly discuss your conception of student anxieties or lack of anxieties
regarding Calculus I.

What is your opinion of the student's attitude toward Calculus I?

132

APPENDIX XII: Questionnaire Given to Calculus Students

 

Name

 

Name of Precalculus instructor (if any)

 

Semester Precalculus was taken (if any)

 

In the following, please check the appropriate response:
1. Compared to what I expected, Precalculus was:

a. much harder b. harder c. about the same
(1. easier e. much easier

 

 

2. So far, compared to what I expected, Calculus I is:

a. much harder b. harder c. about the same
d. easier e. much easier

 

3. Compared to other material you have studied, limits are:

a. most difficult b. somewhat hard c. about average
d. easy e. very easy

4. I have spent the following number of hours per week (outside of class)
studying limits so far this semester:

a. 0-5 h. 6-10 c. ll-lS (1. 16-20 e. Other
(please specify)

In some instances, one can distinguish between degree of difficulty of a
subject and the level of frustration encountered when studying the subject.
With regard to this, the following question is an attempt to discover how
frustrating the study of limits is rather than how hard it is.

S. I find the study of limits this semester:
a. very frustrating b. more frustrating than average

c. average d. less frustrating than average
e. no frustration

 

 

 

133

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N302 HBNBENuoaKm 05 no mfiou no
mweasouo meander new 3339. nomnamom 303.82: NEVA 5&me:

 

134

APPENDIX XIV: Cell Components" of Chi Square Values

for Questionnaire Items

Grouping: Treatment A -- Treatment B -- Control
Critical value (ck = .05): 9. 4

Q (')L2 = 15.44)

 

 

 

 

 

Grouping a and b c d and e
2 17 5
T'R-A 7. 9 11. 9 4.4
4.1 2. 2 .1
5 15 8
T'R-B 8.9 13.9 5.1
1.8 .1 1.6
35 33 11
Control 25. 3 39. 2 l4. 5
3.7 1.0 . 8

 

 

 

 

 

g_5 (38:15.92)

 

 

 

 

 

Grouping a and b c d and e
6 10 8
TR-A 12.1 6.4 5.5
3.1 2.0 1.1
13 4 11
T'R-B 14.1 7.5 6.4
.1 1.6 3. 3
47 21 11
Control 39.8 21.1 18.1
0.0 2.8

 

441.3

 

 

 

*The numbers in each cell are observed values, expected values,

and chi square partial values.

TABLE A. 3

Chi Square Values for Questionnaire Items

 

135

Grouping: Treatment -- Control
Critical value («A = .05): 5.99

 

 

 

 

 

 

 

2
g_2 (12 =13.87) g_4 (7L = 6.97)
Grouping a and b c d and e Grouping a and bl c d and e
7 32 13 27 24 1
Treatment 16. 7 25. 8 9 5 Treatment 22.6 24.1 5.2
5. 6 l. 5 1 3 0. 8 0. 3.4
35 33 ll 30 37 12
Control 25. 3 39. 2 l4. 5 Control 34. 4 36. 8 7. 8
3 7 l . 8 0.6 0.0 2.3

 

 

 

 

 

 

 

 

 

 

 

93 (7&2 =10.28)

 

Grouping a and b c d and e

 

Treatment 26.2 13.9 11.9

 

 

Control 40. 0 21.1 18.1

 

 

 

 

 

TABLE A. 3 (continued)

136

Grouping". Mastery -- Nonmastery -- Control
Critical value (ck = .05): 9.4

2
9_1 ()L =1o.57)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Grouping a and b c d 25d e
8 l4 9
Mastery 8. O 13.7 9. 2
0. 0 0.0 O. 5
10 10 1
Non- 5. 4 9. 3 6. 2
mastery 3. 9 .l 4. 4
16 34 29
Control 20. 5 35. 0 23. 5
l. 0 0.0 l. 3
2
93 ('x = 11.58)
Grouping a and b c d and e
10 8 l3
Mastery 15.6 8. 3 7.1
2.0 0.0 4. 9
9 6 6
Non- 10.6 5.6 4. 8
mastery 0. 2 0. 0 0. 3
47 21 11
Control 39. 8 21.1 18.1
1. 3 O. O 2. 8

 

 

 

‘

 

 

93 (12 = 16.61)

 

 

 

 

Grouping a and H c d and e]

3 18 10

Mastery 10. 0 15. 4 5. 7

4. 9 0. 4 3. 2
4 l4 3

Non- 6. 7 10. 4 3. 8

mastery 1.1 l. 2 0. 2
35 33 11

Control 25. 3 39. 2 14. 5

3. 7 1. 0 0. 8

 

 

 

 

TABLE A. 3 (continued)

 

 

137

Grouping: Delayed mastery -- Other mastery -- Nonmastery -- Control

Critical value ( oL= .05): 12. 59

g_1 (78‘ = 17.57)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Grouping a and b c d and e
6 6 1
Delayed 3. 7 5. 7 3. 7
mastery 1. 6 0. 0 2.1
2 8 8
Other 4. 7 7. 9 5. 3
mastery l. 5 0. 0 1. 4
10 10 1
Non- 5. 5 9. 3 6. 2
mastery 3. 7 0.1 4. 4
16 34 29
Control 20. 5 35. 0 23. 5
l. 0 0. 0 l. 3
2
g (X = 12.62)
Grouping a and b c d and e
0 11 2
Delayed l. 7 7. 6 3. 7
mastery l. 7 l. 5 0. 8
3 6 9
Other 2. 3 10. 5 5.1
mastery 0. 2 1. 9 3. 0
4 9 8
NOD’ 2. 7 120 3 5. 9
mastery 0. 6 0. 9 0. 7
10 51 18
Control 10. 3 46. 4 22. 3
0.0 0. 5 0. 8

 

 

 

 

 

g_2_ (12 = 21.72)

 

 

 

 

 

 

Grouping a and bi c d and
l 10 2

Delayed 4. 2 6. 4 2. 4

mastery 2.4 2.0 0.0
2 8 8

Other 5. 8 8. 9 3. 9

mastery 2.4 0.1 6.7
4 l4 3

Non- 6. 7 10. 4 3. 8

mastery 1.1 l. 2 0. 2
35 33 11

Control 25. 3 39. 2 l4. 5

3. 7 1. 0 0. 8

 

 

 

 

93 (12 =18.53)

A

 

 

 

 

 

 

Grouping a d b c d and
4 6 3
Delayed 6. 5 3. 5 3.0
mastery 0. 3 1. 0 0. 0
6 2 10
Other 9. 0 4. 8 4.1
mastery l. 0 l. 6 8. 5
9 6 6
N011“ loo 6 5.6 40 8
mastery 0. 2 0. 0 0. 3
47 21 11
Control 39. 8 21.1 18.1
1. 3 0. 0 2. 8

 

 

 

 

TABLE A. 3 (continued)

 

 

138

29:23 36:56.55 new moan; phenom EU

a. .< mqmdh
.839, Hanna 0353 Be one .83: 33098 .333 83630 one :3 some 5 33:5: 22....

 

 

 

 

 

 

 

 

 

 

 

 

 

 

#
n . o.N 0.2 o;
v.2. flow «12 w.2 H 33030 5
we mm m A“ women—8 no: “BEEF
h. o.~ n.“ 9.2
o .2. a .8 N .2 m .2 _ 3301.0 5
mm mm 2 2 3:980 3852
gal. IllnIIIouwwE
3.980 383.582 .350 Banana 95.5
N. . w .2 m . v . o .2 m .N
v.3 0 .8 v .2 22 m .2 m.~
we mm m 2 N o H 9:330 5
320.30 you H3552
N. . m .2 m . a . o .2 v .N
e .K N .8 o .2 2.: m .2 m .N 2 33325 5
mm 2 2 w .2 11m? 320.30 H3552
2392582 23222 23.012552 23234 33232
3550 228m 526m 352 82:69 0282 2.8m 0282 @580
o .m o .o N .o in
mm 3 can New 2 2:330 E
, no no v2 mom 3:986 you .8952
o .v o .o m .o n .N
we em «2 o .02 2 man—SRO 5
mm em o2 m2 3:980 H3852
33:00 80:38.5.
22 3mm 2.2 Sam «52 am R2 2mm 2.80

 

 

 

 

 

 

3235. “Bazoém H8 33.5 83$ EU as 338.2888 :00 5x x5292...

 

139

 

APPENDIX XVI: Calculus 1 Test Results and Final Percentages

 

 

 

 

 

 

 

 

Group Student I lWr j Test [ Final
Number J l 2 3 4 Score Percent
NEM 5 18 5 7 91 76
NEM 6 20 2 6 84 72
NEM 15 20 5 7 88 87
NEM 19 20 17 4 7 95 91
NEM 25 20 6 6 99 99
NDM l 20 6 7 100 93
NDM 2 20 18 5 7 90 84
NDM 7 20 5 7 98 83
NDM 10 20 4 6 73 60
NDM ll 18 3 4 80 72
NDM 13 18 8 l 5 54 31
NDM l4 18 20 5 7 86 78
NDM 16 20 17 0 6 83 75
NDM 17 20 15 6 4 81 84
NDM 18 20 20 6 7 98 73
NDM 22 19 20 5 6 74 50
NDM 23 14 20 6 3 67 63
NDM 26 20 20 6 7 100 89
NNM 3 l7 2 6 73 77
NNM 4 10 14 l 7 56 44
NNM 8 20 5 7 91 81
NNM 9 20 13 4 6 61 W
NNM 12 20 12 3 4 57 40
NNM 20 5 13 5 6 58 58
NNM 21 20 20 2 7 76 48
NNM 24 W
SM 27 20 20 6 6 95 87
SM 31 20 6 5 98 87
SM 34 20 20 6 6 97 77
SM 36 20 20 6 6 99 88
SM 37 20 14 6 7 92 85
SM 39 20 6 5 77 78
SM 40 20 6 7 93 94
SM 42 20 4 5 93 88
SM 43 20 20 6 7 96 91
SM 44 20 6 7 91 67
SM 47 20 6 6 88 49
SM 50 20 6 4 82 68
SM 53 18 17 6 5 92 81
TABLE A. 5

Calculus I Test Results and Final Percentages

140

 

 

 

 

Group] Student I Item Number 7 I Test Final
Number 1 l 2 3 l 4 J Score Percent
SNM 28 20 4 7 86 81
SNM 29 20 4 4 80 75
SNM 30 20 3 5 87 73
SNM 32 20 4 6 75 74
SNM 33 20 12 4 7 86 78
SNM 35 20 4 6 83 78
SNM 38 20 15 6 6 82 48
SNM 41 20 20 6 7 98 83
SNM 45 73
SNM 46 20 20 5 6 91 63
SNM 48 20 8 0 6 39 W
SNM 49 20 6 7 90 89
SNM 51 17 6 6 69 73
SNM 52 20 5 7 91 72
SNM 54 18 20 4 7 82 78
C 55 20 4 0 69 73
C 56 18 4 6 67 59
C 57 20 18 6 3 93 W
C 58 6 6 5 74 75
C 59 20 20 5 5 93 82
C 60 20 18 3 5 88 67
C 61 20 20 5 7 99 92
C 62 20 3 6 96 91
C 63 20 18 2 6 86 83
C 64 18 5 5 83 76
C 65 85
C 66 10 4 4 75 69
C 67 17 5 2 90 89
C 68 16 4 7 67 61
C 69 20 14 5 4 80 70
C 70 20 10 6 7 88 92
C 7 l 18 18 5 5 80 75
C 72 20 20 5 7 94 95
C 73 20 20 5 1 83 78
C 74 20 15 5 5 79 75
C 75 20 4 7 94 95
C 76 18 20 3 4 75 68
C 77 6 0 0 61 82
C 78 20 4 6 69 W
C 79 20 18 6 6 97 87
C 80 17 5 2 72 82

 

 

 

TABLE A. 5 (continued)

"

‘ .
$37M-
1' 2

 

141

A 7 - _

 

 

 

 

Group Student 1 Item Number 1 Test Final
Number I l 2 3 4 1 Score Percent
C 81 19 12 4 6 60 65
C 82 20 3 6 89 82
C 83 20 17 6 5 91 85
C 84 20 20 6 6 84 90
C 85 20 18 6 6 92 88
C 86 20 20 6 6 91 83
C 87 12 O 2 25 W
C 88 20 18 6 6 93 80
C 89 20 20 6 6 99 90
C 90 20 20 6 5 96 88
C 91 20 20 6 7 96 86
C 92 20 20 4 6 94 76
C 93 20 18 5 7 83 71
C 94 18 6 7 94 85
C 95 20 20 6 6 87 79
C 96 20 17 6 6 90 73
C 97 20 17 0 6 90 97
C 98 20 S 5 87 82
C 99 20 20 2 6 93 87
C 100 16 18 2 1 56 67
C 101 18 20 6 6 89 74
C 102 w
C 103 15 15 4 6 78 69
C 104 W
C 105 77
C 106 20 6 7 100 95
C 107 10 3 5 80 67
C 108 20 20 6 7 98 95
C 109 20 20 3 2 77 61
C 110 20 6 5 98 83
C 111 20 17 3 7 83 80
C 112 20 6 7 100 98
C 113 20 20 6 3 86 81
C 114 8 S 5 57 60
C 115 18 6 l 78 85
C 116 20 20 4 6 86 78
C 117 4 3 3 69 77
C 118 14 3 5 71 67
C 119 14 14 5 5 72 76
C 120 20 5 7 95 83

 

TABLE A. 5 (continued)

 

142

 

 

 

 

Group Student L Item Number 1 Test Final
Number 1 T 2 fl 4 1 Score Percent
C 121 20 6 7 96 91
C 122 20 16 5 7 86 81
C 123 20 6 3 96 92
C 124 20 6 4 88 85
C 125 16 3 7 68 59
C 126 17 O 2 59 64
C 127 18 16 6 7 94 81
C 128 20 6 6 97 86
C 129 17 5 5 85 82
C 130 20 4 6 88 96
C 131 20 5 5 91 86
C 132 2 l7 6 6 49 W
C 133 20 6 7 100 91
C 134 O 4 7 37 70
C 135 17 20 2 5 85 75
C 136 20 6 6 93 85
C 137 17 3 0 79 71
C 138 75
C 139 20 3 5 89 80

 

TABLE A. 5 (continued)

 

 

143

APPENDIX XVII: Contingency Tables" for Chi Square Test Statistics
Computed for the Sum of Items Two, Three, and Four the
Calculus I Test

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Grouping Score Grouping 1 Score
0-29 30-33 0-29 30-33
2 3 5 22 30 52
NEM 2.7 2.4 Treatment 27.6 24.4
.2 .2 .0382 1.126 1.274 .3939
5 8 13 48 32 80
NDM 6.9 6.2 Control 42.4 37.6
5 .5 .0992 .73 .828 .6061
6 l 7 Total 70 i2 132
NNM 3.7 3.3
1.4 1.6 .0534 X2 = 3.960
2
3 10 13 (l) = 3.841
SM 6.9 6.2 26°05 -
2.2 2.3 .0992
6 8 l4
SNM 7.5 6.6
.3 .3 .1069
48 32 80
Control 42. 7 37. 9
.7 .9 .6107

 

 

 

 

 

 

 

 

Total 70 62 132
X = n.3**

2 _
X.05(5) - 11.07

*Each cell contains observed values, expected values, and chi square
partial sums.

"Only sixty-seven percent of the cells have expected values exceeding
five.

TABLE A.6

Chi Square Values for Calculus I Test

 

L.‘ t _b

I

144

 

 

 

 

 

 

 

 

 

 

‘f‘fi .x- — ’

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Grouping ‘ Score Grouping Score .
0-29 30-33 0-29 30-33
21 31 5 8 l3
Mastery 16. 4 14. 6 Delayed 6. 9 6.1
2. 5 2. 8 .2348 mastery . 5 . 6 . 098
12 9 21 5 13 18
Non- 11.1 9.9 Other 9.5 8.5
mastery .1 l .1591 mastery 2.1 2. 4 .1364
48 32 80 12 9 21
Control 42. 4 37. 6 Non- 11.1 9. 9
. 7 . 8 .6061 mastery .l .l .1591
Total 70 62 32 48 32 80
2 Control 42. 4 37. 6
X = 7 . .7 .3 .6061
1205(2) = 5.99 Total 70 62 132
' 2
'7. = 7.3
Grouping Score ‘ 2
0-29 '30-33 X.05(3) = 7.8
13 12 25

TreatmentA 13.25 11.724
0.0 0.0 .1894

 

9 20 29
Treatment B 15.37 13.62]
2.64 2.981 .2197

 

 

 

 

 

 

48 32 80
Control 42. 7 37. 9 . 6107
. 7 . 9
Total 70 62 132

 

 

X2 = 7. 233

7‘ 2 _
TABLE A.6 (continued)

 

 

BI BLIOGRAPHY

5‘79

 

.‘ A
‘4;-

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