STUDENTS’ UNDERSTANDING OF THE CONCEPT OF CHAIN RULE IN

FIRST YEAR CALCULUS AND THE RELATION TO THEIR

UNDERSTANDING OF COMPOSITION OF FUNCTIONS

A Thesis

Submitted to the Faculty

of

Purdue University

by

James Franklin Cottrill

In Partial Fulfillment of the

Requirements for the Degree

of

Doctor of Philosophy

May 1999

ii

To Cecelia, who arrived when this work was to begin;

To Stephen, whose arrival delayed its completion;

and

To Julie, my partner in love, life and faith.

iii

ACKNOWLEDGMENTS

I wish to thank my colleagues in RUMEC for their support, encouragement, and

assistance as I learned what it means to conduct research in this field. My special

thanks to Julie Clark, Dave DeVries, George Litman, Karen Thomas, and Draga

Vidakovic for their friendship and insights.

I would like to thank Professors Keith Schwingendorf, David Mathews, and Guer-

shon Harel for their willingness to serve on my committee.

I have deep gratitude for Professor Ed Dubinsky and his role in my life. As my

thesis advisor, he helped clarify my thoughts and patiently helped me through the

process. As my colleague, he was a brilliant example of how one can be a great teacher

of mathematics and researcher in education. Also, I thank him for his friendship and

support in my family’s life.

My children’s grandparents have endured years of separation with great patience

and love. I thank them for their support.

Finally, my special thanks to my wife, Julie Cottrill, for her patience and sup-

port in a project that involved relocating, transcribing, and months without visible

progress. Thank you.

iv

TABLE OF CONTENTS

Page

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

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Historical background . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 An overview of the development of calculus . . . . . . . . . . . 3

1.2.2 The Chain Rule . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 The RUMEC study . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4 An overview of this study . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Theoretical Perspective and Research Paradigm . . . . . . . . . . . . . . . 9

2.1 Theoretical perspective . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Schema development . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 Paradigm for research . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.4 Initial genetic decomposition . . . . . . . . . . . . . . . . . . . . . . . 11

3 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.1 On functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2 Students’ understanding of function . . . . . . . . . . . . . . . . . . . 13

3.3 Composition of functions . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.4 Difficulties with differentiation . . . . . . . . . . . . . . . . . . . . . . 15

4 Phase 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.1 Methods: Subjects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.2 Instrument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4.3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4.3.1 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

v

4.3.2 Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.3.3 Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.3.4 Chain rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.4 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5 Results: Phase 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

5.1 Analysis 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

5.2 Analysis 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5.3 An alternate analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

6 Phase 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

6.2 Subjects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

6.2.1 Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

6.3 Instrument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

6.4 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

6.5 Where the RUMEC study left us . . . . . . . . . . . . . . . . . . . . 34

7 Results: Phase 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

7.1 The triad description revisited . . . . . . . . . . . . . . . . . . . . . . 36

7.1.1 Eli [A125] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

7.1.2 Tim [A134] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

7.1.3 Al [A137] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

7.1.4 Ray [A123] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

7.1.5 Peg [A132] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

7.1.6 Jack [A113] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

7.2 Using versus understanding . . . . . . . . . . . . . . . . . . . . . . . 50

7.2.1 Coding the aspects of understanding . . . . . . . . . . . . . . 50

7.2.2 Comparing the codes . . . . . . . . . . . . . . . . . . . . . . . 54

8 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

8.1 Revised genetic decomposition . . . . . . . . . . . . . . . . . . . . . . 56

8.2 Understanding of composition and understanding of the chain rule . . 58

vi

8.3 Comparing instructional methods . . . . . . . . . . . . . . . . . . . . 58

8.3.1 Understanding the chain rule . . . . . . . . . . . . . . . . . . 58

8.3.2 Using the chain rule . . . . . . . . . . . . . . . . . . . . . . . 59

8.3.3 Finding a balance . . . . . . . . . . . . . . . . . . . . . . . . . 60

8.4 Quantitative versus qualitative data . . . . . . . . . . . . . . . . . . . 60

8.5 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

8.6 Instructional strategies . . . . . . . . . . . . . . . . . . . . . . . . . . 61

LIST OF REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

APPENDICES

A Questionnaire from Phase 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 69

B Phase 1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

C Interview Guide for Phase 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 84

C.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

C.2 Review of function and composition . . . . . . . . . . . . . . . . . . . 84

C.3 Classifying the 10 derivative problems . . . . . . . . . . . . . . . . . . 85

C.4 Establishing the notion of chain rule . . . . . . . . . . . . . . . . . . 85

C.5 Extending the notion . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

VITA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

vii

LIST OF TABLES

Table Page

5.1 Descriptive statistics for variables . . . . . . . . . . . . . . . . . . . . 28

5.2 Correlation statistics for variables . . . . . . . . . . . . . . . . . . . . 28

5.3 Regression model with PGPA . . . . . . . . . . . . . . . . . . . . . . 29

5.4 Regression model without PGPA . . . . . . . . . . . . . . . . . . . . 30

5.5 Comparison of courses . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5.6 Comparison of Traditional and any C4L . . . . . . . . . . . . . . . . . 31

5.7 Comparison of first quarter course . . . . . . . . . . . . . . . . . . . . 31

6.1 Matrix of interviews . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

7.1 Aspects of understanding . . . . . . . . . . . . . . . . . . . . . . . . . 55

APPENDICESTable

B.1 Distribution of codes by item . . . . . . . . . . . . . . . . . . . . . . 81

B.2 Student codes in tuples . . . . . . . . . . . . . . . . . . . . . . . . . . 82

B.3 Combined code data . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

viii

ABSTRACT

Cottrill, James Franklin, Ph.D., Purdue University, May, 1999. Students’ Under-standing of the Concept of Chain Rule in First Year Calculus and the Relation toTheir Understanding of Composition of Functions. Major Professor: Ed Dubinsky.

The present study is a follow-up to a study conducted by this author with seven

others. That study (Clark et al., 1997) proposed the use of Piaget and Garcia’s

triad mechanism to describe the development of a schema, specifically with respect

to students’ understanding of the chain rule. This study was designed to collect

data to test this theory. Both quantitative and qualitative methods were employed.

Data were collected via a questionnaire on a group of calculus students’ (n = 34)

knowledge of and skill with function, composition of functions, differentiation and

chain rule. The data were analyzed to investigate how their performance on the

composition of functions items related to that of the chain rule. No strong correlation

is reported. Some of the subjects had experienced a reform calculus course which used

computer experiences and cooperative learning. A comparison was made between the

two groups, although no overall differences are reported. Follow-up interviews based

on the questionnaire responses were conducted with six subjects. These results are

positive in the sense that the triad description fits with the data. We present detailed

descriptions of the Intra-, Inter-, and Trans- levels of the development of the chain

rule schema. Some evidence is presented to support the notion that understanding

of composition of functions is key to understanding the chain rule. The type of

instruction was a factor in how a student performed on these tasks. The differences

between the types seem to be related to the difference between using the chain rule

and understanding the chain rule, and an explanation of these differences is offered.

The need for collecting differing types of data is established. Two students scored

high quantitatively but did not perform well in the interview, while another student

failed the quantitative work but demonstrated a high level in the interview. Finally,

the use of writing as a pedagogical tool is recommended based on the results.

Chapter 1: Introduction

This study looks at a particular concept in calculus, the chain rule, in the first

year calculus course. The chain rule is the underlying concept in many applications

of calculus: implicit differentiation, solving related rate problems, and solving differ-

ential equations. The rule states that if g is differentiable at c and f is differentiable

at g(c), then the composite function f ◦ g given by f ◦ g(x) = f(g(x)) is differentiable

at c and that (f ◦ g)′(c) = f ′(g(c)) · g′(c). Conventional wisdom holds that students’

conception of the chain rule (as with other rules) is that of symbol manipulation.

This conception appears to be a straight-forward manipulation of symbols which can

easily be applied in problem situations. However such a situation carries a heavy

requirement for the function to be given by an expression, fostering students’ tenden-

cies toward instrumental understanding. Even in such symbolic problem situations,

many students have difficulty in recognizing the need to apply the chain rule or in

applying it.

In discussing students’ understandings of the elements of the dihedral group,

Zazkis, Dubinsky, and Dautermann (1996) propose the VA model of mathematical

thinking in which visual and analytic modes of thought are mutually dependent in

problem-solving. The concept of derivative offers another example of a situation in

which a student might alternate between the visual (slope of a tangent to the curve)

and analytic (rate of change) interpretations in building her or his understanding

(Asiala, Cottrill, Dubinsky, & Schwingendorf, 1997).

From a visual perspective, the rule might be interpreted as the derivative of the

inner function compensating for, or scaling, the derivative of the outer. Such a con-

ception lends itself to understanding the rule in relation to the applications mentioned

above. This was the understanding of both Newton and Leibniz as they developed

the calculus (Edwards, 1979). This visual perspective is not presented in current

pedagogical practices, perhaps due to the lack of a reasonable procedure for produc-

ing the graph of a composition given the graphs of the component functions. This

loss of a visual approach may explain the difficulty many students demonstrate when

working with the chain rule. The fundamental nature of the rule demands that the

beginning student must not only be capable of manipulating the symbols, but must

1

2

have a conceptual understanding of the chain rule. This project is an investigation

to describe how students come to understand the chain rule, and via this description,

propose activities which could bring students to the point of being better able to

build this understanding.

1.1 Purpose

The purpose of this study is to attempt to determine the correlation between

a student’s ability to deal with composition of functions and with using the chain

rule successfully. The data is also used to compare two instructional approaches to

the chain rule: a traditional lecture-recitation method and an innovative “reform”

method. Finally, an attempt is made to describe the understanding of the chain rule

concept on the part of the interviewed students.

It is supposed to answer the questions:

• How does a student come to understand the chain rule?

• In what way does the student’s understanding of (de)composition of functions

factor in the chain rule?

• With reference to the results of the RUMEC study (Clark et al., 1997), is

the chain rule concept described well via a schema, and do the current results

support the reported schema?

This study proposes to test the hypothesis: “The chain rule in and of itself is

not difficult for students to apply; rather it is the issue of composition of functions.”

Also, it will be the second attempt of the first cycle through the research paradigm,

which will result in a revised genetic decomposition.

Thus, the overarching purpose of this study is to test APOS Theory (described in

Chapter 2). The theory suggests that certain kinds of mental constructions should

be evident when students work and reflect on chain rule problems. It also states that

these constructions will be organized in a schema that should have certain character-

istics at various levels of development. This study analyzes the data to determine if

the predictions of the theory hold up to scrutiny.

Finally, since APOS Theory is an evolving work, an opportunity to augment or

extend the theory may arise in this work, as it did in the RUMEC study.

1.2 Historical background

In attempting to gain an understanding of how students construct their concept of

the chain rule, it is beneficial to consider its historical development and its proof. The

3

chain rule arose quite naturally in the development of calculus by both Newton and

Leibniz as they worked with infinitesimals. The rule has evolved with the calculus

as Euler brought it to bear on functions and as Cauchy and Weierstrass developed

analysis. The proof has also evolved from a symbolic tautology to an argument based

on the limit of a difference quotient. The historical perspective presented here may

offer an explanation of students’ difficulty with the chain rule.

1.2.1 An overview of the development of calculus

Two types of problems from antiquity served to instigate the development of

calculus. One problem was to rectify a curve, that is, to construct a line with the

same length as that of a given curve. The other was the quadrature of a curve, or

the area beneath it. It was not the case that such problems were unsolved; actually

many specific curves had solutions for both types of problems. The ancient Greeks

used the method of exhaustion and the method of Archimedes (c. 200 b.c.) to find

many volumes and surfaces. The Greeks made use of indivisibles as they dealt with

concepts of infinite processes. What was lacking was a general method for any curve.

(Boyer, 1969)

The work of Oresme, Galilei, and Cavalieri (c. 1635) extended the use of indivis-

ibles as the quadratures of families of curves were obtained. These mathematicians

approached the problems from a geometric point of view. During roughly the same

period, mathematicians such as Descartes, Wallis, and Fermat (c. 1655) were leading

an analytical revolution. The birth of coordinate geometry allowed the creation of

an arithmetic of infinitesimals (indivisibles). Wallis reset the work of Cavalieri in in-

finitesimal analysis. Fermat extended this and invented the method of differentiating

and integrating monomial expressions which are in use today. (Boyer, 1959)

Fermat did not recognize the inverse relation between these processes, nor did

those who followed his work. It was the genius insight of Newton and Leibniz to

deduce from this work (and the work on infinite sums of Gregory) the Fundamental

Theorem of Calculus. They saw and explicitly stated this inverse relation between

differentiating and integrating. Moreover, they showed that area, volume, and related

problems which had been solved by summation were reduced to anti-differentiation.

The distinction between these insights is indicated by the fact that the Fundamental

Theorem is stated in two parts. (Kline, 1972)

Newton (c. 1668) perceived a curve as the path of a moving point and spoke of

fluents x and y as the horizontal and vertical components of this motion. He defined

the rate of change of a fluent to be a fluxion, with the notation x and y. By interpreting

4

increasingly smaller increments in x and y, he found the derivative to be the ratio

of the fluxions which is also the slope of the tangent to the curve. It is interesting

to note that although Newton developed his calculus by working analytically, his

first published account of it (the Principia Mathematica) is presented in a classical

geometric setting. (Rickey, 1987)

The approach to calculus of Leibniz (c. 1673) was to view these infinitesimals

as mathematical objects, which he called differentials. In working on quadratures,

he noted the summation of the area under a curve is related to its anti-derivative.

Leibniz developed a careful notation for the calculus, much of which persists today.

He also derived many of the rules for computing derivatives and integrals, notably

the chain rule. (Kline, 1972)

Following this seminal work in the calculus, the next major advance was that of

Euler (c. 1748) and his incorporation of the concept of function. This was followed

by a long period in which the methods of the calculus were used to solve problems

and were refined. However, the notion of the infinitesimal had not been satisfactorily

defined by Newton or Leibniz. This lack incited Berkeley and Lagrange to call for a

rigorous formulation of calculus in the late 18th century. The issue was settled by

Cauchy and Weierstrass (c. 1821) with the development of analysis as a mathematical

field. (Edwards, 1979)

1.2.2 The Chain Rule

The current formulation of the chain rule, as stated above, is a result of this

development of real analysis. The concept itself—the derivative of a composition

of functions—was intuitively obvious to both Newton and Leibniz. As C. Edwards

(1979) points out discussing the work of Newton:

The tangent and area problems emphasize the importance of systematic

procedures for differentiation (the calculus of y/x, given f(x, y) = 0) and

anti-differentiation (the converse). Newton exploited the facility for dif-

ferentiation and anti-differentiation by substitution methods—equivalent

to what we call the chain rule and integration by substitution —that is

essentially “built into” the calculus of fluxions. (p. 196)

Newton did not set the chain rule out as a theorem or procedure. It was a natural

algorithm employed in many of his examples. In a similar manner, the notation of

Leibniz led him to consider the chain rule as intuitive as simplifying a product of

5

fractions. If z = z(u) and u = u(x), then

dz

dx=

dz

du· du

dx.

Again this process was not even given a name. In Leibniz’ union of geometric and

analytical thinking using infinitesimals, the chain rule is implicitly defined.

The work of Euler and Cauchy precipitated the chain rule in its present form.

By changing the objects of calculus from curves in space to arbitrary functions, the

intuitive nature of the chain rule was lost. It became a theorem to be proved. Cauchy’s

proof of the rule follows:

Now let z be a second function of x, bound to the first y = f(x) by the

formula

z = F (y).

z or F [f(x)] will be that which one calls a function of a function of a

variable x; and, if one designates the infinitely small and simultaneous

increments of x, y, and z by ∆x, ∆y, ∆z, one will find

∆z

∆x=

F (y + ∆y)− F (y)

∆x=

F (y + ∆y)− F (y)

∆y· ∆y

∆x,

then, on passing to the limits,

z′ = y′F ′(y) = f ′(x)F ′[f(x)].

(Cauchy, cited in Edwards, 1979, p. 313)

Edwards notes that Cauchy overlooks the possibility that ∆y = 0 in the work cited.

In fact, Cauchy does not treat the functional relationship of ∆y and ∆x, and so the

proof is false.

Often in mathematics, the proof of a theorem helps to convey the fundamental

concepts on which it is built. That is, by experiencing the proof, a student will

construct a relational understanding of the theorem. Unfortunately, the method of

proof for the chain rule is generally interpreted as an algebraic trick—multiply by 1

(∆y/∆y). Even thoughF (y + ∆y)− F (y)

∆y· ∆y

∆x

is simply Leibniz’s dz/du · du/dx written in terms of difference quotients, the geo-

metric interpretation is lacking. This may be related to our inability to graphically

6

demonstrate the composition of two functions given by their graphs. The difficulties

that students have with the chain rule may be explained by its history, that is, the

function and composition ideas came later and clouded the intuitive idea of Newton

and Leibniz.

1.3 The RUMEC study

In a previous study (Clark et al., 1997), eight members (including this author) of

an academic community of researchers1 working with a common theoretical frame-

work attempted to explore calculus students’ understanding of the chain rule and its

applications. Specifically, the students were asked, as part of a clinical, task-based

interview situation, to solve the following four problems and to explain what they

were doing. The students were prompted to use the chain rule explicitly in the first

problem if it was not chosen as the method of solution. Prompts were given in the

second problem relating to the chain rule if the student indicated difficulty. The

problems were:

1. Let f be the function given by f(x) = (1− 4x3)2. Compute f ′.

2. Let F be the function given by F (x) =∫ sin x

0et2 dt. Compute F ′. Explain what

you did.

3. Let A be a real number. Given that the following relation defines a function,

x√

y + y√

x = A, find its derivative.

4. A ladder A feet long is leaning against a wall, but sliding away from the wall at

the rate of 4 ft/sec. Find a formula for the rate at which the top of the ladder

is moving down the wall.

A preliminary analysis of the responses to these questions indicated that although

almost all of the participants were able to use the chain rule to correctly solve the

first problem, those who had difficulty with the other problems were unable to apply

the chain rule, even when prompted.

Clark et al. (1997)2 attempted to analyze the concept of chain rule based on

an Action-Process-Object-Schema (APOS) theoretical framework which is described

in the next chapter. Using the framework to analyze the mental constructions of

1The group is known as the Research in Undergraduate Mathematics Education Community orRUMEC.2For the rest of the paper, I will refer to this study as the RUMEC study.

7

the students, the RUMEC study set out to describe each of the 41 participants as

working with an action, process, or object conception of the chain rule (at the time

of the interview). In the negotiation of differences between researchers’ analyses, an

issue arose regarding whether a student’s general power rule (as used in Problem 1,

above) was her or his chain rule. The issue could not be resolved within the given

theoretical framework, and so the RUMEC study failed to meet its objectives.

An analysis of the problem situation concluded that the concept of the chain rule

is best described by a schema. Although the framework treated the mental construc-

tion of a schema, it was found to be insufficient to account for the data collected.

Specifically, this insufficiency was that a schema was defined but its development was

not described. The RUMEC study provided the description of schema development

in terms of the chain rule. In this way, it extended the previous APOS framework

(Clark et al., 1997).

The RUMEC study used the interview data to illustrate schema development and

attempted to return to the original objective of the project (describing each of the

41 participants). However, since the data were collected via interviews based on

the previous genetic decomposition, the “right” questions had not been asked. In

particular, no attempt was made to have students discuss the similarity/difference

of Problems 1 and 2. Also, these chain rule problems were at opposite ends of the

spectrum in terms of complexity. The current work is a follow up to the RUMEC

study. Incorporated in its design are a number of examples of problem situations

involving the chain rule, as well as opportunities to explore the students’ perceived

relationships between the situations.

1.4 An overview of this study

This study consists of two phases, one quantitative and one qualitative. Phase 1

involved the collection of data via questionnaires administered to 34 students. Phase 2

used in-depth, task-based interviews with six of the students from Phase 1.

The purpose of Phase 1 is an attempt to determine the correlation between a

student’s ability to deal with composition of functions and with using the chain rule

successfully. Data were collected to establish an individual’s baseline understanding

of function and composition, as well as her or his relative proficiency in mathematics.

Integrated in this phase was a comparison between two methods of teaching calculus

regarding the effectiveness of the instructional treatment of the chain rule.

Students were selected from second quarter calculus by requesting volunteers.

Data were collected on an instrument constructed for this study which assessed the

8

participant’s facility with (a) functions, (b) composition of functions, (c) differentia-

tion, and (d) the chain rule. Each participant was asked to authorize access to her

or his academic record for the researcher to gather data, such as SAT scores and

predicted indices.

The questionnaire items were scored using a 5 point rubric based on general guide-

lines adapted from Carlson (1998). These codes along with the predictor data were

tested for correlational significance. A second analysis explored any differences be-

tween the two groups (traditional versus C4L). Analysis of variance was used to

determine if there exists a significant difference in sample means of the chain rule

scores.

The second phase of the study involves interviews with participants from the first

phase to attempt to describe how these students construct the concept of the chain

rule. The six participants in the interviews were chosen based on their scores in

Phase 1, covering a range of chain rule scores and a range of overall scores. The

interview follows a guide designed to elicit the student’s understanding of the rule,

based on tasks from the previous instrument. This aspect of the study is based on

the following theoretical perspective and research paradigm.

Chapter 2: Theoretical Perspective and Research Paradigm

As Phase 2 of the study is of a qualitative nature, it is necessary to discuss our

theoretical perspective and research paradigm. This discussion is a brief overview; a

complete description can be found in Asiala et al. (1996).

2.1 Theoretical perspective

The APOS theory is based on the following statement:

An individual’s mathematical knowledge is her or his tendency to respond

to perceived mathematical problem situations by reflecting on problems

and their solutions in a social context and by constructing or reconstruct-

ing mathematical actions, processes and objects and organizing these in

schemas to use in dealing with the situations. (Asiala et al., 1996, p. 7)

Three basic types of knowledge—actions, processes, and objects—are observed,

which are organized into structures called schemas. An action is any repeatable phys-

ical or mental manipulation of objects to obtain other objects. It is a transformation

that is a reaction to stimuli that the individual perceives as external. In the case of

function, for example, a student indicates an action conception if he or she is limited

to requiring an explicit formula in order to interpret a situation as a function. A

student in such a case would be unable to do much with the function, apart from

evaluating it at a point. This student would have great difficulty dealing with situa-

tions involving composition of functions (Breidenbach, Dubinsky, Hawks, & Nichols,

1992).

As a student reflects on an action, it is interiorized and becomes a process. The

student perceives the action as part of her or him and has control over it. For func-

tions, this means the student can think of a function as receiving input, performing

some operation, and returning output. This can be done mentally, without actually

performing the operations on the input (Breidenbach et al., 1992).

As a student realizes that an action can be brought to operate on a process,

the process is encapsulated to become an object. The object conception can be de-

encapsulated back to the process as needed. In the case of functions, this encap-

sulation/de-encapsulation arises as one considers manipulations of functions such as

adding, multiplying or forming sets of functions.

9

10

A schema is a coherent collection of actions, processes, objects and other schemas

that is invoked to deal with a new mathematical problem situation. A schema can be

thematized to become another kind of cognitive object to which actions and processes

can be applied. By consciously unpacking a schema it is possible to obtain the original

processes, objects and other schemas from which the schema was constructed. (Note

that this is the description of schema prior to the RUMEC study which extended

it as described in the next section.) A genetic decomposition is an attempt by the

researcher to describe the objects and processes in some set of students’ schemas.

While this description is not unique for any given concept, it is useful in guiding

instructional design and investigations (Asiala et al., 1996).

2.2 Schema development

The triad mechanism, introduced by Piaget and Garcia (1989), distinguishes three

stages in the development of a concept: Intra, Inter and Trans. The Intra stage is

characterized by the focus on a single object in isolation from any other actions,

processes, or objects. With the chain rule, for instance, the student may have a

collection of rules for finding derivatives including some special cases of the chain rule

such as the general power rule and perhaps even the general formula, but might not

recognize the relationships between them.

The Inter stage is characterized by recognizing relationships between different

actions, processes, objects and/or schemas. It is useful to call a collection at the

Inter stage of development a pre-schema. In the case of the chain rule, the student

may begin to collect these special cases and recognize that in some way they are

related.

Finally the Trans stage is characterized by the construction of a coherent structure

underlying some of the relationships discovered in the Inter stage of development. At

the Trans stage of the chain rule, the student would link function composition to

differentiation, and recognize various instantiations of the chain rule as linked in that

they follow from the same general rule through function composition.

It is worth noting that it is only when a schema reaches the Trans stage of de-

velopment that it can properly be referred to as a schema. The reason is that at the

Trans stage the underlying structure is constructed through reflecting on the rela-

tionships between the various objects from the earlier stages. This structure provides

the necessary coherence in order to identify the collection as a schema, that is, it is

a coherent whole. This coherence is deciding what is in the scope of the schema and

what is not.

11

2.3 Paradigm for research

This research in mathematics education is based on Dubinsky’s method (Asiala

et al., 1996) consisting of three components in a cycle: theoretical analysis; design

and implementation of instruction; and data collection and analysis. The theoretical

analysis component influences the instruction component in that the activities and

tasks used in instruction are based on a genetic decomposition. In turn, the imple-

mentation of the instruction influences the empirical data component as those who

have been instructed are studied. Data are collected in as many forms as reasonable

in order to gain insight. Tests and homework, direct observation, questionnaires and

clinical interviews are possible sources for data collection. The data analysis leads

to a revision of the current genetic decomposition; thus the empirical data affect the

theoretical analysis.

On the other hand, the theory influences the empirical data since the data col-

lection instruments are constructed based on the genetic decomposition. These in-

struments are designed to also get at issues involving the “goodness” of the genetic

decomposition. Typically, two genetic decompositions will be presented in a study;

the preliminary one used to design the instruction and the data collection, and the

revised genetic decomposition based on the observations and analysis.

2.4 Initial genetic decomposition

The results of the RUMEC study provide evidence that understanding the chain

rule involves the building of a schema. The schema consists of a function schema

coordinated with a differentiation schema. The student’s function schema would have

at least a process conception of function, function composition and decomposition

(breaking a composite function into two or more component functions). This is linked

to her or his differentiation schema which includes the rules of differentiation at least

at the process level (Clark et al., 1997). Note that a distinction is made between

a differentiation schema and the derivative schema of a student. The former, which

deals with taking derivatives, is a part of the latter which also contains notions of

interpretations and representations of the derivative (Asiala, Cottrill, Dubinsky, &

Schwingendorf, 1997).

The chain rule schema develops through the levels of the triad; Intra, Inter and

Trans. In the first level, the Intra- level, the student has a collection of rules for

finding derivatives in various situations, but has no recognition of the relationships

between them. This collection may include some special cases of the chain rule, and

perhaps even the general formula which is perceived as a separate rule rather than a

12

generalization of the others. The Inter- level, is characterized by the student’s ability

to begin to (mentally) collect all different cases and to recognize that these are related.

At this stage the collection of elements in the chain rule schema is being formed, and

the collection is called a pre-schema. At the Trans- level a student has constructed the

underlying structure of the chain rule. He or she must link function composition and

decomposition to differentiation, and recognize various instantiations of the chain rule

as linked in that they follow from the same general rule through function composition.

The elements in the schema must move from being described essentially by a list to

being described by a single rule (Clark et al., 1997).

Chapter 3: Literature Review

3.1 On functions

Functions are a key component in secondary and undergraduate curricula (see,

for instance, the foreword of Harel & Dubinsky, 1992) and much has been written

on the learning and teaching of the concept. Leinhardt, Zaslavsky and Stein (1990)

present an exhaustive review of the research papers which focus on functions, graphs

and graphing. The studies are organized by three issues: task, student learning,

and teaching techniques. The issue of task is discussed in terms of the action of

the learner, the situation in which the task is set, the variables involved and their

nature, and the focus of attention in the task. The issues of student learning are

presented as intuitions and as misconceptions which the students bring to a task.

Finally, the teaching techniques discussed are entry to the topic, sequencing the tasks,

explanations in the learning, and examples and representations.

Ferrini-Mundy and Graham (1991), Tall (1992), Even (1993), Thompson (1994b),

and Wilson and Krapfl (1994) offer more recent reviews of research on functions.

These note that function is a dichotomous concept—perceived by the student as a

process or as an object (see also Sfard, 1992). Harel and Dubinsky (1992) add that

students start with an action conception of function prior to moving to process and

object. These reviews agree that the concept of function is important to mathematics

and not well understood by our students. These studies and those about which they

report were searched for evidence of explorations of composition of functions and for

discussions of the chain rule. While many of the studies of students’ understanding

of function relate to this study, little was found regarding students’ understanding of

composition of functions explicitly, and no studies were found on their understanding

of the chain rule.

3.2 Students’ understanding of function

Many studies observed students’ understanding of function by presenting ques-

tionnaires of various function situations for the student to interpret (Even, 1988,

1993; Vinner & Dreyfus, 1989; Breidenbach et al., 1992; Dubinsky & Harel, 1992;

Schwingendorf, Hawks, & Beineke, 1992; Hitt, 1993; Ferrini-Mundy & Graham, 1994;

13

14

Lauten, Graham, & Ferrini-Mundy, 1994; Wilson, 1994; and Carlson, 1998). Typi-

cally these studies ask the student to give her or his definition of function, and then

they present many examples of situations to the student. The situations generally

range over graphs, tables, expressions, equations, sets of ordered pairs, parametric

equations, and finite sequences. In some cases, only verbal descriptions of the situa-

tion are provided. The student is then to decide in each case if a function is present

in the situation. Often, the student is asked to relate the answer back to the defini-

tion of function previously given, either at the time of the questionnaire or in a later

interview. These studies conclude that the notions students hold about function are

often different from what they give as a definition. Also demonstrated are the specific

types of functions which present difficulty to students.

Breidenbach et al. (1992), Dubinsky and Harel (1992), and Schwingendorf et

al. (1992) suggest that these difficulties indicate the lack of a process conception

of function. They offer evidence of an instructional approach which seems to help the

students construct this process. Even (1988, 1993), Hitt (1993), and Wilson (1994)

focus on the concept of function found in mathematics teachers. They offer rather

disturbing results as the teachers exhibited very limited conceptions of functions (see

also Vinner & Dreyfus, 1989). Ferrini-Mundy and Graham (1994) and Lauten et

al. (1994) provide descriptions of the students’ function concept and suggest methods

of employing graphing calculators as aids in learning. Carlson (1998) compares three

groups of students at different stages of their collegiate careers. She finds a very

narrow view of function is held by high-achieving second semester calculus students.

She also concludes that the concept of function develops slowly, and this development

is facilitated by reflection and constructive activities.

Other researchers investigated the concept from a problem-solving and graphi-

cal point of view. They describe methods of overcoming difficulties with functions

by using computer software and problem-solving techniques. These reports gener-

ally offer anecdotal evidence as support for their views. Confrey (Confrey & Smith,

1994; Confrey, Smith, Piliero, & Rizzuti, 1991) offers a computer environment called

Function Probe for student use in problem-solving situations. In a similar manner,

Schwarz and Bruckheimer (1990) use an environment Triple Representation Model,

Cuoco (1994) proposes the use of Logo and ISETL, and Goldenberg (1988) employs

the Function Analyzer. These environments allow the student to have control over a

function by switching between representations, exploring covariation, and changing

individual parameters.

15

3.3 Composition of functions

One of the very few papers that discuss the composition of functions is a teaching

methods paper (one which does not present any supporting observations of students)

in which Alson (1992) discussed a qualitative approach to graphing functions. He

defines two operators on the graphs which are essentially composition operators, one

for the domain and one for the range of the graph. His treatment offers insight into

the graphical interpretation of the composition of elementary functions. However, as

an instructional approach, the students must start with a rich concept of function.

Other methods may allow the student to develop this richness while constructing the

idea of composition.

Ayers, Davis, Dubinsky, and Lewin (1988) present evidence that computer experi-

ences induce reflective abstraction as students learn about functions and composition.

Students worked in the microworld of the Unix operating system, using shell scripts

and pipes to represent functions. The authors present a genetic decomposition of

composition and probe the process conception of composition in their investigation.

They indicate that in order for a student to succeed in the task of de-composing a

function—given descriptions of two functions, H, G, find F such that H = F ◦G—he

or she must have a process conception of function and of composition.

Dubinsky (1991) further discussed the issue of composition and reflective abstrac-

tion. Two aspects of reflective abstraction, coordination and reversal, are illustrated

by excerpts of interviews on composition. According to Dubinsky, an understanding

of functions as objects and as processes is necessary for understanding composition

(beyond evaluating each function at specific points with a formula). However, in

her work investigating students’ understanding of inverse functions, Vidakovic (1996)

found students whose concept of composition of functions was based on a process

conception of function but not necessarily on an object conception.

3.4 Difficulties with differentiation

While the search for investigations on learning the chain rule gave no results, two

teaching methods papers on the chain rule are available. Mathews (1989) discusses

the use of a computer algebra system, muMATH, to verify the chain rule. Using

the computer to produce the symbols is expected to help the students understand

the truth value of the rule. Thoo (1995) proposes using arrow (or tree) diagrams as

a mnemonic device for the chain rule. He finds the diagrams especially useful with

functions of several variables. Thoo also believes that these will help internalize the

composition of functions.

16

Studies of students’ difficulties with calculus topics (Orton, 1983; Frid, 1992;

White & Mitchelmore, 1992; Thompson, 1994a; Thomas, 1995; Cottrill et al., 1996;

Vidakovic, 1996, 1997; Asiala et al., 1997) offer insights to misconceptions and ob-

stacles to understanding of calculus concepts. Specific concepts such as graphical

interpretation of the derivative, limit, rate of change, differentiation, inverse func-

tions and the Fundamental Theorem of Calculus (FTC) have been explored. These

reports offer descriptions of the students’ possible mental constructions and ped-

agogical suggestions. Thomas (1995) also used Problem 2 of the RUMEC study

(Section 1.3) in her exploration of students’ understanding of the FTC. She found

that those students who were unsuccessful with the problem were unable to deal with

the area-accumulating function (a function defined through a definite integral with a

variable endpoint). This result is consistent with those of the RUMEC study on the

problem. Selden, Selden and Mason (1989, 1994) approached student understanding

with a problem-solving framework. They found students fail to use calculus strategies

when dealing with nonroutine problems.

Chapter 4: Phase 1

The purpose of this portion of the study is to attempt to determine the correlation

between a student’s understanding of composition of functions and understanding the

chain rule. The data will also be used to compare two instructional approaches to

the chain rule: a traditional lecture-recitation method and an innovative “reform”

method. The comparison is an attempt to establish whether there is a difference in

performance which might be attributed to a more conceptual approach to the topic

(the reform course) versus an instrumental approach. This study proposes to test the

hypothesis: The chain rule in and of itself is not difficult for students to apply; rather

it is the issue of composition of functions.

4.1 Methods: Subjects

The subjects for this study were students at a large, southeast, urban university

who had completed at least two quarters of calculus. A total of 34 students volun-

teered to participate in the study. They were solicited from second and third quarter

calculus classes near the end of the term. Some of the students had been through a

calculus course using cooperative learning, computers and alternatives to lecturing.

The reformed calculus course is known as C4L (for Calculus, Concepts & Computers

and cooperative learning)1. The others had completed a traditional, lecture-based

course. Seven students took the C4L course both quarters, three took C4L in the first

quarter only (then took the traditional course for the second quarter), seven took

a traditional first quarter then C4L for the second quarter, and 17 students took a

traditionally taught course for both quarters. This data is recorded with the variable

C4L, defined in Appendix B.

All of the students were majoring in Arts & Sciences or Education programs; five

were graduate students in Education taking undergraduate mathematics courses for

certification. The students volunteered their time (up to 2 hours) to complete the

questionnaire. In order to make some sort of comparison between the volunteers, an

additional data point was sought. The University computes a predicted grade point

index (PGPA) at the time of admission. The PGPA was collected to compare the

academic strength of each student prior to her or his calculus instruction.1See Schwingendorf, Mathews, & Dubinsky (1996) for a detailed description of the project.

17

18

An alpha-numeric code assigned to each subject was used to identify all of the

written work collected as data. This code was also used on all audio and video tapes

made in the second phase of the study. This code is used in this report to present

data without compromising confidentiality.

4.2 Instrument

Data were collected on an instrument constructed for this study which tries to

assess the participant’s facility with (a) functions, (b) composition of functions, (c)

differentiation, and (d) the chain rule. The items on the written instrument (ques-

tionnaire) are given in the following section along with the coding scheme for each

item. The questionnaire is included as Appendix A.

The questionnaire was piloted with 4 students of known ability (having recently

completed the researcher’s final exam in second quarter calculus) to check for errors

and reliability. As there was no need for major revisions to the questionnaire, the

responses of the pilot study are included in this study.

The data was collected in a piece-wise fashion with individuals and small groups

over the course of a summer session. In each instance, the informed consent form was

read and explained. After the consent forms were signed, the following instructions

were given:

• These items are designed to explore your range of understanding on a number

of topics. Some will be very easy to do, some will be harder. Please do the best

you can.

• Please answer each question as completely as you can. Please do not leave any

answer blank. If you would like to write additional notes with an answer, feel

free to use the back of the page.

• Please do not write your name on any page of this questionnaire. The code

number at the top of this page will serve to keep your identity confidential.

The questionnaires were completed under ordinary test conditions with little interac-

tion with the researcher.

4.3 Analysis

The 20 problems were coded (scored) using a 5 point rubric based on the followinggeneral guidelines adapted from Carlson (1998):

5 Complete response to all aspects of the problem. Indicates completemathematical understanding of the problem’s concept. Includes onlyminor computational errors, if any.

19

4 Responses falling between 5 and 3.

3 Demonstrates understanding of main idea of the problem. Not to-tally complete in response to all aspects. Shows some deficiencies inunderstanding aspects of the problem. Incomplete reasoning.

2 Responses falling between 3 and 1.

1 Attempts, but fails to answer or complete problem. Very limitedor no understanding of problem. Contains words, examples, or dia-grams that do not reflect the problem.

0 No answer. Written information made no attempt to respond to theproblem. Written information was insufficient to allow judgment.

These guidelines were used to construct specific rubrics for each item—a first esti-

mation of each rubric appeared in the proposal for study. Each item was analyzed

using a constant-comparison method (Krathwohl, 1993). In this method, responses

were sorted based on the type of response found in the written work. The resulting

categories were then compared to the general rubric and the initial estimation, and

were given an initial code. More than one category resulting from the sorting oper-

ation would receive the same code occasionally, if the two types of responses both

indicated the same level of understanding. This scheme was repeated as new sets of

data became available. The specific responses were then used as descriptors in the

rubric for the item.

Finally, the entire set of data was re-analyzed some weeks after the last ques-

tionnaire was completed. This last iteration checked the reliability of the generated

rubric. The refined codes were used in the statistical analysis reported in the next

chapter.

In the following discussion, each item from the questionnaire is given. When

applicable, notes on the item follow, stating such things as where it was originally

published, comments on why the item was chosen, and/or issues that arose in the

coding. The rubric for coding that item is described by giving a description of the

typical responses for each level that was observed in the data. If a code was not given

to any response for the item, it is not listed in the rubric. Likewise, no rubric has a

description for the code 0 since that description did not vary between items.

4.3.1 Functions

Item 1

Express the diameter of a circle as a function of its area and sketch its graph.

20

(Carlson, 1998) This item involved covariation in functions. Many students did

not give a graph as part of their answer. Perhaps this was due to misreading the

problem, rather than a misunderstanding.

5 The correct formula was computed and a graph was sketched whichwas reasonably similar to that of the function.

4 The graph was missing; or a circle was sketched in addition to thecorrect formula.

3 Used correct area formula in terms of d, but does not solve for d;used an incorrect formula (such as circumference) and solves for d;or, indicated thinking about a function by use of function notation(with otherwise incorrect answer).

2 The graph was missing; or a circle was sketched in addition to aresponse which was otherwise a 3.

1 Used some circle formulae without function notation.

Item 2

A student has marked the following as a non-function. State whether this student is

correct and why.

y�

x

(Even, 1988, 1993) Items 2 and 3 raised the issue of univalence in functions.

5 Answered “No”. Used a definition of function in the explanation.

4 Answered “No”. Gave a statement about the vertical line test (with-out elaboration).

3 Answered “No”. Restated the situation; or stated that discontinuousfunctions are allowed.

2 Answered “Yes”. Indicated a misunderstanding of the open circle onthe graph; or there was an indication of using an erroneous definition.

1 Answered “Yes”. Any other non-empty response.

21

Item 3

A student has marked the following as a non-function. State whether this student is

correct and why.

A correspondence that associates 1 with each positive number, −1 with

each negative number, and 3 with zero.

(Even, 1988, 1993) It was interesting to note that 8 out of the 34 students literally

read the sentence from left to right, that is, they saw the domain as {−1, 1, 3}.

5 Answered “No”. Used a definition of function in the explanation.

4 Answered “No”. Drew graph and gave a statement about the verticalline test (without elaboration).

3 Answered “No”. Restated the situation; or stated that discontinuousfunctions are allowed.

Answered “Yes” due to interpreting the statement literally left toright.

1 Answered “Yes”. Any other non-empty response.

Item 4

Decide if it is possible to use one or more functions to describe this situation. If yes,

then describe your function(s) briefly. If no, then explain.

x = t3 + t

y = 1− 3t + 2t4

t is a real number.

(Breidenbach et al., 1992) Item 4 was intended to demonstrate whether the student

perceived a process in the situation.

5 Fairly clear description of either a function R → R×R or functionsx(t), y(t).

3 Indicated desire to solve for t.

2 Work for linear equation; substitutes expression for x into y; oranswered “Yes” with no reason given or with disconnected reason.

1 Answered “No” with reason stated.

Item 5

Tim and Donna live 1 km from their school. Usually, they walk to school together.

Yesterday, they both left their houses 10 minutes before school started. Tim started

to walk, but Donna was afraid to be late and started to run. After a while, Tim

22

realized that although he tried to walk faster and faster, he had to run if he did not

want to be late, and started to run. At about the same time, Donna became tired

and had to walk instead of run. They both reached school exactly on time. Which

of the following graphs is Tim’s and which one is Donna’s?

time

dist

ance

t ime

dist

ance

t ime

dist

ance

t ime

dist

ance

t ime

dist

ance

t ime

dist

ance

(Even, 1988, 1993) This item involves covariation in functions.

5 Correct choices.

2 Correct choice for Tim but chose parabola for Donna

1 Indicated need for an expression; or chose two parabolas.

4.3.2 Composition

Item 6

Given two functions v, w such that v(t) = 5t − 6 and w(t) =t

3+ 4, find (w ◦ v)(9).

Explain.

Item 6 was a composition problem to determine understanding of “◦” notation.

5 Answered 17.

4 Found (w ◦ v)(t) correctly, but did not evaluate.

3 Reversed composition; or incorrectly evaluated expression.

2 Subtracted functions then evaluated at 9.

1 Evaluated w(9), v(9) and stopped; multiplied functions; found w(9) ·v(9); solved system of equations.

Item 7

Given that (f ◦ g)(x) = 5√

2x + 3.

23

a. Find f and g that satisfy this condition.

b. Are there more than one answer to part (a)? Explain.

(Even, 1988, 1993) This item required decomposition of a composite function.

5 Two correctly labeled pairs.

4 Misunderstood 5√

notation, but indicated understanding of compo-sition.

3 “No” for part (b); reversed labels (process); or incorrect answer for(a) and “Yes” for (b).

2 correct expressions unlabeled; or wrote identity as g = 1.

1 Left (a) blank; took a derivative; f = 0, g = 0; or g = 1/x.

Item 8

Find k so that g(x + 1) = g(x) + k, given that g(x) = 3x + 5. Explain.

(Carlson, 1998) This item invoked the student’s composition concept and com-

pared it with a functional shift up.

5 Correctly finds k = 3 and explains answer in terms of slope.

4 Pointwise evaluation to find constant without explanation.

3 No explanation or work shown.

2 Error working with g(x+1); used expression g(x+1)+k; or dividedto solve for k.

1 Unable to evaluate g(x + 1); or did not evaluate g(x + 1).

In each of the following two questions, f, g, h are functions whose domains and ranges

are the set of all real numbers, and such that h = f ◦ g.

Item 9

If only the information in the following table were known, would it be possible to find

f(2)? If so, find it and if not explain why not.

x h(x) g(x)

−1 1 −3

4 π 1

π 0 2

(Breidenbach et al., 1992) The last two items of this section required students to

compose and decompose functions given without expressions.

5 Correct answer (f(2) = h(π) = 0) with work.

4 Correct answer with inconsistent or no work.

24

3 Indicated understanding of composition and table but no answer; orwanted line for x = 2 in table.

2 Asked for function/equation/expression.

1 Interpreted as multiplication.

Item 10

If only the information in the following table were known, would it be possible to find

g(4)? If so, find it and if not explain why not.

x h(x) f(x)

−1 1 −2

2 3 1

4 −2 π

(Breidenbach et al., 1992) This problem cannot be solved as stated without as-

suming that f is one-to-one, since the domain of f is given as the set of real numbers,

rather than just the set of x-values in the table. However, this issue did not confound

the subjects in Breidenbach et al. (1992) as the students easily made what was to

them a natural assumption. Likewise, there were no indications that any of these

subjects saw the need for the one-to-one property.

5 “Correct” answer (g(4) = f−1(−2) = −1) with work.

4 Work showed strong attempt but cannot see decomposition.

3 Indicated understanding of composition and table but no answer; orwanted line for x = −2 in table.

2 Asked for function/equation/expression.

1 Interpreted as multiplication.

4.3.3 Differentiation

Compute the derivative of each of the following functions. Show all of your work.

Item 11

f(x) = 11x5 − 6x3 + 8

Item 12

g(x) =3

x2

Item 13

h(x) =(x2 − 3

) (5x− x3

)Item 14

y = 3ex − 4 tan(x)Item 15y = x2 sin(x)

25

5 Correct answer.

4 Minor error such as dropping (−) sign or arithmetic errors; errorwith derivative of trig functions.

3 Errors involving elementary differentiation rules.

2 Did not use appropriate rule; or compound errors with rules.

1 Integrated; or guessed.

4.3.4 Chain rule

Compute the derivative of each of the following functions. Show all of your work.

Item 16

F (x) =(1− 4x3

)2

Item 17

G(x) = 2(5x2 + 1

)4− 4x

(5x2 + 1

)4

Item 18

H(x) = sin(5x4

)Item 19

y = cos3(t)Item 20y = e−t2

5 Correct answer.

4 Minor error such as dropping (−) sign or arithmetic errors; or appliedchain rule but error with derivative rule.

3 Computed f ′(g′(x)) or f ′(g(x))

2 Applied chain rule indiscriminately; or attempted to avoid chain ruleby expanding/rearranging terms (Item 19).

1 No evidence of considering chain rule.

4.4 Data

The data for Phase 1 are presented in tabular format in Appendix B. The codes

are presented by item, by subject as ordered tuples, and by subject as combined

scores.

Chapter 5: Results: Phase 1

These data were analyzed in two ways. First, as one sample from a single popu-

lation, a regression model was used to determine which rating (after PGPA) has the

highest correlation with the chain rule rating. It was expected that the composition

rating would have the most significant coefficient in the model after PGPA. This was

not the case however, as is detailed below.

In the second analysis, the data were split by course (C4L vs. Traditional) and

compared in each category. The comparison between treatments was an ex post facto

study and, since the members of each sample were self-selected as volunteers for the

study, an attempt was made to confirm the similarity of the groups using PGPAs

(Schwingendorf, McCabe, & Kuhn, 1998).

5.1 Analysis 1

The predicted grade point average (PGPA) is computed as an attempt to state the

readiness for a student to attend the university. It was used here to give a measure

of the student’s ability prior to her or his instruction in calculus. The PGPA is

not computed for transfer students or for graduate students. In the case of transfer

students, the cumulative grade point average from the previous institution is used.

There was no PGPA score reported for the five graduate students in the study.

The codes for the five items in each category were summed to create a rating

in that category. Prior to summing, the subsets of codes were inspected as ordered

tuples (Table B.2) in order to decide if such a method was reasonable. This question

arose in observing that ratings in the center of the range for a category might come

about differently and thus tell different stories. For example, a rating of 15 could

come from [3, 3, 3, 3, 3] or from [5, 5, 5, 0, 0]. Should both ratings of 15 be treated

the same way? By returning to the written responses to compare such cases, it was

deemed reasonable to sum the codes. In the example stated, the rating of 15 gave the

same “global” information for the ability of each of the two students in that category.

Also at issue was how to deal with questionnaires with 20–25% of the answers

blank. This latter issue was more easily decided since it involved only three question-

naires. These were excluded from the analysis as being incomplete. Note that the

26

27

directions for the questionnaire specifically stated to leave no item blank. The IDs

for the deleted data sets are marked with an asterisk (*) in Table B.2 and Table B.3.

The descriptive statistics for the five variables used in the regression model are

given in Table 5.1 and the correlation data are in Table 5.2. The means indicate two

pairs of ratings. Function and Composition each have a representative code of a little

more than 3, which would be interpreted as demonstrating understanding of the main

idea of the problem, but showing some deficiencies in understanding aspects of the

situation. The representative rating for Differentiation and Chain Rule is slightly over

4, showing more complete ability in the situation. This difference appears to coincide

with the types of questions in the sections of the questionnaire. Items 1–10 varied

widely in difficulty and in coverage of topics within the headings of function and

composition of function. This variety is a by-product of the existing body of research

in these areas and correct responses might indicate relational understandings of the

situations. The remaining items are of a more instrumental nature, as indicated by

the directions “Compute the derivative of each of the following functions.”

In order to include PGPA in the regression model, only 27 observations are able

to be included. That model is presented in Table 5.3. We note that only Composition

and Differentiation have coefficients significantly different from zero. This result was

predicted by the theory—the genetic decomposition for the chain rule is composed

of the schemas for composition and differentiation. Table 5.4 gives the model using

31 observations without the PGPA score. The difference between models is slight,

although without the PGPA, the Composition rating is no longer significantly non-

zero. Both models account for the variance with very low P-values.

The unexpected result is that the coefficient for Differentiation is larger than that

of Composition. It was hypothesized that the Composition rating would predominate.

Again, this could be explained by the different types of items in the two halves of the

questionnaire.

5.2 Analysis 2

The C4L variable in Tables B.2 and B.3 is a binary code for the first two quarters

of calculus. An “11” represents the student took the C4L version of calculus both

terms. A “10” indicates taking the C4L version first quarter and a traditional course

the second quarter, while a “01” indicates the opposite situation. A “00” indicates

being in a traditional course both terms. The means and standard deviations for

each variable in the regression model is given for each of the four values of the C4L

variable in Table 5.5.

28

Table 5.1Descriptive statistics for variables

PGPA FunctionComposi-

tionDifferen-tiation

ChainRule

Mean 2.73703 14.6129 15.9677 22.6774 21.0967Median 2.8 14 15 24 21Mode 3.1 14 24 24 24Standard Deviation 0.62212 5.13599 5.7531 3.07014 3.36010

Table 5.2Correlation statistics for variables

PGPA FunctionComposi-

tionDifferen-tiation

Chain Rule

PGPA 1Function 0.0089 1Composition 0.20391 0.33574 1Differentiation -0.022 0.10597 0.26548 1Chain Rule -0.1579 0.13552 0.40021 0.671989 1

The data were compared by grouping those students with C4L values greater than

zero, that is, those who had at least one non-traditional course. This was done to

compare groups of almost equal size. Also, this grouping has historical precedence.

In previous studies using two groups, the influence of a non-traditional approach was

seen to show up when investigating topics taught traditionally (see Asiala et al., 1997;

Clark et al., 1997). This comparison is presented in Table 5.6.

Finally, a comparison was made based on the method of their first quarter course,

when the chain rule was introduced. These data are given in Table 5.7. There was

a significant difference for the Differentiation and Chain Rule ratings in favor of the

group who had a Traditional first quarter of calculus. However, these differences on

average are three points out of 25—which can be interpreted as making minor errors

on three problems. Since those items were of a more instrumental nature, this result

merely indicates that those students are less likely to make computational errors in

the test setting.

5.3 An alternate analysis

In order to address the difference found in the types of items on the questionnaire,

two new variables were computed. The Function rating was changed to include only

29

Table 5.3Regression model with PGPA

Regression statisticsMultiple R 0.7978R2 0.6365Adjusted R2 0.5704Standard Error 2.1882Observations 27

Analysis of Variance

dfSum ofSquares

MeanSquare

F Significance F

Regression 4 184.502 46.125 9.6323 0.00011Residual 22 105.3491 4.7886Total 26 289.851

Coeffi-cients

StandardError

tStatistic

P-valueLower95%

Upper95%

Intercept 6.5755 3.7905 1.734 0.0946 -1.285 14.43PGPA -1.133 0.7109 -1.594 0.122 -2.608 0.3408Function 0.0154 0.0965 0.1604 0.8737 -0.184 0.2158Composition 0.1967 0.0927 2.1217 0.0435 0.00444 0.3890Differentiation 0.6163 0.1426 4.320 0.000201 0.3205 0.9121

Items 1 and 2, and a new Composition rating included Items 6–8. Those items were

selected as being the most like the last ten items. That is, they could be considered to

be straight-forward, computational situations dealing with functions and composition

of functions. We note that the original coding of these items was based on assessing

the depth of understanding in the situation, rather than on an instrumental, “right

or wrong” scheme. The items were not re-coded for this analysis.

The use of the new variables did not change the regression model or the results

between groups. This seemed to indicate that there would be no change in results if

one were to construct a new questionnaire using items of the same type for all four

categories to collect new data.

30

Table 5.4Regression model without PGPA

Regression statisticsMultiple R 0.7103R2 0.5045Adjusted R2 0.4495Standard Error 2.4929Observations 31

Analysis of Variance

dfSum ofSquares

MeanSquare

F Significance F

Regression 3 170.910 56.970 9.16684 0.00023Residual 27 167.799 6.2147Total 30 338.709

Coeffi-cients

StandardError

tStatistic

P-valueLower95%

Upper95%

Intercept 3.8272 3.52159 1.08679 0.28578 -3.3984 11.0529Function -0.006 0.09409 -0.0712 0.94363 -0.1997 0.18636Composition 0.1413 0.08663 1.63150 0.11323 -0.0364 0.3191Differentiation 0.6663 0.15379 4.33257 0.000152 0.3507 0.9818

Table 5.5Comparison of courses

00 01 10 11PGPA mean 2.62 2.91 2.67 2.82

SD 0.70 0.35 0.40 0.88Function mean 15.31 13.86 16.33 12.40

SD 6.13 3.98 2.08 4.45Composition mean 15.31 19.43 13.00 15.00

SD 5.16 7.35 3.46 5.57Differentiation mean 23.63 23.00 21.67 19.80

SD 1.26 2.83 4.93 5.07Chain Rule mean 22.00 21.71 20.00 18.00

SD 2.48 2.69 2.65 5.52Overall mean 76.25 78.00 72.20 65.20

SD 10.80 12.74 6.24 15.80

31

Table 5.6Comparison of Traditional and any C4L

Trad (n = 16) C4L (n = 15) t P -valuePGPA mean 2.62 2.83 -0.90 0.19

SD 0.70 0.56Function mean 15.31 13.87 0.78 0.22

SD 6.13 3.89Composition mean 15.31 16.67 -0.65 0.26

SD 5.16 6.43Differentiation mean 23.63 21.67 1.84 0.04 *

SD 1.26 4.05Chain Rule mean 22.00 20.13 1.58 0.06

SD 2.48 3.96Overall mean 76.25 72.33 0.90 1.70

SD 10.80 13.46* significant difference between groups

Table 5.7Comparison of first quarter course

Trad (n = 23) C4L (n = 8) t P -valuePGPA mean 2.73 2.76 -0.14 0.45

SD 0.60 0.71Function mean 14.87 13.88 0.47 0.32

SD 5.51 4.09Composition mean 16.57 14.25 0.98 0.17

SD 6.05 4.71Differentiation mean 23.43 20.50 2.53 0.01 *

SD 1.83 4.75Chain Rule mean 21.91 18.75 2.48 0.01 *

SD 2.48 4.53Overall mean 76.78 67.38 1.98 0.03 *

SD 11.16 12.76* significant difference between groups

Chapter 6: Phase 2

6.1 Introduction

The purpose of this portion of the study is to attempt to describe the understand-

ing of the chain rule concept on the part of the interviewed students. Also, it will

be the second attempt of the first cycle through the research paradigm, which may

result in a revised genetic decomposition.

The discussion and conclusions based on the Phase 1 results will be treated fully

in combination with the results from this interview data. The questions to be inves-

tigated here are:

• In spite of the lack of support for the hypothesis in the quantitative data,

is there evidence that students can be successful with the chain rule without

understanding composition of functions, and to what extent can this success

move toward understanding?

• With reference to the results of the RUMEC study, is the chain rule concept

well described via a schema, and do the current results support the reported

schema?

6.2 Subjects

Students who were interviewed were drawn from the 34 students who had partic-

ipated in Phase 1. Each student was placed in a 3 × 3 matrix which was formed by

partitioning the ratings for the Chain Rule and the Overall ratings into three subin-

tervals (high, middle, low). An attempt was made to interview one subject from each

of the nine subsets. However, only six interviews were conducted. Volunteers could

not be found to represent the cells left empty in Table 6.1. Of the two or three sub-

jects in the three categories, either the student had requested to not be interviewed

on the informed consent form in Phase 1, or he or she declined the invitation during

this phase. Table 6.1 indicates the code number for the student interviewed as an

entry in the matrix.

32

33

Table 6.1Matrix of interviews

Overall High Middle Low

Chain Rule (96–81) (78–66) (63–45)

High (25–23) A113 A134

Middle (22–20) A132 A125

Low (19–1) A137 A123

6.2.1 Procedures

The interviews were audio and video taped with the camera positioned to record

the subject’s actions and written work. The audio tapes were transcribed to written

form and then coded for key events. These were then analyzed according to the

research paradigm described above. The video tapes were logged to ease reference to

key events found in the audio transcripts.

6.3 Instrument

The interview followed a guide designed to elicit the student’s understanding of

the rule, based on tasks from the quantitative instrument. New tasks were presented

to the student to probe the limits of understanding. The students were asked to reflect

on the questionnaire tasks; by asking them for definitions of function and derivative,

domain-range issues in composition, the statement of the chain rule, and a discussion

of any relationships between the 20 problems that the student perceived. The guide

was developed following the initial analysis of the quantitative data.

Then the interview addressed Leibniz’ rule as a way of gauging the robustness

of the student’s concept. A situation with no simple anti-derivative was presented;

“compute F ′ if F is the function given by F (x) =∫ sin x

0et2 dt.” If the need to compute

an anti-derivative was too great, a situation where the integrand has an elementary

anti-derivative was presented to determine if the student can derive Leibniz’ rule from

the result. The text of the interview guide can be found in Appendix C.

6.4 Analysis

The interviews were transcribed from the audio tape recordings by the author and

an assistant. These transcripts were then proofread by the author, who had conducted

the interviews. Each transcript was then coded for key events in the interview and

those events were searched for common issues relating to the topic. Not surprisingly,

34

the events followed the interview questions rather closely, and so most of the issues

found were predicted by the interview guide.

Two new issues arose in the data. We found data that better assessed the inter-

viewee’s understanding of the chain rule than was available in the questionnaire data.

Secondly, by interviewing two students whose data were excluded from the analysis

in Phase 1, we found limitations in employing only one means of data collection in a

study of mathematical knowledge and understanding.

6.5 Where the RUMEC study left us

Clark et al. (1997) introduced the notion of using the triad mechanism (Piaget

& Garcia, 1989) to describe the development of a schema in the RUMEC study

(see Section 2.2, page 10). This mechanism focuses on the relationships between

problem situations and/or between mental constructs that a student brings to bear

on a problem. The relationships may or may not be explicitly perceived by the student

as they are used. The authors found the triad useful, having exhausted themselves in

the attempt to describe the schema in terms of actions, processes and objects alone.

The RUMEC study was able to map the triad levels to the observed data in the

study and gave a genetic decomposition for the chain rule concept based on the triad

(see Section 2.4, page 11). However, it was unable to map the overall response of each

interviewed subject to the triad. The data collected did not indicate clearly if and

how each subject related various problem situations together. Simply put, the data

were collected prior to the researchers’ understanding of the complexity of schema

development. This present study was designed to fill that gap.

The following excerpts from Clark et al. (1997) describe how they interpreted

student responses with respect to the triad in slightly more specific terms than found

in the genetic decomposition.

Students who are at the Intra stage of schema development with respect

to the chain rule see the various rules for differentiation as unrelated. The

students are able to solve some of the problems by simply applying rules

which have been memorized and in some cases, not remembered correctly.

(p. 354)

Many of the students who participated in our study showed evidence of

being at the Inter level of development of the chain rule. They had col-

lected some or all of the various derivative rules in a group and perhaps

35

could provide the general statement of the chain rule, but had not yet

constructed the underlying structure of the relationships. (p. 356)

Once a student’s collection of derivative rules and understanding of func-

tion composition attains a coherence as a schema the student has moved

to the Trans stage of development. That is, he or she is capable of op-

erating on the mental constructions which make up her or his collection.

He or she is now able to reflect on the explicit structure of the chain rule

which these constructions were implicitly containing. (p. 358)

The following chapter reports the analysis of the six interviews based on these de-

scriptions and on the initial genetic decomposition. The triad mechanism is seen in

the observations and the data for each subject is described by a level in the triad.

Chapter 7: Results: Phase 2

7.1 The triad description revisited

Three questions from the interview were written to evoke the students’ relation-

ships between situations involving the chain rule. In Question 6, the student is to

sort the differentiation problems from the previous phase. Later, he or she is asked

to give her or his definition of the chain rule, and in Question 11, relate it back to

the list of problems just sorted. Finally, in Question 15 four problem situations are

presented to find if the student sees the involvement of the chain rule.

While these questions form the basis for the following discussion, the rest of the

last half of the interview is also taken into account when applicable. Certainly, the

success or lack of success on the Leibniz rule question indicates how robust a student’s

schema has become.

In the following discussion pseudonyms are used for ease of readability rather

than the alpha-numeric codes and to protect the identities of those interviewed. The

interviewer is indicated throughout by “I”. The code will be given—in the section

headings with the pseudonym—for cross reference with the data tables.

7.1.1 Eli [A125]

Eli may have been working at the Intra- level. In the grouping questions and in

his definition of the chain rule, he was working on the surface of the situation only.

Eli: Can I have like one question that is in two groups?I: If you want.

Eli: Then probably I would, first I would group 4, 5, 8, 9 to one since they areall trigonometric.

I: Uh-huh.Eli: Then probably group #4 and #10 together because they have e in it, and

#2 just one group cause it has fraction, then #3, 6, & 7 that’s one group,cause you have powers. . .

I: Uh-huh. What about the powers in 1?Eli: Powers in 1? [Mumbling]

I: You said 3, 6, &7 grouped together because they have powers.Eli: Yeah.

I: Right exponents. Do you have exponents in #1?

36

37

Eli: Oh, yeah.I: You have 11x to the fifth. So would that go in there or were you thinking

that or was there a reason that doesn’t go in there?Eli: Um, it’s because 3, 6, &7 all have [pause]

I: Expressions? Which things were you. . . ?Eli: Well, I think [mumbling] they have [mumbling]

I: So you were pointing at #6 when you said multiply it out. . . what would yoube multiplying out?

Eli: Well, like 1− 4x3 times 1− 4x3 I think [mumbling]I: Are you thinking of FOIL?

Eli: Yeah. [Mumbling]I: OK.

Eli: Actually I might need 3 [mumbling] involved with the FOILs. So I wouldprobably put 1, 3, and 6 together.

I: Uh-huhEli: [mumbling]

I: OK, you asked me and I said sure you could. You have some of your groupsoverlapped a little bit.

Eli: Uh-huhI: What if I changed my mind and said, Um, no they can’t overlap. They have

to be disjoint. Can you still group them?Eli: Um. . . yeah.

I: How would you do that?Eli: I would put. . . I’d probably put #4 and #10 as one group and then 5, 8, and

9 as the other group. So a trigonometric group and the e group.I: OK. What about the leftovers then? 1, 2, 3, 6, and 7? Would they be in a

group together or would they each be an individual group? How would theybe treated?

Eli: 1 and 3 as one group and 2 as another group. 6, 7 as another . . .

Eli gave a description of the chain rule but seemed unable to formalize it. This

was consistent with the trouble he had dealing with functions on the questionnaire,

where his combined code was 9/25 with 3 blank responses.

I: OK. Alright. Um the point of the questionnaire was to get you to think aboutand use the chain rule in taking derivatives. OK. Uh, do you remember thechain rule?

Eli: Uh, derivative of inside times derivative of outside. Not really.I: It had to do with insides and outsides.

Eli: No, not really.

38

I: So for instance, #8 is a problem that involves the chain rule. You’d just takethe derivative. Here’s what you wrote down for, well it’s 18 on this paper.The original function was sin(5x4). You wrote down cos(5x4) times 20x3.

Eli: Uh-huh. Cause um the derivative outside times the derivative of inside.I: OK. Why don’t you write that down? Um underneath the blue line that I’ve

drawn on that page. Write down, you can use words or symbols or whateveryou like as best as you can recall what the chain rule says. [pause] So whatdid you write down?

Eli: Uh, I put symbol like A and B then I put um A inside, or B inside of A,then I said derivative of A times derivative of B.

I: So the derivative of A, then what happens?Eli: I just put derivative of A then write the B part down.

I: OK. Um in fact, the last five questions on this list were intended to be chainrule questions and use the chain rule to complete the derivative. OK. Doesyour rule there work for all five of those?

Eli: Uh, I don’t think so.I: What’s wrong with your rule?

Eli: Because it doesn’t deal with the powers. Here it had A and B but, the tenonly had one power.

Finally, the four problem situations also were dealt with superficially. Eli decided

that the first two did not require the chain rule and the last two did. However, he

did not offer any explanation and ended by stating that he did not see it in the last

problem. The interviewer wrapped things up in an attempt to get some response

from him, but Eli was not able to add anything.

I: OK. Good. Um, here’s another sheet of paper. Read each of the follow-ing differentiation situations carefully. Without actually trying to solve theproblem, determine if the solution will involve the use of the chain rule insome way. If so, describe how the chain rule is used.

Eli: [pause] Uh, number 1. Uh, no.I: No. OK.

Eli: [long pause] Uh, I think that the first two, #1 and #2 are. . . have nothingto do with the chain rule, 3 and 4 um is gonna have to use the chain rule.

I: Uh-huh, how does the chain rule get used in part c there, excuse me I meantnumber 3?

Eli: Uh, 3? I have to take the derivative uh, 36πV 2.I: Um-huh

Eli: Then, um, . . . I don’t know. It just seem like I have to use the chain rule,uh, I don’t know.

I: OK, and for #4 does that seem any more clear?

39

Eli: On #4 it says that you have formula of C and it says that F = −32t2. So Iplug in F into the formula of C and I will have to use the chain rule there,uh,. . .Well I can’t see the relationship of the inside and outside function.

I: Well, what variable are we using in C?Eli: What variable? F .

I: I, right. Capital F . So then we’re saying OK, yeah but then you can evenfind capital F in terms of time, in terms of little t. So that’s what is goingon there. So the inside function is all of that, −32t2, that’s going to go infor all of those F ’s.

7.1.2 Tim [A134]

Another student working at the Intra- level was Tim. Like Eli, he grouped based

on the surface characteristics and did little in his second attempt.

I: Yeah I just want you to group these, like if you decide that number 3 andnumber 6 go together, then they are in a group. Read through these anddecide . . .

Tim: Where are the groups?I: Make up groups, and then put them in the different groups.

Tim: Oh, OK. This one [works for 2 minutes] So.I: OK. So you put

Tim: 4, 5I: 4, 5, 8, and 9 in one group. What do they have in common?

Tim: Sine, cosine and tangent.I: Right, OK. And then we have 1, 3, 6, and 7.

Tim: This one is powers, power of 5, power of 6, so I don’t know how to put thesetogether.

I: OK, they are just the leftovers.Tim: Yeah

I: Is there any other way that you could group these?Tim: Ah, yeah. [pause] I could group the e functions together.

Tim did not recall the chain rule by name. In fact, it took much prompting using

an example for him to see what he had done.

I: OK, good. Let’s use the bottom of this page here, I will draw a line. Couldyou write down the chain rule in whatever words or symbols you remember?

Tim: The chain. . . ?I: The chain rule, for taking derivatives.

Tim: You mean y = x2 − 7x + 5 and you take the chain rule for this one?I: No, the chain rule is just a rule we have for taking certain kinds of derivatives.

Do you remember using the chain rule?

40

Tim: Um, y, it was something, um. . . Can I do an example out of the book? Thechain rule for this one? [writes out f(x) = yn, (yn)′ = ny(n−1)]

I: That is actually called the power rule.Tim: The power rule.

I: You have got something raised to a power so you bring the exponent down.You would use that in a lot of these problems here, the power rule. Thechain rule is talking about something else. Let me show you an exampleof a problem where you would use the chain rule, for instance #18 here.The original problem was sin(5x4). [Student is shown written work from thequestionnaire, which has the correct solution to the problem.]

Tim: You mean give the rules of this one, how I worked out this one?I: Yes, how did you find the derivative there?

Tim: Oh, [writes and crosses out f(x) = u(x), f ′(x) = u′(x)] [mumbles] [writesf = (y · x), (y · x)′ = y′ · x + x′y] I don’t know, you know, if you give me anexample of how to do the chain rule, I know how to do products.

I: That is what I am saying, this solution you have right here. . .Tim: Mm-hmm

I: you used the chain ruleTim: yeah

I: to get that solution, which is correct. OK? Does this remind you of the chainrule, then?

Tim: Uh-huh, so what you want me to . . .I: We are starting with sin(5x4) and look at what you wrote down for your

answer, and try and remember how you came up with that idea.Tim: Oh! So first I take derivative of outside, derivative of sine is cosine,

I: RightTim: so then I take derivative of inside, so inside is 5x4, so I write down 20x3.

I: OKTim: That is so easy, you know, I don’t know how to get the something that you

asked me to do.I: You don’t. . . So the question was, what does the chain rule say?

Tim: If there is a function of u, something like that, and take the derivative ofthat, right, first you take the derivative of outside first, then take derivativeof inside.

Tim was able to see the chain rule in the implicit situation, but could not see it

in any of the other three.

Tim: If I do. . . OK, OK. So if I pick one?I: Uh-huh

Tim: This one not chain rule. Hold on, I use the chain rule, yes. This one usechain rule.

I: How would you use the chain rule?

41

Tim: That one has√

y and when I take derivative, I write it as y1/2 and when I takederivative of that one, I get that and that and [writes (y1/2)′ = 1

2y1/2−1 =12y−1/2] So if y is something else, I have to take derivative of y.

I: OKTim: Not really, y, there is something inside y. I have to use the chain rule for

that.I: OK. Right.

Tim: I’m not sure about the next problem. I forgot.I: The name of the next one is a related rate problem. Do you remember that

name?Tim: That has to do with speed? What is the formula? I not clear on the question

you know, ‘find the formula for the rate.’ [pause] This one doesn’t haveanything to do with it. Do you think?

I: OK.Tim: So I don’t know if it has derivative. The next one is . . . [reads problem to

self] . . . power rule. . . No, I not use chain rule.I: OK

Tim: [pause] I don’t think I would use the chain rule for the last one.I: OK. Good.

While it is the case that Tim correctly solved all of the chain rule problems on the

questionnaire, it seems clear that he is not working with a general formula. Rather

he is taking each problem as it comes to him and dealing with it individually.

7.1.3 Al [A137]

Al seemed to be at the Inter- level. While his grouping of the ten derivative

problems did not use the chain rule, as was the case for Eli and Tim, he was able to

connect his notion of the chain rule with the items 16–20 (which are numbered 6–10

on the interview sheet).

I: What I would like you to do is to write down the chain rule, using whateverwords or symbols that you like, and can remember.

Al: OK.I: Feel free to talk out loud.

Al: The chain rule, I know that you have something, you have a function f(x)[mumbling] , so you’re gonna have f(x) [mumbling —pause] that’s 2, youhave something to the power of whatever and you take the derivative ofthat, [has written (x3 +2x)2] that would be 2x cubed plus 2x that the chainrule would be if you take the derivative of that again and multiply it by thatof the inside.

I: OK, so is there an example of that in those problems there?Al: [silence] Yeah, #6.

42

I: #6. Good. So you described it, you didn’t write anything down.Al: No.

I: That’s OK. That’s fine, we got it on tape which is all I’m worried aboutreally. Um, so here’s a point, the last five questions 6–10 were written withthe intention that you would have to use the chain rule to solve them. SoI’m wondering, um, if we compare what you have written down as the chainrule with each of those last five problems, does your statement take care ofit?

Al: Takes care of #7.I: Uh-huh.

Al: Then it takes cares of that because you have cosine of that and the entirederivative of that; which I said is that you have something on the outsidetake the derivative and times the derivative on the inside.

I: OK. That’s not quite what you said before though, right?Al: Right, I said that each have to raised to powers.

I: Right, so now you’re talking about inside and . . .Al: Yeah, I know that you’d have to use the chain rule for that and . . . I think

that I would definitely use the chain rule for that.I: OK.

Al: And, it is a prerequisite condition and I haven’t done chain rule for like ayear now.

I: That’s fine, that’s fine. I was just trying to get whatever you thought of first.And now what you are thinking of second. This is fine.

Al: OK.

He went on to work through the remaining three problems. The main distinction

between his work and that of Eli and Tim was that he seemed to be in control of

making the connections once the chain rule was pointed out.

In classifying the four situations, Al saw the chain rule in the implicit differen-

tiation problem and in the last problem. He was unfamiliar with the related rate

problems and decided that it would not be used. It seemed as though he might have

answered differently had he worked through the solution to the problems.

I: Yeah, alright. What I have is a set of four situations, that are differentiationproblems that you probably dealt with in the first calculus course that youtook. I don’t want you to solve these problems, but if you would read througheach of these situations and decide if the solution would use the chain rulein some way and if so, then how would it be used.

Al: OK. [silence] For one you would use it. You have x times the derivative ofy times the. . . that would be you use the product rule to get the derivativeof that, that is y to the 1/2 times the derivative of y. . .−1/2y to the −1/2times the derivative of y.

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I: And that’s the chain rule?Al: Yeah. You have uh, y in itself is a function and y can be x cubed or you

know t cubed. And the square root of it, and to get to that you have uh,1/2(t3)−1/2 times the derivative of that which we’d write 3t2.

I: OK.Al: For one you’d use it. On to two now?

I: Uh-huh.Al: [pause] I don’t know how to do #2.

I: Um, would it help if I told you that it was a related rate type of problem?Al: Um, no.

I: OK. Alright. What you have is two things that are varying in this situation,the height on the wall and then the distance from the wall on the bottom.And, I’m given information about the rate at which the bottom is changingand I’m asked about the rate at which the top is changing. So what happensis you look for a relationship between those two original quantities. Theheight and the distance from the wall and then you try and go from there toan expression that talks about their rates.

Al: OK, but how would we do that?I: Oh, you want to see the work?

Al: No,I: Well, particularly if you draw a picture um you have a right triangle and so

you might label the height h and the bottom could be x. . .Al: Then you used the triangle 1/2. You use that 1/2, you use that that triangle

thing?I: Uh, I use a triangle thing. Not the area though, I use the Pythagorean

theorem to talk about the sides. We said that the ladder is L feet long so itwould be L2 = h2 + x2.

Al: OK.I: That’s our relationship. That relates these things together and then I would

need to find the rates by taking a derivative.Al: OK. You want me to do three now?

I: Well, in that description, I didn’t say the word chain rule or use the phrasechain rule. Do you think that the chain rule might pop up there or not?

Al: Um, no. [pause] No.I: Um, yeah, so number three.

Al: [student reads to himself—long pause] You just take the derivative of thatand that would put that; you wouldn’t use it there either, I don’t think.

I: OK.Al: I don’t think so. But, I’m not very good at word problems.

I: [laughs] Very few of us are.Al: [chuckles] Yeah.

I: Do you recognize that problem as something that you have done before?

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Al: I’ve probably done it countless times. More than likely.I: OK, but it’s not familiar to you.

Al: Yeah. I’ve done that like probably ten times but. . .I: The second one. . .

Al: Yeah.I: Right.

Al: Cause almost every calculus course has that.I: It’s a classic. OK, how about four?

Al: [pause—reading] This, I don’t think that I’ve ever seen before. I would useit here.

I: You would.Al: Uh-huh. Yeah, because you have −42t2 and put that into F and that’s

squared cubed and then that times that.

So Al had made some connections between situations involving the chain rule, but

is still working toward a cohesive whole.

7.1.4 Ray [A123]

Ray is the most interesting case (he is the topic of section 8.4 below) as he had a

rather highly developed sense of the chain rule but no facility with it generally. This

is due to his making the decision to not memorize the derivation formulas. In the

following excerpt, Ray was attempting to do item 17 from the questionnaire, which

he had left blank.

Ray: This one is messier. I would like to approach this one as a composition.I: 17?

Ray: 17, yeah, I would like to approach this one as a composition of two functionsas two different things, but I don’t remember those rules either. As far aslike the chain rule and except for this one is like 2x4 is f(x) and g(x) shouldbe [mumbles] I would have to use the chain rule to find out what that wasthe derivative was. Oh, boy, foggy memory of the chain rule, I really did relyon the, that, that. I actually had memorized the page where the derivativerules were and then I didn’t have to memorize the derivative rules. Exceptright before a test I would memorize them temporarily, I would cram theminto my RAM space.

I: So, um, the situation right now is that you know that there is a chain rule,and it would apply here for a composition, um, try to dig out as much asyou can.

Ray: It’s like f(g(x)) prime, is g(x) prime. Something like x has derivatives insidethere, different things. . . something along these lines. How ever did you grademy tests? [sighs] I know that you take the derivative of each one separatelyand compose them somehow, either added or multiply them, I think that they

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are added and beyond that that’s all the fog that covers the skyscrapers ofmy mind.

I: That’s called pollution. Um, what other rules do you remember by name?Ray: Let’s see here, the chain rule, the um sum rule is pretty simple, you can take

them separately. Um, the product rule I remember, by name, um that beingthe special case, once again. I don’t exactly know how to do it. There’s arule for division, the quotient rule. Those are the important ones—chain,product and quotient. Then we have like power rule—the easy one it’s likex2; 2x. I remember that one and that really took care of a lot of them. Thepower rule. I used that for the first four of them.

I: Right. Right.

As the interview progressed Ray was able to describe which rules he would use to

solve the problems, provided he could see the formulas. With much interaction on

the part of the interviewer, he recalled the formula for the product rule and the chain

rule. He demonstrated the use of the chain rule on item 18.

Likewise, with the four situations, he was able to see the chain rule in most of the

problems. In both the second and third problems (related rates) he did not see the

need for a derivative; hence no need for the chain rule. Apparently, he had associated

acceleration with rate and with integration.

Ray: OK. [pause] OK, I’m trying to remember . . . got a rate . . . this one doesn’teven scream for derivation to me. Because you have one rate . . .L is goingto be a constant. Um, looking for a second. . . this doesn’t even really screamfor a derivation except there’s too many variables and to get rid of L thatwe’d. . . but no I don’t think that I would use the chain rule on number two Iwould set up some kind of trigonometric relationship to two different sides,time being a constant thing, but distance, the trigonometric relationshipbetween the distances would dictate everything out.

I: How would they dictate them?Ray: Well, in the same amount of time that it sits here, [mumbles to self] co-

sine. . . goes to. . . opposite [mumbles to self] opposite over adjacent is thetangent. . .

I: Right.Ray: um I would probably use the tangent, but L doesn’t help a lot and um, x

velocity over 4 feet per second, and I solve for x to get um, I’m not sure. . . I’dtry to solve for x through some large algebra and that would be the answer.

I: OK. OK.Ray: As the rate, there’s not acceleration there. Which would scream for . . . no.

That would scream for integration, acceleration, but actually. Yeah, yeah.There’s no change here, it is constant velocity so this doesn’t even ask forderivation. Therefore I wouldn’t probably use the chain rule to solve thisproblem. That was number 2. [Reads to self]. . . oh,oh, S cubed. . . solve for

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V . . . it gives. . . I don’t think that I would use the chain rule here because Idon’t really compose V , actually yeah they are, S has V in it. [mumbles toself] Yeah, I’d use the chain rule for finding d(S) and because I would haveto solve for V and S would move into the problem I would have to use thechain rule for finding d(V ) as well, I would use the chain rule twice here.That’s number three.Um [reads #4 to self]. . . hey I remember this problem. . . [mumbles to self]. . . find the expression . . . uh, that’s a composition—I’m talking about fournow—you would substitute −42t2 in for everywhere that you would have Fand that would be an expression for the rate of change of C. Um, [mum-bles to self] well, that’s obviously the composition of functions. [mumblesto self] As I’m trying to find the rate of change, it’s the equivalent of accel-eration. . . [mumbles to self] yeah, I would I don’t think I would even haveto derive here. . . this is question number four. . . but both of these. . . I wouldthink that I would have to use the chain rule because I’m putting one functioninto another.

Ray seems to be at the Inter- level. He made the connections between the various

situations, but he was doing so with what might be considered semi-objects. That

is, one might consider a complete idea of a rule as being the totality of knowing the

formula, how to apply it and when to apply it. Many students seem to focus on and

excel at the first two to the exclusion of recognizing the need in context. Ray, on

the other hand, had focused on the latter two, choosing to look up the formula when

needed.

7.1.5 Peg [A132]

Peg indicated a rather strong Inter- level understanding as she grouped differenti-

ation problems. She used the chain rule as an initial criteria, selecting two of the five

chain rule problems. The remaining problems were classified as being trigonometric

or exponential. When asked for a second way to sort, she kept the chain rule as a

main criterion.

I: So is that five different groups?Peg: [pause] Yes. Five different groups.

I: OK, what was the discriminating features?Peg: OK, One is like the most straight forward, where it is just strictly using the

uh, what’s the name of that rule? The power rule.I: Uh-huh.

Peg: It’s just straight out the power rule. Uh, two and three are pretty muchexactly the same thing except for two, the expression needs to be rewrittento use that rule. 6 and 7 is also just the power rule except for you have uh,uh, you have a power rule and then it’s a chain rule.Uh, 4, 5, 8, & 9 all have

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trigonometric uh terms in them to where then you also have to know theirtrig functions and then 10 is the e function which also has it’s separate rule.

I: OK. When you went through that. . .Peg: And that’s pretty much from my point of view which was from the easiest

to the most difficult. Although, I guess that the e function is not really thatdifficult; it’s just knowing that it, it’s just not as, it’s not difficult it’s justdifferent because it just doesn’t seem to follow the format as all of the others.

I: Uh-huh. Um, was there any other way that you considered grouping these?Peg: [pause] Uh. . . in the time frame, I just went with the first thought that popped

into my mind. I’m sure that if I was to sit here longer, I could think of otherways to do it.

I: OK. What might be other criteria?Peg: [pause] Uh, the other ones would have been, may be, the ones that just had

single terms as opposed to having two terms that have to be differentiated.Or anything that that has a chain, or pulling out anything that has the chainrule and knowing that also have to add more terms to the final expression.

Peg gave a very clear statement of the chain rule, using function notation. She

augmented this with a comment about inside and outside things. But her notion of

chain rule and composition were apparently driven by parentheses.

I: So the last couple of questions that I have for you, write down the chainrule using whatever words or symbols that you like. It’s as best as you canremember it. You have here like examples of it. Number 6 and 7 which is16 and 17 on our papers here. You used the chain rule in your answer. So ifyou want to use those to help spark some ideas. . .

Peg: [pause] [mumbles to self] [pause] OK.I: Read that for the tape.

Peg: OK. Uh, when you have the derivative of f(g(x)) it’s f ′ g(x) times g′(x). It’sbasically you take a derivative of the outer term and I’m using outer termbecause it’s just the way that I look at it, composed with the inner term andthen multiply it by the derivative of the inner term.

I: OK. Great. Um, actually the last five statements here on the list, 6 through10,

Peg: Uh-huh.I: were intended to be, to use the chain rule and you named 6, 7, & 8 as using

the chain rule. Do you see the chain rule being used in 9 and 10?Peg: Yes, I do, but I had pulled them out separately because they had the trigono-

metric functions also.I: OK, no that’s fine. Does your rule that you have written there at the bottom,

does it apply or does it take care of all six of those or uh, all five of thosecases?

Peg: [pause] Uh. . . in the way that I would look at it, the way that I look at theproblem, it does. I look at the outer term and the inner term which would,

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the outer term that I’m looking at is being either how you solve strictly forthe power rule or for the trigonometric functions and then going inside andactually you know. . .

She hesitated to include cos3(t) in the chain rule group since t was in the parentheses

alone (or e−t2 due to lack of parentheses). She had attempted to solve it by rewriting

it as a triple product. This is consistent with the codes she received in Phase 1 on

composition and her lack of success with the final interview problem (discussed in the

next section).

I: OK. [pause] You did a very interesting thing when you solved 19.Peg: I don’t think that’s correct.

I: I don’t think that it’s very far wrong actually. But what you did is ratherthan use the chain rule, you expanded the expression. The expression wascosine cubed of t. And so you wrote down three products of cosine t whichis cosine cubed.

Peg: Uh-huh.I: And then, you took a derivative from there.

Peg: Even with that, I didn’t finish it did I?I: No, you needed to do the other product.

Peg: Yeah, but, yeah, I made it to that.I: But, my question is, if you can recall, this was a long time ago, um, was

there something that didn’t say chain rule to you in that problem?Peg: That would be the one that I’m still the least clear on still today because it’s

not as straight as the chain rule because the uh, the variable t doesn’t haveanything associated with it like all of the other ones, that there’s somethinginside the parenthesis to differentiate. Where this one. . . so would it be 3cosine squared t uh times minus sin? Is that the correct answer.

I: Uh-huh.Peg: Oh.

I: That’s exactly it.Peg: But I, it’s just not as obvious of the chain rule as all of the others.

Peg was no longer familiar with implicit differentiation or related rate problems.

After being reminded, she does not see the chain rule since there is nothing inside the

other. She did see the chain rule in the last problem although the interviewer had to

point out the expression for F in the paragraph.

Overall we see Peg making connections based on a structure, the use of parenthe-

ses. She gave a general rule (again written with parentheses) but had some difficulty

recognizing problem situations which fit her rule. Peg may be working at the Trans-

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level, that has been built on a weak structure. Her understanding of the chain rule—

discussed in the next section—is rather strong, but the data is equivocal regarding

the triad level.

7.1.6 Jack [A113]

The one example of the Trans- level is Jack, who not surprisingly had the highest

overall score of those interviewed. Jack grouped (and then partitioned) the ten differ-

entiation problems based on levels of difficulty. He clearly saw the chain rule as one

of the criteria along with the power rule, product rule and trigonometric rules. He

stated that the trig rules are the most difficult, indicating that the other rules have

reasons behind them, but the trig rules must be memorized.

Jack: 10, I would probably put, I don’t know, I would be tempted to put 10 in agroup by itself just because with it just being e it’s basically the chain rule,and the chain rule and the power rule together. That’s it. That one is done.Um, 7 and, let’s see 7 requires it, 7 requires product rule and chain rule, andthe all powerful power rule. . .8 just requires product rule. I know, no I probably would just go ahead andgroup 4, 5, 8, & 9 all together because they have the trig function and becauseI mean, the trig functions are the only derivatives that can throw you off realeasy if you don’t know them, because chain rule, if you understand chain ruleand you understand the use of things like exponentials and logarithms thenyou aren’t gonna get messed up bad on chain rule. You aren’t gonna getmessed up bad on the power rule if you know simple mathematics. You’renot gonna get messed up too bad on product rule as long as you rememberto keep everything straight. But, the trig functions, you know, you got, Ican’t even think of them at the moment. I haven’t used them in a while, butit’s like you know one it’s the other and it’s just negative and one it’s justthe other, period. And it’s like if you forget that sine, the derivative of sinejust by whether or not it’s got a sign change in it, you just messed yourselfup big time and you’re gonna get a wrong answer because they cycle; and ifyou start off on, off with the wrong derivative of it then you have messed upthe cycle already. Then, no matter, if you know all the others you are gonnabe messed up anyway. So. . .

In the next questions, Jack gave the textbook answer for the chain rule. He

elaborates on his answer describing inside and outside functions, as opposed to Al

and Peg who spoke of inside and outside terms. And finally he outlined a proof of

the rule.

Jack: Um, huh, [pause] let’s see how would I write that down? OK. Let’s see.[pause—writing] Um, that’s just the way that it runs through in my head.[had written f(g(x))′ = f ′(g(x)) · (g′(x)) · x′] Without any words.

50

I: OK.Jack: That’s just the way that I think of it.

I: OK. Read that for the tape.Jack: Um, you take the derivative, essentially, whenever you use the chain rule you

are essentially looking at a function that has got another function within itum, I don’t know, it’s sort of like doing a, handling a composite, um andin order to take the derivative of that composite you have to first take thederivative of the outside function and not even do anything with what’s theinside of it, the function that’s on the inside of it, you take the outside, it’sderivative first and leave the inside function alone and then multiply that bythe derivative of the inside function, and then multiply it by the derivativeof the variable, or however many times you have to break it down. Becauseyou can have a huge function that’s got a lot of stuff inside of it and you’dhave to do the chain rule several times to get the x variable.

I: OK.Jack: So, I mean you could have, that’s just like a simple composite f of g, but

you could have, if you have like h(f(g)) then you’d have to do the derivativeh with f of g inside of it and then the derivative of f with g inside it andthen the derivative of g and then derivative of x.

I: OK.

In the four problem situations, Jack saw the chain rule being used in all of them.

His explanation for each indicated thinking about the functions in the situation.

Overall, he had as well established an understanding of and facility with the chain

rule as one can hope for in a first year calculus student. The Trans- level is not so

evident by the preceding quotes, but is more clear in his work on the Leibniz rule

problem which is discussed in the next section.

7.2 Using versus understanding

A working hypothesis of this study is the need to understand composition of

functions as a condition for understanding the chain rule. While the first phase results

did not add evidence to this claim, the claim was not refuted since the questionnaire

items did not assess understanding of the chain rule, rather they could only indicate

success (or lack of success) with the rule.

7.2.1 Coding the aspects of understanding

The interview data provide us with sufficient information to assess the student’s

understanding of the rule. In order to discuss understanding, we have noted three

aspects from the interview:

1. Did the student mention the chain rule prior to question 10 of the interview?

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2. Did the student arrive at a coherent statement of the chain rule? With or

without a prompt?

3. Was the student successful with the Leibniz rule problem? With or without a

prompt?

These questions were coded as an ordered triple with codes Yes or No for the first

aspect and codes Unsuccessful, Prompted, or Successful for the second and third

aspects. The following excerpts illustrate the codes for each subject which are sum-

marized in Table 7.1 below.

Eli received the code [N, U, P]. As we observed in the excerpts given in the previous

section, Eli’s statement of the chain rule never indicated that the derivative of the

outside function was then composed with the inside function. When working on the

Leibniz problem, he insisted on finding an antiderivative. The interviewer provided

him with the simpler Leibniz problem (H(x)) and had done most of the talking up

to this point, getting Eli to work out the closed form of H:

I: Alright. OK, so now that’s H(x). That’s just computing it all out. Right?You just did this side, it’s still H(x). So I could put an x in and take a sinof it and do that other stuff. What’s H ′?

Eli: [pause] So you take derivative of this?I: Uh-huh

Eli: So this is the chain rule again.I: Always. [pause] OK, so you have. . . Oh, you didn’t label it H ′. But H ′(x) is

4 times sin(x) squared times the cos(x). Right? Now we started off with oneof these deals where it’s defined in terms of its definite integral and you wereable to take the antiderivative and evaluate it at the endpoints and come upwith that, and then take the derivative and you get this. Now back to theoriginal expression up here, do you see a way to jump straight to that answerwithout finding an antiderivative?

Eli: Yeah.I: What is it?

Eli: You just um, replace t by sin(x) then just multiply the derivative of sin(x).I: OK, then let’s go back to the first one that I asked you. F (x) is the definite

integral from zero to the sin(x), et2 dt. What’s the derivative of F?Eli: The derivative? The derivative of. . .

I: The derivative of F at the very bottom here like you found H ′ here.Eli: [mumbling] [writes F ′(x) = e(sin x)2 · (cos x)] OK.

Tim’s code is [N, P, U] where the first two are seen in the excerpts above. In the

last problem, he recognized the fundamental theorem of calculus (FTC) version of

the situation (G(x)), but could not make a connection back to the Leibniz version.

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I: I am asking you to find F ′ where F is the function given by F (x) = theintegral from 0 to the sin(x), et2 dt.

Tim: It is the inside, the anti-derivative [writes F ′ = et2 ].I: Let me show you one other one, here is a function G(x) which is defined

almost the same way, zero to x of et2 dt. What’s G′?Tim: Can I do this? [works and mumbles]

I: Whoa, whoa. You have taken a derivative there, not an anti-derivative.Right?

Tim: Oh! Sorry, I’m sorry.I: Let me just help you out here. There is no anti-derivative of et2 .

Tim: Oh? It is a trick.I: There is no simple anti-derivative that we can just write down on paper.

So your method there, while it would probably work, if you could take theanti-derivative of that, it won’t work here.

Tim: [pause—working] [attempts to find an anti-derivative] Yeah, it does not work.I: But can you answer the question? Can you figure out what G′ is if you know

this is how G is defined?Tim: [1.5 minute pause] I’m lost!

I: Really?Tim: Yeah, I think G′(e) = et2 .

I: OK. G′ of . . .Tim: G′(e) = et2

I: G′(e)? is et2 .Tim: Uh-huh

I: OKTim: Is that right?

I: Let me ask you this, when you answered this first one, you jumped right toit and said ‘oh, F ′ is the insides’.

Tim: Same thing here.I: They are the same. So it doesn’t matter that there is a sin(x) up here?

Tim: Oh?I: This one goes 0 to x, this one goes 0 to sin(x). So I am wondering if that

makes a difference in the answer?Tim: So, yeah, yeah, hold on. Sine . . . [works and mumbles] [writes −1 ≤ sinx ≤ 1,

0 ≤ sinx ≤ 1] Yeah, I think it doesn’t matter for that.I: It doesn’t matter.

Tim: No.

Al got [N, P, U]. In his solution to the Leibniz problem, he quickly stated that the

derivative is found by substituting sin(x) in for t. Then when working through the

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FTC problem, he wanted to use the chain rule and multiply by 2x. After the prompt

to correct G′, he did not go back to modify his original answer.

I: OK. Great. What I’m asking you for here is compute F ′ if F is the functiongiven by F of x = the definite integral from zero to the sin(x), et2 dt.

Al: I’d take et2 and just put the sin(x) squared in there, and then you’d prettymuch have zero and it would be this right here.

I: That’s the derivative? That’s F ′?Al: Yeah.

I: OK, just clarifying.Al: OK.

I: Let me ask you a similar question. This one is called G, it’s basically thesame except um, the definite integral goes from zero to x. What would um. . .

Al: Just change the t to the. . .x squared.I: OK, so G′.

Al: times 2x, and over here you’d take derivative too. I don’t know . . . what I’msaying, you’d take the derivative, uh . . .[end of side one]

I: So you’re wondering do you take the derivative. . .Al: Take the derivative of x2 or not.

I: [talking together] Take the derivative of x2 or not.Al: Yeah.

I: Alright. Do you remember seeing situations like this where you have thefunctions defined through definite integrals?

Al: Uh-huh.I: OK, it was part of the fundamental theorem of calculus.

Al: Oh, [mumbling] I never. OK, I know this part was. . . right and this part wasright.

I: OK, ex2you’re sure about, and you’re wondering about, um, the 2x part.

OK. [pause] So, suppose I tell you that you don’t do 2x in this problem. Theanswer is just ex2

. That comes from the fundamental theorem of calculus,which essentially says that if you have a function defined in this manner thederivative of that function in this case G′ is just the inside written in termsof x.

Al: Uh-huh. Yeah. And that would be the answer.I: Without the 2x?

Al: Yeah.I: Right. Now what I want you to do is to go back and look at this other

problem.Al: Uh-huh.

I: And see, you have e to the sin(x) squared, and the question is, are you stillcomfortable with that?

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Al: [pause] Yeah.I: OK.

Al: Yeah.

Ray and Peg received codes of [Y, S, P]. Ray’s first thought on the Leibniz problem

was, since a definite integral is a number, then the answer is 0. It only took a small

prompt for him to see the function involved. When he saw the FTC version of the

question, he recalled the rule and quickly gave the answer. He then saw that the

original question involved a composition and struggles a bit before coming to the

answer for F ′.

Peg mentioned the chain rule when confronted with a composition problem (that

is not a derivative problem). So the connection between the two is strong. Peg

recognized that the situation involved a theorem that she once knew, but denied

the FTC. The interviewer prompted her through the FTC and also pointed out the

composition.

Jack got a [Y, S, S]. Jack indicated a cohesive schema for the chain rule as he

attempted to deal with the Leibniz problem, which he did not recognize. To deal

with the unfamiliar, he wanted to compute something, so the interviewer provided

him with a simpler Leibniz rule problem with a monomial integrand. Jack worked

through the integration, then needed a little prompt to label his answer (the function

H). When asked to go back to the original problem, he was able to generalize the

use of the chain rule and to arrive at the correct solution.

7.2.2 Comparing the codes

The results of the codes for understanding are presented in Table 7.1. It is sorted

according to a lexicon order on the codes. Also presented are the combined codes

from Phase 1.

Two patterns in this table are of particular note. First is the support for our

working hypothesis. With the exception of Peg, the scores increase for the Compo-

sition codes as one is more successful at understanding. We note that Peg left items

9 and 10 blank on the questionnaire because she did not want to deal with the ta-

bles. If we exclude the codes for items 9 and 10, the Composition column becomes

[6, 9, 9, 9, 15, 15]T which also lends support to our hypothesis.

The second pattern is the third aspect code and the C4L variable. Those who were

able to deal with the Leibniz rule problem were all in a C4L section at some point.

Both of these points seem to say something, but must be understood in context.

These patterns exist in these six students’ results only and do not generalize to any

55

Table 7.1Aspects of understanding

Code C4L Function CompositionDifferen-tiation

Chain Rule

Eli [N, U, P] 11 9 7 16 20Tim [N, P, U] 00 4 11 22 25Al [N, P, U] 00 19 12 24 19Peg [Y, S, P] 10 14 9 25 21Ray [Y, S, P] 11 18 21 19 1Jack [Y, S, S] 01 15 24 25 24

other population, not even the Phase 1 group. It would be unwise to attempt to draw

any strong conclusions from this data.

Chapter 8: Discussion

The analysis of the data is consistent with the genetic decomposition for the

concept of the chain rule based on the triad, which was presented at the start of

this work. Some evidence is presented to support the notion that understanding of

composition of functions is key to understanding the chain rule.

The type of instruction was a factor in how a student performed on these tasks.

The differences between the types seem to be related to the difference between us-

ing the chain rule and understanding the chain rule, and an explanation of these

differences is offered.

The need for collecting differing types of data is established. Two students scored

high quantitatively but did not perform well in the interview, while another student

failed the quantitative work but demonstrated a high level in the interview. Finally,

the use of writing as a pedagogical tool is recommended based on the results.

8.1 Revised genetic decomposition

The overarching purpose of this study was to test APOS Theory, specifically

the proposed use of the triad mechanism to describe the development of a student’s

schema. For the six participants in the interviews, we claim that it does. The inter-

view data show students working at all three levels by discussing how they appear

to be making connections as they approach problem situations which involve the

chain rule in the solution. After presenting evidence of perceiving relationships in the

situation, a working definition of understanding the chain rule as it appears in the

data was constructed. This construct presents a parallel story as the six subjects are

ranked in understanding along the same lines as the triad levels they demonstrate.

The students who have moved into the Inter- level demonstrate some ability to ex-

tend their understanding of the chain rule, as well as the ability to discuss the rule

coherently.

The interview data did not provide evidence which required the modification of

the genetic decomposition given in Section 2.4. So the description of the schema for

the concept of chain rule developed by Clark et al. (1997), including this author, has

the support of this study with the six interviewed students. Clark et al. concluded

that the chain rule schema

56

57

must contain a function schema which includes at least a process concep-

tion of function, function composition and decomposition. The function

schema is linked to a differentiation schema which includes the rules of

differentiation at least at the process level. (p. 359)

Clark et al. (1997) further stated that this schema is developed through the three

levels of the triad. In the Intra- level, the student is able to deal with many types of

problems as distinct cases. Some students received high codes for their work on the

chain rule section of the questionnaire. But their concept of the rule is insufficient to

discuss the chain rule in any detail or to extend it beyond their familiar collection of

rules.

At the Inter- level, the student begins to reflect on these various cases and builds

connections relating them. Students in this study not only used phrases such as

“inside function/term/stuff” and “outside function/term/stuff”, but also used them

as the criteria when presented with new situations. In other words, they were using

their perceived relationship (or more general idea) for the chain rule when asked if

the chain rule would appear in the solution to a problem situation.

A student working at the Trans- level has acted on the relationships from the pre-

vious level to construct a coherent whole, a system by which he or she can determine

what situations are part of the schema and what are not. This investigation used the

Leibniz rule problem as a way to test the robustness of the subjects’ schema (or pre-

schema, if at the Inter- level). This problem features a composition with a function

defined through an integral, which pushes most students’ function schema to the limit

(Thomas, 1995). The problem was presented with simpler cases on hand in order to

help the students progress to the point of needing the chain rule. The simpler cases

were intended to help the student deal with a function defined through an integral

or to give the student a chance to compute something if the need to do so was great.

The analysis of the responses to this problem was focused on how the chain rule was

incorporated into the problem, once the composition in F was established.

This genetic decomposition is not “revised” in the sense that it has changed as a

result of the data analysis. It has however been revisited with new data and found

to be consistent with that data. The triad mechanism is useful in describing the

development of this schema, and will likely be as useful with other schemas. This

mechanism extends APOS Theory by augmenting our understanding of how schemas

might come to be built in the minds of our students. In particular, this genetic

decomposition will be used as a basis for the design of future studies involving the

58

chain rule and for the development of instructional strategies, such as those presented

below.

8.2 Understanding of composition and understanding of the chain rule

Can we conclude that understanding of composition of functions is key to under-

standing the chain rule? The results from the first part of this study were largely

inconclusive. The students performed well on the differentiation problems and the

chain rule problems. There was not the same degree of variation as in the function

and composition sections, and this did not allow significant correlations to be found.

This is a problem of assessment; the difficulty in measuring (depth of) understanding

in a written instrument. Since the questionnaire data did not measure understanding

with respect to the chain rule, the evidence does not contradict the hypothesis.

There is a small amount of supporting evidence in the Phase 2 data. As seen

in Section 7.2, for the six subjects who were interviewed, there exists a correlation

between understanding the chain rule and the codes for (three of the five) composition

items. A new study is needed to address this hypothesis. That study might employ

interviews based on the interview guide for this study with a larger sized sample. Since

the focus would be on understanding, collecting data with a questionnaire alone is not

advised (see Section 8.4). A questionnaire could be useful for screening participants

for interviews and for setting the stage for the interview tasks, as it was for this study.

8.3 Comparing instructional methods

The comparison between the traditional and reform methods is an account of using

versus understanding, of instrumental versus relational. In the interview setting,

the students who had experienced some reform methods were more successful in

extending and discussing the chain rule. In the quantitative data, the traditional

group performed better in the differentiation and chain rule sections, in the sense of

statistical significance.

8.3.1 Understanding the chain rule

In Table 7.1 some evidence is found that the students who had experienced C4L

instruction demonstrated better understanding than the others. Eli is the exception

to this statement, but it fits with his role as an outlier in the quantitative phase and

with his lack of cooperation in the qualitative one. This comparison can only be made

with these six (or five) students and so must be seen as an example, rather than as

a trend. On the other hand, a current review of a series of studies in which student

performance data is available indicates that, on average, students who had instruction

59

based on APOS Theory performed at the same level or above those students who had

traditional methods of instruction. (Clark, Dubinsky, Loch, McDonald, Merkovsky,

& Weller, 1999). The interview evidence of this study is also consistent with that of

Meel (1998).

8.3.2 Using the chain rule

The questionnaire data did give us information on how well the students were able

to use the chain rule. Tables 5.6 and 5.7 both indicate the students in the traditional

course scored higher in the chain rule problems. In the first table, the C4L group

was taken as C4L > 0, which cut the participants into almost equal sized samples.

In that case, the mean Chain Rule score for the Traditional group was 22.00, a 9.3%

increase over the C4L group. In the second table, the groups were formed based on

the type of instruction for the first semester. In that case, the mean of the Traditional

group (21.91) was statistically significantly higher than that of the C4L group (18.75),

16.9% higher.

These results are also consistent with previous studies. Heid (1988) found that

students from traditional calculus courses performed slightly better on skills based

problem situations than students from reformed courses. Dubinsky and Schwingen-

dorf (1991) found the slight differences between groups favored the reform group.

Palmiter (1991) found that the reform students performed better than the control

on both conceptual as well as computational tests. However, her results may have

been skewed since the treatment group was aware of the study and the control group

was not. For our results, there are two global factors and one local factor which may

explain the difference. Globally, a traditional course by its nature emphasizes skill

acquisition. Typically, a student in a traditional course will complete 60–75 problems

in a week, with most of the problems being computational over a small range of top-

ics. Another global factor concerns C4L in particular and reformed calculus programs

in general. In order to utilize technology as a tool for calculus, time has to be spent

getting familiar with it. C4L pays a price in its methodology, and part of that price

is less time for practicing skills.

In addition, the implementation of the C4L course that these students experienced

was new to the institution, as the instructors has just joined the faculty. This local

factor involved changing student expectations for the mechanics of a course, dealing

with a commuter student body (having come from a residential situation), and net-

working problems inherent in a new computer laboratory. There is no practical way

60

to measure the effect of this factor on the difference in the Chain Rule scores between

courses1.

8.3.3 Finding a balance

These results are not surprising but do present the question of our teaching ob-

jectives. If a conceptual understanding is the goal (and how can it not be?), then

this evidence indicates a focus on developing a schema is necessary, as was done in

the reform method seen here. Of course, Ray is an example of going too far away

from the instrumental knowledge to the conceptual. However, it is certainly easier

to fix this than the reverse. Once a conceptual basis has been established—once a

schema has been constructed—it is merely a matter of practice and drill to augment

the understanding with skills. The use of quizzes on computational items or the use

of gateway examinations might help balance any over-emphasis on the conceptual

understanding by a particular methodology.

8.4 Quantitative versus qualitative data

In the quantitative phase, three sets of data from the questionnaires were excluded

from the analysis because there were too many zeros in the code data—which was

interpreted as incomplete questionnaires. Two of these students were interviewed in

the second phase, Eli and Ray.

In the previous chapter, the case was made that Eli should have been excluded

from the quantitative analysis for non-participation. In fact, he was fairly non-

participatory in the interview, too. The decision to exclude Ray from the quantitative

analysis was justified. However, it was not the case that he was not participating—he

was just caught off guard. Had he been told what to expect when he came to fill out

the questionnaire, he may have prepared and done very well as he had in his calculus

classes.

In this set of interview participants, two individuals with high scores in the Chain

Rule codes are found: Tim got 25 and Al got 19. Based on this data alone, we might

have concluded that these two “know” the chain rule or even “understand” it. Yet

both have some difficulty extending the concept. Tim even had difficulty expressing

the rule with prompting. At the same time, Ray got a 1 for the Chain Rule in the

written work. This would have to be interpreted as a failure to understand the rule

in a test situation.

1Over the intervening couple of years, this factor has diminished—especially in the perspective ofthe students.

61

The qualitative data paint almost the opposite picture for these three students.

Which is the accurate representation of reality? One conclusion of this study is the

necessity for collecting multiple data types when attempting to describe a student’s

understanding of a concept. The question raised is, how can we as teachers assign

grades based on written exams and homework only?

8.5 Limitations

The results of the study may be limited in their potential to be generalized in two

ways. First, since the subjects were volunteers, there was no control over the number

of subjects for the study nor over the range of abilities of students. The issue might

be addressed by trying to get n ≥ 50, a goal which was unattainable in this study.

With such a number, the spectrum of abilities should be filled out. The statistical

methods used in Phase 1 might have been extended by deleting variables from the

model or throwing more powerful models at the situation. That this was not done

came from a certain sense of purity; the bad data is just accepted as is. The model

was based on the genetic decomposition of understanding a concept. However, the

data which was collected reported a measure of using the concept. It also came from

the fact that only this set of students was being described, rather than an attempt

to describe the entire population of first year calculus students.

The second issue is also about self-selection. Since the comparison of the two

methods took place after the term, no pairing of students across treatments was

possible. Students at the host institution are free to choose whatever section of a

course they desire. Thus other factors (or student characteristics) may be involved

in whether the student chose to take a Traditional or a C4L course. Schwingendorf,

McCabe and Kuhn (1998) discuss statistical methods that have been developed to

treat this issue, which arises often in health and industrial studies. One such method

is factoring in the students’ PGPA scores, which seems to have accounted for any

natural differences between the classes in this study.

8.6 Instructional strategies

A student’s concept of the chain rule appears to be a schema which is devel-

oped through the triad mechanism. The components of this schema are function

de-composition and differentiation. While the students of this study seem to have

62

the differentiation well in hand, they appear to need to move from a focus on “in-

side/outside” structure based on parentheses to a perception of composition of func-

tions. To aid in the progression through the triad, we should give students opportu-

nities to reflect on a variety of situations in order to build some connections and then

to reflect on those connections.

Short writing assignments may be the tool to take care of both of these needs.

By requesting an explanation which identifies the functions involved in a chain rule

problem, along with the answer, the student will have to approach the situation func-

tionally, rather than symbolically. Likewise, short essays that compare and contrast

banks of problem situations will foster the reflections necessary.

Regular writing assignments will also address the assessment issue raised above.

The written discussion can allow the instructor to see the student extending the

concept to new situations. Wahlberg (1998) found evidence of both learning on the

part of the student and enriched assessment on the part of the teacher using writing

assignments on the topic of limits. While writing assignments take time to grade,

they are a reasonable alternative to scheduling, conducting and transcribing personal

interviews with your students.

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63

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APPENDICES

Appendix A: Questionnaire from Phase 1

Code:

Questionnaire for Calculus Topics Study

Jim Cottrill

Directions:

• These items are designed to explore your range of understanding on a number

of topics. Some will be very easy to do, some will be harder. Please do the best

you can.

• Please answer each question as completely as you can. Please do not leave any

answer blank. If you would like to write additional notes with an answer, feel

free to use the back of the page.

• Please do not write your name on any page of this questionnaire. The code

number at the top of this page will serve to keep your identity confidential.

69

70

1. Express the diameter of a circle as a function of its area and sketch its graph.

2. A student has marked the following as a non-function. State whether this

student is correct and why.

y�

x

71

3. A student has marked the following as a non-function. State whether this

student is correct and why.

A correspondence that associates 1 with each positive number, −1

with each negative number, and 3 with zero.

4. Decide if it is possible to use one or more functions to describe this situation.

If yes, then describe your function(s) briefly. If no, then explain.

x = t3 + t

y = 1− 3t + 2t4

t is a real number.

72

5. Tim and Donna live 1 km from their school. Usually, they walk to school

together. Yesterday, they both left their houses 10 minutes before school started.

Tim started to walk, but Donna was afraid to be late and started to run. After

a while, Tim realized that although he tried to walk faster and faster, he had

to run if he did not want to be late, and started to run. At about the same

time, Donna became tired and had to walk instead of run. They both reached

school exactly on time. Which of the following graphs is Tim’s and which one

is Donna’s?

time

dist

ance

t ime

dist

ance

t ime

dist

ance

t ime

dist

ance

t ime

dist

ance

t ime

dist

ance

6. Given two functions v, w such that v(t) = 5t−6 and w(t) =t

3+4, find (w◦v)(9).

Explain.

73

7. Given that (f ◦ g)(x) = 5√

2x + 3.

(a) Find f and g that satisfy this condition.

(b) Are there more than one answer to part (a)? Explain.

8. Find k so that g(x + 1) = g(x) + k, given that g(x) = 3x + 5. Explain.

74

In each of the following two questions, f, g, h are functions whose domains and ranges

are the set of all real numbers, and such that h = f ◦ g.

9. If only the information in the following table were known, would it be possible

to find f(2)? If so, find it and if not explain why not.

x h(x) g(x)

−1 1 −3

4 π 1

π 0 2

10. If only the information in the following table were known, would it be possible

to find g(4)? If so, find it and if not explain why not.

x h(x) f(x)

−1 1 −2

2 3 1

4 −2 π

75

Compute the derivative of each of the following functions. Show all of your work.

11. f(x) = 11x5 − 6x3 + 8

12. g(x) =3

x2

76

Compute the derivative of each of the following functions. Show all of your work.

13. h(x) =(x2 − 3

) (5x− x3

)

14. y = 3ex − 4 tan(x)

77

Compute the derivative of each of the following functions. Show all of your work.

15. y = x2 sin(x)

16. F (x) =(1− 4x3

)2

78

Compute the derivative of each of the following functions. Show all of your work.

17. G(x) = 2(5x2 + 1

)4− 4x

(5x2 + 1

)4

18. H(x) = sin(5x4

)

79

Compute the derivative of each of the following functions. Show all of your work.

19. y = cos3(t)

20. y = e−t2

Appendix B: Phase 1 Data

The following tables summarize the distribution of codes for each item in the

questionnaire. In Table B.1, the data is presented in the form n(a, b) where n is the

number of responses receiving that code, a is the number in the C4L group (C4L codes

11, 10, 01) and b the number of traditional students (C4L code 00). The maximum

value for a or b is then 17. The other tables give the codes by each student’s ID,

which was used to protect the identities during the analysis. The (*) next to an ID

indicates a questionnaire which was considered to be incomplete.

The C4L variable used in Tables B.2 and B.3 (and in Table 5.5) is a binary code for

the first two quarters of calculus. An “11” represents the student took the C4L version

of calculus both terms. A “10” indicates taking the C4L version first quarter and a

traditional course the second quarter, while a “01” indicates the opposite situation.

A “00” indicates being in a traditional course both terms.

80

81

Table B.1Distribution of codes by item

5 4 3 2 1 0

1 4 (1,3) 8 (3,5) 1 (0,1) 10 (7,3) 4 (3,1) 7 (3,4)2 12 (4,8) 3 (3,0) 8 (5,3) 3 (1,2) 8 (4,4)3 7 (3,4) 3 (2,1) 15 (8,7) 3 (1,2) 6 (3,3)4 2 (0,2) 8 (4,4) 9 (4,5) 9 (5,4) 6 (4,2)5 27 (14,13) 4 (2,2) 2 (1,1) 1 (0,1)6 20 (9,11) 3 (2,1) 2 (2,0) 1 (0,1) 7 (4,3) 1 (0,1)7 17 (11,6) 3 (1,2) 7 (2,5) 1 (1,0) 4 (2,2) 2 (0,2)8 15 (8,7) 3 (1,2) 3 (1,2) 3 (1,2) 8 (5,3) 2 (1,1)9 8 (6,2) 3 (1,2) 4 (3,1) 13 (3,10) 3 (1,2) 3 (2,1)10 2 (2,0) 6 (3,3) 5 (4,1) 11 (4,7) 3 (0,3) 7 (4,3)11 34 (17,17)12 23 (12,11) 5 (2,3) 3 (1,2) 2 (1,1) 1 (1,0)13 22 (11,11) 6 (2,4) 3 (1,2) 1 (1,0) 1 (1,0) 1 (1,0)14 21 (7,14) 8 (6,2) 4 (3,1) 1 (1,0)15 25 (9,16) 2 (1,1) 2 (2,0) 3 (3,0) 2 (2,0)16 28 (12,16) 4 (3,1) 1 (1,0) 1 (1,0)17 12 (5,7) 14 (7,7) 3 (2,1) 2 (1,1) 2 (1,1) 1 (1,0)18 23 (10,13) 6 (3,3) 2 (1,1) 1 (1,0) 1 (1,0) 1 (1,0)19 14 (5,9) 4 (3,1) 1 (1,0) 13 (6,7) 2 (2,0)20 21 (9,12) 5 (3,2) 3 (2,1) 1 (1,0) 3 (1,2) 1 (1,0)

82

Table B.2Student codes in tuples

ID C4L [Items 1–5] [Items 6–10] [Items 11–15] [Items 16–20] Overall

A111 01 [4, 5, 5, 2, 5] [5, 5, 1, 5, 4] [5, 5, 5, 1, 2] [5, 3, 5, 5, 5] 82A112 01 [4, 3, 3, 1, 5] [4, 5, 5, 5, 5] [5, 5, 5, 5, 5] [5, 5, 5, 5, 5] 90A113 01 [1, 5, 4, 0, 5] [5, 5, 5, 5, 4] [5, 5, 5, 5, 5] [4, 5, 5, 5, 5] 88A115 01 [2, 1, 1, 1, 5] [5, 5, 5, 5, 5] [5, 5, 4, 5, 5] [5, 4, 5, 4, 5] 82A121 11 [2, 4, 0, 2, 5] [5, 5, 5, 5, 4] [5, 5, 5, 5, 5] [5, 5, 4, 5, 5] 86A122 01 [5, 1, 3, 3, 2] [5, 5, 5, 5, 2] [5, 5, 5, 5, 5] [5, 5, 5, 0, 5] 81A123* 11 [4, 3, 3, 3, 5] [5, 5, 5, 3, 3] [5, 5, 5, 3, 1] [1, 0, 0, 0, 0] 59A124 11 [1, 1, 0, 0, 5] [3, 1, 1, 3, 3] [5, 5, 5, 4, 4] [5, 4, 5, 4, 3] 62A125* 11 [0, 4, 0, 0, 5] [1, 1, 4, 1, 0] [5, 3, 2, 4, 2] [5, 4, 5, 3, 3] 52A126 10 [2, 5, 3, 3, 5] [5, 5, 5, 0, 0] [5, 4, 1, 3, 3] [5, 4, 2, 2, 4] 66A127 00 [5, 5, 3, 5, 5] [5, 5, 5, 5, 4] [5, 5, 5, 5, 5] [5, 5, 4, 5, 5] 96A128 00 [5, 5, 5, 3, 5] [5, 3, 1, 2, 2] [5, 4, 5, 5, 5] [5, 5, 5, 5, 5] 85A129 10 [2, 4, 4, 2, 5] [5, 3, 1, 3, 3] [5, 5, 5, 4, 5] [5, 5, 5, 2, 5] 78A130* 00 [0, 5, 5, 0, 0] [0, 1, 0, 2, 2] [5, 5, 4, 5, 5] [5, 1, 5, 2, 4] 56A131 00 [0, 3, 0, 1, 1] [5, 5, 1, 2, 2] [5, 5, 5, 5, 5] [5, 5, 5, 4, 5] 69A132 10 [2, 3, 3, 1, 5] [1, 3, 5, 0, 0] [5, 5, 5, 5, 5] [5, 4, 5, 2, 5] 69A133 00 [4, 1, 3, 1, 5] [1, 3, 4, 2, 2] [5, 5, 4, 3, 5] [5, 3, 5, 5, 3] 69A134 00 [0, 1, 0, 1, 2] [1, 3, 5, 1, 1] [5, 3, 5, 4, 5] [5, 5, 5, 5, 5] 62A135 00 [4, 2, 1, 3, 5] [2, 1, 3, 2, 0] [5, 5, 3, 5, 5] [5, 4, 5, 2, 1] 63A136 00 [1, 3, 5, 0, 5] [1, 0, 2, 4, 0] [5, 5, 5, 5, 5] [5, 4, 4, 5, 4] 68A137 00 [4, 5, 3, 2, 5] [5, 0, 4, 2, 1] [5, 4, 5, 5, 5] [5, 2, 5, 2, 5] 74A138 00 [4, 1, 1, 3, 5] [4, 3, 2, 1, 1] [5, 4, 5, 5, 5] [5, 4, 5, 2, 5] 70A139 00 [2, 5, 3, 2, 5] [5, 5, 5, 2, 2] [5, 5, 5, 4, 5] [5, 4, 5, 5, 5] 84A140 01 [0, 1, 3, 1, 5] [1, 2, 1, 0, 0] [5, 5, 4, 4, 2] [5, 2, 3, 5, 4] 53A141 01 [2, 3, 3, 1, 2] [5, 5, 0, 4, 3] [5, 5, 5, 4, 5] [5, 4, 4, 4, 1] 70A142 00 [5, 5, 5, 5, 5] [5, 5, 5, 4, 3] [5, 5, 5, 5, 4] [5, 4, 4, 2, 5] 91A143 11 [0, 5, 5, 3, 5] [4, 5, 3, 2, 2] [5, 1, 5, 5, 5] [4, 4, 5, 2, 5] 75A144 00 [2, 5, 3, 2, 5] [5, 5, 3, 3, 4] [5, 5, 4, 5, 5] [4, 5, 5, 2, 5] 82A145 11 [2, 3, 3, 0, 1] [3, 5, 2, 2, 2] [5, 2, 0, 4, 1] [2, 1, 1, 2, 4] 45A146 00 [2, 3, 3, 1, 5] [5, 4, 5, 2, 2] [5, 3, 3, 5, 5] [5, 5, 5, 5, 5] 78A147 00 [3, 1, 0, 2, 5] [5, 5, 5, 2, 0] [5, 2, 5, 5, 5] [5, 5, 3, 2, 5] 70A155 00 [4, 5, 4, 3, 5] [5, 3, 5, 5, 4] [5, 5, 5, 5, 5] [5, 4, 5, 5, 5] 92A156 00 [0, 2, 3, 2, 2] [5, 4, 1, 2, 2] [5, 5, 4, 5, 5] [5, 4, 5, 5, 1] 67A157 11 [1, 2, 5, 2, 5] [1, 4, 1, 2, 2] [5, 4, 3, 3, 3] [4, 3, 4, 2, 2] 58

83

Table B.3Combined code data

ID C4L PGPA FunctionComposi-

tionDifferen-tiation

ChainRule

Overall

A111 01 2.70 21 20 18 23 82A112 01 2.30 16 24 25 25 90A113 01 3.40 15 24 25 24 88A115 01 3.10 10 25 24 23 82A121 11 2.40 13 24 25 24 86A122 01 3.00 14 22 25 20 81A123* 11 3.00 18 21 19 1 59A124 11 2.40 7 11 23 21 62A125* 11 2.80 9 7 16 20 52A126 10 2.60 18 15 16 17 66A127 00 3.90 23 24 25 24 96A128 00 23 13 24 25 85A129 10 2.30 17 15 24 22 78A130* 00 10 5 24 17 56A131 00 2.90 5 15 25 24 69A132 10 3.10 14 9 25 21 69A133 00 1.20 14 12 22 21 69A134 00 4 11 22 25 62A135 00 15 8 23 17 63A136 00 14 7 25 22 68A137 00 3.00 19 12 24 19 74A138 00 2.50 14 11 24 21 70A139 00 2.80 17 19 24 24 84A140 01 2.80 10 4 20 19 53A141 01 3.10 11 17 24 18 70A142 00 2.30 25 22 24 20 91A143 11 3.80 18 16 21 20 75A144 00 3.20 17 20 24 21 82A145 11 3.70 9 14 12 10 45A146 00 1.80 14 18 21 25 78A147 00 3.10 11 17 22 20 70A155 00 2.30 21 22 25 24 92A156 00 2.40 9 14 24 20 67A157 11 1.80 15 10 18 15 58

Appendix C: Interview Guide for Phase 2

C.1 Overview

The interview will follow a guide designed to elicit the student’s understanding of

the rule, based on tasks from the previous instrument. New tasks will be presented to

the student to probe the limits of understanding. The students will be asked to reflect

on the questionnaire tasks, possibly by asking them for definitions of function and

derivative, domain-range issues in composition, the statement of the chain rule, and

a discussion of any relationships between the 20 problems that the student perceives.

The guide will be written following the piloting of the written part. This will also

then be piloted with the pilot subjects.

The interview will also address Leibniz’ rule as a way of gauging the robustness of

the student’s concept. A situation with no simple anti-derivative will be presented,

such as “compute F ′ if F is the function given by F (x) =∫ sin x

0et2 dt.” If the need

to compute an anti-derivative is too great, a situation where the integrand has an

elementary anti-derivative will be presented to determine if the student can derive

Leibniz’ rule from the result. The data from these situations will be analyzed and

compared to that of Thomas (1995).

The interviews will be structured with four basic components, although each com-

ponent will be modified to fit the student.

C.2 Review of function and composition

The student will be presented with a copy of her or his questionnaire from Phase

I and will be asked to discuss two function situations; one which he or she dealt with

successfully and one less successfully. The same will be done for the composition

questions. The goal will be to get a definition of function from the student. The

student will be allowed to ask about any of the first ten problems.

1. Here is a copy of your work from the questionnaire for question ??. Look it

over and describe how you solved the problem.

2. Would you change anything in that answer?

3. Repeat (1) and (2) for another function and two composition problems.

84

85

4. Questions 2–3 dealt with whether a situation can be a function. What is your

definition of function? Omit if the definition has been established.

5. Do you have any questions about the first 10 questions?

C.3 Classifying the 10 derivative problems

The student will be asked to group in any way that makes sense the ten derivative

problems. It is expected that he or she will use the elementary rules as a guide. This

may help determine if the student sees all five of the chain rule problems as a single

class.

6. Here is a copy of the last 10 problems on one sheet. The directions for all of

these were to compute the derivative of each of the following functions and to

show all of your work. I would like you to group these 10 problems in whatever

manner makes sense to you. You can have as many of them in a group as you

wish and you can form as many groups as you want.

7. Describe how each of the groups were determined.

8. Is there any other way that you considered grouping these? If so, describe it as

well.

9. If no answer given has disjoint sets: What if I required that your groups are

disjoint? Can you still group them?

C.4 Establishing the notion of chain rule

The student will be asked to state the chain rule. The questionnaire problems

will be used to get as general a statement as the student can agree with. The student

will be asked to discuss the proof (or need for proof) of the chain rule as well as any

intuitive notions of the rule that they have.

10. Write down the chain rule using whatever words or symbols that you like.

11. The last five questions were intended to be solved using the chain rule. Compare

each of those situations with what you have written. Does your statement deal

with each or can you modify it so that it does?

12. What is the key idea in the chain rule?

13. Do you recall how you learned the chain rule for yourself? Did you use mnemonic

devices or any sayings or tricks to keep the rule straight?

86

14. In mathematics, theorems need to be proved using more elementary definitions

and theorems. Does the chain rule need a proof? If so, what can you say about

the proof; if not, why not?

C.5 Extending the notion

The student will be asked to solve the Leibniz Rule problem from the RUMEC

study. A simplified version of the problem will be offered if needed to find a solution.

Finally, a set of word problems from a first semester calculus course will be presented.

The set will include a related rate problem and an implicitly defined function problem.

The task will be to identify any situations that require the use of the chain rule in

the solution, discussing in detail how the rule might be used. The student will not

be expected to solve the problems; however if a solution is required to complete the

task, one will be provided to the student.

15. Read each of the following differentiation situations carefully. Without actually

solving the problem, determine if the solution will involve the use of the chain

rule in some way. Describe how the chain rule is used.

(a) Given that the following relation defines y as a function of x,

x√

y + y√

x = 42

find its derivative.

(b) A ladder L feet long is leaning against a wall, but sliding away from the

wall at the rate of 4 ft/sec. Find a formula for the rate at which the top

of the ladder is moving down the wall.

(c) The volume V and the surface area of a sphere are related by the equation

36πV 2 = S3. Find dS/dV , the rate of change of the surface area with

respect to the volume when the volume is π√

6 ft3.

(d) The following formula give a reasonable estimate of the Heat Capacity C

of a certain element at very low Fahrenheit temperatures F :

C = 0.171F 3 + 236F 2 + 10, 800F + 16, 600, 000

Suppose that the temperature F is found at time t according to the equa-

tion F = −42t2. Find an expression for rate of change of the Heat Capacity

with respect to time.

87

16. Compute F ′ if F is the function given by F (x) =∫ sin x

0et2 dt. Freely offer the

fact that there is no elementary anti-derivative of the integrand. One simpler

situation is G(x) =∫ x

0et2 dt. Another is H(x) =

∫ sin x

04t2 dt. Have all three

on separate pages.

17. Finally, do you have any questions?

VITA

88

VITA

Jim Cottrill was born in Akron, Ohio on 1 February 1963, the son of Wayne and

Paulette. He was married to Julie Ann Vetter, daughter of Ron and Judy, on 15 June

1991 in Chillicothe, Ohio. They have a daughter Cecelia, born in 1996, and a son

Stephen, born in 1998.

Jim attended the University of Akron where he earned a Bachelor of Arts degree

in Secondary Education, in the fields of Mathematics and Chemistry, in 1986. He

taught mathematics, computer science and chemistry for the Akron Public Schools.

In 1992, he was accepted into the doctoral program at Purdue University. After

teaching at Georgia State University, Jim received his Doctor of Philosophy degree

in Mathematics Education from Purdue University in 1999.

He has co-authored three articles in the Journal of Mathematical Behavior : “Un-

derstanding the Limit Concept: Beginning with a Coordinated Process Schema”

(1996); “The Development of Students’ Graphical Understanding of the Derivative”

(1997); and “Constructing a Schema: The Case of the Chain Rule” (1997). His re-

search interests are in undergraduate mathematics education and in the training of

pre-service teachers.

Jim has accepted a position as Assistant Professor in the Department of Mathe-

matics at Illinois State University in Normal, Illinois beginning August 1999.