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Stuck on convention: a story of derivative relationships

Ami Mamolo &Rina Zazkis

Published online: 14 March 2012

#Springer Science+Business Media B.V. 2012

Abstract In this article, we explore the responses of a group of undergraduate mathematics

students to tasks that deal with areas, perimeters, volumes, and derivatives. The tasks

challenge the conventional representations of formulas that students are used to from their

schooling. Our analysis attends to the specific mathematical ideas and ways of reasoning

raised by students, which supported or hindered their appreciation of an unconventional

representation. We identify themes that emerged in these responses and analyze those via

different theoretical lenses—the lens of transfer and the lens of aesthetics. We conclude with

pedagogical recommendations to help learners appreciate the structure of mathematics and

challenge the resilience of certain conventions.

Keywords Convention .Transfer .Aesthetics .Derivative relationship

1 Introduction

In this article, we explore university mathematics students’responses to tasks that deal with

derivatives as related to perimeters, areas, and volumes of different shapes. Students’ideas

of derivative have been explored from a variety of perspectives. For example, Bingolbali and

Monaghan (2008) observed that an instructor’s priorities regarding rate of change or tangent

line aspects of derivatives shaped students’developing concept images. They noted that

mathematics majors tended to have tangent line aspects of derivatives prioritized in the

classroom and that this impacted their performance on problems that related to rate of

change aspects of derivatives. In another study on students’concept images of calculus

Educ Stud Math (2012) 81:161–177

DOI 10.1007/s10649-012-9391-0

A. Mamolo (*)

York University, 4700 Keele St., Toronto, ON M3J 1P3, Canada

e-mail: AMamolo@edu.yorku.ca

R. Zazkis

Simon Fraser University, 8888 University Drive, Burnaby, BC V5A 1S6, Canada

e-mail: zazkis@sfu.ca

concepts, Przenioslo (2004) emphasized students’most frequently used conceptions or key

elements. She found that students’concept images tended to be conceptually disconnected

and that the key elements were ones that were most useful in solving problems, such as the

algorithms for calculating limits. Lithner (2003) studied students’responses to textbook

exercises and observed “very few considerations of intrinsic properties and relational

understanding”(p. 51) during students’engagement with conventional calculus problems.

Lithner further noted that “students do not seem troubled by their lack of understanding of

the main ideas”(2003, p. 52).

In complementing this research, our study focuses on an unconventional use of variables

for familiar formulas in derivative-related tasks and investigates how students react to such a

change and whether they proceed to implement a similar one. The importance of considering

unconventional representations was emphasized by Zazkis (2008), who illustrated that

challenging conventions can aid in developing richer mental schemas. In this article, we

explain the (unconventional) mathematics embedded in the tasks. We then continue with the

story of how the tasks were presented to students and what were their responses. Subse-

quently, we identify themes that emerged in these responses and analyze those via different

theoretical lenses. Our analysis considers an aesthetic component in exploring unconven-

tional representations, and we connect this to flexibility in transferring knowledge and to

appreciating structural aspects of mathematical relationships. However, we begin our story

by sharing with the reader how the tasks emerged from our prior research.

1.1 Mrs. Violet’s connection (how this research emerged)

Our interest was spurred by one of Mrs. Violet’s teaching situations, which is discussed

in detail in Zazkis and Mamolo (2011). Briefly, Mrs. Violet is a high school teacher

who was confronted with a thoughtful observation and question from one of her

students. The student, having recognized that the derivative of the area of a circle

equals its circumference, wondered why such a relationship did not hold in other cases,

such as for a square.

Mrs. Violet was aware of the geometrical explanation for this relationship and was able to

show how a similar relationship between area and perimeter can exist in the case of a square.

In the next section, we provide details of the mathematics, elaborating upon Mrs. Violet's

explanations. Her response, being influenced by her awareness of the inherent underlying

structure of mathematics, focused on an analogy between a circle’s radius and the length of

half a side of a square. Her reasoning by analogy, supported by geometrical and numerical

arguments, laid the foundation for a powerful teaching and learning moment.

Our interest in Mrs. Violet was, and continues to be, twofold. It was (1) her ability to

make the connection to the case of the square and (2) her ability to translate her understand-

ing to an accessible-for-her-student explanation. While we discuss (2) in a previous paper

(Zazkis & Mamolo, 2011), Mrs. Violet’s connection continues to attract our curiosity. We

wondered what specific experiences helped Mrs. Violet extend a familiar relationship in an

unconventional context. We also wondered if other individuals were likely to make a similar

connection.

We shared our excitement about Mrs. Violet with a colleague—a mathematician, Dr.

Burgundy. To our disappointment, Dr. Burgundy was much more nonchalant about Mrs.

Violet’s connection than expected. While taking a derivative with respect to a square’s

“radius”(as demonstrated below) might be unconventional, Dr. Burgundy was of the mind

that students with a substantial knowledge of mathematics, calculus in particular, would

readily make such a connection. With a healthy dose of skepticism, we thought: let us see.

162 A. Mamolo, R. Zazkis

1.2 Mathematical connection

The relationship between area and circumference of a circle is noteworthy as the derivative

of the area of a circle equals its circumference. This is expressed symbolically as

dA

dr ¼ðpr2Þ0¼2pr¼C, where Ais the area, Cis the circumference, and ris the radius.

Zazkis, Sinitsky and Leikin (in press) referred to this as the “derivative relationship.”As

derivative is an operation on functions expressing relationships between independent and

dependent variables, the derived numerical relationship is obvious. However, the reason for

it requires an explanation. The gist of this explanation makes use of the limit definition and

its accompanying geometrical representation (Fig. 1).

Here, prþhðÞ

2pr2describes the difference in areas between the circle with radius r+

hand the circle with radius r, that is, the area of the “ring”of width haround the circle with

radius r. The change in this difference approaches the circumference of the inner circle as

happroaches zero. A similar explanation can be extended to the derivative relationship

between the volume of a sphere with radius rand its surface area:

lim

h!0

4

3prþhðÞ

34

3pr3

h¼lim

h!04pr2þ4prh þ4

3ph2

¼4pr2

:

It is also easy to see that V¼4

3pr3and V0¼34

3

pr2¼4pr2¼SA, where Vstands for

volume and SA for surface area of a sphere. Further, while the derivative relationship

between area and circumference of a circle can be seen as “common knowledge,”it is

tempting to conclude that a similar derivative relationship does not hold for a square.

Symbolically, for a square with side s,areaA,andperimeterP: A0s

2

,P04s,and

dA

ds ¼2s6¼ P. However, this lack of analogy is inconsistent with a mathematical sense of

structure. As such, an analogy is sought and achieved by considering the area and perimeter

of a square with respect to half of its side. That is, if a side of a square is 2w,then

P¼42wðÞ¼8w;A¼2wðÞ

2¼4w2, and dA

dw ¼8w. So the desired derivative relationship

holds. The symbolical manipulation also has a geometrical reason behind it as illustrated in

Fig. 2(this is similar to the explanation that Mrs. Violet offered her student).

Of note is that the choice of w, or half-side, as a variable for the area and perimeter

formulas for a square is analogous to the choice of radius in a circle, as wis the radius of the

inscribed circle. Zazkis, Sinitsky and Leikin (in press) showed that with similar analogous

choices the derivative relationship holds for all regular polygons and convex regular

polyhedra. To emphasize the importance of variable choice, note that if we use for a circle

its diameter Din considering the formulas for area pD

2

2

and circumference (πD), we

would not obtain the derivative relationship, as dA

dD ¼pD

2

2

0¼1

4

pD2

0¼1

2

pD6¼ P¼pD.

Fig. 1 Derivative relationship in a circle

Stuck on convention: a story of derivative relationships 163

Based on our interactions with Mrs. Violet and Dr. Burgundy presented above, and the

mathematical analysis in this section, we designed tasks that are presented in detail in the

next section. We were interested in examining the following:

1. How do participants explain the derivative relationship for a circle? (Task 1A)

2. Do they recognize the derivative relationship for a square on their own? (Task 1B)

3. How do they react when introduced to the derivative relationship for a square?

(Task 2A)

4. Are they able to extend the derivative relationship to a cube? (Task 2B)

We continue by describing the tasks, their implementation, and participants’responses.

2 The story of the tasks and their implementation

To satisfy our curiosity (and test out Dr. Burgundy’s expectations), we sought volunteers to

answer two written questionnaires, comprised of our tasks, at an interval of 2 weeks apart.

The first questionnaire took approximately 20 min and was intended to establish (a) how

participants explain the derivative relationship for a circle and (b) whether they recognize the

derivative relationship for a square on their own. Participants were informed that the topic of

the questionnaires would be derivatives, and that the follow-up would be based on the

general trend of responses. The specifics of our focus on the derivative relationship were not

disclosed. The participants were 32 upper-year undergraduate university students all of

whom were studying towards a major or minor in mathematics, with at least two courses

completed in calculus.

Times were arranged during which participants convened to respond to the written

questionnaires. They were encouraged to answer thoughtfully and honestly, take their time,

ask clarifying questions, and to explain their thinking as much as possible. Participants were

informed that we were not concerned with the accuracy or correctness of their responses, but

rather with their ways of reasoning and evoked ideas. All questionnaires were addressed

individually and without the use of external aids (e.g., textbooks, Google, etc.).

2.1 On the familiar and the unfamiliar

The first questionnaire is presented in Fig. 3. What we found, in short, could surprise Dr.

Burgundy. While participants were generally comfortable with the derivative relationship for

the circle, none of them was able to make the connection that Mrs. Violet had. Of the 32

participants, 28 were able to give support for why the derivative of the area of the circle

equals its circumference either with limit or derivative computations (22 of 28) and/or with

Fig. 2 Derivative relationship in a square

164 A. Mamolo, R. Zazkis

verbal explanations that made use of the provided diagram (12 of 28). Some common

themes to participants’responses included (1) derivative as rate of change, (2) loss or gain

in (infinitesimal) area, and (3) uniform change in area. These themes are exemplified in the

excerpts below:

Julie-1A: This makes sense, because if you add an infinitesimal amount of area to a

circle, it would be ‘like’adding a circumference around the existing circle. This

actually works because you are extending the radius (all around the circle) by an

infinitesimal amount, thus drawing a ‘new’circumference around the circle.

James-1A: Looking at the diagram with rbeing the radius of the circle, area0πr

2

, now

adding a very thin length to rto get r+h you increase the area of the circle to

prþhðÞ

2

. If the area added is very little,it’s like adding the circumference of the

circle. This makes the rate of change the circumference of the circle. The derivative

(being the rate of change of a function) of the circle’s area is therefore the circumference

of the circle. Circumference acts like an even ‘layer’being added to the area of the circle.

Of the 22 participants who included computations in their responses, five did not provide

any accompanying explanation and 13 provided irrelevant or erroneous explanations. For

instance, Lauren, who found the limit and computed the derivative, wrote:

Lauren-1A: Working backwards, the integral of a slope may be the area of what is under

or above the x-axis…so the derivative of the area becomes the function of the slope.

Responses to task 1B regarding a possible derivative relationship for the square were, in a

word, conventional. Thirty-one of 32 participants claimed the derivative relationship for the

square did not exist or was not possible; the large majority (26) based their conclusions on

the conventional formulation for the derivative of the area of a square with the length of a

side as the variable. Of the “not possible”responses, 14 were based solely on computations.

Of the participants who included verbal explanations for why the derivative relationship

does not hold for a square, 15 included diagrams, nine of which were analogous to the circle

construction. Despite the analogous diagram, participants argued that the lack of a “uniform

ring”around the square negated the derivative relationship. Responses alluded to problems

with the diagonal and with the corners. For example:

James-1B: No, because unlike the circle when you extend the side length of the square

you’re not gaining equal area around the perimeter of the square. The ‘area’gained on

During a calculus class, one student noticed that when working with the

circle, the derivative of the area formula yields the formula for

circumference. That is, . The student asked why

this relationship held for the circle, and not in other cases such as with

the square.

1A. Use the diagram to show why the derivative of the area of a circle

yields the formula for the circumference.

1B. Is it possible to represent the derivative of the area of a square as the

formula for its perimeter? If so, explain how. If not, explain why not.

Fig. 3 Questionnaire 1, tasks 1A and 1B

Stuck on convention: a story of derivative relationships 165

the sides is clearly h

2being added, but at the corners h

ﬃﬃ2

pis being added. Therefore the

perimeter is not the rate of change of the square’s area since area is not evenly

dispersed when extending the perimeter of the square.

Other participants searched for meaning in their computations, connecting the numbers

found in their calculations to features of the square that are “accounted for”:

Julie-1B: Area of square0l

2

; Perimeter of square04l;d

dl l26¼ perimeter. For a square,

this is not possible. As I’ve illustrated on the left, extending linfinitesimally would be

‘like’attaching two sides to the square. This is why d

dl l2¼2l, and not 4l.

Christina-1B: It is not possible to represent the derivative of the area of a square as the

formula for its perimeter because knowing that, A0s

2

,A′02s. The derivative would

only allow us to add up 2 sides of the square, whereas we would need to add up all

four sides in order to find the perimeter.

As mentioned, 31 of 32 participants claimed the derivative relationship was not possible,

leaving one participant claiming that the alluded relationship was possible. However, this

participant based his response on a flawed calculation, mistaking the perimeter of the square

to be 2s, where swas the length of a side.

The only possible pointer in the desirable direction came from Lauren:

Lauren-1B: The area of a square with side sis s

2

. The derivative of the area is 2s. The

formula for perimeter is 4s. Since 2s≠4s, the derivative of the area of a square cannot

represent the formula for its perimeter. Unless there is another formula for the area of a

square…

Lauren’s response is worth noting as she seemed to suggest that there could be “another

formula”though she did not attempt to seek such a formula or create an analogy with the

circle construction in task 1A. Her description of derivative in terms of slope (see Lauren-

1A) may have hindered the possibility of extending the analogy.

2.2 Planting the seed

In light of the overwhelming trend in responses to task 1B, we designed a follow up to “plant the

seed”of analogy and see what might grow. As such, we presented participants with the second

questionnaire, part of which is shown in Fig. 4. As before, participants addressed the tasks

individually, in writing, and without the aid of external resources. The questionnaire, which

included tasks 2A (below) and 2B (see Section 2.3), took approximately 30 min to address.

In addition to planting the seed of analogy, we sought to explore what specific features of

the “alternative approach”participants would attend to (e.g., the variable), what they would

find convincing or insufficient (and why), and whether any of the arguments given in

response to task 1B would persist.

Of the 32 participants, 17 indicated that the alternative approach was valid, 10 indicated it

was flawed, and the rest were unsure. As before, conclusions were based predominantly on

computations (be they accurate or flawed). While we found the specific features of partic-

ipants’computations interesting, our attention was drawn much more powerfully by the

conceptual justifications provided.

There were nine participants (of 17) who reasoned about why the derivative relationship

was valid for a square, while the remaining drew conclusions solely from computation.

These nine participants attended to the rate of change of the area between the two squares,

166 A. Mamolo, R. Zazkis

with some noting that the change would be equal for all sides since awas being treated as a

“radius”:

Julie-2A: VALID. This alternative approach is valid because measuring aas a ‘radius’

defines the smaller square from the inside to the outside. This makes a significant

difference. The result of this is that when ‘radius’ais increased, all sides of the square

increase equally. This is in contrast to the last approach, where (if I remember

correctly) extending the length of a square’s side simply ‘extended’one of its corners.

Most interestingly is that the ‘radius’ais half the length of a side. The result of this is

that the growth of the square is evenly distributed to all sides of the square, rather than

just one direction of length and one direction of width. This is why it works.

Participants who argued that the alternative approach was flawed tended to acknowledge that

the computation would work for the “inner square,”or sometimes for both squares, but took

issue with either: (1) the corners or (2) generalizability. In contrast to participants such as

Julie-2A, who reasoned that the change in area was uniform due to the radius analogy,

James-2A found the corners to be problematic:

James-2A: FLAWED. The approach may seem valid because it leads to the right

answer of the perimeter. [But] the rate of change is greater than the perimeter. Rate of

change of area is not the perimeter because of corners. Distance out is more at the

corners than the sides.

Participants who took issue with the generalizability of the argument tended to focus on

how the square was parameterized, arguing that if ahad been chosen differently, then the

argument would not work. (Of course this is also true for the circle. Recall our interpretation

of formulas for a circle in terms of its diameter presented in a previous section.)

Christina-2A: FLAWED. This approach definitely works on the following 2 squares

above, where we have the area and perimeter as the derivative of area. However, it

would not work in all such cases. For example if we let a full side length of the square

equal to ainstead of ½ a side length, we wouldn’t get the derivative of area to give you

the formula for perimeter. A0a

2

,A′02a.P≠2a,P04a. Therefore this approach works

for this case but is flawed if we label the square differently.

Other participants believed the argument held for the specific case provided, but that it

lacked sufficient explanation, or would not generalize to other shapes:

Consider the following argument and diagram:

Imagine a as an analogy to the radius of a circle. In this way, we can

describe the perimeter and area of the inner square as a function of a:

, and .

Similarly, the area of the outer square can be described by

.

Then we can express the derivative of the area of the square as its

perimeter in the following way:

Reflect on this “alternative approach.” This approach is (circle one):

VALID FLAWED NOT SURE

If you circled VALID, please explain why this approach makes sense.

If you circled FLAWED, please identify the flaw.

If you circled NOT SURE, please explain why you are unsure.

Fig. 4 Questionnaire 2, task 2A

Stuck on convention: a story of derivative relationships 167

Peter-2A: FLAWED. [The argument] did not explain ‘why.’For example, if we say a

side of a square is t.P(t)04t,A(t)0t

2

.A′(t)02t≠P(t). There needs to be more

explanations on a value h, why we need it and what hrepresents.

Margo-2A: FLAWED. I identify this as flawed as it may work for a specific case, the area

of the circle into the perimeter of a circle. But it will not work for the area of a triangle.

Haley-2A: NOT SURE. This example looks valid where the derivative of the area of

the square does in fact equal the perimeter of the square. I’m unsure because I don’t

feel confident enough with these examples to generalize it to every case. If this method

is in fact valid, I would feel more comfortable with a general statement or theorem, and

I would like to understand why it does hold for every case. Proving by example is

always difficult for me to accept.

In a similar vein to the responses which indicated that the argument was flawed,

participants who circled “unsure”all acknowledged accuracy in the provided computation,

but questioned its general applicability. This uncertainty or lack of confidence with the

alternative approach was also exemplified by the nine participants who changed their mind,

circling an answer and then crossing it out in preference of another position.

Erik-2A: .VA L I D . I a m this is valid because I am

convinced. I am this is valid for all squares and its radius.

2.3 Stuck on convention

Task 2B appeared on the back of the page of task 2A and is presented in Table 1. The

majority, 22 out of 32, were able to answer part (a) correctly. Those who could not extend the

derivative relationship to a sphere either did not know the corresponding formula for the

volume or for the surface area. Responses were by and large computational. Since the case of

a sphere was part of the repertoire for most participants, we focus on their engagement with

part (b). In total, nine of 32 participants could extend the argument, or at least the

computation, to the case of a cube; all of whom had responded “valid”to task 2A. Most

of these participants attended to the change in volume between the two cubes:

Bailey-2B: By visualizing, let’s imagine a cube. Now create another cube with tiny

increase in sides. Then subtract big cube by small cube. Then you will be left with a

shell of cube with nothing inside. That thin shell can be considered as finding SA.

Other participants made sense of their computations with the alternative variable (2a)by

breaking down (correctly or erroneously) aspects of the cube:

Valerie-2B: V¼lwh¼s3

:V¼2aðÞ

3¼8a3

:V0¼24a2. On every cube there

are 12 edges that are of equal length, in this case s;sis equal to 2a. By taking 2a

2

[sic]

out of V′024a

2

, we get 12a

2

which described the 12 edges.

Table 1 Questionnaire 2, task 2B

Consider the derivative of the volume of

(a) a sphere

(b) a cube

How does it relate to the surface area?

168 A. Mamolo, R. Zazkis

Valerie-2B went on to describe the “individual surface area a

2

”and how “4 of those (a

2

) fit

within every face of the cube and since there are six faces, 4× 6024 which makes the

derivative of the volume of a cube in regards to avalid as well.”Arguments which attended

to the number of faces also emerged for why the derivative relationship would not hold for a

cube:

Christina-2B: The volume of a cube is s

3

, where srepresents the [length of the] side.

VðcubeÞ¼s3,V0ðcubeÞ¼3s2. The surface area of a cube is SAðcubeÞ¼6s2. The

derivative of the cube corresponds to only 3 faces of the cube. Therefore the derivative

of the volume of a cube gives you the formula for the surface area of only 3 faces of

the cube (therefore half the surface area of a full cube). Therefore the derivative of the

volume of a cube is half the surface area of a full cube.

Responses that denied the derivative relationship for the cube were again primarily based

on computations—15 of the 23 participants responded solely with computations, all of

which used the standard formulas exemplified by Christina-2B. Of the responses that

included justifications, the most common objection was with a disproportionate “gain”or

“loss”in surface area due to the corners:

James-2B: The cube’s V doesn’t relate as nicely to SA as the sphere, since the corners

again are a source of error. The rate of change in volume of a cube is bigger than its

surface area due to the corners gaining more volume than the sides when adding

layers. It’s not an ‘even’distribution of the volume over the SA unlike the even layers

of the sphere.

An interesting response was Julie-2B’s, who reasoned that the derivative relationship does not

hold for similar reasons as James-2B, yet she acknowledged that with a different model, it could:

Julie-2B: The derivative of the volume of a cube, on the other hand, is not equal to the

SA of the cube. This is for the same reasons as last time’s approach. As we’ve just

explored with the square, adding to the length of a cube does not make it ‘grow

evenly’like it would with a circle or a sphere. The derivative of the volume of a cube

is 3l

2

, where l0length. We know that l

2

is the area of a face and that a cube has 6 faces,

thus the SA of a cube 06l

2

, which 3l

2

is half of. I think that we end up with the

derivative being half (just like last time) because when V0l

3

and we add to lto make

the cube bigger, only one of two directions in each degree of freedom ‘takes the

expansion’. This is why it doesn’t work, while a ‘radius-based’model would.

Despite acknowledging that “a‘radius-based’model would”achieve the desired derivative

relationship, Julie-2B did not proceed to consider such a model. In what follows, we identify

several major themes in participants’responses across the four tasks.

3 The “naked eye’s”interpretation

The story, as presented above, follows the sequence of events as they unfolded. Before

analyzing the results, we summarize the frequencies of occurrences of participants’

responses (see Table 2) and identify several themes apparent to a “naked eye.”

Through our naked eye, there were three trends which most captured our attention:

1. Participants’focus and reliance on computation. This was exhibited by both the

number of responses in each of the tasks for which a conclusion was drawn primarily

Stuck on convention: a story of derivative relationships 169

from computation, as well as by an implicit level of confidence in a solution by

computation.

2. Participants’resistance to or lack of confidence with the alternative approach (task

2A). A lack of confidence in the alternative approach was exemplified by the number of

“unsure”responses, as well as the number of participants who circled more than one

answer. Acknowledging the first trend, we saw several participants concede that the

approach was valid for the “inner square”but then draw a conclusion of “flawed”or

“unsure”from an inaccurate computation with the area of the “outer square.”Other

participants appealed to the lack of a ‘uniform ring’around the square. We also noted a

trend of “won’t work for all cases”for various reasons.

3. Participants’inability or unwillingness to reason by analogy or extend the argument.

The most striking illustration of this is the inability of most participants to extend the

derivative relationship for the square, when they found it to be valid in task 2A, to the

Table 2 Summary of participants’responses to the tasks

Task 1A:

Could explain derivative

relationship

28/32 Used computation only 5/28

Used computation and justification 17/28

Justification, no computation 6/28

Could not explain derivative

relationship

4/32

Task 1B:

Possible 1/32 (based on flawed

computation)

Not possible 31/32 Used computation only 14/31

Used computation and justification 12/31

Justification, no computation 5/31

Task 2A:

VALID 17/32 Used computation only 8/17

Used computation and justification 6/17

Justification, no computation 3/17

FLAWED 10/32 Used computation only 6/10

Used computation and justification 3/10

Justification, no computation 1/10

NOT SURE 5/32 Used computation only (flawed) 2/5

Used computation and justification 3/5

Changed answers 9/32 From VALID to FLAWED 4/9

From FLAWED to UNSURE 3/9

From UNSURE to VALID 1/9

From FLAWED to VALID 1/9

Task 2B:

Derivative-relationship

exists for a cube

9/32 Used computation only 5/9

Used computation and justification 2/9

Justification, no computation 2/9

Derivative-relationship does

not exist for a cube

23/32 Used computation only 15/23

Used computation and justification 8/23

170 A. Mamolo, R. Zazkis

case of the cube in task 2B. We also note that while approximately half the participants

identified the derivative relationship as valid in task 2A, very few drew conclusions

based on analogical reasoning as opposed to computational results.

These naked-eye observations give a sense of what happened, but they do not answer the

question of why this happened. For this, we seek explanatory lenses. We found the notions

of transfer and an extended perspective on aesthetics useful in order to analyze participants’

responses in an attempt to interpret the why behind the what. We turn to these notions in the

following sections.

4 Transfer

In a traditional or “classical”perspective on transfer, the focus is on the tasks presented to the

participants and the similar structural features of the “learning task”and “transfer task”

(National Research Council, 2000). However, the traditional view of transfer has been

repeatedly criticized when applied to research in education for dissociating cognition from

its contexts and socially situated activity (Gruber, Law, Mandl & Renkl, 1996; Lave, 1988).

Context is often treated as the task presented to students, and the structure of such tasks is

often analyzed independently of how students construe meaning in the situation (Cobb &

Bowers, 1999; Tuomi-Gröhn & Engeström, 2003). Furthermore, researchers using a lens of

traditional transfer have been more successful in showing how people fail to apply knowl-

edge learned in one situation to another than they have been in fostering successful transfer

(McKeough, Lupart, & Marini, 1995). Recent work in mathematics education suggests

different approaches to transfer, one of which is referred to as actor-oriented transfer

(Lobato, 2006,2008). It is to this approach that we turn our attention.

4.1 Actor-oriented transfer

The focus in actor-oriented transfer (AOT) is on what the learner sees as similar between two

tasks, rather than what the researcher/expert identifies as similar. From this perspective,

transfer is defined as the generalization of learning or more broadly as the influence of prior

experiences on learners’activity in novel situations. In other words, by adopting an actor’s

perspective, we seek to understand the ways in which people generalize their learning

experiences rather than predetermining what counts as transfer using models of expert

performance (Lobato, 2006). Thus, the following discussion of what is transferred we

focus on the prior experiences that influenced participants’interpretations of the tasks.

A well-known example from Schoenfeld’s research (1985,2011) describes a situation in

which students did not use their relevant and recently reviewed knowledge (a proof for a

theorem) in a new geometry task. He analyzed this behavior as “context bound”(2011,

p. 30), that is, the context shapes the way the task is interpreted, the associated goals for

solving, and the knowledge evoked. Schoenfeld suggested that “the students developed

certain understandings of the rules of the game”(p. 30; italics in the original) and that these

rules are invoked according to students’interpretation of the task. This explanation is

applicable to participants’responses to our tasks and, although the language is different,

we see this as strongly connected to AOT. In particular, individuals’prior experiences

establish expectations of what are the “rules of the game”and accordingly influence their

interpretations of novel situations as they attempt to generalize their learning experiences.

This perspective ties into all three of the trends observed by our naked eye.

Stuck on convention: a story of derivative relationships 171

Participants’focus and reliance on computation With respect to the first trend, we note that

the predominant experiences in mathematics classrooms, and in particular calculus class-

rooms, involve computational solutions to problems. These experiences foster (a) an expec-

tation for computation and (b) confidence in the reliability of computation. Indeed, as

Margo-1A concisely put it: “I think that with the diagram you gave us you would like to

see the limit form of the derivative, with respect to r. ”Recall that the specific prompt given

in task 1A was to “use the diagram to show why,”and made no reference whatsoever to

computation via limit or otherwise.

The influence of prior experience regarding the prominence of computations contributes,

in our perspective, to confidence in the reliability of computation. Such confidence can be

interpreted in participants’attempts to seek meaning behind the computations by introducing

plausible justifications for certain results. For example, Christina-1B wrote that “The

derivative would only allow us to add up 2 sides of the square, whereas we would need to

add up all four sides in order to find the perimeter.”In an attempt to justify her conclusion,

Christina-1B explained her calculation by focusing on a plausible reason for a coefficient of

2 in the derivative of the area. Similar justifications were observed in responses to task 2B,

where, again, Christina-2B attempted to explain the coefficient of 3 in the derivative of the

volume of a cube by arguing that the derivative “corresponds to only 3 faces of the cube.”

Such justifications were observed not only in participants’(incorrect) refusal of the deriv-

ative relationship for a square or cube, but also in support of it. Valerie-2B sought to explain

the coefficient of 24 in her volume derivative by factoring in two different ways: first to

correspond to the 12 edges of a cube, then to correspond to the six faces of the cube. While

her second observation—considering the surface area as the sum of six areas of the cube’s

faces—is an appropriate interpretation of 24a

2

, her first observation—connecting 12a

2

to the

12 edges of the cube—is irrelevant to the task and appears to be a forced attempt to seek

meaning in numbers.

Participants’resistance to or lack of confidence with the alternative approach With respect

to participants’responses to task 2A, we interpret via AOT an implicit desire to transfer

familiar knowledge regarding the derivative of the area of a square. This resistance and lack

of confidence is well exemplified by participants who conceded that the alternative approach

was valid, but then drew conclusions of “flawed”or “unsure.”For instance, Margo-2A wrote

“I identify this as flawed as it may work for a specific case.... But it will not work for the area

of a triangle.”Margo-2A’s rejection of the derivative relationship was not based on the

argument itself, but rather on her skepticism regarding its generalizability. Incidentally, the

argument does hold for an equilateral triangle, as well as every regular polygon (Zazkis,

Sinitsky & Leikin, in press). Like Margo-2A, other participants criticized the argument as

not being generalizable, claiming it “worked, but”and noting that the relationship would not

hold if “we label the square differently”(e.g., Christina-2A). They seemed to respond to

whether or not the relationship A′0Pwas true, as opposed to whether or not the argument

was valid. We suggest that those participants’prior experiences with and expectations for

derivative questions were transferred and subsequently influenced their interpretation of our

task, since if the variables were changed (e.g., if the square were labeled differently), then

the argument itself would also be different (though no participant acknowledged this).

Skepticism regarding generalizability also emerged as some participants considered the

specifics of what was presented. For instance, Haley-2A remarked that while the “example

looks valid,”she was “unsure”citing she “would feel more comfortable with a general

statement or theorem”and “would like to understand why it does hold for every case.”

Haley-2A perceived the argument as “proving by example”and found it “difficult…to

172 A. Mamolo, R. Zazkis

accept.”She seemed to transfer her understanding of what constituted a valid mathematical

justification—general statements, theorems, proof—which are commonly emphasized in

university mathematics classrooms. However, she failed to realize that the argument pre-

sented in task 2A was not “proving by example,”but rather was a “general statement.”We

interpret this omission as resistance toward the derivative relationship as the structure and

styling of the argument paralleled what is familiar to calculus textbooks and lessons. In a

similar vein, Peter-2A claimed the argument in task 2A was flawed as “There needs to be

more explanations on a value h, why we need it and what hrepresents.”Peter-2A supple-

mented his argument with the conventional formulation of perimeter and area of a square

without realizing the analogy. Interestingly, what “hrepresents”was not an issue for Peter in

task 1A, where he used it conventionally in a limit calculation to explain the derivative

relationship for the circle. Indeed, in both arguments “what hrepresents”is the same. Peter-

2A’s dismissal of the derivative relationship speaks to his desire to transfer familiar formulas

and computations of area and perimeter of squares.

Participants’inability or unwillingness to reason by analogy or extend the argument In this

section, we focus on what participants transferred when moving from the square argument in

task 2A to conclusions regarding the cube in task 2B. As previously mentioned, the majority

of participants (23 of 32) reasoned that the derivative of the volume of the cube was not

equal to its surface area. All of these participants fell back to the familiar V0s

3

and SA06s

2

formulas for their justifications, transferring traditional interpretations of volume and surface

area. Notably, of these 23 participants, 17 drew conclusions from computation only. Further,

recalling task 2A, of the 17 participants who accepted as valid the argument for the square,

nine were not able to extend the argument or computation to the case of the cube. This is in

accordance with Schoenfeld’s(2011) observation that the “tacit but strong lesson[s] they had

learned from their classroom experience with such problems”(p. 30) can overshadow

recently acquired knowledge.

Turning our attention to the eight participants who included explanations with their

computations for why the derivative relationship did not hold for the cube, we observed a

transfer of ideas raised in task 1B. For example, James-1B who reasoned that the derivative

relationship did not hold for the square because “you’re not gaining equal area around the

perimeter of the square”reasoned similarly in task 2B, noting that “due to the corners…it’s

not an ‘even’distribution of the volume over the SA”(James-2B). The problem with corners

was also a deciding factor for James-2A, who stated that “the approach may seem valid”but

refused to accept it “because of corners.”

The most striking example for us was Julie-2B, who transferred her objection to the

derivative relationship to the case of the cube despite recognizing the analogy with the

“radius-based model.”Julie-1B wrote that the derivative relationship was not possible

because “extending l[the length of a side] infinitesimally would be ‘like’attaching two

sides to the square.”She reasoned later that “measuring aas a ‘radius’” allows “all sides of

the square [to] increase equally”and that the “result of this is that the growth of the square is

evenly distributed to all sides”and “this is why it works”(Julie-2A). Recall that in Julie-1A,

her explanation for why the derivative relationship for a circle “makes sense”and “actually

works”hinged on the idea of a uniform increase in area.

Julie-2A further observed that her response in task 2A of “valid”was “in contrast to the

last approach”in task 1B. In her response to task 2B, Julie acknowledged both approaches

though she seemed to prioritize her original reasoning, that is, the familiar formulation

experienced in prior work with areas and volumes of squares and cubes. Recalling the

excerpt quoted above, Julie-2B wrote:

Stuck on convention: a story of derivative relationships 173

The derivative of the volume of a cube, on the other hand, is not equal to the SA of the

cube. This is for the same reasons as last time’s approach. …I think that we end up

with the derivative being half (just like last time) because …only one of two

directions in each degree of freedom ‘takes the expansion’. This is why it doesn’t

work, while a ‘radius-based’model would. (Julie-2B)

It is interesting that while Julie-2B recognized that “a‘radius-based’model would”work,

she seemed to assume that the question posed (“How does [the volume of a cube] relate to

the SA?”) was asking for a particular representation of the formula for volume or surface

area of a cube. In addition to transferring her prior knowledge regarding conventional

approaches (in using conventional formulas) and her new knowledge regarding an alterna-

tive approach (in claiming that a radius based approach would work), Julie seemed to

transfer her expectations regarding what knowledge is prioritized (e.g., conventional over

alternative).

A more general analysis of participants’reasoning about tasks 2A and 2B can also be

viewed in terms of noticing and appreciating the consistency of structure and the complete-

ness of an argument within mathematics. Such features are part and parcel to an aesthetic

sensibility of mathematics, which we explore in the next section.

5 Aesthetics lens

The adjective “aesthetic”is often associated with “beautiful,”however Sinclair (2006)

defines it more broadly as “associated with fundamental form of pleasure”(p. 3). In

mathematics, “aesthetic”is connected with elegance and sense of “fit,”completeness of an

argument, cohesion of ideas, and consistency of structure. Further, Sinclair (2006) mentions

connectedness, apparent simplicity, and surprise among the aesthetic qualities of problems

that attract mathematicians. While the aesthetic tends to be neglected in mathematics class-

rooms and textbooks (Sinclair, 2006), it has been acknowledged by mathematicians as

having great value to the discipline. Accordingly, it has been described as “the driving force

which makes mathematical thinking function”(Papert, 1980, p. 192).

Sinclair (2006) identified three roles that the aesthetic plays in mathematical inquiry:

motivational, generative, and evaluative. The motivational relates to the role aesthetic plays

in attracting and stimulating individuals to work on certain mathematical problems. Sinclair

reflects on a problem which seemed to invite her to explore further, and describes being

motivated by the aesthetic appeal of making a connection to establish the significance

behind a problem, and a feeling of surprise in finding “hidden treasures”in a familiar

situation. While the motivational role is connected by Sinclair to an individual’s attraction to

a problem, the generative role pertains to aesthetic qualities of reasoning and solving

problems. In this sense, aesthetic sensibilities are credited with “generating new ideas and

insights that could not be derived by logical steps alone”(p. 43). Aesthetic sensibility guides

certain choices and decisions while solving or exploring problems, and brings to focus some

ideas or relationships over others.

Sinclair describes motivational and generative roles of the aesthetic as belonging “to

more private, evolving facets of mathematical inquiry,”while the evaluative role “operates

on mathematicians’finished, public work”(p. 43). In particular, the evaluative role relates to

the aesthetic value of mathematical products—e.g., of proofs (rather than proving) or of

solutions (rather than solving). Further, the evaluative role of mathematical aesthetic con-

cerns judgments made about which aspects of a solution are the most significant. Sinclair

174 A. Mamolo, R. Zazkis

cites the aesthetic appeal of “seeing previously disparate ideas fit together”(p. 55) and

catching a glimpse of how particular results could belong to a more general context.

In our own engagement with the derivative relationship, our aesthetic experience can be

described by appreciation for a thoughtful argument presented by Mrs. Violet. This fostered

a sense of wonder that motivated us to explore other shapes. In doing so, the structural

components of exploring derivative relationships with different shapes inspired solving

strategies which generated a search for broader connections and consistency. Unfortunately,

our participants’experiences did not resemble our own. In what follows, we suggest that

participants’aesthetic sensibility towards mathematics was not sufficiently developed to

notice the significance of the relationships and structure exemplified in exploring the

derivative relationship for a square.

5.1 The absence of participants’aesthetic sensibilities

The general trend of participants’responses to task 1B demonstrates a lack of appreciation of

structure, connections, and relationships, and suggests an absence of the motivational role of

the aesthetic. Participants readily concluded that the derivative relationship was impossible

based on their prior knowledge and made no attempts to explore the possibility of a

relationship between the case of the circle and the case of the square. A lack of appreciation

of a general structure in mathematics can also be interpreted from participants’resistance to

accepting the alternative argument presented in task 2A. This is exemplified in the insistence

that the approach works for the small square, but not in general (e.g., Christina-2A).

Participants did not seem to feel invited to verify whether or not the approach really would

fail for a triangle (e.g., Margo-2A), nor did they attend sufficiently to their computations to

notice that their conclusions were supported by erroneous calculations. Aesthetic sensibility

as a drive to seek consistency or coherence was largely absent in our participants’responses.

Similarly, we recall an interesting response to task 1B, in Lauren-1B’s observation that “the

derivative of the area of a square cannot represent the formula for its perimeter. Unless there is

another formula for the area of a square…” In her statement, Lauren-1B seems to wonder about

the possibility of “another formula”that would allow the derivative relationship to hold. We

connect this sense of wonder to the motivational role of aesthetic sensibility, though it seemed to

fall short of an invitation to explore further. While Lauren’s aesthetic sensibility may have

alerted her awareness of the possibility for an analogy, the motivational role, which would

manifest itself in an attempt to seek such a formula or to create an analogy with the circle, did

not stimulate the generative role or prompt her to explore the corresponding connections.

We also note a lack of aesthetic appreciation in terms of the evaluative role. Specifically,

in response to task 2A, none of the participants showed surprise or wonder at the result

presented. Even those who accepted the argument as valid did so without noticeable

emotional reaction. Generally, the argument was accepted based on computation rather than

on the sense of structure offered by the analogy. Participants did not seem to be attracted to

the apparent simplicity of the situation and had little sense of connectedness that would have

extended the argument used in case of a square to the case of a cube. As such, surprise that

may contribute to the evaluative role of an aesthetic experience was insufficient to activate

its motivational and generative roles. Aesthetic sensibility is cyclical in nature—the evalu-

ative role spurs the motivational role, upon which the generative role is contingent, which in

turn produces an artifact that elicits the evaluative role, and so on. A lack of aesthetic

appreciation in evaluating the derivative relationship for the square manifested itself in a lack

of motivation to extend the relationship to a more general context. While participants were

invited to explore the derivative relationship in a broader context with task 2B and its

Stuck on convention: a story of derivative relationships 175

application to the cube, very few participants tried to do so. Again, they fell back to prior

knowledge, with little consideration for structure, coherence, consistency, and generalizabil-

ity. The immediacy of routine computations not only guided their thinking, but also seemed

to obscure the bigger picture and the significance of establishing such connections. There

was no “sense of fit”for the majority of participants, nor was there any indication that these

participants sought “fit.”

6 Summary and conclusion

We started this article by sharing our experiences with Mrs. Violet and Dr. Burgundy. We

appreciated the connection made by Mrs. Violet, who explained the derivative relationship

for a circle and extended her explanation to a square. We hesitated over the reaction of Dr.

Burgundy, who suggested that similar explanations should be in the repertoire of every high

school teacher or even every person who took a university calculus course. This experience

initiated the study presented in this article.

We presented tasks based on the derivative relationship to a group of years 3 and 4

university students studying towards a major or a minor in mathematics, all of whom had at

least two courses in calculus. The results suggested that none of the participants was able to

generalize for a square the derivative relationship evident in a circle. Further, when such a

relationship was presented, only about half of the participants considered it as valid, and

very few were able to extend the argument to a cube. We offered several naked-eye

observations of participants’experiences and then analyzed these observations via two

different theoretical lenses: the lens of transfer and that of aesthetics.

The choice of looking at the data with multiple lenses was influenced by Simon (2009)

who highlighted the advantages of considering a situation through different lenses, where

“each lens affords a different view of the same situation”(p. 484). Considering these

advantages, we offered an analysis through two theoretical lenses, which provided an

explanation to the phenomena observed with our “naked eye.”On one hand, based on an

actor-oriented transfer approach (Lobato, 2006,2008) we attended to prior experiences

that shaped participants’engagement with the tasks. On the other hand, based on an

aesthetic approach towards mathematical experience (Sinclair, 2006), we suggested that it

was a lack of aesthetic appreciation of mathematical structure that prevented participants

from noticing the analogy. Further, in our perspective, these two lenses are not disjoint: we see

the lens of aesthetics as related to the lens of actor-oriented transfer—if aesthetic appreciation is

not sufficiently developed in prior schooling experiences, then it will not be a component of

what is transferred in a novel situation. We acknowledge that other theoretical lenses may

provide alternative interpretations and additional insights, by taking into account different

constructs, such as epistemological obstacles and mental imagery (Schneider, 1991).

Our contributions can be seen on several arenas. First, we offer an effective combination

of two theoretical lenses. Second, we extended research on understanding derivatives by

focusing on unconventional use of variables in familiar formulas. Further, we highlight

student difficulties in moving away from conventional representations. The importance of

considering unconventional representations, and in such challenging basic assumptions, was

posited by Zazkis (2008) as a vehicle towards constructing a “richer or more abstract

schema”(p.154) and “understand[ing] better what has been already understood”(ibid). In

considering conventions, Zazkis focused on other-than-ten bases for representing numbers

and other-than-Cartesian coordinates for graphing functions. We add here other-than-

standard use of variables in familiar formulas for perimeter, area, surface area, and volume.

176 A. Mamolo, R. Zazkis

Moreover, we recognize an aesthetic component in considering the unconventional: we

consider the flexibility in accepting the unconventional and acknowledging the analogy with

the conventional as part of an individual’s aesthetic experience of appreciating the over-

arching structure of mathematical concepts and relationships.

Our multilens analysis also implies an apparent pedagogical suggestion: Mathematics

education at any level should highlight aesthetic components of the learned material to help

students see the consistency, elegance, coherence, and structure of mathematics, and, as

such, help students see its beauty. Then, the motivational and generative roles of aesthetic

experiences could result in actor-oriented transfer of such prior experience and ultimately

contribute towards successful problem solving.

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