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# Stuck on convention: A story of derivative relationships

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## Abstract and Figures

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.
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Stuck on convention: a story of derivative relationships
Ami Mamolo &Rina Zazkis
Published online: 14 March 2012
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 lensesthe 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
derivatives as related to perimeters, areas, and volumes of different shapes. Studentsideas
of derivative have been explored from a variety of perspectives. For example, Bingolbali and
Monaghan (2008) observed that an instructors priorities regarding rate of change or tangent
line aspects of derivatives shaped studentsdeveloping 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 studentsconcept images of calculus
Educ Stud Math (2012) 81:161177
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 studentsmost frequently used conceptions or key
elements. She found that studentsconcept 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 studentsresponses to textbook
exercises and observed very few considerations of intrinsic properties and relational
understanding(p. 51) during studentsengagement 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. Violets connection (how this research emerged)
Our interest was spurred by one of Mrs. Violets 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 circles 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. Violets 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 colleaguea mathematician, Dr.
Burgundy. To our disappointment, Dr. Burgundy was much more nonchalant about Mrs.
Violets connection than expected. While taking a derivative with respect to a squares
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 ringof 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 ¼2sP. 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

pDP¼pD.
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?
4. Are they able to extend the derivative relationship to a cube? (Task 2B)
We continue by describing the tasks, their implementation, and participantsresponses.
2 The story of the tasks and their implementation
To satisfy our curiosity (and test out Dr. Burgundys 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 participantsresponses 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 likeadding 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 newcircumference 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,its 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 circles area is therefore the circumference
of the circle. Circumference acts like an even layerbeing 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-axisso 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 possibleresponses, 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
ringaround 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
youre not gaining equal area around the perimeter of the square. The areagained 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
perimeter is not the rate of change of the squares 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 l2perimeter. For a square,
this is not possible. As Ive illustrated on the left, extending linfinitesimally would be
likeattaching 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
,A02s. 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 2s4s, 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
Laurens response is worth noting as she seemed to suggest that there could be another
formulathough 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
seedof 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 approachparticipants 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-
ipantscomputations 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
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 radiusais 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 squares side simply extendedone of its corners.
Most interestingly is that the radiusais 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 wouldnt get the derivative of area to give you
the formula for perimeter. A0a
2
,A02a.P2a,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)02tP(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. Im unsure because I dont
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 unsureall 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 validto task 2A. Most
of these participants attended to the change in volume between the two cubes:
Bailey-2B: By visualizing, lets 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 V024a
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 computations15 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 gainor
lossin surface area due to the corners:
James-2B: The cubes V doesnt 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. Its not an evendistribution of the volume over the SA unlike the even layers
of the sphere.
An interesting response was Julie-2Bs, 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 times approach. As weve just
explored with the square, adding to the length of a cube does not make it grow
evenlylike 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 doesnt work, while a radius-basedmodel would.
Despite acknowledging that aradius-basedmodel wouldachieve the desired derivative
relationship, Julie-2B did not proceed to consider such a model. In what follows, we identify
several major themes in participantsresponses across the four tasks.
3 The naked eyesinterpretation
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. Participantsfocus 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. Participantsresistance 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
unsureresponses, 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 squarebut then draw a conclusion of flawedor
unsurefrom an inaccurate computation with the area of the outer square.Other
participants appealed to the lack of a uniform ringaround the square. We also noted a
trend of wont work for all casesfor various reasons.
3. Participantsinability 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 participantsresponses to the tasks
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
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
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
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 classicalperspective on transfer, the focus is on the tasks presented to the
participants and the similar structural features of the learning taskand 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 learnersactivity in novel situations. In other words, by adopting an actors
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 participantsinterpretations of the tasks.
A well-known example from Schoenfelds 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 studentsinterpretation of the task. This explanation is
applicable to participantsresponses to our tasks and, although the language is different,
we see this as strongly connected to AOT. In particular, individualsprior experiences
establish expectations of what are the rules of the gameand 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
Participantsfocus 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 participantsattempts 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 observationconsidering the surface area as the sum of six areas of the cubes
facesis an appropriate interpretation of 24a
2
, her first observationconnecting 12a
2
to the
12 edges of the cubeis irrelevant to the task and appears to be a forced attempt to seek
meaning in numbers.
Participantsresistance to or lack of confidence with the alternative approach With respect
to participantsresponses 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 flawedor 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-2As 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, butand 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 A0Pwas true, as opposed to whether or not the argument
was valid. We suggest that those participantsprior 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 unsureciting she would feel more comfortable with a general
statement or theoremand would like to understand why it does hold for every case.
Haley-2A perceived the argument as proving by exampleand found it difficultto
172 A. Mamolo, R. Zazkis
accept.She seemed to transfer her understanding of what constituted a valid mathematical
justificationgeneral statements, theorems, proofwhich 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 hrepresentswas 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 hrepresentsis the same. Peter-
2As dismissal of the derivative relationship speaks to his desire to transfer familiar formulas
and computations of area and perimeter of squares.
Participantsinability 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 Schoenfelds(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 youre not gaining equal area around the
perimeter of the squarereasoned similarly in task 2B, noting that due to the cornersits
not an evendistribution 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 validbut
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 likeattaching two
sides to the square.She reasoned later that measuring aas a radius’” allows all sides of
the square [to] increase equallyand that the result of this is that the growth of the square is
evenly distributed to all sidesand this is why it works(Julie-2A). Recall that in Julie-1A,
her explanation for why the derivative relationship for a circle makes senseand actually
workshinged on the idea of a uniform increase in area.
Julie-2A further observed that her response in task 2A of validwas in contrast to the
last approachin 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 times 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 doesnt
work, while a radius-basedmodel would. (Julie-2B)
It is interesting that while Julie-2B recognized that aradius-basedmodel wouldwork,
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 participantsreasoning 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 aestheticis often associated with beautiful,however Sinclair (2006)
defines it more broadly as associated with fundamental form of pleasure(p. 3). In
mathematics, aestheticis 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 treasuresin a familiar
situation. While the motivational role is connected by Sinclair to an individuals 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 mathematiciansfinished, public work(p. 43). In particular, the evaluative role relates to
the aesthetic value of mathematical productse.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 participantsexperiences did not resemble our own. In what follows, we suggest that
participantsaesthetic 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 participantsaesthetic sensibilities
The general trend of participantsresponses 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 participantsresistance 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 participantsresponses.
Similarly, we recall an interesting response to task 1B, in Lauren-1Bs 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 formulathat 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 Laurens 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 naturethe 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 fitfor 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
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 participantsexperiences 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 participantsengagement 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 transferif 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 individuals 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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Stuck on convention: a story of derivative relationships 177
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