Some Calculus 2 Students Seem to Prefer Procedural Approaches to
Exercises over Conceptual Ones
Mary Shepherd
Abstract: This paper presents the results of a study performed during the Spring 2006
semester. Students in a typical Calculus 2 course, in an interview situation, were
presented with two problems that would require a conceptual approach as opposed to a
procedural approach to handle. When available, students seemed to prefer to arrive at an
answer based generalization from examples where a procedural approach was available
to a conceptual approach. Even when no readily available examples existed, some
students seemed to trust a possibly wrong calculation over a conceptually derived at
answer.
I. Introduction
Based on “seminar” with John and Annie Selden during the Spring 2006 semester, I was
intrigued by the ideas of how and/or when students use conceptual knowledge versus
procedural knowledge when approaching a problem or exercise. There has been
considerable written about conceptual versus procedural knowledge in terms of
approaching mathematical problems, but does not seem to be agreed upon definitions for
these terms. Conceptual knowledge has been described as “knowing that” something is
true or “knowing why” something is true. Procedural knowledge has been described as
“knowing how” something is true.
In some sense, procedural knowledge is accessed sequentially (following a specific set of
steps in order) and conceptual knowledge is accessed more randomly. Even though these
definitions attempt to distinguish conceptual and procedural knowledge, as one tries to
use any of the definitions proffered in the literature, it becomes clear that these are not
necessarily distinct, non-intersecting ways of classifying mathematical knowledge. For
instance, does number fact 3 + 2 = 5 represent procedural or conceptual knowledge?
In this paper I will first describe a framework for relating conceptual and procedural
knowledge. Then I describe the preliminary study and its purpose, it’s main results,
directions for future research and some implications (but mostly questions) for teaching.
II. Development of Proceptual knowledge framework:
I propose a model that relates conceptual and procedural knowledge and describes in
some sense how these become integrated. It is not necessarily the only model and may
not even be the best one, but it does serve to model and explain how the students in this
study used their knowledge to solve relatively simple conceptual exercises.
This model is based to a large extent on the procept idea of David Tall and others
working with him (Christou et al, 2005, Tall and Vinner, 1981, Tall et al, 2000, Tall,
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2004). This is a brief description of that idea. In the world of mathematics, where
symbols are widely used, the symbol acts as a pivot between a process and a concept
(Tall et al, 2000). For instance, 3 + 2x represents both the process of multiplying by 2
and then adding 3 and the concept that this is an expression all on its own. This idea of a
symbol representing both a process and a concept has led to the word procept.
Mathematicians seem to be able to pivot effortlessly between the process and concept
meanings for symbols. This seem to lead one to believe there is some type of difference
yet also relationship between a process and a concept.
If one extends this procept idea beyond symbols to the realm of “knowing”, one might
obtain a model that looks something like this. One may have procedural knowledge of
how to, say, take the derivative of some nice function at a given value. One may have
conceptual knowledge that the derivative of a function at a given value represents the
slope of the tangent line at that point. Many who have taught this calculus concept are
aware that in the same class there are students who can take a derivative without having
any idea what it represents and there are those who seem to understand what the
derivative represents, but have great difficulty finding even the derivative of a simple
function such as a polynomial. What seems to happen as students progress through this
material and approach the material from several points of view is that they eventually (we
hope) develop a flexibility and comfort level with both the procedures of taking
derivatives and the concepts underlying and connecting this part of the calculus. We, as
teachers, are trying to help them reach what might be called a proceptual understanding
or level of knowledge, where one not only pivots between procedural and conceptual
knowledge more or less effortlessly, but also can work in either the procedural or
conceptual realm confidently and fluently. At this level there is a connectedness between
several concepts and procedures and some “strategic knowledge” of when to use what
bits of procedural or conceptual knowledge to accomplish the mathematical task at hand
efficiently. This is seen in the research of Weber (2001) and also in the novice-expert
studies of Chi (1981).
A simple model of the knowledge paths and nodes that could occur might be something
like this. First a student is introduced to a concept (such a derivative at a point by a limit
of slopes of secant lines) with a perceived difficult procedure (limit of a difference
quotient). Then a “simpler” procedure is given which the student learns but does not
connect it with the concept (or only dimly connects).
The units of knowledge are isolated at first. Asking or posing different type questions to
students might show them operating in only one or the other of the boxes. For instance, a
student asked to find the tangent line to a curve at a point may attempt to use the limit
definition, or may draw a picture and attempt to estimate the slope. And when asked to
find and interpret a derivative at a point, may be able to calculate the derivative easily,
but be unable to interpret the resulting number.
Connections between the two areas of knowledge result (one hopes) as students progress
through the calculus. Students come to know that when asked for a tangent or normal
line, a derivative will be necessary. So a connection is made between the conceptual to
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the procedural realms of knowledge in this area. Similarly, the derivative can represent a
rate of change and this is added into their conceptual knowledge, thus other conceptual
knowledge areas begins to be linked with these two concepts of derivative.
Likewise, we teachers want students to create a link from the procedural to the
conceptual knowledge. This might be in a case where some function that does not have
an easy derivative (or has no derivative) is given (such as finding the derivative of f(x) =
|x| at x = 0), or where no specific expression for a function is given at all (maybe a graph
or table of values is all that is given), but a student is still asked to find or estimate its
derivative or rate of change at a point. A procedure seems to be called for, but no
procedure is readily available, so a student must access his/her conceptual knowledge of
derivative in order to answer this type of question. When the link from procedural to
conceptual knowledge is established, students might experience an “ah-Hah” moment or
stating something like “That’s why we do this!”
When students can move (more or less) effortlessly and with confidence between the
conceptual and procedural aspects of their knowledge we would have the simplified
model in Figure 1.
Conceptual knowledge
(a web of connections)
Procedural knowledge
(a linear set of steps)
Figure 1: confidence in the linking of procedural and conceptual
knowledge.
Going back to the procept idea related to symbols, Tall et all (2000) indicate there is a
spectrum of performance given for carrying out mathematical processes related to
symbols. One first learns a procedure, and if he/she remains procedure based and
progresses, can eventually do routine mathematics accurately, but not necessarily
efficiently. Some learners gain flexibility in the number of procedures they know and
can choose from several procedures to accomplish some process. (For example: the
process of differentiating (1 + x^2)/x^2 can be accomplished by several different
procedures.) At this added level of sophistication, students who progress can perform
mathematics flexibly and efficiently. At the highest level, students operate on a
proceptual basis where they think about the mathematics in a compressed and
manipulable way moving easily between process and concept.
In this description, the links between process and concept exist, but we are not
necessarily at what I would call a proceptual knowledge level. At the proceptual
knowledge level, there is fluidity in moving around in the conceptual knowledge realm
where its web of relationships and randomness is exploited. To become fluid at this
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level, one seems to need what Weber (2001) called “strategic knowledge”, or knowing
how and when to use procedural and/or conceptual knowledge. This also might have
been what was lacking in the Selden et al (2000) about calculus students who had all the
background to solve some moderately non-routine problems, but couldn’t seem to pull it
all together.
As procept idea is extended to conceptual and procedural knowledge, there may be
several processes linked to a single concept and several concepts linked to a single
process, but at the conceptual level, there will be additional connections with other
concepts, theorems and formal ways of thinking about mathematics.
III. The study
Motivation. When students are presented with a “problem” to solve, can they flexibly
choose which of these types of knowledge they should call on? In many calculus
classrooms, one would suspect that procedural knowledge is stressed so that would be
called upon first. Many exercise and test questions have the form: find this derivative,
evaluate that integral, etc. Students seem to feel comfortable with this type of question.
They know exactly what to do. This would be a case of using procedural knowledge.
Some students will develop to a process or even procept level in working with the
symbols with a number of procedures available to answer various questions. Even when
the instruction has been motivated by conceptual ideas—limits of slopes of secant lines
for the derivative, limits of Riemann sums for definite integrals—instruction might
quickly progress in the eyes of the students to “here are the shortcuts for calculation
purposes.” Even many application problems (such as related rates and optimization) can
be procedural in essence.
There are other potential questions that require a more conceptual approach. For instance
a graph of a function might be given and a student might be asked to graph the derivative
of that function. No explicit functions are given, so students can not go to a well known
procedure. Many students in a traditionally taught calculus class struggle with this type
of problem at first.
The questions for this study: This is a preliminary study to get an idea about what
students do when faced with a conceptual problem. For the purposes of this study a
conceptual problem is either one that does not have specific functions/numbers with it so
that a procedural/process approach is not readily available or one in which a conceptual
approach is much more efficient than a procedural approach. I asked these 4 related
questions.
(1) What do students do?
(2) What tendencies are noted?
(3) Do they attempt to supply examples on which they can “do the process?”
(4) Do they have any confidence in a conceptual approach?
Since this is just a preliminary study, I wanted to look at a conceptual problem for which
a procedure or set of steps is not available directly and at a problem which would need
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some conceptual thinking to approach at first, but for which a totally conceptual approach
was much more efficient than a procedural approach. Conceptual questions are in some
sense always non-routine since there is no readily available procedure (routine) to follow.
The students in this study had been exposed to conceptual problems similar to those used
in the study thoughout their course of instruction. I did find it difficult to find conceptual
questions in the Calculus 2 curriculum for our school.
The students and the questions asked. During the Spring 2006 semester, I asked the
students in my calculus 2 class to take part in an experiment. The students came into my
office in pairs and were given two conceptual problems (see figures 5 and 6 below) on
separate sheets of paper. They were told to work on the first one individually, and then
talk to each other about it, and see if their answer changed. Then they did the same on
the second problem. At the end of the interview, I went over each problem with the
students as part of the teaching for the course. These were problems that appeared in the
exercise sections in their text. Twelve students participated in the study in 6 pairs. Two
other students did not participate. One never attended class and another was ill during the
time of the study. The six pairs of conversations were recorded and transcribed. Each
session was under 25 minutes in length. In addition, the written papers of the students
were collected.
The class was taught from a traditional text, Larson et al (2002). The students had been
assigned similar conceptual exercises in the past in this class (and also in Calculus 1, if
they had had me as three of the students had).
The first problem was presented entirely in a conceptual form. See Figure 2. The graphs
of two functions are given. They appear to have the same arc length on the given interval
and f1 is always greater than or equal to f 2 on the interval. No expressions are given for
the functions. The problem asked, “When each is revolved around the x axis which
surface of revolution has the greater surface area?”
Figure 2: problem 1 given
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To solve this first problem conceptually, the students would need to recall some (but not
necessarily all) of the following concepts (as they had been presented in the course).
(1) how the concept of surface area is developed with partitioning the interval and
using the summing of the areas of the frustrums used to estimate the area on the
subintervals,
(2) a visualization of the two surfaces
(3) how a definite integral is the limit of a Riemann sum
(4) the formula for surface area and why it works.
(5) Some of the properties of integrals (for instance that if f1 ( x)  f 2 ( x) on some
interval [a, b] , then ab f1 ( x)dx  ab f 2 ( x)dx )
(6) Arc length idea and formula.
The second problem (see Figure 3) involved trying to find the limit at infinity of a
rational expression where the numerator was just given as p (x ) where p (x ) is a
polynomial (no specific polynomial is given), and the denominator was an exponential
function 2 x . This problem was conceptual in the sense that no explicit polynomial
function was given, even though the exponential function was given explicitly. Although
this problem can be solved procedurally once an argument about the polynomial is made
by using l’Hopital’s Rule, it is much easier to solve it conceptually. In a sense, this one
could send mixed messages to the student. In some sense, this question is similar to one
posed in Zazkis and Campbell (1996) where when students were given the factored form
of a number, such as m  3 2  5 3  7 2 , and then asked if the number 7 divides it or if the
number 39 divides it, some students would multiply out the numbers first, then divide by
whatever was asked before answering instead of directly using their conceptual
knowledge relating “divides” and “is a factor of”.
Given that p (x ) is a polynomial, evaluate the following limit, if it exists.
p( x)
lim x
x  2
Figure 3: problem 2 given.
Some of the conceptual ideas the students needed for this second problem include:
(1) What a limit at infinity means about the growth of these functions
(2) What a polynomial function is
(3) L’Hopital’s Rule
(4) The growth rates of each of the types of functions given
IV. Results:
Analysis of responses to the first problem: I present first a table (Table 1) that shows
how the students initially responded on their own, finally responded, and the number that
changed their answers upon discussion with the other student in the interview.
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Answer given
f1
f2
Both are same
Number of students
giving each initial
answer
9
1
2
Number of students
changing answer
from initial one
4
1
1
Number of students
giving each final
answer
7
1
4
Table 1: numbers of students giving various responses to first question.
For this problem the following four approaches and behaviors were noted and will be
discussed below. These are in no particular order and there is no claim that these are the
only approaches that might be employed. Some students used more than one approach.
The numbers after the approach indicate the number of students that used that approach at
some time during the interview.
(1)
Some students do try a conceptual approach (5).
(2)
Some students when thinking conceptually become distracted by details in
another concept that seems not well understood (5).
(3)
Some students try to visualize the problem and think visually (8).
(4)
Some students wanted something to calculate (7).
In general, the students placed very little faith in any approach that did not have a
calculation involved.
Five students seemed to employ some conceptual thinking. A typical comment from
Frank1 captures the basic conceptual approach of the students who seemed to employ a
conceptual approach.
Frank: I think that f1 would have the greater surface area. Because looking at
the graph, it,…like it, …between a and b, it’s curve is greater than f 2 , so, when
you revolve it around it would be wider than f 2 which would create a greater
surface area.
Five of the students, some of whom appear to approach the problem conceptually
initially, get “distracted”. For instance, Charles and Chris initially compare the integrals
for the surface area conceptually, but are then distracted by the arc length formula within
the surface area formula.2
1
2
Names have been changed. Names starting with the same letter were in the same group for this study.
The formulas for surface area of a surface of revolution and arc length given in the textbook used are:
Surface area =
b
b
a
a
2  r ( x) 1  [ f ' ( x)]2 dx and Arc length =  1  [ f ' ( x)] 2 dx where in each case
y  f (x) has a continuous derivative on the interval [ a, b] and r (x ) is the distance between the graph of
f and the axis of revolution.
Conceptual/Procedural
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Charles: Well,…that’s like the arc length formula. The only think they add to
the arc length formula is 2 and then r (x ) . So the 2 s are going to be the same
in both formulas. The only thing that’s going to be greater is r (x ) . It’s just this
distance …to all points in which f of 1
Chris: has a greater r
Charles: has a greater r (x ) than…at most points except for a and b. So when
you revolve it around the x-axis, your r (x ) would be greater for f of 1 than at f of
2.
But, after some further discussion along the same lines, another “worry” surfaces which
seems to indicate that there is not a complete conceptual understanding of arc length and
the formula used in procedural calculations of both arc length and surface area:
Charles: The only thing I’m worried about is that arc length is 1  f ' ( x) 2 . And
how can these have the same derivative, f1 and f 2 ? That’s the only way their arc
length could be the same. Because arc length is…is square root of 1  f ' ( x) 2 . So
the 1 (one) would be the same, the square root would be the same. How
would…how would their derivatives be the same? … Oh, unless they’re just
inverses of each other, one’s negative, one’s positive. And then when you square
it you’d get the same thing.
Eight students used some type of visualization in their arguments, and frequently had
difficulty coming to a correct conclusion. Darcy had stated originally, and written, that
both “have the same because it is the same arc, just bent different ways. They both are
revolved on the x-axis. The surface are of f1 would equal the surface area of f2.” Her
partner, David originally has an answer of f1 with a conceptual argument.
David: I thought…I know that the…the curve is the same…the length of the
curve is the same, but when you rotate it around the x-axis, I think..I don’t know if
the surface area stays the same, because I think this one (f1) might travel farther is
what I said. So when you sum up surface area, maybe it goes more distance, so ,
therefore you get more…
But his conceptual argument does not convince him or Darcy. Soon after the comment
above David describes very visually (and physically with lots of hand motion) how these
surfaces are like bracelets, one is indented and the other bent in the opposite direction but
covering the same amount of area on your arm. After his own discussion he changes his
mind to say the two have the same surface area. After further discussion, they decide to
say f1 has the larger area, but both claim they have no idea how to prove it.
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Seven students either state a desire for functions, or attempt to find functions for the two
curves given. David asks Darcy as he looks up the formula for surface area, “Could we
solve it?” and Darcy answers, “I don’t think we can solve it because there are no
numbers.” Toward the end for the first problem interview, David tries to find an example
to follow. Similarly, as Ellen and Edward are discussing the arc length portion of the
surface area formula, Ellen tries to supply actual functions.
Ellen: Let’s say this was a part of, like…say this is a part of x 2 and this is a part
of negative x 2 . Because then it’s flipped, then it had … negative doesn’t matter,
because it’s squared.
And this argument convinced Edward that because the arc length could be the same, the
surface area would be the same.
In summary, for this first problem, clearly some of the students did not feel at all
comfortable with conceptual reasoning. Alice states: “It would just be hard because you
don’t have any numbers.” Three times in his interview, David expresses a desire for
functions or numbers. First, right after his conceptual argument given above for choosing
f1 above, he continues:
David: I suppose we could solve…we could do a little problem like this and see.
Like if we set this to 1 and 3 (for a and b) and then rotated…we did…we solved
this surface area for this one and did the exact same one with…I don’t know how
to get the curves to be exactly the same, though.
What was convincing to the students? In most cases, for this first problem, it was not a
conceptual argument. Four of the students argued entirely visually. Only one, Andy,
came to a correct solution with a conceptual argument, but even his partner was not
convinced and in fact changed her answer from f1 to f 2 after their discussion. The other
eight students seemed to use both a visual approach and looked up the formula for
surface area. Five of these were distracted is some way using the formula and four of
them clearly were not convinced by their initial conceptual argument. The three who did
not appear to be distracted by something in the formula or something in the visualization
argued correctly from a conceptual viewpoint. Seven students appeared to want a
calculation/procedure to be convinced, but none was available. It would appear that for
this problem, some students can move from the procedural to the conceptual knowledge
level, but do not have enough connections at the conceptual level to move around
flexibly, nor to trust a conceptual argument.
Analysis of responses to second problem. Table 2 below contains a summary of the
responses of the students to the second problem presented.
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Answer given
Number of students Number of students Number of students
giving each initial
changing answer
giving each final
answer
from initial one
answer
0
4
10
1
1
1
0
Limit DNE
2
2
No answer
5
5
0
Table 2: numbers of students giving various responses to second question.
For the second problem the following three approaches and behaviors were noted and
will be discussed below. Again, these are in no particular order and there is no claim that
these are the only approaches that might be employed.
(1)
Give a conceptual approach that was confirmed with a calculator (1).
(2)
Chose a typical polynomial so that a procedure could be followed (using
l’Hopital’s Rule (8).
(3)
Distracted in the process of using l’Hopital’s rule (2).
Again, there was a lack of conviction in an answer when no calculation was done.
Only one student, Edward had a conceptual approach.
Edward: Well, p (x ) can be a polynomial, which means it’s just something to a
power in the equation raised to a power. Um.. 2 x is going to get much larger much
quicker than anything on the top ever will. At the beginning it might not be that
way, but in the end it will be. Two to the 100th is going to be bigger than 100
cubed or 100 squared or 100 to the 10th for that matter. It doesn’t matter how
large the polynomial is. The bottom is still going to keep getting infinitely larger,
which anything divided over anything really large in the end turns into zero.
Edward was convinced of his answer, but still used his calculator to check his reasoning
by calculating 2100 compared to 10,000 (or 100 2 ).
Eight of the students chose some polynomial, usually a quadratic one and applied
l’Hopital’s Rule. Most of the students were convinced by their arguments since a
calculation had been involved when applying l’Hopital’s Rule. Even though most only
used one polynomial, it was not clear whether they generalized from that one example or
believed that one example was enough to prove the result for all polynomials.
Additionally, at least two students did not take the derivative of 2 x correctly, although
their errors did not seem to affect their conceptual reasoning toward an answer.
At least two students had some distraction when applying l’Hopital’s Rule which might
indicate a procedural as opposed to conceptual understanding of the rule. Charles
became distracted in applying l’Hopital’s rule when he used the polynomial x 3  x for
his calculations and claimed that at infinity this was the indeterminant form    .
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Maybe the most telling example of needing a calculation or procedure in order to be
convinced came from Andy. His final answer was that the limit did not exist because one
would still end up with the   indeterminant form no matter how many times you apply
l’Hopital’s rule. He did not see or recall that the derivative of a polynomial decreases the
highest power present. Therefore, his incorrect calculation had a greater power to
convince him than his correct conceptual argument.
Andy: I was thinking this top was approaching infinity, but I mean just looking
at it without applying any rules or anything…the bottom’s going to, looks like it
grows faster than the top. So, I think it would approach 0 but, I can’t see how to
prove it.
In the second problem there was not a readily available visual/physical approach. Most
students created examples to give them the ability to use a procedure and do a
calculation. Of the two who employed a conceptual approach at some time in their
discussion, Edward was convinced after doing some calculator calculations, but Andy
remained so unconvinced because he could not see a calculation available that he stayed
with the wrong answer of the limit not existing.
For both of these problems, there seems to be a chasm in students’ minds between the
convincing power of using conceptual and procedural knowledge.
V. Summary:
The common results from the two problems used in this preliminary study indicate that
some students will attempt to use conceptual knowledge to solve conceptual problems.
The students in this study seemed to still want a procedure over using only conceptual
knowledge. Some were easily distracted when other conceptual knowledge needed was
still seen as calculation based or was not yet integrated well. Most were not convinced
by a conceptual argument. The students in this study were working at becoming flexible
in choosing between procedural knowledge and conceptual knowledge, but most seemed
to prefer procedural knowledge. Procedural knowledge appeared to be more convincing
to them.
In answer to our four initial questions, for these problems students tried different
approaches to the two conceptually presented problems. Some attempted to visualize,
some looked for formulas or examples. Many expressed a desire for numbers or
expressions so that procedures could be followed. Some attempted a conceptual
approach, but most have little confidence in an answer arrived at conceptually as opposed
to procedurally. Although these responses were observed, this is definitely not an
exhaustive list of the responses students could have had.
Procedural knowledge seemed to be more convincing. Even at the conceptual level, if
there was a “nearby” procedure, these students were more apt to use a nearby procedure
and look through their weak web of conceptual knowledge to find a more efficient way to
answer the questions posed. In Weber’s words, these students seem to have a weak sense
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of strategic knowledge. Growth in mathematical maturity as seen in Weber (2001) and
other novice/expert studies (Chi 1981), indicates there is a strengthening in the network
of conceptual knowledge as strategic knowledge increases. Research by Jerome Epstein
would indicate that conceptual understanding of calculus 1 concepts remains weak in
traditional lecture based courses. (Epstein, 2007).
VI. Further Questions and Implications for teaching:
Students need practice working in the conceptual realm of their knowledge. I suspect
that only by working in it can they attain some flexibility. The students in this study had
seen similar conceptual problems, yet I suspect many students in other calculus 2 courses
have not. I also suspect students need support and scaffolding to gain confidence in
learning to work more in the conceptual realm of knowledge. I don’t think this mean
that our introduction of an idea from the conceptual point of view needs “help” or more
emphasis. But I think as students start learning procedures, more needs to be done to
help them tie them back to the conceptual ideas from which the procedures sprang. And
students need help in tying together conceptual ideas—creating more hooks and paths at
the conceptual level.
A second question arises. How do we usually test students in these classes? Are our tests
mostly procedural? Do students fail because they can’t do or recall the correct
procedure? Do those who fail procedural based tests have a conceptual understanding
that is not tested?
Is procedural knowledge even necessary for learning and using conceptual knowledge
fluidly? For example, I suspect that most college faculty no longer remember the
procedure for finding a square root by hand. I know I have. But has this diminished our
conceptual understanding of the square roots? Is there even a correct balance between
conceptual and procedural knowledge?
References:
Chi, M.T.H., Feltovich, P.J., and Glaser, R., “Categorization and representation of
physics problems by experts and novices.” Cognitive Science, 1981, 5:121-152.
Christou, Constantinos; Pitta-Pantazi, Demetra; Souyoul, Alkeos and Zacharaides,
Theodossios, “The Embodied, Proceptual, and Formal Worlds in the Context of
Functions”, Canadian Journal of Science, Mathematics and Technology Education, April
2005, (5:2), 241-252.
Epstein, J., personal email 3/2/2007.
Larson, R., Hostetler, R.P., and Edwards, B.H., Calculus of a Single Variable, 7th edition,
2002, Houghton Mifflin Co.
Conceptual/Procedural
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Selden, A., Selden, J., Hauk, S., and Mason, A., “Why Can’t Calculus Students Access
their Knowledge to Solve Non-Routine Problems?”, CMBS Issues in Mathematics
Education, 2000, 8:128-153.
Tall, D., “Thinking Through Three Worlds of Mathematics”, Proceedings of the 28th
Conference of the International Group for the Psychology of Mathematics Education,
(2004), 4:281-288.
Tall, D., Gray, E., Bin Ali, M., Crowley, L., De Marois, P., McGowan, M., Pitta, D.,
Thomas, M., and Yusof, Y., “Symbols and the Bifurcation between Procedural and
Conceptual Thinking”, Canadian Journal of Mathematics, Science and Technology
Education, (2000) Vol. 1, No. 1 pp 81-204.
Tall, D. and Vinner, S., “Concept Image and Concept Definition in Mathematics with
particular reference to Limits and Continuity”, Educational Studies in Mathematics,
(1981) 12:151-169.
Weber, K., “Student Difficulty in Constructing Proofs: The Need for Strategic
Knowledge”, Educational Studies in Mathematics (2001) 48:101-119.
Zazkis, R., and Campbell, S., “Divisibility and Multiplicative Structure of Natural
Numbers: Preservice Teachers’ Understanding”, Journal for Research in Mathematics
Education, (Nov. 1996) 27:540-563.
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