White & Mitchelmore’s paper
“Conceptual Knowledge in
Introductory Calculus”
(JRME, 1996)
Synopsis
Annie Selden
Department of Mathematical Sciences
New Mexico State University
Tucson Calculus Workshop
April 18, 2009
Some Author Definitions
• W&M use abstracting for “the process of
identifying certain invariant properties in a
set of varying inputs.”
(Skemp,1986)
• One goes from generalizing 
synthesizing abstracting.
(Dreyfus, 1991)
2
• Abstracting is a move to a higher cognitive
plane. One goes from interiorization 
condensation  reification. (Sfard, 1991, 1992)
• The first two are operational because they
are process-oriented. Reification is the
“leap” from an operational mode to a
structural mode, in which a process
becomes an object. Dubinsky (1991) calls
this leap from dynamic process to static
object encapsulation.
3
Examples
• Differentiating is a process that students learn to
do. Differentiation in the sense of a
differentiation operator is an object.
• Integrating is a process that students learn to
carry out. Integration is an object.
• Often the symbolism refers to both. In algebra,
2 + 3x can be considered as the process of
adding 2 to the product of 3 and x. It can also
be considered as the object that is the result of
that process, so that 2 + 3x is a “thing” (object)
one can work on.
4
Procepts
• Tall has introduced the term procept for
the amalgam of process, resultant object,
and the common symbol used to represent
both. Gray & Tall (1994) hypothesize that
successful mathematical thinkers can think
proceptually, i.e., can deal comfortably
with symbols as either processes or
objects.
5
• Once an abstraction has occurred, the
generalizing  synthesizing  abstracting
sequence can be repeated. It is a feature
of advanced concepts that they are often
based on several repetitions of the
sequence.
• Each repetition leads to a higher order of
abstraction and further away from “primary
concepts,” those that are formed from
direct experience.
(Skemp, 1986)
6
Different Contexts of Calculus
Problems
• The context of a calculus application
problem may be a realistic or artificial “real
world” problem situation, or it may be an
abstract, mathematical context at a lower
level of abstraction than the calculus
concept to be applied.
• W&M only consider problems that can be
solved using algebra and symbolic
calculus.
7
Solving Application Problems
1. First translate the situation from the context to the
abstract level of calculus. This calls on conceptual
knowledge because one needs to identify appropriate
calculus concepts (such as derivative) and the
relationships between them.
2. Next, solve the abstract (mathematical) problem. This
may require only procedural knowledge.
3. Finally, translate the solution back to the original
context, which may require the same conceptual
understanding as the first step.
(Tall, 1991a)
8
Some more author definitions
• W&M refer to the definition of new
variables and the symbolic expression of
relations between them as (algebraic)
modeling.
• The selection of a calculus concept and its
expression in symbolic form they term
symbolization.
• For them, modeling and symbolization
together constitute translation.
9
• The use of symbols to represent changing
quantities is crucial.
• In research on the teaching and learning
of algebra, it has been observed that the
meaning of the letters is often neglected
so many students only learn manipulation
rules without reference to the meaning of
the expressions involved.
(Kuchmann, 1981;
Eisenberg, 1991; Wagner, 1981; Kieran, 1989; Booth; 1989)
10
Justification for the Study
W&M say,
• “It is a matter of some interest to find out
whether students who aspire to the
advanced mathematical thinking involved
in calculus have an adequate concept of
variable [italics mine].”
11
The Study
• The sample consisted of 40 1st-year, fulltime university math students. A
prerequisite for entry to the univ. math
program was a satisfactory result in the
final h.s. exam for a math course that had
a large component of calculus.
• None of the students finished in the top
10% in the exam; most were between the
50th and 80th percentiles.
12
• Conceptual calculus was taught by W for
4 hrs/wk for 6 wks, as half a semester
course, following Burns (1992) in which
rates of change are investigated using
graphs of physical situations.
• The secant was average rate of change,
the tangent instantaneous rate of change,
and the derivative was defined as the
instantaneous rate of change.
13
• A preliminary study had suggested the crucial
step in successfully solving calculus application
problems was identifying an appropriate
derivate.
• Items1,2,3,4 were constructed in four versions
(A,B,C,D) so the manipulation in each version
was essentially the same. [See Table 1
(Handout).] Each version involved successively
less translation. Version A required translating
all rates to appropriate symbolic derivatives,
whereas version D had all information in
symbolic form.
14
• Table 1 (Handout) gives Items 1-4 across
and Versions A-D down.
• The four versions allowed the translation
steps in each item to be isolated.
• Algebraic modeling was not required in
Item 1; it was rather obvious in Item 2A; it
was substantial in Items 3 and 4 in both
versions A and B.
15
Procedure
• The 40 students were tested 4 times –
before, during, immediately after, and 6
weeks after the calculus course. They
were divided into 4 approximately parallel
groups of 10, based on their algebra
performance the previous semester.
• The students were unaware of these
groupings.
16
• 4 tests of 4 questions each were
constructed, containing one version of
each of the 4 items.
• Each version of each item was on 1 and
only 1 test.
• Each test had only one question in each
version. For example, one test might have
items 1A, 2B, 3C, and 4D.
• Tests were given in cyclic fashion to each
of the 4 groups over the 4 data collections.
17
Interviews
• Interviews were conducted to clarify and
expand on written responses.
• 4 students per group were selected at the
start of the research – they were
interviewed within 3 days of each of the
written data collections.
• Interviews established that students were
unaware they were answering different
versions of the same 4 items each time.
18
Results
Table 2 -- # of correct responses
Time 1
Item
1
2
3
4
Time 2
Time 3
Time 4
ABCD
ABCD
A B C D
A B C
D
0 0 11
0 1 1 1
1 2 2 3
0 3
3
1
1 1 24
0 2 7 4
4 9 6 7
5 4
6
7
0 1 12
1 1 2 2
1 3 4 6
2 2
2
6
0 0 07
TOTAL 1 2 7 14
out of 40
1 1 0 9 2 1 5 8
2 5 10 16 8 15 17 24
1 1 3 10
8 10 14 24
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Authors’ Interpretation
• Performance on version A at Time 1
shows the students could not initially apply
their knowledge of calculus, although they
had similar items in high school.
• The general pattern of difficulties for
across the 4 versions confirms they were
correctly ordered in terms of the amount of
translation (difficulty?) required to solve
the items.
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• The improvement in # of correct
responses at Times 3 and 4 (after
instruction) was substantial, but was only
slightly more than 50% for Item 2.
• The # who improved suggests that
teaching was a positive factor.
• There follows a discussion of Items 1 & 2
(rates of change) and Items 3 & 4
(maximization). I concentrate on the
former.
21
Rates of Change
Table 3 # correct symbolizations and correct
solutions (in parentheses) to Versions A & B of
Items 1 & 2_________________________
Item Time 1
1
0(0)
2
4(2)
TOTAL 4(2)
Time 2
1(1)
2(2)
3(3)
Time 3 Time 4
3 (3)
4 (3)
18(13)
18 (9)
21(16)
22(12)
out of 40
22
• Items 1 & 2 required only trivial modeling
and symbolization was required in only
Versions A and B.
• In the more complex Item 1, few were ever
able to correctly symbolize, but those who
did were almost always correct.
• In the less complex Item 2, during Times 1
and 2, almost all could correctly
symbolize, but only slightly more than 60%
could get a correct solution.
23
Dominant Errors
• In Item 1, the dominant error was to
substitute v = ½ c before differentiating.
(No surprise here.)
• In Item 2, the dominant error was students’
inability to correctly use V = 64.
Examples:
Student 3 left the answer at -6x2.
Student 4 gave 2 answers, one for V=x3 and
one for V=64.
24
When interviewed,
• Student 5 said: The V=x3 and V=64 at the
same time confused me. I didn’t know
which one to use.
• Student 6 said: Is the 64 the starting
volume?
25
Table 4 -- # responses out of 40 showing the
dominant error in Items 1 & 2 (all Versions).
________________________________________
Item Time 1 Time 2 Time 3 Time 4
1
21
17
8
8
_2
7
7
11
15_______
• The decrease in dominant error in Item 1
(substituting before differentiating) and the
increase in Item 2 seemed to result from
instruction as the students became more
aware of the need for a derivative.
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• In Item 1, most students symbolized an
incorrect derivative in the last two data
collections instead of substituting first.
• In Item 2, more students symbolized the
correct derivative in the last two times,
providing more opportunities for making
the dominant error (substituting before
differentiating).
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Authors’ Discussion of Items 1 & 2
• The main problem seems to have been an
underdeveloped concept of variable.
• Other errors suggested this confusion was
part of a manipulation focus, in which
students based decisions about which
procedure to apply on the given symbols
and ignored the meanings behind the
symbols.
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Examples of Manipulation Focus
When interviewed,
Student 7 said: I couldn’t see how to get the
t’s out of the v’s.
Student 8 said: You have to differentiate,
but there is a v and a c, and they’re both
given. I don’t know which one to use.
Student 9 said: There is a change so I
thought of dm/dv because v was the only
variable there.
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• Being able to symbolize derivatives
involves forming relationships between
concepts and should be indicative of
conceptual knowledge.
• Although Item 2 is non-complex, the
formulations of relationships can be on
visible symbols alone and does not require
a sound conceptual base.
• There is a similar analysis of the 2
maximization problems, which I will skip.
30
W&M’s General Discussion
• Responses to the 4 items strongly suggest
a major source of students’ difficulties in
applying calculus lies in an
underdeveloped concept of variable.
• Students frequently treat variables as
symbols to be manipulated rather than
quantities to be related.
• Students have a manipulation focus.
31
Examples of a Manipulation Focus
•
•
•
Failure to distinguish a general relationship
from a specific value (e.g., the difficulty with
V=64 in Item 2).
Searching for symbols to which to apply known
procedures regardless of what the symbols
refer to (e.g., substituting first in Item 1).
Remembering procedures solely in terms of
symbols used when they were 1st learned
(e.g., the x, y syndrome in Item 4).
32
• Students showing a manipulation focus
have a concept of variable limited to
algebraic symbols – they have learned to
operate with symbols without regard to
their contextual meaning. W&M call such
concepts abstract-apart.
(Mitchelmore, 1994; Mitchelmore & White; 1995;
White, 1992; White & Mitchelmore, 1992)
33
• Tall (1991a) calls the accumulation of new
rules, learned by rote and added to
existing knowledge without any attempt to
integrate the rules with old ideas,
disjunctive generalization. This agrees
with W&M’s abstract apart.
• The important characteristic is that
abstract-apart ideas are formed without
any true abstraction, or even without any
generalization.
34
• W&M call concepts that are formed by a
generalizing  synthesizing  abstracting
sequence abstract-general.
• Only abstract-general ideas can be linked
to conceptual knowledge.
• In practice, there is a continuum, rather
than a dichotomy between abstract-apart
& abstract-general.
• Abstract-apart concepts are rarely
completely devoid of meaning.
35
• The generality of an abstract-general
concept depends on the variety of
contexts from which it has been
abstracted.
• The two extremes are often confused and
W&M believe it is important to distinguish
them.
• Most readers would probably say that the
D versions of Items 1-4 are abstract, by
which they mean these items are removed
from reality.
36
• The two different views of abstraction lead to
different views of what constitutes mathematics,
and hence, how it should be taught.
• When symbols represent abstract-apart
concepts, as in the manipulation focus, they are
not related to any mathematical objects that
could give them meaning.
• Relationships between symbols are superficial,
i.e., they are based only on what the symbols
look like; and learned rules can only be applied
on the basis of the visible symbols available,
37
• An abstract-apart concept might be
adequate to deal with routine symbolic
procedures
• But symbolizing a rate of change in
complex problem situations requires the
existence of abstract-general concepts
because the symbols used to represent
general variables and derivatives have to
be related to the specific variables and the
rate of change that occur in that situation.
38
• Students prefer to learn in an abstractapart fashion and become comfortable
with decontextualized problems.
• Abstract-apart ideas are easier to learn
because they are limited to a purely
symbolic context (sometimes only x and
y).
• All decontextualized problems can look
very similar to students, and the
appropriate procedures are easy to
formulate.
39
• Success in such a narrow context can lead
to a sense of satisfaction.
• By contrast, learning abstract-general
concepts requires the formation of links
among a wide variety of superficially
different contexts.
• This takes longer and is more intellectually
demanding, but the learned relationships
can be used to solve more diverse
problems.
40
• This study illustrates the significance of
encapsulation (Dubinsky, 1991) or
reification (Sfard, 1991, 1991) and the
importance of thinking proceptually (Gray
& Tall, 1994) in the formation of abstractgeneral concepts.
• Students who can use defined variables
but cannot identify and define their own
variables are using symbols to express the
process of relating one variable to another.
41
• Such students are still at the condensation
phase of developing their concept of variable.
• BUT, students who can create variables to solve
complex problems have reified variables.
• Student comments for Item 3 such as: “I knew I
wanted time and had some distances and
speeds. So I looked for a connection and got
time and distances so gave x for the distance.”
show that the student was using variables as a
tool/object.
42
• Such students never made manipulation
errors, which is consistent with the view
that reification can only occur after
extensive successful experience using
variables in the operational mode.
43
W&M’s Implications
• In this study, the only detectable result of
24 hrs of instruction that were intended to
make the concept of rate of change more
meaningful was an increase of
manipulation-focus errors in symbolizing a
derivative.
• Most of the students had an abstract-apart
concept of variable that blocked
meaningful learning of calculus.
44
• These findings parallel those reported in
school algebra research.
(Booth, 1989; Eisenberg, 1991; Kieran, 1989, Kuchemann, 1982)
• M&W find this a “most disappointing
result.”
• They conclude that a prerequisite to
successful study of calculus is an abstractgeneral concept of variable at or near the
point of reification.
45
• Even a concept-oriented calculus course
is unlikely to be successful without this
foundation (of variable).
• Students probably need to spend a
considerable amount of time using algebra
to manipulate relations before they can
achieve a mature concept of variable.
(White & Mitchelmore, 1993)
46
• It is unrealistic to attempt to provide
remedial activities within a calculus course
for students with an abstract-apart concept
of variable.
• Either entrance requirements for calculus
should be more stringent in terms of
variable understanding, or an appropriate
precalculus course should be offered.
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THE END
Comments/Discussion?
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