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Continuing my series of summaries of articles that have appeared in the
International Journal of
Research in Undergraduate Mathematics Education (IJRUME), this month I want to briefly describe three studies that
address issues of conceptual understanding. The first is a study out of Israel that probed student
difficulties in understanding integration as accumulation (Swidan and Yerushalmy, 2016). The
second is from France, exploring student difficulties with understanding the real number line as a
continuum (DurandGuerrier, 2016). The final paper, from England, explores a method of
measuring conceptual understanding (Bisson, Gilmore, Inglis, and Jones, 2016).
Integration as Accumulation
To use the definite integral, students need to understand it as accumulation. In particular, the
Fundamental Theorem of Integral Calculus rests on the recognition that the definite integral of a
function
f, when given a variable upper limit, is an accumulation function of a quantity for which
f describes the rate of change. Pat Thompson (2013) has described the course he developed for
Arizona State University that places this realization at the heart of the calculus curriculum.
We know that students have a difficult time understanding and working with a definite integral
with a variable upper limit. The authors of the IJRUME paper suggest that much of the problem
lies in the fact that when students are introduced to the definite integral as a limit of Riemann
sums, they only consider the case when the upper and lower limits on the Riemann sum are fixed.
The limit is thus a number, usually thought of as the area under a curve. Making the transition to
the case where the upper limit is variable is thus nonintuitive.
The authors used software to explore student recognition of accumulation functions based on
righthand Riemann sums. They investigated student recognition of how the properties of these
functions are shaped by the rate of change function. The experiment involved a graphing tool,
Calculus UnLimited (CUL), in which students input a function and the software provides values
of the corresponding accumulation function given by a righthand sum with Δx = 0.5 (see Figure
1). Students could adjust the upper and lower limits, in jumps of 0.5. The software displays the
rectangles corresponding to a righthand sum. Students were not told that these were points on an
accumulation function, merely that this was a function related to the initial function. They were
encouraged to start with a lower limit of –3 and to explore the functions
x^{2},
x^{2} – 9, and then cubic
polynomials, and to discover what they could about this second function. Students received no
further prompts.

Figure 1: The CUL interface. Taken from Swidan and Yerushalmy (2016), page 33. 
Thirteen pairs of Israeli 17year olds participated in the study. They all had been studying
derivatives and indefinite integrals, but none had yet encountered definite integrals. Each pair
spent about an hour exploring this software. Their actions and remarks were videotaped and then
analyzed.
One of the interesting observations was that the key to recognizing that the second function
accumulates areas came from playing with the lower limit. Adjusting the upper limit simply adds
or removes points, but adjusting the lower limit moves the plotted points up or down. Once
students realized that the point corresponding to the lower limit is always zero, they were able to
deduce that the yvalue of the next point is the area of the first rectangle, and that succeeding
points reflect values obtained by adding up the areas of the rectangles. Rectangles below the
xaxis were shaded in a darker color, and students quickly picked up that they were subtracting
values. Seven of the thirteen pairs of students went as far as remarking on how the concavity of
the accumulation function is related to the behavior of the original function.
This work suggests that a Riemann sum with a variable upper limit is more intuitive than a
definite integral with a variable upper limit. In addition, it appears that students can discover
many of the essential properties of a discrete accumulation function if allowed the opportunity to
experiment with it.
Understanding the Continuum
The second paper explores student difficulties with the properties of the real number line and
describes an intervention that appears to have been useful in helping students understand the
structure of the continuum. Mathematicians of the nineteenth century struggled to understand the
essential differences between the continuum of all real numbers and dense subsets such as the set
of rational numbers. It comes as no surprise that our students also struggle with these distinctions.
The author analyzes the transcripts from an intervention described by Pontille et al. (1996). It
began with the following question: Given an increasing function, f, (
x < y implies
f(
x) ≤
f(
y) )
from an ordered set S into itself, can we conclude that there will always exist an element s in S for
which
f(
s) =
s? The answer, of course, depends on the set. The intervention asks students to
answer this question for four sets: a finite set of positive integers, the set of numbers with finite
decimal expansions in [0,1], the set of rational numbers in [0,1], and the entire set [0,1]. In the
original work, this question was posed to a class of lycée students in a scientific track. Over the
course of an academic year, they periodically returned to this question, gradually building a
refined understanding of the structure of the continuum. The author’s analysis of the transcripts
from these classroom discussions is fascinating.
DurandGuerrier then posed this same question to a group of students in a graduate teacher
training program. In both cases, students were able to answer the question in the affirmative for
the finite set, using an inductive proof or
reductio ad absurdum. Almost all then tried to apply
this proof to the dense countable sets. Here they ran into the realization that there is no “next”
number. The graduate students, given only an hour to work on this, did not get much further. The
lycée students did come to doubt that it was always true for these sets. As they began to think
about the “holes” these sets left, they were able to construct counterexamples.
The continuum provides the most difficulty. The lycée students were eventually able to prove that
it is true in this instance, but only after being given the hint to consider the set of
x in [0,1] for
which
f(
x) >
x and to draw on the property of the continuum that every bounded set has a least
upper bound.
Measuring Conceptual Understanding
The last paper in this set addresses the problem of measuring conceptual understanding. We know
that students can be proficient in answering procedural questions without the least understanding
of what they are doing or why they are doing it. But measuring conceptual understanding is
difficult. A meaningful assessment with limited possible answers, such as a concept inventory,
requires a great deal of work to develop and validate. Openended questions can provide a better
window into student thinking and understanding, but consistent application of scoring rubrics
across multiple evaluators is hard to achieve.
The authors build a solution from the observation that it is far easier to compare the quality of the
responses from two students than it is to compare one student’s response against a rubric. They
therefore suggest asking a simple, very openended question, scored by ranking student
responses, which is achieved by pairwise comparisons. As an example, to evaluate student
understanding of the derivative, they provided the prompt,
Explain what a derivative is to someone who hasn’t encountered it before. Use diagrams,
examples and writing to include everything you know about derivatives.
The 42 students in this study first read several examples of situations involving velocity and
acceleration (presumably to prompt them to think of derivatives as rates of change rather than a
collection of procedures) and were then given 20 minutes to write their responses to the prompt.
Afterwards, 30 graduate students each judged 42 pairings. The authors found very high interrater
reliability (r = .826 to .907). In fact, they found that comparative judgments appeared to do a
better job of evaluating conceptual understanding than did Epstein’s Calculus Concept Inventory
(Epstein, 2013).
Similar studies were undertaken to evaluate student understanding of
pvalues and 11 to 12year
olds understanding of the use of letters in algebra. Again, there was very high interrater
reliability, and in these cases there were high levels of agreement with established instruments.
This approach constitutes a very broad method of assessment, but it does enable the instructor to
get some idea of what students are thinking and how they understand the concept at hand. It can
be used even with large classes because it is not necessary to look at all possible pairs to get a
meaningful ranking.
Conclusion
The three papers referenced here are very different in focus and goal, but I do see the common
thread of searching for ways to encourage and assess student understanding. After all, that is what
teaching and learning is really about.
References
Bisson, M.J., Gilmore, C., Inglis, M., and Jones, I. (2016). Measuring conceptual understanding
using comparative judgement.
IJRUME. 2:141–164.
DurandGuerrier, V. (2016). Conceptualization of the continuum, an educational challenge for
undergraduate students.
IJRUME. 2:338–361.
Epstein, J. (2013). The Calculus Concept Inventory  measurement of the effect of teaching
methodology in mathematics.
Notices of the American Mathematical Society, 60, 1018–27.
Pontille, M. C., FeurlyReynaud, J., & Tisseron, C. (1996). Et pourtant, ils trouvent. Repères
IREM, 24, 10–34.
Swidan, O. and Yerushalmy, M. (2016). Conceptual structure of the accumulation function in an
interactive and multiplelinked representational environment.
IJRUME. 2:30–58.
Thompson, P.W., Byerley, C., and Hatfiled, N. (2013). A Conceptual approach to calculus made
possible by technology.
Computers in the Schools. 30:124–147.