Calculus students’ understanding of area and volume in
calculus- and non-calculus contexts
f(x) Abstract
Researchers have
documented difficulties that
elementary school students
have in understanding area
and volume. We know very
little about older students’
understanding of these
concepts. This study develops
descriptions of calculus
students’ understanding of
area and volume concepts in
non-calculus contexts.
Knowing what calculus
students understand about
area and volume will help us
understand student
difficulties with calculus
applications such as integrals
and Volumes of Revolution.
Research questions:
What do calculus students
understand about area in
non-calc contexts?
What do calculus students
understand about volume in
non-calc contexts?
x=a
Calculus
students…
Allison Dorko
Masters of Science in Teaching
University of Maine
•Not all students excel at
calculus.
•Calculus courses have low
enrollment rates; low
retention rates; and some
students demonstrate a lessthan-ideal understanding of
calculus topics (Steen 1987),
sometimes getting by with
just memorization.
•There are a few core ideas
of integration that some
students do not understand,
among them (1) taking a limit
of Riemann sums to find area
under a curve and (2) volume
of revolution (Orton 1993).
•Elementary school students
struggle with computations,
conceptual understanding,
and units associated with
various spatial measures
(Lehrer 2003).
Tasks & Subjects
Results
•Multivariable calculus students
do calculations correctly and have
expert-level conceptions of area
and volume
What is the volume of the object?
•198 introductory calc students
•43 multivariate calc students
•Written tasks:
•Computation
•Short answer
b
Area under the curve = 
a
f (x)dx
•Introductory calculus students
are relatively adept at area
calculations but struggle with
both volume computations and
the conceptual aspects
•Consistent with findings from
about elementary school
students’ understanding of area
and volume
x= b
References
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Additional Selected Tasks
Compute the area of the
rectangle.
12 cm
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Research Companion to Principles and Standards for School Mathematics. Reston, VA: National Council of Teachers of
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J. J. Kaput & E. Dubinsky (Eds.), Research Issues in Undergraduate Mathematics Learning: Preliminary Analysis and Results
(Vol. 33, pp. 29-45). Washington, DC: The Mathematical Association of America
4 cm
Compute the area of the
circle.
5 in
Monk, S. (1987). Students' understanding of function in calculus courses. Unpublished manuscript. [This is very similar to
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What types of things can we
use area to measure?
Standards for School Mathematics, pages 289-303. Reston, Va.: National Council of Teachers of Mathematics.
What types of things can we
use volume to measure?
Strauss, A. & Corbin, J. (1990). Basics of qualitative research: Grounded theory procedures and techniques. Newbury Park,
CA: Sage.
What units are areas
measured in?
White, P. and Mitchelmore, M. (1996). Conceptual Knowledge in Introductory Calculus. Journal for Research in
Mathematics Education Vol. 27, No. I, 79-95
Zandieh, M. J. (2000). A theoretical framework for analyzing student understanding of the concept of derivative. In E.
Dubinsky, A. H. Schoenfeld & J. Kaput (Eds.), CBMS Issues in Mathematics: Research in Collegiate Mathematics Education
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What units are volumes
measured in?
What does area mean?
What does volume mean?