Transcript Dorko, A. (2012). Calculus students` understanding of volume in non

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Calculus Students’ Understanding of Volume
in Non-Calculus Contexts
Allison Dorko
RiSE Center
University of Maine
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Introduction & Rationale0
Calculus in general

Calculus courses are essential to math and science and
are often prerequisites for undergraduates in STEM
fields.
 Calculus students often do not do as well as
instructors might like (CBMS, 2000; Jencks & Phillips, 2001; Bressoud, 2005).
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In many topics, calculus students are procedurally
competent but lack a rich conceptual understanding
(Ferrini-Mundy & Gaudard 1992; Ferrini-Mundy & Graham 1994; Milovanović 2011; Orton 1983; Rasslan
& Tall 1997; Rosken 2007; Thompson & Silverman 2008).

The lack of a rich conceptual understanding creates issues
later
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Introduction & Rationale1
Understanding of volume
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Volume plays a central role in calculus
 Optimization; volumes of solids of revolution; etc.
 These topics use derivatives and integrals, two topics with
which calculus students struggle (Zandieh, 2000; Orton, 1983).
 Volumes of solids of revolution are one of the most difficult
topics for students (Orton, 1983) but it is not known what makes
this so difficult.
 Studies have found that elementary school students
struggle with volume understanding, often finding surface
area instead of volume (Battista & Clements, 1998).
 There exists anecdotal evidence from calculus instructors
that some calculus students also have issues with surface
area and volume.
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Optimization & Volume of Solid of
Revolution
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Introduction & Rationale3
Prior knowledge and new knowledge

Prior understanding may affect learning of new calculus
concepts

Function (Carlsen, 1998; Monk, 1987)

Variable (Trigueros & Ursini, 2003)

The premise for my study is that something similar may
be occurring with student understanding of volume and
the understanding of calculus topics.

Knowing more about what issues calculus students have with
volume may improve the teaching of volumes of solids of
revolution, multiple integration, etc.
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Research Questions
1.
How successful are calculus students
at finding volume?
1.
Do calculus students find surface
area when directed to find volume?
(yes)
1.
If calculus students find surface area
when directed to find volume, what
thinking leads them to do so?
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Advanced Organizer
 Research
about students’ understanding of volume
 Research
design
 Findings
- Two stories:
 1. Finding surface area instead of volume
 2. Continuum of amalgam formulae
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Literature: Elementary School
Students’ Understanding of Volume
•Literature Review
•Research design
•Findings
• Surface area
• Continuum of
amalgam formulae

We don’t know much about how calculus students
understand volume

We do know something about how elementary school
students understand volume.
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Battista & Clements (1998)
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
Students counted cubes on the faces, often doublecounting edges/corners
Specifically, some elementary school students
find surface area when directed to find volume.
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Research Design
•Literature Review
•Research design
•Findings
• Surface area
• Continuum of amalgam
formulae

Cognitivist framework (Byrnes 2001; Siegler
2003)

Participants: 198 calculus I students
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Data collection: two phases



Written surveys
Task-based clinical interviews
Data analysis
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
Grounded Theory (Strauss & Corbin, 1990)
Made use of other researchers’ findings about
area and volume (e.g., Battista & Clements, 2003;
Izsák 2005; Lehrer, 1998; Lehrer, 2003) and
anecdotal evidence from calculus instructors
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Research Instrument
•Literature Review
•Research design
•Findings
• Surface area
• Continuum of amalgam
formulae
+ Categories of Student Responses
•Literature Review
•Research design
•Findings
• Surface area
• Continuum of amalgam
formulae
Category
Found Volume
Found Surface
Area
Other
Definition
Stated magnitude is
the correct magnitude
of the object’s volume,
or magnitude is
incorrect for the
object’s volume but
the work/explanation
is consistent with
volume-finding (i.e.,
multiplication or
appropriate addition)
Stated magnitude is
the magnitude of the
object’s SA or the
student
work/explanation
contains evidence of
SA-like computations,
such as addition. To
allow for
computational errors,
stated magnitude may
or may not be the
actual magnitude of
Student found neither
volume nor surface
area
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Found Surface Area: Triangular
Prism
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•Literature Review
•Research design
•Findings
• Surface area
• Continuum of amalgam
formulae
Findings
Found . .
.
Rect. Prism
(n=198)
Cylinder
(n=198)
Volume
98%
86.9%
77.9%
28.6%
1.52%
5.1 %
13.9 %
71.4 %
0.5 %
8.0 %
8.2 %
0%
Surface
Area
Other
Tri.
Prism
(n=122)
Trap.
prism
(n=7)
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What thinking leads students to
find surface area?
•Literature Review
•Research design
•Findings
• Surface area
• Continuum of amalgam
formulae
Reason 1: Some students think that adding the areas of the
faces of an object finds the measure of the object’s volume.
Reason 2: Some students understand the difference
between area and volume, but mix the formulae together.
I call this mixed formula (e.g., V=2πr2h) an amalgam
Volume of cylinder = πr2h
SA of cylinder = 2πr2 + 2πrh
Area of circle = πr2
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Continuum of Amalgam Formulae
•Literature Review
•Research design
•Findings
• Surface area
• Continuum of
amalgam formulae
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Students used a myriad of formulae to find
volume. I sorted these based on their surface
area and volume elements.
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Students’ volume formulae are useful for
diagnosing their ideas/conceptions about
volume
Correct volume
Incorrect
volume, no SA
element
SA and volume
elements
Surface area
r2h
(1/3)r2h
(1/2)r2h
(4/3)r2h
rh
(1/2)rh
h*d*r
2r2h
2πrh
2r + rh
r2 + 2d
2r2 + 2rh
2r2h +2rh
2r2h +dh
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•Literature Review
•Research design
•Findings
• Surface area
• Continuum of
amalgam formulae
Answers to Research Questions
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(1) How successful are calculus students at computational volume problems?
 Somewhat successful; shape-dependent
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(2) Do calculus students find surface area when directed to find volume?
 Yes, approximately at the same rate as elementary school students
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(3) If calculus students find surface area when directed to find volume, what
is the thinking that leads them to do so?
 Reason 1: Some students think that adding the areas of the faces of an
object finds the measure of the object’s volume.
 Reason 2: Some students understand the difference between area and
volume, but mix the formulae together.
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Difficulties calculus students have with volume-finding are similar to
difficulties elementary school students have.
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Viewing student formulae on a continuum of 2D, and 3D elements may help
us diagnose their ideas about volume (and then design appropriate
instruction)
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Instructional Implications and
Suggestions for Further Research
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Instructional implications: create opportunities for students…
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To revisit and strengthen their understanding of prerequisite
topics in conjunction with the study of new content
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To deepen their understanding of the connections between the
dimensions of objects, the formulae for measurements of them,
and the units associated with various spatial measures
Suggestions for further research:
Does the Surface Area -Volume Amalgam interact with these
students’ understanding of calculus topics that make use of
these concepts?
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Optimization
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Volumes of Solids of Revolution
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Works Cited
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Battista, M.T., & Clements, D.H. (1998). Students’ understandings of three-dimensional cube
arrays: Findings from a research and curriculum development project. n R. Lehrer & D. Chazan
(Eds.), Designing learning environments for developing understanding of geometry and space
(pp. 227-248). Mahwah, NJ: Erlbaum.
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Lehrer, R. (2003). Developing understanding of measurement. In J. Kilpatrick, W. G. Martin, & D.
E. Schifter (Eds.), A Research Companion to Principles and Standards for School Mathematics.
Reston, VA: National Council of Teachers of Mathematics
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Monk, G.S. (1989). A framework for describing student understanding of functions. Paper
presented at the Annual Meeting of the American Educational Research Association, San
Francisco, CA.
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Oehertman, M., Carlson, M., & Thompson, P. W. (2008). Foundational Reasoning Abilities that
Promote Coherence in Students’ Function Understanding. In M. Carlson & C. Rasmussen (Eds.).
Making the connection: Research and teaching in undergraduate mathematics education. (pp.2741). MAA Notes #73. Mathematical Association of America: 2008.
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Orton, A. (1983a). Students’ understanding of differentiation. Educational Studies in
Mathematics, 15, 235-250.
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Siegler, R. (2003). Implications of cognitive science research for mathematics education. In
Kilpatrick, K., Marting, G., and Schifter, D. (Eds.) A Research Companion to Principles and
Standards for School Mathematics. P. 289-303. Reston, VA: National Council of Teachers of
Mathematics.
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Strauss, A. & Corbin, J. (1990). Basics of qualitative research: Grounded theory procedures and
techniques. Newbury Park, CA: Sage.
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For more information…
Email: [email protected]
Reading material: Calculus Students’ Understanding of Area and
Volume in Non-Calculus Contexts (masters thesis; Dec. 2011)