Transcript STUDENT UNDERSTANDING OF AREA AND VOLUME IN SELECTED CALCULUS CONCEPTS Thesis research Fall 2009 to present STUDENT UNDERSTANDING OF VOLUME OF REVOLUTION Fall 2009 Fall 2009 Research  Research.

STUDENT UNDERSTANDING OF
AREA AND VOLUME IN SELECTED
CALCULUS CONCEPTS
Thesis research Fall 2009 to present
STUDENT UNDERSTANDING
OF VOLUME OF REVOLUTION
Fall 2009
Fall 2009 Research
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Research questions:
 How
do students think about a volume of revolution
problem as they are working on it?
 How does the thinking of students who understand
volume of revolution differ from the thinking of students
who don’t?
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Participants: four fall 2009 MAT 127 students
Methodology: audio-recorded task-based interview
To refresh your memory:
Emergent Categories
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Correct answer; visualized a slice but not final solid
Correct answer; visualized a slice and final solid
Correct answer; no visualization
Incorrect answer; visualized a slice but not final solid
Incorrect answer; visualized a slice and final solid
Due to the small number of subjects, the researcher thinks it possible
that a study with a greater sample size might yield a few more
categories, those categories being various permutations of the above.
Summary of Coded Data
Student Task 1
Category
Task 2
Category
Alex
Visualized shell
Correct answer
1
Visualized washer
Correct answer
1
Bobby
Visualized shell and solid
Correct answer
2
Visualized washer
and solid
Incorrect answer
5
Scott
Visualized wrong solid
Visualized disk (fine for his
solid)
Incorrect answer (the solid
he pictured had an infinite
volume)
2
No visualization
Memorized formula
Correct answer
3
Josh
Visualized shells
Incorrect answer
4
Visualized washer
and solid
Incorrect answer
5
Findings
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Students visualize a slice and/or the final solid
while working on a volume of revolution problem
 All
students visualized a slice
 Over a total of eight tasks, students visualized the solid
in four cases
 Cognitive variability (Siegler, 2003): using multiple
thinking strategies when solving problems of the same
type (p. 293)
 Some
other
students visualized the solid in one problem but not the
Continued
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Students generally were able to express volume as
a sum of an infinite number of infinitessimally thin
slices, but struggled with finding the correct radii to
use in integration.
The apparent difference between students who
answered the tasks correctly and students who did
not was the visualization of a slice, used to find the
radii, but not visualizing the final solid.
My findings made me wonder
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We know that students have difficulties reading
information from graphical representations; ie, what
we [mathematicians] see in a graph is different
from what they [students] see. Does this account for
the trouble in finding the radius?
Apart from the issue of what to integrate and why,
how do calculus students understand area and
volume? Does this affect their understanding of
calculus concepts using area and volume, such as
Riemann sums and volume of revolution?
CALCULUS STUDENTS’
UNDERSTANDING OF AREA AND
VOLUME
Current research (Fall 2010)
Discussion of Reading
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I had two questions while reading this paper.One: Does it state somewhere in this
paper what the definition of array is? I realize that this is out of a book of some
sort so maybe the definition is somewhere else within the book?
ARRAY – an order or an arrangement – like a Rubik’s cube
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Two: One of the categories they observed during their research that students feel
into was named Category D. Where "Students explicitly use the formula LxWxH
with no indication that they understand it in terms of layers." Can one make the
assumption that students don't understand them in terms of layers?
My guess is that the researchers probed the students, asking questions to reveal if
they saw L, W, H as layers or if they were just multiplying.
Battista, M. and Clements, D. (1998). Students' understanding of three-dimensional
cube arrays: findings from a research and curriculum development project.
Discussion of Reading
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I think that introducing 3-d cubes in an early age is very important because
it is a big part of the way of thinking...
I agree. I really liked the article and I also think it is important to introduce
3-D to younger kids. I think there are a lot of things that could be
introduced in lower grade levels.
Also, I can't help but think that spatial visualization issues that never get
addressed in lower levels must come back to haunt students later on in
organic chemistry where it's all about considering 2-D structures on a page
and making them 3-D mentally and manipulating them. . .
Battista, M. and Clements, D. (1998). Students' understanding of three-dimensional
cube arrays: findings from a research and curriculum development project.
Discussion of Reading
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I enjoyed reading this article and I am looking forward to the
class discussion to learn more about what pieces of this article
you took to bridge the gap from its focus on elementary
content to calculus content. What are the requirements to be
able to generalize claims from differing levels of content?
My reply: There is not much research (any?) about CALC student
understanding of area and volume! What a perfect idea for a
study!
Battista, M. and Clements, D. (1998). Students' understanding of three-dimensional
cube arrays: findings from a research and curriculum development project.
Rationale
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Anecdotal evidence from University of Maine
calculus professors suggests that calculus students do
not understand area and volume.
Example: the volume of a classroom
Perhaps if students do not understand the concept
of area and the concept of volume, they do not
understand the applications of integration that
involve area and volume
Mini Lit Review
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In order to understand area and volume, students must understand and be
able to represent the structure of space and the tools to measure it in some
way (Example from my reading)
Students often use length measures for measurement of other spatial
content (area, volume, angle)
Students do not tend to understand the units associated with various spatial
measures
Students need to understand making subdivisions; that those subdivisions
are equal units; and if they are, a count represents the measure
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.
Mini Lit Review: Area Measure
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Students treat length as a spacefilling attribute
Students will mix units freely (ie:
give them squares, triangles,
circles and ask them to find the
area of a large rectangle – they
will use all 3)
Students will fill a shape’s area
with the units that most resemble
the shape. (Not shown: grid)
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.
Mini Lit Review: Area
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Resemblance: students want the unit to look like/be a similar
shape to the form
To understand area, students must be able to restructure the
plane
Students have trouble seeing the area of a form to be a
composition of other forms
Students see square units as things to be counted, not as
subdivisions of a plane
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.
Mini Lit Review: Volume
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Students now have to be able to coordinate units of
measure in three dimensions
Students try to: count faces or subdivisions of faces;
count subdivision of layers and multiply by layers;
multiply area*height or area*area
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.
Researchable questions
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What do calculus students understand about area?
What do calculus students understand about volume?
How do calculus students explain the idea of a
Riemann sum?
How do calculus students explain the idea of finding
the volume of a solid by using a Volume of
Revolution?
Methodology
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Give written tasks to a couple (2-4) calc classes
Students can give permission to be contacted for
interviews
Analyze written tasks; make categories
Contact 1-2 students from each category for
interviews
Interviews will be over the written task + a couple
other problems (array of cubes; Riemann sum; VoR)
Try my tasks!
Find the area of the rectangle.
12 cm
4 CM
Find the area of the circle.
5 in
What does area mean?
What type of things can we use area to measure?
What unit is area expressed in?
Find the volume of the box.
Find the volume of the cylinder.
8 in
3 in
What does volume mean?
What type of things can we use volume to measure?
What unit is volume expressed in?
What does this computation tell you?
b
ò
a
f ( x) dx
I would like to use my time to talk about my tasks and
my researchable questions.
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What do calculus students understand about area?
What do calculus students understand about volume?
How do calculus students explain the idea of a
Riemann sum?
How do calculus students explain the idea of finding
the volume of a solid by using a Volume of
Revolution?