Post on 22-Nov-2015
description
Student Understanding of Definite Integrals Using Graphical
Representations
Rabindra R. Bajracharya
MST Candidate,
December 7, 2012
Committee Members
Dr. John R. Thompson (Advisor)
Dr. Natasha M. Speer
Dr. Michael C. Wittmann
Outstanding student award 2012
Internship at the Jax Laboratory
Student teaching at J. Bapst HS
Curriculum design & implementation in two HS
Publications in PERC (finalist) and RUME proceedings, MST thesis
TRUSE, GSG, PERLOC grants
8 poster presentations, 3 talks
AAPT, PERC, TRUSE, RiSE, conferences
PHY 107, 121, 122 TA experience
SMT courses, TA workshops
Acknowledgements
Dr. John R. Thompson (Advisor)
Dr. Michael C. Wittmann
Dr. Natasha M. Speer
RiSE Center Faculty
Physics Faculty
RiSE & Physics Staff
UMaine PERL Members
Physics & MST colleagues
Research participants
Funding sources
Special Thanks to
Professor Susan McKay
What is Definite Integral?
Area f(x1)x + f(x2)x + f(x3)x + f(x4)x + f(x5)x + f(x6)x + f(x7)x
b a x
f(xi)
f(x)
y
x
Riemann Sum
b a
f(xi)
f(x)
y
x
Riemann Sum
b a
f(xi)
f(x)
y
x
Definite Integral = Limit of Riemann Sum
b a
f(x)
y
x
Failure to recognize definite integral as limit of Riemann sum (Orton, 1983; Bezuidenhout et al., 2000; Sealey, 2006)
y
x
f(x)
b a
Math Education Research Findings
Confusion between derivative and integral (Eisenberg, 1992;Ghazali et al., 2005)
y
x
f(x)
b a
?
Math Education Research Findings
Difficulty with negative integral (overgeneralization of area) (Orton, 1983; Thompson, 1994; Bezuidenhout and Olivier, 2000)
f(x)
x o a b
y
Math Education Research Findings
Vel
oci
ty
Time
Kinematics Dynamics Electrodynamics
Electrostatics Thermodynamics Quantum Mechanics
Time
Forc
e
Time
Cu
rren
t
Position
Elec
tric
fie
ld
Volume
Pre
ssu
re
Position
||2
Definite Integral in Physics
Pre
ssu
re
Volume Vi Vf
Work done on the gas =
The cartoon is highly exaggerated.
Work in an Isothermal Process
Students have difficulties with physics concepts that involve definite integrals
Pre
ssu
re
Volume
Process #1
Process #2
Is Work for Process #1 greater than, less than,
or equal to that for Process #2? Explain.
McDermott et al., 1987 Beichner, 1994
Meltzer, 2004
Nguyen, 2011
(x) vs. x
(x)/A(x) vs. x
A(x) vs. x
(x).A(x) vs. x
Physics Education Research Findings
Is Work for Process #1 greater than, less than, or equal to that for Process #2? Explain.
Background: Research on P-V Diagrams
Meltzer, Am. J. Phys. (2004)
W = = Area under the curve
Pre
ssu
re
Volume
Process #1
Process #2
Intro. level: Loverude et al., 2002; Meltzer, 2004
Upper level (UMaine): Pollock et al., 2007
Common incorrect response
W1 = W2 Intro. level: > 25% Upper level (UMaine):
~ 50 %
Common incorrect
reasoning Same beginning and ending states, so works are same.
Interpretation Incorrectly assumed work as a function of state
Is Work for Process #1 greater than, less than, or equal to that for Process #2? Explain.
Background: Research on P-V Diagrams
Meltzer, Am. J. Phys. (2004)
W = = Area under the curve
Pre
ssu
re
Volume
Process #1
Process #2
Pre
ssu
re
Volume
Process #1
Process #2
Compare the works during two processes
Background
Compare the magnitudes of the integrals
Pollock et al., PERC Proceedings (2007)
z
y
Path 1
Path 2
Background: Physicsless Physics Question
> 25% students: Integrals are equal
Specific Difficulties1
Incorrect or inappropriate ideas
Flawed patterns of reasoning to specific questions
Research Question and Perspective
1Heron, Proc Enrico Fermi Summer School PER (2003)
How do students understand the aspects of definite integrals that are relevant
for the understanding of physics concepts?
Pre
ssu
re
Volume Vi Vf
y
x
f(x)
b a
Work =
Instrument Design
a b
y
x
f(x)
g(x)
Analogous math graph
z
y
Path 1
Path 2
Physicsless physics graph
b. Is the integral I2 positive, negative, zero or is there not enough information to decide? Explain.
a. Is the integral I1 positive, negative, zero or is there not enough information to decide? Explain.
Written Survey 1
c. Is the absolute value of the integral I1 greater than, less than or equal to the absolute value of the integral I2, or is there not enough information to decide? Explain.
g(x)
y
x a b
f(x)
Written Survey 1 Second semester calculus based introductory
physics (N = 97)
Multivariable calculus (N = 97)
Grounded Theory for Data Coding1
Examination, Comparison, Breaking down, etc.
Putting back together in new ways
Selecting the core category and filling in categories
Data Collection and Coding
1Strauss & Corbin, 1990
Categorization using Grounded Theory
Categorization using Grounded Theory
Categorization using Grounded Theory
x
y
b a
f(x)
Area under curve
Area above x-axis
Area accumulated under curve, etc.
Function is in first quadrant
Function is positive, etc.
Function increasing
Graph concave up
f(b) > f(a), etc.
Area under the curve
Position of the function
Shape of the curve
x
y
a b
f(x)
Student Reasoning Category
39% 34%
19% 22%
9% 14%
9% 4% 4%
16% 16%
2%
Qa Qb Qa Qb Qa Qb
No reasoning
Other
Shape
Position
Area
Survey 1: Sign Comparison Results
Positive Negative Zero/NEI
Student Response
g(x)
y
x a b
f(x)
42%
5%
4%
6%
4%
21%
5%
Greater than Less than Equal to NEI
Student Reasoning
No reasoning
Other
Endpoint
Position
Area
Compare the magnitudes
of and .
Survey 1: Magnitude Comparison Results
g(x)
y
x a b
f(x)
z
y
Path 1
Path 2
Compared to > 25% for Physicsless question
x
y
b a
f(x)
Area under curve
Area above x-axis
Area accumulated under curve, etc.
Function is in first quadrant
Function is positive, etc.
Area under the curve
Position of the function
Shape of the curve
x
y
a b
f(x)
Student Reasoning Category
Function increasing
Graph concave down
,,
etc.
x
y
b a
f(x)
Area under curve
Area above x-axis
Area accumulated under curve, etc.
Function is in first quadrant
Function is positive, etc.
Area under the curve
Position of the function
Shape of the curve
x
y
a b
f(x)
Student Reasoning Category
Incomplete Picture of Student Understanding of
Definite Integrals
Function increasing
Graph concave down
,,
etc.
Misuse of: Derivative Curvature [f] The FTC
Follow-up Interviews
g(y)
f(y)
b a y
z
Area
Position
Shape
Follo
w-u
p Q
ues
tio
ns
Semi-structured individual interviews 45-60 minutes N = 7 (NPhy = 4, NCalc = 3)
a b
g(y)
y
f(y) z
f(y)
y a b
c
z
Follow-up on Area Reasoning
Backward integration Integration of negative function
b a
g(y)
z
y
f(y)
z
f(y)
y b
o
a a b
f(y)
f(y) f(y)
g(y) g(y)
g(y) g(y)
Follow-up on Position Reasoning
Function in different quadrants
z f(y)
y a b
g(y)
b a
g(y)
y
f(y)
z
Follow-up on Shape Reasoning
Tipped down curves Changed concavity
Interview Results for Negative Integrals
f(y)
y a
b
c
z
Conflicts with the role of area Misuse of the Fundamental Theorem of Calc Use of physics to make sense of integrals
a b
g(y)
y
f(y) z
3 (out of 7) students - sign is positive
Conflicts with the role of area
If you counted this way [moving his hand from right to left] or you count this way [moving his hand from left to right across the diagram] and you keep the dx the same, you should find the same area, right? - Simon
If you wanted to find area, it would always be positive, whereas if one did only math, one could get a negative sign. - Freddie
a b
g(y)
y
f(y) z
f(y)
y a b
c
z
y f(x)
x a b
Survey 2
7% (NPhy = 80, NCalc = 33) of students determined the sign as positive based on area reasoning (area is always positive)
y
x
f(x)
g(x)
b a o
a
f(x)
y
b
g(x)
So, if this value [pointing to the endpoint of his curve] is larger than this value [pointing to the starting point], that should be a negative value. - Freddie
Misuse of the Fundamental Theorem
...and then this way [right-to-left] its going to be negative work because its compressing and so, like thats how I know which direction to go in is by like an intuitive knowledge of what I am doing with this integral. - Abby
Use of Physics to Reason About Negative Integrals
In order to get negative area it is not... conceptually, looking at like a plot of land, it would be an impossibility. However, we are looking at something like a voltage; voltages can very easily go negative - Freddie
a b
g(y)
y
f(y) z
f(y)
y a b
c
z
g(x)
y
x a b
f(x) z
y
Path 1
Path 2 Pre
ssu
re
Volume
Process #1
Process #2
Students have difficulties with the various aspects of integrals that are relevant to physics concepts (negative integrals in particular)
Difficulty with backward integration
Misuse of the Fundamental Theorem of Calculus
The use of physics contexts seems to help students make sense of definite integrals
Inconsistencies in representations across physics and mathematics can lead to difficulties
Conclusions
Emphasize consistency in notation and representations in calculus and physics
More emphasis on negative integrals
Address the notion of area
Put more emphasis on roles of function and direction of integration for sign
Implications for Instruction
Exploration of other physics contexts (kinematics, electrostatics, etc.)
Use of various representations such as algebraic, verbal, graphical, numerical, etc.
More detailed investigation of student understanding of the FTC
Implications for Future Research
Thank you !