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 !