Comparing Student Use of Mathematical and Physical Vector Representations

1
Comparing Student Use of Mathematical and
Physical Vector Representations

• Joel Van Deventer
• MST Thesis Defense
• June 5th, 2008

2
Introduction
Physics Education Research Laboratory
3
Introduction
Physics Education Research Laboratory

• A number of studies have been done on student
  understanding of vectors.
• Various studies have also investigated student
  performance with physics concepts across
  different contexts.
• Our research combines both to investigate
  student performance on isomorphic math and
  physics vector tasks.

4
Current Research Focus
Physics Education Research Laboratory

• How does student performance on isomorphic math
  and physics vector tasks compare?
• What tools are students using in math and
  physics contexts to accomplish vector tasks?
• What tools are students missing in order to be
  successful?

5
Previous Research
Physics Education Research Laboratory

• Student understanding of vectors
• Many students have been shown to
• Treat vectors as scalars during vector
  operations1,2
• Confuse vector addition and subtraction
  algorithms1,3,4
• Treat vectors as fixed objects in space during
  vector operations3

1 Flores and Kanim, Am. J. Phys. 72 (4), 460-468
(2004) 2 Gagatisis and Demetriadou, Inter. J.
Math Educ. Sci. Technol., 32 (1), 105-125
(2001) 3 Nguyen and Meltzer, Am. J. Phys. 71 (6),
630-638, (2003). 4 Knight, Phys. Teach. 33, 74-78
(1995).
6
Previous Research
Physics Education Research Laboratory

• Context Sensitivity
• Results from many investigations vary1
• The extent to which student responses change
  depends greatly on how the contexts differ
• Shifts in correct student responses between two
  contexts have ranged from less than 1 to 69 in
  various studies1
• These variations may be due in part by context
  sensitivity of knowledge in general, which is
  described by many cognitive frameworks2

1 A review of the research on context sensitivity
and additional findings are given by Stewart,
Griffin, and Stewart, Phys. Rev. ST Phys. Educ.
Res. 3, 010102 (2007) 2 Many articles have been
published pertaining to the context dependence of
knowledge please see Edward F. Redish, A
theoretical Framework for Physics Education
research Modeling Student Thinking, The
Proceedings of the Varenna Summer School, "Enrico
Fermi" Course CLVI, (Italian
Physical Society, 2004)
7
Methodology
Math
Physics
End of Semester
Physics Education Research Laboratory
Free Response Tasks (FR)

• Math questions created Then isomorphic physics
  questions.

Instruction

• Topics on MC Quizzes
• Graphical vector addition and subtraction (1D and
  2D)
• Algebraic expressions for vector magnitudes
• Vector components
• Dot and cross products

Math
Physics
Beginning of Semester

• Students self-selected from a calculus-based
  introductory mechanics course
• Interviewed after all instruction on kinematics
  and dynamics
• Students were given and completed one question
  at a time all math questions first, followed by
  the physics.
• Asked to think aloud

• Same course, only accelerated pace
• Pre and post-test data collected
• Each student took the math version first,
  followed by the physics version

½
½

• Same introductory mechanics course
• Quizzes were randomly distributed students
  only received either the math or physics
  version.

11 Interviews (Int. sample)
270 Students (Fall sample)
Physics
Math
Physics
Math

• Questions revised and additional questions added
  
• Two isomorphic multiple choice quizzes created
  from FR tasks
• Responses from interviews used to create
  distracters in MC.

All
Multiple Choice Quizzes (MC) Math/Physics
Revised Multiple Choice Quizzes (MCR)
30 Students (Summer sample)
Tasks influenced by Nguyen and Meltzer, Am. J.
Phys. 71 (6), 630-638, (2003).
8
Methodology Analysis
Physics Education Research Laboratory

• Comparisons of
• Overall Performance on math and physics vector
  quizzes
• Individual question performance
• Individual question response distributions

• Illustrate consistency of student responses
  between quiz versions

9
Methodology
Fall Sample
Physics Education Research Laboratory
(N 270)
Different Students
Pre-lecture (N 200)
Post-lecture (N 70)
Lecture on vectors and vector operations in a
pure math context
t
1st week of classes
Semester Starts
10
Analysis Math and Physics Vector Quiz
Comparisons
Fall Sample Overall Performance
Physics Education Research Laboratory
p 0.002
p 0.044
N105
N99
N34
N33
N105
N108
11
Analysis Individual Question Context Dependence
Fall Sample
Physics Education Research Laboratory
12
Analysis
Two Dimensional Vector Addition
Math
Physics
13
Analysis
Two-Dimensional Vector Addition
Physics Education Research Laboratory
Fall Sample Performance
Performance is dependent on question context for
the post-lecture sections. (Chi-square for
independence p0.02 )
N105
N99
N34
N33
N108
N105
14
Analysis
Two-Dimensional Vector Addition
Common incorrect responses
Math
Physics
Pre-lecture
Student responses are dependent on question
context (Chi-squared for independence, plt0.001)
44
43
40
20
20
15
Analysis
Two-Dimensional Vector Addition
Math
Physics
Common incorrect responses
Post-lecture
Student responses are dependent on question
context (Chi-squared for independence, p
0.045)
62
33
16
Analysis
Two-Dimensional Vector Addition
Math
Physics
Common incorrect responses
End of Semester
Student responses are independent of question
context (Chi-squared for independence, p gt 0.05)
62
59
17
Analysis
Two-Dimensional Vector Addition (Summer Sample
N30)
Physics Education Research Laboratory
18
Analysis
Two-Dimensional Vector Addition (Summer Sample)
Physics Education Research Laboratory
19
Analysis
Two Dimensional Vector Subtraction
Math
Physics
20
Analysis
Two Dimensional Vector Subtraction
Physics Education Research Laboratory
Fall Sample Performance
Student performance is dependent on question
context at the end of semester. (Chi-square
for independence p0.011 )
N105
N99
N34
N33
N108
N105
21
Analysis
Two Dimensional Vector Subtraction
Math
Physics
Common incorrect responses
Pre-lecture
lt10
lt10
15
15
Student responses are dependent on question
context (Chi-squared for independence p0.019)
54
40
15
10
lt10
22
Analysis
Two Dimensional Vector Subtraction
Math
Physics
Common incorrect responses
Post-lecture
Student responses are independent of question
context (Chi-squared for independence pgt0.05)
56
39
10
15
10
23
Analysis
Two Dimensional Vector Subtraction
Math
Physics
Common incorrect responses
End of Semester
Student responses are dependent on question
context (Chi-squared for independence plt0.001)
56
39
17
10
15
24
Analysis
Two Dimensional Vector Subtraction (Summer Sample)
25
Analysis
Two Dimensional Vector Subtraction (Summer Sample)
26
Other Findings
Physics Education Research Laboratory

• Students performed well with identifying
• x and y components
• algebraic expressions for vector magnitudes
• Students performed poorly on dot and cross
  product questions
• Performance lt 30 across the board

27
Conclusions
Physics Education Research Laboratory

• How does student performance on the isomorphic
  math and physics vector tasks compare?
• Generalizations
• Some students sometimes treat some isomorphic
  math and physics vector tasks differently.
• No general trends were found between individual
  questions.
• student performance
• student response distributions
• Consistent with previous research

28
Conclusions
Physics Education Research Laboratory

• How does student performance on the isomorphic
  math and physics vector tasks compare?
• More specifically
• Overall student performance
• similar before instruction
• different after an initial lecture on vectors in
  a math context
• different after a semester of instruction
• It seems that many students do not have coherent
  sets of ideas concerning vectors and vector
  manipulation, and apply their ideas
  inconsistently across different vector tasks and
  contexts.

29
Conclusions
Physics Education Research Laboratory

• What tools are students using in math and physics
  contexts to accomplish vector tasks?
• The majority of students seem to know the steps
  of vector addition and subtraction algorithms
• Many misapply these algorithms when solving
  vector tasks (e.g., using addition in a
  subtraction problem)
• Consistent with previous research
• What tools are students missing in order to be
  successful?
• Almost all students are missing appropriate
  tools to evaluate vector products, especially
  with regard to the directionality of these
  products.

30
Implications

Physics Education Research Laboratory

• We should not assume students can quickly learn
  to use vectors and apply them in a physics
  context.
• It seems more explicit instruction on vectors is
  needed in our introductory courses, however it is
  unclear
• what type of instructional approaches are needed
• where they should be placed in the curriculum
• in what context they should be taught

31
Further Research

Physics Education Research Laboratory

• Performance on the math vector quiz after initial
  instruction seems to predict the upper bound of
  performance on the physics vector quiz after a
  semester of instruction.

32
Further Research

Physics Education Research Laboratory

• Performance on the math vector quiz after initial
  instruction seems to predict the upper bound of
  performance on the physics vector quiz after a
  semester of instruction.

33
Acknowledgements
Physics Education Research Laboratory

• Advisors
• Michael Wittmann
• John Thompson
• Eisso Atzema
• Members of UMaine PERL
• Course instructors and coordinators