1

Assessing College Students’Retention and Transfer

from Calculus to Physics

Lili Cui

Physics Education Seminar

Monday, Feb 13, 2006

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Transfer of Learning

Transfer is often defined as the ability to apply what has been learned in one

context to a new context 1

1 Byrnes (1996)

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MotivationStudents need to apply what they have learned in one class to another class.

Students must be prepared to apply what they have learned in school to the real world.

Transfer = Problem solving

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Various TransferStudent’s World

‘Real’World

School

Other Courses

Physics Courses

Other Physics Courses

One Course

Classroom

Exam

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Research QuestionsTo what extent do students retain and transfer their calculus knowledge when solving problems in introductory physics?

What difficulties pertaining to the transfer of calculus do students have while solving these problems?

What strategies may help students overcome these difficulties?

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WORKING MEMORY

READ-OUTFILTER

LONG TERM

MEMORY

Read-outProblem Info.

Priming

Information in a physics problem

What type of knowledge?Self-made ORFrom ‘authority.’

External Inputs

Asso

ciat

e

ActivatedPrior Knowledge

Activ

ate

PriorKnowledge

ActivatedEpistemic

Mode

Activ

ate

Epistemic Mode

Control

Control

Model of Transfer 2

Control

2 Rebello, Zollman, et. al. (2005)

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Model of Transfer : Focusing inTransfer is the creation of associations between read-out information & prior knowledge

The association is controlled by other factors e.g. learners’epistemology, motivation etc.

Read-outInformation

PriorKnowledge

Association? Control

3 Redish (2004)

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Two Kinds of Associations

Assigning the value of a problem variable to a knowledge element.

e.g. Line of charge extends from x=0 to x=5

Associations between two different knowledge elements.

e.g. Integral of Electric field is the Electric potential.

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Theoretical Framework (1 of 3)

Two kinds of problem solving

‘Horizontal’Applying a pre-constructed schema to a problem.

Associations between read-out problem variables & elements of schema.

‘Vertical’Constructing a new schema to solve a problem.

Association between knowledge elements to create schema.

schema

Problem variables

New knowledge elements incorporated in schema, others

are discarded

A “schema” is a mental model for

solving a problem.

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Theoretical Framework (2 of 3)

‘Horizontal’ Transfer

‘Vertical’Transfer

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Theoretical Framework (3 of 3)

Focus on “Innovation”9Focus on “Efficiency”9

Choosing, using and constructing multiple internal representations6

Limited internal representations in different contexts6

Asses: ‘Prep. for Future Learning’7Asses: ‘Sequestered Prob. Solv.’7

Involves Inductive reasoning: ‘Model development’10

Involves deductive reasoning: ‘Model deployment’10

Uses “interpretive” knowledge8Uses “applicative” knowledge8

Ill-structured, non-traditional probs6Structured, traditional problems6

“High Road”4, “Class A”5 Transfer“Low Road”4, “Class C”5 Transfer

‘‘VerticalVertical’’ Problem SolvingProblem Solving‘‘HorizontalHorizontal’’ Problem SolvingProblem Solving

44 Salomon & Perkins (1989)Salomon & Perkins (1989) 55 diSessa & Wagner (2005)diSessa & Wagner (2005) 66 JonassenJonassen (2003)(2003)

77 Bransford & Schwartz (1999)Bransford & Schwartz (1999)

99 Schwartz, Bransford & Sears (2005)Schwartz, Bransford & Sears (2005) 1010 Hestenes (1987)Hestenes (1987)

88 Broudy (1977)Broudy (1977)

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Research Questions – ReframedTo what extent do students retain and transfer their calculus knowledge while problem solvingproblem solving in introductory physics?

What difficulties pertaining to the transfer of calculus do students have while problem problem solvingsolving?

What strategies may help students overcome these difficulties?

Have students retained their calculus Have students retained their calculus schema to solve calculus problems?schema to solve calculus problems?

Can they associate their physics problem Can they associate their physics problem variables with their calculus schema? variables with their calculus schema?

Can students appropriately activate their Can students appropriately activate their calculus schemas in physics problems?calculus schemas in physics problems?

Can students deconstruct and Can students deconstruct and reconstruct their schemas to solve a reconstruct their schemas to solve a physics problem? physics problem?

PHASE I: PHASE I: ‘‘HorizontalHorizontal’’ TransferTransfer

PHASE II: PHASE II: ‘‘VerticalVertical’’ TransferTransfer

PHASE III:PHASE III:Instructional StrategiesInstructional Strategies

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Research Plan

‘Horizontal’ Transfer

‘Vertical’ Transfer

Study I-1: Quantitative: Exam problems

Study I-2: Qualitative: Textbook-like problems

Study II-1: Qualitative: ‘Contrasting Cases’ 9

Study II-2: Qualitative: ‘Jeopardy’ Problems 11

Phase I

Phase II

9 Schwartz, Bransford & Sears (2005)

11 Van Heuvelen & Maloney (1999)

Phase IIIInstructor Interviews

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Participants

Students2nd semester, calculus-based physics

Electricity and Magnetism

Faculty, Instructors and TAs Physics

Mathematics

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Phase I: Study I-1 Quantitative Students’ Exam Performance

Calculus-based physics studentsN=147 for Fall 2004

N=269 for Spring 2005

Three exams were collected each semester

Develop individual rubric to measure physics correctand calculus correct in every calculus-based physics problem

Calculate the Pearson Correlation between students’calculus and physics performance

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Example of Exam and According Rubric

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SummaryPhase I: Study I-1

Statistically significant correlations between math and

physics performance

Possibility of ‘horizontalhorizontal’ transfer from calculus to physics

Interview to further examine‘horizontalhorizontal’’ transfer

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Phase I: Study I-2 Qualitative Interview, Fall 2004

Participants8 students, volunteer, male,

Most second year in college

Major in ME

Two sessions

For each session:About one hour long

Solve two physics problems

Solve isomorphic calculus problems

General questions about calculus background and application of their calculus knowledge in physics

1) E field caused by a half-circle charge distribution

2) Electric potential caused by changing E field

3) B field caused by a non-constant current distribution

4) Induced current caused by moving of the loop in a changing magnetic field

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Results : Study I-2 (1 of 2)

Self-confidence in calculus knowledge retention

Realization that calculus is required in physicsCorrelation Transfer

Is the knowledge learned in calculus class enough for physics class?

Yes (7 of 8)

No (1 of 8)

“have done it so many times…”“remember well…”,

“just easy integrals…”, “using it all the time…”

“Not hard…”, “have not come across many situations that I have no idea what the math means over there…”

“because it would teach you the basic mathematics, but at some point, I need

them to teach me the different aspects as what’s going on here (physics question)…

although I am satisfied with my math, I think it is not enough to help me with

physics…”

“Use a lot of calculus in physics, more than use physics in calculus”“The math is kind of foundation of physics, do not understand math, you can not do physics”“Physics talks why to solve it, math talks how to solve it. You apply the formula to physics”

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Lack of confidence in setting-up physics problems

Added question: Without calculus knowledge, it is possible to set up the physics problem?

Yes (2 of 5)

Do not know (1 of 5)

No (2 of 5)

Students’ self-reflections are consistent with our observations.

Role of calculus???

Results : Study I-2 (2 of 2)

“formula are all involved calculus, if I do not know, I will not understand the meaning of

physics at all…”“Although the set up part is basically physics, you still need certain math. Like the 20% is

math in the set up process …but you could not know what to do…”

“I am not confident if I set up the problem right or wrong…”

“so many numbers and constants to taking account, I get confused, I lose objective of

what I am actually looking for… ““as soon as I set it up, there is no problem”

“can set it up the relations…”“you do not need to do real calculation. So set it up is usually a physics thing. You can still

understand qualitatively…”

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Phase I: Conclusions‘Horizontal’ Transfer

Students are able to retain their calculus schema.

Students have difficulty associating their physics problem variables into their calculus schema.

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Phase II: Study II-1Interview, Spring 2005

ParticipantsFive males, three females

Various majors

For each session (two sessions total):Similar format as Study I-2

Do not solve pure calculus problem

Contrast use of “integration” vs. “summation”

Focus on exploring the origin of difficulties

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Results : Study II-1 (1 of 4)

Consistent with Study I-2 results

When to use integration in physics problems When problems were similar to the examples they had seen in the text (4 out of 7 interviewees)

Could not explain why use integration

Could not solve the contrasting cases

Use integration in terms of adding up the infinitesimally small elements (3 out of 7 interviewees )

“...you can not add up an infinite number…then I used integrate”

“Because it is the example in the book….i do not know

the reason”; “I just know there is integral involved, I

do not know why”

Have difficulty deciding when to activate certain schema

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Results : Study II-1 (2 of 4)

Difficulties when applying integralsDetermining the variable of integration

Deciding the limits of integration Students usually did not realize they used the wrong limits

Origin of difficultiesPhysics class (majority)

Calculus class

“I know how to integrate it, but it is just figuring out what to integrate,

that is the hard part…”; “…which is really general, but what is ds, what

should substitute to…” ; “These are all constants, I do not know what

should I integrate…”

“…not really to do with my math class. just what variable you put there, cause when I got something to integrate, I know how to integrate it, but it is just figuring out what to integrate, that is the hard part, getting to the part..”“probably from math, because the

concept of physics is pretty simple. Because you can see the concept, I

understand them well. …well, it is not physics is that hard, math is that hard, it

is putting them together is hard, it is writing a equation for what I

understanding is hard.”

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Results : Study II-1 (3 of 4)

Students prefer to use pre-derived algebraic relationship over calculus relationship

unaware of whenwhen to use integration

Is use of calculus in physics just “plug and chug”?Yes (6 of 8)

No (2 of 8)

riE

πμ2

0= enclosedidsE 0μ=∫

“more confidently use algebra expression to go straight rather than understand this (calculus)”

“ I have to understand or I will be confused”

Again, have difficulty deciding when to activate certain schema

“I do not need to understand it, just how to do it. And I was doing good this way in calculus”

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Results : Study II-1 (4 of 4)

To facilitate transfer from calculus to physics

Learning how to set-up physics problems

Focus on understanding

Course sequencing

More ‘word’ problems in calculus

“even in calculus, I had to understand why the

differentiation of s2 equal to 2s…”; “why integration and

differentiation works.”

“I do not think they need to go through all the integration

steps, but they need to show how to set

it up, show the different

varieties…so you know what is changing … “

“…in word problem, you need to think about what integral you want

to se up, so they can do that in calculus, that would be helpful, so when you go to physics, you are

learning new material, like electricity, but you already know

calculus.”

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Phase II : Study II-2Interview, Fall 2006Participants

11 males, 1 female

Volunteers: Selected to represent different exam performance

Covered the required concept 2-3 weeks ago

For each session (two sessions total)Similar format as previous interview

Solve four ‘Jeopardy’ problems

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Physics ‘Jeopardy’ Problems11Van Heuvelen & Maloney, (1999)

Work backwardEquation physical situation

Graph physical quantity

Why use ‘Jeopardy’ problems“Students can not use formula-centered, plug-and chug problem solving method, they must give meaning to symbols in the equation”11

“Help students to learn to translate between representations in a more robust manner”11

Look closely at how students understand the calculus-based equation in physics

Possible strategy to facilitate the transfer process

Difficult!

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Sample ‘Jeopardy’ Equation Problems

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

×××

⋅×× ∫ −

−−6

022

210229

)105(cos)105)(102()1099.8(2

π

θθm

dmmCCmN

( ) ( )

R

rdrrJR

π

πμ

2

20

0 ∫ ⋅

Construct an appropriate physical situation that is consistent with the following expressions.

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Sample ‘Jeopardy’ Graph ProblemsGiven the graph of E field, what is the electric potential at different points?

-25

-20

-15

-10

-5

0

5

10

15

20

25

0 1 2 3 4 5 6 7 8

x (m)

E (N

/C)

0

1

2

3

4

5

0 1 2 3 4 5 6

x (m)V

(KV)

Given the graph of potential, what is the E field in different regions?

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Results: Study II-2 (1 of 2)

While solving ‘Jeopardy’ equation problems …Most tried to use units to find the physical quantity.

Some figured out the geometry from variables of integration, others could not.

Most appear to use pattern matching: Cannot explain why.

Most believe ‘Jeopardy’ problems would help understanding.

“I take all the units and convert to what variable

that they look for”

“I look for pieces of terms that I recognize,…, they will

tell what kind of problem they are, I just tend the recognize forms, like

derivative…” ; “I do not know why those formula work, I just use them”

Have difficulty deconstructing and reconstructing their schema

“Just working things backward, you have to understand it better, because if you just start with everything given and plug in the formula, you might get better out it, you might understand it better, but this way

you will understand it really well, because you have to know where is everything

come from”

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Results: Study II-2 (2 of 2)

While solving ‘Jeopardy’ graph problems …

About half could recognize that…slope of graph ⇒ differentiation

area under curve ⇒ integration

Cannot explain why.“this is kind of thing that I have known so long and

I could not explain”

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Phase II: Conclusions

‘Vertical’ Transfer

Students have difficulty deciding when to activatewhen to activatetheir calculus schemas.

Have difficulty deconstructingdeconstructing their schemas and constructingconstructingnew schemas based on the problem scenario.

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Phase III : Faculty InterviewParticipants

Physics2 Faculty

4 Studio Instructors

Mathematics2 Faculty

2 Teaching Assistants

30 minutes interviewExpectation and outcomes of course

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Results: Phase III (1 of 4)Mathematics faculty

Experienced in teaching calculus

Expectation after ‘Calculus-I’ & ‘Calculus-II’How to do integration and differentiation (4 of 4)Some applications (find maximum/minimum) (2 of 4)

Satisfied with course outcomes

Aware calculus is used in other subjectsEspecially physicsLimited examples of applications

“…I do not have enough background to actually know where

they are generally used”

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Results: Phase III (2 of 4)Mathematics faculty

Limited use of “Word” Problems10% - 20%

Instructor own reflection

Reviewed past exams (from 2001-2005)

Interested to hear what physicists feel

“students told me that they even do not want to try…something I never understood myself, cause that is the problem that you encounter in everyday life, but for some reason, translate a word problem into a mathematical problem is the big step”

“they do not do well on the word problems, so, as far as on exams, I mean I was trying to put some on them, but I do not make the exam too hard”

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Results: Phase III (3 of 4)Physics faculty

Experienced in teaching physics

What do you expect from the calculus class?Basic calculus

Do simple mechanically differentiation and integrals

Conceptual understanding

Strategies to overcome students’ conceptual difficulties

Give many examples (time constraints)

Emphasize concepts

Use Visualizations

YES

NO

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Results: Phase III (4 of 4)Physics faculty

Suggestions for math department

More “word” problems to develop problem solving skills

Focus on conceptual understanding over mechanically doing calculus

“I would be happier if the mathematicians put more

emphasis on the theoretical basis of calculation, in terms of the exercises, more emphasis

on simple problems”

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Phase III: ConclusionsTo facilitate transfer from calculus to physics

MathematicsFocus more on understanding

More “word” problems

PhysicsMore step by step “setting-up”

Different problem representation, e.g. Jeopardy

AdministratorCourse sequence

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SummaryPhase I: Phase I: ‘‘HorizontalHorizontal’’ TransferTransfer

Students appear to retain their schema to solve calculus problems.Students have difficulty associating their physics problem variables into their calculus schema.

Phase II: Phase II: ‘‘VerticalVertical’’ TransferTransferStudents have difficulty deciding when to activate appropriate calculus schemas.Students have difficulty in deconstructing their schemas and constructing new ones based on the problem scenario.

Phase III: Instructional StrategiesPhase III: Instructional StrategiesMath : Focus on understanding, word problems.Physics : Focus on setting up, other problem types.

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Future PlansInvestigate transfer across other domains e.g.

Physics to Engineering

Other disciplines

Are issues similar or different?

Develop instructional strategies to facilitate students’ ‘horizontal’ and ‘vertical’ transfer from calculus to physics.

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Characteristics of Instructional Strategies

Balance ‘horizontal’ and ‘vertical’ transferFollow an ‘Optimal Adaptability Corridor’9

Adapt proven pedagogical strategies e.g.Small steps of Model Development followed by Model Deployment.10

Scaffolded learning in Zones of Proximal Development.12

Emphasize multiple representations‘Zone of Proximal

Development’ is the range of concepts that a student can learn with assistance of a

teacher

12 Vygotsky (1978)10 Hestenes (1987)

9 Schwartz, Bransford & Sears (2005)

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In Terms of Our Framework…

Horizontal

Ver

tica

l

What we currently try to do…What we shouldshould try to do…

Mathematics

Mathematics

Physics

Physics

ModelDevelopment

ModelDevelopment

Model Deployment

Model Deployment

‘‘Optimal

Optimal

Adaptability

Adaptability

Corridor

Corridor’’

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99 Schwartz, Bransford & Sears (2005)Schwartz, Bransford & Sears (2005)

Mathematics

Mostly ‘horizontal’

Scaffolded learning inZone of Proximal Dev.

Phy

sics

Mos

tly

‘ver

tica

l’

1010 Hestenes (1987)Hestenes (1987) 1212 VygotskyVygotsky (1978)(1978)

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Thank You!

Lili CuiPhysics Education Research Group

Department of PhysicsKansas State University

lili@phys.ksu.eduhttp://www.phys.ksu.edu/personal/lili