Cognitive Development at the Middle-division Level

THE PROBLEM (TASK ANALYSIS)

THE PARADIGMS PROJECT

http://physics.oregonstate.edu/portfolioswiki

National Science Foundation•DUE-9653250, 0231194 •DUE-0088901, 0231032

Oregon State University•Department of Physics•College of Science•Academic Affairs

We would like to thank members of the Paradigms team, especially Len Cerny, Tevian Dray, Barbara Edwards, David McIntyre, Janet Tate, Drew Watson, and Emily van Zee.

The Paradigms in Physics Project at Oregon State University has reformed the entire upper-division curriculum for physics and engineering physics majors. This has involved both a rearrangement of content to better reflect the way professional physicists think about the field and also the use of a number of reform pedagogies that place responsibility for learning more firmly in the hands of the students. The junior year consists of short case studies of paradigmatic physical situations which span two or more traditional subdisciplines of physics. The courses are designed explicitly to help students gradually develop problem-solving skills We have developed many effective classroo activities that are documented on our wiki. Along the way we are also learning what it takes to design and implement large-scale modifications in curriculum and to institutionalize them.

Corinne A. ManogueElizabeth Gire

BEHAVIORISM – describes learning by focusing on the behaviors of students. Anything that a student does is described in terms of behaviors, including thinking and learning. Behaviorists do not consider abstract constructs (such as the mind) in their analyses.

COGNITIVISM – describes thinking by positing the existence of mental states that are manipulated during thinking. Cognitivists infer the structure of cognitive entities from experiments and observations of students.

SITUATIVISM – describes knowing by considering that the actions of students are affected by the context (social, cultural, physical) in which the students' perceive themselves to be. Situativists view knowing as determined by both the person and the context. Learning is identified by students' increasingly effective performance across situations, rather than by the accumulation of knowledge.

RESEARCH LENSES

ReferenceGreeno, J. G., Collins, A. M., & Resnick, L. B. (1996). Cognition and

learning. In D. Berliner and R. Calfee (Eds.), Handbook of Educational Psychology (pp. 15-41). New York: MacMillian.

ACTING OUT CHARGE DENSITIESStudents place themselves around the classroom to model various charge density distributions (linear, surface, and volume) while building their conceptual understanding of the idealizations involved in going between discrete and continuous representations of charge.

DRAWING EQUIPOTENTIAL SURFACES

Students are asked to draw equipotential surfaces on whiteboards for various charge distributions.

VISUALIZING POTENTIALS

Using a computer algebra system, students explore different ways of visualizing a scalar field in three dimensions.

STAR TREK

Using a Star Trek scenario as a premise, students discuss how to specify the distance between two objects (Captain Kirk and Mr. Spock)

POWER SERIESStudents use a computer algebra package to plot the first several terms of a power series expansionAnd visually compare their approximation with a plot of the function

MODES OF COGNITION

POTENTIAL OF A RING OF CHARGE

Students working in small groups calculate the electrostatic potential due to a ring of charge. Sums go to integrals.

PREPARATORY ACTIVITIES

FOLLOW-UP ACTIVITIES

POTENTIAL OF A POINT CHARGEStudents recall the formula for the potential due to a point charge. Class discussion focuses on strategies to choose the correct formula.

Many of these cognitive modes and icons were first introduced in:

ReferenceReinventing College Physics for Biologists: Explicating an Epistemological

Curriculum , E. F. Redish and D. Hammer, Am. J. Phys., 77, 629-642 (2009).

We have used the following icons to classify the tasks and problems according to a mode of cognition required to successfully complete that portion of the task or that fails when students have a particular problem.

recognizing patterns

fleshing out formulas

applying learned mathematics

choosing foothold ideas, choosing a principle/iconic formula

restricting the scope

applying a principle to a specific case

sense making

translating representations, harmonic reasoning

seeking coherence

shopping for ideas

probing and refining intuitions

playing the implications game

employing a safety net

PCK

QQ

(x,0,0)

D

WRAP-UP (WHOLE CLASS DISCUSSION)

2 2

0

1

4

Q QV x

x D x D

kqV

r

0

1

4

Q QV x

x D x D

1 1

0

11 1

4

Q D DV x

x x x

2

0

1 21

4

Q DV x

x x

0

1

4i

i i

qV r

r r

Two charges + Q and – Q are placed on a line at x=+D and x=-D, respectively. What is the fourth order approximation of the electrostatic potential, V, valid on the x-axis, for |x|>>D?

Start with an “iconic” equation – the potential due to a point charge.

Recognize that the superposition principle applies.

Recognize that the in the iconic equation is the distance between the source and observation points 'r r

r

Choose a coordinate system and draw a diagram.

Choose a coordinate label (x, 0, 0) for the point at which the potential will be evaluated.

Evaluate the distances in the denominator for this specific case.

Recognize from the geometry that the denominators should be expressed with absolute values, especially when x is negative.

Recognize that the denominators have something to do with known series.

Decide what to factor out to put the denominators in the form of “one plus something small and dimensionless”.

Implement known mathematics from a memorized power series.Simplify, group terms.

In the whole class wrap-up discussion, students:•practice presenting their ideas,•compare with examples from other parts of space,•compare with limiting examples, •explore symmetry.

Pedagogical Content Knowledge is the teacher knowledge associated with how the students interact with the content. Listed in this column are VERY common student problems that come up as students are working on the activity.

Most students do not know what to do with the absolute value signs, especially when x is negative. Many just drop the absolute values. This topic can be a source of rich class discussion during the whole-class wrap-up.

Believing that the only way to find a power series is using successive differentiations.

The failure to recognize that

Not recognizing that is a common power series.

Not knowing how to do the algebra to put in the form of

11x D

x D

1p

x

1p

x

1

x D

Students often claim not to know how to get started. Often, the difficulty lies somewhere in the process of translating an abstract, coordinate-independent, algebraic representation, through a geometric representation, to a coordinate-dependent, algebraic representation on which the students can “do math.”