Making proofs clickClassroom response systems in transition-to-proof courses

Robert Talbert, Grand Valley State UniversityImage: http://www.flickr.com/photos/moto/

Calculus 1

Calculus 2

TRANSITION TO PROOF

Linear algebra Modern algebra Geometry

High school math

Calculus 1

Calculus 2

TRANSITION TO PROOF

Linear algebra Modern algebra Geometry

High school math

7.) Question 7Responses

(percent) (count)Answer 1 4% 1Answer 2 48% 12Answer 3 48% 12

Totals 100% 25

7.) Question 7Responses

(percent) (count)Answer 1 4% 1Answer 2 48% 12Answer 3 48% 12

Totals 100% 25

Peer InstructionBefore class: Information

transfer

Multiple choice question on

essential concept

Individual thinking w/ no interaction

(1 min)

Significant differences?

Yes 1st vote

NO

Instructor minilecture to set up

concept(5-8 min)

VOTE

Pair off, convince others you're right

(2 min)

Instructor-facilitated discussion

Yes2nd vote

Instructor debrief via minilecture

(< 5 min)

Repeat with the next essential

concept

= Active student engagement

Using peer instruction to teach proof by contradiction

Essential concepts for the lesson:

If the negation of a statement is false, the statement is true.

The negation of a conditional statement is a disjunction.

The beginning of a proof by contradiction is to assume the negation of the statement to prove.

The end of a proof by contradiction is to arrive at an absurdity, thereby showing the negation of the original statement is false.

To prove P → Q by contradiction, the first step is

(A) Assume P

(B) Assume Q

(C) Assume ¬Q

(D) Assume P ∧ ¬Q

(E) I don’t know

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(2 students lose attendance credit for the day...)

“Confessions of a converted lecturer”Peer Instruction: A

User’s Manual

Clicker-aided peer review of writing samples

Criterion Descriptors

Mathematical correctness

Correct calculations; correct statement & application of definitions

Logical soundness All steps shown and justified; conclusions follow standard rules of logic and are correct; counterexamples are valid

Written clarityAssumptions are explicit and clear; argument has a

discernible flow; correct grammar and spelling used; writing guidelines followed

Write in groupsRead/rate

individually w/o interaction

Vote, discuss, suggest

improvements

Slides: http://slidesha.re/uO30Xz

Twitter: @RobertTalbert

Blog: http://chronicle.com/

blognetwork/castingoutnines

Email: talbertr@gvsu.edu

THANK YOU