Making proofs click Classroom response systems in transition-to-proof courses Robert Talbert, Grand Valley State University Image: http://www.flickr.com/photos/moto/
Dec 18, 2014
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: [email protected]
THANK YOU