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[Presentation given at the AMS/MAA Joint Meetings, Boston, MA on 1/4/2012.]Transition-to-proof courses, designed to prepare students from calculus and other lower-level courses for the methodologyof upper-level mathematics, are often dicult for students in several ways. Students who are used to purely algorithmicapproaches to mathematics experience culture shock at the more open-ended and uncertain mathematical world that suchcourses introduce. The elements of communication and writing often play a much larger role in these courses than inearlier ones. And generally, these courses signal a major change in the way students conceive of the study of mathematics,which can make further study of mathematics stressfully forbidding.Technology can help students make this transition. In particular, classroom response systems, or "clickers", openup the classroom to a range of pedagogical approaches that can help students learn mathematical abstraction andgood mathematical writing practice. In this talk, we discuss some instances of clicker-enabled pedagogy in the author'sCommunicating in Mathematics class, including peer instruction, and peer review of writing samples.
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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
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