[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 methodology of upper-level mathematics, are often dicult for students in several ways. Students who are used to purely algorithmic approaches to mathematics experience culture shock at the more open-ended and uncertain mathematical world that such courses introduce. The elements of communication and writing often play a much larger role in these courses than in earlier 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", open up the classroom to a range of pedagogical approaches that can help students learn mathematical abstraction and good mathematical writing practice. In this talk, we discuss some instances of clicker-enabled pedagogy in the author's Communicating in Mathematics class, including peer instruction, and peer review of writing samples.

1. Making proofs click
Classroom response systems
in transition-to-proof courses
Robert Talbert, Grand Valley State University
Image: http://www.flickr.com/photos/moto/

2. High school
math
Calculus 1
Calculus 2
TRANSITION
TO PROOF
Modern
Linear algebra Geometry
algebra

3. High school
math
Calculus 1
Calculus 2
TRANSITION
TO PROOF
Modern
Linear algebra Geometry
algebra

4. 7.) Question 7
7.) Question 7 Responses
(percent) (count)
Answer 1 4% 1
Answer 2 Answer 1 48% 12
Answer 3 Answer 2 48% 12
Answer 3 Totals 100% 25

5. Before class:
Peer Instruction
Information
transfer
= Active student engagement
Instructor
Multiple choice Individual thinking
minilecture to set up
question on w/ no interaction
concept
essential concept (1 min)
(5-8 min)
VOTE
Instructor- Pair off, convince
Yes Significant Yes
facilitated others you're right
2nd vote differences? 1st vote
discussion (2 min)
NO
Instructor debrief Repeat with the
via minilecture next essential
(< 5 min) concept

6. 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.

7. 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...)

10. Peer Instruction: A
User’s Manual “Confessions of a converted lecturer”

11. Clicker-aided peer review of writing samples
Criterion Descriptors
Mathematical Correct calculations; correct statement & application of
definitions
correctness
All steps shown and justified; conclusions follow standard
Logical soundness rules of logic and are correct; counterexamples are valid
Assumptions are explicit and clear; argument has a
Written clarity discernible flow; correct grammar and spelling used;
writing guidelines followed
Read/rate Vote, discuss,
Write in groups individually w/o suggest
interaction improvements

17. Slides: Twitter:
http://slidesha.re/uO30Xz @RobertTalbert
Blog:
Email:
http://chronicle.com/
talbertr@gvsu.edu
blognetwork/castingoutnines
THANK
YOU