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Making proofs click: Classroom response systems in transition-to-proof courses


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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 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.

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Making proofs click: Classroom response systems in transition-to-proof courses

  1. 1. Making proofs click Classroom response systems in transition-to-proof coursesRobert Talbert, Grand Valley State University Image:
  2. 2. High school math Calculus 1 Calculus 2 TRANSITION TO PROOF ModernLinear algebra Geometry algebra
  3. 3. High school math Calculus 1 Calculus 2 TRANSITION TO PROOF ModernLinear algebra Geometry algebra
  4. 4. 7.) Question 7 7.) Question 7 Responses (percent) (count)Answer 1 4% 1Answer 2 Answer 1 48% 12Answer 3 Answer 2 48% 12 Answer 3 Totals 100% 25
  5. 5. Before class: Peer Instruction Information transfer = Active student engagement Instructor Multiple choice Individual thinkingminilecture 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 youre right 2nd vote differences? 1st vote discussion (2 min) NO Instructor debrief Repeat with the via minilecture next essential (< 5 min) concept
  6. 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. 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
  8. 8. /(0&(1%2345$67(.+&8-97:(.;+<% =&*+*+%09*>>>?@9&*+1(+%A 35& 7& 6&!"#$%&()*+,-%.+* <& ;& :& 9& 8& 4& 3& 5& !""#$%&)*+,& !""#$%&&.*/& !""#$%&& !""#$%&(& (-& )*+,&(-& 0%12%"3& 4& 5& 6& 7&
  9. 9. /(0&(1%2345$67(.+&8-97:(.;+<% =&*+*+%09*>>>?@%7(.-1(+%A 38& 35& 37&!"#$%&()*+,-%.+* 36& 34& 8& 5& 7& 6& 4& !""#$%&)*+,& !""#$%&&.*/& !""#$%&& !""#$%&(& (-& )*+,&(-& 0%12%"3& 4& 4& 3& 35& (2 students lose attendance credit for the day...)
  10. 10. Peer Instruction: A User’s Manual “Confessions of a converted lecturer”
  11. 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
  12. 12. Slides: Twitter: @RobertTalbert Blog: Email: talbertr@gvsu.edublognetwork/castingoutnines THANK YOU