Understanding and Constructing Proofs: Two Design Experiments Annie Selden John Selden New Mexico State University Department of Mathematical Sciences
[email protected] [email protected] March 26, 2010 University of Toledo Colloquium 1
This work was done in collaboration with our other team members Kerry McKee Milos Savic Funded in part by Educational Advancement Foundation
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Prologue • What is qualitative research in mathematics education? • What is a theoretical framework? • What is a design experiment?
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Cobb, et al. (2003) describe five features of design experiments. They are: 1. Develop local theories about learning. 2. Highly interventionist – researchers do more than observe. 3. Intended to foster learning. 4. Have an iterative design – develop and test conjectures. 5. Speak to practitioners’ problems.
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Outline of our talk • The first design experiment • Aspects and types of proofs • Our view of behavioral schemas or habits of mind • Examples of student work
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• • • • • •
The second design experiment Theoretical perspective Supplement description Surprising student difficulties A supplement proof Effect on the students
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First Design Experiment • A one-semester, 3-credit, special topics course for prospective and beginning math graduate students. • Meets 2 times/week for 75 min. • Purpose is to teach proof construction • Significant modification of the R. L. Moore Method (Mahavier, 1999)
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• Notes with statements of theorems, definitions, requests for examples, but no proofs and only minimal explanations. • No lectures. • Students work outside of class and present their proofs at the blackboard. • We read and check each proof, “thinking aloud” so students can see what we are checking. 8
• We offer, sometimes extensive, criticism and advice. • Halfway through the course, we have students validate each other’s proofs. • Course has practical value because professors assess students’ understanding by asking them to prove theorems. • The course covers sets, functions, a little real analysis, abstract algebra, and topology. • We are describing our 5th (Fall 2009) iteration of the course. 9
• Everything is video recorded: both the classes and our planning sessions. • Field notes are taken. • This information is analyzed in planning sessions between class meetings in order to influence students’ learning trajectories. • Later, we often reanalyze the data for research purposes.
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Aspects of proofs We treat proof as having two parts or two aspects: • the formal-rhetorical part and • the problem-centered part.
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The formal-rhetorical part of a proof is the part that one can write based only on logic, definitions, and sometimes theorems, without recourse to conceptual understanding, intuition, or genuine problem solving. Students sometimes call this the “set-up.” We call the remainder of the proof the problemcentered part and it does require conceptual understanding and genuine problem solving. (Selden & Selden, 2009) 12
Example of the “set up” Theorem. The identify function on the reals is continuous. Proof: Let f be the identity function. Let a be a real number. Suppose e>0. Let d = ___ . Note d>0. Let x be a real number. Suppose |x – a|0 there is a positive integer N such that for all n>N we have |an-A|0. As {an} converges there exists an Na such that for all i> Na,|ai-P| Nb,|bj-P|N. Case 1: Suppose n is even. Then |cn-P|=|an-P|
