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The Trouble with 5 Examples SoCal-Nev Section MAA Meeting October 8, 2005. Jacqueline Dewar Loyola Marymount University. Presentation Outline. A Freshman Workshop Course Four Problems/Five Examples Year-long Investigation Students’ understanding of proof.
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The Trouble with 5 ExamplesSoCal-Nev Section MAA MeetingOctober 8, 2005 Jacqueline Dewar Loyola Marymount University
Presentation Outline • A Freshman Workshop Course • Four Problems/Five Examples • Year-long Investigation • Students’ understanding of proof
The MATH 190-191 Freshman Workshop Courses • Skills and attitudes for success • Reduce the dropout rate • Focus on • Problem solving • Mathematical discourse • Study skills, careers, mathematical discoveries • Create a community of scholars
Prime Generating Quadratic Is it true that for every natural number n, is prime?
Observed pattern: If 4 divides n, then n! ends in zeros. Counterexample: 24! ends in 4 not 5 zeros.
Where do the zeros come from? From the factors of 10, so count the factors of 5. There are Well almost…
Fermat Numbers • Fermat conjectures (1650) Fn is prime for every nonnegative integer. • Euler (1732) shows F5 is composite. • Eisenstein (1844) proposes infinitely many Fermat primes. • Today’s conjecture: No more Fermat primes. =
The Trouble with 5 Examples Nonstandard problems give students more opportunities to show just how often 5 examples convinces them.
Year-long Investigation • What is the progression of students’ understanding of proof? • What in our curriculum moves them forward?
Evidence gathered first • Survey of majors and faculty
Respond from Strongly disagree to Strongly agree: If I see 5 examples where a formula holds, then I am convinced that formula is true.
Faculty explanation ‘Convinced’ does not mean ‘I am certain’… …whenever I am testing a conjecture, if it works for about 5 cases, then I try to prove that it’s true
More evidence gathered • Survey of majors and faculty • “Think-aloud” on proof - 12 majors • Same “Proof-aloud” with faculty expert • Focus group with 5 of the 12 majors • Interviews with MATH 191 students
Proof-Aloud Protocol Asked Students to: • Investigate a statement (is it true or false?) • State how confident, what would increase it • Generate and write down a proof • Evaluate 4 sample proofs • Respond - will they apply the proven result? • Respond - is a counterexample possible? • State what course/experience you relied on
Please examine the statements: For any two consecutive positive integers, the difference of their squares: (a) is an odd number, and (b) equals the sum of the two consecutive positive integers. What can you tell me about these statements?
Proof-aloud Task and Rubric • Elementary number theory statement • Recio & Godino (2001): to prove • Dewar & Bennett (2004): to investigate, then prove • Assessed with Recio & Godino’s 1 to 5 rubric • Relying on examples • Appealing to definitions and principles • Produce a partially or substantially correct proof • Rubric proved inadequate
Multi-faceted Student Work • Insightful question about the statement • Advanced mathematical thinking, but undeveloped proof writing skills • Poor strategic choice of (advanced) proof method • Confidence & interest influence performance
Proof-aloud results • Compelling illustrations • Types of knowledge • Strategic processing • Influence of motivation and confidence • Greater knowledge can result in poorer performance • Both expert & novice behavior on same task
How do we describe all of this? • Typology of Scientific Knowledge (R. Shavelson, 2003) • Expertise Theory (P. Alexander, 2003)
Typology: Mathematical Knowledge • Six Cognitive Dimensions (Shavelson, Bennett and Dewar): • Factual: Basic facts • Procedural: Methods • Schematic: Connecting facts, procedures, methods, reasons • Strategic: Heuristics used to make choices • Epistemic: How is truth determined? Proof • Social: How truth/knowledge is communicated • Two Affective Dimensions (Alexander, Bennett and Dewar): • Interest: What motivates learning • Confidence: Dealing with not knowing
School-based Expertise Theory: Journey from Novice to Expert 3 Stages of expertise development • Acclimation or Orienting stage • Competence • Proficiency/Expertise
Implications for teaching/learning • Students are not yet experts by graduation e.g., they lack the confidence shown by experts • Interrelation of components means an increase in one can result in a poorer performance • Interest & confidence play critical roles • Acclimating students have special needs
What we learned aboutMATH 190/191 • Cited more often in proof alouds • By students farthest along • Partial solutions to homework problems • Promote mathematical discussion • Shared responsibility for problem solving • Build community
With thanks to Carnegie co-investigator, Curt Bennett and Workshop course co-developers, Suzanne Larson and Thomas Zachariah. The resources cited in the talk and the Knowledge Expertise Grid can be found at http://myweb.lmu.edu/jdewar/presentations.asp