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

The MATH 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

Regions in a Circle What does this suggest? #points #regions ?248??

Prime Generating Quadratic Is it true that for every natural number n, is prime?

Count the zeros at the end of 1,000,000! N!# ending zeros 4!0 8!1 12!2 20!4 40!9 100! !249

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) F n is prime for every nonnegative integer. Euler (1732) shows F 5 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.

5 Examples: Students & Faculty

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

R&G’s Proof Categories 1 Very deficient answer 2 Checks with examples only 3 Checks with examples, asserts general validity 4 Partially correct justification relying on other theorems 5 Substantially correct proof w. appropriate symbolization

Students’ Level Relative Critical Courses LevelProgression in the Major 0Before MATH 190 Workshop I 1Between MATH 190 & 191 2Just Completed Proofs Class 3Just Completed Real Variables 41 Year Past Real Variables 5Graduated the Preceding Year

Level in Major vs Proof Category Student Level R&G’s Proof Category

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

Mathematical Knowledge Expertise Grid

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 about MATH 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