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The Trouble with 5 Examples SoCal-Nev Section MAA Meeting October 8, 2005

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The Trouble with 5 ExamplesSoCal-Nev Section MAA MeetingOctober 8, 2005

Jacqueline Dewar Loyola Marymount University

- A Freshman Workshop Course
- Four Problems/Five Examples
- Year-long Investigation
- Students’ understanding of proof

- 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

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.

From the factors of 10,

so count the factors of 5.

There are

Well almost…

- 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.

=

Nonstandard problems give students more opportunities to show just how often 5 examples convinces them.

- What is the progression of students’ understanding of proof?
- What in our curriculum moves them forward?

- 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.

‘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

- 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

- 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?

- 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

- Insightful question about the statement
- Advanced mathematical thinking, but undeveloped proof writing skills
- Poor strategic choice of (advanced) proof method
- Confidence & interest influence performance

- 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

- Typology of Scientific Knowledge (R. Shavelson, 2003)
- Expertise Theory (P. Alexander, 2003)

- 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

3 Stages of expertise development

- Acclimation or Orienting stage
- Competence
- Proficiency/Expertise

- 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

- 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