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

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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

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
• 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

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
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
• 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