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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 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

### 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

### Regions in a CircleWhat does this suggest?

Is it true that for every natural number n,

is prime?

### Count the zeros at the end of 1,000,000!

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

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

### 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