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


Regions in a CircleWhat does this suggest?


Prime Generating Quadratic

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.


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


Students’ Level Relative Critical Courses


Level in Major vs 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


MathematicalKnowledge Expertise Grid


MathematicalKnowledge 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 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


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