The Trouble with 5 ExamplesSoCal-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 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?

#points 1 2 3 4 5 6

#regions ? 2 4 8 ? ?

Prime Generating Quadratic

Is it true that for every natural number n,

is prime?

n2 −n+41

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

N! # ending zeros

4! 0

8! 1

12! 2

20! 4

40! 9

100! 24

1000! 249

Observed pattern:

If 4 divides n, then n! ends in zeros.

Counterexample:

24! ends in 4 not 5 zeros.

n

4−1

Where do the zeros come from?

From the factors of 10,

so count the factors of 5.

There are

Well almost…

106

51+

106

52 +106

53 + ...+106

58

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.

22n

+1=

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

5 Examples Convinces Me

0%

20%

40%

60%

80%

100%

0 Sems 1-2 Sems 3-4 Sems >4 Sems Faculty

SDDNASA

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

Level Progression in the Major

0 Before MATH 190 Workshop I

1 Between MATH 190 & 191

2 Just Completed Proofs Class

3 Just Completed Real Variables

4 1 Year Past Real Variables

5 Graduated the Preceding Year

Level in Major vs Proof Category

Student Level

0 0 1 1 2 2 3 3 3 4 4 5

R&G’s Proof Category

1 4 4 5 5 5 4 5 5 5 4 5

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 GridAffective Acclimation Competence Proficiency

Interest

Confidence

Cognitive Acclimation Competence Proficiency

Factual

Procedural

Schematic

Strategic

Epistemic

Social

MathematicalKnowledge Expertise Grid

Affective Acclimation Competence Proficiency

Interest

Students are motivated to learn by external (often grade-oriented) reasons that lack any direct link to the field of study in general. Students have greater interest in concrete problems and special cases than abstract or general results.

Students are motivated by both internal (e.g., intrigued by the problem) and external reasons. Students still prefer concrete concepts to abstractions, even if the abstraction is more useful.

Students have both internal and external motivation. Internal motivation comes from an interest in the problems from the field, not just applications. Students appreciate both concrete and abstract results.

Confidence

Students are unlikely to spend more than 5 minutes on a problem if they cannot solve it. Students don't try a new approach if first approach fails. When given a derivation or proof, they want minor steps explained. They are rarely complete problems requir

Students spend more time on problems. They will often spend 10 minutes on a problem before quitting and seeking external help. They may consider a second approach. They are more comfortable accepting proofs with some steps "left to the reader" if they hav

Students will spend a great deal of time on a problem and try more than one approach before going to text or instructor. Students will disbelieve answers in the back of the book if the answer disagrees with something they feel they have done correctly. S

Cognitive Acclimation Competence Proficiency

FactualStudents start to become aware of basic facts of the topic.

Students have working knowledge of the facts of the topic, but may struggle to access the knowledge.

Students have quick access to and broad knowledge about the topic.

ProceduralStudents start to become aware of basic procedures. Can begin to mimic procedures from the text.

Students have working knowledge of the main procedures. Can access them without referencing the text, but may make errors or have difficulty with more complex procedures.

Students can use procedures without reference to external sources or struggle. Students are able to fill in missing steps in procedures.

SchematicStudents begin to combine facts and procedures into packets. They use surface level features to form schema.

Students have working packets of knowledge that tie together ideas with comon theme, method, and/or proof.

Students have put knowledge together in packets that correspond to common theme, method, or proof, together with an understanding of the method.

StrategicStudents use surface level features of problems to choose between schema, or they apply the most recent method.

Students choose schema to apply based on a few heuristic strategies.

Students choose schema to apply based on many different heuristic strategies. Students self-monitor and abandon a nonproductive approach for an alternate.

Epistemic

Students begin to understand the common notions 'evidence' of the field. They begin to recognize that a valid proof cannot have a counterexample, they are likely to believe based on 5 examples, however, they may be skeptical at times

Students are more strongly aware that a valid proof cannot have counterexamples. They use examples to decide on the truth of a statement, but require a proof for certainty.

Students recognize that proofs don't have counterexamples, are distrustful of 5 examples, see that general proofs apply to special cases, and are more likely to use "hedging" words to describe statements they suspect to be true but have not yet verified.

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