Transcript Rich Representations of Student Learning

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
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 Circle
What 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,
n  n  41
2
is prime?
Count the zeros at the end of
1,000,000!
N!
4!
8!
12!
20!
40!
100!
1000!
# ending zeros
0
1
2
4
9
24
249
Observed pattern:
If 4 divides n, then n! ends in
Counterexample:
24! ends in 4 not 5 zeros.
n
1
4
zeros.
Where do the zeros come from?
From the factors of 10,
so count the factors of 5.
There are
10 6 10 6 10 6
10 6
 2  3  ...  8
1
5
5
5
5
Well almost…
Fermat Numbers
• Fermat conjectures (1650) Fn = 2  1
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.
2n
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
100%
80%
SD
D
N
A
SA
60%
40%
20%
0%
0 Sems
1-2 Sems
3-4 Sems
>4 Sems
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
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
4
5
5
5
4
5
R&G’s Proof Category
1
4
4
5
5
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
Mathematical
Knowledge Expertise Grid
Affective
Acclimation
Competence
Proficiency
Acclimation
Competence
Proficiency
Interest
Confidence
Cognitive
Factual
Procedural
Schematic
Strategic
Epistemic
Social
Mathematical
Knowledge Expertise Grid
Affective
Interest
Acclimation
Competence
Proficiency
Students are motiv ated to learn by
Students hav e both internal and external
external (of ten grade-oriented) reasonsStudents are motiv ated by both internal
motiv ation. Internal motiv ation comes
that lack any direct link to the f ield of (e.g., intrigued by the problem) and
f rom an interest in the problems f rom the
study in general. Students hav e greater
external reasons. Students still pref er
f ield, not just applications. Students
interest in concrete problems and
concrete concepts to abstractions, ev en
appreciate both concrete and abstract
special cases than abstract or general if the abstraction is more usef ul.
results.
results.
Students are unlikely to spend more Students spend more time on problems.Students will spend a great deal of time
than 5 minutes on a problem if they They will of ten spend 10 minutes on a on a problem and try more than one
cannot solv e it. Students don't try a new
problem bef ore quitting and seeking
approach bef ore going to text or
Conf idence approach if f irst approach f ails. When external help. They may consider a
instructor. Students will disbeliev e
giv en a deriv ation or proof , they want second approach. They are more
answers in the back of the book if the
minor steps explained. They are rarely comf ortable accepting proof s with some
answer disagrees with something they
complete problems requir
steps "lef t to the reader" if they hav f eel they hav e done correctly . S
Cognitive
Factual
Procedural
Acclimation
Competence
Proficiency
Students hav e working knowledge of the
Students start to become aware of basic
Students hav e quick access to and
f acts of the topic, but may struggle to
f acts of the topic.
broad knowledge about the topic.
access the knowledge.
Students hav e working knowledge of the
Students can use procedures without
Students start to become aware of basic
main procedures. Can access them
ref erence to external sources or
procedures. Can begin to mimic
without ref erencing the text, but may
struggle. Students are able to f ill in
procedures f rom the text.
make errors or hav e dif f iculty with more
missing steps in procedures.
complex procedures.
Schematic
Students hav e put knowledge together in
Students begin to combine f acts and Students hav e working packets of
packets that correspond to common
procedures into packets. They use
knowledge that tie together ideas with
theme, method, or proof , together with
surf ace lev el f eatures to f orm schema.comon theme, method, and/or proof .
an understanding of the method.
Strategic
Students choose schema to apply based
Students use surf ace lev el f eatures of
Students choose schema to apply based
on many dif f erent heuristic strategies.
problems to choose between schema, or
on a f ew heuristic strategies.
Students self -monitor and abandon a
they apply the most recent method.
nonproductiv e approach f or an alternate.
Epistemic
Students begin to understand the
Students recognize that proof s don't
common notions 'ev idence' of the f ield.Students are more strongly aware that hav
a e counterexamples, are distrustf ul of
They begin to recognize that a v alid v alid proof cannot hav e
5 examples, see that general proof s
proof cannot hav e a counterexample, counterexamples. They use examples to
apply to special cases, and are more
they are likely to believ e based on 5 decide on the truth of a statement, but likely to use "hedging" words to describe
examples, howev er, they may be
require a proof f or certainty .
statements they suspect to be true but
skeptical at times
hav e not y et v erif ied.
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 about
MATH 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