Multi-stage Mathematics
Instruction
Jeff Knisley
East Tennessee State University
MAA-Southeastern Section, Spring 2005
I enjoy teaching…
 Teaching is its own reward.
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The satisfaction of seeing students progress
The joy of studying and sharing mathematics
for a living
The mutual benefit of the student-teacher
relationship
 Favorite Quote: They won’t care what you
know until they know that you care.
But are they learning anything…
 Performance doesn’t always imply learning
 In the 70’s, a student was taught MACSYMA
but not Calculus
 Student “Aced” a series of MIT Calculus Tests
 I have often wondered…
 Are they learning, or are they just good at
taking my tests?
 How close are they to understanding a
concept well enough to put it to good use?
 How can I make mathematics and problem
solving less frustrating for them?
How do students learn math?
 Early on, I felt I had to have an answer to this
question in order to teach at all.
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Reviewed Math Education Research
Explored Cognitive and Applied Psychology
Had some experience in Artificial Intelligence
 I combined the research, some observations,
and some simple experiments into a
“macro-model” of how a student learns math
Outline
 Mathematics Education, Applied Psychology
 What the Experts say about learning
 Some observations and simple experiments
 A “Macro-model” for mathematical learning
 As a guide for implementing new curricula
 Not an exact description of mathematics students
 Using the model to identify “Best Practices”
 As an indicator of what works and what doesn’t
 As a guide for using Technology in mathematics
What the Experts Say…Math. Ed.
 Individual Learning Styles
 Some students are visual learners, some learn by
synthesizing ideas, some learn by imitation
 Each student has a preferred learning style
 Preferred style used to construct concepts
 Kolb Learning Model
 Those who learn by building on previous experience
 Those who learn by trial and error
 Those who learn from detailed explanations
 Those who learn by implementing new ideas
 Much of this from R. Felder, Engineering Education
What the Experts Say…Psychology
 Learning Models
 Different people associate new ideas to old ones
at different rates
 Different people memorize information at different
rates
 Individual Differences in Skill Acquisition
 Give subjects simple “air traffic control” game
 First they discover simple heuristics—how to land
planes, how to create holding patterns
 Their game abilities improve in “jumps” as they
develop strategies that allow sophisticated actions
What the Experts say…AI
 Heuristic reasoning
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Associates a pattern with an action
Closest Pattern determines method used
Criteria for “closest” often yields incorrect result
Is knowledge without understanding
 Heuristics = Rote Learning
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Reduces learning to a set of rules to memorize
Replaces comprehension with association
Example: Heuristic Reasoning
Problem: Simplify
Pattern
ab  ac
a2
a 2  b2
a  b 2
x 4  4x 2
Action
ab  c 
a
a  b a  b 
a 2  2ab  b 2
Heuristic yields Incorrect Result: x 2  2 x
Observation: Math Learning Types
 Kolb Learning styles translated into math
 Allegorizers: They prefer form over function,
and often ignore details
 Integrators: They want to “compare and
contrast”
the known
with the
unknown.
Allegory
(noun):
figurative
treatment
 of
Analyzers:
Theyunder
desirethe
logical
explanation
one subject
guise
of
and detailed descriptions
another (Webster)
 Synthesizers: They use known concepts like
building blocks to construct new ideas.
 Other models yield similar Math Styles
Experiment: Pythagorean Theorem
 Procedure:
2
2
2
First
Step
:
Prove
a

b

c
 Present an example right triangle with sides
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given and hypotenuse unknown
Prove the Pythagorean theorem and use it to
determine the hypotenuse
3
3?c" 3
Measure the three4 sides of3bthe
" triangle and
show it satisfies Pythagorean theorem
Distribute paper with right triangle with
unknown hypotenuse
1 ( and rulers)
2 a"
4
Expected Observations
 Allegorizers: (@ 15 % of sampled students)
 They reduce learning to a set of “Case Studies”
 They look in the text for a worked example
 Integrators: (@ 60 % of sampled students)
 Ruler is known, Hypotenuse is unknown
 They use the ruler to measure the hypotenuse
 Analyzers: (@ 20 % of sampled students)
 They use the Pythagorean theorem
 They seem to want a logical explanation
 Synthesizers: (A handful, at best)
 Use theorem and explore to find a 3-4-5 triangle
Observation: Topic implies Style
 Style is a function of student and topic
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A student may be an analyzer in Linear Algebra
Same student may be an allegorizer in Statistics
 We resort to Heuristics when all else fails
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Math Ed research shows that even the best
students fail to understand limits
Students pass tests on limits by resorting to
heuristics—memorization and pattern-based
association
Definition of the Macro-Model
 Students acquire new concepts by progressing
through 4 stages of understanding
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Allegorization: A new concept is described in
terms of existing knowledge (i.e., intuitively)
Integration: Comparative analysis is used to
distinguish new concept from known concepts
Analysis: New concept becomes part of existing
knowledge. Connections and explanations follow.
Synthesis: New concept is used as a “building
block” to establish new theories, new strategies,
and new allegories
The Importance of Allegories
 Learning begins with allegory development
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New concept stated in a familiar context
Allegory is description within the given context
 Insufficient allegorization prevents learning
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Failure to Allegorize forces a Heuristic Approach
Some “good students” have sophisticated
heuristics
Example: Chess without Allegories
Valid moves for a
given token are
Each player
B1
C1
D1
E1
F1
D1C1
B1
determined by
receives 8 “A”
A1
A1
A1
A1
A1
A1
A1A1
token’s type. Each
tokens, 2 each of
player attempts to
“B,” “C,” and “D”
capture the other’s
tokens, and 1 each
F token.
of “E” and “F”
tokens
immobilize
A2 A2 A2 A2 A2 A2 A2 A2
(capture is figurative language)
B2 C2 D2 E2 F2 D2 C2 B2
Discussion: Learning Chess
 Context is Medieval Military Figures
 Game pieces themselves are allegories
 Pawns are numerous but have limited abilities
 Knights can “Leap over objects”
 Queens have unlimited power
 “Capture the King” is the allegory for winning
 Colors are allegories
 White Versus Black
 “Battles” take place when 2 pieces occupy the
same square on the checkerboard
Components of Integration
 Student now understands allegorically that there is a
new concept to be acquired
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Places a label on the new idea – i.e., a definition
Definition places concept into a mathematical setting
 Compare and Contrast
 How is new concept like known concepts?
 How does new concept differ from known concepts?
 We often neglect this stage
 Visual Comparisons are the most powerful
 Technology can be used to produce comparisons
Tangent Planes
Which plane, A and B, is tangent to the surface
Analysis of a New Concept
 The new concept takes on its own character
 Explanations and origins are developed
 Techniques for use of new concept are developed
 The new concept becomes one of many
characters
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Connections to existing ideas are established
Sphere of influence becomes well-defined
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What known concepts are related to new concept?
How are known concepts modified by the new concept?
 Analysis desires that a great deal of relevant
information be delivered quickly(i.e., lectures)
Synthesis and Problem Solving
 New idea becomes a tool
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To create allegories for new ideas
To create new versions of existing knowledge
To solve problems and prove theorems
 Strategy development
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New concept and known concept are
combined into sophisticated constructions
New concept is used to solve problems
Applications are desired and explored in depth
Identifying Best Practices…
 Too often, instruction is directed at analyzers
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Lectures and techniques
Integrators and Allegorizers are lost/confused
Synthesizers may get bored and fall behind,
making them allegorizers for later material
 Most Students forced to use heuristics
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Integrators and Allegorizers memorize rules
Analyzers often apply heuristics anyway
Example: Studies have proven this for Limits
Example: Uncertainty Principle
 Physics student asked me to explain
Heisenberg Uncertainty to him from a
Mathematical perspective
Mathematical: If A and B are self-adjoint and
AB – BA = I,
where I is the identity operator, and if f is in
dom(A) ∩ dom(B) vector with ||f ||=1, then
||Af || ||Bf || > ½
 The amazing thing is that non-commutivity of A
and B implies the lower bound
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Example: Heisenberg Uncertainty
 Proof: Define
Q(t) = || (A+itB)f ||2
Expand to show that
Q(t) = ||Af ||2 + <i(AB-BA)f, f >t + || Bf ||2 t2
Q(t) > 0 implies that
4 ||Af ||2 ||Bf ||2 > |<i(AB-BA)f, f >|2 = 1
 Main Example: (Af)(x) = xf(x), (Bf)(x) = f ′(x)
over L2(R)
Heisenberg Uncertainty
 Mathematically, Uncertainty is tricky
1 if
f  x  
0 if
1  x  1
| x | 1
f is differentiable a.e., but cannot allow f in dom(B)
because || Bf || = 0
 So how to explain the mathematics of uncertainty
to a physics student?
Using the Model
 Allegory: relating derivative to multiplication operator
Adjustable points
Slope depends on ||Bf||
steeper
shorter
||Af||
Area
under
= 1=
||Af||
= curve
||Bf||
Area under the curve = 1
 Integration: Interactive applet where they attempt to
construct a function that minimizes uncertainty
Best Practices… Defining Roles
 Role of the Teacher
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Allegorization: Teacher is a story-teller
Integration: Teacher is a guide*
Analysis: Teacher is an expert
Synthesis: Teacher is a coach
 Role of Technology
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Integration = hands on student exploration
Technology for integration
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After Introducing a concept
Before extensive lecture on the concept
Example: Exponential Growth
 How to introduce the exponential (and later,
logarithmic growth) to a biology student?
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Standard:
2t  h  2t
  0.6931 2t ,
h
so there is 2<e<3 such that
3t  h  3t
 1.0986  3t
h
et  h  et
 et ,
h
 The fact that y=et satisfies y′=y does not mean all that
much to a biology student
Using the Model…
 Allegory: Birth/Death processes
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A population growing at a rate of k% per hour
does not reproduce all at once.
Instead, reproduction takes place many times
per hour
Exponential growth is a birth process with a
constant % rate in which reproduction takes
place arbitrarily many times per hour
Introductory Systems Ecology…
 Integration:
 Divide time interval [0,t] into n short periods, where n is a
very large integer
 Having n generations of reproduction means n periods
where each period has length h = t / n
 “Probability” of reproduction in each time period is kh,
which is % rate scaled over period
 Simulation: Start with P individuals and let each
reproduce over 1st period with probability kh. Repeat for
all n time periods
(http://faculty.etsu.edu/knisleyj/biomath/birthdeath.htm)
Introductory Systems Ecology…
 Start with P0 individuals
 After 1st period: P1 = P0 + kh P0 = P0 (1+kh) individuals
 After 2nd period: P2 = P1 + kh P1 = P0 (1+kh)2 individuals
 After nth period: Pn = Pn-1 + kh Pn-1 = P0 (1+kh)n individuals
 n arbitrarily large means n approaching ∞
 Definition: The exponential function is defined
 kt 
e  lim 1  
n 
n

n
t
and from this we can derive all properties of the exponential.
Best Practices…Technology
 Technology as intermediate assessment
 Multivariable Calculus = All quizzes are Maple
worksheets
(http://faculty.etsu.edu/knisleyj/multistage/quiz5.mw)
 Intro Stats
 1200 students per semester in our gen ed course
 4 stage instruction
 Lecture – Applets – Computer – Assessment
 Lecture and Assessment are traditional
 Applets prepare for graded Minitab activities
 Technology for extended projects
Technology as aid to Student Research
 Allegory – Introduction to research problem,
where problem is an extension of known result
 Integration -- Student uses Maple, NetLogo,
etc. to reproduce or simulate known result
 Analysis – Predict an answer to problem using
technological extension of known result
 Synthesis – Proof or otherwise solution of
given research problem
Research Examples:
 M.P. began by reproducing a well-known
agent-based army ant raiding pattern model
en route to agent-based model of division of
labor in social insects.
 P.C. began with simple implementation of
classic Neural Net algorithm en route to using
neural nets for data mining micro-array data
 A.T. began with simple curve-fitting algorithm
en route to proving a version of the C.H.
theorem for a class of elliptic operators
Summary
Allegory (intuition) that leads to integration, perhaps
via technology and interactive assessment, so
that they are ready for lecture-based analysis
that fleshes out the concept, and then can begin
to synthesize their own ideas
But students can’t create their own allegories or
coach themselves as synthesizers. Thus, the
model ultimately predicts the necessity of the
mutually-beneficial student-teacher relationship.
Thank you!
Any Questions?