Interactive Mathematics
Program (IMP)
Goals of IMP
Motivate students to engage with
mathematics
Help students become powerful
problem solvers
Powerful Problem Solvers
From the accountant who explores the consequences of
changes in tax law to the engineer who designs a new
aircraft, the practitioner of mathematics in the computer age
is more likely to solve equations by computer-generated
graphs and calculations than by manual algebraic
manipulations. Mathematics today involves far more than
calculation; clarification of the problem, deduction of
consequences, formulation of alternatives, and development
of appropriate tools are as much a part of the modern
mathematician’s craft as are solving equations or providing
answers.
– Everybody Counts, National Research Council, 1989, p. 5
Goals of IMP
Motivate students to engage with
mathematics
Help students become powerful
problem solvers
Prepare students for the future
The Future
We are currently preparing students for jobs that
don’t yet exist using technology that hasn’t yet been
invented in order to solve problems we don’t even
know are problems yet.1
1The
Jobs Revolution: Changing How America Works,
Richard Riley, 2004.
Principle 1: List of Concepts
and Skills
• Concepts and skills selected and kept in mind
• Examples:
– Write proofs and/or explanations of thought
processes
– Use the distributive law to rewrite algebraic
expressions
– Explain why division by zero is not well defined
Principle 2: Organized around
big problems
• Five big problems a year for 4 years
• Skills taught in smaller problems inside
the big problems
• Rational: Motivate and problem solving
Abby and Bing Woo have a small bakery shop that makes
cookies. They make only two kind of cookies: plain and
iced. They need to decide how many dozens of each kind
of cookies to make for tomorrow.
They are limited by the following things:
the amount of ingredients they have on hand;
the amount of space available in their oven; and
the amount of preparation time.
How many dozens of each kind of cookie
should Abby and Bing make, so that their
profits are as high as possible?
Principle 3: Active Involvement
• To motivate
• The proof of the Pythagorean Theorem
Proof by Rugs
b
a +b
c
a +b
c
b
a
a
a +b
Al’s Rug
a +b
Betty’s Rug
1. Are the areas of the two rugs the
same?
2. How do the two rugs demonstrate that
the Pythagorean Theorem holds in
general?
Principle 3: Active Involvement
• To motivate students to engage with
mathematics
• The proof of the Pythagorean Theorem
• Used to motivate definitions
Example: regression
Two Suggested Solutions
Student A said that the function f given by the equation
f(x) = 40 + 8x approximated the data well. So student
A predicted that on April 18, Mr. Dunkalot would have
280 foot-pounds of strength and would be strong
enough to play.
Student B said the function g given by the equation
g(x) = 55 + 6x approximated the data well. So
student B predicted that on April 18, Mr. Dunkalot
would have only 235 foot-pounds of strength and
would not be strong enough to play.
Your Questions
1. Which student’s function seems to you to fit the
data better, and why?
2. Do you have a function that you think fits the data
better than either of these? If so, what is it?
3. Develop a mathematical procedure by which you
might judge when one function fits data better than
another.
Principle 4: Abstractions
introduced concretely
• Through stages over time
Regression
• By hand with fettuccini
• Intuitively with graphing calculators
• Constructing a procedure
• Using the built in facility on a calculator
Principle 4: Abstractions
introduced concretely
• Through stages over time
• Using physical objects
• With metaphors
Alice Metaphor for
Exponential Growth
[Alice] found a little
bottle . . . with the
words “DRINK ME”
[Alice] found in it a
very small cake, on
which the words
“EAT ME”
Principle 5: Multiple
Representations
• Deeper understanding by seeing
different perspectives
• Accommodates different learning styles
• Can apply more widely to new problems
20 = 1
•
•
•
•
•
•
Through the Alice metaphor
By a numerical pattern
Graphically
Deductively
Then present the definition
Finally, a reflection
20 = 1: Number Pattern
25 = 32
24 = 16
23 = 8
22 = 4
21 = 2
20 = ?
20 = 1: Graphically
Qu i c k T i m e ™ a n d a
d e c o m p re s s o r
a re n e e d e d to s e e th i s p i c t u re .
20 = 1: Deductively
23 • 20 = 23
8 • ? = 8
Negative Reflections
Write a clear explanation
summarizing what you have
learned
about
defining
expressions that have zero or
a negative integer as an
exponent.
Explain, using examples, why
these definitions make sense.
Give as many different
reasons as you can and
indicate which explanation
makes the most sense to you.
20 = 1
•
•
•
•
•
•
•
Through the Alice metaphor
By a numerical pattern
Graphically
Deductively
Then present the definition
Finally, a reflection
WHY ALLTHIS???
Why All This
• Equity issue to include more students in
problem solving
• People who could make valuable
contributions to society are being
excluded from math knowledge
• Evidence says the top students are not
being harmed and are gaining more