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?

y

x

feasible

region

profit

line

Get the Point?

In solving problems like the cookie problem, it is helpful to know how to find the point where two lines intersect.

Your goal is to discover a me thod of doing this, besides guessing or using graphs, when you are working with the equations of two straight lines.

.... .

7. As a group, develop and write down general directions for finding the coordinates of the point of intersection of two equations for straight lines without guessing or graphing. Make your d irections easy to follow, so someone in middle school could follow them.

Principle 3: Active Involvement

• To motivate

• The proof of the Pythagorean Theorem

Proof by Rugs

   

1. Are the areas of the two rugs the same?

2. How do the two rugs demonstrate that the Pythagorean Theorem holds in general?

a

b

a + b

a + b

c

Al’s Rug

b

a

c

a + b

a + b

Betty’s Rug

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

DRINK ME

[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

QuickTime™ and a decompressor

are needed to see this picture.

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