Interactive Mathematics Program _IMP_ by malj

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									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

                    a+ b                             a+b
     c
 b                                 c
                           b


     a                         a
          a+ b
                                         a+b
         Al’s Rug
                                       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 ic kTime ™ and a
                  dec omp re ss or
        are n eed ed to se e thi s pi cture.
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

								
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