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Saturday, June 30, 2018

Exploring Task Design

Wondering what will come of picking a few random secondary math topics.
  1. Area of a Circle - The area of a circle is pi(r^2):  How many 1 x 1 units fit inside of a circle?
  2. Pythagorean Theorem - Given a right triangle with sides a, b and hypotenuse c, a^2 + b^2 = c^2:  
    1. The Egyptians used the 3,4,5 fact to make buildings square and many other peoples knew about the relationship over the past few thousand years.
    2. I think this theorem really captures the essence of mathematics and what mathematicians do.  How in the world did anyone come up with that relationship?  It seems the best answer to this would be an interest in the study of triangles and relationships between side lengths.  Yet, SO many applications!  What can we discover about the properties of triangles?  are there any relationships between side lengths? angles?  As a mathematician embarks on this quest, this surprising result emerges.  Then they seek to prove the relationship they observe generally.
  3. Exponent Product Property - a^n * a ^m = a ^(n+m):  
    1. This operation also seems to get at some of the core ideas behind the work of mathematicians.  So much work just to get to this result.  What kind of foundational work led up to this discovery?  How can we simplify relationships between varied operations or functions?  What kinds of properties do operations have?
    2. Using this operation with Natural Numbers, this almost seems definition-like.  
What can be learned from this exercise?  Students can't experience what it's like to be a mathematician without building theorems or properties from scratch.  The Pythagorean Theorem just can't be taught in a couple of days.  It has to be part of some larger picture and exploration.  The question(s) asked need to be a natural outgrowth of previous explorations.

This is a tall order.  In order to provide such an environment, the teacher needs to have a strong background in the development and progression of mathematics, or at least the designer does.   

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