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.   

Designing Curriculum to Align with Current Research

I came to the realization today of why I have struggled so much to create quality math instruction.  In my previous attempts to design tasks, I have reasoned through the lens of traditional teaching and learning.  This has made it almost impossible to design tasks according to current educational research.  I have been trying to reconcile the two.  It can't be done.

I discovered this as I reasoned through what I would call a great task for the 21st century math classroom (the open math task below):

• Traditional:  What is the area of a 6 x 8 rectangle?
• "Real" math:  Why is the area of a 6 x 8 rectangle 48?
• Open math: How many rectangles can you find with an area of 48?

I would categorize the open math task as a skill development task.  A conceptual development task might be: How much space is contained inside of a rectangle?  How many 1 x 1 squares fit inside these rectangles?  Any rectangle?  Why will the are of a rectangle always be length x width? 

Both types of tasks are necessary for students to grapple with for the construction of knowledge.  Concept development tasks come first, as they do not require desired know-how to complete.  Students may or may not find something.  It is an exploration.  Skill development tasks require repeated application of know-how to complete.

Traditional math education compiles all of the theorems, definitions, tools, and properties that have been discovered and arranges them in a supposed sequential order of difficulty.  The teacher then tells the pupil what was discovered and prepares countless exercises for acquisition of that knowledge.  It doesn't work.  Why?  Because what the student is doing is not even remotely similar to what actual mathematicians do or have done.

Why all those pesky definitions to memorize?  What's the point in knowing that a^2 + b^2 = c^2?  How could anyone have discovered something so seemingly random?  Why did mathematicians even ask the questions they asked that led to discovery?  What do mathematicians do?

I believe that every theorem and property answers a question or multiple questions.  These are the questions that give meaning and life to mathematics.  It seems an impossible task to teach others mathematics without actually engaging in the work of mathematicians.  

Once the question(s) that could or should be asked is/are discovered, how then does the teacher use this to improve mathematics instruction?  Students should experience some of the real angst that mathematicians have felt through the ages.  Careful scaffolding can reveal some of the questions that could be asked as they learn how to think like mathematicians. 

In subsequent posts I will explore the creation of some secondary math tasks.