The Global Picture

Today the graduate student team met. We began with a problem from Maria Terrell's Good Questions project, very slightly modified.

Put down your pencil or pen, and put away any paper. You have 10 seconds to answer.

Imagine that there is a rope around the equator of the earth. Add a 40 foot segment of rope to it. The new rope is held in a circular shape centered about the earth. Then the following can pass beneath the rope without touching it:

(a) amoeba
(b) ant
(c) you
(d) giraffe

We recommend thinking about it before reading the answer.

We used this problem as an example to think about the process of teaching. As we talked about this, several key issues arose.

• Intuition/instinct vs. Reality, where Reality is defined as mathematical rigour. (The fun in math comes from being "tricked".)


• Teaching math vs. "Doing" math. There's a huge difference between being able to do a problem and being able to teach another person how to do the problem. Teaching something and learning something are related but distinct processes.
  


• How to identify pieces of information that are organized. When a student gives a reason, how do you frame what the student has said in terms of (a) what data the student is using (b) how that student has related them?
  


• How to transfer philosophy into action. We have all these lofty goals; how do we bring them to fruition?
  


• How changing the way problem is presented affects learning process. How might a student's reaction changed if we had
  
  
  
  • told them the circumference of the earth ?
    
  
  
  • told them the heights of each of the creatures?
    
  
  
  • imposed a different time limit?
    
  
  
  • allowed writing utensils?
    
  
  
  
  Would a student's learning be affected if we told them the answer without letting them think further about the problem?
  

With regard to the last issue, Terrell collected data from 17 small calculus sections and tracked the sophistication of questions posed by the instructors to their students as well as how many times per week the Good Questions were used, with and without peer discussion. She found that among her sample, the students who on average suffered the most were those whose instructors used Good Questions frequently without peer discussion. These students fared worse on average than those whose instructors barely used the Good Questions. (The students who fared best on average were those whose instructors used the Good Questions frequently with peer discussion.)