Sunday, July 3, 2011

What do we want students to do?

As I finished up last week's post, I was wondering about different kinds of tasks we could ask students to do besides solve traditional or not-so-traditional problems (numerical, proofs, whatever).   Of course, few math tests consist primarily of "problem solving" in the narrower, Polya-type sense of "attacking novel mathematical situations or questions"; most of the time, tests and quizzes ask students to "solve problems" in a somewhat-more-general sense that includes working out answers to exercises similar to ones done in class or on homework.  But in general, prompts look something like
Find all values of w such that...
Draw the graph of a function with the following properties ...
 At what time will ...
 If P, Q, R, and S are points on quadrilateral ABCD such that ... prove that ... 
Find a polynomial with the following properties, or prove that no such polynomial exists.
 Compute the probability that ...
As I was thinking about assessing mathematical knowledge, I began to wonder why virtually all test and quiz questions are of these types.  It's not that hard to dream up different types of prompts.  For example:
Write an explanation of how to determine the degree of a polynomial based on its graph, including any uncertainty in the final answer.
Place isosceles trapezoids in the quadrilateral hierarchy drawn below, and explain your choice.
Explain the following statement: Given enough trials, any event with probability greater than 0, no matter how small, eventually occurs.
 An ironing board has two legs of fixed length, one of which is free to slide along the length of the board, but both legs are attached at their midpoints as shown in the diagram below.  Explain why this setup guarantees that the board is parallel to the floor.
What is the derivative of a function?  What is its importance and how is it computed?  What information about a function can you get by examining values of its derivative?  Explain using symbolic, numeric, and graphical examples, and including one example of motion.
All of these prompts demand a fair amount of mathematical knowledge, and the ability to construct coherent arguments.  And--unlike most traditional math prompts--they get at skills and concepts in a way that might actually be useful to someone not involved in mathematics at a professional level.  But -- okay, I'm contradicting myself slightly here -- coming up with these prompts is not trivial: it takes creativity, insight into what students will be able to do, time on the exam for students to work on them (so, not the 29th question on a 30-question hour exam), and time for thoughtful grading and assessment.  Yet they are worthwhile:  the last question is part of the best test I ever gave, the final exam for a two-week calculus course I taught for elementary school teachers (as part of the SESAME program at the University of Chicago), and what made it great was that it simultaneously allowed every participant to demonstrate at least some knowledge, while differentiating between teachers who knew a lot and teachers who knew only a little.

So here's the question.  Why we almost-entirely-confine ourselves to the first kind of "problem"?  Is it because
  • We think that the most important thing for students to do is to work out these kinds of questions, or
  • We want students to do other kinds of reasoning, but we don't know how to assess it, or
  • [my sinking gut feeling] We haven't really thought carefully about what we might want students to be able to do besides "solve problems," or whether this is the only (or even most important) set of skills and outcomes for students?
Thoughts, please, in the comments.  And guys, I know it's summer, but Blogger tells me we have substantially more followers than I can count on one hand.  So please do comment.

== pjk