What is a function?

What is a function?  I’ve been thinking about this question a lot recently.  Mainly, it’s been on my mind, because I’m cynical about how many of our graduating seniors (at a high performing high school/early college) could answer the question in a rigorous, or even intuitive, manner.

This has triggered a number of thoughts about the teaching that goes on in my school and in many other good schools:

  • Teachers are missing the big picture.  Even at my school that is not bound by the New York State Regents exams, we get so wrapped up in covering the material that’s “supposed to be covered” in Algebra I, Algebra II, etc.  We, as teachers, need to identify a few big, important concepts that we hope students will have some understanding of when they graduate.  Then, we need to ask ourselves exactly where in the curriculum students will have the opportunity to think about these big picture topics in a meaningful way.  
  • Procedure is part of the problem.  Students won’t remember most procedures thirty years from now, and I don’t really care if they do.    However, I do want my students to have a thoughtful answer to the question “What is a function?” thirty years from now.  In this light, I’m starting to have issues with topics like the “vertical line test,” because it teaches student that the procedural question “Is it a function?” is more important than the the deep question “What is a function?”
  • x’s and y’s are part of the problem.  They are the mathematical standard variables to use in most textbooks, but I think a student’s understanding of functions (and of most math) gets conjoined with x’s and y’s in an unhealthy way.   I’m interested in changing some of our questions to look like:  “b = 5.  Is b a function of the variable m.  Explain your reasoning.”
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