A confluence of events this week prompts this post. The first was an email about a workshop on “Advanced Geometry” that I’ll be going to in Santa Fe this sumer. The tagline for the meeting was
The most common misconception about mathematics is that it is aboutnumbers. In fact, mathematics is about thought, and no one knew that better than the greatest geometers of all time: Euclid, Apollonius, Descartes, and Lobachevsky.
I wrote back to the workshop leader and said that I wished more math teachers agreed with that sentiment. His reply was
I think that is one of the things that makes it a frustrating subject to teach because when a kid is struggling with math, they are struggling with the most basic elements of thought.
And yet, I have a number of colleagues who’ve said over the years that they liked math “because it has answers.” The “finding an answer” aspect of math is, to me, somewhat like studying grammar and thinking you’ve learned the essentials of literature…
The second event was an interaction with one of my best Differential Equations students this morning. I had a question on a midyear-exam review sheet, and he came in, saying, “This question seems simple, but when I tried to answer it I couldn’t. It’s not as simple as it looks.” So, I asked him, “What do you do when you see a problem you can’t do?” One of his colleagues sitting nearby said, “Ask the teacher.” His classmates laughed, and I bopped the smart-ass on the back of the head, whereupon he, too, collapsed in laughter.
The question essentially said, “You know how to get to C given A and B. But what if you were given C? How could you find A and B? And are the A and B you found the only ones that would work, or would A’ and B’ perhaps also give C?” So, in essence, it was a “working backwards” problem.
No-one in the room had a good answer to the student’s implicit question of “how do you do this question?” So, I said, “Well, you know the general process you’d use to work out this problem. So, pick a specific example, work it out the way you know, and then try to see the correspondences between what you know how to do and what you were asked to do.” The really good student just stared at me blankly for a moment. I said, “Well, if you don’t like that approach, how else could you do it.” Student:”I don’t know.” Me: “When you’re lost, I think simplifying the problem, or doing a specific example and trying to generalize, or reasoning by analogy are very powerful, general teacchniques.” He finally said “OK.”
Granted, I often used to have students in the precursor course to Differential Equations, so they’d had a year to get used to thinking in these sorts of way before they came to this course, but I am still pretty appalled that a very good student–one of the best–in our top-of-the-line math program could be reduced to the “I don’t see how to do this problem right away, so you have to help me” mindset so easily.
Given that most people honestly don’t need more than Algebra I in their adult lives, why are we teaching the large majority of students topics/algorithms they’ll never need to know and not spending more time teaching them “how to think mathematically”? Or even just think in general?