Monday, December 13, 2010

"Reasons to teach what we teach"

I've got a long post up on Education and Statistics on the reasons for including a concept in the mathematics curriculum and how those reasons should affect what we teach and how we teach it.

Here's an excerpt:
Where a topic appears on this list affects the way it should be taught and tested. Memorizing algorithms is an entirely appropriate approach to problems that fall primarily under number one [Needed for daily life]. Take long division. We would like it if all our students understood the underlying concepts behind each step but we'll settle for all of them being able to get the right answer.

If, however, a problem falls primarily under four [helps develop transferable skills in reasoning, pattern-recognition and problem-solving skills], this same approach is disastrous. One of my favorite examples of this comes from a high school GT text that was supposed to develop logic skills. The lesson was built around those puzzles where you have to reason out which traits go with which person (the man in the red house owns a dog, drives a Lincoln and smokes Camels -- back when people in puzzles smoked). These puzzles require some surprisingly advanced problem solving techniques but they really can be enjoyable, as demonstrated by the millions of people who have done them just for fun. (as an added bonus, problems very similar to this frequently appear on the SAT.)

The trick to doing these puzzles is figuring out an effective way of diagramming the conditions and, of course, this ability (graphically depicting information) is absolutely vital for most high level problem solving. Even though the problem itself was trivial, the skill required to find the right approach to solve it was readily transferable to any number of high value areas. The key to teaching this type of lesson is to provide as little guidance as possible while still keeping the frustration level manageable (one way to do this is to let the students work in groups or do the problem as a class, limiting the teacher's participation to leading questions and vague hints).

What you don't want to do is spell everything out and that was, unfortunately, the exact approach the book took. It presented the students with a step-by-step guide to solving this specific kind of logic problem, even providing out the ready-to-fill-in chart. It was like taking the students to the gym then lifting the weights for them.
This is a bit of a work in progress so if you have any relevant experience in mathematics education (and, yes, experience as a student definitely counts), I would greatly appreciate it if you went by and joined the conversation.


  1. Are these the cross-hatch grid charts where you record clues provided by the prompt, cross off impossible combinations, and use elimination to match up the remaining traits? I loved those! I went through three or four books of them in - was it second grade or fourth grade? - including the charts to fill in. I think it depends on the age of the student and whether they've been exposed to visual methods to solve logic problems before.

    I used to tutor on the weekends, ages 10 through 18, and the mathematics aids we used stressed symbolic language. The idea was to start early with problems - including logic problems - that were represented visually, and then to use minimal amounts of English to explain the rules of each new exercise. The exercises build upon previous exercises in a logical way, so the student figures out the rules without having to ask the tutor how the problem is done. Well, in theory, anyway. Young kids have a pretty good grasp on visual notation, so you want to retain that as they move up in the ranks, gradually adding formal mathematical language. Too much English gets in the way.

    Starting at a higher grade level, you might be forced to use more English. But I'd still think that provided charts would be perfectly fine, as long as the students spent more time actually reasoning through the problems than struggling to recall rules. And if you then varied the problems slightly, without providing the new kind of chart, that might get the students thinking more creatively, as well.

  2. I might have been a bit harsh here, but this was a class of high school juniors and seniors.

    If I were teaching this as a class activity, I would probably start by asking the students how they could represent the information visually. Then I would give them feedback and a few hints until we got an appropriate chart (which would probably end up looking a lot like the one in the book). Once we had the chart we would fill it out as a group.

    This is the long way of doing the problem but it gets the students in the habit of thinking about the best way to represent information in order to solve a problem. It also gives them a sense of accomplishment and independence that you can never get from simply following the steps your teacher gave you.