Now, I'm not taking sides in the perennial "pure-versus-applied" debate, although Dan Meyer's blog last week includes the important point that the new-traditional view "kids are interested by applications, not abstractions" is simply false:
Traditional (false) view of the relationship
between engagement and abstraction
Actual relationship between
engagement and abstraction
What makes an activity engaging isn't how "real-world" it purports to be. It's a combination of factors, including:
- Challenge level
- Immediacy and quality of feedback
- Stakes--low but nonzero is better, especially in the short-term
- Visibility of progress towards ultimate mastery
(Yes, I did mean both of the last. Things that are completely new are often off-putting; things that are the same every time are drudgery. So you need to provide novelty within familiar frameworks, or something like that ... but that's another blog post.)
Math that's been reduced to algorithms for students to memorize and apply with perfect precision on tests whose scores "will follow you for the rest of your life" has none of these qualities. So of course it's not interesting.
What is interesting is thinking about problems, trying to carve them into meaningful abstractions, attacking them again and again with slight gains, etc. Doing problems that seem familiar, but with a twist: this time I'm not asking for the hypotenuse given two legs, but for a leg; or for a right triangle with integer sides and a particular leg; or for the dimensions of a rectangle, given its area and the fact that its sides and diagonal are all integers .... Or this time I'm wondering: suppose I approximate the square root of 2 as 1.414. How bad an error will I get for the hypotenuse of a 10m-10m right triangle? 100m-100m? 1000m-1000m? If it takes a minute to walk 100m, how long do the legs have to be before the error is five minutes of walking?
All those are abstract problems, with varying degrees of "real-world" relevance. I'm not claiming they're great questions--I just came up with most of them right now--but they avoid most of the pitfalls above, and if done in class, in a problem-solving environment (i.e. where the kids have been exposed to the fact that math problems don't always yield on the first attempt, several times), would be a heck of a lot better than a page full of Pythagorean theorem practice problems. Some of them lead to deep questions: are there numbers that can't be sidelengths of a right triangle whose other sides are integers? Are there numbers that can only appear once as members of pythagorean triples?
Math is at its heart a process of abstracting and connecting ideas through logic and generalization. We worry that most kids can't do these things, and so we often do our best to avoid engaging in such practices overtly, or even requiring kids to engage in them at all. But that's exactly the wrong approach. Engaging kids in the process of actually abstracting, connecting, reasoning, generalizing, conjecturing, and applying--rather than just practicing spitting out formulas--now that would make math class interesting. It would make math class about math.