On Open-Ended Non-Questions

I came across this wonderfulness today in my Google Reader, and it made me pretty excited! (I know, I'm a total geek.) I love the idea that you'd give the kids an open-ended situation and leave it up to them to show as much understanding as they can about the situation. Seems like something that works extremely well in physics -- a subject whose goal is to encourage kids to gain a multi-layered understanding of everyday situations using a combination of various "laws" and models. It's also really exciting to me because this process highlights (to the teacher as well as the kids) that conceptual understanding of the same situation will continue to build, as you accumulate deeper knowledge on how to drill down further into pertinent details. For example, take a situation where a kid is told that a hot air balloon flies through the air. At the beginning of the year, they might only be able to say that it's because the air inside the balloon is less dense than the surrounding air. A week later, they might be able to say that this is because the air is warm and warmer air is less dense. Another week later, they might be able to say that the warm air is less dense because the molecules have more kinetic energy and therefore create pressure and expand the volume of the balloon. Another week later they might be able to explain the heating mechanism and why hot air doesn't escape through the hole of the balloon. And even later, they might be able to predict what temperature the balloon would have to have in order to carry a certain amount of weight. (I'm not actually totally sure if they can do those calculations AND I'm pretty sure you'd teach those concepts at a faster pace. I'm just using those steps to roughly illustrate how an idea about the same situation progresses over time.)

It sounds like something I would like to try in my own classes, but two difficulties immediately come to mind:

1. Is it necessarily applicable to a math classroom? Although open-ended explorations in math are indeed possible and interesting, a vast majority of the math processes we practice in our classroom encourage kids to eliminate cluttering information that they don't need. Whereas physics encourages the learner to take something very simple and expand it into something complex and multi-layered and to consider all factors involved, math (in my mind) is more like a funnel that gets rid of all of the cluttering details and focuses in on only the most important details. Even much of the WCYDWT stuff is inherently begging a certain question to be asked, and then more or less inherently requiring some specific math strategy (with some real-world messiness, of course).

Or am I missing something here and my view of the goals of math teaching is really much too narrow? (I'm thinking out loud here. Feel free to jump in. My mind really isn't made up about this one way or the other. After all, it's clear that the early mathematicians never limited themselves to thinking about the fastest way to get from point A to point B.)

Anyway, personally, I think there is always value in "playing around" aimlessly with a problem, even if it doesn't immediately lead you to something productive. Sometimes questions arise that way and other times solutions arise when you least expect them to, just because you've meandered your way through most of the issues. Past a certain age, all the problems that are worth solving are not solvable in your head anyway, so some playing around is usually necessary and giving kids these non-questions encourages that line of open-ended thinking.

2. Let's assume that my first point was moot and that this technique is entirely applicable to the math classroom. The beauty of this process, as I have described above, is that you can give the kid the same situation at the beginning of the year, the middle of the year, and the end of the year, and their understanding should continue to build upon itself and to encompass all existing knowledge, plus brand-new knowledge. What types of problems would I be able to give that tie together all of the things they would learn in the course of a year, so that they could demonstrate understanding at all different levels? (Here I feel that physics has another natural advantage -- a bunch of the ideas you learn in physics are all interrelated.)

I have no easy answers. I'm going to have to think about this one, so that I could possibly implement something like it next year.

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PS. Speaking of good test problems, I recently put a question on my regular Geometry exam that nicely incorporated some old and new concepts all in one shot. The question gave the kids 3 different y=mx+b equations to graph, and asked them to find the angles inside the triangle that is formed from these 3 lines. In order to do that, they needed to know: 1. how to graph lines (old knowledge), 2. how to use pythagorean thorem/distance formula to find edge lengths of the triangle (old knowledge), 3. how to use inverse trig to find the angles within the triangle (new knowledge). Synthesis!