Learning Strategies

Three different experiences in Algebra II today have caused me to rethink the value of projects. Although I’ve always had a theoretical appreciation of project-based learning, I’ve also always had grave doubts about placing a whole lot of emphasis on projects. These doubts have arisen from several sources:

• Sometimes very little (if any) actual math is learned. It happens too often that kids devote most of their attention to how the project looks or to ancillary materials, rather than to learning mathematics.


• Sometimes, as suggested in my earlier post on homework, the bulk of the work on a project is actually done by a parent, a friend, or a tutor, not by the student who is receiving credit for the work.


• Sometimes I have very little control over the structure and content of the math that is learned, so it seems less valuable than the tightly organized material of regular worksheets, assignments, and related activities.

In consequence, I tend to assign very few projects and don’t usually count them very heavily.

So what happened today? The first experience concerned our recent test rather than the project that the students are currently pursuing. I quote a question here in its entirety:

ABC News reported that “Australian polar scientist Professor Patrick Quilty thinks he has a pretty cool idea. He wants to move Antarctic icebergs around the world for use as a source of water... Professor Quilty reckons it can be done by wrapping icebergs in huge, and he means huge, plastic bags and towing them to places like Africa where water is a scarce commodity.”

The bags are useful because the iceberg would start melting on its way from Antarctica to Africa, and the towing would take a long time. (Icebergs are very heavy!)

Suppose the amount of water in an iceberg decays exponentially at the rate of 4% per month, so that the volume is always 96% of the previous month’s volume.

1. You need to tow a small iceberg, with a volume of one million cubic meters, from Antarctica to Mali. You have decided not to use Quilty’s bagging idea; you’re just going to accept the loss
   of the melted water. The journey will take eleven months. How many cubic meters of water will remain in the iceberg when it arrives in Mali? [Be sure to show all your work clearly.]


2. Why is it unrealistic to assume that there’s a constant decay rate all the way from Antarctica to Africa?


3. One-point extra-credit question for those who are good at geography: Why will you have particular difficulty towing an iceberg to Mali (rather than, say, to Morocco)?

Two different students asked, “What does this have to do with math?” (I’m sure many more than two were thinking the same thing, but were reluctant to verbalize the question.)

My response was that one of our main concerns in math is making mathematical models of real-world phenomena, and it’s impossible to judge the accuracy of the model unless you understand the real-world constraints. I don’t know whether this satisfied anyone, but IMHO it’s clearly correct.

The second experience also came from the test rather than the project. Many students (not all, but a clear majority) found the test unreasonably difficult. They were made very uneasy about questions that required them to think differently (or is it “think different”?), such as one where they had to find the half-life of a radioactive substance based on a couple of data points. It’s not at all surprising that they are far happier when given a worksheet that contains 12 equations, where they are explicitly told to solve the first four by Method A, the next four by Method B, and the next four by the method of their choice (though that last part makes some people nervous, especially since a couple required Method C).

The third experience did come from the project and relates to the first. You’ll see if you look at the project description (as I said earlier, I cannot take credit for writing this scenario) that the requirements are slightly fanciful but still are most definitely a combination of pure math, applied math, and non-math. Even the fanciful parts definitely qualify as “real world.” Some students enjoy working on this (in their groups of three), others want more direction (“answer these questions by Method B...”), and some just grumble and put up with it. But the same issue arises here as happened with the melting iceberg: what does this have to do with math? A couple of students verbalized this question explicitly.

I guess the reason I find this so frustrating is that the very same students also ask, “When will I ever use this in real life?” Apparently we can’t win. If we just give them an algorithm to carry out, they will do it cheerfully, but then they think it’s irrelevant. If we give them something relevant to think through, they complain that it’s not math.

Nevertheless, I have been extremely pleased with the way almost all of my students have been grappling with the challenging questions of the project. We have been devoting a lot of classtime and a little bit of homework time to this endeavor, and it’s clearly making most of these Algebra II students think much more deeply about exponential functions, decay, logarithms, and so forth. I have absolutely no doubt that the activity is a worth-while learning experience, well worth the amount of time devoted to it. That eliminates concern #1 listed above. Doing the work mostly in class eliminates concern #2. And having a tightly structured list of requirements for the project eliminates concern #3.

I think I’m going to go further in this direction.

Labels: math, teaching and learning, Weston