Walking through parking lots is interesting to me. Say what you will, but it occurred to me to bring it up in one of my math classes. What’s the shortest path through a parking lot?
Many shorter paths exist by walking diagonally through the lot.
To decide that this is absolutely the best path, mathematically, might take quite a bit of work. I actually never got around to figuring out how I would represent it mathematically – it’s just so much easier to plug it into Google Maps and use the measurement tool. In the field, you can kind of just pick a direction and things work out ok. The hours of work to represent and solve this as a math problem are not worth the effort – the penalty of guessing and being a little wrong is, what, 15 seconds?
And yet, a lot of math problems – especially the “applications” or “real world” problems – ask students to do a tremendous amount of work to get an answer that they could have guessed at in a few seconds.
The Mystery of the Paper Box
“You’re making a box out of paper,” the textbook explains. Your hands clench and your knuckles whiten. Should the box have a top, or will it be open? How much paper will you get? How much does the paper cost… per square inch?!?
“The box has no top.” You hold your breath, waiting for the next blow. “You have a piece of paper that’s 8.5 x 11 and you’re going to cut four squares out of it.” Your mind is racing. The words form in your mind almost before you can read them off the page: “What are the dimensions of the box with the largest possible area?”
When you flip to the back of the book, you see that the publisher recommends creating a formula that represents the area of the eventual box in terms of the length of the sides of the squares you’ll cut out. Find the derivative of that equation, find the zero of that derivative, and use that zero to determine the final dimensions. The best solution in many students’ minds, however, is to guess a length between one and three inches. You’d probably be within 10% of the perfect answer, and you’d finish ten minutes sooner. If you suggest that a different solution is best, you’d better have a very apparent benefit analysis ready, or your students will stop trusting you, check out of your class, and start asking “how much are these problems worth in our grades?”
The benefit of learning about derivatives (or skill xyz) is not apparent when the penalty for guessing is so low. One solution some books attempt is to making the units bigger. A farmer is mowing his field, and wants to find the best path, given that his baler can hold 5 tons of hay. Now we’re not talking about 15 seconds: this could save a full day of work! This is much more interesting, except:
- By guessing, the farmer can get pretty close to the optimal solution right away. After a few harvests, he or she will have refined it to the best path. Your computation forgot to factor in the turning radius of the tractor and the soggy ground in the southwest corner, so he or she would probably have to make these adjustments anyway.
- It seems possible that none of your students will ever harvest hay in their entire lives.
My Best Solution (TL;DR)
The goal is to make the benefit of learning <skill x> apparent to students so that they will want to learn it. The best solution I’ve ever found is in showing students how to program a computer to do all of the textbook problems for them. Geogebra is a programming environment that students can use effectively after a few minutes of training, and in my experience people love making computers do work for them. The math comes to the forefront because now we’re making paper boxes out of a piece of paper of any size. Now the students aren’t learning math to figure out what size to make a box (BORING) but instead they’re learning math to figure out how to get a computer to do their homework (FUN). They’re not doing work by hand for twenty similar problems (TEDIOUS), they’re generalizing and finding patterns that the computer can make visual for them (ILLUMINATING). They’re not talking about farmers or engineers doing some problem (IRRELEVANT AND OVERLY SIMPLIFIED) but they’re learning to identify patterns and make computers work for them (REAL-LIFE DETAIL, OVERWHELMINGLY RELEVANT).
David Cox has a lot of posts about Geogebra, and I’ve written a few other articles explaining different parts of geogebra too. If you teach math or any kind of numerical analysis, I implore you to learn to use Geogebra or another programming environment and bring it into your classrooms. As long as the electricity and lithium hold up (fingers crossed) programming is going to be hugely important to your students. As an added benefit, programming will let you go farther and deeper in your classes because students will get that delicious, stone-cold, 100% accurate feedback that only a computer can give.