One fundamental problem: in Algebra 2 there are few good examples and fewer (zero?) relevant to my students’ daily lives. The solution, so far, has been elusive; wrapping a context around the equation being studied or finding a situation that you can help your students explore in a mathematical way are my best attempts so far.
Let me extract what I mean from that jungle of punctuation and parenthesis. My goal is to help my students see how they can superimpose mathematical models upon their reality to help them do stuff. Unfortunately, at the level of math that I am supposed to be teaching, their skill level with the stuff is actually still too low to be useful in any kind of exploratory way – they know just enough to do what I’m doing and see that it works. That is, they aren’t going to stumble upon the formulation of a logistic population model by themselves, but if I show them such a model they can understand what it means and how to manipulate it.
If we take this too far, we run in to the problem of being overly specific in our instruction and robbing our students of the chance to be creative or even inquisitive. When we are talking about volumes of circular cylinders we have to narrow it down for them, and often our only way of doing so is to narrow it down all the way. ”You see, kids, since , (or even just ) and , the volume is… I mean, what is the volume, kids? And why do you all look so bored?”
But what choice do we have? Real cylinders don’t actually follow this law, when measured by students – they fall somewhere within 5 and 500%, depending on the quality of measuring going on. Also, measuring real cylinders takes like five hundred hours. Are you going to let your kids use an entire hour to get the five data points they need to see a quadratic pattern? Even if you know some of those points will be completely wrong?
Computer models offer us the opportunity to narrow the problem space without narrowing it right down to the nub of the solution. Without going so far as to say that every math teacher should be able to program a computer (I’ll do that later), I’d like to extol some of the benefits of modeling problems with something like geogebra.
Consider the following geogebra applet, which took me about 10 minutes to cook up (I’m a geogebra whiz, though). Drag the bottom right corner of the cylinder to change the radius of the cylinder.
Here, I can let my kids explore! There is a confined space that I have carefully constructed for them and within that space there are no rules. They can do whatever they want. In this universe I can ask questions like “Can you find a cylinder with a volume of 2000?” or “what happens when you double the radius of a cylinder?” I can ask them to make a graph of volumes against radii, and they will enjoy doing it (you may have to trust me on this if you haven’t used an interactive program like geogebra with your class yet).
And from this problem space they can actually figure out the formula for a cylinder with a height of 10. They can try many experiments very quickly, and with satisfying proficiency. When they make the graph of volume vs.
height radius, they will do so with a speed that lets them really enjoy that the shape it makes is a parabola. And the discussion taking you from cylinders with heights of 10 to cylinders of height 20, to cylinders of height 5, to cylinders of height h will be delighting because they not only know what you are talking about but freaking invented what you are talking about.
I haven’t even talked about the best part yet, you guys. The purest virtue remains unsung, if you can believe it. So, here I come with the biggest freaking bell-ringing mallet I can find: the feedback kids get from their computers is absolutely unjudging, unbiased, unhelpful, instant, and 100% correct. Kids don’t have relationships with their computers. Kids don’t ask computers for help, and computers mercilessly avoid volunteering any. There is no “Clever Hans” effect happening with computers. Kids can try an idea and if they’re wrong their computer shows them so without the slightest hint about why. So, guess where that why has to come from?
Whew. More on this later (with more examples, hopefully).
PS: As I was writing this post, one of my students submitted an honors project from last month (due any time in the next month). Drag the points around. Do you have any doubt that this student thoroughly understands how to graph an exponential function through two points? She created this file from scratch. Double-click a and b to see how she did it.
PPS: For teachers with students more or less skilled than mine: you can adjust any problem space tighter or looser, right? If your kids don’t understand graphing or squaring yet, let them vary the height of the cylinder instead of the radius. If your kids were talking about volumes of cylinders with their parents when they were 8, let them vary the height and the radius at the same time. The basic benefits remain.
PPPS: Bonus points for anyone who can concisely explain how to make a cylinder like that in geogebra.