Mathematical Modeling: The Glass is Half-Full

Where is the horizontal half-way line?

By the time they get to me, the students in my calculus class have been given a chance to master a whole lot of math. Typically, though, they haven’t been exposed to many situations where the main challenge of the task is figuring out which types of mathematical tools will best model a problem, and thus, drive the method used to solve the problem.

Take, for example, the problem described here. This is a wonderful, challenging little task that seems so fascinatingly simple and yet becomes quite complicated quickly.

The task: draw a single horizontal line on the cup that represents the half-way line by volume. Almost as soon as I asked the question, you could see the wheels start spinning in the heads of the students.

“Well, it looks sort of like a cone, but it doesn’t have a point.”

Some took measurements and prepared to “calculate” it. But did they need a formula? How would they find it? What the heck kind of shape is this thing anyhow? How can we be sure out measurements were accurate?

Making sure measurements are accurate

Some were going to draw it onto paper. But how to they model it? Can they use a 2-D cross section? Which cross section do we use? What do we do with it now that we have it?

Modeled as a two-dimensional shape

Some figured to guess and check. I mean, the half-way is probably going to be somewhere in the middle. There’s not THAT many different values, really. But what do you guess? Do you have to guess more than one thing? How do check your guess to see if it is right?

And all groups had to deal with the question: When you have finished up the work on the page or on the calculator, how do you accurately transfer it back to the cup?

So, let’s get it out into the open: being able to mathematically model a problem that exists in your hands with math that has always existed in a book is not something that comes naturally to most people. In addition to the content, students need to practice modeling the mathematics. They need to learn what the book math looks like when it is in their hands. Jo Boaler (Stanford Math Ed Professor) has a wonderful line about this.

“…students do not only learn knowledge in mathematics classrooms, they learn a set of practices and these come to define their knowledge. If students ever reproduce standard methods they have been shown, then most of them will only learn that particular practice of procedural repetition, which has limited use outside the mathematics classroom” (pg. 126, see bottom for proper citation).

It’s as if the math experiences we are giving most students in class are the kinds of experiences that will do them the least good outside class.

Authentic mathematical modeling requires giving students a task with a simple and clear goal and then letting them decide what kind of math will help to complete the task. The decision is an incredibly important part of the process. They must recognize the variety of available tools, have to choose which ones will work, and of those ones, which one is the best choice.

The #MTBoS is doing a great job of providing tasks of this kind of a nature (Like here, for example… or here, as another example). They are plentiful, well-done, and FREE! Give them a shot and see what happens. We owe it to our students to provide them opportunities to take the math off of the pages of the book and let them see what it looks like when it shows up in their hands.

Reference

Boaler, Jo (2001). “Mathematical Modelling and New Theories of Learning.” Teaching Mathematics and Its Applications. Vol. 20. No. 3, p 121-128.