Why (most) word problems are not modeling

…and why somebody might try to convince you they are.

It’s even in the Core Standards. This is taken out of context—but not very far:

A model can be very simple, such as writing total cost as a product of unit price and number bought… (Common Core, p 72)

Seriously?

Okay, I could make a case for it, but I won’t.

I’m becoming more convinced that the real hallmark of modeling is simplification  (see this post for more).

Modeling is not simply using math on real-world problems, though that is a Good Thing; you can model to help with pure math as well. And I bet we could find good real-world problems that don’t involve modeling.

But back to simplification. The key element (I believe this afternoon anyway) is taking something and using math in a way that makes it simpler, less complicated than the thing itself. We model to make things tractable. We can handle the model even when the thing it represents it too complicated. If it’s a good model, it captures the essence of what we’re looking at; and exactly what that means may depend on the specific context.

  • We might model a hexnut as a hexagonal prism with a cylindrical hole, and use that geometrical model to find a volume. We avoid the threads and the easing on the corners: they’re too complicated—but we hope our model captures the essence of the hexnut.
  • We might model some messy data as a line or a curve. We can’t make a reasonable prediction from the mess, but we can with a function: just plug in a value and calculate.
  • We might explore the behavior of a system of linked differential equations by creating a numerical model, a system of difference equations we can evaluate on a computer. It’s conceptually simpler (for the computer at least), so we sacrifice some precision for tractability.
  • We might even take all the complexity of Americans and do a Census. When we do, we create a data model: the structure for the information we will collect. We have only approximately captured the people’s information (this is the Census, right, not the NSA). We hope our data has the essence that we need to know—but there is a huge amount of detail that we have ignored. Like the threads, like the deviations in the scatter plot, like the inaccuracies in the numerical model.

What does this have to do with word problems?

Suppose we ask, if Eduardo buys four cans of orange juice for $2.49 a can, how much does he pay altogether?

There is math here, no question. We can argue whether it’s real life.

But it doesn’t involve simplification. All of the information is present. There is no model, and no need for one.