Are We Teaching Tricks or Just Different Math?

On technology and high-stakes assessments.

I will never ever close my DMs on Twitter:

Dan, do you ever cry at night when you find out about how teachers are misusing Desmos to teach their students “tricks” to get them through state testing?

This has come up a few times recently. Here is what is happening. The majority of state-level end-of-course exams in the United States include a button that opens up the Desmos Graphing Calculator, which is cool for lots of reasons, maybe the biggest of which is that students can use that calculator for free during the entire year.

Another reason that's cool is the Desmos Graphing Calculator will very quickly produce a graph when you type a function into it and it is very easy to type a function into the Desmos Graphing Calculator.

So if you're preparing students to answer this item from one particular state's EOC exam, you have two common options and then one that begs the question.

1. Teach students to factor trinomials—for example 3+n-2n^2 = (1+n)(3-2n) in the numerator. Making a one from (1+n)/(1+n) gives us (3-2n) as an equivalent expression.

2. Teach students to multiply binomials—for example multiplying the denominator (1+n) by each of the four options until you get the numerator.

3. Teach students that graphs are great. A graph is a picture of all of the (x,y) pairs that make a statement about numbers true. So if you graph the question and then graph each of the four options, the correct option is the one that produces the same graph. (Mumbles about exceptions.)

Or here.

You can teach students solution methods that involve substituting variables or eliminating variables or evaluating points (toss all four points into the system to see which one is true in both equations) or, again, that graphs are great?

The person who DM'd me on Twitter is frustrated that teachers are teaching students that graphs are great here and I'm trying—honestly I'm trying—to get upset. But I don't think this is a situation where students are learning tricks—a series of steps which produce correct answers while making mathematical ideas less comprehensible. 

Teachers are using technology here to make certain mathematical ideas more comprehensible to students. Big ideas, I'd argue. Ideas about equivalence and representation, for example. And, yes, they're minimizing others, but I'd argue those ideas are smaller. Ideas about symbolic manipulation, for example, even if they’re ideas that the people who wrote this assessment perhaps meant to assess.

You're welcome to share your perspective in the comments here, but this seems like yet another moment where advances in technology are inviting us to reconsider the kinds of mathematics students need for a rich intellectual life. Let's not turn down the invitation.