Legal and Useless Mathematics

Students need more of this in their lives.

Did you know that when you multiply an inequality by a negative number you need to reverse the direction of the sign to make sure all the right numbers are still solutions? Wild, right? Wrong! It is not wild, not to the kids I taught last week, most of whom were still trying to move “operations on inequalities” from the category of “magic mathematical incantations” to “really real things I can do to things that are real to me.”

I went in with a lesson the premise of which was, basically, “Can I help y’all understand a surprising fact about snorfblatters?” and the students informed me in so many words that, “Literally every fact you can tell me about a snorfblatter will be surprising to me.” They needed more experiences with snorfblatters, not a surprising fact about snorfblatters. Snorfblatters need to become real (by which I mean real in your mind) before you can find them surprising.

But I did have exactly one (1) win. When students are in the “magic mathematical incantations” mode, they often have the feeling that not only is there exactly one right answer to every question but there is exactly one right set of steps to get there.

Xavier was trying to help me solve 5x < 15 in front of the class and he told me to divide both sides by 3, probably rushing a couple steps ahead of himself. “Okay great what happens to both sides of the inequality,” I asked. He caught himself pretty quickly and told me to divide by 5 instead.

But it felt useful to ask the class at that moment, “Was Xavier’s move legal?” Of course, the idea that math might be legal or illegal was sufficient to draw a bunch more eyes to Xavier’s moment. We decided that, not only was Xavier’s move legal but that there were lots of possible legal moves.

This is an important fact! Students often think that every math problem offers one correct set of useful steps and then infinite incorrect steps. But we can disentangle those attributes and find steps that are correct and useless and even incorrect and useful!

When you’re trying to transform one shape onto another, go ahead and translate that shape left and then right the same amount. It’s useless but fine!

When you’re working out a mathematical argument, go ahead make some statements that are true to the math but not necessarily moving your argument along at maximum efficiency. No one’s gonna throw you in math jail!

Essentially, if we’d like students to think flexibly about mathematics—understanding all the different correct steps and when to apply them usefully—we can’t ignore opportunities to demonstrate the flexibility of mathematics itself.

BTW

• I am chastened to learn I wrote about this ten years ago as well. Incredible comments over there, particularly on the matter of “incorrect and useful math.”

• Bethany Lockhart Johnson and I are committing the entire upcoming Math Teacher Lounge season to a study of math anxiety–how it’s defined, what causes it, and what we can do about it. The first episode dropped this week.

2023 Apr 10

The comments downthread are truly fantastic and depict teachers who are utterly at home in their craft and their knowledge of math. A few highlights.

Peter W:

Yes!!!! I've been trying to find meaningful ways to work this into my curriculum. I have a sequence where I ask kids to... "Do the same thing to both sides of the equation such that..." with all types of non-standard goals. 2x + 5 = 12. Do the same thing to both sides of the equation until the left side is 10x + 10. Multiple paths, and a familiarity with the moves you can do to both sides of the equations. (And the practice, say, of multiplying both sides by 2, which doesn't come up a lot if you are just solving for x).

I used to get to completing the square (say, x^2 + 12 + 31= 2), and students would say... "Wait, how can you just add 5 to both sides of the equation? Where did the 5 come from?" Now students are more comfortable with the idea that they can do whatever they want as long as they do it to both sides.

Bion Shelden:

When a student proffers a "move" that the rest of the class thinks is the wrong move, I never challenge it, as long as he has phrased it as needing to be done to both sides. I just do it—even if it leads to some bizarre fractional coefficient—and ask someone for the next "move". If someone tries to say the previous move was "wrong", I inform them that anything they know how to do is permitted, except for dividing by zero. I think acting nonchalant about these "flubs" reduces anxiety about being willing to participate. What I have *not* done is use the term "legal" for this concept, but now I will.

Leandra Fernandez:

I'm not sure I agree that picking a number is "incorrect"- I think you're saying it's mathematically incorrect if you choose a number greater than or equal to 3. But if I chose 1 then it is correct! And I'd argue it's even moderately useful in getting me closer to a solution. I think I'd argue that choosing a "wrong" number (like 3) is actually quite useful as well, when you're starting out at least.