The Makings of Mathematical Mistakes

In my years of teaching math, I've amused myself by taking note of the types of mistakes students make. (Yeah, okay. I'm easily amused.) You can pretty much figure out the type of mistake by watching my reaction.

The "Fell Through the Cracks" Mistake

This usually happens in complex, multi-step problems. The student does all the hard stuff right but overlooks something. More often than not, it's losing a negative or mistyping something in the calculator.

R.C.'s Reaction: I just point wordlessly at the paper or calculator and wait while the student looks, ponders, then says, "Oh! Oops."

The "You Know Better" Mistake

Another "careless" variety of mistake. Can I tell you how many times I've asked what four-squared is only to hear, "Eight. No—wait! Sorry. Sixteen."

R.C.'s Reaction: Students often catch those without any help from me. When they don't, they get my 'Did you seriously just say that?' look. If that's not enough, they get a verbal, "Really?"

The "You're Still Learning" Mistake

This happens when students are mostly getting a new concept but aren't quite there yet. OR ... when they have to apply something they learned previously that hasn't quite solidified.

R.C.'s Reaction: Usually I ask them to explain their thinking first, then ask some follow-up questions until they see the wrong turn. Sometimes a neighboring student will try to tease the other about the mistake, at which point I remind them that they made the exact same mistake two minutes ago when I was helping them.

The "Someone in Your Past Failed Both of Us" Mistake

I teach high school math, which naturally relies on concepts learned over several years before arriving in my class. Sometimes we're working on some complicated algebraic thing and I realize some/all of the students have a problem with an underlying principle. (Fractions, anyone? Or measurement conversions?)

R.C.'s Reaction: What can I do? Go off to the side of our work and make up a simplified example (i.e., non-algebraic addition of fractions), quickly refresh the kids' memories on that, and parallel it to the problem at hand.

The "Back the Truck Up" Mistake

These mistakes on the part of the student tell me that I screwed something up as the teacher. Didn't explain clearly, allowed for a massive misconception to take root, etc. Sometimes I even did something just plain wrong.

R.C.'s Reaction: Confess to the class that I made a boo-boo, very clearly indicate where we went wrong, and emphasize the proper way to move forward.

Some people might say it's a teacher's job to eliminate mistakes and a student's job to avoid them. I don't agree with that. Mistakes are great! They're how we learn. (Well, so they're great as long as we learn from them.) And one thing to keep in mind is that I have extremely small classes, and I've taught most of my students for more than one year, some for up to five straight. My reactions to the "Fell Through the Cracks" and "You Know Better" varieties are done in an environment where the students and I are able to laugh off mistakes without embarrassment. (And where I'll accept "It's calculus on a Monday morning," as an excuse for the careless mistakes as long as they keep trying.)

I know some people who were always terrified to volunteer information in class, certain they'd make a mistake. I was one of them. Now, I'm okay with making mistakes in the classroom. Still working on being okay with it in the rest of my life.