Other than Desmos, the most profound impact to my personal math development has the been Nix The Tricks, an excellent work assembled by Tina Cardone and the great people of the math teacher community. This project has been kicking around for a while, innocently started as a Google doc that went on for pages and pages. The strong opinions presented in that document show just how fed up some people are with the way math is taught. We all had the teacher that taught us a trick, we all know a colleague who relies on this stuff, and we've all had the student who brings a stubborn misconception because an instructor took the easy way out.
I am very guilty of a number of the short cuts analyzed by Nix The Tricks. I was taught some of them and in your first years of teacher you fall back on the way you learned. After all, you turned out alright.
Speaking for myself, I know why I relied on some of these short cuts in my teaching. You have that student who doesn't like math. No matter how many times you attempt an explanation, it doesn't click. And in frustration, the little shortcut surfaces. This quick fix will get Frustrating Student off my back right now. In the moment, a fair exchange. But what happens 2 months from now when they don't get it again? Will there be a trick to bail you out? Or will the prior use of tricks bring down the whole house of cards?
In my approach to Algebra II this year I knew we were going to tackle a ton of equation solving. Lots of moving terms, unlike denominators, multiplying both sides, taking the log of both sides, you name it. In the end, I wanted students with a super strong foundation. No set of tricks was going to make this possible.
I set two goals: never say "cancel" and never say "cross multiply." Try as hard as possible to prevent the students from saying those words either. If I force them to be proper about it early the growing pains will be worth it.
This small commitment made a big difference. Early on I made sure to slow down explanations. Every time we needed to clear a denominator I showed it as an operation of multiplying both sides by the relevant terms. Every time we took a square root I mentioned the presence of two solutions. Forcing myself to rely on the mathematically valid reasoning slowly influenced the students. They raised the level at which they explained concepts to each other. I didn't hear anyone looking for an easy way out. They wanted to do it my way.
By the time we ready for logs, no one broke a sweat when we had to take the log of both sides. By the time we were ready for rationals, no one broke a sweat when we're multiplying both sides by some nasty binomial. It made sense to them. It was totally easy.
This process became cyclical. As I saw my students rise to the challenge, I felt an obligation to them and the professional community to continue to focus on valid, universal mathematical explanations for things. Need a percent in decimal form? Divide it by 100. Say it with me, kids. I don't care what anyone taught you about moving decimals points around.
Every math teacher has room for growth. It's not just making lessons more entertaining, its making them truthful and shortcut free. If you are deliberate in your pursuit, the students will rise to the challenge. Tricks capitulate to student frustration. They cause problems that could last for years.
Beyond growth, every math teacher needs to stop pointing fingers. High school points to middle school. Middle school throws up its hands at elementary. If you are a math professional in any position to influence math education in your district, get those teachers Nix The Tricks. Have the meeting, don't take "I'm not a math person" as an excuse, and see if you can't stomp out the hungry crocodile once and for all.
Nix The Tricks is available (and updated regularly) for free as a PDF. Paperback copies are available for $9.99.