Nothing gets a group of science teachers riled up like a mnemonic device that allows students to hold on to misconceptions or keeps students from advancing beyond them. The “density triangle,” for example, gets a lot of hate from teachers, myself included.
The argument is that students will use this crutch to solve problems without understanding the math that makes it work. The approach works in a narrow set of cases, namely for equations of the form y = mx, and it keeps students from practicing their algebra skills, such as rearranging equations, in the science classroom. The density triangle also keeps students from deep conceptual understanding.
This Tweet, responding to “Why are triangles bad for learning equations?” gave me pause a while back:
I think those two statements summarize the debate nicely. I personally don’t teach the density triangle in my classroom, but if a student brings it up because a previous teacher taught it, I politely tell them it’s a valid approach. I’m willing to meet them where they are, so if the density triangle is where they are at in their math lives, they can keep using it.
I will tell these students, however, that they have come a long way in their math skills since they learned that trick. I challenge them to grow further and try thinking of it other ways too. By the time we get to molarity in the spring, they’re much more comfortable with proportional and algebraic reasoning. No one needs the equivalent “molarity triangle.”*
Cross-multiplying is a related trick that gets me. A different mnemonic device that has bugged me in chemistry is this one for writing the formulas of ionic compounds…
I saw this on a student’s paper a while back and went to rant to my husband. I described how students get stuck on all these crutches, don’t get the underlying concepts, and don’t grow beyond them. To my shock, he disagreed.
You gotta understand that my husband, a mathematician,* will not let any claim, no matter how innocuous, go unchallenged. He’s always trying to see where an idea will break, what circumstances will cause a proposal to fail. I was so sure that he would want students to be able to solve equations with “legit” algebra and would extend this expectation to chemistry formulas.
Instead, he argued that it doesn’t matter if students don’t progress past these tricks if they don’t need to right now (reminiscent of the Tweet above). Shortcuts like the crossing-charges trick allow for fast processing. “What’s the formula? How do we find it?” Once these questions are readily answered, when students are secure in the what and how, you can talk about why they work.
It takes a lot of rote work to reach deep understanding. (Though rote work alone won’t get you there!) My husband reminded me of the challenges I faced teaching algebra to students who didn’t know all their multiplication tables. They couldn’t do the upper level stuff because they didn’t have a strong foundation.
Three phrases stuck with me that day:
WHAT and HOW before WHY
ROTE before CONCEPTUAL
and his favorite…
Fake it till you make it
The process he described is the opposite of what I would do in a traditional chemistry lecture: explain the concept and why it works, then use the “why” to explain the problem-solving approach. For example, teach about ions, common charges, and forming neutral ionic compounds (the WHY), then use that overload of information to introduce the process for writing chemical formulas (the WHAT and HOW).
My husband blew my mind with that conversation. Two hours later I was still trying to process what felt like a paradigm shift. I’m still working through this and debating myself constantly.
Then today I met a student over Zoom to talk about some struggles she was having with the material we learned before spring break – writing ionic formulas and writing chemical reactions – and I watched what my husband described in action.
It was incredible. I had to apologize to the student and pause to take notes a couple of times. I’m so glad it was Zoom recorded because I’m going to need to go back and listen to it again.
Because that crutch is just a model. What do we do with models? We deploy them until they break. And then we replace them with better models.
Come back next time and I’ll tell you about that conversation.
Do you help students grow past using these mnemonic devices? How do you do it? Why? Let me know in the comments!