I want to talk about a misconception I’ve seen this last week that I’m struggling with. I’m calling it the difference between arithmetic and algebraic thinking.
I teach 8th grade math, and we’ve spent two weeks thinking about 3-D geometry. One basic idea is to be able to find the volumes of spheres, cylinders and cones (in fact, that is the entirety of the official Common Core 8th grade 3-D geometry standards). Kids are pretty solid at the basics with these formulas.
First curveball: I ask them to calculate in terms of pi. Apparently, this is the first time they have been asked to do so. They’re flabbergasted. I literally had students asking me if they could write their answer as a decimal instead. I showed them how much easier it is to do things in terms of pi. They still wanted to write their answer as a decimal, and sometimes still did it when I asked them to work in terms of pi.
In looking at student work, I found a divide. Students who were comfortable manipulating pi generally started with the formula for volume, substituted, and calculated from there. Students who did not want to use pi generally started multiplying left to right, without showing the formula. For instance, when finding the volume of a cylinder, they would start with 3.14 times the radius squared, multiply it out, then directly below multiply by the height. This showed two misconceptions:
1. The commutative property of multiplication
2. The idea of substituting into a formula
With significant perseverance and strongarming, I convinced students that writing answers in terms of pi wouldn’t hurt them, and we got through it. Then, I started to mix things up. I gave them the volume of a sphere and asked for the radius. I told them a cylinder and cone had the same radius and the cylinder was twice as tall; how many times greater was its volume?
They were no stunned into complete mathematical paralysis. But the questions I got as we worked through questions was fascinating. For instance, on going from the volume of a sphere to its radius:
Student: So you multiply by 3/4 and then take the cube root?
Me: No, you use the formula!
Or, on the cone/cylinder question
Student: So you multiply by 3?
Me: No, you use the formula!
Sarcasm aside, I’ve been seeing what I call this fundamental misconception about arithmetic vs algebraic mathematics.
Students who think arithmetically think of each problem as a puzzle to solve. The puzzle is: what operation is it? Is it multiplication? Is it division? Is it the square root? Is it really really complicated, and you have to square, then square, then add, then square root? (see if you can figure that one out). This can work well through much of middle school mathematics. Many proportions can be memorized as arithmetic questions if you do enough. Fractions are just multi-step arithmetic problems. But students need algebraic thinking as well, and it can be hard to create an intellectual need for variables if so many questions we want students to solve can be answered quickly and efficiently without.
I’ll leave you with one link from Hung-Hsi Wu: