Don't Memorize Formulas! Understand Their Structure

I hate the notion of memorizing formulas. Formulas do not need to be memorized. They need to be understood. A major focus of the high school mathematics standards is the idea of “seeing structure in expressions.” This key idea extends down into the lower grades as well. What does this mean? It means you understand what each piece of the expression or equation means or does. You can read an expression or equation almost like you would read a sentence.

In honor of π Day earlier this week, let’s use some of the relationships of π with circles to illustrate this. If we think back to how we learned circumference and area of circles back in school, there’s a good chance it came down to being told the formulas straight out and then shown how to plug values in to use the formulas in a few examples. Then we were given a ridiculous amount of skill based problems to practice. This is a horrible way to teach this... or, well... anything in mathematics really. Students should figure out formulas and concepts on their own through exploration and sense making... with the teacher questioning and guiding.

Let’s look at circumference. Give a class of students a bunch of circular objects. They measure the diameters and record them. They then wrap string around the objects, cut the length, straighten them out and measure. Now they compare the circumference with the diameter. What do they notice? Through examining all the data collected and some discussion, they notice that the circumference is always a little more than 3 times the diameter. As they look more closely at the data, they eventually get more specific and see that the circumference is precisely π times more than the diameter. The diameter can be wrapped around the circle π times. They then translate that sentence into a formula, C = πd.

Rather than being told, they make sense of the relationship and build the formulas themselves. And since they built it themselves, they don’t need to memorize it. They had an “a-ha”! They understand the concept and the formula is just a mathematical way of showing the concept. If you get the meaning, you don’t need to memorize formulas... because the idea makes sense.

Similarly, they can explore area of circles. They are directed to make a square out of the two radiuses (or radii) as shown in the one picture. They find the area of the square (radius times radius) and then painstakingly find the area of the whole circle by counting grid squares. They compare these two areas and once again see that the area of the circle is π times the area of the square. The square made by the radii fits into the circle exactly π times, hence A = πr².

Memorizing is typically never necessary if there is sense making. If you understand the concept, you can build the formula, because you understand the relationships and structure.