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.

Fair Sharing: Developing Division

 Zoe, my daughter in Kindergarten, was working on a math problem yesterday. She had to share 8 cookies between 2 people. She split each cookie in half. Was she wrong? Technically no. But I asked her, “if you and I were going to share 8 cookies, would you break each cookie in half and share them that way?”

“No, I wouldn’t do that,” she replied.

I pulled out 8 double sided counters (those round little red and yellow chips). “Show me how you would share the 8 cookies between us.”

“One for me. One for you. One for me. One for you...” She continued until she had two piles of 4. “We each get 4 cookies!”

This is a concept known as fair sharing. This is how division begins in early grades. In many cases, children may even learn this before grouping (multiplication), as sharing is more of a pressing notion than making groups. 😁

When time comes to really dig into division later in elementary school, children have had plenty of work with place value and multiplication under their belts, so their first experience may look something similar to the first picture. Can they divide by subtracting “groups” from the number. For 1,375 ÷ 5, they likely know that 5 × 100 = 500 and 5 × 10 = 50, from work with multiplication and place value, so they may start by removing those groups first. Work with money in prior grades will likely bring 5 × 20 = 100 to mind as well. Eventually, when they add up that far right column, they see that 1,375 ÷ 5 = 275. This looks like a real ugly and clunky approach, but it allows them to make sense of the whole idea of division/long division. Again, we don’t want memorization of mindless procedures, we want sense making . This builds toward the long division procedure (and builds mental math in the process).

Eventually we work toward methods like those shown below, more “formal” long division algorithms. Now we are attempting to find the largest possible grouping to subtract out each time. Students will likely already be working toward this as they gain more “finesse” through exploration with the prior methodology (picture 1). I personally prefer the “box method." It takes up less paper and I think it allows for a better visual understanding of the place value aspects of division.

What's the Deal with Multiplication Arrays?

Let’s talk multiplication. I’ve seen people losing their minds over seeing children using the multiplication array on the bottom, and there is no reason.

If we go back to the early grades multiplication is often introduced through grouping. Children concretely group things to multiply. “How many marbles are there if we have four groups of three marbles?” Eventually we want to move to more representational and abstract approaches, and like with subtraction, we want to build off of the heavy place value emphasis from Kindergarten and First Grade. That’s where that multiplication array comes in. Children know that 47 is 40 + 7 and that 23 is 20 + 3. Multiplying each of the pieces is much easier since they have worked with single digit “times tables” and multiplying by 10s is also easy. And the four numbers can be easily added. 40 x 20 = 800, 40 x 3 = 120, 7 x 20 = 140, 7 x 3 = 21, 800 + 120 + 140 + 21 = 1,081. Comparing to the standard algorithm at the top, the square array is the same approach, but it is easier for the children to see what is actually happening. It’s a conceptual stepping stone to the standard algorithm so they can make sense of multi-digit multiplication using place value.

The arrays also help build mental math, which as I said before, is an important goal. With a problem like 25 x 12, starting with the array allows them to get to point of doing something like “25 x 12 is 25 x 10 and 25 x 2. 25 x 10 = 250 and 25 x 2 = 50. 250 + 50 = 300.” Again, looks like a lot written, but takes bit a few second in your head. The array visual helps improve mental multiplication.

Lastly, progression in math is important. Eventually students get to algebra and “more abstract” math in later grades. Arrays are used to multiply polynomials, for example, as in the last pic. Having already seen and used arrays for multiplication allows the student to make a conceptual connection to the algebra. They see multiplication is the same in all facets. We’re building coherence.

These arrays are not new math or a new way to multiply. They are a tool for understanding and visualizing the concept of multiplying and for building mental math. (And they’ve actually been around a long time.)

Is the Standard Algorithm Always Best?

I always see comments from parents on the way their children learn math. The way math is taught now should look rightfully different. The way math was taught in the past focused too much on memorizing procedures and steps. Conceptual understanding and application often took a back seat for procedural fluency. All three are equally important in math, but conceptual understanding and application should be prioritized first in my opinion. If someone understands those two pieces well, the procedural piece will likely come easily... or even on its own without instruction on it. The standard procedure we were all taught isn’t even always the best way.

An example... borrowing. Who likes borrowing in subtraction? It’s annoying. If someone has number sense, a conceptual understanding of how numbers work, it’s unnecessary. Take the example below. 1000 - 843. The whole borrowing from zero thing that was always taught is cumbersome. That’s a lot of borrowing and time to subtract with the standard algorithm. Let’s ponder the idea of differences for a moment. 10 - 6 = 4, 9 - 5 = 4, 8 - 4 = 4, 7 - 3 = 4. What do we notice? When both numbers are reduced by 1, the difference is still 4. Shifting both numbers by the same amount preserves the difference! So forget borrowing; use that number sense. Instead of 1000 - 843, let’s reduce both by 1 and do 999 - 842 instead. Much faster! 157!

Also, those arrow strings you may see elementary students doing. We in math education call that “counting on.” It is a way to visualize subtraction using all the place value they worked with in Kindergarten and first grade, and it’s also how the brain can easily process subtraction mentally. Part of the goal is to build better mental math. Now it looks like a lot on paper because children need to write it out and visualize it at first. But over time they can REALLY quickly in their heads do... “843 plus 7 is 850. 50 more is 900. 100 more is 1000. So, 100 + 50 + 7 is 157.” That seems like a lot written out, but that takes mere seconds in one’s head.

The “standard algorithm” isn’t always the best approach. Conceptual number sense is key.