By the time the students get to my pre-algebra class, they’ve learned "facts" about multiplication and division that lock them in, and actually make it harder to learn the next bits. The misconceptions or limitations in their thinking show up when we begin to solve equations that involve fraction answers or operations. I wonder if we shouldn’t start differently at the beginning. From simplest to most subtle, these are three of the misconceptions that cause the most trouble:

I will tackle each one in its own post.

1) There are some numbers that don't go into other numbers. This shows up when we get to 5x = 13, and the students tell me “5 doesn’t go into 13.” We use bar models before we use algebraic notation, so the problem looks like this:

- There are some numbers that don’t go into other numbers.
- Multiplication and division are opposites.
- Multiplication is the same forwards and backwards.

I will tackle each one in its own post.

1) There are some numbers that don't go into other numbers. This shows up when we get to 5x = 13, and the students tell me “5 doesn’t go into 13.” We use bar models before we use algebraic notation, so the problem looks like this:

And with the problem visually represented this way, the students can conclude for themselves that dividing by 5 will get them the value of one of the unknown units. They will happily answer when the division comes out evenly, but when it doesn’t, some will report “I don’t know what to do.” (Have you noticed this? Sometimes, rather than say “I know I have divide by 5 but I don’t know how” the students’ brains seem to back out of what appears to be a dead-end, and deny that it ever existed in the first place. … Makes me wonder about my own invisible dead-ends.)

I have learned to frontload work with division problems with fraction answers at the very start of the year so this doesn’t happen. We use things like granola bars and sandwiches and we also use bar models to explore cutting things up. The students need to see what it means to divide 13 into 5 equal shares, or, for 5x = 3/7, how to divide 3/7 into 5 equal shares.

In a later post, I will talk about the problem with seeing 5 • 3 as the same as 3 • 5, but it comes up here. There is another way to interpret 13 ÷ 5... or more accurately, 13 ÷ 5 can also represent this question: how many times does 5 go into 13? Again, this is an easy question that becomes more challenging once fractions are involved. The students need time to explore what it means for 5 to go into 13 two and three fifths times.

And here is the rub: one might ask why bother with all of the exploration and confusion? Why not just teach the students how to multiply and divide fractions and be done with it? Because that's not mathematics (see NIxtheTricks). The symbols represent mathematical situations and mathematical thinking. If students can only apply the rules to a problem you wrote on the page, and they can't think about it for themselves, or use the rules appropriately in a situation or word problem, then they haven't really learned anything useful. Division and multiplication describe what we do to stuff, and the symbols need to represent those operations, not replace them with math facts. We will see just what those operations are in the next post.

I have learned to frontload work with division problems with fraction answers at the very start of the year so this doesn’t happen. We use things like granola bars and sandwiches and we also use bar models to explore cutting things up. The students need to see what it means to divide 13 into 5 equal shares, or, for 5x = 3/7, how to divide 3/7 into 5 equal shares.

In a later post, I will talk about the problem with seeing 5 • 3 as the same as 3 • 5, but it comes up here. There is another way to interpret 13 ÷ 5... or more accurately, 13 ÷ 5 can also represent this question: how many times does 5 go into 13? Again, this is an easy question that becomes more challenging once fractions are involved. The students need time to explore what it means for 5 to go into 13 two and three fifths times.

And here is the rub: one might ask why bother with all of the exploration and confusion? Why not just teach the students how to multiply and divide fractions and be done with it? Because that's not mathematics (see NIxtheTricks). The symbols represent mathematical situations and mathematical thinking. If students can only apply the rules to a problem you wrote on the page, and they can't think about it for themselves, or use the rules appropriately in a situation or word problem, then they haven't really learned anything useful. Division and multiplication describe what we do to stuff, and the symbols need to represent those operations, not replace them with math facts. We will see just what those operations are in the next post.