Bringing Addition to a Multiplication Party

I just realized that two things I had thought to be quite different might, actually, be really similar.

First, a series of mistakes my 4th Graders make when they use addition thinking for multiplication problems:

Second, my 4th Graders’ thought that multiplication by a negative would make a number positive, but smaller:

Yesterday a couple of 4th Graders asked, “Wait can you multiply by a negative?”

Any guesses as to what prompted this question?

Kids had been working on a multiplication puzzle and (accidentally) gotten themselves into a position where they needed to solve ___ x 20 = 10. If positive numbers make multiplication bigger, then shouldn’t negative multiplication make things smaller?

What is this mistake? Why should multiplication by a negative make a number smaller, but positive?

Here’s what I’m realizing: it comes from the thought that positive/negatives have opposite effects in multiplication/division. Which isn’t true, but it is true that positives/negatives have opposite effects in addition/subtraction.

The relevant opposites when it comes to multiplying aren’t positives/negatives, but instead numbers greater/less than 1. To draw the contrast really clearly, when it comes to adding the relevant opposites are numbers greater/less than 0.

This is not some out-there and abstract idea, though. When kids work with negative numbers they regularly reveal an understanding that positives and negatives should have opposite effects, as with 3 – (-5):

We talk a lot about opposite operations, but do we talk enough about opposite numbers? We talk a lot about negatives as opposites to positives, but do we talk enough about numbers less than 1 as opposites to numbers greater than 1? How much of learning is trying to figure out the limits of thinking like addition?