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?

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Great connection! And this shift from additive thinking to multiplicative thinking is really the hallmark of upper elementary mathematics. Not always a smooth ride as we all know. Our work with fractions is more of the same.

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My high school kids are struggling with the leap from multiplication to logs and exponents.

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The “old fashioned” slide rule helps, but it can be achieved with logarithmic graph paper as well.

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