Michael and I can’t stop gushing about the Project Z resources and how they have sharpened our thinking about teaching and learning integers.

One of the most eye-opening pieces of their work is a set of videos of 1st and 2nd graders solving math problems such as ___ + 5 = 3 and 2 - ____ = 6. It’s amazing to listen to little kids talk through the exact same ideas that our 7th and 8th graders struggle to understand.

One amazing moment occurred in the second video on this page. Violet, a 2nd grader, correctly answered -4 to the problem 2 - ___ = 6. She giggles nervously at her answer, but when the teacher prompts her to explain her reasoning, she (adorably) says:

“Because it always goes the other - anegative number to me, when you’re adding or subtracting it, it goes the other way than it usually goes with a positive number.”

That is basically a summary of big ideas 2 and 3 from my previous post. In fact, her phrasing may even be easier to understand than my own!

But I think a great deal of her success is due to the format of the question. I am going totally off of intuition here, but I don’t think she would have gotten the correct answer as easily if she had been given the problem 2 - (-4) = ____.

When students are given a standard problem such as 2 - (-4) = _____, I think two thoughts run through their heads:

This problem is different than 2 - 4 because one of the numbers is negative

Subtraction makes numbers go down

Except in their heads, I think it sounds a lot more like

This problem is differ- **Subtraction makes numbers go down!!!!!!!!!**

There is an enormous imbalance between the new, dissonant math problem and the old, well-worn grooves that addition and subtraction have made in students’ minds. This is why integer addition and subtraction are so much harder to teach than multiplication and division.

With multiplication and division, you are just adding a rule about negativity on top of the existing structure of multiplication and division facts that kids already know. But with addition and subtraction, what is happening in the problem is the *precise opposite* of what students have seen for years and years. Imagine the nightmare it would be if multiplication and division worked this way as well: Imagine explaining that 10 * (-5) = -2

But with the open number sentences, the problem directly confronts students’ preconceptions about addition and subtraction. Look at a problem like 9 + ____ = 4. This problem *insists* that addition can make a number smaller. It’s up to the student to figure out which number has that effect.

This creates a subtle but important change in the sequence of teaching integers. Typically in my classroom, I present the following idea:

Adding a negative number makes the answer smaller

But the open number sentence turns this single idea into a two-parter.

Part 1) Sometimes addition makes the answer smaller

Part 2) This happens when you add a negative number

I like the open number sentences such as 9 + ____ = 4 because it drops kids right into the space between Part One and Part Two. They first have to grapple with the idea that addition can make an answer smaller. After all, it’s right there in front of them!

Then they have to decide what sort of number would have that effect. Students as young as Violet with an intuitive understanding of negative meaning “opposite,” so it stands to reason that they would gravitate toward negative numbers as an answer.

This is the part of my post where I want to disclaim again: I am 100% speculating based on my personal experience and intuition about teaching integers. I have no idea if these sorts of problems have a meaningful effect on the way kids think about integers. So I will be seeking out research on this topic. If I don’t find any, I might have to make some of my own...