Integers

[Anonymous]

Hello! I was wondering if you could point me in the direction of any research on why operations with integers were taken out of Grade 6 standards? I am a math coach and several of my teachers would like some reason or some research about why operations with integers are not taught in grade 6.

Thanks so much,
Adrienne

[Bill McCallum]

We looked at many sources, including national reports, well-regarded state standards, and standards of high achieving countries. There is no agreement on where to put operations with integers. Singapore puts them in Grade 7. Massachusetts started with addition and subtraction in Grade 6, excluding subtraction of negative numbers. Curriculum Focal Points puts them in Grade 7. And so on.

As for research, the way you ask the question makes it sound as if you think that the placement of operations with integers in Grade 6 is itself research-based. But is it? How would you design a research experiment to determine the correct grade placement of operations with integers, or any other topic? Doesn’t it depend on what else you are doing in Grade 6, and what you have been doing in Grades 1–5, and what you plan to do in Grades 7–12? The research would have to look at the entire sets of standards. There has been some such research, for example the research of William Schmidt and Richard Houang, but they don’t have conclusions about specific grade level placement of specific topics. Rather they address large scale properties of the standards, such as coherence. My guess is that you could have a coherent set of standards which places operations with integers in Grade 6 and one which doesn’t; and you could have an incoherent curricula which do the same two things. The important thing is not the exact grade level placement but the coherence.

[Anonymous]

Thanks! This is the message we are telling our teachers! I am glad I am on the same page as you! Teachers just wanted a firm “research-based” answer. But I will continue to share your answer!

[Bill McCallum]

You are very welcome! Sorry if I sounded a little bit crabby in my last answer. The term “research-based” sometimes does that to me.

[lhwalker]

I was just watching your Khan video for multiplying and dividing negative numbers. My remedial Algebra students gain solid traction with multiplying negative numbers like this:

-3 lost 3 friends
2(-3) lost 3 friends twice
-2(-3) the opposite of losing 3 friends twice.

The added benefit is that “opposite” connects with -x.

The emotional connections help with retention.

[Bill McCallum]

Lane, I think this is a useful way of helping students remember the rules, and that’s especially needed for remedial students. There’s a bit of sweeping under the rug going on here, because if you spell it out, you are saying that

(opposite of 2) times (opposite of 3)

is the same as

opposite of (2 times (opposite of 3)).

We can actually prove this using the distributive property, because that property tells me that

(opposite of 2) times (opposite of 3) plus 2 times (opposite of 3)

is the same as

((opposite of 2) + 2) times (opposite of 3)

which is just zero times (opposite of 3), namely zero.

But if I add two numbers and get zero, they must be opposites!

Of course, I’m not suggesting that you have to go over all this with your remedial students!

[Author]

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