**What does this have to do with mathematics?**

I had a question at the beginning of the day on Thursday, which I shared through Twitter.

Gotta admit. Starting the day at #beyondtextbooks wondering whether @discoveryed knows exactly how hard writing *math* curriculum is.

— Christopher (@Trianglemancsd) March 28, 2013

The question got louder in my head as the day progressed. From my perspective, a tremendous amount of time was being invested in designing the platform for a mathematics textbook-of-the-future while not very much evidence was being presented that any of our work reflected *knowledge of mathematics for teaching*.

My worry continued to deepen that we were designing a better platform for delivering Khan Academy content.

Considering that my critique of Khan Academy has nothing whatsoever to do with the platform, and everything to do with the pedagogical content knowledge of the instructional designer, this was fast becoming a problem.

So I sought out some sympathetic ears in a lull in activity. I hit Frank Noschese and Chris Harbeck with a vulgar version of this question: *What in the world does this have to do with mathematics?*

Angela Maiers took me up on this question by arguing that, essentially, *Mathematics has nothing to do with this, and that’s the way it should be.*

In the end, it turns out that the two of us had very similar concerns. An example helped to bridge the gap. That example follows.

—

At heart, multiplication is about same-sized groups. Whether you write *five groups of three* as 5×3, 3×5, 5(3) or some other way, multiplication structure is about some number of same-sized groups.

We can use multiplication to count the water bottles in this photograph because they are arranged in an array—rows and columns.

But *many children do not count things this way*.

We can know this by observing children as they count. It is quite common for children to count an array by circling around the outside, or even in a seemingly haphazard order. Even very skilled counters may not notice the unique structure of an array.

If they do not notice this structure, they cannot use it.

If they cannot use the multiplication structure of an array, they miss out on an opportunity to use arrays to develop the commutative property of multiplication. One view of the array below is as five groups of three. The other is as three groups of five. The array makes those groups for you, and it suggests that *a* groups of *b* will *always *be the same as *b* groups of *a*.

The array support the general argument that *a*•*b*=*b*•*a* for all whole numbers.

If they cannot use the multiplication structure of an array, they miss out on an opportunity to use arrays to develop the associative property of multiplication. One view of the collection of shoes below is as four groups of three. A different view is as four groups of six.

If you see four rows of three, then we can express the total number of shoes as (4•3)•2. If you see four rows of six, we can express the total number of shoes as 4•(3•2). Of course these are equal—each of them correctly counts the number of shoes on the shoe rack.

Therefore, (4•3)•2=4•(3•2).

And again, the deep connection between (1) multiplication, and (2) the structure of rows and columns suggests a more general argument.

There was nothing special about 4 rows, nor about 3 pairs, nor about the fact that these were pairs. Anytime we have A groups of B groups of C, we can compute either (A•B)•C or A•(B•C).

That is the associative property of multiplication.

### So What?

But what can we *use *this property for? What *good *is it?

For one thing, it’s good for mental math.

Quick: what is 6×60?

If you are like most of us, you unconsciously multiplied 6•6, then by 10. You used the fact that 6•(6•10)=(6•6)•10. You used the associative property of multiplication.

And Javier, in an IMAP video, uses it to figure 5•12. Go there and watch for it.

Did you catch his implicit use of the associative property?

He knows that:

Or dig this. What is 35×16?

Use the associative property twice:

35x(2×8)=(35×2)x8=70×8=(10×7)x8=10x(7×8)=10×56=560.

This is about number sense; it’s about the numerical relationships that form the heart of mathematics.

But it’s also about the inner working of paper and pencil computation. Let’s say you want to multiply 35×16 by the standard American algorithm. Then you would, at some point, say to yourself “3 times 6 is 18”. But that 3 doesn’t mean 3. It means 3 tens. The fact that you can treat it as a 3 is due to the associative property of multiplication.

Division, by contrast, is not associative. (a÷b)÷c is not the same as a÷(b÷c). This explains why we do not operate digit by digit in the standard long division algorithm.

—

There is much, much more.

Contrast with what Sal Khan has to say about the associative property of multiplication.

Khan knows this property. But he does not know (1) that an array is an important representation that can help to establish this property, (2) that children need to be taught to see the multiplication structure of an array, (3) that—at 1:55 in the video—*he is using the associative property *to do the computation 12•30.

Et cetera, and on and on.

This video demonstrates my concern perfectly. Too much attention to delivery method (exercises! badges! energy points! sympathetic narrator!) and not enough attention to mathematics, not enough attention to how people learn mathematics.

### Bringing it home

And—to be frank—if Discovery Education doesn’t have someone paying extremely careful attention to all of this throughout their beyond-the-textbook writing process, they’re not going to produce something that will have an impact on mathematics teaching and learning in this country.

But if they do? Perhaps the sky is the limit.

I have been through a brainstorming/prototyping process before that was very much like Thursday’s session. That other one didn’t have the same attention to the possibilities of electronic student materials that this one did. If Discovery can get both parts of this right, they could create some exciting stuff.

I believe they *want *to do that. I really hope they can.

“Too much attention to delivery method (exercises! badges! energy points! sympathetic narrator!) and not enough attention to mathematics, not enough attention to how people learn mathematics.”

This is an important piece that Discovery need to do properly. I hope they find a balance that works for teachers and students. Math is different that arithmatic. Math can be taught effectivly through the techbook and using the connections of the network. Seeing examples of student created work and using these examples to energize teaching and learning will go along way.

Great post. Thanks

Quickly browsed through Discovery Education website & watched a couple of snippets of video. Am I missing something? I didn’t see any evidence at all that would lead me to believe they have useful ideas for curriculum.

Even for folks whose ideas I like, it sure seems like a lot of them present the same examples over and over again. Which, I think, is a reflection of your point about how difficult it is to write good curriculum.

Fantastic write up. I’m not sure how much time you had to see our Science and Social Studies Techbooks, but I think it’s pretty clear that the !!!!!’s are definitely secondary to curricular needs.

We brought together an incredibly diverse group because we wanted to get as many perspectives as we could in one room. Believe me, our curriculum team is well aware of the challenges in front of them. But that’s why we wanted as many ideas as we could. Some will be incorporated. Considering the number of ideas people came up with, most probably won’t. Our team knows that they have to tread a line between what’s ‘shiny’ and what’s going to be effective for teachers to use with students. And that line isn’t thin or grey.

That said, as you’ve already pointed out, there’s a reason why you were in the room. And this post demonstrates that reason pretty darn well. Yes, I will be sharing it with the team. And on behalf of them in advance, thanks for helping us make sure that we get it right.

Sorry dude. Wrote a response on your last post before reading this one.

A

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