Tag Archives: Beyond the Textbook

Beyond the Textbook wrap up

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.

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.

A common counting sequence for a child who does not use the rows-and-columns structure of an array

A common counting sequence for a child who does not use the rows-and-columns 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 ab=ba 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.


How many shoes? (Credit to my student, Marissa Brown, for the photo. She submitted it for a class assignment.)

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:


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.


Beyond the textbook

In an odd and interesting turn of events, I find myself sitting at MSP waiting for my flight to Washington, DC to attend Discovery Education’s second Beyond the Textbook conference.

The invitation arrived by email out of the blue about a month ago. I recognized the conference from Audrey Watters’ participation last year, but was very unclear on how they found me.

So I asked.

The reply (edited mostly for length) came from Steve Dembo:

I believe real innovation comes from cross-pollination.  I’m a big believer in the Medici effect, which is why I want as diverse a group as possible…It’s not an easy task by a long shot and I spent days researching people outside the folks I know personally or by reputation already.  I don’t know the exact path that led me to you, but I did spend a lot of time finding one person I respected, someone like a Dan Meyer for example, and then searching through the people he followed on Twitter and that he referenced on his blog.  And from those people, I’d do the exact same thing, looking both for ‘influencers’ as well as people who I thought had interesting ideas that might make a meaningful contribution to the conversation.

So where do you fit in?  You used to be a middle school teacher and we’re going to try to key in on 7th grade.  You’re a professor and have a background in curriculum design already.  And most importantly, people that I trust and respect, trust and respect you.  And I refused to hold your Spartan connection against you (I’m a Hawkeye).

That’s impressive. I have certainly met a lot of people who claim to use social media to do something besides communicate directly with their audience. But this guy is clearly the real deal.

As I think about the upcoming two days, I find myself thinking about the idea of a textbook.

I suppose the category of publishing into which Connected Math fits is Textbook, but I don’t really think of it that way.

It has been years now since I abandoned textbooks in my math courses for future elementary teachers.

And I am designing a new Educational Technology course at my college this semester, which I will teach in the fall. I began the formal planning for that course by digging into a couple of the publisher texts for these courses.

I lasted about 15 minutes before abandoning that part of the work.

Textbooks conjure up—for me—pre-digested second hand material. Textbooks assert, summarize and question.

Then I read Mindstorms and found a text that I could use to introduce major ideas in the course.

So for me, it is important for us to remember that going “Beyond Textbooks” does not have to mean abandoning books, nor texts.

For me, going beyond the textbook means identifying the resources necessary for introducing the material students need to grapple with, and to support them in their efforts.

My students would be happy to tell you that I have not found this perfect set in any of my courses, and that some of my courses are in better shape than others. But none of my students (I hope) would claim that they are bored by the text materials on offer, or that the courses based on these materials consist of regurgitated second-hand knowledge.

So as we think about moving beyond the textbook, let’s celebrate the possibilities for student engagement and the liberation of students’ minds. But let’s not forget that texts and books are resources for building knowledge and questions.

In late developing news, Frank Noschese showed up with Heads or Tails Double Stuf Oreos as a gift. Good man.