Tag Archives: math

The Triangleman Decimal Institute [TDI]

In recent weeks, I have written several times about decimals and their treatment in curriculum. In discussions surrounding that writing, it has become clear to me that everyone involved in children’s learning of decimals can both learn and contribute to the learning of others.


Which is why I am excited to announce…

The Triangleman Decimal Institute

For seven weeks, starting Monday, September 30, I will invite all interested parties to an online conversation about decimals and learning decimals.

Each Monday, I’ll have a new post here to launch and focus our discussions. Comments will be closed in order to move the discussions to more productive venues (see below).

You may participate in any way that you like, including the following:

  1. Self study. Read at your leisure. Discuss with yourself, your colleagues, your spouse and/or your Australian Labradoodle.
  2. Twitter. I invite you to use the #decimalchat hashtag to respond, argue, offer evidence and discuss.
  3. Canvas. It is no secret that I love this LMS. I have established a course in Canvas. The course is public, free and you may self-enroll. We will mainly use the discussion forums there, which function MUCH better than WordPress comments for our purposes. I will establish a new discussion forum there for each week’s post, but students (i.e. you) can also create discussions.

You may come and go as you please.

My promise to you is to keep myself on the schedule in the syllabus below and to engage to the extent possible in the discussions on Twitter and Canvas.


Come join us for some or all of the following.

Week 1 (Sept. 30): Decimals before fractions?

Week 2 (Oct. 7): Money and decimals.

Week 3 (Oct. 14): Children’s experiences with partitioning.

Week 4 (Oct. 21): Interlude on the slicing of pizzas.

Week 5 (Oct. 28): Grouping is different from partitioning.

Week 6 (Nov. 4): Decimals and curriculum (Common Core).

Week 7 (Nov. 14): Summary and wrap up.

There will be no grades, tests or tuition. Just the love of knowledge and the collective passion of teachers wanting to do their best.

See you in class on Monday!

Note from Canvas:

This course has enabled open enrollment. Students can self-enroll in the course once you share with them this URL:  https://canvas.instructure.com/enroll/MY4YM3. Alternatively, they can sign up at https://canvas.instructure.com/register  and use the following join code: MY4YM3


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.

Why study math?

As usual, I was the house rabble-rouser at my institution today. Someone sent around a link to an article (subscription required, unfortunately) titled “Why study math?”

Reading the piece from the perspective of my inner middle-schooler, I was unimpressed. It felt to me like a rehashing of the usual vague unsubstantiated claims about transferring problem-solving skills and learning to reason. And also this:

Learning math develops stick-to-it-ness, defined as dogged perseverance or resolute tenacity, and develops perseverance, resilience, persistence, and patience. Students have opportunities to develop their work ethic in my math class by not making excuses, not blaming others, and not giving up easily.

Um, teacher? Do we have even one shred of evidence to support these claims?

So I wrote up my own reasons for studying math. Here they are:

I have two reasons people should study math.

First is that there is a set of very practical quantitative and spatial skills that are necessary for informed participation in society. Access to these skills ought to be both a civil right and an obligation.

The second is that there are many bodies of knowledge that we have agreed as a society are important; to be educated means knowing and having experienced certain things in the arts and sciences. In this way we pass on our culture.

I see these reasons as being quite different from more generalized claims about reasoning and problem-solving skills. An important part of the difference is that my reasons invite conversation and debate about exactly what mathematics we should teach.
If the practical skills are a major reason we impose math on students, we need to inspect the curriculum pretty closely to make sure we’re teaching the right ones. In a technologically advanced society, the long division algorithm for multi-digit decimals is pretty hard to defend from this perspective, for instance (to say nothing of polynomial long division!)

And if passing along mathematical knowledge and ways of thinking are culturally important, we ought to design curricula that give students experiences with mathematical ways of knowing. I would argue that our standard curriculum K-12 and through calculus does a pretty poor job of this.

But if we appeal vaguely to reasoning skills and stick-to-it-iveness, there is no further conversation. We tell students, “Take our word for it-studying this will make you better at that, and you’re gonna need that. So study this now and don’t ask any further questions.”

And while I don’t know everything about how to teach critical thinking, I know that when we talk to students this way-either explicitly or implicitly-we devalue it.

Just sayin’ (literacy and numeracy)

It now seems to be widely accepted in literacy circles that kids should be encouraged to read. And that it does not matter very much what they read, they should be encouraged to read. Fiction, non-fiction, classics, graphic novels. It doesn’t matter as long as they’re encountering new vocabulary, important narrative structures, etc. They just need to read. In school or out of school, they just need to read.

We’re not there yet in mathematics.

In math, we think they need to work on these number facts at this grade level, and those number facts at that grade level. Identify this shape at this grade and sort those shapes at that grade.

We have free reading time in class every day in elementary school. But free math time?

I’m just sayin’.

TURD: When am I going to use this?

With thanks to Dan Meyer for finding this particular Truly Unfortunate Representation of Data [TURD]:

It is becoming clear that online colleges are major sources of TURDs.

A few of the TURD-lectable features in this one:

  • I can make neither head nor tail of the horizontal location of each bar. If it’s supposed to be according to the x-axis label, then why do we ALSO have the little bar-codey bars within? The legend says those count the number of concepts per subject.
  • What’s up with the little black triangles?
  • I get that vertical location of the bar tells me about salary. Does vertical width of bar mean something? (E.g. compare veterinarians to firefighters.)
  • Do agricultural workers really use that much more math than automotive mechanics? Seems dubious

One last point (and it was Dan’s original point in posting it in the first place): Garbage in/garbage out in terms of information and purpose. You could fix the critiques above and the graphic still stinks; was any student ever convinced to keep on keepin’ on in math class by the poster version of this?