# Tag Archives: multiplication

## Standard algorithms unteach place value

I found a page full of computations sitting around the house this evening. Naturally, I picked it up and gave it a look.

Griffin (10 years old, 5th grade) had been doing some multiplication in class today. Somehow his scratch paper ended up on our couch.

Here is one thing I saw.

Me: I see you were multiplying 37 by 22 here.

Griffin (10 years old): Yeah. But I got it wrong so I did it again with the lattice.

Me: How did you know you got it wrong?

G: I put it in the answer box and it was wrong.

It turns out they were doing some online exercises. There is an electronic scratchpad, which he found awkward to use with a mouse (duh), plus his teacher wanted to be able to see their work, so was encouraging paper and pencil work anyway.

I was really hoping he would say that 37 times 22 has to be a lot bigger than 202. Alas he did not.

Anyway, back to the conversation.

Me: OK. Now 37 times 2 isn’t 101. But let’s imagine that’s right for now. We’ll come back to that.

G: Wait. That’s supposed to be 37 times 2? I though you just multiplied that by that, and that by that.

He indicated 7 times 2, and then 3 times the same 2 as he spoke.

Me: Yes. But when you do that, you’ll get the same thing as 37 times 2.

A brief moment of silence hung between us.

Me: What is 37 times 2?

G: Well….74.

Let us pause to reflect here.

This boy can think about numbers. He got 37 times 2 faster in his head than I would have with pencil and paper. But when he uses the standard algorithm that all goes out the window in favor of the steps.

THE STEPS WIN, PEOPLE!

The steps trump thinking. The steps trump number sense.

The steps triumph over all.

Back to the conversation.

Me: Yes. 74. Good. I like that you thought that out. Let’s go back to imagining that 101 is right for a moment. Then the next thing you did was multiply 37 by this 2, right?

I gestured to the 2 in the tens place.

G: Yes.

Me: But that’s not really a 2.

G: Oh. Yeah.

Me: That’s a 20. Two tens.

G: Yeah.

Me: So it would be 101 tens.

G: Yeah.

I know this reads like I was dragging him through the line of reasoning, but I assure you that this is ground he knows well. I leading him along a well known path that he didn’t realize he was on, not dragging him trailing behind me through new territory. We had other things to discuss. Bedtime was approaching. We needed to move on.

Me: Now. We both know that 37 times 2 isn’t 101. Let’s look at how that goes. You multiplied 7 by 2, right?

G: Yup. That’s 14.

Me: So you write the 4 and carry the 1.

G: That’s what I did.

Me: mmmm?

G: Oh. I wrote the one

Me: and carried the 4. Yeah. If you had done it the other way around, you’d have the 4 there [indicating the units place], and then 3 times 2 plus 1.

G: Seven.

Me: Yeah. So there’s your 74.

This place value error was consistent in his work on this page.

Let me be clear: this error will be easy to fix. I have no fears that my boy will be unable to multiply in his adolescence or adult life. Indeed, once he knew that he had wrong answers (because the computer told him so), he went back to his favorite algorithm—the lattice—and got correct answers.

But I want to point out…I need to point out that this is exactly the outcome you should expect when you go about teaching standard algorithms.

If you wonder why your kids (whether your offspring, your students, or both) are not thinking about the math they are doing, it is because the algorithms we (you) teach them are designed so that people do not have to think. That is why they are efficient.

If you want kids who get right answers without thinking, then go ahead and keep focusing on those steps. Griffin gets right answer with the lattice algorithm, and I have every confidence that I can train him to get right answers with the standard algorithm too.

But we should not kid ourselves that we are teaching mathematical thinking along the way. Griffin turned off part of his brain (the part that gets 37 times 2 quickly) in order to follow a set of steps that didn’t make sense to him.

And we shouldn’t kid ourselves that this is only an issue in the elementary grades when kids are learning arithmetic.

Algebra. The quadratic formula is an algorithm. Every algebra student memorizes it. How it relates to inverses, though? Totally obfuscated. See, we don’t have kids find inverses of quadratics because those inverses are not functions; they are relations. If we did have kids find inverses of quadratics, they could think about the relationship between the quadratic formula:

$x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}$

and the formula for the inverse relation of the general form of a quadratic:

$y=\frac{-b \pm \sqrt{b^2-4ac+4ax}}{2a}$

Calculus. So many formulas (algorithms) that force students not to think about the underlying relationships. If we wanted students to really think about rates of change (which are what Calculus is really about), we might have them develop a theory of secant lines and finite differences before we do limits and tangent lines. We might have Calculus students do tasks such as Sweet Tooth from Mathalicious (free throughout October!). There, students think about marginal enjoyment and total enjoyment.

On and on.

This is pervasive in mathematics teaching.

The results are mistaken for the content.

So we teach kids to get results. And we inadvertently teach them not to use what they know about the content—not to look for new things to know. Not to question or wonder or connect.

I’m telling you, though, that it doesn’t have to be this way.

Consider the case of Talking Math with Your Kids. There we have reports from around the country of parents and children talking about the ideas of mathematics, not the procedures.

Consider the case of Kristin (@MathMinds on Twitter), a fifth grade teacher, and her student “Billy”. Billy made an unusual claim about even and odd numbers. She followed up, she shared, we discussed on Twitter. Pretty soon, teachers around the country were engaged in thinking about whether Billy would call 3.0 even or odd.

But standard algorithms don’t teach any of that. They teach children to get answers. They teach children not to think.

I have read about it. I have thought about it. And tonight I saw it in my very own home.

## Question 5

### “Multiplication is just repeated addition.” Explain why this statement is false, giving examples.

Now this is when things get sticky.

It is a strong and presumptuous claim to say what an idea is.

In recent years, I have come to an understanding of why repeated addition is not the strongest foundation on which to build the idea of multiplication. But that is a far cry from making claims about what it is.

You see, any rich enough mathematical idea has multiple meanings. What is subtraction? Is it the inverse of addition? Is it the distance between two points on a number line? Is it takeaway? Subtraction is all of these, sort of.

And what is a fraction? The only definition of a fraction that is good enough to withstand the test of time and mathematical scrutiny is that a fraction is an equivalence class resulting from the equivalence relation,

$\frac{a}{b}=\frac{c}{d}$ if and only if $a\cdot d=b\cdot c$

Is that what a fraction is?

No. But I am off task.

I suspect that my answer may vary from some others out there. (Although perhaps it will not.)

Repeated addition is shaky ground for establishing multiplication because it doesn’t capture the unique structure that multiplication represents.

There is additive structure, and there is multiplicative structure. Additive structure is about comparisons and changes involving the same units. Apples plus apples gives apples. Miles plus miles give miles.

Multiplicative structure is about comparisons and changes involving different units. Hours times miles per hour gives miles. Three different units; one of them a unit rate. Always.

These are related structures but they are different.

Multiplicative structure is captured better by this idea: $A\times B$ means A groups of B (I am pretty sure I first ran across this particular characterization in Sybilla Beckmann’s textbook for math courses for elementary teachers). A, in this interpretation, is expressed in one unit. B is the unit rate (things per group). The product is expressed in a third unit.

This difference shows up in the following conversation between a mom and her daughter as they count the number of things in this array of meatballs.

Image from The New York Times.

Maya counted the top and bottom, 4 + 4 = 8. Then she counted L and R. 3 + 3 = 6. 8 + 6 = 14. 14 + 2 in the middle = 16. When I asked her why, she said, “Because you double count the corners when you count an array.”

She asked me to count so she could show me how. I counted 4 across the top and 3 down the side. “See, Mommy! You’re counting the corner one twice.”

Why do we count the corner one twice in this scenario? This seems to violate a fundamental principle of counting—one-to-one correspondence. One number word for each object, and one object for each number word.

The answer is that mom really did not count the corner meatball twice. The first time, she counted the meatball to establish that each row has 4 meatballs. The second time, she counted the rows. There are 3 rows, so there are 3 groups of 4 meatballs.

Much, much more on arrays in many places in my writing. Especially these:

Beyond the textbook wrap up (or What does this have to do with mathematics?)

Twister (on sister site, Talking Math with Your Kids)

## Summertime (or anytime) reading recommendations

A friend asked for tips on getting started understanding some new domains in mathematics teaching the other day. An experienced high school teacher, he wants to know more about  elementary and middle school topics, especially fractionsplace value and multiplication and division algorithms.

For obvious reasons (mainly that I won’t shut up about these topics), I was on his short list to ask for recommendations.

It occurred to me that others might be interested in this particular brain dump. So here it is, lightly edited. Enjoy.

Fractions. Entry level stuff on this is Connected Mathematics. In particular, Bits and Pieces 1Bits and Pieces 2, and Comparing and Scaling. Any version of these units is fine. Work the problems from the student edition; have the teacher edition there for guidance.

I made major progress on understanding student thinking when I constrained myself to using only ideas that must have come earlier (i.e. in elementary school) and to those that had been previously developed. When I tried to appreciate the problems on their mathematical merit, or to build connections to my undergraduate mathematics knowledge, I didn’t make much progress that was useful to working with kids.

Then turn to Extending Children’s Mathematics (written by the Cognitively Guided Instruction team—CGI—and published by Heinemann). There is a lovely research perspective that should give you new ways to think about the CMP stuff.

More advanced perspectives are to be found in the work of the Rational Number Project (RNP), and there’s Susan Lamon’s book, Teaching Fractions and Ratios for Understanding. For contrast, read Hung Hsi Wu’s Math for Teachers curriculum. For extra credit, write a comparative analysis paper reconciling Wu’s work with CGI and with RNP; argue which has the greater influence on the Common Core fractions development.

Conspicuously absent from these recommendations is the “Essential Understandings” series from NCTM, published relatively recently. I find the writing style of these texts hard to process. Others may recommend them, and if so, perhaps you ought to take them more seriously than I have been able to.

Place value. There is an oldish JRME piece by Karen Fuson, the CGI folks and another research team about place value. It’s a seminal piece and totally worth your time. There is no one book I can recommend; my exploration of the conceptual landscape of place value has been idiosyncratic and informed more by small pieces of others’ research work combined with my own classroom experience and experiments. Most of that is documented on this blog.

The “Orpda” number system that Cady and Hopkins wrote about (and which I bastardized as “Ordpa”) was foundational to these explorations. Short, short article but the ideas opened a whole new space for me in thinking about what it means to learn place value.

The Young Mathematicians at Work book on number sense, addition and subtraction is pretty good. But those articles and the blog are better starting points.

Multiplication and division algorithms. I am trying to recall how I came to know the algorithms I know. I have to say that these steps I cannot really retrace.  I am loathe to recommend digging through Everyday Math for them, because things are so diffuse; it’s hard to get the right book in your hand in that curriculum to learn any one particular thing.

The Kamii piece I recommended a while back is good. It was published in the 1998 NCTM Yearbook on algorithms. Sybilla Beckmann’s Mathematics for Elementary Teachers book is good, too.

But looking back at my standard algorithm diatribe last week and trying to think about what small set of resources would prep someone else to build a similar case (or to counter it), I am less clear than I am about fractions or place value. I do not know what this says about my knowledge, nor about the topic.

## Math Peeps at Play

I have test driven these photographs and questions with a 6-year old, an 8-year old, a 43-year old and a classroom full of 19—40 year olds. Good conversations were had with all populations. I turn them over to you. Use them for the forces of good, not evil.

### Associative and distributive properties

How many Peeps in this picture?

Do you see 4 boxes of 12?

Or do you see 12 sets of 4?

The first could be notated $4\cdot\left(3\cdot4\right)$

The second could be notated $\left(4\cdot3\right)\cdot4$

That these two are equal is an instance of the associative property of multiplication.

There are, of course, other ways to view these guys, and to notate how you see them. The mathematics doesn’t live in the Peeps, it lives in the interactions we have around the Peeps.

Careful discussion and notation will demonstrate the associative property and/or the distributive property in each of the pictures below.

### Which is more?

In each of the following images, are there more purple Peeps or more pink Peeps? Of special importance is this question: How can you know without counting?

In the meantime, these guys are still hanging around my office. Got any other arrangements you’d like to see?

I know, for instance, that I wish I had a fifth box so I wouldn’t have repeated 4’s in that first picture.

## 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

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:

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.

## The multiplication machine

My goal with most Talking Math with Your Kids posts is to demonstrate how easy it is to do this at home.

That is not the point of this post.

I do not pretend that the average parent is ready for the conversation I document here. This is graduate level parental math talk. I’m just showing off here. Sorry about that.

But I do hope what comes across is the pure joy of a child exercising her mind. The video runs five minutes. I promise it’s worth your time. Transcript with commentary follow.

Here’s the set up.

We have this multiplication machine. It’s a multiplication table with spring-loaded buttons. You push the button and it pops up; you read the product off the side of the button.

Tabitha was playing around with it one day and declared that it had a bunch of wrong answers. She showed me what she was talking about and I quickly diagnosed two things:

1. She thought it was supposed to give sums, and
2. We were gonna have a good old time with this.

I set up my iPad to record the conversation and away we went.

Tabitha (5 years old): If Griffy used this, he must have gotten his homework a little bit wrong.

Don’t you think so?

Cause it’s lying.

Isn’t it?

I set the stage for our conversation by recapping its beginning. I chose my language carefully. I didn’t want to lie to the kid, so I spoke of “6 and 1” not “6 plus 1”.

Me: OK. So we’ve established that 6 and 1 gives us 6.

And we expected 7 and 1 to give us 8 but it didn’t; it gave us 7.

What do you think 8 and 1 is gonna give us?

T: Eight.

Me: And what should it give us?

T: Eight. Because that’s the right answer.

Wait. Has she picked up on what’s going on so quickly?

Me: Wait.

T: There’s seven and then one would be eight.

Got it. She’s still on 7×1. Let’s reset.

Me: Right. But what about when you do 8 and 1? What should that be?

T: Nine.

Me: And uh…What do you think it’s…

What do you think this is gonna tell us, based on what we saw from the other ones?

Do you think it’s gonna say 9? It got all these other ones wrong. You think it’s gonna say 9 here?

T: No.

Me: What do you think it is gonna say?

T: Eight.

Now we’re on the same page. She thinks it’s supposed to give sums, so she expects 9, but with some support she is able to apply the pattern that the machine is giving us one less than the right answer.

Me: Why is that? Why do you think that?

T: Because one less than than the one over here. This [7×1] was seven, so I think it’s gonna say 8.

Aw god! This is so…

Me: What’s this one [9×1] gonna say?

T: Nine.

Me: What should it say?

T: Ten.

Sure enough.

T: Awww! This thing is a liar!

Now it’s time to change things up. We have worked our way down the left-hand column. So she has done 7+1, 8+1, 9+1. She has one more than nine nailed down. Can she as easily do nine more than one? Does she know that addition is commutative?

T: It’s 2.

Oh wait. Oh yeah, it should be…

T: Six.

She is banging these out as quickly as she did the others. She seems to be using the commutative property of addition. Dig it.

Me: But?

T: It’s five.

This thing’s a liar!

T: This thing is a total liar.

T: It has to be four.

I have no idea what she means by this. She clicks before I can ask.

T: Some of these are right.

Her brother, Griffin (8 years old) wanders into the room. The jig is up, I fear, so I chase him off.

Griffy check out this one.

Check out this one. It should be 6 but look; it’s 5.

This thing is a total liar.

Me: All right, except this one it got right.

T: Yeah, but what about the other ones?

T: Five!

It’s six. But it’s supposed to be five.

I don’t think she noticed that this wrong answer is one more than the one she expected, while the previous ones had been one less.

Me: Interesting.

T: Six!

Me: What do you think it’s gonna say? Wait wait. It should be 6. What do you think it’s gonna be instead?

T: Five.

Of course it’s not five. It’s eight.

T: What? That’s totally not right!

Me: OK. Umm…Let’s try a hard one.

What should…

What should this one be [2×6]?

T: Eight.

HEY!

Is this [6×9] [unintelligible]?

Thirteen!

She clicks. It’s not thirteen; not even close. But she’s not really sure about 6+9 anyway.

T: What was it supposed to be?

Me: It was supposed to be 15.

T: WAAAA!

Me: Fifty-four?

I have to think quickly. How can I ramp this up?

It occurs to me that she may think these are arbitrary wrong answers; that this machine is some sort of random bad answer generator. I want to pique her curiosity by demonstrating that there is something else going on.

So I propose working backwards.

Me: Ooo. OK. Let’s play…are you ready for this?

You choose one. Choose a hard one from down in here.

You press it.

Tell me what it says; not what the problem is, but what the answer is and I’ll see if I can tell you what the problem was.

She chooses a problem. Out of the corner of my eye, I can see that she is focused somewhere near the bottom of the machine.

T: Thirty-six.

Me: Thirty-six. So I think you did 9 and 4.

T: Yes.

Is that right?

Me: Nine plus four is not 36.

By this time, Griffin has wandered back into the room.

Griffin: Nine plus four is 13.

Me: So how…so how did I know? How did I know what it was?

If 9 plus 4 is not 36, how could I have gotten that right?

T: Because you’ve seen it before.

A reasonable explanation.

Me: You think I have this whole thing memorized?

T: Yeah.

Me: Here, do one more.

Do a really hard one.

What did you get?

T: Uh…I don’t know what…

G: [whisper] Sixty-four

T: Sixty-four.

Me: Sixty-four. I think you did 8 and 8.

T: Yes. How did you know?

Is that right?

Me: It’s not what 8 plus 8 is.

But here I’ll tell you…

The iPad ran out of memory at this point.

But that’s OK. I told Tabitha about the difference between a multiplication symbol and the addition symbol. I told her that multiplication is about groups. So 2×4 means two groups of four. We talked about how many things would be in two groups of four.

And then the moment passed and we were on to other things.

Several people have observed that they would love to have audio of the conversations I report under the heading Talking Math with Your Kids. I agree that this would be helpful. But here’s the rub. These conversations are a natural part of our day, and they have to be natural.

My kids have no interest in being show ponies. Oh, they’ll show off for the recording device, but it won’t be natural. Observe Griffin as a young lad in this video, for instance.

He’s totally making faces for the camera and watching himself in the monitor. (You may also note that the spinach is washed; and please forgive the praise style—I know better now!) This mugging behavior has only gotten worse with time.

I do have some ideas for getting good audio, but these will require funds (Do you have a couple thousand dollars for a good cause? Tweet me! We’ll talk!) So in the meantime, we’ll continue the transcribed conversations.

Today’s conversation is a brief one, but I want to make a comparison. The discussion in the following video is not a natural one. (Tip o’ the hat to David Wees for the find.)

Here’s what I mean. The woman discusses cookies because she thinks they will interest the child in question, not because cookies are already under consideration. The question of multiplication (or repeated addition—I have no interest in that distinction here) doesn’t arise naturally either. It arises in the context of putting four chocolate chips on each of three cookies.

To be clear, I have no problem with any of this. But it’s different from the kinds of conversations I am hoping to encourage. The ones I hope to encourage go more like this…

Tabitha (5 years old) and her mother made cookies from one of those frozen-cookie-dough-school-fundraiser things over Thanksgiving weekend. These cookies were stored (unwisely) in a transparent Pyrex container on an open shelf. This led to a desire on Tabitha’s part for a cookie before dinner.

Tabitha: How many cookies can I have? One or two?

Me: Zero. You can have zero cookies.

T: A half?

Me: No. I said zero.

T: Zero whole ones and a half cookie?

Me: Zero halves.

T: And a quarter?

This is a natural conversation about cookies. The opportunity to turn it into a mathematical one was Tabitha asking, How many can I have? I could have played the role of rule enforcer and replied, We don’t eat cookies before dinner; you may have one for dessert. Responding with zero in answer to her question gave her some mathematical wiggle room to play with. And we are far enough along in this talking math adventure that she’s going to play with it nearly every time.

For the record, Tabitha and I have spoken about zero before. And several times, we have had conversations about fractions.