1. Send kids to basement to get suitcases.
2. Keep suitcases on first floor.
3. Send kids upstairs to get one type of item at a time. E.g. Three pairs of underpants. Then three pairs of socks. Et cetera.
4. Kids throw each type of item in the suitcase.
5. Repeat steps 3 and 4 as often as necessary.
6. Done.

Seriously. It’s awesome.

I made an observation with Tabitha partway through.

Me: Isn’t it strange how a pair of socks is two socks, but a pair of underpants is only one thing?

Tabitha (six years old): Yeah. It should “a pair plus one” because there are three holes.

Me: Wow. I hadn’t thought of that. So how many holes does a shirt have?

T: Three….No four!

Me: How do you figure?

T: The one you put your head through, the arms, and the head hole.

If you are like me, you may be a bit behind the curve on her language here. “The one you put your head through” is the one that ends up at your waist once your shirt is on. I had to think about this for a moment.

A few days later, I was curious to probe her thinking a bit further. She was getting dressed (a process which is always slow, and occasionally very frustrating for the parents):

Me: Do you remember how you said a pair of underpants has three holes and a shirt has four?

T: Ha! Yeah!

Me: I was thinking about that and wondering whether there are any kinds of clothing that have one hole or two holes.

T: Socks have one hole!

Me: Oh. Nice. Sometimes Daddy’s socks have two holes, though.

T: Yeah. When they’re broken.

By this time, she finally has the underpants on and her pants are being slowly pulled on.

Me: Wait. You need socks!

She goes to her dresser and proceeds to sort through the very messy sock drawer.

T: There are no matches.

I find what appears to be two socks balled up together.

T: No! Those aren’t socks! Those are for putting over tights to keep your legs warm.

We look at each other.

Big smile.

T: Those have two holes!

Griffin must have been thinking about the wonders of the State Fair as summer slowly (oh, so slowly!) unfolded on our fair state. He asked a question at breakfast one recent morning.

Griffin (eight years old): How tall is the Giant Slide?

Me: Good question. I would guess…40 feet. What’s your guess?

G: 45 feet.

OK. That’s a mistake. We should have written our guesses down privately to avoid influencing each other. Oh well.

Me: Let’s look it up.

Google returns nothing useful. It does return this awesome video, though, which we watch together.

Me: I found lots of information mentioning the Giant Slide, but nothing on its height.

G: Measure it yourself, then!

Me: Good idea. How should we do that?

G: We’re gonna need a lot of tape measures put together.

This will be a summer project for us: Measuring stuff without putting a ruler next to it. I’ll report on our progress in this space.

Me: Tabitha, do you still think zero and half are the same? Or have you not thought about that in a while?

Tabitha (six years old): I think…Half isn’t a number. I mean, it’s made of numbers put together, but it’s not a number.

Me: What is a number?

I love this question. How people answer it can be revealing. I asked a version of it of Griffin when he was in Kindergarten.

T: is a number.

Me: Oh?  is a number, but not one-half?

T: Yeah. But it doesn’t really get used.

Me: What do you mean by that?

T: Well, people say, 1, 2, 3, 4, 5, 6, but not .

Me: Oh. So when we count count, we skip over ?

T: Yeah.

We are both silent for a few moments, thinking.

T: Zero, too. People don’t count starting at zero. They say 1, 2, 3…

Me: Yeah. Isn’t that funny?

T: It should go half, zero, 1, 2, 3…

It seems clear that has indeed been thinking about that conversation. She is struggling with the betweenness of ; that it expresses a number between 0 and 1.

A Fruit Roll Up weighs 0.5 oz & is a 12.5 by 11 cm parallelogram.

A Fruit by the Foot weighs 0.75 oz & is rectangular. One dimension of this rectangle is 2.2 cm. What is the other dimension?

(Be sure to state your assumptions, and any other information you draw upon in your solution.)

Problem 2

There are now Cheez-Its BIG. They claim to be “Twice as Big” as ordinary Cheez-Its. One serving of regular Cheez-Its consists of 27 crackers and weighs 30 grams. One serving of BIG Cheez-Its also weighs 30 grams.

(A) How many crackers should one serving of BIG Cheez-Its contain?

(B) How many does it contain?

Me: Do you know what 12÷2 is?

Griffin (8 years old): 6

Me: How do you know that’s right?

G: 2 times 6 is 12.

Me: What about 26÷2?

G: 13

Me: How do you know that?

G: There were 26 kids in Ms. Starr’s class [in first grade],  so it was her magic number. We had 13 pairs of kids.

Me: What about 34÷2?

G: Well, 15 plus 15 is 30…so…19

Here we see the role of cognitive load on mental computation. Griffin is splitting up 34 as 30 and 4 and finding pairs to add to each. Formally, he’s using the distributive property: .

He wants to choose and so that .

But by the time he figures out that , he loses track of the fact that  and just adds 4 to 15.

At least, I consider this to be the most likely explanation of his words.

My notes on the conversation only have (back and forth), which indicates that there was some follow-up discussion in which we located and fixed the error. The details are lost to history.

Our conversation continued.

Me: So 12÷2 is 6 because 2×6 is 12. What is 12÷1?

G: [long pause; much longer than for any of the first three tasks] 12.

Me: How do you know this?

G: Because if you gave 1 person 12 things, they would have all 12.

Let’s pause for a moment.

This is what it means to learn mathematics. Mathematical ideas
have multiple interpretations which people encounter as they live their lives. It is (or should be) a major goal of mathematics instruction to help people reconcile these multiple interpretations.

Griffin has so far relied upon three interpretations of division: (1) A division statement is equivalent to a multiplication statement (the fact family interpretation, which is closely related to thinking of division as the inverse of multiplication), (2) Division tells how many groups of a particular size we can make (Ms. Starr’s class has 13 pairs of students—this is the quotative interpretation of division) and (3) Division tells us how many will be in each of a particular number of same-sized groups (Put 12 things into 1 group, and each group has 12 things).

This wasn’t a lesson on multiplication, so I wasn’t too worried about getting Griffin to reconcile these interpretations. Instead, I was curious which (if any) would survive being pushed further.

Me: What is ?

G: [pause, but not as long as for 12÷1] Two.

Me: How do you know that?

G: Half of 12 is 6, and 12÷6 is 2, so it’s 2.

Me: OK. You know what a half dollar is, right?

G: Yeah. 50 cents.

Me: How many half dollars are in a dollar?

G: Two.

Me: How many half dollars are in 12 dollars?

G: [long thoughtful pause] Twenty-four.

Me: How do you know that?

G: I can’t say.

Me: One more. How many quarters are in 12 dollars?

G: Oh no! [pause] Forty-eight. Because a quarter is half of a half and so there are twice as many of them as half dollars. 2 times 24=48.

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 1, Bits 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.

I settled on situations involving fractional values of unit rates, such as the following.

If of a lawn takes of an hour, how much can I mow in one hour?

Before we begin, remember that if the problem were about 2 lawns in 3 hours, we would easily and naturally divide by 3. Only the numbers have changed, so the mathematical structure remains the same and we need to find .

Click each image to see it full size. If you’re into this sort of thing.

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And so I fear it is with the standard algorithm.

I have heard it said that the use of this phrase (repeatedly) in the Common Core State Standards was a compromise (although I cannot find a source for this—leave any breadcrumbs you can find in the comments, won’t you?) It would satisfy some parties who believe that the standard algorithm is an essential seawall against the encroaching fuzzy math tide, while leaving the precise nature of the standard algorithm unspecified would appease those who argue that alternative algorithms are helpful in developing and maintaining children’s number sense.

But if a compromise owes its precise nature to the fact that different parties will interpret the terms of the compromise differently, has there really been a compromise? Have we really made an agreement when we disagree about its meaning?

What is an algorithm?

Karen Fuson and Sybilla Beckmann, in their “Standard Algorithms in the Common Core State Standards” cite a CCSSM Progression document.

In mathematics, an algorithm is defined by its steps, and not by the way those steps are recorded in writing.

Hyman Bass, in his article from Teaching Children Mathematics, “Computational Fluency, Algorithms, and Mathematical Proficiency: One Mathematician’s Perspective” agrees.

An algorithm consists of a precisely specified sequence of steps that will lead to a complete solution for a certain class of computational problems.

So far, so good. We have accord on the meaning of algorithm.

What is the standard algorithm?

The definite article in the phrase the standard algorithm seems to be important to the alleged compromise I referred to.

Here, for example, is Hung-Hsi Wu on standard algorithms.

[T]he essence of all four standard algorithms is the reduction of any whole number computation to the computation of single-digit numbers.

Wu states the following steps for the standard algorithm (.pdf) for multidigit multiplication.

To compute say 826 × 73, take the digits of the second factor 73 individually, compute the two products with single digit multiplier— i.e., 826 × 3 and 826 × 7 — and, when adding them, shift the one involving the tens digit (i.e., 7) one digit to the left.

He explicitly allows for moving left-to-right, as well as inclusion of zeroes instead of shifting. But explicit attention to place value in the process of working the algorithm seems to be proscribed.

Contrast this with the following figure (click for full-size version) from Fuson and Beckmann.

This figure is labeled “Written methods for the standard multiplication algorithm , 2-digit x 2-digit”. Note in particular methods D (lower left) and F (upper right). Method D shows that we are thinking 6 x 9 tens as we work the algorithm. Method F suggests that we are thinking 6 x 90 as we work.

But wait. The lattice method is an example of the standard algorithm?

Recall that an algorithm is defined by its steps. In Wu’s standard algorithm, you may proceed from left to right, or from right to left; either is acceptable. The lattice has both left/right and up/down steps, and you may do the single digit multiplication steps in absolutely any order.

I cannot imagine that Wu would count the lattice as a standard algorithm, and I seriously doubt he would count partial products (method D) in that category.

All of this got me thinking about whether there are any non-standard algorithms for multi-digit multiplication in the viewpoint that Fuson and Beckmann present. Pretty much every multiplication algorithm I know is in that Fuson and Beckmann figure. Every one except the Russian Peasant Algorithm, that is.

an alternative

I have argued that the compromise of using the standard algorithm but not specifying the standard algorithm in the Common Core is problematic because different people mean different things by it. The lattice is explicitly counted in the standard algorithm by Fuson and Beckmann, but our agreement on what constitutes an algorithm (a precisely defined series of steps) implies that the lattice constitutes a different algorithm from (say) partial products. Both cannot be the standard algorithm.

But here is an alternative. What if Common Core, instead of using the language of the standard algorithm used the following construction: an algorithm based on place-value decomposition.

In this case, 5.NBT.B.5 would read:

Fluently multiply multi-digit whole numbers using an algorithm based on place-value decomposition.

This construction would seem to include all of the algorithms in Fuson and Beckmann’s figure; it would make clear that the Russian Peasant Algorithm does not count; and it would be more transparent than the standard algorithm.

Until and unless I receive cease-and-desist notifications, I will go ahead and use this version in everything I do.

For your convenience, I have rephrased the various citations below. You can thank me later.

4.NBT.B.4 Fluently add and subtract multi-digit whole numbers using an algorithm based on place-value decomposition.

5.NBT.B.5 Fluently multiply multi-digit whole numbers using an algorithm based on place-value decomposition.

6.NS.B.2 Fluently divide multi-digit numbers using an algorithm based on place-value decomposition.

I came in partway through the conversation.

Rachel: I’m going to cut you a small slice of this apple.

Tabitha (6 years old): Do I have to eat the whole thing?

R: The whole apple? No.

T: No, the whole slice!

R: Yes!

If you are unaware of the fun we have had with units around our house, you may wish to check out our discussion of brownies, and (of course) the following.

I tried to keep my work neat so others could follow it. If this sort of thing amuses you (as it obviously does me), then you’ll want to take a few minutes with the larger versions of these images. If it does not amuse (and I cannot begrudge anyone this), then you’ll just want to move along; there’s nothing here for you today.

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