## What is “the standard algorithm”? [#algorithmchat]

Richard Skemp wrote, in “Relational Understanding and Instrumental Understanding,” about faux amis—those pesky words in other languages that look like words you are familiar with, but which mean something else entirely. Skemp argues that the word understand is like this—different people use it to mean completely different things. This leads to misunderstanding

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.

## A kindergartener on units [Talking math with your parents]

The following conversation took place in my house the other day. Tabitha (6) had been informed by her mother that she (Tabitha) needed to eat something healthy before eating a chocolate-covered donut. I was—and remain—ignorant of the origins of this donut.

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.

## Common numerator fraction division [#algorithmchat]

My future elementary teachers explore the common denominator fraction division algorithm at the end of the semester. Reading their work got me thinking about common numerator fraction division, and about what sense I could make of the symbols that result.

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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It is perhaps not widely known that I love good Mexican food, and that—with the assistance from afar of Rick Bayless—have developed a number of specialties de casa.

Among these specialties is tostadas, which I make starting with corn tortillas. A bit of oil and 10—15 minutes in the oven makes them crispy. We build from there.

The tortillas fit nicely in a 3 by 3 array on my favorite cookie sheet. There are four of us in the family. You can see where this is going, I am sure.

Griffin served himself a second tostada the other night.

Tabitha (six years old): Griffy’s having another one?!?

Me: Yes. There’s a second one for you, too.

T: How many did you make?

Me: Nine.

T: That’s not a fair number!

Me: What would be a fair number?

T: One where everybody can have the same amount.

Me: Right. But how do you know 9 isn’t a fair number? And what would be one?

T: I don’t know.

Griffin (eight years old): Eight would be. Or 40.

Me: Oh! Forty! Then we could each have 10. Would you like to eat 10 tostadas, Tabitha? But then I would need to buy a second pack of tortillas.

T: [Silent, but her eyes get big and she nods vigorously.]

G: Or 20. Or 12.

The final count is 2 tostadas each for Mommy and Tabitha, and $2\frac{1}{2}$ tostadas each for Daddy and Griffin. Along the way, I promise Tabitha a taco if she finishes her second tostada and is still hungry. This strikes her as fair.

The article, “Standard Algorithms in the Common Core State Standards” by Karen Fuson and Sybilla Beckmann, published in the National Council of Supervisors of Mathematics journal last fall, was recommended to me this weekend.

It’s a weighty one, and relevant to conversations we have had on blogs and on Twitter in recent months, so I didn’t want to read it alone. I asked who was in for a reading group and got quite a few responses.

The article is available through Beckmann’s website (scroll way down to the “Some Other Papers” heading).

I have no experience organizing this sort of thing, but it seems that a hashtag is appropriate. I have investigated the matter and #algorithmchat is both clear on Twitter and communicates at least part of our purpose.

I considered trying to organize synchronous discussion, but it seemed too controlling and impossible to establish. So I vote we discuss by hashtag on Twitter. Anyone who ends up being moved to go long form can include include the #algorithmchat hashtag in a tweet to their post.

I have not read the article yet. It was passed along to me  by a colleague with whom I was  leading a professional development session. She really appreciated the comprehensive nature of the piece (again—it’s a long one).

I have respect for the work of both authors. Fuson’s clear research-based descriptions of what children have to do in order to understand “number” has been very helpful in the work I do with elementary teachers, and I used Beckmann’s Math for Elementary Teachers book for a few years in my courses, where I found it to be the best of the available formal textbooks for these courses. I no longer use a textbook for these courses, but if I needed to, I’d go back to hers for sure. I met Beckmann at a conference a few years ago and I found her thoughtful and open to conversations about learning (not always the case in mathematicians writing textbooks, I have found).

It will probably be midweek before I can carve out time to read the piece and weigh in. In the meantime, I encourage you all to dig in as you are able, say ‘hi’ on Twitter and pass along your longer tidbits in the form of blog posts, and (if you are so inclined) interpretive dance.

Oh, and invite your friends, relatives and enemies to the party. This will be fun.

## Zero=half

A propos of nothing the other day, Tabitha asked a strange question.

Tabitha (six years old): Why are zero and half the same?

Me: They aren’t.

T: Like seven is one more than six, but zero and half are the same. They’re both nothing.

Me: One half? if you have half of something, that’s more than nothing.

T: But half, the number, that’s the same as the number zero.

Recall that last fall, she was not convinced that one-half was a number at all.

She now accepts that one-half is a number. But she hasn’t really dealt with the idea that there are numbers between other numbers. She is doing a bit of beautiful kindergarten logic here. Her premise is that there is only one number less than 1, namely 0. She has also accepted that one-half is a number less than 1. Therefore, one-half and zero are the same.

And—rightly—she is suspicious of this conclusion. The logic is sound, but it doesn’t make sense.

I go to work on that first premise.

Me: Oh. I see. Well, one-half—the number—is between zero and one.

I draw this picture, which I feel is certain to be totally unconvincing.

I was writing upside down. Forgive the crummy 2′s. Note the complex fraction. Take that, Common Core!

But then again, we hadn’t talked about one-half being a number since October. That last conversation seems to have been fermenting all this time, so maybe this one will do the same.

To be continued, I am sure.

## Planting Seeds with Tabitha (or, The Pigeonhole Principle)

We were planting seeds the other day. Indoors. This is Minnesota, after all.

Over the course of many years of gardening I have worked out a system. Yogurt containers, potting soil and these lovely clear IKEA containers.

The IKEA boxes are a recent innovation. They keep soil moisture high (yet have enough volume to allow the plants to breathe), and they let me move plants inside and out according to the ever-varying spring weather (it was 80° on Sunday this week, and it snowed on Wednesday).

Sorry for the digression. Back on task.

We were planting tomato seeds by poking holes into the soil, placing one seed in each hole and covering the seed. We had discussed how deep to make the holes; that the depth corresponding to Tabitha’s first knuckle is not at all the same as the depth corresponding to my own, et cetera.

Tabitha (six years old): How many holes should I put in this one?

Me: Five. Put one in each hole.

I hold out my hand with several seeds for her to take.

T: But there’s more seeds than holes.

Me: So what?

T: So then they’ll be crowded.

This is her line of reasoning, not mine. I had not been at all concerned with trying to offer the precise number of seeds she would need. I had simply shaken some from the pack into the palm of my hand.

But since she started it, I develop a plan. I am going to do my best to get her to state the pigeonhole principle.

Me: But what did you say about the seeds and holes?

T: There are more seeds.

Me: And what are the consequences of that?

T: You said the plants wouldn’t grow as well if there are two in the same hole.

So close! She is using the pigeonhole principle, but I cannot quite get her to state it.

So I do.

I tell her about pigeons and pigeonholes.

We proceed to a lovely (and thoroughly uninformed) discussion of the mechanics of sending messages by carrier pigeon. She wonders, for instance, about how to send a message to your friend, since the carrier pigeon’s unique skill is to fly home from anywhere, but not vice versa. We deduce together that you must need to borrow your friend’s pigeon.

Oh, and those tomato seeds? Brandywine.