Tag Archives: ccss

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.

Screen shot 2013-05-22 at 9.22.54 AM

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.

Reading group [#algorithmchat]

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.

Guess the temperature

Griffin and I play a little game called Guess the Temperature. It goes about how you would expect. We step outside on the way to his bus. I ask him to guess the temperature. If I don’t already know, I get to guess after he does. If I do already know, I don’t cheat; we just remark on how close his guess was.

In Minnesota, this means we get to study integers.

Me: Griff, guess the temperature.

Griffin (eight years old): Two below zero.

Me: It’s three degrees above.

G: So I was off.

Me: Not by much, though. How much were you off by?

G: [muttering to himself, then loudly] Five degrees!

Me: How did you know that?

G: It’s two degrees up to zero, then three more.

Let’s pause for a moment here. You know how I just won’t shut up about CGI (Cognitively Guided Instruction)? It’s because they’re right. Children know mathematics before it is formally taught.

Consider the grade 6 (for 11-year olds) Common Core Standard 6.NS.C.5

Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.

Griff pretty much has this nailed down and is making progress on grade 7. But no one has formally taught him how to subtract integers. He reasons his way through a problem by making sense of the relationships in the context. He can find 3-(-2) without knowing keep-change-change.

But it’s not just Griffin. CGI demonstrated that children—all children—develop mathematical models of their worlds that precede instruction, and that instruction sensitive to these mathematical models is better than instruction that ignores them.

Back to our conversation.

Me: So what if it 10 degrees out, and you guessed 3?

G: [quickly] I’d be seven off.

Me: Right. How do you know that?

G: Ten minus three is seven.

Me: Nice. Subtraction. Do you know that you can always express the difference between your guess and the actual temperature with subtraction?

So in that last example, you subtracted your guess from the actual temperature. You could do that with your real guess today.

So three minus negative 2 is five.

G: [silent]

By this time we were nearing the bus stop. I had offered this tidbit as an intellectual nugget to chew on, rather than a lesson I expected him to absorb. But that is what it means to have instruction be sensitive to children’s mathematical models.

I don’t think they mean that

Thanks to John Golden (@mathhombre) for the find.

From the Common Core State Standards Progressions document on the 6—8 Expressions and Equations standards:

The “any order, any grouping” property is a combination of the commutative and associative properties. It says that sequence of additions and subtractions may be calculated in any order, and that terms may be grouped together any way (p.5).

Performance assessment: ratios

It’s time for a performance assessment.

This is not multiple choice.

If you have been reading along, you know that I advocate talking math with your kids as the mathematics equivalent of reading with them 20 minutes a day.

Furthermore, you have surely read each of my thousands of words on Common Core’s vision of ratio, rate and unit rate.

And yesterday I proposed a bit of alternate text for the Progression on Ratio and Proportion.

Your assessment task is this.

The task

Imagine you have a 12-year old daughter. She has been learning about ratios and is assigned the task of finding real-world applications of them, as found in the media. She comes across an article that interests her. You strike up a conversation about the excerpt below.

Defend or critique any of the following claims:

(1) The Common Core Progression on Ratio and Proportion will be helpful to you as a parent in discussing the relationship between this passage and her homework.

(2) The distinctions being made (among ratio, rate and unit rate) in the Common Core Progression are useful and meaningful for interpreting this passage.

(3) The discussion proposed in yesterday’s post will be helpful to you as a parent in discussing the relationship between this passage and her homework.

(4) The distinctions being made in yesterday’s post (among ratio, rate and unit rate) are useful and meaningful for interpreting this passage.

The passage

Joe-urban discusses parking at urban grocery stores:

However, David Taulbee, Architectural Manager of Publix, notes that parking at many of their urban stores is full only at peak times, so that sacred parking ratio of five per thousand is called to question, particularly if the store has other parking options nearby like shared, on-street or bicycle parking.

(N.B. That’s five parking spaces per thousand square feet of retail space.)

Your work will be scored on the basis of relevance and the use of evidence. It will not be scored on the extent to which you agree with the scoring committee’s views on these matters.