# Tag Archives: common core

## 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.

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.

## 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.

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.

## Ratcheting back the rhetoric on Common Core

Bill McCallum writes in the comments here at OMT:

I don’t think that effort [to define terms such "ratio" and "rate" that CCSS leaves undefined] deserves quite the ridicule it is receiving here, but never mind, the criticism will be taken into consideration nonetheless and inform the final draft. I’ll only say that if I had a dollar for every time someone told me the answers to all these questions were obvious, I’d be a rich man. Of course, the “obvious” answers are mutually self-contradictory. This seems to be an area where it is very difficult indeed to find common language, and where emotions run high.

Fair enough. I’m happy to tone things down a bit.

I do need to observe that no one here at OMT (least of all me) has suggested that the answers to the questions at hand are “obvious“. I agree that they are not obvious at all, and I agree that it is very difficult to find common language with respect to these ideas. (Although this last bit is tricky; if we all use the same words but mean different things by them, are we speaking a common language?)

No, my critique is not at all that Common Core has failed to state the obvious definitions.

My critique is that I see no evidence that Common Core-either the Standards or the Progression on rational number-take into account research on how children learn this content, nor do they seem to coincide with everyday uses of these terms. In the era of No Child Left Behind and “evidence-based practice”, I find it troublesome that results of important research work such as that in the Rational Number Project or Cognitively Guided Instruction don’t seem to form a basis for either document.

I find it surprising that there are no research references in the Progression.

In my work with Connected Mathematics, I have many times had teachers ask for definitions of rate, ratio, fraction and rational number. As writers, we have hashed out these ideas many times as well. Answers are not obvious and reasonable people can disagree.

But we sort of agree that answers to these questions ought to be consistent with, and explain relationships to, uses of these terms in mathematics and the world. I don’t understand why this isn’t the starting place for the Progression.

ratio is a multiplicative comparison of two quantities (usually both are non-zero). Conventionally, we use the term “ratio” to apply to part-part comparisons, but this need not be the case.

We can express ratios in several forms. If there are 5 girls for every 3 boys in a certain class, we say that (1) the ratio of girls to boys is 5 to 3, (2) the ratio of girls to boys is 5:3, (3) the ratio of girls to boys is $\frac{5}{3}$, (4) there are 5 girls for every 3 boys.

The fraction notation $\frac{5}{3}$ is problematic in early ratio instruction because children may confuse it to mean that $\frac{5}{3}$ of the students are girls. Children are accustomed to fraction notation being reserved for part-whole relationships; for this reason the notation should be saved for later instruction.

The term rate suggests change. We tend to talk about a “ratio” in static situations where the values remain constant but a “rate” in a situation where the quantities are changing. In the girls and boys situation above, it would be correct to say that there is a rate of 5 girls for every 3 boys, but this feels awkward. If students were enrolling in a school and there were 5 girls enrolling for every 3 boys, the term “rate” is a more natural fit.

unit rate is a rate where one of the quantities being compared is 1 unit. If we enroll five girls for every three boys, this is not a unit rate. We could say that there are $\frac{5}{3}$ girls per boy enrolling at the school, or $\frac{5}{3}$ girls for every boy. For every (non-zero) unit rate, there is a reciprocal unit rate. So we can also say that there are $\frac{3}{5}$ boys per girl.

What counts as a unit varies. When computing a “unit rate” for buying pop, we could compute the cost per ounce, the cost per can, the cost per six-pack or the cost per case of 24. Which of these is considered a unit rate depends on our choice of unit.

To summarize the discussion, ratios and rates are different mainly in connotation. Each expresses a multiplicative relationship between two numbers (as opposed to an additive relationship, for which we use the term “difference”). Unit rates are important forms of rates because of their intimate connections to algebraic and calculus ideas such as slope and rate of change.

It’s just a first stab at discussing these terms in this context, but is consistent with common usage and focuses the discussion on the main idea that is important at this level-rates and ratios are about multiplication relationships, which are the heart of proportional reasoning.

Going back to the lawn example that has been the focus of discussion here as well as over at the Common Core Tools website, this would suggest that “7 lawns in 4 hours” is a rate (there is change involved, and it’s not a part-to-part relationship), and that there are two unit rates: $\frac{7}{4}$ lawns per hour and $\frac{4}{7}$ hours per lawn.

Again, I am not claiming that these relationships are obvious. But if a couple of important goals for the Progressions work are (1) clarity and (2) usefulness for teachers, professional development and curriculum development, I think my proposal above is an improvement over the present document.