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.

We have to call this stuff out

Strange start to the day.

First was Kate Nowak noticing the hate Vi Hart routinely heaps on math teachers. As happens at 2:04 in the video below. Note especially her tone here.

Then the following comes across my desk (click on it to see the full size version).

The first line reads, “Statistics show that more students are coming from high school with weaker mathematical backgrounds than ever before.”

No citation. No argumentation. Just Statistics show.

I have replied with the following:

Wow. Gotta admit that first line is provocative.

What evidence are you basing this claim upon?

Thanks,

Christopher Danielson

I’ll keep you posted.

In the meantime, we need someone with some mad video skills to make a “Sal and Vi Hate Math Teachers” video in the spirit of “Hollywood Hates Math“. I’ll help with the archival work. Who’s in?

Khan’s kindness

Say what you will about Sal Khan (and I have certainly said a lot), but he communicates a tremendous amount of patience with his students.

I watched his video on “Basic Addition” the other day.

He begins with the assumption that the viewer has absolutely no equipment for finding the sum 1+1.

This bears repeating. He assumes absolutely no knowledge of the meaning of the addition symbol in the expression 1+1. None.

As he does so, Khan is patient, supportive and encouraging. He does not condescend and he even apologizes for the word basic in the title of the video-worrying that his viewer may be put off by the term.

When I think of the culture of many math classrooms, in which students don’t ask questions out of fear of looking stupid, or in which instructors use words such as trivial and obvious without apology or concern for the effect these words can have on learners, I get a glimpse of what people find so appealing about Khan’s videos.

Khan gives permission to not know. He reassures the viewer that it’s OK to still be figuring things out. And of course he is happy to repeat what he just said as many times as the viewer likes. Just stop and rewind. The calm, patient demeanor never changes.

The field could learn from Khan’s kindness.

 

The Teaching Gap

The Teaching Gap is a funny book. Not funny “ha ha”. Funny in terms of how it has been perceived.

It was published in about 1999 and the professional conversation turned to “Japanese Lesson Study”. Initiatives were initiated. Presentations were made. Etc.

And in fact, Japanese Lesson Study comprises about the last third of the book. But it’s not what the book is about. The Teaching Gap is about how assumptions and values play out in classroom practice. And it’s about how things don’t have to be the way they are in math classrooms because they are quite different in other countries’ math classrooms.

But it’s not about lesson study. Here is why I care.

I frequently recommend The Teaching Gap to colleagues. If the message they take away is that I am recommending Japanese Lesson Study, then my mission is not accomplished. When I recommend the book, I am recommending an insightful study of classroom teaching that forces readers to think critically about their own classroom practice.

We are all ready for critical thinking about our classroom practice.

We are not all ready for Lesson Study.

In fact, as a group, American teachers are tremendously far from being ready for lesson study.

Consider the rhetoric around Khan Academy. According to a recent Time Magazine piece, Khan says “He doesn’t use a script. In fact, he admits, ‘I don’t know what I’m going to say half the time.’ But the low production values of Khan’s videos are part of what makes them so effective.”

How does Khan plan? Same article: “Khan begins by doing two minutes’ worth of research on Google, looking for graphs that affirm what he remembers from his econ class in college, then flips through a few pages in a 4-in.-thick economics textbook sitting on his desk and clicks a button to start recording.”

And simultaneously, Khan has received millions of dollars to do more of this. He is hailed as “brilliant” (Newsweek) and “The new Andrew Carnegie” (Time).

If we’re ready to use Khan Academy as a primary instructional medium, we are not ready for lesson study.

At about 1:50 in the video below, a math teacher extols Khan Academy for the diagnostic data it gives him on students.

I should go back and think about what I’m really asking my students to do. Whether that’s something that’s too high of a level for 80% of the class, or something that’s too low of a level for the class.

That reflection is nowhere close to what lesson study would produce. Lesson study would involve planning with other teachers, teaching (with others observing) and a discussion afterwards of how students’ ideas played out in the lesson. The teachers would discuss the lesson in minute detail. Conclusions would be drawn about how the examples used should be different, what questions should be asked next time, whether the context in the lesson supported student reasoning, etc.

The rhetoric in US schools tends to focus on the speed with which material is covered, not on nuanced details of lesson structure.

That teacher in the video? I’m sure he’s very good at what he does. But he’s not questioning whether a linear trajectory through a set of skills is the best description of what it means to learn mathematics. Instead, he seems to uncritically accept this proposition. And that’s where a lot of us are, frankly.

It’s why Khan Academy is hailed as such a revolution. Khan Academy is a friendly tool for doing something better. But that thing it does? It’s the wrong thing to do. So being better at it doesn’t matter very much.

That teacher needs to read the first two-thirds of The Teaching Gap. Like all of us, he needs to critically analyze his practice in light of the ideas presented there. He needs to talk with colleagues about what he concludes.

Then he can come back in a couple of years, read the last part of the book and form a lesson study group.

So I continue to recommend The Teaching Gap. We desperately need to know how things could be different. But when I recommend it, I will continue to point out that the main message of the book is in the first two-thirds. Because as a field, we’re not ready for lesson study.

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).