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.
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.