## Subtraction: What is “the” Standard Algorithm?

One common complaint amongst anti-reform pundits is that progressive reform math advocates and the programs they create and/or teach from “hate” standard arithmetic algorithms and fail to teach them. While I have not found this to be the case in actual classrooms with real teachers where series such as EVERYDAY MATHEMATICS, INVESTIGATIONS IN NUMBER DATA & SPACE, or MATH TRAILBLAZERS were being used (in fact, the so-called “standard” algorithms are ALWAYS taught and frequently given pride of place by teachers regardless of the program employed), the claim begs the question of how and why a given algorithm became “standard” as well as how being “standard” automatically means “superior” or “the only one students should have the opportunity to learn or use.” It strikes me that it is almost as if such people are stuck in some pre-technological age in which we trained low-level white collar office workers to be scribes, number-crunchers who summed and re-summed large columns of figures by hand, etc. The absurdity of seeing kids today as needing to prepare for THAT sort of world is evident to anyone who spends any time in a modern office, including that of a small business. Desktop and handheld calculators are commonplace. So are desktop and laptop computers, not to mention tablets or smartphones in shirt pockets running Desmos, Wolfram|Alpha, etc. There is a need for people to understand basic mathematics, but not to be fast and expert number-crunchers in that 19th-century sense.

Thus, it seems reasonable to ask what should be an obvious question: if the goal is to know what numbers to crunch and how (what operations need be used) to crunch them, and, most importantly, to correctly interpret and make decisions based upon the results of the right calculations, and further if it is glaringly obvious that the actual number-crunching itself is done faster and more accurately by machines than by the vast, vast majority of humans can reasonably expect to do, why would any intelligent person be obsessing in 2017 over the SPEED of an algorithm for paper and pencil arithmetic? For the big argument raised for always (and exclusively) teaching one standard algorithm for each arithmetic operation seems to be speed and efficiency.

I have argued repeatedly that the efficiency issue is only reasonable if one fairly assesses it. And to do that is to grant that a student who misunderstands and botches ANY algorithm is unlikely to be performing “efficiently” with it. Compared with a student who uses even a ludicrously slow algorithm (e.g., repeated addition in place of any other approach to multiplication) accurately, the student who can’t accurately make use of the fastest possible algorithm is going to be taking a long time to arrive at the right answer which will be reached, if at all, only after many missteps and revisits to the same problem. For that student, at least, the “algorithm of choice” is not efficient at all. So finding one that the student understands and can use properly would by necessity be preferable. But not, apparently, in the mind of ideologues. For them, there’s one true way to do each sort of calculation and they are its prophets.

Of course, I’m not favoring teaching alternate algorithms because I dislike any particular standard one or feel the need to “prove” that, say, lattice multiplication is “better” than the currently favored algorithm. On the contrary, I’m all for teaching the standard algorithm. But not alone and not mechanically, and not at the expense of student understanding. Indeed, from my perspective, it’s difficult to understand why it is necessary to mount a defense for alternative algorithms in general, though any particular one may be of questionable value and might need some justifying or explaining. If anything, it is those who hold that there is a single best algorithm that is the only one that deserves to be taught who need to make the case for such a narrow position. In my reading, I’ve yet to encounter a convincing argument, and indeed most people who hold that viewpoint seem to think it’s glaringly obvious that their anointed algorithms are both necessary and sufficient.

What compounds my outrage at the narrower viewpoint is the fact that it is based for the most part on woeful historical ignorance. Previously, I’ve addressed the question of the lattice multiplication method, which has come under attack from various anti-reform groups and individuals almost certainly because it has been re-introduced in some progressive elementary programs such as Everyday Math and Investigations in Number, Data, and Space. The arguments raised against it are very much in keeping with above-mentioned concerns with speed and efficiency. Ostensibly, the algorithm is unwieldy for larger, multi-digit calculations. The fact is that it is just as easy to use (easier for those who prefer it and get it), and while it’s possible to use a vast amount of space to write out a problem, it’s not required that one do so and the amount of paper used is a social, not a pedagogical issue. But please note that I said RE-introduced, and that was not a slip. The fact is that this algorithm was widely used for hundreds of years with no ill effects. Issues that strictly had to do with the ease of printing it in books with relatively primitive technology and problems of readability when the printing quality was poor, NOT concerns with the actual carrying out of the algorithm, caused it to fall into disuse. Not a pedagogical issue at all, and with modern printing methods, completely irrelevant from any perspective. Yet the anti-reformers howl bloody murder when they see this method being taught. The only believable explanation for their outrage is politics. They simply find it politically unacceptable to teach ANY alternatives to their approved “standard” methods. And their ignorance of the historical basis for lattice multiplication as well as their refusal to acknowledge that it is thoroughly and logically grounded in exactly the same processes that inform the current standard approach suggests that bias and politics, not logic, is their motivation.

### Subtraction algorithms

I raise all these questions because I had my attention drawn to a “non-standard” algorithm (actually two such algorithms and some related variations) for subtraction. Tad Watanabe, a professor of mathematics education whom I’ve known since the early 1990s posted the following on a mathematics education discussion list:

Someone told me (while back) that the subtraction algorithm sometimes called “equal addition algorithm” was the commonly used algorithm in the US until about 50 years ago. Does anyone know if that is indeed the case, and if so, about when we shifted to the current conventional algorithm?

I couldn’t recall having heard of this method, and so I was eager to find out what he was talking about. Searching the web, I discovered an article that repaired my ignorance on the algorithm: “Subtraction in the United States: An Historical Perspective,” by Susan Ross and Mary Pratt-Cotter. This 2000 appearance in THE MATHEMATICS EDUCATOR was a reprint of the article that had originally appeared several years previously in the same journal. It draws upon a host of historical sources, the earliest of which is from 1819. And there are other articles available online, including Marilyn N. Suydam’s “Recent Research on Mathematics Instruction” in ERIC/SMEAC Mathematics Education Digest No. 2; and Peter McCarthy’s “Investigating Teaching and Learning of Subtractions That Involves Renaming Using Base Complement Additions.”

The Ross article makes clear that as far back as 1819, American textbooks taught the equal additions algorithm. To wit,

1. Place the less number under the greater, with

units under units, tens under tens, etc.2. Begin at the right hand and take the lower figure

from the one above it and – set the difference

down.3. If the figure in the lower line be greater than the

one above it, take the lower figure from 10 and

add the difference to the upper figure which sum

set down.4. When the lower figure is taken from 10 there

must be one added to the next lower figure.

In fact, according to a 1938 article by J. T. Johnson, “The relative merits of three methods of subtraction: An experimental comparison of the decomposition method of subtraction with the equal additions method and the Austrian method,” equal additions as a way to do subtraction goes back at least to the 15th and 16th centuries. And while this approach, which was taught on a wide-scale basis in the United States prior to the late 1930s, works from right to left, as do all the standard arithmetic algorithms currently in use EXCEPT notably for long division (which may in part help account for student difficulties for this operation far more serious and frequent that are those associated with the other three basic operations, it can be done just as handily from left to right.

Consider the example of finding the difference between 6354 and 2978. Using the standard approach, we write:

### Equal additions

### Left to right subtraction?

I will not discuss or describe in detail the Austrian algorithm other than to say that it doesn’t feel “right” to me. That’s not saying it’s “wrong,” but rather that I can’t see it as one I would use. And here is one major difference between me and the reform-haters: that doesn’t mean I wouldn’t revisit it or wouldn’t show it to teachers, and perhaps if I saw a particular student or class for whom it might prove helpful, I’d teach it. My “taste” isn’t the issue, but rather keeping a large number of options available for my practice and for my students. I suppose that’s just not very “efficient” of me.

Finally, it bears noting that there are references in the above-mentioned articles to research on the use of these algorithms, and at least some reason to think that equal additions should be looked at again very seriously by mathematics teacher educators and K-5 teachers. If you read the historical treatment of subtraction algorithms in the US, you’ll likely note how much chance and arbitrariness there can be in how one particular algorithm comes into fashion while others fall into disuse. I see no firm evidence for the “superiority” of the current most commonly-taught algorithm, and there is clearly a history of it’s causing difficulties for particular students. Would the universe collapse if we were to teach both? Even more, would it collapse if we didn’t rush to teach it right away, but rather, as has been proposed by more than a few researchers and theorists on early mathematics education, let students play and invent their own algorithms first, before trying to steer them toward one or another of our own? Sadly, the anti-reformers amongst us, the activist educational conservatives who are constantly trying to narrow rather than open up K-12 education, believe that there’s always one best way to do everything. And not coincidentally, that way always turns out to be the one they learned as a child. That, more than anything, is why I think it reasonable to call the not-so-traditional math that they push on everyone “nostalgia math.” It’s not that what they learned is better. It’s just what they learned back in simpler times when life was easy and there were no Math Wars and no one like me to suggest that their emperor is stark naked.

Thanks for this common sense article. Especially appreciative that it is not about cheerleading for CCSS or against CCSS. It just expresses sound mathematical concepts.

Thanks for the kind words, John. Rest assured that the words “Common Core State Standards” never entered my mind at ANY point in the creation of that blog piece, nor do they, in general, these days. I have written a good deal about them over the last decade and remain convinced that:

1) National curriculum standards are a bad idea, doomed before they are actualized, and will never be the right way to improve mathematics education (or other areas of education;

2) even if I were wrong about my first point, the high-stakes standardized testing that came with the Obama Administration’s attempt at national standards ensured that the whole endeavor would hurt kids;

3) the close ties between various corporate interests and the creation and implementation of the standards was a further guarantee of doom for kids;

4) despite the wide-spread paranoia about particular aspects of the math standards (I won’t get into the even wider and deeper paranoia about what sorts of things were recommended (never mandated) for kids to read), the Standards for Mathematical Practice were sound and in fact are the source of much of the backlash against the overall math standards by some of the usual suspects from the Math Wars (e.g., R. James Milgram, Sandra Stotsky, and most of old Mathematically Correct/HOLD cabal); that should be taken as an indication to progressive educators that there are, in fact, things worth preserving and even fighting for in the CCSS-M.

5) unfortunately, the combination of the first three points above ensures that the waters are far too muddied for most people to see the wisdom of my fourth point. And so we slog on towards another few decades of pathetically weak math teaching in US K-12.

Thank you for your comments. As a new teacher about to start teaching Algebra I for 7th graders, I am concerned that the methods of addition, subtraction, and specifically division that I learned as a child under the European (Spanish) system 40 years ago are a bit different than the algorithms used in the U.S. I don’t consider any method superior to any other but it comes naturally to me to teach it the way I did it in Europe so I am learning new methodologies (this is especially challenging when doing long division as dividend and divisor are inverted.

I taught some Haitian nursing students around 1990 who learned to set up long-division the way you describe. Apparently, that is common in France and former French colonies. Didn’t know anyone taught it like that in Spain. They told me they liked the “American” way better. This was my first experience teaching math classes, so I was too inexperienced to “risk” taking class time to have them show their approach to the whole class and elicit reactions from other students. Probably, too, I was afraid I wouldn’t understand what they were doing. That was foolish of me: I missed out on what could have been a wonderful opportunity for me and the students.

That said, I’m not sure that the method you’re talking about is really computationally different so much as it simply is organized differently. If you teach it to students, what will be important is for them to have a chance to react without your biasing them one way or another in advance. Give them a chance to compare and contrast various methods and ask them to discuss similarities and differences, and which they prefer and why. Don’t state in advance that one of them is “right” or the way you were taught. That will maximize the likelihood of getting honest reactions from them.

Cool explanation. If you want to try practicing what you just learnt, try playing the game https://play.google.com/store/apps/details?id=speed.maths.calculation.mathstronaut

Thank you for your insight! I really like how you stress that in today’s day its more important that students focus on the process of understanding mathematical concepts in place of following set theories to reach a certain product quickly. I’m in an education program and hoping to teach primary. I was wondering how you would implement these kinds of concepts in a classroom? To flesh out some different ideas, would you start by placing a subtraction problem on the board and encourage inquiry and discussion about how we (as a class) could solve it? Or would you start by introducing one method of subtraction, get the students familiar with it and then proceed to ask if any students have come up with a different way to reach the same outcome?

I’m a fan of what I’ve seen in videos from Japanese elementary classrooms: get as many student ideas on the board, don’t give ANY feedback as they come in, then ask the class to discuss what’s gone up. There’s no set way of conducting such a conversation because you won’t know for sure what suggestions will emerge (even if you’ve taught the topic many times in previous years). Sometimes in explaining an erroneous method (“buggy algorithm”), a student will catch the difficulty. Sometimes it will take another student to point out places things to get stuck or go wrong or simply fail to make sense to her/him. The more you can let students raise concerns, express confusion, and hash things out without offering an opinion yourself, the better. But you can expect to need to help things move forward by asking probing questions should the conversation bog down.

The alternative you suggest in my experience will lead some, possibly most, students to feel that you’re starting with the ‘right’ one. If you lead these sorts of explorations frequently and have established that you can’t be read so simply, then perhaps that won’t be a problem down the line with other algorithms or problems. But at the beginning of the year, in particular, you want to be very careful about allowing them to try to “read” you as a way to avoid having to think for themselves. I’ve seen video of a few teachers in NYC who had a real gift for not giving much, if anything, away to such an extent that students tell each other not to bother trying to draw out answers from the instructor before they’ve struggled over a given issue themselves. It’s very gratifying to see that. These were high school kids, but I believe similar classroom cultures can be built in K-5 (to the long-term advantage of students) if the teacher is committed to that approach and uses it consistently from the start. Dan Meyer’s motto, “Be less helpful” is perhaps the best, most cogent advice I’ve seen in mathematics education/teacher education in my 30 years of teaching that subject (I started teaching English 45 years ago).

Hope that helps.

thank that help!

Thank you for your insight! I have never thought about that left to right subtraction. I also like how you emphasize students comprehending the math instead of just memorizing procedures. That helps students retain information in the long run.