When the Standard Algorithm Is the Only Algorithm Taught

Standards shouldn’t dictate curriculum or pedagogy. But there has been some criticism recently that the implementation of CCSS may be effectively forcing a particular pedagogy on teachers. Even if that isn’t happening, one can still be concerned if everybody’s pedagogical interpretation of the standards turns out to be exactly the same. Fortunately, one can already see different approaches in various post-CCSS curricular efforts. And looking to the future, the revisions I’m aware of that are underway to existing programs aren’t likely to erase those programs’ mutual pedagogical differences either.

Of course, standards do have to have meaningful implications for curriculum, or else they aren’t standards at all. The Instructional Materials Evaluation Tool (IMET) is a rubric that helps educators judge high-level alignment of comprehensive instructional materials to the standards. Some states and districts have used the IMET to inform their curriculum evaluations, and it would help if more states and districts did the same.

The criticism that I referred to earlier comes from math educator Barry Garelick, who has written a series of blog posts that aims to sketch a picture of good, traditional pedagogy consistent with the Common Core. The concrete proposals in his series are a welcome addition to the conversation math educators are having about implementing the standards. Reading these posts led me to consider the following question:

If the only computation algorithm we teach is the standard algorithm, then can we still say we are following the standards?

Provided the standards as a whole are being met, I would say that the answer to this question is yes. The basic reason for this is that the standard algorithm is “based on place value [and] properties of operations.” That means it qualifies. In short, the Common Core requires the standard algorithm; additional algorithms aren’t named, and they aren’t required.

Additional mathematics, however, is required. Consistent with high performing countries, the elementary-grades standards also require algebraic thinking, including an understanding of the properties of operations, and some use of this understanding of mathematics to make sense of problems and do mental mathematics.

The section of the standards that has generated the most public discussion is probably the progression leading to fluency with the standard algorithms for addition and subtraction. So in a little more detail (but still highly simplified!), the accompanying table sketches a picture of how one might envision a progression in the early grades with the property that the only algorithm being taught is the standard algorithm.

The approach sketched in the table is something I could imagine trying if I were left to myself as an elementary teacher. There are certainly those who would do it differently! But the ability to teach differently under the standards is exactly my point today. I drew this sketch to indicate one possible picture that is consistent with the standards—not to argue against other pictures that are also consistent with the standards.

Whatever one thinks of the details in the table, I would think that if the culminating standard in grade 4 is realistically to be met, then one likely wants to introduce the standard algorithm pretty early in the addition and subtraction progression.

Writing about algorithms is very difficult. I ask for the reader’s patience, not only because passions run high on this subject, but also because the topic itself is bedeviled with subtleties and apparent contradictions. For example, consider that even the teaching of a mechanical algorithm still has to look “conceptual” at times—or else it isn’t actually teaching. Even the traditional textbook that Garelick points to as a model attends to concepts briefly, after introducing the algorithm itself:

Brownell et al., 1955

Brownell et al., 1955

This screenshot of a Fifties-era textbook is as old-school as it gets, yet somebody on the Internet could probably turn it into a viral Common-Core scare if they wanted to. What I would conclude from this example is that it might prove difficult for the average person even to decide how many algorithms are being presented in a given textbook.

Standards can’t settle every disagreement—nor should they. As this discussion of just a single slice of the math curriculum illustrates, teachers and curriculum authors following the standards still may, and still must, make an enormous range of decisions.

This isn’t to say that the standards are consistent with every conceivable pedagogy. It is likely that some pedagogies just don’t do the job we need them to do. The conflict of such outliers with CCSS isn’t best revealed by close-reading any individual standard; it arises instead from the more general fact that CCSS sets an expectation of a college- and career-ready level of achievement. At one extreme, this challenges pedagogies that neglect the key math concepts that are essential foundations for algebra and higher mathematics. On the other hand, routinely delaying skill development until a fully mature understanding of concepts develops is also a problem, because it slows the pace of learning below the level that the college- and career-ready endpoint imposes on even the elementary years. Sometimes these two extremes are described using the labels of political ideology, but I have declined to use these shorthand labels. That’s because I believe that achievement, not ideology, ought to decide questions of pedagogy in mathematics.

Jason Zimba was a member of the writing team for the Common Core State Standards for Mathematics and is a Founding Partner of Student Achievement Partners, a nonprofit organization.

12 thoughts on “When the Standard Algorithm Is the Only Algorithm Taught

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  3. [I have edited this comment to remove comments that I thought were either too personal or irrelevant to the original post.–Bill McCallum]

    All that said, my question becomes this: if you really believe in the idea of changing the game for math education in this country, how can you allow that it *suffices* to just teach the standard algorithms. That is tantamount to saying that as far as doing anything really new in US math education, the authors of the Common Core Math Content Standards were “just kidding. Please go on with business as usual.”

    Some of the biggest failures thus far I’ve seen in the Common Core have revolved around a failure to adequately prepare teachers or the public for what changes are coming and why … . I mean actually explaining why doing a particular thing in mathematics classes at a certain point is important mathematically; how it adds to and deepens students’ understanding and facility; how it breathes life back into mathematics education for as many children as possible.

    Part and parcel with that issue is the decision to make wholesale changes in all grades at once, rather than to roll things out one or at most two grades per year. That would have allowed us to gather data necessary to revise what’s done next year, and then feeding more data into the loop for training new cadres of teachers for the new grade(s) being added. I think trying to do it all at once undermines the entire project.

    Finally, the strong tie between high stakes testing and the standards continues to be a huge error, one that is clearly undermining public confidence in the standards and raising objections to them from parents and educators. As long as these tests and the standards are intertwined, they are doomed to fail. …

    • A few thoughts on this:

      First, if you read the table accompanying Jason’s post, you will see that there is more than “just teach[ing] the standard algorithms.” There is also sense making and mental computation.

      Second, the Progressions documents do a lot of what you ask for in making connections among the standards and explaining how they fit together across grade levels and how they might work with children.

      Third, I agree a gradual rollout would have been better. But that’s not the way anybody ever does anything in this country! And you talk as if there was someone who could have decided to do things differently, but a distinctive feature of the US education system is that there is no such someone, not even any such organization. In fact, it is barely a system at all. The rollout was up to the individual states who adopted the standards (with the federal government mixing in).

      Finally, on assessment, I agree that assessments are currently tethered to too many high stakes. I think the standards have caused people to face up to this problem. I do think good assessments administered wisely and humanely are necessary.

  4. I’ve always been confused about what the “Standard Algorithm” for subtraction is. I remember quite clearly when I was in second grade – many, many years ago – struggling with subtraction. My mother tried to help me using the standard algorithm she had been taught. It was not the same as the one that I was learning. So, what is standard? According to whom? What makes my second grade teacher’s algorithm right and my mother’s algorithm wrong?

    • There are two variants on the algorithm for subtraction: one where you “borrow” from the next place to the left (really an unbundling of a unit from that place) in order to be able to subtract at the place you are at, and one where you give yourself an extra 10 units at the place you are at and compensate by adding a unit to the subtrahend in the next place to the left. Whether you call these different algorithms or different variants on the same algorithm is a matter of opinion. I think the first one is what most people would regard as standard in the US; other countries use the second one.

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  6. But what about the work of Van de Walle that encourages delaying standard algorithms? We have decades of literature and research to back the use of alternate algorithms and because parents and traditional teachers who are unfamiliar with the research are loud and angry, we need to back pedal?

    As a math educator, I am disappointed that Mr. Zimba is encouraging the sentiment that teachers who teach only standard algorithm out of a textbook are still doing their jobs. So much for getting at those mathematical practices by adhering to procedures without understanding.

    • The math standards allow for approaches in which the standard algorithm is first introduced in grade 4, and they allow for approaches in which several different algorithms are taught for each operation. On the other hand, the standards also allow for approaches in which the standard algorithm is introduced in grade 1, and in which only a single algorithm is taught for each operation. Which of these options to pursue, and how best to pursue them consistent with the standards as a whole, are not questions answered by the standards, but rather questions that must be considered by curriculum designers and educators.

      My progression sketch shows one of the possible worlds. To see it, click the link in the post that says “accompanying table.” One will see that concepts are prominent in the progression, that there are questions about the procedure and why it works, that it is to be taught in such a way that it makes sense to students, and that there is also attention to mental computations like 6012 – 13 or 400 – 388, for which the standard algorithm is probably both slower and less reliable than a readily apparent mental strategy. These aspects of the progression show that it is not about “adhering to procedures without understanding.”

      • Some of the biggest controversies in mathematics education are not settled by the standards, precisely because they are big controversies, and should be settled by the field, not by fiat. What the standards do provide is a common measuring stick by which the adherents of different camps can compare their approaches and decide if one is better than the other, or if both are equally good but different.

  7. We are making the assumption then that all elementary teachers are equipped with the mathematical background knowledge and understanding to connect place value to procedures. In my experience, this is not the case due to the fact that they have only taken one or two methods courses in college.

    Most of my colleagues are intimidated by mathematics and went into elementary education because they didn’t think they would have to teach “hard” math.

    The message that teaching only standard algorithm meets the standards, robs children of mathematics and enables teachers to not build their content knowledge in the discipline.

    I do not disagree with the progression. I am an ardent supporter of the standards. However, students who work in procedures only are ill-equipped to mentally compute numbers because they lack place value understanding.

    While I see the need to placate the masses, we should also recognize that our students and parents deserve better than to be happy. They deserve to be mathematically literate.

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