Understanding why algorithms work

Children need to learn algorithms to understand math, write Alice Crary, a philosophy professor, and W. Stephen Wilson, a math professor, in the New York Times.

. .  it is true that algorithm-based math is not creative reasoning. Yet the same is true of many disciplines that have good claims to be taught in our schools. Children need to master bodies of fact, and not merely reason independently, in, for instance, biology and history.

Mastering an algorithm requires “a distinctive kind of thought,” they write. It’s not “merely mechanical.” In addition, algorithms are “the most elegant and powerful methods for specific operations. . . . Math instruction that does not teach both that these algorithms work and why they do is denying students insight into the very discipline it is supposed to be about.”

Some commenters claimed math reformers advocate a “balanced” approach that includes algorithms, writes Barry Garelick in Education News. He is dubious.

I am reminded of a dialogue between a friend of mine—a math professor—and an public school administrator.  My friend was making the point that students need basic foundational skills in order to succeed in math. The administrator responded with “You teach skills. But we teach understanding.”

. . . The reform approach to “understanding” is teaching small children never to trust the math, unless you can visualize why it works. If you can’t “visualize” it, you can’t explain it.  And if you can’t explain it, then you don’t “understand” it.

According to Robert Craigen, math professor at University of Manitoba, “Forcing students to use inefficient procedures that require ham-handed handling of place value so that they articulate “meaning” out loud in every stage is the arithmetic equivalent of forcing a reader to keep his finger on the page and to sound out every word, every time, with no progression of reading skill.”

The power of math, however, is allowing for exploration of concepts that cannot be visualized.  Math is what takes over when our intuition begins to fail us.

Garelick, who’s launched a second career as a math teacher, links to a 1948 math book’s illustration of different ways to do mental multiplication:

Figure 2 (Source: Study Arithmetics, Grade 5)

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  1. That cartoon is very similar to the way that Singapore Math teaches its ‘mental math’.

  2. Headbone arithmetic can be dangerous.

  3. I’ve always done math this way without being taught, and just considered myself a divergent thinker.

    • Yes, good point. When students are taught the standard algorithms they LEARN the ‘strategies’ by osmosis, or by their own natural cleverness, built up over much experience with arithmetic, as was the case with you. However, when students are taught only the strategies, they come away not knowing the standard algorithms. Thus they have only (i) ad-hoc methods that work in (ii) special cases, are (iii) inefficient, if usable at all, in arbitrary problem, and (iv) do not support a general understanding of the operations, because the lack of the standard algorithm results in not having a central organizing method.

      As I say in my article, I am fine with sequences of lessons that introduce various “strategies” as part of a learning continuum whose central goal is the algorithms, and for which they provide scaffolding for important steps in developing those algorithms. This is what is missing in their “fix”, which still treats the algorithms as an afterthought; i.e., standard algorithms are treated as a side dish, with the strategies being the main course.

  4. Don’t try it while operating heavy machinery.

  5. Using the Distributive Property to break down both numbers and doing FOIL on it, then putting the pieces together works nicely too. Like the last panel, but digested a bit better, also lets you take advantage of perfect squares or two multiple of tens.

  6. It’s also helpful if you know your multiplication tables out to 20×20 (as it was in my father’s day). You can then scale those factors up to 3 and 4 digit numbers.

    It’s the same as what they say about cameras: the best camera is the one you have with you when the opportunity for a picture strikes. The best calculator is the one you have with you (your head).

  7. “You teach skills. But we teach understanding.”

    Skills drive understanding, not the other way around. Understanding is a many layered thing. You can have a simple place value understanding, an algebraic (distributive property) understanding, a scientific notation understanding, a base value understanding, and a linear space understanding. Conceptual understanding is not enough, and full understanding only comes from seeing and mastering all of the different possible problem variations – something that textbooks and nightly homework are all about. It’s amazing that the low expectation pedagogical silliness of K-6 gives way in most schools to a steady diet of direct teaching, textbooks, and individual nightly homework in high school. Half of the K-12 world has rejected reform math, and that’s the part that prepares kids for STEM careers, assuming they survive K-6 with help from parents and tutors. Ask the parents of the best students what they did at home in K-6. Why do schools send home blanket notes to parents asking them to practice math facts with their kids? That’s incredible. Some schools even plan math nights to help parents do their job.

    When I tutor high school students in math, their seemingly rote abilities are really just incomplete development of skills. I can explain meanings, and the students can explain the meanings, but if they don’t actually do all of the problem variations in the homework assignment, whatever meaning they have in their heads will fail them. They have to do all of the problems individually. It’s my mantra. Do and understand all of the problem variations in the homework assignments every night and you will get an ‘A’ in the course. When I tutor someone, we go over both understanding and skills, but concepts and one or two problems do not understanding make. After each session, I stress that they really have to go home and do each problem.

    Too many reformists see skills only as rote and a matter of speed. They are wrong. You might be able to weasel a defense related to the standard algorithms for arithmetic, but once you get to fractions and algebra, conceptual understandings leave you nowhere. K-6 math reformists really just want cover for low expectations. They increase the ability range with full inclusion and then claim that what they promote is more rigorous and helps create critical thinking and understanding. Baloney. Our schools have been using Everyday Math for over 10 years, and before that, they used MathLand. It’s been 20 years. Where are the results? Still, reformists haul out the bogeyman of traditional math. They push reform math because that’s all they know. That’s all they were directly taught in ed school. They cannot think outside of their tiny little philosophical box.

  8. I agree with Steve that few people can understand mathematics without a lot of wallowing in the mud. It’s when you wallow in the mud that you realize you don’t really understand it like you thought you did.

  9. I recognize the word “algorithm” and had a general idea of what it means, but it wasn’t good enough to understand what people were really saying here. The linked article provides a great explanation: “A mathematical algorithm is a procedure for performing a computation. At the heart of the discipline of mathematics is a set of the most efficient — and most elegant and powerful — algorithms for specific operations. The most efficient algorithm for addition, for instance, involves stacking numbers to be added with their place values aligned, successively adding single digits beginning with the ones place column, and “carrying” any extra place values leftward.”

    I would think by now that the math community would know which algorithms cover 80% of common situations in which we’d need them (the 80-20 rule) and concentrate on teaching those.