Faulty Logic: Math Wars

If you haven’t seen it yet, two professors (one philosophy, one math ed) just published an opinion piece in the NY Times online blog, decrying the state of the reform movement in math education.  These reformers, they warn, are trying to get away with not teaching the standard American algorithms for computation.  These upstarts and their “numerical reasoning” may destroy us all.

“At stake in the math wars is the value of a “reform” strategy for teaching math that, over the past 25 years, has taken American schools by storm. Today the emphasis of most math instruction is on — to use the new lingo — numerical reasoning. This is in contrast with a more traditional focus on understanding and mastery of the most efficient mathematical algorithms.”

It may have been a mistake to go on to browse the comments.  The top comment right now reads,

“I am a mathematician, and I am indeed outraged and baffled at the move against teaching the standard algorithms in grade school…”


“Anyone who would propose that one can teach “mathematics” without the standard algorithms is simply innocent of any notion of what mathematics is. He or she is proposing to “educate” students in a subject while withholding from them the definitions and meanings of the nouns and verbs one uses in thinking and talking about mathematics.”

Oh dear.

The authors argue that the value of standard algorithms is that “algorithm-based calculation involves an important kind of thinking.”  In order to know whether you have mastered the algorithm, they say, you need to understand it on more than a procedural level.  I think all math teachers would agree with that.  However, using algorithms doesn’t produce such understanding understanding any more than repeatedly transposing sheets of music from one key to another would produce the ability to make music (see A Mathematician’s Lament for more on this wonderful analogy).

So how do we get students to understand the algorithms they use? (And by algorithm, the authors don’t seem to mean ‘method’ but the much more specific Standard American Algorithm, or The Algorithm, as if it is self-evident precisely which methods the real ones are). The reformers, they say,

“insist that the point of math classes should be to get children to reason independently, and in their own styles, about numbers and numerical concepts. The standard algorithms should be avoided because, reformists claim, mastering them is a merely mechanical exercise that threatens individual growth.”

The reform math teacher’s criticism about a traditional approach is actually not that The Algorithm threatens individual growth, but that it  doesn’t produce any growth.  Students don’t come to understand something because they heard someone else describe what they understand. They will absorb and remember what they actually are asked to do, not what they’re asked to watch or listen to. That’s the real foundation of an inquiry-based classroom.  If students actually make math – that is, grapple with numbers and operations in an authentic way – then they will be able to interpret and make sense of the math that others do, including the standard methods.

The biggest mistake the authors make, though, is when they say that the standard algorithms

“are the most elegant and powerful methods for specific operations… Our best representations of connections among mathematical concepts.”

That’s wrong in at least three ways.  First, the fact that these methods are the most common doesn’t make them most elegant and powerful.  That’s like saying that fast food is superior because its so widely available. These methods are the ones that prevailed and flourished in our country during a completely different time, when people had no alternatives but to do computation to the nth decimal number by hand.

Second, other countries and cultures use different algorithms for basic computation and those methods are no less elegant.

Finally, and most importantly, the standard American algorithms are hugely inelegant in many cases.  Use the standard algorithm to multiply 999 x 123.  It’s ridiculous.  Any fifth grader could tell you its a better idea to multiply 1000 x 123 and take away one group of 123.

Maybe this opinion piece is a gift – a throwing down of the gauntlet for math educators.  I challenge all of us to respond to this.  I invite you to read the opinion piece and try – in the comments.  50 words or less. Without profanity.  Unless absolutely necessary.

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7 thoughts on “Faulty Logic: Math Wars

  1. Paul Bogdan says:

    Bloom taught that there are two levels of learning: knowledge and understanding. Algorithms are often used to bypass understanding and solve with only knowledge. A good example is the algorithm “is over of equals % over 100″. This algorithm will solve some very complicated percent problems while leaving its user devoid of any understanding about what they are doing. We math teachers need to engage the students and leave them with more understanding, not less. Dan Meyer’s three act lesson idea is both a great method and explanation for what needs to occur in a math class.

    • Ana Fox says:

      Well said, Paul. This is a great example.
      The crux of the issue seems to be this idea that procedural mastery (knowledge) can some how create understanding. Susan Jo Russell has said, “There are decades of data showing that U.S. students have learned to compute simple problems successfully through traditional instruction but are not able to use this learning as a basis for solving more complex problems.” Whereas three-act-math and other inquiry-based approaches do provide the basis for solving more complex related (but not identical) problems.

  2. jeannev2013 says:

    Here’s an excerpt from Jo Boaler’s book “What’s math got to do with it” that is appropriate: “Mathematics is a performance, a living act, a way of interpreting the world. Imagine music lessons in which students worked through hundreds of hours of sheet music, adjusting the notes on the page, recieving checks and crosses from the teachers, but never playinging the music. Students would not continue with the subject because they would never experience that music is.”

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