Friday, November 17, 2017

What is Fluency in Math?

When I was a junior in college, several of my friends spent a semester studying in Heidelberg, Germany. I decided against going because knowing German or being willing to learn it was a prerequisite for the program. One of my friends, Scott, thought this was funny because the only German he knew/learned was, "Bitte, haben Sie einen Plattenspieler?" which translates to "Please, do you have a turntable?" I thought this was pretty funny, too, and I liked the way it sounded, so I also memorized the question. Clearly, this didn't make me or Scott fluent in German. In fact, unless I'm talking to some hipster familiar with record players, my question probably doesn't even make any sense to him or her anymore. I often share this memory when a discussion about fluency in math turns toward the necessity to be fast and accurate with facts or some procedure. While speed and accuracy might result from fluency, speed is neither a requirement of nor a support for fluency. 

Consider the actors playing Klingons on Star Trek: Discovery: they correctly deliver their lines without any pause. Are they fluent in the Klingon language?

Perhaps (some Trekkies are), but more likely they are professionals that have memorized a script. They are not responding to something said by another Klingon actor, and I doubt many of them would be able to improvise new dialog. 

Is Siri fluent in mathematics? She is certainly fast and accurate.

But ask her how she arrived at her answer, and she is at a lost.

What does it mean to be fluent? Fluent speakers demonstrate flexibility in their language usage - the ability to play with words and use figurative phrases. Certainly, there is also a need to know the words and grammar associated with the language. However, this alone isn't enough to be fluent in the language.

And, in my opinion, knowing alone is not enough to be fluent in math, either - no matter how quickly you can arrive at the answer.

So, how am I wrong? I'm looking for pushback before I tell this story again. Make some comments and make me smart.


  1. I think the metaphor between language fluency and mathematics fluency can sometimes get a bit strained. Fluency to me mostly means the ability to communicate back and forth (you don't have to be verbally sophisticated or use figurative phrases). But even under the setup you posit: knowing a single sentence in German didn't make you fluent. Knowing a single answer in math isn't going to make you fluent in this domain either.

    But there's a line: at a certain point knowing enough words/sentences in German and being able to understand them is basically fluency.

    Likewise, knowing all the basic facts and being able to communicate with them i.e. do problems represents something more than a single one.

    Thinking about this from another angle. In computer science there is the Turing test to determine if a computer program is an artificial intelligence. If it can answer all the questions from an interrogator and you can't tell the difference between it and a human then it passes. Looking at kids and judging fluency is somewhat similar. If they can answer all the basic questions it doesn't matter how or why they did so, they are indistinguishable and fluent.

    Note: being fluent doesn't mean you say intelligent things just that you can communicate back and forth. So I think when we ask for flexible mathematical thinking and problem solving its another level beyond fluency. Perhaps this is a good time to introduce numeracy into the conversation.

  2. The thing about your German knowledge back in college wasn't that you were only fast and accurate -- it's that you only knew that one sentence! Even if you could perfectly explain why that sentence meant what it does it wouldn't have helped -- you'd still only know one sentence of German.

    (And, more to the point, how well can you really explain why a sentence means what it does if you only know one sentence?)

    The math-version of your German story is a kid who literally only knows like 7 x 9 = 63 and no other multiplication.

    The Klingon analogy works better for me. Of course, though, nobody is trying to teach Klingon to the actors.

  3. Maybe it means proficient. Tried to connect it to a review and tutorial center I'm working with where some of my colleagues just misused the words.