Mathematical Fluency Has A Constrained And Dangerous Definition in Education
Spoiler alert. I am not fluent in mathematics.
If you find that shocking, then you probably have adopted a well-intentioned — but in the end, limiting and dry — definition of mathematical fluency.
In fact, I will never be fluent in mathematics. That’s okay. Nobody ever has been…
Society and education sends mixed messages on the “language of mathematics” all the time. On one hand, it wants to say that mathematics is the language of the universe(correct), and then on the other it reduces the fluency of mathematics to something that can be put in a box and wrapped up(questionable distillate at best).
Mathematics is a complex language. The interest/curiosity/need to learn this language is first and foremost. If that’s not there, why the hell are you learning this language in the first place? It’s also a slippery slope towards this:
The love for language — the love for the language of mathematics — has to be put in place first. Or else, you’re building a house a cards with fluency. Why do we do “review” every September in every grade in mathematics? Once you learn a language, you don’t forget it?
Alfred North Whitehead — mathematician, eduator, writer, and philosopher — more or less said all learning will be doomed or be inert if the phase of romance was not attended.
Kids forgot the math they learned because they simply don’t care — and, they don’t care about whether we care about that.
All this dialog about math fluency just goes on, indifferent to the attitudes, alienation, and anxieties that students have about mathematics. We teach mathematics in a setting of compliance, rules, procedures, and orders.
Yup. Really, really going to want to learn the language of mathematics in this setting.
The whole season on Math Teacher Lounge was devoted to “math fluency”. It got off to a cracker of a start by having Jason Zimba bat lead off, thinking he hit a home run, when it was an infield single — as the definition didn’t come close to the scope of language fluency that is described in the comment bubble in the lower left of the picture below. Language is social, linked to history/traditions, and spoken widely and enthusiastically. The optics also doesn’t help the narrative that mathematics is centered by whiteness.
Mathematics has never been about that, but we still want fluency?
Watch this clip of Jodie Foster speaking French. The smile on her face. The comfort in her responses. The anticipation of the next question. This is the level of fluency that should be in mathematics.
I think fuller and rich fluency in mathematics involves — at the very least — these ideas.
- Factual Fluency
Who gets to say what defines factual fluency? Knowing the sum of consecutive odd numbers gives you square numbers is a fact? Somehow that gets eclipsed by something as inert as knowing 6 x 7 is 42. Knowing that 2 x 3 x 7 is 42 is better, as you can create a “family” of math facts for 42 by multiplying any two numbers and then writing it the third one(ie 2 x 7 is 14, so 14 x 3 must be 42). So, let’s be even careful of which math facts are important and why.
2.Procedural Fluency
To solve for x, involves a series five algebraic steps. This would be seen as procedural fluency in algebra. Whether you get the correct answer is irrelevant, as long as you know the algebraic procedure here, starting with cubing both sides.
3. Conceptual Fluency
Same question, but having conceptual fluency/understanding of what is going on bypasses the laborious procedural fluency if you understand right away that the number under the cube root sign has to be “8” and you have “5” so far. Continuing that line of thinking will allow you to envision “9” under the root sign, and now you are essential solving 2x + 1 = 9.
One of the best talks about the importance of weaving in the above fluencies comes from Junaid Mubeen(Ph.D in mathematics from Oxford)
4. Historical Fluency
Most of us have very little fluency on the thematic development of mathematics through every race, culture, and civilization. As such, guess, what, we deprioritize it — because we ourselves are not experts.
We become gatekeepers of not just the definition of mathematical fluency, but of mathematics in general. Not too surprisingly, this gatekeeping and myopic view of mathematics has created the absurd Science of Math.
Here mathematics is finally reduced to its ugliest version — a compliant tool to be tested over and over and over and over and over again.
5. Contemporary Fluency
What I mean by this is are students being made aware of the current landscape of mathematics? What are the biggest unsolved problems(out of the thousand or so which are out there)? Which ones, if solved, will change our lives? Where does AI sit with learning mathematics?
Again, I share with you a great video by Junaid Mubeen.
We have reduced mathematics to pedagogical rubble, where we give more attention to the buzzwords of pedagogy than with the actual mathematics to the point that the quality of the mathematics becomes irrelevant.
Mathematics can hardly be seen as a beautiful, romantic language then.
If anything, current math education makes it sounds synthetic and having a voice like Hal9000 from 2001: A Space Odyssey.
Mathematics is the language of the universe. Be curious to begin to learn it.
Nobody has ever done anymore with learning mathematics…
