On The Importance of Mathematical Content Knowledge for Teaching

As Bob Dylan sang,

“I must know my song well/before I start singing.”

So for musicians, as it is for teachers. Just stepping up and “winging it” when you are introducing a mathematical concept is not good enough. It’s not good enough for kids, and we can do better. Whereas once upon a time we probably thought, “Primary grade math is easy”, we now have a much greater knowledge of the developmental paths that kids take in their learning, with all twists, turns, conceptions, and misconceptions.

Mathematical knowledge and teaching knowledge are inseparable, intertwined, and joined permanently in what we call teaching practice. We should not minimize either one, or the other, although I will say this:

-we know how to teach.

I put mathematical knowledge ahead of pedagogy these days, because our daily practice is of teaching. We are highly skilled at the “how”, but sometimes we need more work on the “what”.

The what is mathematical content knowledge for teaching, which must necessarily included content knowledge. I have learned more about mathematics in the last three years than in the rest of my life put together. I am humble in the face of this vast body of knowledge. I have much left to learn.

In Deborah Ball’s formulation, mathematical content knowledge for teaching is knowing what, knowing how to teach it, and also starting to get a feel for what kids might think about as they work on mathematical tasks.

Experience is a great teachers, and I guarantee you, when you start looking at adding fractions, some kids will add both numerators and denominators.

You will see:

1/2 + 3/4 = 4/6

Ask yourself why. What knowledge of fractions would this kid be missing? How can you help them understand fraction notation, and more importantly, the relationship between part and whole in that relationship?

There is a whole line of learning you can follow back to probably grade 1, where the kid first encountered “one half”. What did “one halfness” mean to them then? What went wrong along the way? Is this just a procedural problem, or something bigger?

To “diagnose” what’s happening here, we need to know a lot about fractions, a lot about how kids learn fractions, and to find out what this kid was thinking. All of these things are tied together. That’s what makes teaching so complex.

Here are a few more potential examples:

  • a kid counts well, and tags objects, and has a sense of the cardinal size of sets. They struggle to skip count by 2s. What do you do?
  • a kid is struggling to add and subtract integers. You started your lesson with no visuals. What should you do?
  • a kid in grade 6 can’t read decimal numbers. How far back in their place value understanding should you look?

Consider for yourself your top priority. Is it knowing mathematics, or is knowing how to teach mathematics? It has to be both- knowledge leads to the skillful pedagogical actions that teachers take every day of their careers.

Watch kids at work. Watch and listen.
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