For the past month or so, I’ve been participating in the Not A Book Study look at Cathy Fosnot’s work on constructing multiplication and division.

This has been a tremendous opportunity to make great connections with other math educators across Northern Ontario, and really deepen my knowledge of how students construct mathematical concepts. I’ve been searching for something deep to sink my teeth into for a while, and I find this book - and all the discussion surrounding it in the #notabookstudy - has been rich and extremely thought-provoking.

As a secondary math teacher, I learned a lot in teachers’ college about tips and tricks to teach math through the intermediate and senior years. But we didn’t learn very much about how students learn math. How do they make the jump from thinking additively (counting 3 + 3 + 3 + 3 + 3 + 3 + 3) to thinking multiplicatively (7 x 3)? How do they approach the concept of division? How do they go from being able to problem solve in a specific context, to a generalized one, eventually using variables and equations?

Needless to say, I’m learning a lot about learning by reading this book, and this will certainly enrich how I approach teaching mathematical concepts in the future.

But I find I’m struggling with another aspect of teaching math as I continue learning about how we learn - finding the balance between discovering the concepts and practicing the skills.

In secondary, there is very little discovery (I’m generalizing here, based on my experience teaching). Most of what the students learn is taught to them in a very procedural way, with perhaps a bit of inquiry or hands-on activities, but almost never where the students are discovering the math themselves. And the students certainly aren’t owning the math.

On page 48, Fosnot says “Mathematics cannot be learned through transmission…” and I agree - students need to understand what the mathematics is telling them in order for them to remember what they learn. And yet we, as teachers, are still merely transmitting information.

However, in elementary (again, a generalization from what I’ve seen, mostly in primary and junior), there is a lot more discovery of the math, with lots of emphasis on concept construction, but not as much procedural learning. While this can promote a better understanding of what the math means, and a better growth mindset and approach to problem solving, the drilling of facts or repetition of procedure doesn’t seem to be, pardon the pun, part of the equation.

Partially because of this, in secondary we are seeing students who struggle more and more with basic math facts. That struggle leads to frustration, cancelling out any gains that may have been made from understanding the math initially discovered in earlier years.

Can students understand where the math comes from AND become procedurally efficient?

**I’d like to think so, but we as teachers need to work on that balance.**With respect to automaticity, Fosnot says “The issue here is not whether facts should eventually be memorized, but how this memorization is achieved: by rote drill and practice, or by fostering on relationships?” (page 86).

The former is shallow but “quick;” the latter is deep but takes time. And if “It is not up to us to decide which pathways out students will use [as they move toward constructing understanding]” (page 18), how can we ensure each student has the time to properly develop their thinking AND their automaticity?

Wherein lies the balance? My journey continues...