Dylan Kane’s recent post about prerequisite knowledge has me wanting to tell you a story from my very first year in my very first full-time classroom job, which I *think* I’ve never related on this blog before, although I’ve told it IRL many times.

It was the 2001-2002 school year. I taught four sections of Algebra I. I was creating my whole curriculum from scratch as the school year progressed, because the textbook I had wasn’t working in my classes, or really I guess I wasn’t figuring out how to make it work. Late late in the year, end of May/early June, I threw in a 2-week unit on the symmetry group of the equilateral triangle. I had myself only learned this content the prior year, in a graduate abstract algebra course that the liaison from the math department to the ed department had required of me in order to sign off on my teaching degree, since I hadn’t been a math major. (Aside: that course changed my life. I now have a PhD in algebra. But that’s another story for another time.)

Since it was an Algebra I class, the cool tie-in was that you can solve equations in the group, exactly in the way that you solve simple equations with numbers. So, I introduced them to the group, showed them how to construct its Cayley table, and had them solving equations in there. There was also a little art project with tracing paper where they drew something and then acted on it with the group, so that the union of the images under the action had the triangle’s symmetry. Overall, the students found the unit challenging, since the idea of composing transformations is a profound abstraction.

In subsequent years, I mapped out the whole course in more detail beforehand, and once I introduced that level of detail into my planning I never felt I could afford the time to do this barely-curricular-if-awesome unit. But something happened, when I did it that first and only time, that stuck with me ever since.

I had a student, let’s call her J, who was one of the worst-performing (qua academic performace) students I ever taught. Going into the unit on the symmetry group, she had never done any homework and practically never broke 20% on any assessment.

It looked from my angle like she was just choosing not to even try. She was my advisee in addition to my Algebra I student, so I did a lot of pleading with her, and bemoaning the situation to her parents, but nothing changed.

Until my little abstract algebra mini-unit! From the first (daily) homework assignment on the symmetry group, she did everything. Perfectly. There were two quizzes; she aced both of them. Across 4 sections of Algebra I, for that brief two-week period, she was one of the most successful students. Her art project was cool too. As I said, this was work that many students found quite challenging; she ate it up.

Then the unit ended and she went back to the type of performance that had characterized her work all year till then.

I lavished delight and appreciation on her for her work during that two weeks. I could never get a satisfying answer from her about why she couldn’t even try the rest of the time. But my best guess is this:

That unit, on some profound mathematics they don’t even usually tell you about unless you major in math in college, was the single solitary piece of curriculum in the entire school year that did not tap the students’ knowledge of arithmetic. Could it be that J was shut out of the curriculum by arithmetic? And when I presented her with an opportunity to stretch her mind around

- composition of transformations,
- formal properties of binary operations,
- and a deep analogy between transformations and numbers,

but not to

- do any of any numbers bigger than 3,

she jumped on it?

Is mathematics fundamentally sequential, or do we just choose to make it so? I wonder what a school math curriculum would look like if it were designed to minimize the impact of prerequisite knowledge, to help every concept feel accessible to every student. – Dylan Kane

Acknowledgement: I’ve framed this post under the title “Re-invitation”. I’m not 100% sure but I believe I got this word from the Illustrative Mathematics curriculum, which is deliberately structured to allow students to enter and participate in the math of each unit and each lesson without mastery over the “prerequisites”. For example, there is a “preassessment” before every unit, but even if you bomb the preassessment, you will still be able to participate in the unit’s first few lessons.

If we treat mathematics as a tower, then imperfect lower floors limit how high the tower can go.

But, nah, you got it.

I see some of the same, not so dramatic, in my combinatorics elective.

The same conceit underlies our choice to teach our non-calculus seniors a full marking period of matrix arithmetic.

I get some of the effect in mini-electives on graph theory and set theory and even number theory.

But… it’s not only that. There is a second effect here. I am good a arithmetic. Really good. And I don’t remember ever being as excited about mathematics as my first exposure to groups, and the symmetries of a triangle was one of my two favorites.