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## My Month Teaching Summer School & The Curse of Content Knowledge

I begged some middle and high school teachers in Berkeley, CA, to let me teach summer school with them this month.

Three reasons why:

• I knew some of my professional muscles were atrophying, and I can only strengthen them in the classroom.
• I knew our ideas at Desmos benefit enormously when we test them regularly in classrooms.
• I knew that for me (and for everyone on my team at Desmos, FWIW) classroom teaching is psychologically satisfying in ways that are impossible to reproduce anywhere except the classroom.

So I rotated between four classes, helping high school students with mathematics that was at their grade level and below, for the most part using Desmos activities.

This was my longest continuous stretch of classroom teaching since I left classroom teaching nearly ten years ago and I learned a lot.

Two truths in particular would have been very hard for me to understand ten years ago.

## One: content knowledge is such a curse.

The more math I understand and the better I understand it, the more likely I am to evaluate student ideas for how well they align with mine.

“Which one doesn’t belong?” we asked the class on an opener.

1. 5x – 5 = 20
2. 5x = 25
3. 5x – 15 = 10
4. -5x + 10 = -5

One student said that B didn’t belong because “it’s the only one with two variables.”

I knew this was formally and factually incorrect. 25 isn’t a variable. It became very tempting in that moment to say, “Oh nice – but 25 isn’t a variable. Does anybody have any other reasons why B doesn’t belong?”

“There are two of something in B. Does anybody know a name for it?”

My content knowledge encourages me to evaluate student ideas for their alignment to my level of understanding rather than appreciating the student’s level of understanding and building from there.

You can see that tendency in some of the responses to this tweet:

Those students understood the absolute difference between the denominator and the numerator (two shots missed) but not the relative difference (two shots missed when you took 38 is better than when you took 20). They needed more experience at a particular level of mathematics.

Perhaps you and I both know a formal algorithm that would help us get an answer to this question (eg. calculating common denominators; calculating a percentage) but simply explaining that algorithm would conceal some very necessary mathematical work under the attractive sheen of correctness. Explaining that formal algorithm would also tell students that “The informal sense you have made of mathematics so far isn’t even worth talking about. We need to raze it entirely and rebuild a different kind of sense from the foundation up.”

I blundered into those moments periodically in my month teaching summer school, most often when I understood my own ideas better than I understood the ideas a student was offering me and time was running short. In each instance, I could tell I was contributing to a student’s sense that her ideas weren’t worth all that much and that math can’t be figured out without the help of a grownup, if even then.

## Two: content knowledge is such a blessing.

I was able to convert my mathematical content knowledge from a curse to a blessing every time I convinced myself that a student’s ideas were more interesting to me than my own and I used my content knowledge to help me understand her ideas.

(Shout out to grad school right there. If nothing else, those five years cultivated my curiosity about student ideas.)

Here is a truth about my best teaching I learned last month in summer school:

Make yourself more interested in the sense that your students are making rather than the sense they aren’t making. Celebrate and build on that sense.

Celebrate it because too many students feel stupid and small in math class (especially in summer school) and they shouldn’t. The teacher time out helped us understand the student’s thinking, but try to understand what it’s like for a student to hear the big people in the room take her ideas so seriously that they’d bring the class to a stop to discuss them.

Build on that sense because it’s more effective for learning than starting from scratch. This is why analogies are so useful in conversation. Analogies start from what someone already knows and build from there.

I don’t think I understood that truth when I left the classroom a decade ago. My content knowledge was high (though in many ways not as high as I thought) and I was less curious in understanding my students’ ideas than I was in the attractive sheen of correctness.

All of which makes the real tragedy of my month teaching summer school the fact that I’ll likely have to wait until next summer to put this experience to work again.

BTW. Max Ray-Riek’s talk 2 > 4 is a beautiful and practical encapsulation of these ideas. Watch it ASAP.