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.

- 5x – 5 = 20
- 5x = 25
- 5x – 15 = 10
- -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?”

Instead, the teacher and I called a time out and talked in front of the class about the sense the student *had* made, rather than the sense she *hadn’t yet* made.

“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:

I'm still living in yesterday's classroom dilemma. A student says that both players are equally good because they both only missed two shots. What do you do? #iteachmath pic.twitter.com/OxIzRXvjuI

— Dan Meyer (@ddmeyer) June 27, 2018

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.

**Featured Comments**

After eighteen years, it’s becoming very apparent that I’m not very helpful as a teacher if I can’t/don’t understand the way a student is making sense of something.

Whenever I find myself going down the road of trying to “fix” a student’s thinking, I pause and then ask a question like, “What do you mean by …” or “Can you say more about …”

This past year, I was only teaching our refugee population in middle school. Since moving away from California, I hadn’t really encountered the needs of language learners, and people with interruptions in their education in a while. My muscles, too, have atrophyed. Luckily, I have spent many hours learning from the #iteachmath community, using visuals to illicit information, and subtracting the clutter in problems to open up scenarios for discussion. I thought this would be great, because I just wanted then to know they can solve problems. I learned so much about what I did not know about student needs, and about how students approach problems that are unfamiliar to them, when they can’t express themselves fully, and when they are trying to build on the few things that are familiar in their toolkit. This empathy with our students is something we all do daily, but naming it and focusing in it, rather than our own agenda, is the complicated and powerful design of teaching.

Your second truth is where I applied my most energy. I put in way more time, most of my time, into figuring out what sense they were making, and helping them to realize the same for themselves. For most, it was at least half-way through, if not three-fourths, for them to begin seeing what my goal for them was. They began caring for their learning, and caring for each others’ learning!