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?”

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:

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

Ann Arden:

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.

Pam Rawson:

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 …”

Annie Adams:

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.

Marc Garneau:

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!

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About 
I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. More here.

19 Comments

  1. Steve Leinwand

    July 26, 2018 - 7:34 am -
    Reply

    Thanks Dan. Powerful insights perfectly timed for our reflections on how we might approach our teaching and listening a bit differently come August and September. Reminds us all too of the power of “wrong” answers and “mistakes” to uplift or demoralize.

  2. Reply

    Really thoughtful post. I found the way you phrased that a student’s ideas were “more interesting than my own” particularly helpful. Early in my teaching, I thought it was most helpful to show how *I* knee the math.

    Featured Comment

    After 18 yrs, 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.

    Curiousity about students’ ideas is so important. Thanks for sharing! Hope you now have some well deserved post-summer school R&R!

  3. Reply

    Dan, as always, you can take something that’s been rattling in my head for a couple of years, and distill it down into something that lets focus on it more clearly, and see parts of it that I’ve been missing.

    Thanks again.

  4. Reply

    Thanks for modeling what reflective teaching looks like, Dan. The other problem with having deep content knowledge as a teacher is that I often forget how difficult it is for many students to learn something for the first time. Because those ideas or procedures have become automatic, or maybe I didn’t struggle to learn that thing in the first place, it’s easy make assumptions about what the student might be thinking. But it’s pretty simple: If I want to actually know what my students are thinking in that moment, I have to ask them.

    Featured Comment

    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 …”.

    And if you have a positive, listening classroom culture, then other students might respond to the original idea. Making sense of ideas is so much more powerful when it happens among the students. Yes, it takes time. But, as you say, if we teachers don’t understand how our students are trying to make sense of the mathematics, then we can’t help them to move forward. Paraphrasing the Aaron Burr character in “Hamilton,” teachers need to talk less, listen more, and then ask questions.

  5. Luc Leavenworth

    July 26, 2018 - 2:16 pm -
    Reply

    Removing my ego from teaching was such a game changer. My attitude about a lot of things (grades, achievement, and “behavior issues”) was put into perspective. I haven’t looked back since.

  6. Kathleen Herring

    July 26, 2018 - 2:23 pm -
    Reply

    Great post. I have been musing lately on the importance of curiosity as an agent of learning, and observing how many people go through their day in automatic mode and don’t seem curious. And I worry how to help my students be more curious. This post answers that – by being more curious myself about what they do know. Nice!

  7. Reply

    I completely agree. But isn’t it true, that teachers know what student’s thought based on their answers? What do you think of a multiple choice platform where you get feedback on your (probable) mistake, instead of just the right answer with the correct procedure? Just being curious…

    • I haven’t seen an example of such a platform that successfully builds on the student’s informal level of understanding.

      Even if it correctly predicts the student’s overgeneralization or mistake, the phrasing is extremely important. When students have to exert mental effort trying to connect their idea to the computer’s phrasing of their idea, that’s effort not spent learning math.

  8. Reply

    Thank you for sharing your insights, Dan, you’ve helped me realize this: for me, making sense of student thinking and using that understanding (both theirs and mine) to move their learning forward is the most exciting/interesting/energizing/satisfying part of teaching.

  9. Reply

    Featured Comment

    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.

    I appreciate your naming this.

    • Thanks for sharing, Annie. I featured your comment above. I’d love to read or hear more about your experiences meeting the needs of your refugee students.

  10. Javier Taylor

    July 27, 2018 - 7:05 am -
    Reply

    Hey Dan I really enjoyed reading your post. I work in the university setting so I admit it is nice to get out into a 7-12 classroom and teach. I work with a program called “upward bound” which is about a month long (basically the month of June). I also get a chance to work on the mathematics research project I am currently pursuing.

    Nice!

    I noticed that perusing unresolved mathematics allows me to appreciate my students pursuit of mathematics in the classroom.
  11. Marc Garneau

    July 27, 2018 - 9:33 am -
    Reply

    Thanks for sharing your reflections – I always appreciate the raw truth you share. I, too, just finished a month of summer school teaching. This is the third year I’ve done this – 2 classes of remedial (ie. they failed) and I’ve got 40 hours to help them ‘pass’. Like you, I do it to for the sake of my professional muscles – and putting pedagogy into action over a ‘long’ term instead of drop-in lessons.
    Both your truths resonate strongly with my experience. Getting ‘through’ a whole course’s content in a third of the time – ugh. And what to say when students cannot distinguish a side from an angle in a triangle!?! My only recourse was to focus more on bigger ideas – and that was a blessing.

    Featured Comment

    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!

    This took huge amounts of energy, but it was worth it beyond measure. It also helps me to appreciate HUGEly the work that classrooms teachers do ALL YEAR – I only did this for 3 weeks.

    • Love that second-to-last paragraph. By middle and upper grades I find students have a conception of themselves as mathematically incapable that’s very difficult to disrupt, especially in three weeks. I’m glad your persistence paid off.

  12. Reply

    For the past year I have been teaching pre-college math at a community college for students pursuing a wide range of degree options including HS 21+. I fell into the job by accident, my only math teaching experience having been at a University.

    This post articulates so well EXACTLY what I have learned from teaching the diverse population of students who take my classes.

    Featured Comment

    It’s easy to throw a loaded question out there, and when someone doesn’t answer in the way you expect, you just say NOPE and move on. When I learned to slow down and try to understand WHY the student answered in the way they did I started to learn where and how they struggle. I also learned that when I slow down and explore student thinking the whole class benefits, because guess what? There’s a good chance there were other students thinking something similar.

    Also, I believe stopping to explore student thinking creates a sense of community in the classroom where students feel safe enough to express their confusion. The students I teach often have an incredible amount of baggage from competitive, threatening classrooms. Showing them I am listening reduces barriers and gives them confidence that they can attain mastery of the concepts.

    Thanks for putting into words what I have been struggling with and embracing and loving all year with my students!

  13. Reply

    Hi Dan,

    Thanks for your insights. Reflecting upon interactions is incredibly powerful. A filmed lesson (or a week) of your experience in the classroom would be invaluable for dialogue with colleagues to collaboratively enhance student experiences.

  14. Reply

    I spent some time last year looking at how to adapt the “Yes, and…” method of improvisation to the math classroom. My teaching has evolved to a point where I feel strongly that it’s important to validate a student’s mathematical thinking even when it isn’t “correct” or doesn’t match what I was hoping for next.

    I just LOVE your bold comment above because it found words for what I was trying to express. I am an instructional coach these days and intend on making this idea a point of emphasis this year. Thank you for putting it to better words than I have been able to so far.

    I will also second your recommendation for Max’s 2 > 4 ignite talk. I shared it with all my teachers last year because it’s so awesome. I won’t spoil his fantastic turn of phrase, but if anyone reading this hasn’t watched it they most definitely should.

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