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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.

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!

## Rough-Draft Thinking & Bucky the Badger

Many thanks to Ben Spencer and his fifth-grade students at Beach Elementary for letting me learn with them on Friday.

In most of my classroom visits lately, I am trying to identify moments where the class and I are drafting our thinking, where we aren’t looking to reach an answer but to grow more sophisticated and more precise in our thinking. Your classmates are an asset rather than an impediment to you in those moments because the questions they ask you and the observations they make about your work can elevate your thinking into its next draft. (Amanda Jansen’s descriptions of Rough-Draft Thinking are extremely helpful here.)

From my limited experience, the preconditions for those moments are a) a productive set of teacher beliefs, b) a productive set of teacher moves, and c) a productive mathematical task – in that order of importance. For example, I’d rather give a dreary task to a teacher who believes one can never master mathematical understanding, only develop it, than give a richer task to a teacher who believes that a successful mathematical experience is one in which the number on the student’s paper matches the number in the answer key.

A productive task certainly helps though. So today, we worked with Bucky the Badger, a task I’d never taught with students before.

We learned that Bucky the Badger has to do push-ups every time his football team scores. His push-ups are always the same as the number of points on the board after the score. That’s unfortunate because push-ups are the worst and we should hope to do fewer of them rather than more.

Maybe you have a strong understanding of the relationship between points and push-ups right now but the class and I needed to draft our own understanding of that relationship several times.

I asked students to predict how many push-ups Bucky had to perform in total. Some students decided he performed 83, the total score of Bucky’s team at the end of the game. Several other students were mortified at that suggestion. It conflicted intensely with their own understanding of the situation.

I wanted to ask a question here that was interpretive rather than evaluative in order to help us draft our understanding. So I asked, “What would need to be true about Bucky’s world if he performed 83 push-ups in total?” The conversation that followed helped different students draft and redraft their understanding of the context.

They knew from the video that the final score was 83-20. I told them, “If you have everything you need to know about the situation, get to work, otherwise call me over and let me know what you need.”

Not every pair of students wondered these next two questions, but enough students wondered them that I brought them to the entire class’s attention as Very Important Thoughts We Should All Think About:

• Does the kind of scores matter?
• Does the order of those scores matter?

I told the students that if the answer to either question was “yes,” that I could definitely get them that information. But I am very lazy, I said, and would very much rather not. So I asked them to help me understand why they needed it.

Do not misunderstand what we’re up to here. The point of the Bucky Badger activity is not calculating the number of push-ups Bucky performed, rather it’s devising experiments to test our hypotheses for both of those two questions above, drafting and re-drafting our understanding of the relationship between points and push-ups. Those two questions both seemed to emerge by chance during the activity, but they contain the activity’s entire point and were planned for in advance.

To test whether or not the kind of scores mattered, we found the total push-ups for a score of 21 points made up of seven 3-point scores versus three 7-point scores. The push-ups were different, so the kind of scores mattered! I acted disappointed here and made a big show of rummaging through my backpack for that information. (For the sake of this lesson, I am still very lazy.) I told them Bucky’s 83 points were composed of 11 touchdowns and 2 field goals.

Again, I said, “If you have everything you need to know about the situation to figure out how many push-ups Bucky did in the game, get on it, otherwise call me over and let me know what you need.” The matter was still not settled for many students.

To test whether or not the order of the scores mattered, one student wanted to find out the number of push-ups for 2 field goals followed by 11 touchdowns and then for 11 touchdowns followed by 2 field goals. Amazing! “That will definitely help us understand if order matters,” I said. “But what is the one fact you know about me?” (Lazy.) “So is there a quicker experiment we could try?” We tried a field goal followed by a touchdown and then a touchdown followed by a field goal. The push-ups were different, so now we knew the order of the scores mattered.

I passed out the listing of the kinds of scores in order and students worked on the least interesting part of the problem: turning given numbers into another number.

I looked at the clock and realized we were quickly running out of time. We discussed final answers. I asked students what they had learned about mathematics today. That’s when a student volunteered this comment, which has etched itself permanently in my brain:

A problem can change while we’re figuring it out. Our ideas changed and they changed the question we were asking.

We had worked on the same problem for ninety minutes. Rather, we worked on three different drafts of the same problem for ninety minutes. As students’ ideas changed about the relationship between push-ups and points, the problem changed, gaining new life and becoming interesting all over again.

Many math problems don’t change while we’re figuring them out. The goal of their authors, though maybe not stated explicitly, is to prevent the problem from changing. The problem establishes all of its constraints, all of its given information, comprehensively and in advance. It tries to account for all possible interpretations, doing its best not to allow any room for any misinterpretation.

But that room for interpretation is exactly the room students need to ask each other questions, make conjectures, and generate hypotheses – actions that will help them create the next draft of their understanding about mathematics.

We need more tasks that include that room, more teacher moves that help students step into it, and more teacher beliefs that prepare us to learn from whatever students do there.

2018 May 23. Amanda Jansen contributes to the category of “productive teacher beliefs”:

Doing mathematics is more than answer-getting.

Everyone’s mathematical thinking can constantly evolve and shift. Continually. There is no end to this.

Everyone’s current mathematical thinking has value and can be built upon.

An important role of teachers is to interpret students’ thinking before evaluating it. Holding off on evaluating and instead engaging in negotiating meaning with students supports their learning. And teacher’s learning.

Everyone learns in the classroom. Teachers are learning about students’ thinking and their thinking about mathematics evolves as they make sense of kids’ thinking.

The list goes on, but I’m reflecting on some of the beliefs that are underlying the ideas in this post.

2018 May 26. Sarah Kingston is a math coach who was in the room for the lesson. She adds teacher moves as well.

## A High School Math Teacher’s First Experience Teaching Elementary School

At a workshop in Alameda County last month, I made my standard request for classroom teachers to help me make good on my New Year’s resolution. I assumed all the teachers there taught middle- or high-school so I said yes to every teacher who invited me. Later, I’d find out that one of them taught fourth grade.

As a former high school math teacher, this was NIGHTMARE MATERIAL, Y’ALL.

I mean, what do fourth graders even look like? I’m tall, but do I need to worry about stepping on them? What do they know how to do? Do they speak in complete sentences at that age? Clearly, what I don’t know about little kids could fill libraries.

I survived class today. I used a Graham Fletcher 3-Act task because I’m familiar with that kind of curriculum and pedagogy. (Thanks for the concierge support, Graham.) A few observations about the experience, which, again, I survived:

Children are teenagers are adults. Don’t let me make too much of my one hour of primary education experience, but I was struck hard by the similarities between all the different ages I’ve taught. People of all ages like puzzles. They respond well to the techniques of storytelling. Unless they’re wildly misplaced, they come to your class with some informal understanding of your lesson. They appreciate it when you try to surface that understanding, revoice it, challenge it, and help them formalize it. I handled a nine year-old’s ideas about a jar of Skittles in exactly the same way as I handled a forty-nine year-old’s ideas about teaching middle schoolers.

Primary teachers have their pedagogy tight. Ben Spencer (my host teacher) and Sarah Kingston (an elementary math coach) were nice enough to debrief the lesson with me afterwards.

I asked them if I had left money on the table, if I had missed any opportunities to challenge and chase student thinking. They brought up an interesting debate from the end of class, a real Piagetian question about whether a different jar would change the number of Skittles. (It wouldn’t. The number of packages was fixed.) I had asked students to imagine another jar, but my hosts thought the debate demanded some manipulatives so we could test our conjectures. Nice!

Also, Spencer told me that when he asks students to talk with each other, he asks them to share out their partners’ thinking and not their own. That gives them an incentive to tune into what their partners are saying, rather than just waiting for their own turn to talk. Nice! As a secondary teacher, I felt like a champ if I asked students to talk at all. Spencer and his primary colleagues are onto some next-level conversational techniques.

Primary students have more stamina than I anticipated. No doubt much of this credit is due to the norms Mr. Spencer has set up around his “Problem Solving Fridays.” But I’ve frequently heard rules of thumb like “children can concentrate on one task for two to five minutes per year old.” These kids worked on one problem for the better part of an hour.

The pedagogy interests me more than the math.

This sentiment still holds for me after today. I just find algebra more interesting than two-digit multiplication. I’ll try to keep an open mind. Today was not an interesting day of math for me, though it was a very interesting day of learning how novices learn and talk about math.

I’m probably not wacky enough for this work. Mr. Spencer greeted his students by calling out “wopbabalubop!” to which they responded “balap bam boom!” Really fun, and I don’t think you can teach that kind of vibe.

Loads of algorithms, and none of them “standard.” Graham’s 3-Act modeling task asks students to multiply two-digit numbers. I saw an area model. I saw partial products. Students used those approaches flexibly and efficiently. They were able to locate each number in the world when asked. I didn’t see anyone carry a one. Everyone should settle down. This is great.

I expected the experience would either kill me or convince me I should have taught primary students. This one fell somewhere in the middle. I’m excited to return someday, and I was happy to witness the portability of big ideas about students, learning, and mathematics from adult education to high school to elementary school.

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## You Can’t Break Math

We were solving linear equations in Ms. Warburton’s eighth grade class last week and I learned (or re-learned, or learned at greater depth) a couple of truths about mathematics.

B suggested they multiply by two as a “fraction buster.”

(One small pleasure of guest teaching is trying to identify and decode the vernacular of each new class. I heard “fraction buster” more than once.)

R asked, “But do we multiply this by two or the whole thing by two?”

If you’ve taught math for a single day, you know the choice here.

You can tell them, “You multiply the whole thing by two.” That’d be helpful by the definition of “helpful” that includes “completing as many math problems as correctly and quickly as possible.” But in terms of classroom management, I’ll be doing myself no favors, having trained B and R to call me over whenever they have any similar questions. More importantly, I’ll have done their relationship with mathematics no favors either, having trained them to think of math as something that can’t be made sense of without an adult around.

Variables like x and y behave just like numbers like -2 and 3.” I said. I wrote this down and said, “Try out both of your ideas on this version and see what happens.”

After some quick arithmetic, they experienced a moment of clarity.

In the next class, students were helping me solve 2x – 14 = 4 – 2x at the board. M told me to add 2x to both sides. One advantage of my recent sabbatical from classroom teaching is that I am more empathetic towards students who don’t understand what we’re doing here and who think adding 2x to both sides is some kind of magical incantation that only weird or privileged kids understand.

So at the board I was asking myself, “Why are we adding 2x to both sides? What if we added a different thing?”

Then I asked the students, “What would happen if we added 5x to both sides? What would break?”

Nothing. We decided that nothing would break if we added 5x to both sides. It wouldn’t be as helpful as adding 2x, but math isn’t fragile. You can’t break math.

BTW

• I haven’t found a way to generate these kinds of insights about math without surrounding myself with people learning math for the first time.
• One of my most enduring shortcomings as a teacher is how much I plan and revise those plans, even if the lesson I have on file will suffice. I’ll chase a scintilla of an improvement for hours, which was never sustainable. I spent most of the previous day prepping this Desmos activity. We used 10% of it.
• Language from the day that I’m still pondering: “We cancel the 2x’s because we want to get x by itself.” I’m trying to decide if those italicized expressions contribute to a student’s understanding of large ideas about mathematics or of small ideas about solving a particular kind of equation.

Here is an awesome sequence of comments in which people savage the term “cancel,” then temper themselves a bit, and then realize that their replacement terms are similarly limited:

I have a huge problem with ‘cancel.’ It’s mathematical slang, and I’m OK with its use among people who really understand the mathematics. But among learners it obscures the mathematics and leads to things like “cancelling terms” in rational expressions.

I think the word “cancel” is misused in math when teaching students. We are not canceling anything we are making ones and zeros.

I never say cancel. I’ve worked hard to eliminate it from any teaching I do. Same with cross multiply, never say it. Instead I say “add to make zero” or “divide by or multiply by the reciprocal of to make one”.

We use “cancel” to mean too many things and so they use the term anytime they want to get rid of something or slash something out. The basis for my concern: when I ask kids “why can you do that?” they often can’t explain.

However, when it comes to squaring a square root, what is most accurate to say? I don’t correct the kids in that case, and I tell them that cancel means the same thing as “undo”.

If the point is to be rigorous, “apply the inverse” is more rigorous and technical than “cancel”.

More miscellaneous wisdom on language in mathematics:

In the topic of “get x by itself”, I’ve started saying “We want to say what x is. What would that look like?” They usually say “It would look like x = something”. Then they’ve chosen what the final equation ought to look like for themselves.

I wonder what would happen if we had an equation and then asked them to find out what, say, 2x+1 was.

Even in my Advanced Algebra 2 classes I have started using the phrases “legal” and “useful”. In the original post adding 5x to both sides was definitely “legal” just not as “useful”.

One of my refrains is that an algebraic step is “correct, but not useful.” Inspired by my dance teacher, I also talk about how a particular procedure is lovely, just not in today’s choreography (which is geared towards solving for an unknown/simplifying a trig expression/ finding intercepts, etc).

## Great Classroom Action

Tracy Zager illustrates a key feature of some of my favorite math tasks: their constraints are simple, but they create paths for complex thinking and ever more interesting questions:

I think my name is worth \$239. Beat me? Haven’t figured out my \$100 strategy yet.

Lisa Bejarano is a recipient of our nation’s highest honor for math teachers, so when she admits “I have no idea what I am doing” and starts sketching out a blueprint for great classrooms, I tune in:

Now, beginning with the first day of school, I intentionally work at building a unique relationship with each student. I make sure to find reasons to genuinely value each of them. This starts with weekly “How is it going?” type questions on their warm up sheets and continues by using their mistakes on “Find the flub Friday” and through feedback quizzes. I also share a lot of myself with them. When we understand each other, my classes are more productive. I still make plans, but I allow flexibility to meet my students where they are.

David Cox describes “a difficult thing for students to believe”:

Once students begin to believe that the way they see something is the currency, then our job is to simply help them refine their communication so their audience can understand them. Only then does the syntax of mathematics matter.

“Help me understand you.”

“Help me see what you see.”

Kevin Hall thoughtfully deconstructs his attempts to teach linear function for meaning, and includes this gem:

Once you introduce the slope formula, slope becomes that formula. It barely even matters if today’s lesson created a nice footpath in students’ brains between “slope” and the change in one quantity per unit of change in another. Once that formula comes out, your measly footpath is no competition for the 8-lane highway that’s opened up between “slope” and (y2–y1)/(x2­-x1).

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