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!

A tool called Graspable Math found an audience on Twitter late last week, and a couple of people asked me for my opinion. I’ll share what I think about Graspable Math, but I’ll find it more helpful to write down how I think about Graspable Math, the four questions I ask about all new technology in education. [Full disclosure: I work in this field.]

1. What does it do?

That question is easier for me to answer with basic calculators and graphing calculators than with Graspable Math. Basic calculators make it easy to compute the value of numerical expressions. Graphing calculators make it easy to see the graphical representation of algebraic functions.

Graspable Math’s closest cousins are probably the Dragonbox and Algebra Touch apps. All of these apps offer students a novel way of interacting with algebraic expressions.

Drag a term to the opposite side of an equality and its sign will change.

Double click an operation like addition and it will execute that operation, if it’s legal.

Drag a coefficient beneath the equality and it will divide the entire equation by that number.

Change any number in that sequence of steps and it will show you how that change affects all the other steps.

You can also link equations to a graph.

2. Is that a good thing to do?

No tool is good. We can only hope to figure out when a tool is good and for whom and for what set of values.

For example, if you value safety, an arc torch is a terrible tool for a toddler but an amazing tool for a welder.

I value a student’s conviction that “Mathematics makes sense” and “I am somebody who can make sense of it.”

So I think a basic calculator is a great tool for students who have a rough sense of the answer before they enter it. (ie. I know that 125 goes into 850 six-ish times. A basic calculator is perfect for me here.)

A graphing calculator is a great tool for a student who understands that a graph is a picture of all the x- and y-values that make an algebraic statement true, a student who has graphed lots of those statements by hand already.

A basic and graphing calculator can both contribute to a student’s idea that “Mathematics doesn’t make a dang bit of sense” and “I cannot make sense of it without this tool to help me” if they’re used at the wrong time in a student’s development.

The Graspable Math creators designed their tool for novice students early in their algebraic development. Is it a good tool for those students at that time? I’m skeptical for a few reasons.

First, I suspect Graspable Math is too helpful. It won’t let novice students make computational errors, for example. Every statement you see in Graspable Math is mathematically true. It performs every operation correctly. But it’s enormously helpful for teachers to see a student’s incorrect operations and mathematically false statements. Both reveal the student’s early understanding of really big ideas about equivalence.

In one of their research papers, the Graspable Math team quotes a student as saying, “[Graspable Math] does the math for you – you don’t have to think at all!” which is a red alert that the tool is too helpful, or at least helpful in the wrong way.

Second, Graspable Math’s technological metaphors may conceal important truths about mathematics. “Drag a term to the opposite side of an equality and its sign will change” isn’t a mathematical truth, for example.

It’s a technological metaphor for the mathematical truth that you can add the same number (3 in this case) to both sides of an equal sign and the new equation will have all the same solutions as the first one. That point may seem technical but it underpins all of Algebra and it isn’t clear to me how Graspable Math supports its development.

Third, Graspable Math may persuade students that Algebra as a discipline is very concerned with moving symbols around based on a set of rules, rather than with understanding the world around them, developing the capacity for conjecturing, or some other concern. I’m speaking about personal values here, but I’m much more interested in helping students turn a question into an equation and interpret the solutions of that equation than I am in helping them solve the equation, which is Graspable Math’s territory.

These are all tentative questions, skepticisms, and hypotheses. I’m not certain about any of them, and I’m glad Graspable Math recently received an IES grant to study their tool in more depth.

3. What does it cost?

While Graspable Math is free for teachers and students, money isn’t the only way to measure cost. Free tools can cost teachers and students in other ways.

For instance, Graspable Math, like all new technology, will cost teachers and students time as they try to understand how it works.

I encourage you to try to solve a basic linear equation with Graspable Math, something like 2x – 3 = 4x + 7. Your experience may be different from mine, but I felt pretty silly at several points trying to convince the interface to do for me what I knew I could do for myself on paper. (Here’s a tweet that made me feel less alone in the world.)

Graspable Math performs algebraic operations correctly and quickly but at the cost of having to learn a library of gestures first, effectively trading a set of mathematical rules for a set of technological rules. (There is a cheat sheet.) That kind of cost is at least as important as money.

2018 Aug 8. Elizabeth Hernandez writes in the comments:

One thing I might add to the section about cost is that it is so important to find out how student data is being used. Resources that are labeled as “free” often make students and teachers pay with their data. That is unethical if the vendor doesn’t provide information about what data they collect and how it is used. Graspable Math is a no-go for me because I can’t find their terms of use or privacy policy. The only information I saw about data collected was one vague sentence when I click login.

I spent nearly as much time searching Twitter for mentions of Graspable Math as I did playing with the tool itself. Lots of people I know and respect are very excited about it, which gives me lots of reasons to reconsider my initial assessment.

While I find teachers on Twitter are very easily excited about new technology, I don’t know a single one who is any less than completely protective of their investments of time and energy on behalf of their students. Graspable Math may have value I’m missing and I’m looking forward to hearing about it from you folks here and on Twitter.

BTW. Come work with me at Desmos!

If you find these questions interesting and you’d like to chase down their answers with me and my amazing colleagues at Desmos, please consider applying for our teaching faculty, software engineering, or business development jobs.

2018 Jun 11. Cathy Yenca pulls out this helpful citation from Nix the Tricks (p. 54).

2018 Jun 11. The Graspable Math co-founders have responded to some of the questions I and other educators have raised here. Useful discussion!

The Desmos Teaching Faculty Is Hiring!

2018 Jun 16: We will close this particular posting Saturday, June 23, 11:59PM Pacific.

You should share that link with anyone who might be a good fit for the work. Alternately, if you think you’re a good fit for the work, you should guard that posting with your life, share it with nobody, and start thinking about your cover letter.

Why you should apply is really simple:

Desmos is the best place to do great work in math edtech right now and for the foreseeable future.

Here are six reasons I’m pulling out of muscle memory. I’m not even thinking about them. Ask me in ten minutes and I’ll give you six more just as fast.

• Teachers and students love our work. Check our Twitter feed. Also we just wrapped up a pilot study of 44 teachers using our activities and the results exceeded all of our expectations.
• Desmos folk are enormously talented in their own areas, curious and humble in all the others. So while my team didn’t come to Desmos having studied the same fields as our software developers and product designers (or vice versa) we’re conversationally fluent in each other’s work and humble about the limitations of that fluency. That disposition results in extremely enjoyable and productive collaboration.
• Great work-life balance. Startups are notoriously unfriendly to families but all of the full-time folk on my team have a couple of kids or more. Each one will tell you they love Desmos’s flexibility to do their best work at negotiable hours and locations.
• Everyone at Desmos is really satisfied with how we handle meetings and remote work, according to an internal company survey earlier this month. That’s uncommon.
• Strong financial position. While other edtech companies take on as much venture capital as they can, mortgaging their ability to make important decisions for themselves, Desmos has worked hard to minimize its reliance on outside investment. The result is that my team has had time and freedom to make decisions, first, based on what works for math students and, second, based on what we can sell. (Example: we decided to invest heavily in making our graphing calculator accessible to vision-impaired students because we thought that reducing an impediment to mathematical thinking sounded like a really good idea. Afterwards, we turned that work into contracts with eighteen states with more on the way.)
• Glassdoor reviews that speak for themselves.

We’ve spent several years tuning up our model for math, education, and technology. We studied it over the last three months and various aspects have clicked right into place. Demand is heating up for that work so we’re looking for people to help us build.

So please check out the posting and think about applying or sending it to someone you know.

BTW. If the formatting of the job posting seems atypical, it’s because we spent a lot of time discussing Lever’s blog series on reducing hiring bias. It would be easy to write a list of required credentials based on our mental profile of an ideal candidate. But that mental profile would be extremely susceptible to implicit and explicit biases. Lever received more responses from a more diverse group of candidates when they focused less on their credentials and more on what they’d need to know for the work and what they’d do at different milestones in their first year.

For example, an earlier draft of our posting required “at least five years of teaching experience at grades 6-12,” which isn’t bad as far as credentials go, but we realized it’s really just a proxy for the first four bullets beneath “What you should show up ready to teach anyone on your first day.”

• What a day in the life of a public middle- or high-school teacher looks like in the United States.
• The major challenges of technology integration in US classrooms from the perspective of both students and teachers.
• What separates a great math lesson from a lousy math lesson.
• What separates great classroom technology from lousy classroom technology.

We’re grateful to Lever for opening up their hiring practices to the public.

Two New Interviews with Yours Truly

I have two interviews out right now that I want to bring to your attention – a PDF and a podcast, so pick your medium! Both sets of interviewers were fantastic – well-researched and probing – and I did my best to rise to the occasion.

First, Ilona Vashchyshyn interviewed me for the May / June issue of Saskatchewan Mathematics Teachers’ Society magazine [pdf]. Ilona read my dissertation and her questions drew together themes of online math edtech, mathematical modeling, and Math Teacher Twitter. Here’s me:

The consistent theme in my participation [in online math teaching communities] is the fact that my thoughts always seem perfect to me until they escape the vacuum seal of my brain. Once they’re out in the world, in a blog post or a tweet, that’s when I realize how much work they need. I can’t get that feeling any other way.

Second, I have an hour-long interview out today with Becky Peters and Ben Kalb of the Vrain Waves podcast. They dug deep into my blog’s back catalog and asked questions about decade-old posts I had forgotten about. Halfway through, they asked about the role of memorization in learning mathematics. I confessed to them that I never want to be stuck driving on the road at the same time as anybody who derides the value of memorization in learning mathematics. I know a bunch of you will disagree with me there, so listen to the piece, and then tell me what you think either here or on Twitter.

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.