Teaching right triangle trigonometry, presented as a series of problems

How can a decimal be a ratio?

  1. Sketch a tower that has a 2:3 height to width (h:w) ratio.
  2. Sketch one with a h:w ratio of 5/4.
  3. Sketch one with a h:w ratio of 0.4, 0.45, and 1.458.
  4. My tower has a h:w ratio of 1.00001. What can you tell me about my tower?

Special (“Famous”) Right Triangles (Chapter 10, Geometry Labs)

  1. Two of my right triangle’s sides are 5 cm and 5 cm. Use the Pythagorean Theorem to find the third side.
  2. Scale the “famous” half-square to find that third side more quickly.
  3. Two of my right triangle’s sides are 1 cm and 2 cm. Why can’t I be sure how long the third side is?

Which is steeper? Part I

Screenshot 2018-04-22 at 8.50.44 PMScreenshot 2018-04-22 at 8.50.49 PM

Screenshot 2018-04-22 at 8.57.53 PMScreenshot 2018-04-22 at 8.58.11 PM

Screenshot 2018-04-22 at 9.01.40 PM

Meet Tangent

  1. Play with the tangent button on your calculator a bunch. If you find anything cool or interesting, write it down.
  2. Tangent takes an angle and gives you the slope of its ramp. Make a smart guess: what’s tan(40)? tan(80)? tan(1)?
  3. Check out this table. What do you notice? What questions do you have?


4. Suppose we graphed this thing, slope/tangent as a function of degrees. What would that graph look like?

5. What are some possible sides for a 35 degree ramp?

Which is Steeper? Part II

New steepness compare_ angle vs. slopeNew steepness compare_ angle vs. slope (1)New steepness compare_ angle vs. slope (2)

Draw two ramps whose tangents are very close, but not quite, equal. Extra points for cleverness.

Moving Between Degrees and Tangent

  1. What’s the first slope that is unskiiable?


2. Try this Desmos activity about slanty hills.

Solving Ratio Equations

  1. Solve: x/5 = 1.3
  2. Solve: 5/x = 1.3
  3. Which of these is more difficult for you to solve?

Scaling, Ratios, and Solving Triangles (Geometry Labs, Chapter 11)

  1. Figure out as many things as you can about the triangle in this diagram.

Unit Triangles - 1.pptx

2. This one too.Unit Triangles - 1.pptx (1)

3. All of these as well.Unit Triangles - 1.pptx (2)

4. Check out these 20 degree ramps.

Screenshot 2018-04-22 at 9.11.48 PM.png

5. How tall is the top ramp? How wide is the bottom?

6. Which way should you spin this to make the problem as nice as possible? What angle of incidence do you prefer? What’s your height, your width?


7. (Ala Geometry Labs, Chapter 11): Given one hypotenuse and one leg of a right triangle, what other parts can you find?

Sine is a Trig Functions Also, I Guess

  1. Sine!

And then Cosine

  1. Cosine!



Reposting: Thoughts about Future of NCTM Conferences I had at NCTM Nashville (in 2015)

[I originally wrote and circulated this as a Google document, which had the benefit of making it easy to update. Over the past few years, lots of people offered interesting comments on that doc, so definitely check those out. Now that the document has been stable for a few years, I thought I’d repost it on this site to make it slightly easier to find.]

  • Fundamentally, I want NCTM conferences to be places where long-lasting professional relationships are formed. I do not want it to be a place whose primary purpose is for people go to sessions.
  • Overall, the quality of NCTM sessions is mixed. Once at Nashville a group of us found an empty room to sit around and chat in because we didn’t see any sessions that we wanted to go to.
  • I’m not sure that I even want more higher quality sessions to attend, though. The mix-and-match nature of session attendance doesn’t really excite me as an opportunity to learn about teaching.
  • I loved the MTBoS booth. There were moments of community around that booth. People go there so that they can talk to people they’ve never talked to before, we played with toys and I met some new people.
  • The MTBoS booth was like a small island of community in a den of icky educational consumerism. I really dislike the sales-pitching of the exhibition floor.
  • On Thursday afternoon I left a session and felt exhausted. I had a weird hankering for some math (I had been working on a problem on the plane) and I realized how little math-doing there was at these conferences. Isn’t that a shame?
  • I went from there to the MTBoS booth and played with Christopher Danielson’s math toys. I saw a crowd gathered around the booth, I saw people waiting for a turn to play with his tesselating turtles or his pattern-making machine.
  • Once NCTM reorients itself towards fostering community, I think it’s going to start seeming very important to figure out how to create spaces for doing math together with other people.
  • I love books. Usually when I walk into a bookstore, I have a hard time leaving without buying something. I walked out of NCTM without buying any books.
  • I went to a bookstore in Chicago a few months ago. I pulled off four books from the shelf, settled in a corner and flipped through them. Others were doing the same. Some people were talking to each other about their selections. It was a space for loving books.
  • The NCTM bookstore is another missed opportunity to make a communal space, I think.
  • I noticed that people congregate around the outlets outside of sessions. People end up sitting there. Any space like this is a chance to help people form connections.
  • I think NCTM is going to start including more formal social events, and this is good. I think NCTM is going to start providing more online spaces for presenters, and this is good too. But the real goal needs to be making sure there are nooks and crannies throughout the conference where people can come together around some shared experience.
  • I’m sure there are things like “fire code” that I’m not considering, but is there any good reason not to have a few rooms where you let we inmates run things? A place to chat, a place to take a group of people and sketch some things out. I’m talking about making sure there are open rooms with tables where people can continue a conversation.
  • As a speaker, now: there are always people who want to talk at the end of a session. It’s sometimes tricky to know where to go. I wish I could just say, “Here’s where I’m going to be if you want to continue the conversation.” In that way I could sort of pitch a more extended experience.
  • In short: yes, formal social events; yes, improved web experiences; but also, NCTM sub guides in advance of conferences; a hangout area with “hosts” to help make connections; a “Do Some Math” area with volunteer facilitators; spaces to go after a session; spaces to go instead of a session; spaces to read and fall in love with books together; fewer speakers; more sessions that are carefully vetted for quality; more places to play with toys; “Post your favorite math problem on an index card and glue it to the wall!”; invite Zome to take over a conference room; more spontaneity, more community and more math.

Added on 2/28/16

  • On twitter [https://twitter.com/dandersod/status/703939312526757888] there was an interesting conversation about whether teachers ought to be given the keynote presentation slots.
  • Keynote speakers play a role in attracting people to NCTM conferences, and so it makes sense to choose keynote speakers whose names are recognizable. I think it’s lamentable that classroom teachers aren’t recognizable names in math education. There’s a status hierarchy with teachers at the base level and consultants, academics, CEOs and journalists all hovering above us classroom folk. On one hand, this is only natural: the work that it takes to build up a personal brand, recognition and influence has very little to do with teaching children. If you’re interested in being well-known enough to influence education widely, that is a journey that will probably lead you out of your classroom.
  • This is a shame, though, because academics, consultants, CEOs, journalists are not doing the work of teaching, and so they often get it wrong. They often gravitate to issues that aren’t at the heart of the practice, or their thinking doesn’t develop in the way it might if they were forced to test their ideas over the course of years of working with children. There is no replacement for developing ideas while being a classroom teacher. Math education is worse off for not having high-status teachers who are able to speak and write with authority about math education.
  • (To be clear, there is also no replacement for visiting and seeing many different classrooms when it comes to making generalizations about teaching. And doing research well is immensely challenging but it enriches the profession. I don’t think the world should exclusively be run by k12 classroom teachers. That would be its own sort of disaster.)
  • So, what are we going to do about it? Thrusting teachers into the big lights wouldn’t fix anything, I think. True, it might raise the status of some teachers such that they could draw in people the way Jo Boaler’s name does. But could that status really be sustained while remaining in full-time classroom work? How do you develop talks and build an attractive brand without missing enormous amounts of time for conferences?
  • (The exception to this rule seems to be Jose Vilson. It seems that the laws of gravity don’t apply to Vilson, I don’t know how he does it. Truly amazing!)
  • I don’t want to advocate for some sort of requirement for keynote or featured presentations to include k12 teachers. Instead, I want NCTM to create infrastructure for gradually raising the profiles of classroom teachers. I think this could be done with the artful combination of fellowships, researcher-teacher partnerships that result in joint publications, awards, mid-level speaking profiles, and a million other things that I’m not smart enough to think of.
  • If there is a systemic critique I would make of NCTM, it’s that it’s entire leadership structure reflects a PD orientation that goes from researchers to PD providers to coaches and then to teachers, as recipients. By this I mean that board service is nearly impossible to pull off while being a classroom teacher, and that the model of the conferences seems to be of maximizing traditional PD delivery (even when it’s delivered by teachers). One thing that we’re seeing from the internet and the MTBoS is that this is just one model of how teachers like to develop professionally. Creating more opportunities at conferences for teachers to interact in ways that classroom teachers might find more natural — like teaching mathematics and talking and writing about practice — would benefit the status of teachers.
  • (To this, David Wees would add modeling and rehearsing teaching techniques. Wouldn’t that be a cool thing to do at a conference!)

An idea for teaching quadratics to 8th Graders

I haven’t seen a curriculum that develops quadratics quite in this way, but I’m having trouble giving up on this approach and going with anything else I’ve found. What do you think? Here’s how the unit would go:

FIRST TYPE OF EQUATION: y = (x + a)(x+b)

Step One: Learning to solve (x + a)(x + b) = c equations by treating them as multiplication equations. An important idea is that these equations can have 1, 2 or 0 solutions.

Screenshot 2018-04-02 at 7.44.30 PM.png

Step Two: In particular, learn how to efficiently solve (x + a)(x + b) = 0 and other similar, non-quadratic equations.

Screenshot 2018-04-02 at 7.47.45 PM.png

Step Three: Study these types of equations as functions. Check out what the zeroes represent in y = (x + a)(x + b).

Screenshot 2018-04-02 at 7.48.43 PM

(I’ve already done these activities in class while experimenting during the week before spring break.)

Step Four: Make generalizations about the graphs of these equations — about where the line of symmetry is, whether is curves up or curves down.


Step Five: Check out a new type of equation x^2 + b = c or x^2 - b = c. (I mean that b is non-negative.) Learn to solve these equations by using what you already know about solving linear equations, with the new twist of taking roots of each side. And notice that sometimes these equations have 2, 1 or 0 solutions, and learn precisely what sorts of equations will have

Step Six: Graph these new equations, y = x^2 + b or $y = x^2 – b$ especially in the case when b is square. All that stuff above about lines of symmetry, zeroes, etc., study that but for these equations.

Step Seven: Big idea time. There are two equivalent ways of expressing many of these quadratic equations. No factoring, no multiplying binomials yet. Just notice: some of these y = (x + a)(x + b) equations produce the same graphs as y = x^2 + c! (Mostly when c is a square.) Let’s give arguments for why this is true, arguments about the zeroes, the lines of symmetry, and that these two equations share a vertex.


Step Eight: Learn to multiply binomials, like (x + a)(x + b), and become equipped with a new algebraic way of doing the work of recognizing equivalent quadratic functions. Here we’ll especially focus on a difference of squares, (x + a)(x - a).

Step Nine: Teach the rest of your quadratics unit at this point — including whatever other factoring you need to teach — while frequently asking the question “Will these equations produce the same graph or nah?”


This all seems to me a nice way to gradually build quadratics knowledge. If pushed on my design principles, I’d say that (a) I’m trying to be sensitive to the fact that the different types of equations that fall under ‘quadratics’ are of widely varying complexity and (b) I’m trying to make sure not to teach a connection between two mathematical objects before students have a chance to really become familiar with the different mathematical objects. (In other words, students would see lots of equations in factored/standard form before trying to connect them via multiplying or factoring one into the other.)

Is there any curriculum that structures a unit in a way that even roughly resembles this? I can’t develop too much of my own curricular stuff given my teaching load (four different courses: 3rd, 4th, 8th and Geometry) but I would love to try teaching this upcoming quadratics unit in something like this way.

Any materials or approaches you’ve seen for quadratics that resemble this? Can anyone talk me out of this approach? Where would I run into trouble, if I went against better judgement and developed my own materials for this unit?

Some new thoughts on hints


A couple years ago I gave a talk (above, and here) about hints.

(What’s the deal with cardigans? Seems like they were totally in for a few years and now (for a fashion-forward fellow) they are practically unwearable. Maybe they were never in, I don’t know.)

Thanks to a conversation with Dave today, I started thinking again about hints, and I think I have something to add about hint-giving.

So, picture yourself in a classroom, a kid waves a hand and gets your attention. ‘Can I have some help? I’m totally stuck.’

You walk over: tell me your symptoms? when did it start? and then?

(My point being, you assess the situation…)

OK but once you have a sense of what’s going on, I think there are roughly three possibilities I experience and three corresponding suggestions I can make for how to react.

Possibility One: I need to teach the student something, so I sketch a quick example.

When an individual student needs to learn something new in the middle of the task, it’s never ideal.

I used to handle these moments by trying to nudge the kid along with questions about the task at hand. I’ve come to think that this is a mistake, and now I try to avoid it. Instead, I try to quickly write a related problem with a relevant, worked out solution.

Here’s a snapshot of what I mean. This student called me over because she was totally stuck on solving -1.7x = 4.3x + 3.6, and after some gentle questioning I saw what she meant. Knowing a bit about what this kid was already comfortable with and where she could go next, I quickly wrote a totally different problem [-2x = 5x + 7] and wrote a solution exemplifying what she might do for the problem at hand. (She still made mistakes, which I highlighted. Sorry for the overstuffed picture.)

Screenshot 2018-03-28 at 8.29.34 PM.png

I do this for two reasons.

The first is that, when one aims to nudge kids along at the task at hand, one runs the risk of ending up in a Polya-esque recursive suckhole of questions, digging the student deeper and deeper into their own brain, until the math is buried beneath piles of questions.

I call this ‘Polya-esque’ because when I think about the sort of interactions I try to avoid one-on-one, I often think of this passage from How to Solve It:

If the teacher, having watched sharply, cannot detect any sign of such intiative he has to resume carefully his dialogue with the students. He must be prepared to repeat with some modification the questions which the students do not answer. He must be prepared to meet often with the disconcerting silence of the students…

“Do you know a related problem?”


“Look at the unknown! Do you know a problem having the same unknown?”


“Well, what is the unknown?”

“The diagonal of a parallelpiped.”

“Do you know any problem with the same unknown?”

“No. We have not had any problem yet about the diagonal of a parallelpiped.”

“Do you know any problem with a similar unknown?”


That’s issue #1 with focusing on the task

The second is that I worry that it’s very hard for a person to learn something from thinking about just one problem. I want to leave students with a chance to think about a whole problem, not just the little scraps that I didn’t solve for them via nudges. (These conversations just are “better-luck-next-time” interactions.)

Ideally, a kid has more than just one example and one practice problem to learn something new. Then again, ideally a kid isn’t missing crucial knowledge that’s keeping them from doing math in class. We’re well past ‘ideally.’

So, the first situation is that a kid really needs to learn something new. In the heat of the moment, I don’t try to feed them each step or to weave a series of vague questions into a meaningful discovery. Instead, when a kid is stuck because they need to learn something, I try to teach them that thing.

Possibility Two: I need to help the student make a connection, so I remind them of a similar problem.

This is the territory that is closest to what I was describing in my talk. If I’ve done a nice job with my teaching, the kids have some memorable examples, ideas, problems or techniques to refer back to when trying something on their own. That way, when a kid tells me that they’re stuck but I don’t think they’re missing something crucial, I can lead with…

  • Remember the diagram we were studying at the start of class? This problem is actually really similar to that one.
  • So this is a complex area problem, and there are always basically two options: add some lines to cut the shape up, or use negative space. Which do you want to try here?
  • I see you solved this equation by adding two to both sides. Why not do something similar here?

Of course, this only works if the students have some prior knowledge. I often lead with this and see if I get a catch. If I don’t, maybe I’ll start thinking about Possibility One.

Possibility Three: I need to help a student realize that they can handle this on my own, so I redirect the student back to the problem.

Sometimes the only thing a student doesn’t know about some math is that they know it. (Which is something that they need to know.) In situations like these, my job is to either reassure the student that they’ve started down a good path, that they aren’t breaking math, or to deflect the question in a way that puts the work back on the kid.

I feel as if there isn’t much more to say about this possibility — it’s the one that math educators generally love to talk about, because it’s the most fun. And, come on, it is fun. How cool is it that the following interaction actually works, ever?

Student: Hey I’m totally stuck.

Teacher: OK what if you weren’t stuck?

This absolutely works, but only sometimes. To get roughly precise, it only works (roughly) a third of the time, because there are two other possibilities.

Other moves that are fun when we’re just trying to redirect attention back to the problem:

  • What have you tried so far?
  • What haven’t you tried?
  • Why did you write this?
  • Talk for a bit about this.

There’s not much magic here. When a student is in a situation like this it’s often just about getting them back into the problem. They probably got nervous about something and stopped early. I do that all the time when I’m doing math, it’s totally normal. Lending confidence is one of the many little things that a teacher can do.


Cardigans or not, I do need more sweaters, though at this point I could easily just wait until the fall.

Hints are a nice slice of teaching. Not too much, not too little. And it’s this interaction pattern that I have so many times a day. Question, response, question, response. It’s so easy to experiment and try out a slightly different pattern of response. If it works, I’ll likely try it again, and again, and then one year passes and then it’s another, and slowly a new pattern of interaction has replaced the old.

Some people that I talk to don’t like the connotations surrounding the word itself, ‘hint.’ I get that, and to that I offer three replacements that correspond to three possibilities. When a kid is stuck, in most cases I respond with either encouragement, a reminder, or a new example. If that’s simpler to talk about, then let’s stick to that.

Talking about faith, politics and gun control with David Cox

David Cox (@dcox21) was one of the people that taught me how to teach, i.e. he was a blogger about math teaching in 2010. I still remember trying one of his lessons in my algebra class — it worked like a charm, and I’ve enjoyed his writing immensely. Online, David often tweets about politics, and just as often he ends up tangling with liberals over any number of issues. I asked David if he wanted to dig deeper into his politics, and I was thrilled that he agreed. We ended up talking about faith (mostly Christianity) applied to politics, obligation, coercion, self-sacrifice and (content warning:) guns. 


So, gun control. The position I’ve seen you take is that reducing the number of guns won’t reduce the number of mass shootings in the US. But there are other reasons why we might want to reduce the number of guns — to reduce the number of accidental deaths and injuries, for instance.

Is there a kind of gun control that you could get behind?

I tend to think that gun control is a Trojan Horse for people control.  In other words, the current conversation is around the AR-15, but the Virginia Tech shooting involved only hand guns.  If we begin to say that the AR-15 can be banned, then the conversation will shift to other firearms that can act like the AR-15.  Which, by the way, most hunting rifles can do the same damage as the current firearm under debate.

As for background checks, I think they are important and anyone with a record of violence should have their rights to weapons limited.  To what extent, I’m not sure because so much of this falls under a slippery slope argument.

What about mental health?  Well, obviously we don’t want people with certain mental health issues to have access to weapons.  However, we then have the problem of determining what constitutes a mental health issue and who’s making the determination.  I heard a story the other day where a woman sought help at one point in her life for an eating disorder. She was later flagged when she applied for a concealed carry permit because that disorder was considered a mental health issue.  

I realize that this ends up a big tangled web that ultimately results in me saying, “I don’t know.” What’s your opinion on this?

“I don’t know” sounds like where I’m at too. When look at the arguments it gets hard quickly, in the same way that all policy questions gets hard when you think about them.

I think if I turn off my rationality and just go with my emotions, I end up favoring strict restrictions on guns. Like, guns are weapons, right? They’re designed to either very badly hurt or kill something that’s alive. What do we gain by having so many guns around? I get that a lot of people hunt…but maybe people shouldn’t kill animals for fun. There are other ways to spend time, you know?\

But then again — injecting just a bit of rationality — guns aren’t really part of my life or culture. I don’t really know anybody who owns a gun, hunts, or shoots for fun. (Come to think of it, I did fire an M-16 and a Desert Eagle in Israel on a high school class trip. Haven’t we all?)

Is this intellectual for you, or are guns part of where you come from?

It’s both intellectual and cultural.  I live in a rural part of California. My more progressive friends in the Bay Area call this part of California, “Western Nevada.”  We’re basically that part you drive through to get from Los Angeles to San Francisco if you want to avoid the bad traffic along the coast line.  Oh, and we grow all of your food.

But anyway, guns are definitely part of the culture here even though my household doesn’t have guns.  I had a BB gun as a kid and have fired .22 caliber rifles and handguns, mostly to shoot at squirrels that were bothering the almond trees.  So, we have a lot of hunters around here and I’d guess more of my neighbors have a gun in their home than don’t.

So, I suppose for me personally, this is more of an intellectual exercise. I don’t want kids to die in mass shootings.  I don’t want gang members to die in Los Angeles, Chicago or Baltimore. I don’t want people shooting each other. However, I still can’t find a way around the fact that people tend to be safer if they have a way to defend themselves.  

I’m not sure what I think is more cool, firing a crazy powerful weapon or taking a class trip to Israel.

It was definitely a weird trip.

One thing I don’t get is what exactly you want guns to keep people safe from. Guns don’t address any of my safety fears, and I’m scared of a lot of things. I think about terrorism too much. I worry about getting hit by a car. I’m scared of mass shootings. But I can’t see how my having a firearm could help me much in any of the things that I’m scared of.

You say guns tend to make people safer, but safer from what? What sorts of things are you afraid of, that a gun in the house could help with?

I feel safer in my neighborhood just knowing that my neighbors may have guns.  The very idea that a homeowner may be armed is a deterrent for potential break-ins and a thief has to weigh that threat against the possible gain of breaking into my home.

Just to test your intuitions: do you think you’d feel safer walking around in NYC, where I live, knowing that lots of people have guns at hand?

Isn’t NYC, like, a city? No, I don’t do cities. Seriously though, that’s a tough question.

What do you think is going on here? Why does NYC feel different?

Honestly, my first thought was that large percentage of those carrying guns in a city would be more likely to be doing so illegally. That’s a tremendous bias on my part. If you told me that the same percentage of gun carriers were doing so legally as in my community, I’d actually feel safer.

I know that you think that you aren’t, like, anybody should be able to get a gun whenever they want one, and that’s consistent with the distinction you’re making here.

One thing that I’m really interested in is how your faith influences your political views, on guns and other issues.

For example, I read people like Elizabeth Bruenig who see Jesus’ legacy as supporting leftist politics and a democratic socialism. They share your faith, but end up at a really different place politically. Is there a version of your own Jesus and Christianity that you can recognize in someone like Bruenig?

Ok, I’m sure I don’t have a total handle on Bruenig’s worldview except to say that she believes that Christianity should be radical and revolutionary.  I agree with this sentiment 100%. I think where we may diverge is in how this may show itself in the world.

It seems like she believes that since Christianity is always concerned with the poor, vulnerable and oppressed then our governmental systems should reflect this.  Again, I don’t disagree, but I’d imagine we’d disagree on what this may look like. She states in this piece that true love can’t be coerced.  So, the question is this: How do we care for the poor, vulnerable and oppressed?  Do we see the government as the most efficient and effective way to care for those who need it? If her answer is socialism, then I’d wonder how that squares with free will and love–assuming we define love as willing the good of the other.

I don’t quite get this. Where is the tension between socialism and love?

If socialism is instituted by a government, then it takes the choice away from the individual.  If an individual lacks free will, then any action is no longer an act of love, but an act of coercion.

Surely an act of coercion can also be an act of love, no? I think of teaching, or even parenting. Isn’t it an act of love when I keep my son from running into the street? Is there any reason why gun control couldn’t be an act of love too?

Yes, you can use coercion to keep your son from running into the street and it would be an act of love.  But is your son acting out of love when he does’t run into the street for fear or Dad’s consequence? I’d say not.

Along the same lines, if I choose to sell all of my belongings, give to the poor and join a commune, then that’d be a tremendous act of love and my new community would have many socialist characteristics.  However, it was done by choice not by force.

So the government might be acting out of love if they were to coerce me to share more of my income, or to keep me from having easy access to a gun. But then the government would be making it impossible for me to then give away my income or refrain from guns out of love, myself. Once the government steps in I can only act out of obligation, not out of love.

It’s really interesting to me the way you, and even Bruenig, can draw such a straight line between religious values and politics. It’s something that feels somewhat foreign to the way I relate my Judaism to politics.

For a lot of Jews, we identify as minorities. I think that’s especially true for traditionally observant Jews. And so even if I think that something like modesty is a Jewish value, there’s room for me to hold back from asking that everyone adhere to my values. I have my values, but I don’t necessarily feel like my religious values are always relevant for talking about policy. Though there’s inevitably some part of my religion that influences my worldview anyway…

But do you feel any of that distance between yourself and the world? Do you ever feel, when talking about guns or anything else, that you reach a point where you say: this is what I believe and what my religion calls for, but I don’t expect anyone outside my community to care about that?

I feel that distance all the time.  It’s quite a tension, actually. When I discuss politics, I try to use language that all parties can agree upon.  In other words, even though my religious beliefs form my worldview, I can’t impose that on others. So, when talking about politics, I try to keep my arguments to rule of law, logic, science, etc. because those ideas are common.

You mentioned guns specifically, so I’ll say that I don’t personally want to have a gun in my home.  However, I don’t believe that point of view should be imposed upon others — that would go against the second amendment and any other local/state laws.  I do believe people have an inherent right to self-defense, so my point of view will be formed by that reality.

However, this idea of self-defense creates a tension internally since my faith (and I’d imagine yours as well) is founded on self-sacrifice.  So, I have to wrestle with this idea of what I’d do with what I could reasonably expect from others.

Does that make sense?

I think your faith is founded on self-sacrifice in a way that mine is not. I think if Judaism can really be said to be founded on any one thing at all, it would have to be something like the giving of the Torah at Mount Sinai, and that’s a story about God and Jews cutting a deal — you’ll be my special people, but with special responsibilities — which isn’t really all about self-sacrifice. More of a win-win for both parties.

Here’s what I’m hearing, though. You’re saying that there’s a tension between (a) people should have the right to defend themselves and (b) but maybe people need to accept the possibility of being defenseless, for the sake of some greater good. Is that right?

Yes.  That’s exactly right.  

But to your contrast between Judaism and Christianity, aren’t those special responsibilities sort of a self-sacrifice?  I mean that in the sense of sacrifice now for blessings to come?

I don’t think that’s quite how the notions of responsibility work in Judaism. There is a notion of heaven and a reward in the world to come (don’t mean to entirely downplay that) but people just talk a lot about your obligations, your mitzvot, in this world. I don’t think self-sacrifice is something I feel a lot in my life. It’s more that it’s a way of life that we and our community adhere to completely.

Your point of view actually reminds me of a really sharp piece by David Foster Wallace after 9/11 titled “Just Asking.” Here’s a juicy quote:

“…what if we decided that a certain baseline vulnerability to terrorism is part of the price of the American idea? And, thus, that ours is a generation of Americans called to make great sacrifices in order to preserve our democratic way of life—sacrifices not just of our soldiers and money but of our personal safety and comfort?”  

I can see how Bruenig, or someone like her, could take those feelings of self-sacrifice and end up with a religiously inspired socialism. Shouldn’t we be willing to make tremendous sacrifices for the greater good of protecting the poor, creating a just world, the sort that constitutes the ultimate vision of Christianity? And maybe someone like David Foster Wallace, a sort of religious secularist, is preaching from a similarly Christian set of values.

Anyway, I like that way of thinking that you describe: that there’s a difference between what you feel obligated in versus what you can reasonably ask of other people.

I can appreciate the DFW quote.

He says, ‘what if we decided…’  I often wonder who the “we” is in the context of “we decided”.  It’s one thing for an individual or group to choose sacrifice for the greater good and it’s another to impose the sacrifice on others.  People will often cite Acts 2:44-45 as the context for Christian-based socialism.

And all who believed were together and had all things in common; and they sold their possessions and goods and distributed them to all, as any had need

But this isn’t coerced; it’s a choice.

Is democracy coercion, though? Like, is all taxation a coercion? Are police and safety coercion? I would say “yes and no” and so socialism would seem the same to me. If the country democratically decided to adopt socialistic policies I have a hard time seeing how it’s different than any other law or government function.

I think at this point, we’d probably have to define socialism.  I mean, do we consider higher tax rates for higher earners to be socialist in that it is a form of wealth distribution?  I think for me, it really comes down to the 10th amendment. The constitution outlines very limited powers for the federal government and the rest of those powers should go to the states.  California could be considered a fairly socialist state compared to, say, Texas. I’m ok with that. I choose to stay in CA.

Now, if you’re asking me if I want to live in Venezuela, then no thanks…hard pass.

I wonder: do you find yourself at odds, ever, with people who share your religion? When you’re in dialogue with people in your community, or internet-people who are Christian, do you ever have a chance to use your particularly Christian language? Does that help you understand each other, or is it just another set of words to use that feel more natural but ultimately can be just as confusing and difficult to hear each other with?

Yes, I think there are times when I’m at odds with people who share my religion.  When it comes to matter of faith and morals, though, we have the Magisterium of the Church  to formally define things. So, basically as long as I’m in line with the teaching of the Church any disagreement isn’t with me but with the Magisterium. However, there is still some place for personal interpretation.  Take this for instance:

2425 The Church has rejected the totalitarian and atheistic ideologies associated in modern times with “communism” or “socialism.” She has likewise refused to accept, in the practice of “capitalism,” individualism and the absolute primacy of the law of the marketplace over human labor. Regulating the economy solely by centralized planning perverts the basis of social bonds; regulating it solely by the law of the marketplace fails social justice, for “there are many human needs which cannot be satisfied by the market.” Reasonable regulation of the marketplace and economic initiatives, in keeping with a just hierarchy of values and a view to the common good, is to be commended.

Complete socialism is to be rejected but so is unfettered capitalism.  There’s probably a pretty broad gap between the two that we can discuss.

We started down this path by thinking about socialism and Jesus, and the distinction between a society trying to decide on radically communitarian policies versus Jesus and his followers voluntarily making the decision to eliminate personal ownership.

It’s a pretty cool distinction you’re making, and one that I think completely changes the emotions of the discussion. When I think e.g. about the Tea Party, I think of a sort of bumper-sticker-morality that is all about individual possession and a sort of worship of freedom of movement, unrestricted by the government or anything else.

What’s cool about what you’re saying, if I get you, is that actually personally you reject that individualist ethos. It’s not a healthy way to live, and it’s not living in the model of Jesus. At the same time, it’s not necessarily a great idea for the government to impose this way of living on everybody…

Does this sound right? If you think it does, I wonder how you think this relates to the different denominations in Christianity. Do you think that your take is a particularly Catholic one? I admit near-total ignorance on intra-Christian issues, but I know that there are holy orders in Catholicism and that various protestant groups are said to have more individualistic perspectives on faith and society.

Yes, I think you’ve nailed my point of view quite well. I think the Tea Party is an interesting example.  I’ve gone back and forth with the same “worship of freedom” sentiment. Maybe my indecisiveness has to do with my backstory…

I wasn’t raised Catholic.  I grew up in a conservative family and my early Christian experiences were more evangelical non-denominational.  From that perspective, the Christian’s relationship with Jesus is wholly personal. This is likely where the worship of freedom comes from.  After all, any evangelical non-denominational fellowship will have it’s roots in Protestantism. And what are they protesting? The Catholic Church.

So on one hand, I have a history of believing that my relationship with Jesus is entirely personal and rooted in my own individual understanding and on the other, I have this newer belief in the teaching authority of the Church.

So, I think for the Catholic and non-Catholic Christian the goal is still to submit to God.  For the Catholic this submission is both within the context of the Church and individually but for the non-Catholic the submission is based on individual understanding.

When is it helpful to make a bunch of different problems look the same?

I’m really fascinated by Craig Barton’s idea of problems that have the same surface features, but whose deep structure is different. He has started a website to collect them, and has started taking submissions.

Here is Craig’s explanation of what the thought behind these problems is:

What I needed instead were a new set of problems – ones where the surfaces were similar, but the deep structures were very different. By exposing students to problems like that, I would ensure that they learned to recognise not just the similarity between problems, but also the differences between them.

I love this idea. Here’s an example of one of the “SSDD” (=”same surface, different deep structure”) activity that Craig created:

Slide18 (1).png

There are now lots of these types of 4-sets of these problems on Craig’s website. As I scrolled through some of them, I found myself with questions. Here are some of them:

  • What makes for a good SSDD activity?
  • Is it important that the four pictures resemble each other precisely?
  • What sort of thinking does a student have to do with similar “surfaces” that they wouldn’t equally have to do with four unrelated problems?

I set out to make a SSDD activity myself to mess around with some of these ideas. Here is what I came up with, intended for my geometry students:

Same Surface Different Deep

Along the way, I tried to ask myself “would this work just as well with four separate diagrams?” For a lot of what I tried, it did.

The thing is that four separate, unrelated problems call on students to think about deep structure just as much as four different questions about the same diagram, I think. When I thought about reasons to keep the surfaces similar, I came up with two possibilities.

First, students often get confused between two different prompts that often come with the same diagram. This has been happening all week with my kids and arcs. They learned to find arc degree measures first, and they often don’t realize that a question is asking them to find arc length. For that reason, I tried to include a problem that asked for arc degree measures and another one calling for arc length.

Second, an important idea in geometry is that the same diagram might have different assumptions associated with it. We want to reason about what can be guaranteed by the information we have at hand; this version of proof isn’t about observing what happens to be true of a given diagram. So I think it’s helpful to show students that different problems can use the same diagram but represent two different sets of information, depending on what else is given. For that reason I tried to contrast two cases, one where the diagram is known to be symmetric and one where we lack any such info.

I think that’s my takeaway for now about these SSDD problems. There isn’t always a tremendous difference for the student between problems that look different (and are different) and problems that look the same (and are different). In fact, I think part of what’s fun about problem solving practice is playing around with a variety of problems that look (and are) different — the variety can provide a sort of buzz.

SSDD problems do seem like a helpful tool to use when there are important contrasts to make between things that look awfully similar at first. I think my best practice resources already incorporate some of these, but Craig’s identification of this as an activity type is very helpful to me. I’m adding it to my mental bucket of practice formats.

A little graph theory

Basically, some graphs are the same. Basically.

Like these two:

Screenshot 2018-03-12 at 8.24.45 PM

And if you don’t believe me, pretend that you tangled the right graph. You end up with something basically identical to the left one.

Screenshot 2018-03-12 at 8.25.18 PM

Straighten out both of these, and you get just a straight line, or a chain. That’s another way of seeing that they’re both (basically) the same:

Screenshot 2018-03-12 at 9.00.08 PM

Here is another pair of graphs. They’re also basically the same, i.e. isomorphic!

Screenshot 2018-03-12 at 8.25.34 PM

I like imagining swinging around the parts of these graphs to convince myself that they really are the same.

Screenshot 2018-03-12 at 8.28.10 PM

I took the above examples from the truly fantastic Introduction to Graph Theory by Richard Trudeau. I found it lying around the math department office and have been carrying it around since. (Though I get why they changed it, the original title was “Dots and Lines” which is awesome.)

Here are a few more of Trudeau’s puzzles. In each pair, are the graphs isomorphic (i.e. basically the same)?

Screenshot 2018-03-12 at 8.28.59 PM.png

Screenshot 2018-03-12 at 8.28.26 PM.png

Screenshot 2018-03-12 at 8.28.44 PM.png

You can check yourself by playing with the diagrams digitally, trying to drag the points around to change their appearances. Here are links to all of the diagram pairs I’ve so far shared:







I love the idea of opposites in math, and there is a great way to think about what the opposite of a graph should be. The fancy term is “complement” but I like thinking of every graph as having an “anti-graph.” Here are some examples:

Screenshot 2018-03-12 at 9.54.10 PM

If you overlay the graph and its anti-graph, the result should be a completely connected graph. Meaning, a graph’s complement should consist of just the edges that are missing from the original.

Now, here is an AWESOME question: are any graphs the same as their anti-graphs? Are any graphs their own opposites? One last way of putting the question, to maximize googleability: are any graphs self-complementary?

The answer is, definitely! Mess around with the graphs in the image above to see what I mean:


One way to start looking for self-complementary graphs is by thinking about the number of edges that a graph with n dots can have, if it is going to be (basically) the same as its anti-graph. After all, the complement can’t have more edges than the original graph…

And then it’s fun to think about how many vertices (dots) a graph can have if it’s going to evenly split its edges between the graph and its complement. For instance, if you have 6 vertices there is a maximum of 15 edges — so there’s no way any graph with 6 vertices could be self-complementary, because there’s no way for a graph and its complement to have an equal share of 15 edges.

It’s fun to look for both of the 5-vertex graphs that are self-complementary.

It’s fun to ask how many graphs that look like empty rings (i.e. a regular polygon) are self-complementary. There’s at least one…

And those are all the fun things that I know about self-complementary graphs. I know it’s not a ton, but nearly all of it can be shared with young children.

This is my post critiquing National Board Certification for Teaching


This hardly seems worth writing, except that so few people write about this stuff.

Six, maybe seven years ago, I started thinking about what it would take for me to teach in public schools. I had already been teaching for a couple years, and the idea of taking time off of teaching to get a teaching degree…I couldn’t convince myself it was financially feasible, and it seemed like it would be a bore, compared to teaching.

Somewhere along the line I tossed off a doomed application to NYC’s teaching fellows program. I remember writing something, like hey, you could use a teacher with some experience, I need a teaching degree, you scratch my back I scratch your’s. Dear applicant: no. 

For a while NY had an independent pathway towards certification that seemed possible, but then they discontinued it.

I kept on reading this bit of the certification website, making sure I wasn’t misunderstanding it: “An applicant who possesses a National Board for Professional Teaching Standards (NBPTS) certificate may obtain an Initial New York State certificate in a comparable title through the National Board Pathway.” This seemed like exactly what I needed.

So, four years ago, I started the process. They were revising the NBPTS portfolios, so I could only do it one bit at a time.

The math test was my first encounter with a Pearson Testing Center. I tried to prepare for the exam by cramming some calculus that I was rusty on. The entire test day was surreal. Went into a surprisingly small office in midtown Manhattan. I was imagining that it would be like when I took the SATs, that a whole crew of stressed out teachers would be sitting for an exam simultaneously. Nah, it’s more like a self-service gas station. Put your belongings in a cubby. Sign here. Here is your computer. Here is your sheet of plastic and a dry-erase marker. Boop. Time’s up. Have a great day.

Component 2 was my first experience with the written stuff. It was then that I learned my most important NBCT lesson: how to condense text.

How little I knew about condensing text when I began NBCT! This is from my first draft of my C2 written commentary:

Screenshot 2018-03-03 at 9.12.57 PM.png

Awful, right! I mean, look at all that space. Here is what I ended up submitting, after getting feedback from a couple NBCT geniuses:

Screenshot 2018-03-03 at 9.13.15 PM

I passed C2. The next year was C3, the video portfolio. This was annoying because you couldn’t do any preparatory work until you had the video, and the little camera that I had set up would constantly run out of battery in the middle of the lesson.

The hardest thing about the videos was that you needed them to provide evidence for exactly what NBCT was assessing you on. I felt like it was hard to capture a video that gave them exactly what I wanted. Here is the feedback I got from NBCT after I received my passing score (3.375) on the portfolio:

Screenshot 2018-03-03 at 9.20.30 PM.png

OK, yes, there is irony in the quality of the feedback that NBCT gives you. Good luck parsing any of that. I just read “evidence of insight on your future instructional practices” three times to figure out if I can figure it out — not yet.

That left Component 4, which was no question the worst component. It’s sort of a mess. There are three parts, each calling for exactly the right kinds of evidence, and the three parts have very little to do with each other. It’s like three mini-portfolios glued together. I hated it, but I did it, and it’s done.

It’s done — I passed.

If a teacher tells me that they are NBCT, I think I know something about that teacher. They’re hard-working, because NBCT is a lot of work. They are likely ambitious, probably not on their way out of the profession.

All this I know because NBCT was a ton of work. I can’t imagine a teacher going through this without something pushing them — either a financial incentive or something internal.

So I know they’re hard-working and committed to teaching, but that’s pretty much all that I know. Nothing about the NBCT process gives me any confidence that it was assessing the quality of my teaching in any sense at all.

I have a couple friends who have been on the other end of things, assessing candidates. I believe them when they tell me there’s a clear difference in quality between different candidates. But having done all the work, I have trouble seeing exactly how you can tell the difference between a candidate who just didn’t understand the prompts and someone whose teaching meets the standards. Because it was really hard to figure out what the prompts were calling for — that was a lot of the work.

Maybe I’m just in a grouchy mood. Even though I love working at my school — public school is going to have to wait — I’ve been feeling a bit down lately.

It all feels sort of bad right now. Writing’s bad, I won’t even edit this piece. Bad at math. Kids hate math, though kids like class. Small apartment, we try not to flush in the AM because it might wake up the kids. Kitchen’s small, fridge is small, always catching mice.

Education can be so, so dumb so often, math education in particular. The dumb stuff is the most lucrative. Teachers seem to love this stuff, though, so what am I doing? All the people I knew teaching math six years ago are off doing other stuff.

But I got this certificate, and now I’m NBCT, and I also have a letter from NBCT saying “your voice matters,” so there’s that.


A quick shout out to proteacher.net. The people on there are the best. If you have questions about NBCT you should absolutely hop on there and make an account. If you’re starting NBCT, you should go there and make an account. The people there were just ridiculously generous with their time and it’s a lovely corner of the internet of teachers. That’s my only useful piece of advice for NBCT.

Learning is Weird


There I was, helping Samantha with some subtraction, when I hear another kid nearby — Lena — cracking up, really losing it. Lena was laughing, and though I try to ignore her, she’s laughing persistently. Lena turns and looks at me with a huge, ridiculous smile across her tiny little third grader face.

“It’s just zero!” she says.

“Yep,” I say. I force a smile.

“It’s just zero!” she says it again. I try to grin convincingly back, as my mind races. What’s so funny?

“Haha, that’s right,” I replied, hoping that I sounded sort of like a human does when they get a joke.

For context, here is what Lena was working on: a big-fat subtraction worksheet. Here is a sampling of some of the hilarious problems I’d included on the page:

120 – 30


21 – 2

Don’t forget:

110 – 60

And this classic:

8 – 3

You may also notice that this list of uproarious problems seems a bit on the easier side for third graders. For Lena (and Samantha) it was not. Subtraction has been coming exceedingly slowly for these kids — much slower than their multiplication, actually. It’s February, so we’re not anywhere near the finish line. Even so, I’m beginning to start to anticipate to realize that my time with my students is, ever so slowly, slipping away. I want these kids to have a good year next year in math, to be happy about school. I don’t want this to gnaw at me over the summer.

Anyway, Lena is cracking herself up so I have to go over and see what she’s up to. I look at her page. Suddenly, I’m in on the joke.


You see Lena subtracts digit by digit, because someone taught her to do that. I don’t know exactly what to say — it’s not wrong, and she is so shaky with so much subtraction. It gets her in trouble with problems like 17 – 8, because she brings the 1 down unnecessarily. Still, it’s something to work with.


But the thing is that she really needs to focus on each digit with all her attention. She can’t yet take that step back to see the problem as a whole. So there she is, with 251 – 251. Carefully, slowly, she considers each digit:

2 minus 2 is…0.

5 minus 5…0 as well.

1 minus 1…wait a second…

And there you go, there’s the joke, it’s just zero.


Ooh, by the way, Samantha is pretty interesting too.

Samantha also does that column-by-column thing, and it serves her well until she gets to problems like 125 – 50, since you can’t take away 5 from 2.

She started the year trying to borrow in these situations, but she really lost all sense of gravity as soon as she got permission to mess with the numbers. She’d do some of the weirdest things I’d ever seen with subtraction — I can’t remember them, they’re so weird. All I remember is that a bunch of times she would proudly shove a piece of paper in front of me and with, like, innocent puppy eyes, ask, “Is this right?”

And 100% of the time the paper would look like this:

125 – 50 = 972

Seriously! It was all over the place.

My take is that Samantha’s brain is just overloaded when she tries to keep track of all the parts of these problems. Every stage of it requires understanding and attention. She uses a strategy to compute 12 – 5, to take away 1, to realize that this leaves 0, to turn the 2 into a 12, to realize that this is, you know, subtraction so it should make things smaller, etc., etc.

I don’t think she should be going all-in on borrowing yet, not until she has a bit more knowledge to rest on.

But what do we do for her? Samantha asks for lots of help, and until recently I’ve been a bit stumped about how to help her.

I think I might have figured it out, though. The other day Samantha comes over to me, once again stumped on a problem. Her paper looks like this:


I have a false start, going into some totally different strategy for subtracting. Whatever, she gets that far-off stare, she can’t deal with all of it. It’s another way of thinking — it’s not her way of thinking which — for better or for worse — is column-by-column subtraction.

I think, and then I have an idea. She can, I know, subtract two-digit numbers — it’s laborious, but she can do it. So I write an example next to the problem on her page. How about this, I say?


OK, this actually makes sense to her! She uses it to work on the original problem. I offer to give her some more questions to practice — she completes each, surprised that she’s handling the problems correctly.

Is there more to notice here? Sure there is. She should know that the “32” in “324” means 320,  she should know how to handle 320 – 150 without drawing little lines, and down the line I sure hope that 32 – 15 doesn’t take quite so much out of her.

But has she learned something? By any fair reckoning, of course she has.


Math class should be joyous, they say, full of laughter and insight. I agree! But it seems that a lot of people in education go further, as they’re eager to point you to the source of classroom joy. See this? It’s a picture of kids smiling while studying math. Want it? You’ve got to try instructional practice X, Y and Z.

I promise, you, though, that kids and learning are weirder than that. You’ll plan for fun, and they’ll hate it. The next day you’ll run out of fresh ideas, open a new browser window, type in www dot kuta software dot com slash free dot html, print out worksheets with answer keys, sort of just push them over the desks until each kid has a sheet nearby, then mumble incoherently for a couple of minutes when all you’d really like to say is “here is this, I’m sorry, please do it” and you’ll brace for the worst…

…and that will be the day when everyone is having a blast with math, even Tobias, which is surprising because Tobias has just been sitting there quietly since October when he broke up with Julia, and like you told his mother it’s been very tricky to get him to open up, but there he is chatting about exponent rules with Harry, and he seems alive and (to be honest) happy in a way that you haven’t seen him in a long time.

(In case you missed it, we moved from third to eighth grade with that last bit.)

All of this is to say that joy and humor in a classroom can come from where you’d least expect it — depending on what you expect.

And Samantha? Well, people will also tell you that you need to listen to the ideas of students, to truly build on their thinking, not to override their thinking but to build on it.

I agree. But what does it look like to build on how your students think? What if your student thinks about a problem in a way that isn’t just wrong, but wrong in the wrong way? It’s not just that her technique is incomplete, but it feels like a trick, like a machine that was designed to perform half the job, like a car that can only turn left?

I’m not always sure that I understand the difference between procedural and conceptual knowledge, but I think Samantha’s case is clear. She has a (half-working) procedure married with a not-quite-there-yet conceptual grounding. Is this a time to accept what she knows and to develop it? Or to dismiss her approach and bring her back to square one, conceptually speaking? Is this an exception to the rule — a time when we shouldn’t build on what she knows, but should instead sort of veer around her structures and start construction on a new lot?

Learning is weird — it will surprise you. Procedures can be a start. Subtraction can be hilarious. Go ahead, come up with a theory about how all of this works, but be ready to find out that something entirely different gets the same results. Share what you’ve found, and then also have the humility to know that something quite different might work as well.

I love being able to laugh about math with kids, and learning how kids think is just about my favorite part of this job. I love that so many people in education want classrooms to be joyous places where children feel understood — I want that too. But if you find yourself setting terms on how this can happen or what this looks like, please proceed with caution: it doesn’t look just one way.

Talking About Diversity in Education with Marian Dingle

Marian Dingle is one of my favorite bloggers, though she’s only written two blog posts — I hope for many more! The most recent of the two is a couple of things at once, including an expression of the idea that people have this somewhat strange, absolutely fundamental desire to be understood by someone else. Her first post asked a simple, basic question: why should we care about diversity? That question forms the basis of this conversation. Marian has been teaching for 18 years in Maryland and Georgia and tweets thoughtfully from @DingleTeach.


I thought we could start by talking about your blog post about TMC [the conference] and diversity, and your question — why should we be more diverse? Do you feel like MTBoS [the community] or TMC is any closer to answering that question for itself than it was last summer, when you asked it?

Because TMC meets only once a year, answering the question has been difficult. I’ve participated in online communication with a subset of folks, but it’s just not the same as face-to-face discussion with a larger group.

I’ve also had several fairly deep online conversations with people one-on-one, and people are in various stages of comfort in answering the question. I do wonder how much of a concern the question is for the majority of MTBoS, though.

I don’t know if you’re counting me in your tally of people you’ve had fairly deep conversations with, but I’ve felt that you’ve pushed me to think more clearly about diversity in a lot of ways.

For example, there was a certain point in this past year when I said to myself, “Diversity matters because it’s a fancy name for affirmative action, and affirmative action is a good thing.” But when I shared that with you, I think you essentially told me, nobody wants to feel like they’re an affirmative action case — they want to feel valued for who they are.

Do you think there’s a way for a focus on diversity without making people feel like affirmative action cases?

My point was that everyone wants to feel valued. I’d much rather be sincerely invited for my value than my presence tolerated.

It’s all about the why. There are plenty of educators at TMC who do not work or teach with any people of color. It’s important to me that they, or other non-persons of color, are really clear of their reasons. It’s unfair for people of color to have to prove that they belong in a group of self-described like-minded mathematics educators, when we have to do this in nearly every other facet of our lives.

Have there been times when you’ve been in school and thought, this is something that just wouldn’t happen if diversity were better in this space?

Maybe it’s a bum question. Feel free to pass on it. Or to tell me the question you wish I had asked.

Question: Tell me about a time when you felt your ability or competence as an educator was questioned because of your color. Was it an issue of diversity?

In my ninth year of teaching, I began teaching third grade in an affluent, mostly White school and district with few teachers of color. Every January, students were tested for eligibility for the gifted program, which began in fourth grade in math. Both teachers and parents could recommend students for testing. While my competence had been questioned in different ways, it peaked when I recommended “too many” students, which included students of color — this per the gifted teacher and my teammate.

Some of those kids did end up qualifying for the program, and I later discovered that the gifted teacher had been asked to vacate her position since the students were performing poorly. Fate intervened, and I was asked to assume her position. Whispers ensued followed by interrogation about my qualifications from colleagues. Apparently, this was a coveted position.

To avoid a similar situation with parents, my principal agreed to a meeting with parents to introduce me and a brief question and answer. That summer I held a math boot camp for the fourth graders, and parent conferences for all. It was my attempt to make them comfortable with me, but also to gain insight into their expectations. The vetting process continued with the fifth graders, who also asked me if I was in a gifted program as a student, and if I’d skipped any grades. I was also given math problems to do.

Things went well, but my experience highlights what we have to go through. No one asked me to do any of the “extras”, but I knew they had to be done to be considered equal. After I proved myself, students and parents saw my worth, and the program grew. Had there been more teachers of color before me, that road may have been easier. It’s never easy being first.

I want to get back to the idea of a quota, and how that could be problematic. Suppose that this school — or TMC, or some other space in education — decided that they wanted to recruit more Black or Hispanic teachers, and so they created a quota. I hear you as saying that this would create a situation where the teachers filling the quota feel they have to prove themselves, to prove that they belong.

Did I get that right? You said earlier it’s all about “the why.” How then does “the why” fit into this?

As far as TMC, I’m saying that even before we hit the quota stage, there has to be an articulated reason for the quota in the first place. What’s the benefit? For whom? Is it for political correctness? I think some have an answer to this, but most don’t. At least not an answer that they are comfortable with. That’s the work – to really get at *why* this is a goal. Does that make sense?

If it’s about a school, then there’s a history of eliminating and limiting the number of teachers of color, and yes, a quota would seem to remedy that. Still, if it is only about filling that quota, it’s not sustainable. Experiences similar to mine and much worse will likely ensue. Some forethought should be given about retention.

I think I see what you mean. Like, if the purpose of a quota was “we have a lot to learn from Black educators” then that diversity policy would not put those educators on the spot. Like, the whole premise of the policy would be that Black educators are valuable, so those educators wouldn’t have to prove themselves. But if the purpose was something that didn’t affirm the value of those educators, there would be a situation where people have to prove themselves. Is this sounding like your take?

In part. But even if it’s “we have a lot to learn from Black educators” then there is the question of who is benefiting. What exactly can be learned? Why can that knowledge be obtained from teachers of color? And who does that benefit? Care should be taken with that line of reasoning.

Sometimes answers to the “why diversity” question can border on exploitation, and it’s a kind of exploitation which is pretty pervasive in schools now. For example, Black male teachers are often assigned discipline duties because “they are good at it” and Black educators are given more students of color because “we are so good with them”.

So, to come to TMC and be expected to “teach how to teach Black kids” is another form of what we experience daily. No thanks.

Right. “Hey, we really want you to come to this conference so that you can teach us about how to handle diversity and racism issues.” That seems to me a close parallel to the exploitation you’re talking about in schools.

Can you imagine a “why” that would justify a demographic quota for you? The alternative seems to me for diversity to be more about a culture change that indirectly changes the demographics of a community or a conference, which feels a bit more slippery to me.

Honestly, I don’t think I’m comfortable with a quota, which is not to say that one isn’t needed. What I’d like to imagine is that it happens naturally. That is, I wish it would happen without being forced. Yes, we need to have uncomfortable conversations, but then after the smoke clears, it should get easier. Naive, perhaps.

I am still advocating for articulating a reason why we believe in diversity. And then, if it’s needed, we should take steps to make that happen, a quota being one option among others.


[Marian sends me Through Our Eyes: Perspectives and Reflections from Black Teachers.]

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This looks really cool and interesting. Does it all resonate with you?

I just skimmed it but pretty much. I’d add that I was once in an all-Black space: administration, staff, students, families. For 3 years, my children and I lived without racism directly affecting me in the workplace or my children in their schooling. I’ve missed that kind of freedom.

This reminds me of something I first learned about from Dana Goldstein’s Teacher Wars, that a lot of Black teachers left the profession after Brown v. Board of Ed, because before the decision they had been teaching in all-Black schools. Mandated integration also greatly expanded the opportunities for racism.

Right. Black teachers were not hired at White schools — explicitly told they weren’t qualified to teach White students. That began the Black teacher shortage. Black schools were closed, Black teachers were fired, and Black students entered White schools.

I know that this isn’t the same because Jews and Black people are in vastly different social situations in the US, but college was my first time out of an all-Jewish environment, and my first teaching job was in an all-Jewish school. I really like my school now, but my wife teaches at an all-Jewish school and all the time I’m like, ok, how amazing would it be to teach in a place where I don’t have to explain myself? Do you ever think about going back to work at an all-Black space?

I completely understand that feeling.

I won’t go back to that particular space, due to other reasons, but I have imagined what it would be like somewhere else. At this point in my life, I see the value in being that person in my students’ lives that they’ve not seen before. It’s so much more than breaking a stereotype. My pedagogy is most valuable in a space where there are multiple cultures represented. That’s my strength.

It seems to me that the thing you’re doing for your students — being the person they’ve never seen before, challenging stereotypes — it’s so valuable. But would it be OK for a school to ask you to play that role? I think probably not, and for the reasons we’ve talked about earlier, that this feels like a justification for diversity that comes close to exploitation: “Come to our school/conference, so that you can break our stereotypes!” It seems like the role you’re describing is so important, but it can only come from the choice of the educator. If it came from the school, conference or organization, it would feel exploitative.

And then I suppose it gets back to what you were also describing above: ideally, spaces in education should transform themselves so that educators naturally want to play these roles. But I’m left with the question, is it possible to engineer a natural development like this?

That’s the question precisely. If racism is structural and systemic, is there any manner of individual pursuits, even if coordinated, that are sufficient to tackle it? My gut says no, yet I persist anyway. It’s human to use what agency we do have, right?

I want to not have to explain/prove my humanity, yet I do. Quite the conundrum.

So you can employ quotas, but then there’s the chance that people are made to feel like they don’t belong. The alternative though seems bad too, which is to let things take their natural course. I think in the case of an organization like TMC, that means embracing non-White educators when they come along and trying to be welcoming, but not really doing much to change the fundamental demographic facts of the organization.

But being welcoming is no small thing. The work begins with attending to personal bias, and really taking a hard look at what is happening in your work space. Maybe that involves looking at equity. Do all students have access to opportunities? Do all colleagues? Or maybe if you are in a non-diverse space, analyze that. Is it by choice?

There’s a lot to be said for making people feel welcomed and considered. I think that’s a matter of being human, and not a big ask. But not ever having the conversation will never get us there. There are a few talking about these things, perhaps more than I know. I hope so.

What I hear you saying is that you actually think “welcoming” is a very good place to focus the attention of a diversity conversation.

Yes, precisely.

One last thing I’m wondering — are things simpler when there has been clear discrimination in the past? Like, suppose that TMC was 100 years old and had a policy at some point of not accepting Black educators to the conference. Would that provide a clear “why” for a quota, justifying the policy without making people wonder if they really belong?

I think we’re past having to have a historical reason we can point to (i.e. exclusion policy) in order to bring other voices in. That gets into another dialogue of assigning blame that’s not helpful. We know what we have now, so the questions become: Do we want to change it? Why do we want it changed? How will we do that? In that order.