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!



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?

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.

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.

My beef with Sunil Singh’s piece about math and math education


If you’re just joining us: I wrote this, some people liked it, others did not. Sunil Singh, whose post I was critiquing, understandably didn’t like what I had to say.

(He also didn’t like how I said it. Admittedly, I was a bit obnoxious. But Sunil’s rhetoric was not kind either and, dammit, sometimes two wrongs do make a right. Happy to deescalate the rhetoric, though.)

As I see it, Sunil’s core argument in his post goes like this:

  • People, in general, hate math
  • It’s math education’s fault
  • The problem is that math education has deviated from the purposes and values of math (as identified by Francis Su)
  • Math needs to take back math education — in particular, mathematicians need to direct math education

I wanted to make sure I was understanding the argument correctly — it seems to check out with Sunil.

Now, the goal here is to reconstruct Sunil’s argument in a way that he would recognize — maybe, if we do a really good job, in a way that he would recognize as even clearer than his own version.

Towards that, let’s try to clarify: what exactly would it mean for mathematicians to “take back” math education? This gets clearer from Sunil’s examples of where math education has gone wrong. He calls out homework as a major problem (“homework is on life support”). He wants to get rid of grades. He wants to radically change assessment. He thinks math education has been infected by financial and political forces. He says math education is not operating in the best interest of children.

The point being that it’s not only classroom teaching that is making children hate math — it’s everything, the whole system. Mathematicians need to be in charge of all this.

And who are these “mathematicians”? Am a mathematician? The problem, as Matt Enlow points out, is that while the term “mathematician” can mean “someone who gets paid to do mathematics” or “someone who has received an unusual amount of training in mathematics,” it can also mean “a lover of mathematics, someone committed to the discipline.” (The exact same difficulty surrounds words like “artist” or “educator.”)

Sunil Singh isn’t a professional working mathematician, so let’s assume that he means to use “mathematician” broadly.

His argument, all together, therefore goes like this: If mathematicians — people who love and truly get mathematics — had control of math education, there’s no way it would look the way it does. There would be no homework. There would be none of this testing to decide who is worthy or unworthy of more math. There would be no grades. The universal values of mathematics would be the focus of education: play, justice, truth, beauty and love. Students would have a chance to learn math that truly interests them, not the garbage we throw at them in algebra classes. They’d learn the “dream team” of beautiful math, not a hierarchy of topics on this inevitable, dreary march to nowhere. 

This, as I understand it, is Sunil’s argument.


I disagree with basically all of it.

Not because the status quo in math education is ideal to me. I don’t like grades, I think they’re overall bad for learning. I think the math curriculum is over-stuffed and deserves a healthy pruning. Our current testing regime in the US is nuts, and my experience with NY’s Regents exams have all been frustrating.

But here’s the thing about mathematicians and math education: you don’t have to sit and wonder what a math education designed by people who love and truly get math would look like. It’s not some mystery. University math departments are designed and filled exclusively with people who love math so much that they’ve signed up for a lifetime of studying and teaching it.

You know what’s coming next, right? Because university math courses are, on the whole, taught far more poorly than k-12 math courses. Where were all the mathematicians when they were designing entrance exams to decide which Calculus section you get to sign up for? Where is the play and exploration in Abstract Algebra classes? Why is the dominant pedagogy notes and lecture? What on Earth is the deal with a class like Math 55?

And it’s also not some sort of mystery what professional lovers of math advocate for when it comes to k-12 education. They call for more rigor, they worry that their students are coming with weaker skills than they used to, they criticize textbooks for having ever-so-fuzzy definitions, and, not to put too fine a point on it, they aren’t exactly lining outside of the statehouse asking k-12 educators to ditch all that algebra.

The way I see things, Singh’s mistake is in thinking that math education is some deviation from the desires of mathematicians and lovers of mathematics. I mean, yes, k-12 math education absolutely is a deviation from a particular vision of math that Singh articulates. But if you look at university teaching and if you look at the rigor, precision, and gate-keeping that mathematicians frequently call for in k-12 education, you can see that this isn’t a deviation; it’s a reflection of what a major segment of the math-loving population wants out of math education.

Singh’s piece was written as a polemic against math education. This is entirely unfair, though, as math education and math culture are codependent. The issues with math education are equally issues facing the broader mathematical community.

Mathematicians and lovers of mathematics love to hate on math education and its deviations. But it’s the lovers of mathematics who have set up the system that we have. They protect it and extend it into higher education. It’s painful to see that some of the problems we have in math education can even spring from that love — from a desire to protect mathematics, or a desire to spread mathematics.

And realizing all of this is a way to realize that Singh’s diagnosis is incorrect. The problems with math education come from the competing desires that non-mathematicians along with mathematicians want from k-12 classrooms. Mathematicians may want students to be exposed to the beauty of math, but they equally want to find the gifted few who can enroll in their college classes, and they want those students to be well-prepared. Lovers of mathematics want to make sure that the discipline — which they love — is well-understood and used by the populace. And they want to make sure that engineers, doctors, accountants, NSA hackers, software designers, everyone is given a solid grounding in math. And, like the rest of us, they want to think that everyone gets a fair shot at any of those jobs.

You can’t improve math education without understanding what exactly is going on. Singh’s vision of mathematics isn’t universal among lovers of math, even among those who really know and get it. (Unless we say no true lover of math would disagree with Singh’s vision, which is totally cheating.)

We’ll never improve math education in our current system by trying to blow it up, and I think that would only make things worse. I have a great respect for those who operate outside of the world of math education who try to spread their love of the field more widely. But here in the world of math education, we’re all trying to figure out how to help kids deal with the mess that mathematicians and everyone else have left us.

And, actually, we’re making some progress. So ease up on the attack on math education.

Should I see my son’s misconceptions?


Yosef turns three tomorrow — happy birthday, kid! My sister got him some new puzzles for his birthday, and that’s how we spent a big chunk of the afternoon.

This is his first foray into “big kid” puzzles. We had no idea he was ready for them, and he can do a lot of it on his own, though he always asks for help. (Like every three year-old, he likes attention from grown-ups.)

While he’s been playing, I’ve been watching and trying to make sense of how he’s thinking. As far as I can tell, his main strategy is to match the pictures of pieces: fish goes with fish, yellow with yellow, etc. He also has an eye for missing pieces — meaning, he matches holes with pieces that are congruent to the holes.

One thing that’s fascinated me: he doesn’t really notice the difference between edge pieces and interior pieces. Here he is, trying to stick an edge piece into the middle of the puzzle.

IMG_0760 (1)

I got curious, so I started asking him about potential fits. Could this piece go here?


He never mentions the shape of the piece, or the way that it would partly stick out. When I asked him about this piece he only mentioned the color. When I swapped out the yellow piece would another green-sea colored one, he would try to smoosh it into the hole. Only then would he tell me why it wouldn’t work — “It doesn’t go in the hole.”

Now, I honestly don’t care how well or poorly he solves puzzles. But learning stuff is fun, and I was curious whether I could help him see the difference between edge and inside pieces.

“Look Yosef,” I said. Just like in that picture, one of those inside pieces was along the top row of pieces, protruding out of the frame. I put my finger at the top of the puzzle on the top left side of the puzzle, and I slowly dragged my finger to the right. “My finger can just keep going, going, going…until it bumps into this. Bonk! This piece doesn’t belong!”

Yosef laughed. “Bonk!” he said. “Bonk!”

“But look Daddy. If my finger goes like this” — he loops down into the interior of the puzzle, far below the top row of pieces, slowly meanders up until it reaches the false piece, right under where my finger had bumped — “if it goes like this, then it doesn’t bump.”

Which was true! Had to cop to that.

He returned to the puzzle. He matched pictures — dolphin into dolphin, clownfish to clownfish — and every so often mystified me by quickly intuiting where a piece went. He also continued to shove edge pieces all along the inside of the puzzle.

I wasn’t lying when I said that I didn’t care how he plays with these puzzles…but doesn’t it just kill you to watch someone painstakingly — delicately with care — try like seventeen different ways of putting an inside piece into the side of a puzzle?

I mostly keep my mouth shut and let him have fun. He’s clearly not seeing edge pieces yet, which is interesting, but obviously fine.

Every once in a while though, I nudge at his understanding. “Pass me an edge piece,” I’ll say, hoping that he starts thinking of edge pieces as a distinctive category. If he asks me to fill in part of the puzzle I’ll talk aloud about my thinking: “This piece couldn’t go here because it doesn’t have a straight side.”

I have no idea if this stuff is connecting with him. Learning to see shapes in different ways is messy and slow. My little two-second nudges won’t make much of a difference to him — right up until he becomes ready for them, and then they might.


It’s pretty clear to me that there are things about shapes and puzzle pieces that Yosef doesn’t yet understand. He’s three. Of course there are. But how should I think about his understanding? In some quarters of the math education world, the answer is contentious.

Brian Lawler is someone who has been incredibly patient with me on Twitter, as we’ve gone back and forth discussing his positions on the nature of mathematical knowledge, teaching and learning. He passionately believes that any talk of misconception is not only wrong-headed, but also the act of labeling someone as holding a misconception is harmful to that person. Ditto for a smattering of other terms that imply that the other person’s thinking is worse than your’s, or on the way to some better understanding — this includes talk of alternate conceptions, early conceptions, preconceptions.

Rochelle Gutierrez likewise asks us to refuse to talk of misconceptions.

These scholars aren’t talking about me doing math with my kid — they’re talking about the ways math teaching can beat down kids in a lot of school situations. Still, their arguments are that thinking in terms of misconceptions or even not-there-yet conceptions is harmful — even violent — to a math learner. Their framework should apply to me doing a puzzle with my son too, I think.

Now, it doesn’t seem entirely accurate to me to say that Yosef has misconceptions about puzzles or shapes. It’s not like he actively thinks that edges don’t matter — he just doesn’t see the difference between edge and interior pieces yet. Yet he has so many amazing things in his little-kid brain that help him put pieces together. He absolutely has a conception of shape that is letting him have a blast with puzzles, and he loves doing them with me. I love playing puzzles with him. I love him.

Is it still harmful for me to think and talk about the things he doesn’t yet see?

I try to be a good father to my children. This is not always easy with a three-year old, but I really do try. I think I’m getting more patient — hopefully in time for the baby’s turn at toddlerhood — and I try hard to give Yosef room to play with toys the way he wants to play with them. I certainly don’t want to visit “intellectual violence” (as the phrase goes) on him by getting all up in his face about the right way to do a freaking 48-piece puzzle. I’d rather save our conflict for when he’s dropping a block on his baby sister’s head.

At the same time, part of our play is learning. The kid wants to put the puzzle pieces in on his own. He wants me to help. He likes learning new things — he’s a kid, he’s deeply curious about the world. The world includes mathematical language. Every time we put on his shoes we end up doing a whole routine about left/right: No, that’s not left. It’s right. No, not right, right. Right. Left. That’s right. His conception of left/right is relatively weak — it’ll get stronger.

Kids love improving their conceptions of the world, if they really get a chance to feel like it’s their own conceptions that are improving. Humans are curious creatures, and we like improving. There are a million ways for this to go wrong and to turn into abuse — in a lot of schools, this is happening.

In a lot of schools and homes, though, it isn’t. I don’t think it’s inherently abusive to see your child’s misconceptions or to help them see things in a new and richer way. It can be, of course, and that abuse needs to be detailed and discussed.


Some people might disagree with the above, but not many. The real question is a linguistic, or even a strategic one:

Does refusing to talk of ‘misconceptions’ cut down on the abuse?

Pretty much everyone I talk to online seems to think that this is a good way to chip away at the problem of abusive teaching practices. The first way this might chip away at the problem: the word “misconception” itself connotes the bad teaching practices. You can hardly use the word without being further nudged towards abuse — or you might nudge a colleague to abuse. If you eliminate the word, you eliminate the subconscious priming of yourself and of colleagues.

The second way: Changing your pedagogy is hard, and it’s easy to forget your principles. The refusal to talk of “misconceptions” is a relatively easy change to make, but it’s like a red string around your finger. It will remind you of your commitment to the proper pedagogy, and you’ll continuously improve as a result.

I actually think there really is something to that second thing, but I also think it’s incredibly risky for the cause of good pedagogy to tie it to refusing to use the word “misconception.”

It’s because my kid’s conception of shape really does have room to improve.

We see misconceptions in children because it really is true that there’s stuff that they don’t yet know. Noticing this doesn’t have to be an act of violence — in fact, I don’t think that it usually is. Usually it’s like me playing with my son and noticing there’s stuff he doesn’t yet know how to do, even as my mind is blown because oh my god my son is into puzzles! When did our baby turn into a kid?

Is it good pedagogy to ask people who don’t already see their pedagogy as abusive to forswear from using words that they use all the time? Isn’t this exactly the sort of “intellectual violence” that we’re being urged to refrain from? Shouldn’t we start with the way people actually see the world, rather than asking them to use language that is not their own?

Seriously: imagine what a teacher whose well-meaning administrator announces that they don’t want to hear any talk of misconceptions any longer, that this is now school policy. Is that good pedagogy?

There is real injustice and unkindness in this world, and I have no clue how to fix it. I think a focus on which words are allowed or not is a tactical mistake. Take any word that you associate with fear, abuse, pain; there are others out there who associate those same words with love, play and growth. To bridge those gaps we need to talk with each other and find a common language. That can only happen if we agree to use each others’ words.

Weird, Sloppy Rant about Giftedness


People who know me or my work in the goyishe world sometimes ask me how my traditional Jewish education — which mostly involves learning to carefully analyze texts —  influences what I do in math education. “You’re super-duper pedantic but you read things closely,” they say. “Isn’t that a result of a life studying Talmud?”

The answer I tend to offer is something like: Who knows? I have a lot of friends from yeshiva, but most of them aren’t nearly as annoying as I am. So, really, who’s to say? Besides, I also studied philosophy, and philosophers aren’t exactly the most easy-going people of all-time either. Maybe philosophy is why I’m such a pain in the ass.

Some people have stories about how their upbringing or education has made them who they are today. But memory is a funny thing; it’s hardly a reliable witness. If we’re honest, how sure can we be about what made us who we have become? All this sort of backwards-speculation is just guessing, and shouldn’t be taken too seriously.


OK, throat cleared, let’s speculate.

I was a good student, but I wasn’t a quote-unquote genius. That’s partly a matter of personality but it also accurately reflects the fact that nobody was ever, like, dude, Michael is breaking the system!

But, yes, ok, I was smart in school and made to feel that way by grades, peers, teachers, etc. I knew there were things I could do that others could not. The things people tell me I’m good at now are things that people were telling me then: that I ask good questions, that I read carefully, that I write clearly (if not quite, you know, beautifully).

Sarah HANNAH Gómez, in her tweets, says there’s a problem with gifted education. She was gifted, told she was smart, but never told to engage with classwork, to push herself, to really do anything at all. She says this is endemic to education and that teachers need to figure out ways to engage their most gifted students.

Here’s what I’m trying to say: in my yeshiva we were taught that we had an obligation to learn, and that obligations were a big deal. “Obligation to learn” means there’s optional Talmud class on Wednesday nights, and also on Sunday afternoons, and also on Thursday nights, and aren’t you going to stick around for it? Aren’t you a serious student?

There were silly parts of this culture, as there are of any culture. Kids trying to out-macho themselves by staying up late, attempting pious acts of learning into the early morning. For some kids it caused a lot of stress, when they were tracked into a middle shiur or out-shined by their classmates. There were stupid competitions about who could learn the most, and there was also a ridiculous award ceremony every year to honor the school’s top students.

(Though, I should add, being a “top student” didn’t mean you were a genius. It mostly meant that you took your studies seriously, logged a lot of hours, and also were a moral exemplar in the eyes of your teachers.)

I can’t imagine a gifted student at my high school somehow getting the message that he didn’t have to learn. That you had an obligation to learn was pretty much the whole point of the school.

You might wonder what our non-Jewish studies classes looked like, whether the same verve was applied to these other classes.

Based on what I saw, nah.

A lot of my other high school classes were a joke. There was not an obligation to e.g. know the Roman emperors or get really good at balancing chemical reactions. I remember reading a lot of textbook chapters during my free period, right before a 30-question multiple choice test.

(Many teachers used the same tests each year, and there was a shady tradition of kids saving the answers and inputting them into TI-83s, which they might get away with using on some test. This worked best for Mr. Rosenbaum’s AP Biology class, since you could often legitimately ask to use a calculator to help with genetic probabilities. Mr. Rosenbaum told us he was mystified why so many of us nailed the multiple choice but put no effort into the essay.)

Looking at my life since yeshiva, though, I think somehow I got bonked with the weird idea that there is an obligation to learn anything, especially if you can. I think I can thank my yeshiva for that idea, and I think that’s the sort of education that Sarah HANNAH Gómez wished she had received.


The yeshiva is an educational institution rooted in scarcity. Your towns and cities needed scholars and rabbis, but how many people could the community support? You need wealth to support equality of opportunity. Post-Holocaust, though, many have found that wealth.

Yeshivas today hold together two ideas side-by-side: the historical belief that some students really are iluys, savants, and are destined for greatness, and the more democratic belief that each student has an obligation to reach their own potential.

My read of the culture is that the drive for equity is subservient to that towards serving giftedness. The reason for equality of opportunity is because everyone has an obligation to explore their own giftedness — the difference between local and global maxima.

(A story that has become popular: Reb Zusha lies on his deathbed, shaking in fear of the conversation he’ll have after he dies. “When I get to Heaven they won’t ask why I wasn’t like Moses; they’ll ask why I wasn’t more like Zusha.” One must imagine himself like Zusha, terrified.)

American schools, as Gómez points out, are largely not like this at all. In fact, they’re sort of the other way around, which makes sense for an institution borne out of plenty, not scarcity. American public schools exist for the least among us. (Initially, out of concern that left unschooled they would rip society to shreds.)

American public schools are the mirror image of yeshivas. The drive to teach gifted students comes out of a drive for equity, the belief that schools should teach everybody.


So, which is a better system? Should giftedness be subservient to equity? Should equity be subservient to giftedness?

The popular answer is that schools can achieve both, that neither concern has to be subservient to the other.

The way that plays out in yeshiva is that there’s a universal obligation to study — and therefore teach — each student to their ability. But no such obligation exists in mainstream culture.

I don’t really know how teachers, in general, think about the needs of the few vs. the needs of the many, the majority of class.

I know, for me personally, I experience this as a tension in my classrooms. I both want to help every student (I really do believe in an obligation to learn) while also making sure that gifted kids get to develop their gifts.

When I say I experience this as a tension, I mean that my efforts in one direction get in the way with my efforts in the other. There is no synthesis, no one way to teach that gives each student what, ideally, they would get.

I think Rochelle Gutierrez describes this well as the “inherent contradictions of teaching mathematics from an equity stance”:

Although teachers must recognize they are teaching more than just mathematics, they also have to reconcile that fact with the idea that, ultimately, they are responsible for helping students learn mathematics. Teachers who are committed to equity cannot concern themselves with their students’ self-esteem and negotiated identities to the exclusion of the mathematics that the students will be held responsible for in later years. Yet preparation for the next level of mathematics must also not be the overriding feature of a teacher’s practice. In answer to which of the two foci are important (teaching students or teaching mathematics), I would answer “neither and both.” It is in embracing the tension…”

That tension I feel as a teacher is the same tension I feel about myself as somebody with gifts. (Trite but true: we all have some.) You have to know that your gifts really are gifts — you really are gifted — also, nobody gives a shit about your gifts. You have an obligation to learn, and everyone has that same obligation. The more time you spend wondering if maybe you really are special, the less likely you are to do anything of value. This is the old growth mindset mantra, and it’s true, but it should only be concerning if you actually do want to do something of value.

So I think there’s maybe no way to solve this cleanly in mainstream US schools. The main thrust of classroom teaching is the need to reach everyone; gifted students are just another everyone. At the same time, there really are gifted students and they really do have different needs. And every inch in one direction takes away an inch in the other. As Labaree puts it, from the perspective of schools and teachers someone has to fail,

The tension is real, but I do think there’s something that would have helped a student like Gómez. Parents, teach your children: there is an obligation to learn.


Doodling the Axioms of Set Theory

Our son is almost three, and he’s starting to really like to draw. He’s also getting to the age where, if he doesn’t have something to do, he’ll tear our apartment to shreds, so most Sunday afternoons we dump a lunchbox of crayons and whatever paper we have on the floor. It’s arts and crafts time, Yosef!

Now, don’t get me wrong, I like watching my kid draw as much as the next parent. (“It’s a fish? No, a dog? Oh, wow, that’s terrific.”) But, what can I say, I get a bit bored.

More to the point, drawing with crayons is so much fun. Arts and crafts time is great. So I draw along with him. And very often I find myself trying to doodle some math.

Lately I’ve been studying a book called Classic Set Theory. It’s been really working for me. It has great exercises, clear organization, oodles of historical context. It’s subtitled “For Guided Independent Study,” and it’s making me wonder why every math textbook isn’t for guided independent study too.

Set theory and logic was my way back into math. I was a philosophy major and had all sorts of worries about whether I could handle a college math class. Then I took a philosophy class about Frege, Russell and Wittgenstein, figures who stand at the creation of mathematical logic and set theory.

In that class, I was finally starting to understand how I had to study this stuff — line by line, ask myself lots of questions, don’t leave anything fuzzy. (Which is more of less how I know to study Talmud.) And I was realizing that if I put in this sort of effort, and if I was honest with myself about what I got and what I didn’t, I could understand some (if I do say so) ridiculously abstract stuff at at least a basic level.

“At some level,” because there was still a lot of stuff that I didn’t get. Since college, set theory and logic has been some of the math that I’ve read the most on. But I still haven’t felt like I really understood set theory, as I hadn’t been able to make much headway in any set theory text.

This is part of why Classic Set Theory is so much fun for me to read. Finally, I feel like I’m starting to get this stuff. Finally!

Here’s the question I found myself asking while drawing with my son today: what could the axioms of set theory* look like if you drew them?

The axioms of Zermelo-Fraenkel set theory with the Axiom of Choice, I mean. I’d love to understand some of the other set theories better. During that class I took in college we studied type theory as Russell’s attempt to patch up the contradiction he discovered in Frege’s system, but I think I only ever had a surface-level understanding of what this meant.

There are nine axioms in all. I know that explanations work better when the objects to be explained are doing stuff, so I tried to make the axioms as active as possible in the drawings. In practice, this means I interpreted the axioms as little machines, rather than as rules or laws. (Axioms have always been presented to me as rules, and until the last couple of days that’s always how I’d always thought of them.)

The toughest one to visually represent was Replacement. The ones I’m most worried about having misrepresented are the Axiom of Infinity and of Foundation. Honestly, all of them are probably flawed in some way. There might be mathematical errors or misinterpretations — as long as you’re nice about it, would you point those out to me?

But I’m not sharing these as resources or to make some point about teaching. I’m sharing these out of celebration, and a bit of relief, that I might be finally figuring out how to learn some math on my own.

Extensionality, Empty Set, and the Axiom of Pairs
Axiom of Separation
Power Set Axiom
Union Axiom
Axiom of Replacement
Axiom of Infinity
Axiom of Foundation
Axiom of Choice