Starting to think about tracking


I think my school does a nice job handling tracking for math classes.

When kids are in K-2, they stay in their classrooms and get taught math by their classroom teachers. But starting in 3rd Grade, kids split up out of their homerooms for math and get taught by a math specialist.

From 3rd through 8th, we teach six sections of math class for each grade. Three of these classes are “regular” pace class; the other three are “accelerated.” At the end of each year each grade’s math teachers and administrators get together with a huge stack of notecards and make classes for the upcoming year. We think about a lot of things — which kids would do well together, whether a class has a nice mix of personalities, and whether a kid would do better in a regular or accelerated section. And, by high school, students get to pick their pace, and kids absolutely do move between the “tracks.”

For the last few years I’ve been teaching an accelerated 4th Grade class, and it’s definitely not an easy class for me to teach (though consensus here is that accelerated tends to be easier teaching). The spread of interest and abilities is still high. (As you’d expect it to be at a school that has ~50% of students in accelerated sections.) There are two things that I find challenging. First, I don’t have nearly as many curricular resources for the accelerated 4th Grade as I do for the regular pace classes. Maybe you think it’s a social problem, maybe you don’t, but I have way more curricular tools for a struggling class than one that’s ready for more.

And the other thing is that I feel a real responsibility in this class for the kids who come in seeing math as their “thing.” There’s a special responsibility to make sure that these kids are challenged and engaged in my class since there’s nowhere else for them to go.

It’s sad but true: there’s more than one way to fall out of love with math.

Overall, I think it’s very good that my school has half the sections accelerated and half not. This gives us flexibility to make classes that we want, and it avoids some of the ways that tracking can make more problems than it solves.


If we’re looking for a good example of bad tracking, look no further than the first school that I taught at.

Here’s what happened each of the three years I spent there. The school would put students in 9th Grade sections based on what they’d heard from the middle schools. The top two sections were “honors,” and they’d study Geometry. The bottom two sections were taking Algebra 1. Once that placement was made, the rest of their high school enrollments in math were more-or-less locked in.

I used to teach the bottom two 9th Grade algebra classes, 9C and 9D (as they were so lovingly called). At the start of the year the classes would be about the same size, maybe 18 and 18 kids. Slowly, though, the 9D kids would figure out where they’d been placed. They’d tell their parents, the parents would call the school, immediately the kids would be moved to the 9C class, which would typically blossom into a lovely group of 26-28 students, leaving the remaining 8-10 stragglers behind.

I really did love the kids in 9D, but WOW that was a hard class to teach. Thanks to this artificial selection process, all the kids in that class were there because either (a) they didn’t care what section they were in or (b) their parents didn’t but mostly (c) they had diagnosed learning needs that weren’t being met, because it was an under-resourced private school.

I sometimes fantasize about going back to that school and teaching that class again. It wouldn’t be fun, but it really nags at me. Could I do better now, if I tried again? I do know so much more about teaching now, but it’s not a class that sets up a teacher for success. If I’m honest with myself…I don’t know if I’d be any better.

The class was very hard to teach. I don’t want to say unteachable, because there were good kids in that room that needed a good teacher. There were also numerous behavior problems, really all the time, just sometimes punctuated with learning. This says something about me in my first few years teaching; it might say something about me now. I’ll never really know.

But it certainly says something about the school.

I taught other low-track classes at this school — 10D, 11E. The Regents exam was far out of their reach, for the most part. We’d have a couple of passes out of each group, but it wasn’t a realistic goal for most students. The environment in the classroom was often out of control, and the school overall had this reputation for barely contained chaos.

I think I did alright there, but this is just the reality. There are hundreds, thousands of schools like it. And while tracking was clearly not the major problem at this school, there was no question who the losers and winners of this arrangement were.


How common are my experiences? How do they fit into the bigger picture?

I’ve been looking into the research on ability-grouping (within a class) and tracking (making classes by ability), trying to make sense of the state of things.

The story of this field is pretty interesting. It’s a field with a million meta-analyses — even a meta-analysis of the existing meta-analyses! All these reviews exist because there hasn’t been much first-order research since the early 1970s or so. So everyone is bootstrapping their analyses on top of the same old studies. If this is making you think that the evidence base isn’t particularly strong here, you’re getting the picture.

While there isn’t an incredibly strong research base here, there is evidence and even a sort of consensus. Tom Loveless does a nice job reporting on this for the Fordham Institute in a report titled Making Sense of the Tracking and Ability Grouping Debate. Loveless, as others do, frames the research around a debate between two researchers, Robert Slavin and James Kulik.

For those seeking a summary, here’s a condensed version of their debate:

Robert Slavin: booo tracking, you have no evidence

James Kulik: yay tracking, actually we do have evidence

Robert Slavin: no, that’s just evidence from gifted and accelerated programs that are poorly controlled, they no count

James Kulik: no they count

Robert Slavin: yo also I find tracking morally repulsive

James Kulik: really? i like it

That’s sort of it. If you like words, here’s Loveless’ summary:

Kulik finds that tailoring course content to ability level yields a consistently positive effect on the achievement of high ability students. Academic enrichment programs produce significant gains. Accelerated programs, where students are taught the curriculum of later grades, produce the largest gains of all. Accelerated gifted students dramatically outperform similar students in non-accelerated classes. Slavin omits studies of these programs from his analysis. He argues that the gains, though large, may be an artifact of the programs’ selection procedures, that schools admit the best students into these programs and reject the rest, thereby biasing the results.

Loveless is correct to point out that this debate is intractable, though, because Slavin actually finds tracking morally problematic and ugly. The burden of proof for Slavin is on schools that want to track, which explains why he can be so opposed to Kulik, even though they don’t seem to disagree very much at all about what the research shows:

Three things are striking about the Slavin-Kulik debate. First, the disagreement hinges on whether tracking is neutral or beneficial. Neither researcher claims to have evidence that tracking harms achievement, neither of students generally nor of students in any single track. Second, accepting Slavin or Kulik’s position on between-class grouping depends on whether one accepts as legitimate the studies of academically enriched and accelerated programs. Including these studies leads Kulik to the conclusion that tracking promotes achievement. Omitting them leads Slavin to the conclusion that tracking is a non-factor. Third, in terms of policy, Slavin and Kulik are more sharply opposed on the tracking issue than their other points of agreement would imply. Slavin states that he is philosophically opposed to tracking, regarding it as inegalitarian and anti-democratic. Unless schools can demonstrate that tracking helps someone, Slavin reasons, they should quit using it. Kulik’s position is that since tracking benefits high achieving students and harms no one, its abolition would be a mistake.

Loveless seems to be taking a compromise position in all this. “The research on tracking and ability grouping is frequently summarized in one word: inconclusive,” he writes. Since the research is inconclusive, he recommends a live-and-let-live strategy. Schools should have the freedom to choose their tracking structure, he says, but they need to be aware of the ways that each model can fail.

Tracking’s issues are well-known these days. Loveless calls for high-standards for the lowest tracks, and for ending what’s sometimes called the “teacher tracking” of putting the least skilled teachers with the lowest tracks. There need to be clear pathways out of the lowest tracks, a real effort to make sure that there’s room for students that start in one place to end up in another.

Untracked schools have problems of their own, though. “On the political side, anti-tracking advocates need to assuage the fears of parents that detracked schools will sacrifice rigorous academic training and intellectual development for a dubious
social agenda,” he writes, and this seems sensible to me also. The really ambitious students in my accelerated 4th Grade class do have needs, and their parents are legitimately concerned about meeting them.

If this seems wishy-washy and balanced, well, sometimes things just shake out that way.


If you think about it, isn’t it sort of weird that tracking doesn’t have clear and measurable benefits for the top groups in the research? Think about it. How often should ability grouping help strong students? Like, roughly, what percentage of the time should the top-group academically benefit from tracking?

I’d say 100% of the time. Roughly.

Teaching kids more stuff because they’re capable of learning more stuff is the single simplest idea in education I’ve ever heard. There is nothing to it. It’s just teaching more. Add to that the way they’re isolated from some of the toughest-to-teach students in the school, and this seems like it ought to be the clearest slam dunk on the educational menu.

So why should there be anything other than the clearest possible data signalling this? I’m not asking, why hasn’t the educational establishment recognized the evidence? I’m asking why the data isn’t super-clear. Why isn’t there a huge effect? Why isn’t it unambiguous in every single case that top-students benefit from ability grouping?

Two explanations:

  • Ability grouping is not necessarily acceleration. Some teachers don’t use a “top” class as a chance to do anything differently at all. You meet the standards? Great, you’ve met the standards. Let’s chill. (Or, let’s look at cool things on the side that don’t accumulate as knowledge.)
  • The skills students learn in a top group maybe aren’t measured.

Thinking about my own teaching, I think both of these things are probably happening in a lot of “top” track classrooms. I certainly do try to cover topics in my accelerated class that I don’t in my regular paced classes. But sometimes it’s just that we cover the same material but without as much stress, because there isn’t a clear vision I have of what an accelerated class ought to look like. I get little help from the available curricular resources, which are really all about fleshing out support for struggling students over kids who are ready for more.

I’m not complaining about this, mind you, but I think it’s true. There are probably a lot of teachers out there who aren’t making significant curricular changes between their tracks.

The second thing is true also. I try very hard to avoid racing ahead in the “standard” sequence of arithmetic skills with my accelerated class. The easiest way for me to handle an accelerated class would be to just march through the curriculum, teaching 5th, 6th, 7th grade standards to my 4th Graders. But this could create problems for the kids and my colleagues. If I unilaterally decide to teach e.g. fraction division, then I’m stepping on the toes of the 5th or 6th Grade teacher, who now has a handful of bored kids who are skilled at this because I decided to keep marching. 

OK, so the department should make a decision. But once you just set a class off accelerating through the curriculum, you’ve suddenly created a track that is relatively impenetrable to kids who start out of it. Somehow, they’ll have to catch up to join, and that’s going to have to happen outside of class. The only way to get ready for an accelerated class would be to be accelerated already, an unsavory Catch-22.

I think what I try to do is to flesh out the standard, grade-level topics with things that don’t have a strong showing anywhere in the standards. Right now my 4th Grade class is taking a deep dive into probability, a topic that only sort of shows up again in the 7th Grade Common Core standards. Earlier in the year I shared a bit of graph theory. We studied angles at a depth that will only really show up again when they’re studying diagrams in high school geometry.

My dream would be to have a curriculum that had a clear vision for what kids who are ready for more could dive into, beyond the grade-level expectations. There would be to keys to making this work. First, the additional skills would have to actually build and develop throughout the year — we want to equip accelerated students with something useful that builds their mathematical knowledge. But we also want a fresh start each year or so, so that kids can move in between the tracks without requiring some sort of catch-up.

I think something like this would give clarity and purpose to classes that otherwise have no choice but to plow ahead in the standard sequence.


On the margins, should US schools have more or less tracking? I think the answer is probably “better tracking.”

Even Slavin, opponent of tracking, admits that there is evidence for certain kinds of tracking in the elementary years, especially something called the “Joplin plan.” (Named after Joplin, Missouri, the district that gets credit for its invention.)

Joplin-style tracking cuts across grade-levels. A school might have an hour for reading instruction, and each student in the school would go to a classroom they’ve been assessed as ready for. So a 4th, 5th and 6th Grader might be reading similar books together in the same room, working on the same vocabulary. It’s a kind of limited breakdown of the age-grading system, really an artifact of the early 1960s.

Slavin, opponent of tracking, calls the experimental results of studies of the Joplin plan “remarkably consistent” and in support of the program.

Which makes sense, right? This is the simplest possible educational idea: teach kids more when they seem to be ready for more. And, as an extra bonus, since the kids are heterogeneously grouped for most of the day, you don’t run the risk of creating really problematic tracks that lead to wildly varying places. By the nature of the plan, there is curricular guidance for kids who are ready for more. This should work 100% of the time.

And Slavin put his money where his mouth is, co-founding Success for All, a school improvement program that has something very much like the Joplin plan as its cornerstone. (His co-founder is Nancy Madden, another Johns Hopkins education professor. Madden and Slavin are married.)

And, ironically, Slavin’s program has been critiqued for its use of tracking. (Also for its use of scripted lessons, which will never make teachers happy.)

It seems as if the situation is that Slavin’s preferred sort of tracking would be good for students and good for equity and mobility. He’s a noted critic of tracking and ability grouping, and is deeply aware of all the traps. Success for All reassesses students every two months, and students are expected to move between groups. This is the form of tracking with the strongest research pedigree. And yet it comes in for criticism.

What’s confusing about this to me is that we aren’t a country that is shy about grouping students by ability. Loveless notes this: “Ability grouping for reading instruction appears nearly universal, especially in the early grades.” In the elementary years, this is usually within-class grouping, e.g. red group sits at this table and blue is on the rug, etc. But by the time students reach high school, the near universal pattern is separately tracked classes, more like what my old school did.

Here’s the puzzle: why do some forms of grouping and tracking attract more ire than others? Is it just a matter of the devil we know vs the one we don’t? Familiarity breeds begrudging acceptance? I don’t know.

But looking at programs like Success for All and thinking about what happens in the math classrooms that I’ve seen, it seems to me that purely from the standpoint of mathematical learning, there is probably a better way of doing things. Here, as bullet points, are my takeaways from all this, with the most doable items near the top:

  • We could use curricular materials that go beyond the standards for each grade level of math, so that classes who are ready for more can dig in without dashing through the standards.
  • Something like Slavin’s reading plan could be useful in elementary grades. Keep the heterogeneous groups, in general, but assess students every couple of months or so to place them in a class that’s right for them. (If we could clone Success for All minus the scripted lessons it would probably be more popular. Though I’m sure there’s probably something lost when we do that too.)
  • Maybe, even in the upper grades, it would be helpful to split the year into two halves, with an ability grouping move in the middle. Or maybe it wouldn’t help at all. But it might be interesting for a school to try something like that and see how it goes. Maybe?

But all these are speculative recommendations. Overall, I don’t get the sense that there is a huge gap between research and practice because (as Loveless notes) there isn’t a great deal of clarity from the research.

Instead, there are promising ideas with research support (like Slavin’s). This doesn’t exclude the possibility that there are other good ideas out there, and it seems likely to me that if a school or parent body thinks that tracking or untracking is necessary for their students, they’re probably correct.


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 [] 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 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.