Excerpting Freddie de Boer’s defense of the SAT

Just read the whole thing here, but I’m disposed to agree with much of it. (Though check the bottom of the post for a research piece that makes me doubt my instincts here.)

The SAT does not enjoy a good reputation among progressives. Arguments against the use of the test, as well as its analog, the ACT, abound. Both are widely derided as tools of elitism, rejected as culturally biased, and denounced for dehumanizing test takers.

I understand the intuitive feeling that we should not reduce human potential to a test score. And the major testing companies (and nonprofit organizations like the Educational Testing Service, which basically function like companies) are not particularly sympathetic entities. But if you believe in equality and a more level playing field in college admissions, you should defend the SAT.

Coaching doesn’t work so well:

Critics of standardized tests often complain that affluent students have greater access to test prep materials and coaching. This is indeed a concern, but the research here is clear: coaching services produce far smaller gains than those advertised by the big test prep companies, which routinely claim triple-digit improvements.

2006 meta-analysis found that students retaking the SAT after coaching resulted in, on average, an increase of about 50 points on a 1600 scale. That’s not an insignificant number. However, as the researchers point out, we can expect some of that gain to occur simply through increased familiarity with the test and, for lower-scoring students — the type most likely to retake the test — regression to the mean. More recent research found that, after using statistical controls to compare similar students, the combined effect of coaching on a 1600 point scale was about 20 points.

Other metrics for judging admissions are easier for the well-off to game:

Detractors of entrance exams often argue for more “holistic” methods of evaluating students than tests, pushing for greater emphasis on student activities, college essays, and letters of recommendation. They argue that these things allow them to select students that are more than just grades and test scores and build a diverse student body. As Jennifer Finney Boylan put it in a piece decrying the SAT, the only way to fairly choose between applicants is “to look at the complex portrait of their lives.”

But this reasoning goes directly against the stated goal of equality. It should be obvious: affluent parents have far greater ability to provide opportunities for extracurricular (and frequently out-of-school) activities than less affluent parents do.

According to this Malcolm Gladwell piece (caveat lector) extra-curriculars were brought in to college precisely to exclude groups (like Jews) who were academically high-performing but not the ideal candidates for admission:

The difficult part, however, was coming up with a way of keeping Jews out, because as a group they were academically superior to everyone else. Lowell’s first idea—a quota limiting Jews to fifteen per cent of the student body—was roundly criticized. Lowell tried restricting the number of scholarships given to Jewish students, and made an effort to bring in students from public schools in the West, where there were fewer Jews. Neither strategy worked. Finally, Lowell—and his counterparts at Yale and Princeton—realized that if a definition of merit based on academic prowess was leading to the wrong kind of student, the solution was to change the definition of merit. Karabel argues that it was at this moment that the history and nature of the Ivy League took a significant turn.

The admissions office at Harvard became much more interested in the details of an applicant’s personal life. Lowell told his admissions officers to elicit information about the “character” of candidates from “persons who know the applicants well,” and so the letter of reference became mandatory. Harvard started asking applicants to provide a photograph.

And to provide evidence that grades can be gamed, Freddie cites this:

Research by Michael Hurwitz and Jason Lee found that, from 1998 to 2016, the average high school GPA rose from 3.27 to 3.38. That may not sound like much, but distributed over millions of students, it’s a large increase. What’s more, the phenomenon is concentrated at the top.

That said, maybe Freddie is wrong. I was talking to researchers on Twitter who found that grades are superior to the SAT for predicting success in college, precisely because they measure more than the SAT, like the ability to ask for help, etc.

So, I don’t know if Freddie is right on this. I need to think more and try to put the pieces together.

What can “I’m not good at this” mean?

Here’s a quick note from the field.

Class today was largely about some fraction arithmetic, and one student was having trouble with it. In class, a couple times, this student said “I’m just not good at this” or “This doesn’t make sense” and “I’m bad at calculating stuff.”

But, all through class, the student stuck with it — I mean that they tried problems on their own and asked questions. And there was definite progress. The student was becoming able to handle this type of question on their own, and starting to make sense of things.

At the end of class I told this student that I thought they were getting the hang of this. I saw a sort of relief pass over this student. Then the student told me:

  • They had a hard time with this topic
  • But this student sort of blames previous teachers
  • This student recently signed up for extra math that is more like our class’
  • The math of our class is the “only math they’re good at”

So what did this student mean when they said “I’m not good at this” etc. in class? In retrospect, it was a statement of fear and anxiety precisely because it was the exception to the rule, at least the rule in our class. This student feels generally competent in our class, so much so that they’ve signed up for similar, optional math next year, and was feeling nervous that this good thing was in danger.

This doesn’t contradict the standard story that people tell about kids saying that they’re “bad at math.” (Especially because this student’s statements weren’t that they were bad at math, in general. It was more specifically about arithmetic and algebra.) But I do think that it illustrates one way that these statements can be an attempt to express something subtler than what they’re usually credited for.

Three things seem important to me about this case:

  1. The student clearly didn’t think that “being bad at arithmetic” was a fixed quality, as this student was simultaneously asking for help. This goes against the “fixed mindset” interpretation, I think.
  2. The student clearly doesn’t, in general, think that they’re bad at all math. These expressions of frustration can be about anxiety that a student’s good thing is under threat. So while it’s never a good thing to hear a kid say that they’re bad at something, it can be a local, specific issue rather than a global one about their status in math class.
  3. The student didn’t think that it was socially OK to be bad at arithmetic or bad at math. This wasn’t a student being proud of being bad at math, and it wasn’t a sign of anything but their own frustration.

I’d be curious to hear other people’s stories about students who make “I’m bad at…” statement in class, to learn more about the different contexts in which kids say this. I’m sure, at the end, we’ll find that they always come at moments of frustration. But I suspect that if a kid is saying that they’re bad at math in class, there’s more to that story, and maybe it’s a sign that (paradoxically) some fundamental things are working for that student in class.

Growth Mindset Roundup

From Marginal Revolution, “Growth Mindset Replicates!”:

In other words, a small, positive effect. But this small effect is coming from a small intervention, two online survey/interventions of 25 minutes each that could be easily scaled to the entire country or even worldwide. We have come a long way from the “mindset revolution” but who am I to discount a marginal revolution? Moreover, the average effect hides heterogeneity, the effect was bigger on the students who needed it most.

Some past opponents of mindset see this as the death of mindset. David Didau has this sort of take in “The nail in Growth Mindset’s coffin?”  He says the true parts of mindset are obvious, the false parts the result of magical thinking:

What this might suggest is that students who have previously underachieved improve when told that if they took more responsibility and worked harder they might do better, and that good behaviour makes a positive difference to any intervention. Neither of which are all that surprising.

This seems to me somewhat unfair. It’s good to have research to know the degree and extent to which obvious things can be shown to have an effect. And, as Didau accurately reported, there had been a number of failed replications in the past. This is decidedly not a failed replication.

There are two new papers out on mindset to keep track of. The first is a huge, carefully done experiment. The other is a huge, carefully down meta-analysis of previous studies.

I haven’t read either carefully yet, but I’ve found it interesting to follow researchers discussing the papers. My impression is that methodologically they hold up to scrutiny. Here are some sample tweets:

Re the meta-analysis:

I don’t entirely stand by this tweet any longer — things don’t feel so confusing now:

What are the educational implications of all this? I think that the claims about the power of mindset interventions to really have a huge impact on learning now have clearly been contradicted by our best research.

It’s hard to know how to talk about this. Smart teachers and educators always knew that these interventions couldn’t have a huge impact on kids, especially if the rest of the classroom pieces weren’t there. That said, not everyone is smart about this, and there was a time early in my teaching life when I believed the over-simplified story about mindset that I heard.

Jo Boaler and YouCubed are pretty clearly going farther than what evidence dictates. I wish they wouldn’t, and am powerless to stop them, and it makes me sad that they don’t seem to care. Defenses of them seem to come down to “well it’s not true but it’s a net-good message so spread it far and wide!” This is something that goes against all my instincts. I don’t do well with this sort of utility calculation.

Anyway, for an picture of exactly the sort of thing that the evidence does not support now (if it ever did) you can check out this video from Boaler and YouCubed, about how believing in yourself has been scientifically proven to change how your brain works and improve your achievement. Blech:

Likewise, I don’t see support for the sort of mindset interventions that are built into the first week of the New Visions math curriculum. There’s some good math in there, and maybe it’s good to talk about growth mindset during that math, I don’t know. It depends on how time-intensive the mindset stuff is, I think.

Where are we headed? Growth mindset is just going to be another high-level name for describing good teaching. It matters, but as a goal, or a value that connects a lot of disparate elements of teaching practice.

It feels like the mindset story is coming to a conclusion with these big, careful studies.

A Jew thinks out loud about Farrakhan

I don’t know a ton about Nation of Islam. A few years ago I read Manning Marable’s biography of Malcolm X, and I left that book with an enormous admiration for Malcolm X, feeling like we’d lost a really great American. And if only for that, I’m prejudiced against Louis Farrakhan.

The other piece of context I have for Farrakhan is Freddie de Boer’s excellent essay in Harper’s, titled “The Charmer.” Here are a few of the choice bits of that essay that pertain to his anti-Semitism:

To speak of what Farrakhan gets right about racism is to risk lending credence to all he gets wrong. For just as he has been correct in his indictment of white supremacy, so too have his critics been correct in their indictments of him. Farrakhan’s ample critical gifts have always come packaged with abundant bigotries: hatred of Jews, mockery and fear of homosexuals, denigration of women. I have heard occasional attempts to deny Farrakhan’s anti-Semitism, but these defenses approach lunacy. Farrakhan’s distrust of and anger toward Jews are as central and constant as any other aspect of his philosophy; anyone reading over his speeches for examples of anti-Semitic rhetoric will quickly find herself spoiled for choice. Jews control the banks, they control the media, they control the government, Israel knew 9/11 was coming.

Why then does Farrakhan fear Jewish reprisals? For no good reason, it seems, other than his palpable anti-Jewish paranoia, along with the anti-Semite’s tendency to see one of the world’s most oppressed peoples as the shadowy driver of all events.

So let’s take for granted that Farrakhan’s anti-Semitism is not subtle or interesting in and of itself. It’s not a phenomenon that demands nuance. It just is: he’s an anti-Semite.

OK, but Tamika Mallory, national co-chair of the Women’s March, was in attendance at Farrakhan’s most recent speech. And, apparently, it’s not an isolated thing, as she’s posted about Farrakhan before on social media:

Here is her defense, it seems:

If I understand, she’s saying that whatever Farrakhan’s faults, he is an important leader. She doesn’t share his anti-Semitism (or, presumably, his deep homophobia and anti-feminism!) but she won’t refuse to associate with him or to praise him.

OK, so what’s the generalizable principle here? I think it’s: Don’t refuse to associate with or listen to or offer praise to someone who plays an important, positive, communal role, even if they have deep, disgusting faults. 

Now, is her invoking this principle hypocritical? I don’t know much about Mallory, so I can’t say. It’s certainly true that e.g. Aziz Ansari didn’t get this sort of benefit of the doubt from the left, but then again that might be entirely consistent with the principle above. Ansari is, at the end of the day, an artist. Farrakhan’s faults — his paranoia and hate — may run deeper than whatever it was that Ansari was guilty of, but then again maybe it gets overridden by the Million Man March? I don’t know.

And, besides, I actually agree with that principle to a point, and wish it was more widely followed.

Is it anti-Semitism for that principle to be selectively invoked for Farrakhan but not for others? I’m not sure. Remember, Farrakhan isn’t just a hater of Jews. He’s homophobic, among other things. So the claim that Mallory and other leftists wouldn’t defend Farrakhan if he had issued hate speech about some other group…Farrakhan hates a lot of people. I’m not sure this argument really flies.

(If it does fly, it’s because of the intensity and persistence of his anti-Semitic speech, even compared to his homophobia or anti-feminism.)

So, where does that leave us? What’s at stake?

I think 95% of the emotion at play here is about whether Jews get to be counted as the oppressed or oppressors in the US today. Of course the answer could be both, and intersectionality would call for that, but in practice intersectionality is a cherry-on-top nuance over a baseline judgement: oppressed or oppressor?

Lots of Jews generally want to be seen as oppressed because of, you know, all of Jewish history.

Leftists want Jews to be seen as oppressors because they’re mostly white, and especially because of anti-Zionism.

Mallory could easily apologize for Farrakhan’s anti-Semitism, but doesn’t want to, because Jews (as oppressors) have it coming for them, just as white people do in Farrakhan’s rhetoric.

Jewish critics want to hold Mallory accountable to leftists and progressives to force a reckoning, and to extract a concession that Jews are victims of oppression.


And where do I stand? I think this is some of the danger of thinking too much about identity and status. Is there anti-Semitism in the US that we want to fight against? Is there an actual plan on how to protect Jews from hate crimes?

I want to avoid the sort of metaphysical issues that talk of anti-Semitism usually devolves into. I’m interested in protecting Jews from the sort of things I’m afraid of: shul shootings, terrorism, street harassment. I’m not interested in fighting anti-Semites, I’m interested in defeating anti-Semitism, and it’s not clear to me that the two are the same project.

So Farakkhan is a loony, but I don’t see much good coming from trying to take down Mallory for her association with him. The Women’s March isn’t about to add an anti-anti-Semitism plank to their work anyway.

Why are companies trying to look liberal?

It would be one thing if companies like Delta, Dick’s, Hertz were uniformly liberal institutions. Then it would make sense why they were publicly and loudly signaling their break with the NRA. And it would be another thing if the country were uniformly liberal — then it would make clear economic sense for these companies to signal their liberalism.

What’s weird is why these companies are performing liberalism in a country with a Republican congress, a Republican president, Republican domination of state and local government, etc. Why are big companies (minus Hobby Lobby) taking a stand against the NRA?

Here are all the ideas that I am aware of, either from reading or making things up:

  1. If a company or a brand has a national reach, they want to please their national customer base. Gun control is a nationally popular issue — it’s only the geographic clustering of NRA-types that keeps us from having gun control.
  2. Maybe liberals — because they’re younger — have greater buying power, so companies are more interested in courting them on divisive issues.
  3. Increasingly, liberals are well-educated and those with less education prefer Republicans. So increasingly the decision-making employees of these companies are liberal and therefore seek to make liberal stands.
  4. Douthat’s take: companies want to signal social liberalism to protect themselves from government interference.

I recently saw Josh Barro’s response to Douthat, and Barro seems to prefer a mix of 1 and 2 and 3. I’d love to read more about this, if you have recommendations.

P.S. Is this also how we get awesome movies like Black Panther?


Thinking about Francis Su’s “Mathematics for Human Flourishing”

Francis Su:

What I hope to convince you of today is that the practice of mathematics cultivates virtues that help people flourish.  These virtues serve you well no matter what profession you choose.  And the movement towards virtue happens through basic human desires.

I want to talk about five desires we all have.  The first of these is play.

Yeah, so the first of these fundamental human desires is play, and the whole list looks like this:

  1. Play
  2. Beauty
  3. Truth
  4. Justice
  5. Love

The way Su sees things, if you use math to chase these desires then you end up cultivating virtues. Here are some of those virtues:

  • Hopefulness
  • Perseverance
  • Joy
  • Transcendence
  • Rigorous Thinking
  • Humility
  • Circumspection

(Justice and Love are desires that don’t end in virtues for Su, it seems, as he doesn’t name any virtues in those sections of his talk. Instead, pursuing love is supposed to enable your other pursuits because it results in seeking meaningful human connections: “Love is the greatest human desire.  And to love and be loved is a supreme mark of human flourishing.  For it serves the other desires—play, truth, beauty, and justice—and it is served by them.”)

(So, then what’s the role of Justice? Maybe for Su it’s sort of like a necessary consequence of valuing all these other things. To truly desire all these other things is to desire them for everyone, and that necessarily is the pursuit of justice? I’m just making stuff up here, I’m not sure what Su thinks.)

Anyway, basically none of Su’s talk resonates with me. I don’t mean that I think he’s wrong.* I mean it doesn’t resonate — it doesn’t hit my frequency, make me hum.

* Well, I guess I don’t have any confidence that the study of math itself can impart any of these virtues. I don’t know if he’s claiming that they will, though I think he is.

OK, fine, but then what do I think?

One of my favorite games is “long response to short thing OR short response to long thing.” Let’s play “short response to long thing”:

Michael Pershan’s Version of Math for Human Flourishing:

Five Fundamental Desires:

  1. Understanding
  2. Belonging
  3. Growing
  4. Teaching
  5. I’m not sure but I came up with four

Virtues that one MIGHT develop in math, but I make no commitments about how frequently or reliably even an idealized version of math education could promote these in our students, since virtues are BIG things and math is just ONE thing:

  • Humility
  • uhhh this is also hard
  • I’m totally stuck

Maybe not “virtues,” but ways in which I think math makes my life richer:

  • Math helps me know that, to really understand something or someone, I need to give it my full attention.
  • It gives me an arena in which to grow.
  • It gives me questions to share with others.
  • Is there a name for that thing where you’re walking down the street and you can’t not see parallel lines or tessellations or similar triangles? The math curse? I love that.
  • I love theory. I love that thing where you take a perfectly conventional idea and flesh it out, completely. I recently read that topology came, in part, in response to Cantor’s proof that the line and the plane were equinumerous. But clearly there is a difference between the two — how can we capture that? “How can we capture that?” is one of my favorite questions to ask.
  • I love that math gives me things to help other people think about.

Responding to Criticism from @blaw0013

I wasn’t sure whether to respond to this or not. I want to be the sort of person that gives people stuff to think about, and (just like in the classroom) there’s a point where you have to step back and give people a chance to speak.

But: “deny joy of and access to maths for many”? It’s an interesting criticism, one that I have a lot of thoughts about.

I don’t see micro-skills as denying joy and access to students. And I think it’s partly about seeing joy in maths as something that happens in the abstract versus something that happens in the context of school.

If you think “abstractly” about what joy in math involves, your mind would probably start thinking about the sort of math that is joyous and exciting, the very coolest stuff that math has to offer. You would think of noticing surprising patterns, of unusual theorems, the endorphin release of cracking a puzzle.

Francis Su is the current leading expositor of this side of math, the beautiful, joyous, elegant side:

Pursuing mathematics in this way cultivates the virtues of transcendence and joy.  By joy, I refer to the wonder or awe or delight in the beauty of the created order.  By transcendence, I mean the ability to embrace mystery of it all.  There’s a transcendent joy in experiencing the beauty of mathematics.

If you think abstractly, and ignore the context that students of math actually encounter math in, then you’d look at something like “micro-skills” as just the opposite of all this. And yet I think if you look at the reality of students’ lives (instead of a radical proposal for what students’ lives should be) then I think you can see where joy comes into the picture.

Yesterday I gave students a no-grades quiz in algebra. A student who, I had been told at the start of the year, frequently struggles in math, has been having a lot of success lately. She knew exactly how to handle both of the systems of equations that were on this short quiz, but she got stumped at one of the resulting equations:

-1.7x = 4.3x + 3.6

I didn’t know what to say when she got stuck, exactly, but I was fairly confident that this was an example of a micro-skill that she was missing.

She and I agreed that she’d like me to write a little example on the side of her page, so I wrote this:

-2x = 5x + 7

[I drew some arrows going down from each side labeled “+2x.”]

0 = 7x + 7

My student read the example and then exclaimed (in a way I can only describe as “joyous”), Oh wait, you can make 0 there?!

You can! It’s very cool, and to the mind of a child learning algebra it’s surprising, elegant, beautiful, joyous. This is what I’m talking about — not treating the moments when kids get stuck as “forgettings” or “bugs” in some universal algorithm, and instead thinking of them as opportunities for students to prove mini-theorems, try mini-strategies, learn mini-skills.

And to treat these as moments lacking joy is also to ignore the major impediment to joy in a classroom: feelings of incompetence, worries about status, anxieties about math.

I’m no psychic and my students’ story isn’t mine to tell, but she showed all appearances of being happy and relieved when she understood how to go about solving this problem. How could she experience this, given that she was dealing with the drudgery of a micro-skill? Well, part of it is that (it’s easy to forget) things that are drudgery to teachers are often rich, problematic (in a good way) terrain for students.

But part of it is that these are children in school, surrounded by other children in school. Joy can’t be separated from that social context. Students can’t experience joy if they don’t feel competent, and conversely there is joy in competence. I see this every day.

If, like me, you care both about helping kids experience joy in math and joy from competence in math (hard to separate) then you need to find opportunities in your teaching to do both. The above is how I’m currently thinking, and I’d be interested to read Brian’s take on all this — maybe he and I can find a way to write up a case that illustrates the different choices we’d make in a situation like this. I love the idea of collaborations to resolve differences.

Where did working memory come from?

This is a quick note, before I forget the last couple things that I read.

Where did working memory come from? Here’s my picture so far:

  1. There’s a limit on how much random stuff you can remember at once. You don’t need science to know this; you just need random stuff that you have to remember. I assume this has been known forever.
  2.  People had different names for this. William James called it primary vs. secondary memory. Others were calling it long-term vs. short-term store. What was controversial was whether these constituted just two facets of the same memory system, or whether they were two totally independent memory systems.
  3. Evidence for independence comes largely from patients with brain damage. These patients either are amnesic, but do perfectly fine with short-term memories, or else they have greatly impaired short-term memories but otherwise function and learn OK. This suggests independence.
  4. Question: does short-term memory constitute a working memory for functioning and long-term memory? In other words, is short-term memory necessary for learning, reasoning and comprehension?
  5. Baddeley and Hitch do a battery of experiments to show that impairing short-term memory with verbal info does (modestly) impair learning, reasoning and comprehension. This is evidence that short-term memory does constitute a working memory system.
  6. But the thing is that performance was only modestly impaired by their experiments, so there must be more to the cognitive system than what their experiments uncovered. (Their experiments almost entirely used span tasks that ask people to remember a bunch of stuff, random numbers, letters, etc.)
  7. They then go beyond their experiments to make a guess about what the structure of working memory might be. They propose that there is a capacity to just passively take in verbal info, up to a point. Beyond that point, a “central executive” has to take active steps to hold on to info. Thus working memory limitations come both because the passive store has a limited capacity and also because the central executive can only do so much at once. The span experiments fill up the passive store, and force the active executive to do something. This allows task performance to go OK, but at a cost in performance. They also guess that there is a spacio-visual store that is entirely parallel to the phonological store.
  8. This takes us up to, like, 1974 or something.


Here’s what I think I do differently

I just read a really interesting post called ‘Applying Variation Theory.’ It’s by a teacher from the UK who I don’t yet know, Naveen Rizvi. The core mathematics is familiar territory to teachers of algebra. To factor an expression like x^2 + 10x + 16, you can ask yourself “what pair of numbers sum to 10 and multiply to 16?”

Answer: 8 and 2.

Therefore: x^2 + 10x + 16 = (x + 8)(x + 2)

I actually first learned about this from Dan Meyer, and for my first few years teaching quadratic factoring I used his “Diamond Problems” as a factoring lead-in:

Screenshot 2018-02-25 at 7.48.45 PM.png

In her post, Naveen talks about intentionally designing problems of this sort to draw out this underlying structure — that the factors of “c” have to sum to “b.” Factoring then becomes a quest to search the factors of “c” for things that sum to “b” (or vice versa).

Her main points is a good one, which is that if you keep everything else the same, and then vary just one thing, that thing will draw a lot of attention. Here is how she uses these ideas to design a practice set of quadratics, one that isn’t so unlike Dan’s:


My experience of teaching this topic is that, even knowing the relationship between c and b, these problems can be very difficult for students.

Why? What makes these factoring problems hard? Part of the reason has to do with fact automaticity, to be sure. While there are many topics in algebra that a student can handle without automaticity, this is definitely one of the times when things get much hairier for kids if they don’t know e.g. quick ways to find all the ways of multiplying to 60.

But take a closer look at some of these problems, and you can see that there is more going on than just knowing the relationship between b and c and knowing your facts. Consider one of these problems from Naveen’s page, which I just chose randomly:

a^2 + 29a + 28

This is a problem that I can imagine my students having some trouble with, at first. Not because the facts are difficult or because they don’t know the relationship…it’s just that the solution (a + 1)(a + 28) might not occur to kids. One thing I’ve noticed is that a lot of kids don’t think of 1 x N for a while when they’re searching for ways to make N. This makes sense — it’s so computationally straightforward, they don’t spend a lot of time thinking about multiplying by 1. It can slip under the radar.

Now, might a kid working on Naveen’s problem set become familiar with this tiny nugget or structure, that if the “b” term is one off from the “c” term, you should try sticking “1” into one of your binomials?

A student might make this generalization from the examples in the practice set, but this would essentially amount to learning by discovery. Which absolutely happens sometimes, but especially since this activity isn’t structured around giving kids instances that would cue-up that generalization…it probably won’t happen for most kids.

Now, the question is whether this sort of “micro-strategy” is a good use of classroom time. Maybe it’s too narrow a class of problems to be worth making it an instructional focus, I don’t know. Maybe you just go for the main strategy, and hope that kids are able to apply what they know to this little side-case.

But then again, maybe you give a quiz and kids end up mostly baffled by this problem. Teaching is full of surprises — this could happen.

That’s when I say, OK, let’s design a quick activity that would focus entirely on this micro-skill. Maybe a mini-worked example, or maybe a string of mental math factoring problems ala Naveen or Dan’s that puts that entire “variation theory” focus on this one, specific corner of the mathematical landscape — just the one that the students in your class need.

a^2 + 8a + 7

x^2 + 14x + 13

j^2 + 176j + 175

q^2 - 14q + 13

s^2 - 11s - 12

And then I call this “feedback” and don’t spend a lot of time writing up stuff in the margins of their quizzes.

There’s something nice about these little micro-skills. For one, it’s an alternate way of thinking about what’s leftover after you’ve taught the “main” skill. (Meaning, it’s not just that kids are forgetting what you’ve taught and need to be reminded — it’s that there are little corners of the mathematical world that haven’t yet been uncovered for kids.)

One thing I’ve been struggling with has been trying to figure out what exactly my pedagogy involves that’s distinctive. It’s not about the activities I tend to choose or design — since I’m pretty boring in that regard. I’m not an amazing motivator of people. Kids like me, I think, but not in the “oh my god he was my best teacher” way. More like “he’s nice.” (“Being nice” is an important part of my pedagogy.)

Though I still don’t have a snappy way to put it, I think that this is part of my story:

  • I’m really curious about how kids think
  • So I try to use that to come up with a more systematic understanding of how they think about different types of problems, especially when it’s something that people typically think of as constituting a single “type” of problems (e.g. factoring quadratics, solving equations, adding fractions)
  • While teaching a topic I try to figure out which types of problems the kids understand how to handle, and which they don’t yet
  • Then I focus in on a micro-skill for handling one of those little types of problem and I teach it with a short little activity followed-up by practice, in place of feedback

I don’t think any of this is exciting or inspiring, and I don’t really think I can make it so. There really is something here, though.

I think the exciting action comes in the second of those bullet points, in describing the mathematical landscape in a way that’s pedagogically useful. One of my favorite things in math education is Carpenter et al’s Cognitively Guided Instruction. I’ve moved away from some of the pedagogy it has grown into, but the problem breakdown is my paragon of pedagogically useful knowledge. It’s what I always come back to.

I sometimes wish there was more to what I do, and I also wish I didn’t have such a hard time figuring out how to describe what it is that I’m into. I’ll be doing a thing with teachers this spring that will give me another shot at refining my little message. I sometimes get jealous when I see all the cool things that people make and share online. I’ve never been cool, and none of what I’ve written here is cool either, but it’s what I’ve got.

(Last year’s version of this post: here.)