## Review: The Weil Conjectures by Karen Olsson

The Weil Conjectures by Karen Olsson

I.

I liked that this was a book full of questions, and I also liked the questions. Here are some of the questions she asks in the book:

• Why do people like math?
• Why did I (the author Karen Olsson) like math when I studied it in college? Even though I was an aspiring novelist?
• Why was Simone Weil — philosopher, writer, mystic — attracted to mathematics?
• What’s the deal with Simone’s relationship with her brother (famed mathematician) Andre?
• Where do mathematical ideas come from?
• What do (the author Karen Olsson) get out of abstract math now that I’m no longer swimming in it?
• Do analogies for abstract mathematical ideas do a person any good if the math itself isn’t accessible to them?

It wouldn’t be fair to Olsson or the book to reduce it to a neat set of answers to those questions. The book is structured so as to provide an experience that is a lot like the experience of learning abstract math. The Weil Conjectures suppose a connection between two mathematical domains — topology and number theory — and Olsson wants (as far as I understand) to create a literary experience that is analogous to the search for such domain-bridging mathematical connections. So she lays out the Weil biography, her own memoir, mathematics and writing about mathematics for the reader. And I think she really succeeds — as the book goes on it feels a bit like learning some deep bit of theory.

So it’s not fair to reduce the book to a neat set of answers because the book is primarily about the experience of reading it. Some books are like that, and that’s fine. But she does answer some of those questions in interesting ways, and memory is necessarily structured, so it’s worth trying to say a bit about what Olsson says about mathematics itself.

II.

Q: What do people love about abstract mathematics?

A: Attraction to the unknown itself.

That, I think, is as close as we can get to Olsson’s answer in brief. The mathematician is someone who desires to create unknowns and to put obstacles in the way of their knowing so that they can search for answers. So for instance we hear about a Kafka story (“The Top”) about a philosopher who seeks enlightenment by hoping to catch spinning tops in mid-spin (whatever). On this story Anne Carson says “he has become a philosopher (that is one whose profession is to delight in understanding) in order to furnish himself with pretexts for running after tops.” That’s what a mathematician is — they love the chase.

Q: Is this so different from what a writer does?

A: No.

And so there is a connection between Olsson’s 2.5 years studying mathematics in college and her life as a writer.

I love this take on mathematics — that it’s about this love of being in the dark and searching for light. So it’s more about finding light than the light itself, if that makes sense.

This should be seen (I think) in contrast with writers who make much of the beauty of mathematics, or the search for beauty. Olsson is good on this. Twenty years after finishing her degree she decides to go back and watch some online lectures for an Abstract Algebra class. She finds it remote and foreign, but also:

And still, it was beautiful. I’m ambivalent about expressing it that way — “beauty” in math and science is something people tend to honor rather vaguely and pompously–instead maybe I should say that still, it was very cool. (This is something the course’s professor, Benedict Gross, might say himself, upon completing a proof: “Cool? Very cool.”) A quality of both good literature and good mathematics is that they may lead you to a result that is wholly surprising yet seems inevitable once you’ve been shown the way, so that–aha!–you become newly aware of connections you didn’t see before.

Still, the mathematician’s next move is to plunge themselves into darkness. This comes from a desire towards something that cannot be grasped.

This part was hard for me to understand. A key for Olsson seems to be Anne Carson’s Eros the Bittersweet but the theory isn’t entirely clicking for me. “A mood of knowledge is emitted by the spark that leaps in the lover’s soul,” she writes but I don’t quite get. Olsson’s take: “It’s not the knowledge itself, not consummation but the mood, the excitement when you are on the verge of grasping.”

What I understand Olsson to be saying is that the main fun of math isn’t the understanding but the feeling that understanding might be near. And that explains the pleasure we non-experts get out of mathematical analogies. There’s nothing unusual about the idea that analogies give us the thrill of desire — what’s more novel is saying this isn’t so different than the usual state of a mathematician.

For this point she goes to the act of mathematical creation itself, and how unpredictable it is:

What does mathematical creation consist of? asks Poincare, who blazed his way through a large territory of mathematics and physics by relying on his remarkable geometric intuition. It requires not only the combining of existing facts but the avoiding of useless combinations: making the right choices. The facts worthy of study are those that reveal unsuspected relationships between other facts. Moreover, much of this combining and discarding and retrieving goes on without the mathematician’s full awareness, occurring instead behind the scrim of consciousness.

Since the mathematician is dependent on their unconscious associations, mathematical discovery is not entirely in their control. In fact, many suspected relationships don’t work out. And so the mathematician spends most of their time afflicted with that same desire for the unreachable that afflicts we, the non-technical lay audience, who only get analogies.

Analogy becomes a version of eros, a glimpse that sparks desire. “Intuition makes much of it; I mean by this the faculty of seeing a connection between things that in appearance are completely different; it does not fail to lead us astray quite often.” This of course, describes more than mathematics; it expresses an aspect of thinking itself–how creative thought rests on the making of unlikely connections. The flash of insight, how often it leads us off course, and still we chase after it.

It’s a neat picture, I think!

I’ve left out all the connections to mysticism, the biographical details of Simone and Andre, and (nearly) all the connections to writing, but that’s in there too. Again, very neat stuff.

III.

I also learned from this book that Brouwer retired early and practiced nudism, that Flannery O’Connor didn’t particularly care for Simone Weil’s writing, and of Hadamard’s fascinating book “The Psychology of Invention in the Mathematical Field.”

One last good quote: “Honestly I think I understand anyone else’s dislike of math better than I understand whatever hold math has had on me.”

## Some interesting things I learned from Robert Pondiscio’s book about Success Academy

Success Academy was originally supposed to be a copycat of Robert Slavin’s Success for All program.

Rich person Joel Greenblatt got interested in Slavin, and gave him a call.

“I gave him a call and basically said, ‘I’ve looked at your results, and you’re getting forty or fifty percent of kids reading, but it’s not ‘success for all,'” Greenblatt recalled telling Slavin. “‘I’m just a guy with money. Is it money?'”

Yes, Slavin replied; its just money–if the money is spent in the right way. But it’s difficult to maintain tight control in most public schools. The two agreed to try to find a school they could convince to put “Success for All on steroids,” as Greenblatt put it.

They took over a public school, got a charter, recruited Moskowitz, one thing led to another, etc., here we are today.

Eva Moskowitz hates too much direct instruction.

More than most charter school leaders of her stature, Moskowitz consistently sees and talks about schooling through a child’s lens. Too much direct instruction–explaining the steps to solve a math problem, for example, rather than guiding kids as they struggle and strategize–sets her off. “If you see a teacher, ‘Blah, blah, blah, blah,'” she says, pantomiming extreme boredom. Her tone has been restrained, her deliver pedantic, even when discussing charged topics like race, cheating the Charlotte Dial video. But now she grows animated, even agitated. “Who has seen a kid when their teacher keeps them on the rug for twenty-five minutes? They’re five, for God’s sakes! You try sitting on that rug for twenty-five minutes listening. That is an educational crime!”

I should note that the book is basically Pondiscio’s account of a year he spent observing Bronx 1, a Success Academy elementary school. It is focused entirely on the elementary years, which is too bad. They operate a lot of middle schools (and a few struggling high schools) and you don’t leave the book with any sense of how they work. The direct instruction is one thing, but another is the ra-ra spirit (Ford-ham! Stand! Up!) which every middle schooler I have ever met would hate.

Math at Success Academy is…progressive? Progressive-ish?

Here is the Kindergarten teacher at Bronx 1 speaking to parents:

Writing “27 + 14” on the classroom smartboard, she warns parents to expect “a totally different world of learning math. We all learned math this way,” she says, doing the sum while narrating her work: “Seven plus four is eleven; put the one here and carry the one.” She stops and turns back to the room. “What does ‘carry the one’ mean? I love that you’re giggling, mom,” she says to one parent. “Before I came here, I had no idea why I carried that one. I knew I couldn’t put it here“– she writes “11” below the stacked “7” and “4.” “I did a procedure. No one ever taught me the real reason.”

For now, she begs parents not to teacher their children “procedures.”

This is not a lone teacher. (The book makes clear that there are no lone teachers at SA.)

It’s also reflected in their curriculum which (believe it or not) is built on TERC!

The math curriculum is cobbled together from different sources: an off-the-shelf curriculum called TERC; Contexts for Learning, a “conceptual math” approach pioneered by Catherine Fosnot, an education professor at City College of New York; and “a variety of things we found on the Internet,” according to Stacey Gershkovich, who oversees math at Success.

That’s pretty progressive stuff! Though Pondiscio emphasizes that they do drill facts a lot.

I will quote extensively, because I was surprised by this whole passage.

In the fourth-grade wing, Kerri Lynch is teaching fractions. Because of its test scores, it is commonly assumed that math lessons at Success Academy resemble a Chinese cram school’s, with instruction focused exclusively on “drill and kill” to prepare kids for high-stakes tests. However, Moskowitz, the daughter of a mathematician, is a proponent of a conceptual approach; she derides “direct instruction” and other standard explanatory pedagogies, where kids learn and practice algorithms and formulas, as “math by card tricks.”

Sure, sure you say. Show me the classrooms.

Lynch’s students sit on assigned spots on the rug. “One thing we’ve worked on is to be able to compare fractions to landmark fractions, such as one half or one whole,” she begins. “Today, when I put up the two fractions you’re comparing, I don’t want you to show any work. I only want you to write if it’s less than, greater than, or equal to. Just the symbol.” With those minimal instructions, Lynch writes 1/8 and 1/10 on the board and watches as her students bend silently over their whiteboards. “I’m seeing that some of us very quickly know it,” she observes. “Go ahead and turn and talk with your partner. What knowledge of fractions did you apply to solve this question?” The chatter rises as Lynch circulates, asking questions, drawing students out, and listening to their explanations, making mental notes about which students she will ask to “share out” with the class.

As the discussion goes on, the explanations gradually grow more economical and precise…Lynch raises the rigor. She writes 3/4 and 7/8 on the board. With no common denominator or numinator the answer is less obvious, but she offers a hint: “Matthew, what you said about ‘closer to one whole’ might be helpful. Turn and talk with your partner.” The room breaks into passionate arguments: “They’re equal!” “Seven-eighths is more!” “Only one piece is left to get it to one whole!” “No they are equal! They’re both only one pieces to a whole!” “But this one’s a smaller piece to a whole.”

Lynch’s ears perk up. “What did you just say? They’re both one away from a whole? How much further away from a whole? I want you to share that,” she says to Matthew.

Jo Boaler would be proud! (Steve Leinwand is.)

Eva Moskowitz is fond of saying that Success Academy is “Catholic School on the outside, Bank Street on the inside.” That is not true, because all the behavioral reinforcements and super-obsessive rules are very not Bank Street. But you can sort of see what she means.

Success Academy uses Fountas and Pinnell’s Leveled Reading System, and they generally spend a lot of time on reading skills (something researchy-types don’t care for)

Success Academy follows the “leveled reading’ system developed by reading researchers Irene Fountas and Gay Su Pinnell, which starts at Level A early in kindergarten and reaches Level Z, usually by the end of eighth grade.

They also use it in a way not recommended by Fountas and Pinnell, which is to give each kid a specific and highly publicized reading level. There’s a scene where a kid excitedly goes around telling teachers that she is “Level L” and again I wonder what their middle school looks like because this culture? Doesn’t work for older kids, right?

Systematically building background knowledge is not an explicit aim of Success Academy’s ELA curriculum. I had long assumed that its ELA curriculum played a significant role, even a dominant one, in the network’s standardized test results. But when the network made its curriculum available for free online in 2017, the response among experts who emphasize the importance of knowledge-rich curriculum was muted. “Kindergartners spend all but one of their seven units supposedly developing skills–like the ‘skill’ of reading nonfiction–and those at higher grade levels get only one or two more units per year that are content-based,” observed education journalist Natalie Wexler, who noted that the lessons posted on the website “sound very similar to the fruitless exercises that are found in classrooms across the country. Teachers jump from one topic to another, using content merely as a delivery mechanism for skills, and then students are sent off to ‘practice’ the skills on books they choose themselves.”

These are the most important quotes in the book.

There’s a meeting for parents who either won or were declared “likely” to gain an admissions spot for their kid. The meeting asks parents to consider whether Success is right for them.

Then, almost as an afterthought, Reeder mentions transportation. Success Academy does not offer buses. For some parents the logistics of getting a child to and from school present an even bigger challenge than complying with culture demands, reading logs and homework. Every Wednesday, children are dismissed at 12:30 so that staff can attend their professional development sessions. “That’s something else you gotta keep in mind, “Reeder adds. “‘Will I be able to pick up my child on time? Will I be able to have somebody to support me with that?’ School lets out at 3:45 but every Wednesday is “12:30 no matter what.” She hits “12” and “30” hard, hammering the point home. “And we do not have after-school, so you guys have to figure that out.”

The meeting lasts just under an hour, but it opens a portal into the model and culture that explain in no small part the network’s consistent results across its schools. Suddenly it all makes sense: The common criticism leveled at Moskowitz and her schools is that they cherry-pick students, attracting bright children and shedding the poorly behaved and hardest to teach. This misses the mark entirely. Success Academy is cherry-picking parents. Parents who are not put off by uniforms, homework, reading logs and constant demands on their time, but who view those things as evidence that here, at last, is a school that has its act together. Parents who are not upset by tight discipline and suspensions but who are grateful for them, viewing Success Academy as a safe haven from disorderly streets and schools. Charter schools cannot screen parents to ensure culture fit, but the last hour in the auditorium is a close proxy for such an effort, galvanizing disciplines and warning off the indifferent and uncommitted. A the same time, there is something undeniably exclusionary about it. If you don’t have the resources to get your child to school by 7:30 and pick her up at 3:45  — at 12:30 on Wednesdays — Success Academy is not for you. Literally.

Pondiscio makes much of research showing that parental factors make a big difference on the success of low-income students. Parental involvement, two-family homes, strong religious faith are all factors that help. The claim is that Success Academy is essentially selecting for these families in low-income neighborhoods.

Nobody likes talking about it in this book, but also nobody seems to deny it. It’s the unspoken but obvious, glaring fact about Success Academy. It’s a point neatly summed up towards the end of the book:

One former Success Academy school leader was philosophical about all this. “Is it really such a bad thing that this is basically an elite private school that admits by lottery?” he asked. “It’s the first time folks in the inner city have had that chance.”

So it’s not curriculum, not the teaching, not the teachers? Not even the test prep? It’s just selection effects?

Well, mostly selection effects. There are a few other things in the secret sauce. The thing that’s well-known is the intense and systematic behavior management. What I didn’t know about was the entirely sensible division of labor. Principals focus entirely on teachers and students — they have an “ops” person for administrative stuff. The curriculum may not have impressed Pondiscio, but its existence does impress him. The curriculum meets whatever minimal threshold it needs to be to free teachers up to focus on going over student work, calling parents and working with kids. That’s good, in general!

But, yeah, mostly selection effects.

Oh, and my school got a shout out.

The oft-heard refrain at Success to “put the lift on students” and to socialize learning, encouraging children to work collectively in pairs or small groups, is the kind of teaching one would expect to see at Bank Street, Saint Ann’s, or any of the progressive private schools beloved by affluent New Yorkers.

That doesn’t exactly nail the culture of Saint Ann’s but whatever! Not his point.

This is a good interview.

The book was a very good read and very much worth reading. Interesting, provocative, rooted in entirely realistic school observations, it’s definitely worth checking out.

## Discovery learning vs. not discovery learning

I.

I think at this point, if you’re reading a math blog, you probably have an opinion about the place of discovery (or inquiry or guided inquiry or problem solving or whatever) versus fully guided instruction (or direct instruction or Direct Instruction or explicit instruction or Explicit Direct Instruction).

(By the way, Ed Realist does a nice job trying to clarify the terminological situation here.)

But the thing is that it is difficult to talk about this in a way that is clear and accessible. I was thinking about this while reading Jasmine’s latest post, which lays out what cognitive science researchers say on the matter. Jasmine and I are on the same page, and she is faithful to the researchers, but I felt myself inclined to express these views in a slightly different way. Not necessarily even better; just different.

(By the way, Jasmine is a first-year teacher and new blogger. She’s on the blogroll.)

So here is how I would put it:

Every mathematician and scientist, as far as I can tell, is clear about just how messy their research is. I am very fond of this account from mathematician Andrew Wiles:

Perhaps I could best describe my experience of doing mathematics in terms of entering a dark mansion. You go into the first room and it’s dark, completely dark. You stumble around, bumping into the furniture. Gradually, you learn where each piece of furniture is. And finally, after six months or so, you find the light switch and turn it on. Suddenly, it’s all illuminated and you can see exactly where you were. Then you enter the next dark room…

You think it’s true…then it’s not…then you waste a morning trying to prove something that in fact is not true and not strictly necessary for proving the thing you actually care about. Then you feel despair, so you take a break and do something else. A week later you come back and you feel stupid — the thing is now obviously true, and you know why — and that feels good! But that’s just Part 1. So on to Part 2…

Here’s the question, and it’s a fundamental one: do you think it’s a good idea to put your students in this situation, or not?

If you say “yeah! kids need to learn how to do this sort of thing” then you will be a fan of discovery and inquiry and problem solving and etc. If you say “wait, no, this doesn’t sound like a good way to make kids feel” then you will strongly dislike discovery and inquiry.

I feel as if that’s almost all there is to say. It pretty much comes down to that.

II.

There is of course a bit more, though. It’s probably easiest to present it in terms of a dialogue. Basically, cognitive science has a bunch of counter-arguments to arguments in support of the “yeah!” view above. Here’s how the dialogue goes.

Q: You don’t like discovery/inquiry/asking students to do math the way research mathematicians and scientists do?

A: That is right, I do not.

Q: But how will students learn to do research-y things if you don’t teach them?

A: “Do research-y things” is not really a skill. Neither is “creatively problem solve” or “think mathematically.” We don’t have evidence that any of these things can be taught to students, except alongside particular mathematical or scientific content. The things you really need to do research-y things that can be taught is a tremendous store of flexible, sturdy knowledge. That’s the best thing you can do to give your kids a leg up.

Q: But that’s demotivating! It’s boring to learn a discipline that way, and the genuine ways of learning are more motivating.

A: Bad teaching will always be demotivating, but there are lots of examples of the “boring” approaches being highly motivating. One way you see this is when an intervention measures affect, i.e. how kids feel about a thing.

But honestly if kids aren’t motivated, they won’t learn, and we have evidence that the more explicit approaches help kids learn. Shrug.

Q: So you think that kids never need a chance to apply their knowledge?

A: Yo, I did not say that.

Q: Yeah you did.

A: No, I did not. Here’s what I think. There’s evidence that when a student has less experience with something, they need a lot of explicit instruction about how to do that thing. Worked examples are a really, really sturdy format for people with little experience in a thing. If a student has never learned how to factor quadratic functions, a good way to start can be to show them examples of factoring quadratic functions. Then, ask them to use the example to solve a problem. And then show another example, and then give them some more practice. And then mix-up the practice, or ask them to apply what they’ve learned in a new context.

And then, the next day, do more stuff like that.

And then on the third day, maybe ask them to solve some problems on their own, and see how that goes.

And if it’s going well, who knows! In a week or so, maybe they’ll be ready to apply these skills to a challenging problem in class. Or maybe it’ll take a few weeks. The point is that as kids get more experienced with a set of skills, they are more ready to take on challenges.

Q: Thank you for saying “challenges,” I like challenges.

A: No problem. The point is where you start. And that’s genuinely controversial! But we believe (see: evidence) that starting with fully explicit instructions like worked examples gives newbies the help that they need. Starting a unit with a vague activity that students aren’t sure how to handle isn’t giving them the help they need.

Q: Does this take into account motivation?

A: No and yes.

No, it doesn’t take into account motivation. Do you have some amazing, super-motivating activity that kids love and that will super-charge a unit of study? Do you start a unit on quadratics with this amazing activity that helps the whole thing get started on a great note? No, the evidence does not take this into consideration. It just notes that it’s hard to find clear evidence of a learning benefit of this sort of thing.

But, yes, this does take into account motivation, because in the long-run there isn’t really any evidence that motivation is easily separable from achievement. So ultimately something like a worked example does a lot of good for motivation, because it helps struggling students understand the material and participate as an equal in your classroom.

Michael Pershan: Can I step in here for a second?

Q: Sure.

MP: I would only add that though there is no evidence for this, I do think a certain amount of variety is healthy in a classroom. Like, kids do get bored if you do the same thing day after day. But, two things. First, if you’re starting with something like fully worked examples and moving to interesting, challenging practice, your kids are getting variety. Second, go ahead, take a day and do something interesting and different. Variety is good! There are lots of interesting practice formats, though we don’t talk enough about that.

That’s all I want to add here.

Q: OK, but here’s the thing. I just want kids to be able to think like mathematicians/scientists in school. That’s the goal I care about. That’s what I think is most valuable. And I don’t even necessarily care if it is helping them do that stuff in the future. You tell me that these skills can’t be taught — OK! You also tell me that there is no real benefit to their skills from these kinds of experiences — that’s OK too! All that I want is for kids to be doing something meaningful in school. Yes, I want to make sure their test scores are OK and they can get into college, but beyond that I want kids to do something they care about.

There are two answers to this last question, which is what I think this discussion sometimes comes down to.

1. There comes a point where people just disagree on what they value. It’s hard to know what to say by the time someone gets to this point of clarity about what they care about.
2. It’s a mistake to assume that regular, “boring” school is not meaningful. And it’s also a mistake to assume that regular, “boring” learning is not meaningful. As I’ve written in the past, mathematicians ask for help all the time, and a lot of cutting edge work is simply focused on understanding things, not on solving a particular problem.

But I guess I’d point out that there are three things going on.

There’s a certain picture of what research mathematicians and scientists do and what their culture is like.

There’s a view about what is most effective for learning and motivation.

And then there is a view about whether it sounds like a good idea to put kids under the conditions of researchers in class.

And cognitive science research is relevant for the second, the efficacy question. And there is a value question, about what you think is worth doing in school.

But for me the decisive point is that the work of learning skills and knowledge is meaningful, and you can see this also in the culture of mathematicians and scientists. It’s just not right that learning skills isn’t meaningful to students.

## How do you make yourself nicer?

I’m not, like, some expert on philosophy. But I have a degree in it! I took a bunch of ethics classes in college, ethics is great. I love ethics. You can quote me on that.

Here’s a question we never studied and nobody ever asked in an ethics class: how do you become a better person?

We talked (often endlessly) about what it means to be a better person, sure. How do you know if you’re a good person or not, what is a good action, is there really such a thing as good/bad. (Philosophy is all about those italics.)

How a person actually goes about the actual process of going from an OK person to a better person along any dimension is, apparently, not a question of philosophy at all. I can’t name a single philosopher I read who says much about it at all. Maybe that’s a limitation of my reading, maybe not, I don’t know, but doesn’t that say something?

I am also a religious person and was raised religious, but I’m only interested in the secular version of this question. Of the religious texts I’ve studied there are a few that tackle the question of moral self-improvement, but it’s still pretty vague. The most explicit Jewish writing about becoming more ethical is from the mussar movement and they are…intense and weird? Though sometimes insightful?

So let’s say that you woke up one morning and realized you were a jerk. You’d been denying it for years, but it’s time to get honest. You’re mean to people, and for no reason at all. You get angry — not “murder” angry but not-nice angry, and with some frequency. You have relationships, but it’s increasingly clear that these are people who tolerate and recognize your flaws. Because you’re not all bad — but you’re mean, and a bit of a jerk.

You realize this, and make a commitment to do something about it. It’s time, gosh darn it! (You’re working on your language, too.)

What exactly would you do about it?

[Oh, by the way, please nobody mention The Good Place to me. It’s good, I like it.]

Here are the only plausible things I can think of that might help you be nicer.

1. Keep a diary about your efforts. But I have no idea how that would help. Maybe it would slow down your quick thinking so your slow brain could take control…is that how people get better? Their rationality just lashes out at the emotions, until emotions grow docile and tame? Look, maybe.
2. Maybe you need a role model? I’ve sometimes felt myself be nicer to others after reading a biography about a really good person. For a few days that person lives in my head, asking questions like “What would Abraham Lincoln do?” or “Can you imagine what Gandhi would think of that?” The problem is that after a few days I go back to normal. But maybe there is a way to keep their presence around? If you wanted to be nicer, maybe spend some time each morning thinking about a really really nice person? I don’t know, I’m making this up.
3. Put up sticky notes everywhere? But everything eventually loses its force and fades into the background.
4. Study ethical texts with regularity? Not because they’ll help you manage complicated moral issues (though maybe they will) but because it’s a way to keep things fresh and interesting while still giving you a regular reminder that you’re aiming to be better?

I have no idea. I don’t think I’m a jerk, but I’d like to be a better father/husband/son/friend/teacher/person, sure. I don’t know how much I care — probably not enough. Most of us don’t care, though we probably should, as Eric Schwitzgebel argues.

I’ve looked around, there seems to be practically no empirical work on any of these questions. Moral psychology is the name of the field to look in, but they steer clear of the developmental question. I assume there’s a good reason, it veers too closely to “in this study I started a cult”, or it’s unstudiable for some other reason.

But this seems like a shame for our jerk, and for all of us too. Don’t we want to study the actual process of moral improvement? For are we not all jerks, wishing just a bit that we were a little better?

## Equations and Equivalence in 3rd Grade

So I was stupidly mouthing off online to some incredibly serious researchers about equivalence and the equals sign and how it’s not that hard of a topic to teach when — OOPS! — my actual teaching got in the way.

I had done the right thing. In my 3rd Grade class I wanted to introduce “?” as a symbol for an unknown so I put up some equations on the board:

15 = ? x 5

3 + ? = 10

10 + 3 = 11 + ?

And I was neither shocked, nor did I blink, when a kid told me that the last equation didn’t make any sense. Ah, I thought, time to nip this in the bud.

I listened to the child and said I understood, but that I would like to share how it does make sense. I asked whether anyone knew what the equals sign meant, and one kid says “makes” and the next said “the same as.” Wonderful, I said, because that last equation is just saying the left side equals the same as the right side. So what number would make them the same? 2? Fantastic, let’s move on.

Then, the next day, I put a problem on the board:

5+ 10 = ___ + 5

And you know what comes next, right? Consensus around the room is that the blank is 15. “But didn’t we say yesterday that the equals sign means ‘the same as’?” I asked. A kid raised her hand and explained that it did mean that, but the answer should still be 15. Here’s how she wanted us to read the equation, as a run on:

(5 + 10 =  15 ) + 5

Two things were now clear to me. First, that my pride in having clearly and decisively taken care of this issue was misguided. I needed to do more and dig into this more deeply.

The second thing is that isn’t this interesting? You can have an entirely correct understanding of the equals sign and still make the same “classic” mistakes interpreting an actual equation.

I think this helps clear up some things that I was muddling in my head. When people talk about the need for kids to have a strong understanding of equivalence they really are talking about quite a few different things. Here are the two that came up above:

• The particular meaning of the equals sign (and this is supposed to entail that an equation can be written left-to-right or right-to-left, i.e. it’s symmetric)
• The conventional ways of writing equations (e.g. no run ons, can include multiple operations and terms on each side)

But then this is just the beginning, because frequently people talk about a bunch of other things when talking about ‘equivalence.’ Here are just a few:

• You can do the same operations to each side (famously useful for solving equations)
• You can manipulate like terms on one side of an equation to create a true equation (10 + 5 can be turned into 9 + 6 can be turned into 8 + 7; 8 x 7 can be turned into 4 x 14; 3(x + 4) can be turned into 3x + 12, etc.)

When a kid can’t solve 5 + 10 = ___ + 9 correctly or easily using “relational understanding,” this is frequently blamed on a kid’s understanding of the equals sign, equivalence or the particular ways of relating 5 + 10 to __ + 9. But now I’m seeing clearly that these are separate things, and some tend to be easier for kids than others.

So, this brings us to the follow-up lesson with my 3rd Graders.

I started as I usually do in this situation, by avoiding the equals sign. I find that a double arrow serves this purpose well, so I put up an arrow relationship on the board:

2 x 6 <–> 8 + 4

I pointed out that 2 x 6 makes 12 and so does 8 + 4. Could the kids come up with other things like this, I asked?

They did. I didn’t grab a picture, but I was grateful that all sorts of things came up. Kids were mixing operations nicely, like 12 – 2 <–> 5 + 5, in general it felt like this was not hard, kids knew exactly what I meant and could generate lots of ideas.

My next move was to pause and introduce the equals sign into this conversation. Would anyone mind if I replaced that double arrow with an equals sign? This is just what the equal sign means, anyway. No problem, that went fine also.

Kids were even introducing great examples like 1 x 2 = 2 x 1, or 12 = 12. Wonderful.

Then, I introduced the task of the day, in the style of Open Middle (R) (TM) (C):

Yeah, I quickly handwrote it with a sharpie. It was that sort of day.

I carefully explained the constraints. 10 – 2 + 7 + 1 was a true equation, but wouldn’t work for this puzzle. Neither would 15 – 5 = 6 + 4. And then I gave the kids time to search for solutions, as many as they could find.

Bla bla, most kids were successful, others had trouble getting started but everyone eventually had some success. Here are some pictures of students who make me look good:

Here is a picture of a student who struggled, but eventually found a solution:

Here is a picture of the student from the class I was most concerned with. You can see the marks along his page as he tries to handle things like 12 – 9 as he tries subtracting different numbers from 12. I think there might have been some multiplying happening on the right side, not sure why. Anyway:

The thing is that just the day before, this last student had almost broken down in frustration over his inability to make sense of these “unconventional” equations. So this makes me look kind of great — I did it! I taught him equivalence, in roughly a day. Tada.

But I don’t think that this is what’s going on. The notion of two different things being equal, that was not hard for him. In fact I don’t think that notion is difficult for very many students at all — kids know that different additions equal 10. And it was not especially difficult for this kid to merge that notion of equivalence with the equals sign. Like, no, he did not think that this was what the equals sign meant, but whatever, that was just on the basis of what he had thought before. It’s just a convention. I told him the equals sign meant something else, OK, sure. Not so bad either.

The part that was very difficult for this student, however, was subtracting stuff from 12.

Now this is what I think people are talking about when they talk about “relational understanding.” It’s true — I really wish this student knew that 10 + 2 <–> 9 + 3, and so when he saw 12 he could associate that with 10 + 2 and therefore quickly move to 9 + 3 and realize that 12 – 3 = 9. I mean, that’s what a lot of my 3rd Graders do, in not so many words. That is very useful.

So to wrap things up here are some questions and some provisional answers:

Q: Is it hard to teach or learn the concept of equivalence.

A: No.

Q: Is it hard to teach the equals sign and its meaning?

A: It’s harder, but this is all conventional. If you introduce a new symbol like “<–>” I don’t think kids trip up as much. They sometimes have to unlearn what they’ve inferred from prior experiences that were too limited (i.e. always putting the result on the right side). So you’re not doing kids any favors by doing that, it’s good to put the equations in a lot of different forms, pretty much as soon as kids see equations from the first time in K or 1st Grade. I mean why not?

Q: If kids don’t learn how equations conventionally work will that trip them up later in algebra?

A: Yes. But all of my kids find adding and subtracting itself to be more difficult than understanding these conventions. My sense is that you don’t need years to get used to how equations work. You need, like, an hour or two to introduce it.

Q: Does this stuff need to be taught early? Is algebra too late to learn how equations work?

A: I think kids should learn it early, but it’s not too late AT ALL if they don’t.

I have taught algebra classes in 8th and 9th Grade where students have been confused about how equations work. My memories are that this was annoying because I realized too late what was going on and had to backtrack. But based on teaching this to younger kids, I can’t imagine that it’s too late to teach it to older students.

I guess it could be possible that over the years it gets harder to shake students out of their more limited understanding of equations because they reinforce their theory about equations and the equals symbol. I don’t know.

I see no reason not to teach this early, but I think it’s important to keep in mind that in middle school we tell kids that sometimes subtracting a number makes it bigger and that negative exponents exist. Kids can learn new things in later years too.

Q: So what makes it so hard for young kids to handle equations like 5 + 10 = 6 + __?

A: It’s definitely true that kids who don’t understand how to read this sort of equation will be unable to engage at all. But the relational thinking itself is the hardest part to teach and learn, it seems to me.

Here is a thought experiment. What if you had a school or curriculum that only used equal signs and equations in the boring, limited way of “5 + 10 = ?” and “6 x ? = 12” throughout school, but at the same time taught relational thinking using <–> and other terminology in a deep and effective way? And then in 8th Grade they have a few lessons teaching the “new” way of making sense of the equals sign? Would that be a big deal? I don’t know, I don’t think so.

Q: There is evidence that suggests learning various of the above things helps kids succeed more in later algebra. Your thoughts?

A: I don’t know! It seems to me that if something makes a difference for later algebra, it has to be either the concept of equivalence, the conventions of equations, or relational thinking.

I think the concept of equivalence is something every kid knows. The conventions of equations aren’t that hard to learn, I think, but they really only do make sense if you connect equations to the concept of equivalence. The concept of equivalence explains why equations have certain conventions. So I get why those two go together. But could that be enough to help students with later algebra experiences? Maybe. Is it because algebra teachers aren’t teaching the conventions of equations in their classes? Would there still be an advantage from early equation experience if algebra teachers taught it?

In the end, it doesn’t matter much because young kids can learn it and so why bother not teaching it to them? Can’t hurt, only costs you an hour or two.

But the big other thing is relational thinking. Now there is no reason I think why relational thinking has to take place in the context of equations. You COULD use other symbols like double arrows or whatever. But math already has this symbol for equivalence, so you might as well teach relational thinking about addition/subtraction/multiplication/division in the context of equations. And that’s some really tricky, really important mathematics to learn. A kid being able to understand that 2 x 14 is equal to 4 x 7 is important stuff.

It’s important for so many reasons, for practically every reason that arithmetic is the foundation of algebra. I can’t list them now — but it goes beyond equations, is my point. Relational thinking (e.g. how various additions relate to each other) is huge and hugely important.

Would understanding the conventions of the equals sign and equations make a difference in the absence of experiences that help kids gain relational understanding? Do some kids start making connections on their own when they learn ways of writing equations? Does relational understanding instruction simply fail because kids don’t understand what the equations their teachers are using mean?

I don’t know.

## High, Holy Days: A Playlist

A lot of you have been asking where my Elul/Rosh HaShana/Aseret Yemei Teshuva/Yom Kippur playlist is. “Is it ready yet?” people ask. “You promised.”

Well, it’s not quite done. I’m still tinkering with it. But it’s as ready as it’s ever going to be. Here it is, on Spotify.

You want me to what? Explain it? That defeats the whole point of a playlist. It would be reductive to go song by song and explain its presence and purpose. I mean, seriously.

Still, there is what to say.

We open as the month of Elul does, with the arrival of the Infanta heralded by the shofar. The Queen is in the field, Elul is in the sky, and the question is what you’re going to do about it. Mad Men, indeed.

It’s time to start asking the big questions. Turn off your mind, relax — but not too much. You need to rethink things, to pay attention. It is not dying, but it’s not not dying either. Because there are certain things to keep in mind when you hear the shofar. Everybody here is a cloud. Don’t forget. If I’m alive, next year.

Sinnerman, Troubleman, Man, it doesn’t matter what you’re called. It matters what you are.

Here’s the deal about “Who By Fire?”: I don’t like any of the versions on Spotify. This is one of those times when a song has a single correct version, and it’s the version with the saxophone.

The whole thing doesn’t work with that Mediterranean guitar intro/accompaniment nearly as well. I think it’s just a fundamental difference between guitar and sax. Sax is a horn, guitar is a string instrument. Your guitar can do a lot of things (e.g. it can weep) but it’s not powered by breath and it never will be. To put it another way, guitar is your siddur but sax is your shofar.

This is the version that should be on the playlist. If it were a mixtape I’d have ripped it, etc.

After this you get to go down to the river for three songs. This is tashlich, but it can also be the mikveh before Yom Kippur if you want to keep things moving roughly chronologically. (I originally tried to make this thing match the chronology more closely. It was a mess.)

From “Get By”:

This morning, I woke up
Feeling brand new and I jumped up
Feeling my highs, and my lows
In my soul, and my goals
Just to stop smokin’, and stop drinkin’
And I’ve been thinkin’ – I’ve got my reasons
Just to get by

And you get one last wordless prayer. Then, the whole thing ends, and we’re on to a new year. Will there be feasting and dancing in Jerusalem this year?

But pat yourself on the back for a second — you have made it to a new year, this year.

May you be written in the book of life. Ketivah v’chatimah tovah.

## Dear Aunt Sally: The exit tickets are all over the place!

Dear Aunt Sally,

My students had a hard time with my lesson on solving equations. How do I know? I gave a short “exit ticket” and the results were…mixed. There were three questions:

• 3x + 5x = 56
• 3x + 7 – 5x = 45
• 6x – 5 + 11x + 17 = 63.

The main issue is combining like terms. (I’ve included some pictures of student work for you, Aunt Sally.)

That said, some students in each class definitely did understand the material.

One more issue. In most of my classes, students didn’t struggle with the first problem. But in each of my classes, some students did, and in one of my classes very few students got that first one correct either.

What do I do to help the students who didn’t understand the material, and what do I do with the kids who did?

Ryan in Florida

***

Dear Ryan,

You have described one of the perpetual struggles of teaching. How do I balance the needs of the many versus those of the few? I suppose back in Neanderthal times the fellow charged with teaching youth to spear a mammoth came home to the cave feeling similarly. “Some of those kids get it,” he’d say. “But what about the one who used the flat end of the spear? He’s going to starve to death. He would really benefit from getting back to the basics. Maybe I’ll split the group in half?”

Thank goodness we aren’t Neanderthals. We are modern teachers! This means we have access to the wonders of modern technology. I of course mean the blackboard and copy machine.

My favorite way to follow-up on these sorts of quizzes is with examples, so I put together three such activities for you:

And then some mixed-up practice, to take another step in the right direction.

A good example activity, in my experience, can seem so simple so as to hide its design. And in fact the actual design of the student work was mostly straightforward. The choices to make are mostly ones I made long ago — to prefer simplicity, to subtly use arrows and lines, and to make the work as much like a student’s as possible while making the work as clear as possible to read.

(The mistake, in particular, closely follows the design of an Algebra by Example mistake.)

Most of the work here is in narrowing in on a specific type of problem that is worth including in the examples. (The simplicity of the format works well with the more complex task of narrowing in.)

Here is a rule I tell myself while looking at a student’s mistake: Every mistake points to a family of situations very similar to the mistake that the student doesn’t yet know how to handle. Find that family, and teach it!

In this case I don’t assume the mistakes on the second problem (3x + 7 – 5x = 45) point not to issues “combining like terms” in general. I assume instead that the mistake points to issues where the like terms are separated visually in the equation and where subtraction is involved. Those things are distinctive and make the equation more difficult to solve correctly — that is the family that I focus in on for the example.

How the examples are used is your decision, Ryan, but here is how I would do it. Begin class by showing just the example, and ask students to silently study it. (They can let you know when they’ve read it all with their thumbs.) When finished, you can ask them to answer the explanation questions on their own or with a partner. If you want to interject with an explanation, by all means, but often students are ready to dive straight into the practice problems.

There are three examples/non-examples I’ve made. Use as many as your class would benefit from.

Then, there are four practice problems. Those are important because they are mixed practice. If we see each of these examples as pointing to a micro-skill, a small family of problems within the broad category of “two-step equations,” then these problems are interleaved in the practice set. That can be useful! Your students will have to think back and remember what they did with the examples.

Teachers of equation-solving know that there are always more mistakes that need to be addressed. The difference between good and bad teaching of this topic, as I see it, is whether the teacher can get specific and point to precisely what is hard for their students. Subtracting, different visual formats, handling a variable with coefficient 1 (x as 1x), and so many other little things — there will always be more of these. Some teachers just repeat and repeat and repeat without getting any more specific. Instruction should get more specific as practice gets more general.

So, keep at it! Pretty soon your young charges will be scattering the landscape with Woolly Mammoth carcasses, so to speak.

-Aunt Sally

## A Syntax of Geometrical Figures in “De Aetatibus Mundi Imagines”

Holanda represents the intelligible reality of the Holy Trinity through a “hypothetical” syntax of geometrical figures:

“Starting from a perfect circle, three triangles merge in the abyss, provoking a strange sensation as much of movement as of immobility. Alpha and Omega are inscribed on the first equilateral triangle, perfectly inscribed in the circle.”

Spectacle and Topophilia by David R. Castillo

Bonus images:

## Introducing Stable Distributions

The story so far:

• There are lots of ways to put two functions together and get a new one out of the process.
• Addition is one of these ways. Multiplication is another.
• If you add two Gaussian functions together, you don’t get another Gaussian function.
• If you multiply two Gaussian functions together, guess what? You get another Gaussian function.
• Convolution is another way of combining two functions.
• If you convolve two Gaussian functions together, guess what? You get another Gaussian function.

In the last post I tried to explain why convolving two probability distributions produces the distribution of the sum of those variables. And that sum is guaranteed to also be a bell curve, as long as the distributions its made of come are distributed normally as well.

It’s worth stewing for a moment on what that means, because a lot of things that we care about can be thought of as sums of random variables.

One example is height. Take, for example, the height of a forest. I know nothing about the biology of forest height, but I did find a few figures online. (Yes, random searching.)

Here they are.

Some of these plots look normal. Then again, some of them don’t. I don’t really care! Let’s pretend that forest height truly is normally distributed. Why would that be?

The thing is that there are a lot of factors that go into how tall a forest is height. There is underlying genetic variation in the families of trees that make a forest. There is underlying randomness is the environmental conditions where a forest grows, and “environmental conditions” is itself not a single factor but another collection of random variables — rain, soil, etc.

If we were going to make a list of factors that go into forest height how long do you think it would be before we had an exhaustive list? Hundreds of factors? Thousands?

Given all this, “forest height” really isn’t a single random variable. It’s a collection of random variables, not the way we think of a single coin flip.

(AND IS A COIN FLIP EVEN REALLY A SINGLE RANDOM EVENT??? OOOOOOOH DID I BLOW YOUR MIND?)

This all feels very complicated. How is it that a forest’s height has such a lovely normal distribution?

Given the math in the previous post we have a very tidy answer: forest height is thought of as a sum of random variables. Even if there are hundreds or thousands of underlying random variables that sum up to forest height, as long as all (most?) of them are themselves Gaussian, so will their sum, represented by the convolution of all those thousands of distributions.

(Note: we did not prove that convolution preserves the Gaussian nature of a distribution for an arbitrary number of distributions. But if you take two of those distributions they make a new Gaussian, and then pair that with another distribution, and then another, etc., so on, it’s a pretty direct inductive argument that you can convolve as many of these as you’d like and still get a Gaussian at the end of things.)

So convolving is nice because it gives us a tidy way to think of why these incredibly complex things like “forest height” could have simple distributions.

But there’s only one problem: who says that you’re starting with a Gaussian distribution?

Convolution does not always preserve the nature of a function. To pick an example solely based on how easy it was for me to calculate, start with a very simple function:

$f(x) = 2x$

Define this only on the region $[0,1]$ so that it really can be thought of as a probability distribution. Then, convolve it with itself.

(Hey, why is this only the first half of the convolution? Because for the life of me I can’t figure out how to visualize the second half of this integral while limiting the domain of the original function. Please, check my work, I am actually pretty sure that I’m exposing an error in my thinking in this post but hopefully someone will help me out!)

In any event, what you get as a result of this convolution is certainly not linear. So linearity of a distribution is not preserved by convolution.

What this means is that whether convolution can explain the distribution of complicated random variables that are the sum of simpler random things really depends on what the distribution we’re dealing with is. We are lucky if the distribution is Gaussian, because then our tidy convolution explanation works. But if the distribution of the complicated random variable was linear (what sort of thing even could) then this would present a mystery to us.

But is it just Gaussian distributions that preserve their nature when summed? If so, this would be pretty limiting, because certainly not every distribution that naturally occurs is a normal distribution!

The answer is that it’s not just Gaussian distributions. There is a family of distributions that is closed under convolution, and they are called the stable distributions.

***

I started trying to understand all of this back when I tried to read a paper of Mandelbrot’s in the spring. Getting closer!

This post doesn’t follow any particular presentation of the ideas, but a few days ago I read the second chapter of Probability Tales and enjoyed it tremendously. There are some parts I didn’t understand but I’m excited to read more.

## Another way to smoosh two Bell Curves together

Previously, we looked at one way of combining two Bell Curve (i.e. Gaussian distributions) together to make a third — multiplication.

There are other ways to do this, though. The best known (and as far as I can tell, the most important) is convolution. So, here are two Gaussian distributions, and what you get when you convolute(?) them:

Formally, this process takes two functions — $f(x), g(x)$ — and then produces a new distribution defined in the following way:

$(f * g)(t) = \int_{-\infty}^{+\infty}f(x)g(t-x)dx$

I have been struggling with this idea for several months, but just a few days ago I made some progress.

Convolution is often described as a blurring process — it blurs one distribution according to another one. That’s how Terry Tao describes it in this Math Overflow post:

If one thinks of functions as fuzzy versions of points, then convolution is the fuzzy version of addition (or sometimes multiplication, depending on the context). The probabilistic interpretation is one example of this (where the fuzz is a a probability distribution), but one can also have signed, complex-valued, or vector-valued fuzz, of course.

I have a hard time seeing the “blurring” in the images above. To really see it, I have to change the initial functions. For example, consider the convolution of a Gaussian and linear function (with restricted domain). Before convolution…

…and after.

Just playing around with the calculator a bit more, here is another before/after pair.

I first learned about convolution a few months ago, and it was explained to me in terms of blurring. The thing about this “intuition building” metaphor is that I’ve been sitting with it since then, and it hasn’t helped me feel comfortable with it at all. It was only last week when I came across the far more prosaic meaning of convolution that things started to click for me. Because besides for whatever blurring convolution represents, it also represents the sum of two independent random things.

(What follows is lifted from this excellent text.)

Suppose you have two dice, both six-sided, both fair. There is an equal chance of rolling 1, 2, 3, 4, 5 or 6 with each die — a uniform probability distribution. $P(x) = \frac{1}{6}$ whether $x = 1, 2, 3, 4, 5, 6$.

What would the distribution of the sums of the rolls look like?

The calculations start relatively simply, but check out the structure. For example, this is the calculation we have to do to find the chances of rolling a a 3:

$P(1)P(2) + P(2)P(1)$

The chances of rolling a 4:

$P(1)P(3) + P(2)P(2) + P(3)P(1)$

The chances of rolling a 5:

$P(1)P(5-1) + P(2)P(5-2) + P(3)P(5-3) + P(4)P(5-4)$

The chances of rolling n:

$P(1)P(n - 1) + P(2)P(n - 2) + ... + P(n-1)P(n - (n- 1))$

So! Using the language of summation, we can summarize this process as so:

$P(sum = n) = \sum P(k) \cdot P(n - k)$

We might as well take the last step of calling this process “convolution,” because it’s just the discrete version of the integral from above!:

$(f * g)(t) = \int_{-\infty}^{+\infty}f(x)g(t-x)dx$

***

Mathematically, there is a lot of interesting stuff to continue exploring with convolutions. Not all convolutions are defined, there’s a connection to Fourier transforms (another thing I don’t understand yet), there are discrete problems to solve in the text (what about different dice?), and so on and so on.

Briefly, though: why didn’t the blurring metaphor help me? I don’t think it’s such a mystery. It’s because while blurring is easy to understand, that image was totally disconnected from the underlying calculation. Why should that complicated integral be related to the process of blurring?

Now, I have a better understanding. (Blurring X and Y sort of is like finding the probability of X + Y.)

In school math, a lot of teachers bemoan their students’ lack of conceptual understanding, and it’s generally felt that procedural understanding is more obtainable. In my life as a learner of mathematics I usually feel it’s the other way around. When I read articles in Quanta Magazine or books about mathematics I haven’t yet studied I can usually follow the exposition but am left feeling a bit empty. Yes, I can follow the metaphors (“Imagine a number as a little bird; those birds fly together in flocks; but what happens when a bird has children? do they rejoin the flock? where? etc.”) but what have I learned?

Popular exposition of mathematics is maybe more difficult than exposition of other subjects simply because of the necessity of some kind of metaphor that brings the abstract to life. What results is a kind of understanding, but something far from the whole thing, and the best mathematical exposition also leaves me feeling jealous of those who can reach past the metaphors and grasp the thing itself.

That’s not to say that math exposition for popular audiences isn’t valuable — it is! Most people aren’t ever going to reach that deeper, unified understanding. I certainly won’t, most of the time! But for convolution, I feel a step closer.