Yet Another Thing that Humans and Viruses Have in Common

When an epidemic rages through a population, at first it faces no immunity at all. The disease constantly encounters fresh meat. There is nothing that can stop it.

Eventually, the population gains some immunity, yada yada yada, the disease doesn’t spread as easily and the epidemic slows down. There is a point when the disease, on average, infects just one more person per infected person. You have heard this before. It is the “herd immunity threshold.”

What is more surprising is that even once the population has hit this immunity threshold, the epidemic continues to grow — for a time. Epidemics have a kind of momentum that pushes infections even past this threshold. If you know about this already, you probably learned about it the way I did: from reading experts discuss the COVID-19 pandemic.

I immediately liked the “momentum” but found myself having a difficult time thinking precisely about it. Then, while reading about mathematical ecology the other day, I learned something that helped it all snap into place for me: this happens with people too.

The notion of population momentum makes a lot more sense to me in a human case. Probably if I was a virus the epidemic case would be easier, but I am what I am. Wikipedia has a great exposition of it, including this handy chart:

In the first generation, the fertility rate is 4 and the 200 fertile people give birth to 400 children — some pretty robust population growth, given the age distribution of the population. Then, at time = 1, the fertility rate drops and parents have only two kids each, merely replacing the fertile population as the old population ages out (dies). Even though the fertility rate has dropped, there are still the result of the previous fertility boom at t = 0. Those 400 children are going to have two children each, and that’s going to help the population grow for a bit longer. Soon enough, though, the fertile population will just be replacing itself.

This phenomenon was first described by Nathan Keyfitz in 1971. He directs the idea to policymakers who are reluctant to offer contraception for fear that their countries will stop increasing in population. “In some countries hesitation in making contraception available is rationalized by the view that the country is not yet “full,” he writes. “Concern that total numbers will taper off prematurely is misplaced.” He goes on to explain how to calculate the total “ultimate” population once fertility reaches replacement levels.

It’s this exact same phenomenon that governs the growth of a virus, even after (say) a vaccine is introduced that brings the rate of infection down to 1. I find it interesting that some of the same population dynamics govern both humans and viruses. It suggests to me that a path towards better educating others about epidemic dynamics would be to start with human stuff.

How To Tame a Function

If you grew up around animals, you probably know a bit about their reproductive cycles. Seeing as I did not, I have been slow to learn what I know about the Birds and the Bees of the birds and the bees. In the absence of really any first-hand contact with animal life, I have had to resort to books for my basic education in animal reproduction.

But what I’ve learned has deepened my understanding not just of the nitty-gritty biology facts but of the fairly abstract mathematics of chaos.

Here’s a bit of biology I didn’t know: the rules of mating for domesticated animals like dogs and cattle are different than they are for their wild cousins. The big difference is the timing. Wild animals often have a narrow reproductive window during a certain time of the year. But the biological changes that come along with tameness somehow also bend the rules of pairing off.

Here is how it’s described in the thrilling “How to Tame a Fox (and Build a Dog)”:

“All wild animals breed within a particular window of time each year, and only once a year. For some, that window is as narrow as a few days and for others it’s weeks or even months. Wolves, for example, breed between January and March. The window for foxes is from January to late February. This time of year corresponds to the optimal conditions for survival; the young are born when the temperature, the amount of light, and the abundance of food offer them the best odds for a successful launch into the world. With many domesticated species, by contract, mating can occur any time during the year and for many, more than once.”

This is of also course true of domesticated apes, i.e. us, the human people.

Imagine an experiment in population growth. We take a small group of wild animals to a protected island. These wild animals have abundant food and no predators on the island. A pretty sweet deal, all said. We set these wild beasts loose to eat, drink and … you know, have fun.

Well, they do have fun, and the population grows. But this is a wild species with an extremely narrow mating season. They can only reproduce once a year. But when they do, they give birth to big broods. This is a “nonoverlapping generation,” and its mathematics happens in nice, even steps. We can calculate the size of each generation one step at a time:

Pop_{n+1} = Pop_{n} \times r

But as Thomas Malthus pointed out way back at the turn of the nineteenth century, a good thing can’t last. If the population grows like this, it will quickly use up all of the resources in this island paradise. In which case, the population will be unable to continue to grow.

One of the first people to put Malthus’ ideas into math was Pierre Verhulst. He described “logistic growth” (intended to echo with unrestricted “logarithmic growth”) as a simple (if arbitrary) way to slow down overpopulation. The key is that the environment can only handle so much of a species — its “carrying capacity,” K — and each additional individual in the population slows the population down by an equal amount.

Pop_{n+1} = Pop_{n} \times r \times (1 - \frac{Pop_{n}}{K})

(Ben Orlin has a very clear exposition of the logistic in “Change Is The Only Constant.”)

You may be familiar with the logistic’s famous S-shape:

However, take care! The S-shape curve is not “wild” logistic growth, which happens in strictly nonoverlapping discrete generations. No, the S-curve is the tame, domesticated growth of Golden Retrievers, Angus Cattle, and American People who reproduce with more flexibility. You could even say, in the case of humans and other such species, that their populations are continuously increasing.

Ah, no worries though. Discrete functions are just like continuous functions, minus the continuity. They’re the dots without the lines. They aren’t meaningfully different, are they?

To be sure, sometimes the discrete function makes a nice smooth S, gliding into the carrying capacity without much fuss:

Thank you to the Desmos user who had the patience to write all those compositions of functions.

Then again, sometimes our island of wild animals with nonoverlapping generations ends up bouncing around the carrying capacity, each year their population crashing or rising above that set parameter:

It all depends on that growth rate. For some values of the growth rate, this oscillation actually converges on that carrying capacity, resulting in nice agreement between the discrete and continuous cases:

Ah, but pump that rate of increase up high enough and you get the real fun, which is utter chaos:


The population almost goes extinct for a minute there, but then experiences a nice bout of exponential growth. That’s chaos for you.

Animal species themselves can be wild or domesticated. It turns out that functions can be domesticated as well. In this case, it’s the continuous version of this function that is the tamer, better behaved variety.

There are indeed animals that are best modeled by the continuous function. There was a time when some people were very gung-ho about the logistic. In the 1920s Raymond Pearl declared the continuous, nice S-shaped logistic to be the “true law of population growth.”

It’s not. Robert May pointed out in the 1970s that the discrete case, though extremely simply, exhibits a huge range of behavior. “Their rich dynamical structure,” May writes, “and in particular the regime of apparent chaos wherein cycles of essentially arbitrary period are possible, is a fact of considerable mathematical and ecological interest, which deserves to be more widely appreciated.” The wildness of the discrete function may be a better fit for species that have nonoverlapping generations — many insect populations give birth in strict, discrete steps, like cicadas that emerge only ever thirteen years.

Chaos is in some ways a very abstract phenomenon, but learning more about ecology and population growth has made it very real for me. I’d seen the logistic map many times in the past, but I don’t think I’ve ever quite understood it until I connected it to its origins in animal populations. I’m left with a lesson for my teaching: there are some mathematical ideas that just work better when you learn them in their biological, natural setting.


Steven Strogatz, “Nonlinear dynamics and Chaos”

Charles J. Krebs, “Ecology: The Experimental Analysis of Distribution and Abundance”

After three days of distance learning, here’s what I now understand

I feel like we all know the issues with online learning. If not, we all will soon enough. No need to make this post all about the problems, though it would be strange not to mention them at all. Here are the main ones:

  • A social environment is more motivating than being stuck in your bedroom alone
  • You can explain stuff, and people can ask questions, but it’s really hard to know what’s going on for kids who aren’t asking questions
  • I am also taking care of my two young biological children
  • You can’t tell what’s on kids’ pages as they are writing
  • My own two children are absolutely going to destroy our two-bedroom apartment
  • It is very difficult to show anything in to the teacher while asking a question
  • The two-year old is hitting the five-year old in the face over and over with her Elmo figurine

So, what do you do? Justin Reich had been very generous in talking through some ideas with me, and he helped give me a way to think about it. “Organize around individual check-ins,” he wrote. “When I talk to the best full time virtual school teachers, they say they spend a considerable portion of time following up with individual kids.”

Designing materials for a lesson like this is not easy. If you start with individual check-ins, there’s a nasty chain of deduction that pretty quickly leads you away from normal classroom practice:

  • You can only check-in and tutor individual students unless most students have something productive to do
  • You’re going to need a few days to individually check-in with all the kids you need to
  • That means that everybody needs something that they can do pretty-much independently for several days

It’s not really easy to design materials for that.

Well, I should say that the difficulty really depends on the group. I have a high-flying 8th Grade class that is seeming relatively easy to design for. I’ve used ASSISTments (buggy, but good) to assign practice problems from Illustrative Math, with Desmos extension problems. That’s been smooth.

But, yeah, all my other classes have been harder.

The way I’m seeing it right now, you’re looking for materials that have three qualities:

  1. The assignment lasts a couple days. This reduces planning time, and also gives me more of a window for checking in and helping my students with the assignment.
  2. The materials include clear examples and support. This is how kids are going to do the assignment more independently. Fawn just wrote a very nice post about making worksheets that start with models and then fade the support, teaching kids what correct answers look like.
  3. There are challenges at the end that are optional. Because some kids will finish things in a day and want more math, and they deserve it. If they want it. If they want to go outside and take a walk while maintaining nice physical distancing conduct, they should do that too.

The hardest, hardest thing so far has been finding time to put materials together while taking care of my own two biological children. My wife is also teaching her middle school students from home, and every evening we’re scraping together a schedule that lets us both do our jobs. I’m really so grateful to be getting a paycheck at all right now, I can’t really complain. But it’s not easy for me to put nice-looking things together.

So, here it is, a sloppy thing that I’ll share. It’s the resources that I gave my high school geometry students as their assignment this week, and it’s been going OK.


The document is here.

We were studying similarity before school closed. I had gotten as far as practicing setting up and solving proportions.

Starts with an example:

Screenshot 2020-03-18 at 8.29.20 PM

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Two things about the example:

  • I talked it through with the whole-group on Zoom, which I might as well not have done. I had to work it through one-on-one with the kids who I knew would need it. Still, it was worth the five minutes to reassure some kids that they understood this so that I could focus on the individuals. I’d say the whole-group is worth it, but not worth a huge investment of time.
  • A lot of my kids don’t have printers so they’re reading my worksheets on their phones, so I’ve been trying to write in big, clear fonts. Not sure if it’s helping, but it’s the rationale behind those huge letters.

After the example comes practice. I send kids into breakout rooms and tell them they can leave the computer and work alone if they want, but they can also collaborate. All I ask is that they remain more or less around so that I can call them back if I need to, and so far that’s been OK.

Screenshot 2020-03-18 at 8.29.52 PMScreenshot 2020-03-18 at 8.30.00 PM

Then another example:

Screenshot 2020-03-18 at 8.30.22 PM

Then, ideally, would have been some indirect measurement practice except it’s BEEN ONE HELL OF A WEEK and I put what I could find into the worksheet even though it’s not a perfect match:

Screenshot 2020-03-18 at 8.30.30 PM

Then, the extension problems:

Screenshot 2020-03-18 at 8.30.36 PM

And also:

Screenshot 2020-03-18 at 8.39.51 PM


I did something similar for my 4th Grade class, and it went better than Day 1 did.


Screenshot 2020-03-18 at 8.40.55 PM

Practice (this time with more fading of the supports):

Screenshot 2020-03-18 at 8.41.34 PM

Extension challenge:

Screenshot 2020-03-18 at 8.41.43 PM

And it went better, in the sense that I was able to have some mathematical conversations with people and figure out what they understood, what they didn’t understand, help people, that’s the whole point of this job, right? The key, when you strip everything else away, is figuring out what people understand and helping them understand it?

But you’re basically flying blind in this distance teaching because you can’t see what anybody is doing at a given moment. Maybe there’s some digital tool that can help give you a better picture of what kids understand and can do, but without that the main thing is the individual conversation. This is a structure that tries to plan around that.

Please, share if you’ve got something better. Please.

Oh, and remember how we used to share resources online and but then we stopped because blogging died? Can we bring blogs back, if just for a few months?


Addendum, 3/19/20:

There is a way to sort of do whole-group instruction that opens up a tiny window into how students are thinking. It uses the chat box.

You can change the chat settings so that all chat messages are sent privately to the host of the meeting. At first, I did this because I wanted to cut down on random chat chatter. But then I realized it’s a private way that students can respond to questions.

Here is a routine I’m finding useful during whole-group discussions:

1. Ask a question. I state it, and if possible I also write the question in the chat box. That way, everyone receives the question even if they missed it when I said it.

2. Then, everyone types in their answer to the question. As the answers come in, I acknowledge receipt. “Thanks, Emma.” “Got it, Jake.” I’ll comment on wrong answers without calling individual students out. “Careful, if you’re writing (x – 2)(x -3) that’s factored form.” I can also address individual students: “Tommy, what you wrote is fine.”

3. Then, I’ll share the answer. I’ll take questions, rinse and repeat for however much whole-group time I’ve decided on for the lesson. Attention spans for this vary by class and by age, of course. More than 15 minutes is probably pushing it, I think.

Crossword Puzzles from Beast Academy

I do love the Beast Academy books. My 3rd Graders are working on multiplication, and the Beast Academy books have these crossword puzzles. The kids love them:

Screenshot 2020-01-16 at 4.00.00 PM

(Why do the kids love them? Oh, I don’t know. If interest = “this is new” times “I can do this” then I guess this has enough going on that it feels new. And the puzzle is self-checking, which probably validates that “I can do it” feeling. That’s all I’ve got.)

Every puzzle can both be solved and studied. I’ve made it a habit to encourage my students to ask questions about the puzzles we solve, and I usually do this by sharing a question or two that I have.

My question was, can you make this puzzle using only multiplication?

And my kids’ questions were:

  • Can you make one of these crosswords only using subtraction?
  • Are the blanks really necessary?
  • Could the puzzle be smaller? Larger? Could it be 5 x 4? 8 x 2?
  • Could it be shaped like a path?

The next class, I gave students some blank crosswords and asked them to see if they could fill in the blanks in a way that worked.

Screenshot 2020-01-16 at 4.14.35 PMScreenshot 2020-01-16 at 4.13.47 PM

Which was interesting. But the kids wanted more, so I sat down to make more crosswords in the original style. The original puzzles always include mostly multiplication, and then one addition and one subtraction equation, always on the right and bottom sides.

Screenshot 2020-01-16 at 3.59.52 PM

So I set out to make a few puzzles in this style. I started filling in the boxes, and got stuck. Then I tried again — still not working. I started to get that familiar good/bad feeling that happens with math. It’s the feeling of “oh this is harder than I thought” but also “there might be something here!”

Screenshot 2020-01-16 at 4.18.22 PM

Over lunch, I interrupted two of my colleagues and recruited them into the problem. (I was happy to return the favor after they’ve done the same to me so many times.) We filled out the crossword with variables.

Screenshot 2020-01-16 at 4.22.06 PM

Using these variables, the puzzle is only possible if ac+bd =ab-cd. My colleague pointed out that you might factor this a bit and then solve for d:

d = \frac{a(b - c)}{(b + c)}

A few things about that equation:

  • It means that the whole puzzle is determined by just three of those variables.
  • d is a whole number, so (b + c) needs to go into a(b – c).

This is not a ton to work on, but suppose that the sum of b and c is chosen to be a prime number. It clearly won’t go into (b – c). So that means a will have to be a multiple of b + c.

Screenshot 2020-01-16 at 4.35.06 PM

That seems to work!

Screenshot 2020-01-16 at 4.37.50 PM

This leaves me with a bunch of questions, though. Does this characterize all the possible crossword puzzles? I feel like this finds one specific way of getting a crossword that works here, but is it really the only way? Also, I haven’t really thought about whether I could use any multiple of b + c. I think I can, just because it’s worked whenever I’ve tried it so far, but it would be better to understand why.

There’s a math textbook that I like that makes the case that there is significant mathematics that has been developed by teachers, just for the sake of having nice examples to give to students. I always like when that sort of thing happens, a nice mathematical surprise that appears sometimes when you remember to look for it.

On Knowing by Sarah Ruhl

From playwright Sarah Ruhl:

I worry that choosing the essay form implies that I know something. Because today, (it is, mind you, an extremely hot day,) I feel that I know next to nothing. Recently I met a mathematician who described himself as “the world’s leading expert on absolute and total ignorance.” Today, I stand with the mathematician. But the mathematician, while an expert in ignorance, also believed firmly and enthusiastically in the concept of progress.

And yet I must not believe in progress because I make theater. And theater, by its very nature, does not believe in progress. Because we must constantly go back to go forward. And theater, by its very nature, does not believe in absolute knowledge, because there are usually two characters talking and they usually believe different things, making knowledge a relative proposition. But increasingly in the American theater idiom we are led to believe that plays are about knowing. Or putting forward a thesis. Today, I stand humbly with the mathematician. I am not the world’s leading expert on absolute and total ignorance. But the importance of knowing nothing is underrated.

“An expert in ignorance” — this is a great perspective on mathematics, I think. It relates to some of my favorite strands of Karen Olsson’s The Weil Conjectures.

I’m loving Sarah Ruhl’s book 100 Essays I Don’t Have Time To Write, which I first learned of from Austin Kleon.

Review: The Weil Conjectures by Karen Olsson

The Weil Conjectures by Karen Olsson


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.


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.


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.”

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.

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.


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.

Screenshot 2019-09-03 at 10.14.15 PM.png

(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:

Screenshot 2019-08-25 at 1.28.42 PM

Screenshot 2019-08-25 at 1.28.55 PM

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…

Screenshot 2019-08-25 at 1.46.17 PM

…and after.

Screenshot 2019-08-25 at 1.46.27 PM

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

Screenshot 2019-08-25 at 2.05.54 PM

Screenshot 2019-08-25 at 2.06.14 PM.png

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.

Gaussians are making Gaussians

Let f and g be Gaussian distributions.

Screenshot 2019-08-23 at 12.01.12 PM.png

Go ahead, add them. You don’t get another Gaussian distribution.

Screenshot 2019-08-23 at 12.04.42 PM.png

Well, of course not. They don’t have the same mean. So set the means equal.

Screenshot 2019-08-23 at 12.06.48 PM.png

That’s no better. The sum of f and g is still very much not Gaussian.

Screenshot 2019-08-23 at 12.07.58 PM.png

So, that’s no good. But of course it failed — just look at those visuals!

What about multiplication? Here’s what the product of two Gaussian distributions with equal means looks like.

Screenshot 2019-08-23 at 12.10.36 PM.png

That looks much better!

In fact this is true: the product of two Gaussians distributions remains a Gaussian function. The only proofs I know of dive into some algebra — I like this one — but the core idea is that multiplying exponents is additive. That’s what keeps it all in the Gaussian family.

So consider two Gaussian functions, one with a mean \mu and the other with a mean at 0 (for a touch of simplicity):

f(x) = \frac{1}{\sqrt{2\pi}\sigma_f} e^{\frac{x^2}{2\sigma^2_f}}

g(x) = \frac{1}{\sqrt{2\pi}\sigma_g}  e^{\frac{(x -\mu)^2}{2\sigma^2_g}}

Their product will look like this:

f(x)g(x) = \frac{1}{2\pi\sigma_f\sigma_g} e^{\frac{x^2}{2\sigma^2_f}+\frac{(x-\mu)^2}{2\sigma^2_g}}

Making common denominators and adding through:

f(x)g(x) = \frac{1}{2\pi\sigma_f\sigma_g} e^{\frac{\sigma^2_g x^2 +\sigma^2_f(x -\mu)^2}{2\sigma^2_f\sigma^2_g}}

Might as well expand that exponent a bit and summarize:

f(x)g(x) = \frac{1}{2\pi\sigma_f\sigma_g} e^{\frac{(\sigma^2_f +\sigma^2_g) x^2 -2\sigma^2_f x \mu + \sigma^2_f \mu^2}{2\sigma^2_f\sigma^2_g}}

And then you can divide the numerator and denominator by (\sigma^2_f +\sigma^2_g) and you’ll end up with a quadratic trinomial in x. You can always express that quadratic trinomial as (x - M)^2 somehow or another.

(Brief but important nit-picky note: that would make the product of two Gaussian function, but the scale factor on the left side of the expression is off, so it’s not a Gaussian distribution. You’d have to scale the product of two Gaussian distribution in order to get another Gaussian distribution.)

Is this useful? Is this significant, in some way? I don’t know. Apparently it’s useful in applying Bayes’ Theorem, but I know nothing about that.

One thing I do know is that it makes for some fun visuals.

Screenshot 2019-08-23 at 1.09.12 PM.png