Teaching Yourself and Others

In his wonderful new book “Proof and the Art of Mathematics”, Joel David Hamkins asks a question about irrational numbers that stopped me in my tracks:

Prove that the irrational real numbers are exactly those real numbers that are a different distance from every rational number.

I reacted to this problem in three stages:

  1. OK, sure.
  2. Wait what?
  3. Oh cool! Why?

It took me a moment to understand what he was asking, then a moment longer to think about how to approach a solution. In the end, what helped me make sense of it was a problem that I’ve used to teach fractions to my 3rd and 4th Graders. These sort of connections between the math of young students and of sophisticated adults is mathematically exciting. It gives me the same sort of “oh man that’s beautiful” buzz that mathematicians sometimes use to describe their realizations.

The question comes from Marilyn Burns’ fraction books: Put two fractions on the number line, maybe 1/4 and 1/2. What number is exactly halfway between them?

Lots of students would say “1/3.” This isn’t how fractions work, though. (The jumps from 1/4 to 1/3 to 1/2 aren’t constant, an early example of non-linear growth.) This problem often catches my students in stages just as the irrational number question caught me: Oh, it’s obvious! Wait what now?

The question mark is half of the distance from 1/4 to 1/2. That makes it 1/8 away from 1/4, and 1/8 + 1/4 = 3/8.

It’s possible to generalize this result. I have good memories of pre-teen students at math camp, huddled around a chalkboard and trying to express this result algebraically.

The actual solution to this isn’t particularly important at the moment. The point is that you can always find the distance between two fractions, and that distance can always be expressed as a fraction. And half of that distance? Again, a fraction. Pick the lower place on the number line, jump ahead by that half-distance and what do you get? Fraction, fraction, fraction. It’s fractions all down the line. As long as you start with two fractions, you get one in the middle.

Fractions are also known as “rational numbers.”[1] So let’s take another step closer to the original problem, which was about the distance between an irrational number and any rational numbers.

Start with a number on the number line. If it’s rational, then you can write it as a fraction. Then reach out in one direction to another fractional/rational number. What’s waiting for you in the other direction?

Again, you can compute the answer using the distance between 2/5 and 1/2, What’s significant is that this other number will also be … wait for it … a fraction. No way out, they’re everywhere. You’ll be able to express the distance between 2/5 and 1/2 as a fraction, you subtract that distance from 2/5 and … you get the picture. Fractions, across the sky. But what does it mean?

This time, start with an irrational number – something like the square root of 2 – and stick it in the middle slot. Then, stretch out in one direction to reach a rational number. What sort of number is waiting for us an equal distance in the opposite direction?

Suppose for a moment that, as in the 4th Grade question, the other number turns out to be a rational number. Then at the left and the right are rationals. Remember the pre-teen version of the question: if the left and right numbers are expressable as fractions, there is a formula for finding a rational number exactly halfway between them. And that would mean the square root of 2 is rational. Which, no. So! That other number is irrational.

Another, more precise way of saying the above: Let m be an irrational number and suppose it is the same distance to two rational numbers. Then m is exactly halfway between two rational numbers, and based on Marilyn Burns that too is rational…hey no way, man, it’s irrational! So irrational numbers are a different distance away from every rational number, which means if a rational number is on an irrational’s right side, there isn’t a rational number an equal distance to the left. And vice versa.

(Strictly speaking this is only half of what Hamkins asked us to prove. The other half is showing that every number that is a different distance to every rational number is irrational.)

A lot of mathematics seems obvious in retrospect, and this was one of those times for me. That’s also the case for the connections to what I teach my elementary and middle school students. Duh, fractions are fractions and this was a problem about fractions. (Fractions, EVERYWHERE. Let that haunt you.)

What I find wonderful is that when you’re teaching you never know what seeds you’re planting. That’s supposed to be a truism about teaching kids, but it also seems true when applied to yourself. Every new idea you share with someone is an idea that might be useful to you, the teacher, in some new and unexpected context. In a very real sense, teaching is also sometimes teaching yourself.

[1] There is probably a nit to pick about rational numbers being more precise a term than fractions in this context, but I’m going to juuuuuust slip right away from that conversation if you don’t mind.

People Actually Really Like Mathematics

Sure, people tell you that they don’t like math. They’ll say they’re bad at it, that they hate it. That they’ll do anything to avoid it, that the very thought of it gives them the heebie-jeebies.

Don’t believe them for a second, though, because actually? Everybody likes math. The proof is that even people who profess to hate mathematics do a lot of it by choice, for fun. The issue is what people see as “doing mathematics,” and how disconnected what they enjoy is from what happens at school.

There may be other examples of popular mathematics, but mostly I’m thinking of games and puzzles.

Mathematics is the art of necessity, but not everything is necessarily true. Some things just happen to be true. My name is “Michael” though my parents could have named me “Marvin” or “Melrose.” Other things have to be true; they’re forced to be the way they are. These are the inescapable facts of existence, the ones we can’t wriggle out of. So it’s not some sort of coincidence that ever since I’ve had children, I’ve been a parent. And you may not be shocked to learn that ever since I’ve been a resident of New York City, I’ve lived here. Much as I wish at times, there’s no escaping your home as long as you live there.

One of my favorite games to play with students is Mastermind. It’s a game of code-breaking and logic. It’s a game that puts you in direct contact with necessity and possibility, and I use it in class to help students grasp the difference between those two kinds of reasoning.

It’s not a “math” game. It’s a board game, the kind of thing my friends’ parents would have in their basements when I was a kid. But it’s all about logical necessity. That’s not science. It’s not a sport. If anything deserves to be called math, these sorts of games do. What else would they be?

And once we’re on the lookout for logical necessity, it’s all around us. It’s printed in the newspaper (KenKen, Sodoku), sold at the toy store (Guess Who?), and built into our computers (Minesweeper). The reason it’s everywhere? People love this stuff. They love how it makes them feel and think. It’s mathematics; they like it.

The point of course is that very few people recognize this in their school mathematics, which is dominated by a different experience: that of carefully following steps.

Now is that so bad? Absolutely not, people also love carefully following steps. They love assembling LEGO cars and (even if they won’t admit it) IKEA furniture. There’s joy in successfully executing a tricky procedure…

…but it’s not quite the same as thinking logically, is it? And why can’t people experience both in school?

I’m getting a bit too close to preaching to the choir for my own personal comfort. What I’m trying to emphasize is that there’s no reason to define “mathematics” as an activity that is identical with the math that people study in school. That’s entirely artificial. True, all definitions here are pretty artificial. Still what’s distinctive about mathematics to me is its obsession with what is logically guaranteed to be true. And that’s an obsession shared by billions of game players and puzzle solvers around the world.

Mathematicians and educators can sometimes be heard asking, how do we get more people interested in mathematics? And I think the answer is, mostly they already are. The question is whether we and our institutions are interested in their mathematics. And mostly, we aren’t.

The Kind of Help My Son Wants

For the last few days my morning has begun at 6:00 on the dot. That’s when my son swings open the door to our bedroom and asks for help with a math question. Fielding questions from a small child who wakes you up and demands help an inch away from your face seems like a good way to sharpen your classroom skills. It’s like some bizarre emergency preparedness drill. This is a million dollar teacher-training idea, but for you I give it away for free.

On the other hand, all that time in the classroom is decent prep for dealing with some of these parental duties. And it gives those of us with young children an N = 1 perspective on a perennial teaching question: what are kids looking for when they ask for help? And I think the answer is that in that moment what they want most of all is to understand. They aren’t primarily interested in having their own original thinking validated. And the implication is that telling kids they’re wrong is mostly trouble when the kids aren’t able to quickly grasp what was incorrect.

This is a bit of a story, but stick with me.

When my son barged into our room this morning, he told us he had two math questions he needed help with, 16 x 9 and 16 x 12. I told him that this was great, but to be quiet so that he didn’t wake up the baby.

Once I picked up the baby (baby woke up) I told my son than I thought 16 x 10 would be a good place to start, since it was so close to 16 x 9. I asked, do you know 16 x 10? He told me it was 160.

Fantastic, now we’re cooking. This is a conversation I’ve had approximately ten billion times in my life as a 3rd and 4th Grade math teacher. I took the next step. Would 16 x 9 be bigger or smaller than that, I asked? He thought and said, smaller. Yes!

I went in for the kill. (Uhh so to speak.) 16 x 9 is smaller than 16 x 10. I asked, how much smaller? This is the hard part, the part that puts a lot of pressure on one’s conceptual understanding of multiplication. If 16 x 10 is ten groups of 16, then we can take away a 16 from 160. If 16 x 10 is sixteen groups of 10, then we have to think about what happens when all those 10s turn to 9s. Without practice thinking this way, kids tend to shrug and guess.

That’s what my son did. He said that we should take ten away from 160. He had zero confidence at all in this. He asked if he was right, and I said it wasn’t.

Now what? I knew that the thing to do would be to draw a picture. Unfortunately, I was horizontal and undressed. I started talking about groups of sixteens … I was shut down by my son who, after all, is six years old and who I recently overheard telling his kindergarten teacher that he is inspired by “vehicles.”

The boy started complaining that he already knew the answer anyway. Cool, I said. So you know the answer is 144. But then he got really annoyed at me, telling me I had spoiled the problem for him. And then I tried to walk this back. I realized that his own fragile competence was on the line. Also he gets loud when he’s worked up and the clock now read 6:01.

Anyway, when he came in he had also said he wanted help with 16 x 12, and I casually changed the subject to that problem. I dragged my sorry butt out of bed and to the table. I poured him a bowl of Cheerios and grabbed a piece of paper and a marker. Then I drew a rectangle in green. I told my son I was going to draw a picture of 16 x 12, but start with 16 x 10. Then I asked him which was bigger, 16 x 12 or 16 x 10, and then how much I would have to add to the green rectangle to make up the difference. This all went much better.

Why did our talk about 16 x 12 go better than 16 x 9? Clearly things started going south when I told him that subtracting 10 wasn’t right. But I also corrected him on things while we were talking about 16 x 12. He told me the wrong dimensions of that green rectangle at first, for instance.

There were two differences when we talked about 16 x 12. First, he was loading up on breakfast cereal and riding a blood sugar high. Second, he understood what I was talking about when I corrected him. That’s thanks to the image. A wonderful thing about people is we turn correction into self-correction. You can’t help but take some credit for recognizing your own mistake, even when someone else points it out.

But when I corrected my son the first time he had no idea why he was wrong, and because of that he freaked out a bit. It’s OK to tell my kid that he’s wrong, but if he doesn’t understand why it’s not going to go well. And it’s the same with students. Don’t focus on whether you’re honoring their original thinking or not — they wouldn’t ask for help if they didn’t want access to your expertise. Focus instead on whether you’re connecting what they don’t yet understand to what they already do.

Is this all there is to it? No. The same boy I’ve written about above just asked for help building something with his toy magnets. He was trying to build a porch, but it kept collapsing. I had an idea and showed it to him. I have to say, my solution was pretty clever. I’m great at toys. The boy took a look at it and decided it was “stupid” and “bad” and that I was “dumb.” Only some of which is true! (It was a great porch.)

It’s not all about understanding, but a lot of it is.

What if the world decided that every kid needed to learn how to solve crossword puzzles?

It was March 2020, we were stuck indoors, and I was looking for something to take my mind off of [aaaaaaaaaaaaaaaaah]. My wife is a big crossword fan, so I subscribed to the New York Times Crossword. And now? I’m hooked!

I’m not very good though. For the uninitiated, Monday is the easiest puzzle of the week. Each day is supposed to get a bit harder. Saturday is the hardest, arguably — Sunday is bigger but not trickier. I can get through Monday and Tuesday all on my own, but by Wednesday I almost always need help, either from my wife or “autocheck” (which tells you if you’ve made a mistake).

This has inevitably led me to this urgent question: What if instead of math and reading, every school in the country was tasked with teaching kids to solve crosswords?

In A Mathematician’s Lament Paul Lockhart has a similar thought experiment about music. “A musician wakes from a terrible nightmare. In his dream he finds himself in a society where music education has been made mandatory,” he writes. Lockhart is a present-tense dystopian and he teases out the likely result of mandatory music: drills, memorized rules, joyless schooling in “music.”

But that’s not what I’m interested in for this post. Sure, school would probably ruin crossword puzzles. But what if you really did want to help someone get better at solving them? What sorts of activities would you do?

Here is what I came up with.

Learning Activity #1: Learn the Most Useful Words

Schools currently teach vocabulary, and if we’re going to reorient the educational system around crossword puzzles we’re going to have to learn some new vocab. But which words?

Good news: not all words are equally likely to appear in a crossword. There are various online rankings of the words that most frequently appear in the NY Times puzzles, with “ERA,” “AREA,” “ERE,” “ONE” and “ELI” topping the list.

Those are fairly common words (sort of) but there are other ways of finding the most important words to teach our alt-world students. Noah Veltman has defined something called “crosswordiness,” which measures how much more frequently a word appears in crosswords than it does in books, in general. Topping Noah’s list are the words “ASEA,” “SMEE,” “URSA” and “SNEE.”

From Noah’s website

The best bang for our buck would be to design a curriculum that takes students through as many of these “crosswordy” words as possible. We would need to think carefully about how to group them in meaningful ways, but once the words and their meanings are introduced there are a lot of ways to practice.

The most crosswordy words must have dozens, scores — even hundreds of clues that hint at them. Students would need to practice these clues outside the context of the puzzles. Flashcards, quizzes, matching activities — whatever it takes to teach the crosswordiest words.

Learning Activity #2: Complete the Word

As a sort of experiment in self-improvement, Max Deutsch tried to spend a month getting much better at the NY Times crossword. One of the tools he created to do this (he’s a programmer and entrepreneur) was a “guess the missing letters” tool. (He calls it “a less exciting version of Hangman.”) Why is this useful? He writes:

For example, if I know a particular six-letter answer is I _ O N I _ , would I be able to guess, just based on word shape and my knowledge of the English language, that the last letter is most likely a C, giving me I _ O N I C, and then subsequently guess that the second letter is also a C, giving me ICONIC?

This seems useful. Let’s get every student in our alternate universe to do this too. We’ll also want to coach them in using this technique to push forward when they’re stuck on a puzzle. Which leads us too…

Learning Activity #3: Complete the Puzzle

Without a doubt, you have to try puzzles to get better at solving them. But if you always practice with blank puzzles, you’ll disproportionately practice the beginning of solving them. Why? No matter where you get stuck, you always will start the puzzle. If you are successful at the first stages of solving the puzzle then you’ll get practice on the middle and end of the game. But if you don’t get very far, you’ll still practice the beginnings — just never anything deeper. (Teachers: sound familiar?)

To get around this, teachers would assign students partially completed crosswords to solve. There is a limited amount of strategy that goes into solving a crossword. Starting with a puzzle that has already been partially filled in would give us a chance to talk about choices with kids.

We’d show students puzzles such as these and say things like: using the partially completed corner, complete 1 and 14 across, and then as many other clues as you can.

Of course, there are a lot of other things we would probably do. We’d ask kids to solve puzzles together on whiteboards, and we’d ask them to make their own puzzles at times. We’d have worksheets of puzzles and we’d give our honors classes 100 x 100 puzzles as extra credit. We’d field phone calls from parents worried that they tried to do a Tuesday — a Tuesday! — with their daughter and she wasn’t able to do any of it. There would be crossword tutors (“I have an Ivy League degree in Applied Puzzle Engineering”) and kids would write “LEE” on their finger nails before the big test.

Best of all, we’d get to read Op-Eds about whether kids really needed to learn crosswords these days: “Are Crosswords Outdated?” After all, in the 21st century most puzzles you need to solve don’t even have words at all…

In other words, this is an educational world that would likely be as dumb as our own. Which makes me wonder whether I really should try to get better at crosswords or not. Maybe part of why I enjoy them is because I don’t really care how good I am at them. So what if I suck? In life you need have things that you don’t have to be good at.

But, I have to tell you, I can’t stop looking at that list of “crosswordy” words.

Some books I loved in 2020

It feels like giving up to write a post like this with a week left to go in the calendar year. But what’s so bad about giving up? Anyway, I’m in the mood for comfort reading at the moment. That means I’m flipping between Kate Atkinson’s Life After Life, Rob Sheffield’s Dreaming the Beatles and Susanna Clarke’s The Ladies of Grace Adieu. No reason to put those on this list, though.

I’m typically a big subway reader which … well, you know. Even this fall, when I was back teaching in-person, I became an e-bike commuter unless the weather was truly awful. Now I’m on paternity leave for a few weeks, which demands its own sort of reading habits — morsels rather than feasts, is what works for me while caring for a baby.

OK, whatever, here are some books I loved reading in the order I read them.

The Power of the Dog, Don Winslow

Amazon.com: The Power of the Dog (9781400096930): Winslow, Don: Books

I think this review nails the situation with Winslow’s flawed but wonderful writing about America’s War on Drugs. “The book is trope-heavy, stereotype-heavy, occasionally (okay, often) one-dimensional,” which is true for all of Winslow’s writing. But why is it great? “Because it is a huge, meticulously researched book that comes at the end of a series 20 years in the making. Because it is a book that eschews flowery language for precision and quick action. Because the internal monologues of quiet men are often the ways we are given to understand their internal histories.”

As an aside, when my older son was born six years ago, I was reading The Jordan Rules by Sam Smith in the waiting room. This summer I was rereading Neil Gaiman’s Sandman when we had to rush into the hospital for my son’s birth. And three and a half years ago when my daughter was born I was reading Winslow’s The Cartel, a sequel to Power of the Dog, in the hospital.

Silas Marner, George Eliot

Silas Marner by George Eliot: 9780375757495 | PenguinRandomHouse.com: Books

I love moralizing fiction. What makes people bad? What makes them good? How do they get better (or worse)? I find it so easy to see myself as Silas Marner, emotionally wrapped up in myself and shut off from the concerns of others. In yeshiva we studied musar to reflect on our moral situation; Silas Marner is water drawn from the same well with a different vessel. Thanks to my wife for promising that I’d love this one.

Incidentally, this was the first book I was able to read to completion after lockdown back in March. Man, what a shitty year this has been in so many ways.

Ninth House, Leigh Bardugo

Ninth House (Alex Stern #1)

Do you hate Yale? I don’t, but maybe Leigh Bardugo does. Or maybe it’s just the moneyed elite that she hates, seeing the crimes she imagines them committing.

There are a lot of books that are “our world but magic” but this one is particularly audacious in the world it imagines. A friend who knows my enthusiasm for Stephen King lent this one to me. My advice? Make sure your friends know that you like Stephen King.

How To Tame a Fox (and Build a Dog), Lee Alan Dugatkin and Lyudmila Trut

How to Tame a Fox (and Build a Dog): Visionary Scientists and a Siberian  Tale of Jump-Started Evolution: Dugatkin, Lee Alan, Trut, Lyudmila:  9780226444185: Amazon.com: Books

I was so annoying while reading this book. I read long passages of it aloud to my wife. I couldn’t shut up about it to anyone I saw outside. “How are you guys holding up during these challenging times?” my friends would ask, and I’d reply with a long rant about all the physical traits that domesticated animals show.

OK, how do I get you to read this book? Put it like this: what does it mean to be domesticated? There is a real biological sense in which people are simply domesticated apes. But how does domestication happen? Who kicks off the domestication process?

This book tells the story of a group of Russian scientists who set out to study this question. Their experiment: to see if the famously undomesticated fox can be domesticated simply through selection for tameness towards humans. The experiment had to be carried out in secret thanks to the Soviet Union’s adoption of Lysenkoism, which held that natural selection to be a Western lie.

Biology is so cool. Science is wonderful. Keep it up, humans.

A Theory of Jerks and Other Philosophical Misadventures, Eric Schwitzgebel

A Theory of Jerks and Other Philosophical Misadventures by Eric  Schwitzgebel: 9780262539593 | PenguinRandomHouse.com: Books

Schwitzgebel has long been one of my favorite philosophy bloggers. His vision of philosophy is much broader than the as some grand debate between various -isms. He asks the best questions. Should we be surprised if ethicists are unethical? Is it OK to aim for moral mediocrity? What does it take for a machine to trigger our ethical impulses?

Each of the chapters is about blog-long, which feels like the perfect amount to bite off for each of these philosophical reflections. Just enough to read in one sitting and keep in the back of your mind for the rest of the day.

Let’s Talk About Love: A Journey to the End of Taste, Carl Wilson

The writing here is at times dense but it’s also lively and accessible. The question here is easy to state but hard to answer: if taste is subjective, what does it mean to have good taste?

More to the point, if Celine Dion sucks then why do so many people love her music?

I loved this book, but it’s not for everyone. You need to have a taste for theory and Wilson’s arguments are subtle and not straightforward. I think he comes out on the idea that artistic quality is always relative to a community. Within a community, there is something more or less objective that can be called “taste,” but that’s because the community has its own reference frame. Switch to another community and the rules of taste change. When a gap in taste between the critic and some other group shows up, that’s an opportunity to learn what makes that community tick such that they love the art that they love — really, a critic should do this.

This book is funny, big-hearted and big-brained. I should probably reread it in 2021.

A Series of Recently Published Famous Books that Everybody Likes that I Also Liked

Piranesi, Susanna Clarke

Station Eleven, Emily St. John Mandel

Circe, Madeline Miller

The Three-Body Problem, Cixin Liu

Exhalation, Ted Chiang

Station Eleven: Mandel, Emily St. John: 8601422213614: Amazon.com: Books
Madeline Miller - Circe
Amazon.com: The Three-Body Problem (9780765382030): Liu, Cixin, Liu, Ken:  Books
Exhalation: Stories: Chiang, Ted: 9781101947883: Amazon.com: Books

Some books that I didn’t love quite as much as the others, but still loved: The Great Secret, Jack Handey’s The Stench of Honolulu, George Bernard Shaw’s Man and Superman, Kitchen Confidential, Small Fry, Red White and Royal Blue, Very Short Intro to the American Revolution.

It’s impossible for me to review this objectively, but my friend Dr. Chavi Karkowsky wrote a truly remarkable book about her life as a doctor helping women with their high-risk pregnancies and deliveries.

High Risk, Chavi Eve Karkowsky MD

Even though it’s not really possible to be objective … the book is objectively great! And it’s not quite as terror-inducing as you might think. Chavi focuses on the systems — systems that save lives but also take on a life of their own and can feel at times oppressive for both doctors and patients both.

And here was my favorite more technical read of the year, one that I’m still in the middle of:

Ecology, Charles J. Krebs

Ecology: The Experimental Analysis of Distribution and Abundance | 6th  edition | Pearson

Like all textbooks this one costs 7 million dollars, but you can get the international edition used for far cheaper. (And if you are a determined googler … well, the internet contains all sorts of things, let’s leave it at that.)

This text has been a bit of a revelation to me. From start to finish, it’s mathematical. Are species more abundant when the species covers more territory? What happens when species compete with each other? How do populations grow?

The math is just accessible enough for me that I don’t need to break my head over it and can just enjoy the science. I’m looking forward to reading more of it and perhaps thinking about how I can bring some of it back for students.

Helping Students Make Good Mathematical Arguments

How do you teach someone how to make a good mathematical argument? Here’s my theory:

  1. Make sure they understand the stuff being argued about
  2. Show them what a good argument looks like
  3. Help them make good arguments on their own

I think I did a better job of this in this year’s geometry class that I have in the past, so I’ll share what I did.

First, I made sure that students understood what was being argued about. This is a plea for teachers not to teach mathematical concepts or skills through proof — a problem I see happening more at higher levels of math instruction. There’s simply too much that is new for students in a proof. It’s not the time to make sure students also understand every else.

In my class this year this happened both at the level of the course and that of the lesson. I started the unit with lots of time working with the sorts of diagrams that would feature later in proofs. But I also did this within a lesson. If I wanted to show students an example of a proof, I would make sure to launch the lesson with an activity that just involved analyzing that diagram.

Here is a diagram, I’d say. I just want you to tell me as many pairs congruent sides or angles in it as you can, and tell me how you know:

Then, after making a nice list of all the congruencies in the diagram, I’d preview the structure of the proof (thanks, Catrambone):

I would then model writing a proof along these lines using the congruencies the class found at the start of the lesson.

There’s a bit more that needs to happen at this stage to make sure students have a chance to really think about this. There are a few things that I’ve tried at this stage — asking students to turn and explain each step of the proof; removing a step from the proof and asking students to explain what it needs to be; asking everyone to take notes from memory with some or all of my proof erased. I’m still working on getting this stage of my little routine down.

Following this, I shared a new diagram and asked students to once again find congruencies. When I did this in class I used this new diagram and this is what students observed:

They came up with those questions as well. I asked the class to prove that the inner triangle was equilateral, using the structure that they had studied in that first proof.

So, the first stage is making sure students know what the proof is talking about. The above is how I’ve done a better job of that this year.

The last step is trickier than it may seem. Yes, it’s “obvious” that helping students make good argument is important for learning how to do exactly that. But the emphasis I’d put is on help and (especially) good. Because in my experience, there’s strong gravity that pulls students towards arguments that are intuitively good but not up to the rigorous standards of mathematical proof.

The issue is basically this: everyone knows a lot about shapes. In fact this is not an issue — this is fantastic. I love hearing people’s ideas about shapes. But if you don’t provide the right kind of help, you’ll confuse and anger your students. The reason is because your job, as a mathematics teacher, is to teach a very particular form of mathematical argument. And students will have many wonderful ideas about shapes all their own. In past years I fell into a tricky spot where my students were baffled why their own extremely reasonable arguments for why a pair of triangles must be the same size and shape aren’t acceptable as proofs.

In case you haven’t heard these things live and in person, the human eye is attracted to symmetry and change. If you ask a group of people why a pair of lengths are the same, they will be drawn to arguments that incorporate those sorts of features. “Those lengths have to be the same because the diagram is symmetric. If it weren’t symmetric then when you moved that angle over there, that other angle would have to change also. And then the whole diagram would be broken.”

What’s the issue here? Nothing, really. It’s beautiful. It’s just not rigorous enough to pass muster in a geometry class. And while some people might be tempted to push on students — how do you KNOW that it’s symmetrical? how do you KNOW that angle will change? — my experience is that this only angers teens, who grow disgruntled that their own reasoning has been invited to the party (finally!) and then rejected as unacceptable.

Far better, I now believe, to help students make GOOD arguments, where “GOOD” is defined as “whatever is acceptable from the point of view of the course.”

In practice, this means that I have started to include a lot of visual choices for students as they are asked to form justifications. Here is a snap shot from a problem set I assigned in class:

There are now choices. Your argument will look like one of these four choices — always. Learn to use them, and they’ll become part of your language of justification. (Thanks, self-explanation literature.)

In short, students’ choices are constrained. To put it one way, especially in the early stages of learning, students are allowed to choose a wrong argument but they are not allowed to make a bad argument. (Again by bad I don’t mean “bad,” just “not ideal within the context of this course.”)

I think I’ll be working on improving my teaching of proof until the day they drag me feet first out of a classroom, but these two ideas feel like progress to me.

Two Kinds of Creative Lives

I always enjoy Austin Kleon’s writing, and this morning’s blog post on his morning routine was no exception. He’s writing about an artist’s routine for sparking daily creativity:

My method is cribbed from The Sedaris Method: write things down all day in a pocket notebook, then wake up the next morning, fill out my logbook, and then write longhand about yesterday.

When I don’t know what to write about I answer “The Best Thing” prompt or draw until I feel like writing.  (This morning I wrote about banana bread and palm trees.)

While Kleon is talking about being a writer and an artist, you might extend his recipes and routines for any sort of creative work — that could include mathematical research, making sense of a dense research literature, or problem solving. And it could even encompass learning, which I’ve long thought of as a creative act all in itself.

What are the ingredients to this sort of creative life?

  • Regularity
  • Routine
  • Dedicated time to producing something at the start of the day

… and so on.

In the world of creativity advice any excuses tend to be dismissed as just that — excuses. I have no doubt at all that this is a recipe for a happier creative life. But let’s be real for a moment: you can either have a full-time job, be an engaged parent, or spend an hour every morning with an open notebook, but I don’t see how you can do all three.

Of course, the life of the full-time writer and artist is not without its own potential distractions. I don’t mean to minimize that. And I also have no doubt that if you want to make the most of your creativity, the full-time artist is the way to go. If you want to get good at anything it takes time and sacrifice. It may not be financially comfortable.

But there’s another kind of creative life out there that many of us live on the margins of our family and work life. Art or learning never gets to take priority in this life. So what sorts of habits and routines support a creative life that’s spent between the cracks of work and family?

I don’t know. I mean, I have the dumb advice that I could give a younger version of myself. (Write on the subway. The family never owes you time for art. Leave things behind. Let yourself get obsessed and bored. Yes, keep a journal. Take the long view. Skip a day. Skip a week. Skip a month. Skip a year.) But my point is not that I have any advice on how to live this sort of life — it’s just that full-time artists are in no position to offer it.

I’d like to hear more about creativity from the part-time poets, the bus-ride novelists, the teacher painters, the subway mathematicians and Saturday night musicians. Because when it comes to life and creativity, that’s most of us.

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.

The Generalized Logistic Function and Pandemic Modeling

The logistic function was invented by Pierre Verhulst to represent exponential growth that levels off. To do this he chose the simplest thing he could think of: each additional “birth” knocks down the growth rate by an equal amount.

Exponential growth:  \frac{dP}{dt} = rP

Logistic growth:  \frac{dP}{dt} = rP(1-\frac{P}{K})

But what if the leveling off happens faster? Or slower? What if population growth really slows down after the first few generations? What if it only levels off when environmental resources really get strained?

OK, no problem: we can use basically any formula to moderate this exponential growth. The “generalized logistic function” adds a power to the logistic function that’s basically a shrug of the shoulders and a degree of flexibility. “Go ahead,” it says, “do whatever you want. As long as it starts exponential, approaches the carrying capacity, and is shaped like a nice ‘S’.”

Generalized Logistic growth: \large \frac{dP}{dt} = rP (1 - (\frac{P}{K})^n)

This function can tolerate a bit of funkiness, compared to the vanilla logistic. Note how with these parameters there’s a bit of a weird asymmetry as it approaches the carrying capacity.

https://www.desmos.com/calculator/stk54ocp5e

This generalized logistic function might better fit some S-shaped data. It adds another parameter, which is to say it introduces another degree of freedom. This amounts to wiggle room for researchers who use it. But without any particular reason to think the function should be one way or the other, it all amounts to guessing.

It’s this sort of guessing that can sometimes get you in trouble as a mathematical modeler.

Early on in the COVID-19 pandemic there was a clear need for information about the virus. What sort of spread should governments expect? Were hospitals at risk of overcrowding? How much death might the world be facing?

In the confusing weeks of March and April, one group of mathematical modelers filled the void and gained prominence above all others: IHME. They created a clear website that made predictions as to when the United States would experience shortages of hospital beds and ventilators.

But before long, the predictions of IHME came under fire from the epidemiological community. The headlines didn’t pull any punches: “Influential Covid-19 model uses flawed methods and shouldn’t guide U.S. policies, critics say.”

What were the issues? The article names a few. But the fundamental problem was this: IHME was just fitting data to curves, and any curve would do.

At the start of the pandemic, the IHME group was using a bit of software called “CurveFit” to make predictions. As you might guess, the software tries to find functions that best fit the given data. So far so good! The IHME group began with the generalized logistic function and looked for the best generalized logistic for existing COVID data.

Then, their plans changed. “We first tried building the analysis using the sigmoidal function,” they write, “we then discovered that the ERF error function provided a better fit to the data.”

Did the ERT error function fit the existing data better? I’m sure it did. But what does this function have to do with population growth? Absolutely nothing, is the answer. It’s just a shape. And for the researchers, that’s what the generalized logistic function was as well — just a shape, not an attempt to capture the underlying dynamics of a new viral epidemic. More from the April article:
“[IHME] doesn’t even try to model the transmission of disease, or the incubation period, or other features of Covid-19 … It doesn’t try to account for how many infected people interact with how many others, how many additional cases each earlier case causes, or other facts of disease transmission that have been the foundation of epidemiology models for decades.”
The message here for mathematical modelers seems to be that there is more to modeling than fitting the data. Unless we have some inkling of why a model should have the shape that it does, we should have absolutely no confidence that a fit is anything but a fluke. Then again, there’s another message here too: modeling is hard. The last paragraph of that article praises more conventional epidemiological models for their more sensible predictions:
“A different, data-driven model from researchers at the University of Washington predicts “about 1 million cases in the U.S. by the end of the epidemic, around the first week in June, with new cases peaking in mid-April,” said UW applied mathematician Ka-Kit Tung, who led the work.”
By early June about 14 million people had been infected with the novel coronavirus. IHME is hardly the only modeling group whose predictions veered wildly from reality, and it’s hard to blame anyone for that. It’s never going to be easy to make predictions about unprecedented times.

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. https://www.desmos.com/calculator/rrrqivja2w

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.

Sources:

Steven Strogatz, “Nonlinear dynamics and Chaos”

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