Why I am not a teacher

(This is in response to Why CS graduates don’t teach which didn’t sound quite right to me.)

Why don’t I quit my job and become a teacher? I’ve thought about it, but

• Money.

• It seems like public school teachers have to spend a lot of time just fighting active sabotage—whether it’s the legislature, school boards, textbook publishers, administrators, parents, students, other teachers, etc. In a lot of these stories everyone comes off badly. I hate politics.

• I’m used to being empowered to make whatever changes are necessary so that I can do my job. Teachers can’t even begin to do that. They don’t control the curriculum, class sizes, external interruptions, disruptive students, prerequisites, and most importantly they have no control over the goal (test scores). It sounds awful.

• There’s a major “classroom management” component of teaching that I would find stressful and probably unrewarding. I don’t expect I’d be any good at it either. (Money aside, this is the biggest sticking point. I probably just can’t do the work.)

• Here in Tennessee, to teach in a public school, I would have to enroll in education courses. This isn’t a huge deal, but it’s basically a lot of unpaid overtime and I’ve never heard anybody say anything nice about that kind of part-time program. Maybe I’d be pleasantly surprised. (Go on, tell me I’d be pleasantly surprised.)

Bottom line, being a programmer is easy: it’s mostly just programming. I love teaching more than I love programming. If being a teacher were mostly about teaching, I might have to give it a shot. I wish it was. But it’s just not.

…Is it? Go on, tell me it is.

Opposing thoughts on teaching

Some thoughts on teaching by Bret Victor.

I used to think that to be a good manager of engineers, you first had to be a good engineer. I could name several particular managers in support of that theory, but that’s anecdotal evidence, right? And believe it or not I have a few counterexamples too. Now I think that management is many things, and there is more than one way to be great.

Bret says that to be a good teacher of mathematics, you must first be a good mathematician, scientist, or engineer. I think the claim is way too strong, and it’s a good thing, too, because we need many times more math teachers than there are mathematicians, scientists, and engineers who want to teach. It’s easy to suspect that Bret, who has an MS in electrical engineering, is harboring a romantic notion here. The structure of the essay isn’t encouraging—five anecdotes followed by a lot of undirected personal incredulity and vague analogies.

Look, if I have to choose between a teacher who “lives math” and one who can tell when a student is perplexed and find another way to explain it, I’ll pick the latter every time.

Of course you have to actually understand the material quite well to be a good teacher. Not just well enough to pass a test on it! You have to know it well enough to know what is actually interesting about it, to invent good demonstrations, to turn it around when you need to explain it a different way, to inspire kids to turn it around in their own minds, to recognize when a student gets it. But you don’t just have to know math well enough to do those things. You also have to actually do those things.

Many engineers are horrible teachers. Many math teachers who are obsessed with teaching, and not math so much, are great. Teaching is many things.

A story with some scary parts

My four-year-old and I decided to collaborate on a book. I was surprised when she decided the story should have some scary parts in it. Like a dragon, she said. Or the dark, I said. Like a dark cave, she said.

Here's what I ended up writing.

The snow fell harder and harder. I remembered that polar bears dig dens in the snow. So to escape the biting wind, I begin to dig.

Suddenly the ice and snow collapsed. I slid. I fell.

Where was I? It was very dark. I was in a cave of ice.

I thought I saw two large, pale, gleaming eyes. Trembling, I crept closer. What was it?

It was a dragon. It saw me. I was afraid. Then I saw that the dragon was trapped in the ice, frozen in place.

The dragon's scales over its heart were warm. I could feel the heart beating, very faintly, very slowly. Thump. Thump.

I stayed there a long time.

The storm passed. I came home safely. That was one year ago. Now I am returning to find the cave again. The zoo wants the dragon for their collection.

As we got near the end, I asked her if the heroine should set the dragon free from the ice. Her eyes grew wide and she smiled and said no in an small tense voice.

Later J. read it and said it was pretty good. I wonder if he felt it was scary. The uneasy feeling I get from this story comes from all the questions it leaves open. Is the dragon a person, an animal, or a monster? Is it safe to take it to live in a zoo? Is it humane? To me those are the scary parts.

Mathematicians in training

First off, if you haven't played Set, you really should, because you're just the sort of person who would love it. It's a clever idea elegantly executed, and it just happens to be great competitive fun. I taught the kids to play, and the four-year-old has a definite edge on the six-year-old. I couldn't be more pleased.

We had a bit of a drive yesterday, and along the way we gave the kids some analogies to puzzle over. You know the sort of thing: “Ice is to water as rock is to what?” This turns out to be engrossing and surprisingly fun. (I might try it with adults sometime. I sense opportunities for nerd humor.) Here the kids were not evenly matched at all. The four-year-old would get the easy ones (cow : moo :: pig : x) but would guess random related words on the harder ones, apparently with equal confidence. The six-year-old saw more clearly what the game was about, so he was able to bring his greater general knowledge to bear.

Why is this post titled “Mathematicians in training”? Well, Set is a transparently mathematical game. There are 81 cards because 81 is 3⁴. The deck is the Cartesian product of four three-element sets. They form some kind of algebraic structure with extraordinary symmetry (of a kind I don't really know anything about—it's not a group—such that I'm tempted to get completely sidetracked here). But the kicker here is, the gameplay itself is mathematical. As far as I can tell, the only good strategy is to try to prove there are no sets.

Analogies are just little homomorphisms, which is to say, structure-preserving transformations. The idea that deep sameness is more interesting than superficial differences is more important to mathematics than numbers.

On the surface it seems like analogies are less mathematical than Set. Appearances can be deceiving.

(P.S. Figured it out. The sets in Set are the cosets of cyclic subgroups of (Z/3Z)⁴. The symmetry I was referring to above was that after you erase the underlying group operation, there's no privileged element. There are isomorphisms on the deck of cards, preserving the sets, mapping any given card to any other given card.)

Milestone

I probably learned about variables from playing around with a Commodore 64 when I was about the age you are now. But I didn't see them used in mathematics for many years, until they were finally introduced, in about 7th grade, as a tool for solving problems. Take a problem, write down the equation, putting variables for the unknown quantities, and then you have something you can solve.

A little while ago I realized that this isn't the only way, or even the most important way, that variables are used in math. Variables are used to write laws.

The other day you were getting a shower, and I told you that letters could stand for numbers, that you can use letters to write rules about numbers. The letter could stand for any number, and the rule would always be true. I wrote in the condensation on the glass:

A + 0 =

Then I stopped and said, well, A plus zero equals what? You said zero. I said I didn't think that was right, because what if A was seven? Seven plus zero equals zero? So then you said A. And I couldn't be sure but I thought you really got it. That's right, I said. I'm sure you could tell I was very pleased. Actually I was surprised and excited.

Later I wrote some math pages for you to solve. The first one said,

Here is a rule:

0 < A

Do all the numbers follow this rule?

If there is a number that doesn't, that's called a counterexample. It means the rule is false.

In math, a true rule is always true, for all numbers.

The other one had some mathematical statements on it and asked you which ones were true.

You surprised me.

(Click for a larger image.)

You got them all right. I asked you about the rule on the first page, A < 0, and you had to look up what the < symbol meant, but then you told me right away that it was false, because zero isn't less than zero.

I was amazed. I asked your mother, “Did you see those pages J. did today? What does this mean?” She wasn't surprised. “It means first-graders can learn pre-algebra,” I said, insistent.

“Some can,” she said.

She is half right: you are special; you are bright; and you are interested. But I know there are millions of special, bright, curious kids like you in this country, and I think by and large their schools are selling them short. You sure are lucky you've got me, kid. But not as lucky as I am to have you.

Lockhart's Lament

Lockhart's Lament (PDF, 25 pages) starts out like this:

Everyone knows that something is wrong. The politicians say, “we need higher standards.” The schools say, “we need more money and equipment.“ Educators say one thing, and teachers say another. They are all wrong. The only people who understand what is going on are the ones most often blamed and least often heard: the students. They say, “math class is stupid and boring,” and they are right.

and ends up like this:

How sad that fifth-graders are taught to say “quadrilateral” instead of “four-sided shape”, but are never given a reason to use words like “conjecture”, and “counterexample”. ...

Mathematics is about problems, and problems must be made the focus of a student's mathematical life. Painful and creatively frustrating as it may be, students and their teachers should at all times be engaged in the process—having ideas, not having ideas, discovering patterns, making conjectures, constructing examples and counterexamples, devising arguments, and critiquing each other's work.

The author is a bit crazed, but that just makes it more fun to read. In the unlikely case that you somehow got here while thinking math is stupid and boring, or if you've ever found yourself teaching a stupid, boring math class, take a look.

P.S. As the previous post maybe suggests, I've only recently discovered how to learn math by making conjectures and trying stuff, which is what Lockhart recommends.

In unrelated news, J. is pretty sharp at finding lines of symmetry. I need to give him a circle to play with and see what he says. (evil chuckle)

P.P.S. I got this link from humph, who is also a one-of-a-kind teacher (but not crazed).

The School Mathematics Project

JJ had me look at a set of old mathematics textbooks, and I found this.

4.1 Division and repeated subtraction

We can write 7 + 7 + 7 + 7 + 7 + 7 + 7 + 7 + 7 = 7 × 9 = 63.

(a) What is 63 - 7 - 7 - 7 - 7 - 7 - 7 - 7 - 7 - 7?

(b) What is 63 ÷ 7?

(c) Explain the connection between the last two questions.

(d) If you were to work out 65 - 7 - 7 - 7 - 7 - 7 - 7 - 7 - 7 - 7, what would you find? How would you give your answer?

4.2 Division of a whole number by a whole number

Example 11 (Method I)

If you were asked to work out 5489 ÷ 12 by finding out how many times you could subtract 12 from 5489, you wouldn't be very pleased!

	5489
-	12
	5477
-	12
	5465
-	12
	5453
-	12
	5441
-	12
	5429
-	12
	5417
	⋮

This is just the start. It would certainly take a long time. However, as you will have realized, there are quicker ways of doing this division.

(Method II)

12	)5489		Consider 5400. There are more than 400 (but less than 500) twelves in 5400. Let us subtract 400 of them all at once.
	4800	(400 twelves)
	689		Now consider 680. There are more than 50 (but less than 60) twelves in 680. Subtract 50 of these all at once.
	600	(50 twelves)
	89		Finally, we know that there are 7 twelves in 89 which if we subtract them leave us with a remainder of 5.
	84	(7 twelves)
	5

So we have subtracted (400 + 50 + 7) twelves and have 5 left over.

5489 ÷ 12 = 457,   remainder 5.

If we were dividing in order to find the answer to a ‘fair shares’ question, we would write

5489 ÷ 12 = 457 ⁵/₁₂

You will probably have recognized this method. Why?

I'll stop there. What struck me as cool about this is that it takes long division, a complex procedure which most students learn by rote, and at once (a) explains why it works (b) makes it seem simple and obvious.

The example is from SMP Book C, published 1969 by Cambridge University Press. JJ has the whole series. They seem quite good, relative to what I recall from grade school. The approach is conversational with a lot of questions. Very few paragraphs are more than a few lines long. There are exercises but no “word problems”. The books are printed in black and red ink. There are no photographs or sidebars. The subject matter is richly mathematical: very little arithmetic, which must have been a separate curriculum; but in the first few books (hard to tell but they appear to be directed at students 12-15 years old) there are chapters about things like relations, directed graphs, symmetry, counting possibilities, why a slide rule works.

The SMP stands for School Mathematics Project, a British nonprofit. They're still making mathematics textbooks.

What is a noun?

But what about earthquakes and concerts and wars, values and weights and costs, famines and droughts, redness and fairness, days and millennia, functions and purposes, craftsmanship, perfection, enjoyment, and finesse?

—Ray Jackendoff, Foundations of Language: brain, meaning, grammar, evolution

I learned in school that a noun is a word that names a person, place or thing.

A few years after that, the definition changed. In hindsight this seems creepy. It happened twice. I don't remember any explicit discussion or even acknowledgment of the change. We would do nouns one way one year, and when that time came around the next year, we would have different textbooks with a different definition. I remember the changes: first “event” and later “idea” were added to the list ...bringing nouns like earthquakes and purposes in from the cold, I guess. We regret the omission, etc.

I didn't know this until a couple days ago, but linguists apparently consider this whole approach to parts of speech hopelessly, fundamentally broken. Morally bankrupt, in fact. That a child is taught the ”person, place or thing“ definition approximately once every 12 seconds preys on the linguist's soul. It causes him to make awkward scenes at parties. Even the funny papers are bristling with painful reminders of this horrible truth.

I never noticed before, but there is a problem or two with this whole “person, place or thing” thing. All the most common words for people (you, I, he, she, they) and things (this, that, these, those, it) are pronouns, while all the most common words for places (here, there, in, out, up, down, to, from, and on and on) are adverbs and prepositions. All the other definitions I learned for parts of speech are bogus, too. I learned that “action words” are verbs; but homocide, defenestration, and touchdown are all nouns. (So is pirouette. My wife didn't believe me.) I learned that prepositions tell about relationships, particularly spacial relationships; but proximity and distance are nouns (and cover and surround are verbs!). I learned that words that describe properties of things are adjectives; but weight, beauty, shape, and color are nouns.

So what is the definition of a noun, exactly? Well, I'll tell you. I don't know. Strangely, I don't think linguists like to say! Here's a pretty good near miss by Geoffrey K. Pullum, writing in Language Log:

The way to tell whether a word is a noun in English is to ask questions like: Does it have a plural form (the terrors of childhood)? Does it have a genitive form (terror's effects)? Does it occur with the articles the and a (the terror)? Can you use it as the main or only word in the subject of a clause (Terror rooted me to the spot), or the object of a preposition (war on terror)? And so on. These are grammatical questions. Syntactic and morphological questions. Not semantic ones.

A bit vague, isn't it? That's way above average, though. Here's an honest attempt; it starts with “A noun is a member of a syntactic class…”. Until I edited it, Wikipedia's article on nouns started, “In linguistics, a noun or noun substantive is a lexical category which is defined in terms of how its members combine with other kinds of expressions.”

There's an interesting twist to how all this gets bootstrapped in the toddler brain. All the first words you learn are nouns, words for people and things in your little one-year-old world. You'll be able to put words together into sentences before you master any pronouns. That is, at the time when you're learning the basic grammar of the language, there is a semantic distinction between the nouns you know and all other words. The values and weights and costs come later.

Mozilla 401

If you know any computer science professors at 4-year colleges, please forward this to them!

Yesterday I learned that there's a professor at Seneca College (David Humphrey) teaching a course called Topics in Open Source Development. It's a fourth-year undergrad course. Students do projects for Mozilla, the open source web browser.

Mozilla contributed guest lectures from seven core developers and mentoring for the students. (!)

I met David yesterday. He has taught the course a few times now, to a total of about 100 students, and he has interesting things to say about it, which I won't try to reproduce here. But man. This is the kind of killer course I wish they had at CBU when I was there. Every CS department needs an internship-like course where students do real projects. They must be a pain to develop and run. This is one of the coolest I've ever heard of and more "real-world" than most.

If you're interested in possibly teaching this course at your school, please contact David or me (jason.orendorff@gmail.com). You can also watch the guest lectures online, view the student project list, or read David's Mozilla development crash course.