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 ring Z[i]

This is a little self-portrait, "The Artist Trying to Learn Abstract Algebra", probably of no interest to anyone else.

I read an introduction to rings (in Gallian, fifth edition, which I enthusiastically recommend). Now I'm trying to come up with some conjectures and prove or disprove them before I start on the exercises. (This book has great exercises, and doesn't bother teaching anything in the text that it can teach in an exercise.)

I figured maybe every ideal of the ring Z[i] is a principal ideal generated by some element of the ring. This morning I think I have the proof. It's a consequence of Z[i] being enough like the integers to support Euclid's algorithm.

That in turn is a consequence of Z[i] having something like integer division. You can define a well-ordered metric M on Z[i] such that M(0) < M(a) where a is any other element; and for any a and nonzero b, there exist a quotient q and remainder r such that a = bq + r and M(r) < M(b). That the domain of M is well-ordered implies that Euclid's algorithm terminates.

Z[i] also has something like prime and composite elements. For example, 5+i can be factored into (1-i)(2+3i). I wonder if these two properties are actually the same thing.

I think the ideals of Z[i] generated by "prime" elements are prime ideals.

A storytelling game

I've played this game a few times now and have really enjoyed it. I've only tried it with two players. It might work with more.

Rules

Each player starts by jotting down a very brief story outline: just five lines. Each line should be seven words or less. It's OK to steal the outline of a familiar story, as in the example below.

1. a girl in red
2. a wolf with a plan
3. the wolf eats grandma
4. the wolf's disguise is unconvincing
5. someone saves the day with an axe

Or of course you can make up your own. You do not have to stick to fairy tales.

Each player passes their story outline face-down to another player. (No peeking!) Then the game begins.

From here on out, it's very simple. All the players are going to cooperate to tell one big story including all the elements in all the story outlines. On your turn, look at the first line of the story outline that someone placed in front of you. Suppose it says ‘a lonely house on a hill’. Tell the first little bit of the story. Keep it brief, and be sure to work in a lonely house on a hill.

Then it will be someone else's turn. Perhaps they will read: ‘a girl in red’. They will pick up the story where you left off, adding a girl in red.

Take turns for five rounds.

The end.

A few game design notes

I haven't played many storytelling games. This game mostly tries to avoid the mistakes of Once Upon a Time, which I greatly enjoyed back in the day, but which has some flaws. It features actual competitive gameplay and a winner. But you don't win by telling the best story. You win by getting all the cards out of your hand.

The fixed deck of character, setting, and item cards in Once Upon a Time got boring after just a few games. At some point, the cards stopped helping. Instead I would find myself trying to scrape together yet another storyline involving a shepherdess, a ring, and a disguise. (I have not played Nanofictionary, but it seems like it might have the same problem.) It was usually fun anyway. But I think having the players instead supply fresh material for each game might help. And story outlines seem better than cards in other ways. They naturally provide characters and exposition early in the game, plot developments in the midgame, and endings at the end. They can revisit previously introduced elements instead of constantly adding more folderol to the story. And they're fun to make.

The endgame in Once Upon a Time was especially unsatisfying: as soon as a player could play all the cards from his hand, he would turn over his Happily Ever After card and crash-land the story into the overly specific, predetermined ending printed on it. This would be something like, ‘...and for all I know, they may be dancing still.’ Constraining the last words of the story just seems like a mistake. In this game, everyone gets to tell an ending, and when it's your last turn of the game, you know it. So every player gets their crowning moment, and loose ends tend to get tied off.

Stoicism, Christianity, and Mother Goose

I read the Handbook of Epictetus. It's very brief, just a few pages really. I'll quote a few paragraphs that should make it clear what Stoicism is about. (I'm quoting a recent translation by Nicholas P. White which I really like. The translations I found on the Web seem stilted, or florid, by comparison; though Higginson isn't bad. Of course you can try the original Greek.)

Some things are up to us and some are not up to us. Our opinions are up to us, and our impulses, desires, aversions—in short, whatever is our own doing. Our bodies are not up to us, nor are our possessions, our reputations, or our public offices, or, that is, whatever is not our own doing. (1.)

You are foolish if you want your children and your wife and your friends to live forever, since you are wanting things to be up to you that are not up to you, and things to be yours that are not yours. (14.)

If you are fond of a jug, say “I am fond of a jug!” For then when it is broken you will not be upset. If you kiss your child or your wife, say that you are kissing a human being; for when it dies you will not be upset. (3.)

Do not seek to have events happen as you want them to, but instead want them to happen as they do happen, and your life will go well. (8.)

The best of it is extremely well said, but the content is troublesome. If the sample above does not convince you that you should prepare yourself to be unperturbed when your wife and children die, then the rest won't either. Still, given that you're reading this, something of Stoicism is very likely in you. It's in Western culture.

I guess none of the quotes I picked addresses the evident problem of whether a Stoic may act, or whether he must be distant and docile at all times. The Handbook doesn't seem to offer a head-on answer. Modern Christianity's interpretation of Stoicism does, though.

God, grant me the serenity

to accept the things I cannot change;

the courage to change the things I can;

and the wisdom to know the difference.

—Reinhold Niebuhr, around 1944.

The wisdom, that is, to know what is up to us and what is not up to us. It is exactly this wisdom that is on offer in Epictetus: nothing is up to us except how we see things and how we comport ourselves. I think I like the modern Christian philosophy, demanding as it is of courage and wisdom, better. But then, I tend to like messy, perilous things in principle even when they are not so enjoyable in practice.

The Wikipedia article on the serenity prayer offers this lovely postscript. I have no idea why this rhyme is not a widely-known classic.

The philosopher W.W. Bartley juxtaposes Niebuhr's prayer with a Mother Goose rhyme (1695) expressing a similar sentiment, but without comment:

For every ailment under the sun

There is a remedy, or there is none;

If there be one, try to find it;

If there be none, never mind it.

Sonnets

Once I wrote two lines of what would have been an awesome sonnet.

Shall I compare you to my friend Matt Jones?

You are more lovely and not half so drunk.

In San Francisco, while everyone else was napping, I managed to sneak out to City Lights Books. There I stumbled on Sonnets by Giuseppe Gioacchino Belli, translated by Mike Stocks. Translated. Imagine translating sonnets. I opened it up and read one, and it was pretty good, so I bought the book.

I emerged unconvinced that translating sonnets is possible or sane. I don't know what I expected. The translations are real sonnets that rhyme and scan, an amazing technical achievement already, and they have a nice spontaneous feel. Not everything works. Many lines, inevitably, are awkward. The English slang Stocks uses is so different from what I grew up with as to sound inauthentic (a real shame).

Oh but what a pleasure to be introduced to Belli. He wrote these sonnets in Rome in the 1830s. If half of them are as true to life as they feel, it was a city alive with outrageous characters, illicit sex, and poor anger management. The best ones are little candid character sketches, slightly or heavily satirical, of harried mothers, old men, annoyed girlfriends, corrupt priests, and so on. There's one from a furious stutterer. Reading them is like eating cupcakes while falling in love.

I'll post one of the translations that I like, with the warning that this one is atypically clean.

A Very Roman Pastime

The treat that us lot liked the most when small,

the biggest thrill, the real McCoy, the biz,

was finding new-built homes and palaces

and using lumps of coal to trash the walls.

So here we'd doodle numbers, little sums,

and Gordon knots and those of Sollymom,

and there some Lotto stuff; and then move on

to filthy words and pricks and twats and bums.

Or else we'd take a stone or nail or stick

to gouge the plaster out, and draw a pic

so deep we'd hit the bricks and stuff below.

Those were the days all right, my God. Although,

that said, I like to dabble still, it's true...

and when I see a nice white wall, I do.

At the moment, a few more translations are posted on the book's web site.

Recently I learned...

• Your browser uses the public suffix list to determine whether two web sites may share cookies. This is not very robust but better than the previous strategy.

• If you take a long strip of paper, fold it in half as many times as you can, and unfold it, it'll make an approximation of the fractal shape called the Heighway dragon.

• If you take two fractions, say 1/2 and 1/3, and add the numerators and denominators, you get the mediant, in this case 2/5. I don't know much about the mediant, but it is linked to Ford circles in a way I don't really understand, and I'm told mediants give a startling way of approximating the value of continued fractions.

• glibc's qsort only does an actual quicksort as a last resort. If there's enough memory, it does a merge sort.

  This is old news, and I kind of figured it was the case, but I never looked at the source code before.

And I was reminded that complex numbers are key to quantum mechanics, something I missed when I was writing about complex numbers a year or two ago.

We've been unpacking books that have been in boxes for a year and a half. It's like meeting old friends. I read the epic of Gilgamesh, the book of Joshua, and Six Easy Pieces. I love small books.

Learning in games vs. applications

I was talking to one usability expert and he was describing how they measure task completion. Did the user press the buttons in the right order? Their ideal app resulted in new users completing tasks 100% of the time. This isn’t exploratory learning. You need to be able to fail and explore the possibility space of a particular tool. Through repeated failure and success, users build up robust skills that can be applied successfully in a wide variety of situations.

—Danc, Mixing Games and Applications (PDF)

Huh!

Well, maybe that's just what you'd expect a game designer to say about application design. Amusing slides anyway. The blog is Lost Garden.

Handlebars

Hey, listen to this song:

Listening to the first 15 seconds, I didn't think I was going to like it. The reasons I like this song are interesting enough to write about at length, but... oh, just hit the button.

In which something is beeping

Several years ago I predicted that in the future there will often be a mysterious beeping which no one can identify or locate. This has happened to me several times since then, but tonight was special. Something kept going BEE-doo loudly, about once every two minutes. I suspected the smoke detector and my laptop before tracking the noise to a cell phone.

It was 3 AM. My house is full of guests trying to sleep.

J. listens to a story

I started to read J. the story of Moses last night. As a kid I probably scorned story bibles, but someone gave us this one when J. was a baby, and flipping through it I was pleasantly surprised. Now I think you can probably do a whole lot of deleting and clarifying without hurting much. You do lose the texture of the Bible. But we've already read the kids D'Aulaire's Greek Myths (miraculously told without any hint of debauchery) and Julius Lester's version of the Br'er Rabbit stories (retaining occasional cruelty but excising the stories' roots in a culture enslaved). So much for texture.

Still, just setting the scene for the story of the little baby in the bulrushes requires a whole new vocabulary of evil and suffering. J. did not know the words slave, taskmaster, or anguish. Also I forgot that the first thing of note Moses does is murder someone.

When J. is surprised, he doesn't look surprised. Just very thoughtful.

I think we'll proceed. J. likes being challenged. And, though I am not religious, there's a message here. The Israelites are not a particularly well-behaved people. Their relationship with God is anything but smooth. Sometimes they are stubborn and refuse to do as God tells them. Sometimes God abandons them to the consequences of their actions. Sometimes He punishes them rather arbitrarily (it seems to me). At one point they get a “time out” that lasts forty years. He never stops loving them.

Did I just write a whole paragraph comparing myself to the Almighty? I'm just saying I can relate. I expect J. can relate too.

J. writes a song

The handwriting is our babysitter's.

Junk and your unconscious mind

It's wonderfully easy to contribute on the Web, and as an unfortunate side effect of this essential fact, hoaxes, cranks, and general nonsense abound. I'll euphemistically call this stuff “junk”. Because so much of what you see online is junk, smart people such as yourself develop a finely-tuned junk detector. This is fine—in any case it's important to have one if you plan to use the Web for anything serious.

But your junk detector is probabilistic, factoring in grammar, habits of speech and writing, vocabulary, the writer's opinions and personality, whether the page has pictures of kittens—anything but the actual argument itself, because the whole point of the junk detector is to avoid wasting the time of reading it. In other words it's like the worst possible use of ad hominem, a logical fallacy. You guess as much as you can about the author, then judge the value of the page based on that. I see no good way around this. Consequences:

Your main way of evaluating the quality of Web pages is subconscious.

The junk detector is not as accurate as actual critical thought.

False positives mean the reader misses out and the writer fails to connect (making good writing skills more important now than ever before).

False negatives mean you may be duped: the junk detector doesn't protect you from lies, logical fallacies, or really sophisticated forms of “junk”. By the time you decide to read the whole page, the junk detector is done working. Another, smarter junk detector had better kick in!

All of this applies in the non-Web world, too, but the Web is so full of junk, and it's so hard to avoid altogether, that the cheapest possible junk detector is highly rewarding and can instill a false sense of confidence.

Structure

On the Web, alternative reading to whatever you're looking at is never far away. There are even links in most Web pages, forever calling you to random-walk. The result, for the reader, can be a haphazard adventure of reading, interesting at every point but without overall purpose.

The result for writers is that time spent organizing thoughts is usually wasted—nobody wants to read all that. Instead, you write one thought per day in a blog, or contribute to sites like Wikipedia, which generally rejoice in the Web's random-walk nature.

Sometimes it is an unexpected pleasure to open a book and follow the development of a big idea over many chapters. I got this kind of feeling from the mathematics textbooks I mentioned a few days ago.

It's weird for me to even be saying this, because I like that the Web is deeply interconnected and wild. But the Web doesn't seem to generate good content with large-scale structure—the kind of stuff that I find most rewarding to read.

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.

The very best of jorendorff?

I like Language Log, but I would like it even better if there were less of it.

Wouldn't it be keen if there were a site where you could enter the URL of any blog, and it would give you back a feed containing only half the entries—the best ones, according to whatever metric of popularity the service could find (links, diggs, whatever).

I proposed this on IRC, where mhoye and humph reacted with a definite meh. (Note: All these chat excerpts are edited to give the illusion that there's a single coherent conversation going on.)

<humph> what's popular and what's interesting to me are not often the same for me

<mhoye> humph++

<mhoye> That sounds like a good way to be drowning in mediocrity, for sure.

<mhoye> jorendorff: Apply your theory to popular music.

...

<ted> so your theory is that if you like a blog enough to subscribe, you would like it even more if you only got the absolute best posts?

<jorendorff> ted: my theory is that "absolute best posts" means something

<mhoye> God, no.

<mhoye> See also, "absolute best music", "absolute best paint color."

I failed several times at explaining why I think this. Let me try again here.

Simple ratings systems are common on the Web. Some, like the Slashdot comment ratings (“Score: 5, Insightful” and such) perform very well. Others, like online restaurant guides, are useless. Ratings work when users agree on what's good and what's bad. On Slashdot, the worst posts are pretty content-free. Subjective tastes don't even really enter into it. Restaurants are a different story. In the case of music, mhoye's example, I'm sure any two people can find plenty to disagree on. But:

<jorendorff> mhoye: do you have a favorite band?

<mhoye> Not just one!

<jorendorff> mhoye: I'm struggling to get you guys to engage on any specific example :(

<mhoye> Jorendorff: Ok, here. "Entertainment", by "Gang Of Four".

<jorendorff> mhoye: excellent - what are your favorite songs off that album?

* mhoye picks "I Found That Essence Rare" and "Anthrax"

Both of mhoye's picks are among what the Apple Store calls the “TOP SONGS” from that album. Both are mentioned in Apple's review. Maybe mhoye picked them because they're the best tracks on the album.

Counterexamples abound too. We could settle this scientifically by sampling a blog's audience, having those people rate posts for a while, and seeing how closely their ratings correlate.

Instead, let's play a silly game. See if you can stand to read these two entries from my old writing journal: Zen in space and the swoon. I believe one of those is about as good as I can write and the other is flat-out bad. I furthermore immodestly claim that those are two different things! And I think you might agree with me on which is which. We'll see (if you're willing) in the comments.