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Thursday, November 30, 2006

The Big Over Easy & The Fourth Bear

On August 8 I wrote about the unusual novels of Jasper Fforde in his Thursday Next series, which could basically be described as science fantasy. Recently I finished the first two novels in his Nursery Crime series, set more-or-less in the same universe as Thursday Next, except that they’re mystery fantasy (fantasy mystery?) rather than science fantasy. The first, The Big Over Easy, is about Humpty Dumpty, not New Orleans as you might expect. You probably never wondered who killed Humpty Dumpty, but now you can find out. It’s all dry political satire, and you’ll have a lot of fun reading it.

The second in the new series is The Fourth Bear. Did you ever realize that there’s something wrong with the traditional story of Goldilocks? How could the baby bear’s porridge be “just right” when it was the smallest in both volume and surface area? It should have been too cool. However you figure out, it should have been the mama bear’s porridge that was just right. There must have been...you guessed it from the title...a fourth bear! I won’t spoil the story for you; suffice it to say that it all makes sense. Well, sorta... as long as you suspend disbelief and accept the premises of the story. If you like Lewis Carroll and Douglas Adams, you will like Jasper Fforde.

PS: I recommend listening to these in the audiobook versions, which are particularly effective for both novels.

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Tuesday, November 28, 2006

Sudburyopoly

I am told that there is an unfortunate preposition in the game of Sudburopoly, where the first half of the slogan of Lincoln-Sudbury Regional High School is misstated as “Think of yourself,” rather than “Think for yourself.” I know a few students who would make the same mistake about Weston.

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Sunday, November 26, 2006

Connecting math with other disciplines

From time to time we try to connect our math teaching with other disciplines. Often this happens naturally — physics examples in precalculus, biology examples in Algebra II, etc. It’s no coincidence that both of these other disciplines are scientific ones: most people naturally think of the sciences as the most fertile ground for applications of mathematics. At some high schools, including Weston, the Math and Science Departments share an office, since it’s natural for the administration to expect a fruitful and regular exchange of ideas between math and science teachers. As I’ve suggested in some previous posts, there are occasionally times when math and science teachers view the world differently, but still it’s true that math teachers are more likely to share a weltanschauung with science teachers than with English or history teachers.

But what about disciplines other than science? What about English, for example? We usually think of high-school and college English as embodying both literature and writing, and it’s easy to think about writing in math classes: for the past 25 years the concept of “writing across the curriculum” has been an emphasis in the United States, and some teachers go so far as to have their math students write actual books. Literature is a tougher connection, though there are a number of resources that are helpful in that area, including Mathematics and Children’s Literature lessons, the Bridging Literature and Mathematics Kit, and many references in the Humanistic Mathematics Network Journal.

It surprises some observers that the two disciplines with which I most often make connections in my math classroom are actually history and music. But both seem rather obvious to me. (Of course they seem obvious to me, now that I’ve been using them for so many years.) Exploring mathematics (or anything) in historical context is always a useful way to provide understanding and perspective, and I try to do that in many ways, both formal and informal. At B.U. Academy we even had students read an entire book about math history, Journey Through Genius, with various chapters spread out through three years according to their mathematical relevance. And I won’t get into the musical connections in this post, but relationships between math and music have long been well-known, going back as far as Pythagoras. We’ll save discussion of that for another post.

Finally, some readers may wonder why I haven’t mentioned linguistics. It’s only because that’s not a high-school discipline, nor is it one that most students are familiar with. But indeed it’s useful to view mathematics as a language (that’s not all it is, but it is a language). Occasionally I’ll make a big deal of this connection, but usually I just slip it in as part of some other activity or assignment.

Administrators want us to make interdisciplinary connections. There are plenty of possibilities in the math classroom.

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Friday, November 24, 2006

The Rule of Four

I recently read The Rule of Four, a truly fascinating novel co-written by first-time authors Ian Caldwell and Dustin Thomason. Fascinating to me, at any rate — your mileage may vary. If you’re interested in Latin, linguistics, typography, academic mysteries, the Renaissance, and cryptography, you’ll be interested in this book. If you want to limit yourself to traditional mysteries, you’ll probably be disappointed by it.

As often happens in novels of this sort, it can be difficult to determine what’s fact and what’s fiction. This is a meta-book: a book about a book. It’s about the Hypnerotomachia Poliphili, a genuine novel published in 1499 by the renowned Aldus Manutius. Although its modern publisher claims that it is “[o]ne of the most famous books in the world,” I suspect that few of today’s readers will have heard of the Hypnerotomachia. Fortunately, MIT Press has put the entire 1499 book on the Web.

The Rule of Four is too complex for me to summarize it adequately in this space. Just read the description at the Random House site in order to see what it’s about. Then read The Rule of Four, and follow that up by reading The Real Rule of Four, a slim volume by Joscelyn Godwin, who will help you disentangle fiction from fact. Do it now!

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Thursday, November 23, 2006

Leaving the answer blank

If you don’t know the answer, should you leave it blank or should you guess? An argument can be made on each side:

On the one hand, many tests (such as the SAT and the AMC) penalize random guesses by giving more credit for a blank answer than a wrong one. On a multiple-choice test this penalty can be enforced either by subtracting a fraction of a point for a wrong answer or by adding a fraction of a point for a blank. Depending on the size of the fraction, one can calculate when it’s worth guessing — typically when at least two of the choices can be reasonably eliminated. While this sort of scoring system is rare for open response questions, it’s not unheard of. For instance, a professor who wishes to remain anonymous writes this:
In large first year classes I normally give students 40% for a blank exam and will take marks away for nonsense. They need to learn not to waste the TA’s time.
On the other hand, we teachers regularly tell students that almost any answer is better than none. If you put something down, you might earn some part credit; if you put nothing down, you’ll earn nothing. The MCAS, for example, is scored that way; it’s always better to guess.

Seeing what a student is thinking is never a waste of my time. Of course we don’t want students to write nonsense. But we want them to be willing to take risks. If we take the attitude of the professor quoted above, we are implicitly endorsing the immortal words of Homer Simpson: “If something’s hard to do, then it’s not worth doing.”

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Wednesday, November 22, 2006

Using Scheme in the so-called real world?

This is a follow-up to yesterday’s post about replying to student questions concerning applicability of a math topic in the “real world.” At Weston we get a similar question with regard to our selection of Scheme as the Math Department’s programming language of choice in regular math classes and in Intro to Programming. There are admittedly extremely few commercial applications written in Scheme or closely related languages, and it’s unlikely that a student will find a summer job programming in Scheme. So why do we teach it?

As I type this, the very question looks odd to me. In an academic high school, we don’t teach what we teach because of its short-range utility in the world of employment. “Go to a vocational school for that,” we reply somewhat snobbily. But, just as I said yesterday in regard to logarithms, that answer is rightfully perceived as non-responsive. Among the real reasons we teach Scheme in a Math Department are the following:
  • It has almost no syntax rules. Basically, you’re thinking in algebra: the Scheme world consists primarily of functions and variables. The big ideas of inputs and outputs, domain and range, and composition of functions all flow automatically from working in this language.

  • The learning curve is very reasonable: in a very short amount of time, one can learn to write an interesting program.

  • Looping in Scheme is based on recursion.
But students still ask for real-world applications, since school is apparently not part of the real world (as I pointed out in yesterday’s post). Here are a few:
  • Continue (conference management software)

  • A bunch of games, packaged with DrScheme, which is the version we use at Weston

  • Margrave (a security policy analyzer)

  • MzTake (a debugger), about which Shriram Krishnamurthi comments as follows:
    (The only sensible thing to do when faced with a Java program is to switch to Scheme.) The Java debugger was selected as one of the award papers at a software engineering conference of people who have no affection for, and perhaps a slight antipathy to, Scheme. That's because smart people recognize good ideas. You should ask your students how they fit that description. (-:
  • Abstrax (hard to describe, but definitely commercial: see their website)

  • See also Functional Programming in the Real World, for a list of more applications.
And here are some comments from a couple of leaders in computer science education, posted on a newsgroup that is relatively public, so I believe it’s OK for me to quote them.

From Boston:
I am sincerely sorry that my colleagues told you that Scheme is useful. I also won’t mention that Disney controls virtual rides with it, that the Air Force controls telescope batteries, that the Navy runs its weather service for all carrier borne jet fighters, that Motorola has an ordering system for packaging hardware, that .. oh, I didn’t mean to.

If your kids are so smart that they know now already what is useful and what’s not, you are wasting your time on them. Tell them to leave the class now. You don’t want to waste their time with stupid, useless, filthy parentheses.

P.S. Oh, and don’t tell them that with Scheme you can write a six line program and get a movie of a rocket launch in the very first lesson (instead of public static void(String argv[]) { System.out.println("hello world"); }), and everyone can understand it too. We wouldn’t want to keep them in class and waste valuable minutes on them.

;; Beginner, world.ss
(define (y-coordinate-for-rocket-at t) (- 300 (+ 20 10 (* 8 t))))
(define (create-picture t) (place-image 150 (y-coordinate-for-rocket-at t) (empty-scene 300 300)))
(define (tock t) (+ t 1))

;; run program run:
(big-bang 300 300 (/ 1.0 20) 0) ;; 20 frames per second
(on-redraw create-picture)
(on-tick-event tock)
...

The question has come up several times this semester and of course in years before. Here is how I really answer it:

“We have chosen the least popular programming language that we could find for this course. We believe that is also completely useless as a practical language and has no practical applications whatsoever. If you find one, please do not share this with the class or we have to look for an even more useless language.

Your first (and second and third) course in this beautiful subject should not be about a language, a sheer notation. Sure, you will need to master some of its vocabulary and grammar but the focus is on systematic problem solving. This is what you will take away from this course and it is something that you will remember for years to come. It will change your life. It’s more useful than mathematics in middle school and most high school grades. It’s as useful as English.

So hang in there and wait for a few weeks. If you find that this is challenging even though you know 2 or 3 programming languages already, that’s the intention. Because this is the pure essence of programming, unmarred by notation issues of whether ‘;’ is a separator or a terminator, of whether it’s object-oriented or class-based, etc.”
And from Waterloo (Ontario, not Belgium):
Scheme is a vehicle for ideas. My students in week 4 of a first-term, first-year course sail through concepts that students using Java in the mainstream courses are beating their heads against in week 6 of the next semester. I put bonus/challenge/enhancement materials on assignments in weeks 6 through 8 that provide simpler looks at topics with which students struggle in third- and fourth-year courses. The practical side of this is that Scheme is great for prototyping, getting a quick sense of whether an idea is feasible. There’s a response I sometimes give, along the lines of Matthias’s sarcasm but slightly more gentle, which goes like this: “Ever visit a gym? You were wasting your time. I’ve never seen a Nautilus machine or a treadmill outside a gym. Barbells — repeatedly lifting two balanced weights connected by a narrower piece of metal — completely contrived and artificial....”

While we can point to commercial systems that use Scheme, we cannot allow the debate to be framed in these terms. That is not the only metric, nor the most important one.

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Tuesday, November 21, 2006

Using math in the so-called real world?

So what do we say when we hear that all-too-familar question, “When am I ever going to use this in the real world?” [Grammatical footnote: logically speaking, that sentence should have two question marks at the end, one before and one after the closing quotation mark. But the official rules of American punctuation prevent me from doing that. Why? I suppose it’s because ?”? looks so ugly.]

Anyway, back to our question. My take on it is that the student who asks it rarely wants to listen to an answer. Usually it’s an indirect way of saying, “This is boring/confusing/hard to understand. Why do I have to learn it?” In other words, the subtext is a plea for more meaningful math (or whatever the subject is). But our tendency, as teachers, is to take the question seriously — as we should — and therefore to try to answer the question that was asked — rather than the question that was meant. We want our answers to be honest and to be perceived as responsive, but it’s hard to do both. So we say something like this: “You will need logarithms in surprisingly many situations in the future, such as chemistry, calculus, statistics, business courses, and even college psychology courses.” That’s somewhat honest: it’s the truth, but it’s not the whole truth. And it’s somewhat responsive: it does take the question seriously, but it provides an unsatisfactory answer to it, especially since the implication of the question is that school is not part of the real world.

Why is the answer unsatisfactory? It’s mostly because it merely postpones the question. Telling a student that she will “use” a concept in a later course only encourages her to ask the same question (quite appropriately) when taking that course. A similarly unsatisfactory answer, one that most teachers are understandably reluctant to give, is that logarithms will appear on the MCAS and the SAT II. Motivating a topic by citing its possible appearance (and it is only possible) on a standardized test satisfies no one and definitely does not answer the question. The reality is that future courses and standardized tests do constrain what we teach, but that’s not really what the student is asking.

So, what’s the honest answer? To my mind, the honest but brutal answer is to refuse to accept the terms of the question. The honest answer has two parts to it:
  • I don’t know, and you don’t know, exactly what you’re going to be doing in your future life. So don’t reject a topic just because it’s not included in your current plans. Maybe you’ll become a pharmacist and will need to understand reports of pharmaceutical tests... I can’t say whether you will use logs, but it’s better to leave the door open than to close it.

  • More important, our reason for selecting a topic is not that the content of that specific topic will be useful to you later on. Yes, it might turn out to be useful, but the most important life lessons you learn from studying any worthwhile topic, whether it be logarithms or fractals, polynomials or proofs, are those big, hard-to-test lessons: how to approach a difficult subject, how to reason quantitatively, how to organize knowledge, how to solve problems, how to explain your solutions, how to present your analysis, and so forth.
That’s what I’d like to say. That’s what I believe. But how many students would accept that as the honest answer it is?

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Monday, November 20, 2006

Case of Lies

Catching up on posts about recent reading:

I highly recommend Case of Lies, by Perri O’Shaughnessy, especially if you are interested in math or linguistics. If you’re not, it’s still a solid mystery, well above average for the genre even though there are a lot of holes in the plot.

“It seemed to be about two different things, math and murder,” complains one reviewer. Well, yes — but he forgot to add, “...not that there’s anything wrong with that.” Math and murder: sounds like an excellent combination to me, especially when most of the math is primarily number theory and cryptography. The rest of it includes probability, combinatorics, and the history of math, taking off from the famous Bringing Down the House: The Inside Story of Six M.I.T. Students Who Took Vegas for Millions, by Ben Mezrich. There are a few errors, but not enough to spoil the effect. Let’s look at a few carefully chosen quotations from the book — chosen from many possibilities to highlight the math and a bit of linguistics, not to be in any sense representative of the entire novel:
He told Elliott about Sanskrit. Linguistics wasn’t about languages, it was about logic. Pop showed him how to diagram Sanskrit grammar so the little x’s and ys added up to a sentence... Subject plus verb plus direct object equals a sentence. In English, anyway. English moved like a number line, marching to the right. But there were other languages that put the direct object first, or even the verb. X still described the subject, y still described the verb. Math described language; wow!
 
To him these four numbers were as real as rocks, more real, alive in some sense. But what were they? What was a number? Where did numbers come from? Had humans invented them or discovered them? Where did they go? He thought they followed a line toward some far infinity where a little breeze sprang up and supported them.
 
“You can’t divide by zero, Elliott. It’s a rule.”
“Why is it a rule?”

“Because the rest of arithmetic won’t work otherwise. You just have to accept it.”

“I thought math was supposed to be logical.”

“It is.”

“Then how come multiplying by nothing wipes out a number?”

“Talk to me after class.”
 
“I know what an exponent is,” he boasted. “I know what a square root is. What’s the square root of minus one?”

“This is way beyond third-grade arithmetic,” Mr. Pell said. “Who told you to ask me these questions?”...

“Nobody. My pop. He’s a Sanskrit scholar. What’s the square root of minus one?”

“You know what? I bet your father already told you the answer, told you it’s an imaginary number with its own number line.”

“Egg-zackly. So if you can set up a brand-new number line for negative square roots, why can’t you set up a new number line for one divided by zero?”
 
“Then somebody said, ‘Let’s try that triangle out with a side that measures a single unit, a one,’ ” Pop said. “They tried it out. And a devil sprang out! Because one squared plus one squared equals two. Therefore the hypotenuse was the square root of two.” He leaned toward Elliott and said in a chilling theatrical whisper, “And that number couldn’t exist.”

“Wow!” Elliott said.

“That thing, that square root of two, couldn’t be described as an integer or a ratio. It completely contradicted the beautiful universe the Pythagoreans had constructed. Now they had a choice — to accept this ugly thing into their system and work with it, or to try to suppress the fact that it existed. To lie about it, because the Pythagorean religion could not encompass something as ill-formed, as unlocatable as this.”

“So what did they do?” Elliott’s mother asked.

“They swore the whole brotherhood to strict secrecy. This secret made a mockery of their beliefs. Now their religion was based on a lie.”
 
He read everything he could about the attempts to find a formula to predict the primes. The geniuses of mathematics, the smartest people who ever lived, had tried to understand the primes, and been defeated. Some had lived long, quiet lives, but many who flirted with the primes had fallen while very young: Gauss, who left math forever in this twenties; Ramanujan, the vegetarian Brahmin who died at thirty-two; Gödel, who starved himself to death; Nash, teeting on the edge of the void most of his life; Grothendieck, still alive, cloistered in a hut in the Pyrenees, obsessed with the devil; Turing, who killed himself at forty-one by eating a cyanide-laced apple.

And the greatest of them all, in Elliott’s mind at least, Bernhard Riemann, who died in Italy at thirty-nine. Because of pleurisy, the books said, but Elliott figured he had died because the heat in him had died. Riemann had simply gone as far as he could. He had found a possible order in the primes and given the world a direction in the Riemann Hypothesis. It made sense to die then.
 
“...As soon as I had that, I could factor really large non-prime numbers, too, as a simple corollary. Are you understanding any of this?”

“I’m not sure. But you sound autthoritative. That’s half the battle.”

“Maybe for lawyers. You only have to be convincing. Not in math. In math you have to be right.”
 
“Well, really big numbers can’t be factored — nobody can find the primes they’re made of — even with today’s computers. So a company called XYC invented a method of encoding financial and other information using that fact, so information couldn’t be hacked as it traveled from one Web site to another. The code lets you type in your credit-card number for certain eyes only.”
Not exactly standard material for a mystery!

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Sunday, November 19, 2006

Tech Model Railroad Club and MIT Museum

Barbara and I spent an enjoyable couple of hours yesterday visiting the MIT Museum and the Tech Model Railroad Club (TMRC). Go see both of them!

MIT’s small museum is currently showing five exhibitions:
  • a fascinating collection of holograms — always amazing to look at, and claiming to be the world’s largest such collection

  • an informative multimedia show about robotics and AI, complete with actual models, videos, and info about the real people behind the development of the robots (no, it’s not science fiction)

  • a well-done exhibition about Harold “Doc” Edgerton, the renowned inventor of the strobe, including a large blow-up of his famous milk drop photo — worth seeing if it isn’t old hat to you

  • a captivating and truly amusing set of kinetic sculptures by Arthur Ganson (are they art? are they engineering?)

  • a wonderful history of MIT called Mind and Hand, of interest not only to MIT folk but also to anyone whose life includes education or technology
Actually, there were six exhibitions. Somehow we missed Tech’ing it to the Next Level. Obviously I’ll have to go back to see it if it’s still there (I don’t know how we could have failed to notice it), since it’s all about educational innovation using technology.

Now, onto the model railroad layout. TMRC is well-known in certain circles as being the originator of the term hacker —not in the current sense to which the media have perverted the term (for which cracker is the approved term), but in the sense of “a person who enjoys exploring the details of programmable systems and how to stretch their capabilities, one who programs enthusiastically (even obsessively) or who enjoys programming rather than just theorizing about programming, one who enjoys the intellectual challenge of creatively overcoming or circumventing limitations.” It’s no coincidence that these meanings arose out of TMRC, whose exhibit suggests “MIT” as much as it suggests “model railroading.” In contrast to the majority of the highly detailed layouts that one is likely to see elsewhere, this one emphasizes the technology more than the illusion of reality. Exposed wiring is a virtue, not a defect. Seeing the scaffolding is fine. Roads don’t need drains, mailboxes, or fire hydrants. But there are many cool high-rise buildings, including one that lets the viewer can play Tetris on it (the controls turn the lights in the building’s rooms off and on appropriately, producing a Tetris game through the windows). There are excellent demonstrations of how to handle tracks and roads on multiple elevations. And don’t forget the ads for the breakfast food for which MIT has been famous for decades: Apple Gunkies, “rhomboidal pellets of true fruit goodness.”

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Saturday, November 18, 2006

The Winter's Tale

I’ve never read it; I had never seen it before last night.

The Weston High Theater Company is currently performing one of Shakespeare’s less well known plays, The Winter’s Tale. It’s very definitely worth seeing, with several outstanding performances and a fascinating mix of comedy and tragedy. Though officially listed as a comedy, the entire first half is very serious indeed, and even in the second half there is considerable attention to the tragic flaws of the protagonist, King Leontes, convincingly portrayed by Quinton Kappel. Laura Caso’s moving characterization of his queen, Hermione, along with appropriately over-the-top performances by Matt Doyle and Robert Slotnick, are aided by a strong cast of a couple of dozen other students.

“Hermione,” eh? The name can’t help reminding one of Harry Potter. This is no coincidence:
J K Rowling modelled the character of Hermione on herself... Hermione’s name is best known from Shakespeare’s play The Winter's Tale. That is a complicated story, but she is a Queen, who is wronged by her husband. Through her strength of character, her patience, and fate, all eventually ends well.
In the Weston performance an excellent technical crew provides all the components that are necessary for supporting the actors — such as sets, costumes, music, and especially Peter Freeman’s effective lighting. Tonight will be the last performance of this psychological “comedy,” mostly very dark but also containing a great many lighter moment. Exit, pursued by a bear.

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Friday, November 17, 2006

Sig figs

Science teachers — and science textbooks — generally insist on careful attention to significant figures. Math teachers — and math textbooks — generally pay no attention to them. Here are two representative examples:
  • Our Algebra II textbook contains a word problem in which a speed is given as 40 mph. Never mind the the details of the problem; the answer in the Teachers’ Edition is given as 199.5 miles. Why not 200, I hear you ask.

  • Many math contests tell students to round answers off to four digits after the decimal point, regardless of the given information. Why four? Well, why not four?
Science teachers will correctly point out that there is no excuse for the 199.5 or the four-digit requirement. The former implies much more precision that the data justify; the latter might imply either more or less, depending on the data.

So why don’t we math teachers follow the rules? There are several good answers. One is simply that the information given in a problem rarely comes from actual measurements. No one clocked the speed of the car and recorded it as 40 mph — not to mention our ignorance about the precision of that figure anyway, which might represent 4.0×102 or perhaps just 4×102, or even possibly 4.00×102. From a mathematician’s point of view, 40 is pretty much just a pure number and is therefore not susceptible to the sig figs rule. I still don’t think that that justifies an answer of 199.5 rather than 200, but at least there’s some rationale there.

Oh — look what I’ve just done. I wrote 199.5 and 200 without units! Shame on me.

Actually there’s a good reason why math teachers only pay lip service to units and rarely pay attention to them. But we’ll save that issue for a different post. Let’s wrap up sig figs in this one.

Anyhow, the second reason is that the rules simply don’t work very well. Here’s are excerpts from a typical statement of “the rules”:
When measurements are added or subtracted, the answer can contain no more decimal places than the least accurate measurement.

When measurements are multiplied or divided, the answer can contain no more significant figures than the least accurate measurement.
Note the subtle difference between the two rules, which we miss if we read too quickly.

Let’s try to apply these rules to a concrete example given by John S. Denker: “4.4 × 2.617 – 9.064”. According to the rules, we should express the answer of 2.451 as 2.5. But each of the three numbers is taken to be a measurement where the least significant digit is uncertain. So the first one, for example, might represent anything from 4.35 to 4.45 [we’ll ignore issues of when a digit of 5 rounds up and when it rounds down]. Taking one extreme, we might really have 4.35 × 2.6165 – 9.0645. At the opposite extreme, we might really have 4.45 × 2.6175 – 9.0635. We’ll do all three calculations:
  1. 2.451
  2. 2.317
  3. 2.584
Our approved answer of 2.5 suggests that the “real” answer lies between 2.45 and 2.55, and yet both the second and third possibilities lie outside that range.

Finally, things really go haywire when we encounter sensitive dependence on initial conditions. For instance, Nagai Tosiya examines the function f(x) = 4x(1 – x). In generation 2, as a result of prior calculations we might be rounding to four sig figs, so 0.15360 and 0.15357 would both be expressed as 0.1536. But in the 15th generation the former value results in an answer of 0.13561, the latter in an answer of 0.00180. Sig figs can’t save us! We need all the precision we can get. In fact, we need more than we can get. The sig figs rule provides the comfort of all clearly expressed rules — but it’s cold comfort when it can’t give correct answers. That’s the most important reason why we math teachers pay little attention to significant figures.

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Thursday, November 16, 2006

Upcoming model railroad shows

Two upcoming model railroad shows:
  1. Tech Model Railroad Club, Saturday, November 18, at the MIT Museum, 265
    Massachusetts Avenue, Cambridge MA, Room N52–118, 2–5 and 7–10. Free!

  2. National Model Railroad Association HUB Division, New England Model Train Expo, Saturday-Sunday, December 2–3, 10–4 at Best Western Royal Plaza Trade Center, Marlborough MA. Not free (but not expensive, especially for kids).
I will be at both, and I hope to see some people I know. Brief reports will follow.

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Wednesday, November 15, 2006

Teaching linguistics in high school

At Lincoln-Sudbury in the 1970s I taught a rotating sequence of linguistics courses along with my primary assignment of teaching math. As my undergraduate and graduate work were in linguistics, it was a natural fit. In my ten years teaching in Weston, I’ve never taught a linguistics course, but last month I proposed a one-semester Multidisciplinary elective, Introduction to Linguistics. Here are some excerpts from the proposal:
This course would be aimed at students in grades 9–12 who are interested in languages, love words, and believe that the scientific approach to learning is a valid one. It needs to be a multidisciplinary course: while the content would focus on foreign languages, it would also strengthen students’ understanding of English, and the approach (though not the content) is that of science. As a course that is offered in almost every college but very few high schools, it would look good on students’ transcripts (always a plus in Weston).

The course would address questions like these:
  1. What patterns are found in all languages, and why?

  2. Why is English used so widely? Why do so many Americans think that there’s no need to know any language other than English?

  3. How does it happen that the Irish and the Pakistanis speak related languages, even though their countries are so far apart? Why do the Austrians and the Hungarians speak unrelated languages, even though their countries are next to each other?

  4. How did Latin evolve into French and Spanish (and Italian and Portuguese and Romanian…)? Why is it false to say that English is descended from Latin?

  5. What's a dialect? Are Mandarin, Cantonese, etc., actually separate languages, or are they really just dialects of Chinese? Are there different dialects of English as well; if so, are they regional or class differences?

  6. How can we understand language better by thinking about it scientifically? What kinds of data are used in linguistics? How do we turn patterns into conjectures and conjectures into conclusions?

  7. Surprisingly, linguists will tell you that it isn’t true that the vowels of English are a, e, i, o, u, and sometimes y. Why not?

  8. Why is sign language really a language?

  9. How did language first originate in the human species? Can other creatures use language, or is it limited to humans? Is there a “language facility” in the human brain that enables virtually all children to learn language?

  10. How can we know about really ancient languages, or other languages that were only spoken and not written? What is the connection between spoken and written languages anyway? Is there such a thing as a primitive language?

  11. Do the Eskimos really have 47 different words for snow? Why does it matter?

  12. As languages develop over the centuries, do they become simpler or more complex?

  13. Do young children learn to speak primarily from their parents or their peers?

  14. Why do some English words come from Latin, some from Greek, some from French, some from Old English, and so forth?

  15. How important is word order in different languages? For example, do German verbs really go at the end of the sentence? Is it true that words can go in any order in Latin?
I don’t know whether this will fly or not.

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Tuesday, November 14, 2006

How do you get the gas company's attention?

So we park our car in front of the house, get out, and smell the distinctive odor of natural gas. Sniffing around, we conclude that it’s pretty clearly coming from the middle of the street. We go in and call the gas company. They say they’ll send someone out within the hour.

Indeed a couple of gas company employees arrive an hour later. They insist on checking out our basement, and of course detect no gas there. Their meter shows gas in the street, but none in the house. They go away and apparently file a report.

A couple of days later, the same sequence of events happens. Check the basement with their meter, etc., etc. Still no sign of a real resolution. We still smell gas.

Then a neighbor calls. Third time with the same rigmarole. One man tells us that repeated phoning is the only way to get action. But no conclusion.

Finally, we call a fourth time. This time they dig a big hole in the middle of the street, determine somehow that there’s a leak in the pipe leading to our neighbors’ house across the street, camp out in their truck all night, and leave in the morning after covering up the big hole with a steel sheet.

Now all is well. But did it really have to take four calls over a two-week span? This could have been really dangerous, after all.

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Monday, November 13, 2006

Numb3rs and the MAA

The Mathematical Association of America (MAA) has generally been enthusiastically positive about the well-known television show, Numb3rs. So has the National Council of Teachers of Mathematics (NCTM). These reactions are to be expected: both organizations want to promote interest in math, and Numb3rs surely has promoted such interest. After all, “we all use math every day.”

But in the November 2006 issue of Focus, the newsletter of the MAA, mathematician Alice Silverberg, writes a very negative essay about the show, even though she’s a consultant for it. Three excerpts:
If you’re watching Numb3rs because you think you’re learning some mathematics, or because you think you’re watching mathematics as it’s actually used in the real world, be warned: you’re not. Getting the math right and getting it to fit with the plot are not priorities of the Numb3rs team.
...
[Producer] Cheryl [Heuton] was very generous with her time... in which she mostly explained why talking with mathematicians would be a waste of their time.
...
I have concerns about the violence, the depiction of women, and the pretense that the math is accurate....
In between these excerpts, Silverberg decries the “excessive violence” of the show — she must not watch much television — and the depiction of women. She claims that women on the show are portrayed only as love interests and sex objects.

I guess this counts as a refreshingly different viewpoint.

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Sunday, November 12, 2006

Should we rename all the Washington Streets?

So the Cambridge City Council wants the MBTA to rename Lechmere Station because they have just discovered that mid-18th-Century local resident Richard Lechmere was a slave owner. By that theory, should they petition to have all the Washington Streets renamed?

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Saturday, November 11, 2006

Ah, that explains it

Some thoughtful reflections on the election — from Grover Norquist, best known for being head of Americans for Tax Reform, as well as being a distinguished alumnus of Weston High School and a close Republican colleague of Karl Rove:
Nobody thinks that Karl is in charge of the occupation of Iraq... I haven’t heard any complaints about him.

Bob Sherwood’s seat [in Pennsylvania] would have been overwhelmingly ours, if his mistress hadn’t whined about being throttled... The lesson should be, don’t throttle mistresses.

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Friday, November 10, 2006

Yearbooks and gender

What is it about yearbooks that makes boys unwilling to serve as editors? Year after year, when I look at the list of Weston High School yearbook editors, what do I see? Pulling out four recent yearbooks at random, I find the following:
  • 13 girls, 0 boys
  • 13 girls, 0 boys
  • 12 girls, 1 boy
  • 13 girls, 0 boys
But it’s not just in high school that boys have this problem. In the Saturday Course at Milton Academy, the sixth-graders put out a yearbook, and a couple of dozen kids sign up to work on it each year. Once again, it’s usually 100% girls and 0% boys.

OK, perhaps this is a recent phenomenon. So I dig up a 1978 Lincoln-Sudbury yearbook: 9 girls, 2 boys. I suppose that’s a tad better, but not much. (And one of the boys was a cousin of mine, so maybe he doesn’t count.)

Let’s go back even further. No use going all the back to my own 1965 yearbook, since P.A. was all male at the time, but I find the 1970 L-S one — from my first year of teaching! — which will surely be different. But no, it’s once again 9 girls and 2 boys. And to complete the stereotyping, one of the yearbook editors was also the only girl on the Math Team.

What’s going on here?

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Thursday, November 09, 2006

Former students

Now that I’ve been teaching in Weston for almost ten years, I seem to be running into a surprising number of former students of mine (from Lincoln-Sudbury, but now through other connections). Weston isn’t exactly far from either Lincoln or Sudbury, so maybe it isn’t so surprising. And only some of these are Weston connections anyway; others I hear on the radio or see on stage.

Most prominent — at least in certain circles — are John Flansburgh and John Linnell, a.k.a. They Might Be Giants. From this band we can go in either of two ways: to another performing artist, the comedian Paula Poundstone, or to John Flansburgh’s brother, Sky, who I think holds the record for having taken the largest number of courses from me. (That was before he changed his name to Paxus Calta.)

You may be noticing that all four former students whom I’ve named so far are rather ... shall we say? ... counter-culture. But if you know Lincoln-Sudbury from the ’70s, you’re not surprised.

Rather different are the former L-S students who are now parents of students of mine at Weston: Meg Cowe, John Batter, and Wendy Spector. (There are more L-S alumni among parents of my Weston students, but those three were actually my students.) All three occupy prominent positions within the town of Weston. Coincidence? I think not.

Let me know if I’ve inadvertently left anyone out.

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Wednesday, November 08, 2006

Mathematica and Maple

At Weston High School we’re considering the use of mathematical symbol-manipulation software such as Mathematica or Maple. Our theory is to pick one of these for a trial run for a year — just one copy per teacher, for use in planning lessons and in class demos. Then we will reconsider whether they would be appropriate for college use. The arguments on each side are pretty clear: this kind of software brings extra power to students and lets them accomplish things that they couldn’t do without it, but it also risks erosion of techniques that many people consider to be basic skills. For example, why learn to factor if the computer will do it for you?

One possible response is to acknowledge that maybe the computer should do it for you. After all, we no longer teach the pencil-and-paper square root algorithm, and its loss doesn’t seem to have damaged anyone.

This isn’t just an argument about math teaching. The whole notion of what skills are basic changes from generation to generation. We can no longer do a lot of things that our great-grandparents could do, but there are a lot more things that we can do. Isn’t it more important in this day and age for me to be able to fix my computer when it stops working than to make candles or construct wagon wheels? Who knows what basic skills are anyway? I have an intuition about them in math, but my intuition is based at least as much on my own training as it is on a reasoned consideration of what’s basic in this decade and the next. A local tutor objects to our inclusion of cryptography in our high-school math curriculum, thinking that it’s a frill that replaces important things like the measure of an angle inscribed in a circle. Her argument? The SAT includes a question on the latter but nothing on the former. But surely there are more authentic ways of determing what’s important than the College Board’s outdated ideas of what ought to be on the SAT. Cryptography may well be the single most important application of mathematics today, regardless of what the College Board says.

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Tuesday, November 07, 2006

Continuity and retention in math classes

I’m sure we’re not alone in finding that there’s distressingly little retention from year to year in our math classes. One of the big differences between honors and non-honors (“college-prep”) classes is that most students in the former can be expected to retain a majority of what they learn from year to year. Not all students, and not all topics, but still it’s pretty good.

But in the regular classes, a majority of students can’t find the equation of a line through two points unless they have recently reviewed the process. We’re all familiar with the mentality of learning something until the next test and then forgetting it, and this definitely isn’t just in math class — how many people can remember the specific dates they memorized in history class last year, or the details of the characters in a novel they read in English?

It’s not just a year-to-year problem, either. There are howls of protest if a question on a test refers back to something studied two months ago. We can rationalize this lack of retention by claiming that our students are still learning the important big ideas, that they’re still learning how to learn, but that’s cold comfort if they don’t learn anything past the next test.

Or maybe things aren’t that bleak. After all, Weston students do very well on the SATs and extremely well on the MCAS, so they must be retaining something. Are we just being unrealistic when we wring our hands over the inability of a precalculus student to simplify a fraction correctly?

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Monday, November 06, 2006

Where have I been?

Oh no! A month and a half have gone by since I have last posted!

I am determined to resume posting right away...

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