First of all, this afternoon I had already been intending to comment on an op-ed piece from this morning’s New York Times, titled “How to Fix Our Math Education.” And I was going to relate it to a comment by Frank Baker, candidate for Boston City Council from the Third District (where I live). But I didn’t actually finish the post before dinner, and then I had to leave immediately in order to go hear all seven of the candidates [...]]]> City council candidates and math education? Those are two utterly unrelated topics, aren’t they? But there turns out to be a connection.

First of all, this afternoon I had already been intending to comment on an op-ed piece from this morning’s New York Times, titled “How to Fix Our Math Education.” And I was going to relate it to a comment by Frank Baker, candidate for Boston City Council from the Third District (where I live). But I didn’t actually finish the post before dinner, and then I had to leave immediately in order to go hear all seven of the candidates participate in a candidates’ forum. So let’s discuss all three topics: the op-ed piece, the candidates’ forum, and the connecting link: Frank Baker.

The op-ed piece, by the distinguished Sol Garfunkel and David Mumford, argues for a major change in the emphasis of high-school mathematics. Their views are basically correct, so go read their entire essay. Here is a brief excerpt so you can see what their claim is:

This highly abstract curriculum is simply not the best way to prepare a vast majority of high school students for life…. Imagine replacing the sequence of algebra, geometry and calculus with a sequence of finance, data and basic engineering. In the finance course, students would learn the exponential function, use formulas in spreadsheets and study the budgets of people, companies and governments. In the data course, students would gather their own data sets and learn how, in fields as diverse as sports and medicine, larger samples give better estimates of averages. In the basic engineering course, students would learn the workings of engines, sound waves, TV signals and computers. Science and math were originally discovered together, and they are best learned together now.

The reason I say that their views are only “basically” correct is that such a curriculum would not be sufficient for students preparing for any career that requires a traditional mathematical background. Even though only a small minority of college-students study science, engineering, and pre-med courses, they need to be well served, and it would be folly to think that we can identify exactly who those students will be when they’re in tenth grade (although many countries do exactly that). We could just offer Garfunkel and Mumford’s solution to non-honors students, but who wants to increase the difference between honors and non-honors courses? That wouldn’t be good either. Some kind of a combination is needed, where students could move more in one direction or another in their last two years of high school, and where the ideas presented in the op-ed piece are incorporated into the traditional program as well. The first of these solutions is what we do at Weston High School, where we offer three semesters of “Applied Discrete Math Concepts” to those who want something different from the traditional pre-calculus and calculus courses. While many of the students who elect this course tend to be our weaker math students, that is far from universally true, and some excellent math students even take Applied Discrete Math in addition to precalculus.

Now let’s move to the apparently unrelated candidates’ forum. We have a surprisingly large field this year, as our long-term councilor, the hard-working Maureen Feeney, is retiring after serving many years. Of the seven candidates who are vying to succeed her, two are clearly Republicans (even though they might not admit it) and I won’t comment on their ideas or their presentations at the forum. The Boston City Council is officially non-partisan, but the other five candidates are clearly Democrats. Three of them gave pretty weak presentations tonight, leaving only the remaining two, whose supporters are coincidentally the only ones whose supporters you will see in this picture:

It may be a little difficult to tell from the picture, but this crowd on both sides of the street was almost entirely white, a bad sign in racially mixed Dorchester. Inside the hall I counted close to 300 people in the audience, of whom 98% were white. So much for “racially mixed.”

Finally, how are these two topics related? The connection is a paragraph last week in the Dorchester Reporter:

Baker suggested the city bring back trade schools and attempt to replicate popular schools like the Richard Murphy K-8 School. “Why aren’t we looking at that and trying to apply it to other schools?” he said.

But here’s the dilemma. On the one hand, college isn’t for everyone, and vocational programs can do a lot to keep kids in school and teach them useful skills. On the other hand, it seems unbearably classist to say that Weston should have a college-preparatory program and Boston should be oriented toward life skills. I don’t know how to resolve this dilemma. Weston is so college-oriented that “college-prep” is the name for our lowest level of courses! Everything that isn’t honors or AP is college-prep. Students who want vocational training can go to Minuteman Career and Technical High School, which offers an excellent program — but it is socially deprecated in Weston and kids don’t want to be separated from their friends. So basically everyone in Weston is assumed to be college-bound. Boston, of course, is far more diverse. Even if it doesn’t bring back trade schools as an option, the kind of math curriculum proposed by Garfunkel and Mumford would at least be a start.

Stay tuned for a post on one project in particular, a spectacular children’s book on fractals.

So here’s the follow-up, or at least a preliminary follow-up. I still haven’t figured out the ideal way to take pictures of the eighteen 11″-by-14″ hand-drawn and hand-lettered pages of The Fractal Adventure, written and drawn by my students Anna, Ali, and Eye. The pages are too big for my scanner, and I suspect that I need stronger ambient light in order to take high-quality digital photos of them. For the moment, though, here are [...]]]> In last month’s post about our Fractal Fair, I made the following promise:

Stay tuned for a post on one project in particular, a spectacular children’s book on fractals.

So here’s the follow-up, or at least a preliminary follow-up. I still haven’t figured out the ideal way to take pictures of the eighteen 11″-by-14″ hand-drawn and hand-lettered pages of The Fractal Adventure, written and drawn by my students Anna, Ali, and Eye. The pages are too big for my scanner, and I suspect that I need stronger ambient light in order to take high-quality digital photos of them. For the moment, though, here are a couple of less-than-ideal images of pages 4 and 5, so you can at least get the idea:

And here’s the class, minus those who left early for vacation and those who were unwilling to appear in the photo:

Naturally I loved the resolution of this difficulty. She was 100% correct, though that certainly wouldn’t have happened all the time. A test should tell [...]]]> A colleague who does not teach in our Math Department was tutoring one of my students. Not being familiar with our mildly unusual Honors Geometry course, she found that she herself did not know how to do the last problem on his test. “But I figured it out,” she then reported. “Looking back at the earlier problems, I could see that they told a story, so I understood what the last problem must be all about.”

Naturally I loved the resolution of this difficulty. She was 100% correct, though that certainly wouldn’t have happened all the time. A test should tell a story. It should have a theme; a beginning, a middle, and an end; a conflict and resolution; and a plot that exhibits a well-defined arc. A test should tell a story, but all too often it doesn’t, whether it’s one that I wrote or one that someone else wrote. Occasionally I write a test that literally tells a story, one in which the student has to fill in some blanks and solve some problems along the way, but that distresses too many kids who are unaccustomed to that form for a math test. Most of the time the story can be discerned only by reading between the lines, but I hope it’s still there, at least on most tests.

Not only should a test tell a story, but a course should as well. A couple of years ago a colleague commented on an Algebra II course at another school by saying something like this: “It isn’t a course; it’s a collection of topics.” Unfortunately all too many Algebra II courses suffer from this failing, which is one of the many reasons why I tend to prefer precalculus over Algebra II. Precalculus, at least as I teach it at Weston, definitely tells a story, with all of the parts I outlined above: theme, beginning, middle, end, conflict, resolution, plot, and arc. Algebra II gets only halfway, though one of the attractions behind the decision to spend the fourth quarter on cryptography is that it truly helps to complete the story of Algebra II, with lots of attention to functions, inverses, matrices, exponents, representations, and of course real-world applications.

…So now you know why “chance favors the connected mind.” Keep that in mind, no matter what organization(s) you’re a part of. I think I’ll show it to my [...]]]> Today we participated in an intense professional development (PD) program and worked on our preparation for NEASC accreditation. NEASC work is often frustrating but often useful as well (more on that later). Today’s PD was quite interesting. The main part of the day’s program opened with “Where Good Ideas Come From,” a great video by Steven Johnson. Drop whatever you’re doing right now, click on that link, and watch the entire four-minute video…

…So now you know why “chance favors the connected mind.” Keep that in mind, no matter what organization(s) you’re a part of. I think I’ll show it to my students.

After that video, as well as other more-or-less related introductory material, we split up into small groups to participate in parallel sessions in which various “professional learning communities” presented progress reports. (You have to keep up with the current jargon, you know.) I chose to attend sessions on Metco, mathematical discussions, and iPads. All were well worth it:

• The Metco presentation was the first time I had heard a report on the entire program, K–12. The central question, how to improve Metco students’ success in math, was of course relevant and important.
• The “mathematical discussions” presentation concerned a project in fourth and fifth grades revolving around Suzanne Chapin’s work, which was reported to result in “increase in civility and logical thinking.” Because the presentation consisted almost entirely of video shot in real classrooms, the findings were powerful and convincing; “they’re articulate and they try to use precise language” was the observation of one teacher.
• Finally, the iPad presentation concerned a pilot project in which an entire class of seventh-graders have been given iPads for second semester to use in science, social studies, and English/language arts. While I teach none of those subjects, the relevance to me is that I was recently approved for participating in an iPad project of my own for one month in Algebra II. So stay tuned for my results and my comparison with what I saw in the report from the seventh-grade teachers.

In the afternoon we watched the film Race to Nowhere and discussed it in small groups. This “documentary,” in the style of Michael Moore, is definitely worth seeing despite its obvious biases. It makes a strong case that our students are being stressed out and pressured to focus entirely on a race to attend the most prestigious colleges. That is certainly true for many students in Weston. But I have several problems with the film: math teachers are always the bad guys (too much math homework drives a girl to suicide???); scenes of black students in Oakland were obviously edited in, probably in response to the otherwise white-suburban bias; scenes from The Blue School in New York City made unfounded generalizations about all other schools; the film claims that 95% of American high-school students cheat, which seems unbelievably high; they also claim that American schools aren’t preparing independent thinkers, which strikes me as a gross over-generalization. Almost everything in the movie is worth thinking and talking about, but I wish it had been more balanced.

I only wish the audience had been more consistently respectful. As Bob’s introducer, I happened to be [...]]]> Professor Robert Devaney of Boston University gave two excellent talks to our precalculus classes (consisting mostly of juniors, with a sprinkling of advanced sophomores and freshmen) on Tuesday. His talk to the college-prep classes (”Precalculus Part One”) focused on the use of geometric transformations to create fractals which in turn could become artificial but convincing landscapes in movies. This combination of pure and applied math was a stunning example of real-life applications of what appears to be a highly theoretical piece of pure mathematics.

I only wish the audience had been more consistently respectful. As Bob’s introducer, I happened to be sitting in front, where I got to see a non-representative sample of the audience. The kids near me were not only respectful but were also attentive and engaged. They were duly appreciative when the apparently random activity of the Chaos Game turned into the highly regular Sierpinski’s Triangle, and when Barnsley’s Fern emerged out of chaos. But teachers in the back of the room reported a different cohort there: kids using cell phones, sleeping, talking, etc. Since students sat where they pleased, the distribution was certainly not coincidental. But the question to me is why this audience was so extremely different from the honors math students (see next paragraph). Of course it’s easy to claim that students in honors classes are almost always better behaved than those in non-honors classes, as those who don’t want to take a subject seriously are unlikely to sign up for an honors class. And there is indeed a certain measure of truth in that observation. But it’s clearly not the whole truth. For instance, my non-honors Algebra II class is far more respectful, polite, and better behaved than my D Block Honors Geometry class. I wonder what accounts for these differences; is it merely the chance distribution of students?

The talk to the honors classes was almost entirely about the Mandelbrot Set, although it had to involve some necessary preliminaries about Julia Sets. The students were attentive and  learned a lot from this presentation, including some surprising interpretations of “how to count” and “how to add.” Although I had heard almost all of this many times before, there was one important nugget that was brand new to me: how to insert sliders into Excel spreadsheets. The resulting dynamic graph became a wonderful tool for visualizing (and therefore understanding) the chaotic effect of varying a single factor when looking at the orbit as a function is iterated. I will have to try using that myself some time.

Also, as a follow-up, yesterday’s Fractal Fair was extremely successful. Almost all the projects were solid, many were excellent, and we got a lot of visitors of all ages. Stay tuned for a post on one project in particular, a spectacular children’s book on fractals. Here are a few photos, taken by the school librarian:

After 35 minutes, a fellow senior [...]]]> Weston High School is a great place! With a student body of only 748 students, we had about 150 show up for a chess match of all things! What a delightfully geeky experience. The context was a fund-raiser for the chess club, in which chess player extraordinaire Alex Kaye challenged students to beat him in chess. He played 27 opponents simultaneously, moving rapidly from board to board around in a circle as the onlookers cheered. While it certainly wasn’t silent as a chess match is supposed to be, there was great excitement and school spirit.

After 35 minutes, a fellow senior finally won. Everyone had a great time. Here are a couple of pictures showing what is was like:

Finally, Weston received the coveted Stage Manager’s Award–given not by the judges but [...]]]> Last night I saw the Weston High School Theater Company’s excellent performance of The Lie That Binds. What? You’ve never heard of this play? That’s because it was written collaboratively by the cast and crew — namely, the students in the Theater Company. Click on the link for more information, including photos of the production. You will also see there that Weston was one of the winners of the preliminary round at the Mass Educational Theater Guild Drama Festival. Congratulations to all involved! I particularly like this paragraph:

Finally, Weston received the coveted Stage Manager’s Award–given not by the judges but by the host school to the stage manager of the school that is the “best guest” — friendly, efficient, organized and professional in their approach to the one-hour technical rehearsal and the festival day itself.  Stage Manager Irene Droney accepted the award on behalf of the whole company.

And now a segue to something that sounds entirely unrelated: Laura Lippman’s novel, Life Sentences, which I just finished reading. Lippman is best known for her Tess Monaghan mysteries, which I have unaccountably failed to review, though I reviewed two of her other novels five years ago — on April 13 and April 24, 2006. Life Sentences is a hybrid, a cross between a mainstream novel and a genre novel, and I definitely recommend it. The connection here is that the crux of the stories is a set of lies that may (or may not) hold a family together, and what happens when the lies come out. The lie involves adultery and a dead child, partly involving the family of a prominent politician — close to the story of The Lie that Binds. Although it is possible that someone in the Theatre Company has read Life Sentences, I think it’s unlikely. It’s more probable that this is an old literary theme that awakens especially whenever real life gets close to it.

Actually, when I watched The Lie that Binds, I wasn’t thinking of Life Sentences (since I hadn’t finished the Lippman book until today) but of Henrik Ibsen’s The Wild Duck. We read this provocative play in the wonderful eleventh-grade English class in which I was a student many decades ago. We also read five or six other Ibsen plays, and The Wild Duck is one of three that have stayed with me ever since. Again I have no reason to believe that anyone in the Theatre Company has read or seen The Wild Duck, but some of the thematic similarities are striking. Like all of Ibsen, the play feels quite modern, even though it was written well over a hundred years ago. As in both of the newer works — the Lippman novel and the Weston play — an initial lie is based on adultery, and then more lies are piled on. Eventually the truth turns out to be worse than the lies. Of the three works, only Lippman’s manages to have a happy ending.

At the assembly on March 3, Thursday, the non-profit organization Invisible Children will be presenting their new documentary based on a former child soldier.  Then a speaker from Uganda, a former child soldier, will share his experience and thoughts on the use of children in conflicts.  There will be a brief Q & A with the audience at the end.

This calm description turned out to be something of an understatement. Both the moving documentary and [...]]]> At school yesterday we had a special assembly sponsored by our local Amnesty International chapter. Here is the official description we were given ahead of time:

At the assembly on March 3, Thursday, the non-profit organization Invisible Children will be presenting their new documentary based on a former child soldier.  Then a speaker from Uganda, a former child soldier, will share his experience and thoughts on the use of children in conflicts.  There will be a brief Q & A with the audience at the end.

This calm description turned out to be something of an understatement. Both the moving documentary and the talk by the two speakers highlighted not only the role of child soldiers but also the massive death and destruction caused by Joseph Kony’s terrorists in Uganda and Congo. The audience of Weston High School students was engaged and respectful.

And yet…and yet…I have a couple of reservations. Why is Kony doing these horrible things? One of my students asked the speakers that, and did not get a satisfactory reply. Of course the Wikipedia article is not to be trusted (unlike math and linguistics, this is the kind of topic for which one needs to be suspicious of Wikipedia); saying that Kony wants to establish a government based on the Ten Commandments is bizarre at best. Second, both the festive aspects of parts of this well-made movie and the attempt to get everyone to “stay silent for 25 hours on April 25th” seem orthogonal to the problem and its possible solution. Maybe I’ve just been out of college for too long.

Finally, how does one do anything for 25 hours on a single day? Inquiring minds want to know.

Weston High School is pleased to announce a new series of talks for the whole community led by our esteemed faculty members on a broad array of topics and expertise that extend outside the classroom!

My talk was called “Making Order Out of Chaos: A Conversation about Linguistics.” We had 53 attendees, an excellent turnout for a fairly technical presentation, and I was delighted by the audience’s enthusiastic response. Everything went very well, though we ran out of time near the end and I had to skip a [...]]]> Yesterday evening I delivered the first lecture in our new Beyond the Classroom series, described as follows:

Weston High School is pleased to announce a new series of talks for the whole community led by our esteemed faculty members on a broad array of topics and expertise that extend outside the classroom!

My talk was called “Making Order Out of Chaos: A Conversation about Linguistics.” We had 53 attendees, an excellent turnout for a fairly technical presentation, and I was delighted by the audience’s enthusiastic response. Everything went very well, though we ran out of time near the end and I had to skip a detailed slide that would have added ten more minutes. I also promised the audience that I would post my list of recommended resources right here. The missing slide needs some considerable commentary — it’s definitely not a standalone piece — so let’s start with the recommended resources (four books, three websites):

• Hofstadter, Douglas. Gödel, Escher, Bach: An eternal golden braid (Basic Books, 1999)
• Jackendoff, Ray. Patterns in the mind: Language and human nature (Basic Books, 1995)
• Pinker, Steven. The language instinct: How the mind creates language (Harper, 2007)
• Yang, Charles. The infinite gift: How children learn and unlearn the languages of the world (Scribner, 2006)
• Ethnologue (www.ethnologue.com)
• Language Log (languagelog.ldc.upenn.edu/nll)
• Popular Linguistics (popularlinguisticsonline.org)

I also recommended Wikipedia as a rich source of surprisingly reliable information about linguistics (and math), even though one wouldn’t want to trust it for areas like history, politics, and biography.

The slide I had to skip was a summative list of some of the various branches of linguistics. My plan was to build it up line by line. Here is the finished result, where the black type represents notes on my intended commentary:

Here is that intended commentary:

The list goes from the smallest level of detail to the largest level to the biggest picture. For example:

• Phonetics is the study of individual speech sounds. I already talked about Turkish vowels, where we had descriptions like “high back rounded vowel.” Through phonetics you can learn about how a French accent differs from an Italian one, or how automated speech recognition is possible.
• Phonology is the study of speech sounds in context, such as which pairs of sounds can distinguish words in a particular language (or dialect). For example, the words merry, marry, and Mary are all clearly distinct to my New Jersey ears, but my wife hears them as identical. (She comes from far western New York state.)
• Morphology (yes, I know, I’ve skipped one) is the study of how the components of words are put together. Examples include things like plural suffixes, tense markers, etc.
• Morphophonology (now we can back up) is the bridge between phonology and morphology, as you might expect. For instance, although the plural in English is usually spelled with an “s,” it is sometimes pronounced like a “z.” Why? And when? Similarly, Turkish /ler/ vs. /lar/ or English past tenses in /d/ vs. /t/.
• Syntax is the study of how words are built up into phrases and phrases are built up into words. (Gee, just like math, isn’t it? Terms are built up into expressions and expressions into equations…) Examples include my discussions of basic sentence order, such as SVO, transformations of basic order, and the ways tenses are formed in languages like Chinese that don’t use morphology (suffixes, etc.).
• Semantics is the study of meaning. For example, Chomsky’s famous sentences, “Colorless green ideas sleep furiously,” is syntactically impeccable but semantically anomalous.
• Pragmatics is the study of how language is actually used in practice. For instance, when a student comes into the Math Office and asks, “Do you know where Mr. McLaughlin is?” I may decide to be an obnoxious mathematician and say “Yes.” Of course that is not truly responsive to the intended meaning, even though it is literally correct.
• Finally, there are many subfields such as historical linguistics, comparative linguistics, etc. And there are interdisciplinary fields in which linguistics is combined with other disciplines, such as psycholinguistics, sociolinguistics, computational linguistics, etc.

Making order out of chaos: A conversation about linguistics

“Linguistics? What’s that?” This is the usual response I get from students when they hear that I majored in linguistics.

“It’s the scientific study of language,” I reply. “Linguists look for patterns, solve puzzles, develop hypotheses, and test those hypotheses.”

BSP^*: Come hear my talk on linguistics at 7:00 PM on Tuesday, February 1, at the Weston Public Library! Here’s a description:

Making order out of chaos:
A conversation about linguistics

“Linguistics? What’s that?” This is the usual response I get from students when they hear that I majored in linguistics.

“It’s the scientific study of language,” I reply. “Linguists look for patterns, solve puzzles, develop hypotheses, and test those hypotheses.”

As an example, let’s examine some data from Kurdish, a language you probably know nothing about, even though it’s spoken by over 16 million people. (Yes, you’ve heard of the Kurds in Iraq, but do you know anything at all about their language? No? I thought not. I don’t either — but I know what to look for.)

---

Here are six sentences in Kurdish, along with their English translations in the wrong order. Try to match them correctly.

1. Ez h’irç’ê dibînim. 2. Tu dir’evî. 3. Tu min dibînî. 4. H’irç’ di’eve. 5. Ez dir’evim. 6. Tu h’ireç’ê dibînî.

A. You see the bear. B. You see me. C. The bear runs. D. You run. E. I see the bear. F. I run.

---

Could you figure out the puzzle? If so, translate the sentence “H’irç’ mîn dibîne” into English. What did you learn from trying to solve this puzzle? Some of my students noticed that the word “tu” closely resembles a word in Spanish, French, and Latin. Is this just a coincidence? Why on earth should Kurdish resemble these far-away languages?

Maybe there’s a reason…

At Weston High School we care about global awareness. Linguistics reinforces that awareness. 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? How do linguistic connections relate to other sorts of connections?

We can also learn a lot right at home. English too is a world language. You’re probably fluent in English, but you may be surprised to hear that it isn’t true that the vowels of English are a, e, i, o, u, and sometimes y. Why not? Doesn’t every language have the same vowels? The answer is “no.” We’ll talk about why the question itself is misleading.

Is there anything that all languages share? This time the answer is “yes.” We’ll look at some examples and their significance.

Finally, you may be wondering how and why a linguist became a math teacher. Does linguistics really have anything to do with math? Come to this talk, and you’ll learn a lot about linguistics, a little about math, and at least one Big Idea about the strange connection between the two.

*Blatant self-promotion

(Context: He had been solving a problem at the board and made a negative remark, causing one of his classmates to say that he was damaging the learning environment in the class. The above quotation was his response. He meant it seriously.)

Honors Precalculus at Weston definitely does tell a story. I was thinking about the themes of that story today, and I realized that a big one is the idea of expanding a domain to go more broadly and more deeply into a topic. We begin the year with a review of right-triangle trigonometry, where the domain of the sine and cosine functions is the interval from 0° to 90° (exclusive). We then expand it to the inclusive interval, then to obtuse angles, [...]]]> A course ought to tell a story. If it doesn’t, it’s just a collection of topics, not a course.

Honors Precalculus at Weston definitely does tell a story. I was thinking about the themes of that story today, and I realized that a big one is the idea of expanding a domain to go more broadly and more deeply into a topic. We begin the year with a review of right-triangle trigonometry, where the domain of the sine and cosine functions is the interval from 0° to 90° (exclusive). We then expand it to the inclusive interval, then to obtuse angles, and then to all angles. Through the use of the unit circle and the switch to radians we have a domain that consists of all real numbers. At the end of the year we’ll expand the domain to complex numbers, through the use of infinite series.

In the meantime, we are turning to complex numbers. Over the years our notion of “number” has expanded from whole numbers to rational numbers to non-negative rational and irrational numbers to real numbers and now to complex numbers. Eventually we’ll break out of the idea of “number” altogether and will explore different infinities.

In the area of polynomials we’ve moved from linears to quadratics, and later this year we’ll explore cubics and beyond.

I’m sure fractals can fit this theme also, but that’s for another day.