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.
Several different threads have recently been coming together under the heading of “inverting the classroom.” The basic idea is that modern technology has let some of us come to the conclusion that the traditional model of the classroom has it all backwards:
- Students currently spend a lot of class time in a group of 25 (or 35, or even more, depending on the school) listening to a teacher lecture to them. They could just as well watch a lecture at home — on YouTube, say — where they could pause whenever necessary and watch difficult material many times.
- Students currently get most of their practicing done at home — it’s called homework, after all — where there is no teacher there to help them.
My department head has been fighting this model for several years now, mostly by devoting 90% of his class time to helping kids as they work individually or in groups. Homework becomes classwork.
But that won’t work for most teachers and most classes. I have no idea what percent of class time is typically devoted to lecturing; in my case I would guess 30%, but I fear that the national average is more like 80%. In any case, all of that time could be better spent. The most well-known example of the getting-the-lectures-at-home-through-technology point-of-view is Khan Academy, which offers over 2000 free videos giving short lectures on topics ranging from simple arithmetic to quantum mechanics. I’ve only watched four or five of these, but the ones I’ve seen look pretty good. I think I’ll recommend some of them to my classes this fall. They have their use.
So what’s the downside? The first issue is that “lecture” is usually a misnomer. When I lecture in class, I try to pause to let students ask questions, I constantly look at them to see whether they seem to be understanding, I give them quick exercises to work on, I vary my pace according to my audience. Sometimes we even have a discussion as part of a lecture, or as a follow-up to it. None of that is possible with a pre-recorded video. That worries me.
The second issue is that the public in the current political climate will use this model as an excuse to fire teachers, decrease class time, and increase class size. If that happens, it will be exactly backwards in a different way: after all, if class time is to be used primarily for getting individual help from the teacher, then we’ll need smaller classes and more teachers.
The third issue is that inverting the classroom will make it more difficult for a teacher to create discovery learning opportunities. If I have a carefully staged series of questions that are all designed to let my geometry students figure out a certain theorem, I don’t want them to be watching a video on that theorem ahead of time. This difficulty can be overcome by using a significant portion of classtime for such a purpose, but teachers who think that homework is primarily for practice will have trouble implementing that idea.
Finally, the fourth issue is that the whole approach atomizes a course into bite-size chunks of facts, all taught in a way that can’t possibly integrate into the story-line of a course. If I’m teaching trigonometry, for example, I may want a particular lecture to use radians rather than degrees, arcsin rather than sin-1, and so forth. I may want a linear equation to be y = a + bx, not y = mx + b. Sometimes I even make words up, for well-thought-out pedagogical purposes. How do I control all this if my students are watching Salman Khan give his pre-recorded lecture? The philosophy behind this sort of lecture seems to be that teachers are fungible, but we aren’t.
So…I have many doubts. It’s not that these issues can’t be resolved, but they won’t be unless we put a lot of thought into finding solutions. Inverting the classroom is a great idea, with wonderful potential, and it’s definitely worth pursuing — but only if it’s done right.
Be sure to watch Weston vs. Hamilton-Wenham in the quarterfinals of High School Quiz show, to be broadcast tomorrow night, 4/24, at 7:00 PM on Channel 2!
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:
My D Block class was even more hyper and less focused than usual today, since it was the afternoon of the last day before vacation. Some of them really wanted to have a cat-drawing contest, so I gave in and let them do that. Almost (but not quite) everyone joined in, drawing cats on the whiteboard. Here’s Zoe’s second-place drawing, followed by Natalie’s first-place entry:
And here’s the class, minus those who left early for vacation and those who were unwilling to appear in the photo:
Congratulations to the Weston High School Math Team for coming in fifth in the state at the Massachusetts State Math Playoffs in Shrewsbury on Monday! We have just learned that those results have qualified us to enter the New England playoffs in Canton on 4/29, so stay tuned…
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.
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.
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:
  
If you’re in or around Weston on Wednesday, come to our Ninth Annual Fractal Fair! It’s from 10:00 to 12:15 in the Weston High School Library. The exhibits and presentations, by 50 Honors Precalculus students (mostly juniors), will focus on ideas of iteration, recursion, fractals, and chaos. Although these are primarily mathematical in nature, many of them will also have tie-ins to other subjects, especially science and art. We’re all used to seeing athletic, musical, artistic, and drama performances — but how often do you get to see exciting math exhibits from high school students???
Be sure to watch High School Quiz Show tonight: Channel 2 at 7:00! The match is between Weston and Woburn, starring Mir Bokhari, Grace Huckins, Jon Birjiniuk, and Matthew Chernick, as seen left-to-right in this brief promotional video.
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:
I have posted the slides from my linguistics talk, but I’m not sure how useful they are without audio. The talk, after all, was an oral presentation accompanied by slides, not a visual presentation accompanied by audio. So I’m going to try to overlay an audio track. In the meantime, you can find the slides here.
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 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.
Congratulations to the Weston High School Math Team for moving on to post-season! We will be competing in the State Meet on April 1 because of our current standing, which is #3 in the state among medium-sized high schools: at this point Worcester Academy has 810 points, Winchester has 797, and Weston has 657. It will be a challenge to become state champions, but it’s not out of reach!
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.
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
Quotation from one of my students: “I hate it when people don’t like negative attitudes.”
(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.)
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.
Why do so many of my students use incorrect names for various polygons? They claim that they are merely recalling what they have been taught; maybe this is so, maybe not. I suppose there are two major possibilities:
- They are remembering incorrectly.
- They really were taught incorrectly.
Since this is Weston, I would prefer to believe it’s #1…but I have to admit that it might be #2, even in Weston.
Of course we shouldn’t just throw around the claim that certain names are incorrect without producing an argument for what the correct names are. Some of my students want to look in Wikipedia or count Google hits, but those methods lead to popularity contests, not truths. As I said in an earlier post, you can usually trust Wikipedia for mathematical information, but names occupy a middle ground between math and English, so Wikipedia is less reliable in this case than with pure math. As a better starting point, here is Wolfram Mathworld’s reasonably authoritative list of names for polygons with n sides:
| n |
polygon |
| 2 |
digon |
| 3 |
triangle (trigon) |
| 4 |
quadrilateral (tetragon) |
| 5 |
pentagon |
| 6 |
hexagon |
| 7 |
heptagon |
| 8 |
octagon |
| 9 |
nonagon (enneagon) |
| 10 |
decagon |
| 11 |
hendecagon (undecagon) |
| 12 |
dodecagon |
| 13 |
tridecagon (triskaidecagon) |
| 14 |
tetradecagon (tetrakaidecagon) |
| 15 |
pentadecagon (pentakaidecagon) |
| 16 |
hexadecagon (hexakaidecagon) |
| 17 |
heptadecagon (heptakaidecagon) |
| 18 |
octadecagon (octakaidecagon) |
| 19 |
enneadecagon (enneakaidecagon) |
| 20 |
icosagon |
| 30 |
triacontagon |
| 40 |
tetracontagon |
| 50 |
pentacontagon |
| 60 |
hexacontagon |
| 70 |
heptacontagon |
| 80 |
octacontagon |
| 90 |
enneacontagon |
| 100 |
hectogon |
| 10000 |
myriagon |
Let’s see what we can do with this list. I make the following observations:
- The very existence of a two-sided polygon sounds doubtful to most people. We’ll discuss this one below.
- Three- and four-sided polygons, being the most common ones, commonly have Latin names (triangle and quadrilateral), even though there are also alternative Greek names, which are very rarely used.
- All other polygons have Greek names. Therefore nobody ever calls a six-sided polygons sexagon or sextagon, and nobody calls a seven-sided polygon septagon, no matter what my students claim.
- For some mysterious reason, the 11-sided polygon is listed here not only as hendecagon (the correct name, from the Greek hendeca, meaning 11), but also with an incorrect alternative Latin-Greek name, undecagon. I see no reason to do this. In fact, another Wolfram Mathworld page makes this observation:
A hendecagon is an 11-sided polygon, also variously known as the undecagon or unidecagon. The term “hendecagon” is preferable to the other two since it uses the Greek prefix and suffix instead of mixing a Roman prefix and Greek suffix.
- Somewhat similarly, but worse, the 9-sided polygon is listed in both the Greek form, enneagon, and the hybrid, nonagon — but here Mathworld oddly prefers the Latin-Greek hybrid to the pure Greek. On their other page, however, they make this observation:
The nonagon, also known as an enneagon, is a 9-sided polygon. Although the term “enneagon” is perhaps preferable (since it uses the Greek prefix and suffix instead of the mixed Roman/Greek nonagon), the term “nonagon,” which is simpler to spell and pronounce, is used in this work.
Even though counting Google hits is a useless way to decide these issues, let’s check them out just for fun:
- 14,900 hits for “hendecagon”; 12,400 for “undecagon.” Hooray!
- 18,400 hits for “enneagon”; 69,500 for “nonagon.” Boo, hiss!
Oh — I also promised a discussion of two-sided polygons, didn’t I? Most people think they don’t exist, so they don’t need to be named. (Unicorns don’t exist, but they still have a name. Hmm….) Actually, however, they do exist: for example, start at the North Pole, draw a line segment along the prime meridian until it reaches the South Pole, and then draw another line segment from the North Pole along the 90° longitude line, also stopping at the South Pole. Voilà: a two-sided polygon! You may think I’ve cheated, since this polygon exists on the surface of a sphere, not on a plane, but it might be worth imagining that you lived on the surface of a sphere, not on a plane… Anyway, I’ve never heard the term digon before; I’ve seen biangle and bigon, however. Be sure to pronounce bigon with a long i, and think of the famous saying, “Let bigons be bigons.” Again we can check Google hits, useless though it may be: 14,600 hits for “digon,” 487 for “biangle,” and 7,330 for “bigon.” Even though “biangle” loses the popularity contest, I suspect that it’s the best choice, since it’s consistent with the general principle: use Latin names for polygons with four sides or fewer, Greek names for those with more than four sides, and hybrid names for none.
In my geometry class today, some students caught sight of a file on my computer named Weston+Yankees.pdf, so of course they wanted to know what it was. It turned out to be a Boston Globe article from six and a half years ago! Here are a few excerpts:
Mark Fishman of Weston is the picture of a happy, well-adjusted 9-year-old… Yet something has happened to Mark during the last two years, a strange and troubling transformation that is sweeping the elementary students of Weston like a bad case of head lice.
Mark and his classmates here are abandoning the Red Sox for the New York Yankees, shunning the Olde Towne Team in a show of rebellion that is upending this quiet community of top-notch public schools, giant homes guarded by sport utility vehicles, and rolling lawns dotted with swing sets.
Mark’s parents say that most of the boys in his class, about 50 children who will enter fourth grade at the Field School this fall, are now waving the pinstriped banner, tossing out their Pedro jerseys, and cheering for Derek Jeter.
Mark, eyeing the Yankees calendar and license plate on the wall, puts the figure closer to 65. The girls in his class, he sniffs, support the Red Sox.
…
Dr. Anita Bohensky, a New York child psychologist, said she was puzzled by the Weston children’s behavior and would need to interview them to render a proper analysis.
She speculated that the children might be exhibiting aspects of early adolescent rebellion or emulating an admired older peer who is a Yankees fan. She said it was unusual for 9-year-olds to rebel against their parents in this way.
I’m particularly amused by the implicit assumption that it’s necessary to interview a psychologist about this phenomenon.
My principal has selected me to give the first presentation in a proposed series of talks to be delivered by faculty and staff; the audience will consist of colleagues, students, parents, and community members. I’ve written a very rough description of what I’m intending to talk about (quoted below), but at a minimum the description needs polishing, and it may need significant revisions. For instance, I already know that I need to include something more about universals of language, I have to show that the presentation will be interactive, I want the focus to be about 90% on linguistics and only 10% on math (which may or may not be evident from the draft), and I have to make it clear that the questions asked in this description are merely examples of the kinds of questions that will be addressed and answered during the talk. So…let me know what suggestions you have!
Here’s the draft description:
Making order out of chaos:
How a linguist ended up teaching math
Linguistics is the scientific study of languages. It involves seeing patterns, putting puzzles together, developing hypotheses. Here’s an example from Kurdish, a language you know nothing about. (Yes, you’ve heard of the Kurds in Iraq, but do you know anything about their language? No??? I thought not.)
Here are six sentences in Kurdish, matched with English translations in random order:
|
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.
|
Can you figure out this puzzle? If so, can you 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” resembles a word in Spanish, French, and Latin. Is this just a coincidence? Why should Kurdish resemble these languages? Maybe there’s a reason…
At Weston High School we care about global connections. Linguistics reinforces those connections. 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?
Of course English is a world language as well. 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? Do all language have the same vowels? Is there anything that all languages share?
Finally, 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 something about the strange connection between the two.
The Weston High School Library recently posted a slide show from Rutgers University explaining why students shouldn’t use Wikipedia. This carefully produced polemic deserves a thoughtful rebuttal; I have endeavored to write one here. Be sure to watch the slide show before reading the rest of this essay.
Those of us of a certain age will remember Hamilton Burger’s frequent cry of “Incompetent, irrelevant, and immaterial!” from the old Perry Mason TV shows. The Rutgers slide show on Wikipedia immediately prompted me to raise the same objection. First, however, I do need to acknowledge the considerable amount of truth in the Rutgers argument. Yes, of course there are many biased statements, inaccuracies, and downright lies on Wikipedia. Yes, it should not be cited as an authoritative source in a formal research paper. Yes, Wikipedia sometimes falls short when we’re looking for accuracy, authority, objectivity, and currency.
But that’s just one side of the picture. The authors of this tendentious slide show fail to meet their own criteria by ignoring the other side! Here, as Paul Harvey used to say, is the rest of the story:
- Every single example in the slide show comes from a narrow range of subjects in which the perceptive reader should immediately be aware of bias. Indeed Wikipedia should not be relied upon for information about history, politics, or biography — subjects where opinion is likely to substitute for facts, whether intentionally or inadvertently. But there are other subjects where Wikipedia is exceptionally reliable. In two fields that I know a lot about, mathematics and linguistics, it is unquestionably the first place to look for accurate information. Go to Wikipedia to find out about vowel harmony in Turkish, but not about political harmony in Turkey. Go there to find out about number theory, but not about the theory of evolution.
This is not to say that Wikipedia is 100% accurate — of course it isn’t. But so-called “authoritative” sources aren’t 100% accurate either. I recently gave an assignment in which I asked my freshmen to comment on some statements about geometry from presumably authoritative sources; these contained errors that I was unable to find on Wikipedia but quickly found elsewhere. Even textbooks are far from immune. I am reminded of the late Richard Feynman’s famous critique of a middle-school science textbook that contained questions like, “John and his father go out to look at the stars. John sees two blue stars and a red star. His father sees a green star, a violet star, and two yellow stars. What is the total temperature of the stars seen by John and his father?” Of course professors like textbooks; they write them, after all. The rest of us should be skeptical of all sources, even textbooks.
- Regardless of the subject, Wikipedia is a fine location for beginning one’s research. It should be your first stop, not your last. I’m reminded of those advertisements in which a dealership or carpet store says, “Shop us last!” Aside from the newfangled use of “shop” as a transitive verb, which I can’t help noticing, I also observe that the slogan makes more sense than the “Shop us first!” that one sometimes sees. Wikipedia is indeed not the place to cite in your footnotes, but shop there first.
- Finally, the authors of the slide show make misleading use of their own sources. Why on earth do they cite Stephen Colbert of all people — not just once, but twice — as an authoritative source? Their whole argument is undermined by quoting a comedian in this role. Furthermore, although it is cute for them to cite the founders of Wikipedia in support of their argument, they do so in a highly misleading way. They quote one statement, “Wikipedia acknowledges that it should not be used as a primary source for serious research,” without emphasizing the word “primary”; of course it’s not a primary source, but it’s a great place to start in order to continue on to those primary sources. Then they quote Larry Sanger as a “co-founder of Wikipedia” as if he were still on board, whereas in reality he has every reason in the world to be bitter and biased; he is hardly an authoritative or neutral source. Finally, they quote Jimmy Wales (who is still very much on board) in a manner that is clearly incompetent, irrelevant, and immaterial: “For God sake, you’re in college; don’t use the encyclopedia.” Go read the original source, and it becomes clear that Wales is talking about encyclopedias in general, not just Wikipedia. He does say that Wikipedia is a good place to start, though you would never know it from reading this quotation that was taken out of context.
A class ought to feel like a community. (Mathematically speaking, that’s what make it a class, rather than a set. Yes, it’s a slightly different meaning of the word “class,” but the resemblance is not a coincidence.) We’ve all been in a class that had the coherence and mutual support that made it a community, and we’ve all been in many classes that were merely a group of students with a teacher. This post is not an analysis of how to turn the latter into the former — entire books have been written on the subject — but is merely a reflection of one small aspect of the issue: why is it so much easier to develop a community feel in an honors class than in a “regular” class? (In Weston the “regular” classes are called “college-prep,” so that’s how I’ll refer to them here.)
Actually, the presupposition behind my question might not be correct. I know from talking with my colleagues that it may depend on the subject, the specific course, and/or the teacher. But it’s definitely true in my experience as a math teacher for 37 years.
It would be easy to predict exactly the opposite generalization. Students in an honors math class — especially at Weston High School or similar schools — tend to be very competitive. Many suffer from academic stress and anxiety. They want to get into the same top-tier colleges, for which there are far more applicants than acceptances. Some of them (but remarkably few, actually) have trouble with social skills. All of these factors add up to a prediction that they wouldn’t function well as a community. And yet they usually do. Why is this?
My guess is that the factors in the previous paragraph may actually create a common bond among the students in an honors math class. They are a self-selected group who are all striving for the same thing. Most of them (certainly not all, but usually enough to create a critical mass) are in the course because they’re interested in mathematics and want to do well. The majority of them have done well in school and find many rewards there. They tend to feel a sense of communality with each other and even with the teacher. If they don’t feel that they belong, there’s a place to go: they can always switch to a college-prep class (even as late as Thanksgiving, at least at Weston).
College-prep math classes, in contrast, contain an enormous variety of students. By their very nature they are heterogeneous, ranging from those who could perfectly well be in honors (but chose not to be) all the way to those who are hanging on by their fingertips. Many students in a college-prep class have just as much stress as their classmates in honors classes, but many of them are overly relaxed. Some have given up. Many haven’t given up at all and do their work diligently. Some care. Some do not. Many are taking math just because it’s required (either an explicit requirement for high-school graduation or an implicit one for acceptance at college). Consider what happens to group work as a result: in an honors class, the large majority of students will accept randomly assigned groups and will work cooperatively with those in the group, but in a college-prep class only a minority will do so.
That, at least, is my analysis of what’s going on. There’s still the puzzling question of why it is different in some other disciplines (assuming that it really is).
The Math Department of Weston High School & Middle School was featured in a report on last night’s NBC Nightly News with Brian Williams! Be sure to watch all the way to the end (it’s only two and half minutes), not only to see two of my colleagues and their students but also to hear the closing remarks:
Rehema Ellis: Every expert we talked to says that if America wants better math test results, there have to be better math teachers. But they’re in high demand, and even with smaller classes and new innovations, to get the best person at the head of the class it’s no surprise that schools will have to make financial investments in those teachers.
Brian Williams: There’s no shortage of evidence out there.
Yesterday I wrote about Penny Noyce’s new book, Lost in Lexicon. What prompted that post was that I was on my way to the official launch party for the book. It was a great success, and I saw a couple of former students there, but I can’t claim to be able to describe it as well as the author can, so check out her own description and photos of the party in her blog.
If you regularly see my Facebook status in your News Feed, you may have noticed that it said “I’m lost in Lexicon right now…” on October 17. This status confused some of my students. One of them asked, “How did you get lost in Lexington?” (Apparently he isn’t a very careful reader.) Another student asked me what it meant:
“Lost in Lexicon is the title of a new book by Penny Noyce, a neighbor of yours from Weston,” I replied.
“Someone in Weston wrote a book????” was her astonished response.
I assured her that there are plenty of people in Weston who have written books.
Anyway, Lost in Lexicon: An adventure in words and numbers is indeed the title of a new book by Penny Noyce. It’s a work of fiction, somewhat in the spirit of The Phantom Toolbooth, aimed at readers in middle school (in my judgment). Of course the real reason I had to get a copy was not that the author lives in Weston (and is the mother of three of my former students), but that the focus of the book is words and numbers, as the subtitle shows. What could be a better combination?
If you know children of the appropriate age (or older, for that matter), suggest this book to them. It’s both fun and informative, and should enhance or kindle interest in both math and language.
Well done, Weston High School Math Team! Congratulations to Alexandra R., Andrew H., Andy Y., Blake W., Caleb T., Daniel P., Jason M., Jonathan B., Julia B., and Pravina S. for a fine performance at this afternoon’s meet at Lexington High School — and special congratulations to freshman William K. for a near-perfect score and to juniors Grace H. and Stephanie P. for perfect scores! Grace and Stephanie earned pizzas at lunch today in recognition of this achievement.
In our department meeting today, we had a guest speaker from the Wellness Department* who talked with us about building closer connections with students, an atmosphere of trust, and greater engagement by our students. All good things, certainly. Part of the pitch was that the few minutes spent on such tasks in each lesson would more than pay off in increased learning, and I do believe that.
So why do I feel uncomfortable about the whole idea? I suppose it’s because it just doesn’t feel like me. Even the initial idea, shaking hands with each student on entering the classroom, feels unnatural; I’m not at all convinced that I could carry it off. The speaker says that his students not only welcome this practice but explicitly ask for it if he forgets. Maybe so, but I can’t see myself doing it.
On the other hand, I don’t want to rule out the idea, either in detail or in the big picture, and I’m willing to try. I’m even willing to sign up for a proposed summer workshop on the subject, if the calendar permits.
* Yes, I know…
|
|
