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:

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.

If I had an extra minute, I’d also talk about how we shouldn’t only show the mathematics that’s useful — and statistics is useful for being an educated consumer and citizen. We could replace a lot of the drudgerous mathematics that’s being taught with math that’s purely fun, with no real promise of “you’re going to use this,” but just “this is beautiful stuff.”

You can go ape over patterns in Pascal’s triangle, in the Fibonacci numbers, in chaos, [...]]]> The great Art Benjamin, whom we’ve had the pleasure of listening to twice at Weston High School, made the following remarks in his TED talk:

If I had an extra minute, I’d also talk about how we shouldn’t only show the mathematics that’s useful — and statistics is useful for being an educated consumer and citizen. We could replace a lot of the drudgerous mathematics that’s being taught with math that’s purely fun, with no real promise of “you’re going to use this,” but just “this is beautiful stuff.”

You can go ape over patterns in Pascal’s triangle, in the Fibonacci numbers, in chaos, in fractals. These things that are just positively inspirational. We don’t make — I mean, I’m listening to this music. It’s inspirational. But I didn’t have to be drilled with how to draw my notes properly and learn all this music theory before I got exposed to that kind of music. I think the same sort of thing could happen in mathematics.

Why not give them a taste of beautiful mathematics in addition to the useful stuff?

He’s right, of course. But what’s most interesting is the interplay among the three different ideas of usefulness, fun, and beauty. Too often we end up with none of the above. Benjamin advocates more statistics and less calculus (and preparation for calculus). That path certainly wins on the usefulness score, though many would question it on grounds of fun and beauty. He cites wonderful examples for those, and we do find that a great many students enjoy studying chaos and fractals, finding both fun and beauty in them. Pascal’s Triangle and Fibonacci numbers are in our curriculum, but we could do more with them, especially if we want students to see their beauty and enjoy studying them.

We held our annual observation of Pi Day on Monday in two of my classes and on Tuesday in the other two (since they didn’t meet on Monday). But one of my students pointed me to a couple of posts claiming that pi is wrong — not wrong in the sense that the ratio of the circumference of a circle to its diameter isn’t actually π, nor wrong in the sense that π doesn’t have the value we think it does (of course it’s the right ratio and does have that value) but wrong [...]]]> This is a few days late, but…

We held our annual observation of Pi Day on Monday in two of my classes and on Tuesday in the other two (since they didn’t meet on Monday). But one of my students pointed me to a couple of posts claiming that pi is wrong — not wrong in the sense that the ratio of the circumference of a circle to its diameter isn’t actually π, nor wrong in the sense that π doesn’t have the value we think it does (of course it’s the right ratio and does have that value) but wrong in the sense that it would be much more useful and pedagogically better to use the ratio of the circumference to the radius. This ratio, called tau (τ) is explored by Vi Hart in her usual inimitable manner. Do watch her video!

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:

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

These videos are “subversive,” as one of my colleagues (approvingly) labels them. The common theme appears to be that math classes in high school are boring, because they tend to focus on minute details rather than the big ideas of interesting mathematics. So how does the bored student react? By doodling, of course. But…as you watch the frenetically paced video, you realize that Hart is actually teaching the very mathematical concept that she pretends to be avoiding by doodling.

I’m not sure which one [...]]]> And the award for coolest math video ever goes to…Vi Hart, for her Doodling in Math Class series.

These videos are “subversive,” as one of my colleagues (approvingly) labels them. The common theme appears to be that math classes in high school are boring, because they tend to focus on minute details rather than the big ideas of interesting mathematics. So how does the bored student react? By doodling, of course. But…as you watch the frenetically paced video, you realize that Hart is actually teaching the very mathematical concept that she pretends to be avoiding by doodling.

I’m not sure which one is my favorite. Perhaps it’s “Infinity Elephants,” which I’ll show to my precalculus class. Or perhaps “Binary Trees,” which would work both for that class and for Algebra II. Do check out all four, actually. And while you’re doing that, I’ll explore Hart’s three online publications, all of which look intriguing:

• “Symmetry and Transformations in the Musical Plane”
• “Computational Balloon Twisting: The Theory of Balloon Polyhedra”
• “Using Binary Numbers in Music”

I’ll let you know about these after I’ve had a chance to explore them.

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.

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 [...]]]> 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:

1. They are remembering incorrectly.
2. 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:

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.

To me, you can’t claim to have solved the Rubik’s cube unless you have invented (on your own) an algorithm that will turn any randomly arranged Rubik’s cube into the correct configuration. [...]]]> Why is it that the phrase “solving the Rubik’s cube” has such a different meaning to me than it does to others? For a long time I was quite puzzled by people who made the implausible claim that they had “solved” the Rubik’s cube. And then I had a sudden insight: they were using the word in a way that was very different from the way I use it.

To me, you can’t claim to have solved the Rubik’s cube unless you have invented (on your own) an algorithm that will turn any randomly arranged Rubik’s cube into the correct configuration. But apparently many people have a far lower bar. They interpret “solving” to be the process of using a known algorithm in order to turn a cube into the correct configuration.

I think of this as the difference between genuine problem-solving and going through an exercise. For example, using the quadratic formula is not problem-solving. Independently deriving the quadratic formula, or manipulating a non-obvious equation or word problem into a form in which the quadratic formula can be used, would be examples of problem-solving.

Student A: Is 1 a prime number?

Student B: No.

Student A: So it’s composite?

Student B: No, it isn’t prime and it isn’t composite.

Student A: Why isn’t it prime?

Student B: Because it only has one factor. Prime numbers have two factors.

Me: That’s true, but it’s not the real reason.

Clearly Student B’s teacher had given him this strange definition. I call it “strange” because it’s unmotivated: why would anyone want to define “prime” in that way? What’s so special about having two factors (or, preferably, divisors)? Why would [...]]]> At Saturday Course we were working with prime numbers, and one fifth-grader asked his classmates a question:

Student A: Is 1 a prime number?

Student B: No.

Student A: So it’s composite?

Student B: No, it isn’t prime and it isn’t composite.

Student A: Why isn’t it prime?

Student B: Because it only has one factor. Prime numbers have two factors.

Me: That’s true, but it’s not the real reason.

Clearly Student B’s teacher had given him this strange definition. I call it “strange” because it’s unmotivated: why would anyone want to define “prime” in that way? What’s so special about having two factors (or, preferably, divisors)? Why would two factors make a number prime, whereas one or three does not? What sense does that make?

The real reason that 1 can’t be prime is that the Fundamental Theorem of Arithmetic (FTA) wouldn’t hold if it were. Number theory guarantees that every positive integer has a unique prime factorization (ignoring order). For example, 40 equals 2³×5 and nothing else. But if 1 were prime, 40 could also be 2³×5×1 or 2³×5×1² or 2³×5×1³, etc. That’s why we exclude 1 from being prime. And there’s an important lesson here: mathematicians often tweak their definitions in order to make theorems work. In the words of…well, I can’t remember who said this…they shoot the arrow and then draw the target.

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 [...]]]> 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.

“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 [...]]]> 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.

I was one of those students. But we were — and are — in the minority. The majority of students find proof-writing difficult, inscrutable, and not very interesting. Worse yet, they find it pointless. This may be especially true when the artificial strictures of two-column proofs are imposed.

One remedy for teachers and learners alike is to read two engaging essays: “What Is Mathematics For?” by Underwood Dudley, [...]]]> Yes, some students enjoy writing proofs. They accept the task as a challenging puzzle, one that provides an agreeable sense of completion once you’ve successfully threaded a path from the given information to the conclusion.

I was one of those students. But we were — and are — in the minority. The majority of students find proof-writing difficult, inscrutable, and not very interesting. Worse yet, they find it pointless. This may be especially true when the artificial strictures of two-column proofs are imposed.

One remedy for teachers and learners alike is to read two engaging essays: “What Is Mathematics For?” by Underwood Dudley, and “Why do we have to learn proofs!?” by Joshua N. Cooper. Both of these make it very clear that the current trend of justifying math by its practical, sometimes even career-oriented, real-life applications is misleading at best and dishonest at worst. The major reason to learn algebra, proofs, and other parts of mathematics is that they teach you to become a critical thinker. I admit that math isn’t always taught that way, but it should be. Treating a proof as a puzzle isn’t dumbing it down, it’s the essence of logical thinking.

More on this later. In the meantime, read the two essays!

The real problem is that statistics can be so misleading. The really relevant criterion is the salary scale, since these figures probably reflect more [...]]]> No, Weston teachers do not have the highest salaries in the state. According to today’s Boston Globe, Weston ranks only 16th in the state in average teacher salaries! At $73,338, we can be compared to a high of $79,444 (Old Colony), though we’re still well above the state median of $61,800. If you look at the alphabetical district-by-district listings, you can compare us to five of our immediately neighboring communities, and we’re higher than any of them:

The real problem is that statistics can be so misleading. The really relevant criterion is the salary scale, since these figures probably reflect more about the average age and teaching experience of the faculty than they tell you about the minimum or maximum salaries in any particular district.