On yesterday’s “Weekend Edition Sunday” on NPR, there was a five-minute segment on “What’s Your Favorite Number and Why?” The interview with British mathematician Alex Bellos is definitely worth listening to; you can find a listen-to-the-story link on that webpage. But it’s also worth reading Robert Krulwich’s follow-up at the same location, especially various contributors’ accounts of what their favorite numbers are and why. The only thing that puzzles me is why nobody seems to have the right answer. It’s 42, of course.
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In last month’s post about our Fractal Fair, I made the following promise:
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:
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 terrific turnout last night at the Driscoll School in Brookline. More than half (!) of the fourth- and fifth-graders (and their parents) showed up for an evening event revolving around Penny Noyce’s Lost in Lexicon. My role was to be the Pi Man, representing the Village of Irrationality. Kids (and often their parents) would measure various circular bowls, dividing the circumference by the diameter in each case. This being Brookline, most of them already knew about pi and expected to get the “correct” value, so the activity tended to turn into the surprise they experienced when the average of their ratios for three different bowls turned out to be less precise than they had expected. One boy decided to measure the entire round table to get a better result. Everyone had a great time, being totally engaged in a variety of activities relating to math and language. What better combination could there be?
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.
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:
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.
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!
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???
BSP*: Come hear my talk on linguistics at 7:00 PM on Tuesday, February 1, at the Weston Public Library! Here’s a description:
*Blatant self-promotion
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:
I’ll let you know about these after I’ve had a chance to explore them.
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:
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:
Even though counting Google hits is a useless way to decide these issues, let’s check them out just for fun:
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.
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:
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.
At Saturday Course we were working with prime numbers, and one fifth-grader asked his classmates a question:
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.
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:
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:
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.
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!
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.
“I make order out of chaos.” This is how an old friend whom I hadn’t seen in years explains her transition from linguistics to statistics, when people think it’s a complete change of field. It’s how she explains it to non-linguists, of course — as I already knew the connection. But the phrasing really resonates with me. I’ve described elsewhere how the search for patterns and abstract generalizations is what unites linguistics and math teaching in my mind, but I rather like the step up the ladder of abstraction implied by “I make order out of chaos.” It also got me thinking about why I like the mystery and science fiction genres in popular fiction. My liking for science fiction is no mystery, so to speak: anyone with a mathematical bent is likely to enjoy the conventions of that genre. But what about mysteries? I’d been thinking about that lately, and it occurred to me that mystery writers also make order out of chaos: the unsolved crime is cognitively chaotic, and the solution creates order out of it. Furthermore, the puzzle that’s often involved bears definite kinship to the kinds of puzzles we solve in both math and linguistics. Just a thought…
“Mental acuity of any kind comes from solving problems yourself, not from being told how to solve them.” So says Paul Lockhart, and I couldn’t agree more. It’s great having cooperative students who will correctly follow directions in solving problems — or should I say exercises — but following directions is a cheap virtue. As Lockhart observes, you don’t develop your mental faculties that way. On March 28, 2008, I wrote a brief laudatory piece about Lockhart’s fascinating essay, which he has now turned into an irritating book, also called Mathematician’s Lament. That’s too bad, as he has a lot of valid things to say. But most readers will be unable to see what’s good because it’s surrounding by so much that’s annoying. In particular, Lockhart seems to take an extreme view in favor of throwing out all curriculum and all direct instruction, replacing everything with student-directed problem solving. I say “seems to take” because in fact that’s not actually his position; it’s just that he gets so carried away with his radical POV that everything else gets lost. So, if you read this book, you need to star the following sentence in particular:
Keep that in mind. It’s just that everything Lockhart discusses is in fact all the way on the other side. Consider, for example, this provocative chapter title: “High School Geometry: Instrument of the Devil.” Certainly some students do like geometry, though Lockhart claims that those students would like it even more if it were taught his way. And surely most adults remember their high school geometry class with something less than fondness. The big complaint about high school geometry — and here I agree with Lockhart — is that the central themes of proof and definition are presented so woodenly. Writing proofs about claims that are obvious feels arbitrary and useless, and yet that’s what most of the early months of geometry are filled with. And the two-column format is arbitrary and restrictive, a peculiar American custom that no real mathematician would ever use. As Lockhart observes, “A proof should be an epiphany from the gods, not a coded message from the Pentagon.” But it’s rare experience in high school geometry for students to spend a long time struggling with a non-obvious problem, then to come up with a non-obvious conjecture, and finally to write a convincing proof that shows how the conjecture connects with other knowledge. That’s how it should be done. A similar issue occurs with definitions:
Hear, hear! All of this, of course, is driven by one’s concept of what math really is. Lockhart is a pure mathematician, viewing problem-solving and puzzle-solving as rewarding for their own sake, and I agree with him there. But his contempt for applied mathematics will do nothing but turn off most of his readers. It’s important for students to understand that applications come after the math is developed and hardly ever motivate the discovery of new mathematics, but it’s also important for them to work with those applications. Some students will be motivated by that, and everyone will learn something that their future teachers will expect. Nevertheless, Lockhart’s characterization of what math really is is spot on:
Unfortunately this characterization flies in the face of so much of what is expected of math teachers and math students. MCAS and SATs and science teachers inadvertently encourage the “collection of truths” misconception, even though they of course also want reason and understanding. Finally, I need to mention the subtitle of Dan Meyer’s blog, dy/dan. The subtitle is simply less helpful. This characterization may seem like an odd one, especially when it’s the subtitle of a blog that’s well worth reading. But Meyer’s resolution to be less helpful is an important one. Like most math teachers, my unthinking inclination is usually to try to be helpful, to answer questions, to point students in the right direction. But Lockhart’s response to a certain question from a student is to observe that “the right thing for me to do as your math teacher would be nothing.” In other words, to be less helpful. That, in the long run, is what will actually be helpful to the student. I just wish that Lockhart had limited himself to a more tempered criticism and had been clearer about taking a balanced approach; he will turn off too many readers who would have a lot to gain from his wisdom if they could only pay attention to what’s good rather than what’s irritating in this provocative book.
Ten years ago, the highly respected mathematician Lynn Arthur Steen wrote an article entitled, “Algebra for All in Eighth Grade: What’s the Rush?” Well, now we know what the rush is…or do we? Steen sets up the issue with a couple of rhetorical questions:
He then proceeds to give several arguments in favor of algebra: it provides access to higher education and jobs, it is the language of the information age, it is the mark of a rigorous education… in short, it is “the key to access in our technological society.” But then come the counterarguments:
So, where does this leave us? Steen’s conclusion is a sensible one: everyone should take algebra, but not necessarily in eighth grade. As the title of his article asks, What’s the rush? The rush is that many states, including California and Massachusetts, are now mandating algebra in eighth grade, which moves the argument from whether we should implement this to how we should implement it; Steen’s four bullet points are real, and passing laws won’t wash them away. This is not to say that all eighth-graders really do study algebra, but Weston is surely not the only system in which Algebra I is simply not even offered at the high school: we expect all incoming ninth-graders to enter with an Algebra I background. I don’t know about other school systems, but Weston has attempted to address all four of Steen’s points, though the frst three are of course easier to remedy than the fourth. Nationwide about one third of eighth-graders study algebra, for better or for worse. Weston, of course, is Lake Wobegon, so all of our students are capable of learning algebra in eighth grade. For much more depth of this question, read Tom Loveless’s article, “The Misplaced Math Student: Lost in Eighth-Grade Algebra,” or his full report from the Brown Center of the Brookings Institution. Here are a few interesting excerpts from this 16-page document:
Loveless then goes on to cite statistics that show that “the typical eighth grader in an advanced math course knows less today than in 2000.” Hmmm…
Anyway, do read the entire article. Here are some excerpts from Loveless’s conclusion:
Fortunately Weston is different. But read the whole article, as I said above. On a slightly different but closely related matter, I need to mention a comment I overheard at the next table at Tavolo: “I don’t understand why kids have so much trouble with algebra. It’s nothing but finding the value of x.” No, that’s not what algebra is about. Sigh.
Some of us can barely remember anything from third grade, but last night at a restaurant in Dorchester I met someone my age who was truly traumatized for life by a single experience way back in third grade. We’ll call her Laura. When she found out that I’m a math teacher, she had to tell me her story. It went something like this:
And to this day — despite success in future math courses, and eventually getting into med school — Laura has a phobia about math.
This is the cue for my students to roll their eyes… Yesterday I got into a heated discussion with another math teacher about an important issue: how to define a trapezoid. He was arguing in favor of the position that a trapezoid has exactly one pair of parallel sides; I was arguing in favor of the position that a trapezoid has at least one pair of parallel sides. We both agree that it’s a quadrilateral. My opponent made several good points:
But I made, IMHO, several better points:
Of course it all leads to a teachable moment — or more than a moment, actually. In my honors geometry class we devoted more than an hour of class and homework time to exploring the ramifications of the two definitions. Then the issue emerged in a question on the next quiz:
And then in a four-part question on the next test:
Bonus: little did I realize that the embedded “if” in Moise’s theorem would confuse some students. Apparently they had never seen a theorem where the consequent preceded the antecedent. So that led to a worksheet for another assignment. Thinking about what you’re learning is a good thing. And, as Humpty Dumpty said in Through the Looking Glass, “When I use a word, it means just what I choose it to mean — neither more nor less.”
This pie chart from Fox News speaks for itself:
At this week’s Math Department meeting, we spent the first 15 minutes or so discussing what we do to help “struggling students” succeed in our courses — particularly what resources we provide. Something was bothering me about the whole discussion, so I waited a few minutes before I said anything. Then I realized what was bothering me: the participle “struggling” was apparently being used as a synonym for “unsuccessful.” This usage has long seemed completely wrong to me. To my mind, I have some students who struggle and do well. I also have some students who are unsuccessful — precisely because they don’t struggle. It all comes down, of course, to the meaning of the verb “struggle.” Let’s see what a couple of reputable dictionaries say about the matter. In each case I’ve selected the appropriate sense of the word:
Linguists, of course, always insist on being descriptive rather than prescriptive, and yet they usually rely on introspection or on the use of a small number of informants. I suppose a more accurate technique in this context would be to survey a large number of people and find out how they use the word “struggle”; I have no idea what we would find, but at least the dictionary definitions make it absolutely clear to me that we should stop using this verb as a synonym for “be unsucessful.” On another front, we spent the next 25 minutes of the department meeting discussing how to solve the equation x2 = 2x. I told my Algebra II class about this, since we’re currently transitioning from quadratic functions to exponential functions, and one of their homework problems called for a comparison between y = x2 and y = 2x. They found it an unlikely topic for a meeting — and they were especially surprised that we were so geeky that the meeting ran ten minutes over before anybody looked at the clock and noticed that we had gone past the announced end of the meeting. By the way, there are three solutions to this equation. One solution, 2, is immediately obvious; a second solution, 4, is not at all obvious until you give it some considerable thought, at which point it “becomes obvious.” The third solution can be estimated by looking at a graph. Finding this solution is left as an exercise for the reader.
A major topic of high-school math is the study of transformations. My colleague, Jim McLaughlin, wants you to know that his desk has somehow undergone a miraculous transformation:
I was recently asked whether a Boston voter should always vote for the full allotment of four at-large City Council candidates, or whether bullet voting made sense. I unhelpfully replied, “It depends.” It occurred to me that I had already dealt with this issue four years ago. So read that link if you want to read the mathematical arguments (actually, not too much math!) for or against bullet voting, depending on the situation.
I don’t know why I never knew about the Cornell Math Explorers Club before now. Its website is a terrific enrichment resource for high-school math students and their teachers, with a wonderful assortment of slightly offbeat topics that are right up my alley for all the programs in which I teach: Weston High School, Crimson Summer Academy, and Saturday Course. Their modules include cryptography, graph theory, probability, set theory, topology, game theory, and the mathematics of voting and elections. I wish I had known about this site earlier. Check it out! |
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