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.

It also got me [...]]]> “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…

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 [...]]]> “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:

If I object to a pendulum being too far to one side, it doesn’t mean I want it to be all the way on the other side.

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:

Definitions matter. They come from aesthetic decisions about what distinctions you as an artist consider important. And they are problem-generated. To make a definition is to highlight and call attention to a feature or a structural property. Historically this comes out of working on a problem, not as a prelude to it.

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:

Math is not about a collection of “truths” (however useful or interesting they may be). Math is about reason and understanding.

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.

How can a subject that for many adults serves as a metaphor for frustration suddenly be the top priority for soccer moms and internet dads? And why do so many parents suddenly demand of their schools and their children something they themselves neither mastered nor loved?

He then proceeds to give several arguments in favor of algebra: it [...]]]> 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:

How can a subject that for many adults serves as a metaphor for frustration suddenly be the top priority for soccer moms and internet dads? And why do so many parents suddenly demand of their schools and their children something they themselves neither mastered nor loved?

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:

• Relatively few students finish seventh grade prepared to study algebra. At this age students’ readiness for algebra — their maturity, motivation, and preparation — is as varied as their height, weight, and sexual maturity. Premature immersion in the abstraction of algebra is a leading source of math anxiety among adults.
• Even fewer eighth grade teachers are prepared to teach algebra. Most eighth grade teachers, having migrated upwards from an elementary license, are barely qualified to teach the mix of advanced arithmetic and pre-algebra topics found in traditional eighth grade mathematics. Practically nothing is worse for students’ mathematical growth than instruction by a teacher who is uncomfortable with algebra and insecure about mathematics.
• Few algebra courses or textbooks offer sufficient immersion in the kind of concrete, authentic problems that many students require as a bridge from numbers to variables and from arithmetic to algebra. Indeed, despite revolutionary changes in technology and in the practice of mathematics, most algebra courses are still filled with mindless exercises in symbol manipulation that require extraordinary motivation to master.
• Most teachers don’t believe that all students can learn algebra in eighth grade. Many studies show that teachers’ beliefs about children and about mathematics significantly influence student learning. Algebra in eighth grade cannot succeed unless teachers believe that all their students can learn it.

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:

At first glance, this appears to be good news… Research also suggests that students who take algebra earlier rather than later subsequently have higher math skills. These findings, however, are clouded by selection effects — by the presence of unmeasured factors influencing who takes algebra early and who takes it late…

The push for universal eighth-grade algebra is based on an argument for equity, not on empirical evidence. By completing algebra in eighth grade… students are able to take calculus in the senior year of high school… From this point of view, expanding eighth-grade algebra to include all students opens up opportunities for advancement to students who previously had not been afforded them, in particular students of color and from poor families. Democratizing eighth-grade algebra promotes social justice.

…

One catch. Course-taking is a means to an end, not an end in itself. Students take math courses to learn mathematics. Will policies mandating algebra for all eighth graders mean that the nation’s students learn more math? Not necessarily…

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…

…Any teacher who stops to teach misplaced students fractions shortchanges the well-prepared students who sit in that algebra class… There will be advocates, despite the data presented here, who will continue to argue for placing low-performing eighth graders in algebra classes. They believe that a more rigorous course is always preferable to a less rigorous one. Many do not believe that students must learn basic mathematics in order to successfully tackle higher-level mathematics… Algebra teachers already feel the strain of such unrealistic expectations.

Anyway, do read the entire article.

Here are some excerpts from Loveless’s conclusion:

One hundred twenty thousand students are misplaced in their eighth-grade math classes. They have not been prepared to learn the mathematics that they are expected to learn… Two groups of students pay a price. The misplaced eighth-graders waste a year of mathematics, lost in a curriculum of advanced math when they have not yet learned elementary arithmetic… Their clasmates also lose — students who are good at math and ready for algebra. These well-prepared but ill-served students also tend to be black and Hispanic and to come from low socioeconomic backgrounds. Teachers report that classes of students with widely diverse mathematics preparation impede effective teaching, that too many students arrive in algebra classes unmotivated to learn… Universal eighth-grade algebra is creating more problems than it solves, with 120,000 students not learning the mathematics that they need to know and hundreds of thousands of their classmates paying an educational price along with them.

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.

A couple of weeks into second grade, Laura’s teacher determined that she was so bright that she should skip a grade, and so Laura was instantly promoted to third grade — with the approval of her mother, but [...]]]> 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:

A couple of weeks into second grade, Laura’s teacher determined that she was so bright that she should skip a grade, and so Laura was instantly promoted to third grade — with the approval of her mother, but the frowning disapproval of the third-grade teacher, whose plans and groups were all messed up by this unexpected child. The third-graders had been adding two-digit numbers with carrying, but of course Laura didn’t know how to do this, since she had missed all but the first two weeks of second grade, not to mention the beginning of third. When she was unable to do the assigned problems, the teacher called her up in front of the room and said to the class, “Laura thinks she’s so smart because she skipped a grade, but in fact she’s stupid. She can’t even add 28 and 47.”

And to this day — despite success in future math courses, and eventually getting into med school — Laura has a phobia about math.

My opponent made several good points: Our current textbook defines the word his way. So do some other textbooks. The common image of a trapezoid has two non-parallel sides. We [...]]]> 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:

• Our current textbook defines the word his way.
• So do some other textbooks.
• The common image of a trapezoid has two non-parallel sides.
• We don’t expect someone to look at a parallelogram and exclaim, “That’s a trapezoid!”

But I made, IMHO, several better points:

• Nowhere else do we define a geometric object in this exclusionary way. We don’t say that a rectangle cannot have four congruent sides. We all agree that squares are rectangles, rhombuses are parallelograms, circles are ellipses, etc.
• Many textbooks, including Moise’s and UCSMP, do define it my way.
• In the software we use, the Geometer’s Sketchpad, it’s straightforward to construct a trapezoid with my definition but not with his.
• Most importantly, the quadrilateral hierarchy should show parallelograms as a subset of trapezoids because theorems about trapezoids also apply to parallelograms.

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:

Always/Sometimes/Never: Under Mr. Davidson’s preferred definition of “trapezoid,” the diagonals of a trapezoid are congruent.

And then in a four-part question on the next test:

1. Define “trapezoid” as the textbook does.
2. Define “trapezoid” in the way that Mr. Davidson prefers.
3. Former Harvard professor Edwin Moise has the following theorem (not definition) in his book: “A trapezoid is a parallelogram if its diagonals bisect each other.” Can you tell which of the two definitions of “trapezoid” must have preceded this theorem? Explain convincingly.
4. Prove Moise’s theorem (using whichever definition you identified in part c).

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 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 [...]]]> 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:

1. to make strenuous…efforts in the face of difficulties… <struggling with the problem>
2. to proceed with difficulty or with great effort <struggled through the high grass> <struggling to make a living>

—Merriam-Webster

1. to be strenuously engaged with a problem, a task, or an undertaking
2. to progress with difficulty <struggled with calculus>

—American Heritage Dictionary

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 = 2^x. 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 = 2^x. 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.

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.

Math content included Fibonacci spirals in nature, and the Unexpected Hanging paradox in its original form. In order to make the latter less gruesome and more relevant to students, it is usually changed to an unexpected quiz, as Jim Loy points out in the linked article. It goes something like this:

I’m going to give you a pop [...]]]> On the other hand, the season opener of Numb3rs — Season Six, which is hard to believe! — was pretty good, even it was skimpy on the math and a bit long on tensions between Charlie and Amita. But this is television, after all.

Math content included Fibonacci spirals in nature, and the Unexpected Hanging paradox in its original form. In order to make the latter less gruesome and more relevant to students, it is usually changed to an unexpected quiz, as Jim Loy points out in the linked article. It goes something like this:

I’m going to give you a pop quiz next week, and I guarantee that it will come as a surprise to you. Of course I can’t give it on Friday, since if you walk into class on Friday and haven’t had the quiz yet, it will no longer be a surprise. So you rule out Friday.

Could I give it Thursday? Since you’ve already ruled out Friday, you will expect the pop quiz when you walk into class on Thursday. Then it won’t be a surprise. So I can’t give it Thursday.

By the same reasoning, I can’t give it Wednesday. Similarly, I can’t give it Tuesday. Or even Monday!

So, I guess I’ll just have to give the quiz right now.

And I hand out the quiz.

Where’s the flaw?

And not basketball — although it’s the right season for that.

I’m talking, of course, about the New England Association of Math Leagues Playoffs, which took place today at Canton High School. I’m not sure yet just how Weston High School did, but we were in sixth place after the individual rounds and just before the final round (the team round, and I don’t have its results yet). In fact, we were close to fifth, so maybe we finished fifth. Or maybe not. At any rate, congratulations to our team for an excellent job: [...]]]> No, not football — too late for that.

And not basketball — although it’s the right season for that.

I’m talking, of course, about the New England Association of Math Leagues Playoffs, which took place today at Canton High School. I’m not sure yet just how Weston High School did, but we were in sixth place after the individual rounds and just before the final round (the team round, and I don’t have its results yet). In fact, we were close to fifth, so maybe we finished fifth. Or maybe not. At any rate, congratulations to our team for an excellent job: Alex Bruce ’09, Katie Hsia ’09, Ernest Zeidman ’10, Jonathan Birjiniuk ’11, Grace Huckins ’12, and Stephanie Palocz ’12!

It turned out that Rachel interpreted the name of the course rather too literally; she saw no reason for us to do anything more than the bare minimum to prepare her for calculus. This was AB Calculus, of course; she wasn’t interested in all the parts of my course that would have prepared her for BC Calculus had she chosen to [...]]]> A few years ago, one of my former students from Honors Precalculus informed me that my course had been “unnecessarily difficult.” An interesting phrase. “What does that mean?” was my puzzled response. Let’s call her Rachel (not her real name).

It turned out that Rachel interpreted the name of the course rather too literally; she saw no reason for us to do anything more than the bare minimum to prepare her for calculus. This was AB Calculus, of course; she wasn’t interested in all the parts of my course that would have prepared her for BC Calculus had she chosen to take it. So I explained that to Rachel, but I also pointed out that every math class has many goals, only one of which is the need to prepare students for the next course in sequence. What else did she get out of precalculus? (I thought she might say “a B minus”…but in fact she just shrugged and repeated her criticism.)

This all made me wonder yet again about the value of feedback from students. I learned something about Rachel, but I learned almost nothing that I could use in rethinking the course. (Actually, I did learn something useful: I realized that I should be more explicit about the goals of the course when talking to students.) Nevertheless, when it came time recently to give questionnaires to students, I of course complied with this requirement as did all Weston teachers. (Question to my current students: They all did, didn’t they?) We were allowed to develop our own questionnaires, and we could share as much or as little of the results as we felt appropriate. I chose to share all the results, both with my students and with my department head, and I’ll present a few interesting excerpts here.

First, as I have already mentioned Honors Precalculus, let’s see what I learned from the 43 students who responded (that’s a 100% return from both sections — one of 23, the other of 20). It’s hard to tally open-ended responses, since a prompt like “The best things about this course so far have been…” can elicit anything from a topic to pedagogy to teaching style. I would surely have gotten different results had I prompted for responses to a particular topic or a particular type of lesson. Nevertheless, it intrigued me that 5 students listed groupwork among their “best things about this course” and 6 listed it among their “worst things about this course.” Hard to know what to do with that. So let’s abandon the four open-ended questions and move on to Part Two, which presented students with a couple of dozen statements, each of which they could agree with or not. Here are a few thought-provoking numbers, showing how many agreed with each statement:

One student agreed with all six of this last set of statements! Where would we find the time? Anyway, the rest of these numbers may be thought-provoking, but it’s not necessarily clear what to do in response.

Now let’s turn to my other course, Algebra II, which is a “college-prep” course at Weston. (As discussed elsewhere, all non-honors courses at Weston are “college-prep” and tend to attract an enormous range of students: by definition all those who choose not to take honors or are unable to do so. Since Algebra II is a graduation requirement, no one escapes it, even those who are weakest at math as well as those few who are not intending to go to college.) I have two sections, one of 21 and one of 24; 41 of the 45 responded. Here are some numbers from those 41 in response to the same statements quoted above:

What do these results tell you (if anything)?

First let’s look at some similarities that go beyond what I hinted at in the first paragraph: For some reason, people who state their uninformed opinions in [...]]]> Just about everyone can speak, so we all have an opinion about language. Just about everyone can count, so we all have an opinion about math. Everyone’s an expert. After reading uninformed opinions about both, I decided to compare and contrast. Off the top of my head I can think of three points of comparison and four points of contrast; if I were writing a thesis on this subject I would undoubtedly find many more.

First let’s look at some similarities that go beyond what I hinted at in the first paragraph:

1. For some reason, people who state their uninformed opinions in public almost always take the conservative side of issues in both of these areas. I suppose that shouldn’t surprise me, given the general tenor of Rush Limbaugh, Fox News, and other instances of talk radio and its television allies. But it’s still striking that your typical layperson wants to preserve or resurrect obsolete grammar “rules,” wants dictionaries to be prescriptive rather than descriptive, thinks that some languages are inferior to others, etc. And it’s equally striking that the same typical layperson advocates “back to basics” approaches in mathematics and thinks that math is all about memorization of facts, rote algorithms, and the like.
2. Members of the general public are confused equally about what linguists and mathematicians do. They think that a linguist is someone who speaks several languages fluently, so why should they ask a linguist for an expert scientific opinion about language in general or English in particular? Similarly, they think that a mathematician is someone who calculates quickly and accurately, so why should they ask a mathematician for an expert scientific opinion on learning mathematical concepts?
3. Because the general public understands neither subject, they are confused about the connections between the two. The fact that I moved from studying linguistics to teaching math sounds like a complete change of direction, not a natural evolution. It’s the rare person who sees cryptology as a connection or bridge between the two, or who understands that in some sense math is a language (only in some sense, I hasten to add), or who can imagine that one can apply mathematics in any way to the study of language.

So much for the similarities. Are there ways in which we can contrast the general public’s opinions about language and math? Sure. Here are a few:

1. People freely profess ignorance about math. When someone at a party asks what you do, and you say that you’re a math teacher, the most common response is, “I was never any good at math.” Rarely are they ashamed. Often they even seem proud of it. But no one admits to be ignorant about reading, writing, and speaking.
2. On what might be the same lines, people at least have some idea that there is such a thing as a mathematician, even if they don’t know what mathematicians do. (“A mathematician is a device for turning coffee into theorems,” according to Alfred Renyi.) But when I tell people that I majored in linguistics, the usual response is either a blank stare or “What’s that?”
3. Perhaps as a consequence of the previous observation, a great many myths about language float around in the popular culture, but there are not nearly so many myths about math.
4. Everyone thinks they can teach English — they can’t, of course — but very few people think they can teach math.

Perhaps in a later post I’ll give some citations to exemplify these seven points, but this will do for now.

We send our top six mathletes to both the state playoffs and the New England. Please recognize the fine [...]]]> Congratulations to the Weston High School math team for their excellent showing in the state playoffs on Friday in Shrewsbury! We finished fourth in the state among medium-sized high schools (the schools with which we compete) and will therefore be going on to the New England finals in Canton in early May. The state playoffs are particularly tough compared to the earlier season since all six individual rounds are totally calculator-free (usually there’s just one such), though the team round allows calculators.

We send our top six mathletes to both the state playoffs and the New England. Please recognize the fine work of Jonathan Birjiniuk ’11, Alex Bruce ’09, Katie Hsia ’09, Grace Huckins ’12, Stephanie Palocz ’12, and Ernest Zeidman ’10 — and wish them the best in Canton!

OK, let’s agree that it wasn’t actually a matter of luck. Here in the reality-based community we look for genuine explanations, not fanciful ones. The fair got rave reviews [...]]]> Seven is a lucky number, so no one was surprised that the seventh annual Fractal Fair at Weston High School turned out to be the best one so far. Of course there were many great exhibits in each of the previous fairs, but never was there such a consistently high level of quality and enthusiasm; there was not a mediocre or poor project among them! And, in contrast to last year, there were no tears.

OK, let’s agree that it wasn’t actually a matter of luck. Here in the reality-based community we look for genuine explanations, not fanciful ones. The fair got rave reviews from many visitors, especially colleagues and parents of students. While my co-teacher and I would love to take credit for that, we can’t: we don’t deserve more than a tiny fraction of the credit. Yes, we did tweak the requirements a bit — for example, each group had to come up with a question that would let them assess whether visitors and listeners got the “Big Idea” of their project, each student had a structured opportunity to see their classmates’ projects, and we scheduled the fair at a better time — but the lion’s share of the credit has to go to the students. They consistently produced projects that were interesting, accurate, attractive, and thorough. No group was excessively ambitious (in the past there were always some who bit off more than they could chew); no group settled for making sloppy posters that were marred by typos or spelling errors. Every group was able to explain their work, and visitors noted their enthusiasm in doing so. Congratulations to all!