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Saturday, March 31, 2007

Non-violent video games?

My Saturday Course students are designing and programming their own video games. These fifth-graders participate in an intensive introduction to programming, but their “Create Your Own Computer Game” course is also billed as a class in which kids create their own computer games. Truth in advertising, you know. Look at the title. They learn pretty much by doing, with short lectures and demos by me and otherwise undergoing hands-on experience, supported by my coaching and the assistance of a seventh-grader from Brookline.

Not surprisingly, the typical fifth-grader designs a game in which you have to shoot down alien space-ships. Or, somewhat less violently, you are piloting a spaceship and have to avoid being shot down by the enemy. Or perhaps you are a turtle trying to cross the road while avoiding being hit by fast-moving cars and trucks; sound familiar?

At the non-violent end of the spectrum, one student is imitating Breakout — remarkably easy in Microworlds, though definitely not trivial for a fifth-grader with only a few hours of training. Most surprisingly, another girl is writing a nutrition game in which a runner has to avoid unhealthful foods. How unusual!

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Friday, March 30, 2007

Meridian Academy photo show

A small but excellent exhibit of photographs by middle-school students at Jamaica Plain’s Meridian Academy was on display for the past two weeks at the JP Art Market. The official description read as follows:
Meridian Academy students will show their photography in an upcoming exhibit at a local art gallery, the JP Art Market. The exhibit will consist of photographic essays representing each student’s perspective on topics such as poverty in Ecuador, the whimsical side of England, and the bustling complexity of the Forest Hills T station.
The students’ own commentaries were particularly interesting, especially since some chose to discuss their photos as works of art and others chose to write about their social significance. And perhaps that combination was the point, whether it’s rural Ecuador or Codman Square.

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Thursday, March 29, 2007

Math is hard at the Home Despot

On the way home from work yesterday, I stopped at the local Home Despot — er, I mean Home Depot — in South Bay in Dorchester, in order to buy some plywood. I picked out a nice sheet, measuring the standard 4 feet by 8 feet, and brought it to the guy with the nice power saw that can cut both horizontally and vertically. “I need three squares, each measuring 30 inches on a side,” I said, helpfully showing the employee a sketch I had made that looked something like this:

Remember when Home Depot used to advertise that their employees were real plumbers and real carpenters, etc.? Well, this guy looked at my drawing and said, “The plywood’s only 96 inches long, you know. I don’t think you can get three 30-inch squares out of that.”

“Trust me,” I replied. “I’ve done the calculations. You can do it.”

So he did it. He made the cuts, and he was startled to find that it was indeed possible. There was even some left over, as he observed with surprise!

Remember back in the ’90s when we math teachers were distressed to hear the new incarnation of the Barbie doll saying, “Math is hard”? (Or perhaps it was “math is tough” or “math class is tough,” depending on whom you believe.) Hmmm...

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Wednesday, March 28, 2007

A Perfect Mess

I’ve just read a fascinating book entitled A Perfect Mess: The Hidden Benefits of Disorder — How Crammed Closets, Cluttered Offices, and On-the-Fly Planning Make the World a Better Place, by Eric Abrahamson and David H. Freedman. If you’ve ever suspected that your messy desk is actually more useful than a pristine work surface, this is the book for you. Those of us who are caught in the paradox of being mentally organized but physically disorganized find much food for thought in Abrahamson and Freedman’s counterintuitive arguments supporting the productivity of disorder. Abrahamson is a professor of management at Columbia Business School, so presumably he knows whereof he writes. As an organization system junkie and a Getting Things Done wannabe with a perennial messy desk, I find the arguments put forth by him and Freedman rather appealing, though not entirely convincing.

I have to go visit a local hardware store discussed by the authors: Harvey’s in Needham. In contrast to Home Depot, Harvey’s appears to be a total mess. But it was cited by Inc. Magazine as the best small business in the country. Freedman writes for Inc., so this is certainly not a coincidence.

My classes are much better when they’re planned than when they’re unplanned. They’re better still when they’re planned but turn out not to follow the plan very well. There’s probably a moral here somewhere...

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Monday, March 26, 2007

Roxbury? Perhaps Dorchester? But Back Bay????

Check out the website for 1010 Mass Ave in Boston:
This six-story building is located at 1010 Massachusetts Avenue in Boston and offers 220,770 square feet of prime office space. Located in Boston's Back Bay, 1010 Mass Ave is within walking distance of key educational institutions such as Northeastern University and Berklee College of Music, and major business hubs including Copley Square, Prudential Center, and Hynes Convention Center.

1010 Mass Ave has high commuter visibility. It is conveniently located near public transportation and has easy access to and from Interstate 90 (Mass Pike).
Well, yes, 1010 Mass Ave does have easy access to the Mass Pike. But “located in Boston’s Back Bay”??? Come now: 1010 Mass Ave is in the Newmarket Square area, which was in Roxbury the last time I looked. Some people might justifiably claim it for Dorchester, since it’s close to the Dot border — but in any case it’s nowhere near Back Bay. It isn’t even one neighborhood over: all of the South End sits between this part of Roxbury and Back Bay!

And what about this claim that they’re “within walking distance of” Berklee College and Copley Square? I suppose it all depends on what the meaning of walking distance is. While I’m perfectly willing to walk two miles, that’s not what most people mean by walking distance.

It must be easier to sell first-class office space if you claim to be in Back Bay.

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Sunday, March 25, 2007

The Plot Against America

I just finished reading Philip Roth’s The Plot Against America. This highly fictionalized autobiography is actually an alternative history, based on the postulate that the Republican convention in 1940 was deadlocked and drafted Charles Lindbergh by acclamation. Lindbergh goes on to defeat Roosevelt’s effort to win a third term, running a campaign based on anti-Semitic and pro-German sentiments, all in the guise of keeping the United States out of the European war. Needless to say, the novel is full of explicit and implicit political content, including some thought-provoking ideas on the subject of the trade-off between liberty and security. The fears of a pro-Nazi government in Washington become all too convincing. “It can’t happen here” — or can it?

On a lighter note, I found that the accounts of growing up Jewish in Newark resonated in various ways. But there’s certainly no need to be either Jewish or from New Jersey to appreciate this book. Despite the political themes, the story is dominated mostly by the fake autobiography and its account of life in the fictionalized Roth family. (I have no idea to what extent this aspect of the “autobiography” is true to life.)

The only strange thing about The Plot Against America is that the linear narrative is occasionally interrupted by flash-forwards, usually for political reasons. I found the break in the sequential flow of the story jarring, and the revealed future was too often a spoiler. Otherwise it’s a terrific novel. Read it!


Friday, March 23, 2007

The one worst method

All right, we know that there isn’t any “one best method” of teaching. That’s one of the reasons why teachers need to be life-long learners: aside from being models for our students, aside from the continual need to learn new content, we can also always improve our teaching techniques. There’s no one best method, but maybe there’s a one worst method.

Yesterday and today Barbara had to attend an intermediate Excel course. The instructor decided not to “burden” the participants by giving them any hands-on experience, offering the lame excuse that doing so would slow them all down to the pace of the slowest learner. So he lectured... and handed out documentation. That’s it. Now I have nothing against doing some paper-and-pencil work when learning about computers — sometimes it’s exactly what an impulsive or confused student needs when trying to write a program or even when planning a spreadsheet — but it needs to be done by the student, not the instructor, and substituting lectures for hands-on work is just preposterous. I know that it’s a cliché to utter the purportedly Chinese proverb, “I hear I forget, I see I remember, I do I understand,” but like most clichés it contains more than just a grain of truth. I’m appalled that any organization that’s too cheap to rent computers would try to justify this decision by claiming that it produces better instruction.

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Wednesday, March 21, 2007

Ends, means, and the content of high-school geometry

Interesting post today in Professor Hirta’s blog. Here are a couple of excerpts:
Got a phone call today from a high school teacher. He was told by his supervisor that he needs to use more manipulatives in class, so he was hoping that I could recommend a manipulative that he could use in geometry class. Nice. More focus by the administration on the means than on the ends.


Also alarming: he described his “basic high school geometry course” as covering topics such as perimeter, area, and volume. When I think “high school geometry” what springs into my mind is Euclidean geometry and proof; I associate perimeter, area, and volume with middle school math.
Let’s look separately at each of these paragraphs:

First, it would be nice to have a little more information about the basis on which the supervisor made his or her comments. Hirta’s response is certainly valid if the supervisor is indeed neglecting the goals. At Weston the administration always tells us — and appropriately so — that we should “begin with the end in mind,” that backward planning is the key to designing both long-term curriculum and one-day lesson plans. Hirta is right to decry “focus by the administration on the means than on the ends” — if that’s really what’s going on. But suppose the supervisor observed the teacher’s class, found out what the objectives were, and only then concluded that using more manipulatives would help achieve those objectives. There would be nothing wrong in that case: an improved means would help achieve the ends. Or maybe the supervisor did no such thing, and I’m just being naive.

The second paragraph deals specifically with the content of high-school geometry. Hirta is right to “associate perimeter, area, and volume with middle school math,” and it hugely bothers me that so many high-school courses merely repeat what was done in middle school. Even in Weston that’s true some of the time, especially in college-prep geometry. The opposing argument, of course, is that 9th-graders don’t remember what they did in 7th grade, so it’s necessary to repeat the material. The high-school course still includes a significant dose of “Euclidean geometry and proof,” but includes review of middle-school math as well. Again, this is at Weston; I have no idea what’s going on at the anonymous teacher’s unspecified high school. Finally, I think it’s unfair to suggest that “covering topics such as perimeter, area, and volume” is automatically a matter of repeating what was done in middle school. It’s entirely possible (and educationally desirable!) to challenge the students with sophisticated area and volume problems that go way beyond what they did in middle school. In that way it’s feasible to sneak in a review along the way, without sacrificing the need to learn something new.

At Weston we’re in a situation where the honors geometry course is a challenge to everyone but the college-prep geometry course is too easy for many of the students. Using differentiated instruction and an appropriate mix of straightforward and challenging problems, the teacher can create a suitable learning experience for both ends of the spectrum: the weaker students can still earn a B if they work hard, and the strongest students can be challenged on their way to an A. Repeating middle-school content is not the answer.

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Monday, March 19, 2007

Global Awareness Day

Speaking of professional development...yesterday was Global Awareness Day in the Weston Public Schools. Unlike Art Day, this was an eight-hour endeavor — very elaborate in planning, development, and conception.

We began with a presentation about the forthcoming visit to Weston by seven teachers from Kasiisi Primary School in Uganda. Only one has ever been out of Uganda (he has a Ph.D. from Michigan State), so we expect a lot of culture shock for the other six. We were also reminded that Weston is not a typical segment of the United States.

Then there was a geography bee, which was a lot of fun; the contestants were divided into four teams, one each from the three levels of schools and one from the central administration. The Middle School won. But the High School came close.

Next was an amazingly diverse choice of seminars run by teachers who volunteered to share their knowledge and expertise. Ordinarily I wouldn’t bother to cite the entire list, but in this case it’s worth doing:
  1. Africa Today
  2. An Afternoon in South America
  3. An Introduction to Multicultural Folk Dance
  4. China’s Minority Peoples
  5. Examining the Chinese Cultural Revolution through the book Balzac and the Little Chinese Seamstress
  6. Global Communications and SKYPE
  7. Global Warming
  8. Globalization of Indigenous Food
  9. Introduction to Mandarin
  10. Making Multicultural Books in the Classroom
  11. Pandemic/Avian Flu — Info for You
  12. Revive Your High School Languages
  13. The VERY Abridged Japan
  14. Uechi Ryu Karate Do
  15. Views of Cameroon
  16. Welcome, Uganda!
  17. What’s in the world’s water?
  18. Writin’ Numbers
I participated in #4, presented by a teacher who had traveled along the Silk Road last summer from Xi An to the border of Pakistan. Everything about the seminar was interesting, but I was most struck by how the Western Chinese terrain, architecture, and people all looked much more Turkish than Chinese. And indeed that’s what they are.

The seminars were followed by an amazing lunch. We were greeted by a cafeteria in which every table had been set with regular place settings — that is, if “regular” includes chopsticks, a large crock pot at each table, and a variety of ingredients. The lunch came complete with documentation at each place, as lunch always should. The menu was listed this way:
  • Vietnamese Pho Chicken/Vegetarian
  • Salad
  • Naan
  • Cookies
But the important part was the comprehensive instructions:

You will need to work together at the table. Each table should be set with a bowl for all diners. In the center of the table you should have a pot of soup (chicken-based), cut-up chicken, noodles, scallions, cilantro, bean sprouts, hot sauce, hoisin sauce, lime, and some bread (naan).
  1. Each person should place any of the following in his/her bowl: chicen, noodles, scallions, cilantro, and bean sprouts... (Those who are vegetarian will need to take their bowls to the front of the cafeteria to get vegetable-based soup.) You can further garnish the soup with hot sauce, lime juice, and/or hoisin sauce. ENJOY! Feel free to take seconds or to ask if you need more.
  2. Each setting should have a salad.
  3. etc., etc.

All of this had been organized and prepared by the Superintendent of Schools, Dr. Alan Oliff, along with the other administrators in the Weston Public Schools!

Anything after this lunch must be an anticlimax. We followed it with 6–12 department meetings and a high school faculty meeting. At the Math Department meeting we discussed a variety of topics relating to global awareness in math — more on this subject later, but I’m skeptical.

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Sunday, March 18, 2007

I, Robot

So why did I bother watching the movie I, Robot? It’s because I rarely read reviews ahead of time, since reviews too often contain spoilers. But I found this movie poorly done, disrespectful to the memory of my hero Isaac Asimov, and generally offensive. Other than that, it was worth watching.

The first big problem was that this movie is just a pile of cinematic clichés, from the loner cop to the out-of-control robots. The latter reveals the second problem: its faithlessness to Asimov’s memory. Asimov always believed in the rationality of humans and our creations. He carefully crafted his three laws so that robots would never turn on their creators (another movie cliché). Sure, I, Robot and other novels and short stories presented numerous personal and scientific problems that humans and robots had to solve, including a great many in which the three laws at first appeared to be violated, but there was never any fear-mongering. I cringe just to think of how the movie has torn apart Asimov’s view of the world. I’m going to go re-read the books now.

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Saturday, March 17, 2007

Pi Day

Three days ago was Pi Day (3/14). Or maybe I should say 3.14 days ago, since that was Pi Day. Anyhow, a lot of the Weston math teachers celebrated it one way or another with our classes, and Kelly of course baked Pi Day brownies (why not a pie?), and two of my sophomores promised to visit my classes at 3/14 1:59 — but they forgot — and one of my physics colleagues sent around the beginnings of a pi mnemonic in the form of a parody of the beginning of Poe’s “The Raven”...

Just a routine Pi Day so far — not that any Pi Day is truly routine.

But then today I received an email from one of my 11th-grade students, in which he provided a link to the entire mnemonic. The full explanation reveals that the text includes an amazing 3835 digits of pi, all in the form of various literary parodies that more-or-less make sense and more-or-less scan. This example of constrained writing is quite a tour de force! Do check it out.

Oh — one more thing. One of my colleagues is eager to hear how it ends. What’s the last digit? Well, it turns out that the mnemonic ends with “The End,” of course.

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Friday, March 16, 2007

When is a math issue really a reading issue?

It was one of those Jungian synchronicities. My department head returned this morning from yesterday’s all-day conference, and he told us about a talk that ascribed many students’ difficulties with math questions (and questions in other disciplines) to difficulties in reading. That sounded plausible to me, at least in the case of average to below-average math students.

A few minutes later I read Jill Walker’s new post in her blog, reporting on her students’ difficulties with close reading.
As Miller said in Repo Man:
A lot o’ people don't realize what’s really going on. They view life as a bunch o’ unconnected incidents ’n things. They don’t realize that there’s this, like, lattice o’ coincidence that lays on top o’ everything. Give you an example; show you what I mean: suppose you’re thinkin’ about a plate o’ shrimp. Suddenly someone’ll say, like, plate, or shrimp, or plate o' shrimp out of the blue, no explanation.
I always seem to remember the next line as, “You think it’s a coincidence, but it’s not”; however, that turns out not to be the next line. Anyway, I think I’ve quoted it before. If so, it’s purely a coincidence.

Anyway, the question arises whether we should really be teaching reading skills as a component of a high-school math class. Walker’s post is on a slightly different topic, dealing with close reading rather than simple reading, and with college students rather than high-school students. But still it’s clearly related. She links to another professor, who calls herself Dr Crazy, so off I went to check out Dr Crazy, despite her website’s irksome horizontal stripes, which make reading a chore rather than a pleasure. And sure enough there’s a direct connection with high-school math (and physics):
Tonight, another reader, who teaches physics, sent me an email asking me to talk more about it because apparently these kids today can’t do story problems. And then today, in handing back papers in my upper-level class, it struck me that I need to do more in class to demonstrate how to do close reading in a more explicit way.


Close reading is not reaction. I don’t particularly care how you “feel” about a text. Nor does anybody else. Nor do I care that it reminds me of when your grandmother had alzheimers. That has nothing to do with what is in the actual text. Nor do I care what you “got from it.” I care about what is there. (This, incidentally, is why I think Dr. Pion’s students may be failing to comprehend the story problems — because they’re used to reacting and not to reading.)

The problem, as far as I can tell, is one of focus. My students don’t necessarily focus intently on what they read. They don’t necessarily read with pen in hand. Or if the pen is in hand, it’s just underlining cool stuff, and not really entering into a conversation with the texts that they read. When I was an undergraduate, I was a talented writer, but I was also the sort of reader who read for “cool passages.” Now, of course I still read for “cool passages,” but now I actually ask myself why they are illuminating and cool. THIS is what missing from the NCLB testing world. There is no need to focus because there is no need for a why. You don’t need to engage deeply because there is no evaluation of deep engagement. And with no evaluation comes no reward.
Anyway, this is just a brief excerpt. The post goes on — and appropriately so — at considerable length. Do read the whole thing. It’s definitely worth thinking about, and possibly implementing. Maybe it will help us not only narrow the achievement gap in mathematics but also help all students do better with story problems.

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Thursday, March 15, 2007

Can we have archaic and read it too?

If you are translating an archaic language into English, should your writing sound archaic? Or should it be readable? Altogether too many amateur translators think the former.

One of my colleagues inadvertently provided a lovely example yesterday. In precalculus class we have been studying cubic functions and other polynomials of lower and higher degree, so the subject of solving quadratic and cubic equations naturally arises. We all know the quadratic formula, and nobody in his right mind would want to memorize the cubic formula — you especially don’t want to memorize the quartic formula — but it would be helpful to gain an historical perspective on all this. (I, for one, firmly believe that mathematical understanding is almost always enhanced by learning the historical context.) So we looked at an excerpt from an ancient Egyptian papyrus:
Find a value eight of which does exceed two of its squares by six. Do apply as a factor the number of squares to the desired excess obtaining twelve.
Decrease by this amount the square formed from half the factor of the number sought. This square has a side of two which should by increased by half the factor of the number sought. Remove from this as a factor the number of squares sought thus obtaining the desired value.
The unknown translator clearly wanted to make his English sound archaic. But he did this at the expense of readability! No actual spoken language ever sounded like this, and Middle Egyptian really was an actual spoken language.

So I took on the task of trying to find the original problem. (I always knew that my two semesters of Middle Egyptian in college would turn out to be useful some day.) Unfortunately I have not yet been able to locate the original, so I can’t honestly offer a more accurate translation. Even a very rough and very free paraphrase is impossible without the original Egyptian text (and, in any case, “traditore tradutore,” as they say). But here it is in algebraic language:
I’m looking for a certain quantity, x, such that 8x is 6 more than twice the square of x. Multiply the number of squares, 2, by the excess, 6, giving 12. Subtract this from the square of half of the multiplier of x. Add the square root of this number to half of the original multiplier of x. Then divide that sum by the original number of squares.
Note how awkward it is to express algebra in words rather than symbols, and even the mixture of words and symbols doesn’t help much. A modern translation is more readable, but still pretty opaque. Converting this text into plain algebra is a good exercise that helps to teach the value of a symbolic notation; see if you can derive the quadratic formula from it! This is, of course, left as an exercise for the reader.

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Wednesday, March 14, 2007

Xerox? photocopy? copy?

Yes, we know that Xerox® isn’t a verb. For years we’ve been carefully trying to say, “I photocopied this document,” instead of “I Xeroxed this document.” But now the photo- prefix seems to be disappearing. Some people are even puzzled at the verb photocopy, asking if I mean copy.

Maybe it’s just because I’m a teacher, but somehow it’s not the same thing; copying a paper has all the wrong connotations. And yet, Xerox’s own website says copier throughout, not photocopier. Where did the photo go? More important, when did it go?

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Tuesday, March 13, 2007

Pi (not π)

I just finished reading Life of Pi, the intriguing novel by Yann Martel. (I owe thanks to my colleague, Donna Gonzalez, for not only recommending this book but also lending her copy to me.) Just to get one thing straight from the start — since I’m a math teacher — the title has virtually nothing to do with the ratio of the circumference of a circle to its diameter. The narrator’s name just happens to be Piscine, or “Pi” for short.

Piscine? Well, yes. Having a protagonist named after a swimming pool is far from the most improbable characteristic of this novel. In fact, the entire story deals with improbabilities, starting with a 16-year-old from India who is simultaneously Hindu, Muslim, and Christian and proceeding to the central and longest part of the story, in which Pi shares a lifeboat with a Bengal tiger for 227 days. While there’s rather too many violence and too much religion for my tastes, I cannot deny Martel’s ability to hold my attention fiercely for the entire length of the book. The writing is intense, the characterization of the protagonist is captivating, and the tale itself is an extraordinary example of dramatic storytelling. Reading this book is an unforgettable experience, so don’t read it if you think it’s something that you might want to forget!


Monday, March 12, 2007

Dorchester described accurately

I know, it shouldn’t be news when a major publication describes Dorchester accurately. It should be the dog-bites-man vs. man-bites-dog thing. But, unfortunately, accurate yet positive descriptions of Dorchester have come to be man-bites-dog stories in the mainstream press.

Then there’s Business Week.

Did I say Business Week? Yep, you heard me correctly. In the March 6 issue, Maya Roney reports writes about Dorchester in a story entitled “America’s Next Hot Neighborhoods”:
...the neighborhood of Dorchester remains affordable even though home values there have increased nearly 60% in the last five years. The median home value is now $331,896, according to Zillow. Almost a separate city in itself, Dorchester is a large and diverse working-class community south of Boston’s center, with many Irish and Southeast Asian immigrants, as well as a significant African American population. Residents enjoy riverfront amenities like beaches and boating, as well as the green space and recreational activities of 527-acre Franklin Park.
I know, she could have mentioned the crime issues and other problems that plague the descriptions of Dorchester in the local media, but everything Roney writes is accurate. It’s a breath of fresh air.

Thanks to the Dorchester Reporter for bringing this to our attention.


Sunday, March 11, 2007

Average grades

What should an average grade be? This question actually has two very different but intertwingled meanings. Some people, when they ask it, are wondering whether the mean (or perhaps median) grade in a school/department/course should be a B or a C or whatever. Others are asking about scaling tests: is it OK for the mean (or perhaps median) grade on a test to be 90%? or how about 30%? If the answer to the first question is, say, B–, then should the 90% or 30% — or whatever — be scaled to a B–?

There is, of course, the persistent myth that C is an average grade. Maybe it used to be, and maybe there are some places where it still is, but real or imagined grade inflation has bumped the mean (or perhaps median) to a B– or B. At Harvard it’s an A–, but that’s another story. As long as people understand the system, it’s totally arbitrary whether the average (all right, I’ll stop saying mean or median, since it’s almost never clear which measure is meant) is B or C or Q — there’s no intrinsic meaning to the symbol, after all. What counts is the relative average when we compare one population with another. For example, studies of the achievement gap often compare grades across racial or economic groups. Another interesting comparison is across departments. At the aforementioned Harvard, for example, grades are significantly higher in the humanities than in math and science. At Weston High School, a recent interdepartmental study showed that 38% of all History grades were A’s, but only 24% of all Foreign Language grades were A’s. Temporarily removing Foreign Language from the study, we find higher grades in History and English than in Math and Science, just as Harvard found. An article in Wildcat Tracks, the school newspaper, attributed most of this discrepancy to differences in the “difficulty” of each department, though one of my former students (from last year’s honors precalculus) is quoted as saying, “Math last year was the hardest class because my teacher was extremely challenging and had high standards.” I take that as a compliment, though I’m not sure that it was intended to be.

Overall, 32% of our Math Department’s grades last year were A’s and another 42% were B’s, thereby making the median a fairly high B. But this leads to the second interpretation that I mentioned in my first paragraph. Some teachers grade by an old-fashioned equivalence of 80–82 for a B–, 83–86 for a B, and so forth. By so doing, they are implicitly promising that the level of difficulty of their questions is such that a minimally competent student will get about 80% on them (assuming, of course, that a B– represents minimal competence, which sounds reasonable to me). I have two difficulties with this traditional but thoroughly arbitrary scale. First, how can a teacher be so sure that the level of difficulty of the problems on every test reflects this matching of 80% with minimal competence? Second, I believe that using this scale is an indication that the students aren’t being sufficiently challenged. In the class described by the student quoted in the previous paragraph, the median grade was still a B, but it’s clear from her comment that there were plenty of challenges and high academic standards. One way to achieve these is to ask some really hard questions while rewarding good work that falls short of 80%: on some tests 75% was worthy of a B–, on others it might be 70% or occasionally even lower.

I do worry, however, when the scale is so extreme that a class median is 30% or 40%. Students may then get discouraged because they think they are being rewarded for inadequate work or because they are able to complete so few problems with successful solutions.

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Friday, March 09, 2007

Being visited by scary reviewers

No, they weren’t actually scary. That’s merely what one of my students thought. “Weren’t you scared?” she asked.

Each department in the Weston Public Schools gets reviewed every ten years or so. This year it was the Math Department’s turn. A group of nine or so outside experts — teachers, college professors, curriculum developers, mathematicians — spends a few days examining our entire K–12 math program. They interview teachers, students, administrators, and parents; read the curriculum; look at copious examples of student work; and visit large numbers of classes. The purpose is to evaluate the program, not the teachers, so why did my student think that the reviewers were scary? I suppose it’s because she had been reading “Among School Children,” the famous poem by William Butler Yeats. (Do high school students still read Yeats?) Or maybe I’m just inventing the connection, but it definitely spurred me to read that poem again. I had forgotten that it contained another connection — with the application of trigonometric and exponential functions to music, which we studied in my precalculus class last month:
World-famous golden-thighed Pythagoras
Fingered upon a fiddle-stick or strings
But that wasn’t what my imagination says inspired my student. It was, of course, the last line of the poem:
How can we know the dancer from the dance?
It’s impossible to separate the program from the implementers of the program, to separate the curriculum from the teacher. They are interdependent.

But the reviewers still weren’t the least bit scary. It’s always helpful to have an outside pair of eyes in the classroom, for they will inevitably see things that I as a teacher don’t notice. A professor from Williams College spent 80 minutes in my precalculus class and then met with me for 45 minutes to talk about it — an energizing and useful experience for me. He was suitably impressed by the focus, energy, and depth of thought shown by this group of students.

Now we’re just awaiting the reviewers’ written report. We look forward to seeing both their commendations and their suggestions for change. Stay tuned.

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Thursday, March 08, 2007

Art Day

The first Wednesday of (almost) every month is professional development day in the Weston Public Schools. Students have a half-day of classes, after which they can go home and the faculty have workshops and the like. Usually these days have a particular theme, such as the achievement gap or writing unit teaching guides.

Yesterday was Art Day.

A week ago, each of us had signed up for our choice of workshops. Being a techie, I naturally selected one that used Photoshop or iMovie — right? Wrong. I figured that I could learn those on my own (not that I’ve done so yet, despite plenty of opportunities). So I decided that this was the right time to take a risk and sign up for an intro to drawing, since I’ve never been able to draw.

It turned out that pretty much all of the eight of us in the drawing workshop were convinced that we couldn’t draw. But the instructor was incredibly encouraging and positive, so nobody felt intimidated. We started with “drawing with scissors” — somehow cutting out an image isn’t nearly as frightening as putting a pencil to a blank sheet of paper — and proceeded to learning to see circles, rectangles, and ellipses in various objects. We then drew some still lifes (still lives?), explored the use of negative space, and moved on to two-point perspective — all in ninety minutes. Along the way we learned about using our right brain instead of the left brain that most of us teachers love to use.

I think it worked. At any rate, I’m no longer convinced that I can’t draw.

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Tuesday, March 06, 2007

Singulars and plurals

I am catching up on reading old posts in Tenser, said the Tensor, which labels itself as “the blog of a graduate student in linguistics. It’s about language, science fiction, computers and technology, comics, anime, and other geekery.” How could I resist? You shouldn’t resist either.

Anyway, the Tensor’s post of February 24 definitely rang a bell with me:
...when borrowing words into English, especially when their number is unclear and they tend to get used as mass nouns, you should invent singular forms for them as if they followed the high-prestige Latin pattern, regardless of their actual language of origin.  Examples:

(First declension)  The warrior class of ancient Japan were the samurai.  Each samura traditionally carried two swords.
(Second declension neuter)  Often for dessert at a Middle Eastern restaurant I will order a plate of baklava.  Generally it comes on a plate containing several pieces, so that each person at the table can have their own baklavum.

If you want to go the extra mile, you can even back-form an irregular third declension singular, as in:

I recommend the tempura. When eating it, be sure to dip every individual tempus in the special sauce provided.  (Extra bonus: round trip Romance-language borrowing!)

Finally, if you’re really feeling ambitious, you can even do Latin-style number concord:

Traditionally, an order of nigiri sushi consists of two pieces.  Each nigirus sushus is a ball of rice with fish or some other food laid on top.

This is all a lovely idea to complement the more usual invention of plural forms. In my household, for example — and in some others I know — we refer to Kleenices. Reversing the process certainly adds a little extra something. (But is it a je ne sais quoi or a lagniappe?)


Monday, March 05, 2007

Math education: an inconvenient truth

It’s hard to know where to begin. What’s wrong with the video “Math Education: An Inconvenient Truth, ” which is primarily an attack on TERC’s Investigations in Number, Data, and Space and other standards-based curricula? Well, let me count the ways.

On second thought, I don’t want to count them, because I’ll be here all day. Let me just mention a few:
  1. Narrator M.J. McDermott complains that Investigations teaches kids to “reason through problems.” As I say, it’s hard to know where to begin. Surely this is a good thing. Surely reasoning through problems is the primary goal of school mathematics learning.

  2. The video’s first example is finding 26 × 31. It goes on to show an “inappropriate” way of reasoning that TERC promotes. Quoting verbatim:
    I know that 26 × 31 = (20 × 31) + (5 × 31) + (1 × 31).
    So how do I find 20 × 31?
    Well, I know 10 × 31 is 310, and I can figure out from mental math that 20 × 31 is twice this, and I can figure out that that’s 620.
    Now I need to find 5 × 31, and I can figure out that 5 × 31 is half of this [points to the 310], so I can figure out from mental math that that’s 155.
    So I add that to the 620 and I get 775 so far.
    So now all I need is one more. I know 1 × 31 is 31, and now I just add the 775 here, and I get 806.
    McDermott correctly calls this method “inefficient,” but surely it demonstrates much more understanding than the standard magical algorithm, and surely the skills involved will have far more payoff in doing algebra.

  3. Ah, next we get to long division — a current favorite in my precalculus class, since we’re doing long division of polynomials in order to factor them. (And later we’ll be doing long division of polynomials in order to find asymptotes.) Many of my students find that their memory of long division with numbers help them with algebra, but more of them find that the work with algebra finally lets them make sense of the traditional algorithm with numbers. Anyway, back to the video. The sample problem is 133 ÷ 6, and — no surprise — McDermott objects to TERC’s avoidance of the traditional algorithm. TERC’s “less efficient” method eventually results in the sentence 6 × 22 + 1 = 133, which is exactly the right form for doing algebra and number theory in high school. But McDermott sarcastically quotes the teaching guide for another series she attacks, Everyday Math, because it concludes that it is “counterproductive to invest many hours of precious class time” to memorizing traditional algorithm for long division. “The mathematical payoff is not worth the cost,” correctly observes Everyday Math.

  4. McDermott laments the lack of math skills among many college students. I’ve lamented that too, but surely it’s not because of TERC and the like. Most of these college students went through traditional math programs!
In any case, it is easier to supplement a reform math curriculum with traditional practice than it is to supplement a traditional curriculum with innovative activities. Both can be done. I could keep going here, but I think I’ll stop there.

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Sunday, March 04, 2007

Blood from a Stone

I recommend Blood from a Stone, by Donna Leon. If you look at the photo on the opening screen of her website, you’ll immediately see what I liked most about this novel: it makes the reader feel that s/he’s in Venice. For some reason I seem to be picking mysteries with a well-developed sense of place lately, and this book is no exception. You can see, hear, and smell Venice as you read it.

That’s #1, but Blood from a Stone has other strengths as well. For one, it deals with issues of prejudice and illegal immigration, familiar to us here but pleasantly jarring in the context of northern Italy. Also, the characters and plot are three-dimensional. The teenagers in the book are a bit too much stereotypically teenagers, but I suppose many teenagers are — and the recurring presence of the protagonist’s family life humanizes him in a way that nothing else could do. The book is somewhat leisurely, so don’t read it if you want a fast-paced mystery. Pick a time when you can be a leisurely and thoughtful reader, and you won’t regret it!

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Saturday, March 03, 2007


We have recently been discussing ethnomathematics in the context of Weston’s global awareness emphasis. Here are some thoughts on this subject:

It’s worth studying number systems other than our own familiar Hindu-Arabic one. Years ago I developed quite a number of activities on different number systems — such as Egyptian, Babylonian, Greek, Chinese, Maya, etc. — including a computer program that probably doesn’t work any more since it was developed for Mac OS 9. I’ve used these activities with a wide range of students from grades 4–11, and they seem to work well. The mathematical ideas that pervade these activities include bases; numbers, numerals, and names of numbers; the concept of zero; alternative algorithms; and unit fractions. These topics are worth studying for many of the same reasons that foreign languages are worth studying. In particular, they give perspective on our own system, which is often so familiar to us that it becomes transparent.

On the other hand, I believe that we are correct in thinking that math is much more nearly universal than it is culturally specific. (The role of proof is the only really deep difference I can think of.) The tiny number of usually superficial differences are important for mathematical reasons, not for cultural reasons. It’s worth knowing that there were parallel discoveries of the Pythagorean Theorem in India, Chinese, and Greece; we don’t have one theorem in one country and a different one in another. The same goes for Pascal’s Triangle. And negative numbers. It’s worth knowing that there are different algorithms for multiplying, but the more important lesson is that all cultures above a certain primitive level of technology do multiply. (I know, it’s not PC to say “primitive”.) Human similarities are more important than human differences. (Likewise, language differences are illuminating, but human languages are more alike than different.) In both math and language, Plato was right. Ethnomathematics is a rich field, but it’s going to teach us much more about appreciating similarities than about appreciating differences.

On the third hand, the real problem with taking a Eurocentric view of math is that it limits our understanding of the contributions of the rest of the world. Years ago I referred to the wall of mathematicians at the Museum of Science as the “wall of dead white European male mathematicians”; it has gotten somewhat better since. I can recommend several excellent resources on multicultural approaches to math, such as the following:
But be suspicious of anything by the late Claudia Zaslavsky!

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Friday, March 02, 2007

The Outlier Effect

Just came back from seeing The Outlier Effect, a one-act play written and performed by Weston High School’s Theater Company. That’s right: not only performed by them, but also written by them. Collaboration by a mere two authors is difficult enough, so collaboration by a large group must be truly daunting. But this was a notably successful effort, avoiding any feeling that it was written by a committee. The writing, the production, and the acting were all excellent. Because there was a 40-minute limit imposed by the state drama festival, the timing and the structure had to be tight. The entire play felt compact and taut without being rushed.

The plot of the play was unusual for a high-school production, though it would seem familiar to anyone who has taken a college psychology course. Playing with the tension between research and clinical psychology, and delving into ethical issues concerning the use of human subjects, The Outlier Effect provided food for thought along with considerable humor. Maddie Redlick was particularly effective as a researcher with a preteen son suffering from PTSD, ably portrayed by Todd Elfman. A cast of 15 other students did a fine job as well. All in all, an ambitious effort that was more than competently executed.

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