Learning Strategies

Rudbeckia Hirta reports that she has a “freakishly competent” college calculus class:

They come to class; most of them do the assigned work; they earn high scores on the assessments.

Whether that situation should be so surprising is another story, but here I want to comment on the real import of RH’s post. She goes on to describe the computational weaknesses of this otherwise admirable class:

Over the past few days several students have come to my office to ask me questions about computations that I did in class. We'd been working with the geometric series, and we've been applying algebraic manipulations to the general term so that it's of the form ar^n-1 (our indexing starts with n=1). We had a problem where the general term is of the form 3ⁿ/4ⁿ⁺¹. To get it to match the form in the book, I used rules of exponents to rewrite it at 3/16 (3/4)^n-1.

Mass confusion. Totally lost. My very successful and accomplished calculus students were unable to follow the algebra. They couldn't see why those two expressions were equal. Working through the problem, slowly, in my office they would ask, "When you multiply, do you add the exponents?" Another student asked, "Is 3ⁿ/4ⁿ the same as (3/4)ⁿ?"

I would like to think that college freshmen who do their work and earn high grades on assessments would understand exponents and fractions as a result of their high-school math, but I guess I’m naive. Or maybe it’s a difference between public high schools in the South and those in the Northeast. Or maybe I’m being naive about the Northeast, and our students wouldn’t be any better.

Labels: math, teaching and learning

In HO scale model railroading, tracks always used to have rails that are 0.100 inches high, even though that’s not strictly to scale. Many model railroaders — mostly those who interpret the word “model” strictly — favor the newer versions, called Code 83 and Code 70 because they’re 0.083" and 0.070" respectively, which are more prototypically accurate than 0.0100". On several of the model railroading newsgroups a fierce discussion breaks out periodically on this subject, usually about Code 100 vs. Code 83. Here are some of the remarks:

I can think of almost no good reason whatsoever to continue to buy, or plan to use code 100 in HO, save for compelling economic reasons, or for trackage that is not modeled and is out of view.

The way I look at it is that code 100 is more then 20% larger then code 83. If I had a boxcar or building that was 20% larger then it should be, it would be out of scale and not look right on my layout. I'm not a perfectionist when it comes to these things, but buying code 83 was a no-brainer for me.

Sure [code 100 track] is more durable, if you like to run your trains on 'I' beams...

Code 100 looks really weird after working with code 83/70/55. You can spot it a mile away in photo of models. No matter the cost or minor (theoretical) structural benefits, don't use code 100. Just my humble opinion.

A common reaction among civilians is to wonder why these enthusiasts care so much, but that’s not really a fair question: after all, I don’t ask why Red Sox and Patriots fans care so much, even though I don’t understand it. My question is to ask whether it’s this difference of 0.017" (about one sixtieth of an inch!) is really so visible to the naked eye. I have trouble believing that it is. On the other hand, code 100 is 20% larger than code 83, so maybe the difference can actually be perceived.

Labels: math, model railroads

I’ve written posts about the Crimson Summer Academy on two earlier occasions: August 20, 2006, and May 23, 2005. Now I’m designing a math curriculum for a program with a somewhat similar basis in New Jersey: the College Prep Program of NJSEEDS. Note their self-description:

The SEEDS College Preparatory Program prepares academically motivated, financially-limited students for admission to selective four-year colleges. The Program includes weekend and summer Honors Classes, cultural enrichment and assistance with the college admissions and financial aid process.

The most interesting thing about both of these programs is their response to the achivement gap problem. Intensive summer programs, small honors-level classes, continued attention during the school year... all of these make a big difference, and I suspect that they’re the only way to make a big difference. Stay tuned...

Labels: math, teaching and learning

Slightly over a year ago, I wrote a post about Getting Things Done (GTD) — how it seemed to me to be “the right thing” and yet I couldn’t make myself actually implement it.

In the intervening 13 months I still believe that I should try it, and I still haven’t in fact tried it. Also, in the intervening 13 months there has been much general discussion of this book, its recommended methodology, and the activities of its author, David Allen. Recently the discussion has focused on Allen’s religion, of all things. I’m sure nobody would care if he were Baptist or Catholic or Jewish, and most Americans probably wouldn’t care if he were Mormon or Muslim — well, I guess there are altogether too many who would care, but I would like to think that Americans are above such religious prejudice. But Allen is none of these; he belongs to the Movement of Spiritual Inner Awareness. As far as I can tell from their website, this religion is a form of new-age Buddhism, and I guess it seems “weird” to mainstream Americans. In particular, it seems like a cult, like Scientology.

Several questions come to mind. Should we care what David Allen’s religion is? Should we perhaps care if it’s a cult, but not if it’s a real religion? What is a cult anyway? Is GTD merely a front for promoting MSIA?

To some Americans, a cult is merely any unpopular non-mainstream religion. Many people outside of the Boston area think that Christian Science is a cult, even though it’s fairly mainstream here. Some think that Hasidic Judaism is a cult, or certain branches of evangelical Protestantism. What about the Moonies? Remember the Hare Krishnas? David Koresh’s Branch Davidians? est? Those last four groups certainly do seem like cults. Our local Twelve Tribes is a cult in some observers’ eyes, but not in others’.

Michael Langone describes the enormous difficulties that arise when trying to define the word “cult.” There is certainly no consensus, but Langone cites a definition that he considers to be reasonable:

A group or movement exhibiting a great or excessive devotion or dedication to some person, idea, or thing and employing unethically manipulative techniques of persuasion and control (e.g. isolation from former friends and family, debilitation, use of special methods to heighten suggestibility and subservience, powerful group pressures, information management, suspension of individuality or critical judgment, promotion of total dependency on the group and fear of leaving it, etc.), designed to advance the goals of the group's leaders, to the actual or possible detriment of members,  their families, or the community.

Some say that MSIA is a cult under the terms of this definition, but I have no way of knowing for sure.

It’s worth reading Brenda No-last-name-given’s recent well-balanced summary of the surrounding issues concerning David Allen and whether GTD is a cult. The opening quotation is particularly nice as an example of biased language: “David Allen admits to being a member of the group.” [That’s not Brenda’s language; she’s quoting someone else.] I have to give more thought to the whole issue. Unfortunately I still don’t know whether MSIA is a cult, and I’m still not sure whether that would disqualify GTD from being a methodology worth following even if it’s true.

Labels: books, life, teaching and learning

I just finished listening to the audiobook version of Prince of Thieves, a crime thriller written by Chuck Hogan and read by Dorchester’s own Donnie Wahlberg. The action takes place in Charlestown, and the sense of place is definitely the strongest characteristic of this novel. It’s in the tradition of Dennis Lehane’s works, especially Mystic River, though it’s complicated by the combination of the old-time working-class townies and the newly arrived yuppies. Wahlberg’s skillful narration only adds to the convincing characterization of Charlestown and its various residents. The whole book is well worth reading (or listening to), but the dynamite opening paragraph may be the strongest:

First, a toast. Raise a glass, solemn now, to the Town, to Charlestown, our one square mile of brick and cobblestone, neighborhood of Boston yet lopped off every map of the city like a bastard cropped out of every happy family portrait. This is the heart of the old eleventh, the district that first sent the Kennedy kid to Congress, the one square mile of America that shipped more boys off to World War II than any other, site of the Battle of Bunker Hill, the blood of revolution sprinkled like holy water over our soil and our souls, turf and tribe and Townie pride, our sacred trinity. But now, look at these outsiders snapping up our brownstones and triple-deckers, pricing us out of our own mothers’ homes, yuppies with the Volvos and their Asian cuisine, their disposable income and contempt for the Church, succeeding where the British army failed, driving us off our land. But sure, we don’t go away so easy. “Don’t fire until you see the whites of their eyes.” That was us, remember? This carnation here, may be a bit brown at the edges, but see it’s still pinned to the tweed lapel over my beating Townie heart. Be a hero now, reach me that jar. We’ll have a hard-boiled egg with this last one. See how she goes down. It’s caps off, gents. Here’s to that towering spike on the hill, the granite battle monument that’ll outlast us all, the biggest fucking middle finger in the world, aimed right at good brother Boston in the 21st century and beyond. To the Town! Here’s how!

Yes, indeed inspired by Dennis Lehane, and perhaps also by a touch of James Joyce.

Labels: books

So here’s the question. Why does everyone like to hear success stories from inner-city, overwhelming minority public schools? I suppose Democrats like to hear these stories because it confirms their beliefs that non-whites without money can be academically successful, and Republicans like to hear them because they are usually examples of spectacular achievement without massive government programs.

But no, that’s not really the question. The real question is whether the stories are even true. I don’t mean that those who tell them are lying; I just mean that the successes may be short-lived and therefore misleading. There was a provocative report on Weekend All Things Considered on February 17 dealing with this issue. Susan Eaton, author of The Children in Room E4, talked about this issue in the context of a successful teacher in “what was supposedly the best school in inner-city Hartford” [interviewer’s words in italics]:

The children in Room E4 had the same problems as many children in high-poverty, racially segregated urban school districts, but...

... Given the conditions in which Miss Luddy was working, she actually achieved remarkable success... Her school got the blue ribbon from the Education Department. She had high test scores... She’s an amazing person, but she’s also an exception. Saintly geniuses are in really short supply. But I think one of the things our American culture does is that we love to focus on the exceptions, the kind of hero miracle teacher who does a few projects with the kids and gets them to achieve at high levels and defies everybody’s expectations...

Those stories are absolutely true, and they’re worth celebrating, but I think we often celebrate them at the expense of understanding the true mess that’s out there and the huge and vast inequalities that show up no place better than our public schools.

If a school like this does show that it can achieve, isn’t that enough? It hasn’t achieved. It achieved at the very small, discrete task of nudging up test scores for a tiny portion of children over four or five years. The year after they won this blue-ribbon award, and they were one of six models for urban education in the country, the test scores plummeted...

The high test scores came at a price of, for example, no recess. Children eight and nine years old, sitting on their butts from 8:30 until 5:00 at night drilling for tests...

Are you making the case that it is not possible to have a high-performing inner-city school over the long haul?... ...No, I know that it can happen, but through all the years of trying, there is no one school system that has been able to achieve any measure of equality with predominantly middle-class school systems. And we’re talking about a huge country here. We focus incessantly on the exceptions, we focus incessantly on the so-called beat-the-odds schools. Some of these stories are true, some of them aren’t; I happened to land in a place where the story was not true, and someone has got to stand up and say it. No one in Hartford wanted to say it. Everybody kept pushing the miracle... Racial isolation, economic isolation, and poverty are still the central issues, and no one is doing anything about it. No one wants to listen, because we all want to believe in miracles.

I’ll have to read the book. I wonder to what extent it’s specific to this one classroom in Hartford, and to what extent it generalizes to urban education elsewhere.

Labels: Dorchester, life, teaching and learning

How could I resist reading a blog entitled “Geoffrey Chaucer Hath a Blog”? Yes, as you guessed, it turns out to be a blog that’s entirely written in Middle English!

Anyway, take a look at it and make a serious effort to read it. Even if your Middle English is rusty (you did read Chaucer in the original in high school, didn’t you?), you will find useful links in the left bar that will get you up to speed and back to the Middle Ages in no time. Yes, I know it sounds strange, but it’s no odder than the Society for Creative Anachronism, is it?

Labels: linguistics, technology

I recently finished reading a fascinating historical novel by David Liss, The Coffee Trader. Now maybe you’re not interested in the formation of the coffee trade in Europe in the 17th Century — though I can’t imagine why not — but you’ll still learn a lot from The Coffee Trader and will enjoy yourself in the process. Though not quite as strong as Liss’s earlier historical novel, A Conspiracy of Paper, this work introduces the reader to the invention of stock-market manipulation four centuries ago. I don’t know how much is fiction and how much is history, but Liss makes the process unfold convincingly before the reader’s eyes. Perhaps most interesting is the cultural setting, combining Christian Netherlanders with a substantial community of Portuguese Jews, who had fled Portugal because of the Inquisition. The inner workings of the Portuguese Jewish community in Amsterdam and its members’ interactions with the Gentiles provides most of the tension and drama in The Coffee Trader — not that stock-market manipulations aren’t dramatic, but somehow they don’t have quite the same emotional impact.

It is obvious that Liss has done a great deal of research for this book — perhaps too much for some readers’ taste. But the details of discovering this marvelous new beverage and the financial shenanigans that swirl around it somehow avoid overwhelming the standard apparatus that makes a novel a novel. The three-dimensional characters are well-developed, there are appropriately persuasive conflicts among them, and the sense of place and time can’t be beat. All the main figures are nuanced, and there are no pat answers. Like all good works of fiction, this one makes you think.

Labels: books

LCSI’s new blog, Thinkering, links to Seymour Papert’s homepage, which in turn links to a four-and-a-half-year-old press release from MIT, which reminded me of our commitment to big ideas in the Math Department of the Weston Public Schools. Such are the ways of hypertext. Anyway, I don’t normally recommend press releases, but do read this one.

Papert has never been detail-oriented. Details are important, but they’re sterile without big ideas. Unfortunately, as Papert points out, several of the fields to which he has devoted his life have lost most of their vitality because they have lost their big ideas:

“I have been through three movements that began on a galactic scale and were reduced and trivialized,” Papert said during the one-man informal symposium in Bartos Theatre on July 9. The three movements — child development, artificial intelligence and kid-friendly computer science — were especially vital and big in the early 1960s, he said.

Dismissing the entire current national educational system as “idea-averse,” Papert said computers themselves could offer children an elementary model for how their own minds work.

The benefits of working with computers could also include a simple and liberating new view of mistakes. “They’re just bugs,” said Papert.

A useful reminder.

P.S.: Those who aware of Papert’s serious accident in Hanoi two months ago, when he was hit by a motorcycle, will be relieved to know that he has apparently been recovering satisfactorily from his brain surgery. He was released from the hospital on January 23 and is reported to be recuperating at home with no permanent effects of the accident.

Labels: teaching and learning, technology, Weston

Ate lunch today at the Blue Fin, a favorite inexpensive Japanese restaurant in Little Tokyo, a.k.a. the Porter Exchange Building in Cambridge. A new sign lists the types of identification that are acceptable for ordering alcoholic beverages — mostly the usual, such as Mass. driver’s license, passport, military ID, and so forth. But then there’s “ID issued by Mexican consulate.” Why only Mexican? Why not the Japanese consulate, say? Is Japanese food a particular favorite of Mexicans? Do Mexican visitors tend to frequent Porter Square? Is there perhaps a Mexican consulate in the same building? (We actually checked. The answer is no.) ’Tis a puzzlement.

If I were more comfortable with modern technology, I would have snapped a photo of the sign to add a visual to this post. But it just never occurred to me that I could have used my cell phone to take a picture. I guess I’m still in the low-tech rut of not thinking of photography unless I actually have a camera with me. Unlike my students, I automatically place my cell in the phone category, not the camera, calculator, music player, or game machine categories. And don’t tell me about the iPhone.

Labels: life, technology

It often happens that an event at the Dorchester Historical Society (DHS) is an informative experience. And it often happens that a DHS event is an enjoyable experience. Today’s presentation by Earl Taylor and John Goff was both.

The title promised a broad sweep of history: “Dorchester from the Period of Settlement to Annexation.” Needless to say, covering 240 years of activity (1630–1870) in a 95-minute talk is not really possible; Taylor and Goff coped with this daunting task by stressing themes rather than giving a comprehensive survey. Rather than spend 24 seconds on each year of history — a technique that I hope no one would seriously attempt — they split both the entire period of years and the time for the talk in half, and then dealt with each half thematically. The result was notably successful.

Goff, a preservation architect and executive director of Historic Salem, gave the first half of the presentation, which dealt with the first half of Dorchester’s pre-annexation history and unsurprisingly focused on architecture. We learned about the log cabin myth and the virgin territory myth: the false ideas that early British settlers in Dorchester lived in log cabins, and the idea that Dorchester was empty until the Brits arrived. Goff showed us the housing of the American Indians (or Native Americans, for those who are mistakenly politically correct) and the timber-framed, post-and-beam houses of the settlers. We learned about wattle and daub (no, I hadn’t known those words either) and its use in the DHS’s Blake House, which now looks tiny but was considered a mansion when it was built. Goff’s portion of the talk completed with a slide show of the progression of architectural styles in 17th- and 18th-century Dorchester.

Then, at the request of an audience member, we had a 5-minute break, which of course lasted 15 minutes and of course it was difficult to get everyone back into their seats. I always have mixed feelings about giving breaks in the middle of long sessions for exactly this reason.

Taylor then took us up to 1870 with a carefully chosen set of themes other than architecture. First came church history, starting with the Puritans, who came here for their freedom to worship (but, as is well known, not others’ freedom to worship). For a long time the history of religion in Dorchester was intimately tied in with the histories of government and education, and education formed Taylor’s second theme. Our own Mather School claims the title of being the first public school in America to be supported by taxes. Of particular interest to the audience were slides showing reproductions of the “Oath of a Free-Man,” and the Bay Psalm Book. The former is notable because swearing this Christian oath was originally a requirement for being a voter; from the latter we read a very awkward rhymed translation of the 23rd psalm. Each of these claims to be the first work printed in North America. After a survey of Dorchester schools we returned to the early days and learned about the wreck of the Elizabeth and Mary in Canada, and were then treated to a whirlwind survey of Dorchester industry and commerce, including a variety of industries that we no longer have, such as chocolate, paper, pewter, and silver-plating.

Labels: Dorchester

This is a follow-up to my earlier post on Second Life (SL) vs. World of Warcraft (WoW). One of my students, Dan Spector, replies to that post by writing the following remarks in an email message to me (quoted by permission):

Contrary to popular opinion, the time investment in World of Warcraft isn’t overwhelming. World of Warcraft was actually developed as a casual MMORPG (massively multiplayer online role-playing game) and Blizzard has devoted a lot of effort to make the game accessible to casual players. That doesn’t mean there isn’t a lot to do; reaching the highest levels of the game does require a significant time investment. Though this may seem contradictory, I think that the distinction between a required time investment and quantity of content is a significant one. Players who devote their time to World of Warcraft do so because it’s enjoyable, not because it’s necessary. Though there is a daunting amount of content, the speed at which it is experienced is up to the player.

...You may be interested in reading about the Corrupted Blood event that occurred in World of Warcraft. The Wikipedia entry (as of 12:19AM, 2/9/06 — time stamps on sources are becoming necessary...) describes it as “a virtual plague that infected characters in the computer game World of Warcraft; it was also the first disease to affect any MMORPG with a significant game effect.” The BBC did a short article on it, also.

...It’s often difficult to overcome the stigma surrounding World of Warcraft, and I think the game offers a lot more than what is mentioned in the media.

As an aside, I would like to claim that the quality of the writing in that message represents what’s typical of Weston High School students’ email. But I would be lying if I claimed that.

Anyway, Dan’s message makes me want to try WoW after all. But then I read Steve O’Hear’s post, which tilts me toward SL. (I don’t have time for both. I thought I didn’t even have time for one, though Dan presents a convincing argument to the contrary.) O’Hear writes that “John Edwards has become the first presidential candidate to set-up-shop in Second Life.” He quotes Jerimee Richir, who is managing the Edwards SL campaign:

Second Life users are a unique audience, in that, they are first adopters. This means that they are extremely adept at creating User-Generated Content (UGC).

While SL users do not have the same numbers as, say, MySpace, they have communication skills, and a desire to communicate, that, I humbly say, exceeds that of MySpace users. For example, and this is just a guess, but I bet that half of Second Life users regularly contribute to multiple blogs. So it is a smaller community, but I would argue it is a more influential community.

So SL campaigns generate more buzz, not because, “the media is stupid” but because Second Life users do more talking.

That certainly sounds intriguing, and makes me want to be part of the SL community (not that I have anything against Obama or Clinton, but I think Edwards is the one to watch).

Labels: life, technology, Weston

We have a problem at Weston High School. Actually, we have more than one — but there’s one problem I want to write about here: every day we have an extraordinary number of students coming late to school and/or late to class. Clearly this has to stop.

Or does it? Maybe we can solve half of the problem by redefining it so that it is no longer a problem. No one is suggesting that it’s OK to come late to class, but some people are proposing that we eliminate half of the “tardy to school” problem by allowing students who have one or more free periods at the beginning of a day to arrive “late” without being “tardy” — assuming, of course, that they are on time for their actual classes.

This is not exactly a new idea. Some high schools have an “open campus,” where students are allowed to come and go as they please as long as they meet their obligations, such as showing up on time for all classes. Others have a “modified open campus,” where some students have that privilege, or where the privilege is limited to the beginning and end of the day (as opposed to leaving school during a free period in the middle of the day). When I was a teacher at Lincoln-Sudbury, we had a totally open campus, and it worked very well. I don’t know what they do now at L-S, but in the ’70s there was no such thing as study halls (if you have a free period, it’s a free period) and there was no particular issue about students’ coming to school at the end of a free first period and being on time for their second-period class. There was no issue within the school, that is. In the outside community there was no issue about this within Lincoln, but many residents of Sudbury had political or moral qualms. Some of them referred to L-S as “the country club,” because it was so unlike the kind of public school that they attended. Often there would be candidates for School Committee reflecting these views. They rarely got elected, and I have never forgotten a particularly memorable candidate who put forth the following proposal: “all students whose grades are below average must be assigned to a study hall during all ‘free’ periods.” When told at a public meeting that he was talking about half of the students (an enormous number to find rooms for, in such a large high school), he was flabbergasted and shouted at the Superintendent-Principal, “What kind of a school are you running if half of your students are below average??!!”

Fortunately he didn’t win the election.

But the political climate was different in the ’70s. As Garrison Keillor points out, “we’re all Republicans now” — but in the ’70s it was OK to be a liberal. This right-wing candidate’s position would be considered middle-of-the-road now, since he was actually advocating a modified open campus! At Weston we “compare ourselves with” a number of other suburban school systems, many of which have modified open campus plans. The two Newton high schools do (for 11th and 12th graders, I believe), and Brookline has it for almost all high-school students, starting midway through 9th grade. In all cases, parental permission is required, and the privilege is revoked if a student gets poor grades, is a behavior problem, misses a class, or is late to class more than a very small number of times.

And therein lies the reason for the success of modified open campus. At Weston (and most closed-campus high schools), students know that the penalty for being late to class is very tiny, and even the penalty for cutting a class isn’t particularly severe. But at schools with complete or modified open campus plans, students don’t want to lose this important privilege, especially since their friends have it. One Brookline parent reports to me that class cutting and tardiness are not a big problem — because of the punishment they entail. And certainly students’ morale improves when they are treated more like adults.

Four questions must be considered:

• Would the School Committee approve such a plan? I have no answer to this question.


• Why do so many students arrive late to school under the current closed-campus system? This question I can answer, at least in part. Teens in general need more sleep than they get, especially in high-stress schools, and it’s well known that getting up early in time to be at school before 7:30 doesn’t really fit their body clocks. So it’s natural for a student with a free first period to want to sleep in and to ask his or her parents to write a note so the tardiness will be excused. And it’s natural for the parents to agree. While this certainly isn’t the only reason why so many kids are late to school, I expect that it accounts for the majority of the cases.


• Where is there to go in Weston? In other words, students who are not sleeping late might arrive at school and then choose to go elsewhere until the first class, but it’s not clear where they would go. There’s an exciting bagel store in Weston Center, I suppose. But we’re not like Newton North or Brookline, where there are things to do within walking distance of the high school. (What happens at Newton South or Lincoln-Sudbury, which are also isolated schools, like Weston?)


• Finally, how would we live with the inadvertent race and class discrimination that a modified open campus would entail? I don’t have any figures (though I suppose I could get them), but it seems clear to me that white or Asian students who live in Weston are more likely to have their own cars than black or Latino students who live elsewhere and attend Weston High School, either as part of the Metco problem or because their parents work for the Town of Weston. Given the absence of places to visit within walking distance of the high school, we would be saying that students who drive to school would be allowed to absent themselves at the beginning or end of the day, but those who take the bus would not be able to take advantage of this privilege. I have trouble with this outcome, though I suppose the same thing happens now with “senior privileges,” our current system by which seniors in good standing and parental permission are allowed to arrive late or leave early if they don’t miss any classes.

Labels: teaching and learning, Weston

Last night, Emily Rooney’s Greater Boston included a segment on truancy in the Boston public schools. One of her guests, Suffolk County Sheriff Andrea Cabral, whom I usually admire and respect, claimed that 50% of high-school dropouts (from Boston public schools) become criminals. In support of this claim, Cabral asserted that when young inmates are interviewed in local jails and prisons, half of them turn out to be high-school dropouts.

Let’s examine this reasoning. Since I don’t have the actual figures, I’ll just give an example. Here is a typical two-by-two table for an imaginary school system:

	graduated	dropped out	TOTAL
criminals	500	500	1000
law-abiders	3500	1500	5000
TOTAL	4000	2000	6000

As I say, these are made-up numbers, but they clearly illustrate a plausible situation where 50% of criminals are indeed drop-outs, but only 25% of drop-outs are criminals. So it’s a classic case of the fallacy of the converse in the context of conditional probability.

Barbara thinks I’m a geek because I exclaimed, “fallacy of the converse!” as soon as I heard Cabral’s claim while we were watching her on TV. Doesn’t sound geekish to me; just sounds normal.

Labels: Dorchester, math, teaching and learning

I have a mixed reaction to Unlucky in Law, by Perri O’Shaughnessy. It’s a decent enough legal mystery, more-or-less in the John Grisham or Scott Turow vein. And it has an undeniably interesting plot, involving the Russian-American community in Monterey.

Never knew that there was a sizeable Russian-American community in Monterey? Neither did I. If the novel is to be believed — and I see no reason why it shouldn’t be — there are some major contrasts with the Russian-American community that I am familiar with. My Russian-American friends and students in the Boston area — primarily Brookline and Newton, to be specific — are almost all Jewish, but the Monterey one is mostly Russian Orthodox, at least in the novel. This is integral to the plot, because of tie-ins between the Russian Orthodox Church and the former Russian royal family. Since I don’t want to give anything away, I can’t be more specific than that, but suffice it to say that this story is really a remake of one that has been written many times before, starting with Dorothy Sayers’s wonderful novel, Have His Carcase, which is the only major work of fiction in which the Playfair Cipher is a major plot point... but I digress...

Anyhow, the feeling of “been there, done that” is the main cause of my mixed reaction to this otherwise satisfactory book. The secondary cause is O’Shaughnessy’s tendency to signal upcoming plot developments, so the reader frequently knows what’s coming next. Nevertheless, if you like legal mysteries, give Unlucky in Law a try; on balance there’s more good than bad there.

Labels: books

Rudbeckia Hirta reports that her college students have trouble understanding irrational numbers:

Most of my students felt that 3 * sqrt(2) / 5 * sqrt(2) was irrational because of the sqrt(2). They didn’t remember that you can “cancel” (hate that word: prefer “divide out”) the sqrt(2) that appears in both the numerator and the denominator. More alarmingly, many of them decided that 3.14159 was irrational, citing only “it can’t be written as a fraction,” and forgetting that 314159/100000 is a perfectly legit fraction

I wonder if my Weston High School students will have the same difficulties. Maybe they won’t, since all Weston students are above average. I particularly hope that they won’t write “proofs” like the ones that Prof. Hirta cites. For example:

We’ll be studying irrational numbers in a few weeks; I’ll let you know.

Labels: math, teaching and learning

As in a great many other high schools, Weston sees more and more students each year taking Advanced Placement courses. Why is this happening? And is it a good thing?

It’s easy to see why it’s happening. Weston students are advised that their transcript should show an “academically demanding program,” which apparently means (among other things) at least three AP courses during high school, not necessarily all in senior year. This is no problem for students in our honors classes, who automatically take an AP math course senior year and most likely also take AP World History as sophomores, AP US History as juniors, at least one AP science and AP language course as seniors, and perhaps something additional such as AP music theory, AP computer science, AP European History, or whatever. So five Advanced Placement exams are not an unusual total for honors students by time they graduate, and even six or seven are far from unheard of.

The problem is that students who do not take honors courses have trouble coming up with the three AP exams that they believe they need in order to get into a competitive college. Maybe they do need those, maybe they don’t, but they feel a lot of pressure to take the “easiest” of the four AP courses offered by the Math Department: Statistics. Now there’s no doubt that the AP exam in Statistics is easier than those in BC Calculus, AB Calculus, or Computer Science — easier, but not easy. Students who have gone through regular (“college-prep”) math courses without getting A’s rarely do well in AP Statistics. It’s not that this exam requires the specific content of honors math courses, but it does require some of the study habits and habits of mind that almost students who are successful in honors math courses pick up along the way. So imagine that you’re a junior getting B’s in college-prep math and you’re trying to select a math course for senior year. You’re currently in Precalculus Part One, so the natural next course in sequence is Precalculus Part Two. Maybe you decide to take that course as well as AP Statistics — a reasonable combination that some students elect, even though the first third of Precalculus Part Two duplicates part of AP Statistics, so it’s not the best combo in the world. If you find AP Statistics surprisingly overwhelming, you can just drop it, right? (Of course you had better drop it before the withdrawal date, or your precious transcript will show the dreaded W.)

But it’s most likely that your senior schedule doesn’t permit taking two math courses, at least if you want to have a life. (Many honors math students don’t worry about having a life, so doubling up on Calculus and Statistics is quite common.) If you’re a typical junior in Precalculus Part One, you may opt to sign up for AP Statistics next year instead of Precalculus Part Two rather than in addition to it. This choice is distressing some of us. In the first place, you won’t be adequately prepared for college calculus the following year. In the second place, if you’re neither in honors math nor getting an A in college-prep math, your chances of doing well in AP Statistics aren’t so great. So what happens? Perhaps you slow down the rest of the class, at least if there are enough of you to form a critical mass. Perhaps you damage our overall AP Exam results, since you probably won’t do well on the exam — not that that’s a noble motive for objecting to this practice. And perhaps you do mediocre work in your chosen math course rather than good work in the course you should have taken: Precalculus Part Two.

What can we do about this phenomenon? How do we get students to sign up for the most appropriate course, when they’re convinced that it won’t get them into Harvard? An AP course may brighten their resume — if they do well in it, and that’s a big if — but it still may be the wrong course to take.

In closing, I seem to have neglected the most important point: is an AP class a good class? Well, sure, it is usually a good class. But it may or may not be better than a given non-AP class. The “AP” label doesn’t guarantee that it’s the best course for you.

Labels: math, teaching and learning, Weston

It’s helpful to read what the Brits say about maths teaching. I’ve recently started reading Mathematics in School, a journal published by the Mathematical Association (MA), which is more or less the British equivalent of our National Council of Teachers of Mathematics (NCTM). I say “more or less” because there are some intriguing differences. For example, the MA describes its membership as follows:

Teachers in primary and secondary schools, further and higher education, advisers, inspectors, practising mathematicians and students. There are also institutional members.

I wouldn’t want to make some sweeping generalization about cultural differences here, but note how the NCTM describes its members:

a dynamic group of more than 100,000 dedicated teachers, educators, and institutions working toward the same goal — improving the teaching and learning of mathematics for all students.

Anyway, Mathematics in School provides articles that complement the NCTM’s Mathematics Teacher — similar but different, so to speak. Again let’s look at how they describe their own journals. From the MA:

Mathematics in School is aimed mainly at teachers of school and college pupils of 10 to 18 years of age and for those working with students who are preparing to enter the teaching profession.

We attempt to attain a balance of articles reflecting this age and ability range and look for pragmatic articles; ready-to-use materials; discursive, possibly philosophical articles; speculative, reflective, and sometimes retrospective pieces... There is also the opportunity to stimulate — and even amuse — otherwise hard-pressed and very busy teachers....

In contrast, the very serious NCTM offers the following “Mission Statement”:

The National Council of Teachers of Mathematics is a public voice of mathematics education, providing vision, leadership, and professional development to support teachers in ensuring mathematics learning of the highest quality for all students. The Mathematics Teacher, an official journal of the National Council of Teachers of Mathematics, is devoted to improving mathematics instruction from grade 8 through two-year and teacher-education colleges. It provides a forum for linking research to practice, deepening understanding of mathematical ideas, and sharing activities and pedagogical strategies....

From these descriptions you can probably come up with a pretty accurate idea of the differences in the two publications’ articles. Reading either one without the other would result in losing something valuable.

Labels: math, teaching and learning

So I’m waiting in line at the deli section of the Stop & Shop at 545 Freeport St. in Dorchester, and the customer in front of me asks the clerk a question:

Customer: Where are we?

Clerk [in a surprised tone]: This is Stop & Shop!

Customer: I know that. But where are we? Stop & Shop where?

Clerk: Oh, well, this is the Neponset section of Dorchester.

Customer: And is that meat really $7.99 a pound?

Clerk: Well...yes, that’s what it says.

Customer: Then why aren’t we in Milton? How can you charge so much in Dorchester?

Labels: Dorchester, food, life

The aforementioned Jill Walker has posted a fascinating article entitled “worksheet for students: am i insane?” No, she’s not insane, although some of my students think she’s misguided. But I think she’s on to something.

Take a good look at her key document: a sample workplan for one week’s worth of work expected from her college students. This exceptionally thorough effort exemplifies all that’s expected in the current American trend toward standards-based education. (I don’t know whether the trend has actually reached Norway or not; perhaps this is merely a parallel development.) At Weston we are encouraged to share our agenda and goals at the beginning of each class; the students are entitled to know where we’re going, why we’re going there, and how we’re going to get there. But Walker does us one better: she outlines the week’s tasks, expected times, and learning outcomes in extraordinary detail. Since her document for college students was inspired by one given to her fifth-grade daughter, I figured I would split the difference and ask my tenth-graders what they think of it. By and large they thought that they would feel micromanaged if I adopted such a system. (They didn’t use that word, but that’s clearly the gist of the majority’s feelings.)

Walker poses the following question:

...my impression is that students don’t do work outside of the classroom because they don’t really know what they’re expected to do. (Is that true?)

I will reply with a resounding yes! Both my honors students and my college-prep students (Westonese for non-honors) have trouble knowing what they’re expected to do and how they’re expected to study.

Walker goes on to wonder...

I’m not sure yet whether this is extra work for me or whether it will help make teaching easier for me.

I suspect that the answer is once again yes — to both of the implied questions. It would definitely be extra work for the teacher (by far the biggest reason why I’m reluctant to try it), and yet it would make teaching easier in the long run. Anyway, read the article and the document, and pay particular attention to the extensive comments to Walker’s post, including her replies to the comments. It’s good food for thought.

Labels: teaching and learning

As Ivars Peterson and others have pointed out, Jackson Pollock’s paintings can be analyzed mathematically as fractals, and they turn out to have a distinctive fractal dimension. As various articles have pointed out, inauthentic (forged) Pollocks have incorrect fractal dimensions.

What does “incorrect” mean in this context? It means that a genuine Pollock painting will have a fractal dimension in a certain range, depending on the year in which it was produced. If a putative Pollock work has a significantly different fractal dimension, it must be a fake.

But then there has been a recent controversy on this subject. The claim that fractal dimension can be used for the detection of forged Pollocks has been called into question. But are the objections valid? The usual argument seems to go something like this:

1. Genuine Pollocks have a fractal dimension in the range of a to b.


2. This so-called Pollock has a fractal dimension of c, where c is not in that range.


3. Therefore this painting is not a genuine Pollock.

This is certainly a valid syllogism — indeed a classical example of Aristotelian logic.

The only trouble is that recent objections — such as the one articulated on “Marketplace” on January 30 — do not follow this usual argument. Instead, they seem to reason something like this:

1. Genuine Pollocks have a fractal dimension in the range of a to b.


2. This so-called Pollock has a fractal dimension of c, where c is in that range.


3. Therefore this painting is a genuine Pollock.

They then proceed to exhibit a counterexample — a fake Pollock with the appropriate fractal dimension — and thereby challenge step #1 of the revised syllogism.

But this is not a valid syllogism. It’s a classic fallacy: the fallacy of the converse. I don’t know whether the error was committed by the Marketplace guest, well-known physicist Lawrence Krauss, or by the article in Nature to which he refers, but Krauss claims that the article supports the following assertion:

...several childlike sketches drawn by one of the authors appeared to satisfy the criteria that had been claimed to distinguish Pollock from mere mortals.

This is akin to claiming that “all genuine Pollocks have a certain fractal dimension” is equivalent to “all works with that fractal dimension are genuine Pollocks.” They’re not equivalent! Finding a fake Pollock with the right fractal dimension in no way demolishes the original claim.

Labels: math

Would I be interested in joining a 3-D virtual world in my copious free time, which doesn’t really exist? Some of my students have been trying to persuade me to play World of Warcraft (Wow). (That was the standard acronym, not a comment.)

Then they say that it would take up all of my time, which is certainly not a selling point. They point out that a fellow senior — who shall remain anonymous to protect the guilty — skipped school on Monday so he could play World of Warcraft all day. Again, not a selling point — not for me, at least.

On the other hand, Jill Walker, associate professor and head of the department of humanistic informatics [sic] at the University of Bergen, is devoted to Wow, is editing a book on it, and has clearly thought deeply about it. That is a selling point. Walker’s remarks, and her readers comments on them, make it tempting to join this community.

On the third hand, there’s Second Life. Instead of immersing oneself in an existing world, you help to create one — at least that’s my take on one of the big differences, though I could be totally wrong. For me the initial selling point for Second Life was the presence of some serious non-techies, such as Judge Richard Posner.

A lot has been written about both of these environments. I’ve particularly been running across Second Life in my sources recently:

• Lawrence Lessig calls Posner “the most influential lawyer of his time” (and “a big fan of the Matrix. And cats.”) and links to an article about his participation in Second Life.
  
  
• The Boston Phoenix says that “Second Life offers students a virtually real education” where “the academic world’s forward-thinking minds are seeing new opportunities for the virtual campus.”
  
  
• And the New York Times writes that Reuters has opened a bureau in Second Life!

So I’m tempted...

Labels: life, technology

Archer Mayor’s well-crafted series of police procedurals has a highly deserved reputation for strong and careful plotting. As a Vermonter, Mayor writes in a style that convincingly evokes the state of Vermont — as much so as Lawrence Block evokes New York. It’s worth reading both authors for a strong sense of place.

Much of St. Alban’s Fire surprises the long-time reader of Mayor’s work by taking place in Newark, New Jersey (“down south,” as one of the characters says). This shift in locale was of special interest to me, since that’s the area where I grew up, living for ten years in Cedar Grove and attending Newark Academy for seven of those years. Anyway, both the Vermont scenes and the New Jersey scenes of St. Alban’s Fire introduce us to well-drawn characters and place us successfully in the world(s) around them. The aforementioned long-time reader will also enjoy the continuing development of the series characters.

And now for the zinger that really caught my attention for personal reasons. Two Vermont cops are on their trip to New Jersey to check out an arsonist who had popped up to Vermont to burn down some barns (and inadvertently killed a boy in the process). Along with their Newark contacts they pick up an informant, head up Bloomfield avenue through Glen Ridge and Montclair to Verona, and turn right on Fairview Avenue. Crossing seamlessly into Cedar Grove, they enter into the campus of the abandoned Overbrook Hospital (not to be confused with Overlook Hospital, though it often is):



Overbrook was finally replaced two months ago by the “state-of-the-art Essex County Hospital Center,” but I grew up on the grounds of the original incarnation of this huge county mental hospital, where my father was first Assistant Director and then Director for many years. Mayor correctly describes Overbrook in some detail, especially the features that made it into a small city in its own right, such as having its own power plant and fire department. And then the car with the police and the informant continues down Fairview Avenue almost another mile, to the opposite end of the huge hospital campus, and turns left onto a semicircular road on which are seven so-called cottages, homes for some of the doctors. They enter one of the ones in the middle... Could it be the very house where I lived for three years as a child? The description certainly makes it sound that way. You think it’s a coincidence, but it’s not...

Labels: books

One of my students came across the Wikipedia article on Happy Numbers and asked about it in precalculus class. This is the sort of topic for which Wikipedia is an excellent source; in fact, if I wanted to know about happy numbers, it’s the first place to which I turned. So we looked at it in class. Ordinarily I don’t like to project a webpage in front of a class if I haven’t vetted it in advance, but this particular topic seemed (and was) harmless. Anyway, check out the article yourself: it’s easy reading and provides some amusing mathematics along the way.

As we were going through the article, and I was helping the students tie it to some of the ideas we have been exploring, I observed that we often talk about families of numbers (primes, squares, and so forth). I pointed out, however, that number families are the reverse of human families, since all unhappy numbers are alike but all happy numbers are unhappy in their own way. Not a single student recognized the allusion, so we had to digress into a two-minute literature lesson. Sigh.

By the way, you should read the Wikipedia article to see why it’s true that all unhappy numbers are alike but all happy numbers are unhappy in their own way.

Labels: books, math, teaching and learning

It is unethical, as I’m sure you know, for a teacher to reveal any individual student’s grade to another student. (Students reveal their own grades all the time, of course, but that’s their decision, not ours.) This principle placed one of my colleagues in an ethical dilemma the other day: student A claimed that student B received a higher grade on a group project and asserted that this was unfair, since they were in the same group. Now we could argue about whether such a discrepancy is inherently unfair — and it’s easy to say that it isn’t — but in this particular case the students had in fact received the same grade. (Presumably student B lied about his grade, inflating it out of one motive or another.) So what does the teacher do? On the one hand, saying “you all received the same grade” is tantamount to revealing B’s grade. On the other hand, telling A “it’s none of your business what B got, just be concerned with your own grade” would perpetuate A’s mistaken belief that B got a higher grade. So what is the teacher to do?

Labels: teaching and learning