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Sunday, July 31, 2005

Doing something about stress

Be sure to read Anand Vaishnav’s article, headlined “Suburban high schools try to ease up on teen stress,” in today’s Globe. My favorite paragraph quotes former Lincoln-Sudbury colleague Charlie Ruopp, now principal of Wayland High School:
Wayland High School tried to hold yoga sessions after school, but it didn’t work. “The kids told us they didn’t have enough time to take yoga because they were too stressed.”
According to the article, Wellesley, Needham, and Wayland High Schools are all making serious efforts to reduce student stess.

There’s another side to this issue:
Not all schools with high college-bound rates and great test scores have taken steps to ease students' workloads. At Brookline High School, headmaster Robert J. Weintraub said he wants his teachers to demand more, not less. The school would not eliminate midyear final exams, for instance, because they prepare students for the sort of studying they will face in college, he said.

“If we’re not going to be rigorous, if we’re not going to be demanding, if we’re not going to apply stress, I’m not sure we’re doing kids a good service,” Weintraub said. “There are so many distractions with computers, cellphones, text messages, TV, music, popular culture, and electronic culture. I’d rather have kids working on math and history and science than text-messaging each other all night long or talking on the phone all night long.”
Both sides are correct. We do need high standards, fewer distractions, and even some stress. But not too much stress, such as I see in many of my students. I can’t write about Wellesley or Brookline — not knowing enough about either — but at Weston we seem to have a bimodal distribution among our students: most of the highly motivating, high-achieving kids are clearly too stressed out, whereas a clear majority of the “average” kids are too relaxed and need a bit more stress.

So what is Weston going to do to about these two groups?

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Saturday, July 30, 2005

Math for democracy

I highly recommend “Mathematics and democracy: the case for quantitative literacy,” published by the National Council on Education and the Disciplines. What they’re calling “quantitative literacy” is very close to what we call “quantitative reasoning” at CSA. This online collection, edited by Lynn Steen, contains 15 articles advocating changes in the standard math curriculum. As Robert Orrill says in the preface:
[T]he consequences of what John Allen Paulos named “innumeracy” (Paulos, 1988) can be profoundly disabling in every sphere of human endeavor — whether it be in home and private life, work and career, or public and professional pursuits. Stating the case in dramatic terms, Lynn Steen warns that “an innumerate citizen today is as vulnerable as the illiterate peasant of Gutenberg’s time” (Steen, 1997). Any such possibility of regress to pre-Enlightenment conditions would be deeply troubling under any circumstances and most certainly is unacceptable in a democracy.
I only wish that the articles pursued their topics in more depth. The entire book contains 115 pages of text, which may sound like a lot but actually only scratches the surface. I suspect that its main purpose to serve as a catalyst for conversation and exploration, and it should serve that purpose well. Take a look at it!

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Friday, July 29, 2005

Straight as an arrow

Too many students (and too many parents, and even some teachers) view the process of learning math as a one-dimensional arrow, in which the courses come in a fixed order and lead inexorably to calculus and beyond. The most successful students feel driven to move faster and faster along this arrow, the goal being to progress to calculus as fast as possible.

Of course this portrayal is an over-simplication of the popular view, but it’s not much of one. It is correct that much of mathematics is sequential and cumulative; many Algebra II topics, for example, cannot be understood without a reasonable grasp of Algebra I. And some topics are not sequential; many people realize that geometry and Algebra II can be learned in either order — after all, some schools do geometry first, some Algebra II. Many also realize that most of geometry could be left out of the sequence altogether, though that would be a pity since so many important lessons about spatial reasoning and logic in general are learned in a geometry course. But if your sole goal is to get to calculus, you could learn the geometry you need in a couple of weeks; why take a whole year?

Wherever you put geometry, a revised arrow is probably the mental model that most people believe in:

But this point of view is only slightly better than the first one. It’s still essentially a one-dimensional model, in which math has length but no depth or breadth.

Students who want to learn more math should be doing just that: learning more math, not moving faster and faster along the single-minded path to calculus. They should be exploring traditional topics in greater depth: there’s always more that’s worth learning about a topic, and there are always challenging problems that are worth attacking. And they should be exploring new topics that the majority of their classmates may never have the opportunity to learn: there are many more worthwhile topics than we can fit into the standard curriculum. So the true arrow of mathematics learning is three-dimensional. If I were more of an artist, I would represent this model effectively in a convincing picture. Maybe one of my students can draw one for me, but in the meantime this sketch will have to do:

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Thursday, July 28, 2005

It ain't fun if it's easy

An article in UMass Boston Magazine, Volume 9 Number 1, introduces Rick Jensen, the new head of the UMass Center for Environmental Health, Science, and Technology:
Roderick (Rick) Jensen sees puzzles everwhere. When his daughter told him in second grade that she didn’t like math because it was hard, the apoplectic father recovered enough to ask her if she liked crossword puzzles and chess. Yes, she said, but wasn’t sure why. “Maybe because they’re challenging?” he suggested. She got the message that math, like all puzzles, wouldn’t be fun if it were easy; she ultimately graduated from Princeton magna cum laude in physics.

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Wednesday, July 27, 2005

Super Size Me vs. Outfoxed

Having recently watched both Super Size Me and Outfoxed — well, only a bit of the latter — I was wondering why I had such different reactions to these two tendentious documentaries. Super Size Me held my attention and kept my interest throughout, even though it was repetitive and obvious. Outfoxed lost my attention and my interest early on, because it was so repetitive and obvious. I think the difference was that Super Size Me told a story: it featured a real character with a genuine conflict. Outfoxed (at least the part that I watched) merely provided tons of evidence that Fox News is anything but “fair and balanced” — more like a legal brief than a story.

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Tuesday, July 26, 2005

Simpson's Paradox

Thanks to Rob Campbell, one of the Harvard students who is working with me as a teaching assistant (“mentor”) in the Crimson Summer Academy, for pointing me to this wonderful example of Simpson’s Paradox: a statistics page at SUNY Oswego demonstrates a scatterplot with a surprisingly negative correlation between high-school SAT scores and college GPAs!

When the scores are disaggregated into the three distinct colleges whose results had been lumped together, we end up with three strong positive correlations instead of one negative one. It’s worth thinking about why the negative correlation appears when the scores are combined.

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Monday, July 25, 2005

Simpsons math

One of the more unusual websites for us math types, simpsonsmath.com, reveals quite a few mathematical connections on The Simpsons. Check it out!

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Sunday, July 24, 2005

Two wonderful mathematical puzzles

I’ve recently been stretching my skills with two new mathematical puzzles, Sudoku and Planarity.

Thanks to the Boston Globe and other papers, Sudoku has now become quite popular. Although the Sudoku page claims that it’s non-mathematical, these puzzles actually involve both mathematical logic and proof by contradiction. They also invite at least two different ways of thinking: either “Where does the 7 go?” or “Which numbers could fit in this cell?” I think the reason that these puzzles are deemed non-mathematical is that the numbers are used purely as glyphs, not for their numeric values; what the page really means is that the puzzles are non-arithmetic. If you haven’t tried sudoku yet, just go to the new Sidekick section in any daily edition of the Globe. It’s my theory, supported by a couple of weeks of data but not confirmed, that the Globe’s sudoku sequence follows Will Shortz’s system for the New York Times daily crossword puzzles: start with an easy one on Monday, then get harder and harder each day until Saturday, which features the most difficult puzzle of the week. Give it a try any day, but maybe you should start with a Monday.

Planarity is a graph theory puzzle. Great for honing those spatial skills as well as mathematical reasoning skills. Unfortunately the point system is time-dependent, so you’re penalized for stopping and thinking (unless you adroitly use the Pause button), but it’s still a wonderful puzzle and well worth trying. Your scores are remembered between sessions, so you can start wherever you left off.

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Saturday, July 23, 2005

Unitarians and Catholics

From the continuing saga of the Library Committee of the Dorchester Historical Society:

I come across an issue of the First Parish Church newsletter from 1956. One page includes an annotated reproduction of an ad that the Catholic Church had placed in the Boston Herald — annotations by the Unitarians, of course. The original ad contained seven photos, with these captions:
  • We browsed around.
  • I was impressed by Father Colleary.
  • I studied the catechism.
  • I prayed for guidance.
  • I made my first confession.
  • I received my first Communion.
  • And now we’ve found happiness in unity of faith.
The main annotation says, “Some Unitarians think there would be more than mild objection were this same kind of publicity published by us.” The seventh photo, which shows a couple walking away along Beacon hill, is annotated “The Path to Freedom,” with the observation that the building they are heading toward is the American Unitarian Assocation (now, of course, the Unitarian Universalist Association, still at 25 Beacon Street).

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Thursday, July 21, 2005

Sam Yoon

It’s a bit hard to believe, but Sam Yoon is the first Asian candidate ever to run for public office in the City of Boston. As an at-large candidate for City Council, he is making the rounds to try to get his name known. He’s going to need a lot of help; Barbara and I got to meet him this evening at a “meet and greet” event held by some friends of ours, and only about 25 people showed up. A former teacher, community activist, and promoter of quality affordable housing, Sam would be a welcome addition to the Council. (My words, though I admit they sound like a campaign commercial.)

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Saturday, July 09, 2005

Public school hours & college curricula

As chair of the Library Committee of the Dorchester Historical Society, I spend a few hours each month working with some other volunteers to catalog our collection of historic documents. Today I catalogued an immense 5-volume history of the local area, which was a true labor of love written from 1894 to 1897 by several members of the Dorchester Woman’s Club. (And I do mean “written”! All but four of the 75 chapters were handwritten in pen and ink; the other four were produced on a “new fangled writing machine,” as Mark Twain described the typewriter.)

Anyway, one of these chapters was an account of early Dorchester schoolmasters, who were, of course, overworked and underpaid. One thing that caught my eye was the description of the hours of elementary and high schools: six days a week, from 7 AM to 5 PM eight months a year, from 8 to 4 the other four months. Let’s see — that’s twelve months of school, isn’t it? Surely there was the occasional holiday or vacation — let’s guess six holidays plus four weeks of vacation — which seems to make an amazing 2632 hours per year! But wait...there was a two-hour lunch break each day, so it’s actually only 2068 hours. Let’s not hear any more complaining about the current 990-hour law, OK?

And surely the top students wanted to continue their education at the newly founded Harvard College. The Dorchester women of the end of the 19th Century described the required Harvard undergraduate curriculum of the 17th as consisting of “Latin, Greek, Syriac, more theology than is taught at Divinity School, a large amount of Hebrew, and a mere bit of mathematics, science, philosophy, and history.”

To conclude with a minor matter of linguistic interest: a Google search for “Women’s Club” yields four times as many hits as “Woman’s Club”.

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Friday, July 08, 2005

Infinite pizzas

An article in this week’s Somerville Journal gives free publicity to a pizza joint in Ball Square, the Urban Gourmet:
...We offer about three dozen toppings with an infinite variety of combinations. ..
Wow!

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Thursday, July 07, 2005

Numb3rs So Far

Now that the first season of Numb3rs is in reruns, I’m finally getting around to watching this show and have seen three episodes at this point. So far, so good. I haven’t yet seen the episode referred to in the review that I cite in my post of June 7, but nothing yet seems to stretch the suspension of disbelief beyond a reasonable point. Most important, from my POV as a math teacher, is their portrayal of real mathematics, at high-school and college level, as something with genuine applicability. This is a revolutionary step for a commercial television network; even on PBS there are almost no mathematical applications that go beyond middle-school math. Putting it in a context that reminds one of Law & Order is a sure way to grab viewers’ attention.

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Tuesday, July 05, 2005

What's wrong with UMass?

Many — probably most — of my Weston students look down on UMass Amherst. Certainly it’s not the choice of our most successful students. I heard one junior disparaging it to the extent that he would consider his life ruined if he had to go there. Maybe they would change their mind, or at least reconsider, if they read the bio of one of my former Lincoln-Sudbury students. Here are some excerpts:

  • B.S. Physics, University of Massachusetts, Amherst (9/84)

  • M.S. Computer and Information Science, University of Massachusetts (5/87)

  • Ph.D. Computer Science, Swiss Federal Institute of Technology, Lausanne. (7/93)
After obtaining his Doctorate in 1993, he moved back to the States where he worked as Principal Software Engineer for Amerinex A.I. in Amherst Massachusetts. In 1994 he joined the faculty in the Computer Science and Electrical Engineering Department at [University of Maryland Baltimore County]. After working two years for Wigitek Corporation as Vice President of Research and Development, he joined Celera Genomics, of Rockville, Maryland.

Dr. Turner also conducted research in the areas of physics-based character animation and object-oriented software architectures for 3D interaction. He was recipient of a 3-year CAREER award from the NSF for development of the Metis toolkit, and has sponsored several Research Assistants, a Visiting Scholar and a Research Associate working in the areas of virtual reality and interactive physics-based animation.

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Monday, July 04, 2005

Such a well-qualified ambassador

It’s nice to know that our president appoints such well-informed ambassadors, isn’t it? Here is an excerpt from a CBC interview with our new ambassador to Canada, David Wilkins:
Q: Have you ever been to Canada before?

A: Ah, many years ago when I was in the Army stationed in Indiana my wife and I visited Canada.

Q: Oh yes, where did you go?

A: Eh, it was, uh, around the uh, the falls area, Niagara Falls, back up in there round uh that area as well as uh going I guess back toward, back West toward, toward Indiana, well obviously above Indiana but I’d have to get out a map to tell you all, it’s been thirty-something years now since we were there but we enjoyed our visit and we cannot wait to get back.
I heard some of this yesterday on NPR’s “Wait, wait...don’t tell me!” For more info, see Voice in the Wilderness.

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Neurological benefits of blogging?

Neurologists Fernette and Brock Eide analyze the Brain of the Blogger. Here are some of their claims:
...our mental activities actually cause changes in the structures of our brains — not only what we think, but how we think as well...

1. Blogs can promote critical and analytical thinking.
...
2. Blogging can be a powerful promoter of creative, intuitive, and associational thinking.
...
3. Blogs promote analogical thinking.
...
4. Blogging is a powerful medium for increasing access and exposure to quality information.
...
5. Blogging combines the best of solitary reflection and social interaction.
...

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Sunday, July 03, 2005

Blogs in the math classroom?

Blogs are public writing. We want our students to publish their work — to take ownership of it, to take personal responsibility for it, to take pride in it. These goals apply not only to English class but also in less obvious contexts, such as math. Check out Writing in the Mathematics Classroom within the Future of Math website. Other than being a reader, I have no connection with this site, which is aimed at elementary and middle schools more than high schools but clearly has a lot of helpful ideas and links for high-school math teachers. If you suspect that blogs might be a useful tool in the classroom, visit this site.

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Saturday, July 02, 2005

Radians

From Trigonometry for Dummies, by Mary Jane Sterling:
Measuring angles in degrees is easier, but measuring angles in radians is preferable when doing computations. The radian is more exact because the radius, circumference, or area of the circle is involved.
Huh?

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Adequate Yearly Progress

A Boston Globe article on October 3, 2004, included Weston High School among the schools cited by the Department of Education for failure to make Adequate Yearly Progress on the No Child Too Far Ahead Act. Sorry, I mean the No Child Left Behind Act. That article got quite a bit of publicity; schadenfreude is always a good way to sell papers.

And a much more recent Boston Herald article (June 24, 2005) said this:
Weston High School topped Boston Magazine’s list of best public high schools last year, and students at Ephraim Curtis Middle School in Sudbury were among the state’s highest scorers on the MCAS.

Yet both schools were labeled as “failing to make adequate yearly progress (AYP)” and face penalties under federal and state No Child Left Behind laws, highlighting a growing problem with classifications that don’t mirror the real quality of kids’ education or performance, state educators and parents say.

“There is something wrong with this picture. It doesn’t make sense,” said Ed Moscovitch of Cape Ann Economics, joining members of a coalition of educators, school administrators and parents in releasing a report based on his finding that 75 percent of the state’s schools will fail to make AYP for two years or more over the next decade.
And then...oops...it turns out that the DOE had made a statistical error. We’re making Adequate Yearly Progress after all. The Herald didn’t know this because there had been no publicity for the correction, of course.

None of this eliminates the Achievement Gap, nor the very real need to do something about it.

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Friday, July 01, 2005

The Achievement Gap

Yesterday I wrote about the pressure to take honors math, but I didn’t even consider the reasons for this pressure. The implied one — which I tried to debunk — was that honors math is necessary in order to get into a good college. But I learned a lot of other reasons today, while participating in a workshop on the Achievement Gap. Some of these were legitimate, while others were based on false premises or illogical reasoning. It’s worth going through this exercise: if you’re a high-school student, or the parent of a high-school student, what do you think are the reasons to take honors math? And which of these stand up under close scrutiny?

Oh... so you wonder what the connection with the achivement gap might be. Whatever the reasons for taking honors math might be, it’s a cause for concern if certain identifiable groups are significantly under-represented in honors math classes. I teach both college-prep and honors math curses, and of course both levels contain a mixture of different ethnicities; you can find an Asian kid in college-prep Algebra II and a black kid in BC Calculus. But... the ratios are disproportionate. The official statistics are misleading, since the DOE, unlike the Census, places each of our many mixed-race students into one bin or another. But anyway, those official statistics say that Weston High School is 5.3% African-American/Black, 12.9% Asian, 2.1% Hispanic/Latino, 0.3% Native American, and 79.5% White. There is no official record of the number of Jewish students among the white students, but my rough count is that 20-22% of Weston High School students are Jewish. Let’s call it 21%, which would leave 58.5% in the white non-Jewish category. (A very surprising statistic, by the way, to the many people who estimate that 95% are in that category! They have a very out-of-date view of Weston.)

When we were discussing a previous Achivement Gap workshop in one of my honors classes, one girl suddenly looked around the room and exclaimed, “Oh my God! Everyone here is either Jewish or Asian!” It turned out, of course, to be untrue: we actually found three students who were neither. (Out of a total of 24.) Nevertheless, these two minority groups made up 88% of this particular section but only 34% of the school population as a whole. (Most honors classes, by the way, aren’t quite so heavily Jewish and Asian. For whatever reason, this particular section was somewhat atypical.) There are cultural reasons — partly stereotypical, but partly true — why Jews and Asians are statistically more likely than other minority groups to choose honors math classes. But what can we do to raise the number of black and Hispanic students who feel comfortable making the same choice — and are prepared to thrive in this context?

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Misanalyzing textbooks

The Washington Monthly publishes an interesting example of the lies about incorrect analysis of math textbooks by right-wing zealots. Not that I’m a fan of very many textbooks, but a little truth would help.

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