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Tuesday, May 31, 2005

Building learning communities

A conference on Building Learning Communities — right here in Weston! I don’t know much about it, but it’s led by Wellesley’s distinguished former tech coordinator, Alan November, and the blurb looks interesting. Stay tuned for more info...

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Chris Lydon redux

Yesterday Christopher Lydon returned to NPR with his new show/blog called Open Source. Day #2 concluded a few minutes ago.

Actually, of course, it was only the broadcast portion that concluded a few minutes ago. The Internet portion — the blog — never ends.

Lydon observes that “a blog is a way of thinking. It’s a way of recording the argument you are having with yourself, admitting that you may be about to be wrong, and ensuring that, when you are wrong, you make your wrongness available as a public record. A blog is Socratic that way; it knows that it does not know.”


Monday, May 30, 2005

An argument from continuity

Two sophomores approached my colleague Josh with a question: “How can we construct a fair 5-sided die?”

Josh posed a prior question: Is it even possible to construct such a die? He fashioned an interesting argument from continuity: Consider two square pyramids with the same height — one with a tiny base (much smaller than the lateral faces) and one with a very large base (much larger than the lateral faces). Clearly the first has a very small chance of landing on its base and the second has a very large chance of doing so. By gradually moving from the one to the other, there must be an intermediate point that makes the pyramid a fair die.

Josh used this teachable moment to explain the difference between an existence proof and a constructive proof. We now know that a fair 5-sided die is possible, but we have no idea how to construct one!

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Intel, security, and Apple

Paul Otellini, the new CEO of Intel, says that he “spends an hour a weekend removing spyware from his daughter’s computer,” according to a Wall Street Journal article about its recent All Things Digital conference. When asked whether a computer user should therefore buy a Mac instead of a PC, he seemed to agree with that idea: “If you want to fix it tomorrow, maybe you should buy something else.”

And security expert Winn Schwartau writes an interesting column entitled “Mad as hell, switching to Mac.” A few brief excerpts:
...the WinTel platform represents the greatest violation of the basic tenets of information security and has become a national economic security risk. I do not say this lightly, and I have never been a Microsoft basher, either. I never criticize a company without a fair bit of explanation, justification and supportive evidence.

I have come to the belief that there is a much easier, more secure way to use computers...

My company has given up on WinTel. We have successfully moved to Mac in less than two days. Think about it: a security-friendly alternative that works and doesn’t require gobs of third-party utilities to safely perform the most mundane tasks. Please follow the details of our experiment at www.securityawareness.blogspot.com.

Too bad the column’s in Macworld. He’s just preaching to the choir.


The new SAT

An interesting column by Mark Franek concerning the writing section of the new SAT includes the following observation:
The writing section is entirely new — 70 percent of it is composed of pesky multiple-choice grammatical questions (where students aren’t writing anything — they’re blacking in ovals), while the final 30 percent is reserved for a persuasive essay that our teenagers are supposed to draft and complete in 25 minutes...

The idea that during the writing of this blitzkrieg essay... “you should take care to develop your point of view, present your ideas logically and clearly, and use language precisely" in under half an hour and under extreme pressure is ridiculous.

The inside quotation comes verbatim from the official SAT prep book.


Sunday, May 29, 2005

The view from college math

Rudbeckia Hirta (a clever pseudonym for a math professor who carefully keeps her true identity hidden) observes:
Due to reasons beyond my understanding, high school math and college math are completely unaligned. The K-12 system sends us students whose knowledge is a mile wide and an inch deep: we get students who are shaky at algebra, frightened of fractions, and unsure of how to find the areas of basic plane figures (and completely unable to accept the idea that it is a reasonable request to ask them to solve non-standard problems where the method of solution is not immediately obvious), but they have been exposed to matrix arithmetic, computations from polynomial calculus and other supposedly “advanced” procedures. You would think that the “college prep” track would prepare students for college, but it doesn’t.

Among the comments to this post is the following:
The problems are many, I think. They range from schools wanting to look good by having X students in Calculus or X students in Algebra in 7th grade or 8th grade. Also, much of the focus in most high school classes is on the use of formulas. If you can plug and chug, you are good to go. There are so few non-standard questions posed to the students that they can't even hope to understand that they need to be able to solve them. (This is not 100% true, but is true a lot!) It’s kind of a broken system in which everyone thinks they’re doing the right thing, but in the long run, they may not be.

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William of Orange

The newest member of our family:

William of Orange

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Friday, May 27, 2005

Where visuals fail

Some of us couldn’t possibly forget the 1969 draft lottery, the new and supposedly “fair” system to pick who was going to be sent to Vietnam. My Algebra 2 class is studying probability and was remarkably interested in learning about the lottery. Everyone wanted to find out what “their” number would have been — even the girls, who of course wouldn’t have been called anyway. They were particularly interested in whether their number was less than 196, since those born on the first 195 birthdays that were “randomly” selected ended up being drafted. After describing the system and pulling some sample birthdays, I showed the class a scatterplot of the numbers drawn for each date of birth:

At first glance this distribution looks fair. In fact, it looks totally random. At second glance, some see a tiny bit of a pattern, but it still looks quite random.

So then we looked at the numbers. Taking the two extremes, we calculated that the probability that a young man born in January would be drafted was only 14/31, or 45%, but the probability that one born in December would be drafted was a shocking 25/31, or 81%.

Usually a picture is worth 1000 numbers, but this is a dramatically counterintuitive example where plain numbers have more impact than a scatterplot.

By the way, my number was 205.

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Thursday, May 26, 2005

Don’t do this

From this morning’s Boston Globe:
A public school teacher fed up with his students’ behavior found a way to berate them in the context of a class assignment. The Jefferson Parish teacher wrote and distributed a two-page essay to his fourth-grade students saying he hated his job, blasting the children’s “animal” behavior, and even identifying some of them by name.
“I’m thinking, you shouldn’t be a teacher if you hate kids,” [said one parent].


Wednesday, May 25, 2005

Do department names matter?

As in most high schools, computer science courses at Weston High School are taught under the aegis of the Math Department. (One of the first exceptions to this rule was Lincoln-Sudbury Regional High School, where we spun off a separate computer department way back in 1979.) Should the department name be changed to Mathematics & Computer Science, just as the History department at Weston is now History/Social Studies?

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High-school math in the 21st century?

What math applications are taught in high school? Principally parabolic arches and widget-manufacturing, of course. This is a bit of an exaggeration, but the principle holds. We teach applications like those rather than cryptography and models of voting. At the Crimson Summer Academy (see my post on 5/23) we have the flexibility to teach the latter two topics, which are both mathematically rich and directly useful. But even at Weston High School, which is generally on the leading edge of high-school math, cryptography is limited to honors students and voting models aren’t taught at all. These, of course, are just examples, but most of the topics presented in COMAP’s For All Practical Purposes strike me as more useful for high school students than most of the so-called applications we currently teach. (The book itself is aimed at college undergraduates and is therefore more appropriate as a resource than as a textbook in high school.)

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Tuesday, May 24, 2005

Martin Gardner for pennies

The Mathematical Association of America is selling a CD containing 15 Martin Gardner books (the entire collection of his Scientific American columns) for a mere $55.95 — or $44.95 if you’re a member!

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Where are the girls?

Why do so few girls sign up for computer programming courses in high school?

High school may be too late. The problem might be starting much earlier. Even in fourth grade at The Saturday Course (see my post on 5/21), the enrollment in Power Programming is overwhelmingly male. One of my colleagues hypothesizes that the problem lies in the course name. Would a different name (such as Build Your Own Computer Game) boost female enrollment?


Math license plates

Winners and other entries in the Math License Plate contest.


Monday, May 23, 2005

Harvard does a good deed

In addition to my day job at Weston High School — and my Saturday job at The Saturday Course — I teach during the summer in an extraordinary program known as the Crimson Summer Academy at Harvard University. In its second year now, this program provides an intensive college-preparatory opportunity for public-school students from Boston and Cambridge. The students, known as Crimson Scholars, make a three-year commitment for the summers that precede their sophomore, junior, and senior years in high school, along with extensive continuity throughout the academic year after each summer. Each Crimson Scholar receives a significant stipend, free room and board, an iBook to keep, and a digital camera and graphing calculator for use during the summer. What’s even more important is that a large number of Harvard students are hired to act as mentors to the Crimson Scholars during the summer and continuing through the year. A typical classroom contains 15 kids, one master teacher, and two mentors — quite a ratio! My job is to develop the curriculum for the Quantitative Reasoning (QR) course and to serve as one of three QR teachers.

The Crimson Scholars take a wide range of courses, including not only QR but also Writing, Science, Photography, Public Speaking, etc. Everyone makes extensive use of their iBooks both inside and out of class. The QR, Science, and Photo classes incorporate a rich technological component; in addition to the obvious software such as Word, Safari, Excel, and Photoshop, students also learn Tinderbox, which Eastgate Systems generously donated to the CSA.

Yesterday I attended the CSA annual reception. which featured presentations in English, Spanish, Chinese, and Haitian Creole. (We have quite a diverse group.) We learned that the program has had an astonishing 100% yield and 100% retention rate: every student who was admitted a year ago accepted the offer, attended last summer, and is returning this summer! Our diverse community has come together as a family... or is it more like a team? That was the issue discussed recently on the CSA electronic bulletin board. It's pleasant to have such an issue!

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Sunday, May 22, 2005

A presentation in 10 minutes or less?

Dennis and Jim and I have been invited to present a paper at the TeachScheme Tenth Anniversary Workshop in Providence on June 11. At first we were allocated ten minutes! Not ten minutes apiece, but ten minutes total.

But wait, a slight revision has arrived in the mail sent to all presenters:
“If you are assigned 10 minutes, you should plan on speaking for six to eight minutes, leaving some time for the audience to ask questions.”
So let’s see...that’s two minutes apiece, or possibly up to two minutes and 40 seconds apiece.

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How to approach a probability problem

If you are dealt a hand of five cards, what is the probability that you have three diamonds and two clubs?

There are (at least) two different approaches to this kind of problem:
  1. The chance the first card is a diamond is 13/52, then the next is 12/51, then 11/50; there are still 13 clubs, so we have 13/49 and then 12/48. Multiply these all together. But since the problem didn’t say that it had to be the first three that were diamonds, we have to multiply by 5C3 since there are that many triples of slots for the diamonds, and then by 5C2 since there are that many pairs of slots for the clubs. I get 0.0085834.

  2. The definition of probability says that I should count the number of ways to get the 5 specified cards and divide it by the number of combinations of 5 cards in the whole deck. So, I need 3 of the 13 diamonds (13C3) and 2 of the 3 clubs (13C2). Divide them by 52C5 and I get 0.0085834.

Years ago I approached problems like this with the first method. Now I use the second, which seems simpler to me. But one of my esteemed colleagues claims that the first method is actually simpler, and that students understand it better.

Now I suppose the truth must be that some students understand one method better, and some understand the other one better. You could argue that that means that we should teach both, but that’s just going to confuse all the weaker students. What should we do?


Saturday, May 21, 2005

Learning in fourth grade — on Saturdays

For about half of the Saturdays each year, I teach in a wonderful program called The Saturday Course. This is an enrichment program for gifted and talented public-school and parochial-school students in grades four through six. Small classes, dedicated faculty, motivated kids — what’s not to love? They come for five or six consecutive Saturdays, taking two courses that meet for approximately 75 minutes each. So each kid gets at most seven and a half hours of math, or computers, or drama, or art, or whatever. You wouldn’t think that would be enough time to get much done, would you? But it definitely is.

Today was the last Saturday of the current session. My Extreme Math class finished up with an investigation of the Tower of Hanoi, having earlier explored permutations, Fibonacci numbers, Pascal’s Triangle, and the Chaos Game. Of course these kids don’t know algebra yet, but when you compare their inquiries into discrete math with the inquiries of 10th- and 11th-graders, it’s remarkable how the similarities outweigh the differences.

Favorite interchange from today’s math class:
“My mother taught this to me, but I don’t understand it,” explained one student who claimed (correctly) that there must be eight odd numbers in the 100th row of Pascal’s Triangle.

“If you don’t understand it, your mother didn’t really teach it to you,” said the classmate in the next seat.

In my other course, Power Programming, kids use Microworlds EX to design and build their own computer games. Seven and a half hours isn’t enough time to learn the fundamentals of Logo, get comfortable with the Microworlds environment, design a game, and create the game — it can’t be done in seven and a half hours, even if you’re in fourth grade — but most kids accomplish an amazing amount even if they need to take their work-in-progress home to complete it. At the end of each Saturday’s program, five or six classes share their work with the entire ensemble of 160 kids and their parents and teachers. Power Programming was one of the classes that shared today. Unfortunately, three of my students had to leave before sharing. But their games came up on the big screen anyway (projected from one of those wonderful ceiling projectors), and who was going to explain them? Amazingly, and totally without prompting or rehearsing, in each case a different classmate stepped forward to volunteer to explain the game. This was especially heartening because students so often view computer programming as a solitary activity and have no idea what their classmates are doing.


Friday, May 20, 2005

Risks and probabilities

We know that both adults and kids are notoriously bad at estimating probabilities. Bruce Schneier (one of the world’s leading experts on security, cryptography, etc.) has this to say concerning risks and probabilities:
One of the things I routinely tell people is that if it’s in the news, don’t worry about it. By definition, “news” means that it hardly ever happens. If a risk is in the news, then it’s probably not worth worrying about. When something is no longer reported — automobile deaths, domestic violence — when it’s so common that it’s not news, then you should start worrying.

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Thursday, May 19, 2005

Kids can’t concentrate? Don’t believe it!

Most exciting event of the week: I’ve been sitting here proctoring MCAS for the past two and a half hours.

Actually, it isn’t exciting (surprise, surprise). But I’ve gotten quite a lot of work done. I don’t think I’m allowed to quote any specific questions on this tenth-grade math non-calculator session, but I can definitely make one observation: adults who say, “Kids today just can’t concentrate; they all have short attention spans,” should observe these students’ work on their MCAS. I have been proctoring a group of 21 sophomores. After 60 minutes they were allowed to leave; three did so. Now, after 150 minutes, six of the original 21 are still working. They’ve been focused and attentive the whole time — and on a math test no less! Who says that kids today just can’t concentrate?

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Wednesday, May 18, 2005

Why start now? First thoughts...

For a year now I’ve been thinking of starting a blog. But there was never a reason to start it today. Tomorrow would do. Or the next day. Or the next day.

So why start one now? The proximate reason was that one of my high-school students decided to write her report about a field trip by blogging it rather than by emailing the report to the teacher who led the field trip. So I had to read her blog, which of course linked to other blogs, and so on and so forth. Several dozen Weston High School students turned out to belong to a webring on Xanga, and of course there are many who keep blogs elsewere (or who have Xanga blogs but don’t list themselves in the webring). And this got me thinking: why haven’t I gotten around to writing my own blog?

It also got me thinking of a number of concerns about blogs written by kids — even kids who are almost old enough to vote. First names are fine, but why do so many of them include their full names, and often even a photo?

Maybe this is an inappropriate worry. After all, the Weston Town Crier — like every local newspaper — includes plenty of pictures of minors, along with their full names.

But my worry isn’t really about predators. It’s a concern that teenagers who publish something for the whole world to read (after all, it’s the World Wide Web) might regret it when they realize that not only their friends but also their teachers, their future employers, and even their parents can read what they wrote. It’s not a private diary, after all. Full names and photos in local papers are almost always there to celebrate achievements in athletics, drama, community service, and the like. They don’t expose a kid’s thoughts — which are barely edited, if at all — to any casual reader.

It’s worth reading some thoughts about this subject, albeit in the context of college students, in Jill Walker’s blog.

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