<body><script type="text/javascript"> function setAttributeOnload(object, attribute, val) { if(window.addEventListener) { window.addEventListener('load', function(){ object[attribute] = val; }, false); } else { window.attachEvent('onload', function(){ object[attribute] = val; }); } } </script> <div id="navbar-iframe-container"></div> <script type="text/javascript" src="https://apis.google.com/js/plusone.js"></script> <script type="text/javascript"> gapi.load("gapi.iframes:gapi.iframes.style.bubble", function() { if (gapi.iframes && gapi.iframes.getContext) { gapi.iframes.getContext().openChild({ url: 'https://www.blogger.com/navbar.g?targetBlogID\x3d12969692\x26blogName\x3dLearning+Strategies\x26publishMode\x3dPUBLISH_MODE_BLOGSPOT\x26navbarType\x3dBLUE\x26layoutType\x3dCLASSIC\x26searchRoot\x3dhttp://larrydavidson.blogspot.com/search\x26blogLocale\x3den_US\x26v\x3d2\x26homepageUrl\x3dhttp://larrydavidson.blogspot.com/\x26vt\x3d53093167121198245', where: document.getElementById("navbar-iframe-container"), id: "navbar-iframe" }); } }); </script>

Wednesday, August 31, 2005

Socially sensitive math?

In our opening Math Department meeting, we all participated in the following activity. First we drew a two-set Venn Diagram, where one circle would contain everyone who was an oldest child in the family and one would contain everyone who was the youngest. We quickly agreed that if you were an only child you would be in the intersection of the sets, and if you were a middle child you would be in the fourth region, outside the two given sets. All was well.

There were several goals to this activity, such as thinking about representation as a “big idea” in algebra and sharing information about ourselves at an opening meeting. In service of the latter goal, we each put ourselves in the appropriate region and described our siblings’ ages and occupations.

But all was not well after all. There were issues. What about half- and step-siblings, for example? Do you straddle the line between two regions? And what about someone who grew up as an only child until he was in college, when his parents adopted his younger sister? (In terms of typical birth-order issues he was an only child for a long time, before becoming the oldest child.) And what about people who just didn’t want to talk about their siblings, or perhaps didn’t even want to acknowledge their existence? Probably all these questions could be discussed amicably and unemotionally in a group of collegial adults, but do these issues rule out using the activity with a group of teenagers? Should math class be a refuge from awkward topics that some kids don’t want to discuss?

Labels: ,

Tuesday, August 30, 2005


Black-eyed Susan reports on the correlation between homework and quiz scores:
So far in calculus, there have been two homework assignments and two quizzes.

The students who have turned in no homework have a quiz average of 52%.

The students who have turned in one assignment have a quiz average of 71%.

The students who have turned in both assignments have a quiz average of 86%.
She teaches in the South, in case you’re wondering how she could have given two quizzes already when it’s still August.


Monday, August 29, 2005

Late to class

“Fed up with students routinely strolling into class well after the bell rings, high school principals across the region plan to crack down on excessive tardiness,” according to an article in today’s Boston Globe.

It’s not clear to me that this is really a new problem. Were there really significantly more students coming late to class last year than in previous years? Weston certainly has a real problem with lateness, especially in first-period classes, and especially with second-semester seniors, but I don’t think it was less severe in the past — unless we put on our rose-colored glasses.

More troubling is the list of proposed remedies, which range from pointless to ineffective. Detention never works.
At Arlington High School, teachers will more uniformly enforce the tardiness policy this year, and will be trained in November on how to talk to students about taking responsibility.

Tardy students may get extra assignments, face phone calls home, or serve detention after school, said the principal, Charles Skidmore. He voiced skepticism, however, that the penalties would be much of a help.

“There are some kids who have the attitude that whether it’s class or parties or school activities, that it begins when I get here. Kids nowadays see time as very fluid and flexible,” Skidmore said.
Extra assignments ? I thought that people realized long ago that it’s counterproductive to make students view homework as punishment. Anyway, Skidmore is right that it won’t help.

I don’t like the rewards approach either: at Needham High School avery student who shows up on time for every first-semester class will get a day off second semester. Then I suppose it’s the teacher’s responsibility to help the student catch up on the work missed during the day off.

The so-called research is also questionable. “Consistent tardiness is often a warning sign of bigger problems, education researchers say. Truant students are two to eight times more likely than others to become juvenile delinquents.” But surely this is mixing up cause and effect.

Labels: ,

Saturday, August 27, 2005

How to describe a circle

MoebiusStripper writes about MathPower 12, an all-too-popular popular high-school mathematics textbook published by McGraw-Hill. In case the student doesn’t already know what a circle is, the text provides the following explanation:
The compact disc player is everywhere these days. Developed initially by Philips and Sony, it first came on the market in 1983. By 1986, over one million CD players were being sold each year. Because of the low-cost laser components, the CD player has become one of the most successful electronic devices to date.

If you trace around the outside of a CD, the result is a circle.

A circle is the set or locus of all points in a plane which are equidistant from a fixed point. This fixed point is called the centre. The distance from this centre to any point on the circle is called the radius.

There could be a lot of interesting problems about CDs, but no: this explanation serves purely to “motivate” the definitions of circle, center, and radius.

Labels: ,

Wednesday, August 17, 2005

Misunderstanding the "Law of Averages"

An article in this morning’s Boston Globe begins like this:
A Fung Wah bus, part of the low-fare passenger line fleet from Boston to New York, erupted in flames on an interstate highway in Connecticut yesterday, sending frightened passengers scrambling off the bus just moments before it became a “charred mess,” police and passengers said.
No one could have calculated the probability that this would happen. But...

The reporters interview “John Quackenbush, 38, a freelance film technician from Cambridge”:
Quackenbush, who uses the bus to commute between work locations, said this will not deter him from taking the bus again. In fact, it will have just the opposite effect, he said.

“What are the odds of this happening again?” he said. “Now I’m safe.”


Monday, August 08, 2005

Standardized tests

In yesterday’s Boston Globe there’s an interview with Bob Sternberg, psychology professor at Yale, president of the American Psychological Association, and newly appointed dean of Arts and Sciences at Tufts. Globe correspondent Peter DeMarco asked him about the use of standardized IQ tests. Sternberg’s reply:
If you grow up in...Weston..., for most of the kids the tests are fairly good measures of the analytical part of intelligence. If you grow up in Roxbury, chances are it’s not going to tell you the same things as it does a kid from Weston. And the reason is that kids grow up with different challenges...
Having taught kids from Weston as well as kids from Roxbury, I had mixed feelings about this claim — until I read I again. When you don’t read it carefully, you realize that it’s a gross over-generalization that contains a kernel of truth. But when you pay attention to Sternberg’s qualifiers — notice “most of the kids” and “chances are” — you realize that his analysis is correct.

Labels: ,

Saturday, August 06, 2005

Sudoku revisited

I now think my theory about Sudoku in the Globe is wrong, or at least needs to be tweaked: IMHO yesterday’s puzzle was a lot more difficult than today’s.


Thursday, August 04, 2005


My current favorite resource on the Internet is the Wikipedia. Considering that anybody can write and edit its entries, I am astonished that this enormous site could be not only so comprehensive but also so reliable. Of course it contains errors, but I haven’t discovered very many yet.

Labels: ,

Wednesday, August 03, 2005


I’m currently reading Quicksilver (William Morrow, Sept. 2003), by Neal Stephenson, previously author of The Diamond Age, Snow Crash, and Cryptonomicon, the last of which I should probably add to my list of favorite books. Quicksilver totals a mere 960 pages, in contrast to Cryptonomicon, which takes a hefty 1168. These novels are primarily historical fiction, with a heavy admixture of science fiction. The stories and some of the characters overlap somewhat — not only in these two books but more so between Quicksilver and two others that I haven’t mentioned yet: Quicksilver is actually only the first novel in the three-volume Baroque Cycle, which also includes The Confusion (William Morrow, April 2004) and The System of the World (William Morrow, Sept. 2004). It’s a “cycle,” but as Stephenson says, “I know everyone’s going to call it a trilogy anyway.”

Quicksilver — at least in the first hundred pages — alternates between mid-seventeenth-century England and early-eighteenth-century Massachusetts. Gottfried Leibniz and Isaac Newton are two of the many characters in it, with a supporting cast that intermingles historical figures and fictional ones. The historical content focuses on the Puritans (on both sides of the Atlantic), the Restoration, the bubonic plague of 1665, conflicts with the Dutch and the French, the tremendous scientific ferment of the era, and other important themes of the times, but of course I’ve been drawn as well to the history of mathematics that also permeates the novel. As with all historical fiction, the reader has to realize that Stephenson intermingles truth and fiction; the following paragraph about the young Newton as an undergraduate provides a lovely example that would capture the attention of any math educator [italics and punctuation as in the original]:
Isaac hadn’t studied Euclid that much, and hadn’t cared enough to study him well. If he wanted to work with a curve he would instinctively write it down, not as an intersection of planes and cones, but as a series of numbers and letters: an algebraic expression. That only worked if there was a language, or at least an alphabet, that had the power of expressing shapes without literally depicting them, a problem that Monsieur Descartes had lately solved by (first) conceiving of curves, lines, et cetera, as being collections of individual points and (then) devising a way to express a point by giving its coordinates — two numbers, or letters representing numbers, or (best of all) algebraic expressions that could in principle be evaluated to generate numbers. This translated all geometry to a new language with its own set of rules: algebra. The construction of equations was an exercise in translation. By following those rules, one could create new statements that were true, without even having to think about what the symbols referred to in any physical universe. It was this seemingly occult power that scared the hell out of some Puritans at the time, and even seemed to scare Isaac a bit.



This page is powered by Blogger. Isn't yours? Made with Macintosh