Sunday, July 31, 2005
Doing something about stress
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.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.
“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.”
So what is Weston going to do to about these two groups?
Saturday, July 30, 2005
Math for democracy
[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!
Friday, July 29, 2005
Straight as an arrow
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:
Thursday, July 28, 2005
It ain't fun if it's easy
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.
Labels: teaching and learning
Wednesday, July 27, 2005
Super Size Me vs. Outfoxed
Tuesday, July 26, 2005
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.
Monday, July 25, 2005
Sunday, July 24, 2005
Two wonderful mathematical puzzles
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.
Saturday, July 23, 2005
Unitarians and Catholics
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.
Thursday, July 21, 2005
Saturday, July 09, 2005
Public school hours & college curricula
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”.
Labels: teaching and learning
Friday, July 08, 2005
Thursday, July 07, 2005
Numb3rs So Far
Tuesday, July 05, 2005
What's wrong with UMass?
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.
- 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)
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.
Monday, July 04, 2005
Such a well-qualified ambassador
Q: Have you ever been to Canada before?I heard some of this yesterday on NPR’s “Wait, wait...don’t tell me!” For more info, see Voice in the Wilderness.
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.
Neurological benefits of blogging?
...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.
Sunday, July 03, 2005
Blogs in the math classroom?
Saturday, July 02, 2005
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?
Adequate Yearly Progress
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.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.
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.
None of this eliminates the Achievement Gap, nor the very real need to do something about it.
Friday, July 01, 2005
The Achievement Gap
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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