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Wednesday, November 30, 2005

Grading on a curve

In her latest post on Learning Curves, Rudbeckia Hirta describes two methods of grading:
Around here there are two schools of thought for grading calculus classes: straight percentages or curving the grades. I favor the former with each letter grade representing ten percentage points. In my class, 90%+ is an A, at least 80% but less than 90% is a B, etc.

I had been arguing with a colleague about which system is better, mine (percentages) or his (curve). The way he assigns grades is to set the mean to the boundary between B and C and then to allow one letter grade per standard deviation. We have been arguing about this all semester. I am convinced that my way is better.
I think they’re both wrong, though her way is better than his. Grading on a curve is destructive and counter-productive. It sets students up in competition with each other, it discourages cooperative study groups (which are known to be exceptionally effective tools for learning mathematics), and it rewards an entire class for not studying. We’ve all had the experience of teaching such a class, just as we’ve all had the experience of teaching a class where almost everyone is motivated and successful. Grades shouldn’t be adjusted up for the first class and down for the second, which is what a curve does. If anything, an inverse curve would make more sense: boost the grades for the class that is working hard, lower them for the class that isn’t.

The counter-argument is usually that a test might turn out to be too long or too difficult, and you don’t want to penalize the students for the teacher’s mistake in judgment. Indeed we do often misjudge length or difficulty, and some compensation is appropriate. But I don’t think a curve is the right solution, since all the objections in the previous paragraph still apply. The correct solution is... well, let’s first look at why I think Hirta’s method of grading is wrong (even if better than grading on a curve), and then we’ll see a common solution to both problems.

Straight percentage grading sounds fair and equitable. It doesn’t have all those negative effects of grading on a curve. But it also assumes that the teacher has somehow managed to control the level of difficulty of the test so precisely that an 80% really does reflect the lowest level of competence (assuming that an A represents excellence, a B competence, and so forth). It discourages writing tests that contain truly challenging problems, since the grades would then be artificially low. It also makes it difficult to decide how to assign points for partial credit, since three points on a four-point problem is automatically below B-level.

The solution to both grading schemes is to adopt a third alternative, which is what I’ve been doing for the past six years. After a test is written, I make a professional judgment about its level of difficulty and I assess the likely cutoff point between excellent and competent work (the lowest A), as well as the likely cutoff point between competent and merely satisfactory work (the lowest B). These serve as anchors for a tentative scale. I then correct the tests, assigning numeric point values only, following a system learned in an Annenberg workshop: student work on a typical 4-point problem is assigned a 4 if it’s perfect or nearly so, a 3 if there’s an error or omission that’s small enough to be explained in a sticky note, a 2 if the student has demonstrated knowledge of how to solve a problem but has not successfully solved it, and a 1 if the student has demonstrated understanding of the problem but not knowledge of how to solve it. Note that a 3 on every problem would result in a 75% total score, so I usually aim for 75% as the lowest B.

But before I commit myself to the tentative scale, I look at a few sample papers that hover on either side of the two cutoffs described above, and I do the same for a proposed D/F boundary. This all leads me to a scale where grades truly convey what they’re supposed to convey, where hard work and cooperation are rewarded, and where a class that doesn’t study is appropriately penalized. The median might be an A, it might be a D. But whatever it turns out to be, at least it’s an honest reflection of the class’s performance. If a test does turn out to be too long or too difficult, there’s a de facto scale in the sense that the cutoffs for various grades will drop, as a student can successfully demonstrate competence with a grade lower than 75%. This means that the sole legitimate excuse for grading on a curve is still preserved, but the negative side-effects are removed. Does that make sense?


Tuesday, November 29, 2005

Dimensional analysis

On tonight’s All Things Considered on NPR, Congressman Mike Sodrel (Republican of Indiana) says:
The information that I get is, like most of my constituents, one-dimensional: it’s flat screen, flat paper. I wanted to see it 3-D.


Monday, November 28, 2005

Student rights

Students in public high schools and middle schools should know their legal rights — as well as the risks they may be taking when attempting to exercise their rights. The Electronic Frontier Foundation has an excellent FAQ on the subject. It goes well beyond quoting from the famous Supreme Court Tinker case:
“Students in school as well as out of school are ‘persons’ under our Constitution,” the Court said, and “they are possessed of fundamental rights which the State must respect...”
They also analyze the much less protective Hazelwood decision, where the Supreme Court allowed censorship that is “reasonably related to legitimate pedagogical concerns.”

...although your opinions are protected by the First Amendment, publishing defamatory content (See our Guide to learn what that is) — even jokingly — may get you in trouble at school, and maybe even get you sued. Other types of speech may also violate the law and put you within reach of the school’s discipline, so read further to see what legal pitfalls you should avoid.
The EFF FAQ also includes an excellent discussion of the risks and pitfalls of blogging, such as the following:
Keep in mind that whatever you post on a public blog can be seen by your friends, your enemies, your teachers, your parents, your ex, that Great Aunt who likes to pinch your cheeks like you’re a baby, the admissions offices of schools and colleges to which you might apply, current and future potential employers, and anyone else with access to the Internet and a search engine. While you can change your blog post at any time, it may be archived by others.
Anyway, read the entire document, since it goes into considerable detail about important issues.


Saturday, November 26, 2005

Vegan firefighters and Starbucks coffee

It’s a good thing to break stereotypes every now and again.

Vegan firefighters? Sounds unlikely. Vegan Texans? Also sounds unlikely. Now we have a vegan firehouse in Texas! They started out as flexitarians, but became vegans over the course of a year. Check out their website.

On a different subject, we all know that the price of Starbucks coffee is higher than Dunkin’ Donuts, right? Well, it turns out that a 20-ounce coffee costs $1.98 at Dunkin’ Donuts, but only $1.89 at Starbucks!


Silber sees the light

Apparently I missed this direct quote from John Silber last month:
I don’t believe in one-man rule.


Friday, November 25, 2005

A half-Chinese Thanksgiving

Thanksgiving dinner was a bit unusual this year. As always, we went to my sister’s house in Somerville — nothing unusual about that. But why was so much of the conversation in Chinese? Let’s see...

You first need to know the cast of characters. The permanent contingent consists of my sister Ellen, her daughters Hannah and Aviva, and of course Barbara and me. For years it has also included Ellen’s housemate Sam, who is ethnically Chinese and comes from Singapore. She speaks English and Mandarin. Hannah, a senior at Brookline High School, and Aviva, a freshman there, have been studying Mandarin for some years now; Hannah also spent the entire second semester of her sophomore year in Xian, China, and is now studying the language privately, having completed Chinese 5 Honors last year and exhausting what BHS has to offer.

Just to add to the mix, they also have a Japanese exchange student staying with them. Yuriko, who is definitely not fluent in English, wanted to experience a traditional American Thanksgiving.

This wasn’t it.

Ellen’s friends Katie and Nicole were also there as usual.

And now we’re up to nine, right? Three more to go: Ellen’s next-door neighbor, Ming, is from Macau by way of Hong Kong — a permanent U.S. resident, but a Portuguese citizen, ethnically Chinese, and a native speaker of Cantonese. He also speaks English and Mandarin. He recently married a woman from Beijing, Lin, who came over here in August along with her seven-year-old daughter, Xiao. They speak Mandarin, as you would expect, and arrived speaking no English. In four months Xiao has of course learned a lot more English than her mother, including important words like “ice-cream”. Yuriko says that Xiao is “rebellious”: she definitely does not behave like a proper Asian child, being loud, strong-willed, and not inclined to obedience. Sounds like a perfect fit for becoming an American child, right?

Oh, did I mention that Ellen and her daughters are vegetarians? So Barbara and I cook the turkey and take it to Somerville every year, to supplement the large number of veggie items prepared by Ellen, Hannah, Aviva, and Sam. One of these items every year is a spicy Chinese green-bean dish. The neighbors next door also contributed to the meal, cooking and bringing excellent potstickers, some wonderful stuffed pancakes, Chinese broccoli, and spicy Cantonese turkey. So the menu included as many Chinese items as American ones. It’s not clear why we even bothered with the turkey (and the associated stuffing and gravy). Maybe we’ll skip it next year.

Not surprisingly, much of the conversation was in Mandarin. I understood about seven words.

But that’s not all that was untraditional here. Ellen’s Thanksgiving dinner is patterned in part after a Seder, so she has a “Harvest Haggadah,” which she and her ex-husband wrote years ago. One of the songs in it is Woody Guthrie’s “This Land is Your Land,” which turned out to be unexpectedly relevant to Ming, who told us that it had been sung by the students in the 1989 Tiananmen Square protest.

To cap off the evening, Aviva had to consult me about some challenging factoring problems that were assigned in her math class. (It’s really a combination of what Weston would call Algebra II Honors and Geometry Honors. The official title of the course is Math 1 Advanced Placement, even though the College Board offers no AP exam in geometry and Algebra II. The justification for labeling it as AP is that it’s part of a sequence that eventually leads to an AP exam in Calculus. Seems like a misnomer to me.) Anyway, try to factor this polynomial:

25x4 - 9 - 4y2 + 12y

You’re tempted to group the first two and factor them as the difference of squares, aren’t you?


Thursday, November 24, 2005

A college perspective

Check out two fascinating posts — one yesterday, one today — from the pseudonymous Rudbeckia Hirta. Both of them lament the state of mathematical knowledge of college freshmen and ask what we high-school teachers are teaching them. Of course her comments don’t apply to Weston alums, since almost all of her students come from one part of one southern state (Tennessee?), but they’re still instructive.

Yesterday’s post was quite specific. It’s short enough for me to quote it in its entirety:
One student responds to a question in three parts:

Question: The probability of winning Pick 3 Lotto is 1 in 1000. What is the probability of not winning?

Student: There is a 50/50 chance.

Question: If you play Pick 3 Lotto every day for a year, what is your chance of not winning?

Student: You have a 1 in 365 chance of winning.

Question: It costs $1 to play Pick 3 Lotto, and you if you win, you get $500. Do you think it’s wise to play this game?

Student: Yes, you only spend $1 and there is a 50/50 chance you will win $500.
Hirta’s follow-up is too long to quote in full, but it opens with this sentence:
I will charitably assume that the high schools in my region are unaware that they produce graduates who, after three or four years of getting A’s in college-prep math, struggle and fail in the lowest-level math classes at my university
She then toys with the idea of asking the students for the names of their high schools and the names of their former algebra teachers, but she rejects it on the grounds that she doesn’t have tenure. Too bad: it would certainly be useful information for the high-school teachers. I would certainly find it helpful to know how my former students are doing in college. I would be very surprised if any former A students were failing in lowest-level college math courses; Weston may have grade inflation, but I think that almost none of our students are in danger of such a fate. In any case, the feedback would be helpful — not only in Weston, but everywhere.

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Wednesday, November 23, 2005


Popco, by Scarlett Thomas. I’m a perfect audience for this book, but I’m obviously not the intended audience for Allison Block, the ALA reviewer on Amazon.com:
Mathematical puzzles. Mind-bending codes. A secret manuscript. And a cake recipe, too. Thomas’ latest (after 2004’s Going Out) has a chronic case of attention deficit disorder. As the novel opens, Brit Alice Butler is en route to a retreat sponsored by her employer, PopCo, a cutting-edge — and slightly creepy — toy company. (Alice takes the midnight train to avoid colleagues — and human contact in general — an early indication that she is a little off-kilter.) It’s no wonder Alice considers herself an outsider; her father disappeared when she was nine, leaving her in the care of her grandparents, two quirky cryptanalysts privy to the whereabouts of a centuries-old buried treasure. Meanwhile, at the company conference, Alice and her colleagues are charged with developing the ultimate product for the teen-girl market. Alice is soon distracted from the task by mysterious encoded messages slipped under her door. Will deciphering them shape her future, or perhaps shed light on the past? Although Thomas’ premise is clever, her digressions into esoteric topics (Godel, anyone?) are likely to leave readers more exhausted than amused.
Clearly I couldn’t stay away from a novel when the review begins with, “Mathematical puzzles. Mind-bending codes.” But what does Block mean when she cites Alice’s desire to avoid human contact as “an early indication that she is a little off-kilter”? Sounds perfectly normal to me. And surely calling Gödel an “esoteric topic” is beyond the pale. (Also, his name is spelled either Gödel or Goedel, not Godel, but I shouldn’t point that out or else Block will think he’s even more esoteric.)

An appropriate analysis would point out not only that math puzzles, cryptography, and Gödel all help to make a novel that the reader can’t put down, but also that Thomas writes in a unique voice that lets the reader know her quirky narrator quickly and deeply. I’m only partway through at this point, so I can’t yet be confident about comparing it with Mark Haddon’s excellent novel, The Curious Incident of the Dog in the Night-Time, but it’s already clear that they have similar sensibilities and similar rewards for the reader. It’s unlikely that Thomas can rise to Haddon’s level, but we’ll see. Stay tuned.


Tuesday, November 22, 2005

The value of projects

Are projects valuable for students in a high-school math class? I suppose the answer must inevitably be, “Sometimes they are, sometimes they aren’t.”

OK, so we need to shift the terms of the question. We should ask, What kinds of projects are valuable? Which students are they valuable for? And how does their value compare to other learning experiences such as lectures, discussions, computer activities, homework, and tests (yes, tests)?

I have a healthy skepticism about projects. (At least I think it’s healthy.) Too often a student spends lots of time on flashy posters and flashy PowerPoint presentations to the exclusion of mathematical content. Too often the student merely cites and rearranges the work of others, usually from websites. (Actually, the work is often not even cited and not even rearranged.) Too often 90% of the work in a group project is done by a single student. And too often a significant chunk of the work isn’t even done by a student at all, but rather by a tutor or perhaps a parent.

And yet...and yet...there are all those times when a student has gained important or even essential knowledge from the process of creating a project. This knowledge might range from new mathematical content to the research process itself to the skills involved in working collaboratively in a group. If there’s a presentation component, everyone can benefit from the experience of standing up in front of a class and trying to explain a mathematical concept. And surely some students benefit from the change of pace and from the different modalities involve in creating and presenting a project — as opposed to homework, tests, etc.

There was one project I still remember back from when I was an 11th-grader. It was an investigation of 4- (and higher) dimensional vectors. I’m hesitant to draw any conclusions from this one example, especially since I don’t want to commit the fallacy of assuming that my students are like me. Not many are, I’m sure.

Perhaps it’s possible to ensure that a large majority of students will experience the potential benefits of project work while minimizing the negatives. Here are a few suggestions:
  1. The project should involve researching and learning some mathematical content that is genuinely new to the student.

  2. A presentation to the class should be required.

  3. After the presentation, all members of the group should be quizzed, both orally and in writing, about the content of the project.

  4. A significant amount of the work should be done in school, in order to minimize contributions by tutors and parents.

  5. The rubric should clearly emphasize content first, clarity of presentation second, and appearance a distant third.


Saturday, November 19, 2005

Differentiated instruction

In this age of No Child Left Untested, our primary goal is apparently a 100% passing rate on standardized tests. But at least there’s a recognition by The Powers That Be that people learn in different ways and at different speeds. So we had an all-too-brief discussion about differentiated instruction during our recent professional development day. Not surprisingly, elementary-school teachers turn out to be (on the whole) much more comfortable than most secondary teachers at implementing differentiated instruction.

Let’s think about some of the ways that we already differentiate our instruction in the Math Department at Weston High School:
  • Most of us offer the opportunity for retakes on tests and quizzes. Second chances are important — for some students all the time, for most students some of the time. We structure retakes in a variety of ways. For example, I weigh the original as 1/3 and the retake as 2/3, to a maximum of 80.

  • We all offer extra help as needed, devoting time before, during, and after school for this purpose. Students who need additional instruction can get it in this way.

  • At the other end of the spectrum, high-achieving students sometimes have the opportunity for additional learning and can sometimes receive extra credit for such learning. (Extra credit is not a substitute for the essential learning in a course. Students doing poorly need to learn or relearn what’s most important, not blow it aside and replace it with a project.)

  • Another excellent opportunity for high-achievers who like math and want to learn more is participation in the Math Team.

  • We provide two levels of most courses — typically honors and college-prep. Instruction is very different at these two levels, thereby providing a rather coarse but definitely effective form of differentiated instruction.

  • The coarse distinction mentioned in the previous item still leaves classes quite heterogeneous, so some teachers create homogeneous groups within a class, at least some of the time.

  • Students with learning disabilities may have IEPs that provide them with SPED tutors, always outside of class and sometimes even right there in the classroom. Many students without diagnosed disabilities hire private tutors as well — a commonplace in communities like Weston.

  • We try to provide multiple modes of instruction with the aim of catching different learning styles. Visual representations help the visual learners, oral descriptions help aural learners, etc.
This list only scratches the surface, but it’s a good start. I do have a couple of questions right away:
  1. Almost every teacher teaches two different sections of the same course. If instruction is differentiated for individuals, it’s inevitably differentiated for sections as well. Under these circumstances, how does one keep the sections synchronized? (Letting classes remain out of sync for more than a day or two is a recipe for confusion or worse.)

  2. How does true differentiated instruction affect grading? Students complain (justifiably) when someone else gets a higher grade for completing easier tasks or for achieving at a lower level. How do we persuade a student who’s already getting an A to accept a more challenging assignment?

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Thursday, November 17, 2005

How people get to this page

It’s a bit puzzling to see how various readers found their way to this blog. Looking at the referrers, I wonder at some of the searches. Here are a few examples:
  • Google: rachel, bartlett, chicago  
  • Google: venn, diagram, about, bubonic, plague
  • Yahoo: what, are, the, numbers, the, 100th, row, pascal, triangle
  • Google: strategies, and, latetoclass
  • MSN Search: strategies, that, close, the, gap, larry, bell
  • Google: internet, safety, high, school, students, xanga
  • Google: use, vectors, with, numb3rs
  • Google: writiing, paragraph, spatial, order  
  • Google: faculty, overheard, students, hack, hacking
  • Google: weston, physics, amherst
  • MSN Search: motivating, students, who, dont, care, allen, meddler
  • Google: puzzle, for, high, intelligence, find, the, probability
  • Google: give, some, detail, this, topic, faith, achivement, health, administration
  • Google: lincolnsudbury, history, department
  • Google: probability, problem, how, many, combinations, letters, and, numbers, license, plates, from, 1986, 2005
  • Yahoo: what, are, the, teachers, strategies, starting, and, finishing, lessons
  • MSN Search: complaints, about, teacher, ego, getting, the, ways, the, learning, experience
  • MSN Search: left, brain, right, brain, and, learning, african, american, males
  • MSN Search: asian, students, learning, strategies
  • MSN Search: math, strategies, boost, cats, scores
  • MSN Search: larry, davidson, nine, essential, component
  • Yahoo: how, can, find, angles, degrees, without, tools, math, oswego, for, grade
  • Google: get, the, barriers, out, the, way, let, people, the, things, they
  • Yahoo: lake, wobegon, days, cryptography, affine
The ones that I don’t mention here were more obvious, but these searches sure are mystifying. How did they lead to this page? And what were some of these people trying to find? (I think the last two are my favorites.)


Wednesday, November 16, 2005

God had a deadline

For the second time this year I came across a link to “The Eternal Flame,” a song that speaks to those of us who believe in the power, efficiency, and mathematical insight offered by the Lisp/Scheme/Logo family of computer programming languages. You may want to have the words in front of you as you listen to the music, in order to read the carefully crafted lyrics while you hear Julia Ecklar’s beautiful singing.


Tuesday, November 15, 2005

Should we use textbooks?

Almost all math teachers (and 63% of teachers of other subjects as well*) distribute textbooks in our courses at the beginning of September. But then four different styles emerge:
  1. Some teachers go through the textbook chapter by chapter — in order — occasionally giving other assignments. They may even (shudder) use the quizzes and tests that accompany the book.

  2. Some teachers use the textbook regularly — maybe only 40%*, maybe as much as 75%* of the time — though not chapter by chapter in order, and they supplement it liberally wherever necessary.

  3. Some teachers write almost all of their own curriculum, dipping into the textbook every once in a while whenever they can find problems that are similar enough to what they would have written themselves.

  4. Some teachers ignore the textbook entirely and might just as well not have handed it out. A variant of this category is the course that simply doesn’t have a textbook at all, so there’s none to hand out.
Let’s face it. Most textbooks are inadequate. They have to be inadequate, since a single book is attempting to address an enormously wide audience, including the statewide textbook-adoption committees in California and Texas. Furthermore, even the best textbook can rob kids of the discovery process, since all the information is there for the taking. From time to time I’ve been in category 4b, even teaching courses in the style of Build-a-Book Geometry (a fine book, well worth reading).

But I’m usually in category 2 for most courses, or occasionally in category 3 for others. The question is, what’s the value of textbooks if they’re inevitably not very good? I suppose the principal answer is that they save time for the teacher and are a comfort to the students. No one who’s teaching three different courses — or even just two courses — has the time to create every bit of his or her materials. And students want to have a source of reference where they can look things up that aren’t in their notebooks, whether they should be or not.

This use is becoming less and less compelling, now that the Internet has become such as good source of reference manual. How is the textbook going to be better than Wikipedia? Seems to me to be unlikely.

A secondary reason is that students need to learn to read textbooks as part of their preparation for college. But how many of us actually assigning reading in the textbook? I’ll do that from time to time, but most often I just assign problems. This is still a valid reason for using textbooks, but not the most compelling one. To my mind the best reasons are that textbooks reduce the likelihood of burning out the teacher (Why reinvent the wheel?) and provide comfort to students and teachers.

* Statistics invented on the spot.


Sunday, November 13, 2005

An unsolicited testimonial

I love my new Neoprene laptop case. It’s soft, it’s extremely lightweight, and it opens up in such a way that it’s remarkably easy to keep the case on while using the computer. What more can I say?


Saturday, November 12, 2005

Stopping to ask directions

An activity in one of my Saturday Course classes this morning was a crypto treasure hunt. Each group of three fifth-graders had to decrypt a cryptogram, leading to a somewhat mystifying plaintext, which in turn took them to a location where they could find the next clue, which necessitated returning to the classroom, etc. There was a story line behind all this, but we’ll save that for another day. Anyway, the groups were formed by alphabetical order, and it happened that two were all-female, one was all-male, and one was mixed. The all-female groups and the mixed group returned quickly, but where was the all-male one? Were they still figuring out their clue (“Down in the dungeon where the heat is controlled”)?

It turned out that the program director had been observing the kids, and these fifth-grade boys were the only group that was unwilling to ask directions! It starts early.


Wednesday, November 09, 2005

Killing five birds with one stone

At a recent Math Department meeting, we discussed the question of whether we should offer more math electives. Currently the only non-AP electives that Weston offers are two one-semester Comp Sci courses, but we’re a small high school and probably wouldn’t get sufficient enrollment for any given additional electives. So here’s my thought:

Suppose we create a new course called Guided Projects in Mathematics or Directed Research in Mathematics or something like that. This would be a loosely structured opportunity for students to pursue work on any math topics of their own choosing. The teacher would be there to advise, to coordinate, to supervise, and to give feedback — but only rarely to provide explicit instruction. The classroom could well contain 16-20 kids working on ten different projects, so each would sign an individualized contract at the beginning of the year. They would have to be proactive and responsible in order to thrive in this environment, since they would be working fairly independently.

This course could kill five birds with one stone, by meeting the mathematical needs of five disparate sets of students:
  1. Those seeking an extra math experience in addition to a regular course. I can easily imagine a dozen different courses that might be valuable and interesting to a few kids, such as Advanced Geometry or Number Theory or Abstract Algebra or Game Theory or Problem Solving or a collection of topics from the For All Practical Purposes book.

  2. Those who have taken AP Comp Sci — or who have taken Intro and don’t want to take AP — and are seeking additional opportunities to broaden and/or deepen their computer programming knowledge.

  3. Those who would otherwise pursue an independent study in math but can’t get much done by meeting only once a cycle.

  4. Those who are on the Math Team and would welcome a loosely structured opportunity to do more practice and learn more math in the process (several other schools offer course credit for Math Team, and this could be a vehicle to do so).

  5. Those who have already taken all our regular courses (we typically have two to four juniors in BC Calculus, our most advanced course).
These sets, of course, are not necessarily disjoint. It would be entirely possible — and even likely — for a given student to mix two or three of these ideas into one year’s worth of math.

To me this sounds exciting and useful. What do you think?

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Sunday, November 06, 2005

The T Word

Why is it politically incorrect to point out that mathematical talent is one of the necessary components of doing well in high-level math courses?

No one would expect that just any random kid could make the varsity football team. We all know that a certain amount of athletic talent — perhaps a lot of athletic talent — is necessary to reach that level. Hard work, of course, is also necessary, but it’s not sufficient.

No one would expect that just any random kid could reach the highest level of musical achievement. Some musical talent — a lot of musical talent — is necessary. Hard work, of course, is also necessary, but it’s not sufficient.

So why is it wrong to point out that there’s such a thing as mathematical talent?

Presumably the answer lies in my long blog entry of 10/27, which is closely related to this one: we want advanced math courses to be “a pump, not a filter,” so we don’t want to discourage anyone from taking them. Even a whisper of the reality of mathematical talent (or lack thereof) will discourage some students. So we pretend that it doesn’t exist, or that it’s unnecessary, or that everybody has it. But, as W.S. Gilbert points out in The Gondoliers,
When every one is somebodee,
Then no one’s anybody.
Hard work, motivation, a good math background, and — yes — talent... all of these are necessary components in order to succeed in advanced math courses.

For more food for thought, read Kurt Vonnegut’s provocative short story, “Harrison Bergeron,” which begins like this:
THE YEAR WAS 2081, and everybody was finally equal. They weren’t only equal before God and the law. They were equal every which way. Nobody was smarter than anybody else. Nobody was better looking than anybody else. Nobody was stronger or quicker than anybody else. All this equality was due to the 211th, 212th, and 213th Amendments to the Constitution, and to the unceasing vigilance of agents of the United States Handicapper General.


Saturday, November 05, 2005

Does the school day start too early?

Tracy Jan’s article in yesterday’s Boston Globe indirectly quotes Brighton High School basketball coach:
To boost attendance, alertness, and academic achievement, Mahoney said, high schools should start later. High schools around the Bay State are considering the idea because federal law now rates schools on attendance and test scores. Scientific studies have found that most teenagers are biologically wired to go to bed and wake up later and that they do better in school if they get more sleep.
The school day at Brighton High School starts at 7:25, five minutes earlier than Weston High School, which has the longest school day in the state (according to thoroughly unscientific claims made by the vast majority of our students).

The article continues:
Next month, Mahoney hopes to persuade his colleagues to vote to start school 20 minutes later, at 7:45 a.m., beginning in January. In Boston Public Schools, a school can change its start time if 75 percent of the faculty agree.

Last year, half of the faculty rejected Mahoney’s request.
Of course the real problem is the bus schedules:
Even the Legislature is considering the idea of giving teenagers more sleep: A pending measure calls for directing the Massachusetts Board of Education to encourage high schools to start no earlier than 8:30 a.m.

Paying for such a change, however, might be problematic, particularly in suburban school systems, which often start high school early so they can use school buses again for younger students. Shrewsbury and Bedford recently scrapped plans to begin high school at 8 a.m. because of the cost.

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Friday, November 04, 2005

Disappearing statistics

A wire-service article appeared yesterday on the Boston Globe’s website, boston.com, but now the article has mysteriously disappeared. Fortunately most of it is still available elsewhere, so we can examine its questionable use of statistics. It begins by reporting some vaguely alarming statistics:
A state Department of Public Health study finds that certain types of cancer are far above the state average in some western Massachusetts communities.
But don’t get too alarmed:
Nine other western Massachusetts towns had cancer rates below state average.
Gee, what a surprise! Some towns had cancer rates above average, and others had cancer rates below average. No numbers, of course, except for an unclear observation that “pancreatic cancer among men in Easthampton is 128 percent above the average.” Given the general level of reporting in this article, we don’t even know whether they really mean 128% above the mean (or is it median?); very possibly they mean 128% of the mean (or median). In any case, we know nothing about the actual incidence of pancreatic cancer or the population of Easthampton. Perhaps the state mean would have predicted 1.3 cases and there were really 3, which is hardly likely to be statistically significant.

I know, numbers don’t sell. But they are essential if we’re to understand the point behind this article. Of course some towns are above average and others are below, but what does that prove?

Interestingly, the sentences omitted from this version include the observations that town A is above average in pancreatic cancer, B in breast cancer, C in stomach cancer, and so forth. No one town is above average in more than one type. This leads us to be even more suspicious about the statistical significance here.

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Thursday, November 03, 2005

Not yet reading

Yesterday we held our regular first-Wednesday-of-the-month professional development activity. This time it was a planning session for an event three months hence — a day based on Tracy Kidder’s latest book, Mountains Beyond Mountains. (This book comes with two slightly different subtitles: the original hard-cover is subtitled Healing the World: The Quest of Dr. Paul Farmer, whereas the paperback is subtitled The Quest of Dr. Paul Farmer, a Man Who Would Cure the World. I’m sure there’s some inscrutable purpose behind the publisher’s decision to change the subtitle.)

Anyway, this book comes enthuasiastically recommended by member of Weston High School’s book club; in fact, “enthuasiastically” is really too mild a word. Published reviews were equally filled with praise. Through the generous support of the Weston Educational Enrichment Fund and an external foundation, copies have been donated to all of our faculty and students.

You can probably read some significance into the fact that Tracy Kidder, George W. Bush, Dick Wolf (“Law & Order”), Bill Littlefield (“Only a Game”), and I all attended the same high school at the same time: Kidder was class of 1963, Bush and Wolf were ’64, I was ’65, and Littlefield was ’66. That means of the 900 or so students attending PA in the spring of 1963, four eventually became famous. Somehow I expected more than a paltry half of one percent.

That’s the prelude. Substance will follow, after I’ve had a chance to read the book. I’ve never been a devotee of reviewing books before I’ve read them.

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Wednesday, November 02, 2005


Why are logarithms so difficult? Algebra students who are consistently competent in other topics often stumble when they get to logs. Sure, they can memorize an algorithm for switching from exponential form to logarithmic form and vice versa, and most of them can use logs to solve an exponential equation as long as a straightforward algorithm can be applied without too much thinking.

But...vary the context a little, and bewilderment sets in. Even the calculator isn’t much help, since it won’t work with bases other than 10 (or e) except by means of a generally incomprehensible formula — yet another thing to be memorized rather than understood. So, we try to provide contexts involving exponential growth and decay, where logs provide the missing link: we know that we always need the inverse when we know the output but not the input, and we know that a logarithmic function is the inverse of the corresponding exponential function, so all is clear.


Perhaps the nub of the problem is that a log is really both a noun and a verb. When we write 2 = log3(9), we are simultaneously saying two different things: 2 is a log, so “log” must be a noun; and we are also applying the process of “taking the log” of 9 — without ever writing the word “taking”! — so “log” is now functioning like a verb.

I really think that this confusion is a major source of students’ difficulties, but I don’t understand why the same confusion doesn’t crop up in the case of powers, which also play the same dual role. In the example in the previous paragraph, we could write the equivalent exponential sentence, 9 = 32, and almost nobody seems to be confused about the fact that we have both the process of squaring and the resulting perfect square. Maybe it’s because we only have numbers in this case — no unfamiliar words like “logarithm.”

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Tuesday, November 01, 2005

An invitation from Tufts Health Plan

The form letter begins as follows:
Dear Rosalita Davidson,

Do you have a plan for retirement? Remember, there’s more to consider than the size of your pension or 401(K). There’s also your health insurance.

Medicare is important. But it may not be enough.


Tufts Health Plan Medicare Preferred HMO may be the answer.
Why are they sending this to Rosie? She’s only twelve years old!

Oh, and there’s another reason why she was surprised to receive this letter:



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