Learning Strategies

Weston, of course, is really Lake Wobegon, where all the children are above average®. Weston’s only public high school has just two levels of math courses: the higher level is called honors, the lower level college-prep. Everyone is above average.

That’s misleading, though not for the reason you might think. Some skeptics leap to the incorrect conclusion that “college-prep” is merely a euphemism for “low-level,” but that’s not true: these courses really are college-prep. Looking at the list of recently graduated seniors, I see products of our college-prep math courses going to Middlebury, Wisconsin, BC, Penn, Tufts, Rochester, Michigan, Johns Hopkins, McGill, and other fine colleges and universities.

So, why do I say the names are misleading? It’s not that college-prep is misnamed; it’s that everyone wants to be in honors. That’s an exaggeration, of course, but it’s still true that honors is no longer for the top 10-15%, the crème de la crème who are excellent and enthusiastic mathematicians. Maybe that’s a good thing; why shouldn’t a wide range of students have this opportunity? And yet...wouldn’t some of these students be better served in a good college-prep class, instead of getting all stressed out by adding honors math to AP history, AP science, honors English, AP Spanish, Theatre Company, chorus, piano lessons, tutoring, and athletics? (I’m not making this up, as Dave Barry would say.)

Labels: math, teaching and learning, Weston

David Allen rants about Casual Fridays. Many of his readers’ comments concern seeing clients in business settings on Mondays through Thursdays. But what about school settings? At Weston High School we’ve had Casual Fridays for many years (though there have been so many new teachers recently that this piece of the oral tradition hasn’t been passed down very effectively). Who are our clients? If our clients are our students, then every day is Casual Friday. Why should Fridays be special? Do I really like wearing ties, as one sophomore asked me?

Definitely not, I told her. (So why do I wear them on Mondays, Tuesdays, Wednesdays, and Thursdays?)

A few Fridays ago, another sophomore said that she had just noticed that a lot of teachers wore jeans on Fridays. “Is Friday Jeans Day?” she asked me.

I wasn’t wearing jeans — in fact, I never do — so it was slightly odd for her to ask me that question. I explained this piece of Weston High School culture to her and pointed out that I never wear ties on Fridays. “Of course,” I added, “there are a lot of teachers who never wear ties.”

“Yeah,” she replied. “Like Ms. Griswold.”

Labels: Weston

I’m examining an Algebra II book that looks really good so far (Discovering Advanced Algebra: An Investigative Approach, by Murdock, Kamischke, and Kamischke — if you’re interested). But a statement in the Note to Teachers got me thinking:

Students will approach each new concept and challenge with mathematical tools that support their particular learning strengths — be they algebraic, numerical, graphical, or verbal.

Maybe I’m taking this too literally. I’m all for using one’s strengths, but how will students ever identify their strengths if they don’t have lots of experiences where they’re required to use approaches that they may consider to be among their weaknesses? Surely a student should spend a considerable amount of time developing greater facility with all appropriate tools.

We promote mathematical growth by saying, “You’ve got to learn this tool well enough so that you become comfortable and competent in using it. Then you can decide whether it’s the best tool for you when approaching a particular problem.”

We don’t promote mathematical growth by saying, “Just play to your strengths.” That way closes doors. We need to open them.

Labels: teaching and learning

It has been so hot and muggy that the cats are just lounging around listlessly. Which means...well, the word seems to suggest that they don’t have any lists. I’ll admit that cats aren’t big list-makers (though they are otherwise fine creatures), but we know that that’s not what the word really means.

Why doesn’t “listless” mean “without lists”?

Naturally I looked it up. It turns out that this “list” is an entirely different word: it’s from the Middle English “list”, meaning ”desire” and cognate to “lust”.

So the cats were really lying around lustlessly, which seems apt.

Labels: linguistics

OK, we’ve heard of online math courses and writing courses, but phys ed???? Even as I type this, I’m listening to a report on NPR about the online phys ed option for students in the Minneapolis Public Schools. Hmmm....

Labels: teaching and learning

Ordinarily I don’t feel very comfortable at parties if most of people are unknown to me. When most of the attendees are friends of mine, I’ll enjoy the party, but my comfort level goes down as the number of strangers goes up.

So why did I feel perfectly OK about today’s annual Open House of the Dorchester Historical Society? There must have been 50 or 60 people there — far too many for my usual comfort level. To make matters worse, Barbara and I were assigned to be bartenders, and what do I know about tending bar? Nothing! That should have made matters worse.

But it actually turned out to be just fine. I have two possible explanations:

• The only drinks we had available were four kinds of wine and three flavors of seltzer, so no particular skills were needed in tending bar.


• The “job” of bartender imposed a structure on the evening, so I didn’t have to circulate and make spontaneous conversation. Actually, the experience had something in common with teaching. Outsiders sometimes wonder how introverts can possibly enjoy teaching, but there’s a simple explanation: the natural structure of a course, a classroom, and even out-of-class interactions mitigates the feeling of “What do I say now?”

Labels: teaching and learning

Three of my sophomore students have volunteered to set up a website for use by our Math Team. They’re designing a database to contain Math Team problems and answers collected from the last n years, typing many hundreds of problems in MathML format, entering them in the database, drawing any associated images, writing a PHP/MySQL program to retrieve and display data, and designing a web site where team members can search for practice problems and check their answers. Quite a production!

Labels: teaching and learning

This note, written by a junior, was attached to the precalculus book she was returning:

Dear Mr. Davidson,

Here is my book. I have enjoyed it thoroughly. I read it when I was feeling down, and it raised my spirits.

“I poured my soul into this book, and I hope the lucky student who has it next year will enjoy it as much as I have.”

The inner quotation was carefully attributed to a classmate, complete with a © symbol. (And there was no smiley or other emoticon to spoil the ironic tone.)

Labels: math, teaching and learning

Check out this interesting visual representation of U.S. income distribution. Try zooming out and out and out...

Labels: life, math, technology

It’s tough to keep kids focused and engaged after they’ve finished their final exams. As I blogged earlier, our math exams were on the 14^th... but school continues until the 23^rd! Everyone knows that tests and grades are all that count, so why should anyone continue learning after the final exam? It’s final, isn’t it?

(Actually, a surprisingly large number of students are still paying attention and learning, but motivation is difficult for almost everyone, especially those who would prefer to cram for next period’s chem exam rather than concentrate on what’s going on in math class.)

Labels: teaching and learning

Weston High School held a math haiku contest, but maybe we should have had a math limerick contest instead. Here are some math limericks about the proof of Fermat’s last theorem (from a contest held at the University of Illinois at Chicago). One limerick has five stanzas, the other the traditional one stanza.

A mathematician named Wiles
Had papers stacked in large piles
Since he saw a clue
He could show Fermat true
Mixing many mathematical styles.

He labored in search of the light
To find the crucial insight
Young Andrew, it seems
Had childhood dreams
To prove Mr. Fermat was right.

He studied for seven long years
Expending much blood, sweat, and tears
After showing the proof
A sceptic said “Poof!
There's a hole here,” raising deep fears.

This shattered Mr. Wiles’s belief
His ship was wrecked on a reef
Then a quick switcheroo
Came out of the blue
Providing his mind much relief.

Mr. Wiles had been under the gun
But the obstacle blocking Proof One
Fixed a much older way
From an earlier day
And now Wiles has his place in the sun.
     By Jonathan Harvey

“My butter, garçon, is writ large in!”
a diner was heard to be chargin’.
“I HAD to write there,”
exclaimed waiter Pierre,
“I couldn’t find room in the margarine.”
     By Everett Howe, Hendrik Lenstra, and David Moulton

Labels: math

A lot of my precalculus students today didn’t like and/or didn’t believe Cantor’s proofs of the denumerability of the rationals and the non-denumerability of the reals. A few articulated their disbelief; most were quiet and attentive, but how many of them were silent disbelievers as well?

Labels: math

You have got to go look at the Visual Dictionary of Famous Plane Curves and study some of the stunning images that Xah Lee has collected. I particularly recommend his collection of sinusoids and his gallery of graphics based on the Witch of Agnesi.

Labels: math, technology

Yes, there really is a product called The Algebrator. Their slogan is, “You Type in Your Homework Problem. Algebrator does the Rest!”

Here is an excerpt from one of their ads. What’s wrong with this picture?

Labels: math, teaching and learning, technology

It wasn’t a great day for taking math exams. I opened up the windows and doors in my classroom at 7 AM to get some cross-ventilation, but when the exam started at 7:35 it was already 83 degrees in my room. By the end of the exam, at 9:45, it was 85 degrees, with a humidity well above 90%. Not a very good environment for learning or for taking exams, is it? Too bad Weston can’t afford air-conditioning.

Labels: teaching and learning, Weston

The pseudonymous Rudbeckia Hirta writes about “the most amusing book ever written about calculus”: The Historical Development of the Calculus, by C.H. Edwards. I know, you don’t think the competition for most amusing calculus book is very stiff, but I’m probably still going to take a look. Or maybe one of my students will review it for me.

Labels: math, teaching and learning

The TeachScheme conference (see my post of 5/22) went very well. Because of some changes in the program, we actually were granted 15 minutes for our talk! I came away with quite a number of interesting ideas, both for the Computer Science courses and for our long-term project of integrating programming into regular math courses. I also saw some ideas that I definitely do not want to pursue, but that’s useful too.

Four interesting ideas that emerged from the conference:

• We can use vectors and trig to model non-hierarchical systems such as swarming behavior — perhaps a good precalc project.


• We should teach structures and animation more clearly and earlier.


• We should emphasize the Design Recipe even more than we do.


• The language levels of DrScheme that are designed for programming courses could be supplemented with language levels that are specifically designed for math courses.

Moreover, we actually learned how to do the first three of these. The fourth, which is surely the most significant, will take some considerable thought.

A few memorable quotations:

“Students want to spend class time thinking, not writiing.”

“Programming is algebra in action.”

“We don’t teach Scheme, we teach design.”

“Computing is combining data and producing more data, not destroying data.”

Labels: teaching and learning, technology

Dennis and I were talking about multiple representations. Multiple representations are one of the Big Ideas that wend their way through all our math courses. A table and a graph and a function machine and a mapping diagram are all multiple representations of the same thing.

But what’s that thing?

A function per se is an abstraction, a possibly infinite set of ordered pairs. The graph doesn’t represent the table; both of them represent something else.

It reminded me of a conversation with a Lincoln-Sudbury parent back in the mid-’70s. He had noticed a disparate array of interests of mine that showed up in the two courses in which I taught his son. These interests ranged from aspects of mathematics to linguistics to cryptography to cartography to musical notation and even to model railroading. “What did all these have in common?” he asked himself — and me.

“I don’t know,” I replied. “What do they all have in common? I’ve never thought about it.”

“Well,” he said, “after pondering it for quite awhile, I realized that they all have something to do with representations. What you’re really interested in is the different ways that abstractions can be represented.”

So a parent of a student helped me understand myself better. Maybe he’s right.

Labels: model railroads, teaching and learning

Is Weston High School the only school that gives out too many awards? Probably not. I counted 14 awards for one of our seniors, and 14 for another as well! And, of course, there were hundreds of others for seniors. And then there are the awards for juniors, sophomores, and freshlings.

I also wonder about the structure of our math awards. Along with prizes for Math Team, American Math Competition, etc., we also award a letter of commendation in each section of each course. This letter is given “for significant mathematical growth during the past year; curiosity and eagerness to learn; contributions to other students’ learning; superior work on daily assignments; effort and dependability; meeting uncertainty and challenges with a positive attitude.” (No, I don’t know why there are semicolons in the absence of internal commas.) Then, out of the pool of letter winners, one student is selected in each course to receive a book award “for excellence in _____.” But look at the disparate criteria. Is it really right that each book awardee is ipso facto a letter-of-commendation winner, when the criteria are so different? Under this system, suppose the student who most deserves the book award for excellence in Algebra II is quiet, sticks to herself, and ignores most homework because she understands it already; she is automatically excluded from the book award because she couldn’t possibly qualify for a letter of commendation.

Labels: teaching and learning, Weston

Please read Graham Cormode’s review of the TV show, “Numb3rs” (which he claims is pronounced “Numbthreers” rather than “Numbers”). Brief excerpt:

Given low initial expectations, it was probably one of the better attempts to show mathematical topics within the context of popular entertainment. It’s best not to get too hung up on all the details, and go along with it to some extent. Still, certain plot points did rankle when there seemed to be significant flaws...

I haven’t seen the show yet myself (other than a three-minute clip that purports to explain the Monty Hall problem).

Labels: math

This week’s New York Times Book Review contains a fascinating Literary Map of Manhattan, preceded by an explanatory article written by Ethicist Randy Cohen. Quoting Meg Wolitzer, Cohen defines his (their?) “cartographic motto”:

a strong sense of specificity, even though everything is made up.

I was immediately reminded of two very different connections. The first, of course, was Marianne Moore’s description of poetry as “imaginary gardens with real toads in them.” The fictional characters on the map live in imaginary gardens, but the convincing details about Manhattan (whether genuine or imagined) are the real toads.

The second connection was with the problems we assign in school mathematics. All too often these imaginary gardens contain concrete or cardboard toads, certainly not real ones. I try to populate my problems with details that are specific enough that they can become real in the reader’s imagination, even when the occasional student takes them all too literally. Here was a problem on my Algebra II test:

Before founding Macy’s department store, John and Mary Macy had five children. Unfortunately they named the first three Casey, Stacy, and Tracy. Then they stopped the rhyming; the last two children were Lee and Chris. You’ve already noticed that it’s impossible to tell from the names which children were boys and which were girls.

1. Find the probability that all five were boys.


2. Find the probability that both Casey and Lee were boys.


3. I don’t know what it’s like in your house, but the Macys were never able to get all five children to show up for breakfast at the same time. Oddly enough, it always turned out that exactly four of them turned up (but not always the same four). Their parents made them line up for breakfast in single file. How many different arrangements of children were possible under these conditions?


4. Eventually, when they retired, John and Mary wanted to pick two of the children to run the store as co-presidents. From how many different pairs could they choose?

I don’t know how real those toads were, but that’s definitely what I was aiming at in my imaginary garden. Similarly, I wrote this problem, inspired by my colleague Josh, on today’s precalculus test:

In the famous movie, “Honey, I Kept Shrinking the Kids,”” Rick Moranis played a biologist who cloned his daughter, Chloe, and then cloned the clone, and then cloned the clone of the clone, and so forth. Chloe was 64 inches tall, but the clone was only 48 inches, and the next clone was only 36 inches, and so forth (each clone was three quarters of the height of the previous incarnation). Their cheerleading coach thought of a unique routine, in which each girl would stand on the shoulders of the next larger girl, and so on down until Chloe was at the bottom. The coach pointed out to Moranis that they could fit an infinite number of clones on top of Chloe within the gym!

1. Find the combined height of Chloe plus the first four clones. (Ignore the fact that each clone is actually standing on her predecessor's shoulders, not head.)


2. What is the minimum height for the ceiling of the gym in order to fit Chloe plus the infinite number of clones?

Math as fiction?

Labels: math, teaching and learning

Sam Hughes is the author of the Venn Diagram cited in my previous post. I also recommend his list of “Things mathematics has taught me”:

1. That there are such things as unanswerable questions — indeed, provably unanswerable questions
2. That Occam’s Razor is fallible
3. To assume nothing
4. That the word “obvious” means “a proof springs immediately to mind”
5. The importance of proof
6. How to reason logically
7. To reason logically
8. How to recognise faulty reasoning and find counterexamples
9. That intuition counts for zilch
10. Point nine recurring equals one

Labels: math, teaching and learning

Perhaps a Venn Diagram would help.

Labels: math, travel

Do check out the NCES Students’ Classroom site. Good stuff — even though a lot of it is in Comic Sans (see yesterday’s post).

Labels: math, teaching and learning

In a column in this week’s Boston Phoenix, Mike Miliard reports on the worldwide campaign by Dave and Holly Combs to ban the Comic Sans font:

...It’s Comic Sans, the goofy, ungainly typeface, meant to mimic handwriting that’s somehow proliferated across the globe... Comic Sans is shoddily designed, with awkward weighting and haphazard kerning...

...The problem has reached critical mass, Combs says, and action must be taken. “I got a funeral announcement in Comic Sans,” she says, pausing to let the words sink in. “I mean, that’s sad! I don’t want a funeral announcement in Comic Sans!”

For more info, see also www.bancomicsans.com.

Labels: technology

My impression is that there are significantly more female teen bloggers than male ones, but maybe I’m wrong. A fascinating study by David Huffaker says that “BlogCensus randomly sampled 490,000 blogs to find 40% male and 36% female, with the rest of the blogs unidentifiable in terms of gender”; on the other hand, he also cites another study that “found there are more females than males in the teen category.”

More important are Huffaker’s conclusions about Internet safety:

...teenagers reveal a considerable amount of personal information in their blogs, including name, age, and location, as well as contact information in the form of an email address, an instant messenger name or a link to personal homepage. The content of blogs typically reflects what is expected to impact a teenager’s life, such as school, intimate relationships, sexual identity and even music...

Males average more emoticons in their posts than females. Males also reveal their homosexuality more often than females, expressing their sexual identity or coming out. Males reveal their location more often than females, while females present a link to a personal web site more often than males.

Labels: teaching and learning

Homework can be counterproductive, according to an article on the physorg.com website. Here are a few excerpts:

Instead of improving educational achievement in countries around the world, increases in homework may actually undercut teaching effectiveness and worsen disparities in student learning, according to two Penn State researchers.

Most teachers worldwide are not making efficient use of homework, said David P. Baker, professor of education and sociology. They assign homework mostly as drill, to improve memorization of material either in math, science or the humanities. While drills and repetitive exercises have their place in schooling, homework may not be that place....

The researchers analyzed data from the Third International Study of Mathematics and Sciences (TIMSS), which in 1994 collected a large amount of data from schools in 41 nations across the fourth, eighth and 12th grades....

Their findings indicated a frequent lack of positive correlation between the average amount of homework assigned in a nation and corresponding level of academic achievement. For example, many countries with the highest scoring students, such as Japan, the Czech Republic and Denmark, have teachers who give little homework. “At the other end of the spectrum, countries with very low average scores — Thailand, Greece, Iran — have teachers who assign a great deal of homework,” Baker noted.

“The United States is among the most homework-intensive countries in the world for seventh- and eighth-grade math classes. U.S. math teachers on average assigned more than two hours of mathematics homework per week in 1994-95,” said LeTendre. “Contrary to our expectations, one of the lowest levels was recorded in Japan — about one hour a week. These figures challenge previous stereotypes about the lackadaisical American teenager and his diligent peer in Japan.”

Labels: teaching and learning