Learning Strategies

As of today, this blog has moved to blog.larrydavidson.com. See you there!

Labels: books, Dorchester, food, life, linguistics, math, model railroads, movies, teaching and learning, technology, travel, Weston

This post is a follow-up to my post of November 30, where I brought up two points that can illuminate one’s views on the big ideas of algebra:

...we discussed the assignment of partial credit for work in solving a problem — more on this later, but it definitely reflects one’s views on what the big ideas are — and whether the study of algebra is distinct from (and prior to) the study of functions...

Partial credit doesn’t sound like a deep issue, but it really is. All you have to do is gather a group of math teachers, give them a student’s solution to a problem, and ask how many points should be assigned. Regardless of whether it’s out of four (where there are only three partial-credit possibilities) or out of ten (where there are nine), there will be significant disagreements; I reached this conclusion from having participated in such activities many, many times, with various groups of colleagues. And I don’t just mean that one teacher will give two out of four and another will give three, or that one will give seven out of ten and one will give five. No, I mean that one teacher will give nine points and one will give zero! And this is in mathematics, which is supposed to be an objective discipline — unlike English, where such disparities might not be surprising.

So, what does it mean when major disagreements surface in this area?

It usually means one (or both) of the following types of differences:

• differences of opinion about what the big ideas are
• differences in what one values

For instance, consider these three examples of student work that we discussed in the seminar in which I participated in November:

1. The problem read, “The sum of three consecutive odd integers is 81. Find the integers.” One student’s solution was like this:

   Let x = 1^st odd integer
   x+1 = 2^nd odd integer
   x+2 = 3^rd odd integer
   x + x + 1 + x + 2 = 81
                    3x + 3 = 81
                          3x = 78
                            x = 26
   The integers are 26, 27, 28

   How many points (out of ten) is this worth? If a big idea is that odd numbers differ by 2, not by 1, then the setup at the beginning of the solution represents a significant error — especially since the student wrote the word “odd” each time, thus showing that s/he didn’t merely skip over the word “odd” in the problem statement. On the other hand, the solution is otherwise correct, the work is clearly shown, and the answers will even check, being three consecutive integers adding up to 81 (ignoring the word “odd” again). If you highly value all the skills of combining like terms, backtracking to solve an equation, and recording a solution, then the solution is worth a lot of points. But if the idea of consecutive odd integers is important, it may be worth very few points. My colleagues rated the solution all the way from one to nine; I gave it a six.
2. Next we have a different student’s solution to the same problem:
   
The grades on this one ranged all the way from zero to ten! Some teachers gave it only a few points — or even zero points — because no algebra was used. But I was one of those who gave it a ten, because not only was the solution correct but it also showed a thorough understanding of what the problem was asking for, and of course the answer was checked. If an algebraic solution was meant to be required, that requirement should have been specified.
3. Finally, here’s a different problem, along with one student’s solution: “Solve 2(x – 10) &ndash (12x – 4) = 20.”
   
   The issue here is that the student couldn’t read his own handwriting (possibly her own handwriting, but the odds are against that): “20” got transcribed sloppily and then read as 26. I gave it a nine out of ten, since that error struck me as a very minor one. But other teachers’ scores ranged all the way down to zero, on the theory that the student had nobviously never checked his answer. My own values are that checking one’s answer is a big idea of algebra, if we mean that it’s important to understand that what we mean by a solution is a number that will satisfy the equation. But failure to check, especially in a time-sensitive situation where there are no instructions to check, strikes me as a very minor offense.

Your mileage may vary.

The second point that I was intending to discuss — whether algebra is distinct from functions — will have to wait for another post. This one is already too long.

Labels: math, teaching and learning

“What do I need to do to get an A?” asks one of my students in an honors math course.

I wish I had a magic recipe. I can say with reasonable confidence that it’s possible to get a B by studying hard, by studying smart, by working hard to understand concepts, by getting enough practice in math skills. But an A? Every student sees some classmates getting A’s by some mysterious method, in some cases working very hard and in other cases magically earning the A with seemingly little effort. Surely there must be a secret recipe that the teacher isn’t revealing.

Of course there isn’t such a recipe. Think of some non-academic endeavors. What can you do to gain a place on the varsity basketball team or the Greater Boston Youth Symphony Orchestra? Sure, working hard is important and necessary (think of the old joke about how to get to Carnegie Hall: “practice, practice, practice”), but unfortunately it’s not sufficient. There’s no way that I’m going to be a top-ranked musician or a top-ranked athlete, no matter how hard I try. And yet, as a teacher, I definitely don’t want to be the bearer of discouraging news. I want my students to work hard, to do their best. But their best might or might not match their hopes. There’s nothing wrong with a B in an honors math course at Weston High School — in fact, I have a couple of students who work very hard and are delighted when they achieve B’s. But, in Lake Wobegon and similar communities, nothing less than an A will do.

So here’s the dilemma: how do I motivate students to achieve their personal best — which, after all, is the aim in music and athletics and other endeavors — without telling them that their personal best might not be good enough? I know how to help students get B’s, but after 34 years of teaching I don’t know how to help them get A’s, at least in honors courses. Some do, most don’t, but it’s not clear what effect I can have on the outcome. I can help a willing, able, and motivated student get an A in a college-prep course, but such a student might well try his or her best and only get a B in an honors course. As I say, there’s nothing wrong with that, but in today’s world of competitive parents and even more competitive college admissions, my point of view won’t be compelling. This is discouraging. The last thing I would do is say ahead of time that any given student is incapable of earning an A. And yet, at some point, one is forced to admit that a particular student is trying as hard as any reasonable person could expect and is earning a B. I don’t know what to say, except to repeat that there’s nothing wrong with that.

Labels: math, teaching and learning, Weston

Strip Search, by William Bernhardt, is an irritating novel.

Why do I say that? Well, it’s not just because Bernhardt portrays math teachers as weird and psychotic, though that’s certainly a major part of it. And it’s not just because the plot is so implausible, though that too is part of it. And it’s not just that the book is riddled with mathematical errors, though of course that definitely bothered me. And it’s not just that the amount of violence is excessive and unnecessarily explicit, though that would certainly put off many readers. No, the most irritating characteristic of Strip Search is that it reads like a “good idea” that someone had. My impression is that someone said to the author, “Here’s a proposal for a novel. Go write it.” Not surprisingly, a coherent novel was not the result.

You may wonder why I started reading this book. In the past I’ve found Bernhardt to be a competent and engaging writer, even if not a memorable one. And I had heard that Strip Search featured a combination of mathematics (equations left as clues at each crime scene) and a major character (Darcy) who’s an autistic savant. No math teacher could resist that enticing combination. Some readers (in customer reviews on Amazon, for instance) were annoyed by the characters and found none of them likeable. Personally I didn’t have that problem, although I can see why others might. But anyone who has taught students who have Asperger’s or autism will find Darcy likeable enough, to coin a phrase. And the detective is no more unlikeable than many a highly flawed protagonist.

You may also wonder why I bothered finishing Strip Search if I was so irritated by it; I’m not one of those people who feel compelled to finish a book once they’ve started it. But I kept irrationally hoping that things would get better, that there would be a good reason for all the flaws. Unfortunately I was wrong, so here is your warning. Don’t read this post any further if you’re intending to read Strip Search, as I can’t write what I need to write without introducing spoilers.

*** SPOILER ALERT*** SPOILER ALERT *** SPOILER ALERT ***

OK, so we have a detective who’s actually a police psychologist (the protagonist) and fits into the genre stereotypes of being insubordinate and an alcoholic. Later she turns to pills. She is a psychologist without a doctorate, and she reaches most of her conclusions by intuition and guesswork. Since she’s also the first-person narrator, I’ve forgotten her name. Oh, that’s right, it’s Susan.

But don’t think that Bernhardt extends genre stereotypes to gender stereotypes. No, we also have Esther Goldstein, a female mathematician who not only teaches math but also has apparently solved the Riemann Hypothesis (misspelled “Reimann” throughout the book). For reasons that apparently stem in some undefined way from an unhappy childhood, she is also reviving the ancient Pythagorean religion, the Brethren of Purity. Unfortunately she also turns out to be a psychotic mass murderer. But then again she is a math teacher, so you can’t expect her to be normal, can you? “Math has been riddled with positively brilliant madmen,” as she explains at one point.

• The sympathetic characters, such as police lab technician Amelia, say things like, “I gave up on math after my second semester of algebra.”
• Susan, even though she presumably has at least a master’s degree in psychology, says, “I hadn’t taken a math class since junior high school.”
• The puzzle expert says, “I’m a word boy. Left brain. Math freaks are a whole different breed. And this doesn’t look like a real puzzle anyway. How can you solve an equation if you don’t have any of the numbers?

Bernhardt’s mathematical errors include confusing variables with unknowns and referring to expressions as equations. For example, on page 161, we have this excerpt:

It was another equation:

Bernhardt’s account of the Pythagoreans’ attitude toward the irrationality of the square root of 2 is also muddled. For instance, Esther, the professional mathematician who’s an expert on the Pythagoreans, says, “The square root of two was a problem with no solution.”

OK. That’s enough. Don’t bother reading the book.

Labels: books, math, teaching and learning

Continuing yesterday’s theme... There has been renewed interest in Larry Summers’s supposed sexist remarks. When Senator Obama (I almost said “President Obama”) announced that he would appoint Summers to be his senior White House economic advisor, bloggers and others revived the old canard that Summers believed that women were deficient in their math and science abilities. For instance, Wendy Hansen in the LA Times wrote as follows:

The notion that boys are better than girls at math simply doesn’t add up, according to a study being published Friday in the journal Science. An analysis of standardized test scores from more than 7.2 million students in grades 2 through 11 found no difference in math scores for girls and boys, contradicting the pervasive belief that most women aren’t hard-wired for careers in science and technology.it

The study also undermined the assumption — infamously espoused by former Harvard University President Lawrence Summers in 2005 — that boys are more likely than girls to be math geniuses. Girls scored in the top 5% almost as often as boys, the data showed.

The trouble, of course, is that Summers did not espouse that position. Summers did observe that there is a gender disparity among the very top mathematicians and scientists (as no one could deny) and proposed that it would be helpful to investigate why: to what extent is it genetic, and to what extent is it societal?

Although this question is precisely what a scientist in a research university should ask, it created great controversy. The very act of asking the question suggested to many people that Summers was assuming that women are less capable than men in math and science.

Summers had great strengths as President of Harvard, especially in his sponsorship of the Crimson Summer Academy (where I teach in the summers, so I can’t claim objectivity) and in his insistence on huge scholarships for low- and moderate-income students. Unfortunately his lack of social skills caused him to lose support in the Faculty of Arts & Sciences (though not in Harvard’s ten other schools), and he was forced to resign. Some say he has Asperger’s Syndrome; he might well, but who knows? Anyway, he has returned to being a professor of Economics at Harvard, and now he is a top advisor to Obama. I don’t believe that he’s sexist, but he clearly has some problems communicating his ideas; nevertheless, he is a distinguished economist with a lot to contribute, and he is an excellent pick for the Obama administration.

Labels: life, math, teaching and learning, Weston

So why is it that the top two mathletes on Weston High School’s Math Team are freshmen girls? And a year young for their grade, at that?

Check out the situation from ten months ago. But it’s only two data points.

Labels: math, teaching and learning, Weston

I recently installed an unusual application on my iPhone: Ocarina. This program turns your iPhone into a four-hole ocarina, with the holes outlined on the iPhone’s touch-sensitive screen. But the really cool thing is that you actually blow into your iPhone to simulate blowing into the ocarina! Try it: it really works!

Many of my students agree that this is really cool, although some adults think that it’s a waste of time. I don’t really understand their point of view, since they are likely to spend their time on useless things like watching football games, but anyway....

The reason that I had to demo this product for my precalculus class is that we have just finished studying the use of trigonometric and exponential functions to model musical sounds, and one of the issues that arose is what the dependent variable represents when graphing an oscilloscopic rendering of a tone. Sure, if Middle C is 262 Hz, we notice that the frequency is 262 cycles per second since the period of the independent variable is 1/262 of a second. But what does the y-axis represent? We say pressure, and we may measure it in pascals or mV, but what does this have to do with the loudness of a sign? The direct analog construction of the iPhone ocarina application — with no intermediate abstractions of digital software — provides a clear understanding of this phenomenon, since the user’s breath blowing into the iPhone moves the membrane of the microphone, illustrating pressure in a literal way.

Labels: math, teaching and learning, technology

Should high-school math classes be teaching Excel? Or, more generally, should we be teaching spreadsheet use — and Excel just happens to dominate the market? We have been exploring these issues at Weston High School.

Certainly the right point of view is to teach spreadsheets rather than Excel, although the dominance of Excel means that it will inevitably be hard to distinguish it from spreadsheets in general. Anyway, I think the real questions are the following:

1. What can spreadsheets add to high-school math?
2. When and in what order should we teach various spreadsheet techniques?
3. Should we teach non-mathematical skills such as formatting cells, creating headers, etc.?

A side point is that knowledge of Excel will be very helpful to our students in college courses and in a great many job situations, so somebody should teach it in high school. I suppose the responsibility falls to the math department by default, even if it isn’t really math, though that conclusion makes me uncomfortable. The only other likely places are the science department — since science courses also do a fair amount with Excel — and the business department, though that wouldn’t touch all students by any means.

There are a lot of issues, large and small, in using Excel. The notion of a variable is quite different from that of a variable in mathematics. Order of operations is important, and that’s almost identical to what we do in math and can therefore reinforce it. Sorting becomes important, and that’s a mathematical concept that comes up a lot in computer science courses but rarely in pure math. Graphing and regression are possible in Excel, but both are clumsy. Sometimes the syntax can be confusing, such as beginning a formula with an equals sign, but getting used to different syntactic conventions is a useful mathematical skill. A great many mathematical functions are built into Excel and can therefore be reinforced when we use spreadsheets. Perhaps most important is the level of abstraction involved in creating formulas that can be dragged vertically or horizontally independent of pre-existing data.

We’ve just completed an Excel activity in my college-prep Algebra II class: Saving for College. My hope is that this activity will not only give some experience with spreadsheets (that’s the secondary goal) but will also reinforce some of the important concepts in exponential functions (that’s the primary goal).

Incidentally, it turns out that many of those who teach Excel only have a very narrow view of this powerful piece of software. (As much as I don’t like Microsoft, I do have to admit that Excel has a great many excellent features and has an enormous number of useful options, many of which I barely know myself.) So I wonder if Weston needs to have a workshop in which we try to plumb the depths of Excel and figure out which aspects of this software will be most useful in teaching high-school math.

Labels: math, teaching and learning, Weston

Earlier this month I participated in a fascinating two-day seminar on The Big Ideas of Algebra, taught by Deborah Hughes-Hallett and sponsored by Teachers as Scholars. Although I undoubtedly talked too much, I figure that that was because I had a lot to contribute. Nevertheless, I learned a number of valuable things from the seminar, and they will be as useful both at Weston High School and at the Crimson Summer Academy. Mostly this was a matter of focusing my attention on things I already knew and believed but wasn’t thinking much about. In particular, I am now convinced than ever that we need to pay a lot more attention to symbolic literacy (understanding the symbols and combinations of symbols used in algebraic expressions) and to the distinctions among expressions, equations, and functions. Because of my linguistics background I have always believed that one important way to look at mathematics is that it is a language — in fact, that’s probably what got me to make a smooth transition from linguistics to math — but I hadn’t thought enough about the implications of that point of view when teaching math.

In addition, among the other matters we talked about in the seminar, we discussed the assignment of partial credit for work in solving a problem — more on this later, but it definitely reflects one’s views on what the big ideas are — and whether the study of algebra is distinct from (and prior to) the study of functions. More on that later as well.

Labels: math, teaching and learning

By this point I’ve taught simplified versions of the RSA algorithm to ten different cohorts of teens: four years’ worth of Honors Algebra II students at Weston High School, juniors for four summers at Crimson Summer Academy, and two years’ worth of college-prep Algebra II students at Weston. Tweaking the details as I’ve gone along, and benefitting from changes in technology, I’ve learned a lot from these experiences.

There are several types of benefits for the students. Some benefits are conceptual, involving understanding ideas about public-key cryptography, ranging from technical questions like how a cryptosystem can use a public key, why that’s necessary, and why it’s secure, to public-interest issues like whether we can trust so-called “secure” financial transactions on the Internet. I could have predicted these benefits; the unit was designed to try to achieve them, after all. And I could have predicted the success of some of the more concrete mathematical benefits as well, since RSA involves exponentiation, prime numbers, modular arithmetic, factoring, representation of characters as integers, and other operations with numbers. But a third type of benefit is more of a surprise: because the technical details are complicated, and even a single mistake can doom the effort to failure, most of my students have been doggedly persistent in paying attention to details and getting them right. Too often we can fall into the trap parodied by Tom Lehrer: “The important thing is to understand what you’re doing rather than to get the right answer.” We give so much partial credit that a student can get a B without ever producing a correct result. Part of the way that my colleagues and I have avoided this trap with RSA is that our sequence of activities and assignments concludes with a two-way exchange of messages: each student sends me a message using my public key, and I reply with a message using the student’s public key. This gives everyone practice in figuring out their private and public keys, enciphering, and deciphering. But fewer than half the kids get it right the first time, since there are so many opportunities to make mistakes. Unlike the usual math problem, they can’t settle for having a couple of points taken off; the message simply won’t work. So they try over and over again — sometimes four or five times — in order to get it right. If they don’t, I can’t read their message, or they can’t read mine.

If you’re interested in checking out how I’ve simplified RSA so it can be studied at the level of Algebra II, take a look at my worksheets, starting at RSA Phase One. (If you keep incrementing the “1” in that URL, you’ll find the next three worksheets at the expected URLs.)

Labels: math, teaching and learning, technology

I want my CSA sophomores to understand many sorts of visual representations of data — tables, charts, graphs, etc. — especially in the context of elections, since we’re applying mathematics to models of voting. This summer, of course, we have a wealth of material to choose from, both prospective (polling data) and retrospective (historical data). Perhaps the best source for both is FiveThirtyEight; Electoral Projections done right; the only downside is the complexity of their representations. Being a teacher, I try to turn this complexity into a teachable moment, but it’s still necessary to figure out how to help kids digest and understand the representations. Here are two examples:

1. One day we looked at FiveThirtyEight’s Electoral History Charts. Here's a small part of one chart, so you can see what I mean about complexity:
   
   
   This is pretty overwhelming even for college students, so it certainly can’t simply be thrown without explanation at kids who’ve just finished their freshman year in high school. After digging into it, you realize that it’s a wonderful (and almost successful) attempt at representing several different dimensions of information in one two-dimensional table — but it does take some work:
   
   
   • It’s clear what’s going on horizontally. Each row represents one state.
   
   
   • The vertical axis is almost as clear; one realizes from context that each column is a presidential election fro 1948 through 2004.
   
   
   • By now we’re all familiar with the symbolism of red states and blue states, so of course the red cells show elections where the state voted for the Republican presidential candidate, and the blue cells where the state voted for the Democratic candidate.
   
   
   • But there are different shades of red and blue: the darker and more saturated colors (such as Maine in 1956 or Massachusetts in 1996) represent landslides. The lighter or less saturated the color, the closer the election was (in that state); really pale colors, almost indistinguishable from white, show races that were so close that they were almost a tie.
   
   
   • Finally, there’s a letter and a number in each cell. These show the winning party and the margin by which it won. For example, in 1984 the Republican candidate (Reagan) took Connecticut by a 22-point margin.
   
   The class is already divided into six groups, each with five students. To get them to dig into the data and to understand the interpretations, I assigned each group a region of the country and asked them four questions:
   
   
   • For each of your states, figure out the average margin by which the Republicans won those states in the past ten elections (1968–2004). One way to do this is to count the “r” numbers as positive, the “d” numbers as negative, and average them. Do this separately for each state in your region(s), and list the results.
   
   
   • If your region(s) contained both positive and negative averages, find the state with the most positive average and the state with the most negative average. List them, along with their averages. If your region(s) contained only positive or only negative averages, find the states with the smallest and largest averages, and list them, along with their averages.
   
   
   • In which year did your region(s) vote most heavily Republican? Find out who the presidential candidates were in that year.
   
   
   • In which year did your region(s) vote most heavily Democratic? Find out who the presidential candidates were in that year.
   
   
   This all took a long time, but it worked out well. Each group then reported their findings back to the whole class, after which we were able to make some horizontal and vertical generalizations: What happened (nationwide) in 1984, where we see a sea of red? What happened (chronologically) to Vermont? And so forth.
   
   I hear you asking what the chart does when a state votes for a third-party candidate. Using the RGB principle, the site colors the cell green. For example:
   
   This, of course, gave us an opportunity to talk about third-party candidates. Students were suitably amazed when they learned how a segregationist candidate (Strom Thurmond) could have won Alabama by a huge margin in 1948, despite what should have been a large number of black voters.


2. These charts are abstract and difficult, partly because they’re so textual. So what about a more visual representation, such as a map? Maps are a great way of presenting this sort of information, though they really have to be snapshots in time and are therefore not suitable for longitudinal historical data. It turned out that the maps we used had a great side benefit: students learned a bit about geography (which isn’t much taught these days), since most of the maps omit the names of the states. Here were two of the maps we used, the one on the left showing the current polls for the upcoming election, and the one on the right showing the results of the 2004 election:
   
   
   These maps let me ask questions like these:
   
   
   • Of the states that Kerry won in 2004, which ones are now likely to switch to the Republican candidate (McCain) according to the map? What is the total of their electoral votes?
   
   
   • Of the states that Bush won in 2004, which ones are now likely to switch to the Democratic candidate (Obama)? What is the total of their electoral votes?

More on simulations later. Stay tuned...

Labels: math, teaching and learning

As I mentioned in my post of four days ago, my sophomores at Crimson Summer Academy (CSA) are currently studying models of voting. While I’m trying to move them away from cuteness as a criterion and toward serious consideration of candidates, my mission is more mathematical than political. So I want my students to learn about the mathematical methods involved in various answers to our Big Question for the summer: “What if nobody gets a majority?” We’re a democracy (more or less), which means that the majority should rule (more or less) except where minority rights are involved. So we study all sorts of real-life voting methods that soon-to-be voters will have to confront:

• simple plurality, as in elections for Massachusetts governor
• two-round runoff, used in much of the South and elsewhere
• preliminary-and-final (very close to two-round runoff), used in elections for Mayor and City Council in Boston
• Plan E Proportional Representation, used in elections for City Council in Cambridge
• the Electoral College, used in elections for president of the United States

The problem is that even with the current heightened interest in Obama, teens still aren’t going to pay much attention to candidates for offices other than president. They don’t care about Cambridge and Boston city councillors. So, how do we grab their attention? What’s something in which they have a lot of interest and about which they have a lot of knowledge? Several years, ago one of my teaching assistants (“mentors” in CSA jargon), himself a Harvard undergraduate, made an excellent suggestion, which we’ve followed ever since: hold an election for three Supreme Musical Artists of the Past Fifty Years.

So that’s what we do. We collect nominations on the first day. Then, at various points throughout the course, we hold elections using the different methods listed above, always starting with the same nominees. Maybe the results will differ, depending on the method. Needless to say, the results among 15-year-olds bear no relation to the candidates for whom I would have voted. In fact, about a third of the nominees were individuals or groups that I hadn’t even heard of. Anyway, with no further ado, here were the winners from the two-round-runoff method, each listed with the number of votes in the second round (stay tune for the results of other methods later in the summer):

37	Tupac Shakur
32	Lil Wayne
31	Chris Brown

And the runners-up in the second round were...

23	Michael Jackson
15	The Beatles
15	Jonas Bros.

Labels: Dorchester, math, teaching and learning

Take a close look at the yellow bars in this bar chart:

Much to my surprise, it turns out that subway fares in Boston have remained nearly unchanged for 110 years when adjusted for inflation, especially when you ignore the short-lived drop to ten cents in 1950. This chart comes from a fascinating document entitled “Report on the Proposed 2007 MBTA Fare Restructuring and Fare Increase,” published by the MBTA in July of 2006.

Why am I so unusually interested in this geeky document? Perhaps it’s because I’m a math teacher — or maybe it’s because I’m building a model railroad based loosely on the T, though I think there’s a lurking variable there that explains both phenomena. In any case, the general public should also become aware of it, especially with all the recent complaints against the MBTA. Many, perhaps even most, of those complaints are certainly justified, but isn’t it interesting that the fare hasn’t actually risen significantly in real dollars in over a century? Using my trusty TI–83 Plus calculator, I computed a linear regression for the data, with years since 1900 as the input values and the fares in 2005 dollars as the output values, excluding the anomalous dip in 1950. The slope indeed turned out to be indistinguishable from zero — actually 0.0013 to two sig figs. Even when I included the 1950 fare it was still only barely higher, being 0.0017. The correlation coefficient wasn’t good, but it’s still quite compelling that the best-fit line should be nearly flat.

Labels: Dorchester, math, model railroads

As I mentioned in my post of two days ago, the sophomore component of the summer course I teach at Crimson Summer Academy focuses on models of voting. Although the emphasis is primarily on applied mathematics, the 2008 course was destined to be more political than the previous summers’ versions. For the first time it was clear that voting would not be merely an academic exercise for these teens who won’t be able to vote for three more years. Ordinarily I try to avoid talking about politics with students, and in the typical math class it’s easy enough to avoid it. But in these circumstances there was no alternative but to take this as a teachable moment and embrace the students’ natural interest in the presidential race. Keep in mind that these are inner-city kids who attend public and parochial schools in Cambridge and Boston, with the largest contingent living in Dorchester and Roxbury.

So I started by polling the students about their political views. Here were the results:

0	John McCain
0	Ralph Nader
25	Barack Obama
4	None of the above
1	Prefer not to say

I guess we won’t invite McCain to speak to the class.

Labels: Dorchester, life, math, teaching and learning

Check out the Lawrenceville School’s amazing math applets! They provide links to class-demonstration applets that range from the unit circle and the sine function through transformations and vector addition all the way to slope fields and Riemann sums — not to mention the ever-popular Generic Applet, from which all the others can be built. Here’s a screen shot of the Sine and Cosine applet just to show one tiny example of what can be done:

Labels: math, teaching and learning

I have just finished reading Double Vision, by Randall Ingermanson. This science fiction thriller has a great concept, but the execution is disappointing. On the plus side, the novel speaks effectively to those of us who have worked in the computer industry, especially if we have any interest in computer science and physics. Knowing something about RSA and factoring certainly helps, but it isn’t necessary. Knowing something about quantum computing might also help — but since I know almost nothing about that field, how could I be sure? Anyway, the idea behind the book is fascinating, and the fact that the protagonist is a computer programmer with Asperger’s makes it fit into my accidental recent theme of Asperger’s Syndrome. However, there’s also the minus side: implausible characterization, poor writing (à la Dan Brown), unbelievable plot, and excessive Christianity. Worse yet, there’s a subsubplot concerning Jews for Jesus, and if anyone can explain to me how that organization is distinct from Christians, please let me know.

Labels: books, math, teaching and learning, technology

Barbara and I spent a few hours yesterday evening in Jamaica Plain. First we walked to the Axiom Gallery, which is hosting an intriguing Math and Art exhibit through April 27 right next to the Green Street T station (confusingly on the Orange Line, not the Green Line). Most of the art is worth seeing for its own sake, but the special appeal for me was the large number of connections with some of the topics that my precalculus class has been studying this year. The sculptures of Bathsheba Grossman are particularly striking. For example:

The works of Kevin van Aelst were also vividly related to what we’ve been studying, as his Dragon Curve, Sierpinski Arrowhead (made of Triscuits!), and Cantor Set made out of a fractal egg all show:
      
I was also intrigued by the works of Keith Peters, which could readily be modeled in NetLogo or StarLogo, even though he apparently used neither, and also by the works of J. Michael James, whose fractal condor was especially beautiful as it swooped around on a large screen.

All in all, definitely a worthwhile experience. I just wish the exhibit had been more extensive, so that I could have justified recommending it to my Weston students. Barbara and I spent 45 minutes there, but I think most of my students would feel done with it after 20 — hardly worth the trip from Weston. But it would be worth the trip from Dorchester, even if we hadn’t already been in JP (where Barbara works).

Anyway, after visiting the gallery, we walked to Cafe D, where we had a pleasant and delicious dinner. Crispy calamari followed by a fish taco and salad for Barbara; arancini followed by duck confit with a cassoulet of braised white beans, portabella mushrooms, and pancetta for me. With wine, tax, and tip, it all came to just under a hundred dollars, which seems to be par for the course these days. It might or might not be worth the trip from Dorchester (on the edge, in my judgment), but, as I said, we were there anyway.

Labels: Dorchester, food, math, teaching and learning, travel, Weston

I mentioned two days ago that I was going to watch N is a Number: A Portrait of Paul Erdős, a documentary that had been enthusiastically recommended to me by my former student, Kelly Mathislife. She writes that N is a Number is the “best movie ever” — although she does admit to a slight bias since Erdős is her hero.

So now I’ve watched it, with help from Netflix, which coincidentally delivered the DVD on Erdős’s birthday! This really was a genuine coincidence, since I had put it in my queue a couple of months ago with no idea when it would rise to #1. Anyway, Kelly knows that I certainly intend no disrespect toward her when I point out that of course she was exaggerating; N is a Number isn’t quite the “best movie ever.” It isn’t even even the best documentary ever. But it’s definitely a well-made, captivating documentary that should be watched by every math teacher, math student, and mathematician. It becomes totally clear that Erdős meets Paul Graham’s criteria that I discussed two days ago: absolute honesty and caring obsessively about his work.

Erdős, who died 12 years ago at age 83, was one of the greatest mathematicians of the 20th Century and certainly the most prolific; he is best known for his peripatetic life style, having had no fixed abode and collaborating extensively with hundreds of other mathematicians wherever he traveled. The movie is successful at vividly letting the viewer know the kind of person Erdős was, portraying him in person and through the eyes of his collaborators. Fortunately the filmmakers were willing to use subtitles extensively, since the accents of various Hungarian mathematicians (and others) could get in the way of ready understanding, even though almost everyone in the documentary was speaking English. As a math teacher, I thought there was a bit too much of an emphasis on anecdotes, but that’s a small cavil; I use anecdotes myself in similar ways, and I recognize that it’s the best way for the film to appeal to a general audience, who wouldn’t want to watch or listen to lots of mathematics.

I want to quote a couple of snippets out of N is a Number. One comes from Ron Graham — another Graham! but no relation to the aforementioned Paul Graham, as far as I know — who has a major role in the movie:

When mathematics appears in print, it’s theorem, proof, theorem, proof, but when we’re doing math it’s a completely different thing. It’s three or four people sitting around with cups of coffee, a pad of paper, throwing ideas back and forth, making a lot of conjectures, most of which turn out to be completely false.

That’s what should happen from time to time in our math classes, but it almost never does, even at Weston, except in last year’s Friday-afternoon optional after-school math get-togethers.

The other snippet comes from Erdős himself:

We’re trying to read the pages of The Book. We don’t create mathematics, we’re just trying to read the pages of The Book.

How Platonist can you get? This is clearly the right attitude toward the mathematical endeavor!

Labels: math, movies, teaching and learning, Weston

As I was reading Paul Graham’s essay, “Some Heroes,” it struck me that I’ve never liked being asked who my heroes are. In his second and fourth paragraphs, Graham reflects on the question itself:

I’m not claiming this is a list of the n most admirable people. Who could make such a list, even if they wanted to?

...

When I thought about what it meant to call someone a hero, it meant I’d decide what to do by asking what they’d do in the same situation. That’s a stricter standard than admiration.

I had never thought of that criterion before, but perhaps it would unstick me. Then I thought of the statement from one of my former students that Paul Erdős is her hero. [Brief aside: it’s tough to get the correct diacritic over that o. The natural tendency is to try for an unlaut — Erdös — especially since umlauts are relatively easy in HTML. But in Hungarian the diacritic looks like a double acute accent rather than an umlaut, producing a character with Unicode ID 0151. Thus you want “&#” followed by “x0151;” in HTML. End of aside.] So I wondered whether Erdős would fit the description in Graham’s next paragraph:

After I made the list, I looked to see if there was a pattern, and there was, a very clear one. Everyone on the list had two qualities: they cared almost excessively about their work, and they were absolutely honest. By honest I don’t mean trustworthy so much as that they never pander: they never say or do something because that’s what the audience wants. They are all fundamentally subversive for this reason, though they conceal it to varying degrees.

More on Erdős after I watch the movie about him. But note that Graham’s characterization is not a definition of “hero”; it’s simply a comment on two of their properties. Graham’s twelve heroes are Jack Lambert, Kenneth Clark, Larry Mihalko, Leonardo da Vinci, Robert Morris, P.G. Wodehouse, Alexander Calder, Jane Austen, John McCarthy, the Spitfire, Steve Jobs, and Isaac Newton. Could I make a similar list (though surely not duplicating any of Graham’s)?

I don’t think so.

But it did make me think about the issue. Which people have influenced me to such an extent that I would consider them to be my heroes? Would I really “decide what to do by asking what they’d do in the same situation”? Would my list consist of people who “cared almost excessively about their work” and “were absolutely honest”?

I suppose Isaac Asimov, Socrates, Charles Darwin, and Bertrand Russell would come to mind first. And maybe Johann Sebastian Bach. And probably Martin Gardner and Noam Chomsky. And it’s a cliché to put one’s mother and father on such a list, but it’s a cliché for a reason, so I will do that as well. And shouldn’t Shakespeare and Ibsen be on the list? And perhaps James Joyce? Well, that’s twelve, but I’m not convinced. This bears more thought...

Labels: books, life, math, movies, teaching and learning

Two and a half years ago I wrote a brief negative review of Bringing Down the House: The Inside Story of Six M.I.T. Students Who Took Vegas for Millions, by Ben Mezrich. I suggested that the account seemed to be fictional (even though it claims to be non-fiction) and that it “alternates between melodrama and tedium.”

Now they’ve gone and made a movie of it, 21. The plot outline on IMDb asserts that the movie is a “fact-based story,” But Drake Bennett’s article about it in the Boston Globe has this comment on the original book:

Bringing Down the House is not a work of “nonfiction” in any meaningful sense of the word. Instead of describing events as they happened, Mezrich appears to have worked more as a collage artist, drawing some facts from interviews, inventing certain others, and then recombining these into novel scenes that didn’t happen and characters who never lived. The result is a crowd-pleasing story, eagerly marketed by his publishers as true — but which several of the students who participated say is embellished beyond recognition.

I haven’t seen the movie yet, but the Globe article certainly makes me skeptical. Read the article, not the book.

Labels: books, math, movies