Learning Strategies

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

Two years ago I promised a review of the dim sum at the then-new branch of Chau Chow in Dorchester, but I don’t think I ever wrote one. So here, at long last, is that review.

Barbara and I ate dim sum there this morning (for what must be at least the sixth time — so you can see that we like it). Chau Chow serves traditional dim sum, where the servers roll carts around the restaurant and you order small quantities (Chinese tapas?) by pointing, not from a menu. There are, of course, both advantages and disadvantages to this system: aside from being authentic and just generally cool, the rolling-cart method has the advantage that you can see what you’re getting; it has the disadvantages that the food can sometimes come in very rapid succession, and you don’t always know what you’re getting, especially when the server speaks little English or very heavily accented English. At Chau Chow there are several servers in this category, but they’re all friendly and willing to try. The food is quite delicious, at least to these moderately educated Western palates. I don’t know what customers at the extremes would think — either the extreme of wanting totally Americanized food or the extreme of wanting nothing familiar. Perhaps both of those groups would be disappointed.

There are several dishes about which Barbara and I can agree: we’re both very fond of them. This morning we had scrumptious pork-and-shrimp shumai, unctuous eggplant that’s probably bad for us, yummy ground pork dumplings, the always delicious fried taro cakes (with a bit of shrimp in them), two different kinds of lovely shrimp-and-scallion dumplings, some not-to-be-missed lobster dumplings (yum!) — all of those were items we both loved. Do you begin to detect any themes there? In addition, we had scallion dumplings (which Barbara liked more than I did, since I’m put off by the flavor that steaming imparts to scallions) and stuffed mushrooms (which I love, but which have a texture that doesn’t appeal to Barbara). Needless to say, this was too much food, so we took quite a bit home to reheat for breakfast and/or lunch tomorrow. The entire bill came to $42.00 including tax and tip, which sounds like a lot for breakfast or brunch but was actually quite reasonable when you consider all the leftovers it provided.

So go to Chau Chow, especially if you have a group of more than two people, so you can sample more choices. Maybe you’ll try the chicken feet.

Labels: Dorchester, food

One day I walk into a Russian grocery store in Watertown, and the owner starts speaking to me in Russian; I don’t understand a word. Another day I walk into a Russian grocery store in Waban, and the employee at the register starts speaking to me in Russian. Waiting to check out a book at the Boston Public Public Library, I can’t understand a question from the next patron in line, because — you guessed it — the question is in Russian. Buying new glasses at LensCrafters, I remark to the optician that I assume from his name that he must be Russian, and he says yes and that it’s clear that I am too.

Do I look Russian? Apparently I do, though I never thought so. I suppose it isn’t surprising, since that’s what most of my ancestry is. But I speak only about 20 words of Russian, and the only phrase that’s really useful to me is, “Я не понимаю.”

Labels: life, travel

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

Instead of being out in the thunderstorm this afternoon, I attended a beautiful concert performance by my fellow Dorchesterite and former student, coloratura soprano Zakiyyah Sutton. (Yes, I had to look it up too. I used to know what coloratura meant, but I had forgotten.) The concert was held at the mostly white Old South Church in Copley Square, but was actually sponsored by the Concert Committee for Young People’s Artistry and Education of their sister UCC Church, the predominantly black Eliot Congregational Church in Roxbury. As you’ll see, this racial distinction turned out to be relevant.

Zakiyyah, who just graduated from the Boston Arts Academy, was a student of mine for two summers at Crimson Summer Academy and will be attending Wellesley College in the fall. She sang an amazing 14 numbers in this concert: ten as solos and four as duets with fellow performer Jamal Hoskins, a tenor, who performed five solos (and, of course, four duets). The Eliot Studio Singers accompanied a few of the songs as well.

By far the best performance was Zakiyyah’s rendition of the aria “Der Hölle Rache” from Mozart’s Magic Flute. This was actually her second song in German, since she had opened the concert with “Bist Du Bei Mir,” attributed to J.S. Bach in the program but apparently actually written by Gottfried Heinrich Stölzel, or so they say. Zakiyyah’s other two pre-intermission solos were Scarlatti songs, both beautifully sung in Italian; she was so convincing in both languages that it was only afterwards that I found out that she actually doesn’t speak either of them.

The second half of the concert was less familiar to me, including a song from Aladdin, one by Stevie Wonder, and several gospel numbers, only two of which I knew. Culturally speaking, I found the performance eye-opening in several ways, starting with the fact that I was in a small minority in the audience (there were probably only six or seven whites there) and I have very little familiarity with the traditions of the black church, both for racial and religious reasons. Sure, I know about them second-hand from books, plays, and movies, but it’s something quite different to be immersed in the black church in person. The differences became vividly evident when Zakiyyah movingly dedicated a song (“His eye is on the sparrow”) to her ailing father and then broke down when she started to sing it. The audience was just so supportive of her, and in a way that no white audience could have been. I don’t meant to suggest that a white audience wouldn’t have been equally supportive, because of course they would have tried to be — they just wouldn’t have been able to show it very well. The interaction between performer and audience was just so meaningful and effective in this context.

About ten minutes later, Zakiyyah got back up and said that she had recovered and wanted to sing the song the way it should be done. Actually, I thought she had sung it perfectly well the first time — she did manage to get through it successfully with the aid of the audience — but I have to admit that the second rendition was truly beautiful, and I’m glad she decided to do it.

In general, the solos were much more effective than the duets, perhaps because Zakiyyah was definitely the stronger performer. But their closing number, a duet version of “Amazing Grace,” was moving and perfect.

I’ll close with a couple of non-musical notes: the opening remarks by Old South Church Associate Minister Quinn Caldwell included an African proverb that definitely resonated with me: “If you want to walk fast, walk alone; if you want to walk far, walk together.” This speaks to me in part because of what it says about teaching. (More about that in a later post.) Also, I was struck by the explicit recognition of two judges in the audience: Judge Leslie E. Harris and Judge Milton L. Wright, Jr.; the latter turns out to be “a gifted singer and writer of a musical production as well as a lawyer.” The musical director from the Eliot Church pointed out that performances like this one show “Roxbury on the good side,” in contrast to what they usually see in court. While Zakiyyah is from Dorchester, the point is still completely valid and contrasts with what we usually hear on the news. Crime is news; music isn’t.

Labels: Dorchester, life

It was too hot to cook today, so Barbara and I went to the Ashmont Grill, along with our friend Cheri. This was our third dinner visit there in the past six months or so (in addition to a couple of brunches). I didn’t post anything about the previous two dinners, since I was waiting for a tie-breaker, and now we have one.

First of all, what happened two visits ago? We had been disappointed at that time, because earlier experiences at the Ashmont Grill had all been wonderful; but that time everything was mediocre. Vegetables were somewhat overcooked, meat was a bit dry and not hot enough, and service was haphazard. Next time would be better, we hoped. And indeed it was. So which was the real Ashmont Grill? I am pleased to report that the one disappointing experience was an anomaly, and all seems to be well. The three of us sampled a variety of items on the menu, and there wasn’t a false step among them: crisp, light calamari, whisked to us from the fryer without spending time under a heat lamp; plump, fresh, garlicky mussels in a red pepper sauce; a top-quality hamburger cooked exactly to order; fall-off-the-bone short ribs in not too much sauce; excellent home-made cole slaw; hot, thick, sinfully rich home-made onion rings; hot, fresh, cornbread; and a nice bottle of Côtes du Rousillon.

The prices, of course, are significantly lower than those at Chris Douglass’s other restaurant, his flagship Icarus, which is one of the very best restaurants in Boston. But it still seems a bit expensive for an informal, low-key, neighborhood-type place. Oh, it’s still definitely worth it, but don’t expect cheap. The food and service are great; now if only the prices were a little lower...

Labels: Dorchester, food

Just getting around to blogging this, but there was a fascinating article a few weeks ago in the Boston Globe, made all the more relevant to me because it mentioned several of my Weston students and was written by the mother of one of those students. Ellen Freeman Roth’s article, headlined “Not your father’s nicknames when teens talk to parents,” explored what kids call their parents and their parents’ friends:

Lisa and Michael Josephson of Old Greenwich, Conn., are Mama Jo and Papa Jo, names coined by their daughter’s friend. Timothy Sweet of Watertown began calling his father “Sweet Man” a dozen years ago on a Boy Scout trip. Sweet likewise has nicknames for his friends’ parents, including “Glenzo” for Glen and “Pina” for Patricia.

Sarah Switlik, 18, a Babson College student from Princeton, N.J., said her mother, Pam, wasn’t thrilled at first when Sarah called her P-Money. “Initially my mom said, ‘Really, Sarah,’ exasperatedly. Now when she’s texting she signs off, ‘Love, P$.’ It makes her feel like one of the girls.”

...

Caroline Gaulin, 22, of Greenwich, Conn., yelled “My bad, G-Dog!” to her father, Dan, during a basketball game to make light of an error she'd made. “After that we started calling him G-Dog,” she said. “Now he loves it.”

Teachers are almost always called by title and surname at Weston, but at CSA we’re all on a first-name basis. These customs run counter to expectations and fly in the face of the customs for naming of parents and parents’ friends, at least based on my predictions. There are probably some interesting class issues here. Although I grew up calling my parents “Mom” and “Dad,” I called all my other relatives and my parents’s friends by their first names: it was Lillian and Leonard, not Aunt Lillian and Uncle Leonard; Luke and Gen, not Mr. and Mrs. Garner. But Barbara grew up more formally, with Aunts and Uncles and surnames with titles. I’ll have to ask my CSA students what they do; I’ll predict big differences between Weston and Dorchester.

Labels: Dorchester, life, linguistics, teaching and learning, Weston

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

After many hours of listening — and I do mean many — I have finally finished the audiobook version of Interred with Their Bones, by Jennifer Lee Carrell. At times I wasn’t sure whether it was worth slogging through to the end, but I like to listen to something when taking a walk, and Interred with Their Bones held my attention sufficiently.

As you can tell, I am mostly unenthusiastic about this mystery novel. I am told that it was inspired by the DaVinci Code, but I guess I’ll never know, since I steadfastly refuse to read any more Dan Brown after suffering through Digital Fortress. They say that Carrell is a better writer than Brown, but that’s not hard, as I’m not the first to observe. In any case, Carrell combines implausible but exciting action with long academic passages about Shakespeare’s lost play Cardenio and the long-standing controversy about the authorship of Shakespeare’s plays. As something of an academic myself, I found these discussions reasonably interesting, though not very suitable to the audiobook format. The action scenes seemed more like a treatment for a subsequent movie. Shakespearean themes run throughout the novel, from the characters' professions as actors, directors, and professors through the Shakespearean locales and staged events intended to mimic Shakespearean scenes to the characters’s names (Rosalind, Kate, Henry, Athenaide). Well, I don’t think that Athenaide is actually a Shakespearean name, but it could just as well be.

This novel belongs to the genre in which it’s hard to tell the good guys from the bad guys. No spoilers here, but suffice it to say that there are several plot twists that are not too severely telegraphed. Some reviewers found Interred with Their Bones to be fast-paced. I did not. I’ll admit that it’s breathless, and maybe that qualifies. Also, there’s far too much wanton killing. Read it if you’re interested in Shakespearean issues and academics, but don’t bother if you’re looking for thrills and actions.

Labels: books

My regular readers know that I teach Quantitative Reasoning (QR) at the Crimson Summer Academy (CSA) over the summer. (If you don’t how what CSA is, read my blog posts from May 7, 2007, and April 30, 2008.) The theme for all courses for the summer is The Student as Citizen, and the particular QR unit for rising sophomores is models of voting. More on that in a later post, since of course it’s especially interesting in this particular summer. Anyway, one girl’s succinct comment on Barack Obama yesterday was, “Obama’s cute.”

I didn’t know how to respond to this observation, but another girl in the class did: “Obama isn’t cute! Deval Patrick is cute!”

I decided to stay out of this conversation. Fifteen-year-old girls don’t see the world in quite the same way as I do.

Labels: Dorchester, life, teaching and learning

Which is a better place to buy soft-shell crabs, the Legal Seafoods fish market at Chestnut Hill or the Roche Bros. supermarket in West Roxbury? The obvious answer is Legal, of course. Everyone knows that fish markets are better than supermarkets, and Legal is well-known for selling impeccably fresh fish (not to mention fresh ads). But the Conventional Wisdom is apparently wrong. I’ve bought soft-shell crabs three times in the past month — once at Legal and twice at Roche Bros. — and the verdict is clear. The Roche Bros. product (at three dollars per crab) was tastier, fresher-tasting, and plumper than the Legal product (at six dollars per crab, though for slightly larger ones, I must admit).

On a slightly different matter, it's unfortunate that the Shaw’s and Stop & Shop markets in Dorchester can’t provide food and service of the quality provided by the West Roxbury Roche Bros., where the food is always good and the employees are both well-trained and friendly.

Labels: Dorchester, food