Learning Strategies

Epiphany:

A comprehension or perception of reality by means of a sudden intuitive realization (definition 3b in the American Heritage Dictionary)

So I guess I was visited by an epiphany this weekend: I realized that Sudoku has a surprising connection to systems of equations. Of course I’ve written separately about each of these — Sudoku on July 24 and August 6, systems on January 29 — but I hadn’t ever thought about any connection between them.

Now, however, as we’re starting our Algebra II unit on systems of equations, I have been thinking about how we characterize different kinds of systems. As is traditional, the Mathematics Teacher article that I described on January 29 characterizes any system of equations as having exactly one of the following descriptions:

• consistent independent
• consistent dependent
• inconsistent

You may wonder why we don’t have four possibilities, splitting “inconsistent” into “inconsistent independent” and “consistent dependent.” Dr. Math gives a fairly good explanation:

“Inconsistent” means no solution. Independent and Dependent both mean there is a solution, so they can’t ever go with Inconsistent because that would be contradictory.

So really there are only three possibilities: Consistent Dependent, Consistent Independent, and Inconsistent.

We ordinarily don’t even use “consistent” with dependent or independent, since once you know what these latter two words mean, you already know they are consistent, so it is enough to say the system is “dependent” or “independent.”

We usually use the word “consistent” when we are more interested in indicating that the system does have a solution, rather than indicating how many solutions it has.

OK, so how does all this possibly relate to Sudoku? What occurred to me over the weekend — while working on a Sudoku puzzle — was that the process of filling in any given cell in Sudoku yields exactly the same three possibilities, and with the same meanings. Consider, for example, the following fragment of a partially completed puzzle (showing the first row and the first column):



From the first column alone, we can determine that the upper-left cell must hold a 5. But the first row gives us no additional information; thus this system is dependent. On the other hand, consider this slightly different variation:



Both the row and the column are needed in order to conclude that the first cell holds a 5 and the last cell in the first row holds an 8.

Finally, if you make a mistake in solving a Sudoku — or if the puzzle was constructed or printed incorrectly — you can end up with an inconsistent system:

Labels: math

So how do you hide a password in plain sight when it doesn’t need to be particularly secret? For instance, imagine that you are using a hard-copy textbook for which the publisher also provides an online version. And the online version is, of course, password-protected. It’s not particularly a secret, since dozens of students can openly have access to it, but you’re not allowed to post it on your website if there’s any indication of what it’s a password to, since people could then use the online version for free without buying the textbook for the outrageous price that math textbook publishers charge. One thing you could do is say in your blog that the password starts with a number that’s 588 more than the course number and ends with Y4T_2DZV, with a hyphen between the two parts. Oh, you also have to replace the underscore with the letter that follows the block when the course meets. This becomes a good example of security by obscurity, since only those who have a right to know will have any idea what you’re talking about. But anyone who forgets the password will know where to find it.

Labels: technology

How do we think about systems of equations (or inequalities)? I know, most of us don’t think about them at all. But teachers and students of algebra certainly do. Whether we call them systems of equations or simultaneous equations, we traditionally think about them in a static way, as the word “simultaneous” suggests. For example,Wikipedia says this:

simultaneous equations are a set of equations where variables are shared. A solution consists of values for the variables which satisfy all of the equations simultaneously.

In other words, all the equations are given at the same time — i.e., simultaneously — and we find a common solution (if we are looking for a solution). If you prefer the more concise definition given by Mathematica, try this one:

Simultaneous Equations

A finite set of equations in the same unknowns of which the common solutions have to be determined.

Far be it from me to say that Wikipedia has better definitions than Mathematica, but in this case it certainly does, since the Wikipedia entry distinguishes between the use of an equation to state a relationship and the use of an equation to determine the value of an unknown. Mathematica ignores the former (and conceptually more important) definition. Far too many students — and even teachers — think of algebra as the study in which variables represent unknowns that have to be solved for, rather than the study of the relations among variables.

But that’s not really the point of this post. Whichever way you characterize equations, you also have to confront what simultaneity means. I always used to believe that it means what it says: we have a static view of a system, whether it’s equations or inequalities; all are true at once, like a snapshot.

But now, after conversations with my department head, I am thinking of a system in a dynamic way: each equation or inequality represents a constraint, and the constraints are imposed in time, not all at once. We have a video, not a snapshot. Consider, for examples, a system of three equations presented in an article by Joseph Ordinans in the February 2006 issue of the Mathematics Teacher:



Ordinans takes the standard emphasis on the geometric description of equations and presents it in a 3-D context: three planes that intersect in any of eight ways match the algebraic conclusions to be drawn from them (consistent or inconsistent, dependent or independent). This is the traditional, static view.

A dynamic view would take the equations one at a time:

1. The first equation provides a constraint. The variables can have (infinitely) many tuples as their values, such as (10, 3, 10) or (-5, 1, 1). But there are also (infinitely) many tuples that do not satisfy the first equation, such as (1, 1, 1). So the constraint partitions the set of all tuples of numbers into two subsets, those that satisfy the first equation and those that don’t. In other words, at this point the first equation represents one constraint. If we take a snapshot, or freeze-frame our video, we can capture the set of solutions so far.


2. Now the second equation provides another constraint. We can further subdivide our solution set to the first equation by finding the subset that also satisfies the second equation. Our video has moved on to its second frame.


3. Finally, the third equation provides a third constraint: frame #3. In this particular case, only the empty set remains. After imposing the third constraint, we find that no tuples satisfy all three.

One thing that I like about the sequential, dynamic approach is that it translates well when we have a system of inequalities, or even a system containing both equations and inequalities. The human brain is not great at considering multiple issues in parallel; most of us tend to think sequentially, rather than simultaneously.

Labels: math

Just got back from the Amherst Railway Society’s annual model railroad show, which is held in...no, not Amherst...Springfield, MA. Barbara wasn’t interested in going, so Meredith accompanied me. Although she isn’t a model railroader, her interests in crafts and technology combine to make model railroading relevant to her.

When we arrived, the first person we ran into was Colby Cousens, Weston High School’s tech support person. (And what’s the probability of that? I doubt that there’s any way to calculate it.) Literally tens of thousands of other fans were attending this huge show, which sprawls over three buildings of the Eastern States Exposition Fairgrounds, generally known as the Big E. The organizers describe the show this way:

...The event features real life railroads and scale model railroads, historical societies, travel agencies, art shows, flea market dealers, importers, manufacturers and photographers. Modelers’ exhibits will display outstanding handiwork on layouts ranging from the tiny Z scale which fits on a coffee table to a monster 80 foot N-Trak system. The Amherst Belt Lines, an HO scale model railroad empire, has become a show highlight. As of the 2005 show layout, It has grown to 15.8 scale miles of mainline (960 linear feet) on 78 modules and has the capacity of multiple train operation. The Southern New England O Scalers will show its huge O gauge railroad with 100 car freight trains and 7 unit diesels.
...
There are 3 buildings with over 4 acres of railroading of all kinds.

We spent three and a half hours at the show. That still wasn’t enough time to see everything, but I think we saw all the exhibits that I needed to see. Anyway, after all that time we were exhausted, so we couldn’t have stayed longer even if it weren’t 5:00 by that point.

Stay tuned to see the effects of what I learned and bought, including a working (animated) model of the famous Citgo sign in Kenmore Square, which was built in 1965 and is therefore perfect for my 1969 Boston layout.

Labels: model railroads

Catching up on last week’s television shows with the wonders of the VCR — soon to be replaced by the greater wonders of TiVo — I just watched the excellent January 18 episode of Law & Order, Heart of Darkness. I don’t know whether the plot was based on a true story, but I doubt it, since there was no disclaimer asserting that it was entirely fictional (normally an indication that it isn’t entirely fictional).

Usually we are concerned about suicide among teenagers — at least that’s what high-school teachers are concerned about — and of course we think about physician-assisted suicide among the elderly. This episode of Law & Order concerned assisted suicide, but not physician-assisted and not of a 16-year-old or an 86-year-old. The subject was a middle-aged journalist who suffered from depression and wanted to die.

Or did he? If it’s OK to assist someone in killing himself, what happens when he changes his mind? And was he in a position to make the decision in the first place, given that he was off his meds. That was because his girlfriend advised it, so what was her responsibility? A lot interesting issues were raised by t his show, made all the more relevant by the recent Supreme Court decision upholding Oregon’s physician-assisted suicide law in the face of the Bush administration’s attempts to meddle in state and personal issues. The Law & Order episode was obviously shot long before the Supreme Court decision, which, in a piece of serendipitous timing, was handed down the day before the episode aired.

Labels: life

Does the name of a course matter? At Weston High School we recently renamed our two-year college-prep precalculus sequence. The first course, taken primarily by juniors but always including a few seniors, used to be called Math 4. What does “Math 4” mean? Not much, actually. Back in the good old days, students didn’t take Algebra I in eighth grade as they do now. So the first math course in high school was naturally called Math 1. Then Geometry was called Math 2, Algebra II Math 3, and Precalculus Math 4. These uninformative names were logical and sequential.

Then Algebra I was pushed into the Middle School, and Precalculus was split into two years. Algebra I, Geometry, and Algebra II were given more informative names: Algebra I, Geometry, and Algebra II. But they were followed, of course, by Math 4 and Math 5.

An unfortunate side-effect was that many students elected not to take Math 5, since it didn’t appear to be the second half of a two-year sequence. And several of us found those names uninformative, to say the least. Not that “Precalculus” is a perfect name, since it tells you almost nothing about the course itself. All it tells you is that it prepares for calculus — in other words, it tells you something about the next course, not about the precalculus course. Nevertheless, it’s better than Math 4 and Math 5. So we have just changed the names of this two-year sequence. The old Math 4 is now called Precalculus, Part One; the old Math 5 is now called Precalculus, Part Two, with Statistics — since it not only finishes Precalculus but also includes a hefty dose of statistics. We are hoping that students who are looking for a strong preparation for calculus will realize that they ought to take both courses of this two-year sequence.

In the honors sequence, by the way, the two courses (except for the stats component) are combined into a single year, the course formerly know as Math 4 Honors (as you would expect). It’s now denoted by a strange symbol — no, that’s not right, it’s now called Precalculus, Honors.

And does it make any difference? I’m hoping so...

Labels: Weston

Currently I’m halfway through reading Playing with Trains: A Passion Beyond Scale, a memoir by Sam Posey. There’s a certain irony to the title. The word “passion” is accurate, for this book is truly about Posey’s deep enthusiasm and passion for building a model railroad layout. He spent 6000 hours on it, so it had better be a labor of enthusiasm and passion! But the word “playing” is largely ironic: the 16 years during which he devoted intense work to building his layout were anything but play.

Playing with Trains is not only about Posey’s passion for layout design and construction, but also about his family; many passages in this book are about model railroading as a family endeavor. Posey also writes about other aspects of his life, including his career as an architect and builder and his simultaneous career as a race car driver and ABC sportscaster.

The most poignant parts of what I’ve read so far concern his battles with Parkinson’s disease, which destroyed his ABC job and deeply affected his attitude toward model railroading. There’s a subtle and unnamed connection between Parkinson’s Disease and Parkinson’s Law in the chapter I just finished reading. (These are two different Parkinsons: James and Northcote.) Knowing that his degenerative disease would probably cut his life short and would almost certainly end his ability to build a model railroad long before he died, Posey vowed to prolong the projected completion date of the layout. But what if he wanted to spend four years and there was only one year’s worth of work? He realizes that the amount of work would expand to fill the time available for its completion, so there was nothing to worry about.

The real question is whether you have to be a model railroader to enjoy this well-written book. I think not. It’s perfectly possibly to become engaged in reading any work that portrays its author’s knowledge and passion, even about a subject in which the reader has no particular interest. This often happens in fiction — the works of Dick Francis come to mind, which captivate all readers, including those of us with absolutely no interest in horse-racing — and it can sometimes happen in non-fiction as well. It does here.

Labels: books, model railroads

On yesterday’s episode of NPR’s Living on Earth, Steven Cherry interviewed Gordon Bell about his project at Microsoft, called MyLifeBits. Bell is in the process of recording everything in his life in digital form:

Gordon Bell has captured a lifetime’s worth of articles, books, cards, CDs, letters, memos, papers, photos, pictures, presentations, home movies, videotaped lectures, and voice recordings and stored them digitally. He is now paperless, and is beginning to capture phone calls, IM transcripts, television, and radio.

Most of the story consisted of an interview with Bell, who is justifiably famous in the computer science community. It was clear from the tone that Cherry’s concluding words weren’t meant to scary, but judge for yourself:

In ten years time, all your life bits will easily fit onto two or three hard disks the size of matchboxes. Your smartphone-sensecam will dangle casually around your neck, snapping away. Can’t remember what you wore on that blind date last Saturday? How many glasses of Chardonnay you drank? Who you called on the phone the next day and what you talked about? Where you were, what you did every minute that weekend? Let’s just open up that database of yours, the matchbox containing your life bits, and take a look.

The dystopian nature of this predicted future reminded me of Jorge Luis Borges’s great short story, “Funes the Memorious,” as did Cherry’s words just before his conclusion:

Frank Nack, a Dutch computer scientist based in Amsterdam, has given a lot of thought to the issues raised by MyLifeBits. Do we always want to have at our fingertips the answer to each and every question about our past? The act of forgetting, he says, makes our life bearable, and is closely related to some essential cultural concepts, such as forgiveness and absolution. Would removing this human imperfection do more harm than good? Would we be as creative? As free?

But Cherry made no explicit or even implicit reference to Borges. If you haven’t read his story recently, or haven’t read it at all, perhaps I’d better quote from it, so you can see the connection. Here is a critical paragraph from the middle of the story, followed by one close to the end:

We, in a glance, perceive three wine glasses on the table; Funes saw all the shoots, clusters, and grapes of the vine. He remembered the shapes of the clouds in the south at dawn on the 30th of April of 1882, and he could compare them in his recollection with the marbled grain in the design of a leather-bound book which he had seen only once, and with the lines in the spray which an oar raised in the Rio Negro on the eve of the battle of the Quebracho. These recollections were not simple; each visual image was linked to muscular sensations, thermal sensations, etc. He could reconstruct all his dreams, all his fancies. Two or three times he had reconstructed an entire day. He told me: I have more memories in myself alone than all men have had since the world was a world. And again: My dreams are like your vigils. And again, toward dawn: My memory, sir, is like a garbage disposal.
...
Without effort, he had learned English, French, Portuguese, Latin. I suspect, nevertheless, that he was not very capable of thought. To think is to forget a difference, to generalize, to abstract. In the overly replete world of Funes there were nothing but details, almost contiguous details.

Yes, “to think is to forget a difference, to generalize, to abstract.”

Or, as Robert Pirsig wrote in Zen and the Art of Motorcycle Maintenance, “data without generalization is just gossip.”

Labels: life, teaching and learning

A fellow Dorchesterite, calling himself Trxckster — yes, the third letter is indeed an x, not an i — quotes visionary Alan Kay in his blog:

Today, science (a concern with what is real) is mixed with mathematics (a concern with what is true) is mixed with engineering (a concern with how something can be made). Each worker in each of these fields also partly works in the other two. Each field has a different temperament associated with it: mathematicians tend to be idealists, scientists realists, and engineers pragmatists.

For those who like graphic organizers — and this is certainly an appropriate context for one — we can put the above into a nice table:

	What is it?	Temperament
Science	a concern with what is real	realists
Mathematics	a concern with what is true	idealists
Engineering	a concern with how something can be made	pragmatists

Labels: Dorchester, math, teaching and learning

Dot!

Dorchester is becoming the city’s hip new destination after dark

says the headline above Johnny Diaz’s big story splashed over the front page of the Living section of today’s Boston Globe. Not Landsdowne Street, not the South End, says the teaser above the masthead on page one: Dorchester (“Dot,” for short, of course).

The article ends with a quotation from a South End resident:

A few years ago, I would have never gone out in Dorchester, because it was so unsafe. Now I go out here more than in the South End. It lacks the attitude. It’s my hangout.

Read the entire article!

Labels: Dorchester

I can’t resist quoting from Rudbeckia Hirta’s post in her blog today:

I have been dared by one of my colleagues to write one of the questions on my calculus exam in the form of a limerick.

This is especially challenging since the first exam only covers a few topics: parametric equations, secant lines and tangent lines, definition of limits, epsilon-delta, and limit laws.

Labels: math

For several months now, I’ve been determined to implement some version of David Allen’s compelling Getting Thing Done. His book by that title was one of those rare self-help books that immediately grabbed my attention and thoroughly convinced me that it was the right way to organize. Many people whom I respect showed convincing evidence on their websites that some version of Allen’s methodology — perhap a significant variation, but still following all his big ideas — was the most effective way to get things done.

So I tried step one, involving a physical inbox and 43 folders.

But I couldn’t make it work, because I couldn’t finish step one and thereby get to the regular weekly routine of the subsequent steps. I determined that the 43 folders aren’t right for me — my life isn’t primarily structured around paper that’s due on specific days or months well in advance — but I was still a believer in the rest of the system.

I still am.

But how do I make it work? How do I get off the dime and get through step one?

Labels: life

Many high-school teachers believe that so-called graphic organizers are helpful to students. Readers who are my age may wonder what a graphic organizer is. According to the North Central Regional Educational Laboratory,

a graphic organizer is an instructional tool used to illustrate a student or class’s prior knowledge about a topic or section of text.

Another definition comes from Enchanted Learning:

Graphic organizers (some of which are also called concept maps, entity relationship charts, and mind maps) are a pictorial way of constructing knowledge and organizing information.

Both organizations cite various examples, some of which are undoubtedly helpful in getting kids to organize and think about their knowledge. Like anything else, a particular graphic organizer might be more useful to some students than to others, but on the whole they are clearly a net plus.

But...

When should the graphic organizers be used? They’re great as class activities, as part of homework assignments, and as review for a test. And they’re surely helpful to students who create a graphic organizer for themselves. But should the teacher be in the business of providing one on a test in order to structure an answer to an essay question or a multi-step math problem? Some teachers say yes, on the theory that it helps students organize their work and therefore write a more successful essay or solution. But shouldn’t it be the student’s responsibility to organize his or her own work? More specifically, aren’t we supposed to be assessing the student’s competence in structuring an essay or solution as part of the objectives of any test?

Labels: teaching and learning, technology

Dorchester is actually getting some high-quality restaurants. A couple of years ago, the Blarney Stone in Fields Corner transformed itself from a typical Irish pub to an excellent restaurant. The new C.F. Donovan’s in Savin Hill serves inexpensive but reliably good food. The new incarnation of the Ashmont Grill (owned and managed by Chris Douglass, of Icarus fame) is a strong presence and a welcome newcomer to the Ashmont area, with many first-rate choices on its menu. And dbar just opened to mixed notices. Barbara and I had an excellent meal there — a bit pricey, but otherwise wonderful — so it would be natural to wonder why our friends’ reports and the review in the Globe weren’t quite so enthusiastic.

It’s all a matter of timing: dbar is something of a hybrid, being a regular restaurant before 10 PM and a nightclub after that time. Apparently you feel rushed if you arrive for a 9-o’clock dinner; the service isn’t calm and perhaps even the kitchen’s attention isn’t on the food. But we had dinner at 6:30, and the service was perfect — attentive, friendly, and relaxed. The food was equally good: great calamari, excellent steak, sushi-grade tuna cooked truly rare as ordered, delicious molten chocolate cake. So go to dbar — but go early!

Labels: Dorchester, food

Try doing a Google search on the pair of phrases “lottery tickets” “tax on the poor”; you’ll find surprisingly few hits. Change poor to stupid and you’ll collect a few more hits, but still only 507 (at this moment). I’ve heard both descriptions from many people, and even seen them on bumper stickers.

Perhaps uneducated would be the most accurate word. Let’s try that...

Only 12 hits now! That’s even more surprising.

On New Year’s Day, the MetroWest Daily News published a table of various data for the towns in their area, including the average lottery-ticket purchases per resident for each town. The lede to the accompanying article contained the following conclusion:

People in some of the poorest towns in MetroWest and the Milford region are among the biggest spenders when it comes to playing the Massachusetts State Lottery, a Daily News examination of lottery statistics found.

Residents of Blackstone and Bellingham, two of the poorest towns in this otherwise mostly affluent area, spent the most money on lottery tickets and could least afford to do so. Weston, the richest, spent the least. No one is surprised by this.

The lottery is supposed to be a good thing, since it helps fund education, and since its “taxes” are voluntary. But whom are we taxing?

Labels: life, Weston

One of my precalculus students (or is it the hyphenated pre-calculus?) thought that he was studying calculus. He figured that precalculus was a kind of calculus, just as differential calculus is a kind of calculus.

What does that prefix “pre-” mean, anyway? To him it meant something like “in advance”: prepaying is paying in advance; to preplan is to plan in advance (in contrast to postplanning, which is much easier); if you preheat an oven, you heat it in advance of cooking; and a pre-approved credit card has been approved in advance of applying. So why shouldn’t precalculus just be calculus in advance?

I, on the other hand, thought of the prefix “pre-” as meaning “before” or “preparatory”: precalculus comes before calculus, or it prepares you for calculus. There are plenty of examples of this meaning as well: prehistory comes before history; a preadolescent will be an adolescent, but s/he isn’t a type of adolescent; and a precancerous condition comes before cancer.

How do we tell the meaning of this prefix when we meet an unfamiliar word?

It’s amazing that anyone actually learns English.

Labels: linguistics

Numbers and Palindromes. No, not numbers that are palindromes: Numbers and Palindromes, the television show and the movie.

I wrote about Numbers six months ago; at that point I had only seen three episodes, and it would have been premature to offer much of a review. Since yesterday’s episode combined some interesting math with a good show, and since I’ve now been watching it for most of a season, it’s time for me to be more forthcoming. I now can feel confident in what was only a tentative opinion six months ago:

Most important, from my POV as a math teacher, is their portrayal of real mathematics, at high-school and college level, as something with genuine applicability. This is a revolutionary step for a commercial television network; even on PBS there are almost no mathematical applications that go beyond middle-school math. Putting it in a context that reminds one of Law & Order is a sure way to grab viewers’ attention.

Yesterday’s episode was a fine example. Based on Bringing Down the House: The Inside Story of Six M.I.T. Students Who Took Vegas for Millions — which I wrote about on October 2, and which was explicitly referred to in the episode — it took the mathematics of blackjack one step further by considering the algorithms and mechanisms behind how the cards are shuffled. Of course, in true Law & Order fashion, it augmented the so-called true story by wrapping it around a murder. Well worth seeing.

Palindromes is also worth seeing, but it’s definitely not for everyone. (Numbers probably is for everyone, despite the emphasis on mathematics. But why say “despite”? Math is for everyone too.) Anyway, although it’s something of a sequel to Welcome to the Dollhouse, Palindromes doesn’t in any way require the viewer to have seen the earlier film first. Director and screenwriter Todd Solondz is always challenging and provocative and never cheerful, so don’t watch either Welcome to the Dollhouse or Palindromes — and especially don’t watch Happiness — if you’re looking for a straightforward or upbeat movie. The protagonist, Aviva, is played by many different actors, young and old, black and white, and — according to both IMDb and Roger Ebert, though I don’t see it — female and male. We’re not supposed to judge people by appearances, so why should that bother us?

Labels: math, movies

I was excited to read about Xerox’s Language Guesser. If you can’t identify a sentence in a foreign language, just paste the sentence into their convenient type-in field, and the intelligent Xerox software will correctly guess the language.

Sounds like a good example of AI. So I typed a sentence in Haitian Creole that I found in Mountains Beyond Mountains:

pa ka konprann bagay ki pa senp

And sure enough, the artificially intelligent software identified it as...

Labels: linguistics

The problem below comes from the Education Records Bureau’s Independent School Entrance Examination (ISEE) for middle school students. How many things can you find wrong with it?

Thanks for Thane Plambeck for the pointer.

Labels: math, teaching and learning

Listen closely...

Labels: math

Just got back from Narragansett, RI, where we attended the wedding of two of my former students. They became high-school sweethearts ten or eleven years ago, and now they’re married! And so we have another B&B to report on (see also my post of December 28): Barbara and I stayed at the Blueberry Cove Inn, a warm and lovely B&B that unfortunately suffers by comparison after the Painted Lady. It’s not that there’s really anything wrong with it, but a room of 200 square feet can’t possibly match a suite of 575 square feeet — and a pretty but ordinary house can’t really match Victorian splendor paired with a billiards room, a jacuzzi, a DVD player, etc. (We also had to pay $25/night more at the Painted Lady than we paid at Blueberry Cove. The difference was well worth it.)

Labels: travel