Applied Math Curricula

This article from a 1993 edition of the Phi Delta Kappan somehow summarizes hundreds of pages of a digest of hundreds of research studies that wasn’t written until 1999 in only seven pages—and a drawing takes up most of the space of the first one. I agree with almost everything in it, except the bit about math movies at the end. Also, I might take issue with some of the suggestions curriculum reform. For example:

In addition, topics not previously explored in traditional curricula must be added. Changes from an agrarian society to a technical/information society demand the literate citizens be familiar with such concepts as mathematical modeling, discrete mathematics, and data analysis. An example of discrete mathematics would be the decision process whereby a street sweeper is routed through a town so that the fewest number of streets possible will end up being swept twice.

Now, sure, we ought to rethink what we teach our kids. The author is absolutely correct. The standard American math curriculum hasn’t budged much in the past century despite radical changes in American society, culture, and technology. To ignore the passage of time is stupid. But I’m not entirely sure what the author is proposing here. To me, she’s suggesting we load up our kids with graph theory and Fourier analysis. This is great if we want all our kids to go into algorithmic optimization, data compression, and signal processing. Maybe that wouldn’t such a bad thing. In my experience, Fourier analysis can be hard. But such a suggestion presupposes that everyone, everywhere will end up in the high-tech sector. Even then, my friends who do lots of computer science [most of whom actually work in finance] don’t rely on mathematics proper so much as things in computer science, and then, that they picked up in college.

Also it’s worth mentioning that we can’t just continue to add things to the curriculum and expect a change in our students’ abilities or understanding of the subject matter. Right now the curriculum is too broad and lacks substantial depth. As is, kids have to memorize lots of seemingly unrelated, mathematical facts. They’re presented in isolation and learnt in isolation. If you’re going to revamp the curriculum, fine: just don’t tack on more and more things and then look for a miracle. But moving on.

The author cites a statement by the Mathematical Sciences Education Board. Following fold, it’s reproduced for your benefit and my scrutiny in part below:

Almost no time is spent on estimation, probability, interest, histograms, spread sheets or real problem-solving, things which will be commonplace in most of these young people’s later lives.

I agree. We pay little attention to any of those things; it’s a shame, too. Probablility and statistics are perhaps the most important “real-life” mathematics we could be teaching. I’ve always found it funny that calculus has dominated as the capstone math course at many high schools. College freshmen use it as a measure of their peers mathematical prowess. You’re especially frightening if you took a multivariable calculus or linear algebra class. Statistics isn’t as nearly frightening. Too bad, because if it were, maybe more people’d offer and take it. I routinely run into Ivy-league educated people who don’t know “correlation is not causation.” I’m a bit worried and confused by their inclusion of “real problem-solving.” I don’t know what it means, but I can guess.

They want our kids to pretend that they’re the CEO of a juice company, and they need to figure out whether to make more cranberry juice or more grape juice according to a number of contraints. Maybe they’ll do the linear programming themselves, or maybe they’ll plug it into a computer. In any case, these sorts of highly contextualized, so-called real world problems are, in researched fact, a bad, bad, bad idea. Not only do they not interest most children—most children are not, nor do they dream of being the CEO of a juice company—these sorts of problems actually hinder transfer of abstract principles and problem solving structures to other types of problems and disciplines. What does go, however, are abstract relationships.

With a little practice and a lot of thought, it’s not hard to come up with motivating questions that live soley within the realm of mathematics. There’s no need to introduce confusing and distracting details from the physical world.

But, if you must, I could see the addition of more mathematics into the school day. What if we had two math classes, but disguised one of them as “real-world problem solving”? Then we wouldn’t have to compromise learning how to think through problems in an abstracted way that promotes the formation and understanding of relationships, and we’d have a venue for an integrated approach science, math, and technology with applications that is so hot nowadays.

Anyway, I’m meeting with the author sometime next week to talk about these sorts of issues in person. I’ll let you know how it goes.

Hello world!

I’ve just begun the migration from the old Manilla server to the new, expedient, and aesthetic WordPress server. [It even correctly identifies and interprets en-dashes, em-dashes, and hyphens! I could not be happier.] As a matter of good cheer—and as a bow to good karma—, I’ve kept the automated post that indicates that things are working properly.

Welcome to Weblogs at Harvard Law School. This is your first post. Edit or delete it, then start blogging!

You’ll notice that this post also has a comment associated with it. That was made automatically, too. Due to the increased ease provided by WordPress, perhaps you, too, dear reader, will start posting comments. I dare you; I double-dog dare you to.


I’m thinking of moving my blog to because they use WordPress software to manage their blogs. While there, I found Plasma Pong [click for a download] Atari Boy‘s blog. It’s pretty fun, but I suggest you use an external mouse rather than the touchpad on your laptop. You’ll need Windows. [Sorry, my Apple-inclined friends.] The music is dramatic. Think that Guinness commercial staged on a train and set to the Carmina Burana-level dramatic.These screen shots are mine. Atari Boy has more here.

I apologize for the impersonal nature of this post. I tried to make up for it with lots and lots of links.

Blizzards, Adventures, and Maps!

The snow didn’t keep me out of Cambridge yesterday [primary because it hadn’t yet snowed]. And that means that Henry and I started our weekly study of Old English. He wants to recite some passage in Beowulf; I want to use the language as a super-secret method of communicating — perhaps exclusively with Henry — no matter where I am, regardless of the company. Also, Old English looks and sounds really funny, and that’s reason enough to learn it, too.

But the snow is keeping me inside today. So, rather than go on a winter adventure, I’m prepping the documentation and presentation of future adventures. There’s a giant road atlas of the whole of the United Kingdom on one of my bookshelves. Rather than throw it out once I returned from Scotland two years ago, I had planned to make a website with scanned pictures, anecdotes, and all the rest, and to organize everything through the maps in the road atlas. Alas, things came up, and the website was never realized. But now! now I’ve signed up for an API key to use Google Maps. I was even able to understand and reproduce their example map [which I can’t display on this server].

Soon I’ll start a blog-type site over at the GSD [where I have more file control, access to web services, and other computerish things; you can find all my map stuff there for now]. With some determination, I’ll be about to start up a GIS-powered Wiki, so that you, too, my faithful and ambitious readers, can add geographically anchored markers with linked online media.

I need to think hard about how to make good use of this technology. Check out a good example which maps out recent BBC News articles here. History classes could make large, interactive maps detailing famous exploration, battle maps, or immigration with seemlessly coupled pictures, documents, videos, and sounds. Another day I’ll rant about why this is a fantastically empowering idea in classroom education. Maybe I’ll stroll down the street back to the LifeLong Kindergarten group at the Media Lab.

PHP, XML, and Google Map API

Perhaps I should’ve taken some CS courses in college. Even still, I’m racing through a couple books on the above subjects (minus the API). But, inspired by Walk Jog Run, I’ve decided to learn how to and think of a way to use the Google Map API here on my blog. There’d be one of those Hello, World! examples accompanying this entry would that the API key sign-up page work. As it doesn’t, you’ll have to be content to wait. But soon, I’ll organize and detail my adventures map-style! Which reminds me, I need to get back to campus to finish up my certificates in ArcScene — a 3D modelling and mapping program in the ESRI ArcMap 9 suite. I’m about half way done and have been for almost a year. My course code is about to expire, and that’d force me to pay to finish it up; heaven forbid I should pay for my education.

For the Sake of Formality.

In the interest of keeping you, my loyal and extended reader base, from becoming too bored, and to validate your checking up on me, I have written a short albeit very boring post. It teeters slightly on the technical side. Please excuse me. I promise a post of worth in the immediate future.

Tonight my father and I discussed some of the merits of XML (extensible mark-up language) with extensions to telephony, public switching, and the .NET standard. This, in turn, prompted me to look up more on XML, and a little about SGML (standard generalized mark-up language). XML is a meta-language and fits nicely in my present trek into meta-physics. Of course, my goal is ultimately in meta-cognition. In the meantime, I should work on some practical skill sets. Hence the computerish talk. Since graduation, I’ve taken on a lot of academic goals for myself, programming being among them.

[Other resolutions include: reading a book I should have already read every two weeks — not the same book — Someone is mugged in Central Park every two hours, and he’s really sick of it; working through the proof of the Atiyah-Singer Index Theorem; learning more relativity, more developmental psych, more epistemology; applying to graduate school.]

In addition to my light XML reading, I decided to check out, on Divia’s suggestion, the Apple Human Interface Guidlines. Mitchel Resnick, my professor at the Media Lab over at MIT for a class on technology and education, would be glad to know that his class had a lasting effect on me. [In his class, I wrote a design brief for a software package to impart a particular mode of thought while developing a geometric intuition. In the conclusion I urged the reader never to implement my design lest it be used irresponsibly. The moral: technology, like alcohol, can be a valued servant but terrible master. Teach first, dazzle the kids with blinking lights later.] If I’m very lucky but not very careful, I’ll end up with a D. Ed in math and cognition.

Also I went to Tracy’s with DJ to hang out with them, Michelle, and some of Tracy’s friends from, as I understand it — and I’m making this part up based on imperfect data — Legal Seafood. Michelle brought me my GameCube power cord, The Twenties by Edmund Wilson, and the navy fleece all of which I had left her place the night before.

In the morning my sister, a handful of her friends, Michelle, and DJ, if he wakes up, are heading Downtown to Filene’s to celebrate the liquidation sale.

Paper Mario may be the end of me.

Putting the Simple in Supergravity.

Despite the title of this entry, I am not going to talk about supergravity outside of this: Cabot recently purchased a book I may’ve used for my thesis. However they processed my order over a week late. I’ve got it with me now, just for kicks — personal edification and so that I can impress you, the reader, and the people who see me with it at lunch, like Luke and Lixin. On my way back from the Science Center, I flipped through the table of contents, I came across a chapter called “Geometrical Gravitational Theories,” which is why I asked the library buy the book in the first place. This book,Geometry, Spinors, and Applications, makes heavy use of — wait for it — spinors. There are lots of books on geometry; tons on applications; a sizable number on both geometry and applications. But there are surprisingly few on geometry, application, and special mention of spinors. And under that chapter on gravity, there’s a subheading: Simple supergravity. It reminded me of a conversation I had with Eda a few weeks ago. She finds that mathematicians are superficially humble, but in an oblivious and therefore endearing way. The idea of supergravity ever being simple is sort like a slap in my face, but in an endearing kind of way.

I quit you now to take up a programming assignment Paul has given me. He has resurrected that automated inspection-announcement-general purpose-web-based-email-program-thing project for me. He doesn’t like me to admit to him that I can’t do things, like, we’ll say for example, program. I’ve got a copy of PHP3: Programming Browser-Based Applications to my right. Somehow I think the supergravity would be simpler.


Nambi, the one who studies wireless finger detectors or something, introduced me to a Firefox browser alternative tonight called Opera. The layout and feel and functionality and everything, everything is better designed, better implemented.

He also taught me how to play a complicated, team variant of the classic kid card game Fish, at which he is very good. Then I convinced everyone to play Bullshit, at which I am very frustrating, though not so good. Nambi, however, is bad. It’s fun to watch.

The Internet is Everywhere.

Last night, Danielle and I headed over to the harbor nearest to the US Coast Guard building in the North End. There is what one might, at first glance, perhaps, believe is a scary alley populated by nogoodnick gansgters. What better place to dispose of a body than the docks next to a government building and behind what seems to be an abandonned public pool? Ah, but here the clovers grow thick and their scent mixes with the brine of the see. Across the USS Constitution and the Bunker Hill Monument shine (until midnight or so, then they shut down the Hill. Same goes for Old North Church, which, no less stunning, resides directly behind the playground — inlandwardly.)

But that’s not really the meaning of this post. What I mean to tell you is this: The internet is everywhere. With my laptop less than twenty-four hours old in hand and the Strawberry Festival Full Moon (also the Rose Full Moon to some) waxed complete, it was time to take this puppy on the road. (At this point it should be noted that Bubbles, Danielle’s iBook came along, too. In fact, I’m writing this post on Bubbles right now. Danielle has Jacobi, my laptop, held captive until she finishes another of her practice GRE tests.) Once we found parking in nearby Charlestown, she and I headed over to the clover covered patch to — you guessed it folks — study math. I hit up an article on the Penrose Inequality (gr-qc/0312047).

Not too long ago Greg Valiant told me that the CIA asks during its interviews for new employees if they have ever wanted to be a florist or had an interest in flowers, something like that. Statistically, he explained, there is a high correlation among those who have and those who suffer from, euphamistically speaking, mental instabilities. It didn’t make a whole lot of sense to me at the time. But I accepted this fact on faith, until now.

Danielle exhibited reservations at first, but once she picked her first clover, she, and they, were done in. The sort of fastidiousness, the calculation, the avariciousness, it all smacked of torture. And that was convenient. During our walk from the car we had discussed the merits of torture, or rather, the lack thereof. But as she ruthlessly reached for the thickest clover, no, now the longest one, oh, this one bent, I’ll grab another one, we both began to understand the wisdom of the CIA.

Of course, as she hunted her next victim, I couldn’t help myself. After all, I had successfully found and logged onto an unsecure local wireless network and emailed my excitement.

By the way, the bouqet Danielle arranged looked nothing short of magnificent. I wish I knew more about plants so that my description could mean more. It looked largely like a purple broccoli, punctuated by the long, pointy closed blossoms of some small lily-like wild plant. The bouquet symbolized all the buzzwords from a syllabus on literature: nature and the city, order and chaos, and, my favourite, beauty and brutality.