Category: Uncategorized

Posted on

Math in Ancient India

I just checked my web server statistics and found that part of my high school research paper on the history of mathematics is getting well over a thousand requests a month.

The topic of that paper is “the usually unrecognized achievements of Vedic and Hindu mathematicians from 2000-300 B.C.” When writing it, I was surprised at how hard it was to find good research on the topic:

In Mathematical Thought from Ancient to Modern Times, [a “comprehensive” summary of] the history of mathematics, [author Morris Kline] included only half a chapter (out of 50 chapters total) on Indian math. Everything he said seemed to sneer at them, put them down, and belittle their accomplishments.

Despite this dearth of understanding, the facts were clear:

Besides using simple arithmetic operations like addition, subtraction, etc., Indians invented the decimal system and the idea of positional notation, both of which are still in use today. They also used the “Pythagorean” theorem and “Pascal’s” triangle long before either of those men were born!

Today, it turns out that if you google “Sulva Sutras” (the title of the web page getting most of the visits), my research paper is the first result, above Wikipedia and everything else! If you search for “mathematics in vedas”, I’m the third result.

True, this seeming popularity may have something to do with spelling inconsistencies — the Wikipedia article uses “Shulba Sutras” and is the first result if you search with that spelling. It’s also true that my paper was written in 1999 and has been on the web since 2003, so has had time to gather links from other websites (which influence Google’s ranking).

But we’re talking about the founding documents of mathematics! The origin of zero! The “Pythagorean” theorem, recorded hundreds of years before Pythagoras! And the most relevant article was written by a fifteen-year-old?

In my research paper’s conclusion (which is mostly too embarrassing to quote), I wrote, “I find it unbelievable how little work has been done in the field…. The vast majority of the work has been done only by Indians. Most of the books on the subject are written in Sanskrit or Hindi [and are ignored by] eurocentric scholars.”

Ten years later, my incredulity lives on.

Posted on

Steve Jobs’ fifth revolution

I think one of the most significant announcements at Apple’s media event today was that the iPod touch now has over 50% market share in the worldwide portable gaming industry — the iPod touch outsells the portable game consoles from Nintendo, Sony, and all other manufacturers, combined. Steve Jobs also said over 1.5 billion games have been downloaded so far to iPod touches alone. “It has become by far the most popular game player in the world.”

Steve Jobs discusses the iPod touch as a gaming platform

It’s widely recognized that Jobs has already revolutionized four industries: personal computers (Mac), digital music (iPod and iTunes), animated films (Pixar), and smartphones (iPhone). I think it’s now safe to add a fifth to that list: portable gaming.

His impact is a revolution both in terms of the new multi-touch user interface for gaming and the App Store platform for game distribution and payment. The major products involved are not just the iPod touch but also the iPhone and iPad.

So what will be number six?

Apple is making some progress on movies and TV shows. However, the studios and cable companies have all the power, and they are terrified about what happened to the music industry. It’s hard to find a path that transitions the industry from cable TV “channels” to browsing and paying for individual shows.

Another possibility for revolution is in textbooks and online education, where iTunes already carries recorded lectures and the iPad has started to inspire a new class of interactive educational content.

Posted on

7″ iPad

I continue to see rumors that Apple will release a 7-inch iPad. The idea is that it would be closer in size to a paperback or Kindle; lighter and less expensive than the current iPad; and easier to fit in a purse.

I’m a bit scared of this vision because it means all of our apps would have to be redesigned for yet another screen size. iPhone apps already do run on the iPad, but they are awkward to use. Scaled-down iPad apps are not really an option because the touch targets would be too small. Apple could use a screen with the same number of pixels as the iPhone 4 (but bigger in size); that way, all retina-display-compatible apps would fit pixel-by-pixel on the device. Still, graphics would look too big and some interactions would still be awkward. In short, redesigning our apps would be necessary for a good user experience.

This redesign will be a lot easier than porting apps from the Mac to the iPad. Still, for the sake of my own sanity, I hope Apple waits a while before introducing the next screen size.

Update: On the other hand, most of our existing Mac apps have to be designed to work well at any screen resolution between the 13″ MacBook and the 30″ Cinema Display. From this point of view, having to support just two discrete iPad sizes should be comparatively easy.

Update 2: Steve Jobs just criticized the 7-inch form factor.

Posted on

Humanizing math

I used to think that math was the perfect subject to teach via computer software (instead of lectures). My rationale was that computers are already good at math; and software-driven customization for each student is most useful for topics that require a solid knowledge of previous topics (e.g. algebra builds on multiplication and fractions).

This TED talk by Dan Meyer challenges those assumptions. He suggests that good math education starts with good discussions. It uses open-ended questions whose answers may be as unpredictable as responses to works of literature. It emphasizes how math relates to intuition and the real world, and deemphasizes arithmetic and equation solving. In other words, it humanizes math, a notion which makes a lot of sense in a world where computers not only compute sums but can easily solve, graph, and symbolically manipulate indefinite integrals. (On a cell phone. For free.)

Indeed, a lot (most?) of cutting-edge science today involves calculations so complicated that it would never even occur to the scientists to complete the math by hand.

So why bother teaching students in detail how to do the things that computers will always be better at? Meyer’s approach focuses on the human side — understanding when and why to apply mathematical tools. It’s not immediately clear how computers themselves will figure into this educational mission.

Posted on

Form Follows Fiasco

We can always use a reminder to keep it simple. This one comes from the thoughtful and amusing textbook Form Follows Fiasco by Peter Blake, published in 1977. (A friend recommended the book, which was not in the public library holdings but was available used on Amazon for about $4.) In this passage, he is discussing one of the problems with construction via prefabricated modules.

Many wonderfully inventive designers spent decades, if not lifetimes, trying to perfect the absolutely perfect, universal joint — the magic mechanical device that would join their modular panels together in wedlock (yet leaving open the possibility of some future disengagement, for the sake of greater post-marital flexibility).

But it was all in vain. The universal joints, the seams, the gaskets, the unbelievably ingenious interlocking connectors — many of them leaked, wracked, delaminated, or experienced some sort of material fatigue. Yet jointitis — a disease increasingly prevalent among theorists in prefabrication — continued to spread. One of prefabrication’s most illustrious pioneers designed a joint to connect two or more wooden panels; it was a miracle of ingenuity, and required little more from the on-site joiners than a doctorate in Chinese puzzling. The pioneer, it seemed, had never been told of an earlier and less sophisticated joint used in wood-framing, known as the nail.

My take on the overarching theme of the book is that there’s something to be said for a little messiness. The straight, clean, orderly, centrally planned structures of Modernist architecture and Modernist urban planning sound good in theory. But in practice, they are expensive to keep straight and pure, so before long they become ugly (stained cement walls). They are also bland and boring because they are so simple (high-rise apartments and suburbia). And they are inefficient because they artificially standardize (modular approaches that are ok for many uses but not great at anything) and require connecting artificially separated functions (rush hour in heavily zoned cities, and the “universal joints” discussed in the above passage).

The alternative is to turn to more practical, locally and organically designed, human-centric (not technology-centric), financially sustainable structures. He points to examples of old wood-and-brick buildings that have been completely repurposed but still work great; vibrant urban centers like SoHo, which was designed organically by new residents violating the zoning laws; and structures such as Grand Central Station which hide all of the technology (trains, subways, electricity, plumbing) to make a welcoming, functional, human-centric space.

As always, there is a balance to be found between order and disorder, predictability and randomness. The Modernist movement was in many ways a reaction against the disorder and uncleanliness of previous eras. Form Follows Fiasco and more recent trends swing back from the extremely sculpted order of Modernist plans to reintroduce what they hope is a healthy dose of messiness.

Posted on

Signal Strength and Leaks

The extreme publicity over the iPhone 4 signal strength issue reminds me of the similarly overblown media coverage regarding leaks in the Frank Gehry-designed MIT Stata Center.

In both cases, the affected products are gorgeous, highly-acclaimed, innovative masterpieces. They’re already famous in their own right. They’re sufficiently innovative that some unforeseen problems are bound to crop up. And apparently, people love to find those flaws.

“That’s what you deserve for spending too much money on that product [which I secretly want and now feel better about not having].”

Of course, these reports leave for the last paragraph the fact that “the iPhone 4 enjoys better reception than any of its earlier models.” The same goes for quotes from Stata Center occupants, who maintain that “it is a joy to work in this building.”

Posted on

Moviemaking, 1995-present

In 1995, I was in a summer program where we made a 45-minute movie from scratch, using a shoulder-mounted VHS video camera. We used no editing equipment because it was too expensive. Instead, we shot the movie in sequence, starting a new clip by rewinding the VHS tape to the end of the previous clip (trying to be as precise as we could). To get background music, we carried around a boombox that played a cassette tape as the scene progressed. Any camera-angle cuts had to go along with breaks in the song. We approximated fade-to-black transitions by gently covering the camera lens with a black cloth. The credits were hand-written on a long piece of scrolling paper.

Considering all of the technological hurdles (and the fact that we were about 10 years old), the quality of the movie is astonishing. Really, the fact that we could inexpensively make any sort of movie at all felt like quite a breakthrough.

Last week Apple introduced the iPhone 4. It not only lets you record video but has an app for editing — including background music, subtitles, and advanced transitions. All of the video is high-definition. And the whole device fits in your pocket.

I wonder what sorts of movies are possible now with just an iPhone and a creative team.

Update: The first such movie has been made (it’s well done, though only one and a half minutes long).

Posted on

The transition to touchscreen computing

In conversations with friends about the iPad, what seems to surprise them most is the idea that Apple will phase out the Mac in favor of touchscreen devices. I think part of the reason for the surprise is that news outlets have focused so strongly on the limitations of the iPad in its current, initial form. I like to remind people that the first Mac was black and white, could only run one application at a time, and had no hard drive. Similarly, the first iPod was enormous by today’s standards, had a small capacity for songs, and only worked if you also had a Mac. What’s most interesting about the iPad is not where it is now, but where it is going.

Despite all of the limitations of the first version, consumers have already purchased over 2 million iPads. This is not just some toy for tech nerds.

And we know from using iPads that the touchscreen interface feels more real than an indirect mouse cursor ever has. That’s what gives iPads the “magic” that Apple commonly cites.

Earlier this week at the All Things D conference, Steve Jobs gave a few hints about where he thinks personal computers are going.

Walt Mossberg: Is the tablet going to eventually replace the laptop, do you think?

Steve Jobs: PCs are going to be like trucks. They’re still going to be around, they’re still going to have a lot of value. But they’re going to be used by one out of X people.

And this transformation is going to make some people uneasy. People from the PC world, like you and me. It’s going to make us uneasy because the PC has taken us a long ways. It’s brilliant. But… and we like to talk about the post-PC era, but when it really starts to happen, I think it’s uncomfortable for a lot of people, because it’s change, and a lot of vested interests are gonna change, and its gonna be different.

So, I think that we’re embarked on that. Is it the iPad? Who knows? Will it happen next year, or five years from now, or seven years from now… who knows? But I think we’re headed in that direction.

Mossberg: Well, you don’t really think it’s going to happen next year, right? I mean, it’s a longer process than that, isn’t it?

Jobs: [pauses, shakes head slowly, brings up his hands] Sure!

Mossberg: [laughs]

(Transcribed from a video of Steve Jobs being interviewed at the All Things D conference.)

Microsoft CEO Steve Ballmer (in his interview at the same conference) dismissed the iPad as a “fad” that was difficult to take notes on during a meeting. Good luck with that, Microsoft.

Update: In the first quarter of iPad availability, Apple sold 3.27 million iPads and 3.47 million Macs. In the previous quarter (ended in March), Apple sold 0 iPads and 2.94 million Macs. And Apple’s iPad manufacturing has not even been able to keep up with demand.

Tim Cook (paraphrased) commented: “If you look at how long it took us to sell the first million iPods, 20+ months vs. one month of iPad, it’s a phenomenal difference. It’s not following the typical early adopter curve and cross into the mainstream. It’s gone straight to mainstream. Our guts tell us this market is very big, and we believe that iPad is really defining the market. We want to take full advantage of it so we are investing enormous time and resources in increasing our capability to getting iPads out to as many people as we can.”

Update 2: (July, 2010) Apple’s success seems to have caught Ballmer’s attention; he now says that producing a new Windows tablet to compete with the iPad is “job one urgency.”

Update 3: (October, 2010) In the second quarter of iPad availability, Apple sold 3.89 million Macs and 4.19 million iPads. A few days later, Apple held a special media event to reassure audiences that they still care about the Mac — and to introduce the new Macbook Airs, which are starting to function more like iPads…

Posted on

Can software tutors reform education?

I just read Disrupting Class: How disruptive innovation will change the way the world learns by Clayton M. Christensen, et al. I’m not sure whether the book’s findings were obvious or profound, but I was fascinated by the theory of disruptive innovations — technological changes that overtake an existing market in ways that make it almost impossible for incumbent organizations to make the transition. The authors use this theory to predict what sorts of education reform are possible.

One way to summarize the authors’ vision is as follows. There are plenty of cases when a students’ learning experience in school is not well matched to the student (for any number of reasons, be they learning style, teaching style, intelligence type, socioeconomic background, etc). In such cases, the affected student would benefit from a personal tutor to fill in the gaps. Personal tutors for everyone is obviously too expensive. But if we can make computer software that performs reasonably well as a tutor, it would be better than nothing. As the software technology matures, it might even be better than a human personal tutor for teaching certain subjects. In fact, some students might notice that actually going to class is unnecessary; they can learn all of the material in a faster and more enjoyable manner via the tutor. If this is true for a large enough number of students, schools would have no choice but to undergo reform in a way that embraces the software tutors and focuses on providing the services that computers cannot, such as coaching, motivation, teamwork, and other emotional skills.

There are a lot of “ifs” in this vision. However, it seems fairly certain that some version of the scenario will come to pass. The authors think that successful software tutors will be based on lesson contributions from the network of teachers, students, and parents who are also benefiting from the system. How far the vision goes depends on how well the new technology is able to visibly and measurably demonstrate effectiveness — a goal that has historically been elusive in attempted education reforms.

Update: If you have any trouble imagining these changes taking place, start with college-level lecture classes. Some students already skip lecture because they learn better from reading and doing problem sets. But why would anyone attend lecture in person if the best teacher in the world can be viewed online at any convenient time in high-definition video, paused and replayed as necessary? Real-life teachers and TAs quickly assume the role of facilitating, grading, and other forms of individualized feedback and assistance.