Thought of the day: The best intellectual training

At the risk of sounding self-serving and elitist, I think the best intellectual training is mathematics and physics. These subjects are the most challenging to learn due to their abstract nature and extremely difficult to pick up as an adult (maybe as difficult as learning a musical instrument or foreign language as an adult). There are many claims that mathematicians and theoretical physicists make their biggest discoveries before the age of 40. Mathematicians and physicists have a reputation for being "smart" and after a long period of reluctance and doubt, I have to agree that this reputation is well-deserved.

If you know math and physics, it's easy to pick up almost everything else. I'm not saying you'll achieve a deep understanding of literature, history, or business, but you'll be able to learn it pretty fast and be decent at it. My friend remarked that the coolest people are the scientists who are the top in their field and interested in everything. Unfortunately, these people are a minority. The rest are rather one-dimensional and dull company. (Her opinion, not mine.)

I think that the other difficult fields to pick up are visual art and music. They are also quite abstract. Artists are trained to "see" in a special way; they can translate what they see into an artistic representation (often translating 3D into 2D). By visual art, I'm talking about drawing and painting, not photography (which is kind of a technological cheat). Musicians innately understand rhythm and scales.

I've heard that philosophy is the best intellectual training if you restrict yourself to humanities fields. I'm not really sure about social science. Those are interesting subjects, but I think if you just want to be a great thinker, you're better off starting with math and physics.

So I guess if I had a child, I would have him/her learn art, music, math, and physics, plus a couple foreign languages.

What a scientist should be able to do

A human being should be able to change a diaper, plan an invasion, butcher a hog, conn a ship, design a building, write a sonnet, balance accounts, build a wall, set a bone, comfort the dying, take orders, give orders, cooperate, act alone, solve equations, analyze a new problem, pitch manure, program a computer, cook a tasty meal, fight efficiently, die gallantly. Specialization is for insects.
- Robert Heinlein, Time Enough for Love

I was thinking about what are the skills a scientist should have. What do you think a grad student should know at the end, after they finish their PhD?

1. Technical skills - mathematics, programming/numerics, lab techniques, etc
2. Writing - ability to write good scientific papers that are clear, concise, and well-motivated
3. Presentations - ability to write good presentations and deliver them well, this is closely related to writing
4. Comprehension - ability to distill the important ideas from a paper or presentation, ability to tell the difference between crappy research and good research
5. Process - (advanced) ability to come up with concrete ways/experiments to answer questions, ability to overcome deadends in research, ability to stay organized, keep good records, and manage other people
6. Community - talking to people including those outside your field, attending seminars/conferences, convincing people your research is important, building a network of trusted friends who you can turn to for feedback and support
7. Creativity - ability to understand the difference between good and bad ideas/questions, (advanced) ability to come up with interesting questions that are soluble
8. Resilience - ability to stay positive and motivated even when the research isn't going very well

I am by no means an expert on this subject.

Improving the public image of science

I think sometimes about how to improve the public image of science. There are a million things going on in people's lives, things that worry them; children have so many ways of occupying their time, whether it's sports, Facebook, or video games.

I want people to believe that science literacy is important the way that reading is. You wouldn't tell someone you can't read. Yet people have no problem saying that they're "not good at math."

I want to see people doing amateur science whether it's on the computer, looking at stars, or performing experiments. I see all these people buying $500+ dSLR cameras. Thanks to the advances in digital photography and the huge drop in the price of equipment, anyone is capable of taking pro level photos if they work at their skills [1].

Some ideas I have

1. Take better photos of scientists. I've never really seen many good portraits of scientists [2]. On that note, why can't we make a documentary or a music video that will convince people that scientists are heroes?
2. Get the public more involved in science. Make them feel like they can make a contribution.  We need more initiatives like Galaxy Zoo.
3. Find ways to get children more interested in science. Maybe high school kids could be allowed to write software for the library. Have kids do Make Magazine projects.
4. Show people how they can use science and math to great benefit in their lives. I have to admit, I don't really know how to do this. I've always liked how you can use statistics to expose cheating in polls and things of that nature.

[1] Not that I'm saying amateurs are actually pros.  The professionals can take a much higher percentage of good photos than an amateur can.
[2] This one from the New York Times is not bad for an environmental portrait.

Link of the day: Khan's Academy

Lately, I've tired of TED talks. They were bold and exciting when they first appeared online (3-5 years ago?), but now people just seem to be selling their ideas whether they are merely good or truly brilliant. It's important to have inspirational meetings, but I think they should also be authentic and realistic.

There is one recent TED talk that I do like very much. Salman Khan, a former hedge fund analyst, spoke about how video can re-invent education. Originally, Khan recorded videos to help tutor his cousins in math. He posted the videos on YouTube and left them publicly available, in case someone else might find them useful. His cousins told him that they preferred their "virtual" cousin on video than the real thing! They found the video less intimidating because they could stop and repeat it without appearing stupid; they could learn at their own pace.  Other people discovered Khan's videos and gave him so much positive feedback that he quit his finance job and started producing videos all the time. (Khan does all the math and science videos, and he hired experts to do the videos on humanities subjects.  The scope of this project is astounding: 2000+ videos.)

That alone would have been an outstanding accomplishment, but Khan didn't stop there.  He tried to track learning outcomes.  He associated each video with a particular concept and made tree diagrams showing which concepts were prerequisites for other concepts.  Khan calls this a "knowledge map."  Students can work on modules.  When they get enough problems from the module correct, they can move onto another module.  When they master the prequisite modules, they can move on to a more advanced module, and so on.

This systematic tracking of the student's progress is invaluable to a teacher in charge of 30 students.  The teacher can see how the class is doing.  Moreover, if a student is struggling with a particular module, the teacher can find another student who mastered it and have that student teach the other one.  Peer learning!  (I discussed this topic in an earlier post about a Harvard professor struggling to teach first-year physics.)  Now, at least one school district (in Los Altos, California) is trying out Khan's system in the classroom.

When the system was used in the classroom, it showed that different people find different concepts easy and different concepts hard.  In Khan's words:

Because every time we've done this, in every classroom we've done, over and over again, if you go five days into it, there's a group of kids who've raced ahead and there's a group of kids who are a little bit slower. And in a traditional model, if you did a snapshot assessment, you say, "These are the gifted kids, these are the slow kids. Maybe they should be tracked differently. Maybe we should put them in different classes." But when you let every student work at their own pace -- and we see it over and over and over again -- you see students who took a little bit extra time on one concept or the other, but once they get through that concept, they just race ahead. And so the same kids that you thought were slow six weeks ago, you now would think are gifted. And we're seeing it over and over and over again. And it makes you really wonder how much all of the labels maybe a lot of us have benefitted from were really just due to a coincidence of time.

I found this very interesting.  I'm guessing that a lot of teachers and coaches know that student learning is much more complicated than "gifted" and not gifted.  It's nice that Khan can actually provide hard evidence establishing this fact.

Khan's work is amazing and inspiring. I do have a few questions. Using technology to tailor education is not a new idea. Why did Khan succeed? Is it because students are more comfortable with technology compared to students of the past? Why is video better than a textbook? A textbook is also non-intimidating and self-paced.  Maybe it's because Khan is a great tutor who is both a talented teacher and entertainer?  (I briefly viewed one of his videos and he seemed funny and charismatic.)  In an ideal world, each student would have a one-on-one tutor.  This isn't realistic.  However, if we have a great tutor like Khan and he makes free videos available to anyone on almost every possible math and science topic from kindergarten to high school, this tutoring database is a pretty good, though imperfect solution.  It's reminiscent of an idea in artificial intelligence.  You can have a computer that isn't smart in the human sense, but if you program it with an astronomical amount of information, it can be very useful.

I think that doing online homework is becoming more popular as teachers realize that there is simply not enough time in the classroom to do everything.  There are a lot of things students can do on their own with a "computerized" tutor.  By "outsourcing" this teaching and doing it outside the classroom, the human teacher has more time to teach things that are hard for computers.  Like having students discuss problems together.  Or showing how many seemingly disparate concepts unify into a larger concept.  Or doing hands-on science experiments.  I know that for first-year physics courses, some universities assign online homework several times a week.  This forces students to read the book and work on problems at home so that the lecturer can spend time explaining concepts rather than writing 20 equations on the blackboard.

What Salman Khan is doing is incredible work and I wish him the very best.

Physics as a subject for a kids comic book and how to get kids interested in science

For several years now, I've wanted to write a physics comic book with the goal of getting kids interested in science. The problem is that I just don't have a good idea how to do it. I need a really great idea because a comic book has to compete with TV, video games, internet, and all the other entertainment children are exposed to in the modern world.

I'm starting to feel like physics is simply not a good subject for a kids comic book. Anything that is abstract like physics will be difficult to pick up quickly and therefore you need to be able to play with it, experiment. Programming is abstract but lots of kids pick it up because you can write code and run it immediately. You can quickly progress to the point where you can make images fly across the screen. You know if the program works, because either the image flies across the screen or it doesn't. Instant feedback. That is fun, exciting, and addictive (in a good way). When you do a physics problem and you get an answer, it's very difficult to know if your answer is correct or if it makes sense. You could do a real physics experiment, but physics experiments are notoriously difficult to do right and require special equipment. I was always amazed in high school at how much equipment we needed to do simple experiments let measuring the velocity of an object moving along a track. When you do a chemistry experiment, you mix two solutions and the color changes. You can see or feel the result qualitatively. Physics experiments require too much precision; you actually have to measure the exact numbers to see if you're doing it right.

I'm starting to think that if you want to get kids interested in science and engineering without much equipment, the appropriate subjects are programming and math. I've already discussed programming. You can come up with all sorts of interesting math problems at all different levels. You can get a sense of whether your answer is correct by plugging in numbers. For geometric problems, you can often solve them by drawing pictures. Best of all, you don't have to worry about equipment failing in your experiments. Mathematics isn't constrained by the physical world, so there are lots of different ideas you can talk about, whereas in physics you are stuck discussing Newton's laws, Maxwell's equations, etc. For older kids who want to do something hands-on, I would recommend electronics. The parts are small and you don't need to go a machine shop.

Link of the day: "Maths busking"

A huge problem in science is how do you get laypeople interested in science? (I'm using science as a blanket term which includes engineering, natural science, and math.)

Most of the popular books and media written about science are only interesting to people who already enjoy science and those people are only a small fraction of the world's population. If we truly want to make science mainstream and boost science literacy, we need a way to reach "regular" people.

Enter "Maths Busking" -- a really cool project whose goal is

Maths Busking aims to show the public the surprising and fascinating side of mathematics through the medium of street performance.

You can also see some video of their street performances.

Link of the day: Looks, 1/f distribution, myth of meritocracy

First, an old link from months ago that I forgot to post: Your looks and your inbox. OK Cupid, an online dating service, does cool statistical analyses of how your profile affects response rate. One idea to take away is that women are very harsh on men's looks. Ouch.

One of the key ideas in my research is quantum noise. So it's neat to find an article claiming that modern movies have a cut frequency that follows a 1/f distribution.

Finally, via Cal Newport and a few links, I enjoyed the following New York Magazine article on why kindergarten admissions tests are worthless. The single most interesting takeaway message is how unreliable intelligence tests are for 4 year-olds.

I wrote to [University of Iowa psychologist David] Lohman and asked what percentage of 4-year-olds who scored 130 or above would do so again as 17-year-olds. He answered with a careful regression analysis: about 25 percent.

Link of the day: Lockhart's Lament

Michael Nielsen posted a link to the wonderful essay "Lockhart's Lament." It was originally written by a mathematician named Paul Lockhart in 2002. He laments the state of K-12 mathematics education in America. I was a bit shocked to realize what a mediocre mathematics education I had, even though I attend some of the top public schools in the country. In retrospect, I realize how rote my math classes were. No wonder I never really liked math as a child. When I went to college, I took physics and my teachers showed me the rich history behind what we were learning, how you could take different approaches to solving the same problem, and how physics was still a growing, changing field. No wonder I became a physicist and not a mathematician.

I feel like even my college math classes were taught in a rather rote fashion. I never really got a feel for how the various facts I learned were inter-related, nor did I understand why these facts were interesting. My analysis teacher mentioned a book called A Radical Approach to Real Analysis by David Bressoud. I wish I had read it; apparently it explains why mathematicians wanted to come up with these obscure concepts like sets and measures.

I really need a good context to understand advanced math. At some point, I just can't handle so much abstraction. I think most people have even less ability to handle abstraction than I do, which probably explains why so many people hate math.

Not that science education is that much better. I remember Bruce Alberts (an author of the famous biochemistry textbook The Cell) saying that he was shocked at how boring his son's high school chemistry textbook was. The only reason that many students learn science and math is because their parents and teachers tell them how important it is. Students probably don't appreciate it until later in life, if they're lucky. I was one of the lucky ones. There are actually a number of prominent physicists working on physics education including Nobel Laureates Leon Lederman, Carl Wieman, and Kenneth Wilson.

Link of the day: Abramowitz and Stegun

The famous reference Handbook of Mathematical Functions by Abramowitz and Stegun is on the web!

If you're stuck on a desert island with internet access:

http://www.math.sfu.ca/~cbm/aands/

If you're stuck on desert island with a laptop but have no internet access, you can get a very large electronic file:

http://www.lacim.uqam.ca/~plouffe (71 MB)
http://www.math.sfu.ca/~cbm/aands/dl/abramowitz_and_stegun.tar.gz (43 MB)

Deconvolve your functions

For some reason, people like to say "deconvolute" instead of "deconvolve" in the context of Fourier analysis. Deconvolute sounds awful, like you're made the problem worse and now you're trying to go back and fix it. Sam Lord discusses the deconvolute vs deconvolve debate in this post. Any comments, linguists?

Link of the day: Math screensavers and exercise videos

I found some cool stuff this morning. First, Chris Lomont has a bunch of math-based screensavers including Mandelbrot sets, knots, and polyhedra. Second, John Sokolowski has an excellent site called Athlete 365 which includes a library of exercises, many illustrated in video.

Link of the day: Imagining the Tenth Dimension

I have a hard time taking string theory seriously because it's so hard to imagine more than three dimensions. But now I can visualize higher dimensions much better, thanks to a flash tutorial at the site appropriately called "Imagining the Tenth Dimension." Click on the sidebar link labelled "Imagining the Ten Dimensions".

Direct product vs. tensor product

Suppose you have two vector spaces. How can you combine them into another vector space?

The simplest way is to use a direct product. For example, suppose I have vector space A with Cartesian coordinates a1 and a2 and vector space B with Cartesian coordinates b1, b2, and b3. The direct product of A and B is a new vector space with coordinates a1, a2, b1, b2, and b3. Basically, the idea is to add the dimensions of A and B together. Notice that A has 2 dimensions, B has 3 dimensions, and their direct product has 5 dimensions. Pictorially we might imagine a direct product as taking the union of vector spaces A and B. Then we get a new vector space where you label the elements of the set by a set of coordinates in A (a1, a2) and a set of coordinates in B (b1, b2, b3).

A slightly more interesting thing we can do is to take a tensor product. Let's call the basis vectors of A, ^a1 and ^a2, and similarly the basis vectors of B, ^b1, ^b2, and ^b3. The tensor product of A and B is a new vector space with basis vector space with basis vectors ^a1 x ^b1, ^a1 x ^b2, ^a1 x ^b3, ^a2 x ^b1, ^a2 x ^b2, and ^a2 x ^b3, where "x" connotes tensor product. So the tensor product is kind of like multiplying two vector spaces. The tensor product space of A and B has 6 = 2 x 3 dimensions.

In physics, a simple example of direct and tensor products is spin-1/2 particles. A single spin-1/2 particle is a direct product of |&uarr> and |&darr> -- dimension 2. However, the Hilbert space of 2 spin-1/2 particles is a tensor product of two single spin-1/2 Hilbert spaces -- dimension 4 = 2 x 2. We see that in general that a tensor product space is larger than its corresponding direct product space. For example, the direct product of 3 spin-1/2 particles has dimension 6 = 2 + 2 + 2, but the tensor product of 3 spin-1/2 particles has dimension 8 = 2 x 2 x 2. That means that there are vectors in the tensor product space that can't be mapped to the corresponding direct product space. We know that spins combine in tensor products because we observe "entanglement." A direct product of 2 spin-1/2 particles would have states like |&uarr&uarr> but not Bell states like |&uarr&darr> - |&darr&uarr>.