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I've often wondered if more people might take to math if it was taught as another language. Maybe I say this because back in the day my verbal scores consistently topped my math scores on standardized tests. But what would happen psychologically if math was framed as the language of nature, complete with it's own dialects, rules of grammar, and alphabets. As for text-books' real-world examples, when I was taking algebra in junior high, I was staring out the door dreaming about designing rockets (early days of the Apollo program), radios, or model airplanes from scratch. I wonder how much more of my attention the teacher would have captured if her examples had to do with subjects like those, even on a simple level, rather than cars passing each other on the freeway. At least they would have touched on the topics of the day. In the end, I got more out of algebra (at least in its simplest forms) as I worked toward my ham-radio license as an adult. I could finally see algebra's practicality as I fiddled with circuit designs!

Physics was my favourite subject in high school, and I went on to do my undergraduate degree in it as well. Although I was somebody who grasped most aspects of mathematics relatively easily, and did and do enjoy working on math puzzles, I wasn't terribly interested in pure mathematics as a subject. For my university calculus courses I just did what I had to do to get through them. My goal was to learn only as much mathematics as necessary in order to do physics. Although my interests in science and mathematics are broader now, it was a physics course (electricity and magnetism) that really taught me calculus in a meaningful way. It's a highly practical and useful tool (developed by perhaps the all-time greatest physicist) for understanding the real world.

It was only when I started learning how to conceptualize situations (an annulus of charge of some variable radii, for example) as functional relationships through the language of calculus that I think I truly grasped the meaning of integration at a deep, conceptual level, and saw it's power as a way of describing a real-life situation, and being able to calculate anything about that situation, once you've translated it into mathematical terms. In many ways, this is very much a language, and having made that connection, through practical applications, I now know that I could use calculus to do the same for any physical phenomenon. So, for me, my experience is the opposite of that study you and Chad have both cited.

The most depressing thing from that academic study article was this:

"Dr. Kaminski and her colleagues Vladimir M. Sloutsky and Andrew F. Heckler did something relatively rare in education research: they performed a randomized, controlled experiment."

Oh noes! Experimentation!

That being said, I think that concrete examples are much more useful in learning how to apply theory than they are in learning the theory.

I've learned and then forgotten calculus about 5 times during the course of my scientific career. So my recollection could be fading. But I seem to recall that the key revelation that I had in geophysics was not learning new and cleverer ways to integrate, it was figuring how to set up the problem so that the resulting equation was something we knew how to deal with.

Of course, this technique requires the ability to be able to perform at least one type of integration. And when it comes to learning how to do the math, I'm not surprised if an algebraic explanation work better than trying to assemble a stack of arbitrarily thin apple slices.

I was dumped in "advanced" math classes beginning in jr. high based on the snap judgment of a well-meaning guidance counselor. My understanding and skills in no way matched those of my peers, and I struggled along thinking I was a dunce at math and that science would never be my thing as a result. In retrospect, I think I would have fared better in the "regular" track. I've seen so many cases in which administrators and instructors assume that an intelligent student will learn regardless of instruction, with unfortunate results. Hey, you're interested enough to seek out a DVD course in calculus, of course that means you'll just kinda "get" how to find that position function.

For a few brief, glorious, months in high school my calculus and physics classes both covered the same content and I actually understood both. Without a concept to anchor the calculus, none of it stuck to my brain. Without a ready ability to work the numbers, the physics all started looking like inexplicable magic. They needed to be together for me to learn either. I remember when the two courses diverged, because that was the point at which I assumed that all of the interesting sciency stuff was beyond me.

I never questioned that assumption until I took an introductory course in acoustics my junior year of college. At which point I promptly signed up for a course in optics and started haunting the physics department. The desire to solve problems that were interesting to me brought the math into focus and I learned. Later, while taking a class on how to teach math to elementary school students, it finally clicked that I wasn't incapable of mathematical thinking. I just needed a chance to explore how it worked before I could practice and master a new concept. There have got to be others in that same position.

I like Pete's idea of math as a language. It makes a lot of sense, and I'd totally take that class. And work on keeping a bilingual household.

Your remarks about the DVD reminded me about my experience at University. At school I'd been keen, and very good, at maths and physics, and pursued astronomy as a hobby. So I ended up going the University of London to study astrophysics. After all, what could be difficult about spending 3 years looking through telescopes, right?

Unfortunately, it was an immense shock. For some reason, at 6th form (UK education tier for 16-17 year olds) I learned about differentiation, but not integration. So I turned up at college and sat in an introductory fluid mechanics course on day one (huh? where's the telescopes?) and saw the professor write some equations on the board. They were gibberish to me. And then he said the immortal words, "and so, obviously, the triple-integral of this becomes...". Obviously. Yeah, right! After that it was just white-noise to me and my promising career as the next Patrick Moore turned instead into year-long party, interrupted by Drama Soc, playing bridge and drinking too much.

It seems a pretty common amongst physicists to mention that they learned calculus from their first mechanics course and multi-variate calculus from their first Electromagnetics class rather then from the math classes designed to teach them those topics. The reason, I suspect, is that in early calculus classes its too easy to do well by blindly manipulating the symbols according to the appropriate rules without understanding the underlying abstractions. Physics courses, on the other hand, force you to attach the abstractions to actual physical processes, and require you to have at least some feel for what those symbols actually mean and the motivation for creating them.

Pete wrote: "I've often wondered if more people might take to math if it was taught as another language. "

I would highly agree with this. In my experience, a lot of the difficulties students have with the math is making a mental connection between the abstract symbols depicted on the board and the often very intuitive ideas that those symbols represent. Once you understand the 'language', a lot of seemingly difficult mathematical statements become almost embarrassingly trivial. When I teach a math-related course, I make a point of emphasizing this observation to the students.

A couple of examples come to mind: the definition of a 'limit' of a sequence, and very physical vector calculus theorems like Gauss' theorem.

I'm a real math-head, so I had no trouble at all of understanding both the abstract concepts and the real world applications.

However, I've since worked sporadically as a math-tutor, and I see this problem all the time. Ask a student what the derivative of x^3 with respect to x is and they'll probably get it right, since it's such a simple formula, but ask them what it means and their stumped. In Sweden, where I live, Math and physics are two totally different courses, and when you get to the point in physics where calculus is extremely useful, for some reason the curriculum doesn't assume that you've learned the necessary tools in math. So instead of teaching basic mechanics in terms of calculus, the teacher basically just gives you a formula. As an example, (s)he'll say that if you want to know the position of a stone thrown up in the air with respect to time, use this formula:

p(t) = -g/2*t^2 + v0*t + p0

So the students use it, but really have no concept of what it means. When helping them understand calculus, I usually explain it step by step ("-g is the acceleration due to gravity, right? That's what you get when you differentiate twice. etc.") and it's like a light-bulb goes if inside their head. Suddenly they get a feeling for both calculus and what that damn formula meant.

Because here's the thing: there's an extremely useful, tangible, real-world application for calculus: basic physics! I mean, for Pete's sake, that's what calculus was developed for in the first place! Don't teach it as two subjects, teach them together, as one! You'll learn more, and both will be easier to understand.

When I was learning to solve a loop equation (electronics engineering) of a circuit, a lot of times we would end up with an integral on one side and a differential on the other. Solving such an equation isn't easy (at least for me). Fortunately we were taught Laplace transforms. We could then use plain algebra to solve, and then use the table to transform back to the original domain. The reason I mention this is that there is usually a diferent approach which may be easier (without being wrong). I have often learned concepts that were difficult to grasp by seeing two completely different approaches to the explanation. Hope something in my ramblings makes sense. May the force be against you! (Since it's dark energy).

Your explanation of your problem puzzles me. I think there's an important distinction to be made here, between the **practical** difficulty of doing a lot of arithmetic and the **conceptual** difficulty of attempting to solve a problem given insufficient information.

Let's say we're given the velocity as a function of time, which we can denote v(t). For the moment, I'm going to assume that this really **is** the velocity, and not just the speed; being a vector quantity, velocity has both magnitude and direction, and speed is the magnitude of the velocity vector. "100 kilometers per hour" is a speed, but "100 kph due north" is a velocity.

Remember doing "absolute values" in algebra class? The absolute value of a number is how far that number is from zero: +1 and -1 both have absolute value 1. Symbolically, this is written with vertical bars:

|-1| = |1| = 1.

Taking the absolute value means forgetting the **direction** you moved away from zero and only remembering the **size** of the step you took. Magnitudes of vectors work the same way:

|100 kph due north| = |100 kph due south| = |100 kph at 35 degrees west of north| = 100 kph.

If you integrate the **speed** as a function of time, you find how far you've traveled, but you won't know where you ended up. To illustrate, imagine I drove for one hour at 100 kph: I've been burning rubber for 100 km, but I might have been making left turns the whole while, so that I've ended where I began. My **velocity vector** has been changing direction, even though it's always remained the same **length**.

OK, now that the terminology is settled, let's do some kinematics.

We know our starting position, x(0), and we're told our velocity function, v(t). If we wanted to figure out how much wear we've put on our car -- the number the odometer measures, which we need to know so we can figure when to change the oil -- we take the magnitude of the velocity function at each point in time to create a new function, the speed:

s(t) = |v(t)|.

Va-va-voom, we integrate the speed.

Suppose we want to figure out where we ended up. We can do this if we know the direction information, i.e., if we have the full velocity function v(t) and not just the speed function s(t). We drove for 1 hour at 100 kph due east, then turned south and drove for 2 hours at 110 kph, etc., etc. Our position at time t is our initial position, x(0), plus the integral from 0 to t of the velocity.

If we **don't** know the direction information, we can't calculate the position function. This is just a fancy way of saying that doing high-speed laps at the Indy 500 will take you to a different place than driving the same speed for the same time in a straight line! It has nothing particular to do with the ease or difficulty of integrating different functions. **That** problem arises because finding integrals, at least the way it's taught in school, is "more art than science" (to coin a phrase). You have to recognize patterns: do we have the derivative of a function divided by the function itself? If so, the answer will involve a logarithm. Can we integrate by parts? Will a clever substitution of variables simplify the problem?

More recently, the mathematicians have figured out how to distill this "art" into something a computer can do, and (for example) often does a pretty good job. I bet Johann Bernoulli would have gotten a kick out of that!

I think Blake is getting at a similar point to the one I was trying to make, namely, that the way integrals are traditionally taught in school involves being able to recognize patterns -- at least, if one is going to successfully apply those principles. (And I discovered that, in fact, my DVD lectures DID cover this... it was just buried in a much-later lecture, so far removed from the initial discussion of integrals that I failed to make the connection until I watched the entire lecture series a second time.)

But the broader point I was trying to make is that most textbook examples necessarily must be simplified to a ridiculous extent, e.g., the car driving scenario is usually presented in the case of maintaining a constant speed, and/or a case involving constant rate of acceleration. Then you can use the nifty "cheat sheet" included in most standard calculus textbooks to find the relevant function(s). Where it really becomes art is when you try to apply it to a real-world situation where -- as Blake says -- you are missing key information.

And that, I think, is why the "real world examples" included in textbooks aren't always successful in helping students learn the abstract principles. They're not actually "real."

Yes, a great many of the "word problems" I remember from my schoolhouse days had clearly been invented to showcase a particular type of equation, rather than to illuminate an aspect of the real world. Feynman's story of evaluating textbooks is quite applicable:

I think to actually figure out why mathematics is hard for people to learn, we have to be careful not to confuse different difficulties they might have. If the problem involves calculating the position of a car, for example, and the given information is a big table of velocities measured at different times, then there's (a) the conceptual requirement of knowing how position and velocity are related, and (b) the arithmetic requirement of doing a whole bunch of calculations. If the given information is an algebraic expression for v(t), then the task becomes (c) the evaluation of an integral by symbolic methods, which is a different challenge then either of the other two. Realizing that the speed as a function of time isn't enough information to reconstruct the position falls under (a); if we were given s(t), our skills in department (c) would let us integrate it, but lacking an understanding of part (a), we wouldn't be able to tell how our efforts failed.

Precisely. How we misunderstand things can be very telling, in that respect. And that's assuming we're being honest about what we don't understand. :) I got As in high school algebra because I was really good at plug-n-chug, and thus my teachers didn't realize how much I was missing in terms of true comprehension. I was much too shy and embarrassed by my failure to tell them, and the end result was far too many years spent suffering from severe math-phobia.

I had a mild version of the same experience in fifth grade, or thereabouts. I went to a Montessori-like private school for three years, which was a great environment and helped me in all sorts of ways, but I don't think they had actually figured out how to teach math. They used a set of classroom materials called "Mortensen Math", which involves a lot of visual manipulation -- shuffling around blocks of squares drawn on the page, which represent terms in algebraic equations and such. It's a nifty conceit, but it didn't really **connect** with anything else. We didn't really even pick up the idea that "x" stands for an unknown number, and you want to find out what x is, or figure out what possible values it might have. There were even Mortensen booklets on "calculus", and in retrospect I realized that the columns and squares we were drawing and the arithmetic operations we were doing to fill in the worksheet-style blank spaces were actually differentiating and integrating polynomials. You can tell a kid that if you see x^n, you fill this other blank with nx^(n-1), but if you don't explain that this operation means finding the slope of a line, and this other one means finding the area under a curve, then it's just no damn good!

The booklets themselves were probably fine, but all such materials have to be employed by teachers who know what the point of the lesson is. In fifth grade, I didn't really know what I was missing; it took about another five years for me to learn enough math on my own that I could tell my formal classes were severely lacking.

Somewhere on the list of the top half-dozen things I'd like to do in the near future is to write a math textbook. This probably goes to show that I am, indeed, intermittently idealistic and perennially geeky.

Hey, I'm a fellow blogger here on the Web, although not nearly as science/math related as you. ;) I'm an artist/designer, and my blog is at: www.lanza-media/blog, if you want to check it out.

Anyway, I'm wondering if I can have permission to use the photo above of that chimpanzee in thought, because it would be a very nice visual addition to my latest entry.


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