More recently, I've been following a conversation at Phi Beta Cons about effective methods of teaching mathematics. The conversation continues, with the continued false choice.
Getting a general notion of how mathematicians think may be interesting or useful to a small fraction of students, but I think most of them would be better served by actually doing some math themselves.It's necessary to do both, as a linked American Heritage article about the folly that became New Math illustrates.
[Proponents] argued that math could be exciting if it showed children the whys of problem solving rather than just the hows. Memorization and rote were wrong. Discovery, deduction, and limited drill were the best routes to arithmetical mastery.The first sentence illustrates that practice, and organizing concepts, both matter. The second paragraph leads to the worst form of deconstruction: until you know the dimensions of the box, it is silly to speak of thinking outside the box. But the teachers didn't know the dimensions of the box. (The generalizations to Marxian or Freudian approaches to literary criticism are left to the reader as exercises.)
In practice, this meant learning how different number systems worked, that the number 9 in the decimal, or base ten, system would be the number 100 in base three. It meant learning about the set, a grouping of things: a beach as a “set” of grains of sand, for example. It meant learning the difference between a number like 7 and its representation the numeral, which could be expressed many different ways—21 minus 14, 7 times 1, VII.
New math became a pejorative term. And because it was difficult to know if trying to understand the structure of math made it any easier, most teachers deserted discovery learning without any pangs.In part, because they had not developed their intuition.
I wrote that previous post, and all of the preceding paragraphs, as a motivation of Book Review No. 6, Paul J. Nahin's An Imaginary Tale: The Story of i. It's been a long motivation, and -- my taste for light reading being somewhat idiosyncratic -- the review itself is going to be heavy going, even by my usual standards. You might want to skip to some other post with the observation that Imaginary Tale is useful at what it does, although its presentation is uneven.
The part of New Math, and discovery learning more generally, that I agree with is that an intellectual discipline is more useful if the learner has an opportunity to confront a challenge from which the value of a principle might be hinted at or observed. Many of the economic education lessons for K-12 work in that way, and my exercises with compounding interest and factoring (a2 + b2) had that approach in mind. (I also produced the complex conjugates a + bi and a - bi ... the dramatic convention of placing a gun on the table in the first act in order to fire it in the third.) Professor Nahin's book, which he claims is not a textbook, might nonetheless be useful as a textbook or as supplemental readings, as he makes an effort to provide some of the intuition that led to complex analysis.
It's an uneven effort. In order to do complex analysis, it helps to have an intuitive understanding of i itself (and I stole the factorization of a2 + b2 from Imaginary Tale) and e plays a prominent role. But economists have a better intuition for e than other applied and many pure mathematicians because of that interest rate connection. It's also useful to understand such things as polar coordinates, the series expansion of the trigonometric functions, vector addition, and linear algebra.
These are things that ordinary mathematics textbooks often throw at the student with little preamble or preparation. Here's how my high school calculus text, the fourth edition of Calculus and Analytic Geometry by George B. Thomas, does it.
Another useful way to locate a point in a plane is by polar coordinates. First we fix an origin O and an initial ray from O. The point P has polar coordinates r, θ, with ...I confess that my knowledge of polar coordinates was limited to being able to transform a Cartesian pair (x,y) into a polar pair with the same origin of r = (x2 + y2)1/2, θ = arctan(y/x), and if I had a polar pair (r,θ) I could do x = r cos(θ), y = r sin(θ). Years later, Buddy Melges gave a talk at the Lake Geneva Yacht Club about speed testing the AMERICA3 fleet where he referred to "polars."
That leads to an (admittedly culturally biased) exercise in developing your intuition. What is the easiest way to keep track of the velocity made good toward a windward mark by a sailboat? Straight into the wind, you go nowhere. At right angles to the wind, you're fast, but you make no velocity good toward that mark. With the wind behind you, your velocity made good is negative. On the wind, if you get too close to the wind, you're slow (that's pinching) and if your angle of attack is less acute, you're faster, (that's reaching) but not making velocity good toward the mark. The art of sailing a boat upwind is to find the right balance between pinching and reaching, so as to obtain maximum velocity made good. (Believe me, I've struggled with that balance, on rare occasions to good effect.)
Mr Melges has excellent intuition for sailing, and he didn't sound too excited about the polars: all that mattered was that he had a close-winded boat. Call that getting good effect more frequently.
The engineers wanted more precision. You sail at varying angles to the true wind and work out the boatspeed and the velocity of the boat is v. So your polar diagram has r = v and θ is your boat's angle toward the true wind, which is your initial ray. On a frictionless lake with a constant wind speed from a constant direction, your polar diagram will resemble a cardioid. (On a real lake with shifts and puffs, you have to do a lot of tests.) With the right knowledge of the laws of physics, the naval architect and the engineer can work out whether the boat's cardioid has the classic form r = k(1 - cos(θ)) where k is a positive constant that could itself reflect the hull shape. And the tacking angle that offers the best velocity made good has the maximum value of v cos(θ), which is easier to observe on the polar diagram with the aid of a see-through square.
For this exercise, it suffices to measure angles in degrees. But for what is to follow, a student has to be comfortable with radians. The idea of a radian is simple enough: it is the angle subtended by the arc of length equal to the radius of a circle. To walk all the way around the circle is thus to travel 2π radians. That may be intuitive enough, but the website I got it from contains a classic understatement.
The reason for this is that so many formulas become much easier to write and to understand when radians are used to measure angles.Somebody must have had reason to invent the concept. For some formulas, it's mandatory.
Although the word "radian" was coined by Thomas Muir and/or James Thompson about 1870, mathematicians had been measuring angles that way for a long time. For instance, Leonhard Euler (1707-1783) in his Elements of Algebra explicitly said to measure angles by the length of the arc cut off in the unit circle.We're about to get to Euler, but before Euler was DeMoivre, and analysts knew that sin(θ) and cos(θ) had infinite-series expansions that only made sense in radians, and the Greek geometers had made a lot of progress with the circular functions. Provide the intuition.
Imaginary Tale provides a lot of intuition about what comes next. The student has to be comfortable with what came before, and more intuition (or some kind of discovery learning) might be helpful. For instance, express the complex number a + bi as a vector. This idea occurred to a Norwegian surveyor, Caspar Wessel, and after more thought, became the Argand diagram of calculus classes. Now the x-axis gives the real coefficient, and the y-axis the imaginary coefficient. (The mind boggles at Flatland's Edwin Abbott introducing a complex sphere -- here's a new twist on Upward, not Northward.) The fun, however, has only begun. Your polar, now, converts the Cartesian coordinates x = a, y = bi into the polar coordinate r = (a2 + b2)1/2, θ = arctan(b/a). But if I give you (r, θ) you must give me back x = r cos(θ), y = i r sin(θ).
Now, expand limn->infinity(1 + iθ/n)n = eiθ = cos(θ) + i sin(θ). (Euler got away with one there, factoring on i and summing two different infinite series.) The Argand diagram and the infinite series expansions of the trigonometric functions and Euler's efforts with e took place separately: are there some classroom exercises for students to be able to see how each of these things might have happened? With all this at hand, consider the product i(a + bi) = -b + ai, convert to the polar form as r = (b2 + a2)1/2, θ = arctan(-a/b): multiplying a complex number by i rotates a vector by π/2 (it's really less complex than tacking a sailboat!) And there's a tantalizing linear algebra trick in the footnotes (page 81, endnote at page 244): let J be a 2x2 matrix [0 -1 : 1 0] where the colon separates each column. Then JJ = -I: a squaring operation that produces a negative identity matrix with no annoying complex numbers.
You're now ready to locate buried treasure on an island where the clues are two trees that are still standing and a gallows that has rotted away, to work out the periods of retrograde movement of the planets, and you can probably best that Very Modern Major General in cheerful use of the binomial theorem, the square of the hypotenuse and mysteries of the hyperbolic sine. "The shortest path between two truths in the real domain passes through the complex domain" (page 70) forsooth!
And thus my griping about the intellectual effort people put into parsing Dan Brown's work. Start instead with Gödel, Escher, Bach: An Eternal Golden Braid, read and understand good histories of e and π, familiarize yourself with the greatest theorems. In particular, dear, reader, if you would teach mathematics (as opposed to providing training in pushing buttons on a five dollar calculator) at any level from the fourth grade up, your intuition must be first rate. (Enthusiasm for the subject is a plus, but enthusiasm without intuition will not produce learning.)
That said, I must call Professor Nahin out on some details of presentation. He tweaks (pages 162-166) an obscure British mathematician, Roger Cotes (1682-1716) of whom no less than Isaac Newton said "If he had lived we might have known something." But what Professor Cotes left us was a difficult to follow manuscript including an expression -iθ = ln(cos(θ) + i sin(θ)) "embedded, in almost unbelievably obscure language" in a 1714 paper. Professor Nahin concludes, "His readers at the time probably simply didn't understand what Cotes meant. Let this be a lesson in the value of clear exposition!" Sometimes the exposition in Imaginary Tale is less than clear, as the presentation sometimes switches from the mathematical conventions I have been using to the forms more common in engineering (where i becomes j because Ohm preempted i (and e, but I digress), and polars become (x2 + y2)1/2/_tan-1(y/x)) for a lot of extra typesetting to no apparent gain in clarity.
(Cross-posted to Cold Spring Shops.)