YOU HAVE TO DEVELOP YOUR INTUITION. Jane Shaw offers a false choice in the teaching of mathematics.

Should students learn about about the real world or about the beauty of abstraction? (And can the normal college student appreciate the latter?)

She links to a Robert Blumenthal essay for the Pope Center that hits at the real problem.

The one area of general education that has been almost totally ignored is mathematics. Most of the required courses are dull, pedestrian, and often repetitive of what students took in high school. At many schools College Algebra is the mathematics course that students must take. However, this course is really second-year high school algebra. It asks of students no more than they were required to do in order to graduate from high school.

The challenge, dear reader, is to develop the connection between the beauty of abstraction and the real world. Consider a simple problem. A bank pays 100% simple interest per annum. If you invest $1 now, after a year you have ... everybody get $(1 + 1) = $2. (I'm using notation to compel your mind to see the solution.) Now suppose the bank pays 100% interest, but it compounds semi-annually. If you invest $1 now, after a year you have $(1 + 1/2)(1 + 1/2) = $2.25. With me so far? How does your answer change if the bank compounds quarterly? Now your $1 compounds to $(1 + 1/4)⁴ = $2.44 (the bank keeps the remainder.) If a really aggressive bank offered to compound your money continuously, what would happen?

Evaluate lim_n->infinity (1 + 1/n)ⁿ.

The number you get, 2.7 1828 1828 45 90 45 ... is the upper bound on what the bank pays out. It's also one of the more useful concepts in applied and pure mathematics, the natural exponential base e.

Perhaps, when Mr Blumenthal makes the case for mathematics in the core, it's developing student intuition that he has in mind.

Students should be offered the opportunity to have an engaging and meaningful experience with important mathematical ideas. It isn’t easy to devise such a course for students whose mathematical background is limited, but it can be done. If we aspire to produce graduates who truly are liberally educated, it should be done.

Admittedly, the student who grasps limit is not the student whose mathematical background is limited, so to speak, but how often does a high school math teacher make a hash of the concept of limit. And how often does the high school or college algebra text simply introduce e (and its inverse, the natural logarithms) without any context at all.

Or consider some tricks with squares. It's relatively easy to see that (a + b)² = a² + 2ab + b². There's a large class of problems for which the factorization a² - b² = (a + b)(a - b) is useful. But what do we do about a² + b² (besides equate it to c²?) Imagine a number i, such that (a + ib)(a - ib) = a² + b². You say i² = -1? Is i algebraic or transcendental?

I'm not sure where that trick fits in Mr Blumenthal's scheme.

Students deal wtih numbers all the time, but unfortunately their only conception of numbers it that they are things you compute with. Our schooling, with its relentless emphasis on solving real-world problems, teaches us that we can use numbers to model all sorts of phenomena and that crunching these numbers can enable us to better understand real-world situations. It is unfortunate that, beyond computational applications, there is little appreciation of all that underlies the concept of numbers.

On one hand, using the result above, one can demonstrate that the product of two sums of two squares of integers can be expressed in two different ways as the sum of two perfect squares. Check it out. Is that a computational application, or the basis for deeper conceptual understanding?

Here's where I think more intuition would be useful. I referred to adding a² to b² to produce c². There has to be some parable one can tell about generating the trigonometric functions. There has to be an intuitive way to get people thinking about walking around the perimeter of a circle in terms of the distance travelled from where you started -- that has to be the basis of the use of radians in the trigonometric identities and in the development of the infinite series expansions of the sine and the cosine and the wonderful result e^iπ + 1 = 0. To introduce equivalences to degrees (one radian is approximately 57^o) confuses the issue. (I admit, telling the navigator to steer course - π/4 lacks the romance of Make Course 135^o True, and that negative sign would drive multiculturalist conscience-cowboys with a smattering of mathematical literacy crazy.) There must, however, be a better way. I've been reading up on the theory of complex numbers recently, and there will be more to come. Study up on the implications of r = 1, θ = arctan(b/a), where, again, I am using notation to compel your mind. There will be a quiz later.