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Teaching the Fullness of Mathematics

Students can see mathematics in all of its fullness, and experience a wonder akin to being dropped into an out-of-the-way part of a foreign country with a phrase book and desire to figure things out. It’s big. It’s strange, and one doesn’t really know what’s going on. But it’s a real place.

Every fall, here in the United States, we lament the woeful underpreparedness of entering college freshmen in the field of mathematics. Proverbial stories abound of students overfilling classes called “College Algebra” or “Pre-Calc” in search of the elementary mathematical competencies that will lead them to introductory Calculus and Linear Algebra classes.

Students are unexcited at the prospect of such remedial learning, and instructors themselves stand just as unstimulated in front of blocky, laconic PowerPoint slides.

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The obvious assumption is that a focus on fundamentals will provide these freshmen with the necessary tools to approach the subjects heretofore deemed gates of entry into Higher Mathematics: An understanding of functions will lead one to grasp what derivatives and integrals are all about. Knowing what a matrix is will allow one to see different and new ways of solving algebraic equations, etc.

But what if this approach makes the mistake of viewing Mathematics from the wrong angle?

If we step back and take a long gaze at the totality of Mathematics, we see a world of mysterious, complicated, and (in most cases) incomprehensible sets of symbols.

Go to your nearest university library and pull an advanced Mathematics text from the shelf. Open it at random. Behold the aesthetic value of the letters, shapes, and signs. Leaf through a few pages. Behold a hidden world of flowing, dense collections of printed ciphers. What is all of that? Even without comprehending anything, one can be made aware of the beauty of the subject.

It is this sense of vastness and sublimity that gives us the power to reshape our thinking about Mathematical learning and teaching.

There is no area of human endeavor more international, more respected, more merit-based, more competitive (Think Math Olympiads), less suffering-of-fools than Mathematics. It’s as difficult to fake as it is to learn, and our math thinking would most likely be the metric used by extraterrestrials in a fly-by evaluation of our intellectual precocity.

With that in mind, why not introduce students to the totality of mathematics right away, during their freshman year? “Open the book,” so to speak, on the whole subject.

Most people don’t really know what Mathematics is because they sit unaware on the shoulders of giants as they fiddle with this derivative or that integral. It’s rote, painful, and diminutive.

What if we taught introductory Mathematics from the top-down?

Imagine a 100-level course that divides the whole of Mathematics into the great areas based on the subject classifications of the American Mathematical Association’s Mathematics Classification System (MCS).

We could have, say, a big swath that covers “Analysis.” Similar modules could comprise “Algebra,” Combinatorics,” “Geometry and Topology,” “Probability and Statistics,” etc.

It’s a taxonomic way of learning that takes advantage of the universal dimensions of the subject and would use as its primary text an encyclopedia or dictionary of the subject such as The VNR Concise Encyclopedia of Mathematics or Zwillinger’s CRC Standard Mathematical Tables and Formulae

If a student knew a little about what Galois Cohomology or a Cokernel or a Mellin Transform was, she could work backward and downward into the more elementary terms and axioms. It doesn’t take much knowledge to get hooked on the mystery. One entry in such a reference text can send one on a historical and definitional journey of discovery.

That way, entering so-called remedial students can see mathematics in all of its fullness, and experience a wonder akin to being dropped into an out-of-the-way part of a foreign country with a phrase book and desire to figure things out. It’s big. It’s strange, and one doesn’t really know what’s going on. But it’s a real place.

We need to take advantage of our student’s sense of awe, and leave the “elementary building blocks” in the hotel lobby off the main road.

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