Thursday, February 10, 2011

Math As More Than Algebra

I recently printed out some preschool concept worksheets to do for fun and learning for my soon-to-be three year old.

Having once majored in and taught math, it's not surprising that I initially gravitated toward the "pre-math" worksheets section. It included an intro to numbers and counting. Below this section was a section for worksheets that dealt with concepts such as "more and less", "small and large" and "same and different". "These sections should be combined under the "pre-math" heading", I thought (because I am a geek and take issue with such things).

If you believe that math is just about numbers, then I am sure you are scratching your head (or, more likely, never read past the title of this post). If, in addition, you cannot easily do arithmetic in your head or failed algebra in high school, you are probably also thinking "I am no good with numbers (and therefore no good at math)" or simply "I HATE math!".

I feel entitled to make these assertions because I am one of those people who failed high school algebra. In teaching and tutoring, I've met many young "haters of math" who have asserted their disdain for the discipline while taking an algebra class. So, why is algebra the culprit behind a youth in rebellion against mathematics, the most fundamental tool humanity possesses?

Well, algebra (actually pre-algebra) is the first time you really hear x = 5. WHAT?! How confusing is that? Confusing because it usually precedes a demonstration of how the rules governing mathematical operations exist independently of the values they are manipulating. It abstracts the grade school arithmetic we have all thought of as defining math, and does so through the use of symbols. It makes math about more than just numbers. It is now about the relationships and operations that exist between quantities.

This conceptual leap is quite challenging at first, probably due to inefficacy in how "pre-math" concepts are taught in American public schools to begin with. And, as is a common tendency of our species, our immediate reaction to something we don't understand is to hate it. However, if your curriculum takes you past high school algebra and into calculus, then you start to see math as describing how things change. How what changes? The position of flea on a dog, the flow of blood exiting a gunshot wound... ANYTHING. And once you understand that, you begin to see how beautiful and powerful of a tool it really is.

If you are in that small minority who agrees and goes on to study a field of mathematics in college, then you may end up taking a course called Linear Algebra. Linear Algebra is algebra applied to vectors, which describe anything having a magnitude and a direction in space. It is heavily used in natural, social and computer science (numerical linear algebra). It was this course that served to completely change the way I thought about...well... everything.

The first few weeks were dedicated to learning the theorems of linear algebra and how to prove them. A theorem, which is different from a scientific theory in that it is deductive and not empirical, has a hypothesis and a conclusion and is simply a statement of mathematical truth. A simple example is Pythagorean's theorem which says that if you square the lengths of two sides of a right-angled triangle and add them, you get the square of the length of the longest side. But what fundamentally changed my way of thinking and reasoning was the proving bit. I think Wikipedia says it best:

"The proof of a mathematical theorem is a logical argument demonstrating that the conclusions are a necessary consequence of the hypotheses".

I have highlighted "logical argument" because it is the exercise of considering a hypothesis and deducing what does and doesn't logically conclude, and repeating this day after day, that sharpens one's ability to reason. This is because "logical arguing"extends way outside the proving of mathematical theorems and into everyday life. Everyone in my class at the time emphatically agreed that, after beginning the course, they were able to catch people in logical inconsistencies more often. And, if they chose to do so and were not particularly shy or diplomatic, could provide an argument that was hard to topple.

If you are still reading this you may not hate math and may even be entertained, so I can go on to what sparked my interest in math and science to begin with. One late night, while I lay in bed watching Nova with my then boyfriend (now husband), a special came on. I can't recall what it was called but it was all about fractal geometry. Now is a good time for a quote from my favorite Mathematician and the father of fractal geometry.

"Why is geometry often described as 'cold' and 'dry' ? One reason lies in its inability to describe the shape of a cloud, a mountain, a coastline, a tree..."

-Benoit Mandelbrot, The Fractal Geometry of Nature, 1977, Ch 1

A key property of any fractal set and the reason I find them so intriguing is "self-similarity". In simplest terms, this is when the smaller parts of a thing look like the whole. Have you ever noticed that this is also a fundamental property of nature? For example, the branches of a tree, if cut from the tree and examined, look like small trees. If you then tear a twig off of the branch and examine it, it resembles the branch and, therefore, the tree. A snowflake is the same way. If you zoom in on a snowflake, it looks similar to the whole snowflake. Below, I list things whose shape can be described as fractal. I will stop short because the list is probably infinite.

Capillaries, nerve cells, clouds, root systems, rivers, lightning, seashells, broccoli, sea urchins...

The power of fractal geometry to describe and simulate so many diverse systems is seriously underexploited. But, the fact that there exist a math to describe something as intangible and complex as beauty in nature, is mind blowing and never fails to move me. Indeed, the "nature of nature", the underlying patterns that tie everything on every scale together, from cells to galaxies, is fractal. So, when people say they "hate math", I wish I could tell them there are such things as fractals. Instead, I hold my tongue because I know how it may received. I hope, however, that more and more people might stumble across these fields and start to think of math as more than just algebra.