Every few days, a new study decrying the sorry state of the mathematical aptitude of American students is published and the powers that be, particularly those in education, wring their hands in exasperation before advocating some new trendy way of teaching math.
The reality is that all of these new pedagogies are doomed to failure because of a fundamental flaw in understanding about what mathematics really is.
When most people think of mathematics, they are really thinking of its most elementary form: arithmetic. Ask the vast majority of Americans what math is about and they will say "numbers." Saying mathematics is about numbers is like saying Shakespeare is about words.
Where would human civilization be if our conception of mathematics never evolved beyond that most elementary understanding?
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The history of mathematics is a rich tapestry of human ingenuity and pursuit of an effective way to understand the world around us. Mathematics is a human discovery, not a human invention. The forces of the universe, the interactions of matter and energy are all explicitly and implicitly defined by mathematics.
With our need to understand an increasingly complex world, humans discovered new approaches to the real components of mathematic: patterns, processes, shape, quantity, structure and change.
At its core, the essence of mathematics is the search for truth that can be verified by irrefutable proof. Epistemologically, the question evolves from what do you know into how you know that a proposition is true or false.
Although recently the dumbing down of the educational standards has corrupted the way it is taught, the first glimpse of true mathematics occurs in the subject that is the anathema of high school students across the nation: plane geometry. The Greek civilization's numeration system was very awkward, yet the Greeks laid the foundations of modern mathematics using compasses and unmarked straightedges to explore and discover basic truth about shapes and the logical arguments to prove or disprove statements about the properties and relationships of and between those shapes.
The Greek philosophy of mathematics has played out repeatedly in the field of human knowledge. Over time, mathematicians have expanded the boundaries of the counting numbers to include, those devil-spawned fractions, zero, negative numbers, irrational numbers like pi, and whole new branches of math such as calculus, topology, graph theory, logistics, and so on.
Although many of the new ideas were conceived as abstractions with no immediate practical applications, people kept finding new ways to use them. For example, most people do not realize that UPS drivers are given a stop-by-stop schedule that has been worked out mathematically to minimize deadheading and unnecessary turnarounds.
Chemistry and genetic engineering rely heavily on understanding the topological structure of molecules and chromosomes. The calculations for the physics of electrical engineering and many other fields are greatly simplified by using the so-called imaginary and complex numbers.
Higher mathematics can produce surprising results. Mathematicians have proven that a completely fair election is impossible when more than two candidates are running. The fair apportionment for the House of Representatives is another aspect of our democratic society that cannot be realized because legislative seats can only be assigned as whole numbers.
Statisticians and actuaries calculate the probabilities of events that control our lives: automobile, medical, and life insurance, epidemiology of disease, the odds of sporting events and winning the lottery. Experts in logistics design the roads and signals that dictate traffic patterns and even such things as how many hamburgers fast-food restaurant keep on hand to minimize customer wait times while maintaining acceptable product quality.
I have hardly scratched the surface of how mathematics shapes our lives, but clearly something is wrong in a society that pigeonholes student mathematical expectation into arithmetic that can be done on a cheap calculator. Over the last few years, student achievement in mathematics has plummeted because they hold the opinion that pushing buttons on a calculator is mathematics.
The ideas of problem solving, creativity, ingenuity, logical thinking and persistence are completely foreign to students today. For these students, an answer that cannot be obtained with a few strokes of a keyboard is not worth the effort.
Unless the public abandons its stereotypical view of mathematicians as a bunch of pencil-necked geeks and ivory-tower academicians, we can expect our students to languish in the cellar of international competitiveness.