Content Knowledge
Mathematics is an elegant science. It took me a considerable amount of time to come to the realization that math is not about pounding out numbers and possessing the ability to do "mental math", after all that's what calculators are for, but rather that math is about molding known material into new and insightful meanings. This is not to say that math is devoid of set rules and structure for few other fields surpass the occasional rigidity of mathematics. However, there is an art to doing math. There are subtleties in the most simple of mathematical concepts that can, and have, lay hidden for hundreds of years and it takes an artist to coax these into the light.
I will not claim to be such an artist for that takes part innate ability and part years of experience, both of which I do not have ample helpings of. I will claim, though, to have a deep appreciation of the subject and a nearly as deep understanding of what mathematics truly means and how it works.
Mathematics begins with questions. How many apples will I have left if I give Susie two of my five? What is the area of a square with sides of length 2? How can I find the cheaper cell phone plan based on monthly cost and price per text message? Why can't we divide by zero? How do we know that the real numbers are infinite?
These questions lead to inquiry in which the mathematician must apply all of her previous knowledge in order to find a suitable solution. Occasionally the inquiry also requires a bit of cleverness and/or luck as well. The outcome is a rigorous solution which contains sound logic and flawless execution.
One project that I got to work on, and which I think best exemplifies the idea of question inquiry, dealt with the countability of the algebraic numbers. Prior to beginning work with the algebraic numbers I had seen and worked through a proof of the countability of the rational numbers. This proof was a clever way of mapping the rational numbers into a pyramid structure which followed a certain method and ensured that no rational numbers could be skipped. This method was discovered by Georg Cantor.
So with Cantor in mind, my partner Danny Dorado and I set out to prove the set of algebraic numbers are countably infinite. Knowing what we knew from Cantor's proof, we felt that the key would be to devise a way to "count" the algebraic numbers without skipping any. Following clues from a textbook we discovered that we could focus on the set of all polynomials. Since an algebraic number is a solution to a polynomial then if we could devise a way of generating finite sets of polynomials then we could use the definition to prove that the algebraic numbers can also be grouped into finite sets.
With this in mind we defined the height of a polynomial as the sum of the highest exponent and the coefficients. So for a given height, we would have a finite set of polynomials and by extension a finite set of unique algebraic numbers. The last step was to "count" them by creating a map onto the integers. The final and formal proof is on page 8 of this PDF file.
Mathematics is not constrained to the confines of the formal proof and would have little meaning to the world if it were. Instead, math has many vehicles which allow sometimes complex concepts to be conveyed to the less initiated in such a way that the overall idea is understood without a necessary understanding of the underlying mechanics.
The clearest picture of this use of multiple vehicles to convey a concept is the use of algebra, graphs, and tables to explain functions. By far the most accessible form of a function is the graph. News programs, magazines, websites, and numerous other information outlets constantly use graphs to convey data to their consumers. Through a simple graphic the viewer can easily see the overlying structure of the data (i.e. increasing, decreasing, fluctuating) and get a "feel" for the underlying function.
The next level of function representation is the table. The table shows the same information as the graph but adds the actual values for a given set of data points. In this instance the consumer can not only gain a sense of behavior of the data but also of the data itself. Is the increase as great as the graph suggests or is the graph's scale misrepresenting the data? The table allows the viewer to see the trees in the forest.
The final level is the algebraic model of the function. Once trained a person is able to view an algebraic function and see in the mind's eye the intricacies of the model. Where are the zeroes, if they exist? Is it constantly increasing or will it bend and change direction at some point? Will it go on forever or approach a limit? These are the types of questions that can be answered from the algebraic model. Not only can we see the trees in the forest but also the ecosystem around each tree and the interplay between the branches.
The important thing is that a complex notion, such as a function, can be represented in multiple ways which allows multiple audiences to gain some level of understanding of the notion. Only by using such explanations does math become applicable to many people's lives and expand the knowledge of society.
One thing that I discovered during my upper level college math courses and which I had never realized before is the large amount of connections between what appear at first to be disjoint topics within mathematics. How does geometry relate to Calculus III? Why are Algebra skills so important to study number theory? These questions, and other like them, lead to interesting investigations and a deeper understanding of both topics and of mathematics as a whole.
One simple example that comes to mind is the notion of completing the square taught in high school Algebra I. In high school I was taught the procedure and understood its use in multiplying binomials, but I always thought it was called completing the square because of the 2nd degree in the exponent. It wasn't until my History of Math course that learned the true origin. The ancient Greeks performed algebra by doing geometry. One of the things they investigated was multiplying binomials. When they did this they formed a square with a missing piece in the corner. In order to "complete the square" they found the missing term.
This may seem like a trivial bit of information, but for me it was a revelation and sparked a quest into the history of mathematics where I found connections of all kinds between geometry and algebra. In my student teaching I have learned about algebra tiles, something I had never seen before, and now I can appreciate the beauty of using such geometric figures to explain algebraic concepts. The interplay between the two subjects adds a richness to both.
As mentioned above in Topic Connections, a discovery in the relationship between algebra and geometry lead me on a quest into the history of math. For several months I couldn't get enough of it. There are such interesting stories and surprises around every bend that math history reads like a modern thriller in some regards.
Take the case of Cardano and Tartaglia. Tartaglia had solved the cubic equation but was keeping it secret, as was the practice of the day. Cardano convinced Tartaglia to tell him the secret and that he wouldn't tell anyone. After Tartaglia told him Cardano quickly published the solution much to Tartaglia's chagrin. Tartaglia spent the rest of his life trying to bring down Cardano.
Although some of the history of math is merely entertaining it has a very important place in education. Math is often seen as a soulless subject devoid of feeling and this is a misconception. By using history we can breathe life into math and students can begin to see that math is done by people and these people have great triumphs, miserable failures, and everything in between...just like everyone else.
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