Sanderson M. Smith
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MATH COMMUNICATION: A NOTE TO SECOND-YEAR ALGEBRA STUDENTS
I do hope that you believe that I want you to learn (not memorize) mathematics, feel good about learning mathematics, realize the importance of mathematics, and become effective readers and writers of mathematics.
As you know, I view mathematics as a language. It is, in my opinion, the language of the universe that we inhabit. The creator of the universe, the force behind the universe, or whatever you think governs the universe, is a mathematician. I don't know if the amazing laws of mathematics are created or discovered, but the fact remains that mathematics rules the universe. If you've been following the math history tidbits, you know that Pythagoras (500 B.C.) and the Pythagoreans were the first to realize this. Plato (400 B. C.) believed in an external world of mathematical truths, and that humans simply discover mathematical laws that already exist in this other world. Some other schools of thought do not accept the external world idea, and argue that humans use their mind to create mathematics. The intuitionist philosopher Immanuel Kant (1724-1804) totally rejected Platonism, arguing that mathematics is not inherent in the physical world, but rather comes directly from the human mind.
What is the nature of mathematics? Is it discovered? Is it created?
Quite frankly, I don't know. What I do believe is that in our modern world, many more opportunities present themselves to those who are mathematically literate than to those who are not.
Here's something I hope you will all think about. As you move on in life, you will probably make great use of word processors as a method of communication. I assume you do it now in courses like English and history. Handwriting, while certainly important, is generally less efficient since it takes longer to produce and is generally less neat. When you use handwriting, you can certainly slop all over the place. When you do this, you are not giving yourself any useful training in communicating. I'm speculating that you do not slop all over the place when you type. (It would be almost hard to do this, unless you typed total gibberish.)
What's my point? Simply this: Train yourself to be neat, and to communicate, when you do your handwritten homework. This training (which requires self-discipline if you are not yet used to doing it) will pay great dividends in your future. (At least I believe that!)
Having recently reviewed your notebooks, it is clear that many of you are making some efforts to communicate in mathematics. Some others don't appear to be making this effort. The latter group seem content to be residents of what I will call Slop City. Now, at this level of math, it is definitely possible to reside premanently in Slop City and do well in a course such as second-year algebra. You can be a Slop City resident for the entire year, and get an A in second-year algebra. If I can't convince you that making efforts to get out of Slop City will pay future dividends for you, then that is my failure, not yours.
We are definitely at a point where you can make efforts to use the language of mathematics properly and effectively. Current problems involve the concepts of direct variation and inverse variation. Below are homework-type problems in which I have indicated both the premise (given information) and a solution. You should be able to read it since I have made an effort to communicate. I remind you that I am using the symbol ==> as "implies." If p and q are statements, then p ==> q can be read "p implies q," or "if p, then q," or "the truth of p necessitates the truth of q." A statement such as 10/2 ==> 5 makes no sense.
Physics: Falling Objects. The distance s an object falls is directly proportional to the square of the time t of the fall. If an object falls 16 feet in 1 second, how far will it fall in 3 seconds? How long will it take the object tofall 64 feet?
The solution and associated reasoning follows:
Premise ==> s = Kt2, where K is a constant.
t =1 when s = 16 ==> 16 = K12 ==> K = 16.
Hence, equation relating s and t is s = 16t2.
In what follows, s unit is feet. t unit is seconds.
When t = 3, we have s = 16(32) = 144 (ft.).
When s = 64,we have 64 = 16t2 ==> t2 = 4 ==> t = 2 (seconds).
Here's another example:
PREMISE: x varies directly as y and inversely as z.
x = 80 when y = 20 and z = 2.
What is x when y = 10 an z = 4?
The solution and associated reasoning follows:
Premise ==> x = Ky/z, where K is a constant.
x=80 when y = 20 and z = 2 ==> 80 = K(20)/2 = 10K ==> K = 8.
Hence x = 8y/z.
y = 10 and z = 4 ==> x = (8)(10)/4 = 20.
Math is power. Math is a language. Use it properly. Say what you mean! Mean what you say!
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