Not All Mathematical Ideas are Created Equal
I was reading a book this week, Mathematics for the Intelligent Non-mathematician. If it was a person, this book would be your grandmother, not terribly exciting but pleasant to spend time with and if you paid attention you were likely to learn something.
Since I use mathematics for my living, you might reasonably wonder why I would be reading this book. The answer is that I believe in considering different perspectives. I’ve never really quite “got” the whole humanities thing. When I took history in school, I was secretly thinking, “They’re all dead. Get over it.” In English class, I was the kind that made teachers throw up their hands in despair. They wanted me to discuss, “The deep meaning of Moby Dick, what do you think it is really about?”
What did I think it was really about. I thought it was about a big white whale, for crying out loud, because it said that on the first page and about seven hundred more times throughout the book. The title? That’s the name of the whale, hello? Apparently, that was not the correct answer and you are supposed to say that it is a metaphor for the universal struggle of man against the sea, or man against himself or for man’s domination of marmots.
As you might guess, the second I had the opportunity for classes in college like Accounting, Calculus and Statistics where the questions had actual answers, like 42, I jumped at the chance. This isn’t to say that I made A’s in all of those classes initially, as that would have interfered with my plan of going to parties at night and sleeping through the morning. This plan was ended through a talk with the dean and some threatening words about losing my scholarship and having to find $20,000 under a mattress. Heck, I didn’t even own a mattress, much less $20,000 to find under it.
So, here I am thirty years after graduation looking at mathematics from a more naive point of view, which brought out a couple of points I had never really given much thought.
The first is that mathematics is the most general thing in the world. You cannot apply psychology to rocks or biology to building a space shuttle or oceanography to orthopedic surgery. However, as the author said, you can count devils or angels, whales or stars. In fact, when I went from being an industrial engineer to studying for my Ph.D. in Educational Psychology I used the exact same equations I had applied to predict which cruise missile would fail testing before launch to predict which child with a disability would die within the next five years. (Yeah, I wasn’t a lot of fun at parties back then.)
The second interesting point was one that is obvious after someone else states it, i.e., some ideas in mathematics are more important than others. For example, it is a fact that the digits in multiples of nine always add up to nine, e.g., 2x 9 = 18 and 1+8 = 9. This is not a key fact on which a lot of mathematics is based. So, this led me to thinking about the ideas in mathematics that I think are crucial and wondering about what other people think.
I always thought that the basic properties of real numbers, such as the distributive property –
A x B = B x A or A+ B = B +A was one of the most fundamental ideas in mathematics.
A second really important idea was the associative property, –
A(B+ C) = AB + AC
and the commutative property is a third
(4A + 2B) + 11C = 4A + (2B + 11C)
Once a student understands these properties, it opens up an enormous number of problems that he or she can now solve.
And that is why I like teaching Algebra.
It is indeed true that not all mathematical ideas are created equally. Some are much more fundamental and essential than others.
The author mentioned the commutative, associative and distributive properties among these.
The commutative property: ab=ba or a+b = b+a.
This concept can be helpful, for example, if a student needs to mentally calculate 12 x 7, but decides it is easier to find 7 x 12.
The associative property: a(bc)=(ab)c or
(a+b) + c = a + (b + c) can be helpful in mentally finding the sum of (23 + 14) + 6, by first adding the last two numbers and adding that sum to 23.
The distributive property of multiplication over addition: a(b+ c)=ab + ac, can enable he student to mentally calculate 7(103) by doing 7(100) and then 7(3), which then becomes the intuitive way for this problem to be done without paper and pencil.
I would add the subtraction of property of equality to the list of essential mathematical ideas, and probably place it very high near the top of the list. It is this property that enables students to solve algebraic equations.
Abstractly, it can be expressed this way: if a=b, a-c = b-c. In actuality, a better way to express it would be if a+c = b + c, then a = b, which is an equivalent formulation.
As must be clear, however, the symbolic notation is rather dry and not exciting, yet the concepts are essential for students to learn, and the earlier the better. Making the concepts visual or kinesthetic does make it interesting and accessible even to grade school students. One way to accomplish this is through the Hands-On Equations approach (I am the author of that program) where the students use game pieces to learn the subtraction property of equality and successfully work with equations such as 4x + 3 = 3x + 9 and 2(2x+3)=x + 9, even in the upper elementary grades. A video demonstration of this method can be found on You Tube or at http://www.borenson.com.
I would like to recommend you a good movie called “Pi”. The main character is a mathematician obsessed on discovering patterns in everyday life, the stock market and in the Torah.
These are the rules of the main character’s life:
(1) Mathematics is the language of nature; (2) Everything around us can be represented and understood from numbers;
(3) If you graph the numbers in any systems, patterns emerge. Therefore there are patterns everywhere in nature.
The movie does a good job a explaing how math
has been used to explain nature and the universe be it the spiral, fibonucci numbers, the golden ratio.