The mathmatics of education
Math is a language with little difference from others. In english I can say, that red ball is larger then yours and we know something about the world without having to experiance it. Mathmatics works the same way. If I write that x+y=3 then we know alot about x and y. In fact, because definitions of x and y are inherent in that equation we know an aweful lot about x and y.
That, though, is litterally childs play. The grammer of math can be extended to many more complex situations with relatively few assumptions. The beauty of this is that if we know alittle something about a number or veriable often we can end up knowing an enourmous amount. So the task of an engineer becomes fitting reality into the mathmatics so that the little that we can say we actually know can be translated into more then we care to know.
The only reason this works is because the gramatical structure of math is so very strict. In english, I can say that pig is flying through the orange sky or that I am terrific beyond comprehension. That flexability is very powerful in terms of description but poor in terms of extrapolating information. There are no rules in english that keep me from telling lies about myself and besides, terrific is a rather vague term that used to mean something quite different. Math though, is timeless and harsh in its rules.
Unfortunately very few of us ever actually learn math. Most of us are those people in spanish class who memorize the important phrases: “Como estas? Muy bien, y tu? Asi Asi…”. Moreover, the higher that one gets in math the more complex the memorization becomes. A good example of this is the line Que sera sera, whatever will be will be. The grammer of that phrase is quite complex compared to typical “Donde esta el bano?” but as long as you and the person you are talking to have memorized its meaning then it doesn’t require knowledge of the subjunctive verb tense. Most of math is like this, expecually for engineers and scientest. We are memorizing ever more complicated song phrases trusting that our collegues have also.
Which is fine. Like most languages, if you are imersed in it long enough the phrases start to make sense as a whole and eventually, you can speak it. Only, instead of a 6 month study abroad to Spain it took 7 years of college *shakes fists*. The problem though is when those fluent in the language attempt to speak to those who are not.
Spoken languages are a majority vocab and a minority grammer. In fact, I can get across an awefull lot of meaning speaking only nouns if I needed to. Math though is the opposite, it is almost entirely grammer and this is what makes it difficult to pick up through immersion. Recognizing vocab words is easy, you hear the word pollo a few times while someone is pointing at a chicken and its hard not to understand what pollo means. Understanding the rules behind partial differential equations is not easy.
This is the crux of the problem with the undergraduate engineering education. The vast majority of the professors have become proficient in math, at least to the degree that they prefer to talk about engineering in those terms. They have good reason, that is what math is for! The students though must perform a mental translation from math to english and then from english to reality and often that does not happen. They develop reflexive responses to problems without ever really knowing what they are doing. It would be like taking spanish for 4 years and never getting beyond conversations like “Como estas? Muy bien, y tu? Asi Asi…”. Its no wonder people get frustrated with engineering.
As the focus of education gets further and further from route memorization the problem will continue to get worse. In the past, engineering students did math problems till it hurt. Indeed, engineering used to be a far more elite major. Now though, with the advent of computers, the emphesis is on understanding the problem and less on solving it. Overall this is a good thing but it robs students of valuable experiance in math that they need to be able to understand their professors.
Math is valuable because it is a uniquely unambigous method of communication. A problem, once formulated mathmatically, has a solution. The difficulty often is not solving the problem but rather formulating it. Granted the quality of the solution will vary based on your mathematical ability but in principle it is true. For the vast majority of the students though the difficulty is seen in figuring out the math problem, the formulation is secondary and besides, you can usually tell which section it came from by the problem number…
What to do? Start teaching math better! Not just college but straight down through first grade. We need to start teaching math like a language and less like… math. How well do you think you would be able to learn a foriegn language if the only exposure you had to it was a text book?
There is a language learning method out there called the Pimsleur Language Learning System. Like I said earlier there are significant differences between math and the spoken language but there is no reason that a similar method, based on cognitive psychology [read:science] cant be developed for math.
Some people are trying, here in Boston and in other places. What we need though is an Apollo program for education. Our knowledge of the brain and developement is not even comparable to what it was 10 and 20 years ago when most of our educational standards and state mandated curriculums were being developed. These curriculums have done nothing but strangle any form of progress that might be made. Teachers no longer have room to experiment for fear of not fitting in some section of the state aptitude test. Failure at the aptitude tests has dissasterous effects on funding.
Somehow, given the current congressional orgy of control, I don’t see that happening.