Take for example the problem of finding the break-even point in a cost-revenue analysis. The problem is to find the production level for which cost equals revenue. This is a problem that may come up in an actual business situation. But what kind of person will need to answer this question? Probably not a person who is only required to know a small amount of algebra to earn their required minimum math credit. So why would we teach this type of problem to everyone, regardless of whether their degree is in business?
If someone needed to know how to do this, it should be taught in a business class. Actually any smart business would have already solved this problem and established a formula so that their numbers could simply be plugged-in to get the break-even point, and no one would actually ever write down an algebraic equation and solve for x as part of their daily routine. There's really no need to teach everyone how to solve the problem if all we care about is having a technique to get the answer. But it seems that there's still a good reason to teach someone how this problem was solved in the first place.
Learning the problem of finding the break-even point is actually a fantastic example of how our minds work, and can demonstrate the process of problem solving. But it is not taught as such, the only emphasis being "you may be faced with this problem some day, and here's what you do." In other words, we have completely removed the education experience and replaced it with job training. Yet it is depressing to students to think that they have to struggle learning all of the language, notation, and mechanics of algebra just so that they may need it for the job. No wonder we as teachers are seeing such depleted morale in our classrooms.
Wouldn't it be better to teach students how to think, not what to think? If you were truly educated, and knew how to think, you wouldn't need to be shown what to do to solve this problem, because you could think for yourself and apply problem solving skills. You would be a human being with a mind, not a robot who does exactly as told. This is the essence of the problem. Job training tells people what to think, education teaches people how to think. Remember the old "give a man a fish..."? As an educator I would prefer to teach people how to think, and thus feed them for a lifetime.
Administrators who need to make "fiscally responsible" decisions realize that they too have a break-even point that needs to be met. Unless a certain "production level" of students is met, the business is operating at a loss. "Come to college, get a great job, and now you're rolling with cash" is their hypnotic chant. What better selling-point than this could be advertised to lure in prospective customers? But the business of college cannot represent itself this way as an institution, so they hide their true money-making agenda under such "mission and values statements" that claim education is the top priority. I would like this to be true, but as a teacher I am required to teach specific things that are not the best for imparting a top-quality education experience. It seems that colleges and their government financiers do this because they don't really want well-educated citizens who can think independently, probing for the truth, just workers who are good at taking orders.
"The mind is not a vessel to be filled, it is a fire to be kindled."
-- Plutarch, Greek Platonist, 1st century C.E.
It surprises me however that, while some colleges do have better ideals and take leading initiatives to increase student success and graduation rates, they don't realize that it is their view of the role of mathematics in the curriculum that creates a reversal in the students' motivation for success. "Why do they make me take these math classes? I just want a good job, I'll never use this stuff", which is what anyone would think who thought they were getting a worm for a meal and now has a sharp hook in their mouth.
All criticism aside, I think the problem is simply that no one but a true mathematician can understand why we need mathematics, and usually a true mathematician does not become a college administrator. I have decided to use my understanding of the original intention of mathematics to campaign for mathematics as education and not as job training.
Mathematics, rightly understood, is completely useless. "If it ain't got no use, why do we need to learn it?" Here I'm merely resounding an echo that causes pain in my ears. Actually I'm playing upon the meaning of the word "use", because in the sense it appears here, it refers to the vulgar use of mathematics, which is all most people can conceive. The vulgar use of mathematics refers to building houses and other such things. This is how you think when your mind is clouded by a false belief that the material world is all there is. We must understand that mathematics was not invented by the Greeks for building houses, but for building temples. This is actually a metaphor that shall now be expounded.
The true enlightened use of mathematics is to liberate the mind from the illusory way of thinking derived by sense experience. It is meant to purify the mind from accepting appearance as reality, and to lead it into a way of knowing the truth as such. Unless one builds knowledge systems after the pattern of mathematical knowing, it is inherent that errors will manifest themselves. Only one who thinks like a mathematican can become aware of the subtleties of reasoning, and can establish truth on a rigorous foundation.
Even if we don't want to become philosophers, although really we should, since if we are not lovers of wisdom then we are not experiencing the fullness of life, there are still good reasons for why true mathematics is a necessary component for every person who claims to be educated. Mathematics teaches us how to think, not what to think. It teaches us reason and logic, problem-solving, and how to discern error from truth. If you can't think for yourself, you aren't free. If you always have to have someone else do the thinking for you, you are still a child mentally. Many of the problems we are faced with in today's world are due to masses of so-called educated people not being able to think for themselves, and relying on elected individuals to do the thinking for them. Much of the problem could be reversed through proper mathematics education, which actually makes you want to think for yourself and not let others do it for you, and soon you learn to love doing this.
Mathematics is full of great use, but this perspective has been drowned out through gradual historical neglect. The use of mathematics for freeing the mind and knowing the truth was established by the Greek philosophical elite, such as Pythagoras, Socrates, Plato, and Aristotle. They knew the difference between the vulgar use of mathematics versus the enlightened use I am promoting here.
Why did Plato have the words "Let no one ignorant of geometry enter" chiseled over the portal to the Academy? Were they busying themselves with calculating land areas and how much wood was needed to build a house? Why did you need to know geometry to hang with the philosophical elite of Athens? My next post will highlight the specific words of the philosophers and show that this was their understanding of mathematics education. I leave you with this sample as an indication of where we're headed:
"My noble friend geometry will draw the soul towards truth, and create the spirit of philosophy."
-- Plato, The Republic
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