It turns out from our collective experience that most undergraduate students don't know how to think about math at all (ourselves included). This doesn't mean that they are incompetent or stupid, it's just that they have not ever developed the skills of critical thinking and self-reflection.
Before talking about math, let me point out a specific example in real life. Over by the Mandeville center, down where the art people have a space to do their thing, there used to be a painting on the ground (someone painted over it). It was a picture of what I've always supposed was Mary and baby Jesus (woman, child, halo). Next to the figure was painted "Do not push your beliefs on others" (or something like that). I am not sure whether the person who painted that appreciates the irony. As a personal philosophy ("I will not try to force you to believe what I believe"), it works just fine. However, it cannot possibly be given as an instruction to someone else in an internally self-consistent manner ("You should not impose your beliefs on me" -- by making this statement, the speaker is trying to impose HIS beliefs on the listener, thus doing what he says one should not do). A lack of critical analysis leads to nonsense.
The same internal inconsistency exists with bland, naive relativistic statements such as "all religions are the same" (different religions have different claims to truth, and they are obviously incompatible) and "you should accept everybody" (the person doesn't accept you because you don't accept someone, but then that person is himself not accepting of everyone). Most of the people I know who talk that way haven't ever really spent time thinking and reflecting on their system of beliefs -- it's much like how Christians who never study the Bible come to false conclusions about how God "should" behave (really, how God "does" behave).
But what does this have to do with math? I'll grant that philosophical reasoning is a bit stickier than mathematical reasoning. At least with mathematical reasoning, we (the math community) have a set of mostly agreed upon fundamental beliefs (axioms) upon which we build mathematical structures. There also isn't a whole lot of room for "personal interpretation" when it comes down to the formalism of mathematics (there is actually room for personal interpretation when it comes to trying to understand math... but that's a different story).
Where was I? Oh yeah, what does this have to do with math? A lack of critical analysis leads to nonsense. The nature of mathematics for a lot of students is that you find some numbers, or a formula, and plug stuff in, and move some terms around, and get some answer. It doesn't matter where the problem came from and it doesn't matter how silly the answer is. It doesn't matter that it's a word problem computing the terminal velocity of a falling object, and that the final answer they got is "t = 10 minutes." The problems have no intrinsic meaning to the students, so their answers often have no intrinsic meaning.
Here are a few examples of relatively simple mathematical processes that a lot of people know they're supposed to do, but don't actually know why it's the right thing to do (other than they were told to do it that way):
- Why do we "carry the 1" in addition? (Even more fundamentally, why do we start on the right side when we add? -- This one isn't about being right or wrong)
- Long division: Divide, subtract, multiply, bring down, repeat... What does this process actually do?
- x^n * x^m = x^(n+m) -- If you're not familiar with typing math, x^n is x to the nth power.
- Kindergarten - Learn to count
- 1st Grade - Learn to add and subtract
- 2nd Grade - Learn to add and subtract in multiple columns
- 3rd Grade - Learn to multiply, introduction to fractions
- 4th Grade - Learn to multiply in columns and do long division
- 5th Grade - I don't even remember what I was supposed to learn...
I was fortunate that my school and my teachers were willing to put in extra effort for the bright students. There was a weekly thing in the library for the more proficient math students (I remember learning about exponential growth there by the chessboard and grains of wheat problem).
My fourth grade teacher borrowed books from the library to keep me occupied while she taught arithmetic to the rest of the class (she got me books on finance, but since I didn't have a good conception of money at the time, the real value didn't sink in until much later... for example, if I understood no-interest government student aid loans and all that stuff, I would have realized that when I started my undergraduate studies, I could have taken out a maximal student loan and earned 5% interest in a CD somewhere and make free money off the government. Even at 3.5% APY on a $10,000 student loan, because I was in school for 9 years and the loan would be without interest would have given me an extra $5500 for free. Then taking the $5500 and continuing to earn 5% APY on it from age 27 to age 65 is an extra $35000 in retirement funds for doing essentially nothing. Of course, I probably could have qualified for more at the start, and probably could have gotten better than 5% return on average (I think the average market growth rate is around 8%), but I think this makes my point about the gap between being able to compute things and being able to critically analyze what the math is saying.
Who is to blame for the inability of students to think critically about their math? It's not the fault of any individual, but it's a faulty system. Teachers teach math this way because this is basically how they learned it. Students learn math this way because they have no other model for learning math. The students grow up, and this way of thinking is never corrected, so it propagates itself to the next generation.
Fortunately, I do believe change is in the works. There are a number of educators who are looking into the failing mathematics system, and while I'm not involved there, I do hope they find true solutions and not false ones like having teachers teach to standardized testing... that just reinforces the same problems.
