I have no idea what any given mathematician might be aspiring to; I simply said they tend to suck shit at conveying any understanding they may or may not have*.
*My experience only
Ah, I see. I understand now.
The reason I thought you were saying otherwise was the line “After talking to them for awhile though, I invariably realize that they really have no idea what they’re doing, and are just good at rote memorization”, which seemed to suggest you recognized not only a failure to convey understanding, but also to even have or care to have it. (As opposed to your own aspiration to a deeper understanding of what things mean, and apparently more complete understanding of physics, astronomy, and biology than people who actually have degrees in those subjects)
I’m not disputing your experience; it’s just oddly unfortunate.
Are these the “real” introductory physics courses, intended for those who might end up actually majoring in the subject, or in some allied field like engineering? I ask, because my my uni had different streams of natural science and math courses, so that liberal arts people could be taught calculus and physics on an intuitive level without most of the rigor.
I think the thing that confused me is that pulykamell was also talking about people far beyond high school algebra, yet it seemed as though you were arguing that he wasn’t talking about the right level of people. Anyway, whatever.
That is often the case.
I guess I should explain that my experience is probably pretty biased at this point. I work at a university. I meet a lot of students who are, for example, going into nursing. A lot of them, for reasons I can’t fully explain the intricacies of (because I don’t know them), will major in something like biology or physics, or the life sciences program we have, but will have already planned a long time ago to get a masters in nursing. That’s an example of how someone can have a degree in a subject and not know or care all that much about.
Maybe I’m just jaded about education, but I don’t think it’s that odd. I’ve said before on this board: My sister was on the honor roll all 13 years of K-12 - I think there were years when she never scored below an A on anything - and I honestly don’t think she could tell you who won WW2, what an organelle is, or the difference between a verb and an adjective. It’s in one ear and out the other for the majority of people, in my experience.
Honestly, I think the English students are even badder.
Old Herman cartoon featuring a kid showing his report card to his father:
“I got six percent in math. Is that good or bad?”
These comment touch upon what I think is the real issue. Up until algebra, math is pretty concrete. It really is mostly a questions of repetition of routines, and in that sense, it is “rote.” And what you’re doing is getting fluent with the routine of the procedures of basic operations.
But once you get to algebra, there’s a major change, (and my understanding is that at this point, many students previously having no problems with math, fall behind). The real point of algebra is to manipulate abstract relations, not concrete quantities. Yes, you can use cases of concrete examples to demonstrate the process, but most teachers (and the older text books) continue to teach it as pure procedure, and just ignore the more important abstractions behind them.
Unfortunately, in the context of most classrooms, a student can get pretty far by simply approaching things solely procedurally, because the traditional practice of math instruction and student evaluation is only to present a whole lot of entirely decontextualized and de-natured routines, which the student is asked to perform over and over. (You know, like, “Here’s an equation, go solve for x. Now do it again with this one. Here’s one that’s a little different–go solve it too”, etc.)
In fact, I think there’s a whole class of student that thrives on this kind of educational approach, because there are no surprises, paradigm shifts or risks involved. It’s a safe, non-threatening way to get by in the educational system.
Let me just add one more data point about mathematics education.
I graduated in 2001 from high school having taken calculus and getting a “4” on the A/B Advanced Placement exam. I will tell you that, looking back on it all, I didn’t learn concepts; I learned formulas and recipes. That’s really how AP has to be, though, because the class has to cover such-and-such material in order to prepare the class for the AP test. You could spend a week or two on deriving the formulas for derivatives using the limit definition, but when all you really have to know for the exam is that d/dx (x^2) = 2x (“just take the exponent and multiply it by x and subtract one from the exponent”), there is a tendency to just do a lot of handwaving.
Even before calculus, my math classes were poorly taught. I “learned” algebra II from a self-professed drunk, for instance. I don’t remember many specifics from geometry class other than it was mostly cookbook-y. Algebra I I don’t think was too bad, but nothing special. Never was any attempt made to make it intuitive. But, I was a young stupid kid then, so a lesson on how cool a proof could be would probably have been lost on me anyway.
Only now, after having to literally teach myself enough calculus, linear algebra, and analysis to get by in my PhD program in stats, do I really appreciate the hows and whys of mathematics. It’s unfortunate that I’ve gotten to where I’ve gotten without a more solid grasp of the stuff. It’s going to severely limit my professional options (not much high-level research). But, I’ve been told that I have a way of communicating statistical concepts that works well for the social sciences (who tend to do lots of handwaving when things turn too mathy, at least from my perspective), so maybe I’ll find a niche teaching.
Math is badly taught in the U. S. And we wonder why we’re lagging behind other countries?
I am twenty four years old this year, and this semester I have a B in introductory algebra. That is a huge accomplishment. When I started I could not add or subtract fractions, could not multiply decimals, could not even do long division. Yes, there are college students out there who have never properly been taught how to do math. There are a lot of middle aged people who have never properly been taught how to do math. There are a lot of children out there, who will never properly be taught how to do math. I’m not quite sure what you’re pitting, here. Math is bad? Math students are bad? I’m not a bad student, I’ve just had a long string of bad teachers and have never been given a reason to care before. Are you pitting me for this? Am I a bad person because I slept through trigonometry and failed pre-algebra in high school? REALLY!
One of the problems I had with Calc was the opposite: we’d have to use it to solve Geometry problems - and I could see the solution just fine (news at 9: you can use geometry to solve geometrical problems!) so my brain just sort of went into lockup. “You want me to go through two pages of equations to find a solution we already know? Really? If that is your sense of humor, get a new one, I’m going back to sleep here.”
A few not-logically-linked thoughts:
I think part of the problem with Math is that, unlike Chemistry or Physics, where you study (some of) the history of the science and can see how this bit led to this other bit, you never get that in Math. Often, people learn more Math from having to use it in Chemistry or Physics than from the actual Math courses: yes, Math is an abstract science (the abstract science), but most people are concrete thinkers. Archimedes, Gauss and Newton weren’t sitting on top of a mountain meditating in their underwear, they were trying to solve concrete, practical problems, but nobody tells you about that.
We speak of “Math”, but “Maths” (as the Brits say) is actually more correct: some people are great at Calculus but can’t do Geometry, some are great at Geometry (which includes Trig) but get lost inside a 2x2 matrix… In other sciences, it’s acceptable to do well in, say, Inorganic Chemistry but barely wing it in Orgo; in Math, people at the HS level are expected to have learned the introductory bits of a bunch of different branches of Maths without any lead up to the whys and wherefores. What’s worse, people often think that those introductory courses are all there is to a specific branch of Maths: how many people have gone through Combinatorials and the Gauss Bell without ever being told or realizing that both are sub-branches of Statistics? How many think that the Gaussian Bell is the only continuous probability distribution there is, instead of knowing that it’s the only one they have been taught? I know nobody told me there was more than one until I got to Applied Stats, midway through college.
Too often, Math teachers aren’t either “Mathematicians teaching” or “teachers who learned extra Math”: they’re someone with a vaguely sciencish background who have been dunked into “You’ll be teaching Calc I and II” “:eek: Mommy? :eek: I want my lawyer! :eek: Help!”
Every single person has some things (s)he is good at, and some things (s)he is bad at. This can include such subjects as map reading (a form of applied math, btw), hand-eye coordination, cooking (another one with several branches)… or different branches of math. Some people haven’t been taught right, some simply wouldn’t ever get beyond a certain point. So long as they don’t insist in spending their lives trying to ram down that wall, it’s all right.
OK, then your experience is very much different than mine. Most people I know by that level are beyond rote memorization, and the professors (at least the ones I’ve had) very much stressed understanding the meaning behind the math, and constructed tests in such a way that the word problems required you to have an understanding of what and why you’re doing what you’re doing, and not just regurgitating formulas for integration and differentiation.
Well, having gone through graduate level physics, I have to chime in here to contrast what you say with my own experience: not one in my (fairly large) program fit your description (graduate student or professor), and IMO none could possibly have gotten very far with the intellectual characteristics you describe. It was a different story in college, where in both high-level math and physics I would guess about 50% of the students didn’t know wtf they were doing (of course this group never went on to grad school). But the other ~50% were nothing like you describe. And again none of the professors met your description. High school, on the other hand, is where a lot of real losers end up, unfortunately…
I find this odd: calculus is one of the easiest maths to visualize. Stands in stark contrast to most of analysis. The fact that you could find anyone to help you is sad indeed. Apparently some standards are lower than others, but where I have tutored in college, all of us would have been able to do at least an adequate job of answering fair visualization questions.
I teach AP Economics, and one thing I have found is that a lot of kids don’t really grok the relationship between a fraction and a percent. They know how to convert a fraction to a percent and vice-versa, but they’ve just memorized the formula, and it’s like they are translating from one language to another, from one set of rules to another. They don’t understand why the rules are different. I spend a surprising amount of time teaching fractions.
Also, if I ask them what 20% of 1500 is, they tend to reach for a calculator. Mental math seems to have disappeared, and I think that’s part of the problem. Figuring out mental math is where I started to really understand it.
Sounds to me like you’d be exactly the kind of math teacher our schools or colleges really need. As lofty as “high-level research” sounds, it’s not necessarily the most intellectually challenging work. Being a good math teacher (like being a good writing teacher) requires much more than knowledge of the subject–it requires a keen awareness of the learning and cognitive processes that operate in what is probably the most complex thing there is: the developing human mind. On top of that, it requires an ability to shape that awareness into effective pedagogy.
“High-level” research–once you get your “security clearance” and bona fides–can be as much as 95% unimaginative, mechanical routine.
As a teacher of undergrad macro, I approve this statement.
I’m rather surprised, you being a chemist or engineer of some kind (I forget which). Sure everyone’s given a formula in geometry class for figuring the volume of a sphere, cone, or what not. But inquiring minds want to know a proof. Or did you mean that in geometry class you went over the Archimedean proofs, or something similar?
OTOH using calculus to prove the volumes does seem like overkill. Once you’ve learned how integration can be used to compute the area under a seemingly odd shaped curve, using it to prove a circle or sphere seems like shooting a fish in a barrel.
I did terribly in math all the way through school, doing OK in HS geometry but having had to repeat Algebra I. I never had second year algebra; then in college I merely had liberal arts calculus much as I described upthread.
Then, several years out of college I came across my father’s college algebra book and learned about progressions, mathematical induction, and equation theory. Something clicked inside me, and armed with the proof for the sum of the first N squares, I set out to prove the volumes of the cone and sphere. My success in doing this, all on my own, gave me tremendous satisfaction, and made me better disposed towards math than I was when I was younger. I’m still not terrific at it; I have a deep-seated aversion to trigonometry. But at least I learned that all math, basically, can be like geometry in that you prove generalities, rather than just the right number to fill the blank on a problem sheet.
Our regional ops manager was totally blown away when I wrote a production planning tool for our warehouses at my last job that would do some rather complex projections of delivery demands and labor needed to meet those demands.
I could even plug in a sick call and instantly recalculate the impact on the production schedule.
excel is a wicked powerful tool in the right hands.