Can you draw me a triangle that has angles 50, 60, 70 and has an area of exactly one square foot?
How about the intersection of a one inch diameter circle and a two inch diameter circle where the area of the intersection is exactly one square inch?
Yep!
Yes, if you let me use the CAD/CAM software on my computer. Pencil and paper, not so good.
In case it needs to be said: There’s no one who doesn’t have trouble with math, at some level, except for those who stay away from the parts they’d have trouble with. That’s why Ph.D.'s in math are hard to get, and why there are unsolved problems that have defied the efforts of generations of mathematicians. In that sense, math is unquestionably hard. Everyone, even professional mathematicians, has some area at some level that they have to struggle to understand. For some people, they hit that level at fractions; others hit it at, say, algebraic topology.
In one sense, that’s one of the good things about math: you’ll never run out of challenges; and there will always be some interesting problem or area of study that’s right at your level, no matter what that level is.
This, plus the fact that math builds on the stuff taught in previous math classes, is the problem. Say you have one bad English teacher who you don’t learn anything from. You’ll have a different English teacher next year. You don’t have to understand Tom Sawyer to be able to understand Huckleberry Finn in another class. If you have a bad European history teacher, that won’t affect your ability to comprehend American history the next year. A bad biology teacher won’t keep you from understanding chemistry the next year. It does work that way with math. If you have a bad Algebra II teacher, you’re likely to have trouble the next year in trigonometry. One bad teacher does have more effect on your ability to understand math than they would in other subjects.
There are fewer math and science teachers out there for every available job than there are, say, English teachers. If you have a lot more candidates than jobs available, you can get the best candidates for the jobs. You can be very choosy about who you hire. You can’t do that as easily if there are more jobs than candidates. You might have to settle for what you can get for a math teacher. The law of supply and demand applies here, same as everywhere else.
Yeah, chemistry and biology always seemed to be asking me to memorize stuff. After a while, I realized that in math I could stop wasting effort memorizing things like (for example) trigonometric identities, and just derive the ones I needed on the fly.
I went to art school, or rather a college filled with artist types. Everybody hated math. One person nailed their partially-burned class notes in the student lounge and denounced the requirement.
Through talking to those types, I gather that 1+1 just doesn’t make 2 in their brains. The process of manipulating one number so that it becomes another number gets lost among a jumble of context-free numerals. Any math skills become very difficult to understand in that environment.
I think that type could get most of geometry if unencumbered by equations and numbers. (Proofs are logic problems, not really number games.)
Actually, I think a more appropriate analogy would be, practice and testing of the ability to run long distances being made the focus of driving instruction…
Yes. Tolerating laziness and stupidity.
Guess there weren’t any architects.
How about you are traveling in a car at 60mph. How much space should you give the car in front of you to avoid smashing into it if he stops suddenly?
I don’t understand why you are asking me this question. But, sure, that’s information a driver needs a sense of. A sense I doubt they would get a better grip on through practice running (as in, with one’s legs…).
This is certainly why my daughter hates math. She missed out on getting her basic numeracy skills (not just counting but having an intrinsic understanding of what numbers are, three or four as a concept). As such, she can do the work given to her by following the steps (21-7=14 by borrowing and counting down for example) but she doesn’t really get what subtraction is or how the numbers feel.
The reality is that she is getting average grades in math and would just proceed on doing the work and getting by if her father and I weren’t math types and couldn’t see the problems she is having are of a different nature.
I suspect that a lot of people get lost at some point in math (fractions seem to be a common one) and never really get over that hump and it just makes everything from then on out more difficult.
Because it’s a math question relevant to driving a car.
I’m curious about this. In what way does her not having an understanding of what three or four are as concepts manifest?
This is a nice idea in theory, but impracticable in reality unless you have infinite time. On a 50-minute trig exam, one doesn’t have time to re-derive every trig identity (there are billions of them) necessary to solve each problem. So you have no choice but to commit a minimum number of things to memory.
The British texts have lots of worked problems, and by studying them, you can learn the correct way to proceed.
The weakness of most American texts seems (to me) to equate flashy pictures and irrelevant verbosity with sound problem solving techniques.
Many algebra texts from the 1930’s-40’s are better than the ones published today-check the authors-most of the bad ones are written by people with education degrees.
You also have no choice but to fail to commit to memory most of those billions of trig identities (there are billions of them!). So I’m not sure why you feel the existence of billions of trig identities is a point in favor of the necessity of memorization of trig identities to efficiently solve trig problems.
Oddly, I might agree to some extent with you in general, but not in this specific example. Naturally, studying and remembering the development of some idea or realization saves you time and effort in trying to reconstruct the same insights afterwards; otherwise, we wouldn’t bother teaching anyone anything. In that sense, memory is important.
But the only trig identity I know by heart is cos^2 + sin^2 = 1 and I did alright on tests (of course, I also had to memorize the meanings of the various named functions in trig, this linguistic knowledge being impossible to derive). What sorts of trig identities are you saying one must memorize for a typical exam?
At any rate, I agree it is certainly possible to construct a exam which one could only hope to pass by the technique of memorization of a sheet of formulas (key to this, of course, is barring access to textbooks or such a sheet during the exam). This demonstrates only that it is possible to construct pointless exams. Surely you do not feel memorizing random lists of trig identities is a useful skill in life or math more generally?
Not a theory - I did it that way fairly often (sometimes not deriving the equation from ground zero, but by remembering the basic idea, like sin(a+b) equals some trig function of a times another trig function of b plus or minus the same functions applied to b and a - I found using a little trial and error to be less of a hassle than memorizing)
Very few people like doing things that they don’t understand or find difficult. We’ve all had an experience of trying to do something that we don’t understand, or understand only vaguely, and that is difficult to do (many adults have had this experience with doing income taxes or dealing with bureaucracy at work). It’s frustrating. Add in not knowing why you should have to do this thing you find frustrating (at least we know why we have to do our taxes, math classes do not, by and large, do a very good job of explaining to kids why they need to know this stuff), and it’s an even worse experience.
That’s the experience you’re likely to have in math class, if (for whatever reason) you didn’t understand what was taught in math class in a previous year.
I can’t think of any trig identity students are expected to be familiar with which is not simply some rephrasing or consequence of either the Pythagorean theorem or the addition identity (for example, every identity on this sheet is that way).
The addition identity is also not unreasonable to hope to memorize. But it’s also perfectly straightforward to derive when trig is emphasized as about rotation instead of triangles. [Specifically, the addition identity is just the breakdown of rotation by the angle x + y into rotation by the angle x followed by rotation by the angle y, along with the definition of cos and sin as the components of rotation; (cos(x) + sin(x) * rotation by 90 degrees)(cos(y) + sin(y) * rotation by 90 degrees) = whatever you get by distributing it out and noting that rotation by 180 degrees is the same as -1]
There’s a stray “also” in that last post which no longer makes sense due to editing… ah well.
With the exception of geometry, my complaint about HS math was that I wasn’t getting the underlying principles of it. It’s like the “teach a man to fish” metaphor: if they you teach the solution of a given pattern of variables and coefficients, you can apply that against other problems in the same pattern; but if they teach you the underlying theorem, you can generalize to a wider class of problems. That’s not to say that they didn’t try to teach me those principles, but for whatever reason I was not grasping them.