How about calculating the amount of carpet needed for a room that is square, with a smaller square at one corner for a closet or bathroom? The identity above becomes self-evident if you diagram it this way, which might help algebraically-challenged people to learn the numeric derivation.
That’s certainly a concrete example of the difference of two squares, and one that I’ve composed and deleted three posts about already since the thread started!
Problem is, it doesn’t really show why we’d *want *the (a+b)(a-b) form. If I’m carpeting a room with a square out of it, I’d find the area of the room (the first square) and subtract the area of the closet (the second square). It would never occur to me, nor can I figure out why it would be advantageous, to measure the edge of the room, add the edge of the closet, and then multiply that by the edge of the room minus the edge of the closet, y’know?
I mean, it’s *neat *that it works out like that, but this is one case where real world application doesn’t really make anything clearer or more desirable to learn.
Except that it would have taught me far better that “squared”, as in 13[sup]2[/sup] has a real world relationship to square, the shape. I seriously don’t think I grokked that until I was in my 20’s.
This must be me. I don’t quite get how would do these calculations without understanding the underlying principles, at least to the depth of understanding how the placement of digits in decimal notation works. OTOH, I don’t consciously have to be aware of the principles every time I need to do math, I just follow the procedures I’ve been taught–maybe I’m not the person you’re thinking of after all. Nevertheless, this has been an issue with other areas of my life. A high level diagram of a software system is of little use to me unless I can learn what each box does, at least down to an architectural level (i.e., how the boxes relate to one another in terms of inputs and outputs).
An earlier post explained that a carpet (a-b) wide and (a+b) long can be cut into two pieces that exactly fit the room missing the closet, so there’s a very practical application - you can order an (a-b) x (a+b) piece of carpet and make one cut to make it fit the area you want carpeted, instead of ordering a a*a area carpet and having to cut out and discard part of it.
Cryptology, including simple cryptograms, is absolutely mathematical. I spent several years out of grad school working in that field.
Oh, gosh, that brings back memories - I went to art school as well. Suddenly > I < was the math genius! When I was taking my class on fabric dyes, where we had to mix commercial dyes to very specific concentrations, I was the only one who had any comprehension of the algebraic formulas used to calculate the precise hues we needed. Suddenly, I was teaching a math class and checking everyone else’s work! Man, I felt so smart! I was a goddamned genius!
Still hated math - but I did love being able to get consistent results. That’s what I mean, it’s just a tool to me, but because it gives me the results I want it’s worth the effort to slog through the learning curve.
But the OP really is “why do people hate math” not “why do people not do math”? Most people in the thread have talked about having to do it even when they hated it. There ARE only so many hours in a day, so how is it lazy to focus your attention and energy on your actual skills rather than waste them on something that just doesn’t work? And I definitely disagree that “not being good at” something then equates to stupidity or laziness.
That’s probably different for each person. What drives me to tears is not “math” but not innately knowing (as those who ARE good at math do) where to apply what equation.
For instance, here’s a simplified example from my own job, I’m trying to figure out how much contaminant is in a tank of product. The lab reports the contaminant in weight, because of how they extract it, their instruments, blah blah blah. However, the tank is in gallons. So I’ve got 3.2 mg/kg of Contaminant A in a 20k gallon tank. How many gallons of Cont A are in the tank?
BUT WAIT, don’t write that check yet. The product has a specific gravity of 0.8 and the contaminant is around 1.2. I can ask my chemist (well he’s not really MY chemist), or I can take 4 hours to attempt to research this, not even knowing how to phrase it in a search engine so that I CAN find some sort of equation to plug stuff into. After about 2 and a half hours my head explodes and I start climbing the clock tower with an Uzi so I can take half the city out. 
I’m talking about, at the very minimum:
Reciprocal identities:
sin = 1/csc
cos = 1/sec
tan = 1/cot
Pythagorean identities (you mentioned the first one, but there are two more):
1 + tan[sup]2[/sup] = sec[sup]2[/sup]
1 + cot[sup]2[/sup] = csc[sup]2[/sup]
Sum and difference identities:
sin(a + b) = sin(a)cos(b) + cos(a)sin(b)
cos(a + b) = cos(a)cos(b) + sin(a)sin(b)
Product identities:
sin(a)sin(b) = (cos(a - b) - cos(a + b)) / 2
cos(a)cos(b) = (cos(a - b) + cos(a + b)) / 2
Period identities:
sin(a+pi/2) = cos(a)
cos(a+pi/2) = -sin(a)
Law of sines:
sin(a) / A = sin(b) / B = sin(c) / C
Law of cosines:
c[sup]2[/sup] = a[sup]2[/sup] + b[sup]2[/sup] - 2abcos(θ)
Heron’s law:
Area = sqrt( s(s - a)(s - b)(s - c) ) where s is half the perimeter (a + b + c)/2
Double-angle identities:
sin(2a) = 2sin(a)cos(a)
cos(2a) = 2cos[sup]2/sup - 1
tan(2a) = 2tan(a) / 1 - tan[sup]2/sup
Half-angle identities:
sin(a/2) = ± sqrt( 1 - cos(a) / 2 )
cos(a/2) = ± sqrt( 1 + cos(a) / 2 )
tan(a/2) = csc(a) - cot(a)
…and that’s just the stuff I can remember to look up.
It is not humanly possible to re-derive all of these from base axioms (SOHCAHTOA, say) and still have time to actually answer all the questions on the test. I don’t care how brilliant a proof-deducer you are, there is a physical limit to how fast your pen can travel upon the paper.
No, but that has nothing to do with the point I was making, which is that deriving every equation of your entire subject at the start of the exam is not a practical strategy for any test given in finite time, whether or not the instructor makes you memorize stuff (as mine did) or provides a cheat sheet. At some point, you simply must know how to do things. There is no getting around this.
How often does a test ask you “Here’s half of a trig identity. What is the specific equivalent other half I’m thinking of?”? Because otherwise, it’s not clear to me what the value of even knowing all these identities with instant recall is; there’s generally more than one way to solve a problem, very little of which is pulling a trig identity out of thin air or blindly applying them to the expressions which appear. The task of figuring out what is necessary in the solution of a test problem usually ends up rederiving any relevant identities anyway, I think. That having been said…
These are the definitions of csc, sec, cot. I granted you the definitions cannot be derived and are purely a matter of memorizing names.
I have neither of these memorized. I suspect neither does Andy L. They are, however, just the standard Pythagorean identity divided through by cos and by sin, respectively. The context has never arisen in which I needed to memorize them separately; if I were confronted with one and felt the need to search for an identity, re-expressing in terms of sines and cosines would make the equivalence to the standard Pythagorean identity apparent right quick.
This is the addition identity I was referring to, the one other identity perhaps worth memorizing, although easy enough to re-derive, as I explained above. It’s not worth it to me to try and remember all the signs of every little thing in every little identity; I’ll just mess up. (Although, having just gone through the derivation above, I’ll note that you should have a - instead of a + on the RHS of the second line).
I have neither of these memorized. I can’t recall ever needing them either, but for what it’s worth, they follow immediately by applying the addition identity to the RHS.
I have neither of these memorized, and always have to pause to rederive the signs (they are just particularly nice special cases of the addition identity, the case where one of the two rotations being combined is a 90 degree rotation).
This was not the sort of thing I was thinking of as a trig identity, but, alright, I do have this one memorized.
I don’t have this one memorized in quite this form, though I do have a very similar identity memorized: (a + b)^2 = a^2 + b^2 + 2ab. I have it memorized not by conscious dint but by having rederived it so often… The law of cosines is in fact just a special case of this familiar algebraic identity, where multiplication is interpreted as dot product of vectors.
I was also not thinking of this as a trig identity, but I do have this one memorized. It’s the one identity on this list I would be stumped for time to rederive.
Special cases of the addition identity. Why would you memorize them separately?
Follow from the double-angle identities. Why would you memorize them separately?
Is it humanly possible to memorize all these identities and pull them out as needed without thinking about where they come from? Well, I suppose so, but does anyone do so? Have you memorized all these identities? Did you find you needed them all memorized to pass your trig tests? How did you know which of the myriad trig identities to memorize and which not to?
I agree; that was the general point I said I was agreeing with you on. But for the specific case of trig identities, I did not feel it rang true. I have not memorized hardly any of the identities you listed, and I suspect you have not memorized most of them either; is there anyone who has?
Basically, it seems to me memorizing trig identities in preparation for a trig test is like memorizing (1 + x + x[sup]2[/sup])[sup]2[/sup] = 1 + 2x + 3x[sup]2[/sup] + 2x[sup]3[/sup] + x[sup]4[/sup] for an algebra test. Sure, it’s conceivable a situation could arise in which going between these two expressions is of use to you, but if your approach to being prepared to deal with such a situation is to hope to have crammed that specific fact already into your memory banks somewhere, you are making the task much more difficult in a way than you need to; you are saving some small amount of time on figuring the math out on the test, but at the cost of an awful lot of time spent on an awful lot of scattershot memorization in test preparation.
Again, I am not denying the value of knowing things instead of constantly rederiving them from scratch in general… I am only denying its value for trig identities in particular. Much more valuable a use of time to work on developing the skill of discovering the trig identities than to worry about the skill of memorizing them, since the latter will probably just lead to not having either skill. At least, in my own experience. I admit, YMMV.
There are problems with how math is taught in schools. Most new mathamatical concepts build upon earlier ones. If the earlier ones are not mastered, it becomes very difficult to learn the newer ones. Math classes are often structured around individual lessons. A concept is taught. Examples are given, and the students take problems home for practice. Then they move onto the next concept and don’t really touch on the earlier one until the test. Maybe they’ll review it again next year after it’s all been forgotten. Some math programs (like Saxon) are very good about including lots of review in the problem sets, but most don’t. They should. A student who gets a C in a math class probably shouldn’t be moving on to the next level.
Every single child who we’ve had come in for math tutoring at the algebra level has had problems with basic arithmatic. All of them. A child who hasn’t automatized addition, subtraction, and multiplication is going to have trouble with division. Then more trouble with fractions. And then algebra is going to fuck all impossible. Some students hit a wall. Others face a slow creep of difficulty.
In all cases where the student cooperates, these problems have been correctable, usually by working with the students on what may seem to be ridiculously simple concepts. Kumon is a good program for this, or one can custom-make a similar program. Younger students catch up rather quickly, and often end up ahead of their peers if they stick with it. I believe that nearly all students of average intelligence or greater are capable of handling algebra by the age of 12. If a child is NOT in a program that stresses automatization of basic arithmatic, I strong suggest the parents supplement. Both Kumon and Saxon are excellent choices for this.
Sometimes the kids have eye problems too, which is a different matter that will often result in reading problems as well. It’s hard to do math when the numbers don’t line up! Depending on the issue, this may be correctable. Again, younger is better. Many optomatrists and opthamologists do a sloppy job checking eye-tracking. I recommend finding someone who does this electronically.
Think about the number four. It has meaning to you and has probably come up frequently in your life. It is an intrinsic meaning and you can see that three is one less, five is one more. You just GET four.
Now think of 145, 672. Different feeling right? You understand the number as a concept but it has little feeling to you. It is not a number that you know and understand well.
It’s like you have a concept of 1, 2, many. You don’t really get any other numbers.
This means you need to follow the rules for basic arithmetic instead of understanding how to do it. Slows her down and leads to interesting answers (like 5+4=3).
Because you have to. Because if you spend time re-deriving them you can not finish the test. I thought I made that clear.
Yes. Anybody who wishes to pass a college trig class and who can not magically pull identities out of their ass in infinitesimal time.
It would not have been possible to pass the tests otherwise. I knew which ones to remember because we knew what kinds of problems would be on the test.
Well, not anymore, because it’s pretty useless knowledge. But at the time I had most of them committed to memory, because I had done enough practice problems to remember them. Even then, I didn’t recall all of them, and had to waste time re-deriving some stuff, and as a result never managed to actually complete any test in that class. (Though I still managed to get a B.)
I just remember the “SOHCAHTOA” mnemonic device we were taught in high school
Well, I can’t speak for all math haters, but SOME mathematicians are responsible for much of the difficulty mathphobes face. Those that were bad and/or abusive teachers that is. They’re not responsible for all of it of course, but they had the responsibility to shape very young minds, and they blew it. BIG TIME.
First of all, there ARE teachers out there who, through ignorance or arrogance ARE abusive to those who don’t instantly pick up on the math. Your statement that Aruqvan, “claimed” abuse is insulting. As to your statement of “some of us have to work”. What would you have those of us that are adults, and have been in our respective careers for 20+ years do? Throw out our lives, jobs, homes, kids so that we can get up to speed and become acceptable humans who know good math? It wasn’t a whine, it was realistic. In order for most people to go back, especially at middle age, and relearn would take more time, money and resources than most working people have.
Again, I don’t think most of us think math is pointless at all. In fact, most people who are self-admitted math haters in this thread, have almost all also stated that they understand the need for math, and almost all have stated that they wished they were better at it..
I think the problem lies, psychologically and society-wise, when the attitude (as espoused by snotty condescending math PSAs not too many years ago) is, if you don’t know at LEAST advanced algebra and quantum physics, you’re going to end up a worthless bag lady, or worse. So people that are already feeling the stress and self-doubt turn it back around on what they see as the author of their pain. Pretty normal psychologically speaking.
The solution is NOT for the math-abled to then even further humiliate and badmouth mathphobes, but to understand that everyone has something that they’re not good at. I’m sure the mathematicians have things they’re not good at, no matter how hard they try. (dancing perhaps? :D).
The mentality that “math is easy, you people are just idiots” is ridiculously unhelpful. Unless, that is, the goal is just to be able to “lord it over” other people. If the goal really IS to have everyone think “math is easy” then the learning process has to change. Bottom line. Because what’s out there now, as is evidenced by the percentage of math haters, isn’t working.
In my experience, relatively few bad math teachers are “mathematicians” - the teachers who destroy students’ interest and self-confidence in math are usually the ones who have been dragooned into teaching it and hate it themselves (especially elementary school teachers, who have to teach all the subjects whether they like them or not).
The MIT Ballroom Dance Team is arguably the second-best in the country (after BYU, which has a much more intensive academic program - it’s purely extracurricular at MIT). 
“Some of us have to work” is Aruqvan’s quote. Does that change your opinion about who was insulting who?
On edit: I wrote an explanation of the misinterpretation of the “Some of us have to work” paraphrase, but Andy L has already covered it, so I’ll just leave the following link: See thread here
Oops. I thought it was a direct quote, not a paraphrase; thanks for the link.