Stupid Math Teachers?

For the discussion of math. Duh.

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aldimond
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tendays wrote:About trigonometry mnemonics - the way I learned it:
The coordinates of a point on a unit circle are (cos t ; sin t), where t is the angle compared to the x axis. I always found that way easier than trying to remember poems... Just need to remember that cos is horizontal and sin is vertical.
And tan t - well, make a line *tangent* to the circle (x=1), and see where it intersects the line of angle t...

I sometimes think of the three triangles. There's the cos-sin-1 triangle, the 1-tan-sec triangle, and the 1-ctn-csc triangle. It helps me remember the identities (e.g. 1+ctn^2(x)=csc^2(x)).
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kira
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Mighty Jalapeno wrote:Showing work is important for other applications (in college, showing work was 90% of the mark), but for high school math, the fact I got the correct answer at all should be enough (given that most other people did show their work, and got lower scores than me).

Why is showing work important in college but not in high school?

Anyway, to play devil's advocate (I also hated showing my work when I was a young'un), it's very hard to tell when students are cheating if they don't show your work. It's harder to copy an answer and the correct work than to just copy down an answer (same as it's harder to copy a short paragraph answer than a multiple choice answer).

aldimond
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Showing work is important, because process is important.

I know a dude that's really quick at math. He once came up with a really quick way to solve some problems in math class (junior high level). He kept answering them in his head and the teacher asked him how he was coming up with the answers. It turned out his process wouldn't always give the correct answer. Without showing the process the teacher couldn't know that.

If I'm writing some code at work and I use an algorithm that doesn't have an obvious purpose I should comment on its meaning. It's more important there than it is in school because my code won't just be tossed out after it's reviewed and checked in. It will be easier to edit and debug in the future if I write clearly and comment. Even if nobody ever has to change that line of code again, someone might expand on the function and want to do something else similar. If I write clearly and comment the other person will have a better understanding and potentially write better code.
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iw
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kira wrote:Why is showing work important in college but not in high school?

At the very least, it's smart for the student to do so because of those two magic words: partial credit.

Gelsamel
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Well again, in highschool 1 mark was awarded for the correct answer (or 2, if there were 2 answers needed). And the rest would be for the working. So an out of 5 question would have 4 marks worth of working in it.

Either way, High School is prep. for Uni, you gotta learn to do yer workin'!
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Mighty Jalapeno
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Gelsamel wrote: Some people CAN'T do calculations perfectly in their head like you! And if they can't, how do they know the answer you've given is correct? Especially if they don't know the method?

Well, the only people who cared was my teacher, and he had the answer key.
Gelsamel wrote:The whole idea of assessment is to see if you've learned what they've taught, not whether you can get the right answer.

I did learn what they taught. I learned how to take an equation, and get the correct answer.

And don't tell me the value of showing my work... I'm an engineer. I have to show upwards of thirty pages of equations just for a single bearing wall in a lot of cases. However, in high school, millions of dollars and dozens of lives weren't on the line... it was just to see if I understood exponents, or trig, or whatever. Almost nothing I learned in high school math had the slightest similarity to anything I learned in college math, and in a lot of cases, showing the high-school math wasn't necessary.

I seem to be getting a lot of flack here, and I'm not sure why.

Gelsamel
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Of course you learned what they taught. But they don't know that from you listing answers.
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Phi
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In high school, it's not just what you put down on your paper, part of how the teacher can know that you know the work is how you act inside the classroom when "learning" the actual stuff. If you prove to the teacher before a test that you know how to work a problem, they are going to be more inclined to think that you know how to do the problem, even if you write a single line of work.

For instance, when we did parameterization in math awhile ago, I'd be somewhat active in class and do things on the board and prove all the stuff. When, on a test, I had to write parameters for, say, an ellipse, I had one circle and one line in my workspace, and then the answer was written down. There was no possible way to explain what went through my head unless I wrote out a paragraph, so I just drew what I was thinking and wrote the correct answer down, and got the full amount of points.

Of course, had the answer been incorrect, I doubt much (if any) partial credit would have been given. For my school, they usually want you to show work (some teachers have made me waste the time on quite obvious math so that I don't get a zero on the problem), but it depends on the teacher's preference as to how much work needs to be shown.

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Consider: You meet a man who can square 3-digit numbers in his head. Fast. For example, you tell him "681" and he answers, in under three seconds, "453761". Naturally, you want to know exactly how he did it. He might have memorized the squares of all natural numbers less than 1000. He might use the multiplication algorithm that everyone learns in elementary school (start from the right). He might have calculated 600^2 + 2*81*6 + 81^2. He might have calculated 600*762 + 81^2. He might have calculated 680^2 + 2 * 68 * 1 + 1^2. He might have calculated -- well, you get the idea.

And the point is that each of these methods is different. It's not enough to know that he can do it if you're a teacher. You have to know exactly what your students are learning. This is especially important because -- in the fictitious example above -- he got it wrong. The correct answer is 463761. And you (playing the role of teacher) want to understand what happened. Did he memorize a wrong digit? Did he forget to carry a 1? Was his method fundamentally flawed? The only way you can find out is by getting him to explain his process -- i.e., to show his work.

And the only way to make sure that you show your work is to provide incentives. That is, you must deduct points for failing to show work. You might protest: "But it's only important if the answer is wrong!" True. But odds are you don't know when your answer will be wrong, so you either always show your work, or you never show it. The teacher wants you to do the former, even if you are correct 99% of the time, because the 1% of mistakes could be due to anything -- from an alpha particle in your calculator to your incorrect belief that (x+y)^2 = x^2 + y^2.

In order to make sure you always show your work, the teacher takes off points for not doing so. It's his/her only leverage.

Mighty Jalapeno wrote:I seem to be getting a lot of flack here, and I'm not sure why.
We are envious that you are better than us at mental math

Woxor
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I've got Fancy Academic Qualifications in mathematics and engineering, and I had never heard of a gradient (other than the vector calculus kind) until this thread. My guess is that it's more of a regional thing; they didn't mention them in the American schools I went to.

In response to the "showing your work" discussion: if anyone is planning on going to graduate school in mathematics (not that that's the be-all and end-all of Mathematical Truth or anything), then get used to showing your work, and get good at it. Advanced mathematics IS the work you show; any proof you'll ever write is nothing more than the work shown in getting from one proposition to the next.

demon
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Showing your work. Sure, showing the process is what you should do. I mean - if you're dealing with transfinite sets, then your work is basically the answer. But I find it gross how often teachers expect kids to write down the obvious arithmetics. You have to learn how to present a mathematically correct response to a question. Write down the necessary assumptions. Comment on known identities and so on. But simplifying expressions is obvious. You just do it in any way you want, no big deal. That's the general principle our teacher employed. As long as the process was clear, you could skip just about all the number crunching. He made this rather clear when he gave my class a quadratic equation test. A test that covered extremely easy material, but a test that most of my class failed, nonetheless. That was because he zeroed the mark for a task if you did something without making sure it is possible first, and people who weren't used to dealing with problems thoroughly and ignored special cases really got a lesson.

Mighty Jalapeno
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Woxor wrote:if anyone is planning on going to graduate school in mathematics (not that that's the be-all and end-all of Mathematical Truth or anything), then get used to showing your work, and get good at it. Advanced mathematics IS the work you show; any proof you'll ever write is nothing more than the work shown in getting from one proposition to the next.

I learned that pretty gosh-darned quickly in college. The question would be 3 or 4 lines, and the test would be four hours long, and half the class wouldn't finish in the allotted time. That was fun. Looking back I wish I had studied more and learned better study-skills in high school. My sister is (and she'll be the first to agree, actually) not as smart as me, but her marks were consistently higher, because where I'd be playing Warcraft II or Duke Nukem or Civ or whatever, she'd be doing homework. I was a lazy bugger, and my first time through college I suffered for it greatly (I was expelled right around the time I dropped out).

The second time through college, I had a reason to actually apply myself, and got the highest average in the class. I accept most of the blame for sucking so bad the first time, but really in high school, I'd have loved to have had a teacher who was... well... good.

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Woxor wrote:I've got Fancy Academic Qualifications in mathematics and engineering, and I had never heard of a gradient (other than the vector calculus kind) until this thread. My guess is that it's more of a regional thing; they didn't mention them in the American schools I went to.

In response to the "showing your work" discussion: if anyone is planning on going to graduate school in mathematics (not that that's the be-all and end-all of Mathematical Truth or anything), then get used to showing your work, and get good at it. Advanced mathematics IS the work you show; any proof you'll ever write is nothing more than the work shown in getting from one proposition to the next.

from a little googling it looks like its used in surveying (they like right angles and 100 gradians in a right angles is neat). every science + engineering use i can think of has always been radians.
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mrguy753
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Twasbrillig wrote:
Vandole wrote:
Twasbrillig wrote:n my grade 10 Math class today, we were learning about finding the size of angles on a Cartesian Plane in excess of 360 degrees.

This sentence made no sense to me. What kind of crazy grade 10 math does BC have?

Um... normal math?

What with the Syrcxrtyx.

Today in math, our teacher was explaining how to find lengths of hypotenuses and side lengths of right triangles on a Cartesian plane, and gave us the equations:

Sin = y/r (r being radius of a circle, the hypotenuse of a right triangle created in a circle on the plane being the length of the radius of said circle)

Cosine = x/r

and

Tangent = y/x.

I looked up to this equation and immediately stated that it's easy to remember, "because the mnemonic is syrcxrtyx. Sir-kuhsir-ticks."

To which uproarious laughter ensued.

AAAAAAAAH! I just learned the EXACT SAME THING this morning. I mean, I kinda already knew it, but i was TAUGHT it today. weird....
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ZeroSum
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In high school I learned (from another student) Some Old Hippie Caught Another Hippie Tripping On Acid for my SOH CAH TOA. Using opposite, adjacent and hypotenuse as the terms.

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ZeroSum wrote:In high school I learned (from another student) Some Old Hippie Caught Another Hippie Tripping On Acid for my SOH CAH TOA. Using opposite, adjacent and hypotenuse as the terms.

That is... win.
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kira wrote:Why is showing work important in college but not in high school?

Well, two answers for you. For one, high school math classes (elementary school ones, too) ask the same problem over and over and over again. Come on guys, I know how to solve a linear system already! I know how to solve quadratics too, and how to factor polynomials with the rational roots theorem, and how to figure out which trig function to use. I demonstrated that the last time you gave me 50 problems to solve. Move on!

In college, there were only a couple classes like that, and I didn't like them much. (Like this combinatorial game theory class, where the answer to every problem was induction on the definitions. Until we got to infinite games, and then it was transfinite induction on the definitions.)

OK, second answer: to some extent, I didn't show my work in college either, and wasn't expected to. See, in high school, the questions were like, solve this equation, or evaluate this integral, and the question ultimately asked for a number or a formula. To make sure I wasn't cheating, or that my algorithm was correct, or whatever, they made me show how I got the answer.

On the other hand, in college, the problems were generally to prove some statement. In some sense this was showing work, since the answer was more complete than a single number, and was evidence of its own correctness. But the pages and pages of scratch work to figure out how to prove the statement (my work) would never be shown; only the half-page final result mattered. That final result might be reworked and polished so that none of the original intuition remained.

evilbeanfiend
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another reason for the showing working thing: often my university physics question wouldn't ask for a numerical answer at all. the question would be of the form

from X derive a fomula for Y

where X was either some formula or 'first principles'
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Dark Ragnarok
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Hm, I used to never show work, cause even if i could solve a lot of problems in my head, it was due mostly to rearranging the problems in a simpler expression. Eventually the problems got a lot more technical, tricky, and more complex. Storing 3-5 separate numbers in my head for references per problem, would leave me being less efficient than at least writing down the numbers before i did the math in my head. Right now I've seemed to find the balance of at least showing how to set the problem up but do half or more of the math in my head so I'll get full credit, but still do the math in my head.

I've only taken high school math courses right, this year, senior year, i took stats, and that class somewhat forced me to find that balance, but i have no idea if that balance can successfully carry on in college. I hope so.

In my opinion, instead of deducting points for not showing work, they should award points to showing it. Not much, but enough to maybe incentivize people to learn to show at least some work if it at least benefits them. It's hard to argue why you need to show work when you have correct answer, versus why you didn't show the work when you could have gotten extra credit.

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In high school, when you are marked 1/5 on the correct answer and 4/5 on the process, they are saying "find the answer, and prove that it is correct". Your work is nothing more and nothing less than a proof that your answer is correct.

Woxor wrote:I've got Fancy Academic Qualifications in mathematics and engineering, and I had never heard of a gradient (other than the vector calculus kind) until this thread. My guess is that it's more of a regional thing; they didn't mention them in the American schools I went to.

Gradients sometimes show up on road signs. Ever seen a picture of a truck, a steep slope, and a percentage like "25%"? That's a gradient.

Woxor
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Yakk wrote:Gradients sometimes show up on road signs. Ever seen a picture of a truck, a steep slope, and a percentage like "25%"? That's a gradient.

I thought that was the tangent of the angle of ascension/descension, expressed as a percentage, which wouldn't be the same as a 100-gradients-to-a-right-angle gradient as listed here.

blob
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zombie_monkey wrote:When I first saw one on a calculator, my dad told me the grad is a French unit for angles and 360 degrees equal 400 grads. Wikipedia confirms this.

In the days before Wikipedia, a young student asked their calculator: "deg. sin 90. grad. arcsin ans ="
And lo, the calculator replied: "100"
And the student was enlightened. The wonders of the scientific method! So they asked their calculator "deg. sin 90. rad. arcsin ans ="
And lo, the calculator replied: "1.570796327".
And the student thought, verily, what is this crap? For that student had not yet studied radian measure and was unfamiliar with pi.

Mighty Jalapeno wrote:And don't tell me the value of showing my work... I'm an engineer. I have to show upwards of thirty pages of equations just for a single bearing wall in a lot of cases.

I would've guessed that having Mathematica verify your equations would be more useful, or something?
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blob wrote:
Mighty Jalapeno wrote:And don't tell me the value of showing my work... I'm an engineer. I have to show upwards of thirty pages of equations just for a single bearing wall in a lot of cases.

If the wall falls down, it'll be in court, so.... I hope the architect reads it!

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Solt wrote:
Tchebu wrote:My grade 8 teacher was genuinely confused when i presented her with the following:

28-36+8 = 35-45+10
4(7-9+2) = 5(7-9+2)
4=5
2*2=5

Good job dividing by 0 there. Apparently she was confused because you didn't learn anything.
I think his point was, that it's a trick 'proof' (obviously), and when he showed his teacher, she didn't get the trick. As in, she was confused, knowing that 2*2 does not equal five, but also not being able to fault the shown proof.

That's how I read it, at least.
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orangeperson
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Torn Apart By Dingos wrote:Further, I haven't memorized any values of cos or sin. I just know they look wavey, and that sin starts at 0, going up, while cos starts at 1, and that they have a period of 2pi, so I always have to draw them if I want, say, sin(pi). If I want sin(pi/3) I have to draw a circle.

This should help

I'm supposed to memorize these numbers simplified, which turns out to go 0, 1/2, sqrt(2)/2, sqrt(3)/2 which just sucks.

This is much better. It clearly shows the sin and cos waves. Also, it's easy to figure out why it works this way by drawing some triangles.
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I don't have em memorized, but I know the unit circle, so if they want sin(pi/6) then I just draw that line on the circle and say, yeah that's about .5, and I know .5 is one of those values for pi, pi/2 pi/3 etc. etc. so that's probably right.
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Torn Apart By Dingos
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orangeperson wrote:
Torn Apart By Dingos wrote:Further, I haven't memorized any values of cos or sin. I just know they look wavey, and that sin starts at 0, going up, while cos starts at 1, and that they have a period of 2pi, so I always have to draw them if I want, say, sin(pi). If I want sin(pi/3) I have to draw a circle.

This should help

I'm supposed to memorize these numbers simplified, which turns out to go 0, 1/2, sqrt(2)/2, sqrt(3)/2 which just sucks.

This is much better. It clearly shows the sin and cos waves. Also, it's easy to figure out why it works this way by drawing some triangles.
But then I have to remember the sequence {0,2,3,4,6,8,9,10,12,14,15,16,18,20,21,22,24}pi/12. It's missing precisely the "non-trivial" primes (if we define non-trivial prime as not being divisible by 2 or 3 (which would include 1 )), which would help me remember it, but I think I'd still need to write out the entire table to get the value, and then the other method would be faster.

orangeperson
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What if you just remember 0, pi/6, pi/4, pi/3, pi/2, and then use the reference angles, remembering the mnemonic device "all students take calculus/craps"? All of the functions are positive in the first quadrant, sin in the second, tan in the third, and cos in the fourth.

So if you get 7pi/6, the reference angle is pi/6, and it's in the third quadrant, so sin is -sqrt(3)/2, and cos is -1/2.

That's how I do it, but whatever melts your butter works.
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orangeperson wrote:remembering the mnemonic device "all students take calculus/craps"?

Our teacher made us use "All Seniors Turn Crazy" which I thought pretty clever.

And just so I feel good about my knowledge, (I know I could just look this up), isn't the other trig functions just written in terms of the other ones inversely to see where they are positive/negative as well?
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Strilanc
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Highschool math always was a delicate balance between "obvious" and "non-obvious" steps.

3x = x + 2
put 2x = 2 first?
or just say x = 1?
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Woxor
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Alky wrote:Highschool math always was a delicate balance between "obvious" and "non-obvious" steps.

3x = x + 2
put 2x = 2 first?
or just say x = 1?

At the other end, the most frustrating thing to encounter when you're proving some kind of result is when the only step you don't fully understand seems like it should be obvious. You can just put it down and hope the grader doesn't notice a jump, but you might be overlooking some huge problem with the proof (and of course you haven't satisfied your own curiosity!).

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God, the amount of proofs that I have submitted to an exam with a ridiculous leap because I can't work out how to prove that bit. Also annoying is when you have a negative sign hanging around... gotta love sign fudges!

On the angle learning, I always just draw the sin and cos curves- you need to learn the basics, pi/6,pi/4,pi/3,pi/2 but you can just then look from the graph. Also sin and cos are just swapped round for pi/6 and pi/3 (at least thats how I imagine it)

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gradients: I'd heard of them. They use them to describe the slopes of hills and trails sometimes. I think they're nicer than degrees, but that doesn't count for much.
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Alky wrote:Highschool math always was a delicate balance between "obvious" and "non-obvious" steps.

3x = x + 2
put 2x = 2 first?
or just say x = 1?

I tend to go for even more abvious, for no actual reason.
3x=x+2
3x-x = 2
2x=2
x=1

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Sounds to me like there's a lot of confusion ITT

In vector calculus, the gradient of a scalar field is a vector field which points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change.

A grade (or gradient) is the pitch of a slope, and is often expressed as a percent tangent, or "rise over run". It is used to express the steepness of slope on a hill, roof, or road, where zero indicates level (with respect to gravity) and increasing numbers correlate to more vertical inclinations.

The grad is a unit of plane angle, equivalent to 1⁄400 of a full circle, dividing a right angle in 100. It is also known as gon, grade, or gradian (not to be confused with grade of a slope, gradient, or radian). One grad equals 9⁄10 of a degree or π⁄200 of a radian.
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Re: Stupid Math Teachers?

A few weeks ago, a friend and I decided to find out what this mysterious unit was, so we plugged in number after number, and got to the 360deg = 400grad after some 'back of the envelope' calculations and ratios. That's what happens when a couple of high school seniors with TI89s get bored.

edit: no internet available, and no one to ask (the end of Japanese class)

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Re: Stupid Math Teachers?

I recall in grade 9 we were learning the basics of multiplying binomials using the infamous FOIL method. Unfortunately by first he meant "The first two parts" giving us something like (x+1)(x+2) = x*1 + 1*x + x2 + 1*2. The next day he tested us over the material, which conveniently had an answer key. When the highest grade in the class was 15%, and it was by one of the poorest students in the class, he had some explaining to do to the headmaster in charge of academics. He was sacked a few months later at the Christmas break.
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Re: Stupid Math Teachers?

as far as stupid teachers, my 5th grade math teacher was rather, um, liberal arts oriented.

We were taking fractions and turning them into decimals and visa-versa. Her method for going from fractions to decimals was to take the denominator, muliply it by the whole number that will get it closest to 100, and then multiply the numerator by the same number.

Now, my mom was a math major, and my dad is an electrical engineer (needless to say, I knew basic math and science before I could tie my show laces) so I raised my hand and asked if it would just be easier and simpler to divide. I was told I was wrong, and that didn't turn the fraction into a decimal. WTF? So, I demonstrated this principle with 1/4, 1/5, 1/2... how it was faster, and more accurate than her method. I was told, again, that I was wrong. Fine. Let her be an idiot.

So, 10 or so minutes later, she starts "showing" us how to make decimals into fractions. She said that .05 is 1/16. Again, I raise my hand, and said that it's not 1/16, that it's 1/20. She said I was wrong because the book said it was 1/16. Obviously a typo. I showed her and the class that it was 1/20 using the methods my parents had taught me. She made me sit outside for the rest of the day.

As far as sin, cos, and tan, this is what my calc teacher in high school taught me for the numbers:
trig.JPG (24.68 KiB) Viewed 9832 times

so, all I have to remember is 0, 1, 2, 3, 4 ... 4, 3, 2, 1, 0, and how to divide fractions.
"YES. DO IT WITH CONFIDENCE" ~fortune cookie

Fafnir43
Posts: 98
Joined: Mon Apr 14, 2008 11:11 pm UTC

Re: Stupid Math Teachers?

Showing work. Forcing people to show their work at high school-level problems is a terrible, terrible idea. Sure, it doesn't showcase understanding of the ideas involved, but nor does mindlessly stepping through a method and quite frankly that's what the vast majority of high school/A-level maths boils down to. What's more, at higher levels of education the ability to mentally complete high school maths problems quickly is an extremely valuable skill. It lets you think a few steps further ahead in a complex problem - a problem that requires more than rote application of a method or formula - than you would otherwise be able to. It's also vital when you actually start reading papers or attending lectures by people who consider rote computation trivial and skip it entirely, a class of people that steadily expands as you go up through the system. By the time you enter research, it includes everybody. In short, this is not a skill that should not be stamped out - it is one that should be nurtured.

Yes, if someone's getting the answers wrong it can be hard to tell why if they don't show their working. And there are some techniques that work in most cases, but not all of them. That's what homework is for. Set the standard sheet of twenty boring questions, then make people redo the questions they get wrong showing their working. That way, you can help the people who need help without penalising the people who are developing useful skills.

Declaration of bias: I'm in the UK. Over here, generally about half of the marks available in A-level (high school equivalent) maths papers are given for working, and cannot be obtained just by giving the correct answer. I believe it maimed me somewhat as a mathematician, not because I rebelled against the system but because I went along with it and thus lost a key opportunity to pick up the skills I now need on a daily basis. Luckily I've managed to pick up most of what I missed, but still...

Also, on gradients: I've never really understood the point, to be honest. I mean, radians are obviously the best for any serious maths because sin(x)/x->1. And degrees seem to be better for simple applications, since 360 has far more divisors than 400, so you can represent most simple divisions of a circle without having to resort to fractions...

Me sleep now.
This post brought to you by the language C and the number F.

Charlie!
Posts: 2035
Joined: Sat Jan 12, 2008 8:20 pm UTC

Re: Stupid Math Teachers?

I've used gradians (which are definitely not the same as gradients) once or twice.

They're used to do solid angle problems. What does that mean? Well, just like a radian is arc length / radius, a gradian is the area of a circle on a sphere (drawn where outside line is some angle away from the center point) / radius. They make the occasional "surface of only part of a sphere" problem verrry easy.