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We don't need new schools out here. We need good teachers who make more than $22,000 a year, and we need lots of them! |
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Marijuana tax revenue distributions to the Colorado Department of Education, 2015-2017 Five out of six of those aren't Brick and Mortar. |
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So for example start out with a general equation containing a term of say the average mass density in a control volume, and end up with an equation having variables that you can measure such as dimensions of the sump tank and weight of the oil. Or start out with an equation requiring a certain molality of solution, and end up with an equation involving weights and volumes. In most interesting engineering cases it's a lot more convoluted than that. And typically you have to be looking at the same diagram as the author or you'll be scuppered. But the goal is always a similar kind of thing. Link stuff together to get to something you can use. |
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Lemme throw out a question and see if it goes anywhere. How well does our system for educating educators work? To what extent are budding teachers taught the nuts and bolts of succeeding in a classroom (the way Normal Schools and Teachers Colleges presumably used to do it), and to what extent are their heads crammed with education "theories" that exist only to help professors get tenure? (Clarification available on request, pending a less foggy state of mind.) |
Just as an aside, you seem to have supposed that modern threories of education are largely bullshit. I think that is wrong, and that the best schools deliver a better education than ever, and they do that by applying principles developed in the last half century.
And as an aside to an aside, when Learer says: "Thing is to understand what you're doing, rather than to get the right answer," he fully understood that understanding is a crushingly more important goal than getting the right answer. He was, as we say in the business, making a joke. |
The new math method he was satirizing in that song is exactly how I do math, for the record.
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New Math replaced Rote Math. No-one who ever did Real Math learned math by rote. Sorry, NO-ONE. EVER. NOR COULD THEY. Of course, the complaint is that few need Real Math. Interesting to note (Lehrer aside (who was joking)), only a non-mathematician would argue that. |
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Learning to understand what you're doing is great...but not in the case of new math. Trying to learn number theory as a 1st grader before you learn simple arithmetic is just dumb. Teach the kids how to do it, by rote if necessary, then teach them why/how it works. And I hope nobody has to live in a building designed by someone who thinks that "trying to understand is a more crushingly important goal than getting the right answer". Or has their taxes prepared by someone who thinks that way. As an aside, here's an interesting quickie article on New Math |
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Fuck it, I will stick my neck out. That song is the only reference you have too. |
BTW: you've never learned anything if you think that effective learning means "getting it right" rather than "developing appropriate techniques".
I suspect you have learned many things, but you've yet to learn that you can look like a boorish cunt flopping your cock on a chopping block like that hoping that no-one will notice. |
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Regardless, though. The Tom Lehrer thing is a distraction. Even if he thought it was the greatest thing since sliced bread, or the worst thing since unsliced bread, he's still only one guy and teaching set theory and alternate bases to kids who can't do simple, absolute beginner's arithmetic is just a self-evidently dumb idea. Quote:
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Look, we may be talking at crossed-purposes. The aims of New Math were good and just -- its implemetation was shoddy and piecemeal. As a consequence it was buried.
Having "kids who can't do simple, absolute beginner's arithmetic" is truly a dumb idea: New Math was designed to address this. In that respect it is perceived to have failed. But I was taught it (I'm a math teacher (those who can, and all that...)), and it continues to be taught in essence, not in the failing schools, but in best. And I agree, Lehrer's opinion counts for nothing if he ain't gonna post it here. |
Great! I'm having trouble converting partial differential equations to finite difference equations. WTfuckingF, over?
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Try taking logs. It mostly works.
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What?!?? How does passing logs help solve...
Huh? Oh. Sorry, my bad. |
It's simple:
For a given prism, of breadth, b, and cross-sectional-area, a, and of mass, m (scaled by a conversion factor, y), and density, d; we have : my/d = ba #Worst maths punny thing EVA |
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Since you're a math teacher (and absolutely no snark is intended), let's say you have a classroom of first graders (I don't know what grade you teach, so you might. :) ). Most of them can count to say, 20. That's the extent of their math knowledge. How and where would you start "Introduction to Math, 101"? Me? I'd start with a number line and use it as a concrete way to show them how adding and subtracting moves things up and down the number line. I might* also teach them number positions (ones, tens, one-hundreds, etc). I would swipe a tool from New Math and say "Here's a pile of red, green and blue buttons. Blues are worth 1. For every 10 blues you get you can trade it for one green and for every 10 greens, you can trade it for one red" or something similar to illustrate that concept. What I would not do is what was done to me: teach me how to count to 100 in base 10, 80 in base 8 and 120 in base 12. And then do exercises like "If there are ten boys and ten girls in a classroom and half of each group are named "Pat", how many kids are in the intersecting set?" I remember these examples personally from either K, first or second grade (I moved to a new school in third grade so I can pin it down for sure to earlier than third). No snark, no sarcasm, how would you introduce kids to math who's only experience is counting to 20? *Don't hold me to this, I don't know if this is too advanced for first graders |
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Likewise, several other examples in this thread point to problems with the DOE, or problematic attempts to modify the educational system, but I've yet to see a convincing argument for scrapping the entire federal entity. Nothing supports the overall benefit to U.S. education inherent in having 50 educational systems working completely autonomously with no centralized oversight or guidance of any sort. |
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Teaching bases other than ten is often used as a tool to develop deeper understanding of the decimal number system. It seems premature to me to tackle alternative bases before some significant understanding of this system has already been achieved. Your experience sounds properly bonkers. However, some notions of set theory (for example) can be appropriately tackled long before mastery of decimal arithmetic. Indeed the game Guess Who? is an object example of set theory in action. Guess Who? is marketed 6 and upwards. Of course, you can play the game having no idea what an intersection of sets is but the game can be made richer with a modicum of abstract notions (union, intersection, etc). Of course, if I was teaching set theory to a six-year old I doubt I'd use any technical language whatsoever but some valuable notions can still be worked with without any hi-falutin' jibber-jabber. Mathematics is not a collection of formulas (though there are tools), nor even a language (though there is a vocabulary). It is a methodology for studying structure wherever that may be (hint: it is everywhere). It is fundamentally abstract in essence and arithmetic is but a tiny little corner of its whole (this drives a number of people mad, they believe we should be training Little Johnnie to count widgets faster than Chinese children can). Here's a nice bit of maths wot I just invented. At a dinner party there are an odd number of people. Prove that at least two people of the same sex* must be sat next to each other. * I have grossly over-simplified the situation in supposing that there are only two sexes and exactly one table. The interested mathematician will go on to generalise the problem for n-sexes in m-dimensions modulo-p. |
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well, yes, but that was never done in any math class I ever attended. Never told the applicability of the formulae (which is why D=R x T sticks in my head--it was probably the only one), just given a list of problems to solve for X. Well, some of that was mildly interesting, but mostly it was either dull has hell or frustrating as all get out. Why do math books leave steps out? Mystery. And don't get me started on pi and the whole geometry/trig thing. Geometry had applicability going for it, but the theorems were horribly taught (I learned who Pythagoras was, but not why his formula was needed etc). And please don't drag word problems into this. I fully understand why word problems are so important--applicability is all with them--but I just couldn't do them. Call it innumerancy or whatever you like, but making "math sentences" beyond D=R xT is beyond me. :sciencefail: anyway, what were we talking about? ETA: To Great Unwashed: Lehrer is beside the point, and yes, I get it was a joke (why would I think it was anything else?) :confused: Then again, tone and context mean a great deal on message boards and can be hard to convey. This is my bottom line: I am pro-DoE, pro-teacher, pro-public education. I had a very good public education, but it took place before Gardner and his 7 types of intelligence (I hear tell there may be 9 now) were popular or even thought of in terms of helping pupils. I was told by a teacher that since I scored so highly in language and verbal skills, that I "must" be good in math. Today we know this is complete BS. I struggled mightily with math (until another teacher--a student teacher, no less--told me that I didn't have to worry about math, because i was a GIRL. Not kidding. Year was 1972). I wasn't getting Fs in math, but as it became more complex, I dropped from a B to a C and then in HS I really struggled. I made it through Trig (nightmare) and purposely took 4 years of HS math so that I would not have to take any college math. And I didn't. Phew! I learned to regroup--I'm not kidding; the penny dropped and I understood it--when I was helping one of my kids with their 3rd grade math homework. So, forgive me if I say that you sound a tad defensive due to being a math teacher yourself. What comes easily to you, does not to others. Frankly, the New Math was horrible when I had it in 2nd grade and I glad I never saw it again (it was a fad and short-lived, thank god). |
Missed the edit window.
I agree with you re maths being all about infrastructure, so to speak, and w/o it, science is not possible. But unlike in math, I can spot underlying themes in English and patterns in behavior and commonalities in the roots of words etc, so it's not just math that hides infrastructure in plain sight. I think I could have enjoyed math more if it had been better taught. I also wish I had taken actual music lessons, not just noodled around and taught myself to "play" the piano, because music helps with math (and vice versa). Or maybe I'm brain damaged in a certain way and just don't get math. I do know I'm not alone in either my mild math phobia or my experience with poorly taught math (I did have some good math teachers, too--which I'm grateful for). Lastly, I was always told that I was going to love calculus, but I never got that far, something I regret. |
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It also taught (though I don't recall the teacher ever saying it that way) that problems can be broken into steps. Once you've shown, for example, side-angle-side congruence in triangles, in future proofs that's all the farther you need to go. All the steps in the original proof just condense down to "thus the triangles are congruent by SAS" and you get to skip doing the whole thing over again. Comes in handy when trying to tackle non-math things, too: If you know that once you're to X that you can replace the outlet, install the showerhead, paint the railing, whatever, now you don't have a giant project in front of you. Get to the point you already know and you've got the rest covered. Algebra/pre-calc/calc are unfortunately just taught as glorified puzzles. "Plug and chug," as my physics teacher called it. Which on the one hand is not too different from the geometry proofs, but the focus was on the results instead of the process so it feels different and repetitive. The teachers and textbooks also aren't too good about relating things back to what you already learned. The way I learned multiplication back in grade school in the 80s, once you got past the rote times tables (1x1 up to 9x9) went something like this: Code:
2 Fast forward to Algebra II and now we have: Code:
(4x + 7)(3x + 5) = 12x^2 + 20x + 21x + 35 Well, if I write the first equation from grade school like this: Code:
(4*10 + 7)(3*10 + 5) = 1200 + 200 + 210 + 35 Code:
47 Well, when a kid gets sent home with their "weird" grids with diagonal lines, that thing above is what they're doing. The grids are just forcing them to line up the numbers right since they're dealing with the four multiplications separately. So the adults see something they don't remember, that the kid can't explain (just like they couldn't explain the way they did it) so they get confused and trash it. Which probably doesn't help the kid learn math too well. |
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TEACHER SALARIES Got it? |
Dude, you should see Utah. :sciencefail:
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I could probably ramble on for several pages and say little with any certainty. This is one long ISTM:
Once upon a time being a teacher carried some status and was considered a respectful profession. Not so any more -- teachers are underpaid, under-trained and over-worked. I cannot help but feel that a great deal of my job is essentially child-minding. But I don't think this is the biggest problem. I could read and write and count and do simple arithmetic when I started school aged 4 having been taught by my mother. But now it seems that a great many parents out-source entirely the education of their kids to the teaching system. I am English but I live and work in France. People tell me my French is good, they are idiots with no standards, my French is shit. But there is never an hour that passes when I am not in a position to correct the French of one student or another. I'm not talking about correcting their use of "le teen speak" but correcting their supposed standard French. Everytime I think about that, my brain freezes and I have to stare off into the distance for a minute. I mean, WTF!? We're essentially talking about young adults. There is a non-invisible proportion of my students who do not readily see that 0.5 = 1/2, and a worryingly large proportion who cannot readily see that 1/2 x = x/2. And woe betide if I ever gloss over a step in a calculation, thusly : x÷1/2 = 2x. The standard French class size is 35. There is no selection before Seconde (grade 11). There is some disagreement among educators if this is a good thing or a bad thing. Personally, I think it is a deeply regrettable that by age 16 there can be such massive disparities in capacity to reason, discuss and calculate -- but while those disparities exist it is hopelessly optimistic to believe that mixed-ability large classes are the way to go. There isn't a classroom in my Lycée that doesn't have a broken chair, or some graffiti, or a window that doesn't shut, or wall-paper partially torn from the wall, or the overhead-projector doesn't work. Schools here (and in the UK) are woefully underfunded (or perhaps mismanaged). No 16-year-old could find gainful employment here -- it's either staying in bed all day playing Les Oiseau En Colére or going to school. You know what's worse than teaching a struggling student? Teaching a disaffected, uninterested (uninteresting) entitled student full of hormones and attitude who has Les Oiseau En Colére on their iPhone. |
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You could throw money hand-over-fist at the teachers in those underserved schools. You'd probably cut the teacher turnover rate some, but you wouldn't cut the rate of teacher burnout. You wouldn't put a dent in the classes where students arrive at school hungry and unfocused because free breakfast and free lunch at school are their only source of food and they don't eat on the weekends. You wouldn't help the kids who didn't do their homework because there literally was no writing utensil in their house. It wouldn't change the classes where teachers have to spend more time teaching the kids how to sit, pay attention, and keep their hands to themselves than they do teaching arithmetic. Some schools have to deal with a large percentage of their student population coming from households where education is not remotely a priority. Teacher salaries don't fix that. |
And the U.S. Department of Education does what, exactly, to make that better?
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By leaving educational funding up to states, we already have the SES segregation referenced above. I'm not saying DOE will fix that, but it's further evidence to me that states aren't going to voluntarily develop/adopt standards that ensure country-wide homogeneous education expectations.
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:fist: |
Ouch!
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Good points made by Great Unwashed, Solfy and Jackie. She's referring to teaching in the USA, especially primary grades. There are almost no male teachers in K-6, at least around here. It's too bad because there is a lot of good a male teacher can do for those young kids. I bet you're a good teacher, GU. I have no answers as to why schooling is no longer a priority for so many families--there was a time when education was seen as the path to betterment, even in my lifetime. I hadn't realized (due to not really thinking about it) that it was affecting other countries as well. Sometime in the late 80s(?) or early 90s, education came to be seen as a product, not a striving. And the Federal Gment stopped supporting state universities, and tuition skyrocketed. I'm vastly overgeneralizing, so I'll stop maundering on.
The DoE may not be able to swoop in and solve all the ills, but leaving the educating of our kids to each state is a recipe for disaster. Our country is too big geographically and in terms of population to not have some sort of centralized oversight. I have no faith that various states will actually even fund education, never mind provide true support to schools. I'd love to see some alternate way of funding public schools than property taxes, but even with that change in funding, like Solfy said, it doesn't solve the problems of homelessness, the working poor, domestic violence and childhood neglect etc. Public schools reflect back our social ills--I am tired of us blaming them for the hard truths. Double J: I remember FOIL, but I can't say I followed any of the rest of what you did. I could probably get it if I applied myself, but one of the luxuries of being 54 is that I no longer have to do as many things that I'd rather not. :) |
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Unless you start volunteering to tutor I'll ignore the grids and diagonal lines. I know why they're doing it but shit's weird, yo. As an aside (that turns out to be the longest part of the post) I'd also argue that the main reason people don't get algebra is because elementary school teachers spend six years acting like an equals sign means "makes" -- "six plus two makes eight" -- when they ought to be stressing that it's an equals sign -- the same on the left as the right. So not, "what does six plus two make" but "what is six plus two the same as". And if some smartass wants to give the answer of "five plus three" then they're right because those two things are the same. The sad fact is, they have the opportunity to teach it this way but don't -- remember the problems that looked like "5 + __ = 8" from back about second grade or so? That's goddamned algebra! But they're just treating it as a way to reinforce the rote addition tables so it's still "five plus what makes eight" instead of starting down the path of "if these two things are the same, what goes in the blank?" And then seventh or eight grade rolls around, everything changes, and people get hopelessly lost because it feels like there's no connection between this and everything before it. |
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123 × 456 = (120 + 3)(450+6) = 120×450 + 120×6 + 3×45 + 3×6 = (100 + 20)(400 + 50) + (100 + 20)×6 + 3(40 + 5) + 3×6 = 100×400 + 100×50 + 20×400 + 20×50 + 100×6 + 20×6 + 3×40 + 3×5 + 3×6 ? simples!* I appreciate your point, but long-multiplication= FOIL is a bit hasty. Personally I distrust nmemonic devices like FOIL (they are somewhat arbitrary (what's wrong with FILO?)), and in this case the distribution law can be understood in and of itself (don't forget it is more important to understand than to get the right answer). Teachers who like their kids to get the right answers teach FOIL, just sayin'. The parallels that you see certainly exist, but students typically do long multiplication before they do algebra, I'm not sure how it would fall in the mind of the naive student to be told that (a + b)(c + d) works like (10 + 2)(30 + 4) (and doubly so if previously they have been taught some other method for long multiplication). Where there are good and/or useful parallels to be drawn a good teacher will make them. I do not wish to doubt your personal reportage but it's even possible (likely?) that such parallels were drawn to your attention but didn't seem relevant or valuable enough for you to retain. * I suppose we'd prefer (100 + 20 + 3)(400 + 50 + 6) but where lies your FOIL now? |
DoubleJ--that is an excellent point! I had never thought of basic math that way, but yes, 5+___=8 IS algebra! For some odd reason, I feel better.
FOIL is taught because it's handy, quick, in every math book known to America. And this is another problem: our system is so focused on the standardized tests and has so many other ills to cope with, there is NO time to spark that interest in a kid by showing alternatives or even encouraging creative thinking, unless you're lucky enough to be in a well-off white suburb near a major metro area, preferably one that is in a liberal state. Then we get something approaching the ideal in American education w/o church/Bible influences (especially science) or politically influenced information (especially history). And even then, it's probably not as open or comprehensive as it needs to be. IOW, there is no perfect system. IMO, we could start right away with the premise that lowering standards does nobody any good at all. |
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The whole point of mathematics is to get the correct answer, isn't it? Otherwise, why even bother? To be sure, everyone goofs occasionally. I'll bet even Einstein occasionally screwed up the balance in his checkbook. But that is not what I have in mind here. If one keeps getting the wrong answer, doesn't that mean that he doesn't understand the process? Conversely, if one understands the process he should, as a matter of course, bloody well get the correct answer. So it makes no sense to say that one is more important that the other. The two are inextricably linked. |
Lehrer made the joke (in some words or other), I just repeated it.
You are right that (by and large) : (understanding implies getting the right answer) and this implies (not getting the right answer implies not undertanding) But notice, (getting the right answer does not imply undertanding) So getting the right answer cannot be the ultimate aim. (It is why "teaching to the test" is intrinsically flawed.) Posing u = undertanding r = right answer we have (u ⇒ r) ⇒ (¬r⇒¬u) but not (u ⇒ r) ⇒ (r ⇒ u) Conflating (u ⇒ r) and (r ⇒ u) is a rather widespread cognitive bias. I believe it is behind Lehrer's joke, it was certainly behind mine. |
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You can drill into people that 3x7=21, but it's also useful to teach them that 3x7 is the same as 7+7+7 is the same as 3+3+3+3+3+3+3. That's how people make the leap from doing math problems in a textbook to recognizing that math is actually a tool for the real world. This is why people end up railing against story problems. "Why do I have to know how much money Consuela had left over after she bought three pencils and a notebook? This is stupid and pointless!" |
I know you kept characterizing the line as a joke, Unwashed, but I just wasn't getting it. Still don't. Looks like I don't have the proper bent of mind to appreciate it.
It occurs to me that one might be able to use a process without actually understanding it, but I'm not gonna go there. . . . |
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Given values for C, p, and n, solve for x: C - 3*p - n = x |
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Original amount of money Consuela had minus three times the cost of a pencil and one notebook equals remaining money.
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I didn't read the quoted text. I'll give myself a 2/10 and a "must try harder"!
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Unwashed: I don't disagree. FOIL is good for what it is: binomials being multiplied, which can be algebraic expressions, or just two-digit numbers. The root knowledge (distributive property?) is what's important. Like you say, there's no trinomial/three digit version of FOIL. And I'm pretty sure that when we got to that point we just multiplied everything out long-multiplication style, lining up terms based on power of x. Which kind of loops it all back around.
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It's not something I'm proud of, in all honesty. But it's like old people (generalizing) and computers: my frustration/quit button is very quickly pressed. The first story problem I ever had to solve, (that I remember) involved the Johnson family, going on vacation. The workbook wanted to know something about the miles per hour or the distance traveled or the speed plus arrival time or some such. This was just not enough information for me. I needed to know more about the Johnson family: where were they going? How many kids did they have? Did they take the dog with them? I wish I were kidding. I had zero interest in solving for whatever number they were asking for. Instead, I built an entire story around "The Johnson family was going on vacation." My second grade teacher was not amused. :) |
:facepalm:
(Actually, I love it.) |
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