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United States24690 Posts
On July 30 2012 00:10 paralleluniverse wrote:Show nested quote +On July 30 2012 00:06 micronesia wrote:On July 30 2012 00:03 paralleluniverse wrote:On July 29 2012 23:59 farvacola wrote:On July 29 2012 23:53 paralleluniverse wrote:On July 29 2012 23:51 farvacola wrote:
I'm not exactly sure why the article makes the connection between flawed primary/secondary school math education and a need to remove certain educational requirements; it would make sense, at least to me, to instead focus on better teaching methodologies, as algebra and many of the "hard" areas of study have been taught in relatively the same manner for many, many years now. As posters above are explaining, I certainly think that Algebra ought to be requisite, as it lends itself to so many endeavors in life, and perhaps the key in improving its teaching is a more vocational or everyday focus in application, and a change from the standard drudgery of problem sets and timed tests. I'm thinking on my own honors math education, where in 8th grade we learned the quadratic formula via song and were required to sing it at the door one day in order to get into class. Simple little changes in teaching technique can do wonders for making subjects such as math more palatable. As I see it, there are simply too many inherent problems in putting forth entire populations of people who are unable to the basic algebraic underpinnings of the economy and their own personal finances. Are you saying that's good or bad? I am suggesting that such an alternative strategy is a good start when thinking on how math might be taught differently than it is now. My school district happened to have one of the best honors programs in the state of Ohio, but from 4th grade through graduating high school, it was readily apparent to all of the honors kids that we were getting the cream of the districts educational crop while the kids in normal classes fell by the wayside as a result of a less successful and less interesting curriculum standard. Singing math is one of the stupidest ideas I've ever heard. Suppose your students were to learn the quadratic formula. Would you give it to them or make them somehow come up with it? There are arguments for both, depending on things like at what stage in their math career they are at. However, let's suppose you needed to give them the formula and teach them how to use it. They need to memorize the formula (barring a cheat sheet). How are you going to get them to memorize the formula? This is the same as asking how you will get students to memorize anything else (it isn't math specific). You may not like your core subject teachers requiring you to sing a song to learn something, but don't think that this has anything to do with math. I would not gets students to memorize the quadratic formula. I would teach them how to solve quadratic equations.
How? Assuming factoring and the like is already covered, and you are going on to problems that require the quadratic formula to be solved.
I would also derive the formula. You've lost the majority of students at this point, lol. I like the idea in certain applications, but not all. Remember that we are talking about public school education in the USA in this thread.
Once you've solved several quadratic equations you'll naturally remember it without any effort specifically on trying to memorize it. Back to my first question of the current post.
Memorizing formulas has zero educational value. Almost completely agree with you.
I've read it, and you are assuming that teachers actually have 100% control over what happens in their classes, which they usually don't.
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On July 30 2012 00:15 Brutland wrote:
saying that algebra isn't useful to everyday living is almost like saying that oxygen isn't useful for everyday breathing. for instance, i need to go grocery shopping, i need to know how many and at what price my needed items are, then how much money i can spend. thats the heart of algebra (Ax+By...)-Z=0.the problem is most people think of math as this esoteric thing instead of as a way to frame the world and a way to make decisions about concrete ideas. hell, people use calc3 when playing baseball or football (3d analysis and force vector combinations intersecting planes of multiple moving pieces). the problem is that math taught by someone who doesn't understand math is worse than useless. it teaches that math isn't applicable, when in reality, it is how the world works.
Nobody uses algebra when doing shopping. No baseball player does calculus. Just because the situation can be analyzed with algebra or calculus doesn't mean anyone does so, or would need to do so.
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On July 30 2012 00:18 micronesia wrote:Show nested quote +On July 30 2012 00:10 paralleluniverse wrote:On July 30 2012 00:06 micronesia wrote:On July 30 2012 00:03 paralleluniverse wrote:On July 29 2012 23:59 farvacola wrote:On July 29 2012 23:53 paralleluniverse wrote:On July 29 2012 23:51 farvacola wrote:
I'm not exactly sure why the article makes the connection between flawed primary/secondary school math education and a need to remove certain educational requirements; it would make sense, at least to me, to instead focus on better teaching methodologies, as algebra and many of the "hard" areas of study have been taught in relatively the same manner for many, many years now. As posters above are explaining, I certainly think that Algebra ought to be requisite, as it lends itself to so many endeavors in life, and perhaps the key in improving its teaching is a more vocational or everyday focus in application, and a change from the standard drudgery of problem sets and timed tests. I'm thinking on my own honors math education, where in 8th grade we learned the quadratic formula via song and were required to sing it at the door one day in order to get into class. Simple little changes in teaching technique can do wonders for making subjects such as math more palatable. As I see it, there are simply too many inherent problems in putting forth entire populations of people who are unable to the basic algebraic underpinnings of the economy and their own personal finances. Are you saying that's good or bad? I am suggesting that such an alternative strategy is a good start when thinking on how math might be taught differently than it is now. My school district happened to have one of the best honors programs in the state of Ohio, but from 4th grade through graduating high school, it was readily apparent to all of the honors kids that we were getting the cream of the districts educational crop while the kids in normal classes fell by the wayside as a result of a less successful and less interesting curriculum standard. Singing math is one of the stupidest ideas I've ever heard. Suppose your students were to learn the quadratic formula. Would you give it to them or make them somehow come up with it? There are arguments for both, depending on things like at what stage in their math career they are at. However, let's suppose you needed to give them the formula and teach them how to use it. They need to memorize the formula (barring a cheat sheet). How are you going to get them to memorize the formula? This is the same as asking how you will get students to memorize anything else (it isn't math specific). You may not like your core subject teachers requiring you to sing a song to learn something, but don't think that this has anything to do with math. I would not gets students to memorize the quadratic formula. I would teach them how to solve quadratic equations. How? Assuming factoring and the like is already covered, and you are going on to problems that require the quadratic formula to be solved. You've lost the majority of students at this point, lol. I like the idea in certain applications, but not all. Remember that we are talking about public school education in the USA in this thread. Show nested quote +Once you've solved several quadratic equations you'll naturally remember it without any effort specifically on trying to memorize it. Back to my first question of the current post. Almost completely agree with you. I've read it, and you are assuming that teachers actually have 100% control over what happens in their classes, which they usually don't.
Completing the square.
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My problem with the article is that it mixes very different points:
1) Math entrance exams in colleges / universities.
2) Math education in colleges / universities.
3) High school math education.
As for for the first two points, I tend to agree that it's useless to both have math exams and teach higher math courses to humanities / history / art students. Actually, at least right now, this isn't the case in Universities here in Austria, and I think also not in Germany (though there might be exceptions? Germans correct me  )
As for school education, I think that the article's focus on Algebra is strange. I think the problem is a more fundamental one. I really liked the text linked by Severedevil.
On July 29 2012 21:28 Severedevil wrote:Subjects are often taught stupid. There's gotta be a greater motivation for learning material than "it's on the syllabus" or "it's on the test." However, the problem does not lie in the difficulty of introducing basic abstraction to arithmetic. http://www.maa.org/devlin/LockhartsLament.pdf
The way schools have developed to function during the last decades works very well to teach facts and test whether children can learn them by heart and reproduce them. This process takes a limited amount of time, which can be accommodated for in the curriculum, and every idiot can do it as long as he has the discipline to do it. History, biology, etc. can be nicely organized in lists and facts: Lists of species, lists of years with associated events. One can memorize this stuff without having to understand a lot. I don't say the subjects are mindless, but they can be taught in a way that there is not much intellectual effort involved apart from memorizing.
It's interesting that the subjects where students fail a lot are exactly the ones where this approach is not very well suited: Languages and math. Considering languages, I feel there are people that intuitively spot errors or have some talent to write texts. While exercise helps, I've always had the impression that there were some students that struggled with this at my school, and others that just didn't, with little change over time. Language subjects, however, tend to include some percentage of 'learnable' stuff (let's call it 'fluff') like history, politics, etc., which help the poor sods that cannot write a decent text. They can instead just learn the fluff and still pass.
Considering math, first you have to understand abstract concepts. How long this process takes, differs from student to student. Second, what is tested at school is usually the ability to do manipulations on equations and numbers in a limited amount of time as error-free as possible. Again, I have the impression that this is a skill that can be exercised, but that there are people who have talent for this, and others that don't. However, no fluff to fall back on.
I have the impression that much of the current math / science education tries to emulate the tried-and-tested 'memorize and reproduce' tactics. The subject has been cut down to facts and formulas, as outlined in Seveverdevil's text. Still, it's not learnable in the above sense. Much is taught by 'cooking recipies' to solve specific examples, but even they are hard to apply without a minimum understanding that is hard to be forced in a fixed timeframe. Additionally, school math tests to a huge extent this narrow skill-set of 'mechanical calculation' that is astonishingly different from what you need when you actually work with this stuff. (I am a physicist and I cannot do extended calculations without flipping a sign or dropping some constant for the life of me. But it doesn't matter much thanks to Python, Mathematica & Matlab )
Also, I feel that this misunderstanding of how to teach math and science is a symptom of a deeper misunderstanding that society on average has of what math and science is or is supposed to be. You can schedule memorizing; you cannot schedule understanding. You can't say 'If I think hard for 2 hours about my problem, I have understood it to 75%.
Therefore, I don't think math is the problem. I'd rather argue that the way we teach other subjects is the issue which only rears its ugly head when it fails to convey math to the students. I don't have a fix for this; But I hope there is a better one than dropping inconvenient subjects.
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On July 30 2012 00:11 farvacola wrote:Show nested quote +On July 30 2012 00:03 paralleluniverse wrote:On July 29 2012 23:59 farvacola wrote:On July 29 2012 23:53 paralleluniverse wrote:On July 29 2012 23:51 farvacola wrote:
I'm not exactly sure why the article makes the connection between flawed primary/secondary school math education and a need to remove certain educational requirements; it would make sense, at least to me, to instead focus on better teaching methodologies, as algebra and many of the "hard" areas of study have been taught in relatively the same manner for many, many years now. As posters above are explaining, I certainly think that Algebra ought to be requisite, as it lends itself to so many endeavors in life, and perhaps the key in improving its teaching is a more vocational or everyday focus in application, and a change from the standard drudgery of problem sets and timed tests. I'm thinking on my own honors math education, where in 8th grade we learned the quadratic formula via song and were required to sing it at the door one day in order to get into class. Simple little changes in teaching technique can do wonders for making subjects such as math more palatable. As I see it, there are simply too many inherent problems in putting forth entire populations of people who are unable to the basic algebraic underpinnings of the economy and their own personal finances. Are you saying that's good or bad? I am suggesting that such an alternative strategy is a good start when thinking on how math might be taught differently than it is now. My school district happened to have one of the best honors programs in the state of Ohio, but from 4th grade through graduating high school, it was readily apparent to all of the honors kids that we were getting the cream of the districts educational crop while the kids in normal classes fell by the wayside as a result of a less successful and less interesting curriculum standard. Singing math is one of the stupidest ideas I've ever heard. Haha I agree with you in a certain sense, but that this approach to teaching Algebra included singing is more a reflection of my old teacher than the sort of changes in math education I'm talking about. Funnily enough, that same class of individuals turned out some very brilliant math minds, and not a single person I've talked to who has taken Mrs. Flahie's honors algebra class has ever forgotten the quadratic formula The point is that this article is basically supporting a pedagogical signal of defeat, as though the very nature of Algebra somehow makes it incommunicable to some people. And while this may be the case in an extremely remote sense, the teaching of math is something that can still be improved on dramatically enough to improve the classroom experience by leaps and bounds.
I might add that here in Finland we are top 3 PISA in math, not because of how good our best are, but because of how massively better our worst are than the average worst. And our worst (the below average I mean) are terrible slackers. The only conclusion to me is that the teaching process somehow manages to get through all the energy drink and disinterest. It's a feat worthy of the admiring. And perhaps studying.
Singing seems a bit.. extreme, not least because musical talent is even rarer than mathematical talent. My point in all this is that this stuff should be doable in America, unless there are some hidden reasons why math is easy for Finns. One such I guess could be the intuitive vocabulary: We talk about "five-angles", not "pentangles" or whatever.
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On July 30 2012 00:18 paralleluniverse wrote:Show nested quote +On July 30 2012 00:15 Brutland wrote:
saying that algebra isn't useful to everyday living is almost like saying that oxygen isn't useful for everyday breathing. for instance, i need to go grocery shopping, i need to know how many and at what price my needed items are, then how much money i can spend. thats the heart of algebra (Ax+By...)-Z=0.the problem is most people think of math as this esoteric thing instead of as a way to frame the world and a way to make decisions about concrete ideas. hell, people use calc3 when playing baseball or football (3d analysis and force vector combinations intersecting planes of multiple moving pieces). the problem is that math taught by someone who doesn't understand math is worse than useless. it teaches that math isn't applicable, when in reality, it is how the world works. Nobody uses algebra when doing shopping. No baseball player does calculus. Just because the situation can be analyzed with algebra or calculus doesn't mean anyone does so, or would need to do so.
Just because jobs don't use the education has ZERO to deal with why you educate. That's an argument for not requiring education at all. That is not an argument against algebra specifically. You can be a perfectly successful citizen while not having a high school diploma. Your options are more limited, but that's how it's supposed to be.
Then you get college students who aren't properly educated in algebra, and they will want to try things like biology and chemistry. Well, then the colleges will have to deal with this piss-poor educational system in algebra. This is shit you should have learned in high school, and we'll be putting way more onus on our colleges. They will be spending more and more time on basic education rather than their actual damn major, because a high school diploma will be completely worthless in terms of what it means in terms of an education.
Lowering standards does not improve education. All it does is make your numbers look better, as more students glaze through the system even if they haven't done any work or have any intelligence whatsoever.
The real problem with our education system is the absolute disrespect our culture gives to teachers. They're paid shit, and now getting blamed for children's bad grades. People think of teaching as an easy job that anyone can do well. Give teachers some damn respect.
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On July 29 2012 23:56 Vega62a wrote:
I understand the basic problem - looking for ways to make education more accessible - but removing a subject because people don't like it or don't do well at it is the wrong way to go about it.
People don't HAVE to be bad at math. Not everybody is going to ace their college calc courses, but basic algebra doesn't really require mental pushups. We are bad at math because we don't care about it, and because we spend most of our lives talking about why we don't care about it.
Think about it. How many times have you asked yourself, or been asked, where you're going to use a math course in the future? We lack a fundamental appreciation for the basic goal of basic math courses: To make ourselves comfortable with numbers, and to gain an appreciation, at a really personal level, for how much they impact our lives.
Maybe you'll never need to use precisely what you learn in high school algebra. But then, you'll probably never need to know why the war of 1812 was fought, either. You can get by without both. But ask anyone why they're learning history, and you've got a decent chance of hearing, "because those who don't remember the past are destined to repeat it." I've heard no such similar slogan for mathematics, and that's not math's fault. It's ours.
Those who don't understand numbers in a world that's run by them are destined to flounder.
Imagine if all those people getting tricked into subprime loans had been mathematically literate enough to whip out a pencil and paper when they were presented with the terms of the loan, and figure out that they probably couldn't afford it. Wouldn't have helped everyone (some of them were just too desperate) but I assert that it would have been a good start.
I have to agree with this. Everything from gas prices to the concept of freedom are determined by numbers. If you don't understand the numbers, you cannot make informed disicions for yourself or for the democracy. Numbers (specially in the form of money) govern everything, and you can make someone without understanding of numbers believe Anything you want (or conversely, they can't be made to beleive anything, not even the truth, because they don't understand the numbers). Which, I suppose, is what we're seeing in politics today.
In a way, very few things you learn at school are directly applicable to anything, but I believe language and maths are directly applicable in "everything". Every day you write on forums, read signs, look at your watch, perhaps tip someone, pay your loan, spend money (maybe you have a credit card aswell), etc etc. You should unserstand the power of your wallet and its weaknesses. And many other things. If you were open to this, you'd be less likely to fail math to begin with. If you were good at these things, you'd also be good at general algebra, which is essentially the same. If you are good at algebra, you have every reason to be good at calculus. If you're good at calculus (and numbers in general) you'll have alot more opportunities. Math has always been math, and if you look to the ones passing it, you can see that math isn't the problem. It is just another subject. There were many subjects I hated. Many subjects I still don't see the value in; but that's part of a general education: You don't close any/many doors while you're still young and still don't know which direction you want to go. This is why your first education is so generalized. You could argue thawt you don't *really* need any of it; but then what is left? You could take away math and still function, you could take away history and still function, you could take away religion and still function, you could take away english (the subject, not your tongue) and still function; but imo, only math and language reach a form of objectively global application to everyday lives. And you shouldn't separate algebra from "basic arithmetic operations". If you can't perform algebra, it is because you can't perform the full variety of basic arithmetic operations. Etc, etc. Algebra is simply the 'language' that ties them together, and can allow for a much greater amount of applications if you take slightly more advanced algebraic classes.
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United States24690 Posts
On July 30 2012 00:19 paralleluniverse wrote:Show nested quote +On July 30 2012 00:18 micronesia wrote:On July 30 2012 00:10 paralleluniverse wrote:On July 30 2012 00:06 micronesia wrote:On July 30 2012 00:03 paralleluniverse wrote:On July 29 2012 23:59 farvacola wrote:On July 29 2012 23:53 paralleluniverse wrote:On July 29 2012 23:51 farvacola wrote:
I'm not exactly sure why the article makes the connection between flawed primary/secondary school math education and a need to remove certain educational requirements; it would make sense, at least to me, to instead focus on better teaching methodologies, as algebra and many of the "hard" areas of study have been taught in relatively the same manner for many, many years now. As posters above are explaining, I certainly think that Algebra ought to be requisite, as it lends itself to so many endeavors in life, and perhaps the key in improving its teaching is a more vocational or everyday focus in application, and a change from the standard drudgery of problem sets and timed tests. I'm thinking on my own honors math education, where in 8th grade we learned the quadratic formula via song and were required to sing it at the door one day in order to get into class. Simple little changes in teaching technique can do wonders for making subjects such as math more palatable. As I see it, there are simply too many inherent problems in putting forth entire populations of people who are unable to the basic algebraic underpinnings of the economy and their own personal finances. Are you saying that's good or bad? I am suggesting that such an alternative strategy is a good start when thinking on how math might be taught differently than it is now. My school district happened to have one of the best honors programs in the state of Ohio, but from 4th grade through graduating high school, it was readily apparent to all of the honors kids that we were getting the cream of the districts educational crop while the kids in normal classes fell by the wayside as a result of a less successful and less interesting curriculum standard. Singing math is one of the stupidest ideas I've ever heard. Suppose your students were to learn the quadratic formula. Would you give it to them or make them somehow come up with it? There are arguments for both, depending on things like at what stage in their math career they are at. However, let's suppose you needed to give them the formula and teach them how to use it. They need to memorize the formula (barring a cheat sheet). How are you going to get them to memorize the formula? This is the same as asking how you will get students to memorize anything else (it isn't math specific). You may not like your core subject teachers requiring you to sing a song to learn something, but don't think that this has anything to do with math. I would not gets students to memorize the quadratic formula. I would teach them how to solve quadratic equations. How? Assuming factoring and the like is already covered, and you are going on to problems that require the quadratic formula to be solved. I would also derive the formula. You've lost the majority of students at this point, lol. I like the idea in certain applications, but not all. Remember that we are talking about public school education in the USA in this thread. Once you've solved several quadratic equations you'll naturally remember it without any effort specifically on trying to memorize it. Back to my first question of the current post. Memorizing formulas has zero educational value. Almost completely agree with you. I've read it, and you are assuming that teachers actually have 100% control over what happens in their classes, which they usually don't. Completing the square.
Okay so if I understand correctly, you would cover completing the square prior to the quadratic formula (different from most programs I'm aware of) because it makes it easier to learn rather than just be given the quadratic formula. Can you explain how students will get from completing the square to the quadratic formula?
We also have a similar problem with completing the square. How would you teach it without simply giving them a list of steps (the algorithm)? In theory you can use derivation, but as I said earlier this doesn't work for every student.
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On July 30 2012 00:10 paralleluniverse wrote:Show nested quote +On July 30 2012 00:06 micronesia wrote:On July 30 2012 00:03 paralleluniverse wrote:On July 29 2012 23:59 farvacola wrote:On July 29 2012 23:53 paralleluniverse wrote:On July 29 2012 23:51 farvacola wrote:
I'm not exactly sure why the article makes the connection between flawed primary/secondary school math education and a need to remove certain educational requirements; it would make sense, at least to me, to instead focus on better teaching methodologies, as algebra and many of the "hard" areas of study have been taught in relatively the same manner for many, many years now. As posters above are explaining, I certainly think that Algebra ought to be requisite, as it lends itself to so many endeavors in life, and perhaps the key in improving its teaching is a more vocational or everyday focus in application, and a change from the standard drudgery of problem sets and timed tests. I'm thinking on my own honors math education, where in 8th grade we learned the quadratic formula via song and were required to sing it at the door one day in order to get into class. Simple little changes in teaching technique can do wonders for making subjects such as math more palatable. As I see it, there are simply too many inherent problems in putting forth entire populations of people who are unable to the basic algebraic underpinnings of the economy and their own personal finances. Are you saying that's good or bad? I am suggesting that such an alternative strategy is a good start when thinking on how math might be taught differently than it is now. My school district happened to have one of the best honors programs in the state of Ohio, but from 4th grade through graduating high school, it was readily apparent to all of the honors kids that we were getting the cream of the districts educational crop while the kids in normal classes fell by the wayside as a result of a less successful and less interesting curriculum standard. Singing math is one of the stupidest ideas I've ever heard. Suppose your students were to learn the quadratic formula. Would you give it to them or make them somehow come up with it? There are arguments for both, depending on things like at what stage in their math career they are at. However, let's suppose you needed to give them the formula and teach them how to use it. They need to memorize the formula (barring a cheat sheet). How are you going to get them to memorize the formula? This is the same as asking how you will get students to memorize anything else (it isn't math specific). You may not like your core subject teachers requiring you to sing a song to learn something, but don't think that this has anything to do with math. I would not gets students to memorize the quadratic formula. I would teach them how to solve quadratic equations. I would also derive the formula. Once you've solved several quadratic equations you'll naturally remember it without any effort specifically on trying to memorize it. Memorizing formulas has zero educational value. If you think math is about memorizing formulas, then I would direct you to Lockhart's Lament, which has been linked a few times already: http://www.maa.org/devlin/LockhartsLament.pdf
I don't think thats what Micronesia or I are getting at, that memorization is to be frowned upon in the teaching of mathematics is a hotly debated issue among educators, and I wouldn't want to pass judgement either way as there are good justifications behind both sides of the debate (I only know this via a close friend's recent completion of his high school math teaching certification). If it makes you feel any better, we also had a game called "quad crossfire" in which we went head to head against classmates and had to speed solve a difficult problem on the board in front of the class. There was yelling, anger, and even a fight once, but the point is that the unique approach to math education in this scenario greatly improved the learning experience for all involved.
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On July 29 2012 22:42 paralleluniverse wrote:Another point of view is the idea of comparative advantage from economics, which basically says that it's better for society if everyone specialized in doing what they're good at and traded for everything else. So we should make mathematicians better at mathematics, and mechanics better at fixing cars, and when a mathematician's cars brakes down, it is more efficient for him to call the service of a mechanic than for him to understand how to fix a car and do it himself. Of course, there are some basic knowledge common for most fields, for example mathematicians need to learn to write English, because that's part of being a better mathematician. Advertising executives need to learn basic math and statistics since it's part of the job, etc. But apart from the basic and necessary skills that are required to be proficient in a profession, it's socially optimal for people to specialize. Thus, to the extent that people do not need to know algebra for their jobs, comparative advantage says it's better for them to learn about things that make them better at their jobs instead of algebra. http://en.wikipedia.org/wiki/Comparative_advantage
I think the level of knowledge before it's more beneficial to specialize is set a bit higher than elementary algebra. And besides, if someone is very mathematically inclined but misses out on algebra which he would find super fun, we may miss out on a mathematician.
Let's face it, high school is more of a "do everything, figure out what you want to study at uni", taking such a huge part as algebra away (or well, make it not necessary) is really stupid.
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Please don't use those lists as a merit of how good a school is. What they go after is funding and publications/citaions. It is compeletely broken in favor for america because of 1) The Legacy system that has been built up in american universities over the years (i.e. rich families dumping money in schools) and 2) The way supervisors in america always get cited when their student publishes a paper.
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I agree in theory, that mathematics should not be required to graduate high school.
But if you make math an elective (it obviously can't be cut entirely), where does it end? Most people won't use 95% of what they learned in high school. You'd need to restructure everything and have high school be entirely elective-based, like college is now.
Again, I'm not opposed to this idea in theory, and there are probably a lot of countries who handle their high schools better than the US does, but for us...I don't feel like we have the resources necessary to make it happen. Teachers are underpaid and schools are underfunded. So, teach kids everything and let them use what they want to.
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On July 30 2012 00:22 micronesia wrote:Show nested quote +On July 30 2012 00:19 paralleluniverse wrote:On July 30 2012 00:18 micronesia wrote:On July 30 2012 00:10 paralleluniverse wrote:On July 30 2012 00:06 micronesia wrote:On July 30 2012 00:03 paralleluniverse wrote:On July 29 2012 23:59 farvacola wrote:On July 29 2012 23:53 paralleluniverse wrote:On July 29 2012 23:51 farvacola wrote:
I'm not exactly sure why the article makes the connection between flawed primary/secondary school math education and a need to remove certain educational requirements; it would make sense, at least to me, to instead focus on better teaching methodologies, as algebra and many of the "hard" areas of study have been taught in relatively the same manner for many, many years now. As posters above are explaining, I certainly think that Algebra ought to be requisite, as it lends itself to so many endeavors in life, and perhaps the key in improving its teaching is a more vocational or everyday focus in application, and a change from the standard drudgery of problem sets and timed tests. I'm thinking on my own honors math education, where in 8th grade we learned the quadratic formula via song and were required to sing it at the door one day in order to get into class. Simple little changes in teaching technique can do wonders for making subjects such as math more palatable. As I see it, there are simply too many inherent problems in putting forth entire populations of people who are unable to the basic algebraic underpinnings of the economy and their own personal finances. Are you saying that's good or bad? I am suggesting that such an alternative strategy is a good start when thinking on how math might be taught differently than it is now. My school district happened to have one of the best honors programs in the state of Ohio, but from 4th grade through graduating high school, it was readily apparent to all of the honors kids that we were getting the cream of the districts educational crop while the kids in normal classes fell by the wayside as a result of a less successful and less interesting curriculum standard. Singing math is one of the stupidest ideas I've ever heard. Suppose your students were to learn the quadratic formula. Would you give it to them or make them somehow come up with it? There are arguments for both, depending on things like at what stage in their math career they are at. However, let's suppose you needed to give them the formula and teach them how to use it. They need to memorize the formula (barring a cheat sheet). How are you going to get them to memorize the formula? This is the same as asking how you will get students to memorize anything else (it isn't math specific). You may not like your core subject teachers requiring you to sing a song to learn something, but don't think that this has anything to do with math. I would not gets students to memorize the quadratic formula. I would teach them how to solve quadratic equations. How? Assuming factoring and the like is already covered, and you are going on to problems that require the quadratic formula to be solved. I would also derive the formula. You've lost the majority of students at this point, lol. I like the idea in certain applications, but not all. Remember that we are talking about public school education in the USA in this thread. Once you've solved several quadratic equations you'll naturally remember it without any effort specifically on trying to memorize it. Back to my first question of the current post. Memorizing formulas has zero educational value. Almost completely agree with you. I've read it, and you are assuming that teachers actually have 100% control over what happens in their classes, which they usually don't. Completing the square. Okay so if I understand correctly, you would cover completing the square prior to the quadratic formula (different from most programs I'm aware of) because it makes it easier to learn rather than just be given the quadratic formula. Can you explain how students will get from completing the square to the quadratic formula? We also have a similar problem with completing the square. How would you teach it without simply giving them a list of steps (the algorithm)? In theory you can use derivation, but as I said earlier this doesn't work for every student.
A good plan would go like this:
1) How should we solve (x+1)^2 - 4 = 0?
2) Now that we've established that 1) is really easy to solve how do we solve x^2 + 9x - 1 = 0
3) The problem reduces to writing 2) in a form like 1).
4) We work out how to do that, and hence solve 2).
5) We apply this to solve ax^2 + bx + c = 0, hence arriving at the quadratic equation.
6) Then I would remark that the procedure in 3) and 4) can be used to solve most quadratics. Alternatively, the formula in 5) can be used.
7) Now ask why this method fails for some quadratics like (x+1)^2 + 4 = 0?
At no point would I impress upon my students all the wonderful applications of the quadratic equation, because there seriously are none. I would present this as a neat math trick.
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On July 30 2012 00:22 micronesia wrote:Show nested quote +On July 30 2012 00:19 paralleluniverse wrote:On July 30 2012 00:18 micronesia wrote:On July 30 2012 00:10 paralleluniverse wrote:On July 30 2012 00:06 micronesia wrote:On July 30 2012 00:03 paralleluniverse wrote:On July 29 2012 23:59 farvacola wrote:On July 29 2012 23:53 paralleluniverse wrote:On July 29 2012 23:51 farvacola wrote:
I'm not exactly sure why the article makes the connection between flawed primary/secondary school math education and a need to remove certain educational requirements; it would make sense, at least to me, to instead focus on better teaching methodologies, as algebra and many of the "hard" areas of study have been taught in relatively the same manner for many, many years now. As posters above are explaining, I certainly think that Algebra ought to be requisite, as it lends itself to so many endeavors in life, and perhaps the key in improving its teaching is a more vocational or everyday focus in application, and a change from the standard drudgery of problem sets and timed tests. I'm thinking on my own honors math education, where in 8th grade we learned the quadratic formula via song and were required to sing it at the door one day in order to get into class. Simple little changes in teaching technique can do wonders for making subjects such as math more palatable. As I see it, there are simply too many inherent problems in putting forth entire populations of people who are unable to the basic algebraic underpinnings of the economy and their own personal finances. Are you saying that's good or bad? I am suggesting that such an alternative strategy is a good start when thinking on how math might be taught differently than it is now. My school district happened to have one of the best honors programs in the state of Ohio, but from 4th grade through graduating high school, it was readily apparent to all of the honors kids that we were getting the cream of the districts educational crop while the kids in normal classes fell by the wayside as a result of a less successful and less interesting curriculum standard. Singing math is one of the stupidest ideas I've ever heard. Suppose your students were to learn the quadratic formula. Would you give it to them or make them somehow come up with it? There are arguments for both, depending on things like at what stage in their math career they are at. However, let's suppose you needed to give them the formula and teach them how to use it. They need to memorize the formula (barring a cheat sheet). How are you going to get them to memorize the formula? This is the same as asking how you will get students to memorize anything else (it isn't math specific). You may not like your core subject teachers requiring you to sing a song to learn something, but don't think that this has anything to do with math. I would not gets students to memorize the quadratic formula. I would teach them how to solve quadratic equations. How? Assuming factoring and the like is already covered, and you are going on to problems that require the quadratic formula to be solved. I would also derive the formula. You've lost the majority of students at this point, lol. I like the idea in certain applications, but not all. Remember that we are talking about public school education in the USA in this thread. Once you've solved several quadratic equations you'll naturally remember it without any effort specifically on trying to memorize it. Back to my first question of the current post. Memorizing formulas has zero educational value. Almost completely agree with you. I've read it, and you are assuming that teachers actually have 100% control over what happens in their classes, which they usually don't. Completing the square. Okay so if I understand correctly, you would cover completing the square prior to the quadratic formula (different from most programs I'm aware of) because it makes it easier to learn rather than just be given the quadratic formula. Can you explain how students will get from completing the square to the quadratic formula? We also have a similar problem with completing the square. How would you teach it without simply giving them a list of steps (the algorithm)? In theory you can use derivation, but as I said earlier this doesn't work for every student.
I would do that, although I'm not the guy you're responding to. If you're at the point where you're learning about quadratics, you know how to factor, and you know how to solve equations that can be factored. When as a class we ran into something that didn't factor, I would point out "well wait a minute, this ALMOST factors, if there were one extra term over here it would. What if we just put that extra term over there... Can we do that? No, of course not, only if we put it on the other side of the equation as well." BAM. You elicit one or two responses from students, and all of a sudden the entire class can "factor" equations that can't be factored, and they've learned completing the square without an algorithm to be memorized. Once we use it a few times, we do it to ax^2+bx+c=0 to find the roots of any quadratic, and we've learned not only why it's possible for the quadratic formula to exist, but also what it is and why it's pretty neat.
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On July 30 2012 00:29 paralleluniverse wrote:Show nested quote +On July 30 2012 00:22 micronesia wrote:On July 30 2012 00:19 paralleluniverse wrote:On July 30 2012 00:18 micronesia wrote:On July 30 2012 00:10 paralleluniverse wrote:On July 30 2012 00:06 micronesia wrote:On July 30 2012 00:03 paralleluniverse wrote:On July 29 2012 23:59 farvacola wrote:On July 29 2012 23:53 paralleluniverse wrote:On July 29 2012 23:51 farvacola wrote:
I'm not exactly sure why the article makes the connection between flawed primary/secondary school math education and a need to remove certain educational requirements; it would make sense, at least to me, to instead focus on better teaching methodologies, as algebra and many of the "hard" areas of study have been taught in relatively the same manner for many, many years now. As posters above are explaining, I certainly think that Algebra ought to be requisite, as it lends itself to so many endeavors in life, and perhaps the key in improving its teaching is a more vocational or everyday focus in application, and a change from the standard drudgery of problem sets and timed tests. I'm thinking on my own honors math education, where in 8th grade we learned the quadratic formula via song and were required to sing it at the door one day in order to get into class. Simple little changes in teaching technique can do wonders for making subjects such as math more palatable. As I see it, there are simply too many inherent problems in putting forth entire populations of people who are unable to the basic algebraic underpinnings of the economy and their own personal finances. Are you saying that's good or bad? I am suggesting that such an alternative strategy is a good start when thinking on how math might be taught differently than it is now. My school district happened to have one of the best honors programs in the state of Ohio, but from 4th grade through graduating high school, it was readily apparent to all of the honors kids that we were getting the cream of the districts educational crop while the kids in normal classes fell by the wayside as a result of a less successful and less interesting curriculum standard. Singing math is one of the stupidest ideas I've ever heard. Suppose your students were to learn the quadratic formula. Would you give it to them or make them somehow come up with it? There are arguments for both, depending on things like at what stage in their math career they are at. However, let's suppose you needed to give them the formula and teach them how to use it. They need to memorize the formula (barring a cheat sheet). How are you going to get them to memorize the formula? This is the same as asking how you will get students to memorize anything else (it isn't math specific). You may not like your core subject teachers requiring you to sing a song to learn something, but don't think that this has anything to do with math. I would not gets students to memorize the quadratic formula. I would teach them how to solve quadratic equations. How? Assuming factoring and the like is already covered, and you are going on to problems that require the quadratic formula to be solved. I would also derive the formula. You've lost the majority of students at this point, lol. I like the idea in certain applications, but not all. Remember that we are talking about public school education in the USA in this thread. Once you've solved several quadratic equations you'll naturally remember it without any effort specifically on trying to memorize it. Back to my first question of the current post. Memorizing formulas has zero educational value. Almost completely agree with you. I've read it, and you are assuming that teachers actually have 100% control over what happens in their classes, which they usually don't. Completing the square. Okay so if I understand correctly, you would cover completing the square prior to the quadratic formula (different from most programs I'm aware of) because it makes it easier to learn rather than just be given the quadratic formula. Can you explain how students will get from completing the square to the quadratic formula? We also have a similar problem with completing the square. How would you teach it without simply giving them a list of steps (the algorithm)? In theory you can use derivation, but as I said earlier this doesn't work for every student. A good plan would go like this: 1) How should we solve (x+1)^2 + 4 = 0? 2) Now that we've established that 1) is really easy to solve how do we solve x^2 + 9x - 1 = 0 3) The problem reduces to writing 2) in a form like 1). 4) We work out how to do that, and hence solve 2). 5) We apply this to solve ax^2 + bx + c = 0, hence arriving at the quadratic equation. At no point would I impress upon my students all the wonderful applications of the quadratic equation, because there seriously are none. I would present this as a neat math trick.
There are none? Math is a tool, tools have applications. Trust me. But you don't need to present it in any other way than simply a tool.
No other high-school subjects require teachers to tell of applications. In some cases they do. Oh you can write a history book, Oh you can create a battery. But these are not applications a typical teenager is looking for. I guess you could claim that with the quadratic forumula you can create technology. You can create the plans needed to create any building, any piece of technology, any weapon, do anything you want in the field of economics. Anything you can fathom. Create rockets and spaceships. But first you will need this formula. This tool. And many others like it.
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His main argument is that algebra is difficult, the whole reasoning is ridiculous. Young people don't necessarily know what they'll want to be doing for the rest of their lives. Removing options by making crucial subjects optional, or not allowing them to try and decide whether they like it is a bad idea. That said, there are plenty of problems with the way math is taught. High school taught me it was difficult and frustrating. Uni taught me it's beautiful and interesting.
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America trying to dumb itself down...
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A social scientist talking about having insufficient evidence and experimentation to back something up. How hypocritical.
If students in high school or college are having a difficult time in an Algebra class (OR any level of mathematics) its simply because their prerequisites are not strong enough. This problem can be alleviated by going back and mastering the prerequisites. These students also most likely have a study regiment that is not based on understanding, and modifying that regiment to induce understanding while studying will help that student excel in whichever topic they choose.
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On July 29 2012 15:16 Integra wrote:
Congrats to the author to finally discover that math, like any other subject, have areas which it cannot be applied too and thus has no real value for. What brilliance!
Agreed, just like nearly any subject one learns in high school. Did I need to learn U.S. or European history, or biology? No, but plenty of other people probably did, and knowing a multitude of different academic subjects gave me a wider variety of options when pursuing jobs and majors in higher education. I like being a well-rounded individual who was presented with a wide variety of subjects early on. Not everyone knows what they're going to grow up to be by age ten.
Some people won't use algebra (and certainly not trigonometry), but many people will. Just like any other course you've ever taken. Strange to pick out mathematics in particular though, as it's the language of science, and the United States is quite behind many countries in math and science. We're not #1 so we don't need it? Psh. This article in the OP is just as valid substituting any other course for algebra.
I have a bachelor's in math, a master's in math education, and I'm pursuing my PhD in math education. I plan on teaching math at the high school and university levels, and I've been tutoring all levels of math, from elementary to college (including standardized tests) for many years now. If I wasn't on vacation right now I'd be all over this thread, but sadly, I'm still away for a few more days. Feel free to shoot me a PM if you'd like to reply to this, as I'll most likely miss your thread responses x.x
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On July 30 2012 00:32 Cutlery wrote:Show nested quote +On July 30 2012 00:29 paralleluniverse wrote:On July 30 2012 00:22 micronesia wrote:On July 30 2012 00:19 paralleluniverse wrote:On July 30 2012 00:18 micronesia wrote:On July 30 2012 00:10 paralleluniverse wrote:On July 30 2012 00:06 micronesia wrote:On July 30 2012 00:03 paralleluniverse wrote:On July 29 2012 23:59 farvacola wrote:On July 29 2012 23:53 paralleluniverse wrote:
[quote]
Are you saying that's good or bad? I am suggesting that such an alternative strategy is a good start when thinking on how math might be taught differently than it is now. My school district happened to have one of the best honors programs in the state of Ohio, but from 4th grade through graduating high school, it was readily apparent to all of the honors kids that we were getting the cream of the districts educational crop while the kids in normal classes fell by the wayside as a result of a less successful and less interesting curriculum standard. Singing math is one of the stupidest ideas I've ever heard. Suppose your students were to learn the quadratic formula. Would you give it to them or make them somehow come up with it? There are arguments for both, depending on things like at what stage in their math career they are at. However, let's suppose you needed to give them the formula and teach them how to use it. They need to memorize the formula (barring a cheat sheet). How are you going to get them to memorize the formula? This is the same as asking how you will get students to memorize anything else (it isn't math specific). You may not like your core subject teachers requiring you to sing a song to learn something, but don't think that this has anything to do with math. I would not gets students to memorize the quadratic formula. I would teach them how to solve quadratic equations. How? Assuming factoring and the like is already covered, and you are going on to problems that require the quadratic formula to be solved. I would also derive the formula. You've lost the majority of students at this point, lol. I like the idea in certain applications, but not all. Remember that we are talking about public school education in the USA in this thread. Once you've solved several quadratic equations you'll naturally remember it without any effort specifically on trying to memorize it. Back to my first question of the current post. Memorizing formulas has zero educational value. Almost completely agree with you. I've read it, and you are assuming that teachers actually have 100% control over what happens in their classes, which they usually don't. Completing the square. Okay so if I understand correctly, you would cover completing the square prior to the quadratic formula (different from most programs I'm aware of) because it makes it easier to learn rather than just be given the quadratic formula. Can you explain how students will get from completing the square to the quadratic formula? We also have a similar problem with completing the square. How would you teach it without simply giving them a list of steps (the algorithm)? In theory you can use derivation, but as I said earlier this doesn't work for every student. A good plan would go like this: 1) How should we solve (x+1)^2 + 4 = 0? 2) Now that we've established that 1) is really easy to solve how do we solve x^2 + 9x - 1 = 0 3) The problem reduces to writing 2) in a form like 1). 4) We work out how to do that, and hence solve 2). 5) We apply this to solve ax^2 + bx + c = 0, hence arriving at the quadratic equation. At no point would I impress upon my students all the wonderful applications of the quadratic equation, because there seriously are none. I would present this as a neat math trick. There are none? Math is a tool, tools have applications. Trust me. But you don't need to present it in any other way than simply a tool.
I can think of many applications of math. But I cannot think of any applications of the quadratic equation that isn't contrived.
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EPT
