|
On July 29 2012 16:59 ]343[ wrote:Show nested quote +On July 29 2012 16:56 thesideshow wrote:On July 29 2012 16:48 ]343[ wrote:
But an actual mathematician is concerned about maps between objects and universal properties rather than explicit constructions (that is, how an object behaves, not how it's written; the written convention is merely for communication's sake), so he reasonably would not see a test of "knowledge of convention" as a test of mathematical aptitude.
Knowledge of convention allows you to read and learn from what others have done. Also, its pretty much required for any form of work or publication, where it needs to be checked or certified or whatever. I would say it's reasonable to accept a "test of knowledge of convention" as a test of mathematical aptitude for any practical purpose. Hmm, let me try to reword that then. Testing knowledge of convention is a test of memory, not a test of mathematical reasoning ability (which is what mathematicians value).
Anyway, more on topic: this (admittedly over-referenced) article has much to say on this issue. The problem isn't that algebra is unnecessary, but that the way it's taught in the US turns people off.
Thanks for your articulate post. The link to your article isn't working.
|
Alright. I'm going to assume that algebra taught in the USA is not significantly different than the algebra up here in BC
http://www.bced.gov.bc.ca/irp/pdfs/mathematics/2008math89.pdf
page 16. Between grades 8 and 9, you're expected to learn simple problems involving variables (5x+3 = ?, x = 4) and linear equations, get introduced to higher level polynomials and similar things, and only additon/subtraction of polynomials and the concept of equality on both sides. If you have trouble with this, it isn't algebra's fault. It's your parent's/previous teacher's faults for letting you get out of elementary school with poor fundamentals.
http://www.bced.gov.bc.ca/irp/pdfs/mathematics/2006prinofmath1012.pdf
In grade 10, you're introduced to the concept of factoring polynomials, multiplication and division of polynomials.
That's all there is to basic algebra. You don't do anything more than basic addition/subtraction/multiplication/division and other manipulation of algebraic expressions until grade 11 math which isn't required to graduate. Assuming that the USA requires nothing more difficult than this, if you have two extra years to learn how to do algebra and you still can't, you're fucking retarded. Yes algebra is necessary and no it should not be excluded. If you can't learn it, it isn't algebra's fault.
|
United States10328 Posts
On July 29 2012 16:58 TheRabidDeer wrote:Show nested quote +On July 29 2012 16:55 ]343[ wrote:On July 29 2012 16:35 TheRabidDeer wrote:On July 29 2012 16:30 paralleluniverse wrote:
What order you solve an equation in is irrelevant.
Consider 2x+1 = 0, under the convention that addition happens before multiplication.
So we want to solve 2(x+1)=0, you can divide by 2 and subtract 1 from both sides to get x=-1. Or you can expand to get 2x+2=0, subtract 2 and divide by 2 to get x=-1. (9 + 3)^2 If you dont use pemdas you could come to the conclusion of 9^2 + 3^2, which is entirely different than 12^2. You could foil it out, but foil uses the same principles of pemdas in that you must know it to use it. Also, if you actually plug a number into your equation you could arrive at the conclusion of: let x = 1 2(1 + 1) = 4 or 2(1) + 1 = 3 How do you know where the brackets go? On a side note: statements like "foil it out" are good examples of how the American math system teaches formulae and algorithms, not mathematics. The american math system has far worse problems than that. All throughout learning math, we are told things are impossible. Then a year later, we learn how to solve what was previously thought to be impossible. I am curious though, how would you teach somebody to do (9+3)(9+3) without telling them about foil? I only ever learned it because of that, and would like another perspective.
The distributive property is certainly necessary for expanding products of multinomials. FOIL is a decent, but over-emphasized way of teaching students how to distribute, though: it's not obviously generalizable to, say, trinomials (what's "inner" now?), and it makes students rely far too much on a mnemonic, which points to memorization rather than understanding.
Instead, schools could, for example, introduce the distributive property geometrically: a rectangle with dimensions (a+b) and (c+d) can be split into four rectangles with dimensions ac, bc, ad, bd. I think this gives a better intuition than simply memorizing "first, outer, inner, last."
On July 29 2012 17:07 paralleluniverse wrote:
The link to your article isn't working.
Oops, fixed.
|
Schools job is to build children to proper parts of the society. Big part of that is of course preparing you for a job, BUT a part of that is also teaching relevant things to your society. History for example: "Those who don't know history are doomed to repeat it". Logic and critical thinking should taught in school too. Why? In my opinion one of the biggest problems in modern society is peoples inability to utilize their brains properly. Learning algebra/math helps you in learning how to think better.
|
As someone who is a math major, I cannot understand what's so hard about algebra. It seems so natural and very logical.
You are given a set of rules to follow. If you follow the rules then you will do everything correctly.
i've taken math classes(graduate algebra) where things are not so natural and trivial. So I do have a sense of not understanding something. But this is simple algebra + - division and graphing polynomial, exponential functions. What is so hard about it?!
|
On July 29 2012 17:10 toopham wrote:
As someone who is a math major, I cannot understand what's so hard about algebra. It seems so natural and very logical.
You are given a set of rules to follow. If you follow the rules then you will do everything correctly.
i've taken math classes(graduate algebra) where things are not so natural and trivial. So I do have a sense of not understanding something. But this is simple algebra + - division and graphing polynomial, exponential functions. What is so hard about it?!
Using brain is like using your muscles: don't expect world record breaking achievements with zero training.
|
I had to learn the multiple proper techniques for throwing a frisbee in Physical Education in high school; some people found it very difficult and it is NOT useful in life.
But I don't really care.
Algebra is definitely more applicable than how to throw a "hammer" or "thumber."
|
On July 29 2012 16:56 andrewlt wrote:Show nested quote +On July 29 2012 16:40 UrsusRex wrote:
Not one person has given a compelling reason how making algebra mandatory improves critical thinking skills. All of you supporting and condeming it are missing the basic problem. The entire world teaches algebra to their students but nowhere has it ever been shown to improve the quality of the people who learn it. All of you talking about tools and learning skills and resonating knowledge do not one shred of evidence for your position beyond asserting it as fact repeatedly. Show me any data than doesn't even imply, just correlate thats all I ask, any data that would link studying algebra to improving learning skills, because if it doesn't do that, we are teaching an irrelevant subject to millions of people. Take your high school subjects and rank them from most useful to least useful to the average student. I can guarantee you that algebra is one of the most useful. You don't need history. Your science classes have even more niche uses than your algebra classes. You don't need literature. What's left of high school education?
You can't even progress further in other subjects without algebra (i.e. physics, programming and such).
It would also make the following years and years of school completely pointless, because you need algebra throughout your whole school life, unless you're planting trees in a tree nursery. Do I need it now at age 30? Fuck no. I haven't had to deal with it seriously since my last test at the university, but other things I have learned through the basics of algebra allowed me to go further in many other areas.
If I'd strip all the knowledge I attained in school away, that I do not need now in my life right now ... hell that would have been quite a lot less years of education, but I would probably have had NO WAY of going to university, no matter the field of study without knowing the basics for anything and starting with something incredibly huge like math from zero is pretty fucked up. There's a reason why mathematics follow you throughout ALL of your school life, every single year. That's not the case with every subject you have in school.
|
Also, math is fun, if you don't like it you suck.
|
On July 29 2012 17:03 Silidons wrote:society likes to remember each and every aspect of their favorite celeb or sports team. doesn't surprise me much. Show nested quote +On July 29 2012 17:01 TheRabidDeer wrote:On July 29 2012 16:59 Eufouria wrote:On July 29 2012 16:35 TheRabidDeer wrote:On July 29 2012 16:30 paralleluniverse wrote:
What order you solve an equation in is irrelevant.
Consider 2x+1 = 0, under the convention that addition happens before multiplication.
So we want to solve 2(x+1)=0, you can divide by 2 and subtract 1 from both sides to get x=-1. Or you can expand to get 2x+2=0, subtract 2 and divide by 2 to get x=-1. (9 + 3)^2 If you dont use pemdas you could come to the conclusion of 9^2 + 3^2, which is entirely different than 12^2. You could foil it out, but foil uses the same principles of pemdas in that you must know it to use it. Also, if you actually plug a number into your equation you could arrive at the conclusion of: let x = 1 2(1 + 1) = 4 or 2(1) + 1 = 3 How do you know where the brackets go? You could only come to the conclusion that (9+3)^2 =9^2+3^2 if you don't know how to expand brackets properly, thats not a problem with order its just wrong. How did you learn to expand brackets properly? Is that a "natural logic"? No, its something you learned. everything was thought of at one point, not learned
"Everything" was thought of by brilliant people who've expanded upon what they learned... To excuse the process of learning by saying that everything was thought of at one point, is absolutely ludicrous.
|
On July 29 2012 17:08 ]343[ wrote:Show nested quote +On July 29 2012 16:58 TheRabidDeer wrote:On July 29 2012 16:55 ]343[ wrote:On July 29 2012 16:35 TheRabidDeer wrote:On July 29 2012 16:30 paralleluniverse wrote:
What order you solve an equation in is irrelevant.
Consider 2x+1 = 0, under the convention that addition happens before multiplication.
So we want to solve 2(x+1)=0, you can divide by 2 and subtract 1 from both sides to get x=-1. Or you can expand to get 2x+2=0, subtract 2 and divide by 2 to get x=-1. (9 + 3)^2 If you dont use pemdas you could come to the conclusion of 9^2 + 3^2, which is entirely different than 12^2. You could foil it out, but foil uses the same principles of pemdas in that you must know it to use it. Also, if you actually plug a number into your equation you could arrive at the conclusion of: let x = 1 2(1 + 1) = 4 or 2(1) + 1 = 3 How do you know where the brackets go? On a side note: statements like "foil it out" are good examples of how the American math system teaches formulae and algorithms, not mathematics. The american math system has far worse problems than that. All throughout learning math, we are told things are impossible. Then a year later, we learn how to solve what was previously thought to be impossible. I am curious though, how would you teach somebody to do (9+3)(9+3) without telling them about foil? I only ever learned it because of that, and would like another perspective. The distributive property is certainly necessary for expanding products of multinomials. FOIL is a decent, but over-emphasized way of teaching students how to distribute, though: it's not obviously generalizable to, say, trinomials (what's "inner" now?), and it makes students rely far too much on a mnemonic, which points to memorization rather than understanding. Instead, schools could, for example, introduce the distributive property geometrically: a rectangle with dimensions (a+b) and (c+d) can be split into four rectangles with dimensions ac, bc, ad, bd. I think this gives a better intuition than simply memorizing "first, outer, inner, last." Show nested quote +On July 29 2012 17:07 paralleluniverse wrote:
The link to your article isn't working. Oops, fixed.
True, I had completely forgotten about trinomials (its been ages since I have had to examine math practices, and most of what I know is just in my mind on how to do).
Though I am honestly not sure if students would grasp it much better with the rectangle. I know just as many people that struggled with a lot of geometry... and it doesnt help that geometry is taught after algebra.
I still want to restructure all of our school curriculums because things seem to be taught in a horrible order that makes very little practical sense for learning. Education starts off fine, but it slips away quickly.
Why do we learn algebra separate from geometry, for example? Why cant we incorporate multiple subjects together?
|
Of course algebra is necessary, who would ever become a scientist in the physics, mathematics, chemisty etc. department if they have never learned algebra? I guess very few people would still try to study one of these subjects as the knowledge/skill gap from school to university would become way too large overcome it. Or Universities would have to begin their lessons with "A function is ..."
Who would ever want future generations of engineers to learn that instead of the usual Technical Mechanics or something like that where they need algebra?
|
On July 29 2012 17:08 ]343[ wrote:Show nested quote +On July 29 2012 16:58 TheRabidDeer wrote:On July 29 2012 16:55 ]343[ wrote:On July 29 2012 16:35 TheRabidDeer wrote:On July 29 2012 16:30 paralleluniverse wrote:
What order you solve an equation in is irrelevant.
Consider 2x+1 = 0, under the convention that addition happens before multiplication.
So we want to solve 2(x+1)=0, you can divide by 2 and subtract 1 from both sides to get x=-1. Or you can expand to get 2x+2=0, subtract 2 and divide by 2 to get x=-1. (9 + 3)^2 If you dont use pemdas you could come to the conclusion of 9^2 + 3^2, which is entirely different than 12^2. You could foil it out, but foil uses the same principles of pemdas in that you must know it to use it. Also, if you actually plug a number into your equation you could arrive at the conclusion of: let x = 1 2(1 + 1) = 4 or 2(1) + 1 = 3 How do you know where the brackets go? On a side note: statements like "foil it out" are good examples of how the American math system teaches formulae and algorithms, not mathematics. The american math system has far worse problems than that. All throughout learning math, we are told things are impossible. Then a year later, we learn how to solve what was previously thought to be impossible. I am curious though, how would you teach somebody to do (9+3)(9+3) without telling them about foil? I only ever learned it because of that, and would like another perspective. The distributive property is certainly necessary for expanding products of multinomials. FOIL is a decent, but over-emphasized way of teaching students how to distribute, though: it's not obviously generalizable to, say, trinomials (what's "inner" now?), and it makes students rely far too much on a mnemonic, which points to memorization rather than understanding. Instead, schools could, for example, introduce the distributive property geometrically: a rectangle with dimensions (a+b) and (c+d) can be split into four rectangles with dimensions ac, bc, ad, bd. I think this gives a better intuition than simply memorizing "first, outer, inner, last." Show nested quote +On July 29 2012 17:07 paralleluniverse wrote:
The link to your article isn't working. Oops, fixed.
The distributive property is also in the definition of a field or a vector space. So, in a sense, you can't argue with it in the same way you can argue against using the order of operations.
Basically, the distributive property isn't merely just notation, it's an actual property of numbers and cannot be changed without changing the concept you're talking about.
|
Math was definately the hardest subject for me in school.
|
Yes, even more harping on the way math is taught being the source of the problem.
Something my old math teacher used to say was: Mathematics is the language of science and nature, it's far more universal than English or German or 'Languages'. It should be thought with the same priority then! Even more actually. If you do 2 units of sciences, 2 units of language etc. ... but 1 unit of math, [especially when you can't do anything interesting in the sciences because that relies on Math] --> ??? You should study the subjects fairly.
Most European systems (at least the French and Swiss) value math more, as I believe they should.
Back to my point of how high-school math sucks. ITs really unfortunate, because it turns people off from studying it, and that also directly limits their 'scientist' throughput. Haha find all the kids that want to do business, then tell them their economics classes, sadly, actually need to do something technical like basic optimization equations. And even those aren't that inspiring.
Solution:
Teach History of Maths topics at high-school level. Its really great to see the development of mathematical thought over time, because you can watch the progression of human thought quantitatively, and see 'where the stuff comes from' which hopefully gives a better grasp of 'what this stuff ~means~'. Also, with History of Maths courses, you don't need to go as far in terms of technical difficulty. But I believe it makes so much more sense to explain the relation between cartesian and polar coordinates, if you introduce it like the founders of those topics discovered it. That gives it a human story - useful for sciences, because you can see when people went 'wrong' and learn to identify that.
Yes there is the drudgery of (even high-school level) of basic complex number algebra and doing problem sets upon problem set of integrals (hmm, I may be stepping beyond US high-school level, \: ), but you really must have that familiarity and grasp of the skill to be able to do fun stuff! I see that now, its beautiful. Lol that doesn't make my summer reading any easier :E
|
On July 29 2012 17:18 KuKri wrote:
Of course algebra is necessary, who would ever become a scientist in the physics, mathematics, chemisty etc. department if they have never learned algebra? I guess very few people would still try to study one of these subjects as the knowledge/skill gap from school to university would become way too large overcome it. Or Universities would have to begin their lessons with "A function is ..."
Who would ever want future generations of engineers to learn that instead of the usual Technical Mechanics or something like that where they need algebra?
No one is saying scrap mathematics. People who are inclined towards physics and mathematics would still be able to do it. But why would a social worker, a radio host or a fitness coach need to know algebra?
And they still need to redefined a function in university anyway, because the high school definition isn't abstract enough. For example, in high school (at least for me), functions were defined as a rule, a domain and a range. In university, the proper definition involves a rule, a domain and a codomain. And the change from a range to a codomain is quite important.
|
Italy12246 Posts
The whole point of high school isn't necessarily to teach you things that you will actively use in life, because you learn things so generically and superficially it can't really be useful. The point of high school is to give you a general culture on a variety of subjects; you can then go deeper into them in college if you are interested, which is when you actually learn what you will use. High school kids are supposed to learn how to reason on different things, not how to solve real life math problems for example.
Also it's fucking high school as long as you do your assignements you will be fine. I have been giving private lessons to high school students struggling in english, math and physics, and each and every time it was simply because they just wouldn't study.
I do agree that math is taught like shit in high school though.
|
On July 29 2012 17:10 toopham wrote:
As someone who is a math major, I cannot understand what's so hard about algebra. It seems so natural and very logical.
You are given a set of rules to follow. If you follow the rules then you will do everything correctly.
You're a maths major, you take it as a given.
That said, it isn't, but I honestly believe that the problem is more mental than anything else. Given the aversion to maths of most people, and the oft repeated mantra of "maths is complicated, maths is hard", it's not hard to see why kids develop such an aversion to it. Making maths "go away" isn't the solution. Getting teachers to properly teach maths at a young age and removing that "Oh maths is hard, I can't do it, I'm not good with numbers" fear is crucial.
Making basic maths non-compulsory or elective, and especially algebra, is a great way to screw up an entire generation. Kids don't make the best choices when it comes to cake now or cake later, and having parents unconcerned with education don't help matters either.
|
I have a bit of issues with the article.
1. "It’s true that students in Finland, South Korea and Canada score better on mathematics tests. But it’s their perseverance, not their classroom algebra, that fits them for demanding jobs." I'd say that's debatable. While perseverance helps with jobs, so do many many other factors, such as mathematical skills. Like all skills, the amount that's beneficial or required will vary on a job by job basis.
2. the article spends a truckload of time talking about how people are struggling too much with algebra. First of all, if other countries are doing it right, maybe the problem is with the way it's being taught, not the fact it's being taught.
I had rather serious problems with long division, fractions, multiplication, and other basic math when I was young — does that mean we shouldn't learn any of that either of many people have the same problem? I could only assume that such a problem was very serious and common 100-200 years ago (or earlier or later), but we get past it and progress.
After basic math though, I excelled very much across most of the remaining math, especially basic algebra. I think it had a fair bit to do with a different environment. I was playing educational computer games, as well as had less distractions such as chatting/fooling around with classmates (since all my best friends weren't with me). Aside from that I also obviously wasn't drinking soda or junk food all the time, which I think would play a large factor in education.
3. It doesn't even address this issue much, which really shows off how biased and/or shallow the article is, but something that's important to discuss is how much algebra/advanced math should be taught? (and how much shouldn't) sure I could see having cubic and greater polynomial functions, exponential functions, quadratic regression, lots of the trigonometric identities and laws, as well as linear algebra/matrices, sets, or calculus as obviously more optional (although I am quite sure some of that already is), but basic elementary algebra is very important, and not that difficult of a concept.
On July 29 2012 17:21 bITt.mAN wrote:
Solution:
Teach History of Maths topics at high-school level. Its really great to see the development of mathematical thought over time, because you can watch the progression of human thought quantitatively, and see 'where the stuff comes from' which hopefully gives a better grasp of 'what this stuff ~means~'. Also, with History of Maths courses, you don't need to go as far in terms of technical difficulty. But I believe it makes so much more sense to explain the relation between cartesian and polar coordinates, if you introduce it like the founders of those topics discovered it. That gives it a human story - useful for sciences, because you can see when people went 'wrong' and learn to identify that.
Yes there is the drudgery of (even high-school level) of basic complex number algebra and doing problem sets upon problem set of integrals (hmm, I may be stepping beyond US high-school level, \: ), but you really must have that familiarity and grasp of the skill to be able to do fun stuff! I see that now, its beautiful. Lol that doesn't make my summer reading any easier :E
Sound's like a pretty good idea. I know when chemistry or biology is taught the brief brief history was also done (not-so-much physics) for me, but not math, and I think it would be useful, and certainly more slow and clear for people to learn.
But why would a social worker, a radio host or a fitness coach need to know algebra? because algebra use is part of life skills — dealing with money, time, ratios, etc. I guess with computers and credit cards people kinda need to know this far less, but it makes for a dangerous dependency (just like if everyone had machines to read for us and people didn't learn how to read; a poor comparison, but still similar)
|
On July 29 2012 17:13 RageBot wrote:
Also, math is fun, if you don't like it you suck.
+1111111
User was warned for this post
|
|
|
|
|
|
EPT
