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The Declining Quality of Mathematics Education in the US

By Coryoth in Op-Ed
Fri Jan 26, 2007 at 07:02:59 AM EST
Tags: math, mathematics, education, science, usa (all tags)
Science

Mathematics education seems to be very subject to passing trends - surprisingly more so than many other subjects. The most notorious are, of course, the rise of New Math in the 60s and 70s, and the corresponding backlash against it in the late 70s and 80s. It turns out that mathematics education, at least in the US, is now subject to a new trend, and it doesn't appear to be a good one.


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To be fair the current driving trend in mathematics education is largely an extension of an existing trend in education generally. The idea is that we need to cater more to the students to better engage them in the material. There is a focus on making things fun, on discovery, on group work, and on making things "relevant to the student". These are often noble goals, and it is something that, in the past, education schemes have often lacked. There is definitely such a thing as "too much of a good thing" with regard to these aims, and as far as I can tell that point was passed some time ago in the case of mathematics.

A couple of prime examples, in terms of textbooks and material for instructors, are brought up and suitably lampooned in a YouTube video by a Washington state weather presenter who encountered, and was appalled by, these particular teaching programs. The material in question is the TERC Investigations "Investigations in Number, Data, and Space", and the University of Chicago School Mathematics Project "Everyday Mathematics". The focus of the YouTube video is on these math programs complete aversion to teaching students the classic methods for performing multi-digit multiplication and division. Indeed, these programs not only fail to teach such a method, they go so far as to actively discourage the method ever being taught, preferring that students didn't learn it outside class either. What sort of methods do they teach? Well, for example, to solve the problem 26×31, a student might use the following approach: we can write 26×31 as 20×31 + 5×31 + 1×31 since 20+5+1=26; Now we know that 10×31=310, and 20×31 should be twice that (620) and 5×31 should be half that (155); so the solution is 620+155+31=806. Note that the student could break the problem up differently, and thus there is no single approach that consistently works on all problems; each new multiplication is an entirely new problem. To be fair the methods they do teach, such as the above, are interesting, and I myself tend to use them (or variations thereon) for quick mental calculation. My complaint is not so much to the methods taught, but to the failure to first provide a solid grounding in traditional systematic algorithms for performing multiplication and division. Indeed, in my view, the real problems run much deeper than this particular symptom.

At this point I should perhaps provide a little background as to who I am to complain. I am a mathematician, currently completing my Ph.D. in mathematics. My interest in math is mostly pure math and philosophy of math, but extends to math education and popular mathematics. I've been a TA for many years and have plenty of experience dealing with students. And I am not alone in my concerns with the current direction of math syllabuses, plenty of other professional mathematicians who actually look into the syllabus are taking issue too.

So what do mathematicians see as the problem? I would say that it is, in essence, that the individuals writing these new math programs have lost sight of the core skills that early math education should be instilling. In the drive to make the material "relevant to the student", what is being taught has become too applied. In the new programs there is a a focus, almost to the point of exclusivity, on teaching mathematics via real world stories using pictures, blocks, etc. Indeed arithmetic is done using blocks, and fractions and fraction arithmetic using "fraction strips". While such props and aids are useful in motivating the mathematics, it should be just a beginning. A key skill in mathematics, if not the key skill, is abstraction: the ability to abstract away from real world objects, and manipulate these abstractions to draw deep results, is vital. Abstraction is fundamental to mathematics; it is what gives mathematics both its power and its scope; it is the mechanism by which higher mathematics is built upon elementary mathematics. Abstraction and abstract thinking is one of the core skills that mathematics education should be imparting - and yet it is completely ignored by these math syllabuses.

Equally, in an effort to nurture students and foster creativity there is an effort to eliminate rote learning, and emphasize that there may be many ways to arrive at a solution, and letting the students invent their own procedures. Often these invented procedures are very problem specific - they may work for the particular problem at hand, but fail to generalize to other cases. Ultimately this, combined with the very visual (as opposed to symbolic) approach results in the students having limited exposure to consistent, systematic, algorithmic approaches. Again, a core skill that mathematics education should imbue, logical structured thought and a systematic approach to dealing with abstract objects, is being ignored. This is particularly poor in light of the ever increasing importance of skills in algorithms and computation brought about by the needs of modern computers.

The real tragedy is that, because mathematics is a heavily layered subject, each new topic building upon the previous ones, once students fall behind catching up can be a nightmare. Indeed, students often meet a rude awakening in late high school or at college when their limited mathematical repertoire fails to provide the necessary tools to fully grasp the next topic. Even worse, by failing to impart the core skills of abstraction, and logical systematic approaches to dealing with abstract objects, we are denying students the very skills necessary to even begin to expand their mathematical toolkit. At its heart mathematics is about abstract and logical thought, and without these core skills no student can hope to succeed in mathematics.

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Poll
Is Mathematics education headed down the wrong road?
o Yes 85%
o No 14%

Votes: 21
Results | Other Polls

Related Links
o YouTube video
o TERC Investigations
o University of Chicago School Mathematics Project
o not alone
o my concerns
o Abstractio n is fundamental to mathematics
o Also by Coryoth


Display: Sort:
The Declining Quality of Mathematics Education in the US | 248 comments (243 topical, 5 editorial, 6 hidden)
My math skills were so bad... (2.37 / 8) (#2)
by mybostinks on Thu Jan 25, 2007 at 06:08:31 PM EST

from high school and middle school that when I went to college I had to take a summer of remedial algebra. It sucked having to do that but it was worth it. I ended up with a tutor that taught it in a way that the light finally came on in my head.

+1 FP

What is the matter with you? (1.78 / 14) (#6)
by United Fools on Thu Jan 25, 2007 at 06:50:06 PM EST

Knowing 1+1=3 is not enough for you? What do you want from us?

We are united, we are fools, and we are America!
a kidney (1.33 / 3) (#174)
by khallow on Sun Jan 28, 2007 at 12:32:50 PM EST

Those sell for a lot on the black market. Maybe a piece of liver too.

Stating the obvious since 1969.
[ Parent ]

Declining? (1.78 / 32) (#7)
by kitten on Thu Jan 25, 2007 at 07:00:22 PM EST

As far as I can tell, math education has always sucked in the US. My experience going through school began with utterly boring, rote reptition of fifty or more multiplication (or division, or subtraction, or whatever we were learning at the time) problems per night. Big surprise -- I was quickly bored by this mindless endeavor and learned to hate math.

In high school I took the usual algebra courses where I was taught such useful concepts as the quadratic formula, something 99% of humanity will never, ever need in their lives -- and because of its uselessness, most people will promptly forget it, just like taking a year of a foreign language and never using it means you'll barely be able to grunt out more than a few words a couple of years later.

During this, I did what was expected: Memorize the formula, look at an equation, plug the numbers in and grind through the process. It was nothing more than performing steps to a dance on command. I had no idea what a quadratic equation really was beyond a vague "something to do with parabolas", nor why anyone would ever use this. Nobody made any attempt to explain, either. It was just something you had to learn, so shut up and learn it.

(I did learn and memorize it, but naturally, today, I couldn't do it if my life depended on it. I, like most people, have never encountered it outside of a classroom, nor any of the other concepts I learned after about sixth grade.)

Despite the bleating assertions by math geeks, there's no evidence that math somehow teaches abstractation or logical thinking. The way it's taught now certainly doesn't, because it's presented as just a bunch of crap to memorize without understanding.

But if the idea is to get kids to start understanding logic, there are better ways than to teach them math and hope that they gain an understanding as a side effect. Pure logic as a class -- why not? It's something people will actually put to use, whether they think of it or not, as opposed to linear regressions, which they will never use.

Though you bemoan the notion that kids mayh never succeed at mathematics, you never stop to pull back and ask why they should. It's a serious question that is rarely addressed: So what if someone sucks at math? Beyond basic four-operation arithmetic and simple fractions/decimals, few people will ever have call for mathematics. If someone grows up without the ability to factor a quadratic equation, what does it matter?

Not everyone likes math. Few have a use for it. Not everyone is good at it. The notion that people should be dragged kicking and screaming through a discipline for which they have no interest and no use strikes me as utterly silly.

And now, to make things worse, you want to take away any attempt at making this (largely irrelevent) material at least seem relevent to the kids, and go back to.. what exactly? You don't seem to offer a solution. All I know is, if we go back to what I had to go through in school -- endless repetition of multiplication problems -- you're going to wind up with a bunch of people who think math is boring.

While I respect your devotion to the discipline, you seem to share a propensity that most mathematically-inclined types have, which is the inability to see that math isn't all that useful to most people. They aren't the ones designing bridges or developing encryption schemes or calculating orbital trajectories or proving theorems or even just measuring areas. They're just cab drivers, politicians, ranch hands, businessmen, lawyers, factory workers, managers, deliverymen. Math is of professional use to a select few, of personal interest to a few others, and useless to everyone else.

If it's not something they'll ever need, and not something they're interested in, does it really matter if they suck at it or not? Does it matter if they even learn it or not? Does it even mean anything to say they "learned" it if they never use it again, and thus forget it after a year or so?
mirrorshades radio - darkwave, synthpop, industrial, futurepop.
On the abstract thought and necessity points. (2.42 / 7) (#10)
by WonderJoust on Thu Jan 25, 2007 at 07:51:37 PM EST

I disagree with you that math doesn't teach abstract thought, but would qualify that by saying it only happens at the college level, unfortunately. By then, trying to grasp complex functions and proofs requires more than a little mental gymnastics.

I do agree, however, that there is a complete lack of effort in the developmental years from teachers to get the ideas and proofs behind these abstractions in the US.

The attitude seems to be, at least in Texas, cater to the lowest standard. If Jimmy doesn't get it, don't try to teach Suzy, she won't get it either. I am deeply greatful my father and computer drilled the idea of abstraction and symbolism into my young mind as soon as they could. I think alot of this stems from using the state standardized test as the measuring stick for class-placement instead of previous preformance.

Actually, in highschool I had one of the single worst techers for Calc imaginable. Totally unable to explain anything, she just knew her 'tricks' (read:memorized methods) and didn't know why (or often if) it was right. It crippled me in college calc and I'm still recovering.

Further more, on the necessity issue, not being able to reason out even simple algebra is REMARKABLY crippling. Talk about limiting your life choices. And, honestly, to be 'good' at algebra requires very little practice; admittedly, it takes more for others, but seriously, I could do double digit multiplication in 4th grade with a shit-ton of adding. It wasn't quick by any means, but it was functional and the step from that to 'real' multiplication' and further, division, was MUCH more natural for me. Of course, these were personal revelations, not something I was taught.

I was appalled, after thinking about it, that division and fractions are taught as seperate subjects (or they were for me). It's the same motherfucking pony with a different saddle.

Ok, I feel I've lost focus and will cut this here.

_________________________________
i like your style: bitter, without being a complete cunt about it.
-birds ate my face
[ Parent ]

Well (2.33 / 6) (#23)
by kitten on Thu Jan 25, 2007 at 08:37:50 PM EST

I disagree with you that math doesn't teach abstract thought, but would qualify that by saying it only happens at the college level, unfortunately.

By which time the kid has -- probably ten years prior -- already decided whether or not he likes math enough to pursue it. Forcing him into even more of it after he's shown that he has no interest in it is counterproductive.

The attitude seems to be, at least in Texas, cater to the lowest standard. If Jimmy doesn't get it, don't try to teach Suzy, she won't get it either.

I grant that, but that criticism is applicable to any scholastic field, unfortunately. Leave No Child Behind and all that -- which translates to, lower your standards to the point where they're already met.

Further more, on the necessity issue, not being able to reason out even simple algebra is REMARKABLY crippling. Talk about limiting your life choices.

This sort of thing is oft-repeated but no realistic examples are ever provided. Will you be the first?

I think a key point you're overlooking here is that by the time algebra gets introduced -- eighth or ninth grade -- it's already pretty clear whether or not this kid has any interest or aptitude for math. The ones that enjoy it, or are good at it (often these go hand-in-hand, of course!) aren't going to complain about attending math classes, and may well enjoy them. The ones that hate math and have no gift for it -- honestly, why bother?

"Limiting life choices" is ridiculous. Dragging someone, kicking and screaming, through a discipline they despise or are horrible at, isn't going to teach them anything except that authority is composed of sadists. :P If that's the lesson you want them to learn, very well.

Otherwise, I say leave the kid alone. Let him develop something he is good at and does have talent for -- art, writing, history, whatever it is.

We have this completely idiotic notion that we need to churn out "Rennassaince Men", "well-rounded" students, but I see no good argument for it. If someone is happy studying history, and hates math, you will accomplish nothing by death-marching him through another four years of math courses. He is guaranteed to forget most or all of it afterwards, anyway. It's a waste of his time, it's a waste of the teacher's time, it's a waste of the other students' time, and it breeds one more crushed student who loathes school.

Finally,

seriously, I could do double digit multiplication in 4th grade with a shit-ton of adding. It wasn't quick by any means, but it was functional and the step from that to 'real' multiplication' and further, division, was MUCH more natural for me.

Well, that's good. You either had a knack for it, or enjoyed it, or both. I'm guessing you probably went into something technical where you could apply your interest and talent. As for me, I hated math, loathed every moment of it, and sucked hard at it. By the time I was 14 and in high school it was blindingly obvious I would never go into anything requiring more than basic arithmetic and fractions -- so what was accomplished by making me take another four (six, counting college) years of it?

Why not let the kids like you, who are good at math, explore that? And the kids like me, who sucked at it, explore something else? Why do we think everyone "can" learn everything, and must?
mirrorshades radio - darkwave, synthpop, industrial, futurepop.
[ Parent ]
I see your point, but... (2.00 / 7) (#34)
by WonderJoust on Thu Jan 25, 2007 at 09:47:54 PM EST

I think you, too, are overlooking a major point: kids don't always know what is best for them. In fact they rarely, if ever, do.

This sort of thing is oft-repeated but no realistic examples are ever provided. Will you be the first?

This really ties into my idea of it's not that complicated, learn it anyway. I know that seems cold and like I'm forcing someone to learn something they may never use, but, barring a serious learning disorder, highschool algebra (in my infinite wisdom) should be something everyone should at least be exposed to. Perhaps not a requirement to pass to graduate, but at least have exposure. I know that doesn't fit into the current curriculum program of US public schools, but I hope you understand what I'm getting at.

Hell, if they don't get 'traditional' math, maybe something like the methods discussed in that video should be presented instead?

Further more, I do believe math as far as multiplication and division is a necessity. With that established, forcing it on students until we deem them old enough to make their own choices, I think, can't be a bad thing. They might suddenly get it all one day or they may never understand 2+2; let's at least force them to try. I get the idea you don't think it's a necessity, so I doubt we're going to come to terms on this.

Finally, I think you're being too broad in your application of "if you suck don't do it". If a kid were to hit a lousy teacher and show no innate ability, this doesn't mean they can't be taught and should therefor be shunned from math. Of course, that is straying into shitty school systems more than the students, so bunnies.

I guess my main problem with letting someone else decide what I/my child should be exposed to is who decides? Who, really, knows what's best? It also seems counterproductive, to me, to say "You didn't get it the first time, you won't get it the second." I understand your point of forcing a jaded student to try is also counterproductive, but on the remote chance they'll get it on round 2, I'm willing to ignore that. I'm simply not willing to place my personal development completely at the mercy of the system (which is who I am assuming would make the call as to who is adept and who isn't).

I agree with you on the other points looks like.

Nice to find someone who doesn't mind being as long-winded as I am. Cheers.

_________________________________
i like your style: bitter, without being a complete cunt about it.
-birds ate my face
[ Parent ]

We sort of agree. (2.57 / 7) (#41)
by kitten on Thu Jan 25, 2007 at 10:01:08 PM EST

I have no problem with exposing kids to a wide variety of subjects. I also have no problem with forcing them to learn things that are of obvious and immediate value -- basic arithmetic being one of them, no matter how much the kid hates it. Knowing how to divide and add and multiply is absolutely a necessity in life -- though I'd also argue that most people would probably pick it up on their own just by virtue of living in a society that revolves around money. But I digress.

Consider the kid entering high school. He's now 13 or 14. He's already gone through something like six or seven years of formal education. By now it should be becoming clear where his strengths lie, even if they're only developing. In some, it may be more subtle than in others, but if the parents and teachers can't tell by now whether this kid has a head for math (or art, or writing, or the sciences), they aren't doing their job.

I understand your point of forcing a jaded student to try is also counterproductive, but on the remote chance they'll get it on round 2, I'm willing to ignore that.

See, I'd be okay even with that. But after two years of high school algebra, enough already -- either the kid is going to get it or he isn't. Forcing him to march through another two years, plus two more minimum in college, isn't helping anyone, least of all him.

My complaint here is that schools do a lot more than "expose" -- to throw topics at the kid and see what sticks. We have this inane idea that if a kid isn't "well rounded" he's going to be a failure, but who says he has to be well-rounded? If he comes out of high school and doesn't know a lick of algebra, so what?

A couple of semesters of exposure to any subject isn't going to hurt anyone, even if they hate it and are terrible at it. You'll never know until you try, right? But there comes a time when it becomes pointless to continue -- this kid just plain doesn't get math, and more, isn't interested in it. Let him go, I say. By forcing him through another three or four years of it, all he's doing is making himself miserable, and dragging down the rest of the students who may be good at math.

Finally,

barring a serious learning disorder, highschool algebra (in my infinite wisdom) should be something everyone should at least be exposed to.

It continues to amaze me that there is no recognized disorder which prevents one from learning math well. For some reason we accept that some people can't process language very well, stick them into special classes where they learn the bare minimum of how to read and write in society, and call it a day. But when it comes to math, if you're bad at it, you're deemed lazy or stupid or both. Frankly, when I look at most math equations, all I see is a disordered jumble of incoherent gibberish. I imagine it's what a dyslexic sees when they try to read. But no amount of protesting on my part would ever have convinced an educator that I had a problem.
mirrorshades radio - darkwave, synthpop, industrial, futurepop.
[ Parent ]
Right. (1.75 / 4) (#53)
by WonderJoust on Thu Jan 25, 2007 at 10:41:49 PM EST

I think we are almost in total agreement. We've found we both know where the compromises happen on either side.

Really, at the center, we're arguing more about the shittiness of the institution. The difference is, you trust the parents and teachers to be an adequate judge of ability and I do not which could either contest a problem in the system or that I'm a pessimistic jackass.

Good talk.

_________________________________
i like your style: bitter, without being a complete cunt about it.
-birds ate my face
[ Parent ]

*fist bump* /in tea (1.80 / 5) (#54)
by WonderJoust on Thu Jan 25, 2007 at 10:59:28 PM EST

You know, fist bump. Asshole.

_________________________________
i like your style: bitter, without being a complete cunt about it.
-birds ate my face
[ Parent ]

Hmm (1.85 / 7) (#82)
by jmzero on Fri Jan 26, 2007 at 11:59:57 AM EST

If he comes out of high school and doesn't know a lick of algebra, so what?

I think this kind of thing manifests itself fairly strongly in a variety of career paths.  Mostly my experience is dealing with programmers who can't think through problems.  I don't think you can be a good programmer unless you've trained yourself to deal with algebra, with equations.  Sometimes this is direct (estimations of runtime complexity) and sometimes less direct (I see a lot of problems with sets and counting, which doesn't require any specific fancy math - just rigorous thinking).  I guess computer programming might be a little too close to pure math to count, but it is a real problem.  People get a random two year certificate in programming in which they learn how to use a variety of programming tools.  They learn how to piece together an application, but they never pick up the big ideas needed to solve complicated problems.  Their careers never really advance (well, other than to management), because they're forever missing some parts of the puzzle and eventually they find problems they can't really solve.

Similarly, I deal with a lot of management types who have an aversion to numbers.  You don't directly need a lot of math to figure out which people are performing well, pick out trends, identify significant variations, or do simple projections (I'm not talking about regression, just "if I sell 10 widget a month, how many will I sell in a year").  All you need to do is divide and multiply and maybe find averages.  However, if multiplying and dividing and averaging is the limit of one's mathematical abilities it seems that these skills can't be properly applied to the problem.  I see a lot of decisions made on vague math that doesn't say what people seem to think it does.

With a little more work in algebra (or calculus, or anything), these simpler skills could have been exercised enough that they're strong enough to actually be used later.  To nail fractions, I think you have to do algebra.  To nail algebra, perhaps you have to do calculus.  I think people need a little more math than they'll ever directly use - push the boundaries out a little bit so that when they contract there's still enough competency to do what they need.
.
"Let's not stir that bag of worms." - my lovely wife
[ Parent ]

Oh well (none / 1) (#242)
by Corwin06 on Thu Feb 08, 2007 at 07:11:07 AM EST

But then, how comes someone who knows he sucks at math takes a career path that's almost exclusively abstract thought?

That's like people with sickle-cell (sp?) trying to become pro-wrestlers. Maybe one extreme case will have some success, but all others will be miserably crushed.

"and you sir, in an argument in a thread with a troll in a story no one is reading in a backwater website, you're a fucking genius
--circletimessquare
[ Parent ]
Oh well (none / 1) (#250)
by russm on Mon Feb 19, 2007 at 01:27:43 AM EST

perhaps because he's been given such a pissweak math education that he has no idea he's not any good at it?

"well I can add and subtract, so I'm set for programming"...

[ Parent ]

I skirted the realistic example. (2.66 / 6) (#38)
by WonderJoust on Thu Jan 25, 2007 at 09:58:03 PM EST

Well, it was there but I deleted it when I re-read it. It sucked.

In retrospect, I agree, it may not limit you severly, but acknowledging you have no math skill and refusing to try on that basis seems short-sighted to me.

So I retract that it WILL limit you and instead propose that if you do happen to pick it up, it will open countless doors for you.

I think we can agree on that.

_________________________________
i like your style: bitter, without being a complete cunt about it.
-birds ate my face
[ Parent ]

and they made me read John Steinbeck (2.00 / 4) (#134)
by Jazu on Sat Jan 27, 2007 at 10:59:06 AM EST

I hate writing about "themes", but I'm not going to argue for eliminating english classes.

[ Parent ]
Difference there is (1.66 / 3) (#181)
by kitten on Sun Jan 28, 2007 at 04:08:52 PM EST

that English (or whatever your native language may be) is of direct and immediate benefit. Anyone, from any walk of life, social status, economic status, profession, race, religion, or creed, can benefit from being able to communicate their ideas simply and effectively to someone else. And when you learn some technique or skill in language, you apply it immediately -- there's no "it'll be useful later" or "well, it'll teach you how to do some other, tangentally related thing" or "it affects everyone" gibberish and hocus-pocus hopefulness that accompanies the same arguments about math.

You should see the emails I get from users. Supposedly educated, professional adults, who can't spell or capitalize. People who ramble on incoherently, are utterly unable to describe the problem, and who cannot follow written instructions because they can't read. It takes us a long, long time to respond to these people, because we can't spare the fifteen minutes it would take to figure out what the hell they're on about. They are, by any measure, functionally illiterate -- they can recognize letters, words, and even read out loud to you, but they're not able to comprehend what they just read.

These people would find much more benefit from a rigorous study of English than they would from understanding quadratic equations. (Or would have, anyway -- I suspect it is far too late for most of them.)

Now, I agree that English could be taught better, but whether you like reading the alleged "classics" or not, there's no doubt that most people who learn how to communicate effectively do so by seeing it in action (as well as seeing examples of poor communication). You won't learn anything from watching someone solve equations, though.

Furthermore, John Steinbeck sucked. I despised every moment of his torturous novel, from the nonstop and unnecessary descriptions of irrelevencies, to the "interludes" which did nothing to enhance the mood or theme, to the skull-crushingly dull subject matter. Ten years later, no one rememebrs what that book was about, except to say "Oh yeah, the damn chapter with the turtle!" Ask anyone -- that's what they'll tell you.

But on the plus side, witnessing such utter crap is very good at teaching what not to do.
mirrorshades radio - darkwave, synthpop, industrial, futurepop.
[ Parent ]
A better comparison is History (2.00 / 4) (#183)
by Coryoth on Sun Jan 28, 2007 at 04:31:45 PM EST

I mean really, who, in their everyday lives, actually needs to know when Columbus sailed, or what particlar year the American Revolution occured, or what the causes of the first World War might have been. Sure, some history professors, but  most people don't need to know those facts to get by in their everyday life - so why teach any history at all? Perhaps you don't think we should. Perhaps you resent learning any history as much as you resent having learned any math. Hopefully not.

History, like math, provides a background of important knowledge that informs our views and opinions about the world. Sure, you can pick out certain math facts, or certain historical minutiae and complain that in your life you'll probably never make clear practical use of this particular bit of knowledge. That's just ignoring the value of the subject by being reductionist (like complaining that you don't need to know how to spell a particular word, so spelling isn't important).

History provides a background of knowledge that helps us better understand ongoing events in our world. Math provides a background of knowledge that helps us to better understand science and the reality of our world. Remembering the particulars is often less important than having seen them in action and knowing what can be done. You have some knowledge upon which to judge the facts of the world that are presented to you rather than simply having to blidly believe everything you are told simply because the information comes from a self-proclaimed expert.

[ Parent ]

You're missing the point. (2.00 / 2) (#200)
by kitten on Sun Jan 28, 2007 at 07:38:09 PM EST

The point is that Steinbeck wrote an entire chapter about a friggin' turtle crossing the road. Seriously. What does that have to do with anything? I don't know. Nobody knows! But that's all anyone remembers from that book.

As an English major I hesitate to use the phrase "turd parade" to describe a seminal work of literature, but Steinbeck leaves me no choice.
mirrorshades radio - darkwave, synthpop, industrial, futurepop.
[ Parent ]
Today in class... (2.50 / 1) (#221)
by Kariik on Thu Feb 01, 2007 at 01:34:21 AM EST

We just finished A Tale of Two Cities. Great book, well written, interesting, historical, all of the above. One slight slight problem....I'm pretty sure I'm one of about 5 people in my class of 30 who actually read it. So, there goes 2 weeks of English, wasted for the vast majority of my class.

And today! Iambic Pentameter! Here are the patterns of stressed words in poetry. On the other hand, no explanation of WHAT a stressed word is or how its used, no mention of when any of these are used (since all anyone's ever heard of is iambic pentameter, as opposed to the other 40 odd combinations. even then, the only reason anyone knows iambic pentameter is because old shaky used it.). It's boring, dull memorization of facts, which are going to be forgotten as soon as this segment of class is over.

I can see that as a large problem in many of the classes I've taken, learning for the sake of learning, instead of learning for the sake of actually improving our knowledge. It's in the curriculum, TEACH IT.

[ Parent ]

Blarg. (none / 0) (#248)
by EngineeringEmo on Tue Feb 13, 2007 at 10:10:17 PM EST

I couldn't handle that book, and didn't make it very far. The best prose manages to remain interesting even when something sombre is being described, but that book just begged to be but down because it wasn't very engaging at all. I'm sure there was some interesting storyline in there somewhere, but I'd much rather read a book on philosophy, or a book by Mark Twain, or someone who could write an interesting book. (That's not to say that philosophers are necessarily very wonderful writers -- Nietzsche wrote book after book of basically him ranting about stuff, but those who chronicle the works, like Antony Gottelieb, are often excellent writers.)

[ Parent ]
You're joking, right? (none / 1) (#247)
by EngineeringEmo on Tue Feb 13, 2007 at 10:04:24 PM EST

I think you're ignoring the fact that many people don't end up learning either, would rather not learn either.

English is just as, if not more, rote than math. I remember taking a Grade 12 English class. I got the highest mark in my class, but hated every second of it. All we had to do was memorize the novel long enough to write the test.

Later, in college, I'd end up taking courses that went back to basics, and actually taught fundamental elements of style and grammar which helped people form effective communication. High school was a waste of time for all the arts.

[ Parent ]

Try reading these two essays (1.33 / 3) (#172)
by Alan Crowe on Sun Jan 28, 2007 at 08:35:01 AM EST

People are always drawing causal inferences from non-experimental data. For example, the Japanese live longer and eat lots of fish. If you change your diet to include more fish, you will live longer too. This essay does a nice job of highlighting the extremely treacherous nature of non-experimental data, but you will not be able to follow it unless you are at ease with arithmetic.

Come the election, tax policy will have an influence on how you vote. But is tax policy presented honestly. Do you have the grasp of arithmetic to follow the essay? If not, are you being manipulated?

Notice that the intellectual stimulus that prodded me to write the tax rates essay was my study of General Relativity. The Schwarzschild solution has a singularity at the event horizon, but there is not in fact a singularity at the event horizon. It is a coordinate singularity, an artifact of the parameterization. So the issue of parameterizations and whether singularities are real or not was lurking at the back of my mind. My essay on tax rates only uses simple arithmetic. Differential Geometry is total overkill, but that is often the way. You usually need to study mathematics way past the actual requirements of a particular application before you can do anything creative.



[ Parent ]
Bad teaching is bad. (none / 0) (#246)
by EngineeringEmo on Tue Feb 13, 2007 at 09:53:24 PM EST

That's nothing. I remember when they first introduced division to us. I didn't understand it in the least, so when it came time to do the problems, I had to ask for help. The thing is, the way they tried to 'fix' my problem was to just repeatedly ask me if I knew the answer. "What's 4 into 9?" "I don't know how to do this." "But how many TIMES does 4 go into nine?" "I don't know how to do this." "But now MANY times does 4 go into 9?" "I don't know how to do this".

If applied mathematics is the thing that made me love math, it was that sort of thing that made me hate math.

[ Parent ]

d00d, get over it. (2.11 / 9) (#11)
by V on Thu Jan 25, 2007 at 07:55:46 PM EST

You suck at math. The only one responsible of your sucking is yuo. Stop blaming the whole world because of yuor failure.

HTH.

V.
---
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[ Parent ]

My fault? (2.11 / 9) (#65)
by kitten on Fri Jan 26, 2007 at 03:32:38 AM EST

Are you sure? Are you sure it wasn't the feminists?
mirrorshades radio - darkwave, synthpop, industrial, futurepop.
[ Parent ]
Yes, YUOR fault.$ (1.00 / 3) (#72)
by V on Fri Jan 26, 2007 at 05:51:55 AM EST


---
What my fans are saying:
"That, and the fact that V is a total, utter scumbag." VZAMaZ.
"well look up little troll" cts.
"I think you're a worthless little cuntmonkey but you made me lol, so I sigged you." re
"goodness gracious you're an idiot" mariahkillschickens
[ Parent ]
It's been a while, kitten (2.16 / 6) (#13)
by debacle on Thu Jan 25, 2007 at 08:11:00 PM EST

It's much worse these days.

And regardless of what you think, learning mathematics is incredibly useful to a large amount of people. Relying on computers is not the way to go, and trusting others to be honest doesn't work either.

How many people get shafted by companies touting deals that are no good, because the victims have no understanding of the basics of mathematics?

He's not talking about linear algebra. He's not talking about ODEs or boolean logic. He's talking about basic mathematics - most kids today can't recreate a 10x10 multiplication table, or divide numbers bigger than ten.

It tastes sweet.
[ Parent ]

If what you say is true (1.85 / 7) (#20)
by kitten on Thu Jan 25, 2007 at 08:25:25 PM EST

then he wouldn't be talking about abstractations and logic, neither of which are skills necessary to memorize multiplication tables and come to a basic understanding of four-operand arithmetic. Any schmuck can do that, and I'd wager that if you left kids alone, they'd learn most of it on their own, simply because it's a valuable and necessary skill to living in this society. Just like they learned how to walk and talk.
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[ Parent ]
If you let kids learn on thier own (2.50 / 6) (#29)
by debacle on Thu Jan 25, 2007 at 08:53:23 PM EST

They become individuals.

Are you sure you want that rampant individualism on your hands?

It tastes sweet.
[ Parent ]

And today (2.83 / 6) (#47)
by kitten on Thu Jan 25, 2007 at 10:22:44 PM EST

debacle explains why all of this discussion is just wanking. Schools aren't there to educate or inform; they're there to prepare you for the workforce by turning you into someone who obeys stupid, senseless orders when asked, and performs silly tasks on command. Thinking is not encouraged.
mirrorshades radio - darkwave, synthpop, industrial, futurepop.
[ Parent ]
As least we agree on some things (2.16 / 6) (#56)
by debacle on Fri Jan 26, 2007 at 12:07:07 AM EST

Billy Joel or Elton John?

It tastes sweet.
[ Parent ]
99% of people who lease a car (2.40 / 10) (#21)
by balsamic vinigga on Thu Jan 25, 2007 at 08:25:46 PM EST

have no interest or understanding of how their monthly payment came up.  Lucky for them there're laws on their side that limit how much they can get fucked.

Math can be a useful tool to anybody who wishes to utilise it. It's sad that it's the status quo to just trust the car dealer to not fuck you. :/

---
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[ Parent ]

You misunderstand kitten's point (1.50 / 4) (#62)
by curien on Fri Jan 26, 2007 at 02:41:51 AM EST

You see, no one needs to know how to calculate monthly payments on loans because some smart guy will write a web page for them to do it. Brilliant! I mean, all those lower-class, lower-educated folks should be used to trusting everyone else to make sure they do OK in life.

--
I'm directly under the Earth's sun ... ... now!
[ Parent ]
Fortunately (2.16 / 6) (#64)
by kitten on Fri Jan 26, 2007 at 03:31:26 AM EST

Calculating that sort of thing is only slightly above basic, four-operation arithmetic. I guess you have to understand what I=PRT means, and then crank out the multiplication.

Now show me Joe Everyman's use for linear regressions or quadratic equations, smartass.
mirrorshades radio - darkwave, synthpop, industrial, futurepop.
[ Parent ]
heres how it is (1.66 / 6) (#90)
by blackbart on Fri Jan 26, 2007 at 12:42:37 PM EST

laid out so that even you can understand it:

Its been demonstrated that most human beings proceed to abstract thought in their early teens, and since you failed to make this jump, you are intellectually stuck in elementary school. You've already demonstrated the point where you stopped developing socially.

"I use this dupe for modbombing and impersonating a highly paid government worker"
- army of phred
[ Parent ]

academics is not about preparing (2.35 / 14) (#17)
by balsamic vinigga on Thu Jan 25, 2007 at 08:20:00 PM EST

you for adulthood.

There's no class on paying your bills, raising a family, coping with job stress, getting laid, financial planning, buying a home, etc etc.

These things you'll learn on your own.

Academia is about taking the time at an early age to expand our mind. To appreciate human intellectual endeaver, achievment, history, culture... to get a summary of how we understand the world today.

You can't strip math of its virtue without crumbling all of academia.  Then what would school be about? How to raise a family? How to be complacent when your boss is wrong?

I don't mean to panit a slippery slope here, but a well rounded thorough education requires math.. people with bad attitudes about it be damned!

---
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[ Parent ]

What a nice fairy tale. (2.30 / 10) (#19)
by kitten on Thu Jan 25, 2007 at 08:23:19 PM EST

But school post WWII at least has never been about "expanding our minds". It's designed to be a place to keep the little snots off the street until they're 18, so the parents can go to work and further the economy. Few, if any, minds were expanded by coloring in pictures of cells and labelling the mitochondria, or memorizing a chronological list of US presidents, or memorizing the funeral speech to Julius Caesar, or making stupid posterboard projects. It's all busywork, but it's got to look really important, otherwise the game would be up. And hence why we insist on filling student's heads with worthless crap like quadratic equations, things we know for a fact they will never use and are guaranteed to forget.
mirrorshades radio - darkwave, synthpop, industrial, futurepop.
[ Parent ]
I agree that K-12th grade public school (1.83 / 6) (#22)
by balsamic vinigga on Thu Jan 25, 2007 at 08:30:13 PM EST

is taught poorly. However, the rote memorization does pay off for college and uni students where academia is still alive and well.

sure doing page after page of quadratic formula plug 'n chug misses the point..  and no teacher does their job correctly and shows how to come up with the quadratic formula.  But the failure of K-12 public school academia is a much larger topic that that has implications on every subject not just math.  But it doesn't do anybody any good to single out math simply because it's one of the most botched.

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[ Parent ]

Heh. (1.71 / 7) (#25)
by kitten on Thu Jan 25, 2007 at 08:44:49 PM EST

I assume you're referring to the "alive and well academia" of modern colleges that seem to be more interested in churning out idiotic "business degrees" than anything else? Where an MBA is lauded and a degree in medieval studies is snickered at because it is "useless"?

But it doesn't do anybody any good to single out math simply because it's one of the most botched.

I think I could make a pretty strong argument that not only is math the most botched, but truly is the most patently useless to bother teaching to people unless they express an interest or talent in it. Everyone, in any walk of life, can benefit from knowing at least a little about where they've come from (history), how the world works (science), and especially how to communicate an idea effectively to someone else (English, or whatever your native language may be).

You don't have to be an expert in any of these fields, nor would I expect anyone to be, but some base knowledge in these topics comes in handy for pretty much everyone, both professionally and personally.

The same cannot be said of most math studies. Almost no one will ever find a use for a quadratic equation, or a linear regression, or finding the third derivative. And those who do? They aren't the ones who would have to be forced into it -- they'll choose to enter such fields on their own.


Finally, even if I grant your point, what you've missed is that after twelve years of "plug n' chug" as you put it, students have pretty much decided they hate this shit. College math may be the most exciting, fascinating thing in the world, but most students, after twelve years of miserable memorization and boredom, will elect never to find out.
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[ Parent ]
Well we'll have to agree (2.25 / 8) (#30)
by balsamic vinigga on Thu Jan 25, 2007 at 08:53:57 PM EST

to disagree..  I'd say having little understanding of math is just as debilitating as having no understanding of history and geography.

The specifics I may agree..  many people may never need to know the quadratic formula..  but getting caught up in such specifics is a strawman argument against math, which, after all, is a study of logic and patterns.  Everybody everywhere encounters logic and patterns going about their every day life.  Those with a solid understanding of math are at an advantage. Of course, because math is so botched people leave school without these skills (even if the aced math) ...  but again math being the most botched isn't a valid argument against math.

---
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[ Parent ]

The difference is (2.71 / 7) (#35)
by kitten on Thu Jan 25, 2007 at 09:50:13 PM EST

that learning (say) English has immediate and direct benefits -- in the sense that the information is applicable right then, without any further ado. Arguments about how math is grounded in logic, blah blah blah, are so common, yet almost entirely without merit -- yes, it may be grounded in logic (though Feinman often disagreed), but it's inane to teach kids something "grounded in" logic and hope that "somehow" they learn to gain logical thought from it.

It'll never happen the way it's taught now, as a bunch of rules to memorize with zero understanding required of how it all works and why. But even if it were taught differently, the hope that you can teach math and, as a sort of side-benefit, logic will mystically be imprinted onto the youngsters, is naive.

If we want to teach logic and pattern recognition there are formalized means of teaching these things directly, no math required. The fact that we still teach mathematics which no one will ever use says to me that we're just trying to make school look really important when it isn't.
mirrorshades radio - darkwave, synthpop, industrial, futurepop.
[ Parent ]
Something we agree on! (2.71 / 7) (#39)
by Coryoth on Thu Jan 25, 2007 at 09:58:53 PM EST

it's inane to teach kids something "grounded in" logic and hope that "somehow" they learn to gain logical thought from it.

Aha! We agree. I think it is utterly inane to teach kids arithmetic by rote and expect them to somehow suck the logic out of it on their own. Indeed, this is even the case at University: students are expected to pick up logic and proofs by osmosis and general contact. But that's the point: sucking all the logic out entirely (as these programs do) is hardly beneficial. What we should be doing is injecting the logic back in. Yes, kids need to learn arithmetic (it is useful), and the methodical processes are also useful, but we should be explaining why the processes work, and teaching the logic that underlies arithmetic - that's one of the things we're lacking.

[ Parent ]

It occured to me (2.42 / 7) (#46)
by kitten on Thu Jan 25, 2007 at 10:19:50 PM EST

that one great example is when we're ten or eleven and learning long division.

What a catastrophe that was! Nobody understood it, and it was presented as this mysterious, new operation. And what was this "remainder" crap? Why not teach us what to do with the remainder to get a proper answer, instead of leaving it there and getting around to it later?

It wasn't until years, years later that I realized long division was basically just a shorthand way of doing a massive series of subtraction problems. If they'd demonstrated it this way, the process would have made sense to me -- and a lot of others, I imagine. Subtraction is easy to understand to a child, and we'd all been doing it for a couple of years. If only the teachers had built on that, instead of introducing a totally new way of doing things that made no sense.
mirrorshades radio - darkwave, synthpop, industrial, futurepop.
[ Parent ]
Exactly what I mean (2.57 / 7) (#55)
by Coryoth on Thu Jan 25, 2007 at 11:46:46 PM EST

Division is, at it's core, an abstraction over subtraction in the same way that multiplication is an abstraction over addition. It's a second order concept - you need to understand "number" as an abstract concept as opposed to constantly visualising it as some number/collection of physical objects. What do I mean by that? Well multiplication is performing a given number of additions of some number - that is we are applying number to numbers themselves. Numbers have gone from describing physical objects, to describing abstract objects - in this case numbers, which must be considered as objects in their own right. Division works the same way as a number of subtractions of a certain number (you are essentially counting subtractions, counting an sbatract object).

Now of course the full depth of that might not be the best thing to explain all at once to kids, but the outline, and the concept of making that abstraction from addition to multiplication, from subtraction to division, and treating numbers themselves as objects in their own right - that we can (and should) teach. Of course actually doing some divisions and seeing how the long division algorithm makes this work - that helps too. It's hands on work with the objects under consideration - which in this case are the abstract objects we call numbers.

If you are taught things as a bunch of disconnected mechanical processes then of course you don't see any logic to it, and you don't see how the process of abstraction is informing it. And that means you aren't really learning math any more than simply memorizing dates is learning history.

[ Parent ]

I never got past that, really. (2.16 / 6) (#68)
by kitten on Fri Jan 26, 2007 at 03:43:01 AM EST

I don't conciously think about it, but if I do, I still consider numbers as representations of some physical quantity. I barely even know what it means to talk about numbers qua numbers in the way you're describing -- makes absolutely no sense to me.

This may go a long way to explaining why I wanted to bash my face into a locker when we got to imaginary numbers.
mirrorshades radio - darkwave, synthpop, industrial, futurepop.
[ Parent ]
heh they shouldn't call them imaginary numbers (1.87 / 8) (#71)
by balsamic vinigga on Fri Jan 26, 2007 at 04:01:21 AM EST

that was what descartes called them when refering to them in a derogitory manner before we realized that they do exist...  er well yeah i mean in that they're no more imaginary than say negative numbers. You can't have -5 apples physically, but you can conceptually..  just like you can have a number of apples whose square is a negative number.. conceptually. That word imaginary trips a lot of students up... and makes the concept seemingly way more dificult than it needs to be :/

---
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[ Parent ]
Magical and mysterious (1.50 / 2) (#173)
by Alan Crowe on Sun Jan 28, 2007 at 08:53:54 AM EST

I just love the way that the square root of minus one raised to the power of the square root of minus one turns out to be just a little more than one fifth. Its a real number. How cool is that?

[ Parent ]
It explains a lot (1.80 / 5) (#89)
by Coryoth on Fri Jan 26, 2007 at 12:41:31 PM EST

I think it explains why you struggled with a lot of mathematics.  Fractions and algebra are abstractions built atop numbers - that is, once you're comfortable with numbers as objects in their own right, then you can abstract over all the different numbers to get algebra in the same way you can abstract over all the collections of different physical objects to get numbers. Later elementary algebra starts to be taken for granted and we make abstractions on top of that - if you never really get a mental grip on the very first step then it becomes increasingly hard to make the further steps that build upon it by the same process. In that case everything becomes mechanical manipulation with no intuitive feel for what's going on, and that makes mathematics frustrating and very difficult to understand.

A lot of people actually fall into the same category as you - and its not because they're stupid (indeed, often they are quite intelligebt individuals), it's that they either don't have the type of mind that makes those leaps of abstraction easily, or (far more common in my experience) they had a poor teacher very early on who failed to really instill the concepts properly, particularly of abstraction, and have been lost ever since.

[ Parent ]

That's not a big intellectual leap (2.75 / 4) (#223)
by itsbruce on Thu Feb 01, 2007 at 03:49:25 AM EST

I don't conciously think about it, but if I do, I still consider numbers as representations of some physical quantity. I barely even know what it means to talk about numbers qua numbers in the way you're describing -- makes absolutely no sense to me.

Given that k5 is alleged to be a site for people with at least some intellectual curiosity, this seems a little like boasting in public that you can't tie your shoelaces and don't see the point.


--

It is impolite to tell a man who is carrying you on his shoulders that his head smells.
[ Parent ]
Sooo... (1.50 / 2) (#58)
by WonderJoust on Fri Jan 26, 2007 at 01:16:10 AM EST

I'm gonna go ahead and show off how ill-equipped I came out of high school... could you linkercize me some examples using long division for mass subtraction?

Thanks. *fist bump*

_________________________________
i like your style: bitter, without being a complete cunt about it.
-birds ate my face
[ Parent ]

I'm the worst person to ask (2.00 / 4) (#63)
by kitten on Fri Jan 26, 2007 at 03:26:46 AM EST

I am sure others here could explain it better -- remember, I'm the math 'tard in these parts.

Still, division at its core is just subtraction. When you ask "What's nine divided by three?" you're asking "How many times can you subtract three from nine?" 9 - 3 = 6, and 6 - 3 = 3, and 3 -3 = 0. So, the answer is 3. You can subtract it three times before there's nothing left. If someone had told us this back in fourth grade or whenever it was, I think we'd all have been better off. I know I would have been.

All the fancy numberwork you do underneath the long division equation is just a way of simplifying that process for larger numbers. I know this to be true but I'd have to break out the old pencil and paper to figure out a way to explain it, and it's 3am and my explanation would suck anyway.
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[ Parent ]
Oh, ok, you're talking the traditional method. (2.00 / 5) (#66)
by WonderJoust on Fri Jan 26, 2007 at 03:39:05 AM EST

I thought you meant there was some crazy correlation in the in the subtraction of the multiples like if you had...

...you know, I've been smoking, we're just gonna let this one go.

_________________________________
i like your style: bitter, without being a complete cunt about it.
-birds ate my face
[ Parent ]

that's just weird (1.33 / 3) (#61)
by livus on Fri Jan 26, 2007 at 02:22:24 AM EST

long division was totally the best thing we ever did in maths before age ten.

(btw if you check you'll find you weren't eleven - at least I hope you weren't).

---
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[ Parent ]

You're insane. (1.60 / 5) (#69)
by kitten on Fri Jan 26, 2007 at 03:49:48 AM EST

I was about ready to set fire to my textbook when we got to that part. It made no sense to me, plus each problem took bloody ages, and I had to do like thirty or forty of them a night. The whole "remainder" thing seriously fucked me up, too -- it made no logical sense that we could just casually dismiss this leftover bit. Even I knew that. Even at that age.

Also, I guess you're right. I think we did this when we were in fourth grade, so that means nine? I barely remember anymore. Much like I barely remember how to do long division by hand anymore. :P
mirrorshades radio - darkwave, synthpop, industrial, futurepop.
[ Parent ]
no, in fact your comment below (1.50 / 2) (#117)
by livus on Fri Jan 26, 2007 at 07:56:07 PM EST

explains what I liked about it. It gave me a physical/spatial understanding of short division.

After we hit whatever-you-call-the-11-class ("junior high"?) we got to do some decent geometry and algebra and I felt a lot better about mathematics.

But back in the under-tens dark ages, where most of my teachers had a completely illogical and uncaring approach to maths (and to much else) long division was about as logical as maths got.

---
HIREZ substitute.
be concrete asshole, or shut up. - CTS
I guess I skipped school or something to drink on the internet? - lonelyhobo
I'd like to hope that any impression you got about us from internet forums was incorrect. - debillitatus
I consider myself trolled more or less just by visiting the site. HollyHopDrive

[ Parent ]

Finalnd's approach to improving math ed. (2.16 / 6) (#119)
by Coryoth on Fri Jan 26, 2007 at 09:01:37 PM EST

This is where I have to point to the Finnish approach to improving math education (and it has worked for them, in that they now finish among the very top of OECD countries in student math achievement): Get more people who have taken (university) math into elementary school, and get more elementary school teachers to learn higher level math.

I think a significant problem with math education is that most people who go on to be elementary school teachers have a very poor understanding of mathematics. That means that what they teach the kids is very much "recipes" straight from the book with little real insight or understanding. That makes it harder to learn and less appealing (because it develops into a lot of grind wor with not much enlightenment). On the other hand if you have early teachers who are teachign arithmetic who actually have a deeper understanding an appreciation then they can help make the logic, connections, and thinking process that informs the methods clear. That makes a big difference to how kids percieve and learn mathematics.

[ Parent ]

That makes a lot of sense (2.20 / 5) (#120)
by livus on Fri Jan 26, 2007 at 09:11:15 PM EST

I remember being deeply frustrated at the general unwillingness to explain the principles behind what we were doing - it was one of those areas where teachers would often get quite angry, particularly if we tried to experiment with other methods.

I realise now that it was probably more that they couldn't, rather than that they refused. I also realise that several of them were quite young and presented this as a decision in order to save face and maintain authority. But at the time it was quite bewildering. The Finnish approach would definately have improved things in this respect.

---
HIREZ substitute.
be concrete asshole, or shut up. - CTS
I guess I skipped school or something to drink on the internet? - lonelyhobo
I'd like to hope that any impression you got about us from internet forums was incorrect. - debillitatus
I consider myself trolled more or less just by visiting the site. HollyHopDrive

[ Parent ]

You've got a point. (none / 1) (#238)
by grendelkhan on Tue Feb 06, 2007 at 01:30:17 AM EST

The moment I realized that college was going to be different from high school was in one of my introductory lectures, where someone asked a question, and the instructor answered it with an informative digression. I realized that he actually knew the stuff he was teaching; he wasn't just reading us a book.
-- Laws do not persuade just because they threaten --Seneca
[ Parent ]
Remainders annoyed me too (1.75 / 4) (#128)
by superiority on Sat Jan 27, 2007 at 12:52:53 AM EST

I learnt to do real division before I ever learnt what a remainder was. I always got the division questions wrong because I would give an answer with a fractional part.

[ Parent ]
Yeah I guess what I'd do is this (2.00 / 7) (#42)
by balsamic vinigga on Thu Jan 25, 2007 at 10:05:29 PM EST

keep K-jr high math the same

then in highschool give them basic practical introduction useful algebra and trig and geometry. Like trig and geometry that would say make quick work out of calculating angles for woodshop or something.

Then in highschool teach the rich and facinating world of math history and an overview of the math topics and what they provide and what they've enabled and accomplished.

Instead of doing useless psuedo series that NOBODY understood in highschool.. People could be learning a layman's overview of zeno's paradox.  The simultaneous discovery by newton and liebniz of calculus...  a layman's overview of what calculus is, and go on to discuss einstein's contributions, hawking's, feynmann's, an overview of topology and linear algebra.  In highschool they need to start putting a facinating human face on math, and prepare them and excite them for what's to come if they pursue academia.

Less technical, more exciting, and directly useful historical context.

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[ Parent ]

Yeah, great plan. (1.83 / 6) (#44)
by kitten on Thu Jan 25, 2007 at 10:16:31 PM EST

One thing you're forgetting though, is that it would require getting competent, well-paid teachers, textbooks that didn't suck out loud, administrators that gave a flying fuck about something other than the football team, a school board that cared about education and not pandering to the parents' whiny demands, and parents that spent more time encouraging kids to learn how to learn, instead of bitching that Harry Potter books are on the library shelves or that the science curriculum offends their religious sensibilities.

Given all that, I'm guessing nothing will ever change. It's easier to just go the way we've been going.
mirrorshades radio - darkwave, synthpop, industrial, futurepop.
[ Parent ]
no doubt $ (1.16 / 6) (#49)
by balsamic vinigga on Thu Jan 25, 2007 at 10:24:56 PM EST



---
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[ Parent ]
This was my HS's problem. (2.00 / 2) (#59)
by WonderJoust on Fri Jan 26, 2007 at 01:24:01 AM EST

I excelled through middle school because once I got the hang of adding, everything up to geometry was a breeze. Algebra just dissects itself in my head. In addition to me, I had some great techers who really enjoyed what they were doing. But we moved when I was 10 and the other school system was miserable.

So I hit highschool and I didn't get it as well and I no longer had a backup of stellar teachers to coach me properly through it.

Not that I'm blaming them, but fuck, it is their job.

I remember, when we first moved, I was in 4th grade and my sister and I actually complained about it being so easy enough that my mom volunteered to truck us the 15 minutes each way every morning.

_________________________________
i like your style: bitter, without being a complete cunt about it.
-birds ate my face
[ Parent ]

The textbook problem. (none / 0) (#239)
by grendelkhan on Tue Feb 06, 2007 at 01:40:24 AM EST

There is a tremendous problem with textbooks. My significant other is taking basic math at the moment, and dropped about a hundred bucks on some textbooks which teach multiplication tables, long division, that sort of thing. Seriously; this should be a solved problem. Public schools, if nobody else, should have taken a single year's worth of textbook cost, and commissioned a public-domain textbook for elementary math, which all students could use. Hell, have a couple other states do the same; variety's a good thing. (I shed no tears for the lost profits of the textbook companies.) But to have students forking over a hundred bucks for a textbook detailing a subject which hasn't changed in over a century? Madness!
-- Laws do not persuade just because they threaten --Seneca
[ Parent ]
(which isn't really math) (1.83 / 6) (#48)
by balsamic vinigga on Thu Jan 25, 2007 at 10:23:11 PM EST

more science/history at that point...  but hey filler for between the bells right..  and excellent prep for real math...

---
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[ Parent ]
Immediate and direct benefits, eh? (none / 1) (#237)
by grendelkhan on Tue Feb 06, 2007 at 12:29:22 AM EST

Look, all of your arguments about why you don't need to learn math apply equally well to English, or history. Reading the classics doesn't have immediate and direct benefits for the average high-schooler.

What are these methods for teaching logic and pattern recognition without teaching math? Are any of them better than, for instance, teaching Euclidean geometry like I learned in tenth grade, where you're given a set of axioms and some rules, and asked to prove or disprove something?
-- Laws do not persuade just because they threaten --Seneca
[ Parent ]

That's a ridiculously simplified view. (2.50 / 2) (#235)
by grendelkhan on Tue Feb 06, 2007 at 12:11:54 AM EST

Where an MBA is lauded and a degree in medieval studies is snickered at because it is "useless"?

Because in your world, there's nothing between job training and absolutely inapplicable academia. Did they even have hard science or social science where you came from?

I agree wholeheartedly that plug-and-chug is demeaning and will sap the will of any bright-eyed student. (I've seen a remedial-math class require eight copies of the 1-12 times table. Yes, the same times table, copied eight times.) However, that doesn't say anything about whether or not people should learn math as a whole.

Without math, the hard sciences are closed off, as well as anything to do with statistics, so that leaves out the social sciences as well. Are these things useless? Is it useless to know just how stupid it is to play the lottery? (The concept of "expected value" comes into play here, but good luck with that if you can't do basic algebra.) Is it useless to understand statistics and how they can be abused?

You don't have to be an expert in any of these fields [history, science, English], nor would I expect anyone to be, but some base knowledge in these topics comes in handy for pretty much everyone, both professionally and personally.

You know, I bet most people could make it through life without knowing what year the Normans invaded England, or Newton's second law (which they'd somehow learn without math, right?), or what a soliloquy is. Gee, it's easy to stomp on a field you don't like by picking an abstract part of it and complaining that it's not useful, isn't it?

Top-of-my-head example of something interesting and/or applicable involving nontrivial math.

Did you ever wonder why tin cans are the shape they are? You can make a cylindrical can shorter and squatter, or taller and thinner, and have it contain the same volume. So, why are they the shape they are? What's the height-to-thickness ratio that gets you the most contained volume per unit of sheet metal? I suppose if you're naturally incurious, you won't really care, but it's the sort of thing you can figure out with calculus.
-- Laws do not persuade just because they threaten --Seneca
[ Parent ]

What kind of teacher did you have? (none / 0) (#234)
by grendelkhan on Mon Feb 05, 2007 at 11:28:41 PM EST

sure doing page after page of quadratic formula plug 'n chug misses the point..  and no teacher does their job correctly and shows how to come up with the quadratic formula.
I don't know where you went to school, but I clearly remember going from factoring quadractics, to completing the square, to deriving the quadratic formula from that. I went to an unremarkable public school, in the US.
-- Laws do not persuade just because they threaten --Seneca
[ Parent ]
Math is more important than you are... (1.93 / 16) (#24)
by GrandWazoo on Thu Jan 25, 2007 at 08:38:09 PM EST

despite what you may think. Had you learned it better, maybe you would have posted a comment worth reading.

I quit when I read

"Despite the bleating assertions by math geeks, there's no evidence that math somehow teaches abstractation or logical thinking."

Next time, at least show a little more intelligence when you write something more than -1 Not Devo or whatever other nonsense that pops into your head at the moment.

[ Parent ]

Ah, I see (2.00 / 7) (#26)
by kitten on Thu Jan 25, 2007 at 08:46:14 PM EST

So you going "NUH UH THAT WAS STUPID" somehow negates my argument. Yes indeed, I have learned my lesson -- thanks for taking the time to present such a well-reasoned and rational discussion.

Idiot.
mirrorshades radio - darkwave, synthpop, industrial, futurepop.
[ Parent ]
you really did entirely miss the boat (1.55 / 9) (#79)
by army of phred on Fri Jan 26, 2007 at 10:15:54 AM EST

with everything you posted. I'm very mediocre at math, having never done calculus for instance, dropped out in the 11th grade, yet I can look at what you posted and know enough, even now as I ring up losers buying blunts in a convenience store, that you have basically entered your adulthood fully uneducated and unable to put even the briefest string of critical thinking together in that attrophied lump you have for a brain.

How does it feel to be owned by a convenience store clerk in a pre college level academic subject?

"Republicans are evil." lildebbie
"I have no fucking clue what I'm talking about." motormachinemercenary
"my wife is getting a blowjob" ghostoft1ber
[ Parent ]

I'm sorry (2.57 / 7) (#28)
by MechaA on Thu Jan 25, 2007 at 08:52:09 PM EST

During this, I did what was expected: Memorize the formula, look at an equation, plug the numbers in and grind through the process. It was nothing more than performing steps to a dance on command. I had no idea what a quadratic equation really was beyond a vague "something to do with parabolas", nor why anyone would ever use this. Nobody made any attempt to explain, either. It was just something you had to learn, so shut up and learn it... ..Despite the bleating assertions by math geeks, there's no evidence that math somehow teaches abstractation or logical thinking. The way it's taught now certainly doesn't, because it's presented as just a bunch of crap to memorize without understanding.

With all due respect, the reason why you don't think that math teaches abstraction or logical thinking is because it seems like you never learned math (most folks don't in school, really, and that's what ought to be addressed.)

Being able to plug numbers into variables is not anything very interesting or instructive or useful. The 'draw' of mathematics is the process of gaining an intuitive understanding of something abstract. The goal of the lesson is not to enable you to solve a quadratic equation - it's for you to understand why the procedures for solving a quadratic equation actually come up with the right answers, and to come to greater general knowledge about the mathematical and geometrical universe through that understanding.

I can personally say that experiencing that kind of revelation primarily in mathematics when I was a child is what made me passionate about learning all sorts of things (and I have spoken to other people who mostly agreed), so whereas mathematics might have seemed useless to you, there is something important there, if it is taught well to the right kind of person.



k24anson on K5: Imagine fifty, sixty year old men and women still playing with their genitals like ten year olds!

[ Parent ]
Ahh (2.50 / 6) (#43)
by kitten on Thu Jan 25, 2007 at 10:12:40 PM EST

I admit that. In fact, it's one of my problems with math education today -- that no one comes out of it learning anything (at least in public schools). I thought I'd made it clear, but if not, let me do so now: The way math is taught in schools is "memorize this formula, plug numbers in, crank out answer."

It makes no difference whether you know what the hell you're doing or not, as long as you can dance on command. That's what passes for math education. It's ridiculous, and of course it doesn't teach logic -- there is nothing logical about memorization if you don't know what it means. I could memorize the Chinese character set without actually understanding Chinese.

So having said that, there are other ways of teaching logic and abstractation. I think it's sort of silly to teach math and hope that "somehow" these other, related concepts rub off on the kids. Frankly, I think a rigorous course of the scientific method would do people a lot more good in terms of being able to break down a problem and find a solution. Why not focus on that? Why do we think proofs and theorems are the way?

Finally,

so whereas mathematics might have seemed useless to you, there is something important there, if it is taught well to the right kind of person.

Precisely. And you were apparently that kind of person, and I was not. There was no point in dragging me through it once that became clear. And you, obviously someone who enjoys math -- you're not the type who would have had to have been dragged through it. :) You'd have gone because you had a knack for it and/or were interested in it.

Also please note I never said mathematics was useless. The fact that we're discussing this on a computer which relies on an intimate understanding of silicon semiconductor physics, electromagnetic interactions, and things I can't fathom, demonstrates that. But it is useless to me and most other people -- who aren't researching these things, designing things, building things. The overwhelmingly vast majority of humanity gets on just fine without knowing any algebra, and ten years after high school, the most that most people can say about quadratic equations is "Hm, I remember learning about those." Ask them to do it, and they'll be helpless. Can you guess why?
mirrorshades radio - darkwave, synthpop, industrial, futurepop.
[ Parent ]
Maybe (2.60 / 5) (#51)
by MechaA on Thu Jan 25, 2007 at 10:31:33 PM EST

I guess that it's very possible that some people are "non-math" people and just somehow don't gather any value from it.  Presumably, I'm not one of them, and I don't understand the kind of human mind that could not derive my sort of enjoyment from learning mathematics, but then again, I don't understand the mind of a religious fundamentalist, someone with ADD, a blind person, a mathematical savant, or someone who can draw something flawlessly when they look at it, and those people sure exist.

That said, how do you know that you're one of them, or that anyone is?  If there's one thing we can agree on in this thread, it's that public mathematics education is, more often than not, in a pretty bad state.  I don't think a great deal of my appreciation for mathematics is attributable to schooling - it was instilled at a really early age probably by my parents and by books I read.  I worry that the label of "non-math" is a misdiagnosis of someone who just never had quite the right sort of moment to pick it up.

I am a big optimist, and I really suspect that almost any person of average intelligence can understand mathematics and enjoy that knowledge if they are taught well.  It becomes as useful or useless then as is having the pleasure of listening to great music or reading a wonderful novel.  

So I am a strong proponent of learning how to teach math correctly, because I think it is terrible that so many people might never experience such a wonderful human art.  Is the status quo of mathematics education better than nothing?  It's difficult for me to estimate.  I don't know how many people are and aren't inspired somehow to really learn rather than memorize (I got out of high school just a few years ago, and I'd personally peg it at about 5% of students who I think really get something out of math classes.)  So that's all I have to say.

k24anson on K5: Imagine fifty, sixty year old men and women still playing with their genitals like ten year olds!

[ Parent ]

Aha (2.80 / 5) (#52)
by MechaA on Thu Jan 25, 2007 at 10:36:10 PM EST

This comment you wrote is exactly what I meant.

That realization is just about the only thing worth teaching in mathematics and it is almost the only reason to ever learn long division in the age of a calculator. I bet that even you would enjoy mathematics if all of your mathematical understanding were built piece by piece on realizations and extensions like that - and that is exactly how "math people" learn the subject.

I hope that's a goal that we can achieve someday with mathematics education.



k24anson on K5: Imagine fifty, sixty year old men and women still playing with their genitals like ten year olds!

[ Parent ]
"Math is hard" (2.22 / 9) (#33)
by I am teh Unsmart on Thu Jan 25, 2007 at 09:44:42 PM EST



[ Parent ]
I'm very sorry... (2.83 / 12) (#36)
by Coryoth on Thu Jan 25, 2007 at 09:51:02 PM EST

...that you had such poor teachers and apparently didn't get a chance to really learn mathematics. When you say

there's no evidence that math somehow teaches abstractation or logical thinking.

That is, I'm afraid, merely an indictment upon those who claimed to teach you mathematics, and says nothing about mathematics itself. Why is that? Because mathematics is abstraction and logical thinking. You say "teach a course in pure logic" but what do you think mathematics is? Seriously, formal symbolic logic was developed as part of mathematics. Indeed, some of the great early texts on the subject referred to the field as "mathematical logic" (and many still do).

Mathematics is the art of abstraction - abstracting away as much as is possible so that statements apply as broadly as possible. And mathematics is the natural playground for logic because such carefully constructed abstractions can be fully prescribed by rules. Mathematical objects are, if you like, toy objects upon which logic always and consistently works.

It's not like I'm suggesting that you be stuck with the horrid education grinding away at a quadratic formula that you don't understand and no-one cares to explain that you were apparently stuck with. I'm suggesting that we keep the core concepts of abstraction and logic in mind, and use mathematics - their natural realm - to teach, to reinforce, to let kids play with those concepts.

But apparently you were mathematically abused as a child by teachers who, I very strongly suspect, didn't understand, nor like, mathematics themselves. So yes, they never bothered to explain what quadratics are, why they are relevant to mathematics, and what the quadratic formula means.  That requires understanding and insight on the part of the teacher. So lacking insight they just made you chug through the drudge work.

You abuse at the hands of teachers is no excuse to inflict it upon everyone else though.

[ Parent ]

Oh look (2.33 / 6) (#37)
by MechaA on Thu Jan 25, 2007 at 09:57:59 PM EST

This guy is sorry too.  We're all very sorry for kitten.

k24anson on K5: Imagine fifty, sixty year old men and women still playing with their genitals like ten year olds!

[ Parent ]
Now accepting donations. (2.50 / 4) (#67)
by kitten on Fri Jan 26, 2007 at 03:39:17 AM EST

I could hold out a sort of Pity Jar or something.
mirrorshades radio - darkwave, synthpop, industrial, futurepop.
[ Parent ]
Killer example to shut-up ignorant poster (1.00 / 3) (#162)
by lyapunov on Sat Jan 27, 2007 at 10:43:48 PM EST


"Despite the bleating assertions by math geeks, there's no evidence that math somehow teaches abstractation or logical thinking."

Let me show the difference between a logical, cartesian mind and a common mind:

http://www.youtube.com/watch?v=Gp0HyxQv97Q

(The common mind cannot distinguish the difference between 0.002 dollars and 0.002 cents, even with the help of a calculator)


[ Parent ]

So you found some complete dumbass (1.50 / 4) (#217)
by kitten on Tue Jan 30, 2007 at 02:07:47 PM EST

And this proves what, exactly?

Not much, except that you're a pretentious twit who uses, in seriousness, phrases like "cartesian mind".
mirrorshades radio - darkwave, synthpop, industrial, futurepop.
[ Parent ]
Examples: (2.75 / 4) (#206)
by Pentashagon on Mon Jan 29, 2007 at 03:46:57 PM EST

While I respect your devotion to the discipline, you seem to share a propensity that most mathematically-inclined types have, which is the inability to see that math isn't all that useful to most people. They aren't the ones designing bridges or developing encryption schemes or calculating orbital trajectories or proving theorems or even just measuring areas. They're just cab drivers, politicians, ranch hands, businessmen, lawyers, factory workers, managers, deliverymen. Math is of professional use to a select few, of personal interest to a few others, and useless to everyone else.

Here's some concrete reasons why mathematics is important to "normal people":

Currently, the world (I generalize, and there are notable exceptions which should be lauded) is involved in a "war on terror" that is numerically foolish. More civilians and armed forces personnel have now been killed in the war than were actually killed in any of the acts of terrorism that supposedly triggered the war. In terms of risk and reward, the war simply doesn't make sense. It would be cheaper and safer for everyone to just buy terrorist insurance and stop messing with rogue nations in the middle east.

In recent American elections many counties have had a greater variance between election outcomes and polling results than would be permissible in any other country. What this means is that there is a very low probability that the election results were actually valid, and most people don't care. The reason is that in their mind they can't understand how polls and surveys actually work and why they can be trusted; they just assume (wrongly) that there's a greater probability that the polls were wrong because they don't understand probability theory either.

The reason most Americans' finances are in poor shape is that they don't understand the mathematics necessary to properly invest their money and avoid debt. If you think the quadratic formula is hard, try working with recurrence relations to describe your income, investments, debts, and expenses. Almost everyone, if asked to calculate the number of payments needed to pay off a loan at a given interest rate with a given monthly payment would be hopelessly lost. People just assume that their car, house, or boat costs $X a month and leave it at that, and hope their 401K will cover them in retirement, or that social security won't be bankrupt.

The number of people who believe that creationism must be true, radioactive dating is unreliable, the fossil record and the grand canyon must be extremely young, and numerous other absurdities are just icing on the cake.

[ Parent ]

Not bad. (2.33 / 3) (#209)
by kitten on Mon Jan 29, 2007 at 10:44:52 PM EST

But I understand all these things and I don't know a damn thing about math. How do you explain that? I'm not asking to be snotty.

Also, a lot of your examples may have light shed upon them with the judicious application of math, but you fail to factor in human emotion and indoctrination. It doesn't take much of a grasp of math to see that American deaths due to terrorism are a tiny percentage of overall deaths, and the number is greatly outshadowed by the number of American deaths due to, say, car accidents or heart disease or even Americans shooting other Americans.

But numbers and math won't convince people. They -- perhaps even "we" -- are accustomed to things like shootings, heart attacks, and accidents. They happen slowly, to a few at a time, and no one makes a big deal out of it. But a one-time attack, particularly as dramatic an attack as the Trade Center, makes people panicked, nervous, fearful. Quoting numbers and statistics at them won't outweigh their fear.

That's just one example. Finances are another -- yes, someone could sit down and crank out I=PRT or whatever the equation is which describes whatever loan system they're using, but the notion of having something cool now is more appealing, and they're willing to cast off the money issue until later. The idea that they'll still be paying for it twenty years later is unfathomable to most people, even if they crunch the numbers.

Just speculating, but I think a large part of my ability to grasp these things, while not really understanding the math, comes from my father's devotion to science (he holds a Master's in physics), and his drumming into me the usefulness of logical and critical thinking and the scientific method. I may not be able to run the numbers the way some of you can, but the process by which one arrives at a sound and valid conclusion is ingrained into me. It's also a lot easier to grasp, I think, than raw mathematics. Perhaps, if we want our future generations to stop being the imbeciles you describe in your post, we should instill in them a reverence for the scientific method, rather than running them through drills and formulas and hoping to god they "somehow" suck logic out of it.
mirrorshades radio - darkwave, synthpop, industrial, futurepop.
[ Parent ]
Number and math won't convince eh... (1.50 / 2) (#210)
by Coryoth on Mon Jan 29, 2007 at 11:48:57 PM EST

You say numbers and math won't convince people, but did you stop to consider that that may be because they don't understand the numbers and the math? People who have a solid grasp of math, for instance, tend to be convinced by solid mathematical arguments, and to trust those arguments over emotional their own reponses. If you don't have the same math background you don't have the same degree of trust in math, and you fall back on the emotional response. Numbers and math only fail to convince people who haven't learned enough math to have come to truly trust it over their intuition and emotion.

[ Parent ]
Abstraction, logic, and mathematics (1.66 / 3) (#211)
by Pentashagon on Tue Jan 30, 2007 at 02:15:23 AM EST

But I understand all these things and I don't know a damn thing about math. How do you explain that? I'm not asking to be snotty.



How well do you understand them? Could you derive the results on your own, or double check the results of someone else? If so, you obviously know more math than you say you do. If not, then how do you know that they are right? Trusting experts can be dangerous, and it basically comes down to who you view as experts. When the general public isn't able to verify things, news reporters and the government and unfortunately even scientists resort to inaccurate generalizations or metaphors that confuse or mislead people. Heck, when the general public doesn't know enough about mathematics and science Fox can air a show about the "fake moon landing" and get away with it.

Also, a lot of your examples may have light shed upon them with the judicious application of math, but you fail to factor in human emotion and indoctrination. It doesn't take much of a grasp of math to see that American deaths due to terrorism are a tiny percentage of overall deaths, and the number is greatly outshadowed by the number of American deaths due to, say, car accidents or heart disease or even Americans shooting other Americans.

It took Ralph Nader to get seat belts and other safety features in cars, not just raw statistics. The reason is that the car companies did have the statistics, and people didn't. They concluded that cheap cars without safety features cost them less to build and made more money, even including the threat of lawsuits. People still haven't figured out why terrorism in America is not a greater threat than the erosion of liberties, mostly because they can't equate the number of people who could possibly have their lives ruined by witch hunts with the number that might be killed due to terrorism. It's difficult to argue persuasively using statistics and probability if the audience thinks 72.8% of statistics are made up on the spot and couldn't check them if they cared. It's just another appeal to authority in the mind of the general public.

Just speculating, but I think a large part of my ability to grasp these things, while not really understanding the math, comes from my father's devotion to science (he holds a Master's in physics), and his drumming into me the usefulness of logical and critical thinking and the scientific method.

The scientific method requires quantifiable results to be testable, and quantification requires mathematics. Knowing how to properly apply statistics to science is a field in itself, and understanding it is vital for valid scientific work. My guess is that you do understand more about math than you think, you just think about it  in terms of logic rather than mathematics. In reality, they are equivalent in a rather beautiful way. Godel numbering allows any statement in logic to be translated into numbers such that the rules of logic are simply arithmetic operations, and proofs can be checked by algorithms. Logic is used to build mathematics out of set theory. It's really all the same bucket of ideas with different representations and rules. If you want to learn about these ideas, a fun and interesting book is Godel, Escher, Bach. It basically takes the reader from set theory through Godel's Incompleteness Theorem with lots of interesting stops along the way. It even offers one of the most reasonable (to me) explanations of what modern art is.

I agree with you that simply presenting the rules of mathematics without an explanation is probably not the best way to do it. The problem is that the steps from pure logic and set theory to mathematics are rather long and complex, and in reality you have to know some mathematics to know why the steps are useful or desirable. What aspects of mathematics did you find most difficult? From the rest of the thread it sounds like equations in general (the quadratic equation in particular) and division pose some of the hardest to understand.

The best example of the quadratic equation I can ever come up with is that of stopping a car to avoid a collision. The most pertinent result of solving the physical equation of motion for a braking car is that the stopping distance is quadratically related to the speed, so that means traveling twice as fast requires four times the stopping distance. It's also why speed limits have fine gradations at the lowest speeds: It takes almost twice as far to stop at 35 MPH than at 25 MPH. It only takes 1/3rd the stopping distance dropping from 25 MPH to 15 MPH, roughly 1/6th the distance at 35 MPH.

The reason long division has a remainder is that division is only a valid algebraic equation when q*d + r = p, where q is the quotient, d is the divisor, r is the remainder, and p is the dividend. Using the equation for division it's easy to see how the remainder is equivalent to the fractional part of rational division: (q*d + r) / d = p / d = q + ( r / d ), which results in the whole number quotient plus a fractional part. This is because q is calculated by the long division algorithm to be the greatest integer such that q*d is less than or equal to p. This means p - q*d is also less than d, which means r is less than d and r / d will always be greater or equal to 0 and less than 1, e.g. the fractional part of the division. Like lots of people said, figuring out that the long division algorithm actually produces a q with the desired property is the hard part to prove or recognize as a child. In fact, it's not true in general because q*0 + r = p has no greatest value of q that satisfies the equation q*d < p. All q satisfy the equation, so the result of division by zero is taken to be undefined.

Hopefully you can see the logic that's used in deriving the above. Mostly, when construction the set of natural numbers from set theory, the axioms and theorems are concerned with proving two things: The set of natural numbers exists and is infinite (every natural number n has a successor,n+1 and no number's successor is zero which gives a basis point for induction proofs but disallows negative natural numbers), and that every operation of arithmetic is a well defined function from a pair of distinct natural numbers to a unique natural number. For every two natural numbers addition, subtraction, multiplication, and division must be well defined so that the answer is one natural number, with the exception for subtraction and division that the result must exist in the set of natural numbers. So subtraction is only valid for a - b if b is less than or equal to a, and for division there must exist natural numbers such that q*d + r = p, and only one q and r are the correct answer for any choice of d and p except d=0. Basically, the rules at this point are so simple that they're pretty much common sense. The power comes from being able to prove every result beyond arithmetic using only those initial rules (probably a couple I didn't mention) and the axioms of set theory. The integers are a quick extension to the naturals using negative numbers, and the rationals and reals and complex numbers follow by keeping in line with the rules for distinct inputs and unique outputs for operators.

How much formal logic have you played with? You might find it interesting to look at how set theory and number theory are built from set theory using logic.

[ Parent ]

Math and logic (1.50 / 4) (#214)
by Coryoth on Tue Jan 30, 2007 at 12:29:22 PM EST

The relationship between mathematics extends further and deeper than you might think. Indeed mny basic principles of logic can actually be shown to be equivalent to purely mathematical statements. For instance, take DeMorgan's law:

¬(p∧q)↔(¬p)∨(¬q)

It can (and has) bee shown that this is equivalent to (among other things) the proposition that every maximal ideal in a commutative ring is a prime ideal (see ring (mathematics)). That is, each statement can be deduced from the other - the mathematical statement implies DeMorgan's law as muh as the other way around. At their heart mathematics and logic are deeply intertwined and essentially inseparable.

[ Parent ]

The rules of logic and mathematics (2.00 / 2) (#224)
by Pentashagon on Thu Feb 01, 2007 at 02:31:37 PM EST

Actually all you need are the axioms of set theory and category theory, and after that all of mathematics is merely a deduction from those axioms. Technically this means that any theorem can be derived from any other by adding the missing lemmas from one theorem to the second and identifying the isomorphisms between the definitions used in the theorems. At the set theory level, definitions are just derivations of axioms themselves, and so isomorphisms should be apparent. Most mathematics proofs are expressed at a higher level where the isomorphisms between definitions can be lost in the notation.

The fact that all of mathematics is described by the 7 (or 8 with choice) axioms of set theory is stunningly beautiful. If you're a mathematical realist, it's even more beautiful because it implies that even though the formal axioms were only discovered in the past couple centuries the relationships between mathematical objects has always existed, and always will. I don't see any reason to doubt that claim, because it's obvious that whatever humans do with mathematics is immaterial to the results, given certain axioms. Finding the axioms is the key that unlocks the mysteries of the universe. Going one step further, I think that the physical universe itself is just a manifestation of mathematical realism, that it exists simply because it and its rules are equivalent to some Set (or category of sets). Physics is just the study of finding the isomorphism between the objects of the universe and mathematics.

[ Parent ]

Logic and math (2.50 / 2) (#225)
by Coryoth on Thu Feb 01, 2007 at 03:49:27 PM EST

I'm not a mathematical realist. I'm not even a logical realist. I'm a logical pluralist. The set theory axioms also rest on axioms of logic, many of which can be profitably denied (the law of excluded middle is a popular target). The point is that you can drop these axioms from your logic, and find purely mathematical results that imply them. For example, it is possible to infer the law of excluded middle from the axiom of choice - indeed even finite choice is sufficient. Thus your mathematics can define your logic (and, of course, your logic can define your mathematics: drop the law of excluded middle and accordingly the axiom of choice and you end up with Constructivist mathematics and non-standard analysis - a quite different, but very profitable field).

I guess if you are a mathematical realist, and see there being "one true mathematics" based on "one true logic" then the interplay between logic and mathematics isn't interesting. On the other hand I am happy to accept that there are many logics, and many mathematics that can flow from them - so to have mathematics actually define and influence your choice of logic is an interesting result.

[ Parent ]

Logical framework equivalence (2.00 / 2) (#226)
by Pentashagon on Thu Feb 01, 2007 at 08:59:12 PM EST

I guess I'm not really sure what to call the theory that I'm describing if not mathematical. As you point out, even logic is not immutable. Perhaps I should call myself a formal system realist? Even though standard and intuitionist logic differ in their axioms and mathematical results, they are not mutually exclusive because either can be embedded within the other using an encoding. Take intuitionist logic and define set theory, and then define the terms, variables, and rules of standard logic in the intuitionist set theory and the standard mathematical results can be derived in a subsystem of intuitionist logic. The class of all formal systems that can embed each other is what I would call Mathematics, although I can't say for sure whether set theory (or any of our human logics) can actually represent all formal system. The question of which logical framework and resulting mathematics is "true" is really just a question of which universe it's used in to describe the physical properties of that universe.

Perhaps all I'm claiming with my concept of modal/mathematical realism is just a silly tautology like "everything that exists exists," but I think it's closer to "everything expressible with set theory exists" which has a little more philosophical value than the first statement.

[ Parent ]

Without back-of-the-envelope, you're one-handed. (none / 0) (#236)
by grendelkhan on Tue Feb 06, 2007 at 12:25:11 AM EST

Skeptical thinking and a finely-tuned bullshit detector are absolutely key to doing good science, which uses principles applicable in one's daily life (consider buying a used car sans skepticism). However, without the ability to do back-of-the-envelope math in order to see if a hunch is way, way off the mark, you're missing a vital tool.

"Running the numbers" isn't just an excercise in self-indulgent dorkery; without it, you're stabbing in the dark for your answers. (Example: based on the expected price of gas and the mileage you drive, is it worth paying such-and-such more for a car that gets such-and-such better mileage?)
-- Laws do not persuade just because they threaten --Seneca
[ Parent ]

Another point... (none / 0) (#245)
by EngineeringEmo on Tue Feb 13, 2007 at 09:49:09 PM EST

Most people either can't or simply don't do difficult math problems in their head, it doesn't matter what their job is.

I do engineering for a living. Tonnes of math. It's ridiculous sometimes. I even end up scaring others because I have to be able to visualize certain mathematical concepts as processes with respect to time. Thing is, in the real world, if you need a calculator, it's just as likely to be close at hand as a pad and piece of paper.


[ Parent ]

My experience (1.91 / 12) (#8)
by QuantumFoam on Thu Jan 25, 2007 at 07:20:01 PM EST

I went to a mediocre elementary school and then moved to one of the best school districts in the state for junior high and high school.

I was in the advanced programs in school from the get-go. We went faster than other classes, but the style of instruction was pretty much the same. I excelled in math at first, but quickly became bored. I started covering topics on my own before we got to them, learning division when the other kids were still on multiplication and so on. I taught myself Calculus in high school and in my personal studies was usually one level of science ahead of the class.

The upside was that I learned how to teach myself things. The downside is that I usually don't learn them the way the authorities want us to, which was fine in high school, but in college has come to bite me in the ass. I've been a senior for four years now...

I often thing I would have been better off being home schooled, but I did learn some important lessons about authority. In the fourth grade I did a report on nuclear weapons that was too technical for my teacher's tastes, and she sent me to the school shrink.

- Barack Obama: Because it will work this time. Honest!

Yeah (2.57 / 7) (#14)
by debacle on Thu Jan 25, 2007 at 08:12:18 PM EST

I loved being told in 12th grade to dumb down my paper on robotics so that the English teacher could understand it.

It tastes sweet.
[ Parent ]
Perfectly sane advice (2.71 / 7) (#165)
by damiam on Sat Jan 27, 2007 at 11:37:40 PM EST

You always need to consider your audience: there's no reason to expect an English teacher to have any robotics background. If you're writing for laymen, explain things in layman's terms.

[ Parent ]
Calculus in High School (1.33 / 3) (#227)
by aralin on Fri Feb 02, 2007 at 04:06:38 AM EST

I taught myself Calculus in high school and in my personal studies was usually one level of science ahead of the class.

That is the problem. The fact that Calculus was not part of standard curriculum in high school. That is why the rest of the world has such a low opinion of the US high school education.

Calculus was the standard part of my high school math curriculum, along with algebra, graph theory, set theory, discrete math, trigonometry, descriptive geometry and other advanced topics. I cannot imagine why is it so difficult to teach those when you've got 12 years to do it.

As math advances even further, how can you as math major get to the stuff on the border of known to work on something novel in your thesis if you don't get the basics by the end of high school?

And if you are not a math major, high school is the end of your math education. How can you get along in other topics, like law, medicine, or social sciences without the problem solving skills  and ability to apply algorithms to find solutions, which you should have learned by studying those topics?


[ Parent ]

Great article! (2.16 / 6) (#9)
by GreenYoda on Thu Jan 25, 2007 at 07:34:46 PM EST

Well written, and the YouTube video was a real eye-opener.

Another significant problem with math teaching might be that the teachers who teach it don't understand it very well.  You might want to check out an interesting book: "Knowing and Teaching Elementary Mathematics: Teachers' Understanding of Fundamental Mathematics in China and the United States", by Liping Ma.  Seeing some of the simple concepts that elementary school math teachers don't understand will shock you.

I doubt it would shock me (2.20 / 5) (#32)
by Coryoth on Thu Jan 25, 2007 at 09:37:38 PM EST

I'm quite well acquainted with how little many primary/public/elementary school teachers know about mathematics. Indeed, I think that one of the problems is that, currently anyway, the people who most often go into primary teaching are people who have a serious distaste for mathematics. The distaste tends to rub off on the kids, and where it doesn't the teacher knows sufficiently little about math that lessons become rote and mechanical, explicitly following whatever lesson plan was given and providing no enlightenment, no zest, no inspiration. Is it a surprise that, consequently, many children get turned off mathematics early?

As I mentioned in another comment, one of the major changes Finland made (which has resulted in Finnish students being some of the best in the world at mathematics) has been getting more elementary school teachers better versed in, and more interested in, mathematics.

[ Parent ]

The University I went to (1.50 / 2) (#75)
by wiredog on Fri Jan 26, 2007 at 08:49:38 AM EST

Was started as a teacher's college (about 100 years ago), and that remains the main focus. So my Cs/ Math Minor could also have been a Math Education minor, without changing any of the math classes, and adding only 2 or 3 education classes. Heck, could've made it a double minor.

Beyond calculus there weren't many of us in the classes. 4 of us (2 men, 2 women) is the theory course (can't remember what it was called, but it was set theory, proofs, boolean algebra (one of the most useful courses in the real world, along with stats)).

In calc it was just assumed you knew your algebra and trig (I'd forgotten it, and sweated bullets re-learning it in a hurry). In later courses it was asumed you knew calc.

No calculators allowed in class. Why use a graphing calculator in calculus class to help draw the graph of the function when a little calculus will give you the maxima, minima, and inflection points?

Sadly, other than stats and that theory course, I've forgotten most of the math through disuse. Sure, I can still integrate exdx, but who can't?

Wilford Brimley scares my chickens.
Phil the Canuck

[ Parent ]

Good article (1.16 / 6) (#15)
by debacle on Thu Jan 25, 2007 at 08:13:23 PM EST

On an important topic.

It tastes sweet.
Just... (1.33 / 6) (#16)
by gr3y on Thu Jan 25, 2007 at 08:17:08 PM EST

one editorial comment: "multiplications and divisions" = "multiplication and division." Otherwise, spot on.

I am a disruptive technology.

mathe vs. music (2.44 / 9) (#18)
by sye on Thu Jan 25, 2007 at 08:20:25 PM EST

I offer a somewhat different view point. Mathematics and music composition used to belong to the same and highest branch of human intelligence. In US, more people can afford to spend their adulthood career in making and playing music which is THE measure of mathematical civility in a developed country. In Asia, GO/围棋 is another purest pursuit on the beauty of abstract representation of human intellect life.

We shouldn't let politicians define what is 'proper' mathematical education for us when we experience injustice to what ought to be abundant and enduring human life at the cost of human reason and human calculation...

~~~~~~~~~~~~~~~~~~~~~~~
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rubbing u ~~> ~procrasti: getaway to HE'LL
Hey! at least he was in a stable relationship. - procrasti
enter K5 via Blastar.in

mindless commentary propagation (2.50 / 4) (#27)
by oilmoat on Thu Jan 25, 2007 at 08:47:32 PM EST

I'm sure you can find this tidbit on bbc from a year ago or so: in the UK A level math applicants are dropping. For gender equity, the grading of math solutions was changed to partial credit versus binary grades for correct or incorrect proofs. Which turned off guys but helped women since they are more what's-the-word, persistent/consistent.
I have IBPND. (I believe in people, not disorders.)
Rote symbolic manipulation (2.33 / 12) (#31)
by I am teh Unsmart on Thu Jan 25, 2007 at 09:25:25 PM EST

isn't mathematics, either. There's also far too much time devoted to learning arithmetic and a lack of rigor in subjects at the secondary level. Thanks to New Math backlash, sets are introduced too late, proofs are frequently elided, and most textbooks feature pictures of children every 1-3 pages while having maybe one page of exposition spread across three pages with worked examples and several pages of clustered rote exercises. Useful subjects such as statistics are often electives, and many people seem to graduate thinking trigonometry and calculus sans proofs are the height of mathematics. When people that aren't spending USD120k to write papers about sentimentality get to college, a year is wasted teaching many of them Calculus without a short bus. Subjects like Linear Algebra, Discrete Mathematics, and Calculus 2 get turned into 'gateway classes,' where a decade of brain rot must be corrected.

There's too much pandering to the laziness of U.S. children in most subjects, because parents don't pay attention to their children if they can help it, and learning cuts into their television and Playstation time.

Meh? (1.80 / 5) (#81)
by jmzero on Fri Jan 26, 2007 at 11:25:59 AM EST

There's too much pandering to the laziness of U.S. children in most subjects, because parents don't pay attention to their children if they can help it, and learning cuts into their television and Playstation time.

I'm sure things vary, but the kids I know do more hours of homework than I think is healthy or necessary.  I think they're working plenty long and hard enough, they're just not given the right work to do or taught with the right philosophy.
.
"Let's not stir that bag of worms." - my lovely wife
[ Parent ]

If any class should be mandatory (3.00 / 2) (#124)
by nightfire on Fri Jan 26, 2007 at 10:48:43 PM EST

It's statistics.

How many ridiculous situations could we avoid if the majority of a country's constituents had a solid foundation in statistics?

Every god damned month we hear in the news about some new plague, tragedy or opportunity that requires urgent action.  Without the basic skills required to evaluate risk, how are people to make sound decisions?

In Ontario, last year, there was a motion to make mandatory the installation of car alarms for all new vehicles sold in the province.  The rationale was "15 people have been killed in car chases" in the last 10 years.

One point five persons per year.  In a province of what.. 10 million?  Where thousands have died due to simple incompetence behind the wheel?

Of course the point was - someone stands to make a lot of money selling car alarms.  But all over the net you could read arguments like "but just one life is too many."

Not even going to discuss things like war, probability of terrorism, or (gasp) smoking lethality.

A solid foundation in statistics would enable people to properly weigh cause and effect, analyze risk, and make sound decisions.

[ Parent ]

This is a surprise? (1.90 / 10) (#40)
by HackerCracker on Thu Jan 25, 2007 at 10:01:06 PM EST

The masses sent to volkschulen aren't supposed to be given anything more than they need to either a) punch a cash register, b) fill out a form, or c) follow orders.

Declining math scores means absolutely nothing--nothing, that is, to people who are running the show.

It's Hauptschule (1.60 / 5) (#77)
by tetsuwan on Fri Jan 26, 2007 at 09:48:56 AM EST

Don't pretend that you know things German.

Njal's Saga: Just like Romeo & Juliet without the romance
[ Parent ]

Volkschule is correct, (1.25 / 4) (#87)
by Joe Sixpack on Fri Jan 26, 2007 at 12:26:37 PM EST

it means primary school in Austrian German.

---
[ MONKEY STEALS THE PEACH ]
[ Parent ]

interesting subject..well done imo..+1FP (1.33 / 12) (#45)
by dakini on Thu Jan 25, 2007 at 10:17:48 PM EST



" May your vision be clear, your heart strong, and may you always follow your dreams."
proofs are gay. (1.42 / 14) (#50)
by the spins on Thu Jan 25, 2007 at 10:29:38 PM EST

suck it, mathboy.

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I've never been much for classroom learning (2.00 / 9) (#57)
by Morally Inflexible on Fri Jan 26, 2007 at 01:05:26 AM EST

But I'm currently trying to change that; I'm going back to school, and will be getting at least my undergrad in math. I'm preparing by reading through the classics; I started with G. Pollya's "how to solve it" (an excellent and extremely accessible introduction to the topic) and am now halfway through book one of "The Elements" (the Heath translation) - We will see how it turns out, but really, I fail to see why we don't use a translation of "the elements" (supplemented perhaps by commentary that is less historically focused than the volume I am reading- also, the references to non-Euclidean geometries probably should be reserved for the AP class.) to teach Geometry; The book seems quite a bit more clear than anything I was given in high school.

I didn't take proofs in high school, so learning those is currently my big goal; It is difficult for me to restrain the 'intuitive jump' and complete the proof. I've got a book titled "how to solve it" around here somewhere; I can't find it, so I can't give you the author.

But really... my problem with high school math was not the curriculum; it was the motivation. I just didn't see why I should put in the effort at the time. I think this is the problem most Americans have with Math.

see, there is no respect for learning; a liberal arts degree is generally considered as good as a math-based degree. Socially speaking (especially in the high-school environment, but also in the wider American pop culture to a lesser extent) math types are at the bottom of the pile. To some extent this is inevitable (I mean, if I'm spending all this time learning about math, that's less time I have to spend learning about being popular, right?) but I think this is worse in America than most other places.

In places with a larger wealth gap (places with a real wealth gap; places where being poor means you have to work hard just to survive.) the advantages conferred by the larger income granted by the math education can overcome this, but in places like America, in high school, at least, it seems irrational for anyone but those that are already too damaged to have a normal social life to pursue a mathematics education at all. This, I think, is the real problem that the US school system has, and I don't see it changing anytime soon.

Now, once you get out of high school, and can choose to largely ignore the people that don't value learning, things change dramatically, but that is non-obvious when you are in high school.

The Declining Quality of Grammar Usage (2.38 / 13) (#60)
by memetomancer on Fri Jan 26, 2007 at 01:39:16 AM EST

I hate to say it coryoth, but your article shows a converse relationship to the inverse of improperly formed English Sentence Construction, or ESC. Your sentence structure, comma usage and paragraphs are all a-whack. It is a common shortcoming of specialists in other fields besides Grammars.

Perhaps I'd even go so far as to say that the individual writing this new math article has lost sight of the core skills that early education should be instilling.

I'm not sure whether my observation illustrates some withering irony or instead bolsters your point somehow, so don't be alarmed if you suddenly realize that all education everywhere has always been horribly terrible. Those idyllic times of yore you hear about are lies. The students may have performed better, but the only applicable difference between now and then lay in attitude, not technique.

That's not decline. That's called reform (1.83 / 6) (#109)
by United Fools on Fri Jan 26, 2007 at 06:34:51 PM EST

We are reforming the language from one of the elites' tools to the language of the common people.

Democratic English.

We are united, we are fools, and we are America!
[ Parent ]

Well, next thing you know, (2.00 / 11) (#70)
by i want plentiful cheap gasoline on Fri Jan 26, 2007 at 03:55:21 AM EST

we'll have pig farmers posting here telling us how important they and their jobs are, making all kinds of demands on the education system and whining about their social status.

DIAF, nullo $ (1.50 / 4) (#113)
by HackerCracker on Fri Jan 26, 2007 at 07:15:03 PM EST



[ Parent ]
sp math - > mathS (1.15 / 13) (#73)
by A Bore on Fri Jan 26, 2007 at 07:37:31 AM EST

It's short for mathematicS you illiterate worthless yank skunks.

So... (2.50 / 4) (#222)
by kitten on Thu Feb 01, 2007 at 03:00:17 AM EST

You have no problem with leaving off the string of letters "ematic", but by God, you're going to demand that S at the end!

Welcome to abbreviation land, old chap.
mirrorshades radio - darkwave, synthpop, industrial, futurepop.
[ Parent ]
the value of mathematics (2.20 / 5) (#74)
by circletimessquare on Fri Jan 26, 2007 at 08:38:17 AM EST

rests not in remembering the quadratic equation. 99.999% of us students will learn how to solve ax2+bx+c=0 in the seventh grade and then never, ever use the skill again in their entire lives. the value of mathematics rests not in many of the bs middle school kids learn (cosine, sine, tangent, etc.)

the value of mathematics is similiar to the value of learning to play a musical instrument, another skill that is quickly lost in adulthood: music and math create intelligent minds. simply put, the value of a math and/ or music education is that exposing young minds to them makes the brain grow. that's really about it. so math is important for kids in an indirect, but very important way

The tigers of wrath are wiser than the horses of instruction.

Schooling isn't for the dull majority (2.00 / 5) (#76)
by A Bore on Fri Jan 26, 2007 at 09:46:15 AM EST

It's for the one or two who have it, and go on to become mathematicians. Everyone has to be taught the same to find these one or two people, it has nothing to do with musical instruments (which need to be played before 5 years old to excel in) or intellectual development.

It's a scattergun approach - teach everone a little of everything and let the best of them take each of their subjects on from then on.

And you solve a quadratic equation everytime you catch a tossed football. So I'd say 99.9999% of students use it every other day.

[ Parent ]
agreed, except last sentence (2.66 / 3) (#80)
by circletimessquare on Fri Jan 26, 2007 at 11:15:34 AM EST

that's like saying every time i fart i play mozart. nobody is doing math in their brain when they catch a football

The tigers of wrath are wiser than the horses of instruction.

[ Parent ]
They are doing math in their brain (2.50 / 4) (#85)
by wiredog on Fri Jan 26, 2007 at 12:18:25 PM EST

Just not consciously. That's one of the problems that the AI/visual perception guys (and neuroscientists as well) get wrapped around the axle about. How do we unconsciously solve problems like that?

Wilford Brimley scares my chickens.
Phil the Canuck

[ Parent ]
so birds are calculating bernoulli's equation? (1.33 / 3) (#98)
by circletimessquare on Fri Jan 26, 2007 at 03:38:50 PM EST

get real moron

The tigers of wrath are wiser than the horses of instruction.

[ Parent ]
You don't have to calculate Bernoulli's equation (1.50 / 4) (#100)
by wiredog on Fri Jan 26, 2007 at 03:53:16 PM EST

to fly. You do have to solve some equations to catch a football.

At least, that's true for computers, and no one has come up with a way to do it without calculating. Calculating intercepts in 3-space is difficult to do quickly, yet we do it automatically and unconsciously.

Wilford Brimley scares my chickens.
Phil the Canuck

[ Parent ]

my head asplode (2.25 / 4) (#102)
by circletimessquare on Fri Jan 26, 2007 at 04:13:55 PM EST

you think football quarterbacks are saying in their head

9.8 m/s squared... 106.34 yards... ... right bicep extend 3.4 inches... apply 4.56 joules of force in 3.95 seconds...

what kind of weird alternate universe do you live in?

man, i must so be getting trolled right now


The tigers of wrath are wiser than the horses of instruction.

[ Parent ]

where are you from? (3.00 / 4) (#111)
by diskis on Fri Jan 26, 2007 at 06:47:39 PM EST

Your universe is the really weird one. My muscles contract. I'm really curious to see how you move.

[ Parent ]
cts misspoke (2.80 / 5) (#170)
by A Bore on Sun Jan 28, 2007 at 05:23:29 AM EST

Instead of "bicep" he meant "pseudopodia".

[ Parent ]
Read my comments man (1.33 / 3) (#202)
by wiredog on Mon Jan 29, 2007 at 08:30:48 AM EST

I said "we do it automatically and unconsciously".

We may not be calculating the way you said it, but some calculations are done. We're probably not using lookup tables.

Wilford Brimley scares my chickens.
Phil the Canuck

[ Parent ]

why would they do that? (1.25 / 4) (#101)
by the spins on Fri Jan 26, 2007 at 03:54:43 PM EST


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[ Parent ]

why would quarterbacks use the quadratic equation? (2.00 / 3) (#103)
by circletimessquare on Fri Jan 26, 2007 at 04:14:27 PM EST

nt

The tigers of wrath are wiser than the horses of instruction.

[ Parent ]
Well, not by doing math (1.50 / 1) (#231)
by p3d0 on Sat Feb 03, 2007 at 08:16:41 AM EST

You solve those problems by working out "the last 100 times I saw the ball move like that, what did I have to do to catch it?"  That's why catching things requires practice.  Nobody is solving quadratic equations in their head to catch a ball.  The brain essentially works by pattern matching.
--
Patrick Doyle
My comments do not reflect the opinions of my employer.
[ Parent ]
If only... (1.50 / 1) (#230)
by cvalente on Fri Feb 02, 2007 at 06:31:37 PM EST

"It's a scattergun approach - teach everyone a little of everything and let the best of them take each of their subjects on from then on."

If that is the case, then why are math courses insanely stupid and try to castrate any kind of thinking involved up until at least one is 16 years old?

It would make more sense to try and provide the best for each field early on, and that doesn't happen at all.

This way no one is happy. Those who just want to learn to get by can't, those you want to understand things can't either. Everyone loses.

[ Parent ]

More than you think (1.00 / 4) (#86)
by Coryoth on Fri Jan 26, 2007 at 12:24:45 PM EST

I suspect you'll find that significantly more than 0.001% of students will use solutions to quadratic equations after the seventh grade. Roughly half of all students who go the college take a calculus or stats course - that's because its required material for pretty much all hard sciences, and even for a variety of soft sciences (I run into a lot of psychology, social science, and management science students who suddenly discover that those courses require a math background).

That, of course, doesn't account for everyone - only a percentage of high school students go on to college and, as noted, only about half of them will take a college math course. So I'm not suggesting that there aren't lots of people who won't use a lot of that material again, merely that there actually is a pretty significant percentage for whom the background mathematics given in high school is required material for their further course of study.

[ Parent ]

real life (1.60 / 5) (#99)
by circletimessquare on Fri Jan 26, 2007 at 03:41:52 PM EST

learning something you don't need in real life but you do need to learn something else later... that you ALSO don't need in real life doesn't mean anything

sometimes i think they should replace all of those algebra and geometry lessons with a lesson in basic finances: "this is a credit card, this is what interest rate means" etc.

much more valuable math lessons

young adults who know the quadratic equation but can't get out of credit card debt is pretty pathetic


The tigers of wrath are wiser than the horses of instruction.

[ Parent ]

depends on what you mean by real life (2.25 / 4) (#115)
by Delirium on Fri Jan 26, 2007 at 07:44:51 PM EST

If someone wants to become, say, a psychology researcher, and therefore needs to know statistics to do a competent job, doesn't that count as needing to know statistics for "real life"? Or do you only include hobbies and around-the-house errands in "real life"?

[ Parent ]
The thing is (1.25 / 4) (#190)
by kitten on Sun Jan 28, 2007 at 07:12:12 PM EST

that pointing to a specialized field and saying "That person needs this one bit of math to do their job" isn't saying much. Those for whom math holds zero interest are unlikely to enter such fields -- or in the case of your psychologist, they will learn just enough to understand that one thing they need, which is hardly a ringing endorsement for putting every single child through years and years of algebra, geometry, trig, and calculus.

mirrorshades radio - darkwave, synthpop, industrial, futurepop.
[ Parent ]
All you "real lifers" are pathetic.$ (1.20 / 5) (#133)
by V on Sat Jan 27, 2007 at 05:40:26 AM EST


---
What my fans are saying:
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[ Parent ]
I studied finance problems in highschool maths (2.00 / 3) (#135)
by Scrymarch on Sat Jan 27, 2007 at 11:53:09 AM EST

A compounding loan, the most common type, is a sum of a geometric progression, which requires reasonably solid highschool algebra to work with. This is usually trained by teaching quadratic equations, which are a useful stepping stone to calculus.

As for the quadratic equation, it was always taught to me as a necessary evil when factorising a quadratic didn't come out. I seem to recall it being most important to those who needed some memorisable content to hold onto in a sea of technique they didn't get.

[ Parent ]

Agreed (none / 0) (#232)
by p3d0 on Sat Feb 03, 2007 at 08:24:56 AM EST

They ought to teach both.  For instance, if they're going to teach us arithmetico-geometric series anyway, why not use mortgages as an example, and show the effect of different interest rates and amortization periods?
--
Patrick Doyle
My comments do not reflect the opinions of my employer.
[ Parent ]
They teach the quadratic formula in grade seven? (1.00 / 2) (#129)
by superiority on Sat Jan 27, 2007 at 01:06:26 AM EST

What is that, somewhere between 11 and 12 years old? In my country it's taught the year before your last at high school. Weird.

[ Parent ]
Varies a lot (1.33 / 3) (#164)
by damiam on Sat Jan 27, 2007 at 11:29:09 PM EST

In the US it's taught as part of an Algebra 1 class, which some kids take as early as sixth or seventh grade (11 or 12yo) and some kids take in their last year of high school (17+) - depends on how smart/stupid you are and how flexible your school is. Probably the most common track has kids taking it in ninth grade (14), with the "gifted" kids doing algebra in eighth grade.

[ Parent ]
Interesting topic.. (2.33 / 3) (#78)
by tweet on Fri Jan 26, 2007 at 10:06:43 AM EST

though not particularly well-written.  Maths can be a tough subject to teach and learn, and I speak from experience.

The hardest, but most rewarding, group I have to teach are medical students on an A-level-style course.  Hard because they have absolutely no flair for the subject and rewarding because they work so hard to improve themselves.  

I'm not sure about the other stuff we teach, but stats is a necessity for them - clinical trials, drug dosage etc.

_______________________________________________
Not everything in black and white makes sense.

Observations (2.50 / 6) (#83)
by LilDebbie on Fri Jan 26, 2007 at 12:13:23 PM EST

As a graduate of one of the best math education programs in the world (or was; my calc prof took time off his summer vacation to hurry a few of us through the program before it followed the lead of the rest of the country), I feel compelled to comment.

First as a bit of background, I started UMTYMP in 8th grade with algebra and finished a standard four semester engineering calc cycle in 11th. My final project for the program was the derivation of the Heisenberg Inequality from Plank's law. I don't mean to toot my own horn so much as to point out the relative difficulty of the program.

The aspects of the program that made it so successful (which, sadly, have largely been abandoned) were actually quite simple and, dare I say, conservative. First, though this wasn't universal, I had the same professor throughout the calculus cycle. Obviously this isn't a tenable solution for most educational institutions, but nevertheless it helped immensely in understanding the material. Second, and this is the most trivially easy change in mathematics education to effect, was a ban on graphing calculators (all calculators, for that matter). Problem sets were designed with minimal arithmetic, allowing us to better focus on the abstract. Later, I would ace the SATs using only a four-function calculator (for time) in a classroom filled with TI-XXs.

Anywho, my $0.02.

My name is LilDebbie and I have a garden.
- hugin -

Sounds like my university (2.50 / 4) (#84)
by wiredog on Fri Jan 26, 2007 at 12:15:37 PM EST

Same professor through calc (only 3 math professors the whole 4 years I was there.) No calculators allowed. In any math class.

Wilford Brimley scares my chickens.
Phil the Canuck

[ Parent ]
Four years ago or so (1.50 / 2) (#95)
by MechaA on Fri Jan 26, 2007 at 02:34:05 PM EST

When I took the SATs, they were designed specifically to not give any advantage to someone using anything more than a four-function calculator, for what it's worth.  Generally the numbers in the problems were simple enough that you could do it as quickly by hand.

k24anson on K5: Imagine fifty, sixty year old men and women still playing with their genitals like ten year olds!

[ Parent ]
Likewise (2.00 / 3) (#96)
by LilDebbie on Fri Jan 26, 2007 at 02:41:45 PM EST

There were a few questions where you could shave off a few seconds by foregoing the long hand. Not that it mattered in the end, but better safe than sorry.

My name is LilDebbie and I have a garden.
- hugin -

[ Parent ]
Greetings from the Elder Days! (2.50 / 2) (#130)
by rigorist on Sat Jan 27, 2007 at 01:44:14 AM EST

Greetings!  I was in MTYMP, too.  But I'm old enough that I was in the second year of the program before they added the "U" and we did algebra and trig at Hamline.

Everything shifted to the University of Minnesota in my third year.

[ Parent ]

It's terrible (1.66 / 3) (#176)
by LilDebbie on Sun Jan 28, 2007 at 01:31:10 PM EST

The year after (why my prof hurried us through) they starting using a text that many of the UMTYMP profs worked on under the auspice of designing a text for use in programs "with teachers lacking in the necessary mathematic education for the subject."

My prof left in disgust the next year. A shame, too. He was a great teacher.

My name is LilDebbie and I have a garden.
- hugin -

[ Parent ]

Aye (1.50 / 4) (#97)
by Aneurin on Fri Jan 26, 2007 at 03:28:57 PM EST

However, as a previous comment mentioned, attitude is the most important thing. The desire to abstract isn't something that tends to be 'taught' in school. My experience of mathematics has been a strange one since the UK is pretty abysmal. In secondary school each teacher had a totally different style which made it somewhat hard to follow-- these styles of teaching fell between complete apathy or total strictness; each missing the point that to be doing something, you must want to be doing it in the first place. At one point, I had three teachers for one class; all of them subsitute and with no ability to relate to the class. No surprises that nobody gave a fuck whatsoever.

Eventually I decided to pick up a textbook and, shockingly enough, teach myself. And my how things have progressed...
---
Just think: the entire Internet, running on jazz. -Canthros

With that multiplication method... (2.25 / 4) (#105)
by sholden on Fri Jan 26, 2007 at 05:33:14 PM EST

It'd be really tempting to make the final exam consist of:

61x83=?

--
The world's dullest web page


Done. (2.00 / 4) (#127)
by acceleriter on Sat Jan 27, 2007 at 12:18:02 AM EST

61 x 83 (60 + 1)(80 + 3) 4800 + 80 + 180 + 3 5063

[ Parent ]
You fail that exam then (2.25 / 4) (#143)
by sholden on Sat Jan 27, 2007 at 07:03:41 PM EST

But 60x80 is two digit multiplication still.

The lattice method was cool, it's intuitive and shows the place value system well enough (contrary to what the video says). Tedious though.

--
The world's dullest web page


[ Parent ]
I disagree (2.00 / 4) (#149)
by MechaA on Sat Jan 27, 2007 at 08:30:58 PM EST

The lattice system is a neat trick, but it takes more thought to figure out what is going on behind the curtain than with the standard algorithm, and in no way is it ever more efficient or easier to execute.

I'm not sure what about it you find intuitive or demonstrating the place value system, but I don't see either.

k24anson on K5: Imagine fifty, sixty year old men and women still playing with their genitals like ten year olds!

[ Parent ]

Don't get me wrong (1.25 / 4) (#159)
by sholden on Sat Jan 27, 2007 at 10:21:56 PM EST

I'd never actually use the thing, but the way it shifts the "tens" digit of a single digit multiplication up into the next diagonal shows place value to me. Of course if you didn't already know it won't teach you, but my first reaction was "neat trick".

I can't see any benefit over the traditional method,  which shows place value even better, and is less tedious and simpler. But compared to the ridiculous "do multiplication by repeated addition with some shortcuts if you happen to know some X*Y values" it's a real honest to goodness algorithm.

The strange thing is that the X*16 - oh that's just X*10+X*5+X  method is what all the people who were good at math did when doing calculations in their heads. That and things of the form: X*98 = X*100 - X*2. But teaching it as the underlying method is just dumb - it's the shortcut for when you don't have pen and paper.

Anyway how are you meant to possibly finish such a  calculation writing in the fogged up shower wall with your finger before it refogs up?

Oh well I guess it keeps the programming jobs reasonably safe from competition from young Americans :)

--
The world's dullest web page


[ Parent ]
the problem with all these 'we're in decline' (1.83 / 6) (#106)
by trane on Fri Jan 26, 2007 at 06:06:03 PM EST

hypotheses, is that average IQ scores have been consistently increasing for several decades.

We don't see that (2.20 / 5) (#108)
by United Fools on Fri Jan 26, 2007 at 06:32:08 PM EST

What we see is more fools keep coming out of the closet and declare for their rights.

We are united, we are fools, and we are America!
[ Parent ]
I hate when the average IQ increases! (2.40 / 5) (#110)
by I am teh Unsmart on Fri Jan 26, 2007 at 06:45:19 PM EST



[ Parent ]
no, thats a problem with the IQ system. n (2.75 / 4) (#116)
by livus on Fri Jan 26, 2007 at 07:48:51 PM EST



---
HIREZ substitute.
be concrete asshole, or shut up. - CTS
I guess I skipped school or something to drink on the internet? - lonelyhobo
I'd like to hope that any impression you got about us from internet forums was incorrect. - debillitatus
I consider myself trolled more or less just by visiting the site. HollyHopDrive

[ Parent ]
There's also an amusing problem with his reasoning (1.00 / 2) (#121)
by I am teh Unsmart on Fri Jan 26, 2007 at 09:17:28 PM EST

Namely that he confuses levels of education with measures of aptitude.

[ Parent ]
not necesassarily (1.33 / 3) (#122)
by livus on Fri Jan 26, 2007 at 09:25:00 PM EST

your interprettation may well be right but I thought he was referring to the aptitude of the people/society who sets the education programme, not the ones who undergo it.

Now that I examine this assumption of mine it seems less likely, thought.

---
HIREZ substitute.
be concrete asshole, or shut up. - CTS
I guess I skipped school or something to drink on the internet? - lonelyhobo
I'd like to hope that any impression you got about us from internet forums was incorrect. - debillitatus
I consider myself trolled more or less just by visiting the site. HollyHopDrive

[ Parent ]

s/ sets /set (2.50 / 2) (#123)
by livus on Fri Jan 26, 2007 at 09:25:34 PM EST

argh.

---
HIREZ substitute.
be concrete asshole, or shut up. - CTS
I guess I skipped school or something to drink on the internet? - lonelyhobo
I'd like to hope that any impression you got about us from internet forums was incorrect. - debillitatus
I consider myself trolled more or less just by visiting the site. HollyHopDrive

[ Parent ]
but (1.33 / 3) (#139)
by trane on Sat Jan 27, 2007 at 03:30:40 PM EST

even if there's a problem with the IQ measuring instrument, it is reasonable to assume that that problem would continue through the years at about the same level - that would be the null hypothesis. So an increase in scores, given the same amount of measurement error, still means that there is probably an increase in the actual IQ. I mean you'd have to get into the statistics to figure out the exact nature of that "probably", but until we do that it's a reasonable hypothesis that IQ is generally increasing, for some reason.

I guess you could argue that we're getting better at measuring IQ, but I don't know if they're using the same tests now as they did before.

[ Parent ]

I don't see this as a good thing. (2.00 / 3) (#141)
by Kasreyn on Sat Jan 27, 2007 at 06:30:03 PM EST

Since American public education is so startlingly lacking in training in ethics, critical thinking, and history. All you're talking about is producing an even smarter crop of amoral Arthur Andersen stooges. I'd say we ought to focus on other defects before we just make people smarter. To quote Einstein, "We should take care not to make the intellect our god. It has great muscles, but no personality."


"Extenuating circumstance to be mentioned on Judgement Day:
We never asked to be born in the first place."

R.I.P. Kurt. You will be missed.
[ Parent ]
but it depends on what is measured. (2.00 / 4) (#146)
by livus on Sat Jan 27, 2007 at 08:23:33 PM EST

We can go into any library in the country, and many book stores, and find books on how to increase my scores in these tests. We have been familiar with the kinds of things required by the tests since childhood; and we have been aware that these things give us "I.Q" and that IQ is supposedly valued by our societies.

Given this it's not at all surprising that the aptitude in what the tests measure has increased.

Compare this to something like pleaching a hedge;  it's also not surprising that aptitude in other things that are not valued in modern society has decreased.

(By the way I don't accept your point that it's reasonable to assume static state in I.Q tests, but that wasn't my main point here.)

---
HIREZ substitute.
be concrete asshole, or shut up. - CTS
I guess I skipped school or something to drink on the internet? - lonelyhobo
I'd like to hope that any impression you got about us from internet forums was incorrect. - debillitatus
I consider myself trolled more or less just by visiting the site. HollyHopDrive

[ Parent ]

to give an example (1.66 / 3) (#147)
by livus on Sat Jan 27, 2007 at 08:27:00 PM EST

my sibling and I used to play a game that involved answering IQ test style questions - we played it a lot because aside from the boring test questions it was quite good. When I first sat an IQ test, it seemed strangely familiar, and I felt I had an unfair advantage because of this game.  

---
HIREZ substitute.
be concrete asshole, or shut up. - CTS
I guess I skipped school or something to drink on the internet? - lonelyhobo
I'd like to hope that any impression you got about us from internet forums was incorrect. - debillitatus
I consider myself trolled more or less just by visiting the site. HollyHopDrive

[ Parent ]
sure but (1.66 / 3) (#208)
by trane on Mon Jan 29, 2007 at 09:28:56 PM EST

on average, it is probably safe to assume that most people who take the tests haven't prepared as you did.

Another theory I've heard explaining the increase is that a lot of the IQ test questions are visual, and we are a more visual society now, given TV and movies and advertising billboards. I don't know if I buy that one...

I'm not sure what is going on but I think that every generation looks at the changes in the education system and thinks that they herald a decline, but they are very rarely right. Maybe the Romans at the end of their time when the Dark Ages were beginning were right but there were other reasons for that such as the Huns and Visigoths and all destroying everything...

[ Parent ]

it's possible (2.50 / 2) (#219)
by livus on Tue Jan 30, 2007 at 07:47:52 PM EST

but if we qualify that by saying a decline in traditional skill and knowledge bases then they're probably right rather than wrong.

I mean, we can (and do) quantify declining levels in given areas (eg maths, grammar, child fitness).

---
HIREZ substitute.
be concrete asshole, or shut up. - CTS
I guess I skipped school or something to drink on the internet? - lonelyhobo
I'd like to hope that any impression you got about us from internet forums was incorrect. - debillitatus
I consider myself trolled more or less just by visiting the site. HollyHopDrive

[ Parent ]

that might be true by definition, though (none / 1) (#240)
by Delirium on Tue Feb 06, 2007 at 03:34:05 PM EST

If "traditional" is "that which was customary previously", then any shift in emphasis will result in a decline in the things that were traditionally emphasized.

[ Parent ]
Therefore it's declining if it doesn't change (1.00 / 3) (#205)
by Pentashagon on Mon Jan 29, 2007 at 03:11:17 PM EST

The article you linked is just an example of evolution. As more people are subjected to intelligence tests, society as a whole selects for educational methods and teachers that increase scores on intelligence tests. It's especially telling that they only tested modern people on old intelligence tests, and not novel new forms of intelligent tests that would most likely show an overall decrease in intelligence due to not being specially trained to take them. For instance, ask anyone about detailed events in history and you'll get markedly lower scores today than 50 or 100 years ago. Since we're doomed to repeat the history we don't remember, it's arguable that there should be some sort of History Intelligence Quotient.

[ Parent ]
You realize (1.33 / 3) (#216)
by kitten on Tue Jan 30, 2007 at 02:05:39 PM EST

that "average" IQ is always 100? If everyone in the world were zapped by aliens tomorrow, and were made five times more intelligent, then the average IQ would still be 100. It's not an absolute number -- it's a relative measurement.

If the "average" scores really have been increasing then the test isn't being calibrated or measured properly.
mirrorshades radio - darkwave, synthpop, industrial, futurepop.
[ Parent ]
Just different (2.33 / 6) (#112)
by tthomas48 on Fri Jan 26, 2007 at 06:53:46 PM EST

I fail to see how this is any worse than the old way. You're still taking an algorithm, breaking down a problem and attacking it in small chunks. Who cares if I line it up vertically and add or subtract in vertically, or if I break a problem into tens and solve it horizontally. I think it's funny that you're even attacking this when as you say this is what most people do in their heads (heads not being so great at keeping 3 or 4 rows of vertically stacked digits in place). And this seems like the essence of algebra. You are basically breaking down a problem and simplifying it before attempting to solve it. I dunno though, I'm not a mathematician.

It's not the technique (2.00 / 0) (#114)
by Coryoth on Fri Jan 26, 2007 at 07:19:57 PM EST

I don't have a problem with the technique - as I said, I use it myself. I have a problem with teaching that to the exclusion of all else, and the way they teach it (which doesn't involve any systematic breaking down into tens, but whatever hodge podge way each and every child wishes to approach it, and which fails to explain why this technique works at all either). As I said, my problem with these syllabuses runs deeper and is more to do with the shift away from abstraction adn logic - not that the old syllabuses contained much of that either, but moving even further away is not an improvement!

[ Parent ]
no math skills (1.00 / 3) (#118)
by Phusion on Fri Jan 26, 2007 at 08:06:54 PM EST

Yeah, my math education was poor at best for as long as I can remember up through high school. I'm in California of course, the ed. is pretty messed up here. I'd blame the hippies, but they'd clobber me with a bong. That wouldn't be too bad.


Fighting The War, On Drugs
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Drug Info, Rights, Laws, and Discussion.
i am not (1.66 / 6) (#125)
by wampswillion on Fri Jan 26, 2007 at 11:25:03 PM EST

a math person.   i hate math. i hate numbers. i use them begrudginly because i have to.   i have since the 3rd grade when they introduced the new math in my elementary school.  suddenly what seemed easy to me up til then, seemed entirely insane.  and i tuned the hell out and just started reading books instead.  words were comprehensible to me, math was not any longer.

fast forward, years and years later, and i am in education.  and i am frustrated with my field.  and so i read a lot about education in my country and in others.  and i read about the difference between math education in japan as compared to the united states.  and you know what seems to make the difference according to what i read?  it's not the amount of time spent in school. it's not the "tracking" that japan does in it's upper grades, it's that they spend an enormous amount of time in the early grades making sure that kids are proficient with the basic concepts and the basic manipulation of numbers.   they don't try to be as broad in scope, they don't try to accelerate students through a curriculum quickly. they don't try to teach them goofy stuff like your example.  they teach simply, then they drill, they drill, they drill, they drill.  until something is "overlearned."  and that gives them a good foundation for higher level skills. if that's where a student chooses to go.  

i sincerely believe that it's when the government gets too involved in education and tries to micromanage what we do and how we do it because "business" wants to see results like kids are coming off a factory line, that we drown the baby in the bathwater.  all the new math came about because of a fear of not keeping up with other countries and getting to the moon or something first.   and while yes, the education system needed (and needs) some overhaul, it seems i watch over and over again, mistaken methods for meeting the challenges.  

i believe whole heartedly that public education needs to be preserved, changed, and strengthened. to not do so, in my mind, spells the absolute end of democracy.  however what the government needs to learn to do is SUPPORT public education, not undermine it by telling it how to do things.   i think THAT is what has created this mish-mash that we have now.

I agree totally (1.00 / 3) (#137)
by SaintPort on Sat Jan 27, 2007 at 01:08:12 PM EST

overlearning the basics of education is like overengineering a bridge... very useful for the long run.

--
Search the Scriptures
Start with some cheap grace...Got Life?

[ Parent ]
Drill this (none / 0) (#229)
by cvalente on Fri Feb 02, 2007 at 06:14:27 PM EST

I suspect your drill, drill, drill means mechanizing stupid things that kids don't understand and emphasising that not only is understanding them irrelevant, it may also be harshly punished (in my parents' generation even physically).

This doesn't create people, this creates stupid drones and computers do that far more efficiently and are less error prone than humans.

You want to follow that path? Great!
I want that sort of thing out of my life and the life of my children.

[ Parent ]

TERC (1.40 / 5) (#126)
by bugmaster on Sat Jan 27, 2007 at 12:06:18 AM EST

Huh... so, instead of having students mechanically memorize "The Standard Algorithm" (tm), and execute it flawlessly on each problem, the TERC book encourages them to actually figure out their own algorithms for doing things -- as though being able to think was actually an important skill ! Oh noes ! Alert the media !

Bah. My fascination with computer science, and math in general, began when I understood exactly why the multiplication algorithm worked. I am grateful to my teachers for this. That's what all math teachrs should be teaching, not robotic multiplication skills.
>|<*:=

Same here! (2.00 / 3) (#131)
by MechaA on Sat Jan 27, 2007 at 01:44:39 AM EST

I remember how pleasant it was when I figured that out in elementary school.

k24anson on K5: Imagine fifty, sixty year old men and women still playing with their genitals like ten year olds!

[ Parent ]
Obviously that's not the way it works. (none / 1) (#228)
by duffbeer703 on Fri Feb 02, 2007 at 05:26:12 PM EST

As a former stats 101 TA at a pretty decent school, I can tell you that maybe 1 out of 50 students thought about thinking about cooking up a new multiplication algorithm.

If you learn a standard method that always works, you have a baseline that the people who want to cook up better ways can move from.

Or better yet, consider that multiplication is pretty much a solved problem, and encouraging people to cook up new ways to do it is a pretty useless exercise.

[ Parent ]

Re: Obviously that's not the way it works (none / 0) (#244)
by bugmaster on Mon Feb 12, 2007 at 10:20:48 PM EST

Or better yet, consider that multiplication is pretty much a solved problem, and encouraging people to cook up new ways to do it is a pretty useless exercise
That's like saying, "object-oriented programming is pretty much a solved problem, and encouraging people to cook up new ways to do it is a pretty useless exercise". Yeah, it's useless in a sense that we have established ways to do it commercially, but it's far from useless for the student himself, who is going to learn how to program, or how to multiply. Encouraging people to think is a problem that we still haven't solved, IMO, but we might as well try.
>|<*:=
[ Parent ]
That's a disingenous comparison (none / 0) (#249)
by curien on Fri Feb 16, 2007 at 08:39:36 AM EST

One is a fundamental numerical operation humans have been doing for thousands of years in sundry disparate domains; the other is a software engineering methodology (itself a domain no more than 200 years old) that is barely 40 years old and only widespread for 20 or so years.

--
Wiser words have never been not said. -- lilnobody
[ Parent ]
proving multiplication (1.60 / 5) (#132)
by cronian on Sat Jan 27, 2007 at 03:25:32 AM EST

Multiplication is complicated, and most leave out the details. First, you have to understand addition. Next, you can get multiplication intuitively through the concept of area. Although, I think it can be tricky to use that to prove the multiplication algorithm.

Axiomatically, you can define multiplication inductively, with something like a*(n+1)=a*n+a, probably by example. However, without an existing knowledge of algebra this is difficult to understand. Furthermore, generalizing to fractions may be quite tough.

Furthermore, all modern multiplication systems rely upon our number system. However, our number system relies upon exponents, which are probably learned after multiplication. It is probably quite difficult to teach exponentiation without a solid grasp of multiplication.

I suppose the best solution is through art. For instance consider the following picture to describe base 2: a ab aabb aaaabbbb aaaaaaaabbbbbbbb ... A similar picture could be drawn for base 10, but the important point is the indivisibility of certain groups. Next, the question becomes how to select so many widgets under such constraints. Understanding this requires a bit of intuition, along with comprehending the continuance ad infinitum. This can be tough to make in itself. Further, it is complicated, because one cannot understand the motivation without understanding multiplication algorithm, and motivation for multiplication.

Story problems potentially provide motivation for multiplication, but like many mathematical concepts I think it is difficult to see use in multiplication without first understanding it.

Properly understand, I think the question of how best to teach mathematics is more a question of art than mathematics. The proportions, ratios, and abstractions are much useful, especially for highly elementry mathematics than axioms or even algorithms. I think the real problem is a lack of artistic mathematical expression.

We perfect it; Congress kills it; They make it; We Import it; It must be anti-Americanism
My math education (2.00 / 5) (#136)
by Rainy on Sat Jan 27, 2007 at 12:52:24 PM EST

I disagree, from my experience. My problem with standard way of multiplication was that I didn't understand why it works and no teacher or a textbook ever tried to explain the "why". The new way of multiplication appeals to me because it's apparent how it works and it is less robotic since you are prompted to experiment and try to use it in a different way, as you mention.

My opinion is that the standard multiplication method and similar teaching patterns killed, or almost killed, any love for math that I had. I'm not a math dummy, I was on a math team in HS and I was one of the top in class. I barely passed Calculus AP in HS. I thought math team was much more fun than regular classes.

When I was first taught standard method of multiplication and similar math techniques, I felt a distinct aversion and repulsion of the method, teacher, classroom, textbook and everything associated with these, including the math itself and the general concept of trying to do things analytically. Sometimes I would find the idea of mathematical insight exciting, as it naturally is, but I always felt that if I wished to do some studies, I'd have to seriously and constantly fight the aversion that accumulated during the school days.

However, I'm not sure what fraction of blame should I assign to the teaching methods similar to multiplication. It's really been a lot of things: the fact that teachers were constantly bored and annoyed while teaching us (boredom and annoyance are contagious), pressure to study math when you are not inclined for some reason, etc.

Perhaps the most important thing is having a good teacher. Other things like teaching methods or textbooks may simply be insignificant compared to that. Every time I learned something in class and out of class was only due to having a good teacher, I don't remember a good textbook or a teaching method playing a large part. A good teacher is someone who has the energy to teach inventively and have a good feel for whether students understand something and to what extent. It's a tough job, and there are very few who are good at it.

Out of three dozen of teachers I may have had, about 3 were very good. I don't think I was particularly unlucky, I bet that's pretty much the usual percentage of really good teachers. And even then I'm not sure I could call them excellent teachers, it's just that they stood out very favorably compared to a lousy average teacher performance. Not that I want to blame the poor lot of them, it's quite likely that if I were a teacher I would be one of the bottom 10%.

I mean, repeating the same things over and over to a class of 30, 25 of which are bored out of their minds and the remaining 5 are slightly interested half the time.. That's just such an inhuman job, isn't it? But, I guess, most jobs are.

But, the hope lives that a new way of multiplying numbers will change that.. I don't see that happening.
--
Rainy "Collect all zero" Day

Understanding and proficiency (1.00 / 2) (#138)
by Coryoth on Sat Jan 27, 2007 at 01:09:54 PM EST

I agree that if you're teachers are not mathematically inclined and just teach you a rote process with no explanation and drill it into you then you won't gain much appreciation for what's going on.

On the other hand the exploratory approach, while in theory giving some idea of what is happening, needs just as much explanation in it's own way. From your point of view the TERC books lack of many actual examples (they do a few practice exercises, but never very many) is probably good - but mostly what it means is:
(1) The students don't actually attain understanding because they don't ever play with enough examples to ever figure out what's going on for themselves.
(2) They certainly never gain any proficiency in multiplication. Remember, they don't learn any times tables, so even single digit multiplication has to be done this way, reducing or relating things to a few "key" numbers like 2, 5 and 10.

Given some of the responses I see that I am getting caught on the fact that while the video (which gt me to have a look at this stuff, and has some interesting points) is hung up on multiplication techniques (and I bother to explain that) that is not what I am complaining about. My issue is that, if you actually look through the material these books cover, and how they cover it, they do an even worse job of providing real understanding, and in so doing they destroy the one thing that the old system (broken though it may be) did impart: proficiency.

Mathematics is a subject that builds upon previous work. Unless you are proficient with last years work, this year is going to be very difficult - and these texts fail to provide any degree of proficiency in the topics that students need to be proficient at for any higher mathematics. What's more they do this still not imparting any greater degree of understanding, and failing to provide kids with the tools they need to develop understandign themselves.

[ Parent ]

Real understanding (2.33 / 3) (#140)
by Basselope on Sat Jan 27, 2007 at 05:56:37 PM EST

Coryoth, thanks for writing an article on this topic. It's an important and timely issue, even if I don't think I agree with you entirely.

I'm a little unsure, however, about what you think "real understanding" is. Could you please clarify?

I think what understanding is is an issue central to the discussion going on here. From the title of your comment and your belief that abstraction is fundamental to mathematics, it's pretty clear that you don't think understanding and proficiency are the same thing. What, then, do you think is the relationship between these two? How can we measure understanding, as you define it? What sort of supports do students need from textbooks -- independent of teachers, peers, parents, etc. -- in order to develop real understanding?

[ Parent ]

Proficiency and understanding (1.00 / 2) (#144)
by Coryoth on Sat Jan 27, 2007 at 07:22:49 PM EST

Proficiency, as I am using the term, is the ability to perform efficiently (ideally without having to think about it), be that performing an arithmetic calculation, rearranging or solving an algebraic equation, or evaluating an integral - proficiency at different skills is required at all levels, often with proficiency at higher level skills requiring mastered proficiency at lower ones: rearranging algebraic formula is only done easily if arithmetic operations are automatic; evaluating an integral is only easy if algebraic manipulations are second nature. Proficiency is flat out skill - the ability to do something efficiently. Ultimately that just takes raw practice at the skill until it is trained. That is something these new texts explicitly avoid doing at all costs.

Understanding, as I am using the term, refers be able to see what operations to apply by knowledge of why they should apply. Understanding is the ability to now what to do, while proficiency is the ability to actually do it. In a sense the authors of the books are fishing for understanding, but the reality is that they are falling far short of it: they seem to lack any real understanding themselves. Teaching understanding is hard, but you can aim to do it.

[ Parent ]

Editorial on TERC in American Journal of Physics (2.00 / 5) (#142)
by glor on Sat Jan 27, 2007 at 06:48:45 PM EST

February 2007; subscription required. This editorial has a stronger focus on how these programs get funded, specifically on the involvement of the National Science Foundation. Some excerpts: the sample problem
Problem: Find the slope and y-intercept of the equation 10=x−2.5.

Solution: The equation 10=x−2.5 is a specific case of the equation y=x−2.5, which has a slope of 1 and a y-intercept of −2.5.

This problem comes from a 7th grade math quiz that accompanies a widely used textbook series for grades 6 to 8 called Connected Mathematics Program or CMP.1 The solution appears in the CMP Teacher's Guide and is supported by a discussion of sample student work.

leads to the conclusion
The root cause is money badly spent. The NSF and corporate foundations maintain a gravy train of education grants and awards that stifle competent mathematics education. Although it is conceivable that ongoing NSF grants for new editions of defective math programs, such as those I have described, will improve matters, that is a poor strategy. It amounts to throwing good money after bad. The most that we can realistically hope for is that the original NSF-funded math programs will eventually rise to the level of mediocrity.

The organization's strategy is analogous to placing in charge of the hospital the surgeon who consistently amputates the wrong leg. School district grant recipients involved in implementing low quality K-12 math education programs gain prestige from their association with the NSF and often gain authority over school district math programs. But the reputation of the NSF is suffering from this association. The National Science Foundation logo, prominently displayed on promotional materials for its math programs, has become a warning symbol for parents of school children. It identifies programs that are best avoided, much like the skull-and-cross-bones symbol on poisons. The NSF should drastically change course, or get out of the business of funding K-12 mathematics education altogether.


--
Disclaimer: I am not the most intelligent kuron.

Yes (1.66 / 3) (#148)
by MechaA on Sat Jan 27, 2007 at 08:27:11 PM EST

So it's a straight vertical line, thus the slope of 1.

k24anson on K5: Imagine fifty, sixty year old men and women still playing with their genitals like ten year olds!

[ Parent ]
Gradient of a vertical line (2.66 / 3) (#152)
by kirin on Sat Jan 27, 2007 at 09:05:33 PM EST

Assuming that slope is the same thing as gradient here in the UK, which can be found by the change in the y value over the change in x value, the slope of x=12.5 must be infinite or undefined. You move an infinite amount of y without moving along the x-axis at all.

[ Parent ]
Correct: the textbook example is wrong (1.50 / 2) (#156)
by glor on Sat Jan 27, 2007 at 09:45:39 PM EST

... in a convoluted and confusing way, no less.

--
Disclaimer: I am not the most intelligent kuron.
[ Parent ]

Ummm... (none / 0) (#251)
by AngelKnight on Wed Mar 21, 2007 at 12:57:53 PM EST

What kind of mind-bending reasoning leads to the identification of "10=X-2.5" as a specific instance of "Y=X-2.5"?  Who edits these books?

[ Parent ]
Stitching tables together (2.66 / 6) (#150)
by Alan Crowe on Sat Jan 27, 2007 at 08:44:45 PM EST

When I was a volunteer tutor with an adult numeracy charity I developed an approach I called Number runs to enrich the experience of learning tables.

Coryoth makes a valid criticism of these

math programs complete aversion to teaching students the classic methods for performing multi-digit multiplication and division. Indeed, these programs not only fail to teach such a method, they go so far as to actively discourage the method ever being taught, preferring that students didn't learn it outside class either.
but he fails to see why this is so bad and fails to see that the core of the criticism applies equally to traditional mathematics education.

The central puzzle of mathematics education is that there is not all that much to learn. Compare learning tables with learning the vocabulary of a foreign language. The ten times table has at most one hundred entries. Only 64 are non-trivial. The commutivity of multiplication reduces that to 36 cases. Learning 36 words of vocabulary is a tiny task, yet learning the times tables looms very large. What is going on?

The maths teacher's problem is to make the facts of his subject mean something to his pupils so that they will stick in his pupils memories. If he can make them sticky at all he will soon find that he has little to teach compared to other subjects.

The methods that Coryoth criticises are reactionary. Educators notice that traditional methods of drilling pupils in traditional systematic algorithms for performing multiplication and division involve vast amounts of drill and produce disappointing results. They react against these methods, and try to use other methods instead.

If the pupil can find a perspective from which rules of arithmetic makes sense to him or her then the traditional methods make sense and are easily learnt. All that is left to do is a modest amount of drill to aquire facility so that the methods may be practised with high accuracy.

The difficulty lies in finding such a perspective. Study of the traditional methods themselves works for some pupils. Working out areas on squared paper can help. Manipulating numbers in the playful way described in the article sometimes unlocks the mystery for a pupil.

Both the reactionary methods criticised in the article and the tradition methods of drilling specific methods have the same central flaw: they do not roam over the range of possible perspectives trying to find vistas from which individual pupils will have their "Aha!" momnent and see what is going on. The different perspectives are mutually illuminating. When a teacher decides to focus on one particular technique wether "traditional" or "modern" he obviously fails the pupils who needed exposure to the techniques he omits, but he also causes difficulties for all his pupils. Whichever technique provides a pupil with a way in to understanding arithmetic, that technique unlocks the whole puzzle, and is more easily apprehended by pupils who have had sight of the whole puzzle and can see what is being unlocked.

The core issue is that there is not very much to be learned, but pupils are unable to learn it at all because it doesn't make sense to them. Intellectual stimulation that raises the pupils level of mathematic sophistication to the point at which it starts to make sense allows pupils to learn in weeks what rote drilling may fail to teach in years. However there is a catch. What kinds of activities are intellectually stimulating?

When I was studying category theory I thought to myself "Oh shit, I'm not getting this." So I settled down to learn axioms for categories, functors, and adjoints by rote. If I couldn't understand it, I could at least be familiar with it, and that would provide opportunities for understanding later.

This applies just as much to arithmetic. If a child is stuck he will not become unstuck by waiting. He has to continue working with the material. Learning some routines by rote drilling them provides material for curiosity. If one can do the necessary carrying for multidigit arithmetic but do not understand, one will soon forget it, or get it muddled and get wrong answers. That brief period of knowing what to do without knowing why to do it is a very promising time for realising why, and once that realisation is attained it will not be lost.



Pushing through... (1.75 / 4) (#158)
by Coryoth on Sat Jan 27, 2007 at 09:54:31 PM EST

You're quite right that sometimes pushing through and simply getting some experience working with whatever you're doing (in math) by simply following the rules for it can be the best approach at times. The deep understanding often comes later, after you've actually been working with whatever it is for some time. There's also the fact that if you can perserve and learn the later material you often find rich connections that start to unlock secrets in earlier material.

I also agree entirely that presenting various ways of looking at things is often key to sparking understanding - I've certainly found this as a TA: indeed, managing to find several different ways to present the same topic until you find one that clicks with the student is fundamental to my approach to teaching. The downside is that that tends to require a certain amount of one on one work that doesn't scale well.

As I say, I have no problem with presenting the TERC "cluster" method of multiplication - but ultimately it's just "one more way" and the standard method, with its consistent success rate and ability to work for all students, provides a solid base. It should be a starting point, letting the kids persevere through to find understanding via presentation of the other methods and the one that eventually clicks.

[ Parent ]

Ossification of standard methods (2.20 / 5) (#171)
by Alan Crowe on Sun Jan 28, 2007 at 07:47:48 AM EST

I found a wonderful book, Capitalism and Arithmetic: The New Math of the 15th Century. It is mostly David Eugene Smith's translation of the Treviso Arithmetic of 1478, plus a commentary to put it in its historical context.

What comes out clearly is that the standard methods were designed for a world in which calculators were human. Methods had to be economical on paper because paper was expensive. It was worth streamlining methods for greater speed even if the slicker method required greater mental dexterity from the calculator. A "reckoning master" would use the method every day so maintaining mental dexterity was not a problem, while greater speed was obviously profitable.

This became clear to me when I tried to teach long division. I remembered from my own school days that this was terribly hard and wondered how to make it easier for my pupil. Consider for example dividing 826 by 469. I taught her to build a multiplication table for 469 by successive addition. If you add an extra rendundant addition at the end, adding 469 onto nine 469's to get ten 469's, well you already know that must be 4690, so you have a check for error half way through. Once you have your table

1  469
   469
2  938
   469
3 1407
   469
4 1876
   469
5 2345
   469
6 2814
   469
7 3283
   469
8 3752
   469
9 4221
   469
  4690

the traditional tableaux

            1.7 6 1
      .____________
4 6 9 | 8 2 6
        4 6 9
        -----
        3 5 7 0
        3 2 8 3
        -------
          2 8 7 0
          2 8 1 4
          -------
              5 6 0
              4 6 9
              -----
                9 1

is much easier to follow. There is no straining your brain to guess how many times does 469 go into 3570. You can see from your table that it goes seven and a bit times, so you can confidently write in the seven and procede with the algorithm.

This is vastly easier. As a child I was not taught to write out a multiplication table first. My school taught us to do the multiplications on demand in the body of the calculation. So it wasn't strictly an algorithm, because you had to guess at each digit by doing enough of the multiplication in your head, and the division tableaux was cluttered with little carry numbers internal to the multiplications. Although writing out the table first makes it easier this comes at the expense of taking twice as much paper and taking twice as long, due to the extra writing.

So it is time to reckon up the loss and the gain. Making it easier to learn is on the face of it a big gain. The extra time and paper do not matter. Nowadays, if we want to do many long divisions we use a power tool, so we care little for the efficiency of our manual method. On the other hand we must be suspicious of making it easier to learn, we don't want to leave out the essential content that we aim to teach.

In this case it is clear that the essential content is still present. We have made the method easier by leaving out the streamlining, which is no longer useful and does obfuscate the algorithm. This reform to the standard method is a clear win.

The wider point is that the world really has changed. We are educating persons for a world of calculators and computers, so we must rethink mathmatics teaching from first principles. I personnally believe that we end up close to where we started, teaching much the same material in much the same way, but I view this as the surprising result of some deep thinking. When I see people justifying the old ways by an appeal to tradition I dismiss those people as fools. The world really has changed and an appeal to tradition is not at all persuasive.

Take credit cards as an example. Electronic computers do the book keeping. This has two consequences. First, there are no jobs sitting in an office doing arithmetic for MasterCard or Visa. Second, with no people to pay, there is not a big arithmetic charge on your credit card bill each month. So credit cards have become wide-spread to the point that it is awkward to do without one. Interest rate calculations become important to every-one: you need to have a credit card and need to be aware of how compound interest can shaft you. Notice the very strange causal link: you need to know about compound interest because there are no longer any jobs to had sitting in an office calculating it.

There is a dramatic shift in our motivation for teaching arithmetic. We are no longer teaching manual methods in order to equip some of our pupils for clerical work. We have a very different goal, almost a New Age/Hippy goal. We want our pupils to have an emotional connection to arithmetic so that the results of calculations have significance to them and can affect their life choices. We are trying to avoid the situation in which persons do important calculations on a calculator and then ignore the results because they have no emotional connection to the glowing figures on the calculator's display and fail to take them seriously.

I advocate teaching manual methods. The idea is that persons who understand how their calculator reached its answer are more likely to take those answers seriously instead of shrugging them off as mysterious and therefore unimportant. If that is the goal, then the new manual methods will be neither as slick nor as difficult as the old manual methods and there will be an emphasis on understanding why they work.

Let me expand on the New Age theme. It is important to use the power of positive thinking, to discard negative thoughts and liberate yourself to do the impossible. On the other hand positive thinking can be positively dangerous. You need to work in a balanced way. The more you work on a positive attitude the more you need to work on your contact with reality. Having a facility with arithmetic, doing the sums, running the numbers, keeps you grounded. When you "attempt the impossible" you attempt things that are metaphorically impossible. You attempt things that impossible for other persons. They are hemmed in by vague apprehensions. They are not able to do the calculations and cannot say with confidence "It sounds stupid, but actually the numbers do add up". And people who "do the impossible" succeed because they do not waste their energy on the truly impossible. They can run the numbers and respect the facts when actually the numbers don't add up.

[ Parent ]

Abstraction is fundamental to software development (2.14 / 7) (#151)
by skyknight on Sat Jan 27, 2007 at 08:49:23 PM EST

More generally, abstraction is fundamental to higher order thinking of any kind. One cannot solve complex problems without a divide-and-conquer strategy of attacking sub-problems, packaging the solutions to those sub-problems tidily, and building in turn larger solutions from them. The argument for concise mathematical notation is the same argument for higher level programming languages. Without the ability to momentarily and selectively ignore implementation details, a complex problem simply becomes impossible to fit within the confines of a human skull, thus rendering it utterly intractable.

It's not much fun at the top. I envy the common people, their hearty meals and Bruce Springsteen and voting. --SIGNOR SPAGHETTI
Only in Java da nut (2.66 / 3) (#212)
by cronian on Tue Jan 30, 2007 at 04:45:02 AM EST



We perfect it; Congress kills it; They make it; We Import it; It must be anti-Americanism
[ Parent ]
online math tutors. (2.50 / 4) (#153)
by joeclark07 on Sat Jan 27, 2007 at 09:22:34 PM EST

Even my son had problem in learning math at school and I was not in a postition to afford to get him math tutor locally. Then , friend of mine suggested me to use online tutor www.tutorswithoutlimits.com or www.tutorvista.com
Begining I was reluctant to use online tutoring serive because I was not sure how some one can teach online from another country and how would my son react to that.
But after some , my son started feeling good about online tutors.
It's very flexible and very cheap.
good luck all
Joe.

You offshored your son's eductation? (1.00 / 2) (#178)
by nlscb on Sun Jan 28, 2007 at 02:44:43 PM EST

Why do you hate America?

Seriously, good for you. Online education is the future. Pay for the testing and guidance, not the overpaid administration and tenured faculty, facilities, professional in all but name sports teams, and useless overpriced labs that you'll never get to touch.

Comment Search has returned - Like a beaten wife, I am pathetically grateful. - mr strange
[ Parent ]

No. It's just a shameless plug ... (n/t) (none / 0) (#241)
by icastel on Tue Feb 06, 2007 at 06:00:12 PM EST




-- I like my land flat --
[ Parent ]
Practicality. (2.00 / 4) (#154)
by TDS on Sat Jan 27, 2007 at 09:28:23 PM EST

There is another reason to use set techniques and "show your working", so the teacher can check you've applied the technique. If they do long multiplication/divison wrong, their error can only stem from (a) applying the method wrongly, and implicitly part of the method is showing your working anyway, (b) making an arithmetical error.

My question is how, if you don't do this, the teacher is supposed to figure out where a kid has gone wrong if they off on some sort of error-prone arithmetical safari.  One would imagine that if they are making mistakes they might not be great at recording everything brilliantly anyway. And if every solution is novel, showing working will be harder anyway, its less of a filling the boxes activity. If you imagine the maths teacher trogging home on the weekend with a hundred exercise books (x 30 problems) to mark, I don't see how they are going to manage it after a point purely in terms of time if not lack of mindreading ability.

A red cross next an answer is utterly worthless in and of itself.

So I think the implication of this problem are actually wider than you say.

And when we die, we will die with our hands unbound. This is why we fight.

The TERC solution... (1.00 / 2) (#157)
by Coryoth on Sat Jan 27, 2007 at 09:46:55 PM EST

TERC solves the difficulty in marking such work by the simple expedient of setting the absolute bare minimum of problems - there is basically nothing for the teacher to mark because the students never really do any exercises. This, of course, is even worse again...

[ Parent ]
Hang on a minute (1.00 / 1) (#168)
by TDS on Sun Jan 28, 2007 at 12:26:58 AM EST

TERC and MIC aren't just RME in disguise is it?

In which case we can mock but the Dutch do OK in stuff like TIMMS don't they.

And I shall wag the finger at you for making the article so Yank-centric when half the world is trying out RME at the moment.

And when we die, we will die with our hands unbound. This is why we fight.
[ Parent ]

welcome diggers (1.80 / 10) (#155)
by blackbart on Sat Jan 27, 2007 at 09:34:55 PM EST

you may wonder who this "kitten" poster is, for more information go to his home page.

"I use this dupe for modbombing and impersonating a highly paid government worker"
- army of phred

Quadratic equation is the gateway to NONLINEARITY (1.50 / 6) (#160)
by lyapunov on Sat Jan 27, 2007 at 10:23:48 PM EST

Why is Bill Gates so much richer than other workers?
Why is it important to stop Global Warming now, instead of 10 years from now?
Why is it that my co-worker/classmate/roommate has so many more sex partners than I do?

These are just a few examples of non-linear phenomena. So many posters take it for granted that the quadratic equation is "useless" beyond 7th grade. What's important is the idea, not how to solve x^2=-1.

The comments above tell me that most poster didn't learn the idea. Believe you me, the quadratic equation affects us everyday; you just have to open your eyes.

-Anonymous Physicist

Simple. (2.00 / 3) (#180)
by kitten on Sun Jan 28, 2007 at 03:19:29 PM EST

Why is it that my co-worker/classmate/roommate has so many more sex partners than I do?

I don't know -- maybe because he's out having a good time on weekends and meeting people, instead of staying holed up inside working on quadratic equations to calculate Bill Gates' fortune?

Once again, just because math is obviously useful (hello, computer!) and affects everyone doesn't mean most people need it or will ever use it. Your examples are the same as saying everyone needs to understand the chemical processes by which oxygen is converted carbon dioxide by the body and how that affects food consumption and metabolism, because it's a part of everyone. Most people just eat the damn food and keep breathing.
mirrorshades radio - darkwave, synthpop, industrial, futurepop.
[ Parent ]
And here's the truth... (1.33 / 3) (#185)
by lyapunov on Sun Jan 28, 2007 at 04:50:29 PM EST

Why is it that my co-worker/classmate/roommate has so many more sex partners than I do?

A common mind's reply:
I don't know -- maybe because he's out having a good time on weekends and meeting people, instead of staying holed up inside working on quadratic equations to calculate Bill Gates' fortune?

A cartesian mind's reply:
The web of human sexual contacts (It's a power law distribution).

In summary, an innumerate person will never grasp how the world really works.

[ Parent ]

Truth redux (2.66 / 3) (#187)
by kitten on Sun Jan 28, 2007 at 06:49:18 PM EST

Your knowledge of English once again shows itself to be rather lacking, I'm afraid. "Sexual partners" means people with whom one has had direct sexual contact with. This isn't the Kevin Bacon game unless you go out of your way to make it one.

It's not like "A has sex with B and B has had sex with C and D" is such a difficult concept that we need Cartesian minds such as yourself to educate us unwashed masses, but I guess when you're a pretentious twahn, you feel the need to assert yourself wherever you can.

Such generators were often used to break the ice at parties by making all the molecules in the hostess's undergarments leap simultaneously one foot to the left, in accordance with the Theory of Indeterminacy.

Many respectable physicists said that they weren't going to stand for this - partly because it was a debasement of science, but mostly because they didn't get invited to those sort of parties.
This was written precisely for people like you.
mirrorshades radio - darkwave, synthpop, industrial, futurepop.
[ Parent ]
Read the paper again, if you dare (1.50 / 2) (#189)
by lyapunov on Sun Jan 28, 2007 at 07:07:51 PM EST

Littel kitten, I'm afraid your command of the English language has languished at the 7th grade-level, which closely parallels your mastery of math (or rather, the lack thereof). With today's newspapers written to a audience assumed to have your level of english proficiency, it is no wonder that you cannot grasp the basics of conceptual, abstract thinking.

Let me translate that to you in proverbial terms:
sometimes, kitten, you have to read between the lines.

I'm afraid that a common, concrete-minded person like you cannot grasp double entendres, a verbal skill that requies abstract thinking.

A skill that you have shown, beyond the shadow of doubt, that you completely lack. Try a math class sometimes.

[ Parent ]

"Again"? (2.00 / 1) (#192)
by kitten on Sun Jan 28, 2007 at 07:15:38 PM EST

You presume I bothered reading it the first time. I didn't. I don't care enough to sit around reading dry academic papers so idiots like you can demonstrate some incredibly subtle point that was "between the lines" as you put it, and then clap for themselves for being so very very clever.

But what do I know? I could read the paper, I guess, but since I don't have your OMFG CARTESIAN MIND like you, Your Worshipfulness, I'd probably just continue dragging my knuckles on the ground like the Cro-Magnon that I am.
mirrorshades radio - darkwave, synthpop, industrial, futurepop.
[ Parent ]
Engrish (2.50 / 2) (#197)
by lyapunov on Sun Jan 28, 2007 at 07:31:12 PM EST

kitten said:
Your knowledge of English once again shows itself to be rather lacking

My knowledge of enligsh is no-good, you say? Let me quote directly from the horse's mouth:

kitten said:
I didn't realize I should be churning out eloquent turns of phrase in offhand comments to K5. This is a casual argument, not a literature competition, nor a powerful speech.

Kitten, your self-contradictions are throughly entertaining. Please go on, I could use a few laughs.

[ Parent ]

Try again, sucka. (1.66 / 3) (#199)
by kitten on Sun Jan 28, 2007 at 07:36:13 PM EST

I was criticizing you for a lack of understanding common, vernacular written words.

You were bitching because I said something "sucks" and that wasn't eloquent.

My point was valid. Yours was just petty.
mirrorshades radio - darkwave, synthpop, industrial, futurepop.
[ Parent ]
to register your vote to ban this user (2.00 / 3) (#169)
by cattleprod of peace on Sun Jan 28, 2007 at 03:24:41 AM EST

click here: www.kuro5hin.org/abuse_report?uid=68339

Missing (1.40 / 5) (#175)
by izzyd3434 on Sun Jan 28, 2007 at 12:37:27 PM EST

Although the author genuinely hopes to improve math education, I think that s/he is simply basing the recommendations off of his/her experiences with math.  And since s/he is a Mathematician and works with college kids in mathematics the "other" side becomes increasingly difficult to see.  Whether you look at our standardized Math test scores or self-reported accounts of how good we are at Math, the results are not good.  The reality is that most kids do not like math (starting with K-6 which is when a lot of kids decide which subjects they are "good" at and which ones they are not).  The reason being that we have traditionally focused on these rote memorization skills that are very boring for someone who is not naturally interested in numbers and not applicable to the outside world which involves people and things and situations, not just numbers written on a piece of paper.  So the move to relating math to the real world and focusing on "thinking" rather than memorizing IS A GOOD THING.  You could argue all day as to how far this should go etc. but this should be the foundation of Math if we hope to produce future adults who can calculate the tip on a dinner bill or give proper change at a store.  These are the capacities in which 95% of people use math in their lives and it seems that lots of people cannot do these simple things....but give them a piece of paper and pencil and some numbers and they can usually figure it out.  I bet that if  you polled adults regarding use of trig or Calculus or algebra skills in their lives it would be virtually 0....yet there are so many real world situations that actually do involve these concepts...but since they were taught as memorization to a particular problem in a classroom, the connection to the outside world is never made.

John
http://www.monomachines.com
 

Too much application (1.75 / 4) (#177)
by Coryoth on Sun Jan 28, 2007 at 02:43:22 PM EST

The issue with making everything be applied practical examples is this: you connect the process with the particular examples. When faced with a realted problem that could have similar methods or processes applied to it, unless it is remarkably similar to the examples you've been taught, you simply won't realise that the process could apply to this case too.

You see the power of mathematics is that it is so abstract. By being incredibly abstract, and containing so very little detail about particularities, its range of application is incredibly broad. By always binding the math to an application the breadth of application granted by abstraction is lost. Moreover, the art of abstraction - of ignoring the particularities of the current situation to generalise to a much wider class of problems - is never properly learned.

Sure, rote, mindless memorization is bad, but then I'm hardly defending that. The suggested "solution" is, in many ways, worse than the problem. If you find that children are bored and uninterested in (and fail to understand the practical value of) history because a school teaches by simply forcing kids to rote memorize a long list of names and dates with no explanation then the solution is not to "make it relevant to the students" by just teaching them the history of their own lives.

[ Parent ]

Pfft, all I need to know is that 2 + 2 = 5 (1.12 / 8) (#182)
by nlscb on Sun Jan 28, 2007 at 04:19:59 PM EST

God damn know it all eggheads.

Europhile Hippies and the Christian Right are the ones who know how to teach.

Comment Search has returned - Like a beaten wife, I am pathetically grateful. - mr strange

This sums it up (1.20 / 5) (#184)
by greenthumbstocks on Sun Jan 28, 2007 at 04:43:37 PM EST

This is a great article on math education.  Teaching K - K12 math in 20 contact hours using a 1898 math book as the text.. http://www.besthomeschooling.org/articles/math_david_albert.html

Three things (1.85 / 7) (#201)
by trhurler on Sun Jan 28, 2007 at 10:38:41 PM EST

1) Most students are never going to learn any math beyond geometry and a bit of trig. I realize this appalls you. It is also true, and probably will always be true, because honestly you have to be somewhat more intelligent than the average bear to go much farther than that in mathematics, period.

2) Young students don't think abstractly much at all, save a gifted few, which is why traditional mathematics programs for them were all about memorization and so on. There's no point in trying to get an eight year old to grasp the abstract notion of, say, functions on integers, and really even of specific ones such as summation - unless he's a rare exception to the rule.

3) The smart students will gain the skills you're talking about pretty much no matter what the curriculum might be - most of them would have figured out a lot of it if left to their own devices with a textbook and too much spare time, in fact.

The obvious solution is that you may as well coddle the average and separate the smart ones. The curriculum for the former should be whatever will allow them to one day balance a checkbook and count change. And for the latter? Almost anything will work, honestly.

--
'God dammit, your posts make me hard.' --LilDebbie

ATTN: DIGGFAGGS (1.00 / 10) (#204)
by local host unknown on Mon Jan 29, 2007 at 02:11:16 PM EST

GO TO HELL, BUTT-PIRATES

When you Limeys (1.50 / 4) (#215)
by kitten on Tue Jan 30, 2007 at 02:01:36 PM EST

can figure out subject/verb agreement, then you can quibble over abbreviations which aren't meant to follow any sort of pattern.
mirrorshades radio - darkwave, synthpop, industrial, futurepop.
Mathmatics (none / 0) (#233)
by Geno on Sun Feb 04, 2007 at 02:02:50 PM EST

I think the first reality that one has to face is: Education is like the measles, just because it's there, and someone is exposed to it, doesn't necessarily mean that everyone exposed to it is going to get it.  There are many who get it, and many more who will not, no matter how you approach it.  Wouldn't it be better to direct those who will not (or cannot) get it to an area that might be more beneficial and productive to their innate abilities?

The Declining Quality of Mathematics Education in the US | 248 comments (243 topical, 5 editorial, 6 hidden)
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