# MathFilter(edout)

February 16, 2006 9:56 PM Subscribe

The Value of Algebra: "

*Gabriela, sooner or later someone's going to tell you that algebra teaches reasoning. This is a lie propagated by, among others, algebra teachers.*"*I confess to be one of those people who hate math. I can do my basic arithmetic all right (although not percentages)*

He can't do percentages, but gets to write for the Washington Post?

*Writing is the highest form of reasoning. This is a fact. Algebra is not.*

Oh, it is dark satire. Funny!

posted by b1tr0t at 10:04 PM on February 16, 2006

Clearly, since I've survived without algebra, it's useless.

I wonder why I can't seem to control my credit card debt?

posted by agent at 10:06 PM on February 16, 2006

I wonder why I can't seem to control my credit card debt?

posted by agent at 10:06 PM on February 16, 2006

*all the people in my high school who were whizzes at math but did not know a thing about history and could not write a readable English sentence.*

Actually, the best math students were almost invariably the most literate. Rarely did you see a respectable humanist who wasn't also capable of handling math.

But Richard Cohen is really the WaPo's worst columnist. That he lacked the capacity to reason well enough to pass algebra doesn't come as a surprise, as he can't reason well enough to write decent columns, either.

posted by deanc at 10:08 PM on February 16, 2006

Well he certainly makes a case for writing being the highest form of reasoning as he stands up and makes the fallacy of composition. Just because it is true for the few doesn't mean its true in all cases.

posted by Rubbstone at 10:08 PM on February 16, 2006

posted by Rubbstone at 10:08 PM on February 16, 2006

Stupid.

That is a great example of the petitio principii fallacy, but doesn't prove much.

There's lots of really bad math classes out there, and feel free to hate on them all you like. The claim that math classes are wastes of time because you don't use algebra, though, is like the claim that because you don't need to quote Shakespeare in your "real" life, English Lit classes are a waste of time. Many lit classes suck too. So?

posted by freebird at 10:10 PM on February 16, 2006

*[...] algebra teaches reasoning. This is a lie propagated by, among others, algebra teachers. Writing is the highest form of reasoning. This is a fact. Algebra is not. The proof of this, Gabriela, is all the people in my high school who were whizzes at math but did not know a thing about history and could not write a readable English sentence.*That is a great example of the petitio principii fallacy, but doesn't prove much.

There's lots of really bad math classes out there, and feel free to hate on them all you like. The claim that math classes are wastes of time because you don't use algebra, though, is like the claim that because you don't need to quote Shakespeare in your "real" life, English Lit classes are a waste of time. Many lit classes suck too. So?

posted by freebird at 10:10 PM on February 16, 2006

As an nth-power algebra failer, I can say this: he's right, you're wrong, suck it haters.

posted by mwhybark at 10:10 PM on February 16, 2006

posted by mwhybark at 10:10 PM on February 16, 2006

*Most of math can now be done by a computer or a calculator.*

That's an extremely narrow definition of math and all too typical of what most people perceive math to be about. If anything, that statement itself makes a strong argument for more math education.

Regarding the rest of the article - sure, one can get along fine in life without knowing where the Ghobi desert is or without knowing who Shakespeare is. But, thats not the point, mathematics is, like those things, part of our shared knowledge, our shared humanity, and one of our grandest creations.

posted by vacapinta at 10:10 PM on February 16, 2006

Waah waah algebra is hard. . . . No, it's not. Although I recognize that other people find it difficult, it's not some terrible spectre.

College calculus is.

posted by jenovus at 10:11 PM on February 16, 2006

College calculus is.

posted by jenovus at 10:11 PM on February 16, 2006

I totally

posted by freebird at 10:12 PM on February 16, 2006

**didn't**write "There**'re**lots" on purpose to make a really awesome point.posted by freebird at 10:12 PM on February 16, 2006

This guy is an amazing idiot. I already sent him a letter.

It's so becoming to be proud of your own ignorance and shortcomings. Good job, Cohen.

posted by teece at 10:13 PM on February 16, 2006

It's so becoming to be proud of your own ignorance and shortcomings. Good job, Cohen.

posted by teece at 10:13 PM on February 16, 2006

He couldn't be more wrong about math & reasoning. Mathematics is the distilled art & science of pure reasoning and pattern recognition.

posted by daksya at 10:14 PM on February 16, 2006

posted by daksya at 10:14 PM on February 16, 2006

What a paltry, incurious, shameful life he must lead if he's actually never used algebra. Twenty-whatever years and he's never even once pondered whether he received correct change, or wondered how long it would take to get to his destination on a long drive, or what mileage his car was getting, not to mention anything more significant.

posted by ROU_Xenophobe at 10:19 PM on February 16, 2006

posted by ROU_Xenophobe at 10:19 PM on February 16, 2006

daksya, I second that. I'll cite the fact that philosophy programs don't teach students to argue by reading arguments alone, but by having a set of well-developed calculi for describing arguments symbolically. This lets us abstract out all of the cruft and crap that muddies prose into something clean and elegant that is much easier to verify.

posted by agent at 10:21 PM on February 16, 2006

posted by agent at 10:21 PM on February 16, 2006

Facts:

1. Writers are mammals.

2. Writers fight ALL the time.

3. The purpose of the writer is to flip out and kill people.

Testimonial:

Writers can kill anyone they want! Writers cut off heads ALL the time and don't even think twice about it. These guys are so crazy and awesome that they flip out ALL the time. I heard that there was this writer who was eating at a diner. And when some dude dropped a spoon the writer killed the whole town. My friend Mark said that he saw a writer totally uppercut some kid just because the kid opened a window.

And that's what I call REAL Ultimate Power!!!!!!!!!!!!!!!!!!

Facts:

1. Writers are mammals.

2. Writers fight ALL the time.

3. The purpose of the writer is to flip out and kill people.

Testimonial:

Writers can kill anyone they want! Writers cut off heads ALL the time and don't even think twice about it. These guys are so crazy and awesome that they flip out ALL the time. I heard that there was this writer who was eating at a diner. And when some dude dropped a spoon the writer killed the whole town. My friend Mark said that he saw a writer totally uppercut some kid just because the kid opened a window.

And that's what I call REAL Ultimate Power!!!!!!!!!!!!!!!!!!

A computer could never compose anything that brilliant.

posted by b1tr0t at 10:23 PM on February 16, 2006

*The L.A. school district now requires all students to pass a year of algebra and a year of geometry in order to graduate... the sort of vaunted education reform that is supposed to close the science and math gap and make the U.S. more competitive. All it seems to do, though, is ruin the lives of countless kids.*

This is all written tongue-in-cheek, right?

posted by Kwantsar at 10:41 PM on February 16, 2006

As an nth-power algebra failer, I can say this: he's right, you're wrong, suck it haters.

As an nth-power algebra failer, I can say this: he's right, you're wrong, suck it haters.

"n-th power"? If you knew some algebra, you'd have been able to solve for the value of n.

posted by juv3nal at 10:42 PM on February 16, 2006

*Gabriela, sooner or later someone's going to tell you that algebra teaches reasoning. This is a lie propagated by, among others, algebra teachers. Writing is the highest form of reasoning. This is a fact. Algebra is not. The proof of this, Gabriela, is all the people in my high school who were whizzes at math but did not know a thing about history and could not write a readable English sentence. I can cite Shelly, whose last name will not be mentioned, who aced algebra but when called to the board in geography class, located the Sahara Desert right where the Gobi usually is. She was off by a whole continent.*

God, what an idiot. Writing is the highest form of reasoning? What the hell does that even mean? Reasoning is the ability to apply the specific case to the general population, or the general population to the specific case. Reasoning is knowing how to systematically attack a problem. Reasoning is useful in so many areas of life, from figuring out which of your kids is lying, to finding your lost car keys, to changing a tire for the first time, to about a billion other things.

Writing is not reasoning, it's fucking art. Which is not to say it's not difficult, or challenging, or anything like that. Knowing where the Gobi desert is? Or where it isn't? That's not reasoning either, that's memorization. An important skill? Sure. Does that mean it's reasoning? No.

Just because something requires skill to do doesn't mean that you are reasoning when you are doing it.

posted by 23skidoo at 10:49 PM on February 16, 2006

I am a theatre dude. I have written plays, improvised, directed, designed, you name it. While all the theatre classes and lit classes and history classes have been invaluable to my art, I credit math - especially linear algebra and 80's style computer science - with most of my understanding of art.

Algebra taught me to look for the underlying pattern in something - be it a work of art, a bank statement that baffled me or a faulty relationship. While it is not the only thing that informs my understanding and approach to a thousand different aspects of life, it is absolutely something that I use every single day even when I don't deal with actual numbers for months at a time (my ability to go for months at a time without dealing with numbers being a prime cause of my credit problems, but I digress).

My point is, algebra taught me to look at the world more closely and deeply. The few times that I have felt the tug of religion have been when I had a stunning epiphany of pattern recognition in some beautiful thing or event.

Yeah, you don't need algebra. Technically, you don't need history, foreign languages, art, science or physical education. But, damn, if it doesn't enhance your life to know it and understand it - even if it is a struggle.

posted by Joey Michaels at 10:53 PM on February 16, 2006 [1 favorite]

Algebra taught me to look for the underlying pattern in something - be it a work of art, a bank statement that baffled me or a faulty relationship. While it is not the only thing that informs my understanding and approach to a thousand different aspects of life, it is absolutely something that I use every single day even when I don't deal with actual numbers for months at a time (my ability to go for months at a time without dealing with numbers being a prime cause of my credit problems, but I digress).

My point is, algebra taught me to look at the world more closely and deeply. The few times that I have felt the tug of religion have been when I had a stunning epiphany of pattern recognition in some beautiful thing or event.

Yeah, you don't need algebra. Technically, you don't need history, foreign languages, art, science or physical education. But, damn, if it doesn't enhance your life to know it and understand it - even if it is a struggle.

posted by Joey Michaels at 10:53 PM on February 16, 2006 [1 favorite]

I can sympathize with him. I was a complete whiz in school all the way up to Algebra. I was baffled that I was baffled by Algebra. I could understand the concepts, even the more complex stuff; I just couldn't execute. It was the first class I ever failed because of inability rather than lack of effort (which I also excelled at). But that just made me work harder at it. And I eventually passed... in my second year. It wasn't required for graduation, and I dropped out anyway. I use algebra often (on preview: what Joey Micheals said).

Having said that, Cohen's conclusion is crap. That is a fact.

posted by effwerd at 10:58 PM on February 16, 2006

Having said that, Cohen's conclusion is crap. That is a fact.

posted by effwerd at 10:58 PM on February 16, 2006

I consistently failed math, never getting beyond algebra one. I aced english and visual art. However, I believe that writing and mathematics are equally valid ways of understanding and describing existence, so yeah... this person is a small-minded moron.

However, I have to say that the way math is taught really needs to be reformed. My high school experience with math was one of drudgery. I hated every single minute of it. It stifled creativity. I realize, now, that a certain foundation needs to be laid before people can explore the bounds of mathematics, but that wasn't at all clear in school. Math is taught as rote memorization, with no specter of anything beyond rote memorization.*

It wasn't until I was in college that I was able to get beyond that and realize I was missing out on a whole hell of a lot. A big part of that realization is due to my reading some of Rudy Rucker's books (fiction and non-fiction). He managed to present math as something fluid. Something that one can explore. I now regret my prior "education" in math, because I tuned out and missed that exploration-enabling foundation.**

Recently, I have discovered a desire to learn how to program. It has been slow, slow learning for me. CSS was an accomplishment. I know, I just know, that if math had been taught in an engaging manner, the road would be a lot smoother.

*The same is certainly true of science.

**"Exploration-enabling Foundation" would make a good band name.

posted by brundlefly at 11:00 PM on February 16, 2006

However, I have to say that the way math is taught really needs to be reformed. My high school experience with math was one of drudgery. I hated every single minute of it. It stifled creativity. I realize, now, that a certain foundation needs to be laid before people can explore the bounds of mathematics, but that wasn't at all clear in school. Math is taught as rote memorization, with no specter of anything beyond rote memorization.*

It wasn't until I was in college that I was able to get beyond that and realize I was missing out on a whole hell of a lot. A big part of that realization is due to my reading some of Rudy Rucker's books (fiction and non-fiction). He managed to present math as something fluid. Something that one can explore. I now regret my prior "education" in math, because I tuned out and missed that exploration-enabling foundation.**

Recently, I have discovered a desire to learn how to program. It has been slow, slow learning for me. CSS was an accomplishment. I know, I just know, that if math had been taught in an engaging manner, the road would be a lot smoother.

*The same is certainly true of science.

**"Exploration-enabling Foundation" would make a good band name.

posted by brundlefly at 11:00 PM on February 16, 2006

In other words, Rudy Rucker should write an Algebra I curriculum.

posted by brundlefly at 11:01 PM on February 16, 2006

posted by brundlefly at 11:01 PM on February 16, 2006

The pride in one's ignorance displayed by the writer...

I had to kill -9 that thought as I went into a while(true){crazyGoNuts(KILL_EM_ALL);} loop.

That being said, I do see some of his points. Algebra doesn't teach reasoning. Math doesn't teach reasoning. Dialectic, Rhetoric, and Logic (formal) teach reasoning.

Math relies on these fundamental tools. It is probably the only place students get exposed to _reasoning_ in the public school system, but it is not taught with a focus on reason but a focus on rote procedural memorization - though the situational ("word") problems require the application of reason.

That's part of what makes math so bloody unattractive to certain people - like me! I love to reason, I'm steeped in it every single day as it IS my job (Systems Architect), but back in school Math was my absolute worst subject. All just memorize this or that rule, apply it over and over again until it becomes reflex, then onto the next rule. You don't get to do anything really freaking cool with it until way later on. I guess you have to walk before you can run, but still, so many students can't even visualize anything BUT endless walking when they think about Math.

Maybe that's changing these days with tools like Mathematica and the rise of Computer Graphics and complex games. Maybe kids these days can see more of what math can really do. Too bad most never get the opportunity.

posted by C.Batt at 11:02 PM on February 16, 2006

I had to kill -9 that thought as I went into a while(true){crazyGoNuts(KILL_EM_ALL);} loop.

That being said, I do see some of his points. Algebra doesn't teach reasoning. Math doesn't teach reasoning. Dialectic, Rhetoric, and Logic (formal) teach reasoning.

Math relies on these fundamental tools. It is probably the only place students get exposed to _reasoning_ in the public school system, but it is not taught with a focus on reason but a focus on rote procedural memorization - though the situational ("word") problems require the application of reason.

That's part of what makes math so bloody unattractive to certain people - like me! I love to reason, I'm steeped in it every single day as it IS my job (Systems Architect), but back in school Math was my absolute worst subject. All just memorize this or that rule, apply it over and over again until it becomes reflex, then onto the next rule. You don't get to do anything really freaking cool with it until way later on. I guess you have to walk before you can run, but still, so many students can't even visualize anything BUT endless walking when they think about Math.

Maybe that's changing these days with tools like Mathematica and the rise of Computer Graphics and complex games. Maybe kids these days can see more of what math can really do. Too bad most never get the opportunity.

posted by C.Batt at 11:02 PM on February 16, 2006

brundlefly, you get out of my head and I'll get out of yours. m'kay?

posted by C.Batt at 11:07 PM on February 16, 2006

posted by C.Batt at 11:07 PM on February 16, 2006

But... but... I like it in here...

posted by brundlefly at 11:09 PM on February 16, 2006

posted by brundlefly at 11:09 PM on February 16, 2006

I'm not good at something, so obviously it's worthless to everyone. I mean if I can't do it, no one else needs it, right?

And people wonder why Americans are falling behind in almost every single academic field.

posted by Talanvor at 11:24 PM on February 16, 2006

And people wonder why Americans are falling behind in almost every single academic field.

posted by Talanvor at 11:24 PM on February 16, 2006

I'm good at math and English and reasoning! Check it out:

Richard Cohen = douchebag

posted by speicus at 11:24 PM on February 16, 2006

Richard Cohen = douchebag

posted by speicus at 11:24 PM on February 16, 2006

He says it much better than I... http://math.ucr.edu/~snelson/whymath.html

posted by dustsquid at 11:27 PM on February 16, 2006

posted by dustsquid at 11:27 PM on February 16, 2006

What's the difference between a troll and a columnist?

Again I forget.

Anyway - nice troll Cohen!

posted by sien at 11:34 PM on February 16, 2006

Again I forget.

Anyway - nice troll Cohen!

posted by sien at 11:34 PM on February 16, 2006

Someone send that whymath.html link to that douchebag. Maybe it'll cause the spontaneous growth of a brain cell and he'll LEARN something.

or maybe not.

posted by Jelreyn at 11:46 PM on February 16, 2006

or maybe not.

posted by Jelreyn at 11:46 PM on February 16, 2006

Mathematics and logic exercise one kind of muscle. Writing and making music exercise another kind. Dwarf-tossing yet another.

Exercising all your muscles makes you stronger.

posted by stavrosthewonderchicken at 11:46 PM on February 16, 2006

Exercising all your muscles makes you stronger.

posted by stavrosthewonderchicken at 11:46 PM on February 16, 2006

Did it ever occur to the principal at Gabriela's school that what she needed to retake was

My sister failed algebra and had to change high schools to get a diploma so she didn't have to retake it. Later on in life she decided to go to college and realized, damn, she needed to take algebra. But she had to work up to it. So she buckled down and tried and found out an interesting thing. She didn't know how to multiply. Somwhere along her educational career (like 3rd grade) she was allowed to progress without being able to multiply. After she learned how to do that, algebra came relatively easily. And of course it did, she's a smart girl.

I bet Gabriela has a similiar problem. Math is a structure, and if you miss a key piece you can never progress.

posted by slickvaguely at 11:49 PM on February 16, 2006

**pre-algebra**, not algebra. You fail a class 6 times, you never should have been in it in the first place.My sister failed algebra and had to change high schools to get a diploma so she didn't have to retake it. Later on in life she decided to go to college and realized, damn, she needed to take algebra. But she had to work up to it. So she buckled down and tried and found out an interesting thing. She didn't know how to multiply. Somwhere along her educational career (like 3rd grade) she was allowed to progress without being able to multiply. After she learned how to do that, algebra came relatively easily. And of course it did, she's a smart girl.

I bet Gabriela has a similiar problem. Math is a structure, and if you miss a key piece you can never progress.

posted by slickvaguely at 11:49 PM on February 16, 2006

Mr. Cohen exhibits what used to be called good old American anti-intellectualism. goAa-i is why computer people are called names that once were reserved for ~~geeks~~ carnival performers who tore the heads off chickens with their teeth.

posted by Cranberry at 11:52 PM on February 16, 2006

posted by Cranberry at 11:52 PM on February 16, 2006

Math

posted by hoverboards don't work on water at 11:56 PM on February 16, 2006

**is**writing. Math is not a string of abstract symbols. Math papers are written using full sentences, with the symbols being integral, grammatical parts of those sentences. You cannot be good at math without being good at writing. Those people who are apparently good at math but can't write a clear sentence are actually**not**good at math. They're simply good at arithmetic.posted by hoverboards don't work on water at 11:56 PM on February 16, 2006

*Math is writing.*

That was really the stupidest part of his column. The idea that math people can't write (or do anything else) is just beyond stupid. It's an idea that has no place being published in a national paper, it's just so dam stupid. It's just sour-grapes, anecdote equals data crap.

*shakes head ruefully*

And for the record Mr. Cohen: all of the smart kids when I was in high school, even if they hated math or english or biology or whatever, got good marks in those fields. That's part of what it means to be smart.

posted by teece at 12:03 AM on February 17, 2006

*Math doesn't teach reasoning. Dialectic, Rhetoric, and Logic (formal) teach reasoning.*

But formal logic is a branch of mathematics. In fact, the philosophical tools you cite adhere to simple standards of logic that are grounded in basic mathematics. Rhetoric without logical standards would be an aesthetic, not a proof system. It might persuade, but it will not prove anything.

posted by kid ichorous at 12:10 AM on February 17, 2006

Algebra teaches reasoning? Algebra doesn't do anything for me personally. Let me write any day and I'll learn something but algebra teaches little if nothing and is not needed later in life.

posted by pancreas at 12:10 AM on February 17, 2006

posted by pancreas at 12:10 AM on February 17, 2006

**Algebra doesn't do anything for me personally.**Let me write any day and I'll learn something but algebra teaches little if nothing and is not needed later in life.All I can do is highlight that sentence.

posted by brundlefly at 12:14 AM on February 17, 2006

Maybe the truth is that there is no one educational subject that ought to be sanctified to the point that a person is reduced to leaving school because of it, (Math or Literature). I'm an engineer and my mother has a doctorate in probability, so I'm not hating on math, but I really enjoy a well written book as well. What I've gathered from her experiences teaching is that math professors hate to teach people who don't want to learn the subject. Physics for Poets is always going to be a terrible terrible class. Maybe if we tried to build our educational system to respect and aid a person going into a mathematic...al field, or a field in language, and even (gasp), people who just are not going to pursue a very intellectual lifestyle, we'd all be a little better off.

I don't want to make people choose their lifetime profession at the age of 15, but IMHO the vast majority of people don't have time to learn to do but a few things really really well. I don't think schools should try to force a mediocre knowledge of all subjects. So I kinda sympathize w/ the girl in the article, but not in the use of her situation to attack the subject of mathematics.

posted by SomeOneElse at 12:14 AM on February 17, 2006

I don't want to make people choose their lifetime profession at the age of 15, but IMHO the vast majority of people don't have time to learn to do but a few things really really well. I don't think schools should try to force a mediocre knowledge of all subjects. So I kinda sympathize w/ the girl in the article, but not in the use of her situation to attack the subject of mathematics.

posted by SomeOneElse at 12:14 AM on February 17, 2006

Arithmetic is NOT math, and algebra and calculus aren't really either... they're just tools. Real math (math that is done by mathematicians) is about proofs and relations and abstraction. The author really displays total ignorance of what math is - the stuff we do on calculators and computers is not math, it is computation.

Mathematical concepts are fundamental to reality - everything about our world is defined by mathematics, from the simplest laws of physics to complex economic and social models. You CANNOT consider yourself educated and informed about the world if you do not understand some level of mathematics. How a columnist for a leading newspaper can think that algebra (one of the easiest subjects to master) is unimportant is beyond comprehension.

Writing cannot teach reasoning, writing is a form of expression, and plenty of good writers display poor logic and reasoning, as this columnist does.

posted by aerify at 12:16 AM on February 17, 2006

Mathematical concepts are fundamental to reality - everything about our world is defined by mathematics, from the simplest laws of physics to complex economic and social models. You CANNOT consider yourself educated and informed about the world if you do not understand some level of mathematics. How a columnist for a leading newspaper can think that algebra (one of the easiest subjects to master) is unimportant is beyond comprehension.

Writing cannot teach reasoning, writing is a form of expression, and plenty of good writers display poor logic and reasoning, as this columnist does.

posted by aerify at 12:16 AM on February 17, 2006

*algebra teaches little if nothing and is not needed later in life.*

You're seriously ignorant if you believe algebra teaches nothing. There is an entire world you are completely ignorant of.

I don't need music or art "later in life" -- I am very glad I was taught the rudiments of these things as a kid. Don't wear your ignorance as some kind of badge of courage -- it's unbecoming.

posted by teece at 12:24 AM on February 17, 2006

*Math is writing. Math is not a string of abstract symbols. Math papers are written using full sentences, with the symbols being integral, grammatical parts of those sentences.*

Hoverboards, this is absolutely true, but most people have never been introduced to math as a formal language. My school, for example, made this transition somewhere between linear algebra and undergrad abstract algrbra - at this point, most of the class was math majors. I wonder if people might have a better relationship with math if proofs and formal language could be taught earlier, instead of focusing on calculus for freshmen and geometry in highschool.

posted by kid ichorous at 1:59 AM on February 17, 2006

*In other words, Rudy Rucker should write an Algebra I curriculum.*

Yes!

posted by sonofsamiam at 5:28 AM on February 17, 2006

In these post-Katrina days I've found myself designing a house. Because my wife is traumatized by fallen trees (ca. 2003), hurricanes, tornadoes (thanks to our local fishwrapper), and whatnot, it needs to be an underground house. You can't use the standard design tables for buildings that have to hold up a 320 psf roof load. So I have been using a

I suppose I could have just plunked down a bunch of money for someone else who knows the math to do it for me, but where's the fun in

posted by localroger at 5:40 AM on February 17, 2006

*lot*of math I thought I'd never see again. Algebra. Statics. Moments of Inertia. A couple of times I have even come*this close*to having to solve an integral, and I may end up needing my rusty calculus skillz yet.I suppose I could have just plunked down a bunch of money for someone else who knows the math to do it for me, but where's the fun in

*that*? Yes you can live a full and productive life without any of a number of skills, but you will never know what you are missing if you do.posted by localroger at 5:40 AM on February 17, 2006

Lit me tell u: writurs r a bunch o' morans n u dun' need no writin' or collum or watever to be vary sucesfull in this weld. writin is just sum conspircy to make u flunk at skool. I nevr neded ritin' skils in me life, n dun think ain't gonna never need.

posted by qvantamon at 5:47 AM on February 17, 2006

posted by qvantamon at 5:47 AM on February 17, 2006

*Typing: Best class I ever took.*

I don't agree with most of the column, but I agree with that sentiment. I took two semensters of typing (yes on real typewriters) in high school. I got, by far, the worst grades of my whole HS career in those two classes, but there (in HS specifically) isn't anything else I learned there that I use more every on a daily basis. <Look!! I'm doing it now!!>

posted by hwestiii at 5:54 AM on February 17, 2006

So, to join the pile-on, I never quite understood people who make a public display of their ignorance. My guess is that he still feels inadequate after all these years and he hopes to resolve some his low self-esteem issues by congratulating other people who have failed in the same way.

He doesn't need to learn algebra, it's true - he needs therapy.

posted by GuyZero at 5:58 AM on February 17, 2006

He doesn't need to learn algebra, it's true - he needs therapy.

posted by GuyZero at 5:58 AM on February 17, 2006

Interestingly enough -- whether a provable causal link between writing and reading and cognition (including the ability to reason) is a hot debate in the academic fields of rhetoric, composition, and literacy.

posted by mrmojoflying at 5:58 AM on February 17, 2006

posted by mrmojoflying at 5:58 AM on February 17, 2006

yeah, this is really sad. I also hope that more logic-oriented / less memory oriented forms of math get taught... I loved geometry in junior high because it was all about proofs, and I found that exciting & interesting. But then algebra came along and suddenly it was all a bore again - I suffered thru until pre-calc but dropped the math curriculum at that point & completely ignored math in college, and it really wasn't until grad school in philosophy that I started realizing how fascinating math is (thanks plato). I would really like to take calculus now...

It's kind of amazing how poor the reasoning skills of yr average citizen are, though. I mean, it does take a lot of work to really come to terms with what logic is and what it means, but even just very basic 'critical thinking' kind of stuff is new to a lot of people... (I teach an adult ed critical thinking class as well as community college intro to philos, so have exchanges with populations perhaps somewhat less well-grounded in this stuff than mefi)

posted by mdn at 6:01 AM on February 17, 2006

It's kind of amazing how poor the reasoning skills of yr average citizen are, though. I mean, it does take a lot of work to really come to terms with what logic is and what it means, but even just very basic 'critical thinking' kind of stuff is new to a lot of people... (I teach an adult ed critical thinking class as well as community college intro to philos, so have exchanges with populations perhaps somewhat less well-grounded in this stuff than mefi)

posted by mdn at 6:01 AM on February 17, 2006

"...whatever Algebra is, there’s no doubt about it, ’tis a true friend to the poor man."

Just a humourous tale from my homeland.

posted by hangashore at 6:01 AM on February 17, 2006

Just a humourous tale from my homeland.

posted by hangashore at 6:01 AM on February 17, 2006

Cohen's column was evidently based on a much longer story from the LA Times: "A Formula For Failure In L.A. Schools" The article reveals what has happened to Gabriela and hints at how she now understands why she should have made the effort to pass. (She reportedly skipped class most of the time.)

The article also considers the efforts administrators, teachers and parents are making to improve standards in the school. Overall, a very interesting article.

One last thing, I would like to invite people post real examples of how they have actually used algebra in their daily lives. I don't think that making change counts. That's just subtraction. Figuring out how many apples you get for $1 at .33 an apple barely qualifies. I'll bet that even Gabriela can figure that one out.

I'm talking about things like quadratic equations, linear equations, adding and subtracting square roots and other radicals, exponents, factoring and so on. I was impressed with what localroger wrote. Surely there must be others who can give such examples.

posted by notmtwain at 6:28 AM on February 17, 2006

The article also considers the efforts administrators, teachers and parents are making to improve standards in the school. Overall, a very interesting article.

One last thing, I would like to invite people post real examples of how they have actually used algebra in their daily lives. I don't think that making change counts. That's just subtraction. Figuring out how many apples you get for $1 at .33 an apple barely qualifies. I'll bet that even Gabriela can figure that one out.

I'm talking about things like quadratic equations, linear equations, adding and subtracting square roots and other radicals, exponents, factoring and so on. I was impressed with what localroger wrote. Surely there must be others who can give such examples.

posted by notmtwain at 6:28 AM on February 17, 2006

My first thought upon reading that article was this quote from Heinlein, which one of the Pharyngula folks put into the comments:

posted by bashos_frog at 6:32 AM on February 17, 2006

**"Anyone who cannot cope with mathematics is not fully human. At best he is a tolerable subhuman who has learned to wear shoes, bathe and not make messes in the house."**posted by bashos_frog at 6:32 AM on February 17, 2006

Mathematics is, like language, a powerful tool for putting handles on the hot pot that is human experience. Like all tools, it inevitably reveals the skill and wit of its user.

posted by sic friat crustulum at 6:43 AM on February 17, 2006

posted by sic friat crustulum at 6:43 AM on February 17, 2006

Gabriela is almost an anagram of Algebra. Coincidence? Or almost a coincidence?

posted by petri at 6:48 AM on February 17, 2006

posted by petri at 6:48 AM on February 17, 2006

example:

I'm writing a web application that, among other things provides the nearest subway station for places. Given latitude and longitudes for all the locations involved, I used trig to write my database query.

ACOS(

SIN(x1.lat_degrees) * SIN(x2.lat_degrees) +

COS(x1.lat_degrees) * COS(x2.lat_degrees) *

COS(x2.long_degrees - x1.long_degrees)

)/360 * 2 * PI() * 3956

posted by bashos_frog at 6:49 AM on February 17, 2006

I'm writing a web application that, among other things provides the nearest subway station for places. Given latitude and longitudes for all the locations involved, I used trig to write my database query.

*Distance between two coordinates (x1 and x2) on a sphere:*ACOS(

SIN(x1.lat_degrees) * SIN(x2.lat_degrees) +

COS(x1.lat_degrees) * COS(x2.lat_degrees) *

COS(x2.long_degrees - x1.long_degrees)

)/360 * 2 * PI() * 3956

posted by bashos_frog at 6:49 AM on February 17, 2006

where 3956 is the radius of the sphere - and yes I know the earth is an oblate, not perfect, sphere - sue me.

posted by bashos_frog at 6:51 AM on February 17, 2006

posted by bashos_frog at 6:51 AM on February 17, 2006

Cranberry, what is goAa-i ?

I'm not convinced that all high school graduates need algebra or trigonometry. Everybody needs business/common sense math. I took high school and college algebra, and was woefully unprepared for using it in the real world. I think logic and ethics should be required.

Cohen's delight in his ignorance is embarassing.

posted by theora55 at 6:52 AM on February 17, 2006

I'm not convinced that all high school graduates need algebra or trigonometry. Everybody needs business/common sense math. I took high school and college algebra, and was woefully unprepared for using it in the real world. I think logic and ethics should be required.

Cohen's delight in his ignorance is embarassing.

posted by theora55 at 6:52 AM on February 17, 2006

Ok here's a an example for you: the other day, my brother writes me to ask how long he needs to make the sides of a collapsing brace for a travel case. You know, those little bars on sliders you snap into a triangle to brace the lid of a case open.

He wanted a somewhat fancy one, with a sliding latch and so on. He'd wrestled with it for a couple of days, playing with bits of metal, measuring them, but was unable to get the action to work smoothly. He was going to manufacturing soon and need a spec to give the shop.

With a couple of quick sketches, he figured out a couple of relationships and with the help of the right triangle formula (sum of squares of the sides equal to square of the hypotenuse) he was able to get a quadratic for his latch lengths. A quick application of the quadratic formula later and he was able to solve in a quarter hour, by algebra, a problem that he could not figure out empirically in a couple of days.

Algebra saves the order for my brother's company. Therefore algebra=$$$

QED.

posted by bonehead at 6:52 AM on February 17, 2006

He wanted a somewhat fancy one, with a sliding latch and so on. He'd wrestled with it for a couple of days, playing with bits of metal, measuring them, but was unable to get the action to work smoothly. He was going to manufacturing soon and need a spec to give the shop.

With a couple of quick sketches, he figured out a couple of relationships and with the help of the right triangle formula (sum of squares of the sides equal to square of the hypotenuse) he was able to get a quadratic for his latch lengths. A quick application of the quadratic formula later and he was able to solve in a quarter hour, by algebra, a problem that he could not figure out empirically in a couple of days.

Algebra saves the order for my brother's company. Therefore algebra=$$$

QED.

posted by bonehead at 6:52 AM on February 17, 2006

Algebra = problem solving

If you don't know or like algebra, you probably also suck at solving problems in general.

posted by HTuttle at 6:57 AM on February 17, 2006

If you don't know or like algebra, you probably also suck at solving problems in general.

posted by HTuttle at 6:57 AM on February 17, 2006

I remember a book I read called Innumeracy, which I foolishly lent to a pal (and have not seen since). One of the interesting things he said was how odd it is that some people are proud of their inability to do math, and that no one would boast of being unable to use basic grammar or spelling.

And I can't believe people arguing that they never use algerbra. I can see arguing that calculus has been largely useless (though it does come in handy for any sort of question you have about physics, which comes up fairly often or should). But all I can think is that people who don't spend at least a little time with math must be profoundly incurious about the world around them, and uninterested in proving any argument they might have. Hell, the most important thing that I learned in math class was probably something that comes AFTER algebra: probability. Figuring out the chance that something will happen is so frequent in my life (from poker to making budgets) that I can't imagine not having that tool.

posted by klangklangston at 6:58 AM on February 17, 2006

And I can't believe people arguing that they never use algerbra. I can see arguing that calculus has been largely useless (though it does come in handy for any sort of question you have about physics, which comes up fairly often or should). But all I can think is that people who don't spend at least a little time with math must be profoundly incurious about the world around them, and uninterested in proving any argument they might have. Hell, the most important thing that I learned in math class was probably something that comes AFTER algebra: probability. Figuring out the chance that something will happen is so frequent in my life (from poker to making budgets) that I can't imagine not having that tool.

posted by klangklangston at 6:58 AM on February 17, 2006

bashos, you're a showoff.. but that's a cool application of math.. funny thing is I was awesome at it till HS.. barely made it through in HS but I got a 700 in Math section of the SAT..

posted by pez_LPhiE at 7:05 AM on February 17, 2006

posted by pez_LPhiE at 7:05 AM on February 17, 2006

The problem is that the punishment for not learning algebra isn't strict enough. Rather then not graduating high school, they should go to jail. Then they'll learn algebra.

See, it has nothing to do with not being able to understand math, it has to do with

And what the hell is someone who can't even do percentages doing being a reporter?

posted by delmoi at 7:09 AM on February 17, 2006

*I confess to be one of those people who***hate**math. I can do my basic arithmetic all right (although not percentages) but I flunked algebra (once), barely passed it the second timeSee, it has nothing to do with not being able to understand math, it has to do with

**hating**it. Hating it probably because they were taught it wrongly in elementary school. I'm certain that other then the severely disabled, there is child on earth who could not be taught algebra.And what the hell is someone who can't even do percentages doing being a reporter?

posted by delmoi at 7:09 AM on February 17, 2006

*In due course, this came to be the way I made my living. Typing: Best class I ever took.*

The only typing class people need to take these days is the one that gets them to type 'you' rather then 'u'.

posted by delmoi at 7:11 AM on February 17, 2006

Like Brundlefly, I think back to my algebra classes in horror. I simply never got it. Geometry was a snap for me, but algebra was simply an enigma. I also have to agree that the main problem was in the way it was taught. Sheets full of problems with no real background as to

I have a feeling if it had been taught more akin to a language, rather than the next step beyond multiplication/division, I may have actually gotten it.

I believe algebra class is where many people fall into great hatred with math.

posted by Thorzdad at 7:12 AM on February 17, 2006

*why or how*algebra works in the way it does.I have a feeling if it had been taught more akin to a language, rather than the next step beyond multiplication/division, I may have actually gotten it.

I believe algebra class is where many people fall into great hatred with math.

posted by Thorzdad at 7:12 AM on February 17, 2006

Don't people use algebra everyday, whether they know it or not? For example... I have 50 cookies and 5 kids. How many cookies should I give each kid?

5 kids times X-number of cookies = 50

Divide both sides by 5

x = 10

That's algebra, no?

posted by Witty at 7:13 AM on February 17, 2006

5 kids times X-number of cookies = 50

Divide both sides by 5

x = 10

That's algebra, no?

posted by Witty at 7:13 AM on February 17, 2006

Hey, if you click the "all blogs" link on the page you can see they apperantly link back to metafilter.

posted by delmoi at 7:20 AM on February 17, 2006

posted by delmoi at 7:20 AM on February 17, 2006

The thing is, she was required to pass a subject (for which there is now a recognised dyslexia) without assistance and without necessity.

I've never used algebra, or a lot of maths in my life. I'm not proud of this, but I can recognise that I simply can't get my head around it.

Why not just let algebra be an elective and basic maths remain the requirement?

posted by katiecat at 7:25 AM on February 17, 2006

I've never used algebra, or a lot of maths in my life. I'm not proud of this, but I can recognise that I simply can't get my head around it.

Why not just let algebra be an elective and basic maths remain the requirement?

posted by katiecat at 7:25 AM on February 17, 2006

I'm still trying to figure out if that column was a joke. It's just...wow.

This thread has got me thinking back on my math education. I was one of those "bad at math" kids until 6th grade, when they gave us all the 7th grade test to place us in different levels. Because the 7th grade test had stuff I hadn't "learned" yet, I was able to use REASON to figure out some of the answers. I was bumped up a level. Before that, math had been all memorizing and timed tests.

Between 6th and 12th grade, I had some good math teachers and some bad ones. The worst was the one that would take away credit (in a very UN-mathematical way...if the homework had 20 questions, and I didn't show work for #20b., she'd give me a 7.5/8--bizarre, I know) if I didn't show the work the way SHE taught it to us...usually some convoluted diagram that was very counterintuitive to me. Once I understood the structure of a problem, I could usually reason it out in my head. To me, that's what algebra was--just boiled down logic. Some teachers presented it that way, others didn't. The drawings that teacher made us do seemed like they were for dense people or babies to me. Of course, now I know we just thought differently, and you can't really teach people to "just do it in your head." But I still hold it against her that she couldn't or wouldn't recognize that I understood it, just not how she did.

I went to NYU, where most kids had already taken AP Calc in high school. I only got as far as pre-calc. I took the test to get exempted from the math credit, and I think I got 100 on it. Some of the questions I didn't know how to do, but I used reason. I've sinced moved on to a career not unlike Mr. Cohen's, and I've forgotten most of the actual "math" I learned in high school. But I can figure things out if I need to. I don't think algebra necessarily teaches reason (especially if your teacher is like mine, above), but I do think it can nurture it.

When I read about Gabriela, all I could think is that she must have had an awful teacher. I honestly don't think there's a kid in America who could not pass algebra in six tries (!) with a minimally qualified teacher. If the teacher didn't have time to give her special attention, she could have at least set up some kind of peer-to-peer tutoring thing, or something! And now that I see that it's a problem for lots of kids in that area, there must be something wrong with how they're teaching it. I understand that algebra is harder for some than for others. But not that hard.

posted by lampoil at 7:36 AM on February 17, 2006

This thread has got me thinking back on my math education. I was one of those "bad at math" kids until 6th grade, when they gave us all the 7th grade test to place us in different levels. Because the 7th grade test had stuff I hadn't "learned" yet, I was able to use REASON to figure out some of the answers. I was bumped up a level. Before that, math had been all memorizing and timed tests.

Between 6th and 12th grade, I had some good math teachers and some bad ones. The worst was the one that would take away credit (in a very UN-mathematical way...if the homework had 20 questions, and I didn't show work for #20b., she'd give me a 7.5/8--bizarre, I know) if I didn't show the work the way SHE taught it to us...usually some convoluted diagram that was very counterintuitive to me. Once I understood the structure of a problem, I could usually reason it out in my head. To me, that's what algebra was--just boiled down logic. Some teachers presented it that way, others didn't. The drawings that teacher made us do seemed like they were for dense people or babies to me. Of course, now I know we just thought differently, and you can't really teach people to "just do it in your head." But I still hold it against her that she couldn't or wouldn't recognize that I understood it, just not how she did.

I went to NYU, where most kids had already taken AP Calc in high school. I only got as far as pre-calc. I took the test to get exempted from the math credit, and I think I got 100 on it. Some of the questions I didn't know how to do, but I used reason. I've sinced moved on to a career not unlike Mr. Cohen's, and I've forgotten most of the actual "math" I learned in high school. But I can figure things out if I need to. I don't think algebra necessarily teaches reason (especially if your teacher is like mine, above), but I do think it can nurture it.

When I read about Gabriela, all I could think is that she must have had an awful teacher. I honestly don't think there's a kid in America who could not pass algebra in six tries (!) with a minimally qualified teacher. If the teacher didn't have time to give her special attention, she could have at least set up some kind of peer-to-peer tutoring thing, or something! And now that I see that it's a problem for lots of kids in that area, there must be something wrong with how they're teaching it. I understand that algebra is harder for some than for others. But not that hard.

posted by lampoil at 7:36 AM on February 17, 2006

*5 kids times X-number of cookies = 50*

Divide both sides by 5

x = 10

Divide both sides by 5

x = 10

That's a rather long way to go to just divide 50 cookies by 5 kids. Most people would use simple division. 50/5=10

And maybe that's where a lot of people had a problem with algebra. It sometimes did certainly look like it was the math version of taking the long way home.

posted by Thorzdad at 7:45 AM on February 17, 2006

Everyone should read Bertrand Russell's The Problems of Philosophy, especially the last chapter The value of philosophy (the copyright has expired, I guess). Anyone who would claim that mathematicians can't write after that would prove themselves a fool. I think Russell would have considered mathematics a part of philosophy.

posted by delmoi at 7:51 AM on February 17, 2006

posted by delmoi at 7:51 AM on February 17, 2006

How many of us ever actually use algebra in real life? And why weren't we taught things we actually do use, numbers-wise? Financial education should be taught--credit cards, debt, interest, loans, how markets work and how they affect prices and everyday life, investment, etc...

If education is to prepare us for the world, why not teach stuff that's actually more relevant to the world? Math is taught the most abstractly, yet is vitally important.

posted by amberglow at 7:54 AM on February 17, 2006

If education is to prepare us for the world, why not teach stuff that's actually more relevant to the world? Math is taught the most abstractly, yet is vitally important.

posted by amberglow at 7:54 AM on February 17, 2006

Interesting level of unselfconscious ignorance in this thread. Several people identify arithmetic as algebra

I'm not sure I can defend a liberal arts rationale for the teaching of mathematics, or anything else, in our public schools because, frankly, our schools do this very poorly. And it's not what almost anyone wants from our schools, even most of those who defend, for example, math as teaching "analytical thought", or whatever. Almost all of these people, when we get down to brass tacks, expect our schools to primarily function as vocational schools. And if our schools are

I hardly know anyone who didn't resent a requirement to take classes which form the liberal arts core of most undergraduate degrees. "It's a waste of my time", they said. And, as a matter of fact, because of the factory-like dehumanized production system which are our universities, it almost always

So what liberal arts tradition are we defending? Does it really exist in the US?

There’s a startlingly large number of college freshmen who, supposedly, made it to Calculus by the time they left high school and yet struggle with College Algebra and are forced to drop back to Intermediate. A large part of the reason for this is because math is typically taught these days as a set of techniques, not as something essential which must truly be comprehended. Make no mistake—those techniques are hugely important, absolutely critical, for those students who will complete degrees which require a mastery of them. But when I look at those students who will not use any of these techniques as a part of their personal or professional lives, and yet who are thought to have a considerable amount of basic math under the belt by the time they reach university—and then demonstrate that they don’t—well, I then wonder at the absurdity of it all. They don't remember it after as short interval of time as a summer break because, sadly, they never really understood it in the first place.

The educational system in the US is very poor, and that includes almost all undergraduate educations. The reason for this is this absurd charade of valuing a liberal arts education while, in fact, we value vocational education. The result is that we manage to screw

Also, surely this writer is correct when he claims that a strong facility with language is far more useful

I strongly support the idea of math as an essential part of a liberal arts education. But I would, of course, being a product of the St. John's College "Great Books" curriculum which includes four full years of math as part of its program. Math is, of course, one of the core liberal arts. And I watched a good number of students who thought themselves haters of math become lovers of math.

But I'm baffled by the claims of the utility of math as it's predominantly being taught to general students in secondary schools and universities. There is very little deep comprehension, while there is a great deal of repitition of the execution of very specific techniques. But will these general students ever need those techniques? Yes, they'll need them for further required math classes that rely upon a facility with those techniques as a foundation to teach even more technique. When it's all said and done, for the general student, what has been accomplished?

In my opinion, math education is the clearest example of how our educational system is failing as a result of a lack of clarity of purpose. That young woman may have benefited from a liberal arts approach to algebra, or she may have benefited from a few semesters spent preparing her for some kind of work. But instead, the school created another dropout.

1. An implication of this misconception is that algebra is platonic, it

2. One outcome that our schools seem quite successul at realizing is the instillation of a deep antipathy toward mathematics.

posted by Ethereal Bligh at 7:59 AM on February 17, 2006

^{1}. Another identifies logic as mathematics. These sorts of conceptual errors, which are really errors of ignorance, demonstrate that those above who defend the teaching of mathematics as a liberal art and not as a set of vocational techniques nevertheless can't tell the difference between the two and seem to only know math as a vocational technique, albeit highly specialized. Irony.I'm not sure I can defend a liberal arts rationale for the teaching of mathematics, or anything else, in our public schools because, frankly, our schools do this very poorly. And it's not what almost anyone wants from our schools, even most of those who defend, for example, math as teaching "analytical thought", or whatever. Almost all of these people, when we get down to brass tacks, expect our schools to primarily function as vocational schools. And if our schools are

*de facto*vocational schools, and are thus because they reflect the wishes of an overwhelming majority, then this guy's argument is more right than it is wrong.I hardly know anyone who didn't resent a requirement to take classes which form the liberal arts core of most undergraduate degrees. "It's a waste of my time", they said. And, as a matter of fact, because of the factory-like dehumanized production system which are our universities, it almost always

*was*a waste of their time. It's taught poorly to too-large classes to indifferent-to-hostile students.So what liberal arts tradition are we defending? Does it really exist in the US?

There’s a startlingly large number of college freshmen who, supposedly, made it to Calculus by the time they left high school and yet struggle with College Algebra and are forced to drop back to Intermediate. A large part of the reason for this is because math is typically taught these days as a set of techniques, not as something essential which must truly be comprehended. Make no mistake—those techniques are hugely important, absolutely critical, for those students who will complete degrees which require a mastery of them. But when I look at those students who will not use any of these techniques as a part of their personal or professional lives, and yet who are thought to have a considerable amount of basic math under the belt by the time they reach university—and then demonstrate that they don’t—well, I then wonder at the absurdity of it all. They don't remember it after as short interval of time as a summer break because, sadly, they never really understood it in the first place.

The educational system in the US is very poor, and that includes almost all undergraduate educations. The reason for this is this absurd charade of valuing a liberal arts education while, in fact, we value vocational education. The result is that we manage to screw

*both*of them up and our students lag behind those elsewhere.Also, surely this writer is correct when he claims that a strong facility with language is far more useful

*vocationally*than is a facility with algebra. Yet, even from a liberal arts perspective, formal reasoning exists first and foremost as an activity deeply associated with language. This is certainly true with regard to history—but it may very well be true with regard to human biology. That young woman would have been far,*far*better served by six semesters of reading and writing.I strongly support the idea of math as an essential part of a liberal arts education. But I would, of course, being a product of the St. John's College "Great Books" curriculum which includes four full years of math as part of its program. Math is, of course, one of the core liberal arts. And I watched a good number of students who thought themselves haters of math become lovers of math.

But I'm baffled by the claims of the utility of math as it's predominantly being taught to general students in secondary schools and universities. There is very little deep comprehension, while there is a great deal of repitition of the execution of very specific techniques. But will these general students ever need those techniques? Yes, they'll need them for further required math classes that rely upon a facility with those techniques as a foundation to teach even more technique. When it's all said and done, for the general student, what has been accomplished?

^{2}In my opinion, math education is the clearest example of how our educational system is failing as a result of a lack of clarity of purpose. That young woman may have benefited from a liberal arts approach to algebra, or she may have benefited from a few semesters spent preparing her for some kind of work. But instead, the school created another dropout.

1. An implication of this misconception is that algebra is platonic, it

*is*mathematics itself—an implication that greatly diminishes algebra as an actual creation/discovery. An implication that is comfortable to Westerners, who are taught nothing at all about the genesis and history of algebra. "Calculus is a grand intellectual achievement!" in contrast to "Algebra is elementary and obvious."2. One outcome that our schools seem quite successul at realizing is the instillation of a deep antipathy toward mathematics.

posted by Ethereal Bligh at 7:59 AM on February 17, 2006

Where did he find this Gabriela? Did he get her permission to use her name like this? Seems pretty humiliating.

I have tutored kids like Gabriela, and helped them to get A's in Math where before they were failing. Really, even someone as dumb as Cohen can learn Math all the way up to calculus, as long as you're careful to watch how they practice and set them straight when they stumble. Not everyone can afford that kind of attention, though, I suppose. It's a shame...

posted by Coventry at 7:59 AM on February 17, 2006

I have tutored kids like Gabriela, and helped them to get A's in Math where before they were failing. Really, even someone as dumb as Cohen can learn Math all the way up to calculus, as long as you're careful to watch how they practice and set them straight when they stumble. Not everyone can afford that kind of attention, though, I suppose. It's a shame...

posted by Coventry at 7:59 AM on February 17, 2006

*When I read about Gabriela, all I could think is that she must have had an awful teacher. I honestly don't think there's a kid in America who could not pass algebra in six tries (!) with a minimally qualified teacher.*

I totally agree. Basic algebra is incredibly simple. The kids who get turned off on math do so because they have bad teachers.

posted by delmoi at 7:59 AM on February 17, 2006

"And what the hell is someone who can't even do percentages doing being a reporter?"

We had to take a unit on percentages as part of my Advanced Reporting class, since so much of a local beat in government reporting covers millages.

The other thing I can say is that I never liked math, really, until I had an amazing Trig and Pre-Calc (we called it Analysis) teacher my junior year of high school. Suddenly, it was like the veil was lifted. I think that probably has to do more with people hating math than anything about the math.

posted by klangklangston at 8:00 AM on February 17, 2006

We had to take a unit on percentages as part of my Advanced Reporting class, since so much of a local beat in government reporting covers millages.

The other thing I can say is that I never liked math, really, until I had an amazing Trig and Pre-Calc (we called it Analysis) teacher my junior year of high school. Suddenly, it was like the veil was lifted. I think that probably has to do more with people hating math than anything about the math.

posted by klangklangston at 8:00 AM on February 17, 2006

I'm going to be the contrarian here and agree with Cohen that you shouldn't have to pass algebra to graduate. Sure, he's over the top (hey, he's a columnist, not a philosopher), and he may in fact be anti-intellectual (I don't read him regularly, so I have no idea), but he's right that algebra, interesting as it can be, is in fact useless in ordinary life (calculating your change? please, what does that have to do with algebra?) and for the vast majority of people is forgotten immediately after they take their final exam. This is not to say it shouldn't be taught, or that it's not important in many ways, just that it's stupid to forbid someone from graduating because they happen to be bad at it (or, as is too often the case, because it's wretchedly taught).

And I say this as a former math major who loved algebra in high school.

posted by languagehat at 8:04 AM on February 17, 2006

And I say this as a former math major who loved algebra in high school.

posted by languagehat at 8:04 AM on February 17, 2006

I was wondering when someone was going to comment on that statement.Math doesn't teach reasoning. Dialectic, Rhetoric, and Logic (formal) teach reasoning.

But formal logic is a branch of mathematics. In fact, the philosophical tools you cite adhere to simple standards of logic that are grounded in basic mathematics. Rhetoric without logical standards would be an aesthetic, not a proof system. It might persuade, but it will not prove anything.

posted bykid ichorousat 1:10 AM MST on February 17

Logic is more than the algebraic expression of logical statements. It is more than boolean math. That is

*symbolic logic*which is the application of Math to the realm of Logic.

See the wikipedia article on the matter (not the only source, the the quickest). I would argue that Math and Logic are intimately tied together, perhaps in a chicken-and-egg coexistence, but that they are indeed separate entities.

posted by C.Batt at 8:08 AM on February 17, 2006

I should have post scripted that last statement with a:

I'm just being a pedantic hair-splitter and don't want to derail this thread.

/ please don't hate me

// the article sucks. Anti-intellectualism sucks. The education system can also suck, especially in the case described in the article.

posted by C.Batt at 8:10 AM on February 17, 2006

I'm just being a pedantic hair-splitter and don't want to derail this thread.

/ please don't hate me

// the article sucks. Anti-intellectualism sucks. The education system can also suck, especially in the case described in the article.

posted by C.Batt at 8:10 AM on February 17, 2006

No muddin' up ma mind with that librul media mathe matics !

posted by elpapacito at 8:16 AM on February 17, 2006

posted by elpapacito at 8:16 AM on February 17, 2006

I did very well in my math classes right up until algebra - then, I just didn't get it. I started falling behind, and my parents got me a tutor. In

A lot of times, the problem is poor teachers, not any lack of aptitude.

And to address one of Cohen's other points - on my PSAT and my SAT my math and verbal scores were 10 points apart, and both in the 99th percentile. (sorry for the bragging tone - there are many, many mefites smarter than me, I know)

There are more math whizzes that can write than there are columnists that can do math - of this I am sure.

posted by bashos_frog at 8:16 AM on February 17, 2006

**one**session she made me see how it worked, and I went back to being very strong in math. Until calculus. I should've got myself a tutor, but I didn't and I wound up taking the class again for a B.A lot of times, the problem is poor teachers, not any lack of aptitude.

And to address one of Cohen's other points - on my PSAT and my SAT my math and verbal scores were 10 points apart, and both in the 99th percentile. (sorry for the bragging tone - there are many, many mefites smarter than me, I know)

There are more math whizzes that can write than there are columnists that can do math - of this I am sure.

posted by bashos_frog at 8:16 AM on February 17, 2006

If you ever try to run a small business, you will use algebra (although you might not call it that.) If you ever saw a piece of wood to fit to another, you will use geometry.

Cohen shows disturbing know-nothing-ism in his column. Gabriela should've been placed in a pre-algebra class after failing algebra the first time, but I have to admit, about the only mention of schools in the news headlines is that announcing more budget cuts. I could believe that the school simply didn't offer it.

posted by telstar at 8:17 AM on February 17, 2006

Cohen shows disturbing know-nothing-ism in his column. Gabriela should've been placed in a pre-algebra class after failing algebra the first time, but I have to admit, about the only mention of schools in the news headlines is that announcing more budget cuts. I could believe that the school simply didn't offer it.

posted by telstar at 8:17 AM on February 17, 2006

C.Batt: I just think it's somewhat hilarious that someone who claims to know anything about logic would make an argument as to whether or not it's a part of mathematics that is so logically incoherent. What necessary condition does Logic not have that makes it not math? Without a formal definition of math

posted by delmoi at 8:26 AM on February 17, 2006

*as well*as logic, you can't make any claims.posted by delmoi at 8:26 AM on February 17, 2006

*That's a rather long way to go to just divide 50 cookies by 5 kids. Most people would use simple division. 50/5=10*

Well, you're right. But all I did, ultimately, was divide as well. I guess the difference is I was trying to point out was how one puts the equation together in their head. So, if I had 5 kids with 10 cookies, how many cookies total do I have?

5 kids times 10 cookies = ?

answer: 50 (arithmetic)

But in my previous example, the way I see it, most people "do the algrebra" in their head (with even knowing it, possibly) and come to the conclusion that they need to divide. They know they have 50 cookies... "how many cookies would I have to give each kid so that everyone has an equal amount"? To me, the arrangment of that thought is more like my first example, which is algebraic.

Visually, I see the kids on one side of the room... a "known", five. I see the kids holding their empty hands out... and "unknown", how many cookies (or X). I see the adult on the other side of the room holding a bag of cookies... a "known", 50 cookies. Therefore, "5 times X = 50", algebra. 'Course, the next step is to know to rearrange that equation by putting the X on the other side of the equal sign, leaving a simple arithmetic equation, requiring you to divide. But getting to that point, you have to use algebra, in it's most basic form.

posted by Witty at 8:30 AM on February 17, 2006

delmoi, you're entirely right to call me out on that. I'm doing a really shitty job presenting the argument. It's especially laughable because I'm making a ludicrously weak statement about the whole system of making strong statements. This isn't the right forum for the discussion though, so I'm going to wimp out, dodge the issue, and concede defeat.

/ this time

// shakes his fist at delmoi and kid ichorous

posted by C.Batt at 8:35 AM on February 17, 2006

/ this time

// shakes his fist at delmoi and kid ichorous

posted by C.Batt at 8:35 AM on February 17, 2006

I was about to post a long explanation of how I interpreted the article, but Ethereal Bligh pretty much covered it for me.

What I took away from the piece was that Gabriela has abandoned her education, at least seemingly, as a result of her inability to pass one algebra class.

So, what I take away from the column is the writer's sadness that someone has given up on school because of repeated failure in one class, in a subject which is arguably not any more or less useful than many other subjects.

I thought that the columnist was trying to say, albeit poorly, that it is important to not give up on education based on difficulty in/with math.

posted by melimelo at 8:35 AM on February 17, 2006

What I took away from the piece was that Gabriela has abandoned her education, at least seemingly, as a result of her inability to pass one algebra class.

So, what I take away from the column is the writer's sadness that someone has given up on school because of repeated failure in one class, in a subject which is arguably not any more or less useful than many other subjects.

I thought that the columnist was trying to say, albeit poorly, that it is important to not give up on education based on difficulty in/with math.

posted by melimelo at 8:35 AM on February 17, 2006

It's true that algebra doesn't teach reasoning;

posted by kindall at 8:40 AM on February 17, 2006

*geometry*teaches reasoning. But you need algebra for geometry. And you need algebra for calculus, and you need calculus for physics, engineering, computer science... etc. First arithmetic, then algebra -- then you can learn basically everything else in math or science. And some of those things*certainly do*have their uses.posted by kindall at 8:40 AM on February 17, 2006

Nobody should have to do anything they don't want to in order to graduate. All you need to do is turn 18 and get a diploma. Don't think math is useful? Don't have to take it! Think literature is pointless? Never read anything!

I hope the many people in this thread who think learning the most basic of algebra skills is useless would also then agree that no one should have to learn evolution either.

posted by ozomatli at 8:41 AM on February 17, 2006

I hope the many people in this thread who think learning the most basic of algebra skills is useless would also then agree that no one should have to learn evolution either.

posted by ozomatli at 8:41 AM on February 17, 2006

I shocked myself and my parents by getting a relatively higher score on the math section of the SAT than the verbal. I hated math, was lukewarm to science, loved history and english, went on to get an english degree. I thought it was wierd, but anecdotally I've met a lot of humanities majors with similar experiences. And yes, the SAT is a bad indicator of pretty much everything (but I guess Cohen is happy there's a written section now) but I think it's interesting.

BTW, Cohen's been a bit unhinged lately. Pee in his Cheerios? That's my guess.

posted by bardic at 8:49 AM on February 17, 2006

BTW, Cohen's been a bit unhinged lately. Pee in his Cheerios? That's my guess.

posted by bardic at 8:49 AM on February 17, 2006

*delmoi, you're entirely right to call me out on that. I'm doing a really shitty job presenting the argument. It's especially laughable because I'm making a ludicrously weak statement about the whole system of making strong statements. This isn't the right forum for the discussion though, so I'm going to wimp out, dodge the issue, and concede defeat.*

Well, the crux of the argument, obviously, is that you and I have different definitions of "mathematics". Mine is probably a bit more encompassing :P

posted by delmoi at 8:57 AM on February 17, 2006

Keep in mind also that Cohen's job is to aggregate what his friends in Washington are talking about and then distill that chatter into a well-written column. The process of reasoning, logical thinking, and puzzling through a problem don't play into it. It therefore doesn't surprise me that he doesn't understand the use of it all.

At the same time, I wonder if he gives similar advice to his children as he does to lower-middle class latinos in Los Angeles.

posted by deanc at 9:09 AM on February 17, 2006

At the same time, I wonder if he gives similar advice to his children as he does to lower-middle class latinos in Los Angeles.

posted by deanc at 9:09 AM on February 17, 2006

Its been said that once you factor all the terms, every equation boils down to a = a.

And that this formula is the sum total of what math can teach us about philosophy.

posted by StickyCarpet at 9:23 AM on February 17, 2006

And that this formula is the sum total of what math can teach us about philosophy.

posted by StickyCarpet at 9:23 AM on February 17, 2006

*That's a bit of a strech, witty.*

OK. I understand. I'm not trying to make something out of nothing. I'm just trying to suggest that quite often, when people do simple math in their heads, they often use more than one step to solve a problem (like algebra does) that could possibly be done in just one (arithmetic).

Here's an example...

What's 60% of 7... When I'm faced with a question like that, to do in my head (and I know this is a super-simple example), I almost never change the % to a decimal and multiply through. What I do is divide 7 by 10 and get .7, then multiply .7 by 6 to get 4.2 (similar to how I would calculate a 15% tip on a restaurant bill... find out what 10% is, take half of that and add it to the original 10%). Now I don't know what kind of math that is exactly, but it's not plain 'ole arithmetic is it?

Now that may seem silly so to some, the "long way" to others... but it's the way I do it in my head and it happens rather quickly, beyond my control. I see the numbers, they shuffle around and out pops the answer. I do lots of math in my head in similar ways. I

**could**visualize myself multiplying that equation out on a piece of paper, like my friend just told he would do, but I find it hard to keep track of what I'm "seeing" that way. The way I described previously is just how it happens, on it's own.

posted by Witty at 9:27 AM on February 17, 2006

*I'm going to be the contrarian here and agree with Cohen that you shouldn't have to pass algebra to graduate.*

Well, as someone who has to deal with school standards on a regular basis, I have to say that "algebra" is worse than meaningless in this discussion. Personally, I'd like to see exactly what competencies are included in "algebra" and what are left out before making that kind of statement.

And in regards to the "not necessary for real life" argument. Well, the purpose of High School is no longer about preparing a person for real life, but preparing a person towards making a set of career choices in post-secondary education or trades. So personally, I've found that algebra was an essential prerequisite for understanding Biology (which is also badly taught in High School as a vocabulary course) and Psychology (via statistics). I've not written sonnets since high school, but I don't doubt that early exposure to poetic forms helped my more literary peers.

posted by KirkJobSluder at 9:28 AM on February 17, 2006

StickyCarpet: For many reasons, the idea you have quoted is just plain wrong. I think Goedel's theorem is a fine counter-example: the math makes an important philosophical point, namely that human cognition cannot be fully represented by any consistent and closed formal system. This is to me a worthwhile and interesting philosophical statement, because that wasn't always clear.

posted by sonofsamiam at 9:34 AM on February 17, 2006

posted by sonofsamiam at 9:34 AM on February 17, 2006

*How many of us ever actually use algebra in real life?*

If you ever ask yourself "How much change should I get back?," you're doing simple algebra.

If you're driving to Toronto and know you've gone 200 miles and wonder when you're going to get there, you're doing algebra.

When you pull into the gas station and wonder what mileage you got on that last tank, you're doing algebra.

When you ask how many points SportGuy needs to do SportsThing, or to raise his SportsStatistic to whatever, you're doing algebra.

Not complex algebra. And certainly not algebra with polynomials. But algebra nonetheless -- you're taking information you have, and rearranging and applying logical processes to it to get information you don't have but that you want.

posted by ROU_Xenophobe at 10:02 AM on February 17, 2006

It always seems to come down to definitions doesn't it? Goood lord, communication can be such a hassle :-)Well, the crux of the argument, obviously, is that you and I have different definitions of "mathematics". Mine is probably a bit more encompassing :P

posted bydelmoiat 9:57 AM MST on February 17

posted by C.Batt at 10:06 AM on February 17, 2006

ROU_Xenophobe - Right... which is what I was trying to explain in my cookies example.

posted by Witty at 10:13 AM on February 17, 2006

posted by Witty at 10:13 AM on February 17, 2006

*calculating your change? please, what does that have to do with algebra?*

You have two knowns: the total due and what you hand over. You have one unknown: your change. A simple equation governs their relationship: Change = What you paid minus the total due. One equation, one unknown.

That's algebra. Very simple algebra, but algebra. That people can't make the conceptual leap from simple problems like this to more complex or more abstracted problems is symptomatic of bad teaching, or a lack of skills at abstraction, or both.

Algebra doesn't mean "Math involving poorly-motivated word problems." It doesn't mean "Math about squared stuff." It means, AFAIC, "Math about using information to discover unknown information."

posted by ROU_Xenophobe at 10:13 AM on February 17, 2006

If you ever have any loans at all, I'd really think you would want to be able to handle exponential equations.

posted by sonofsamiam at 10:18 AM on February 17, 2006

posted by sonofsamiam at 10:18 AM on February 17, 2006

notmtwain:

I think this is pertinent to the discussion of this article. Gabriela didn't fail algebra because it was too hard; Gabriela failed algebra because she rarely went to class and didn't put in much of an effort.

If she had tried with all her might and failed, I would say it was a failing of the system. However, reading the article, while I have sympathy for her situation, I hbelieve that the failure lies with her and not the system in this case.

posted by Joey Michaels at 10:39 AM on February 17, 2006

*Cohen's column was evidently based on a much longer story from the LA Times: "A Formula For Failure In L.A. Schools" The article reveals what has happened to Gabriela and hints at how she now understands why she should have made the effort to pass. (She reportedly skipped class most of the time.)*I think this is pertinent to the discussion of this article. Gabriela didn't fail algebra because it was too hard; Gabriela failed algebra because she rarely went to class and didn't put in much of an effort.

If she had tried with all her might and failed, I would say it was a failing of the system. However, reading the article, while I have sympathy for her situation, I hbelieve that the failure lies with her and not the system in this case.

posted by Joey Michaels at 10:39 AM on February 17, 2006

*the purpose of High School is no longer about preparing a person for real life, but preparing a person towards making a set of career choices in post-secondary education or trades.*

Yeah, I suppose that's true, but how do you know what's going to be useful? Sure, if you're going into the sciences you'll need algebra, but most kids aren't going into the sciences, and if they are they can probably pass it with no problem. If you're going into

*Beowulf*studies you'll need Old English; should that be required to graduate from high school?

You have two knowns: the total due and what you hand over. You have one unknown: your change. A simple equation governs their relationship: Change = What you paid minus the total due. One equation, one unknown.

That's algebra.

You have two knowns: the total due and what you hand over. You have one unknown: your change. A simple equation governs their relationship: Change = What you paid minus the total due. One equation, one unknown.

That's algebra.

Oh for Christ's sake. People sure love splitting hairs here at MeFi. Fine:

*what they teach in algebra class*isn't needed in real life by most people. You can define subtraction as algebra if you want to, but you're just being silly in terms of this discussion.

posted by languagehat at 10:43 AM on February 17, 2006

Joey Michaels:

Well, I don't know that I'd fully agree with this. Ideally, educational systems should get the student invested in learning the subject matter. In some cases I think schools become anti-learning.

I'm not naive enough to believe that an educational system can or should be motivating to everyone, but there are enough areas where they are so succesful at turning students away from a topic that it is a failure of the system.

posted by KirkJobSluder at 10:45 AM on February 17, 2006

*However, reading the article, while I have sympathy for her situation, I hbelieve that the failure lies with her and not the system in this case.*Well, I don't know that I'd fully agree with this. Ideally, educational systems should get the student invested in learning the subject matter. In some cases I think schools become anti-learning.

I'm not naive enough to believe that an educational system can or should be motivating to everyone, but there are enough areas where they are so succesful at turning students away from a topic that it is a failure of the system.

posted by KirkJobSluder at 10:45 AM on February 17, 2006

*Fine: what they teach in algebra class isn't needed in real life by most people.*

If you mean "solving formal mathematical exercises," of course that's true. But the skills of manipulating information to find the information you want out of the interrelationships of the information you have, the skill that should be at the core of algebra, is something you use every day.

Your position is akin to me saying that I don't use grammar because I haven't diagrammed a sentence in 25 years. Of course I use grammar. I don't do formal exercises in grammar, but I use skills derived from those formal exercises anytime I write or speak.

posted by ROU_Xenophobe at 10:50 AM on February 17, 2006

I don't know if I use algebra exactly since, like I said, I never really got it in school, even after I passed it, the knowledge just slipped out of my mind.

I do a lot of prepress automation. So for example, I would have a script that takes a group of multipage PDFs, determine page sizes and page counts, determine which impositions to use, determine which pages go where on each flat (a representation of a larger sheet of paper to hold multiple pages positioned so folding and cutting will result in the correct position of pages for binding), extract the pages as needed, then position each page on the flat accounting for creep and bottling (horizontal, vertical, and rotational adjustments to the page position as required due to paper folding, which increases as more flats are added to a publication). It sure seemed like algebra when I was hacking away at the scripts. But given the 50 cookies and 5 kids example and reaction to it, some might say this is just division. I don't know.

I had recently tried to write a script for Illustrator that would use a selected object to select all objects that the originally selected object overlaps but I quickly gave up after I realized I would need some form of calculus (or maybe trig) to determine the actual shape of the object based on control points and object points and how these effected the actual shape and area of the object. A friend of mine had tried to explain limits to me and my brain would just boggle. I try to read the linear algebra and calculus text books I've got but they're so dense, presume a lot of foreknowledge, and are premised on an instructor that I hardly ever look at them anymore.

And speaking of Illustrator, it introduced me to the idea of matrix multiplication. After finally finding a reasonable explanation of it, I still have no stinkin' clue what it could possibly be used for or how it relates to object transformations.

posted by effwerd at 10:52 AM on February 17, 2006

I do a lot of prepress automation. So for example, I would have a script that takes a group of multipage PDFs, determine page sizes and page counts, determine which impositions to use, determine which pages go where on each flat (a representation of a larger sheet of paper to hold multiple pages positioned so folding and cutting will result in the correct position of pages for binding), extract the pages as needed, then position each page on the flat accounting for creep and bottling (horizontal, vertical, and rotational adjustments to the page position as required due to paper folding, which increases as more flats are added to a publication). It sure seemed like algebra when I was hacking away at the scripts. But given the 50 cookies and 5 kids example and reaction to it, some might say this is just division. I don't know.

I had recently tried to write a script for Illustrator that would use a selected object to select all objects that the originally selected object overlaps but I quickly gave up after I realized I would need some form of calculus (or maybe trig) to determine the actual shape of the object based on control points and object points and how these effected the actual shape and area of the object. A friend of mine had tried to explain limits to me and my brain would just boggle. I try to read the linear algebra and calculus text books I've got but they're so dense, presume a lot of foreknowledge, and are premised on an instructor that I hardly ever look at them anymore.

And speaking of Illustrator, it introduced me to the idea of matrix multiplication. After finally finding a reasonable explanation of it, I still have no stinkin' clue what it could possibly be used for or how it relates to object transformations.

posted by effwerd at 10:52 AM on February 17, 2006

*But the skills ... is*

...

I don't do formal exercises in grammar, but I use skills derived from those formal exercises anytime I write or speak.

...

I don't do formal exercises in grammar, but I use skills derived from those formal exercises anytime I write or speak.

With varying degrees of success, of course. Augh.

posted by ROU_Xenophobe at 10:58 AM on February 17, 2006

*And speaking of Illustrator, it introduced me to the idea of matrix multiplication. After finally finding a reasonable explanation of it, I still have no stinkin' clue what it could possibly be used for or how it relates to object transformations.*

Well, a multi-dimensional object like a vector can easily be handled with the mathematics of vector spaces. Vector spaces can be handled with matrices and matrix addition and multiplication, and matrices can be handled with C arrays.

You can reduce any sort of multi-dimensional manipulations of mutli-dimensional objects to matrix math.

Linear algebra was easily my favorite course I ever took in college, but I had an incredible instructor.

posted by sonofsamiam at 11:05 AM on February 17, 2006

languagehat:

From what I can tell, the concepts and skills taught in algebra are prerequisites for just about any career that requires basic college-level math or science. This includes a large chunk of business, ICT careers, and just about anything in health care beyond purely custodial work. Fundamentally, the only way to avoid needing algebra is to engage in a small set of career fields in the pure humanities.

I would say that you've posed an embarassingly bad false analogy. Algebra is more analogous to senior-level English composition. If you can't master algebra, then you are likely to have problems with calculus, statistics, biology, chemestry, geology, physics, accounting, business, etc., etc.. If you can't master basic English composition, then you are likely to have problems with literature, fine arts, history, cultural studies, speech and communication, language concentrations, etc., etc..

Well, about the only thing that is really "needed" by most people is the ability to read basic street signs and notices. But I'm wondering, how is mentally rearranging the terms of the problem not algebra? (And actually, a lot of people do have problems with that kind of rearrangement.)

Your objections neem to be embarssingly silly and dim, which is surprising.

posted by KirkJobSluder at 11:09 AM on February 17, 2006

*Yeah, I suppose that's true, but how do you know what's going to be useful? Sure, if you're going into the sciences you'll need algebra, but most kids aren't going into the sciences, and if they are they can probably pass it with no problem. If you're going into Beowulf studies you'll need Old English; should that be required to graduate from high school?*From what I can tell, the concepts and skills taught in algebra are prerequisites for just about any career that requires basic college-level math or science. This includes a large chunk of business, ICT careers, and just about anything in health care beyond purely custodial work. Fundamentally, the only way to avoid needing algebra is to engage in a small set of career fields in the pure humanities.

I would say that you've posed an embarassingly bad false analogy. Algebra is more analogous to senior-level English composition. If you can't master algebra, then you are likely to have problems with calculus, statistics, biology, chemestry, geology, physics, accounting, business, etc., etc.. If you can't master basic English composition, then you are likely to have problems with literature, fine arts, history, cultural studies, speech and communication, language concentrations, etc., etc..

*Oh for Christ's sake. People sure love splitting hairs here at MeFi. Fine: what they teach in algebra class isn't needed in real life by most people. You can define subtraction as algebra if you want to, but you're just being silly in terms of this discussion.*Well, about the only thing that is really "needed" by most people is the ability to read basic street signs and notices. But I'm wondering, how is mentally rearranging the terms of the problem not algebra? (And actually, a lot of people do have problems with that kind of rearrangement.)

Your objections neem to be embarssingly silly and dim, which is surprising.

posted by KirkJobSluder at 11:09 AM on February 17, 2006

*If you ever ask yourself "How much change should I get back?," you're doing simple algebra. ...*

Unless all math is algebra, no, i'm not. I use subtraction: the price is $17.84. I give $20. 20-17.84=? (and the cash register shows me what it is anyway, so i don't even have to do that much.)

posted by amberglow at 11:29 AM on February 17, 2006

Well, to expand on what ROU_Xenphobe says. The basic concept behind algebra is that you can rearrange an equation multiple ways depending on your needs. For example, a basic equation used in many different fields.

Rate = Units/Time

In business you might have problems involving cars produced by a factory per week. In social geography you might think in terms of distance. In ecology you might think in terms of stream flow. In social informatics number of messages per day. I have trouble thinking about any field that does not have some form of a rate equation.

With algebra, it's trivialy easy to know how to rearrange the terms to calculate how long it takes to travel 400 miles, or how many cars your factory can produce in a month.

That's all algebra is. The rest is just reinforcement and elaboration.

amberglow:

Well, I'd say that math starts with algebra. Before that, you are just using arithmatic. But yes, you have arranged the problem so you have two knowns and one unknown. This is fundamentally basic algebra. And there are many people who have trouble just understanding that it's a subtraction problem without a worksheet with a big minus sign.

posted by KirkJobSluder at 11:36 AM on February 17, 2006

Rate = Units/Time

In business you might have problems involving cars produced by a factory per week. In social geography you might think in terms of distance. In ecology you might think in terms of stream flow. In social informatics number of messages per day. I have trouble thinking about any field that does not have some form of a rate equation.

With algebra, it's trivialy easy to know how to rearrange the terms to calculate how long it takes to travel 400 miles, or how many cars your factory can produce in a month.

That's all algebra is. The rest is just reinforcement and elaboration.

amberglow:

*Unless all math is algebra, no, i'm not. I use subtraction: the price is $17.84. I give $20. 20-17.84=? (and the cash register shows me what it is anyway, so i don't even have to do that much.)*Well, I'd say that math starts with algebra. Before that, you are just using arithmatic. But yes, you have arranged the problem so you have two knowns and one unknown. This is fundamentally basic algebra. And there are many people who have trouble just understanding that it's a subtraction problem without a worksheet with a big minus sign.

posted by KirkJobSluder at 11:36 AM on February 17, 2006

I hated History in school. All that stuff was in the past. Who needs to know that? What's the point? Maths, Physics, Chemistry.... that's where the excitement is. I went on to college and studied Engineering (which is all just applied maths, at college level).

My point, if I have one, is that if you enjoy the modern life but don't like maths, you better be grateful there's lots of folks out there that do, because most of what we know and use in everyday life wouldn't exist without it. It's not an either/or thing. All the good engineers and scientists I've ever worked with could write, and well.

What's always puzzled me is that it's common and considered socially acceptable to say you were bad at math in school and never liked it and never really got it. I can't imagine being able to say "Reading and writing? I hated that in school, can't do it and don't see the point of it!"

posted by normy at 11:46 AM on February 17, 2006

My point, if I have one, is that if you enjoy the modern life but don't like maths, you better be grateful there's lots of folks out there that do, because most of what we know and use in everyday life wouldn't exist without it. It's not an either/or thing. All the good engineers and scientists I've ever worked with could write, and well.

What's always puzzled me is that it's common and considered socially acceptable to say you were bad at math in school and never liked it and never really got it. I can't imagine being able to say "Reading and writing? I hated that in school, can't do it and don't see the point of it!"

posted by normy at 11:46 AM on February 17, 2006

Witty and ROU_X are emphatically correct about algebra, languagehat and bashos wrong. Making change is algebra. Just because you memorized the simple rule long ago and now just do subtraction or division does not make it any less so.

But some folks are saying that algebra, as taught in school, is useless. It is to most students. You know why? Because if you actually make kids learn reasoning and problem solving along with the algebra, and require them to actually

Student, parents (and to some degree) teachers revolt.

Almost every math major has a proofs course. A very large component of this course is getting students to realize that math is not just a bunch of symbol manipulation, but rather an exercise in logic.

It'd be super cool if you could teach kids that in high school. I guarantee you that more people would understand the utility of algebra -- and if more people had basic competence in

Fixing this problem is not at all trivial. It starts very early (in grade school). It manifests itself in teachers, parents, students, and curricula. For whatever reason, math is REALLY hard for most people. But it's a not simply a matter of bad math teachers* -- those teachers do what they have to. Math takes more work than any other course (except college physics): because of the ignorance that pervades, and tells people that it is useless, very few students or parents are willing to put that work in, so that math is understood.

This is something for which society is as much to blame as bad teaching. It's very, very common for folks to have the attitude that Cohen does. Because math is hard, and most people don't want to put in the work, they blame math. The smarter the person, the more likely they are to blame math as a defense mechanism.

But it not at all true that algebra is useless. It's unused because 99% of the population does not have the ability to use it.

* And the place where bad teaching hurts the most? Grade school. The average teacher of mathematical concepts is just another math-phobe in grade school. So they teach rote memorization and such, because

posted by teece at 11:46 AM on February 17, 2006

But some folks are saying that algebra, as taught in school, is useless. It is to most students. You know why? Because if you actually make kids learn reasoning and problem solving along with the algebra, and require them to actually

*apply*algebra, guess what happens?Student, parents (and to some degree) teachers revolt.

Almost every math major has a proofs course. A very large component of this course is getting students to realize that math is not just a bunch of symbol manipulation, but rather an exercise in logic.

It'd be super cool if you could teach kids that in high school. I guarantee you that more people would understand the utility of algebra -- and if more people had basic competence in

*applying*it, it would be everywhere. As it stands now, the average American is completely innumerate. Even applying simple algebra is left to highly trained professionals, for the simple reason that no one from the general population can even do that.Fixing this problem is not at all trivial. It starts very early (in grade school). It manifests itself in teachers, parents, students, and curricula. For whatever reason, math is REALLY hard for most people. But it's a not simply a matter of bad math teachers* -- those teachers do what they have to. Math takes more work than any other course (except college physics): because of the ignorance that pervades, and tells people that it is useless, very few students or parents are willing to put that work in, so that math is understood.

This is something for which society is as much to blame as bad teaching. It's very, very common for folks to have the attitude that Cohen does. Because math is hard, and most people don't want to put in the work, they blame math. The smarter the person, the more likely they are to blame math as a defense mechanism.

But it not at all true that algebra is useless. It's unused because 99% of the population does not have the ability to use it.

* And the place where bad teaching hurts the most? Grade school. The average teacher of mathematical concepts is just another math-phobe in grade school. So they teach rote memorization and such, because

*they know no other way*. So that by the time the kids get a real math teacher a bit later, they are already way behind.posted by teece at 11:46 AM on February 17, 2006

*"Reading and writing? I hated that in school, can't do it and don't see the point of it!"*

I have a couple of acquaintances who will, if pressed, say something like that. You would never know they are illiterate unless you actually tried hard to engage them on books or something.

(Needless to say, they are extremely poor and their lives would surely be improved by learning to read. Still good people.)

posted by sonofsamiam at 11:53 AM on February 17, 2006

*What's 60% of 7... When I'm faced with a question like that, to do in my head (and I know this is a super-simple example), I almost never change the % to a decimal and multiply through. What I do is divide 7 by 10 and get .7, then multiply .7 by 6 to get 4.2 (similar to how I would calculate a 15% tip on a restaurant bill... find out what 10% is, take half of that and add it to the original 10%).*

This might surprise you, but I couldn't tell you what 60% of 0.7 is without a calculator. I could tell you what 60% of 0.5 (.3), or 0.4 (.24), or 0.6 (.36), or .8 (.48). I can only do a few 'compatible' numbers. Boiling it down I guess I just don't have my entire multiplication table memorized. I should probably try to do that, now that I actually think about it. Hmm...

Anyway, without a calculator I can do stuff like this. So whatever.

*I had recently tried to write a script for Illustrator that would use a selected object to select all objects that the originally selected object overlaps but I quickly gave up after I realized I would need some form of calculus (or maybe trig)*

Hah. That is a very complicated problem with a surprisingly simple solution. All you have to do to find out if a point P is inside a shape S is draw a line emanating from that point that goes on forever. If that line crosses the outline of S an

**odd number of times**then P is inside S. You don't need calc. I guess you might need trig to see if the lines cross, depending on the API and data you're using. This is from a field called "computational geometry" by the way.

*You have two knowns: the total due and what you hand over. You have one unknown: your change. A simple equation governs their relationship: Change = What you paid minus the total due. One equation, one unknown.*

Yeah, but you're basically defining

**all**arithmetic as algebra 2+2? 2+2 = x, solve for x is not algebra just because you threw an x in there.

posted by delmoi at 12:06 PM on February 17, 2006

Well, such equations are certainly a subset of algebra.

posted by sonofsamiam at 12:08 PM on February 17, 2006

posted by sonofsamiam at 12:08 PM on February 17, 2006

*Witty and ROU_X are emphatically correct about algebra, languagehat and bashos wrong. Making change is algebra. Just because you memorized the simple rule long ago and now just do subtraction or division does not make it any less so.*

Could people solve these problems before the invention of algebra? I think the obvious answer is YES, which would imply, to me that it's

*not algebra*.

Like I just said, it's obvious that there is some math that is not algebra, and if you have a definition of algebra that includes all mathematics, then your definition of algebra is wrong.

posted by delmoi at 12:08 PM on February 17, 2006

*if pressed*

That's the thing. Illiteracy is considered an embarrassment and social stigma. Many people who are illiterate know this all too well and don't advertise their handicap. At one time I volunteered at literacy classes for adults. Most of the students were acutely aware of their inability. That's why they were there.

On the other hand, some of those who are mathematically incompetent seem to think it's fine to wear their ignorance as some kind of badge of honor. It's as if making it through their education avoiding any mathematical thought was an achievement in itself.

posted by normy at 12:12 PM on February 17, 2006

*Well, such equations are certainly a subset of algebra.*

Well, not until you add the '=x'. That's like saying "0+0" or even just "1" is an arithmetic statement, but that doesn’t mean that you would say a rock can do some arithmetic because it can add two zeros together.

My point is that even though all arithmetic can be expressed as an algebraic expression, saying that anyone, for example, who doesn’t know algebra also doesn't know arithmetic. Obviously when people say algebra, they mean the part that is not simple arithmetic. So "2+2=x" isn't really algebra, but "2+x=4" is.

posted by delmoi at 12:12 PM on February 17, 2006

That makes sense. So-called "modern" algebra doesn't even necessarily address any sort of arithmetic at all, but you would obviously still call it algebra.

posted by sonofsamiam at 12:14 PM on February 17, 2006

posted by sonofsamiam at 12:14 PM on February 17, 2006

btw, (illiteracy : illiterate) : reading :: (innumeracy : innumerate) : math

posted by delmoi at 12:14 PM on February 17, 2006

posted by delmoi at 12:14 PM on February 17, 2006

*Could people solve these problems before the invention of algebra? I think the obvious answer is YES, which would imply, to me that it's not algebra.*

That's just stupid, delmoi. Elements of calculus existed for centuries before calculus was "invented." That doesn't mean those elements are not calculus. Ditto algebra -- it's expanding upon some of the realization that there was great power in such manipulations.

The above is algebra. Period. Arithmetic is the performance of the subtraction -- understanding the underlying mathematical structure of problem to know to do the arithmetic is algebra.

posted by teece at 12:15 PM on February 17, 2006

*Hah. That is a very complicated problem with a surprisingly simple solution. All you have to do to find out if a point P is inside a shape S is draw a line emanating from that point that goes on forever. If that line crosses the outline of S an odd number of times then P is inside S. You don't need calc. I guess you might need trig to see if the lines cross, depending on the API and data you're using. This is from a field called "computational geometry" by the way.*

Now that I think about it a bit more, all you need to see if two lines cross is the equations of the line and a little, say it with me, algebra. Just check to see if there is an x and y that satisfies both equations. For two curves, check to see if they cross each other twice the same way.

posted by delmoi at 12:16 PM on February 17, 2006

*The above is algebra. Period. Arithmetic is the performance of the subtraction -- understanding the underlying mathematical structure of problem to know to do the arithmetic is algebra.*

It may be "algebra. Period." by some formal definition, but I don't think it counts when discussing these things with normal people. You can take 2+2, convert it to 2+2 = x, then to ∫(2+2) = x*dy, but would you say everyone can do some calculus? In that case, you are vastly over-extending Cohen's argument to complete and absolute innumeracy. Do you think he doesn’t think it's important to be able to figure out what numbers you need to subtract to figure out what change you're due? If not, what is the point of your statements? Do you believe that it's important to learn what is commonly refereed to as algebra, or only to be able to count change and figure out gas mileage?

posted by delmoi at 12:25 PM on February 17, 2006

I wonder if he wants his pharmaceuticals tested by a statistician, or the winner of an essay contest?

Lets give him a bullet proof vest designed by a poet and I'll take the one designed by an engineer.

Mathematics is only important if you want to measure things and stuff like that.

posted by Megafly at 12:28 PM on February 17, 2006

Lets give him a bullet proof vest designed by a poet and I'll take the one designed by an engineer.

Mathematics is only important if you want to measure things and stuff like that.

posted by Megafly at 12:28 PM on February 17, 2006

er, that should really be (∫(2+2)*dy)/y = x. I think.

posted by delmoi at 12:30 PM on February 17, 2006

posted by delmoi at 12:30 PM on February 17, 2006

*In that case, you are vastly over-extending Cohen's argument to complete and absolute innumeracy.*

Not at all. Firstly, I don't give a shit what Cohen thinks. He's an idiot on this matter, and his opinion is nearly worthless.

I'm saying algebra is ubiquitous -- much more so than many realize. Indeed, even the arithmetic you are taught as a third grader has its fundamental roots in algebra.

Yes, you can subtract and divide and add and multiply with the algorithms you are taught, without any knowledge of algebra. But to actually understand why those algorithms work, and why they do what they do, is going to require algebra.

Cohen thinks algebra is totally arcane because he really has no idea what it is.

posted by teece at 12:31 PM on February 17, 2006

*Yes, you can subtract and divide and add and multiply with the algorithms you are taught, without any knowledge of algebra. But to actually understand why those algorithms work, and why they do what they do, is going to require algebra.*

Only if by "algebra" you mean "set theory".

posted by delmoi at 12:34 PM on February 17, 2006

Well, if we're going in that direction, only if by "set theory" you mean "category theory".

posted by sonofsamiam at 12:35 PM on February 17, 2006

posted by sonofsamiam at 12:35 PM on February 17, 2006

*Well, if we're going in that direction, only if by "set theory" you mean "category theory".*

Well, first we have to figure out what we mean by "understand". But the point is there are other ways to understand arithmetic besides an algebraic framework. Basic arithmetic was formulated in the past in set theory long before anyone came up with Category Theory in the 1960s.

If you say "algebra is important" because of a few examples of

*arithmetic*in daily lives, that only requires the ability to count to understand, you actually

*undermine*the idea that

*formal*algebra is an important thing for kids to learn.

Idiots like Cohen can sit there and say "Well, I know how to do change, but I don't know algebra" because he doesn't know the finer points of what mathematical systems can be expressed in other mathematical systems.

You know, thinking about this, the really important thing isn't to teach kids how to do math, but to teach kids how to

*enjoy*doing math and thinking about it (the same way most people enjoy reading). I think all people could enjoy math if taught the right way.

posted by delmoi at 12:46 PM on February 17, 2006

normy: Great point. It seems like a point of acceptable pride to say that one is math or science illiterate. But to say that you zoned out in classes where you read Hamlet or Pride and Prejudice, that you found the history of Europe to be boring and irrelevant, that you took a semester of Spanish or Latin but didn't really like it much, that you can't tell Monet from mayo is to be branded hidebound, crass and uncultured.

The one exception to this is when it comes to the culture war surrounding Evolution vs. Creationism/Intelligent Design, but most often I'm appaled that vocal advocates for Evolution are arguing based on a set of pretty stories from National Geographic, Scientific American, and the occasional pop book by Gould or Dawkins. Their ignorance of Evolution as a quantitative theory leads them to make silly mistakes in their advocacy.

So perhaps advocates of Math and Science need to start turning the tables and say that innumeracy is not something to boldly confess, and that it identifies one as an uncultured boor.

delmoi:

I don't think the claim has been made that all math is algebra. I would agree that 20.00 - 17.84 is not algebra.

But that's not the problem being discussed here in this case. You have an abstract formula for making change:

The values you plug into this abstract formula can vary depending on context. If I'm buying a cup of coffee: $5.00 - $1.60 = $3.40. If I'm buying a camera, $400 - $380 = $20. We know this formula does not work in commercial settings if I don't have enough cash to cover the cost of the item I'm buying.

The fact that we can express this relationship has abstract variables with a known relationship to each other, and can just plug in values from situation to situation is algebra.

Well I'd make the argument (following your analogy) that the concepts covered in algebra are pretty fundamental to basic numeracy, in the same way that the ability to construct (and read) a proper paragraph is fundamental to basic literacy. A person who can only apply route formulas to problems is functionally innumerate, in the same way that a person who can only read street signs and simple phrases is functually illiterate.

Of course one can make the argument that illiteracy is more damaging than innumeracy, and I'd agree. Still however, I'm not convinced that functional innumeracy should be a primary goal of our educational systems.

And I'd just say that the expression of mathematical problems as abstract variables is "algebra" at least according to the "normal people" responsible for writing and implementing the math and science curriculum at the K-12 level.

posted by KirkJobSluder at 12:46 PM on February 17, 2006

The one exception to this is when it comes to the culture war surrounding Evolution vs. Creationism/Intelligent Design, but most often I'm appaled that vocal advocates for Evolution are arguing based on a set of pretty stories from National Geographic, Scientific American, and the occasional pop book by Gould or Dawkins. Their ignorance of Evolution as a quantitative theory leads them to make silly mistakes in their advocacy.

So perhaps advocates of Math and Science need to start turning the tables and say that innumeracy is not something to boldly confess, and that it identifies one as an uncultured boor.

delmoi:

*Unless all math is algebra, no, i'm not. I use subtraction: the price is $17.84. I give $20. 20-17.84=? (and the cash register shows me what it is anyway, so i don't even have to do that much.)*I don't think the claim has been made that all math is algebra. I would agree that 20.00 - 17.84 is not algebra.

But that's not the problem being discussed here in this case. You have an abstract formula for making change:

*Change = Payment - Cost***when**Change => 0The values you plug into this abstract formula can vary depending on context. If I'm buying a cup of coffee: $5.00 - $1.60 = $3.40. If I'm buying a camera, $400 - $380 = $20. We know this formula does not work in commercial settings if I don't have enough cash to cover the cost of the item I'm buying.

The fact that we can express this relationship has abstract variables with a known relationship to each other, and can just plug in values from situation to situation is algebra.

*It may be "algebra. Period." by some formal definition, but I don't think it counts when discussing these things with normal people. You can take 2+2, convert it to 2+2 = x, then to ∫(2+2) = x*dy, but would you say everyone can do some calculus? In that case, you are vastly over-extending Cohen's argument to complete and absolute innumeracy. Do you think he doesn’t think it's important to be able to figure out what numbers you need to subtract to figure out what change you're due? If not, what is the point of your statements? Do you believe that it's important to learn what is commonly refereed to as algebra, or only to be able to count change and figure out gas mileage?*Well I'd make the argument (following your analogy) that the concepts covered in algebra are pretty fundamental to basic numeracy, in the same way that the ability to construct (and read) a proper paragraph is fundamental to basic literacy. A person who can only apply route formulas to problems is functionally innumerate, in the same way that a person who can only read street signs and simple phrases is functually illiterate.

Of course one can make the argument that illiteracy is more damaging than innumeracy, and I'd agree. Still however, I'm not convinced that functional innumeracy should be a primary goal of our educational systems.

*Only if by "algebra" you mean "set theory".*And I'd just say that the expression of mathematical problems as abstract variables is "algebra" at least according to the "normal people" responsible for writing and implementing the math and science curriculum at the K-12 level.

posted by KirkJobSluder at 12:46 PM on February 17, 2006

*Hah. That is a very complicated problem with a surprisingly simple solution. All you have to do to find out if a point P is inside a shape S is draw a line emanating from that point that goes on forever. If that line crosses the outline of S an odd number of times then P is inside S. You don't need calc. I guess you might need trig to see if the lines cross, depending on the API and data you're using. This is from a field called "computational geometry" by the way.*

Now that I think about it a bit more, all you need to see if two lines cross is the equations of the line and a little, say it with me, algebra. Just check to see if there is an x and y that satisfies both equations. For two curves, check to see if they cross each other twice the same way.

Now that I think about it a bit more, all you need to see if two lines cross is the equations of the line and a little, say it with me, algebra. Just check to see if there is an x and y that satisfies both equations. For two curves, check to see if they cross each other twice the same way.

But how would I calculate what the curve is (really a rhetorical question, don't want to turn this into an Ask MeFi discussion; also sorry for the derail)? In Illustrator, when describing, say, an oval, I usually have only four base points and eight control points (two control points per base point). To determine the curve between two base points I need to calculate the offset of the appropriate control point as it relates to its base point and how those control points of each base point relates to each other as they relate to their base points. After determining the curve, I would then need to do the same for the potentially overlapped object and then figure out how to determine if those curves intersect. I don't have points to describe every point along each curve. Worst case example (of a potential overlap that is in fact not) below:

posted by effwerd at 12:48 PM on February 17, 2006

I am reminded of a character in C.S. Lewis's

posted by neuron at 12:53 PM on February 17, 2006

*That Hideous Strength*, a professor of the humanities, who remarked that as a student he did well in "those subjects that didn't require any actual knowledge."posted by neuron at 12:53 PM on February 17, 2006

The control points determine how the line is drawn, I belive illustrator uses beizer curves like most other drawing programs. In that case, the curve is drawn by the formula

Anyway, the braindead way to do it would be to do the 'point inside shape' algorithm for t = 0.1, t = 0.2 t = 0.3 ... t = 1.0, or even smaller increments. That would be slow, but since this dosn't need to be done in real time, you could probably do it, even with 100 steps and it wouldn't be to bad.

Otherwise, you could slove for the equations to see if there is some t

posted by delmoi at 12:58 PM on February 17, 2006

A cubic Bezier curve is defined by four points. Two are endpoints. (x0,y0) is the origin endpoint. (x3,y3) is the destination endpoint. The points (x1,y1) and (x2,y2) are control points.The linked page has the actual formulas for x(t) and y(t). They have a lot of subscripts and superscripts so I'm not going to try pasting them in.

Two equations define the points on the curve. Both are evaluated for an arbitrary number of values of t between 0 and 1. One equation yields values for x, the other yields values for y. As increasing values for t are supplied to the equations, the point defined by x(t),y(t) moves from the origin to the destination. This is how the equations are defined in Adobe's PostScript references.

Anyway, the braindead way to do it would be to do the 'point inside shape' algorithm for t = 0.1, t = 0.2 t = 0.3 ... t = 1.0, or even smaller increments. That would be slow, but since this dosn't need to be done in real time, you could probably do it, even with 100 steps and it wouldn't be to bad.

Otherwise, you could slove for the equations to see if there is some t

_{1}and some t_{2}such that x_{1}(t_{1}) = x_{2}(t_{2}) and y_{1}(t_{1}) = y_{2}(t_{2}) where t_{1}and t_{2}are between 0 and 1. If there is, they cross, if they cross twice, the two shapes overlap.posted by delmoi at 12:58 PM on February 17, 2006

Just because this conversation has involved more than a hundred posts of pissing in the wind and making arguments from ignorance, here are the California Math Standards. The preamble to Algebra I states:

posted by KirkJobSluder at 12:59 PM on February 17, 2006

Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences. In addition, algebraic skills and concepts are developed and used in a wide variety of problem-solving situations.We can digress for another 100 posts about algebra vs. set theory vs. category theory, but this is how algebra is defined for the purposes of classroom instruction, evaluation, and textbook adoption in the State of California (which in turn drives textbook publication across the United States.)

posted by KirkJobSluder at 12:59 PM on February 17, 2006

effwerd: It depends on what sort of curves you are using. Bezier, quadratic, or cubic. If you know which kind of curve, you should be able to find on google a simple algebraic representation of that curve, which you could then use to find the intersection and determine whether a point is inside or outside of the shape.

Note that each closed shape is composed of several different curves, so you have to test intersection for each of them.

on preview: yeah

posted by sonofsamiam at 1:00 PM on February 17, 2006

Note that each closed shape is composed of several different curves, so you have to test intersection for each of them.

on preview: yeah

posted by sonofsamiam at 1:00 PM on February 17, 2006

Right, they might be some other kind of curve, (or you might be able to chose for each curve when you make it).

As an aside, I had no idea that illustrator had a scripting language that let you do things like that. I'm totally going to have to learn it.

posted by delmoi at 1:03 PM on February 17, 2006

As an aside, I had no idea that illustrator had a scripting language that let you do things like that. I'm totally going to have to learn it.

posted by delmoi at 1:03 PM on February 17, 2006

Cool, thanks, delmoi and sonofsamian. I had done some searches in the past and I think since I didn't know exactly what I was looking for, I couldn't know if I found it.

posted by effwerd at 1:04 PM on February 17, 2006

posted by effwerd at 1:04 PM on February 17, 2006

delmoi, AI (and most Adobe apps) have a JS/VisualBasic/AppleScript DOM that lets you get the necessary data out of the app. Then it's just using the math functions in each scripting language.

posted by effwerd at 1:07 PM on February 17, 2006

posted by effwerd at 1:07 PM on February 17, 2006

*That's algebra, no?*

No, it's delicious, delicious cookies.

posted by davejay at 1:10 PM on February 17, 2006

*Only if by "algebra" you mean "set theory".*

Uh-uh. That's just silly. It has nothing to do with anything (set theory can underpin pretty much all of mathematics, if you want to go that way, but that has absolutely nothing to do with what I was saying).

posted by teece at 1:13 PM on February 17, 2006

*My point, if I have one, is that if you enjoy the modern life but don't like maths, you better be grateful there's lots of folks out there that do, because most of what we know and use in everyday life wouldn't exist without it. It's not an either/or thing. All the good engineers and scientists I've ever worked with could write, and well.*

To the first point: of course we should all be grateful there are lots of folks out there that know how to do stuff we don't; it's called specialization, and is the basis of modern society. If we all had to know how to do everything, we'd still be gathering nuts and berries.

To the last point: either you have a limited and unrepresentative sample of friends or you are being disingenuous. It's perfectly obvious there are lots and lots of engineers and scientists, good or otherwise, who can't write worth beans. Why make unsustainable claims that aren't necessary to your argument?

Once again: I like algebra and math in general and think as many people as possible should be taught it (and taught it well, so they don't grow up hating it). I don't think, in the present state of high school math teaching, algebra should be a required subject. And the "all math is algebra" argument is silly and a complete derail.

posted by languagehat at 1:31 PM on February 17, 2006

"

Oh, get stuffed kirkjob.

Some of us are truly ashamed at how cretinous we are at math. "To boldly confess" is sometimes not a boast at all (though it IS a

posted by Jody Tresidder at 1:43 PM on February 17, 2006

*So perhaps advocates of Math and Science need to start turning the tables and say that innumeracy is not something to boldly confess, and that it identifies one as an uncultured boor*. "Oh, get stuffed kirkjob.

Some of us are truly ashamed at how cretinous we are at math. "To boldly confess" is sometimes not a boast at all (though it IS a

**split infinitive**- heh heh!) - but an uncomfortably frank revelation of incompetence.posted by Jody Tresidder at 1:43 PM on February 17, 2006

*And the "all math is algebra" argument is silly and a complete derail.*

Straw men are fun.

posted by teece at 2:01 PM on February 17, 2006

*I don't think, in the present state of high school math teaching, algebra should be a required subject.*

This is dangerous and backwards thinking. Firstly, it blows the problem of math teaching out of proportion. But secondly, it's not the correct way to respond to that problem.

Over the last 10 years, American students going to college have had a marked delcine in reading and writing ability. A hefty chunk (I forget the number, around half) do not have the reading ability to read something and understand it at the college level (meaning, if they don't get info from lectures, these kids are not learning it).

So obviously, as critical reading is taught now, it's not worth teaching. Right? Wrong.

Fix the way kids learn math, if you think that is the problem. Don't ditch it. It's impossible to be a fully informed citizen in a modern democracy without the basics of numeracy. Scaling back algebra and throwing in some prob. and stat. would be good. Teaching kids to

*think*earlier might help. Who knows. But making the overwhelming majority of students innumerate by tossing out algebra requirements would be really stupid.

posted by teece at 2:07 PM on February 17, 2006

Apparently there was a similar fuss in the UK about whether or not the quadratic equation was useful enough to teach. It ended up being discussed in Parliament.

I was looking up the quadratic equation to see how much I remembered of my 8th-grade Algebra class. Fuck-all, that's how much I remember.

posted by dilettante at 2:08 PM on February 17, 2006

I was looking up the quadratic equation to see how much I remembered of my 8th-grade Algebra class. Fuck-all, that's how much I remember.

posted by dilettante at 2:08 PM on February 17, 2006

sonofsamiam:

The idea I paraphrased is from Wittgenstein's Tractatus Logico-Philosophicus, his only complete authorized publication. Poor guy had the misfortune of introducing his magnum opus at the very same conference where Goedel introduced his theorem, thoughoghly eclipsing the Tractatus.

And while Goedel can certainy cause those little neck hairs to stand up and quiver, I will still proudly paraphrase Wittgenstein. He's da man.

posted by StickyCarpet at 2:13 PM on February 17, 2006

*StickyCarpet: For many reasons, the idea you have quoted is just plain wrong. I think Goedel's theorem is a fine counter-example...*The idea I paraphrased is from Wittgenstein's Tractatus Logico-Philosophicus, his only complete authorized publication. Poor guy had the misfortune of introducing his magnum opus at the very same conference where Goedel introduced his theorem, thoughoghly eclipsing the Tractatus.

And while Goedel can certainy cause those little neck hairs to stand up and quiver, I will still proudly paraphrase Wittgenstein. He's da man.

posted by StickyCarpet at 2:13 PM on February 17, 2006

Yeah, I got the Tractatus at home. It bears little on formal systems, although back then, there was much more conflation of formal systems with the philosophical ideas we impose on them.

A wonderful critique of the positivists, who should be critiqued at every turn.

posted by sonofsamiam at 2:25 PM on February 17, 2006

A wonderful critique of the positivists, who should be critiqued at every turn.

posted by sonofsamiam at 2:25 PM on February 17, 2006

If you ever ask yourself "How much change should I get back?," you're doing simple algebra. ...

Unless all math is algebra, no, i'm not. I use subtraction: the price is $17.84. I give $20. 20-17.84=? (and the cash register shows me what it is anyway, so i don't even have to do that much.)

If you ever ask yourself "How much change should I get back?," you're doing simple algebra. ...

Unless all math is algebra, no, i'm not. I use subtraction: the price is $17.84. I give $20. 20-17.84=? (and the cash register shows me what it is anyway, so i don't even have to do that much.)

The algebra in getting change isn't how much you get back, but in terms of what number and denomination of coins you expect to receive.

Being irked that a clerk gives you, say, 10 pennies back instead of a dime already implies a minimizing problem which is even a step beyond basic algebra.

(a+b+c+d+...) <= (a+b+c+d+...) for all a,b,c,d where

change = (.01)a +(.05)b+(.1)c +(.25)d +...

posted by juv3nal at 2:27 PM on February 17, 2006

"for all a,b,c,d where" should be "for all positive integers a,b,c,d where"

posted by juv3nal at 2:34 PM on February 17, 2006

posted by juv3nal at 2:34 PM on February 17, 2006

*of course we should all be grateful there are lots of folks out there that know how to do stuff we don't*

Those without specialist knowledge of some field are generally not proud nor dismissive of it, however. This seems to be acceptable for some people in the case of mathematics, however. I don't know why.

*It's perfectly obvious there are lots and lots of engineers and scientists, good or otherwise, who can't write worth beans.*

By "good" engineer (or scientist, for that matter), I mean one that can communicate their ideas and knowledge. I agree that there are many who can't communicate the concepts of their specialism well in writing for a non-specialized audience. I'd count myself amongst them. But context is the thing. If an engineer or scientist can't communicate with their peers in the context of performing their work, they're not good by any measure. This requires a sound grasp of at least the fundamentals of written language - the basics we're all supposed to be taught in school. Those who can communicate well in both contexts, however, are indeed valuable.

As for an argument, I wasn't aware I was making one.

posted by normy at 2:39 PM on February 17, 2006

*every equation boils down to a = a.*

Well, there are inequalities. But for equalities, sure. It's tautological. The definition of an equation stipulates that both sides of the equal sign are the same value. It's what allows you to solve for the unknown.

posted by kindall at 5:30 PM on February 17, 2006

languagehat:

To the first point, certainly there is a level of specialization required in modern society. However algebra as taught in High School focuses on a very general concept that is a necessary prerequisite for a huge number of subject matter areas practiced in "real life." Personally, I'd place the bar at Calculus.

To the other point, there are a lot of people in the humanities who can't write worth a hill of beans. But I don't hear you making a similar argument against basic English composition. As normy said, being functionally illiterate will kill your career in the sciences.

languagehat:

Wouldn't you say though that a better solution is to fix the way that math is taught?

I have few problems with the idea that algebra should not be a requirement for a high school diploma, provided that we throw out the pretense that a diploma means that the student has mastered a broad set of the skills needed for future education (including vocational training).

As I've posted, "algebra" as defined by many school systems is simply the concept that you can represent numeric quantities with symbols, and express relationships using those symbols. I can't think of much in mathematics that does not involve this form of semiotic trickery to some degree. It is certainly the case that the professional and trade fields that use symbols for numbers is substantially more broad than just scientists and engineers.

Jody T.:

Certainly I was not talking about anyone engaged in this discussion, but it is a tone present in Cohen's argument against math literacy in general. And kindly please get stuffed yourself.

posted by KirkJobSluder at 6:10 PM on February 17, 2006

*To the first point: of course we should all be grateful there are lots of folks out there that know how to do stuff we don't; it's called specialization, and is the basis of modern society. If we all had to know how to do everything, we'd still be gathering nuts and berries.*

To the last point: either you have a limited and unrepresentative sample of friends or you are being disingenuous. It's perfectly obvious there are lots and lots of engineers and scientists, good or otherwise, who can't write worth beans. Why make unsustainable claims that aren't necessary to your argument?To the last point: either you have a limited and unrepresentative sample of friends or you are being disingenuous. It's perfectly obvious there are lots and lots of engineers and scientists, good or otherwise, who can't write worth beans. Why make unsustainable claims that aren't necessary to your argument?

To the first point, certainly there is a level of specialization required in modern society. However algebra as taught in High School focuses on a very general concept that is a necessary prerequisite for a huge number of subject matter areas practiced in "real life." Personally, I'd place the bar at Calculus.

To the other point, there are a lot of people in the humanities who can't write worth a hill of beans. But I don't hear you making a similar argument against basic English composition. As normy said, being functionally illiterate will kill your career in the sciences.

languagehat:

*I like algebra and math in general and think as many people as possible should be taught it (and taught it well, so they don't grow up hating it). I don't think, in the present state of high school math teaching, algebra should be a required subject. And the "all math is algebra" argument is silly and a complete derail.*Wouldn't you say though that a better solution is to fix the way that math is taught?

I have few problems with the idea that algebra should not be a requirement for a high school diploma, provided that we throw out the pretense that a diploma means that the student has mastered a broad set of the skills needed for future education (including vocational training).

As I've posted, "algebra" as defined by many school systems is simply the concept that you can represent numeric quantities with symbols, and express relationships using those symbols. I can't think of much in mathematics that does not involve this form of semiotic trickery to some degree. It is certainly the case that the professional and trade fields that use symbols for numbers is substantially more broad than just scientists and engineers.

Jody T.:

*Some of us are truly ashamed at how cretinous we are at math. "To boldly confess" is sometimes not a boast at all (though it IS a split infinitive - heh heh!) - but an uncomfortably frank revelation of incompetence.*Certainly I was not talking about anyone engaged in this discussion, but it is a tone present in Cohen's argument against math literacy in general. And kindly please get stuffed yourself.

posted by KirkJobSluder at 6:10 PM on February 17, 2006

kirkjob,

Spot on. Consider me stuffed. I've always thought the split infinitive "rule" the worst refuge of scoundrels - it was just a jokey attempt to be snippy:)

posted by Jody Tresidder at 6:42 PM on February 17, 2006

Spot on. Consider me stuffed. I've always thought the split infinitive "rule" the worst refuge of scoundrels - it was just a jokey attempt to be snippy:)

posted by Jody Tresidder at 6:42 PM on February 17, 2006

I'm forced to repeat that a good number of you

No you don't.

No you don't. (Why don't you pick up the

No you're not.

No you're not.

No you're not.

That's such an all-encompassing definition of "mathematics" that it is entirely useless. It's also very poor reasoning (or "mathematics", if you insist).

That's a useless and incorrect definition. That would mean, for example, that geometry is algebra. Or that trigonometry is algebra. Or even that categorical logic is algebra. (Because IIRC, you are among those who want to subsume logic into "math".) Yours is an very wrong statement.

No, your argument is that the technique of diagramming sentences is the essence of grammar. Which it's not. It's an analytical technique. Your argument implies an equivalency between a thing and the analytical approach to a thing. With that equivalency in mind, I'm really fucking good at organic chemistry. Am I doing organic chemistry right now as I breath?

I take it, then, that you are explaining to a

Amberglow is right. It isn't.

No it isn't "all that algebra" is.

No. Teece is wrong.

Delmoi is right.

Yes it does. Calculus is an analytical technique. Few, if any, contemporary mathematicians are the platonists that you are implicitly insisting they are (because you are). Calculus didn't exist before it was invented, and neither did algebra.

Oddly enough, it's

Okay, people. Raise your hands if you've a) been formally taught the history of mathematics; in concert with b) been formally and rigorously taught mathematics

My hand is raised. Who else? No one? Okay, then.

All this so arrogantly and self-confidently spewing of stuff you think you

The emphasized sentence in the previous paragraph powerfully summarizes the very real limitations of what is, essentially, a highly advanced technical eduation—the form of education dominant in the US. Mastery of technique is not equivalent to essential comprehension.

posted by Ethereal Bligh at 7:55 PM on February 17, 2006

**are very, very wrong**. You are excellent examples of how contemporary education instills a false sense of extensive comprehension.*"But you need algebra for geometry."*No you don't.

*And you need algebra for calculus.*No you don't. (Why don't you pick up the

*Principia*—you know, that book where Newton (co)invents the Calculus? Yeah, that one.)*"If you ever ask yourself 'How much change should I get back?,' you're doing simple algebra."*No you're not.

*"When you pull into the gas station and wonder what mileage you got on that last tank, you're doing algebra."*No you're not.

*"When you ask how many points SportGuy needs to do SportsThing, or to raise his SportsStatistic to whatever, you're doing algebra."*No you're not.

*"you're taking information you have, and rearranging and applying logical processes to it to get information you don't have but that you want."*That's such an all-encompassing definition of "mathematics" that it is entirely useless. It's also very poor reasoning (or "mathematics", if you insist).

*"It means, AFAIC, 'Math about using information to discover unknown information.'"*That's a useless and incorrect definition. That would mean, for example, that geometry is algebra. Or that trigonometry is algebra. Or even that categorical logic is algebra. (Because IIRC, you are among those who want to subsume logic into "math".) Yours is an very wrong statement.

*"Your position is akin to me saying that I don't use grammar because I haven't diagrammed a sentence in 25 years. Of course I use grammar."*No, your argument is that the technique of diagramming sentences is the essence of grammar. Which it's not. It's an analytical technique. Your argument implies an equivalency between a thing and the analytical approach to a thing. With that equivalency in mind, I'm really fucking good at organic chemistry. Am I doing organic chemistry right now as I breath?

*"I don't do formal exercises in grammar, but I use skills derived from those formal exercises anytime I write or speak."*I take it, then, that you are explaining to a

*linguist*your certainty that the diagraming of sentences is nothing more than an obvious representation of the very essentials of grammar and that the facility of language implicitly requires the facility with the technique of diagramming sentences—because, as you say, that's what diagramming sentences*is*? You're doing it*right now*?*"Unless all math is algebra, no, i'm not."*Amberglow is right. It isn't.

*"That's all algebra is. The rest is just reinforcement and elaboration."*No it isn't "all that algebra" is.

*"Witty and ROU_X are emphatically correct about algebra, languagehat and bashos wrong. Making change is algebra."*No. Teece is wrong.

*"Could people solve these problems before the invention of algebra? I think the obvious answer is YES, which would imply, to me that it's not algebra."*Delmoi is right.

*"That's just stupid, delmoi. Elements of calculus existed for centuries before calculus was 'invented.' That doesn't mean those elements are not calculus."*Yes it does. Calculus is an analytical technique. Few, if any, contemporary mathematicians are the platonists that you are implicitly insisting they are (because you are). Calculus didn't exist before it was invented, and neither did algebra.

*"It may be 'algebra. Period.' by some formal definition, but I don't think it counts when discussing these things with normal people."*Oddly enough, it's

*not*"algebra" by*some formal definition*, it's "algebra" in the words of a large number of people who are pulling crap out of their asses.Okay, people. Raise your hands if you've a) been formally taught the history of mathematics; in concert with b) been formally and rigorously taught mathematics

*vis a vis*philosophy?My hand is raised. Who else? No one? Okay, then.

All this so arrogantly and self-confidently spewing of stuff you think you

*know*about the*philosophy*of math is exactly like confidently claiming your expertise in linguistics because you can speak. Don't confuse a facility with something with a strong, rigorous comprehension of it.*Don't confuse a facility with an analytical technique with a deep comprehension of its object of study.*The irony here is that the argument of those wrongly defining "algebra" as, well, all reasoning, is that the learning of algebra is an essential component in aquiring the skill of rigorouus analytical thought. If that were true, and assuming you people have a lot of math under your belts, then you are counter-examples to your argument.The emphasized sentence in the previous paragraph powerfully summarizes the very real limitations of what is, essentially, a highly advanced technical eduation—the form of education dominant in the US. Mastery of technique is not equivalent to essential comprehension.

posted by Ethereal Bligh at 7:55 PM on February 17, 2006

Rather than continue to criticize, I'll offer a concise definition of

Descartes's coordinate geometry is a key element in this transition, for obvious reasons.

While it's true that the Calculus

I thought I had a book on the history of algebra in my Amazon wish-list as a result of a reliable recommendation, but I cannot find it. A search on "history of algebra", and looking for something less a narrative (though that's important) than a philosophical and practical presentation, is what I'd recommend.

posted by Ethereal Bligh at 8:16 PM on February 17, 2006

*algebra*. Algebra is the analytical technique used in the study and practice of mathematics which is the first and possibly most important step in the transition to symbolic abstraction and abstracted equation. This transition makes possible great advances in mathematics in two interrelated respects. The technique implies more strongly the abstraction of the process of reasoning and thus is the critical step in the transition from platonism to formalism; and in so doing makes possible entirely new kinds of calculation (mixed units is one very obvious and important example). It is also extremely provocative from the philosophical point-of-view, and in this regard it, too, was greatly influential.Descartes's coordinate geometry is a key element in this transition, for obvious reasons.

While it's true that the Calculus

*can*be done geometrically, as Newton's example proves, it's also tellingly the case that in the actual practice of writing the*Principia*, Newton depended upon algebra—a (then thought) dubious technique only recently imported from the east/middle east via Fibonacci. Once finished with his book, Newton actually went back and "translated" the entire work into a series of geometrical demonstrations in order to satsify contemporary prejudices.I thought I had a book on the history of algebra in my Amazon wish-list as a result of a reliable recommendation, but I cannot find it. A search on "history of algebra", and looking for something less a narrative (though that's important) than a philosophical and practical presentation, is what I'd recommend.

posted by Ethereal Bligh at 8:16 PM on February 17, 2006

*The algebra in getting change isn't how much you get back, but in terms of what number and denomination of coins you expect to receive.*

Why would i even care? And if i did care, it wouldn't matter how many pennies, for instance, i get, but the fact that i was handed a ton of pennies in the first place, instead of the expected mix of change, etc (most cashiers actually mention it, anyway, if they're out of certain change or bills, and apologize for it, i find). All I--and most people--care about is getting the correct change, which is subtraction of the price of something from the size bill handed to the cashier (assuming you use cash instead of a card in the first place).

I could see, tho, something like figuring out a tip actually really being algebra, but most of us use tricks for that kind of stuff, like doubling the tax and adding a little to figure out how to get to 15-20% (which is just addition really, and a little basic foreknowledge of what the tax percentage is).

posted by amberglow at 10:45 PM on February 17, 2006

I take my hat off to the learned and eloquent EB, who says it better than I would have.

posted by languagehat at 5:15 AM on February 18, 2006

posted by languagehat at 5:15 AM on February 18, 2006

EB: Didn't I predict early on that a lot of this discussion will depend on how "algebra" is defined?

You know, perhaps I'm being a bit cranky at waking up and seeing both that my attempt at pulling in actual evidence into this discussion was ignored in favor of what to me is an obvious and inflammatory attempt to poison the well via a straw man in the form of "wrongly defining 'algebra' as, well, all reasoning." From my point of view this statement is so obviously false, clearly false, so over the top in flames that I find it hard to believe that this is an honest misunderstanding and a failure to recognize common ground. I'm going to reluctantly extend the benefit of the doubt and kindly suggest that you step back a bit in your outrage and consider what people are really saying.

Much of your discussion would be great and nice if the FPP had anything to do with history of mathematics, or philosophy of mathematics. But it does not. Instead, we have a FPP that is talking about

Educators of mathematics tend to focus on a third discipline that I feel should have a voice in this discussion: psychology of mathematics. I suspect the basic conflict here is between the view of algebra as a formal system of manipulating symbols, and algebra as a cognitive phenomena in which we manipulate numbers via symbols. While you are correct in saying that from the view of philosophers of mathematics algebra is a formal tookit. I also think it's entirely correct to talk about algebra as the application of cognitive mathematic abstraction. From a cognitive point of view, algebra isn't something that springs from the forehead of arithmetic as a formal discipline, but the application of an abstract form of cognition to the specific domain of numeric quantities.

From a cognitive point of view, there is not much difference between recognizing

Of course there are other methods and domains applying the principle of abstraction, so don't go arguing against a claimed exclusive role for algebra. I will point out that HS algebra is usually the first time where the revolutionary value of this type of abstraction is explored in depth.

The reason why educators tend to talk about psychology rather that philosophy or history is pragmatic, given that history of mathematics and philosophy of mathematics have little to say about the empirical struggles of helping students discover that numeric quantities can be represented and manipulated via symbols.

So with a latch-ditch attempt at cleaning out the poisons thrown it the well by yourself and languagehat. Here are what I see as the core issues of concern:

1: Is it the case that "symbolic reasoning and calculations with symbols are central in algebra?" (This is where your philosophy and history of mathematics can play a big role.)

2: Is it the case that "symbolic reasoning and calculations with symbols" are necessary prerequisites for technical facility in a wide variety of post-secondary trades and careers?

3: Is it the case that "symbolic reasoning and calculations with symbols" are necessary prerequisites for a detailed comprehension of theories in current practice in a wide variety of fields?

4: Should "symbolic reasoning and calculations with symbols" be a requirement for graduating from high school?

5: Is it possible for us to talk about these issues using different lenses and different theoretical backgrounds?

My answers are:

1: Yes.

2: Yes.

3: Yes.

4: Probably.

5: Doubtful given the history of this discussion. But perhaps you could surprise me.

posted by KirkJobSluder at 7:58 AM on February 18, 2006

You know, perhaps I'm being a bit cranky at waking up and seeing both that my attempt at pulling in actual evidence into this discussion was ignored in favor of what to me is an obvious and inflammatory attempt to poison the well via a straw man in the form of "wrongly defining 'algebra' as, well, all reasoning." From my point of view this statement is so obviously false, clearly false, so over the top in flames that I find it hard to believe that this is an honest misunderstanding and a failure to recognize common ground. I'm going to reluctantly extend the benefit of the doubt and kindly suggest that you step back a bit in your outrage and consider what people are really saying.

Much of your discussion would be great and nice if the FPP had anything to do with history of mathematics, or philosophy of mathematics. But it does not. Instead, we have a FPP that is talking about

**school standards**and the primary definition that matters is that used by the**California State Board of Education.**Educators of mathematics tend to focus on a third discipline that I feel should have a voice in this discussion: psychology of mathematics. I suspect the basic conflict here is between the view of algebra as a formal system of manipulating symbols, and algebra as a cognitive phenomena in which we manipulate numbers via symbols. While you are correct in saying that from the view of philosophers of mathematics algebra is a formal tookit. I also think it's entirely correct to talk about algebra as the application of cognitive mathematic abstraction. From a cognitive point of view, algebra isn't something that springs from the forehead of arithmetic as a formal discipline, but the application of an abstract form of cognition to the specific domain of numeric quantities.

From a cognitive point of view, there is not much difference between recognizing

*Change = Payment - Cost*, and**where**Payment > Cost*y = ax +b*, and ax^2 + by + c = 0. There is a huge cognitive jump in this abstraction, and it appears to be strongly limited by human development in that people tend to have a lot of trouble with abstraction in general until they hit adolescence.Of course there are other methods and domains applying the principle of abstraction, so don't go arguing against a claimed exclusive role for algebra. I will point out that HS algebra is usually the first time where the revolutionary value of this type of abstraction is explored in depth.

The reason why educators tend to talk about psychology rather that philosophy or history is pragmatic, given that history of mathematics and philosophy of mathematics have little to say about the empirical struggles of helping students discover that numeric quantities can be represented and manipulated via symbols.

So with a latch-ditch attempt at cleaning out the poisons thrown it the well by yourself and languagehat. Here are what I see as the core issues of concern:

1: Is it the case that "symbolic reasoning and calculations with symbols are central in algebra?" (This is where your philosophy and history of mathematics can play a big role.)

2: Is it the case that "symbolic reasoning and calculations with symbols" are necessary prerequisites for technical facility in a wide variety of post-secondary trades and careers?

3: Is it the case that "symbolic reasoning and calculations with symbols" are necessary prerequisites for a detailed comprehension of theories in current practice in a wide variety of fields?

4: Should "symbolic reasoning and calculations with symbols" be a requirement for graduating from high school?

5: Is it possible for us to talk about these issues using different lenses and different theoretical backgrounds?

My answers are:

1: Yes.

2: Yes.

3: Yes.

4: Probably.

5: Doubtful given the history of this discussion. But perhaps you could surprise me.

posted by KirkJobSluder at 7:58 AM on February 18, 2006

Oh and one more thing:

EB:

If I wasn't trying hard to clean out the well, I'd suggest that this assumption of a false dichotomy between platonism and functionalism reveals a blind spot in your claimed superior liberal arts education. But in the interest of making peace, I'd suggest reading with two alternative views to platonism in mind:

1: A historic view that elements of algebra developed over a long period of time before they were formalized.

2: A cognitive view that algebra builds on some very human ways of thinking about the would we live in that probably originated with the neolithic revolution.

posted by KirkJobSluder at 8:16 AM on February 18, 2006

EB:

*Yes it does. Calculus is an analytical technique. Few, if any, contemporary mathematicians are the platonists that you are implicitly insisting they are (because you are). Calculus didn't exist before it was invented, and neither did algebra.*

The technique implies more strongly the abstraction of the process of reasoning and thus is the critical step in the transition from platonism to formalism; and in so doing makes possible entirely new kinds of calculation (mixed units is one very obvious and important example).The technique implies more strongly the abstraction of the process of reasoning and thus is the critical step in the transition from platonism to formalism; and in so doing makes possible entirely new kinds of calculation (mixed units is one very obvious and important example).

If I wasn't trying hard to clean out the well, I'd suggest that this assumption of a false dichotomy between platonism and functionalism reveals a blind spot in your claimed superior liberal arts education. But in the interest of making peace, I'd suggest reading with two alternative views to platonism in mind:

1: A historic view that elements of algebra developed over a long period of time before they were formalized.

2: A cognitive view that algebra builds on some very human ways of thinking about the would we live in that probably originated with the neolithic revolution.

posted by KirkJobSluder at 8:16 AM on February 18, 2006

*Your position is akin to me saying that I don't use grammar because I haven't diagrammed a sentence in 25 years. Of course I use grammar. I don't do formal exercises in grammar, but I use skills derived from those formal exercises anytime I write or speak.*

Funny thing, though. On the GRE, you don't have to diagram sentences. I had to do algebra and geometry. When had I last taken those subjects? Exactly 17 and 15 years before taking the stupid test. That the GRE includes this points to expectations of our educational institutions, and a certain bias that few question. I'm doing fine, got into a decent doctoral program in my field, but I do not know what the point was.

posted by raysmj at 9:37 AM on February 18, 2006

Oh, and I got through the parts of my program that required statistics just fine. You only had to learn certain formulas and show you could use them to understand the logic of what you were doing on a machine, but otherwise what was important was learning how to use computer stats programs, knowing what formula to use when, and blah blah.

What was terribly hard for me was a class in reading comprehension in a foreign language, but the GRE doesn't test you for that. I'm guessing that's because foreign language skills aren't that highly valued in America?

posted by raysmj at 9:42 AM on February 18, 2006

What was terribly hard for me was a class in reading comprehension in a foreign language, but the GRE doesn't test you for that. I'm guessing that's because foreign language skills aren't that highly valued in America?

posted by raysmj at 9:42 AM on February 18, 2006

raysmj:

Well, just as one example, high school algebra IME is essential for understanding statistics, which in turn is used in any graduate field that does quantitative research (including fields you may not think about like Nursing, Business and Education.) You can always just plug numbers into SPSS and use a guided decision tree to produce some values, but really understanding what those values mean, and whether they are the best tool to represent your data requires some basic understanding of the abstractions behind symbols and functions.

On preview:

Perhaps I'm a bit baffled here, but concepts relating to functions and their relationships to graphic visualizations are listed as algebra according to the California math standards starting with "Algebra and functions" at grade five. To me it certainly seems as if you are using algebra. Or is this an argument similar to the old standby defense of innumeracy that we don't need to teach multiplication because we have machines to do it for us?

posted by KirkJobSluder at 10:10 AM on February 18, 2006

*Funny thing, though. On the GRE, you don't have to diagram sentences. I had to do algebra and geometry. When had I last taken those subjects? Exactly 17 and 15 years before taking the stupid test. That the GRE includes this points to expectations of our educational institutions, and a certain bias that few question. I'm doing fine, got into a decent doctoral program in my field, but I do not know what the point was.*Well, just as one example, high school algebra IME is essential for understanding statistics, which in turn is used in any graduate field that does quantitative research (including fields you may not think about like Nursing, Business and Education.) You can always just plug numbers into SPSS and use a guided decision tree to produce some values, but really understanding what those values mean, and whether they are the best tool to represent your data requires some basic understanding of the abstractions behind symbols and functions.

On preview:

*Oh, and I got through the parts of my program that required statistics just fine. You only had to learn*(emphasis added).**certain formulas**and show you could use them to**understand the logic**of what you were doing on a machine, but otherwise what was important was learning how to use computer stats programs, knowing what**formula**to use when, and blah blah.Perhaps I'm a bit baffled here, but concepts relating to functions and their relationships to graphic visualizations are listed as algebra according to the California math standards starting with "Algebra and functions" at grade five. To me it certainly seems as if you are using algebra. Or is this an argument similar to the old standby defense of innumeracy that we don't need to teach multiplication because we have machines to do it for us?

posted by KirkJobSluder at 10:10 AM on February 18, 2006

KirkJobSluder: I had in my posts that I took stats-oriented courses (in PoliSci) and came through them just fine, all As actually. And I didn't do well on the GRE part. The social sciences also require a great deal of writing, but you're not tested on writing for that, nor are you tested for general knowledge of the world or, as noted, foreign language comprehension of any sort. And you have to know all that to get through most grad school programs.

I use algebra, sure, but I was talking about the point ROU made re sentence diagrams. I don't use the specific algebraic formulas I learned in high school, nor do I use sentence diagrams. I'd forgotten most of those formulas and diagramming techniques, because I don't use them every day. I don't have to use them. I know how to put together a sentence without the diagrams.

You're reading past what I'm saying, frankly, or reading what you choose to read into it.

posted by raysmj at 10:16 AM on February 18, 2006

I use algebra, sure, but I was talking about the point ROU made re sentence diagrams. I don't use the specific algebraic formulas I learned in high school, nor do I use sentence diagrams. I'd forgotten most of those formulas and diagramming techniques, because I don't use them every day. I don't have to use them. I know how to put together a sentence without the diagrams.

You're reading past what I'm saying, frankly, or reading what you choose to read into it.

posted by raysmj at 10:16 AM on February 18, 2006

In short, I'm talking about a societal or institutional bias regarding specific formulas, rather than the big picture of what algebra teaches you or what it should teach you. You *will not* remember the specific formulas or lingo 15 years down the line, unless you work with that lingo and those formulas every day, or you just remember practically everything, including what day you saw so-and-so cute redhead at Wendy's.

posted by raysmj at 10:21 AM on February 18, 2006

posted by raysmj at 10:21 AM on February 18, 2006

raysmj:

I don't see where specific formulas are being specified. Classes of formulas, and generic types of problems, of course. But specific formulas, not really.

The GRE's view of writing is really interesting. Its history as a scanned multiple-choice exam has traditionally excluded some of the more complex forms of writing evaluation. So instead, what you have are things like reading comprehension, vocabulary, and analogies. With the assumption that one's score on these measures correlate well with writing ability.

posted by KirkJobSluder at 10:34 AM on February 18, 2006

*In short, I'm talking about a societal or institutional bias regarding specific formulas, rather than the big picture of what algebra teaches you or what it should teach you.*I don't see where specific formulas are being specified. Classes of formulas, and generic types of problems, of course. But specific formulas, not really.

The GRE's view of writing is really interesting. Its history as a scanned multiple-choice exam has traditionally excluded some of the more complex forms of writing evaluation. So instead, what you have are things like reading comprehension, vocabulary, and analogies. With the assumption that one's score on these measures correlate well with writing ability.

posted by KirkJobSluder at 10:34 AM on February 18, 2006

By the way, I teach at a commuter school/urban u right now, and there's a course for non-trad students that is basically "algebra for older people who haven't had algebra in a thousand years." Non-traditional students don't have to take a similar course in English comp--they take the same English comp that Brittney and Brad the Typical Undergrads do.

posted by raysmj at 10:34 AM on February 18, 2006

posted by raysmj at 10:34 AM on February 18, 2006

KirkJobSluder, I accept your basic point as I understand it: from a cognitive/pedagogical point of view, "algebra" is, in practice, something more broadly defined than "algebra" either as a branch of mathematics or a historical development in mathematics. Well, that's not the best way of saying it, but I get your drift.

Hmm. Maybe I don't accept it as much as I thought (or wanted to) because that seems too expansive. As you say, there are other methods, in theory and practice, to teach abstraced, formal reasoning. Why is algebra special? If algebra

As an example, assuming we want to teach abstracted, formal reasoning because it's an important skill for adults, I'd start with Venn diagrams before most anything else.

And, as I think about it more, I'm also struck with the thought that formal, abstracted reason can be taught in a variety of ways, even though it's very important, and thus I will not accept the contention that algebra is necessary for this purpose...while (here's the kicker) the essential concept in calculus is deeply important in many senses in abstractly comprehending the world around us and I can't think of a substitute. But people aren't clamoring for the universal teaching of calculus. Maybe they should, but they aren't. Why is that?

Frankly, I think it's because this "algebra is essential" idea is accepted and internalized deeply by a great many people. I know it is in my case—I have difficulty taking the position in this argument that I'm taking. I'd like everyone taught algebra. But then I'd like everyone taught some other things that they are not and most people don't advocate.

posted by Ethereal Bligh at 3:43 PM on February 18, 2006

Hmm. Maybe I don't accept it as much as I thought (or wanted to) because that seems too expansive. As you say, there are other methods, in theory and practice, to teach abstraced, formal reasoning. Why is algebra special? If algebra

*is*, broadly defined for these purposes, "abstracted, formal reasoning", or we're defending the teaching of abstracted, formal reasoning, then we're no longer speaking precisely of the algebra that is actually taught and it doesn't make much sense to defend it in this context. It's a red herring.As an example, assuming we want to teach abstracted, formal reasoning because it's an important skill for adults, I'd start with Venn diagrams before most anything else.

And, as I think about it more, I'm also struck with the thought that formal, abstracted reason can be taught in a variety of ways, even though it's very important, and thus I will not accept the contention that algebra is necessary for this purpose...while (here's the kicker) the essential concept in calculus is deeply important in many senses in abstractly comprehending the world around us and I can't think of a substitute. But people aren't clamoring for the universal teaching of calculus. Maybe they should, but they aren't. Why is that?

Frankly, I think it's because this "algebra is essential" idea is accepted and internalized deeply by a great many people. I know it is in my case—I have difficulty taking the position in this argument that I'm taking. I'd like everyone taught algebra. But then I'd like everyone taught some other things that they are not and most people don't advocate.

posted by Ethereal Bligh at 3:43 PM on February 18, 2006

EB:

*sigh* You know, I've posted links to the California State standards for mathematics

Your continued insistence on arguing against straw man definitions that no one has proposed or defended, and your choice to ignore definitions that have been repeatedly linked to and quoted strikes me as a bit of fundamental intellectual dishonesty. What is so hard with addressing the claims that have been made, that you need to create straw man positions to argue against?

Did you actually read what I said earlier? "I also think it's entirely correct to talk about algebra as the application of cognitive

Of course there are other forms of abstraction, music notation for example. But my argument is that the forms of abstraction used in algebra, "symbolic reasoning and (numeric) calculations with symbols" (California State Board of Education) are ubiquitous in fields far removed from the formal discipline of algebra. I can't make the same argument in regards to music notation. Music notation and programming languages contain quite a few common abstractions, but use radically different languages and symbols to describe them.

Among the reference books on my shelf, Venn diagrams are few and far between, while equations and formulas tend to pop-up three and four on a page. In fact, I'm writing a paper where I had to explain how I calculated a certain network statistic, and the simplest way to do it was to express it into an "algebraic" formula. (I wish the authors who introduced the statistic to me had been so kind.)

Which is a good thing, because no one as far as I can tell is proposing that algebra is necessary for this purpose. No one as far as I can tell is proposing that algebra that algebra should be taught for this purpose.

I've laid out what I am arguing in five questions, and my five answers to those questions. Is it so hard to actually go to the California Math Standards and come back with an informed response to those questions?

posted by KirkJobSluder at 5:30 PM on February 18, 2006

*As you say, there are other methods, in theory and practice, to teach abstraced, formal reasoning. Why is algebra special? If algebra is, broadly defined for these purposes, "abstracted, formal reasoning", or we're defending the teaching of abstracted, formal reasoning, then we're no longer speaking precisely of the algebra that is actually taught and it doesn't make much sense to defend it in this context.**sigh* You know, I've posted links to the California State standards for mathematics

**twice**. I've quoted from those standards no less than 6 times. These standards will lay out exactly what is meant by algebra in 5th grade, 6th grade, 7th grade, 8th grade, and in two semesters of high school study.Your continued insistence on arguing against straw man definitions that no one has proposed or defended, and your choice to ignore definitions that have been repeatedly linked to and quoted strikes me as a bit of fundamental intellectual dishonesty. What is so hard with addressing the claims that have been made, that you need to create straw man positions to argue against?

Did you actually read what I said earlier? "I also think it's entirely correct to talk about algebra as the application of cognitive

**mathematic**abstraction. From a cognitive point of view, algebra isn't something that springs from the forehead of arithmetic as a formal discipline, but the application of**form of cognition to the***an*abstract**specific domain of numeric quantities.**" (Emphasis added in the slim hope that you have a problem with reading, and are not engaged in dishonest participation for the sake of sheer perversity.)Of course there are other forms of abstraction, music notation for example. But my argument is that the forms of abstraction used in algebra, "symbolic reasoning and (numeric) calculations with symbols" (California State Board of Education) are ubiquitous in fields far removed from the formal discipline of algebra. I can't make the same argument in regards to music notation. Music notation and programming languages contain quite a few common abstractions, but use radically different languages and symbols to describe them.

*As an example, assuming we want to teach abstracted, formal reasoning because it's an important skill for adults, I'd start with Venn diagrams before most anything else.*Among the reference books on my shelf, Venn diagrams are few and far between, while equations and formulas tend to pop-up three and four on a page. In fact, I'm writing a paper where I had to explain how I calculated a certain network statistic, and the simplest way to do it was to express it into an "algebraic" formula. (I wish the authors who introduced the statistic to me had been so kind.)

*And, as I think about it more, I'm also struck with the thought that formal, abstracted reason can be taught in a variety of ways, even though it's very important, and thus I will not accept the contention that algebra is necessary for this purpose...*Which is a good thing, because no one as far as I can tell is proposing that algebra is necessary for this purpose. No one as far as I can tell is proposing that algebra that algebra should be taught for this purpose.

I've laid out what I am arguing in five questions, and my five answers to those questions. Is it so hard to actually go to the California Math Standards and come back with an informed response to those questions?

posted by KirkJobSluder at 5:30 PM on February 18, 2006

EB: You know something, I'm sick of trying to communicate only to have you make a post where you argue points that are

I ask you, how the fuck do you read, "From a cognitive point of view, algebra isn't something that springs from the forehead of arithmetic as a formal discipline, but the application of an abstract form of cognition to the specific domain of numeric quantities." and come away with "abstracted, formal reasoning".

How the fuck do you read, "Of course there are other methods and domains applying the principle of abstraction, so don't go arguing against a claimed exclusive role for algebra." and come away with, "Why is algebra special?" (Ignoring the fact that I've provided some reasons why algebra is probably worth learning that does not depend on it being "special.")

How the fuck do you have the gall to lecture me about, "then we're no longer speaking precisely of the algebra that is actually taught and it doesn't make much sense to defend it in this context." When as far as I can tell, I'm the only person in this "discussion" who's been willing to crack open the standards to figure out exactly what is taught?

How the fuck do you get "algebra is essential" when my earlier post answered the question of whether algebra should be required for a high school diploma with a weak, "probably."

The kindest answer I can propose to these questions is that you've had this argument before in another context, and your willingness to present a series of specific arguments have rendered you blind, deaf, and stupid towards anything that isn't the exact opposite of your chosen position.

posted by KirkJobSluder at 6:14 PM on February 18, 2006

**the exact opposite of what I explicitly said**.I ask you, how the fuck do you read, "From a cognitive point of view, algebra isn't something that springs from the forehead of arithmetic as a formal discipline, but the application of an abstract form of cognition to the specific domain of numeric quantities." and come away with "abstracted, formal reasoning".

How the fuck do you read, "Of course there are other methods and domains applying the principle of abstraction, so don't go arguing against a claimed exclusive role for algebra." and come away with, "Why is algebra special?" (Ignoring the fact that I've provided some reasons why algebra is probably worth learning that does not depend on it being "special.")

How the fuck do you have the gall to lecture me about, "then we're no longer speaking precisely of the algebra that is actually taught and it doesn't make much sense to defend it in this context." When as far as I can tell, I'm the only person in this "discussion" who's been willing to crack open the standards to figure out exactly what is taught?

How the fuck do you get "algebra is essential" when my earlier post answered the question of whether algebra should be required for a high school diploma with a weak, "probably."

The kindest answer I can propose to these questions is that you've had this argument before in another context, and your willingness to present a series of specific arguments have rendered you blind, deaf, and stupid towards anything that isn't the exact opposite of your chosen position.

posted by KirkJobSluder at 6:14 PM on February 18, 2006

The main argument of the majority of the people in the thread that have defended the mandatory teaching of algebra has been that essentially all mathematical reasoning is algebra and thus for anyone doing mathematical reasoning in any respect, algebra is a very useful skill. They've very confidently defined algebra as something it is not. My primary argument was against theirs.

When you complain:

It's as if you've read no one's comments in this thread but yours and mine. I wasn't setting up straw men, they're right there in this thread.

With regard to that argument, I accept the first portion of it (but not without reservations) and am undecided on the other. I'll grant you that alegbra is by far the best means for teaching symbolic mathematical reasoning. But I'm undecided on your claim that a majority of students graduating from secondary school will find that they need a facility with symbolic mathematical reasoning.

And to that previous sentence, people in this thread have asserted—and would assert again—that almost all mathematical reasoning is "really" algebra, including calculating change in a retail transaction. Surely you can see why I countered that argument.

I'd accept the assertion that symbolic mathematical reasoning is a required skill in a good many disciplines that an undergraduate education prepares students for. But I'd reject the implied assertion that what is good for 2/3 of all college graduates is good for all high school graduates.

When we get down to the nitty-gritty like this, and start really looking at vocational skills, when it becomes hard to justify something that is the

And the bottom line here is that it is absurd to assert that, assuming algebra teaches important life skills, that these skills are more important than a strong facility with written language.

It seems clear to me, with math as perhaps the prime example, that US secondary and university undergraduate education attempts both a technical vocational and a liberal arts education—in the case of math, in the same subject—and as a result manages to achieve neither goal. The solution is either longer secondary education, or the division of education into a vo-tec and liberal arts track at secondary education, or both. A system which cannot teach algebra to a student in six attempts and from which the student drops out, is a broken system.

posted by Ethereal Bligh at 8:56 PM on February 18, 2006

When you complain:

*"Your continued insistence on arguing against straw man definitions that no one has proposed or defended, and your choice to ignore definitions that have been repeatedly linked to and quoted strikes me as a bit of fundamental intellectual dishonesty. What is so hard with addressing the claims that have been made, that you need to create straw man positions to argue against?"*It's as if you've read no one's comments in this thread but yours and mine. I wasn't setting up straw men, they're right there in this thread.

*Your*argument is somewhat more limited in respect to its claims about whatever are the essential concepts of algebra, but then is more expansive with regard to how widely such reasoning is necessary and thus an important skill to teach*all*students.With regard to that argument, I accept the first portion of it (but not without reservations) and am undecided on the other. I'll grant you that alegbra is by far the best means for teaching symbolic mathematical reasoning. But I'm undecided on your claim that a majority of students graduating from secondary school will find that they need a facility with symbolic mathematical reasoning.

And to that previous sentence, people in this thread have asserted—and would assert again—that almost all mathematical reasoning is "really" algebra, including calculating change in a retail transaction. Surely you can see why I countered that argument.

I'd accept the assertion that symbolic mathematical reasoning is a required skill in a good many disciplines that an undergraduate education prepares students for. But I'd reject the implied assertion that what is good for 2/3 of all college graduates is good for all high school graduates.

When we get down to the nitty-gritty like this, and start really looking at vocational skills, when it becomes hard to justify something that is the

*status quo*of US public education, people then typically appeal to a liberal arts utility of a contested skill. If I question how effectively it's taught*as a liberal art*, and specifically complain that it's taught as a set of related techniques that is assumed they will use, then the argument switches again to the various places in which those specific technical skills are required.And the bottom line here is that it is absurd to assert that, assuming algebra teaches important life skills, that these skills are more important than a strong facility with written language.

It seems clear to me, with math as perhaps the prime example, that US secondary and university undergraduate education attempts both a technical vocational and a liberal arts education—in the case of math, in the same subject—and as a result manages to achieve neither goal. The solution is either longer secondary education, or the division of education into a vo-tec and liberal arts track at secondary education, or both. A system which cannot teach algebra to a student in six attempts and from which the student drops out, is a broken system.

posted by Ethereal Bligh at 8:56 PM on February 18, 2006

*The main argument of the majority of the people in the thread that have defended the mandatory teaching of algebra has been that essentially all mathematical reasoning is algebra*

No, that's actually your interpretation, restatement, or straw-man of their argument, implying a much farther-reaching definition of algebra than, I think, any of the people you were arguing against intended.

I, at least, was arguing, or intending to argue, that many everyday occurrences use the same mental or cognitive processes as elementary algebra, in the sense that both follow the same or very similar flowcharts or how-to-do-its. In this sense, figuring out how much to tip or making change is (very simple) algebra to the extent that you're following the same procedure to do them both. If I did so sloppily, fine, I did so sloppily.

This from the arguments in the original article, which said that the dropout would never need to know how long it would take for so many people to mow a lawn when two extra people show up halfway, or anything like that. But that's just wrong -- of course people need to figure out things like that, on a common, everyday basis. People need to figure out how much more paint to buy when they've run out with one coat on two walls and two coats on a third. They need to figure out how many people or hours to assign to a partly-completed project in order to get it finished by the deadline. They need to figure out whether their tank of gas will last them until the next payday, in which case they can go drinking, or whether they need to buy gas now and stay in tonight. All of these problems use the same or very similar internal logic to the problem that Cohen scoffs at. To the extent that they share that same logic, they are the same problem.

Further, at no point did I defend the mandatory teaching of anything. Nothing I have written here has said anything about what ought or ought not be taught in high school or universities. All I have argued is that the idea that "I haven't used algebra" is a very silly argument, because people do in fact use the same processes that they would need to solve simple algebra problems on a day-to-day basis.

You take me to task before for the grammar example, lecturing me that diagramming sentences is not grammar itself, but an analytical technique. But this was exactly my point. In the language you used, there is a difference between a thing and the techniques or exercises that are used to teach (at least in theory) a thing. Saying that you don't use grammar because you haven't diagrammed a sentence would be a profoundly stupid thing to say, precisely because that scholastic exercise is not, itself, the object being taught -- completely irrespective of whether it ought to be taught or not, a subject I, again, said exactly nothing about. Similarly, it would be foolish to say that you haven't "used" algebra* in fifty years simply because you haven't sat down with a page of word problems or a list of equations with SOLVE FOR X at the top in all that time, because you do in fact use the very same algorithms you would use to solve algebra problems on a frequent basis, unless you are a clod. For a third time, I have not said anything about the usefulness of teaching algebra in any school. All I have argued against was the blinkered perception that because you have not sat down to do schoolbook exercises in something, you have not used the skills the schoolbook exercises were intended to impart. Note that this can be true even when the schoolbook exercises were completely useless at imparting the skills or even actively obstructed the process of learning them.

I think you're reading what's posted here with far too critical an eye, as if you were reviewing articles for a journal, seeing nuances and implications that simply aren't there in what was written and what was left out. Like KJS, I think this should have been very obvious from context, and wonder what argument you're really having, and with who.

*in the sense of the simple subject taught in middle and high school, not in the sense of following in the footsteps of al-Khwarizmi.

posted by ROU_Xenophobe at 10:18 PM on February 18, 2006

EB:

Well you know, I've read every comment in this thread, most more than once, and I think you are at best engaged in a biased reading, and at worst engaged in some pretty hefty dishonesty.

Well certainly I would hope it's more limited. Such faery tales as "essential concepts" are best left to philosophers, preferably locked in their own ivory towers. What concerns people interested in pedagogy are which concepts are prerequisites for other concepts. The standards at issue, (which again you show no sign of having even given a casual glance,) are there because of the fairly well-supported claims that those concepts and skills are likely to be necessary to master other concepts and skills down the road.

The people who make these standards ask questions like, "How many of our students get into post-secondary education? How many drop out of post-secondary early? How many need remedial instruction when they get into post-secondary education?" Then they work backwards and say, "If college freshmen need this, what do we need to teach HS seniors? If HS seniors need this, what do we need to teach HS juniors?" And so on back to making certain that kids who go to kindergarden are at least potty trained.

In my opinion, what you've done is countered such a radically stripped down version of the argument that it has little in common with the argument actually made. I'll simply assert this: being able to describe a problem such as "calculating change" as an equation such as

Certainly. And this is a good argument. It is in fact the first honest argument I've seen from you in this discussion. And as I've mentioned before, I'm open to discussion on this issue. I think in an ideal world we would have an educational system based around individual learning needs and goals. I think we need richer evaluation models.

But I'm skeptical as to whether this kind of a revolution will be seen outside of private schools in my lifetime. (And even private schools have to face pressure from high stakes accreditation tests.) Ultimately, if we are talking about algebra as it is actually taught in the majority of US School Systems (that phenomena you give lip service to without actually investigating) you have to deal with performance-based instructional objectives, that include prerequisites that you probably don't consider to be algebra because they are superficially trivial to numerate people, like the equation for "calculating change."

And well, there are needs and there are

Well, personally I see this as another false dichotomy, in that I don't see the two as necessarily contradictory. And in fact, I'm no longer comfortable with my separation of these issues between questions 2 and 3 in my previous post. I'll confess, I'm bad at algebra as a technique. I'm slow, and prone to making stupid mistakes. For me, algebra is a spectator sport that permits me to grasp the brilliance of the Student's t, distribution, Einstein's special relativity, and Quantitative Genetics. Algebra enabled me to make the bold leap to understanding computer programming as a way of shuffling and transforming data using various types of handles. (As opposed to a string of simple LOOP and GOTO statements.)

A large chunk of human knowledge is expressed in the form of statements like "E = MC^2." Other chunks are expressed in other forms like novels, short stories, musical forms, visual art, computer programs and newspaper articles. I'd argue that the ability to read and interpret "E=MC^2" as more than a string of pretty letters is an important form of cultural literacy. Does having this cultural literacy return vocational benefits? I'd argue yes.

And how is that literacy labeled in the current Math content area standards? algebra.

Why is this the bottom line? Has anyone asserted that algebra is more important than a strong facility with the written language? And I'm a person who advocates strongly for the preservation of

If you want to argue about what forms of mathematics should be required for graduation, and what forms should be electives, then argue based on the merits and difficulties of that content area. If it is essential that students be numerate and literate, then we need to make room for both. If we as a culture decide that students should sight-read musical notation, then we need to make room for that as well. If we as a culture decide that it is critically important that well educated and well-rounded students can do Tai Chi long form, then we need to make room for that.

posted by KirkJobSluder at 12:03 AM on February 19, 2006

*It's as if you've read no one's comments in this thread but yours and mine. I wasn't setting up straw men, they're right there in this thread.*Well you know, I've read every comment in this thread, most more than once, and I think you are at best engaged in a biased reading, and at worst engaged in some pretty hefty dishonesty.

*Your argument is somewhat more limited in respect to its claims about whatever are the essential concepts of algebra, but then is more expansive with regard to how widely such reasoning is necessary and thus an important skill to teach all students.*Well certainly I would hope it's more limited. Such faery tales as "essential concepts" are best left to philosophers, preferably locked in their own ivory towers. What concerns people interested in pedagogy are which concepts are prerequisites for other concepts. The standards at issue, (which again you show no sign of having even given a casual glance,) are there because of the fairly well-supported claims that those concepts and skills are likely to be necessary to master other concepts and skills down the road.

The people who make these standards ask questions like, "How many of our students get into post-secondary education? How many drop out of post-secondary early? How many need remedial instruction when they get into post-secondary education?" Then they work backwards and say, "If college freshmen need this, what do we need to teach HS seniors? If HS seniors need this, what do we need to teach HS juniors?" And so on back to making certain that kids who go to kindergarden are at least potty trained.

*And to that previous sentence, people in this thread have asserted—and would assert again—that almost all mathematical reasoning is "really" algebra, including calculating change in a retail transaction. Surely you can see why I countered that argument.*In my opinion, what you've done is countered such a radically stripped down version of the argument that it has little in common with the argument actually made. I'll simply assert this: being able to describe a problem such as "calculating change" as an equation such as

*X = Y - Z when Y*is an objective listed under the heading**Algebra and Functions**in the California State content standards for mathematics. Now, you certainly could argue that it should be under another heading, such as**Foobarbaz**. But that's the taxonomy of objectives that we are currently faced with.*I'd accept the assertion that symbolic mathematical reasoning is a required skill in a good many disciplines that an undergraduate education prepares students for. But I'd reject the implied assertion that what is good for 2/3 of all college graduates is good for all high school graduates.*Certainly. And this is a good argument. It is in fact the first honest argument I've seen from you in this discussion. And as I've mentioned before, I'm open to discussion on this issue. I think in an ideal world we would have an educational system based around individual learning needs and goals. I think we need richer evaluation models.

But I'm skeptical as to whether this kind of a revolution will be seen outside of private schools in my lifetime. (And even private schools have to face pressure from high stakes accreditation tests.) Ultimately, if we are talking about algebra as it is actually taught in the majority of US School Systems (that phenomena you give lip service to without actually investigating) you have to deal with performance-based instructional objectives, that include prerequisites that you probably don't consider to be algebra because they are superficially trivial to numerate people, like the equation for "calculating change."

And well, there are needs and there are

**needs**It's possible to argue that the only thing we really need to teach is how to avoid getting hit by a bus, and where to find a soup kitchen. The rest is just enrichment.*When we get down to the nitty-gritty like this, and start really looking at vocational skills, when it becomes hard to justify something that is the status quo of US public education, people then typically appeal to a liberal arts utility of a contested skill. If I question how effectively it's taught as a liberal art, and specifically complain that it's taught as a set of related techniques that is assumed they will use, then the argument switches again to the various places in which those specific technical skills are required.*Well, personally I see this as another false dichotomy, in that I don't see the two as necessarily contradictory. And in fact, I'm no longer comfortable with my separation of these issues between questions 2 and 3 in my previous post. I'll confess, I'm bad at algebra as a technique. I'm slow, and prone to making stupid mistakes. For me, algebra is a spectator sport that permits me to grasp the brilliance of the Student's t, distribution, Einstein's special relativity, and Quantitative Genetics. Algebra enabled me to make the bold leap to understanding computer programming as a way of shuffling and transforming data using various types of handles. (As opposed to a string of simple LOOP and GOTO statements.)

A large chunk of human knowledge is expressed in the form of statements like "E = MC^2." Other chunks are expressed in other forms like novels, short stories, musical forms, visual art, computer programs and newspaper articles. I'd argue that the ability to read and interpret "E=MC^2" as more than a string of pretty letters is an important form of cultural literacy. Does having this cultural literacy return vocational benefits? I'd argue yes.

And how is that literacy labeled in the current Math content area standards? algebra.

*And the bottom line here is that it is absurd to assert that, assuming algebra teaches important life skills, that these skills are more important than a strong facility with written language.*Why is this the bottom line? Has anyone asserted that algebra is more important than a strong facility with the written language? And I'm a person who advocates strongly for the preservation of

**fine arts**and**physical education**(although phys. ed. really needs some work) in K-12 schools, which have an even weaker vocational argument. Trying to pull the, "this class is more important than that class" BS on me just doesn't work.If you want to argue about what forms of mathematics should be required for graduation, and what forms should be electives, then argue based on the merits and difficulties of that content area. If it is essential that students be numerate and literate, then we need to make room for both. If we as a culture decide that students should sight-read musical notation, then we need to make room for that as well. If we as a culture decide that it is critically important that well educated and well-rounded students can do Tai Chi long form, then we need to make room for that.

posted by KirkJobSluder at 12:03 AM on February 19, 2006

*How the fuck do you have the gall to lecture me about, "then we're no longer speaking precisely of the algebra that is actually taught and it doesn't make much sense to defend it in this context." When as far as I can tell, I'm the only person in this "discussion" who's been willing to crack open the standards to figure out exactly what is taught?*

Was he? I assumed he was arguing with teece. I have no idea what your position is, or what you are trying to say, quite frankly.

posted by delmoi at 12:29 AM on February 19, 2006

The main argument of the majority of the people in the thread that have defended the mandatory teaching of algebra has been that essentially all mathematical reasoning is algebraNo, that's actually your interpretation, restatement, or straw-man of their argument, implying a much farther-reaching definition of algebra than, I think, any of the people you were arguing against intended.

I don't know what thread you were reading, but that is

**absolutly**what teece was arguing.

posted by delmoi at 12:36 AM on February 19, 2006

I appreciate delmoi's comment, and I think he's right, but I am very much not comfortable with the level of angry disagreement in this thread. I'm sure everyone here is arguing in good faith, probably agree on most everything when it comes down to it, and we've just had some of our buttons pushed. That's certainly true in my case.

As an aside, it seems like it's natural and even makes sense to read a thread, notice an argument made by different people, to different degrees, with varying amounts of qualification and subtlety and then respond agressively against "it" as one understands it. But doing so usually causes trouble because whatever assumed generalized argument one thinks is being made, it's going to be a misrepresentation of any given individual's argument. I apologize for making this mistake.

posted by Ethereal Bligh at 1:15 AM on February 19, 2006

As an aside, it seems like it's natural and even makes sense to read a thread, notice an argument made by different people, to different degrees, with varying amounts of qualification and subtlety and then respond agressively against "it" as one understands it. But doing so usually causes trouble because whatever assumed generalized argument one thinks is being made, it's going to be a misrepresentation of any given individual's argument. I apologize for making this mistake.

posted by Ethereal Bligh at 1:15 AM on February 19, 2006

delmoi:

In context, he was responding and engaging in what to me appear to be bad-faith paraphrases of my statements.

My position is simple:

1: We are not talking about "algebra" as a historic artifact, or "algebra" as defined by philosophers of mathematics, we are talking about algebra as it is taught in public school systems.

2: Educators are extremely explicit about how they define a subject area. These definitions are made public, and in recent practice published on the World Wide Web.

3: Attempting to engage in a critique of how mathematics is taught in school without talking about mathematics content standards is insanely ignorant and dishonest.

#3 is really unpardonable in this case. The content standards are not locked up behind a subscription service, and they are written in such a way that minimal technical knowledge is needed to understand them.

posted by KirkJobSluder at 7:00 AM on February 19, 2006

*Was he? I assumed he was arguing with teece. I have no idea what your position is, or what you are trying to say, quite frankly.*In context, he was responding and engaging in what to me appear to be bad-faith paraphrases of my statements.

My position is simple:

1: We are not talking about "algebra" as a historic artifact, or "algebra" as defined by philosophers of mathematics, we are talking about algebra as it is taught in public school systems.

2: Educators are extremely explicit about how they define a subject area. These definitions are made public, and in recent practice published on the World Wide Web.

3: Attempting to engage in a critique of how mathematics is taught in school without talking about mathematics content standards is insanely ignorant and dishonest.

#3 is really unpardonable in this case. The content standards are not locked up behind a subscription service, and they are written in such a way that minimal technical knowledge is needed to understand them.

posted by KirkJobSluder at 7:00 AM on February 19, 2006

My position is simple: I couldn't give a rat's ass about how a content standard defines "algebra". Algebra is what it is and it's not what it's not.

If the subject were French, and we were arguing about whether French should be taught to everyone in secondary schools, I don't think we'd be arguing about the definition of "French". Thankfully, educators, no matter how they may try, may succeed in defining a language as they choose, nor a branch of mathematics.

We are talking about teaching algebra or not teaching algebra. That does not require guidance from a content standard. Sorry.

posted by Ethereal Bligh at 10:10 AM on February 19, 2006

If the subject were French, and we were arguing about whether French should be taught to everyone in secondary schools, I don't think we'd be arguing about the definition of "French". Thankfully, educators, no matter how they may try, may succeed in defining a language as they choose, nor a branch of mathematics.

We are talking about teaching algebra or not teaching algebra. That does not require guidance from a content standard. Sorry.

posted by Ethereal Bligh at 10:10 AM on February 19, 2006

*My position is simple: I couldn't give a rat's ass about how a content standard defines "algebra". Algebra is what it is and it's not what it's not.*

Then you really aren't having any manner of useful discussion with anyone else in the thread.

posted by ROU_Xenophobe at 11:04 AM on February 19, 2006

Man, algebra isn't exactly hard. I mean I can understand people having difficulty getting to learn the concepts, but seriously with a bit of help you should be able to get it.

If you can't get problem-solving down with algebra then you probably shouldn't graduate school. It shows a fundamental flaw in your education. I'm surprised... I think I remember doing algebra in year 8 in our schools... I dunno what the equivalent is in the US or wherever... but our graduation is after year 12, 5 years later.

If you can't get it by the time you graduate, then there's something seriously wrong and you SHOULDN'T be graduating.

This guy thinks Algebra is a waste of time. Some people think Maths is a waste of time altogether. And I've talked with people that think an education of any kind is a waste of time too. *sigh*

They just don't get it. They don't appreciate what they've learned, or at least had the opportunity to learn, at school.

If the specific methods of solving various problems isn't particularly useful in later life, then the problem solving abilities certainly are, and the methods of looking at various problems.

From learning how to write essays in English (yep many people never use that skill either) to calculus in Maths and learning ancient languages like Latin... whilst you may never speak Latin to anyone they're all useful in teaching you how to think and look at kinds of problems in different ways, and equip you with at least some basics so that you might tackle the some of the problems that you encounter in your life. And if not that then at least they should teach or FORCE you to learn how to learn.

I don't get why or HOW people don't see this???

I guess it's cool to be cynical and thankless bastards these days.

posted by Jelreyn at 11:26 AM on February 19, 2006

If you can't get problem-solving down with algebra then you probably shouldn't graduate school. It shows a fundamental flaw in your education. I'm surprised... I think I remember doing algebra in year 8 in our schools... I dunno what the equivalent is in the US or wherever... but our graduation is after year 12, 5 years later.

If you can't get it by the time you graduate, then there's something seriously wrong and you SHOULDN'T be graduating.

This guy thinks Algebra is a waste of time. Some people think Maths is a waste of time altogether. And I've talked with people that think an education of any kind is a waste of time too. *sigh*

They just don't get it. They don't appreciate what they've learned, or at least had the opportunity to learn, at school.

If the specific methods of solving various problems isn't particularly useful in later life, then the problem solving abilities certainly are, and the methods of looking at various problems.

From learning how to write essays in English (yep many people never use that skill either) to calculus in Maths and learning ancient languages like Latin... whilst you may never speak Latin to anyone they're all useful in teaching you how to think and look at kinds of problems in different ways, and equip you with at least some basics so that you might tackle the some of the problems that you encounter in your life. And if not that then at least they should teach or FORCE you to learn how to learn.

I don't get why or HOW people don't see this???

I guess it's cool to be cynical and thankless bastards these days.

posted by Jelreyn at 11:26 AM on February 19, 2006

EB:

And gee, you like to accuse everyone else of Platonism?

If you want your pet subject matter to be taught in a public school, and in most private schools, you are going to have to develop some kind of a content standard: a list of concepts and skills that says you are going to teach this, but leave that to further study.

I think it's meaningless to talk about "subjects" (aka a curriculum) without talking about specific objectives. What are you going to teach in French? Is the goal spoken fluency or translation (I've taken both forms of foreign language courses)? Are you going to focus specifically on grammar? Are you going to include archaic vocabulary and grammar? How much cultural studies and enrichment will be included? Which dialect are you going to teach?

Educators are not defining language or a branch of mathematics, they are defining a curriculum of study, a set of concepts and skills that will be included at the High School level, and a set of concepts and skills that will be left to later study.

Well, this again is a false dichotomy because it assumes that you can't break down algebra into a subset of skills and concepts. A full study of algebra would require several years of work. Given that we can't download all of algebra into a person's head in 10 minutes, we have to consider what gets taught in the first week, the second week, the first semester, the second semester, high school, trade school, university and graduate school.

One of the reasons why I keep suggesting that we look at the content standards is because they provide one way of analyzing a complex field of study as a set of smaller concepts and skills. My suggestion is that if you look at "algebra as it is actually taught" (something you gave lip-service to earlier but then proclaimed you can give a rats ass about) that you would probably find some objectives that you would strongly recommend, and other objectives that you find trivial. You seem to have developed this preconception about the infamy of those standards without actually having seen them, and that's baffling to me.

posted by KirkJobSluder at 12:29 PM on February 19, 2006

*I couldn't give a rat's ass about how a content standard defines "algebra". Algebra is what it is and it's not what it's not.*And gee, you like to accuse everyone else of Platonism?

If you want your pet subject matter to be taught in a public school, and in most private schools, you are going to have to develop some kind of a content standard: a list of concepts and skills that says you are going to teach this, but leave that to further study.

*If the subject were French, and we were arguing about whether French should be taught to everyone in secondary schools, I don't think we'd be arguing about the definition of "French".*I think it's meaningless to talk about "subjects" (aka a curriculum) without talking about specific objectives. What are you going to teach in French? Is the goal spoken fluency or translation (I've taken both forms of foreign language courses)? Are you going to focus specifically on grammar? Are you going to include archaic vocabulary and grammar? How much cultural studies and enrichment will be included? Which dialect are you going to teach?

*Thankfully, educators, no matter how they may try, may succeed in defining a language as they choose, nor a branch of mathematics.*Educators are not defining language or a branch of mathematics, they are defining a curriculum of study, a set of concepts and skills that will be included at the High School level, and a set of concepts and skills that will be left to later study.

*We are talking about teaching algebra or not teaching algebra. That does not require guidance from a content standard. Sorry.*Well, this again is a false dichotomy because it assumes that you can't break down algebra into a subset of skills and concepts. A full study of algebra would require several years of work. Given that we can't download all of algebra into a person's head in 10 minutes, we have to consider what gets taught in the first week, the second week, the first semester, the second semester, high school, trade school, university and graduate school.

One of the reasons why I keep suggesting that we look at the content standards is because they provide one way of analyzing a complex field of study as a set of smaller concepts and skills. My suggestion is that if you look at "algebra as it is actually taught" (something you gave lip-service to earlier but then proclaimed you can give a rats ass about) that you would probably find some objectives that you would strongly recommend, and other objectives that you find trivial. You seem to have developed this preconception about the infamy of those standards without actually having seen them, and that's baffling to me.

posted by KirkJobSluder at 12:29 PM on February 19, 2006

EB:

And I just have to say, ohhh, baby! Nothing advertises the open-minded benefits of a liberal arts education than a perverse insistence on remaining ignorant to the ways people from other disciplines view the world! While attempting to inventing arguments about how those disciplines should do their business.

"Arguing in good faith," my ass.

posted by KirkJobSluder at 12:45 PM on February 19, 2006

*My position is simple: I couldn't give a rat's ass about how a content standard defines "algebra".*And I just have to say, ohhh, baby! Nothing advertises the open-minded benefits of a liberal arts education than a perverse insistence on remaining ignorant to the ways people from other disciplines view the world! While attempting to inventing arguments about how those disciplines should do their business.

"Arguing in good faith," my ass.

posted by KirkJobSluder at 12:45 PM on February 19, 2006

I insist that we look at how algebra is taught in Sweden before we continue this conversation. It might even tell us whether or not "making change" is "algebra". It's astonishing and unforgivable that you'd disregard this one truly relevant point of view. I can only imagine it's a willfull display of your own arrogance and closemindedness.

posted by Ethereal Bligh at 7:12 PM on February 19, 2006

posted by Ethereal Bligh at 7:12 PM on February 19, 2006

EB:

Certainly, even though we are talking about schools in California rather than Stockholm, I think that comparing the ways is which algebra is taught in different school systems can be quite useful. Would you happen to have any English language documentation on this?

Of course, given that we are talking about Schools in California rather than Stockholm, I would dare say that documents published by the California State Board of Education are probably more relevant than documents published by the Swedish Ministry of Education and Research. But by all means, educational research is international in scope, and while I don't have any colleagues in Sweden, I have learned quite a bit from hearing how school systems in Japan, Korea, China, Germany, Turkey and Australia have approached similar pedagogical problems.

I've asked this before, "Is it possible for us to talk about these issues using different lenses and different theoretical backgrounds?" I'll read anything including the trademark on the kitchen sink if you can link it to math education. I object to the claim that content area standards should be rejected from this discussion blindly and with prejudice.

You can't talk about the state of math education without talking about content area standards. You can love content area standards, you can hate content area standards. You can hate specific objectives and love others. But in U.S. public school systems and in many private schools, the content area standards are the law of the land. The majority of lesson plans must be justified as teaching one or more objectives in the content area standards, and there is little time for lesson plans that can't be tied to the content area standards. You don't have to like or approve of the content area standards, but you can't ignore them, you can't dismiss them, you can't refuse to read them and present an argument regarding the state of education in the U.S..

posted by KirkJobSluder at 11:55 PM on February 19, 2006

*I insist that we look at how algebra is taught in Sweden before we continue this conversation.*Certainly, even though we are talking about schools in California rather than Stockholm, I think that comparing the ways is which algebra is taught in different school systems can be quite useful. Would you happen to have any English language documentation on this?

Of course, given that we are talking about Schools in California rather than Stockholm, I would dare say that documents published by the California State Board of Education are probably more relevant than documents published by the Swedish Ministry of Education and Research. But by all means, educational research is international in scope, and while I don't have any colleagues in Sweden, I have learned quite a bit from hearing how school systems in Japan, Korea, China, Germany, Turkey and Australia have approached similar pedagogical problems.

I've asked this before, "Is it possible for us to talk about these issues using different lenses and different theoretical backgrounds?" I'll read anything including the trademark on the kitchen sink if you can link it to math education. I object to the claim that content area standards should be rejected from this discussion blindly and with prejudice.

You can't talk about the state of math education without talking about content area standards. You can love content area standards, you can hate content area standards. You can hate specific objectives and love others. But in U.S. public school systems and in many private schools, the content area standards are the law of the land. The majority of lesson plans must be justified as teaching one or more objectives in the content area standards, and there is little time for lesson plans that can't be tied to the content area standards. You don't have to like or approve of the content area standards, but you can't ignore them, you can't dismiss them, you can't refuse to read them and present an argument regarding the state of education in the U.S..

posted by KirkJobSluder at 11:55 PM on February 19, 2006

*We are talking about teaching algebra or not teaching algebra. That does not require guidance from a content standard. Sorry.*

That makes no sense to me. Every subject taught in schools is given consideration in regards to content... and it's especially important to this converstation. I took a class called "Challenges of Engineering" in high school, available to seniors that were interested in possibly pursuing an engineering degree in college. I think it's safe to say that we only had the time and resources to touch on a few basic concepts and ideas.

I took Algebra I (then Geometry) and Algebra II (then Trig.) in high school. I wonder how they decided where to split the algebra content into separate classes? Algebra I was required for graduation, the latter was only required for a more advance diploma.

posted by Witty at 4:54 AM on February 20, 2006

« Older La Bella Napoli | I Lead a Normal Life Newer »

This thread has been archived and is closed to new comments

Writing is the highest form of reasoning. This is a fact.Can't argue with that reasoning.

posted by dsword at 10:04 PM on February 16, 2006