Friday, March 03, 2006

The Value of Algebra

I am so mad right now.

I devote a small part of each and every day to perusing some of the dopiest, most inane websites on the net. There are about a dozen creationist web sites I monitor on a daily basis, not to mention several very right-wing political sites. I've grown so accustomed to their relentless stupidity that I barely notice it anymore.

But I read a number of other sites that are supposedly more reliable. At those sites I expect to read things that will provide some food for thought, even if, in the end, I disagree with them. So encountering breathtaking, creationist-level stupidity at such a site is just too depressing.

I'm sure you've figured out by now that I am referring to this column, by Richard Cohen, from the February 16 edition of The Washington Post. We consider it in full:

I am haunted by Gabriela Ocampo.

Last year, she dropped out of the 12th grade at Birmingham High School in Los Angeles after failing algebra six times in six semesters, trying it a seventh time and finally just despairing over ever getting it. So, according to the Los Angeles Times, she “gathered her textbooks, dropped them at the campus book room and, without telling a soul, vanished from Birmingham High School.”

Gabriela, this is Richard: There's life after algebra.

In truth, I don't know what to tell Gabriela. The L.A. school district now requires all students to pass a year of algebra and a year of geometry in order to graduate. This is something new for Los Angeles (although 17 states require it) and it is 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. In L.A., more kids drop out of school on account of algebra than any other subject. I can hardly blame them.

I can blame them. I'll bet real money that every one of those kids who dropped out of school on account of algebra had problems that go well beyond difficulties in math. Were these students who were generally doing well in their other subjects but, doggone it, just couldn't pass algebra? Or were these people who didn't take school very seriously, had little in the way of discipline, and were just scraping by in their other, less demanding classes? I'm sorry, but no one of normal intelligence fails algebra six times simply because the subject is just so darn difficult.

I'm glad to hear that LA now requires a year of algebra and a year of geometry to graduate. I'm amazed that only 17 states do likewise.

Incredibly, though, that's not the supid part of the column:

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 time -- the only proof I've ever seen of divine intervention -- somehow passed geometry and resolved, with a grateful exhale of breath, that I would never go near math again. I let others go on to intermediate algebra and trigonometry while I busied myself learning how to type. In due course, this came to be the way I made my living. Typing: Best class I ever took.

It is a truism among mathematicians that mathematics is the only subject that people brag about being bad at. Cohen thinks it's just the cutest thing that he has trouble with percentages. Can you think of any other academic subject where he would proudly joke about how bad he is at it?

The following fantasy conversation plays in my mind from time to time:

NICE PERSON AT PARTY: What do you do?
ME: I'm a mathematician.
NICE PERSON: Oh, (giggles), I was never any good at math.
ME: That's because you're an idiot.

I never say that of course. No, usually I say something tactful like, “You just never had the right teacher.” But it really is irritating when otherwise intelligent and well-educated people act like you're the weird one for being good at math.

The fact is that things like math phobia, or the idea that people's brains are wired differently, or that some people just can't do math, are total bilge. Hostility towards mathematics has nothing to do with any of those things. In reality it is just standard anti-intellectualism. Nothing more glamorous or interesting than that.

Actually, though, we still have not reached the truly dumb part of the column:

Here's the thing, Gabriela: You will never need to know algebra. I have never once used it and never once even rued that I could not use it. You will never need to know -- never mind want to know -- how many boys it will take to mow a lawn if one of them quits halfway and two more show up later -- or something like that. Most of math can now be done by a computer or a calculator. On the other hand, no computer can write a column or even a thank-you note -- or reason even a little bit. If, say, the school asked you for another year of English or, God forbid, history, so that you actually had to know something about your world, I would be on its side. But algebra? Please.

Now that's one of the dumbest things I've read in a while. And remember, I read websites that say things like: “If humans evolved from apes, why do we still have apes?”

First, how on Earth does Cohen know what Ms. Ocampo will need to know later in her life? The fact is that if she wants to pursue any sort of scientific subject she will inevitably need to know algebra to do that.

As evidence that Ms. Ocampo will never need algebra, Cohen presents the fact that he has never had to use it. He's never even rued not knowing it! I have no doubt that Cohen is telling the truth here, but that only proves that Cohen has led an empty, intellectually impoverished, life. It says nothing about what students should learn in high school.

He then presents a standard caricature of what algebra is. He has apparently overlooked the possibility that sometimes you use an unrealistic, contrived situation to illustrate a more general principle.

But the idea that calculators and computers can do “most of math” is really just too much. Calculators and computers do not do math at all. They do computation. It's clear at this point that Cohen thinks algebra, and mathematics generally, is just about manipulating symbols according to arbitrary rules. That's like saying that carpentry is about hammering nails and sawing wood. Mathematics is about the reasoning process you go through in taking the information you have and inferring the things that you need. Algebra and computations are just tools you use in the course of implementing such reasoning.

Incredibly, Cohen then tells us that computers can't reason - not even a little bit. Well, duh! That's the whole point!

And then, to cap off this masterpiece of inanity, Cohen offers up his examples of genuinely useful subjects: History and English. Oh, for heaven's sake. Thirty seconds with a good internet connection is enough to learn any bit of history you might be interested in. And English? Has anyone other than an English professor ever needed to read Dickens or Shakespeare?

Of course, the problem here is that Cohen has a pathetically empty idea of what education is all about. Education isn't job training. Education isn't about teaching you how to use an ATM, or balance a checkbook, or any of those other things you need to know in your day-to-day life. You don't need school to teach you those things.

The reason you learn about history, or read great literature, or study math and science when you're in school is precisely because you won't do these things in your day-to-day life. You don't read Dickens because you think it is going to get you a job someday. You read Dickens because the man could write, and your life will be just a little richer for being shown what the English language can be made to do. You read Dickens because by doing so you realize that the things people worried about a hundred years ago are mostly the same things they worry about today. Likewise for any of the other things you learn in school.

Cohen seems to have this idea that education is about learning a bunch of facts. And if those facts don't materially help you in the course of your professional life, then it was a waste of time to learn them in the first place. When teenagers display attitudes like that we chalk it up to immaturity and the lack of a long-term perspective. Columnists for major newspapers don't have that excuse.

Cohen continues:

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.

And this paragraph isn't even coherent. Writing is the highest from of reasoning? What? The proof of this is that some old classmate of his aced algebra but couldn't find the Sahara Desert on a map? Does this guy understand what reasoning is? Reasoning is what you do when you are told that a hen and a half can lay an egg in a half in day and a half and want to determine how long it will take seven hens to lay seven eggs. Finding the Sahara Desert on a map is just a matter of being in possession of certain facts.

Cohen continues:

Look, Gabriela, I am not anti-algebra. It has its uses, I suppose, and I think it should be available for people who want to take it. Maybe students should even be compelled to take it, but it should not be a requirement for graduation. There are those of you, and Gabriela you are one, who know what it is like to stare at an algebra problem until you have eyeballed a hole in the page and not understand a thing you're seeing. There are those of us who know the sweat, the panic, the trembling, cold fear that comes from the teacher casting an eye in your direction and calling you to the blackboard. It is like being summoned to your own execution.

Why not take that attitude with every subject? Maybe we should just make every subject available to students who want to study it, but then leave it up to the kids to decide what they need to know?

As for the uses of algebra, do I really need to point out that every bit of technology Cohen uses every day was invented by scientists, that virtually every major branch of science has mathematics at its core, and that no matter what branch of mathematics you are using it is certain that algebraic manipulations reside at its foundation? Of all the subjects he could have labelled as useless, Cohen could not possibly have made a worse choice.

As for the rest of this paragraph, I'm getting weepy. When you're not understanding what you are reading in your textbook, the solution is to work harder and ask questions. Everyone has subjects they're not good at. But we don't whine and blame the subject. As for being called to the blackboard, I don't know many algebra teachers who actually do that. And I know even fewer who, when the student says he doesn't know how to do the problem, ridicules the student for his ignorance.

Cohen closes with:

Almost 20 years ago, I wrote a similar column about algebra. Math teachers struck back with a vengeance. They made so many claims for algebra's intrinsic worth that I felt, as I once had in class, like a dummy. Once again, I just didn't get it. Still, in the two decades since, I have lived a pretty full life and never, ever used -- or wanted to use -- algebra. I was lucky, though. I had graduated from high school and gone on to college. It's different for you, Gabriela. Algebra ruined many a day for me. Now it could ruin your life.

Sorry, but I don't think it's unreasonable for a high school diploma to certify that you know a tiny amount of higher mathematics.

As I said, I expect this sort of stupidity and short-sightedness from the creationists and the right-wingers. But Cohen is supposed to be one of the Post's liberal columnists. Yet here he is promoting the very silliness he's supposed to be fighting against. Ugh.


At 6:12 PM, Anonymous Jodi said...

Perhaps Cohen could have made better use of his pulpit by demanding that schools offer students the tutoring and assistance they need to pass subjects they find difficult.

At 7:25 PM, Blogger RandomDNA said...

The question is which century the students want to stop their education at, 12th, 17th, or the 21st. If they want to be of this century, nothing less than differential equation will do.

At 7:35 PM, Anonymous John M said...


You are almost as far off in your post as Cohen is in his column.

Without a decent ability in English, one can't write or express one's self. Or even know where to start in picking books to read.

As to history, a student needs a good understanding of history in order to know where to search and what for on the web.

If all the history one knows comes from rabid right propaganda; then one probably won't even suspect how wrong the propaganda is.

At 7:36 PM, Anonymous Kevin from NYC said...

Here's someone who's given this issue a lot of thought...and who writes about education issues on a regular basis.

He has a sharp take on the whole standards debate

At 7:42 PM, Anonymous Kevin from NYC said...

John M, if you're refering to this....

"Has anyone other than an English professor ever needed to read Dickens or Shakespeare?"

you don't know Jason very well.

At 10:12 PM, Blogger MissPrism said...

Beautifully put.
I love the party conversation idea so much, I'll have difficulty refraining from using it at the next available opportunity.

At 10:56 PM, Blogger RPM said...

It's amazing how much relativity (not like Einstein, but like relative to the people around you) plays into it. I consider myself incompetent at math, and I can get by with rudimentary statistics and basic calculus. I can't do matrix math, diff eq, or any good statistics and I think of myself as a math idiot. Algebra should hardly be viewed as a lower limit of the math knowledge required in any field. Calculus should be the absolute lower level of math knowledge.

At 11:01 PM, Anonymous ArtK said...

Well said. I was going to blog on this one when PZ Myers brought it up, but the op-ed so disgusted me that I wasn't able to face it.

At 11:48 PM, Anonymous Rob said...

I hope you sent an abriged version of this to the Post.

At 1:00 AM, Blogger Evan said...

Math being treated as so special that only gifted people can learn to do it is a "get out of jail free card" for math educators from elementary school up to college.

That's all right Gabriella; only savants and nerds can do math.

At 9:45 AM, Anonymous wolfwalker said...

What's sad about Gabriella's case is that algebra is not that hard to get, when it's taught right.

What's really ironically funny about Cohen's column is that he almost certainly does use algebra every day. The ideas behind basic algebra are so essential that most people use them a hundred times a day without realizing it. If Cohen has ever looked at his gas gauge and figured out whether he had enough gas left to finish his current trip without filling up, he's used algebra. If he's ever looked at a store display and figured out which size of a product is cheaper in the long run, he's used algebra.

At 11:54 AM, Blogger Jason said...

john m-

If you think I disagree with anything you said then you have missed my point.

I was not saying that students should not learn English and history. I was merely pointing out that most of the things you learn in high school are hard to justify on the basis of their utility to your day-to-day life.


Excellent point. I hadn't thought of that!

At 12:04 PM, Blogger Martin Wagner said...

I was never any good at math either (though, unlike the stupid columnist, even I can do percentages), but I don't brag about it. I regret it, because if I had a better understanding of math I'd have a better understanding of physics and other things. I don't need this knowledge for my career, but it is knowledge I'd like to have, just to be more knowledgable.

At 12:45 PM, Anonymous SkookumPlanet said...


There's something missing here. I skimmed through the LATimes series to try to find it, but couldn't. A PBS weekly TV program, California Connected, did a half hour from the Times material about a month ago. They mentioned a startling statistic, implying this was the key. It was the percentage of LA students entering high school with fourth-grade level math skills! I've forgotten it and I'm not going to guess, but I almost fell off the sofa. The point was made that there's no way you can get these kids up to grade-level math skills while getting them through the high school math courses too.

Another algebra in the California schools story. This is from the San Francisco Chronicle, oh, probably in the last couple months. I think it was on the op-ed page. It was the tale of what happened when a group of San Jose teachers formed a charter school designed as a college prep school for low income, English-as-second-language, immigrant kids, primarily Hispanics.

They had been teaching these kids and were convinced the kids could do the work if they were assumed to be incapable, given tools, tutoring, and encouragement, and really pushed to work. It basically worked and it's an inspiring story. But the first couple years were a train wreck. After the first year they had to junk the algebra book. None of the kids could read it. This was a big surprise to the faculty.

As for Cohen's column . . . I'm a writer. I have two writing degrees. I've taught writing to university students. I've worked as a journalist. "Writing is the highest form of reasoning." Well, all I can say is his column proves either he isn't a writer then, or that writing clearly can be some of the lowest form of reasoning. Not real bright, is he?

At 12:50 PM, Anonymous SkookumPlanet said...

Whoops! ...that's "assumed to be capable"

At 3:06 PM, Anonymous dogscratcher said...

Cohen says:
"They made so many claims for algebra's intrinsic worth that I felt, as I once had in class, like a dummy. Once again, I just didn't get it. Still, in the two decades since, I have lived a pretty full life and never, ever used -- or wanted to use -- algebra."

Somewhat off topic, but why is it that whenever someone says something like, "You're trying to make me feel stupid..." they always reject the most parsimonious answer: that they are in fact, stupid?

"Writing is the highest form of reasoning. This is a fact."

Is he sure it isn't just a "theory?"

Wolfwalker makes the best points though: it doesn't seem possible for someone to live their life without in fact doing algebra on an almost daily basis, even if it is just to bake an off-sized batch of brownies.

At 11:06 PM, Anonymous Anonymous said...

The WP post article is of course appalling. I would like to make the following point: I don't consider myself naturally brilliant at maths but have achieved reasonable proficiency as my profession requires (research chemist). I am however always surprised at how bad most people, including other chemists, are at it. Maybe we're just not teaching it right.

At 12:10 PM, Anonymous Ed Hubbard said...

In his original post Jason said:

"The fact is that things like math phobia, or the idea that people's brains are wired differently, or that some people just can't do math, are total bilge. Hostility towards mathematics has nothing to do with any of those things."

As an active researcher in cognitive neuroscience, and in particular in the cognitive neuroscience of number processing, I have to take issue with this statement. In fact, there is a growing body of data showing that, in fact, some people are fundamentally impaired at doing even the simplest of mathematical tasks, such as saying whether 5 is larger or smaller than 4. This impairment gets nowhere near the level of algebra, calculus, or other deficits. It should be noted that these deficits occur in people who have otherwise intact mental abilities. For example, in Turner Syndrome (TS), Fragile X, and low birth weight babies, there are clear impairments in even the most fundamental numerical abilities. This is also true, to a variable degree, in children born with Fetal Alcohol Syndrome (FAS). In low-income areas like inner city LA, FAS might be a problem. To begin to learn about what has been done from a brain based perspective on these questions, go to our laboratory website.

We are now learning that there is an evolutionarily conserved ability to differentiate between different numbers of items, present in non-human animals - it has been demonstrated in mice, house cats, lions, and every species of primate tested (the evolutionary advantage of knowing when one is outnumbered is almost so obvious as to not merit pointing out). Similarly, this ability seems present in limited abilities in new-born infants. Research using functional magnetic resonance imaging (fMRI) and single-unit recordings in macaque monkeys shows that the circuitry for these abilities lies in a region of the brain called the intraparietal sulcus (IPS. Children with quantifiable mathematical difficulties, like children with TS or FAS can be shown to have (on anatomical MRI scans) to have abnormalities in the region of the IPS.

However, this should not be taken to mean that I disagree entirely with Jason's point. A great deal of math phobia does have to do with having a single bad experience, which gets compounded as math education builds on previous experiences in a way that, for example, reading Shakespeare does not depend on reading Dickens. See the wonderful discussion of this point in Brian Butterworth's book "The Mathematical Brain."

The reason I bring this up is to point out that dyscalculia (the technical term for mathematical disabilities, in parallel with dyslexia) is not always simply due to poor teaching, environment or motivation on the part of the student. Rather, there are identifiable brain based differences that make this a legitimate learning disability in much the same way that dyslexia or ADD is.

I believe that, hand in hand with higher standards of mathematics education (which I applaud) we need refined measures of testing for objective definition and quantification of real difficulties in mathematical abilities. In situations where such a specific deficit can be identified, children who have learning difficulties should be able to graduate from high-school without mastering a particular area. However, if we insist on treating mathematical abilities as a sort of “weakness of the intellectual will” for everyone, we will surely be guilty of holding some people to a standard that is precluded to them because of differences in the way their brains are wired.

Was this Gabriela’s problem? I do not know. In fact, it is probable that nobody knows. And this to me is the fundamental problem. Testing for dyscalculia, unlike testing for dyslexia, is not yet standard in schools. But, with a growing understanding of the way that genetic syndromes and even alcohol ingestion during pregnancy can impact on numerical abilities, perhaps we can get past thinking that all people who have difficulty in mathematics are simply stupid, lazy or suffered bad teaching.

Best wishes,

Edward M. Hubbard, PhD
NUMBRA Post-Doctoral Fellow
INSERM Unité 562 - Neuroimagerie Cognitive
Service Hospitalier Frédéric Joliot CEA
4 pl. du Général Leclerc
Orsay, F91401 France

At 12:54 PM, Blogger Jason said...

ed hubbard-

Thank you for the information. Perhaps I should have phrased my statement more carefully. While I am sure that there are, indeed, cases where people are otherwise normal in intelligence but are incapable of learning mathematics, I don't think that's especially common. I think it's far more often that people who are failing math in high school are doing so either because of their own lack of hard work, or perhaps because of poor teaching.

At 4:28 PM, Anonymous SkookumPlanet said...

Re: Ed Hubbard's post and Jason's response.

The actual issue on the ground, that Cohen ignored, is that many kids were being forced into algebra, basically, before they were minimally competent with fractions. This is no different than kids showing up in early high school being expected to read, think about, and discuss adult novels when they read at an early elementary grade level. Imagine both problems in the same kid. Apparently there is a great deal of this across the nation.

I'm 100% behind the trend toward tougher standards and higher levels of classwork. But if you're going to do what California did, the only responsible way to do it is to first assess the actual situation, and if necessary phase in the change in earlier grades so by the time the algebra requirement kicks in these kids have the basic skills to do it. My understanding is there wasn't even a pro forma attempt to do so.

Jason's point that the Cohenesque hostility is BS is right on, but such ongoing stupidity by legislators, administrators, and education thinkers serve to drive youth, and so our future, even further away from education and knowledge and competence. The adults responsible for this boondoggle are the ones with the cognition problems. I include Cohen on that list.

At 12:29 AM, Blogger Alane said...

I also have to take issue with the statement that no one is "wired differently" wrt to math. I know that in the sciences, I have observed a definitely split between "good at statistics", "good at the rest of math", and not good at math, and while the last may be (mostly) due to poor teaching, I do think that people who are good at statistics (as I am) think differently from people who are good at math. There's also a lot of variance among math people in their strengths.

I am also, I believe, dyscalculic (is that the right adjective form). Untested, but given that I consistently switch numbers in my head (but not letters, so algebra doesn't give me nearly as many problems as arithmetic). Dyscalculia is a documented disability.

Do you not believe that reading and language skills come more or less easily to different people?

Better math education would certainly be a help, and I agree with the rest of the post.

At 2:05 PM, Blogger mal said...

that column is one of the most FOOLISH pieces I have ever read. That clown is rationalizing his own intellectual short falls and foisting it off something else.

I struggled thru math, all the way to 3 semesters of calc and stat. It was never easy, but I USE THE MATERIAL regularly.

One of my daughters has a BA in Math and the other is a senior in an engineering program. Both obviously have math, but also english, history, social psych, dancing etc.

There is no such thing as "useless" knowledge

At 9:57 AM, Blogger goss said...

One of the most important issues overlooked by both the writer of the article and the blogger (or simply not mentioned: Education's purpose is not job training. In a democratic society (and even a democratic republic) an educated populace is key to effective self-governance.

If one is simply looking for job training, try a vo-tech school or drop out and apprentice with a tradesman. Then, when you graduate, despite all pleas to the contrary, please don't vote!

At 10:29 AM, Blogger KishCom said...

Great post!
I failed grade 10 math, grade 11 math and grade 12 math. I hated the homework in highschool - that was it. I took a few years off and through physics and some computer programming, I've learned the beauty of math (I'm studying to be a physicist now).
IMHO the problem in school is they teach math like you stated: "just about manipulating symbols according to arbitrary rules". They don't show the wonderful and beautiful things math does.

At 11:14 AM, Anonymous Anonymous said...

Most people can't find a use for Algebra because nobody told them how to use it. How many times does the cable company offer 40% off to new subscribers for the first 3 months. That's only $30 per month! Okay, but how much will it be after the 3 month introductory period?

I use algebra, geometry (sheesh, how do people build things without geometry), and statistics. I don't use Calculus because I don't know how to apply it to anything. I understand the concepts and what limits and integrals really are, but I have no idea how to use it. I asked all FOUR of my Calc teachers that very questions and not one of them gave me an answer.

Sheesh, the writer can't handle percentages? He a maroon.

At 12:37 PM, Anonymous Pope Zach 64 said...

Wish I had read this sooner.

This guy sounds just like a creationist - not only is he ignorant of that which he despises, he's proud of it!

So algebra's useless, huh? I guess people really don't need to know how much money they're wasting every year in credit card interest, or how much money to set aside every month to pay for that gizmo they want to buy (hell, who needs to save - just put it on the credit card!), or how to scale up a recipe for two people into a recipe for four people.

Yup. Algebra has NO utility in everyday life. Just keep on running up those credit cards.

At 12:47 PM, Blogger Tarun said...

The fact is that things like math phobia, or the idea that people's brains are wired differently, or that some people just can't do math, are total bilge. Hostility towards mathematics has nothing to do with any of those things. In reality it is just standard anti-intellectualism.

It seems like you are insecure, and make up for that by hiding behind your 'intellectual capability'. Just like people who are not accepted and loved start to take pride in the romanticized ideas of being a renegade and part of the whole 'punk' image.

I do not doubt that you have mathematical ability, but if you are intelligent then you will realise that some people are not good at math and it is not always their fault, but sometimes has to do with the structure of their brains. Do you think Shakespeare could do advanced calculus, or Einstein could write beautiful poetry if they wanted to?

Everything is possible in theory. It is convienient to blame failure in mathematics on a 'lack of discipline'. Yes, math is a set of rules that govern numbers and with sufficient effort, it can be mastered - but the effort you need to put in is much less than some others need to put in, and that is why it is all right for some people to be bad at math.

People like you believe that scientists are the smartest people on earth. People like you believe IQ is a measure of intelligence. Tell that to an autistic person.

At 1:24 PM, Anonymous Tarun said...

I'm sorry if I was rude in the above comment - its just that I have bad mental associations with the idea that mathematics and science is the height of higher thinking and intellectualism. 'Intellectuals', of all the minority groups should realise that 'superioroty' does not exist. It is a foolish human tendency to try and categorise things and create hierarchies. For example - we think that animals are inferior to us because they are less biologically developed (or rather because they evolved in a different direction; or because they can perform less functions we deem impressive).

I do not believe that proficiency at science is always indicative of a sharp-intellect combined with a strong, disciplined mind.

At 1:27 PM, Anonymous Anonymous said...

I thought this blog post was stupid enough as it is, ignoring as it does everything we know from psychometrics, but then I found this comment:

The question is which century the students want to stop their education at, 12th, 17th, or the 21st. If they want to be of this century, nothing less than differential equation will do.

This of course is utter bilge. Differential equations have literally nothing to do with modern life, unless you're an engineer. And to think that everyone should be an engineer (or a mathematician) is the most self-centered, narcissistic thing I have ever seen.

For the record, I've studied algebra, set theory, ring theory, and mathematical logic. And I'm pretty good at those. But I'm very bad at continuous mathematics (calculus) and you can forget differential equations entirely.

The fact is, there are different skills involved in learning different things. It's been my experience that learning calculus or differential equations involves memorizing a lot of arbitrary crap. I'm not any good at that. I'm good at creating abstractions, I'm good at synthesis and analysis. But I'm not good at rote memorization.

Anyone that's bothered to so much as look at the definition of intelligence knows that it's a multi-dimensional property. It's perfectly possible to score extremely high in some of the dimensions of intelligence (such as pattern matching or logico-mathematical) and score miserably low in the other dimensions (such as short term memorization of arbitrary symbols). Depending on how you score, it might be impossible for you to assimilate certain subject matters.

And even if you can assimilate a subject matter, if your score pattern differs markedly from "average", it will be utterly impossible for you to learn it from a regular school. That's because schools assume the student population is homogeneous and that students have a single learning style. They don't.

And in any case, why exactly is it important to you that everyone know math? Contrary to propaganda, it has nothing to do with critical thinking skills. Our of all academia, the highest proportion of religious people are found in mathematics. The conclusion is inevitable, mathematicians are cretins, right?

The fact of the matter is that math simply doesn't matter for most things. Politics and economics do. Unfortunately, critical thinking about politics and economics is not something the elites will ever encourage teaching.

And just to deflate your massively self-important ego, how much dance do you know? And if you don't know any, is it because you're an idiot?

At 4:18 PM, Blogger ThomasC said...

You really hit the nail on the head here. Math is absolutely essential and an underpinning that unfortunately is beginning to rust in this great nation. I hate hearing teenagers say, "why learn algebra/math - I won't ever use it." And then I propose to them a simple question: If the local tax rate is 7.7%, how much money do you need to bring with to buy an $80 pair of jeans? Sadly, most answer $80.

At 8:30 PM, Anonymous Fred said...

Hey anonymous, no one's going to rip you off on a car loan because you don't know dance. But if you don't know any math, well, then you're at the mercy of others.

Anyway, I don't recall that the original post or the comments have said that math is the most important of all knowledge, or that everyone who doesn't know math is to be spit upon. The point is that while it might not be the end-all-be-all to human existance, it's certainly not valueless, which is what the author in the article in question has said.

At 9:38 PM, Anonymous Ted Dziuba said...

People like Gabriela and Cohen are collectively the fundamental reason that the lottery exists.

At 8:42 AM, Anonymous Pope Zach 64 said...

Anon. & Tarun -

Who the hell said anything about math or science being the pinnacle of human intellectual achievement? Apparently reading comprehension is not your strong suit either.

Of course everyone has different aptitudes and interests. But there is a basic set of knowlegde everyone needs to function at a basic level. Sure, you may not ever have to solve an integral - but who could argue with the notion that everyone should be able to figure (or at least estimate) the tax on their purchases, or how much interest they're paying on their credit card debt?

Cohen also revels in his ignorance. Since when is ignorance something to celebrate? I admit, I don't know diddly squat about dance, nor can I dance my way out of a paper bag. I probably won't ever make the effort to learn dance - but my ignorance of dance is not something I celebrate. And at least I appreciate, to a degree anyway, how dance enriches culture.

Likwise, I recognize that not everyone has the ability or desire to be a professional mathematician or scientist, but a rudimentary understanding is necessary to make informed decisions when voting or making purchases. How many dollars per year are wasted on those ridiculous magnetic bracelets, or herbal penis "ehnancement" potions?

Ignorance of science and math has consequences: financial, political, environmnental. Ignorance of science and math is not something to be celebrated - let alone encouraged.

Perhaps, down the road, Cohen would like to lend Gabriela the money she will need to pay off her credit cards?

At 5:02 PM, Anonymous Jer said...

The comment about dyscalculia notwithstanding, the reason that so many people don't digg math, is just plain mental sloth. The various Cohen-like types get into their petty, stupid and deeply insecure squabling because they actually try to hide their intellectual inferiority. Reasoning, whose master Cohen thinks he is, by virtue of being able to use a keyboard, shares a linguistic root with the latin word for fraction, where an understanding of percentages is based. And if one know some history, they get that this is for a good reason(;)). Fragmenting knowledge and partial, prejudiced understanding of the world outside our minds, is the ugly kind of laziness, the one that has made Cohen such a poor and sloppy stylist even in his own chosen craft. You can put any monkey in front of a typewriter and wait a thousand years, but interesting writing (or good, creative reasoning) at a blink of an eye, from people like Cohen... I don't really think so.

At 9:56 PM, Anonymous Richard said...

Fred said:

Anyway, I don't recall that the original post or the comments have said that math is the most important of all knowledge, or that everyone who doesn't know math is to be spit upon.


But actually, people here are acting as if anyone who doesn't know (or value) math is to be spit upon. And that starts with the original article that replies to Cohen.

People here are acting in a completely hysterical manner about the putative value of algebra. Wa wa people don't know algebra, oh the sky is falling!

Let's be rational for a minute here. What's actually taught in high school math? Polynomials, long division, quadratics, geometry, and so on and so forth. Who the hell needs any of that?

The facts of the matter are that math simply isn't that important. Almost everything that's taught in high school math classes has no relevance whatsoever to the real world.

And even the little that does have relevance to the real world isn't that important. Oh no, people get scammed by lotteries. Well so f-ing what?

Compare that with 100,000 Iraqis dying because Americans don't know history.

Compare that with widespread poverty and disease because they don't know basic social science.

Compare that with a corrupt regime that threatens the existence of the human species because Americans don't know political science.

People who think that math is the be all and end all of knowledge (or even on par with things that are Really Important) have their heads stuck so far up their ....


Thank you all, I am incoherent with rage at your idiocy.

At 1:04 AM, Anonymous Fred said...

Richard said:
Thank you all, I am incoherent

You're right!

At 8:27 AM, Anonymous Pope Zach 64 said...

Richard -

I see you can use some help with your reading comprehension as well.

While I agree with your points about the appalling ignorance of history and social science, I'm sure many others would argue with you. After all, when was the last time you needed to know the details about the Battle of Hastings?

The point is, igorance about anything is not something to be celebrated or encouraged, as Cohen has done.

One of the goals of learning geometry and algebra is try to teach the skill of analytical thinking. What information is needed to solve the problem, what information is superfluous, etc etc.

I'm sure you'd agree that analytical thinking is useful to other areas in life - such as evaluating the self-serving bluster of politicians.

You accuse Jason and many of the posters on this thread of some sort of mathematical "elitism", yet your "who the hell needs any of that" attitude is an elitism of its own. You're saying, that if you can't see its relevence to YOU, then it's crap. Who's the "elitist" here?

Education should teach more than just job training, or everyday skills. If that were true, most of us wouldn't get past using the cel phone and the remote.

At 11:33 AM, Anonymous Richard said...

You fool. History is not a series of dates or battles.

History is the fact that the USA led a deliberate campaign of genocide against the natives, some of them savage and others civilized, on the continent.

History is the fact that the early Americans ethnically cleansed all British influence, murdering thousands and starving tens of thousands.

History is the fact the USA is a barbarian nation that ruthlessly butchered and annexed everyone standing in their way.

History is the fact that the US President at the time wanted to annex the whole of Mexico, not just a great big chunk of it.

History is not knowledge, history is UNDERSTANDING. Understanding which has a great deal of relevance to the modern world.

If one *understands* that the USA has a long history of massacre and conquest, not "liberation" and "democracy" then this has direct relevance to actions, *important actions* in the real world.

There is no comparable thing in mathematics. Geometry? Give me a break, geometry is the least useful of all fields in mathematics. Hell, it's no longer useful even IN mathematics.

You want to teach critical or analytical thinking skills? Fine, teach philosophy. There are plenty of philosophy for kids textbooks out there. Teach logic, teach mathematical logic. But nooo, you want geometry. Yeah, right.

The purpose of an education isn't to have a use in the job market. Nor as you imply, is there no purpose to education at all. The purpose of education is to make you a better CITIZEN.

And mathematics *does not do that*.

The purpose of mathematics classes in high school is *precisely* to prepare you for the job market. The mathematics job market. Quadratics and polynomials are precursors to ring theory, as should be obvious to anyone. Calculus is the precursor to advanced calculus which is the precursor to differential equations.

If an education is meant for more than just preparing you for the job market then we must STOP teaching mathematics and concentrate on things that will actually make you a better citizen. That would be politics, sociology, economics, history and philosophy.

And don't talk about ignorance because it's obvious that you are quite ignorant of the economics and dynamics of knowledge.

Like for example that all knowledge carries with it an opportunity cost. Everything you learn uses up the opportunity of learning something else instead.

And more besides since knowledge carries with it a crushing cost in creativity (Sources of Innovation by Eric von Hippel, page 102). The only thing that prevents mathematics from exerting any inhibiting effect on creativity is the fact that mathematics is so truly useless.

At 1:10 PM, Anonymous Fred said...

If I thought for one second that you were serious I don't think I could stop from laughing. Thanks for that post.

At 1:13 PM, Anonymous Pope Zach 64 said...

Richard -

So learning is a zero-sum affair, eh? The more math you learn, the less room in your brain for history?

Sounds like you're the one who's ignorant of dynamics of knowledge.

OK, quiz kid - what's your formula for the optimum mix of subject material so that one's brain isn't overloaded in any one area?

And who said anything about there being no purpose to education at all? Once again your reading comprehension is lacking.

I believe I advocated a good grounding in all areas of study. You advocate more emphasis on history - hey, I agree! I think the historical knowledge of the average guy is appallingly poor.

But in history classes, I'm afraid they'll teach other things besides the misdeeds of the US - things like the Battle of Hastings, Waterloo, Stalingrad - or do you advocate expunging the history curriculum of all non-US related material?

Once again, anything not of direct use or interest to you is considered useless. Who's the fool?

At 2:12 PM, Anonymous SkookumPlanet said...


I understand your frustration with the posters here, with the attitude that leaks out of a few of them about those unable to master advance mathematics.

But I'm having trouble understanding the immense amount of energy behind your attitude.

For example, your list of historical facts are of a particular type, let's call it America's unpleasant side, that never get taught at primary or secondary level. Perhaps they should be. Why bring these up, though, to make a case that math is over hyped? Also, "History is not knowledge, history is UNDERSTANDING." What do you think of virtually all the high school history classes not teaching this type of material, in terms of "understanding"?

Do you not think that any field of academics, when understood deeply enough, can be a tool for understanding what's around us and how we fit in? Or are there only particular fields that aren't useless?

"And more besides since knowledge carries with it a crushing cost in creativity"

Humans are complex and subtle creatures with enormous variation. I have two degrees in fiction writing. I currently write poetry about, and also paint abstract versions of, landscapes. I'm pretty good at both. I also love science and one of my major hobbies is voraciously reading science. I've been doing this since the third grade.

So, in my own life, I have found the exact opposite to be true -- knowledge increases my creativity. It gives my subconscious more material to work with.

In preparing my schedule for my senior year in high school I was forced to make a choice. I had two open periods, and could take any one of three advanced science classes with a pre-calculus class, or I could take advanced liberal arts [poly sci [kids called it "isms"] and modern lit]. What made my decision was staring down the line at calculus and realizing an education in science meant math. [I did end up taking statistics 3 years later]

I had a recent roommate who let it "slip" that all of this knowledge about science, and the world, in my mind [there's a lot] was "useless knowledge". I told her there's no such thing. There isn't.

First, memory is associational. At the obvious level we perceive daily, but also at a neural level -- "a memory" is really composed of separate elements that are stored in different parts of the brain. Researchers have just recently reported seeing this in lab animals, as it happens. So, everything is connected to everything else. I personally think this is a source for metaphor. The red of an apple is the same red of the fire truck, actually the same few neurons! The things I remember about science, say, I remember because they're associated with so much else. My subconscious makes the decisions what to prune away.

Second, classifying some knowledge as useless means one would have a precise idea what the rest of his or her life would be. This isn't philosophical claptrap; I mean it seriously and literally.

In my late twenties I found myself joyously working the decks of commercial fishing boats in wild, western Alaska, using skills I'd mastered being unjoyously stuck spending Saturday's with my dad doing yard work. The palpable, often overwhelming, spiritual experience of seeing and touching nature's abundance in such utter remoteness now infuses my poetry, which is celebratory in nature. How could I have known?

This is where Joseph Campbell's admonition to "follow your bliss" comes from. How can we know what knowledge we won't need. [I feel sorry for the people who actually manage to construct an unsurprising life.] We can't know, so the idea is to acquire the knowledge and experiences that turn us on the most. Sometimes that puts you on a course to do intermediate things, like math, that don't turn you on. Many kids like Gabriela [but not her, actually] sit in school and with their little sliver of experience of life conclude nothing being offered them is relevant. Hey, they're adolescents. But much of what they are rejecting is there to give them options. They haven't yet figured out how valuable options are.

"... is the fact that mathematics is so truly useless. I was going to suggest that you meant useless to you. But that's not accurate either. Without mathematics you wouldn't be commenting on this blog. There would not be blogs, nor the internet, nor computers, nor modern medicine... Perhaps its invisible to you. But math isn't really useless to you either.

At 12:12 PM, Anonymous Richard said...

Pope Zach,

you continually whine about my reading comprehension but that just makes you a hypocrite. Or it would make you a hypocrite if you were self-aware enough that you are projecting your own flaws on me.

Learning knowledge uses up time. Time is finite. This is what 'opportunity cost' means. There is far, far more to learn than there is time in a single lifetime to learn it. Does that sound enough like a zero sum game to you? It should.

Further, the minimal politico-economic knowledge necessary to become a good citizen takes about 10 years to learn. And you would waste this time on mathematics.

You ask who's the fool. You are the fool. Knowledge that does not have a use to the individual or to society is by definition useless. And clearly, useless knowledge, such as what a cubit is, should not have valuable time and attention lavished on it.

Now, when most *Americans* talk about "use" they mean the job market and money, because the only thing Americans care about is money. I am not an American so I don't care about money above all things. So when I (or any non-American) says 'use' they mean either a practical use, a social use, a citizen use, or just a psychological use of enriching someone's life.

Mathematics does not have any use except for becoming a mathematician. Some mathematics has a use for engineers or statisticians. Some mathematics, like geometry, has no use whatsoever even to mathematicians.

The entire reason why mathematics is so popular in education curricula is because it is not politically controversial. And it is not politically controversial because it is useless. But it actually does have a use, a sinister use.

You see, mathematics is used in order to artificially rank students for the job market. The poor do badly in math classes and they get crappy jobs. The rich do well in math classes and they get good jobs.

The correct use for studying mathematics is to have fun. Exactly like art, ring theory is fun. The key difference is that nobody is compelled to study art, nor is anyone punished for doing badly at it. Nor do people claim that everybody should study every art, nor that talent at art is immaterial, nor that you are lazy, slothful or an idiot if you do not have any talent for art. Yet plenty of people, a truly impressive amount of fools on this blog, say precisely this in regards to mathematics.

The study of mathematics should be for those who have fun studying it. Mathematics does not deserve any cachet or status beyond any other art. There should certainly be no penalty for not studying it.

Mathematics is not like history, politics, or philosophy, the study of which are absolutely necessary for the good development of people and society.

At 12:50 PM, Anonymous Richard said...


I would respond in email because I doubt anyone else is interested in these issues you raise.

There's knowledge which is essential for being a good human being. This includes much of psychology, philosophy, and some politics.

Then there's that essential to being a good citizen. This includes all of politics, economics (real economics not economic theory), sociology, much of psychology, moral philosophy, political & economic & social history (not military history as is currently taught in the militaristic USA).

Then there's knowledge necessary to become a rational actor. These are the natural sciences like quantum physics, other physics, biology, and chemistry.

Then there's knowledge that just enriches your life. This is the arts, literature, and some mathematics. Not all mathematics, because numerical analysis doesn't enrich anyone's life.

Then there's the lowest possible grade of knowledge, which is that knowledge necessary to perform a job. This includes most of mathematics.

Then there is truly useless knowledge such as geometry.

It is simply false that all knowledge has value or that all knowledge is equally deserving of attention. And that's even before accounting for individual differences in talent and interest.

See also

Finally, the relation between creativity and knowledge is very complex.

There are specific circumstances under which knowledge inhibits creativity. Research was done and well documented, so I suggest you actually look at the research. Chapter 8 of Sources of Innovation by Eric von Hippel, it's freely available online.

The long answer is that knowledge inhibits the creation of alter-knowledge in its vicinity. If you know that paperclips are for holding sheafs of paper together, it's just become more difficult for you to use them as lockpicks. Knowledge boosts the creation of consonant knowledge only, not dissonant knowledge. Sometimes it helps, other times it kills.

Knowledge, literally, occupies space in your mind. And by 'mind' I am not talking about the brain, but literally the mind. Knowing a given concept makes it more difficult for a person to learn a different concept which would displace it. The human mind obeys quantum mechanical laws of superposition of concepts only with great difficulty.

At 1:25 PM, Anonymous Richard said...


I've described some of the dynamics between knowledge and creativity at the micro level. You should know that at the macro level, the same dynamic is replicated.

Expert knowledge in a field of endeavour crushes one's creativity in that field. There's a reason for the truism in physics that all revolutionary ideas are discovered by beginners who simply don't know better. And the reason is because it's true. The more you know in a field, the more you are expert in it, the less you can innovate in that field.

This is why many of the best scientists chose to start over in a completely different field after they have exhausted their creativity in their first field. They know too much, and can't innovate, so they switch to a field where they know nothing.

Most innovations are made by accident. People who don't know better, and don't appreciate their ignorance, make the ground-breaking discoveries. So the first thing they do is they destroy the blessed ignorance that enabled them to make their discoveries.

A creative person should choose a field to excel at and become an expert at every other field. In their chosen field, they should jealously guard their ignorance and endeavour to remain a perpetual amateur.

At 3:43 PM, Anonymous Fred said...

You have stated that knowing math (such as 1+1=2) is completely useless. I disagree with that. I can live my whole life without needing to know about American history, but I do need to know how to count money so I can buy food and pay rent.

I would like to ask you this: How do you get by on a day to day basis without knowing or using ANY math?

Do you think it's valueless to teach children how to figure things out like "If this television is priced at $100, but you have a coupon for 25% off, and you have $70, can you buy the television?"

At 1:38 AM, Anonymous Anonymous said...


Thanks for the considered response. I actually agree with much of what you say. Even that "There are specific circumstances under which knowledge inhibits creativity."

Certainly one must make choices about what knowledge to seek out. I didn't mean to say, "...that all knowledge has value or that all knowledge is equally deserving of attention." What I meant to say is our brains have evolved to forget what's not important and, given the proper environment [psychological, social, economic, etc], we can make use of that phenomenon. And, some of this remember/forget brain mechanism can be adapted to foster one's creativity. Indeed, it's possible this is where some important part of creativity is "located".

I am also aware of scientists switching fields in mid-life to encourage a new round of innovative thinking. I even did something similar by becoming a visual artist relatively late in life, albeit in my case it began inadvertently. Still I had to make the decision to pursue it. And visual art was something I was basically ignorant of.

Perhaps our differences derive from the specific fields of knowledge we've focused on in the course of our lives. For example, over the years I've seen several studies about when v

At 1:52 AM, Anonymous SkookumPlanet said...

It wasn't anonymous, it was me. Hit the publish button too soon and aborted too late. Continued...

...about when various professions do their best/most innovative/etc. work. Although it's been a decade or more since I've seen data like this, my memory is that physicists "peak" fairly early in life, mathematicians among the earliest [you'll draw conclusions no doubt], and among the latest to peak are novelists -- late thirties. I'm highlighting this simply to note there are differences.

"A creative person should choose a field to excel at and become an expert at every other field. In their chosen field, they should jealously guard their ignorance and endeavor to remain a perpetual amateur." The first sentence is what a writer does, especially a novelist. Non-fiction writers and journalists have to be a generalists to some degree, but fiction writers especially do. Depending on their larger thematic concerns, novelists may consume knowledge in lots of fields, not specific to a work, and generate their ideas from that accumulated knowledge. But, the second sentence applies differently in that the knowledge in their field is actually the technical means to put together a successful work. There is a corollary in the amount of intuitive craft that's employed, but there are few long-form fictionists who do not deconstructively read other fictionists voraciously to become more more knowledgeable of technique.

Creativity research talks about meta skills [forgot the terminology], and domain skills. Domain skill acquisition varies tremendously with the specifics of the domain. For example, the years of training necessary to pick up the body-mind memory for playing a difficult musical instrument before one can then proceed to master it. But in the arts, if I remember correctly, research shows an average of about ten years of domain skill training is necessary before professional level work begins. In the arts these skills tend to be physical so generally nothing of import gets produced until domain skills are mastered.

"Knowing a given concept makes it more difficult for a person to learn a different concept which would displace it." Yes, but, and perhaps this is more specific to something like the arts, one can literally unlearn concepts, or suppress them enough so other conceptual knowledge may impinge on or supersede it. I've seen people in the process of doing this. In fact, some of formal art training and often much of an artist's technical learning during a career is focused on discovering various means to see a paper clip as a lock pick.

Perhaps I'm making too fine a distinction here, but scientists who switch to another field aren't going where "they know nothing." Switching from sciences to arts, and vice versa, is closer to knowing nothing. Just that there is a great deal of common, acquired knowledge underpinning broad areas in which innovation is valuable.

It's also possible, in the problem area you've pointed out, the accumulation of knowledge might only be a part of the cause. After all, young physicists who innovate generally [I would think, I don't know] have a thorough introduction to their fields, albeit a relatively fresh one. Other factors that might be involved are a change in the nature of their work as they move through career steps, exposure [below], life stages, etc.

I wonder also if it's always necessarily the amount of knowledge that's the primary culprit over time. It could also be a sort of cognitive version of sensory adaptation [I think is the term] -- as when our brain moves recurrent, repetitive noises in our environment into the "background" and they stop consciously registering. In other words, an effect of exposure over time. There's a concept that comes out of Buddhism, Zen I believe, called "Beginners Mind" that attempts to inculcate mental techniques to counter such "exposure adaptation".

"Most innovations are made by accident. People who don't know better, and don't appreciate their ignorance, make the ground-breaking discoveries." I don't disagree with that, but I think it's an insufficient analysis. Again, this may simply be a difference in the ways our lives have brought us specifically to the subject, but one can have all the ignorant accidents one wants, but without, first, a proper context of knowledge for the "accident" and, two, some sort of "watcher of" or "alarm on" one's consciousness, accidents can't turn into discoveries. This may be so obvious to be assumed in many fields of endeavor, but in others it's a conscious, ongoing, often career-long, practice to cultivate these. Indeed, I use a number of accidental or barely controlled techniques in my painting specifically to provide material that's beyond my rather pathetic "knowledge" of art technique.

And if necessary, to the blog readers I provide apologies for this long diversion off-topic.

At 6:33 AM, Blogger Theo said...

I know I'm joining in very late - but I couldn't go past this statement: 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.

Obviously Cohen's experience is universally true. No way any one person could do some maths, and know about history… and be a great writer.

At 9:02 AM, Blogger Theo said...

I've since been moved to post on this mathematical moron.

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At 9:41 AM, Anonymous Anonymous said...

You know, I'm a high school student. I do very well in all my classes, yet I still find Algebra that one thing I can't get around. I've looked around on the Internet, asked for help from people who know the topic, and done everything I can to learn it better. I'm still just BARELY passing.

But I've learned something important. Much of the struggle kids face when dealing with algebra is the teachers themselves. They may be good-hearted, well-intentioned people, but they sure as heck don't know the first thing about teaching methods or how people learn. They can dance with differential equations, eat epsilon-delta proofs for breakfast, and generally do all kinds of things with numbers that are highly confusing to people who don't know the subject inside out.

If you are going to expect kids to learn algebra, at least make sure math teachers are just that: TEACHers. Kid-friendly mathemagicians.

And if you're saying we still have to learn the "proper mathematics vocabulary" because we'll need them in college, then how about the teachers (imagine this) teaching us what they mean instead of a two-minute dissertation about math terms, or none at all.

Just some thoughts for those of you who think that all teens who can't ace algebra classes are either too lazy to get good grades or troubled teens who would have dropped out anyway.

At 5:10 PM, Anonymous Anonymous said...

You guys all sound like learned men of Indostan who went to see the Elephant
(though all of you are blind), prating about an elephant not one of you has seen. How about we just teach everything equally in high school?

Or we could just teach mathematics and be done with it. That'd be great. Everyone in the world would be able to solve the mysteries of the universe..without any life skills.

Or, we could just quit teaching math. Great, no more antacids needed, everyone ceases to use logic, and scams are on the rise.

In all honesty, you guys are pitiful, despite all your higher education and superior intellects. None of you have the whole picture about any of this. Look up "The Blind Men and the Elephant" for more information about getting in touch with your inner selves.

At 5:19 PM, Anonymous Anonymous said...

And before you go telling me you didn't mean for high school to teach ONLY math or NO math, lemme tell you: it's called a hyperbole. A useful writing tool.

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