TCS Daily


Algebra and Its Enemies

By Kenneth Silber - May 8, 2006 12:00 AM

Early this year, Washington Post op-ed columnist Richard Cohen weighed in on a subject about which he, by his own admission, knew nothing. The subject was algebra, and Cohen's column took the form of advice to a young woman who had dropped out of high school after failing in that subject. Cohen advised the ex-student and the public at large that algebra's importance was overblown -- and that he, Cohen, "had never once used it and never once even rued" that he could not use it.

"Most of math" Cohen explained, "can now be done by a computer or a calculator," and moreover it is a "lie" that algebra teaches reasoning. "Writing is the highest form of reasoning," Cohen affirmed, stating that the most valuable class he himself had taken in high school was ... typing.

Cohen's dismissal of a central branch of mathematics got some negative attention from science bloggers. Biologist P.Z. Myers castigated Cohen for complacency and arrogance in advising a young woman to throw away career options and intellectual tools; Myers also noted that the people who design calculators, among many others, need to know algebra. Gary Stix of Scientific American pointed out that Cohen's advice was particularly inapt in light of growing international economic competition. In uncharacteristically heated language, Stix wrote: "No algebra=No calculus=No science=No technology=We're totally *&$#FRTDG!!!!!" (Disclosure: I freelance sometimes at Scientific American and occasionally nod at Stix in the halls.)

Subsequently, science writer Steven Johnson made his own argument that algebra is overrated. In a Time magazine essay, Johnson contrasted studying algebra with time spent online and playing video games, by way of arguing that the latter activities yield more benefits. "In the office of the future," he wrote, "which skill set will today's kids draw upon in their day-to-day tasks? Mastering interfaces, searching for information, maintaining virtual social networks and multitasking? Or doing algebra?" He added: "It's a good bet that 99% of kids will never use algebra again after they graduate from high school." Johnson went on to lament that the testing establishment devotes so much time to algebra and so little to digital skills.

Johnson may have a point that fears about children getting dumber through digital technology are overstated. But his argument neglects some pertinent facts: One is that the online skills he exalts tend not to require much classroom training, while algebra clearly is not something kids will pick up without taking courses and tests on the subject. Moreover, the very technologies in question depend heavily on algebra (and on more advanced math that requires algebra). With a nod to Stix, one might say "No algebra=No computers=No networks=No information age=We're totally *&$#FRTDG!!!!!"

Students (and pundits) who find algebra hard might consider how difficult such math must have been for the people who actually pioneered it. That story is told in a new book, Unknown Quantity: A Real and Imaginary History of Algebra, by John Derbyshire (Joseph Henry Press). Derbyshire, who wrote a previous book on math, Prime Obsession, and is a frequent contributor to National Review, gives an absorbing account of algebra over the millennia, from its rudimentary origins in ancient Mesopotamia and Egypt to its cutting-edge applications in 21st-century physics.

An interesting feature of this history is just how slow progress often was. Babylonians in the 2nd millennium BCE worked out algebraic word problems on cuneiform tablets, and the ancient Greeks handled similar problems with a geometrical approach, but it was only at the time of Diophantus, who lived in Alexandria in roughly the 3rd century CE, that anyone used letter symbols to keep track of unknowns in equations. The brutal death of the female mathematician-philosopher Hypatia in 415 at the hands of a religious mob marked the twilight of math in the declining Roman Empire.

Around 820, the Islamic scholar al-Khwarizmi wrote a book on algebra (the word comes from the Arabic al-jabr, or "completion," his term for adding the same amount to each side of an equation to put it into a standard form). However, al-Khwarizmi and his contemporaries worked on algebra through word problems and geometry. Diophantus' practice of employing letter symbols in equations had vanished into forgotten archives. It was not until the late 1500s, particularly with the work of French mathematician François Viète, that algebraic symbols were reinvented and started to be used in a systematic way.

Such tortuous history, as Derbyshire points out, suggests that symbolic algebra, with its high level of abstraction, does not exactly come naturally to people. He finds this a bit depressing but also inspiring. The remarkable thing is not that it took humanity so long to learn how to do this stuff, but that we can do it at all. No thanks to some pundits, though.

Kenneth Silber is a TCS contributing writer who focuses on science, technology and economics.

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100 Comments

logical carried to its conclusion - - -
Gosh, how stupid of us to teach our kids to walk, after all, we have advanced prams, cars and even motorcycles.

Like fire and jello
This is an argument that will never be satisfactorily resolved. Even in high school, the arts and letters crowd didn't speak with the engineering crowd. Science and techies have always been considered incomprehensible nerds by the arts crowd, while the liberal arts people have been considered airy and irrelevant by the folks with the slide rules in their pockets (well, high school was a long time ago). They speak different languages and use different areas of their brain.

It's obvious that this "two different worlds" theme is behind the ire that surfaces so violently here toward any "left wing Hollywood actor" who uses his/her fame to publicise some cause. How dare they! What do they know! They're only actors. They can't have valid opinions on anything.

Thus Omar Bashir, who merely kills tens of thousands of villagers, comes under none of the criticism that George Clooney does in objecting to the slaughter. He's an ACTOR for crissake!

I'll bet you can find lots of other areas where the gulf in thought modes is apparent.

Oh, and let's not forget the SEGWAY:) -NT
NT

what is FRTDG???? NT
nt

Right brains and left brains
When computers first came to real estate, around 1975, I was a sales agent-- but not a very good one. There wasn't a single natural born sales person who could figure out how to even turn the thing on. They were all afraid of it. I had to patiently tutor each one and get them to get over their fear. They acted like if they pushed "enter" the thing might blow up. Yet they were all superbly skilled at what they did.

Logic
Algebra and literature can both teach logic.
Algegra stretches the mind into on kind of abstraction and good literature will do the same. Especially if you study that literature in Latin or another foreign language.
But it is absurd to think algebra is never used. How do you calculate MPG, or your financial portfollio?
If you depend upon others for that, be prepared to be swindled.

How to think
Algebra should also teach students HOW to think.
Public schools have chosen for the past few decades to teach WHAT to think so it is no wonder algebra is considered useless.
Fortunately home schooled and charter schooled children are entering the world and should do very well in a world where people do not know how to think.

Hemispheres
What's your point?

Excellent point by Marjon
As one who learned math through The Calculus I in high school and entered college with no mathematics requirements at all (and a few credits) because of that, I have been saying for many years that the way mathematics are taught in public school is a heap of rubbish. The administrations (and the typically undertrained teachers) go out of their way to make it the most off-putting subject of them all. For the most part, conceptual questions are discouraged, and there is very little teaching of the history of the process of mathematical concepts' development. The students rotely memorize pre-digested second-hand news and are taught to regard it all as a fax from God.

Valid opinions, late to the party
"It's obvious that this "two different worlds" theme is behind the ire that surfaces so violently here toward any "left wing Hollywood actor" who uses his/her fame to publicise some cause. How dare they! What do they know! They're only actors. They can't have valid opinions on anything."

"I wish the celebs well. Those of us who wanted action on Darfur years ago will hope their advocacy produces more results than ours did. Clooney's concern for the people of the region appears to be genuine and serious. But unless he's also serious about backing the only forces in the world with the capability and will to act in Sudan, he's just another showboating pretty boy of no use to anyone."

http://www.theaustralian.news.com.au/story/0,20867,19056736-7583,00.html

Is this just another left/right brain thing or do actors and leftists celebs need publicity or should they have studied algebra and come this conclusion years ago. Logic and reason do not usually correlate with 'actors'.

Hey Joanie, good post
I would go further and say the dimwith who says he has never used Algebra simply doesn't realize what he is doing. I'll admit, I don't use it often, but I have, and do, use it at time in odd and simplistic ways.

I wasn't the greatest math whiz, but knowing the basics is very important. Thank God for my persistent math teacher, without him I never could have figured out the more advanced calculations to work out transistor parameters in my electronics classes.

Algebra
You guys just don't get it! Why should we learn Algebra?

Well... just about two weeks ago, one of my son's highschool teachers just got through lecturing the class why reality does not exist... that it is just a form of perception and is unique for each individual.

Therefore, why should we learn to think? This is giving me a headache. Anyone got an aspirin?

Exercise for the mind
As a member of the Speakers' Bureau where I work, I often give talks at career days at local elementary schools.

Part of my spiel involves "letting the kids in on a secret." I tell them that after taking algebra in school, 95% of them will never use it again. So why learn it?

OK, how many are on a sports team? Let's see a show of hands. OK, I bet your coach makes you do sit-ups and other exercises, right? Well, no team ever goes out on the field to see which one can do the most sit-ups in a minute. But the sit-ups build up muscles you use in the sport itself...

No Subject
If reality doesn't exist, then why not imagine you've taken an aspirin?
:-)

Teaching math badly
Maybe that's how the teachers learned it. Conceptual questions are discouraged because the teachers have no conceptual understanding of the subject, and they merely pass along their own rotely-memorized second-hand news which they regard as a fax from God.

"No algebra=No calculus=No science=No technology=We're totally *&$#FRTDG!!!!!"
As a Computational Chemist, Engineer, and parent - I use algebra on a daily basis.

If one wants to regress to a totally agrarian society, then by all means, eliminate algebra. Unfortunately that would mean that millions of people would starve, our standard of living would go down to the Sudanese level, and women would have to go back to using dried moss and grass during their menstrual periods - paper and plastic not being available.

However, the reverse is true as we continue to develop technology and improve our lives and quality of life. Unfortunately, any engineer or scientist will tell you that algebra, trigonometry, and calculus are an integral part of their work and cannot be performed in innovative and creative new ways by pocket calculators, PCs or mainframes - at least not yet. Innovation must come from the human brain which must use algebra, trigonometry, and calculus to make innovation (and associated products) real and of practical use.

So, I agree, "No algebra=No calculus=No science=No technology=We're totally *&$#FRTDG!!!!!"

Aspirin
Hey! Guess what? Eureka it works!!!!!!!!!!!!!!!

I can feel it now. That pain relief is surging through my body. I've been touched!

the claim that writting teaches logic and reason
can be disproven by reading Cohen's columns.

too late now
Could have copied all the files to your own area, and kept your local area under your password?

roy mischaracterizes ... again
Nobody has ever said that the actors etc., have no right to air their opinion.

What we have done is show how stupid it is for other people to waste any time listening to such people's opinions.

If you can't understand the difference, you probably believe we are all going to die because of rising CO2. (Probably because some actor told you we were going to die.)

algebra and thinking
I have a 16-year-old daughter struggling with algebra II right now. If her study habits were better, I suspect the struggles would be reduced. I began as an engineer and programmer of primitive computers (so primitive I'm sure no one here has ever seen one), then went on to medicine. Medicine and engineering are quite different. One relies on memory and the other on reasoning. However, memory is not enough unless correlations can be made and quickly. Whatever else algebra does, it teaches thinking, a talent not stimulated by instant messaging and computer games. If the latter two activities were helpful to reasoning, my daughter would have an A in algebra.

fuzzy math
There is a "fuzzy math" (not their term) movement that has taken over many education schools and school systems. They've dumbed down the math curriculum for huge numbers of American kids, including New York City public school dids. See http://www.nychold.com/ , or, for the national scene, http://mathematicallycorrect.com/ . As a liberal, active in the anti fuzz movement, I am frustrated when fuzzies accuse me of being conservative (a mistake the Great Mark never would make).

Two factors that make fuzzy math more common in the US than in Europe is (1) anti-intellectualism in the US -- (book larnin' is for sissies), and (2) elected officials rather than competent technocrats choosing curricula.

Algebra & Thinking
Do you any information about the qualifications of the algebra teacher?

enemies of science
Our president has not made up his mind whether he believes in evolution. The visitor secter at the Grand Canyon is forced to sell books claiming that the canyon was formed in the great flood 4000 years ago. The President of South Africa does not believe that HIV is related to AIDS. If we were to insist on scientific rigor in public life, there would be much bathwater to throw out than fuzzy math.

book larnin'
I respectfully offer an alternative suggestion for fuzzy math and other educational absurdities. There is a left-ward bent in academia that dismisses poor performance in public school kids because so many of them are minorities. Secondly, attempts by kids who want to learn, even if it means being accused of "acting white", are often discouraged through teacher laziness or concerns that failing kids will be "embarrassed." There is also a concern among parents, including me, that university humanities programs are hopelessly contaminated by politics and deconstructionist theories that make such an education useless. I will no longer pay for an education unless it is in the sciences or business. This is a recent development as I have older kids who graduated in the 80s and they did not get the indoctrination that ruined the education of a daughter who graduated three years ago.

Competency
" elected officials rather than competent technocrats choosing curricula."

Who do you suggest is competent? NEA leaders? Education Department professors? These ARE the people who have been deciding curricula.

Let's decentralize it with charter schools and vouchers. Parents will make better choices than administrators. Look at the success of home shcoolers.

Our charter school history teacher won't use any history book written prior to 1970 because they have been PC'd.

KIPP
Check out the KIPP schools.

http://www.kippschools.org/

Why can't all schools operate this way?

Stand and Deliver
"Escalante?s program was already in place when Gradillas came to Garfield, but the new principal?s support allowed it to run smoothly. In the early years, Escalante had met with some resistance from the school administration. One assistant principal threatened to have him dismissed, on the grounds that he was coming in too early (a janitor had complained), keeping students too late, and raising funds without permission. Gradillas, on the other hand, handed Escalante the keys to the school and gave him full control of his program.

Gradillas also worked to create a more serious academic environment at Garfield. He reduced the number of basic math classes and eventually came up with a requirement that those who take basic math must concurrently take algebra. He even braved the wrath of the community by denying extracurricular activities to entering students who failed basic skills tests and to current students who failed to maintain a C average.

In the process of raising academic standards at Garfield, Gradillas made more than a few enemies. He took a sabbatical leave to finish his doctorate in 1987, hoping that upon his return he would either be reinstated as principal of Garfield or be given a position from which he could help other schools foster programs like Escalante?s. He was instead assigned to supervise asbestos removal. It is probably no coincidence that A.P. calculus scores at Garfield peaked in 1987, Gradillas? last year there."

http://www.reason.com/0207/fe.jj.stand.shtml

It can be done!

Minds Are Different
In college (as a 46-year-old student) I saw what I had long suspected. Minds differ in abilities. Those who were good in mathematics would literally vomit if they had to give a speech or write a paper. I saw them break out in sweat at the prospect.

On the other hand, those who were comfortable--and good at--writing papers and giving speeches panicked when time to take an algebra test. I was one of these. It was then that I realized that our brains are wired differently.

In computer programming class, which I almost failed, I could do the logic, but could not write a functional program. The mathematicians had no problem writing the programs, but the writers and speakers did.

I think it is wonderful that there are those who are good at algebra and the field of mathematics. Not everyone is suited for this field, and should not be forced into it. Their time would be better spent preparing for those things that they will do in their lives, not those things that they will fail at, and therefore, avoid.

To you mathematicians--you need to get it! We can't be just like you merely by trying harder!

Excuse
What you say may be true, but that doesn't mean the challenge can't be met.
Toastmasters helps millions loose their fear of public speaking.
And if you really wanted to learn algebra, I believe you could do it.
So many today are not willing to do the work required to learn something and will use your excuse.
It is no wonder real self esteem among students is so poor. They know they are dumb, but society gives them a pass. "You are just not wired that way."

They need some level of exposure and understanding.
There's a minimal level of exposure and competence that everyone needs. Why do we make kids learn about (and write) poetry and read "Jane Eyre" in English class? Because they should see and experience literature and poetry at some level. That's what advanced societies do. That's why they're advanced.
Same goes with Algebra. At some point kids will be adults who must make societal decisions about scientific issues (e.g., the environment). If they have some exposure to mathematical concepts, they at least know something about the level of logic and rigor that scientists use to reach their conclusions. It's better than "believe us because we say it's so and we're smart."

Although personally, I still don't see the point of "Jane Eyre." ;-)

Keep algebra, dump geometry
I think one math course that is still taught in many secondary schools, and which could profitably be discarded, is classic geometry. It's a nice mental exercise, but there is nothing practical or useful about it that cannot be learned as part of analytic geometry. By deleting classic geometry from the usual curriculum (often taught between the first and second years of algebra), students will enter the secopnd year of algebra without having forgotten most of first year algebra, and there will be room in the curriculum to offer more advanced courses, especially for the mathematically talented students. Linear algebra, statistics and probability, and calculus would all be useful, especially with the ability to diagram the behavior of functions using computer programs like Mathematica.

I fear that the main obstacle to such a change, which would advance math education, is that there are many teachers who are not capable of teaching anything beyond the basic curriculum. So we have a vicious cycle, in which the competence of teachers limits the curriculum, and the curriculum limits the attraction of high schools to people who have more advanced math skills. And the losers are students, who are first bored by classic geometry, and are distracted from building a good understanding of algebra and functions that can progress into calculus and differential equations.

You Made My Point
You just made my point for me. I am not looking for a pass, but better use of talent and resources. I didn't learn to be a good computer programmer, but I did decide not to enter the computing field largely because of my college experiences in algebra and programming.

Some Exposure, Yes
I agree to "some exposure". I took an intro physics class at the same college. It was geared to those going into the computer tech field. It was less rigorous than for those who were going into a deeper mathematical or scientific field of study. We were introduced to concepts of physics, did some lower level physics experiments, and given quite a bit of the history of physics. I enjoyed the class.

I made a 97 on my intro to algebra final. I hated the class, and have retained almost nothing from it. The teacher was boring, and acted bored. Several students failed the class. One lady had failed it five semesters in a row. At what point does it make sense to say enough is enough?

Not Anti-Math
By the way, I am not anti-math or anti-science. I happy that we have many techological and other advances as a result of mathematics and science. My point is this: not everone chooses a career in math or science, and not everyone should! There are other roles to be played in society, and all must be filled by competent people. Not all are that competent in science, and by forcing them, you do not make them so.

I took every math and science class that my high school offered back in the 1960s. I passed them all with high grades, earning awards as the "high achiever" in most of them. I struggled always with the math classes, and did not retain much of what I studied then, or in college decades later.

Is it not possible to have a world of mathematicians and non-mathematicians side-by-side? Why does it have to be "either/or"?

Oh, lnow I getr it - - - - -
Glassmen is just teasing us, to drive us nuts by allowing Cohen's tripe exposure on this site. I'm on to him, this is some sort of experiment. Cohen sounds like a public school teacher, the intellectual lees of academia, "instructing" our kids. We'd be better off just teaching them to ride, shoot, and speak the truth, ourselves.

Who's for burning at the stake, or boiling in oil, anyone convicted of belonging to the NEA? Who's for vouchers? Who's for the separation of education and state?

Absolutely correct
And a good opinion it is.

Basic Algebra and Geometry are great for improving you ability to focus as well. I'm not one who likes higher math, but I'm not dumb enough to say you don't need it in your everyday life.

Here's a good one. My brother in law dropped out of school during his junior year. He works in construction and is now the foreman of a cement crew that lays foundations. I once asked him if his pen and paper work to set the angles and levels had any bearing on anything he learned in school. He said, "thank god I got through sophomore geometry, I always believed this stuff was useless in the real world, boy was I wrong."

People who say 'they never use Algebra or Geometry drive me crazy. Trust me, you do indeed use this stuff in the real world, you just may not always realize it.

Rubbish
coltakashi, you're making the same mistake with geometry that Cohen made with algebra. Both are essential, at least for rudimentary knowledge, a "stretching of the mind," as it were.

Have you ever tried to cut crown moulding? Doing this without the rudiments of geometry is well nigh impossible.

There's a point to basic curricula, which includes Art 101 and Algebra 101, and that is to expose kids to many different things. How else will they identify those things in which they have gifts? The English PhD doesn't need to study algebra, but s/he surely should have at some point in the past (and likewise for the Chemistry PhD with poetry, for example).

What point?
Are you saying you back a system of education that tests kids in, say, 6th grade and decides what direction their education will take from there?

Very poor idea. Some of the best composers, writers, doctors and scientists would have found themselves in an environment they were best suited for and would have died of boredom. These people rose to the challenge of their difficulties instead of skirting around them.

You are wrong and this "liberal education" type of thinking is a bad idea. A person getting an education in a field of their choice is why we have colleges. Grade school and high school should try and deliver a well rounded education to everyone. This give them a background from which to decide the direction of their future education.

Logic and reason
"Is this just another left/right brain thing or do actors and leftists celebs need publicity or should they have studied algebra and come this conclusion years ago. Logic and reason do not usually correlate with 'actors'."

I believe what was going through Clooney's mind was to raise consciousness of the issue through publicity-- which is something the value of which Hollywood certainly understands. By keeping this atrocity on the event horizon he keeps people talking about it, and thus spurs politicians to do more about it. And the tactic seems to be working.

I would not knock a tactic that's gaining the effect intended.

you are right...
I own and manage a construction company. One of the things I value in my supervisors and journeymen carpenters is good mathematics skills. I have several men that are proficient not just in algebra but also trigonometry and it is these guys that are the very best at cutting complicated roof systems and layout.

The most difficult young men to teach these skills to are those that have little or no mathematical acuity. I have on several occasions paid for remedial math courses for motivated guys at the local junior college.

most of us can do without quadratic equations but the simple fact that one must do the same thing to
One of my pet complaints is that children are not taught what they need to know to live, but what they need to know to succeed in further education. This seems absurd to me because most will not graduate from college and it would be wasting time if we made it so that they could. But even I think that algebra is valuable. I use it all the time in small ways. I use it in life.

Some other useful things that are not taught that are useful in life:

Principals of probability – so as not to be ripped off
Principals of statistics - so as not to easily fooled
Techniques to do math in our heads - also so we cannot be easily ripped off
Principals of physics - so that we can move big things and so we cannot be easily ripped off
Principals of chemistry - so that we cannot be easily frightened also useful in cooking
Principals of economics - so that we cannot be easily fooled by politicians
The miracle of compounding interest and what it can do to us if we get on the wrong side of it.
Principals of accounting just debits and credits – also so we cannot be easily ripped off

Principals of all these subjects are simple and can be taught to low IQ people, but are not because we laden these subjects down with higher math to make school a better test. IMHO schools should pounding down the simple principals of these subjects. Most of us can do without quadratic equations, the teaching of which serves mostly as and IQ test, but the simple fact that one can/must do the same thing to both sides of an equation is very valuable in life.




Geometry and carpentry
Classic geometry has almost nothing to do with practical measurement of lengths and angles of polygons, but in PROVING certain truths about the relationships between lenghts and angles. Working out the logic of a proof related to internal anges of a line crossing two parallel lines being equal has little to do with knowing the simple fact that is useful in various applications of design and fabrication. As I said, all of the practical applications of geometry are taught in a much more usable fashion, with real metrics, in analytic geometry, where the ability to use algebra to establish the logical foundation for the relationships between lines and angles in a cartesian plane, a cartesian 3 dimensional space, and converting to radial coordinates and wrapping functions that define trigonometric functions, is just as logical and rigorous, but much more straightforward than guessing backward to derive a geometric proof from a few axioms. Classical geometry was created in a world that did not have arabic numerals, so its reasoning was based on drawing lines and arcs with a straightedge and a compass. Remembering the rules of geometry, and deriving them yourself, is almost trivial in difficulty compared to deriving proofs in classical geometry. All sorts of geometric theorems that are special cases can be derived simultaneously through the application of algebra to polygons in a cartesian plane. And classical geometry tells us little about non-polygons (ellipses and parabolic and hyperbolic curves) while analytic geometry makes them easy to derive and describe. Then there is the fact that classical geometry assumes a Euclidean flat plane, and can't cope with curved spaces like the surface of a sphere, ellipsoid, or conic rotations of a parabola, all things that occur in the real world.

Classical geometry is a highly thought intensive tool that yields very limited information about a small class of objects, while analytic geometry allows the straightforward derivation of tremendous information about a large class of objects with much less effort. It was invented by Decartes and others because Euclidean geometry was too limited in its usefulness. Why are we wasting our kid's time learning the math equivalent of a year of ancient Greek, while very few learn Chinese, Japanese or Russian?

Different minds
You may believe that people don't need to know science or math because "their minds are wired differently" but what happens when you have to interpret statistics or probability ? You are choosing to remain ignorant of a very large, albeit difficult, area of human knowledge because you don't like it. Engineers should be able to speak and write and writers should understand some science. Otherwise we can't communicate.

Where is this stuff coming from?
In this country, it's coming from the teacher's union, and the graduates of the mislabled Education Schools.

It's the self same "competent technocrats" that LG wants to turn everything over to.

In most school districts, the board of education has been fully captured by the teacher's union.

I'd add basic logic and rhetoric(sp?)
so that they can spot the logical fallacies in another person's presentation.

Pragmatism wins??
It is obvious that there is a hierachy of knowledge, and algebra and other mathematical and scientific principles are at the foundation. However, the "pundits" have a point about success in the digital age. You don't have to understand assembly language to successfully use a computer--and even to use it exceptionally well and/or creatively. The end user of a multitude of sophisticated technologies is often ignorant of the underlying order, structure and science. The successful use of such technology seems more related to the vision, creativity and intelligence of the individual.

Pragmatism wins??
It is obvious that there is a hierachy of knowledge, and algebra and other mathematical and scientific principles are at the foundation. However, the "pundits" have a point about success in the digital age. You don't have to understand assembly language to successfully use a computer--and even to use it exceptionally well and/or creatively. The end user of a multitude of sophisticated technologies is often ignorant of the underlying order, structure and science. The successful use of such technology seems more related to the vision, creativity and intelligence of the individual.

Pragmatism wins??
It is obvious that there is a hierarchy of knowledge, and algebra and other mathematical and scientific principles are at the foundation. However, the "pundits" have a point about success in the digital age. You don't have to understand assembly language to successfully use a computer--and even to use it exceptionally well and/or creatively. The end user of a multitude of sophisticated technologies is often ignorant of the underlying order, structure and science. The successful use of such technology seems more related to the vision, creativity and intelligence of the individual.

fuzzy math
I disagree with the issue of anti-intellectualism.

I think its more about money. If you tie performance to test scores for people that control the outcomes, the better they perform... the more money they get. If they can't perform... just adjust the curriculum to insure that the outcomes are appropriate.

It's simple economics.

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