Math education

[HINDYhat]

http://www.maa.org/devlin/LockhartsLament.pdf

You really shouldn't TL;DR it, because it's a great text. Regardless, it's about a mathematician and teacher who criticizes the standard math curriculum taught in schools.

He suggests, among other things, that people hate math, or at least see it in the wrong light, because the math that is presented in school is boring and not captivating. As a teacher, he has applied unorthodox teaching methods which sound amazing.

I think low level math education needs to be reformed, because not only do people get almost nothing of relevant value from this kind of math, but also it is boring and lame as shit. If low level math would focus more on problem solving and logic problems, rather than worthless informal introductions to formal concepts with a huge emphasis on computation, I think people would at least develop a useful tool in their brain: the ability to solve problems and reason, as opposed to the ability to plug numbers into calculators.

[Boris_Spider]

Currently, I'm a few pages in (slowly reading it). It reminds me of my dad's paper that he's currently working on. It's essentially a redesign of the entire public school system.

I'm not sure if you're from the states, but over here critical thinking is strongly discouraged. I agree with what you said, but the problem extends to anything that uses math (the sciences) as well.

There's too much emphasis on the solution, rather then the method. All answers are arbitrary; trivial at best.

//\\oo//\\

[Rui]

I read the first few lines, then moved down to see how long it was, and then got engaged again just before the formula of the triangle, 1/2*b*h, and read until the conversation between the two (Italian?) sirs. It's interesting, I might look into it again later.

[Kercyn]

OMG TL;DR!!!!!!111

Reading long papers online is really, really tiring, but since you suggest that we shouldn't "TL;DR" I'll give it a try ;)

[Soul]

I understand that mathematics is pretty much a tool to help you survive in our society yet people keep assisting that it is either, key to the universe (Hawkings) or something else that is useless to learn (Some Republicans). Depends on the person really, for me Math is a tool to understand what the scientist are saying and since science is the Study of Everything.

Sorry, sometimes I get punchy, since science has not studied everything so I can only take it as face value like everything else. Now back to the subject at hand.

I do agree that a reform on K-12 teaching of all subjects should happen but it won't until certain political forces are removed from power. That is the life of this current area, mostly a power struggle in those of the Mental classes, when I mean mental, I do not mean disable or crazy but those who are smart.

[Jazztastic]

There's too much emphasis on the solution, rather then the method. All answers are arbitrary; trivial at best.

I strongly disagree. I have spent my entire mathematical career being told "It's about the process, not the answer", then getting screwed out of 7 points on a test because I was 0.5 off of the answer. All math should focus 100% on getting the answers, as mathematical mistakes cannot be tolerated in the real world. What if an architect makes a mistake, and a building collapses? What if a wall street broker puts the decimal one too far to the right, and thinks he just got his client rich as hell? What happens when a doctor prescribes too much medicine for a patient to take, and the patient OD's later that evening?

Mathematics is a practical science, the purest truth you can find in the world, because everything is rational and makes sense. If a reform was going to happen, it should make students understand how important math really is, and the real life consequences . Schools should also offer more specialized math classes directed at career choices, but at the same time, everyone should be required to take certain courses. For example, an aspiring musican or artist might not ever need to know tangent or cosine, but having knowledge helps round out a person to the fullest and increases the fullness of their life.

Saying answers are arbitrary is ignorant. The saying "Practice makes perfect" is essentially your math class. You practice math over and over and over again and nail down new concepts solidly to the point where you can do them in important real life situations without a single mistake. Imagine where our world would be with more focus on the method rather than the solution.

I think low level math education needs to be reformed, because not only do people get almost nothing of relevant value from this kind of math, but also it is boring and lame as shit.

I respect your opinion, and understand where you are coming from. Math is not a love for everyone, the repetitive methods and dull lectures put even me to sleep sometimes. Gardners multiple intelligence's, one of the most commonly held phsycological breakdowns of peoples personalities, and their strengths and weaknesses, dictates that people who are more logical have a better understanding and a better relationship with math. If you are an artist, again, I don't expect you to have a passion for math. But when I walk through the streets of downtown Portland and see hipster trash fumbling over money, trying to do the simple addition and subtraction to pay for food or shitty, hipster clothes, I die a little bit on the inside.

Another reason why students become disinterested is because math is hard if you slack off early. Math builds on itself, stacking higher and higher. If you get lost on long division, you are going to have a hell of a time doing algebra 3-4. Students will develop a gradual hatred for math, which is just thinly disguised misunderstanding and lack of knowledge. Finally, some people just flat out dislike math, these people are beyond any help, reformation or otherwise.

Students who seem interested in math probably use math on a regular basis. Whether it be something as simple as doing your daily Brain Age on the DS (Love that game :DDDDD), or something more challenging. I use math everyday when I buy groceries, when I edit wc3, when I figure simple stuff out (can't think of any examples, just things that come up in casual conversation). A rather dim-witted cheerleader sitting at my math table last year said "Wow, you're like Google!" after I solved a simple equation for her in 5 seconds, by simply staring at it and doing the numbers in my head.

If low level math would focus more on problem solving and logic problems, rather than worthless informal introductions to formal concepts with a huge emphasis on computation, I think people would at least develop a useful tool in their brain: the ability to solve problems and reason, as opposed to the ability to plug numbers into calculators.

I disagree. The brain shouldn't become your calculator. While you should be able to do any math you learned in elementry-middle school in your head, once you start reaching the high school math, you need a calculator. A calculator is a tool, just like a an artists pencil, or a photographers camera. I do believe that you should be taught how the functions work, and the logic behind them, not just "We press the Tan button, then . . ." Teachers should actually explain the concept of how tangent works in relation to the circle, not just how to use it. At the same time, they *must* have a calculator and be taught to use it. A calculator is nothing without a brain to process the idea and thought to operate it, and using it as the means to an end (the answer) should already show a clear comprehension of the subject, in that it was identified correctly to the extent that the user picked the right formula and method of solving it, and that they came across the correct answer.

Ultimately, to me, math is fine as it is. I believe that no matter the system, teachers abilities to engage students will fluctuate interest and comprehension of math according to that teachers ability. Creative teachers will find ways to be creative, and dull teachers will whip out the textbook and assign problems. Reforming the system will have little to no effect, as you can't create creativity in a person with words on a paper. It's worth a shot, but I'll remain skeptical until I see my befuzzled teacher Mr. Young control his class and engage them.

I know I've posed a lot of points, so I expect a few comments, mostly telling me I'm wrong :)

[Boris_Spider]

I strongly disagree. I have spent my entire mathematical career being told "It's about the process, not the answer", then getting screwed out of 7 points on a test because I was 0.5 off of the answer. All math should focus 100% on getting the answers, as mathematical mistakes cannot be tolerated in the real world. What if an architect makes a mistake, and a building collapses? What if a wall street broker puts the decimal one too far to the right, and thinks he just got his client rich as hell? What happens when a doctor prescribes too much medicine for a patient to take, and the patient OD's later that evening?

Mathematics is a practical science, the purest truth you can find in the world, because everything is rational and makes sense. If a reform was going to happen, it should make students understand how important math really is, and the real life consequences . Schools should also offer more specialized math classes directed at career choices, but at the same time, everyone should be required to take certain courses. For example, an aspiring musican or artist might not ever need to know tangent or cosine, but having knowledge helps round out a person to the fullest and increases the fullness of their life.

Saying answers are arbitrary is ignorant. The saying "Practice makes perfect" is essentially your math class. You practice math over and over and over again and nail down new concepts solidly to the point where you can do them in important real life situations without a single mistake. Imagine where our world would be with more focus on the method rather than the solution.

First off, that statement was covering more than just math and applied towards the american public education system. I don't disagree with you that answers are important, but what you get is weighted far more then how you got it. Memorizing answers doesn't allow for critical thinking, knowing the answers to already solved problems is at best a foundation.

~Knowing what method to use, when to use it, and when not to is more important than the answer. The nice thing about math is their is a multitude of ways to get answers, doesn't mean it's the best method to use. Calculus can be used to solve 2 + 2 = __. It's also much more complex than algebra, increasing the chance that mistakes occur.
mathematical mistakes cannot be tolerated in the real world
Not very often, no. However, in the real world there are redundencies (or should be) set up to check and then double check one's math, be it with a team of people or an array of machines. In fields where a higher degree of precision is required, more steps are in place to ensure the answer is appropriate.

~I agree, math has a wide variety of uses. It is the language of the sciences.

~"Practice makes perfect" does summarize math courses Do you practice the same problem 2 + 3/4 = Z over and over and over again, or do you practice the same method over and over and over again with a variation of numbers X +Y/W = Z? The numbers used are arbitrary, the process used isn't.

Mistakes can always occur. That's why we have redundencies in place to catch them, or formulas that are followed to try to prevent them. They still occur, the phrase for that is: "Shit happens."
Imagine where our world would be with more focus on the method rather than the solution.
A lot further than where it is now. It's rather hard to do in our world where the solution is more important than the method.

//\\oo//\\

I know I've posed a lot of points, so I expect a few comments, mostly telling me I'm wrong :)

Don't worry about it unless you're trying to change other peoples' minds. I'm here to try and get a better understanding of viewpoints besides my own, thus right and wrong are irrelevant.

[HINDYhat]

I understand that mathematics is pretty much a tool to help you survive in our society yet people keep assisting that it is either, key to the universe (Hawkings) or something else that is useless to learn (Some Republicans). Depends on the person really, for me Math is a tool to understand what the scientist are saying and since science is the Study of Everything.
None of which is really what the paper says mathematics is (also, what @ republicans?). If anything, mathematicians decide what mathematics is, and I'd be hard pressed to find a mathematician who would go with any of those definitions.

I do agree that a reform on K-12 teaching of all subjects should happen but it won't until certain political forces are removed from power. That is the life of this current area, mostly a power struggle in those of the Mental classes, when I mean mental, I do not mean disable or crazy but those who are smart.
I don't know what this means.

200

Beef_Is_Back, I've read your post, and I'd like to know if you've read the paper. There's a good reason why the OP is pretty short: it's because I linked the paper.

I strongly disagree. I have spent my entire mathematical career being told "It's about the process, not the answer", then getting screwed out of 7 points on a test because I was 0.5 off of the answer. All math should focus 100% on getting the answers, as mathematical mistakes cannot be tolerated in the real world. What if an architect makes a mistake, and a building collapses? What if a wall street broker puts the decimal one too far to the right, and thinks he just got his client rich as hell? What happens when a doctor prescribes too much medicine for a patient to take, and the patient OD's later that evening?
I'd highly suggest you reconsider thinking what mathematics actually is. It is not, primarily, a tool. I kind of like what Boris_Spider said here, though I think you might've misunderstood it. Or perhaps I understood it in a different way.

Mathematics is a practical science, the purest truth you can find in the world, because everything is rational and makes sense. If a reform was going to happen, it should make students understand how important math really is, and the real life consequences . Schools should also offer more specialized math classes directed at career choices, but at the same time, everyone should be required to take certain courses. For example, an aspiring musican or artist might not ever need to know tangent or cosine, but having knowledge helps round out a person to the fullest and increases the fullness of their life.
Mathematics is not a practical science. It's an art that can be used for practical purposes, just like music is art, yet it is used to lead armies into war (this is mentioned in the paper). Read the paper. It nicely outlines how there aren't really any real life consequences, and how informal introductions to formal concepts (like tangent or cosine) are pretty useless.

Saying answers are arbitrary is ignorant.
I like what Boris_Spider said, though I think I would have phrased it differently. I'd have simply said "All answers are trivial at best."
Really, the answers show no creativity or reasoning skills. I had a quote from the paper to add here but I forgot it. Either way, read the paper.

But when I walk through the streets of downtown Portland and see hipster trash fumbling over money, trying to do the simple addition and subtraction to pay for food or shitty, hipster clothes, I die a little bit on the inside.
Adults who have suffered math education can't add or subtract either. Are you suggesting that we aren't doing enough addition and subtraction?

Another reason why students become disinterested is because math is hard if you slack off early. Math builds on itself, stacking higher and higher. If you get lost on long division, you are going to have a hell of a time doing algebra 3-4. Students will develop a gradual hatred for math, which is just thinly disguised misunderstanding and lack of knowledge. Finally, some people just flat out dislike math, these people are beyond any help, reformation or otherwise.
You're right that students gradually come to hate math more, but I believe, just as is suggested in the paper, that students hate it because it actually sucks.

Take for example music class. I've rarely heard anyone say that they didn't like music class. It's a form of art, and you're (mostly) left to play, learn, and interact with others in creative forms. Why can't math be this way?

Students who seem interested in math probably use math on a regular basis. Whether it be something as simple as doing your daily Brain Age on the DS (Love that game :DDDDD), or something more challenging. I use math everyday when I buy groceries, when I edit wc3, when I figure simple stuff out (can't think of any examples, just things that come up in casual conversation). A rather dim-witted cheerleader sitting at my math table last year said "Wow, you're like Google!" after I solved a simple equation for her in 5 seconds, by simply staring at it and doing the numbers in my head.
I paraphrase from the text: "Many a graduate student has learned that they are not good at math, contrary to what society had been telling them for the past ten years; they are just very good at following directions." This is what you are feeling, and I've felt it before.

The only reason I became legitimately interested in math is because I began asking questions of my own and answering them on my own. This creative work is where all of the fun is. Before then, I truly hated math like everyone else. If you believe that you like math because you cross-multiply several times a day, you're very mistaken.

The brain shouldn't become your calculator.
Exactly. It should be used for greater purposes.

... once you start reaching the high school math, you need a calculator. A calculator is a tool, just like a an artists pencil, or a photographers camera.
No. A calculator is a thing that few mathematicians rely on and all are thought to depend on.

Please read the paper and get back to this.

200

I like a lot of what Boris said but I'm too drunk and tired to comment on it right now.

[Skycraft]

Haven't really read much, but I do agree with what has been said :/ Low level education is really boring after you know the basics. I'll read more on it later when I'm not sleepy.

[Jazztastic]

@Hindyhat, I didn't read the paper, just sharing my personal thoughts on the subject of school reformation and math in school. I'm going to read the first few pages, and if interesting, continue. Here's what I've read, and my thoughts on it.

The first thing to understand is that mathematics is an art.

Mathematics is a form of science, which is a fact. Taken from dictionary.com "A branch of knowledge or study dealing with a body of facts or truths systematically arranged and showing the operation of general laws: the mathematical sciences." Science is based on cold hard fact and logic, something that drives forward the future and shows humanity at its best, the ability to think clearly and soundly, and observe. The first definition of art on dictionary.com is "The quality, production, expression, or realm, according to aesthetic principles, of what is beautiful, appealing, or of more than ordinary significance." Science can be related to art, people see beauty both in method and practice of something so truly wonderous it dazzles the human mind. This is not the problem. People have denied certain mediums the title of art for forever, and art can be anything people consider to be art. If Mr. Lockhart happens to see beauty in art, while my math teacher sees it as a means to an end, there is nothing that can be done about it short of Mr. Lockhart getting a job at a high school or university. Part of the reason our educational system is such a peice of shit right now is because we have all these bleeding heart idealists writing papers bitching about the educational system and no one actually reforming it.

Nevertheless, the fact is that there is nothing as dreamy and poetic, nothing as radical, subversive, and psychedelic, as mathematics. It is every bit as mind blowing as cosmology or physics (mathematicians conceived of black holes long before astronomers actually found any), and allows more freedom of expression than poetry, art, or music (which depend heavily on
physical universe). Mathematics is the purest of the arts, as well as the most
misunderstood.

Assertion vs. evidence. Mr. Lockhart is asserting his viewpoint in a manner that says he thinks he is 100% correct when he is anything but, he gives absolutely no sources, an example that isn't explained in depth enough to the point its validity can be conceived as correct, and he does not even consider a different opinion being possible. How does being dependant upon the physical universe limit the creativity or purity of an art form? I'd also like to point out the logic loophole. "And allows more freedom of expression than poetry, art, or music", followed by "Mathematics is the purest of the arts, as well as the most misunderstood.". How can something allow more creative freedom than itself? In a formal reply, I'd completely disregard these two paradoxing sentence and focus on the first, "Nevertheless, the fact is that there is nothing as dreamy and poetic, nothing as radical, subversive, and psychedelic, as Mathematics". The definition of poetic, again from dictionary.com (my source for all word related topics), is "Possessing the qualities or charm of poetry:". How can anything be more poetic than poetry? How can something full of fact and irrefutable proof be considered dreamy? How can something that follows its own rules, based purely on clear headed thinking, be radical? Taken from a literal standpoint, even a single of Mr. Lockharts paragraphs are filled with such shenanigans and pooey they can't hold a candlestick to the educational board, which sees Math not in an artistic fashion. Anything could be seen in the artistic fashion, I met a man so believing in his own artistic capabilities that he was selling blank canvas for the hefty price of $10,000. When I asked him about what it was, he responded "There was a field here, that was eaten by a cow." Upon further questioning of where the cow was, he replied "You don't really believe the cow would be daft enough to stay here after he ate all the grass, do you?". The moral of the story is that art is such a wide and encompassing subject that EVERYTHING can be considered art. Art is what can be defined as art, nothing more, nothing less. If Mr. Lockhart decides to putt around on the mathematical green all day because he considers it artistic and fun, let him. It's not changing the fact that the overwhelming majority of people in the world DO NOT CONSIDER MATH TO BE AN ART. If he really considers this a problem (which it appears he most certainly does, along with you) he needs to make people see his point of view with evidence, clear fact and reason, sources, and most importantly, LOGIC.

Another paragraph (which it won't allow me to quote, because the website fluked out and for some reason isn't letting me select more than one word at a time, so forgive me on this one for the lack of a proper quote, I'm sure anyone who read the first part of the paper will remember it), which deals with the concept of teachers being taught by teachers who were taught that math is so called "pseudo-mathematics". I find this incredibly ironic, because these "pseudo-mathematicians" are probably the exact same people who taught Mr. Lockhart. How can one man have such an incredible and accurate critique of the exact problem of this nations mathematical stagnation, when he was taught the exact same methods with the exact same focus on the solution rather than the process? If I had to guess, I'd say that Mr. Lockhart suffered every year of math he took in highschool and college, due to the rather deviant nature of his thoughts on the subject. I'm not trying to attack Mr. Lockharts person, I'm just saying that maybe personal experience did have a roll in his rather different view of mathematics.

I can't read this garbage. I moved to the bottom to check sources and I saw his breakdown of classes. His completely honest course catalog was a mere bashing of every single class offered for the study of mathematics. There was NO CRITIQUE, NO HOW TO FIX, NOTHING OF THE SORT INCLUDED IN THAT SECTION. Above it, he had a fake conversation between two people, with one pushing for no boundaries or rules for learning, in a way that students discover their own rules and equations and basically bound from one problem to the next. I have 2 major problems with this. 1, it has taken many absolutely brilliant mathematicians (with probably even more of a love for math than Mr. Lockhart) multiple millenia to establish even the rules we have now (Incredibly short, if you consider that over 2000 worth of knowledge and top tier thinking can be learned in a single lifetime by almost anyone), so what makes anyone think that a kid could conceive this ideas themselves? Toying with math in an uneducated and uninstructed way will not allow students to teach themselves things such as tangent. That is completely ridiculous. Second, this darwinistic approach to teaching should not be allowed. Reaching the levels of an Ayn Rand capitalism (Put forth in the ideas of Objectivism, fascinating author and idea, but not appropriate in a nation were All Are Equal, absolutely brilliant writing though. No better story than Atlas Shrugged) for schooling is pushing boundaries way too far. The Survival Of The Fittest mindset fucked the U.S. economy in a way that almost caused another depression. You want that style of thought in our schools? The weak will perish, the strong will survive, this concept will castrate the minds of our nations youth. Proper education and structure is needed to allow young minds to flourish properly, and reach their full potential. The material and lesson plans have been thought about by the brightest minds in America for the last hundred years, so I'd put more faith in the system than a Mathematics idealist who can't even propose how to fix it.

In personal response to you, HINDYhat, I still thoroughly disagree.

I'd highly suggest you reconsider thinking what mathematics actually is. It is not, primarily, a tool. I kind of like what Boris_Spider said here, though I think you might've misunderstood it. Or perhaps I understood it in a different way.

Mathematics is used primarily as a tool. Everyone but mathematicians use it on a practical basis. If the majority uses it as a tool, it is a tool, despite the teeny tiny minorities cries that it is something higher. If a bananas are art, something grown from God Almighty himself, true art, amazingly beautiful and inspiring in all who behold it, but 99% of people eat bananas, it is a food. Some will consider the beauty of the banana and be thankful of the epiphany granted unto them by the banana, but it does not change the fact that the banana is food. Mathematics is the tool of the masses, an "art" so plagued by the layman that it was descended into worthlessness in the values of society. Wrong. Everyone being able to do it DOES NOT make it any less an art. Anyone able to use their hands can draw a smily face, but does it make it any less a smily face, or any less smily for that matter? Absolutely not.

Mathematics is not a practical science.

I'm so absolutely lost for words right now I cannot even begin to describe how I feel about those 6 words. That's an arguement I'm not willing to go.

I like what Boris_Spider said, though I think I would have phrased it differently. I'd have simply said "All answers are trivial at best."
Really, the answers show no creativity or reasoning skills. I had a quote from the paper to add here but I forgot it. Either way, read the paper.


Answers are not trivial. Without those little answers you'd be starving under the moon right now with no clothes or technology to speak of. Answers are, the real world definition of math. Math is pointless in the literal sense without an answer, it would be just useless problems over and over again with no gain or outcome, other than enjoyment. The people who have been through the wringer of the USA's educational system are taught to not like math, to use the most correct and easiest methods to answer the problem incredibly quickly so they can stop doing math. I don't see a single problem with that, if you enjoy math, go right ahead and write problems for yourself all day.

Adults who have suffered math education can't add or subtract either. Are you suggesting that we aren't doing enough addition and subtraction?


Are you speaking from experience, because every adult I've ever met who got C's or higher on their maths can do addition and subtraction well enough to the point they don't struggle in the supermarket. No we don't need more problems on addition and subtraction, because anyone who paid attention in class and aren't mentally retarded have no problems with them.

You're right that students gradually come to hate math more, but I believe, just as is suggested in the paper, that students hate it because it actually sucks.

Take for example music class. I've rarely heard anyone say that they didn't like music class. It's a form of art, and you're (mostly) left to play, learn, and interact with others in creative forms. Why can't math be this way?


Do you play an instrument? You do realize that to create ACTUAL music at any level besides complete beginner that yo have to practice hard and be directed by a teacher? Being left alone at a drum set, almost everyone would not be able to teach themselves a Joe Morello drum solo, even if given hundreds of years to practice. We are taught by those above us who know, so we don't have to blindly fumble in the dark. You are completely wrong about this one.

I paraphrase from the text: "Many a graduate student has learned that they are not good at math, contrary to what society had been telling them for the past ten years; they are just very good at following directions." This is what you are feeling, and I've felt it before.

The only reason I became legitimately interested in math is because I began asking questions of my own and answering them on my own. This creative work is where all of the fun is. Before then, I truly hated math like everyone else. If you believe that you like math because you cross-multiply several times a day, you're very mistaken.


New ideas come from critiques of old ones. It is rare to find completely new ideas. What happens in school is that we are told how things work, how to do things. We take this concepts and break them down the core, truely understand them. Maybe it's the people who don't do these things that need the help. Critical thinking is the highest level of thinking you can do, and I saw my peers do it every single day in math class. Saying that being good at math is just following directions could be said for ANY class. I hate to drag Hakeems good name into this, but he has a quote that perfectly expresses what I'm driving at:

Given that most of the information we ever consider ourselves to "know" comes from the words of other people, most of what we think we know isn't subjectively founded. Of course, most people aren't liars. If someone lies to you, they generally have a reason for it. If they don't have a reason to lie, that is, if their reason is just for the hell of it, then no real emphasis is placed on convincing people it is factual. In such a case, it is generally readily apparent the liar is lying.

Almost all knowledge is passed by speech or parchment. Does this mean we are just copying them word for word, reciting lines out of a textbook as our own opinion? Absolutely not. Humans take the information received and put their own spin on it, whether it be dislike or agreement, a different viewpoint, whatever. Critical thinking happens all the time, and saying we are just reciting like parrots what we are taught is wrong and almost idiotic to suggest in the first place. Do not try to put yourself in my shoes, I have a very different background than you almost guaranteed, and obviously have a different opinion set. Viewing math as the artist viewed his blank canvas would ruin math for me in such a way that I could never take it back.

No. A calculator is a thing that few mathematicians rely on and all are thought to depend on.

Please read the paper and get back to this.


Saying that mathematicians don't use calculators is like saying that scientists dont use microscopes. Tools are made to make peoples lives easier, and I'd like to see anyone, mathematician or otherwise, say that doing tangent by hand is easier. It isn't. A calculator is a tool, not a machine to replace our minds and steal the maths from our minds. Treat it as such.

And that's it. Call me ignorant for refusing to read the rest of the paper, but I refuse an academic paper with assertions, no evidence, and no sources. He proposes an objectivist school system with no focuses and in such a way that wastes all prior knowledge garnered in the subject. I don't need to read that garbage.

[HINDYhat]

Mathematics is a form of science, which is a fact...
It is, alright. I also agree that nearly everything can be subjectively considered an art. The difference is that mathematicians prove things, and good mathematicians prove important things in beautiful, unique and elegant ways. I like to think that mathematicians determine what math is, and so under this consideration, math is science but also a form of art.

You think that math's purpose, since it is a tool, is determined by how it is most often used, and this happens to come down essentially to computation. Now, I haven't done any research on this, but the internet is a tool and it is extensively used for porn. This doesn't mean that the internet's purpose is porn. Now, you could say that the internet's purpose is to exchange information, and porn is a form of information exchange. Then why not generalize math's purpose to something else?

Part of the reason our educational system is such a peice of shit right now is because we have all these bleeding heart idealists writing papers bitching about the educational system and no one actually reforming
I doubt this statement, but regardless Lockhart is a school teacher and actually applies unorthodox teaching methods himself. Sounds to me like he's actually trying to reform.

How does being dependant upon the physical universe limit the creativity or purity of an art form?
I thought that was kind of the definition of a field's purity: http://xkcd.com/435/
He is just applying the same principle to art.

I'd also like to point out the logic loophole...
It is a sensational text. It is not a letter to the education board. I won't bother replying to each of your questions because you're taking statements literally when you shouldn't be. Please don't generalize what I've just said to the entire paper. That would be silly.

I find this incredibly ironic, because these "pseudo-mathematicians" are probably the exact same people who taught Mr. Lockhart. How can one man have such an incredible and accurate critique of the exact problem of this nations mathematical stagnation, when he was taught the exact same methods with the exact same focus on the solution rather than the process? If I had to guess, I'd say that Mr. Lockhart suffered every year of math he took in highschool and college, due to the rather deviant nature of his thoughts on the subject. I'm not trying to attack Mr. Lockharts person, I'm just saying that maybe personal experience did have a roll in his rather different view of mathematics.
Once you begin undergraduate studies in math, you are taught by actual mathematicians: in this case, people who teach but also do math research for a living. These people can make you unlearn the silly things you were taught before, if you are motivated enough. I'd go so far to say that anyone who has studied at a high level of mathematics would agree with Lockhart on most of his points. Every person I've spoken to, at least, does.

I am at work right now, but I will reply to the rest once I get back home and after I eat etc, so it might take a while.

[Jazztastic]

This is a great topic. I havent flexed my debating muscle in forever.

It is, alright. I also agree that nearly everything can be subjectively considered an art. The difference is that mathematicians prove things, and good mathematicians prove important things in beautiful, unique and elegant ways. I like to think that mathematicians determine what math is, and so under this consideration, math is science but also a form of art.

This is an assertion, and almost entirely opinion. Art can be anything you want it to be, true, and people have different opinions are art, and some consider something to be art, while others don't. They are both correct. This sentence bugged me particularly: "The difference is that mathematicians prove things, and good mathematicians prove important things in beautiful, unique and elegant ways". This sentence can't be taken seriously because things that are important, beautiful, unique, and elegant, are all subject to opinion. What mathematicians consider important probably isn't what I consider important. What mathematicians consider beautiful, unique, and elegant, are probably not what I consider beautiful, unique, and elegant. You yourself have adopted the opinions of most mathematicians, and consider it to be a truly wonderous and beautiful thing, while almost everyone else who isn't a mathematician considers it a tool.

You think that math's purpose, since it is a tool, is determined by how it is most often used, and this happens to come down essentially to computation. Now, I haven't done any research on this, but the internet is a tool and it is extensively used for porn. This doesn't mean that the internet's purpose is porn. Now, you could say that the internet's purpose is to exchange information, and porn is a form of information exchange. Then why not generalize math's purpose to something else?

The beautiful part of the internet is that it could actually be argued in favor of its main purpose being a medium for pornography. For every 1 website that doesn't contain pornography, there are 5 websites that do. Differing opinions put people on different sides of that, and facts allow people to help choose which side they belong to. That's the beautiful part about debating.

I doubt this statement, but regardless Lockhart is a school teacher and actually applies unorthodox teaching methods himself. Sounds to me like he's actually trying to reform.

I have two comments about this. I shouldn't have written that, it was a blind assertion, my mistake. I also wonder how well Mr. Lockharts unorthodox teaching methods work in comparison to others in terms of how well students score in the class and on tests. The only true way to show any improvement in method would be a rigorous testing of his methods over a few decades, if you've taken any serious study of probability you know why (because there might be a weird year, where students were naturally brighter or less bright. If you haven't the foggiest idea of why a single year sampling isn't appropriate, read the book The Drunkards Walk, it's wonderful). Given the time it would take to actually see positive or negative results, this shouldn't even be an option for immediate math redesign. Please please please don't show ignorance and try to deny that we need a test phase and at least 10-15 years to see at least a small amount of data on which we can base the teaching method.

I thought that was kind of the definition of a field's purity: http://xkcd.com/435/
He is just applying the same principle to art.

That isn't the definition of a fields purity, it shows a mans personal opinion of how pure the fields are. In both Mr. Munroes comic and Mr. Lockharts paper, no subjective scale of any kind is given to determine how the objects were placed where they are, nothing weighing math above all others in purity of the scientific field and purity in art. This, once again, is an assertion, a blind opinionated shot in the dark with no factual evidence to back it up. And for future reference, I wouldn't cite XKCD or any other comic, no matter how scientific or intelligent, in any formal debate. It will invalidate your arguement when you cite bad sources.

It is a sensational text. It is not a letter to the education board. I won't bother replying to each of your questions because you're taking statements literally when you shouldn't be. Please don't generalize what I've just said to the entire paper. That would be silly.

Sensational text or not, any kind of scholarly thesis or paper meant for reading needs to be understood in full, and shouldn't be written figuratively or poeticly. Ignoring my questions is also a sign that you, either aren't capable of answering them, or deem them unworthy of being answered. Even viewing math as an art, any critique of art shouldn't be written in such a way that makes the reader confused. Take the Inferno for example. The first 34 cantos of Dante's masterpeice, it has been translated to contain figurative speech and open questions in an effort to puzzle the reader and ask him questions. Also, the language is of such a high caliber and the figurative language is so advanced, that most people can't even follow what Dante is doing. Cliffnotes has a wonderful 80-ish page breakdown of this work of art, and it is written in laymans terms, without any symbolism, any analogies, any wordplay of any kind. This is because the scholarly essay designed to bring concepts clearer are MADE SO THAT THE READER UNDESTANDS, IN FULL, THE CONCEPTS PROVIDED BY THE AUTHOR.

Once you begin undergraduate studies in math, you are taught by actual mathematicians: in this case, people who teach but also do math research for a living. These people can make you unlearn the silly things you were taught before, if you are motivated enough. I'd go so far to say that anyone who has studied at a high level of mathematics would agree with Lockhart on most of his points. Every person I've spoken to, at least, does.


Again, you are asserting instead of providing real evidence. What is your definition of mathematician, because dictionary.com's definition of a mathematician is "an expert or specialist in mathematics.". Someone who teaches mathematics in high school qualifies for this term. My father, who majored in mathematics, but works a job in Kaiser Permanente, is a mathematician under these qualifications. Both you and Lockhart have been rather vague on the term, and it leaves your arguement full of holes. "I'd go so far to say that anyone who has studied at a high level of mathematics would agree with Lockhart on most of his points. Every person I've spoken to, at least, does." Who are these people? How do I know that they have read the source material we are referencing or have even studied at a high level of mathematics, as you claim they did?

Now, for you, I'd like you to respond to these sections, which you hadn't addressed before, due to a time constraint

I can't read this garbage. I moved to the bottom to check sources and I saw his breakdown of classes. His completely honest course catalog was a mere bashing of every single class offered for the study of mathematics. There was NO CRITIQUE, NO HOW TO FIX, NOTHING OF THE SORT INCLUDED IN THAT SECTION. Above it, he had a fake conversation between two people, with one pushing for no boundaries or rules for learning, in a way that students discover their own rules and equations and basically bound from one problem to the next. I have 2 major problems with this. 1, it has taken many absolutely brilliant mathematicians (with probably even more of a love for math than Mr. Lockhart) multiple millenia to establish even the rules we have now (Incredibly short, if you consider that over 2000 worth of knowledge and top tier thinking can be learned in a single lifetime by almost anyone), so what makes anyone think that a kid could conceive this ideas themselves? Toying with math in an uneducated and uninstructed way will not allow students to teach themselves things such as tangent. That is completely ridiculous. Second, this darwinistic approach to teaching should not be allowed. Reaching the levels of an Ayn Rand capitalism (Put forth in the ideas of Objectivism, fascinating author and idea, but not appropriate in a nation were All Are Equal, absolutely brilliant writing though. No better story than Atlas Shrugged) for schooling is pushing boundaries way too far. The Survival Of The Fittest mindset fucked the U.S. economy in a way that almost caused another depression. You want that style of thought in our schools? The weak will perish, the strong will survive, this concept will castrate the minds of our nations youth. Proper education and structure is needed to allow young minds to flourish properly, and reach their full potential. The material and lesson plans have been thought about by the brightest minds in America for the last hundred years, so I'd put more faith in the system than a Mathematics idealist who can't even propose how to fix it.

Answers are not trivial. Without those little answers you'd be starving under the moon right now with no clothes or technology to speak of. Answers are, the real world definition of math. Math is pointless in the literal sense without an answer, it would be just useless problems over and over again with no gain or outcome, other than enjoyment. The people who have been through the wringer of the USA's educational system are taught to not like math, to use the most correct and easiest methods to answer the problem incredibly quickly so they can stop doing math. I don't see a single problem with that, if you enjoy math, go right ahead and write problems for yourself all day.

Are you speaking from experience, because every adult I've ever met who got C's or higher on their maths can do addition and subtraction well enough to the point they don't struggle in the supermarket. No we don't need more problems on addition and subtraction, because anyone who paid attention in class and aren't mentally retarded have no problems with them

Do you play an instrument? You do realize that to create ACTUAL music at any level besides complete beginner that yo have to practice hard and be directed by a teacher? Being left alone at a drum set, almost everyone would not be able to teach themselves a Joe Morello drum solo, even if given hundreds of years to practice. We are taught by those above us who know, so we don't have to blindly fumble in the dark. You are completely wrong about this one.


New ideas come from critiques of old ones. It is rare to find completely new ideas. What happens in school is that we are told how things work, how to do things. We take this concepts and break them down the core, truely understand them. Maybe it's the people who don't do these things that need the help. Critical thinking is the highest level of thinking you can do, and I saw my peers do it every single day in math class. Saying that being good at math is just following directions could be said for ANY class. I hate to drag Hakeems good name into this, but he has a quote that perfectly expresses what I'm driving at:

Given that most of the information we ever consider ourselves to "know" comes from the words of other people, most of what we think we know isn't subjectively founded. Of course, most people aren't liars. If someone lies to you, they generally have a reason for it. If they don't have a reason to lie, that is, if their reason is just for the hell of it, then no real emphasis is placed on convincing people it is factual. In such a case, it is generally readily apparent the liar is lying.

Almost all knowledge is passed by speech or parchment. Does this mean we are just copying them word for word, reciting lines out of a textbook as our own opinion? Absolutely not. Humans take the information received and put their own spin on it, whether it be dislike or agreement, a different viewpoint, whatever. Critical thinking happens all the time, and saying we are just reciting like parrots what we are taught is wrong and almost idiotic to suggest in the first place. Do not try to put yourself in my shoes, I have a very different background than you almost guaranteed, and obviously have a different opinion set. Viewing math as the artist viewed his blank canvas would ruin math for me in such a way that I could never take it back.


Saying that mathematicians don't use calculators is like saying that scientists dont use microscopes. Tools are made to make peoples lives easier, and I'd like to see anyone, mathematician or otherwise, say that doing tangent by hand is easier. It isn't. A calculator is a tool, not a machine to replace our minds and steal the maths from our minds. Treat it as such.


And please, everyone who posts here should have facts, not assertions. Quote the source material, experts in the field, bring up studies, cite definitions, or bring forth the words of others who share your opinion. DO NOT blindly throw out your standpoint without backing it up. Do not include "logical thinking" in your arguement, because what is logical to you, might not be logical to someone else. Do not use assertions, provide evidence to help give your arguements punch. I know that this is more the literary and writing field being applied on the subject, but that is the nature of debating, you can't argue in favor of something if you can't argue correctly. And it really does help to use proper punctuation and grammar, or else you might be misunderstood or have your illogical sentences completely ignored, like Mr. Lockharts were.

[Dr Super Good]

They should just teach mathimatica. Who needs to know what 1+1 is equal to when you can solve an advanced matrix, vector or differential.

[HINDYhat]

This is a great topic. I havent flexed my debating muscle in forever.
:>

By the way, if you doubt Lockhart's credentials, he has a Ph.D. in mathematics, has worked with great mathematicians, and previously taught university level mathematics. This paper is published on the Mathematical Association of America's (MAA) site.

This is an assertion, and almost entirely opinion. Art can be anything you want it to be, true, and people have different opinions are art, and some consider something to be art, while others don't. They are both correct. This sentence bugged me particularly: "The difference is that mathematicians prove things, and good mathematicians prove important things in beautiful, unique and elegant ways". This sentence can't be taken seriously because things that are important, beautiful, unique, and elegant, are all subject to opinion. What mathematicians consider important probably isn't what I consider important. What mathematicians consider beautiful, unique, and elegant, are probably not what I consider beautiful, unique, and elegant. You yourself have adopted the opinions of most mathematicians, and consider it to be a truly wonderous and beautiful thing, while almost everyone else who isn't a mathematician considers it a tool.
We differ here only in that I think mathematicians determine what mathematics is.

I have two comments about this. I shouldn't have written that, it was a blind assertion, my mistake. I also wonder how well Mr. Lockharts unorthodox teaching methods work in comparison to others in terms of how well students score in the class and on tests. The only true way to show any improvement in method would be a rigorous testing of his methods over a few decades, if you've taken any serious study of probability you know why (because there might be a weird year, where students were naturally brighter or less bright. If you haven't the foggiest idea of why a single year sampling isn't appropriate, read the book The Drunkards Walk, it's wonderful). Given the time it would take to actually see positive or negative results, this shouldn't even be an option for immediate math redesign. Please please please don't show ignorance and try to deny that we need a test phase and at least 10-15 years to see at least a small amount of data on which we can base the teaching method.
It is indeed unrealistic to expect any kind of math reform that would make Lockhart and myself happy. He says it himself, "of course what I'm suggesting is impossible for a number of reasons." He ends up simply asking for math teachers to know more about math:
"Why is it that we accept math teachers who have never produced an original piece of mathematics, know nothing of the history and philosophy of the subject, nothing about recent developments, nothing in fact beyond what they are expected to present to their unfortunate students? ... Now I'm not saying that math teachers need to be professional mathematicians— far from it. But shouldn't they at least understand what mathematics is, be good at it, and enjoy doing it?"

That isn't the definition of a fields purity, it shows a mans personal opinion of how pure the fields are. In both Mr. Munroes comic and Mr. Lockharts paper, no subjective scale of any kind is given to determine how the objects were placed where they are, nothing weighing math above all others in purity of the scientific field and purity in art. This, once again, is an assertion, a blind opinionated shot in the dark with no factual evidence to back it up. And for future reference, I wouldn't cite XKCD or any other comic, no matter how scientific or intelligent, in any formal debate. It will invalidate your arguement when you cite bad sources.
I wasn't citing XKCD, I was using it to illustrate what I was saying without having to write it down. If you really want me to cite stuff, then I will, and if you don't trust Wikipedia, then I don't care enough:
"Fundamental science (or basic science, pure science) is science that describes the most basic objects, forces, relations between them and laws governing them, such that all other phenomena may be in principle derived from them following the logic of scientific reductionism."
I feel like this makes math the purest of sciences. Again, his use of pure art is analogous to this concept. It is by no means a formal term.

Sensational text or not, any kind of scholarly thesis or paper meant for reading needs to be understood in full, and shouldn't be written figuratively or poeticly. Ignoring my questions is also a sign that you, either aren't capable of answering them, or deem them unworthy of being answered. Even viewing math as an art, any critique of art shouldn't be written in such a way that makes the reader confused. Take the Inferno for example. The first 34 cantos of Dante's masterpeice, it has been translated to contain figurative speech and open questions in an effort to puzzle the reader and ask him questions. Also, the language is of such a high caliber and the figurative language is so advanced, that most people can't even follow what Dante is doing. Cliffnotes has a wonderful 80-ish page breakdown of this work of art, and it is written in laymans terms, without any symbolism, any analogies, any wordplay of any kind. This is because the scholarly essay designed to bring concepts clearer are MADE SO THAT THE READER UNDESTANDS, IN FULL, THE CONCEPTS PROVIDED BY THE AUTHOR.
This is not a scholarly paper, though it is supported by academic authorities. Your questions were mostly semantic nitpicking. For example:
"'And allows more freedom of expression than poetry, art, or music', followed by 'Mathematics is the purest of the arts, as well as the most misunderstood.'. How can something allow more creative freedom than itself?"
Replace 'art' in the first quote with 'drawing/painting' and all is well.
If you really feel like it is absolutely necessary for me to go through other such nitpickings, then I will, but I don't feel like it because it's beside the argument.

Again, you are asserting instead of providing real evidence. What is your definition of mathematician, because dictionary.com's definition of a mathematician is "an expert or specialist in mathematics.". Someone who teaches mathematics in high school qualifies for this term. My father, who majored in mathematics, but works a job in Kaiser Permanente, is a mathematician under these qualifications. Both you and Lockhart have been rather vague on the term, and it leaves your arguement full of holes. "I'd go so far to say that anyone who has studied at a high level of mathematics would agree with Lockhart on most of his points. Every person I've spoken to, at least, does." Who are these people? How do I know that they have read the source material we are referencing or have even studied at a high level of mathematics, as you claim they did?
A high school mathematics teacher only needs a bachelor's degree in education to teach (at least here in Canada, that is what is required). If I'm to go by definitions, an expert is "a person who has a comprehensive and authoritative knowledge of or skill in a particular area" and a specialist is "a person who concentrates primarily on a particular subject or activity; a person highly skilled in a specific and restricted field." If you think education degrees really fill these roles, then you should meet some education majors. I know this is an assertion, but it's based on my experience as a student and friend.

I can't get the people I've asked to come on THW and swear that they agree with Lockhart's position, but I can tell you that all of my classmates in honours mathematics agree with him, a few graduate students who I've spoken to agree with him, and a few professors as well. Additionally, his paper is officially supported by the MAA, which is "the most widely-read mathematics journal in the world according to records on JSTOR" and etc. etc. etc.

I can't read this garbage. I moved to the bottom to check sources and I saw his breakdown of classes. His completely honest course catalog was a mere bashing of every single class offered for the study of mathematics. There was NO CRITIQUE, NO HOW TO FIX, NOTHING OF THE SORT INCLUDED IN THAT SECTION. Above it, he had a fake conversation between two people, with one pushing for no boundaries or rules for learning, in a way that students discover their own rules and equations and basically bound from one problem to the next. I have 2 major problems with this. 1, it has taken many absolutely brilliant mathematicians (with probably even more of a love for math than Mr. Lockhart) multiple millenia to establish even the rules we have now (Incredibly short, if you consider that over 2000 worth of knowledge and top tier thinking can be learned in a single lifetime by almost anyone), so what makes anyone think that a kid could conceive this ideas themselves? Toying with math in an uneducated and uninstructed way will not allow students to teach themselves things such as tangent. That is completely ridiculous. Second, this darwinistic approach to teaching should not be allowed. Reaching the levels of an Ayn Rand capitalism (Put forth in the ideas of Objectivism, fascinating author and idea, but not appropriate in a nation were All Are Equal, absolutely brilliant writing though. No better story than Atlas Shrugged) for schooling is pushing boundaries way too far. The Survival Of The Fittest mindset fucked the U.S. economy in a way that almost caused another depression. You want that style of thought in our schools? The weak will perish, the strong will survive, this concept will castrate the minds of our nations youth. Proper education and structure is needed to allow young minds to flourish properly, and reach their full potential. The material and lesson plans have been thought about by the brightest minds in America for the last hundred years, so I'd put more faith in the system than a Mathematics idealist who can't even propose how to fix it.
There is a conversation between Simplicio and Salviati which contains the following:

Simplicio: But we don't have time for every student to invent mathematics for
themselves! It took centuries for people to discover the Pythagorean
Theorem. How can you expect the average child to do it?

Salviati: I don't. Let's be clear about this. I'm complaining about the complete
absence of art and invention, history and philosophy, context and
perspective from the mathematics curriculum. That doesn't mean that
notation, technique, and the development of a knowledge base have no
place. Of course they do. We should have both.

As for how the system was designed by bright minds, well a large portion of the bright mathematicians of America agree with Lockhart. Where is your source on your assertion? I personally seriously doubt that the current system was put together by the brightest mathematicians in America.

Math is pointless in the literal sense without an answer, it would be just useless problems over and over again with no gain or outcome, other than enjoyment.
Exactly, that's what math would be, but it would entail creative development, honing of reasoning skills, and enjoyment.

Are you speaking from experience, because every adult I've ever met who got C's or higher on their maths can do addition and subtraction well enough to the point they don't struggle in the supermarket. No we don't need more problems on addition and subtraction, because anyone who paid attention in class and aren't mentally retarded have no problems with them
Yes, from experience. None of them remember what a quadratic formula is either, nor any trigonometric identities (SOHCAHTOA SOHCAHTOA SOHCAHTOA ...), nor anything about logarithms, and very little of anything else. Some are able to do basic arithmetic. I am not suggesting that every adult is unable to do arithmetic, just as you were (hopefully) not suggesting that no hipsters can do arithmetic.

Do you play an instrument? You do realize that to create ACTUAL music at any level besides complete beginner that yo have to practice hard and be directed by a teacher? Being left alone at a drum set, almost everyone would not be able to teach themselves a Joe Morello drum solo, even if given hundreds of years to practice. We are taught by those above us who know, so we don't have to blindly fumble in the dark. You are completely wrong about this one.
Yes, I have played drums for around 5 years now, and I played piano for 4 years when I was younger. I haven't had lessons in either, and I learned initially by following along or reading tabs, just as you said, but eventually I came to playing on my own and figuring things out. Of course, I wasn't innovating, and neither do I expect children to innovate when it comes to mathematics. Hell, I don't expect that from undergraduates or even graduates sometimes. What I can expect is personal discovery.

Almost all knowledge is passed by speech or parchment. Does this mean we are just copying them word for word, reciting lines out of a textbook as our own opinion? Absolutely not. Humans take the information received and put their own spin on it, whether it be dislike or agreement, a different viewpoint, whatever. Critical thinking happens all the time, and saying we are just reciting like parrots what we are taught is wrong and almost idiotic to suggest in the first place. Do not try to put yourself in my shoes, I have a very different background than you almost guaranteed, and obviously have a different opinion set. Viewing math as the artist viewed his blank canvas would ruin math for me in such a way that I could never take it back.
My last paragraph answers a part of this.
In any case, what's great about math is that there are (probably?) infinitely many problems, and it is difficult to find systematic ways to prove them. Some of the brain teasers Lockhart writes as examples are "Are the prime numbers infinite? What is the area of a triangle? Is infinity a number?" etc. Neither of these are solved in the same way, yet each (although possibly not the first one) can be tackled by children and solved using a number of different approaches. There is, however, a systematic way to solve "Find the roots of x2 - 2x + 1" and "Simplify the expression ln(a5b3/c6)." This takes away the wonder, and the critical part of critical thinking.

Saying that mathematicians don't use calculators is like saying that scientists dont use microscopes. Tools are made to make peoples lives easier, and I'd like to see anyone, mathematician or otherwise, say that doing tangent by hand is easier. It isn't. A calculator is a tool, not a machine to replace our minds and steal the maths from our minds. Treat it as such.
I assume you mean "calculating tan(x)" when you say "doing tangent." Your definition of a mathematician is someone who crunches numbers into a calculator, so when I say that few mathematicians use calculators I can see why you think I'm silly. I will define a mathematician as anyone who has regularly done serious academic research in math, and although this seems to disagree with your dictionary definitions, I think it is far more appropriate (I guess I should stop talking here because it won't matter?). This includes applied mathematicians, by the way. In this sense, most mathematicians would think it silly to actually compute tan(x) if it doesn't come out as an algebraic number.

[Jazztastic]

I'm not going to try to retort, I have to applaud you for a well written arguement. You included both sources and even playing along with me, in questioning my own assertions. Nice catch. Everything that I have wanted to say on the subject has been said, and we have conflicting opinions. 'Nuff said.

EDIT: I decided to finish reading Lockharts paper, and one of the sentences you quoted stuck with me

"Why is it that we accept math teachers who have never produced an original piece of mathematics, know nothing of the history and philosophy of the subject, nothing about recent developments, nothing in fact beyond what they are expected to present to their unfortunate students? ... Now I'm not saying that math teachers need to be professional mathematicians— far from it. But shouldn't they at least understand what mathematics is, be good at it, and enjoy doing it?"

I couldn't have phrased it better myself. Everyone who dabbles in the art of math for a living should both fully appreciate math itself as an art, and attain pure joy from doing it. Great debate, I don't think I can contribute anymore though :)

[JetFangInferno]

Beef, the thing is some people who are good at math may not necessarily be interested in its history. Also, there's the difference between theory and application, as some people enjoy crunching numbers but dislike studying the theory, and there are others who enjoy the abstract nature of higher math.

[Silverfish423]

I think critical thinking is one of the most important tools that a person can utilize. The world is advanced through critical thinking, not repetition of what is known throughout the entire global community. This is why critical thinking (the process or method) is more important than the solution.
This is one reason why I think philosophy should be implemented into the public school system. It involves critical thinking and can be directed related to any facet of education (mathematics, science, music, english, etc).
Also something interesting is that some of the first philosophers, or atleast first well-known philosophers, were mathematicians.

[Crazed_seal2]

They should just teach mathimatica. Who needs to know what 1+1 is equal to when you can solve an advanced matrix, vector or differential.

Because in the real world. Dumb people exist.

[Zarathustra]

Math is the language of Nature. Much of the "art" in science seems to be the ability to pick up this language by immersion rather than from a teacher or a textbook.

Most people who do work in or study the "hard sciences" can tell you about some aspect of their discipline that they find particularly beautiful, unique, or elegant. What makes a particular experiment, technique, or proof beautiful is hard to pin down... but the same could be said of beauty in art.

[Rui]

So, after 6 months, I've finally decided to pick up this document again. I've been reading it and I agree that it's quite a refreshing point of view. I concur with much of what he's said, except this little bit:
(...) But if your math teacher gives you the impression, either expressly or by default, that mathematics is about formulas and definitions and memorizing algorithms, who will set you straight?
The cultural problem is a self-perpetuating monster: students learn about math from their teachers, and teachers learn about it from their teachers, so this lack of understanding and appreciation for mathematics in our culture replicates itself indefinitely. (...)
Is it? I think no one goes to study Mathematics in university unless they really like the subject. Do people end their courses thinking it's about memorizing algorithms, formulas and definitions?
Thoughts?

[Edhel-dur]

So, after 6 months, I've finally decided to pick up this document again. I've been reading it and I agree that it's quite a refreshing point of view. I concur with much of what he's said, except this little bit:

Is it? I think no one goes to study Mathematics in university unless they really like the subject. Do people end their courses thinking it's about memorizing algorithms, formulas and definitions?
Thoughts?

Depends on the country.

This problem exists in other fields not only mathematics, such as history for example. I myself know someone who finished the university of History and she just learned what happened and when. But she didn't learn the value of the actions which happened in history. And she isn't expanding her knowledge of history either.

I once went to the psychologist when I was in high-school for career orientation. At the time she was 24 and she just finished the university about two years before. So I went to her, went in messed up, went out messed up.

All I found out was that I incline towards social related stuff at 9/10, and towards art related stuff at about 8/10. That didn't help me one bit. She didn't knew how to guide me through my own thoughts, and my own thought process. She knew nothing about what she was supposed to do. All she knew, was to give me a standard test.

I'm 23 now, and hell am I a better psychologist then that girl. And I didn't even go to university for psychology.

Furthermore a lot of people just do a university for the diploma, especially where I live, since here university is cheap compared to other countries.

At the end of the day it's about two types of people.
Firstly it's about people who chose that university because they like the subject, they like the field of work, they desire to expand their knowledge in that area and they see a potential future behind the horizon.

Secondly it's about the people who do a university just because, they do it because they have nothing better to do, they do it because their parents tells them too, they do it to avoid working, etc... but they only manage to lie to themselves, and at best, become a mediocre person in the respective field.

--------

I haven't read the entirety of that PDF yet, didn't had time and forgot about it for a while too, but I guess I didn't need to in order to answer Rui's question.

[Magtheridon96]

Do people end their courses thinking it's about memorizing algorithms, formulas and definitions?

Memorizing? Math is logic ;D
If you know one or two formulas, you can derive a thousand others!

[Edhel-dur]

Memorizing? Math is logic ;D
If you know one or two formulas, you can derive a thousand others!

Exactly, and since Math has a direct correlation to Logic, just look at how many people are Illogical! Thus they do not understand Math. Thus this kinds of people simply learn all the formulas. If that's not why than it's because they don't actually care about Mathematics.

[Rui]

In my country, there's a certain pressure from our adults for their children to get university diplomas too, but I think people will usually go for areas that they like, even if there is not enough job offer. Maybe that's why there are so many people who, even though having recently obtained a degree, are unemployed.

[Gilles]

The same can be said for education as a whole.

Interesting video I watched a while ago.
Studio Schools (http://www.ted.com/talks/geoff_mulgan_a_short_intro_to_the_studio_school.html)