Slashdot Mirror
Mathematics As the Most Misunderstood Subject
Lilith's Heart-shape writes "Dr. Robert H. Lewis, professor of mathematics at Fordham University of New York, offers in this essay a defense of mathematics as a liberal arts discipline, and not merely part of a STEM (science, technology, engineering, mathematics) curriculum. In the process, he discusses what's wrong with the manner in which mathematics is currently taught in K-12 schooling."
Commenting on something near the top of the list.
Mathematics is the foundation for philosophy, not technocracy. What a better world we'd be in if we were motivated by the former rather than pursuing the latter.
2 + a = 3
a = 3 - 2
a = 1
Know that, and the world is your oyster !!
Math misunderstood because it's hard, and that's why people have misconceptions about it. Understanding of math require considerable effort and concentration which most people tend to avoid if possible.
I can attest that "true" math is very removed from computation. The computational classes are all regarded as the "easy" classes. This is in contrast to the "hard" classes, real analysis and abstract algebra. Being thrown into real analysis after just one quarter of study in proofs is extremely rough going. If proofs were introduced as puzzles or just introduced earlier in education the whole of America would be better off for it.
My own motivations for being in math are for the challenge and because of the lack of concrete answers in calculus. Trigonometric functions especially are always treated as little boxes that magically calculate what you need.
In any case, at least math attracts the curious.
Eat sleep die
I've seen the following link in many a Slashdot thread before, but it certainly bears repeating here: "A Mathematician's Lament" by Paul Lockhart It's mostly known as an insightful critique of what's wrong with K-12 math education, but I've always liked it as an explanation of why people who enjoy math do it in the first place: it's satisfying in an artistic way. I think it would be great if more students saw math as something worth doing for its own sake, like art or athletics, and hey, it lets you do science and engineering too.
In fact, this summary sounds similar enough to "Lament" that I wouldn't be surprised if this Dr. Lewis was inspired by and/or cited it. But this is Slashdot, so I'll let someone else check that out.
"This algorithm runs in constant time. Come on, 2,147,483,648 is a constant..."
I have a cousin who is great at mathematics, and really can see mathematics as an art. Whereas I am happy if I can solve a problem, he will look for an "elegant solution". I had a number of equations that I solved, trying to optimise the buffer size for various input queues. I shown him, and he quickly said that I had the right answer. A day later he came and shown me how he derived an equation that could simply solve all problems of this type. He also generalised it to allow buffer sizes that were complex numbers. The first part was very useful to me, the second absolutely useless - but to him it was all just interesting.
This is one way that mathematics as an art is unlike any other art. It gives useful results. I have heard time and time again about engineers going to the mathematics department of a University asking how they can solve a "new" problem - to be told that the solution had been discovered a century before. I am sure most of these solutions came from someone just wanting to find an elegant way of expressing something without thought of any use. So if its an art and is useful why do so few people follow it?
The answer is obvious, because its hard! In many forms of art you can slap anything down and convince someone that it has value and its art. This may not always have been true, before photography accurate representational art was highly valued - but today someone producing a lifelike portrait will not be values as much as someone slapping their name on an unmade bed! Mathematics has to be right, you can't just slap down a few numbers and call it an equation. This is the basic problem that anyone will have in persuading someone to follow maths for its art, there are a lot easier ways to become an artist.
I wish science in general was considered part of what a learned person has to know. I mean, if you want to pass for an intellectual you have to read your Dante, your Beckett and you at least need to know who Lautreamont was. But, apparently, you can very well get away with thinking that you can suck gravity out of a room the way you suck air, or with not having even heard about string theory. That divorce makes no sense, and it was impossible in the history of ideas till very recently. And Euler's formula is more beautiful than most poems.
This is by far the best defense of mathematics I've ever read. It's a shame that the poor quality of grade school math education has made it necessary, though. Can one imagine a similar essay on any other subject? Only math is so poorly taught.
-- The parenthetical comment "(if it was done right!)" in "Ready For The Big Play" should, of course, be, "(if it were done correctly!)"
-- References in "Cargo Cult Education" to the "south Pacific" should be to the "South Pacific"
-- Also in "Cargo Cult Education", "But of course nothing came. (except, eventually, some anthropologists!)" should be, "But of course nothing came (except, eventually, some anthropologists!)."
Simple that... I honestly can not understand where there can be "beauty" in a mathematical expression that covers the entire blackboard. And more so when the teacher fails miserably to show practical uses for the expression.
Religion: The greatest weapon of mass destruction of all time
I don't know about the rest of the world, but where I live inspiring math teachers are rare. I've had one, but consider myself one of the few lucky. A new approach on teaching mathematics with base in understanding and application would be great. This would of course require much more from the teachers than before. They would actually have to understand the math themselves. Can't really blame the good mathematicians for not wanting to end up as low wage teachers. Maths really are hard too, and all about reasoning and understanding, which you really can't teach, only help by guiding. Even though I'm studying maths at university now, I wouldn't really trust my self to tell a kid what maths really are about (I have my own ideas though, like everyone else).
Also, maths aren't for everyone. While I believe many would enjoy a different approach on the subject, there are enough of those who just don't have any interest in it (or science at all). The learning curve is also much independent, and to include a whole class of students you would have to ignore both those who learn too fast, and those who just don't get it. You can't really hold all your students back to wait for someone to have their own little revelation. So, while I find the current approach on maths somewhat ridiculous, I can certainly see why it's ended up as it is. I feel he really nailed the problem, but what should we do?
I remember been taught differentiation at school – One lesson, lecturer puts a parabolic curve, x=y*y, on the board, and asks the problem, determine the angle of the line
Then, he didn’t say anything else.. Just, for the rest of the lesson, responded with ‘Yes’, ‘No’, or ‘Maybe’. So, after a frustrating 20 minute discussion, trying to work out how the hell to do this problem, someone came up with the idea of adding a ‘little bit’ of x, to x..
We worked out, as a group, the concept differentiation, with only the smallest bit of guidance from the lecturer. This is how things should be taught – allowing people to discover concepts themselves, rather than preaching the correct ways to do things.
"A Mathematician's Lament", an article that's been making the rounds among mathematicians since 2002 (but was only published in 2008), expresses some similar views, and is also a good read.
10 PRINT CHR$(205.5+RND(1)); : GOTO 10
Recommended and relevant reading is "A Mathematician’s Apology" by G. H. Hardy.
Available online at http://web.njit.edu/~akansu/PAPERS/GHHardy-AMathematiciansApology.pdf
Opinions expressed above are mine, and not my employees'.
As a standalone subject Math is too abstract, the relevant areas should be taught along side the subjects where it is applied. I think a lot of degrees could be cut down to a year or two. Cut out all the crap that no one needs to know and make that part of a further research degree. Revolutionize the education system with my ideas, or fall behind when I move to China and they listen to me.
The second part is the "live" Mathematics, i.e. the process of actually doing Mathematics in the sense of figuring something out. That's a slow, arduous, iterative and groping process. Starting with an observation that confuses or amazes us, incrementally and tentatively formulating concepts (definitions, constructs of previously known mathematics), their properties (sometimes axioms but mostly properties of known constructs), drawing inferences from those concepts, seeing if they throw light on the situation, and going back to changing the concepts if they don't).
Where the second part is like mental rock-climbing, the first part is like a list of views that were discovered by rock-climbers but which which can now be reached by cable-car (or bus).
For better or worse, the mental rock-climbing takes more talent and dedication on part of a student than about 75% of them have. And even talented and dedicated students will take thousands of years (about two millennia to be exact) to reinvent Mathematics on their own (so much for "Letting students discover Mathematics on their own").
We therefore tend to teach the finished results because they are (a) enormously valuable insights (b) useful in other subjects, and (c) accessible to someone with a modest amount of perseverance, an adequate memory, and ordinary talent.
The problems really start when people (education boards) fail to distinguish between the two forms of Mathematics and neglect to clearly set out the goals they want education to address. Which then results e.g. in them insisting on letting students memorise the square-root formula for quadratic equations instead of teaching them how to solve a quadratic equation through simple algebraic manipulation (which also gives people a bit of insight in what they're doing) and letting them look up the quadrature formula when they need it.
Hi,
in school mathimatics is mostly execution of algorithms provided by your teacher, learning when and how to apply them. This changes a lot with university. At first, mathematics is a language to be learned. You have to be able to express your problems in a normed language. This is the first art. If you read papers, you can distiguish easily between those peoples who truely have mastered that language and those who don't have. Later on, you learn how to prove things. The interesting things you cannot prove by just applying an algorithm. At that point you need a lot of creativity, which the second art form required by a true Mathemagician.
CU, Martin
...is the loneliest number.
"Meanwhile we have turned the majority of Western humans from independent men into chair-warming consumers singing in lockstep for trinkets."
I suggest you take off your rose coloured glasses and go read some history, in particular just how "free" your average serf was in feudal times and even later. Don't like what your overload or king does? Tough. Complain and you'll probably at best end up homeless or at worst end up swinging from a tree.
People in the west have NEVER been as free as they are now.
So get yourself a fucking clue!
What is being said in the article is almost verbatim true for physics as well. Poorly taught, even more poorly understood by almost anyone (including teachers).
Mathematics is much more fundamental than physics, no doubt. Very good points are made here. But (ironically?), you could replace "math" with "physics" in the article and most of it would be true just as well.
I don't hold hope that especially Americans will ever get this, though, either for math, or for other STEM fields. Because it's "too hard". Being ignorant is just too damn easy, if you ask me.
Heck, I'd be happy if people at least would get math.
Do your own thing. And overdo it!
A tool is a man-made thing (even if it a rock to chip flints with it is selected and used in a way that is man-made). The cave-man who acquires a better hammer rock is naturally pleased and proud of it and will either imbue it with magical qualities (God-Nature you see) or appreciate its qualities as they matter to a flint-maker (weight, hardness, fit in the hand etc.). The latter is just a 'beautiful' as a clever team manoeuvre to score a goal, or the technology that goes into making an affordable, low maintenance, lightweight bicycle. Of course you have to 'know what beauty looks like' - Those ingredients that make you most proud of your tools and achievements.
I don't think anyone was claiming that 'expressions all over the blackboard' were beautiful... ...but the conclusion may be, and the lead-up to it may be a guide for our own explorations.
FWIW here is my analysis of levels:
Somewhere, possibly after school, especially in old age, people need a sense of 'be safe with numbers, statistics and graphs'
My highschool math teacher was a retired NASA programmer. According to her, teaching Mathamatics was about leaching logic and problem solving. If you forgot all the formulas taught in her class, she said, it wouldn't matter. The real skill learned was how to deal with an entirely new mathematical problem. WHY is area "height x width"? How to build your own sort of equations. Sure enough, decades later I have forgotten every single equation I had been taught there, but when faced with a logic problem I'm still able to work it out.
I can very well relate to this post.. the foremost reason for Mathematics being misunderstood is the problem with the way it is taught in schools. Right from childhood you are told to mug up the multiplication tables, formulas and everything is told be "it is like that.. just remember". The flaw is with the education system where stress is not to "understand" and see things logically but on how much can you mug up and pass those tests and get a so-called "good score". The teachers need to be trained to generate interest, talk concepts and not just ask to be ready with the multiplication table the next day for the test.
defense of mathematics as a liberal arts discipline, and not merely part of a STEM
Not part of "STEM?" It is clearly the "M."
Case closed.
In practice, there are two forms of teaching. The first is applied subject matter in school. In this specific case, it is applied mathematics. They give you the calculation tools for describing a relationship and then they expect you to find similar relationships and apply that formula. The goal is to teach the use of a tool. It is no different than teaching one to write a coherent paragraph, communicate in a foreign language, or to be a good citizen in a democracy. Teaching applied mathematics is a necessary element of any school curriculum.
The second is one of discovery. My journey began as a teen, when I read about fractals in an article from Scientific American. Since then I've gone on and explored prime number theories, methods of calculation, the history of these discoveries, and I've gone looking for the blind alleys that may not have been explored as thoroughly as we might think.
We need to recognize that education is not about discovery. It is about teaching a person the tools of modern society. However, in our zeal to teach the applied aspects of these subjects, we need to realize that we are failing to nourish the creative spirit of discovery. Mathematics is no different than reading, writing, civics, history, geography, or language. Learning to write a coherent text does not make one appreciate literature.
Our schools are obsessed with application, not discovery. We spend ridiculous time teaching application, application, and more application. Then we sit and wonder why our children lack the will to explore...
Nearly fifty percent of all graduates come from the bottom half of the class!
This article frustrates me. He talks a lot about some particular thing, claims that it relates to maths, but doesn't really say what particular part of maths it relates to, nor does he get into specifics, nor does he spend much (if any) time on how to improve matters.
Okay, I'll try to explain my confusion with a parable. When I was fifteen, I did a school certificate maths exam. It had a whole bunch of questions, none of which we had ever answered earlier in the year, but somehow the examiner thought I could answer them, and unfortunately I was unable to answer all questions "correctly" according to the examiner.
What does that have to do with mathematics education over the past 25 years? Unfortunately a great deal. We were required to have exams for mathematics, because every subject had exams. The end result was that some people didn't do well in exams, even failing enough to be unable to continue on in their maths education in the next year. The truth is that exams cannot alone be used to evaluate a person's effectiveness as a mathematician. The only way to get around this is to teach mathematics properly, and make sure each person understands maths at all levels.
Ask me about repetitive DNA
2 + a = 3
a = 3 - 2
a = 1
Gosh, you're making quite a lot of assumptions here, and you've changed where the (-2) goes, which is not necessary. Here's a slightly more detailed expansion:
2 + a = 3 [assume addition exists for 'a', follows standard rules of integer addition]
-2 + (2 + a) = -2 + (3) [assume can add -2 to LHS of both sides of '=' without changing equation, define -2 as the additive inverse of 2]
(-2 + 2) + a = -2 + (3) [assume order of addition does not change outcome]
0 + a = -2 + (3) [additive inverse + number is additive identity, 0]
(0 + a) = -2 + 3 [assume shifting brackets doesn't affect outcome of addition]
a = -2 + 3 [addtive identity + number can be represented by number alone]
a = -2 + (2 + 1) [assume 3 is the successor of 2, where the sucessor is generated by + 1]
a = (-2 + 2) + 1 [assume order of addition does not change outcome]
a = 0 + 1 [additive inverse + number is additive identity, 0]
a = (0 + 1) [assume shifting brackets doesn't affect outcome of addition]
a = 1 [addtive identity + number can be represented by number alone]
Anyone want to expand further on this?
Ask me about repetitive DNA
Here's your first assignment!
http://www2.b3ta.com/namethatbeard/
My first Journal Entry ever, in 8 years! http://slashdot.org/journal/365947/aphelion-scifi-fantasy-horror-poetry-webzine
Memorization is needed and sometimes you just have to accept things as the way they are. Example: I understand that a^2+b^2=c^2 is true for right triangles. I don't understand why the areas add up. I just accept it because it works all the time. Another example, I don't know why the volume of a cone has the 1/3 term. I simply don't. I can't derive the formula. But here's the important part: you don't have to be a good engineer or physicist, two math heavy occupations, to be able to derive the formula. They just need to know it. This applies to every other occupation.
Which brings me to the next point- the education spends a lot of time on BS. Formal proofs aren't so important in geometry. It's just tedious work. Another example: those weird boxes we're forced to do back in elementary and middle school. The theory behind doing that is so that kids learn how to write better sentences. The problem is that it doesn't work. Language is understood because children imitate their parents vocabulary and grammar. K-12 cirriculum could be easily cut down at least 2 years with more time spent on the basics and we would turn out better graduates.
Really. Must we contextualize mathematics, or try to talk about what it is or is not? Do we really need to point to a particular cognitive framework as "the reason" why math is not taught "properly?"
To use a slightly loathsome phrase, math "is what it is." Instead of talking about how people should relate to it, I suggest a radical approach: just LEARN it. Teach it for what it is.
I struggled with arithmetic when I was in grade school, not because I didn't understand the rules, but because I kept making mistakes. And my teachers had the wisdom to know that those errors had to be drilled out of me before I could proceed any further. I suffered. I *hated* the tedium. We were asked to multiply two twelve-digit numbers with no assistance from any computing devices or tables; divide four-digit numbers into twenty-digit numbers, until we could do it with 100% accuracy every time. It didn't have to be lightning fast. It just had to be CORRECT.
And when I mastered that skill, it felt fantastic. We moved on to more advanced topics, and each time the teacher made sure we had firmly laid down the next conceptual brick of this vast mathematical edifice we were building for ourselves. It was hard but rewarding. To those critics who might say such an approach would discourage some students, and that some kids just need to be excited by what they learn, clearly you have never really understood what it means to build that foundation. It's got to be ROCK SOLID. No crap about trying to make math "fun" or "interesting" or "relevant." That sort of stuff comes when it comes; they are merely ornaments on the pillars. There's no point in making the structure pretty before you make it sturdy.
So then, how do you get students motivated? It's really quite simple. You challenge them and you force them to bust their asses, and when all their hard work pays off, that sense of accomplishment is better than any drug. To know that you did it on your own, and you have complete confidence in your mastery of the concept, is precisely what must drive them forward. You can't entice them with anything else. You can't try to swaddle the math in some cutesy real-world application, because that is going to be fake, and they know it.
That's the story of how I graduated with my BS in mathematics from one of the most prestigious scientific universities in the world. It was purely the early appreciation for persistence toward understanding mathematics for its own sake. I'm not saying everyone has to keep math "pure." If your goal is to apply it in some other discipline, go for it. But the learning process has to build upon that foundation of math for math's sake.
My older son is in the 2nd grade and is gifted (IQ somewhere around 140). Right now, they're learning simple addition. There's only one problem. He already learned this last year. He was doing complex subtraction with my wife (a teacher) over the summer break. But the class is doing simple addition so that's what he's stuck on.
It gets worse. They're using a so-called "spiral curriculum" this essentially means they learn one way of figuring out that 8+3=11, then learn another way, then a 3rd, 4th and 5th way. My son gets it the first time, yet he has to sit through all of the other ways. He yearns for more advanced math. He asked me about multiplication and division and, when I showed him an example using Legos, he got the concept right away.
He already knows his times tables up to 5 and wants more. But school is boring to him because they don't push him. He isn't being challenged at all. He tends to act out when he's bored too which makes everything more complicated. If you have a child who is falling behind in school, there are resources to help them catch up. If you have a child who is gifted and wants to pull ahead, your kid needs to sit down, be quiet and learn for the fifth time what 8+3 equals.
My sci-fi novel, Ghost Thief, is now available from Amazon.com.
just checked. BS Chemistry, BS Mathematics.
46 & 2
WRT tfa, wtf?
I tried to RTFA, but after the 3rd or 4th "parable" gave up on it: tl;dr.
I would appreciate it if anyone with analytical training and some skill in developing succinct expressions would tackle the material. I sense that there may be a valuable and elegant concept just a layer or two under the dross of the current presentation. But bringing that concept to the surface and expressing it properly requires the kind of trained mind that is product of a sound schooling in the liberal arts. That apparently is very rare among the mathematicians of our time.
So would one of you guys who knows how a metaphor is like a simile care to respond with a Readers Digest Condensed Version of TFA? I am sure that I am not the only /. minion who would appreciate that.
Will
...in this comment 13 years ago; just not so eloquently.
its is applied knowledge and skill of your knowledge its hard bcause you wish not to try. YODA says do or do not there is no try. math teaches persistence in solving. ironic that the USA is losing on world scores and like a failing business its only recourse is laws to restrict and lawsuits that is truth my truth your truth and the real truth
Math is perceived as some sort of torture in school which serves no real purpose. But math is a formal language which can be a very powerful tool to do a lot of things in different fields. It can help to describe things very, very precise. And because certain properties of mathematics have been analyzed and though about so much, a wide variety of procedures and knowledge exist which can help in deduction.
Set-theory is for example widely used in computer science or biology or medicine to describe knowledge in form of classes and properties and sometimes even rules for these classes and properties. But even when you look at a classic point an click desktop OS. These use folders and files to describe structures which are no mare than sets and elements. And in new system people use tags. Tags can be understood as sets or classes (in a mathematical sense).
Math can help you to find the cheapest pizza fungi-salami in town and it can help you to project the revenue of a new product in your store. As you can first monitor previous sales and establish an curve for normal product sales from the point of introduction to the fading out. This can be helpful to find the right time to order new items.
You can use it with all the FB friends data to learn who you should target with advertising. You could evaluate talks of politicians and prove that they are lying which is much better than just having a feeling that they are telling a lie. The difference is, that you can understand the inclined hidden messages in a slogan, you can put the finger on it which is a big difference to that feeling where you will still memorize parts of the message an believe them.
And math is in general a good training thing for you thinking muscle.
This is an interesting essay. Dr. Lewis does a great job of pointing at and philosophizing about the math education problem, but like the math teachers he assesses, he inevitably fails. He sheds light on the fact that math is misunderstood. He also explains how its teaching is flawed and which educational practices have led to this state. However, he fails to provide a way to remedy the problem. I think, concerning the idea of math as a liberal arts subject, that most educated people understand the intellectual growth and logical benefits that come from studying math. He says, "Mathematics is not about answers, it's about processes," I think most people understand that as. And furthermore, I think that may be precisely why a lot people don't like math. By that definition it is inherently tedious.
I've always thought of art as something useful which gives useful results. The only problem with art is that the results are not often predictable or measurable. The way you describe art is the same way a layman describes math. But I'm not here to argue the merits or flaws of math, because I agree with you (it is elegant; it is complex and yet simple), but instead I just want to correct some of your misunderstandings of art.
How does one measure the "usefulness" of a field? By the amount of money it generates (monetary); how many people sees it (popularity); how it can be automated or re-created consistently (made into an algorithm); difficulty to master or innovate ("because it's hard!")?
1. Money - most people don't consider applied arts as an art, especially among established gallery artists, but they undergo the same requirements in art school. They must be able to draw and communicate visually. This may include an illustrator or an actor: one communicates with his product and the other with his body. You can see graphic design everywhere just walking down the aisle of your grocery store. Each packaging was designed by at least one person. If it has photography, then a photographer was subcontracted by the dept and he probably had a small army of assistants. If a model was used, then there was a casting agent, make up artist, and most likely a stylist. Why? Because most people can't do all of these by himself. They're all art in its own right. And if they were truly "useless" as stated, I would much rather pay them nothing. But an assistant cost about $300 a day; a model, $2,000/ day (mid-sized city).
As for actors, they don't make much in the median, except for the top earners in Hollywood who makes the vocation seem attractive. But for every person that succeeds, I think 10,000 fail. But as a whole, from the director to the grip (which I've worked alongside), most of us went to art school in one way or another. But I'd say Hollywood as a whole makes quite a bit from the box office if you don't count their accounting shennaniganism (kudos to math).
2. Popularity - It's everywhere, from every packaging to websites and logos. We just don't notice it anymore because we're visually flooded. IBM's logo(s) designed by Paul Rand is simple and elegant, despite the massive constraints of a corporation and branding guidelines. And if you're think it's easy or cheap, just take a look at the costs associated with rebranding Tropicana or Gap - and its subsequent backslash. In the same way most people don't realize how much math and logic is in a Quartz wristwatch and take it for granted.
3. Consistency – I think that math tends to be very consistent, precise, and elegant in its own mechanical way. I also think (and correct if I’m wrong) that upper levels math tend to be organic in the way that numbers and functions serve as nouns and verbs. It essentially becomes a complex language about the world.
Art is opposite: it starts out as organic and disorganized, but as an artist develops his style, it becomes difficult to be consistent (or rather, it is difficult to develop a style). Granted, anyone can take a photo. It’s easy. Just press the button. What’s hard is making it look good and different, and yet consistent to your style every time. Or take a chef, for example, it’s no surprise he can make the perfect flatiron steak. Even you and I could do it. But what makes him a “pro” is that he can consistently make it right 100 times in one evening, every night. Now add 20 other menu items. It may be rote and mechanical, but to the culinary world, it’s an art.
4. Innovation – Creating a new art style is hard to say the least. To be synonymous to an art movement is the equivalent of earning a Fields Medal. Marcel Duchamp with the Dadaists, and Picasso with cubism (these are the only two that comes to mind). If art were easy, name 5 famous living sculptors. There’s a lock of muck to go through to find a gem, as I’m sur
Know that, and the world is your oyster !!
but what is this for ?!
that's one of the main problems of maths education:
not enough problem solving. Not enough application of the math tools to the real world.
when you remove that, all that is left is rote memorising of formulas (which is as interresting as a phone book) and monkey-training to calculate abstract stuff.
no challenge, not interesting, seems useless. With no incentive to work, it looks difficult, because it's hard to master.
that said math isn't the only affected course, and non-science courses aren't magicslly immune to this.
as a personnal example: dead languages. I studied both latin and old greek.
what's the point one might say ?
well that's indeed what one might ask after the kind of latin teacher we had half of the time.
with them, latin was just about memorising declinations, grammar rules, and translation of a few synthetic sentenses.
on the other hand the greek teachers were passionate about their subject. It was also about civilisation (whic is pretty much damn interresting in it self), linguistics (wich come pretty handy to learn any other language, living or dead), authors and their philosophy.
same difference exist between "please solve the next 20 equations" (math badly taugh) and showing that math is a use ful tool that can come handy, that there's an inner beauty to it, etc.
interstingly enough,the former is how i was taugh math in Switzerland, whereas the later was how my parent learned it in eastern europe, and paid attention that I didn't miss this part.
"Sufficiently advanced satire is indistinguishable from reality." - [Tips: 1DrYakQDKCQ6y52z6QbnkxHXAocMZJE61o ]
Being blissfully ignorant or believing that I am anyway. Look at the above there are like 400+ comments from people who do not know anything who will get enraged that someone might suggest they do not know anything and who will likely mount a staggering defence to prove the contrary. I submit that we know jack shit. We know what other people know and we believe in these people because they are capable fo convincing us that this 'is' a statement of fact, generally rooted in logic. Fans of logic are unwilling to grasp the concept that their terminology their entire pattern of thinking is simply a construction of the general evolution of thought. Humans have to explain everything, label it all and categorize it. Anything that would contradict this pattern of stability is met with vehement denial until someone else (who equally knows nothing) discovers more facts to back up their hypothesis thus debunking earlier 'facts'. lol it really is quite amusing seeing people armed with multiple degrees and flaunting their high IQs believing in what they are saying.. Which, if your a little odd like me, is tentamount (sp) to a child playing with colored blocks and then taking a tantrum when their mom or dad puts them in the proper order. The difference between what we know, and what we dont is as comparable as the difference between apples and astrophysics. This is just my opinion I could be wrong, and I am sure people are gripping their degrees in their hands right now sputtering in contempt. lol but if even one person is going "holy shit thats as plausible as any other philosophy ive heard" than it was worth the 5 minutes it took to write this opinion.
When you dislike the human race as much as I do, Karma:Bad is inevitable lol.
When I was in second grade my best friend and I enjoyed studying mathematics. At one point we'd worked many pages ahead in the class's math workbook, and had fun.
But then Mrs. Cooper, our teacher found out. We got in trouble and had to go stand out in the hallway, and then after eating our lunch stand up with the other troublemakers against the wall instead of getting to go out and play.
Well, I learned my lesson from that. Don't get ahead in math!
"Look, it's either 'true' or 'false'."
"Not necessarily."
if your lysdexic.
Can my karma get any worse than bad? Let's find out!
I am posting this anonymously because I use Math in my job.
Math is a series of formulas and numbers. It is provided by some great Math Genius who tells me that to draw a rectangle on the screen I use this formula. I obtained an advanced degree including DiffyQ and Advanced Stats. What does it mean to me? Squat. A calculator helps me more than a lot of things and why is this? BECAUSE I cannot see beyond the numbers and formulas. I speak to others about this for a long time, How does Math actually describe a Rectangle? How does it describe a Black Hole? To me, Math is an invention to try and talk about things and not really anything else. Math is also assumed to exist since its based on a point which is by definition "Assumed to exist". I can prove 1 does not equal 1. I can do all these cool formulaic tricks. Oh goody, more math.
A train goes from X at Y speed, when does it crash into the Sun? Thats the leap it appears that Physics makes to me. I can generate an algorithm for just about anything and implement it. Does it mean I understand Math? Math is a language. Math describes the Universe. Math is Math. We are engineers and we are expected to love Math. I don't hate it, but I don't love it.
Its more important to learn Logical Fallacy than to discover that a Transform makes derivatives easy. What the heck are derivatives really anyways? Heck, what is a darn x^2 to 2x really mean?
Example: Doing a problem in Theory of Computation that came down to a x^2+2x+b equation. Then determining the answer was... is this a greater than Zero equation? Since no one had told our group before, we were stumped. our Math genius prof (at the time) who had a Phd in Math just threw up a graph and graphed out that of course if the equation is positive its greater than Zero. He broke it down and did it, and did it all like we were supposed to KNOW that we can do something like that. The WHOLE class (64+) were stumped basically because he assumed we were taught Calculus in such a way we can Understand the underlying nature of Math. Nope... and still dont.
Einstien was never taught algebra so he LEARNED Math from the other side, the language and not the forumulaic way we were taught and still are. Yes, Its been 25 years since University for me alone.
To know the difference between a million and a billion. Maybe a semester-long project to fill a swimming pool with a bucket?
"People in the west have NEVER been as free as they are now."
Eh, that's pretty iffy.
It would be more accurate to say that people in the West have never been better off in terms of material wealth, true. We've never had as high a level of technology or cheap access to gadgets or advanced medicine.
But free? I guess it depends on your definition of freedom. We're certainly more free than the Russian serf of the 1700's or the Spaniard under the Caliphate of the middle ages or the Greek and Serbian living under Turkish rule before the 20th century. But the homesteader in 1800's Oklahoma or Nebraska had far more freedom than you'll ever have, simply because the laws that governed him could be read, from beginning to end, in a matter of minutes. He didn't live as long, have cars or the Internet, or run up a huge Mastercard bill. But also he didn't have anyone telling him how fast he could ride his horse, he didn't have a "homeowners association" suing him for the color of paint his chose for his humble home, and the government wasn't trying to "help" him by taking half of what he earned and spending it on services he didn't ask for. He had to face the big bad world all on his own, but they were his choices.
I don't think many people want to go back to a horse and buggy, but at the same time it's patently silly to talk about how free we are when our government has re-defined freedom from "freedom TO" do things, and now regards it's role as "freedom FROM" things, "protecting" us like a nanny looks after a child.
Life is hard, and the world is cruel
Math and Physics are beautiful in their generality. It's difficult to fully appreciate the generality, however, without having seen a lot of specific examples. That's why it's important to complete so many problem solving exercises. Math and Physics are taught that way but you really have to do the homework to grasp the subjects. It hurts the brain sometimes but it's worth it. Dr. Lewis attributes misunderstanding of math to poor teaching practices but I attribute it more to human laziness.
The type of advanced symbolic logic you mention is a relatively late development.
The massive movement toward emulation of physics in philosophy is older. It goes back, at least, to Newton and the success of the mechanistic world.
More people should read The Glass Bead Game by Hesse. Not to sound like a prat, but then this is obvious...
Can someone with Free Will invalidate Free Will? I'm asking this because of the observed belief by many behavioralists (starting with that bastard Watson) that there is no such thing as Free Will and everything is deterministic.
Or, can a successful proof for Hard Determinism be made?
Here's to hot beer, cold women, and Glaswegian kisses for all.
Here a new subject for you: tomatoes vs rainbows. Go
Double Rainbow, what does it mean?
Surprised nobody has posted this yet.
For all intensive purposes, "whom" is no longer a word. That begs the question, "who cares"?
The only thing you've shown is that such a simulate function cannot exist, or at least cannot complete execution in time to affect the state of the world at T1. This follows from the fact that in order for the simulate() function to be a true simulation then it must also simulate itself. Put another way, if simulate() were to compute an accurate result, then it must include its own effect on the future state of the world. In essence it's a little paradox machine.
So much for your plan of replacing philosophy with computer science.
That's a remarkably dense collection of bullshit you've managed to put into that post, and it would be hard to untangle it all. I'll just scratch at the surface:
Are you adequate?
In philosophy a bunch of people agree that some one was/is a great philosopher and so they give more value to a statement from such person. The credibility flows from the speaker to the statement.
This is what always drove me up the wall in my philosophy classes. I remember reading Descartes' Meditations on First Philosophy and thinking "Ok, sure, the arguments that I could be misled about existence are decent, and sure, I have to exist, but everything past his second meditation is refutable". Maybe it's a matter of not having been born in an age where god is taken for granted (or at least those with opposing viewpoints aren't killed/tortured/ridiculed), but the arguments are just plain weak.
You've reached an odd combination of (a) getting the point and (b) missing the point that you've gotten the point. What you've stated is, very succinctly, what the subsequent Western philosophical tradition thought about Descartes ever after. When you read all those other post-Cartesian philosophers, it helps enormously to understand that they more or less agreed with Descartes' first meditation, but thought the subsequent ones were weak...
Are you adequate?
I think it depends on where you go to school. Where I went, you could get either a BA or BS in math. If you took math + a bunch of liberal arts stuff, you got the BA. If you took math + a bunch of science stuff, you got the BS.
Oh, and your exercise in hippie-punching regarding environmental engineering just makes you look uniformed. There's plenty of biology, chemistry, physics, math, statistics, and geology/oceanography/meteorology involved in that sort of a degree program. It's not all about smoking dope and communing with nature.
The problem I see with the way I was taught math (and my kids even today) is that it is taught without context. In the Karate Kid, whats-his-name was assigned seemingly pointless tasks. Although each task taught an essential skill that would eventually be used to kick the punkass loser's butt, it was not clear at the time and even discouraging.
Math is taught the same way...a seemingly endless series of tasks that have no tanglible result other than to say yes, you got it right.
Perhaps it would be better to have math taught in a project manner. Show the cool things that can be done with it such as a computer game or airplane simulation or climate simulation. etc. then explore the math behind it and start with the basic mathematical concepts required. That way there is no "I will never use it" because, yes, you will...and soon.
When Fascism comes to America, it will call itself Anti-Fascism, and tell you to give up your guns.
Reminds me of this essay: http://www.maa.org/devlin/LockhartsLament.pdf
I completely agree about needing a basic understanding of reading and writing math first. I haven't seen the inside of a math classroom in many years, and my work doesn't involve a lot of "heavy" math, but I use basic algebra at work fairly regularly. Recently, an acquaintance of mine in a college algebra class commented that she understood most of the class really well except for the "word problems." I hadn't heard that phrase in so long that I had forgotten it existed. All the math I do is "word problems" and it is utterly useless to me otherwise. It astounded me that someone could consider themselves good at math when they didn't even have the basic skill of being able to transcribe back and forth between math and english.
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I think the problem is that we teach "how" and not "why". We tell kids that pi = ~3.14, but we don't tell them that it is how many times the diameter of a circle will wrap around the circle. And if we do, we don't tell them why that's important. We tell them what they need to know to do the math, but not the underlying concepts they need to understand the math.
Coder's Stone: The programming language quick ref for iPad
I think a lot of the frustration the GP feels with philosophers is that they spend a lot of time arguing about these subjects without even coming to a common understanding of what they mean by their terms. If they would start by rigorously defining "be-a" and "has-a" (which in practice is wrapped up in the process of defining the object hierarchy) they could spend a lot less time going in circles. As it is, they're talking past each other.
Another issue is the fundamental untestability of some of the propositions involved. Some of these arguments, at least to the layman, appear to be nothing more than assertions - or at least, assertions rather poorly clothed in reasoning that uses (again) ill-defined terms. And if there's no way to validate the reasoning, what are we really doing here? For example, Plato says that actual objects are just instantiations of true forms (speaking of problems that Computer Science has solved... object-oriented programming, anyone?). Other philosophers say that's not true. So, who's right?
I tend to find these questions interesting myself, but there's certainly appears to be at least an element of mental masturbation involved in some of this.
I believe this is true as far as it goes - hunter-gatherer societies were on average better nourished and did less work than early agricultural societies. But the variance in availability of calories to the hunter-gatherers was a lot higher - to the extent that if you were a hunter-gatherer, you were actually significantly more likely to starve to death during lean times than if you did agriculture.
Unfortunately I can never remember where I read this, so no link.
The mindset of the teacher reminds me of the Harry Chapin song "Flowers Are Red."
Teachers that are that narrow minded should be transferred to places where they can't do any damage to students. Perhaps a prison environment would be best for them. They could at least try to help some of the people they screwed over.
Some in their 50s or so may remember "New Math", which was an attempt to teach elementary math with more emphasis on the underlying theory. It's now widely considered to have been a disaster. The author of the original article seems to date from that era.
One of the approaches to fundamental mathematics is to start with axiomatic set theory and build up from there. (That's not the only approach; one can also start with the Peano axioms and build up to set theory via lists, as is done in constructive Boyer-Moore theory.) This is minimalist and elegant (which is why mathematicians like it) but it requires considerable theoretical development before you get to addition. Teaching kids arithmetic that way was a disaster.
Euclid's approach to axiomatic geometry is like that, too. There's a lot of abstract logical structure that has to be built up before you can do anything. That's how math was taught up to 1900 or so, and 7th grade geometry is still often taught that way.
That's the "liberal arts" approach to mathematics. It's an intellectual exercise forced onto little kids. Even if you use advanced mathematics in your work, it's very rare to need either axiomatic set theory or axiomatic plane geometry.
A completely different approach can be found in some math courses given during WWII courses to soldiers who needed to do technical work. These were utterly practical. Trigonometry was taught with direct applications to surveying and static structural analysis. After that trig course, you could calculate the size of the beams required for a truss bridge. The calculus course covered subjects like the ballistics of big guns. (I especially liked the "tables method" of integration, which taught you how to use those tables of integrals in the back of the book.)
There's a mindset in math teaching that math is about "puzzles". It's not. (Mathematics in England at the university level went off into that dead end for a century, with rated "wranglers" and "senior wranglers", until Hardy kicked them out of it.) But the school version of mathematics overstresses puzzles, because they're easy to assign and grade. That's a bigger problem than the "liberal" aspect.
For a non-puzzle curriculum, see PSSC Physics, which was taught in the 1960s. Lots of little experiments which required some calculation and data analysis.
it's so misunderstood :( Don't worry mathematics, people don't understand me either. Want to go get a drink?
One of the biggest problems with the 'spiral curriculum' is that it seems to assume that all students are on the same track with the same quality teaching throughout the country. If you happen to change schools, or have different quality teachers, or even use different systems, you are up a creek without a paddle.
Even worse, some of the books students use don't even have examples of what they are trying to show because they assume that was learned in an earlier cycle. That makes it difficult for parents to help students when the students get stuck.
Take your liberal arts and STUFF THEM..
Rather than argue the fine points of your Gedankenexperiment, I imagine it's sufficient to point out that were we to do so, we would be practicing precisely the thing you hoped to make obsolete: philosophy.
The notation doesn't translate well in text, so I'm supplying a link.
Can you explain this one?
Another puzzle.
Enjoy!
http://homepage.smc.edu/nestler_andrew/simpsonsmath.htm
... isn't it? I mean, confidence building can't replace real learning.
...
Women's educational expert Melanie Upfoot addresses the children at school.
Upfoot: For too long, there's been an anti-woman bias in math. Boys are aggressive, obnoxious, and never let us be heard. From now on, I'm splitting the school in two, separating the boys and the girls forever.
4. Melanie Upfoot begins teaching her first class in the all-girls classroom.
Upfoot: Now, let's buckle down and do some math.
Lisa: Yes!
[The teacher turns on an electronic device that plays soft music and projects colorful mathematical symbols all around the classroom.]
Upfoot: How do numbers make you feel? What does a plus sign smell like? Is the number 7 odd, or just different?
Lisa: Are we gonna do any actual math problems?
Upfoot: "Problems"? That's how men see math, something to be attacked - something to be "figured out."
Lisa: But
Upfoot : Uh-oh, Lisa, it sounds like you're trying to derail our self-esteem engine.
5. Lisa peers through the window to the math class in the all-boys classroom.
Teacher: Now boys, who can tell me the volume of this snowman. Anyone?
Martin: Just add the volume of the spheres! We know the radii....
Lisa: He forgot the volume of the carrot nose: one-third base times height! Oh math, I have missed you!
Skinner: No girls allowed!
Lisa: Assistant Groundskeeper Skinner, don't you think it's wrong that I can't get the best math education because I'm a girl?
Skinner: [sighs] I don't have any opinions anymore. All I know is that no one is better than anyone else, and everyone is the best at everything.
6. Lisa: Mom, the girls' school is a joke, and I'm not allowed to take the boys' math.
Marge: When I was in school, I loved math. Until....
[flashback to Marge studying with a calculus book on the beach]
Homer: Hey, Professor Von Hubba Hubba - wanna hop in my dune bug and erode some beach?
Marge: I'd love to. But I've got my calculus final tomorrow.
Homer: C'mon, baby, the only math you need is You + Me = Forever.
Marge: Oh, Homie. [She leaves with him.]
[Present day] Marge: Since then, I haven't been able to do any of the calculus I've encountered in my daily life. But that's not going to happen to you!
Math is a joke, let's make up theories and formulas and prove them..
IMO anyone can make a theory up and prove it over time.
1+1 does not = 2
"Education is built up with facts, as a house is with stones. But a collection of facts is no more an education than a heap of stones is a house."
Oddly enough, the text book my Philosophy 101 class used was "Mathematics and logic for digital devices" by James T. Culbertson.
Of course, I was in a technical college at the time and the person teaching the class also taught mathematics and computer science.
Reading all the way through the article, the article concludes with the author claiming that government would be better if it was run by people like him.
Surprise, surprise.
Contribute to civilization: ari.aynrand.org/donate
Pure mathematics is tedious in the extreme.
It should be taught for what it is from day one - a means of modeling the behavior of the things around us, rather than the totally abstracted gobbledegook that it is at the moment. No wonder the kids lose interest, they can't see any practical use for it.
The most understood subject is no where near math. Since schools teach only fabrications, nothing near reality... History is he most understood. 90 % of what is taught is bunk, bs, and fables for making one or another side (mostly libs) to look better than they came close to being. Thanksgiving - the story of thanksgiving is not near true as taught in schools. ( a proof that socialism is worthless and capitalism works) The words Liberal, Communist, Socialist, Nazi are treated as antonyms not synonyms Most teaching are by people from book compiled with an agenda, that has NOTHING to do with history Math is not even distinguished from Arithmetic. Logic is shunned until the thinking patters as mashed by the bull sh*t they substitute.
And if you believe that, just like the GP, then you're wrong. Very wrong. Not to mention...
"fucking encode whatever paradox they're trying to create in a object hierarchy, and be done with it"
This is a retarded statement. Whether the paradox is represented as a symbol based language (mathematics) or spoken language (philosophy), if they both maintain the same analytical rigor, it makes no difference. The same results can be achieve either way, the language is superfluous, the only difference being the persons/minds ability in using the language for that analysis. In fact, if the symbol based language obscures insight from all but those with an extreme competence in the subject, then it's not necessarily a good language for this discussion. You'll probably agree, unless you're one of those Lojban fuckers!
Have you met my friend Kurt Gödel? He's got quite a lot to say about this very topic.
If you want to read this in a fun, easy to read, well written book, then get Gödel, Escher, Bach. I've only just started reading it, and don't have a background in math (I'm almost retarded with it), but do have a background in programming, and it very quickly explained these complex ideas. Brilliant book. His idea (Godel's and the application Hofstadter comes up with) is so simple, yet so complex, and has application in almost everything.
Can't wait to study more maths!
This is my footer. There are many like it, but this one is mine.
I don't see anything "merely" about science and technology; it is what our entire society is based on. Lawyers, philosophers, politicians, judges, priests and other decision makers and influencers should be required to demonstrate a reasonable degree of understanding of logic, algebra, calculus, chemistry, biology, astronomy, and physics. Anybody in those positions who doesn't shouldn't be getting a degree or professional certification.
Yours is the clearest explanation I've found so far, so I've copy/pasted it to my site (with attribution).
Thanks.