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."
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.
let A=1, B=1
A^2=B^2 because A=B, so
A^2=AB and
A^2-B^2=A^2-AB , next we factor
(A+B)(A-B)=A(A-B) , divide like terms
(A+B)=A
substituting our variables for their values we learn that
2=1.
Dude, this is too trivial. You cannot divide by (A-B), because a-b = 0.
You cannot even shock a highschool kid with that lame attempt. At least try derivations or something.
Error: Division by zero at line 50
No sig today...
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
Basic math is easy enough for nobody to have an excuse for not knowing it.
No sig today...
(A+B)(A-B)=A(A-B) , divide like terms
Divide by zero error! After this point, every conclusion is invalid since the results are undefined.
Depressingly, some people (adults as well as kids) would not spot that.
Those who can make you believe absurdities can make you commit atrocities. - Voltaire
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..."
This is exactly the kind of thinking that has got us into the mess we're into now.
Learning math is just as difficult as learning any other subject or content material. Deciphering poetry, learning programming, studying psychological theory, and learning calculus all involve concentration, study, and struggle from the learner. No one is born knowing any of those things, therefore they all must be learned. The entire point of the OP is to say that the way we go about teaching math is wrong and that people need to reconceptualize how they teach the information because it doesn't make sense to the learner. In the end, its all difficult to some degree. It's when you have that "A-Ha!" moment, it clicks, and you get it. But if you have some terrible algebra teacher who doesn't understand advanced math or someone who doesn't care that you learn, only that you can complete problems 1-50 in a mechanic fashion, then of course it's going to seem difficult (or more difficult than it should be).
Carl Sagan quotes get you an automatic +5 on all posts.
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.
The way math is taught in schools is atrocious. Most math texts that I've used with 5th and 6th graders emphasize learning processes and methods for solving a set of problems. The texts do not hold all of the blame, however. The texts are written to follow state and national standards. The standards are written in such a way to emphasize process and not necessarily apprehension of greater concepts. For example:
5th Grade Level Expectation 1. Differentiate between the term factor and multiple, and prime and composite (N-1-M)
While these vocabulary items are important and these skills are definitely useful, learning this skill in isolation (which most texts teach) is pretty useless as students do not connect these skills to a greater picture.
A revision of mathematics standards and teaching methods will go a long way to improving the quality of mathematics education. A holistic approach that includes some wrote learning of basic skills and lots of real application problems. Real application problems are not word problems. How many "real" word problems have you had to solve in the last ten years?
Some texts such as Every Day Math from the University of Chicago does a much better job at integrating all sorts of skills and teaching in a much more holistic method. It includes some excellent modeling exercises, games that rely on a real understanding of mathematical principals for mastery and interesting lessons. But even the best text can't help a kid if they don't have a good teacher that really understands mathematics. Watching an uniformed teacher try to explain what a prime number is, or a different method for division (such as repeated subtraction) is painful. They simply can't do it. Unfortunately, in my experience most of the teaching candidates that were in my classes thought that math was "hard" and "didn't really matter." They scraped by with the lowest possible scores in the required math classes and one even told me she "wasn't going to bother teaching math." While this is pure anecdotal evidence, the declining math scores in the US show that we really do suck and producing math teachers.
The problem stems from bad math teachers badly teaching math which of course leads to more poorly instructed math teachers. Placing a real emphasis on reading and mathematics, with highly qualified and well-supported specialists is the only way we're going to solve this problem. Unless we have some real political will akin to that found during the space race, we're not going to solve this problem any time soon. We'll just keep cranking out kids that think that math is done by computers and a few nerds that wave their magic math wand over problems to find solutions.
This one's tricky. You have to use imaginary numbers, like eleventeen... --Hobbes
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!)."
Actuallly:
- if A=B, then result is undefined
- if A!=B, then B is 0
Cheers,
G.
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
(1+1)(1-1)/(1-1)=1(1-1)/(1-1) which is, of course
1 * 0 / 0 = 1 * 0 / 0 which is
Excuse me, but last I checked 1+1=2, meaning you should have written:
2 * 0 / 0 = 1 * 0 / 0
But don't let me get in your way of being a fucking prick.
Other problems with your post (including your FAIL at line breaks) are left for others to marvel at.
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.
1st line assumption: "let A=1, B=1".
Thus, second case is a contradiction.
We know where leadership by an anti-intellectual "strongman" who scapegoats minorities and likes boisterous rallies goes
You're right, I'm sorry. Let's be friends!
"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'.
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.
Not necessarily so. I'm given to understand - though I'm not a mathematician myself by any means - that the problem is not so much maths is difficult as teaching is difficult.
While it's relatively easy to teach a subject to someone who's been blessed with a pretty innate grasp of it, it's damn difficult to teach that exact same subject to someone who doesn't have such a grasp.
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
Math [is] misunderstood because it's hard, and that's why people have misconceptions about it. [The u]nderstanding of math require[s] considerable effort and concentration which most people tend to avoid if possible.
Look, that's just flat wrong. When I was in grade school, the same people who wouldn't do their math homework would then go to the gym and shoot baskets for three hours every day. When I asked why, they would say "math is hard, it's either right or its wrong, and to be any good at all takes considerable effort." When I told them I felt the same way about shooting a basketball, and if they spent the same amount of time in a math book as they do in the gym they'd be stars at math, all I would get was funny looks. I never could understand how such people would work so hard at learning one thing -- basketball -- that they'd sweat, be out of breath, and have to take a shower afterwards, and then turn around and say that learning math was hard!
Like learning anything else (including basketball), if learning math is hard, you're learning it incorrectly, and need better instruction.
"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'
yes, but seven ate nine
new sig
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.
the excuse is that it's pretty much pointless to know, much like basic grammar or how to make a bomb from fertiliser.
This is a joke. I am joking. Joke joke joke.
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.
In the same spirit, although somewhat more "advanced" (whatever that means), we are going to solve the following equation:
x = 1 + 3x
now, substitute x for 1 + 3x, and we get
x = 1 + 3(1 + 3x) = 1 + 3 + 9x
Do this again and again, until you can't see the end:
x = 1 + 3 + 9 + 27 + 81 + ...
and you'll notice that it looks familiar, in fact, looking in an old calculus textbook we find the formula:
1 + p + p^2 + p^3 +p^4 + p^5 + ... = 1/(1-p)
along with some stuff about Zeno's paradox that we didn't read. Anyways, using this, we have:
x = 1 + 3 + 3^2 + 3^3 + ... = 1/(1-3) = -1/2
and lo and behold! If you put that back in the original, it works!
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!
Part of the problem is that persons capacity of abstract thinking varies. Some people just cant understand it beyond simple computation. The curriculum is made for the lowest common denominator at primary school level, based on the idea that everyone just need to learn to compute, plus a little bit extra for those who may want to know more. And that continues until the end. There are things that shouldn't be thought this way. Whole high school math for example. But at that point, its too late to switch. Kids, all of whom have never learned the reasons behind math would just flunk... So its stretched out a little bit more and those of us who go to study science subjects in uni are then in for a shock.
... unpleasant to the point of rebellion. So I tried to understand as much as I could. If I needed a formula I had forgotten, I usually managed to construct it from vague memories and logical connections and test it mid exams. Understanding math was made surprisingly hard at times. For example, by pure accident I saw an old hand drawn picture with the geometric proof if the Pythagorean Theorem. I was one amazed ninth grader. Why wasn't that picture in the schoolbooks? Not all math can be visually proofed, but why on earth is the proof that can be represented in the single picture hidden away?
May be an education that filters people by their apparent capacity is the answer... Or maybe its fine as it and those of us that care about the why-s can manage by educating ourselves.
I had a different impairment. I have serious problems remembering things I don't understand. Formulas that didn't make sense were forgotten and doing anything without understanding was
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
Dude, +6 for you is not enough
Exactly. The biggest problem in mathematics today is that nobody bothers to show practical uses for it. It is much easier to learn something when you explain what it is for and what you can use.
Religion: The greatest weapon of mass destruction of all time
Learning math is just as difficult as learning any other subject or content material. Deciphering poetry, learning programming, studying psychological theory, and learning calculus all involve concentration, study, and struggle from the learner
Learning (advanced) math takes a lot more concentration, study and struggle, than most other things. Therefore learnig math is not equally difficult than these other things, it's much harder.
After certain level of complexity and abstraction, it also requires the kind of abstract thinking not everybody is capable of. IOW, after certain level, learning math becomes exponentially more difficult, until ultimately it becomes impossible to learn a new and yet more complex thing. Not so with most other things, learning new things is not exponentially more difficult.
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
Learning math is just as difficult as learning any other subject or content material.
You think so?
One of the things I promised I would never do is to assume that my area of study is harder than another's. Or, assume that because I do not grasp something easily it is a difficult subject.
So I started as an art major in college and was a straight A student for a dozen classes. Lots of that was spent sitting by the lake with a guitar in my hand hoping that some girl would think I was cool (didn't work). Well, there was also an hourly visit to the campus gallery to look at art and write some bullshit story about it. Most amazing was the ability to take my *opinion* and get an 'A' grade because I mentioned shamanism or diversity or Warholian ethos or myth cycles. Cool thing? It was my *interpretation* so could never be wrong.
I switched from Art to English (there's was an English major chick who I had this mad crush on). You know those art history papers I wrote? Amazingly I could use the same themes in English. Same approaches too.. Look at the political clime at the time (hey that rhymes), make a statement that all art is political (because the professor wants to rebellious against the others and say that art is not motivated by beauty) and voila, an 'A' paper. In one paper I managed to roll together gothicism (not black fingernail gothicism, but you know, the rebellion against the void type), the rise of photography, Godel, and Eastern mysticism into one 3,000 word essay. cool thing? It was interpretation and could never be wrong. Even the historical parts -- you know, Dante's influence on Tolkien or Conrad's on Vonnegut -- were so broadly argued that it could never be wrong....
Funny thing though... when I finally switched to computer science and mathematics major that shit didn't fly. I had to work my ass off to pull down those grades. For the first time in my college career, I got a C in a class. Up to that point it was all A's. MAC1101 was easy enough. So were the first couple calculus courses. Calc II was tough, but I scraped by. Calc 3 wasn't so bad. Differential equations kicked my ass. I still remember dreading the tests. I still remember trying to figure out how to find vector fields and tensors and determine saddle points and lots of stuff that I've forgotten. I read my notes from back then and I can't make heads or tails of it. If my "interpretation" of the math was incorrect, my answers were marked wrong.
So is one harder than another? I don't know. But I can tell you that it's certainly easier to get a high grade in an English or Art class than it was for Mathematics. At least for me.
Here, let me show you an even more beautiful mathematical paradox:
We try to solve this equation: x^2 - x + 1 = 0
We do that by adding x - 1 on both sides: x^2 = x - 1
We multiply both sides by x: x^3 = x^2 - x
Add 1 on both sides: x^3 + 1 = x^2 - x + 1
Recognize the first equation in the right side: x^3 + 1 = 0
Subtract 1 on both sides: x^3 = -1
Take the cube root on both sides: x = -1
Check the answer: (-1)^2 - -1 + 1 = 0
Have fun!
In my opinion, (from my experience helping a friends kid with their homework quite a while ago), the real problem starts even earlier than that.
Basic mathematics/arithmetic is only hard if you don't understand numbers, and this is where I think the root of the problems lie. The problem I ran into (and have seen a lot of evidence elsewhere to support it - (number posters etc. form 1-100)), is that kids are taught to count from 1 to 10. (I'm in the UK btw. so YMMV). The problem with that is, of course, that our decimal system works from 0 to 9. It can be tricky to teach kids that, (especially when they use their fingers so much for 10), but I think it's really important - everything else they learn will be based upon such a thing. As soon as they understand 0 to 9, (and have them work out the symbols of 10 for themselves, if possible), basic arithmetic becomes much easier to work out - especially on paper, which then makes it easier to visualize and do it in their heads...
'Stupidity is an often fatal disease' - R. A. Heinlein
Well, yes...and no. Math - in efficient representation (i.e. as numbers and operation symbols) is fairly abstract. Math also is not as ingrained as an entertainment. Complex storytelling likely goes back hundreds of millenia, and forms the basis for survival of early human tribes. Complex mathematics - hell, even some high school math - only goes back a couple hundred years. Math is not as ingrained in our beings as language.
The second - and perhaps more pertinent - is that the vast majority of teachers are women. And not just women - women who chose a career which values personal interaction with children. While there are people who excel at many fields, very few actually excel in more than one or two areas of knowledge. Imagine how well the average college-level mathematician would deal with teaching 8 year olds all day long. They would find ways to work math into the curriculum, and the language arts, or history, or some other "soft" subject would probably get a smaller and smaller portion of the day.
Now put someone who isn't naturally inclined to math in that spot. She (and I'm using the female as that is the bulk of teachers) will play to her strengths - usually language arts. Math was "hard" for her as a concept when she was a student, so she's not going to be a solid a teacher.
My wife is an accountant, but she struggles trying to help my 8yo daughter with math. She knows the stuff easily enough, but she doesn't think about math every day. She deals with rules, organization, and basic arithmetic in a linear fashion. As an engineer, I speak and read simple math every day. I understand some higher math (I struggle with tensor notation, some matrix theory, stochastic processes and functions, and other higher math - but I don't use that in my job). More importantly, I can see both the way the math worksheet wants the child to come to the answer, and the path my daughter is taking to solve the problem. Unlike my wife, I can guide her along her path to understand the concept. It makes homework "easy".
Sometimes she understands the way the worksheet wants her to do it - often after working it in a very round-about way, and me walking her back from the answer to the problem statement. She doesn't want to spend any more time than is necessary, but until she understands how circumspect her process is compared to the foreign one being taught, she'll be stubborn about it. It's simple human nature.
Many, if not most, teachers don't have the background to help students the way I help my daughter. Even those that do, don't have the time to spend with every student. I know how much time it takes for mine to really get it, and by most measures she's in the top 10% of students in math. They've tried lots of ways to teach math over the years. Just because we find it fun, or useful, or easy, doesn't mean that using another method is going to make it easy for "the rest."
Is it just my observation, or are there way too many stupid people in the world?
Learning math is just as difficult as learning any other subject or content material. Deciphering poetry, learning programming, studying psychological theory, and learning calculus all involve concentration, study, and struggle from the learner.
You can end up in the human learning equivalent of dependency hell much easier in math than in any of your other examples, in fact I'd go so far as claiming its the worst possible tree of dependencies of any intellectual subject. With the possible exception of western philosophy, or at least thats a close second.
"Science flies us to the moon. Religion flies us into buildings." - Victor Stenger
I tended to suffer from this phenomenon but the opposite way around ; I didn't go to an American college (being a UK student), but at school, in subjects where creative interpretation was involved I would end up with huge periods of mental paralysis where my brain was trying to work out the right answer. As you've just pointed out, there wasn't one.
Quite the opposite in science subjects. Maths, on the other hand, also got too much after calculus when doing things like binomial expansions. Since I didn't need it for my university entrance requirements, I dropped it.
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
It's funny you mention the "Everyday Math". I live in the Chicago suburbs, so it seems every person I know that has a kid in grade school in the area is using this series of books for math. I have gone over my sisters house and there have been times where I have been asked to help my niece with her homework because my sister didn't know how to solve the problem since it is so different than how we were taught. Since I have a MSEE and have taken just a few math classes, and even I have to think about how to explain it to here without going over her head. So my first response is to ask where is the text book. I figure this is a good start to see what has been covered so far. She is unable to provide one, because it seems that everything is a worksheet.
I still remember the problem. It goes something like this: "People are on the phone in the US for a total of 123 billion minutes each day. If there are 200 million people with phones, what is the average amount of time each person is on the phone". I am sure for you and I this is a trivial problem, but here is what I have issue with it. First, I asked my niece if she knew what an average was. She said no. Ok, not a big deal so I tried to explain to her what an average was. My second issue with the problem was found out from trying to help her again. I asked her if she knew what factoring was. A blank stare is all I received for that. I didn't even bother asking about scientific notation*. So that leaves the only practical way to solve this is with long division, which I don't necessarily think that is wrong but to me detracts from what they might actually be trying to teach.
So that leads me to my issue with this problem and with "Everyday Math" in general. What is that they are trying to teach in this problem? Is it to explain the concept of an average? Is it how to do tedious long division? I can understand wanting to cross apply different math techniques to solving problems, but if neither concept has been taught then is either concept really being taught by the cross application? If anything, my impression from Everyday Math based on what I have seen is a removal of formalism to math. I know, how boring is it to talk about the associative property or commutative property, but if you have those tools under your belt then everyone is speaking the same language when it comes to math. And given that, then I think I could have helped her to solve this problem in a more algorithmic way. The idea of an average on the other hand needs to be taught in a different way than in just a word problem for kids her age (btw, she is now is 6th grade). Something as easy as rolling a die 10 times and summing the value and dividing by the total number of rolls. Or measuring the height of all the kids in class to find the average, and so on.
Now back to other issue of bad teachers. I would love to be a teacher of either math or physics, but guess what? They won't pay me as well to be a teacher as I can make being an engineer. Not that it is all about the money, but it certainly is an important factor especially when I can pull in twice what an average teacher makes. Since schools are so heavily unionized (at in Chicagoland) there is no way the union would allow disparate rate for math or physics teachers than the others. I think if you fix this problem, then I think there will be better recruiting of good teachers.
*Now I know this is slashdot, so I expect some snarky comments about her maybe not being that good in math. That's fair, but I know that on her report card she had a B+ in math recently, so that would put her at above average in the class. Then again, this could be a whole other discussion on grade inflation in the US.
Do you know what "merely" means?
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.
http://en.wikipedia.org/wiki/Divide_by_zero#Fallacies_based_on_division_by_zero
~ In Trust, We Trust ~
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.
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.
What is the practical use for poetry, or for painting? Do you think these subjects should be banned from schools because there is no practical use?
Le français vous intéresse?
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.
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!
Exactly. The biggest problem in mathematics today is that nobody bothers to show practical uses for it. It is much easier to learn something when you explain what it is for and what you can use.
You know, I would say that one of the biggest problems in mathematics teaching today is that they try too hard to show practical uses for it. This ends up with textbooks in a tangle to present "practical" problems for the kids to solve with the resulting emphasis being a meat-grinder approach to problem solving, where you just dump whatever you have in one end, turn the handle, and hope for the best, with little or no understanding of what the hell is going on. Mathematics is about abstraction, generalisation, and logic. A lot of mathematics is about solving a problem not because it presented any practical interest, but simply because it seemed interesting in its own right. By pandering to students need for practicality all you are really doing is killing their mathematical curiousity. They don't actually learn any mathematics, just a bunch of formulas and a hodge podge selection of tools for solving problems with little or no ability to generalise those tools to any problem outside the neatly defined box. Worse, they don't ever actually learn that mathematics is anything other than bunch of formulas and a hodge podge selection of tools for solving specific arbitrary problems.
So yes, some practical examples can help at times, but that is in no way the problem with mathematics teaching right now.
Craft Beer Programming T-shirts
"Look, it's either 'true' or 'false'."
"Not necessarily."
The problem with math instruction is that faults in that instruction accumulate, until the student is rendered incapable of further progress.
The math we ask most students to master, say up to basic calculus, isn't really very hard. Geometry is probably the most fundamentally difficult topic, but we typically teach it in a pretty self-contained way. The problem with something like calculus is that if a student's grasp of a single prerequisite skill falls short, say in factoring polynomials, his advancement in calculus is crippled. Because we don't precisely measure and characterize a student's accomplishments beyond ridiculously coarse figures ("he's got a B average in pre-algebra"), we set students up for increasingly certain failure as the number of otherwise "minor" or "narrow" holes in his mathematics education increases.
I believe that we should *never* take a statement like "I'm no good at math" at face value. What would be more accurate in most cases is to say, "I'm not adequately *prepared* to progress in math."
Post may contain irony: discontinue use if experiencing mood swings, nausea or elevated blood pressure.
I wonder if a big part of the problem is that the breakdown of the concepts is wrong; someone above talks about teaching counting starting with 0 instead of 1, because it forces and understanding of where 10 is coming from. With that understanding of 0 and 10, understanding 100 and 1000 and 1 million is easy, without it, it is nearly impossible (or at least, the big numbers will continue to be confusing and mysterious).
So the standards are written to check if a student has memorized the definition of a factor and that they know that zero is nothing, but do they check to see if the student has internalized zero as an abstract concept?
(The idea I am sort of reaching for here is that lots of abstract concepts depend on other abstract concepts, and people/students that lack an internalization of the basic abstractions have little hope of ever understanding the higher ones, whether they have learned some definitions by rote or not).
Nerd rage is the funniest rage.
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
It's funny you mention the "Everyday Math". I live in the Chicago suburbs, so it seems every person I know that has a kid in grade school in the area is using this series of books for math. I have gone over my sisters house and there have been times where I have been asked to help my niece with her homework because my sister didn't know how to solve the problem since it is so different than how we were taught.
This is one of the great weaknesses of Everyday Math. If the teacher doesn't use skill sheets (a method for tracking student comprehension and achievement) and ensure that every kid knows how to do the homework and doesn't send home the parent letters that accompany each unit to ensure that teachers understand what is being taught, parents are going to struggle to help students. This comes back to being a good teacher and ensuring that you aren't just "presenting" material, but rather actually teaching.
I figure this is a good start to see what has been covered so far. She is unable to provide one, because it seems that everything is a worksheet.
Ideally, there isn't a need for the textbook to come home with the kids for younger kids as skills are presented in tiny chunks and the worksheets should be spiraled reviews of already mastered concepts. Again, if kids aren't actually taught and they aren't on the road to mastery, they're going to struggle with the homework.
I still remember the problem. It goes something like this: "People are on the phone in the US for a total of 123 billion minutes each day. If there are 200 million people with phones, what is the average amount of time each person is on the phone". I am sure for you and I this is a trivial problem, but here is what I have issue with it. First, I asked my niece if she knew what an average was. She said no. Ok, not a big deal so I tried to explain to her what an average was.
Once again, the teacher probably didn't make sure that her kids knew how to average. Averaging is a HUGE part of EDM as is factoring and division. I don't want to rag on a teacher I've never observed, and I certainly have made similar mistakes, but it sounds like once again this is a failing of the teacher and not necessarily a failing of EDM.
So that leads me to my issue with this problem and with "Everyday Math" in general. What is that they are trying to teach in this problem?
They were probably trying to review finding an average and the process associated with that.
I can understand wanting to cross apply different math techniques to solving problems, but if neither concept has been taught then is either concept really being taught by the cross application? If anything, my impression from Everyday Math based on what I have seen is a removal of formalism to math.
EDM is less formal than the way I was taught, but there is definitely some good thinking behind it. Once again, if the teaching is good, there is a logic, rhyme and reason behind the way the material is taught. If the teacher just passes out the homework and continues along blindly without checking in with the students, you end up with a situation like you had with your niece.
I know, how boring is it to talk about the associative property or commutative property, but if you have those tools under your belt then everyone is speaking the same language when it comes to math.
Everyday Math does struggle with this as well. It could be stronger here. It doesn't put a lot of emphasis on terminology, but it does do alright in general, when once again taught properly.
Just like so many other things, the best tools in the world aren't the only factor in determining the quality of the product produced. Having good tools just helps you spend less time faffing around and more time getting the job done. A master of education like Jamie Escalante (of Sta
This one's tricky. You have to use imaginary numbers, like eleventeen... --Hobbes
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.
If you learn basketball you have a 0.001% chance of becoming a multimillionaire , or you'll be another poor modern peasant. But at least you have a chance.
If you learn math your career will either be asking if they want fries with that, or possibly a few short years before your job moves to the east and you'll be another poor modern peasant. No chance at all.
Not quite. If you learn basketball instead of math, you have a 99.999% chance, using your numbers, of not becoming a multimillionaire and of not holding a quality job, since you also didn't learn math.
If you learn math, you'll at least be more employable than all but 432 of the guys that learned basketball (that being the number of players in the NBA), since they didn't learn math.
I note in passing that the value of a 0.001% chance at $10 million is only $100. Math helps in making career decisions, too.
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.
Now go back to the original and look at the first line.
Slightly disreputable, albeit gregarious
As a former math major who has always struggled to learn physical skills, I sympathize. OTOH, a million years of evolution emphasized the things the people in the gym were doing: run, throw, hit the target. Not to mention that many biologists credit the success of our distant ancestors on the African plains with their ability to sweat, thereby remaining cool during intense and extended physical activity. The opportunity to sit still and solve more abstract problems has only existed for the last few thousand years.
In my experience (about to finish up the coursework part of a mathematics PhD), math doesn't get more difficult. It does certainly get more abstract and sophisticated, but you come into each successive course with a more abstract and sophisticated background, so it's not actually any harder. I can't say that I had any more trouble understanding singular homology of a topological space (second-year algebraic topology) than I did with the derivative of a real-valued function of one real variable (high school calculus). The chain rule from calc 1 was just as hard for me to master as anything in my graduate classes. It's all the same--hard when it's new, obvious later.
Yo dawg, I heard you like the Ackermann function, so OH GOD OH GOD OH GOD
The problem with something like calculus is that if a student's grasp of a single prerequisite skill falls short, say in factoring polynomials, his advancement in calculus is crippled
It's not as bad as it sounds. Everyone I know (I'm in a PhD math program) has this or that area that they never quite got when they were supposed to. When it bites us later on, we just go back and learn it properly--with all the other stuff we know now that we didn't know then, it's easy, and we wonder how it wasn't obvious in the first place..
Yo dawg, I heard you like the Ackermann function, so OH GOD OH GOD OH GOD
5 out of 4 Americans have trouble with fractions.
There are no karma whores, only moderation johns
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?
First, thank you for being a conscientious Math teacher, and being willing to talk about it! I bet your students' parents love you. My hat is off to you.
And I agree with your point that Math concepts should not only be taught in isolation, but in real application problems. You seem to really understand what is important here.
Although it is only of secondary importance to this thread, I am going to take issue with Every Day Math. That curriculum is hideously flawed in its mathematical approach, and no mathematician would ever condone the sloppy, ambiguous way that problems are posed and answered. I happily defer to your professional judgement on all aspects relating to child development, learning and teaching methods, etc. My indictment is of the actual math content and the unnecessarily sloppy wording used to describe it.
I have spend countless hours attempting to teach my children real math concepts, which are in direct conflict with the content and methods in Every Day Math. And at the same time, I have to balance this against the desire for them to provide the expected answers to homework and test questions.
Mike
---
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 first equation doesn't have a real number solution hence the substitution in step 5 is incorrect.
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.
In the essay linked in this post, the author specifically discusses the distinction between training and education. Perhaps you should [re]read that section?
Rhapsody in Numbers
Let me try to explain why this appears to work but doesn't. The problem is with this line:
We multiply both sides by x: x^3 = x^2 - x
When solving an equation, there is an assumed logical progression. Suppose you want to solve:
Then, you want to find the set S1={x: x is a solution of (1)}. You do this by transforming the equation repeatedly until you get to a form from which it is easy to derive the solutions. But when you make a transformation of the equation, you need to think about what the set of solutions is after the transformation. Let proposition P1 = x is an element of S1. (Similarly Pn for Sn). If, as the next step, you write:
you are implicitly stating that:
(<=> means "if and only if") If you then write:
x^3 = x^2 - x, (3)
the set of solutions has changed: -1 is introduced as a new solution. In this case, this is because (2) was multiplied by x, which is not a non-zero constant, and thus the meaning of the equation has changed. Logically, you are now stating that:
In other words, if you find an x for which P2 is true, then P3 will also be true for that x, but not the other way round.
Normally when you solve an equation, you implicitly create a progression P1<=>P2<=>...<=>Pn. From this, if you can see that Sn is the set of solutions for (n), then going back by implication from Pn to P1 you can conclude that Sn=S1. However, if the chain is broken and you write P1<=>P2<=>...<=>Pj=>P(j+1)<=>...<=>Pn, you can only conclude that S1 is a subset of Sn. However, because you are missing an implication from P(j+1) to Pj, you cannot say that Sn=S1.
There are many operations that potentially change the set of solutions, such as multiplication of both sides by zero, squaring both sides, and others. At every transformation, you must make sure that the solutions stay the same. In solving other problems, the logical progression can become more complex and then cannot be implicitly assumed like this. Generally, it is always a good idea to know precisely what you are stating in terms of logic.
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.
Beautiful trick. There's exactly one point where the problem is introduced, and it's one of the subtler ways you can go wrong with solving an equation algebraically... Hint: Consider the problem, step-by-step, in the complex plane. Every quadratic has either two solutions or is a perfect square. This one's not a perfect square. At one point, we turn it into a cubic - which necessarily has either three or two solutions. This one has three. Where'd the extra solution come from?
it's so misunderstood :( Don't worry mathematics, people don't understand me either. Want to go get a drink?
Maybe I don't understand what is being debated here, but the equation reduces as follows:
(A+B)(A-B)=A(A-B)
A^^2 - B^^2 = A^^2 - AB
- B^^2 = - AB
B^^2 = AB
B = A
I see no "divide by zero" errors. Someone give me a whoosh, I can take it.
I come here for the love
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.
I'm not sure I could count almost any of that as agreeing with my grandparent post. :-/ I count "justifications" as part of the nuts-and-bolts, and I'm largely down on memorizing random stuff.
We know where leadership by an anti-intellectual "strongman" who scapegoats minorities and likes boisterous rallies goes
It's not just the div by zero. You also flipped a sign between step 3 and 4.
a^2 = ab
does not proceed to
a^2 - b^2 = a^2 - ab
(of course, they are both = 1, so doesn't really matter, though it's not usually how this misproof works.
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!
Or just bring both terms with x to the same side for:
x - 3x = 1
-2x = 1
-2x(-1/2) = 1(-1/2)
x = -1/2.
Bingo.
While these vocabulary items are important and these skills are definitely useful, learning this skill in isolation (which most texts teach) is pretty useless as students do not connect these skills to a greater picture.
It's hard enough --and sometimes not possible-- to connect to the greater picture without willfully ignoring it. You have to learn a lot of mathematics before it starts to connect. You're not gong to be ready for a course in real analysis until you can manage calculus pretty fluently ( and analysis illuminates why various parts of calculus actually work. ) So let's make it harder still and simply not even try.
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!
Missing two roots in "Take the cube root..."
>> roots([1, 0, 0, 1])
ans =
-1.0000
0.5000 + 0.8660i
0.5000 - 0.8660i
The first equation has two solutions. But the equation x^3+1=0 has three solutions. You chose the one solution that didn't apply. The other two work just fine.
>> roots([1,-1,1])
ans =
0.5000 + 0.8660i
0.5000 - 0.8660i
I like the problem though!
"Uh... yeah, Brain, but where are we going to find rubber pants our size?" --Pinky
"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.
Since A = B, dividing (A + B)(A - B) and A (A - B) by (A - B) is undefined.
Excellent Explanation AC.
To give a simpler example of the same fallacy:
X = -1
squaring both sides we get.
X^2 = 1
Taking the square root we get.
X = 1
Substituting back into the original equation we find:
1 = -1
I often don't like the choices people make, but I like the fact that people make choices. That's why I'm a conservative.
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.
How about this one?
Y = X.
Y is just the sum of x ones. Y = 1+1+1+1...
The derivative of 1 is zero. The derivative of a sum is equal to the sums of the derivatives of the components.
Therefore dY/dX = 0+0+0+0...
Therefore the derivative of X is zero.
Okay, I'll bite too.
x = 1 - 1 + 1 - 1 + 1 ... ... ...
= (1 - 1) + (1 - 1) +
= 0 + 0 +
= 0
But by associativity of addition,
x = 1 - (1 - 1) - (1 -1) - ... ...
= 1 - 0 - 0
= 1
Therefore, 1 = 0
Now back to other issue of bad teachers. I would love to be a teacher of either math or physics, but guess what? They won't pay me as well to be a teacher as I can make being an engineer./quote>
There is no such thing.
Teachers are teachers of students, not of math or physics. The job you think you would love does not exist.
For those who don't know why this is incorrect:
A-B = 0, and division only makes sense if you are dividing by something which is non-zero.
(1+1)x(1-1) = 1x(1-1) is in effect saying 2x0=1x0, which is correct. Removing zero from both sides by dividing by the zero makes no sense. :-)
I haven't drilled down further than 5 Karma points, so hope no one has already explained this. Hope I'm not being redundant.
Sure enough, the cow costume was hanging up next to the superhero outfit and sailors uniform. (S,Spud)
Some texts such as Every Day Math [sic] from the University of Chicago does [sic] a much better job at integrating all sorts of skills and teaching in a much more holistic method.
... Have you actually tried teaching using that curriculum? A few posts up and down from your post, I see plenty of complaints about Everyday Math (including a few from parents not knowing its name and complaining about the "spiral curriculum"), and few defenses. The only eager defense I've heard about it was from kids who like the tedious diagonal grid style of multiplication (if someone else builds the grid), especially the ones that are too special-ed to multiply numbers the normal way.
I think the biggest problem is that Everyday Math really depends on excellent teaching, and on students being ready and eager. If teachers and parents don't understand the material, and if the kids don't care, then Everyday Math is a big waste of time. Furthermore, it teaches confusion by introducing all sorts of weird terms and symbols that have no meaning when you graduate from it.
A good math teacher, given free rein, will teach well no matter the assigned curriculum. A bad teacher should at least teach something useful when given a traditional curriculum. Cue standard arguments about teacher freedom and how horrible bad teachers actually can be.
In my own experience (a volunteer tutor), the schools in San Francisco switched to Everyday Math last year. For the first year, the teachers slavishly tried to spiral, going into statistics then arithmetic then factors like some Russian dance, and everybody got dizzy. My workload seemingly quadrupled, because I was the only tutor in the organization with the mathematical background to keep up. This year, the teachers are taking a more moderate approach, concentrating on single subjects and breaking the spirals apart. Everybody's less confused, but the kids especially love how few problems they solve on each assignment. A few teachers are supplementing the Everyday Math curriculum with traditional drill assignments.
How many "real" word problems have you had to solve in the last ten years?
I like word problems. (The kids hate them.) Being able to translate issues into solvable equations and back is a very useful skill. Granted, most often it comes in terms like, "Which block of cheese will give me the best value?" rather than "Two trains are speeding at each other. One is traveling at 25 mph and the other at 40 mph. How long until they collide?"
Have a nice time.
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.
Thanks for the correction, you are right of course.
You just concluded 1 = 1 which is perfectly valid. The original "proof" divides both sides by (A-B), which is zero, and gets a result of 2 = 1. This is not valid because once you divide by zero, results are undefined.
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.
Ahhhh, an excellent, creative and paradoxical problem. Thank you. First I dissect and disprove U17, then present the reconciliation of the paradox. TO U17: The trouble with math is someone can write an excellent reasoning with advanced techniques and notation. However, if there's a single critical mistake, the whole thing is spoiled. Who would want to drink filtered purified, cold refreshing water, after someone put in a drop of used motor oil? It's perfectly acceptable to multiply by a non zero variable or expression. This spoils U17's otherwise beautiful reasoning. Multiplying by an expression which IS zero, (x = 0) can create extra solutions. Check x =-1 on the step P3, (-1) ^3 ?=? (-1)^2 - (-1), -1 ?=? 1 +1 doesn't work even after we multiplied by x. The extra solution must have come from somewhere else. SPOILER ALERT- THE RESOLUTION OF THE PARADOX IS BELOW. Consider working on the question a while, per Lockhart, then after you've worked on it for a while. Try to rediscover the challenge & creativity of math, then come back here to read. Hint: The truth is x^3 = -1 contains the solutions. We need to delve into complex numbers though. This particular equation has 3 solutions in the set of complex numbers. http://www.wolframalpha.com/input/?i=x^3%20%3D%20-1&t=ff3tb01 x = -1 is real, and reals are a subset of complex numbers. http://www.wolframalpha.com/input/?i={x%20%3D%3D%20%28-1%29^%281%2F3%29}%2C%20{x%20%3D%3D%20-%28-1%29^%282%2F3%29}&t=ff3tb01 x = (1 +- i * root 3) / 2 are two solutions. (should See DeMoivre's theorem (convert to polar coordinates or see the graph), multiply it out by hand or check with WolframAlpha.com to confirm x^3 =-1 And the two complex solutions are our answer. http://www.wolframalpha.com/input/?i=x%3D%3D+1%2F2+(1-i+sqrt(3))%2C+x^2+-+x++%2B+1%3D The original reasoning has two oversights. A. x = -1 is an extraneous solution akin to y = 5, y^2 = 25, y = +- 5 and assuming y can be -5 or +5. Apparently when taking a cube root, we need to check for all solutions if they are extraneous (extra) solutions which rules out x = -1. B. Failure to consider complex solutions when taking a cube root. (Not commonly taught, but hidden right next to the complex properties which are commonly taught) Aughh, this properly formatted text doesn't display properly in the preview screen.
Ahhhh, an excellent, creative and paradoxical problem. Thank you. First I dissect and disprove U17, then present the reconciliation of the paradox. TO U17: The trouble with math is someone can write an excellent reasoning with advanced techniques and notation. However, if there's a single critical mistake, the whole thing is spoiled. Who would want to drink filtered purified, cold refreshing water, after someone put in a drop of used motor oil? It's perfectly acceptable to multiply by a non zero variable or expression. This spoils U17's otherwise beautiful reasoning. Multiplying by an expression which IS zero, (x = 0) can create extra solutions. Check x =-1 on the step P3, (-1) ^3 ?=? (-1)^2 - (-1), -1 ?=? 1 +1 doesn't work even after we multiplied by x. The extra solution must have come from somewhere else. SPOILER ALERT- THE RESOLUTION OF THE PARADOX IS BELOW. Consider working on the question a while, per Lockhart, then after you've worked on it for a while. Try to rediscover the challenge & creativity of math, then come back here and read. The truth is x^3 = -1 contains the solutions. We need to delve into complex numbers though. This particular equation has 3 solutions in the set of complex numbers. http://www.wolframalpha.com/input/?i=x^3%20%3D%20-1&t=ff3tb01 x = -1 is real, and reals are a subset of complex numbers. http://www.wolframalpha.com/input/?i={x%20%3D%3D%20%28-1%29^%281%2F3%29}%2C%20{x%20%3D%3D%20-%28-1%29^%282%2F3%29}&t=ff3tb01 x = (1 +- i * root 3) / 2 are two solutions. (should See DeMoivre's theorem (convert to polar coordinates or see the graph), multiply it out by hand or check with WolframAlpha.com to confirm x^3 =-1 And the two complex solutions are our answer. http://www.wolframalpha.com/input/?i=x%3D%3D+1%2F2+(1-i+sqrt(3))%2C+x^2+-+x++%2B+1%3D The original reasoning has two oversights. A. x = -1 is an extraneous solution akin to y = 5, y^2 = 25, y = +- 5 and assuming y can be -5 or +5. Apparently when taking a cube root, we need to check for all solutions if they are extraneous (extra) solutions which rules out x = -1. Every Day Math B. Failure to consider complex solutions when taking a cube root. (Not commonly taught, but hidden right next to the complex properties which are commonly taught)