Here`s why you suck at math..
Page 1 of 1 [ 13 posts ]
18 Oct 2011, 9:33 am
Arousing read..
Mathematicians Lament
Quote:
Mathematics and Culture
The first thing to understand is that mathematics is an art. The difference between math and
the other arts, such as music and painting, is that our culture does not recognize it as such.
Everyone understands that poets, painters, and musicians create works of art, and are expressing
themselves in word, image, and sound. In fact, our society is rather generous when it comes to
creative expression; architects, chefs, and even television directors are considered to be working
artists. So why not mathematicians?
Part of the problem is that nobody has the faintest idea what it is that mathematicians do.
The common perception seems to be that mathematicians are somehow connected with
science— perhaps they help the scientists with their formulas, or feed big numbers into
computers for some reason or other. There is no question that if the world had to be divided into
the “poetic dreamers” and the “rational thinkers” most people would place mathematicians in the
latter category.
Nevertheless, the fact is that there is nothing as dreamy and poetic, nothing as radical,
subversive, and psychedelic, as mathematics. It is every bit as mind blowing as cosmology or
physics (mathematicians conceived of black holes long before astronomers actually found any),
and allows more freedom of expression than poetry, art, or music (which depend heavily on
properties of the physical universe). Mathematics is the purest of the arts, as well as the most
misunderstood.
So let me try to explain what mathematics is, and what mathematicians do. I can hardly do
better than to begin with G.H. Hardy’s excellent description:
A mathematician, like a painter or poet, is a maker
of patterns. If his patterns are more permanent than
theirs, it is because they are made with ideas.
So mathematicians sit around making patterns of ideas. What sort of patterns? What sort of
ideas? Ideas about the rhinoceros? No, those we leave to the biologists. Ideas about language
and culture? No, not usually. These things are all far too complicated for most mathematicians’
taste. If there is anything like a unifying aesthetic principle in mathematics, it is this: simple is
beautiful. Mathematicians enjoy thinking about the simplest possible things, and the simplest
possible things are imaginary.
For example, if I’m in the mood to think about shapes— and I often am— I might imagine a
triangle inside a rectangular box:
4
I wonder how much of the box the triangle takes up? Two-thirds maybe? The important
thing to understand is that I’m not talking about this drawing of a triangle in a box. Nor am I
talking about some metal triangle forming part of a girder system for a bridge. There’s no
ulterior practical purpose here. I’m just playing. That’s what math is— wondering, playing,
amusing yourself with your imagination. For one thing, the question of how much of the box the
triangle takes up doesn’t even make any sense for real, physical objects. Even the most carefully
made physical triangle is still a hopelessly complicated collection of jiggling atoms; it changes
its size from one minute to the next. That is, unless you want to talk about some sort of
approximate measurements. Well, that’s where the aesthetic comes in. That’s just not simple,
and consequently it is an ugly question which depends on all sorts of real-world details. Let’s
leave that to the scientists. The mathematical question is about an imaginary triangle inside an
imaginary box. The edges are perfect because I want them to be— that is the sort of object I
prefer to think about. This is a major theme in mathematics: things are what you want them to
be. You have endless choices; there is no reality to get in your way.
This is what a piece of mathematics looks and feels like. That little narrative is an example
of the mathematician’s art: asking simple and elegant questions about our imaginary creations,
and crafting satisfying and beautiful explanations. There is really nothing else quite like this
realm of pure idea; it’s fascinating, it’s fun, and it’s free!
Now where did this idea of mine come from? How did I know to draw that line? How does
a painter know where to put his brush? Inspiration, experience, trial and error, dumb luck.
That’s the art of it, creating these beautiful little poems of thought, these sonnets of pure reason.
There is something so wonderfully transformational about this art form. The relationship
between the triangle and the rectangle was a mystery, and then that one little line made it
5
obvious. I couldn’t see, and then all of a sudden I could. Somehow, I was able to create a
profound simple beauty out of nothing, and change myself in the process. Isn’t that what art is
all about?
This is why it is so heartbreaking to see what is being done to mathematics in school. This
rich and fascinating adventure of the imagination has been reduced to a sterile set of “facts” to be
memorized and procedures to be followed. In place of a simple and natural question about
shapes, and a creative and rewarding process of invention and discovery, students are treated to
this:
“The area of a triangle is equal to one-half its base times its height.” Students are asked to
memorize this formula and then “apply” it over and over in the “exercises.” Gone is the thrill,
the joy, even the pain and frustration of the creative act. There is not even a problem anymore.
The question has been asked and answered at the same time— there is nothing left for the
student to do.
Now let me be clear about what I’m objecting to. It’s not about formulas, or memorizing
interesting facts. That’s fine in context, and has its place just as learning a vocabulary does— it
helps you to create richer, more nuanced works of art. But it’s not the fact that triangles take up
half their box that matters. What matters is the beautiful idea of chopping it with the line, and
how that might inspire other beautiful ideas and lead to creative breakthroughs in other
problems— something a mere statement of fact can never give you.
By removing the creative process and leaving only the results of that process, you virtually
guarantee that no one will have any real engagement with the subject. It is like saying that
Michelangelo created a beautiful sculpture, without letting me see it. How am I supposed to be
inspired by that? (And of course it’s actually much worse than this— at least it’s understood that
there is an art of sculpture that I am being prevented from appreciating).
By concentrating on what, and leaving out why, mathematics is reduced to an empty shell.
The art is not in the “truth” but in the explanation, the argument. It is the argument itself which
gives the truth its context, and determines what is really being said and meant. Mathematics is
the art of explanation. If you deny students the opportunity to engage in this activity— to pose
their own problems, make their own conjectures and discoveries, to be wrong, to be creatively
frustrated, to have an inspiration, and to cobble together their own explanations and proofs— you
deny them mathematics itself. So no, I’m not complaining about the presence of facts and
formulas in our mathematics classes, I’m complaining about the lack of mathematics in our
mathematics classes.
If your art teacher were to tell you that painting is all about filling in numbered regions, you
would know that something was wrong. The culture informs you— there are museums and
galleries, as well as the art in your own home. Painting is well understood by society as a
6
This medium of human expression. Likewise, if your science teacher tried to convince you that
astronomy is about predicting a person’s future based on their date of birth, you would know she
was crazy— science has seeped into the culture to such an extent that almost everyone knows
about atoms and galaxies and laws of nature. But if your math teacher gives you the impression,
either expressly or by default, that mathematics is about formulas and definitions and
memorizing algorithms, who will set you straight?
The cultural problem is a self-perpetuating monster: students learn about math from their
teachers, and teachers learn about it from their teachers, so this lack of understanding and
appreciation for mathematics in our culture replicates itself indefinitely. Worse, the perpetuation
of this “pseudo-mathematics,” this emphasis on the accurate yet mindless manipulation of
symbols, creates its own culture and its own set of values. Those who have become adept at it
derive a great deal of self-esteem from their success. The last thing they want to hear is that
math is really about raw creativity and aesthetic sensitivity. Many a graduate student has come
to grief when they discover, after a decade of being told they were “good at math,” that in fact
they have no real mathematical talent and are just very good at following directions. Math is not
about following directions, it’s about making new directions.
And I haven’t even mentioned the lack of mathematical criticism in school. At no time are
students let in on the secret that mathematics, like any literature, is created by human beings for
their own amusement; that works of mathematics are subject to critical appraisal; that one can
have and develop mathematical taste. A piece of mathematics is like a poem, and we can ask if
it satisfies our aesthetic criteria: Is this argument sound? Does it make sense? Is it simple and
elegant? Does it get me closer to the heart of the matter? Of course there’s no criticism going on
in school— there’s no art being done to criticize!
Why don’t we want our children to learn to do mathematics? Is it that we don’t trust them,
that we think it’s too hard? We seem to feel that they are capable of making arguments and
coming to their own conclusions about Napoleon, why not about triangles? I think it’s simply
that we as a culture don’t know what mathematics is. The impression we are given is of
something very cold and highly technical, that no one could possibly understand— a selffulfilling
prophesy if there ever was one.
It would be bad enough if the culture were merely ignorant of mathematics, but what is far
worse is that people actually think they do know what math is about— and are apparently under
the gross misconception that mathematics is somehow useful to society! This is already a huge
difference between mathematics and the other arts. Mathematics is viewed by the culture as
some sort of tool for science and technology. Everyone knows that poetry and music are for pure
enjoyment and for uplifting and ennobling the human spirit (hence their virtual elimination from
the public school curriculum) but no, math is important.
The first thing to understand is that mathematics is an art. The difference between math and
the other arts, such as music and painting, is that our culture does not recognize it as such.
Everyone understands that poets, painters, and musicians create works of art, and are expressing
themselves in word, image, and sound. In fact, our society is rather generous when it comes to
creative expression; architects, chefs, and even television directors are considered to be working
artists. So why not mathematicians?
Part of the problem is that nobody has the faintest idea what it is that mathematicians do.
The common perception seems to be that mathematicians are somehow connected with
science— perhaps they help the scientists with their formulas, or feed big numbers into
computers for some reason or other. There is no question that if the world had to be divided into
the “poetic dreamers” and the “rational thinkers” most people would place mathematicians in the
latter category.
Nevertheless, the fact is that there is nothing as dreamy and poetic, nothing as radical,
subversive, and psychedelic, as mathematics. It is every bit as mind blowing as cosmology or
physics (mathematicians conceived of black holes long before astronomers actually found any),
and allows more freedom of expression than poetry, art, or music (which depend heavily on
properties of the physical universe). Mathematics is the purest of the arts, as well as the most
misunderstood.
So let me try to explain what mathematics is, and what mathematicians do. I can hardly do
better than to begin with G.H. Hardy’s excellent description:
A mathematician, like a painter or poet, is a maker
of patterns. If his patterns are more permanent than
theirs, it is because they are made with ideas.
So mathematicians sit around making patterns of ideas. What sort of patterns? What sort of
ideas? Ideas about the rhinoceros? No, those we leave to the biologists. Ideas about language
and culture? No, not usually. These things are all far too complicated for most mathematicians’
taste. If there is anything like a unifying aesthetic principle in mathematics, it is this: simple is
beautiful. Mathematicians enjoy thinking about the simplest possible things, and the simplest
possible things are imaginary.
For example, if I’m in the mood to think about shapes— and I often am— I might imagine a
triangle inside a rectangular box:
4
I wonder how much of the box the triangle takes up? Two-thirds maybe? The important
thing to understand is that I’m not talking about this drawing of a triangle in a box. Nor am I
talking about some metal triangle forming part of a girder system for a bridge. There’s no
ulterior practical purpose here. I’m just playing. That’s what math is— wondering, playing,
amusing yourself with your imagination. For one thing, the question of how much of the box the
triangle takes up doesn’t even make any sense for real, physical objects. Even the most carefully
made physical triangle is still a hopelessly complicated collection of jiggling atoms; it changes
its size from one minute to the next. That is, unless you want to talk about some sort of
approximate measurements. Well, that’s where the aesthetic comes in. That’s just not simple,
and consequently it is an ugly question which depends on all sorts of real-world details. Let’s
leave that to the scientists. The mathematical question is about an imaginary triangle inside an
imaginary box. The edges are perfect because I want them to be— that is the sort of object I
prefer to think about. This is a major theme in mathematics: things are what you want them to
be. You have endless choices; there is no reality to get in your way.
This is what a piece of mathematics looks and feels like. That little narrative is an example
of the mathematician’s art: asking simple and elegant questions about our imaginary creations,
and crafting satisfying and beautiful explanations. There is really nothing else quite like this
realm of pure idea; it’s fascinating, it’s fun, and it’s free!
Now where did this idea of mine come from? How did I know to draw that line? How does
a painter know where to put his brush? Inspiration, experience, trial and error, dumb luck.
That’s the art of it, creating these beautiful little poems of thought, these sonnets of pure reason.
There is something so wonderfully transformational about this art form. The relationship
between the triangle and the rectangle was a mystery, and then that one little line made it
5
obvious. I couldn’t see, and then all of a sudden I could. Somehow, I was able to create a
profound simple beauty out of nothing, and change myself in the process. Isn’t that what art is
all about?
This is why it is so heartbreaking to see what is being done to mathematics in school. This
rich and fascinating adventure of the imagination has been reduced to a sterile set of “facts” to be
memorized and procedures to be followed. In place of a simple and natural question about
shapes, and a creative and rewarding process of invention and discovery, students are treated to
this:
“The area of a triangle is equal to one-half its base times its height.” Students are asked to
memorize this formula and then “apply” it over and over in the “exercises.” Gone is the thrill,
the joy, even the pain and frustration of the creative act. There is not even a problem anymore.
The question has been asked and answered at the same time— there is nothing left for the
student to do.
Now let me be clear about what I’m objecting to. It’s not about formulas, or memorizing
interesting facts. That’s fine in context, and has its place just as learning a vocabulary does— it
helps you to create richer, more nuanced works of art. But it’s not the fact that triangles take up
half their box that matters. What matters is the beautiful idea of chopping it with the line, and
how that might inspire other beautiful ideas and lead to creative breakthroughs in other
problems— something a mere statement of fact can never give you.
By removing the creative process and leaving only the results of that process, you virtually
guarantee that no one will have any real engagement with the subject. It is like saying that
Michelangelo created a beautiful sculpture, without letting me see it. How am I supposed to be
inspired by that? (And of course it’s actually much worse than this— at least it’s understood that
there is an art of sculpture that I am being prevented from appreciating).
By concentrating on what, and leaving out why, mathematics is reduced to an empty shell.
The art is not in the “truth” but in the explanation, the argument. It is the argument itself which
gives the truth its context, and determines what is really being said and meant. Mathematics is
the art of explanation. If you deny students the opportunity to engage in this activity— to pose
their own problems, make their own conjectures and discoveries, to be wrong, to be creatively
frustrated, to have an inspiration, and to cobble together their own explanations and proofs— you
deny them mathematics itself. So no, I’m not complaining about the presence of facts and
formulas in our mathematics classes, I’m complaining about the lack of mathematics in our
mathematics classes.
If your art teacher were to tell you that painting is all about filling in numbered regions, you
would know that something was wrong. The culture informs you— there are museums and
galleries, as well as the art in your own home. Painting is well understood by society as a
6
This medium of human expression. Likewise, if your science teacher tried to convince you that
astronomy is about predicting a person’s future based on their date of birth, you would know she
was crazy— science has seeped into the culture to such an extent that almost everyone knows
about atoms and galaxies and laws of nature. But if your math teacher gives you the impression,
either expressly or by default, that mathematics is about formulas and definitions and
memorizing algorithms, who will set you straight?
The cultural problem is a self-perpetuating monster: students learn about math from their
teachers, and teachers learn about it from their teachers, so this lack of understanding and
appreciation for mathematics in our culture replicates itself indefinitely. Worse, the perpetuation
of this “pseudo-mathematics,” this emphasis on the accurate yet mindless manipulation of
symbols, creates its own culture and its own set of values. Those who have become adept at it
derive a great deal of self-esteem from their success. The last thing they want to hear is that
math is really about raw creativity and aesthetic sensitivity. Many a graduate student has come
to grief when they discover, after a decade of being told they were “good at math,” that in fact
they have no real mathematical talent and are just very good at following directions. Math is not
about following directions, it’s about making new directions.
And I haven’t even mentioned the lack of mathematical criticism in school. At no time are
students let in on the secret that mathematics, like any literature, is created by human beings for
their own amusement; that works of mathematics are subject to critical appraisal; that one can
have and develop mathematical taste. A piece of mathematics is like a poem, and we can ask if
it satisfies our aesthetic criteria: Is this argument sound? Does it make sense? Is it simple and
elegant? Does it get me closer to the heart of the matter? Of course there’s no criticism going on
in school— there’s no art being done to criticize!
Why don’t we want our children to learn to do mathematics? Is it that we don’t trust them,
that we think it’s too hard? We seem to feel that they are capable of making arguments and
coming to their own conclusions about Napoleon, why not about triangles? I think it’s simply
that we as a culture don’t know what mathematics is. The impression we are given is of
something very cold and highly technical, that no one could possibly understand— a selffulfilling
prophesy if there ever was one.
It would be bad enough if the culture were merely ignorant of mathematics, but what is far
worse is that people actually think they do know what math is about— and are apparently under
the gross misconception that mathematics is somehow useful to society! This is already a huge
difference between mathematics and the other arts. Mathematics is viewed by the culture as
some sort of tool for science and technology. Everyone knows that poetry and music are for pure
enjoyment and for uplifting and ennobling the human spirit (hence their virtual elimination from
the public school curriculum) but no, math is important.
18 Oct 2011, 11:41 am
Yeah, I've been trying to say this for awhile now; not just about math but about our education system. It's too bad our culture looks at it differently. As a result, our education system suffers drastically because the arts suffer drastically. There is no creativity or imagination in the way we approach a certain subject.
There are too many "how to's" and "one absolute answer (at the back of the book)" in the way our society approaches education.
In such a stimulating time period, people are being conned into exchanging their aspirations with a dull, corporate-like way of living. It's no surprise students (such as myself) are becoming more and more discouraged by college (as you can see in numerous threads in this forum). There's no inspiration or solid reasoning to pursue education. At least, not when the only incentive in our society is based on capital gain as a means to controlling life.
20 Oct 2011, 8:52 pm
Unfortunately, that is the way to learn technical subjects. It's impossible to learn advanced mathematics without committing the basics to memory. Addition, subtraction, multiplication tables, long division, quadratic formula, integrals, laplace transforms... The more advanced mathematics builds upon the fundamentals. Since mathematics requires so much background knowledge (and some people just cannot remember that much), the focus is on learning the fundamentals and drilling problems. Unless there are separate classes for the "gifted" and the "average" student, mathematics will be taught the same way.
Most people are bad at drawing, photography, music, and other arts just as most people are bad with mathematics. But the visual and performing arts are easy to understand since they can be communicated through sensory experiences. How does one communicate mathematics? Proofs, theorems, and logic. Mathematics simply requires far too much background knowledge that most people are incapable of possessing.
22 Oct 2011, 2:06 am
fiooo wrote:
long division,
What exactly is the point of teaching the long division algorithm? It simply doesn't get used later and isn't necessary to understand the concept of division. Also: given that we have plentiful and extremely cheap calculators and computers, and that most people carry a cell phone with a computing processor in it all the time, why should we learn a pen-and-paper algorithm to perform a computation that the computer does much faster and more accurately?
People used to be taught a method of calculating square roots in school as a standard thing. It was quite labor-intensive as a calculation with pen-and-paper, since it iterated and converged on a solution. The method was kind of a neat little thing (although actually doing it would be incredibly tedious), but they stopped teaching it when calculators came out. They ought to do the same with long division, which is just tedious and not even a neat little thing.
Quote:
The more advanced mathematics builds upon the fundamentals.
Sometimes. And this depends on what you call the fundamentals. I would call set theory, logic, the abstract idea of mappings (aka functions) and so forth the basics, not long division and whether you happen to have memorized your multiplication tables.
Quote:
Since mathematics requires so much background knowledge (and some people just cannot remember that much),
I don't think it's really about memory storage space. People can remember the stats of hundreds of baseball players without trying. It might be a lack of interest, or bad teaching, or an inability to handle abstraction, or something else, but I don't think ability to memorize is the big problem.
Quote:
the focus is on learning the fundamentals and drilling problems.
I don't think drilling really helps, except to get people proficient with certain mechanical calculations. Mechanical calculations are almost always better done by a computer, which never gets tired or misremembers a formula, and is much, much faster as well.
I won't disagree completely, though, knowing how to add and multiply easy numbers in your head is a useful skill. But that isn't so much math as it is calculations.
_________________
"A dead thing can go with the stream, but only a living thing can go against it." --G. K. Chesterton
23 Oct 2011, 12:11 am
Ancalagon wrote:
fiooo wrote:
long division,
What exactly is the point of teaching the long division algorithm? It simply doesn't get used later and isn't necessary to understand the concept of division. Also: given that we have plentiful and extremely cheap calculators and computers, and that most people carry a cell phone with a computing processor in it all the time, why should we learn a pen-and-paper algorithm to perform a computation that the computer does much faster and more accurately?
To expand upon your point, what is the point of teaching obsolete techniques and technology in school? Schools still have us write code for the bubble sort in computer science class, derive equations in physics that have been derived before (some of which are experimentally determined to be very inaccurate models), solve complicated integrals using tables and/or obscure integration techniques, etc. I still remember when one of my classmates asked my computer science professor what is the point of writing those functions and algorithms that were in the C++ standard library. His answer was "as a mental exercise." Similarly, my calculus teacher in high school said that the point of solving those complicated integrals when Mathematica would be more efficient was to "improve your algebra skills." So using the same logic, the purpose of long division is to introduce the concept of a slightly more complicated algorithm (compared to multiplication) to students and to improve their arithmetic skills.
But I do agree with you that long division, drilling problems, and repetition do make people lose interest in math. The quick learners get bored from from the repetition and the slow learners get frustrated from their inability to solve the problems. I never liked math in school because my teachers didn't like math and it was evident in their teaching.
Maybe your idea of math is different from mine because your education is so different from mine. I have never taken a "real" math class with proofs and abstract ideas. Maybe that's why I never appreciated mathematics as something that is "beautiful." I see math as a tool to quantitatively describe something.
23 Oct 2011, 11:38 pm
fiooo wrote:
Schools still have us write code for the bubble sort in computer science class,
This I can understand. Bubble sort really isn't a good algorithm, but it makes a good example of a bad algorithm and a good excuse to talk about big O notation.
It's also dead simple to code. Two nested for loops, a compare, and a swap and you're done. Anyone who's even half-decent at programming can write an implementation down that works the first time as fast as he can write. That's actually a good thing when you know what size your input is going to be and it's really small.
There was a little sorting routine that I needed to be able to sort exactly 5 things with as part of a much larger program. It was in C and I was being graded partly on speed, so I was trying to squeeze performance out of every little bit. I could have just written down a bubble sort as fast as I could type, but I decided to be tricky and just write down the comparisons and swaps necessary for it, since it was so small. I never did figure out my bug, but when I threw in the towel and did a bubble sort, it worked the first time.
Quote:
I still remember when one of my classmates asked my computer science professor what is the point of writing those functions and algorithms that were in the C++ standard library. His answer was "as a mental exercise."
Depending on what it is, there could be a very good reason for doing that. Knowing how the library (probably) solves a problem means you can estimate how fast it will be, which is going to affect how fast your program is if it's in the inner loop. Sometimes you need a function that is similar to what's in the library, but is less general, so that's another place you can optimize if you need to.
Knowing why certain input functions are bad can be explained better if you can see the (mostly short) code for at least one of the routines.
And doing that sort of thing really is good mental exercise, unlike cranking out 50 long divisions by hand.
Quote:
Similarly, my calculus teacher in high school said that the point of solving those complicated integrals when Mathematica would be more efficient was to "improve your algebra skills."
That's just BS. It never helped my algebra skills.
Quote:
So using the same logic, the purpose of long division is to introduce the concept of a slightly more complicated algorithm (compared to multiplication) to students and to improve their arithmetic skills.
I wouldn't mind if they actually just used it to introduce the idea of an algorithm. But they don't.
Quote:
The quick learners get bored from from the repetition and the slow learners get frustrated from their inability to solve the problems.
And the medium learners get a little bit of both, so nobody's happy.
_________________
"A dead thing can go with the stream, but only a living thing can go against it." --G. K. Chesterton
25 Oct 2011, 5:28 pm
End of thread:
http://en.wikipedia.org/wiki/Reform_mat ... _standards
http://en.wikipedia.org/wiki/Principles ... athematics
Quote:
This idea that mathematical truth can be both creative and objective
suggests a new way of looking at the implications of G¨odel’s
Incompleteness Theorem. If mathematics is inherently creative it
need not follow a single path of development. By following an ever
increasing number of paths it is possible to develop mathematics
in a way that is not subject to the limitations G¨odel’s result implies
for single path processes. This is how the mathematically capable
human mind was created. Biological evolution has followed an ever
expanding number of paths without selecting a single “correct” or
“true” path.
Divergent creative processes like biological evolution always involve
tradeoffs between diversity and concentration of resources.
By exploring the mathematics of creative divergent processes it
may be possible to find boundary conditions for this tradeoff that
optimize creativity. Such mathematics may be broadly applicable
to many human endeavors.
In the long run mathematics will need to be extended through a
creative divergent process if it is to avoid stagnation. There will be
an ever expanding number of possible incompatible extensions to
mathematics and schools will develop pursuing each of these with
no expectation of any final resolution. The expectation will be for
ever expanding diversity.
suggests a new way of looking at the implications of G¨odel’s
Incompleteness Theorem. If mathematics is inherently creative it
need not follow a single path of development. By following an ever
increasing number of paths it is possible to develop mathematics
in a way that is not subject to the limitations G¨odel’s result implies
for single path processes. This is how the mathematically capable
human mind was created. Biological evolution has followed an ever
expanding number of paths without selecting a single “correct” or
“true” path.
Divergent creative processes like biological evolution always involve
tradeoffs between diversity and concentration of resources.
By exploring the mathematics of creative divergent processes it
may be possible to find boundary conditions for this tradeoff that
optimize creativity. Such mathematics may be broadly applicable
to many human endeavors.
In the long run mathematics will need to be extended through a
creative divergent process if it is to avoid stagnation. There will be
an ever expanding number of possible incompatible extensions to
mathematics and schools will develop pursuing each of these with
no expectation of any final resolution. The expectation will be for
ever expanding diversity.
Quote:
Math wars is the debate over modern mathematics education, textbooks and curricula in the United States that was triggered by the publication in 1989 of the Curriculum and Evaluation Standards for School Mathematics by the National Council of Teachers of Mathematics (NCTM) and subsequent development and widespread adoption of a new generation of mathematics curricula inspired by these standards.
While the discussion about math skills has persisted for many decades,[1] the term "math wars" was coined by commentators such as John A. Van de Walle[2] and David Klein.[3] The debate is over traditional mathematics and reform mathematics philosophy and curricula, which differ significantly in approach and content.
While the discussion about math skills has persisted for many decades,[1] the term "math wars" was coined by commentators such as John A. Van de Walle[2] and David Klein.[3] The debate is over traditional mathematics and reform mathematics philosophy and curricula, which differ significantly in approach and content.
Quote:
Critics of reform
Those who disagree with the inquiry-based philosophy maintain that students must first develop computational skills before they can understand concepts of mathematics. These skills should be memorized and practiced, using time-tested traditional methods until they become automatic. Time is better spent practicing skills rather than in investigations inventing alternatives, or justifying more than one correct answer or method. In this view, estimating answers is insufficient and, in fact, is considered to be dependent on strong foundational skills. Learning abstract concepts of mathematics is perceived to depend on a solid base of knowledge of the tools of the subject.[5]
Supporters of traditional mathematics teaching oppose excessive dependence on innovations such as calculators or new technology, such as the Logo language. Student innovation is acceptable, even welcome, as long as it is mathematically valid. Calculator use can be appropriate after number sense has developed and basic skills have been mastered. Constructivist methods which are unfamiliar to many adults, and books which lack explanations of methods or solved examples make it difficult to help with homework. Compared to worksheets which can be completed in minutes, constructivist activities can be more time consuming. (Reform educators respond that more time is lost in reteaching poorly understood algorithms.) Emphasis on reading and writing also increases the language load for immigrant students and parents who may be unfamiliar with English.
Critics of reform point out that traditional methods are still universally and exclusively used in industry and academia. Reform educators respond that such methods are still the ultimate goal of reform mathematics, and that students need to learn flexible thinking in order to face problems they may not know a method for. Critics maintain that it is unreasonable to expect students to "discover" the standard methods through investigation, and that flexible thinking can only be developed after mastering foundational skills.
Some curricula incorporate research by Constance Kamii and others that concluded that direct teaching of traditional algorithms is counterproductive to conceptual understanding of math. Critics have protested some of the consequences of this research. Traditional memorization methods are replaced with constructivist activities. Students who demonstrate proficiency in a standard method are asked to invent another method of arriving at the answer. Some teachers supplement such textbooks in order to teach standard methods more quickly. Some curricula do not teach long division. Critics believe the NCTM revised its standards to explicitly call for continuing instruction of standard methods, largely because of the negative response to some of these curricula (see below).
Those who disagree with the inquiry-based philosophy maintain that students must first develop computational skills before they can understand concepts of mathematics. These skills should be memorized and practiced, using time-tested traditional methods until they become automatic. Time is better spent practicing skills rather than in investigations inventing alternatives, or justifying more than one correct answer or method. In this view, estimating answers is insufficient and, in fact, is considered to be dependent on strong foundational skills. Learning abstract concepts of mathematics is perceived to depend on a solid base of knowledge of the tools of the subject.[5]
Supporters of traditional mathematics teaching oppose excessive dependence on innovations such as calculators or new technology, such as the Logo language. Student innovation is acceptable, even welcome, as long as it is mathematically valid. Calculator use can be appropriate after number sense has developed and basic skills have been mastered. Constructivist methods which are unfamiliar to many adults, and books which lack explanations of methods or solved examples make it difficult to help with homework. Compared to worksheets which can be completed in minutes, constructivist activities can be more time consuming. (Reform educators respond that more time is lost in reteaching poorly understood algorithms.) Emphasis on reading and writing also increases the language load for immigrant students and parents who may be unfamiliar with English.
Critics of reform point out that traditional methods are still universally and exclusively used in industry and academia. Reform educators respond that such methods are still the ultimate goal of reform mathematics, and that students need to learn flexible thinking in order to face problems they may not know a method for. Critics maintain that it is unreasonable to expect students to "discover" the standard methods through investigation, and that flexible thinking can only be developed after mastering foundational skills.
Some curricula incorporate research by Constance Kamii and others that concluded that direct teaching of traditional algorithms is counterproductive to conceptual understanding of math. Critics have protested some of the consequences of this research. Traditional memorization methods are replaced with constructivist activities. Students who demonstrate proficiency in a standard method are asked to invent another method of arriving at the answer. Some teachers supplement such textbooks in order to teach standard methods more quickly. Some curricula do not teach long division. Critics believe the NCTM revised its standards to explicitly call for continuing instruction of standard methods, largely because of the negative response to some of these curricula (see below).
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Principles and standards
Main article: Principles and Standards for School Mathematics
Quote:
The momentum for reform in mathematics education began in the early 1980s, as educators reacted to the "new math" of the 1960s and 1970s. The work of Piaget and other developmental psychologists was shifting the focus of mathematics educators from mathematics content to how children best learn mathematics. [3] The National Council of Teachers of Mathematics summarized the state of current research with the publication of Curriculum and Evaluation Standards in 1989 and Principles and Standards for School Mathematics in 2000, bringing definition to the reform movement in North America.[4]
Reform mathematics curricula challenge students to make sense of new mathematical ideas through explorations and projects, often in real contexts. [3] Reform texts emphasize written and verbal communication, working in cooperative groups, making connections between concepts, and connections between representations. By contrast, "traditional" textbooks emphasize procedural mathematics and provide step-by-step examples with skill exercises.
Traditional mathematics focuses on teaching algorithms that will lead to the correct answer. Because of this focus on application of algorithms, the traditional math student must always use the specific method that is being taught. This kind of algorithmic dependence is de-emphasized in reform mathematics.[5] Reformers do not oppose correct answers, but prefer to focus students' attention on the process leading to the answer, rather than the answer itself. The presence of occasional minor errors is deemed less important than the overall thought process. Research has shown that children make fewer mistakes with calculations and remember algorithms longer when they understand the concepts underlying the methods they use. In general, children in reform classes perform at least as well as children in traditional classes on tests of calculation skill, and considerably better on tests of problem solving.[6][7][8][9]
A common misconception is that reform educators do not want children to learn the standard methods of arithmetic. As the NCTM Focal Points make clear, such methods are still the ultimate goal, but reformers believe that conceptual understanding should come first. Reform educators believe that such understanding is best pursued by allowing children at first to solve problems using their own understanding and methods. Under guidance from the teacher, students eventually arrive at an understanding of standard methods. Even the controversial NCTM Standards of 1989 did not call for abandoning standard algorithms, but instead recommended a decreased emphasis on complex paper-and-pencil computation drills and greater attention to mental computation, estimation skills, thinking strategies for mastering basic facts and conceptual understanding of arithmetic operations..
Reform mathematics curricula challenge students to make sense of new mathematical ideas through explorations and projects, often in real contexts. [3] Reform texts emphasize written and verbal communication, working in cooperative groups, making connections between concepts, and connections between representations. By contrast, "traditional" textbooks emphasize procedural mathematics and provide step-by-step examples with skill exercises.
Traditional mathematics focuses on teaching algorithms that will lead to the correct answer. Because of this focus on application of algorithms, the traditional math student must always use the specific method that is being taught. This kind of algorithmic dependence is de-emphasized in reform mathematics.[5] Reformers do not oppose correct answers, but prefer to focus students' attention on the process leading to the answer, rather than the answer itself. The presence of occasional minor errors is deemed less important than the overall thought process. Research has shown that children make fewer mistakes with calculations and remember algorithms longer when they understand the concepts underlying the methods they use. In general, children in reform classes perform at least as well as children in traditional classes on tests of calculation skill, and considerably better on tests of problem solving.[6][7][8][9]
A common misconception is that reform educators do not want children to learn the standard methods of arithmetic. As the NCTM Focal Points make clear, such methods are still the ultimate goal, but reformers believe that conceptual understanding should come first. Reform educators believe that such understanding is best pursued by allowing children at first to solve problems using their own understanding and methods. Under guidance from the teacher, students eventually arrive at an understanding of standard methods. Even the controversial NCTM Standards of 1989 did not call for abandoning standard algorithms, but instead recommended a decreased emphasis on complex paper-and-pencil computation drills and greater attention to mental computation, estimation skills, thinking strategies for mastering basic facts and conceptual understanding of arithmetic operations..
Infoseeker
Deinonychus
Joined: 6 Mar 2011
Age: 37
Gender: Male
Posts: 359
Location: Metro Detroit area, MI, US
27 Oct 2011, 7:49 am
I don't even want to imagine what kind of mind can find the trends and patterns that lead to the logarithms that help convert signals from a device, like a camera, into actual image. Or convert the signal from a strange thermometer probe into an accurate temperature estimate.
I knew right away in college I did not have a tendency with math. All those solid state devices and their logarithms....
Though, I love playing with the devices that depend on them! 
Lots of fun devices in a biology research lab.
_________________
Age: 27
27 Oct 2011, 11:02 pm
Ancalagon wrote:
fiooo wrote:
long division,
What exactly is the point of teaching the long division algorithm? It simply doesn't get used later and isn't necessary to understand the concept of division. Also: given that we have plentiful and extremely cheap calculators and computers, and that most people carry a cell phone with a computing processor in it all the time, why should we learn a pen-and-paper algorithm to perform a computation that the computer does much faster and more accurately?
People used to be taught a method of calculating square roots in school as a standard thing. It was quite labor-intensive as a calculation with pen-and-paper, since it iterated and converged on a solution. The method was kind of a neat little thing (although actually doing it would be incredibly tedious), but they stopped teaching it when calculators came out. They ought to do the same with long division, which is just tedious and not even a neat little thing.
Quote:
The more advanced mathematics builds upon the fundamentals.
Sometimes. And this depends on what you call the fundamentals. I would call set theory, logic, the abstract idea of mappings (aka functions) and so forth the basics, not long division and whether you happen to have memorized your multiplication tables.
Quote:
Since mathematics requires so much background knowledge (and some people just cannot remember that much),
I don't think it's really about memory storage space. People can remember the stats of hundreds of baseball players without trying. It might be a lack of interest, or bad teaching, or an inability to handle abstraction, or something else, but I don't think ability to memorize is the big problem.
Quote:
the focus is on learning the fundamentals and drilling problems.
I don't think drilling really helps, except to get people proficient with certain mechanical calculations. Mechanical calculations are almost always better done by a computer, which never gets tired or misremembers a formula, and is much, much faster as well.
I won't disagree completely, though, knowing how to add and multiply easy numbers in your head is a useful skill. But that isn't so much math as it is calculations.
you need long division when dealing with equations.
TheMatrixHasYou
Snowy Owl
Joined: 8 Apr 2011
Age: 28
Gender: Female
Posts: 160
Location: Having dinner with Alan Turing's adorable ghost.
03 Nov 2011, 5:14 pm
TheMatrixHasYou wrote:
Quote:
you need long division when dealing with equations.
....like polynomial long division.
I'm not sure dividing polynomials the long division way is really necessary, but even if it is, why not teach polynomial long division as one thing, rather than teaching something else first?
Quote:
Weird thing is, I can do this, but don't remember how to do long division. 
If you think about it, the way we write numbers is really a polynomial. In base 10, x=10.
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