Why is the transition from multiplication tables to algorithms flawed?

[Crosson]

The problems with mathematics education cannot be fixed until the general public stops using these terms synonymously. Today the news (Good Morning America - ABC) reported an autistic savant who, although he is mentally disabled in many ways, could "solve complex mathematics". I was very interested until I saw that "solving complex mathematics" meant to this network news outlet "multiplying large numbers mentally"!

Mathematics is the science of theorems. It is true that some theorems pertain to arithmetic, but this does not mean that arithmetic (the basics of which everyone agrees that everyone should learn) should be considered as part of mathematics any more so then writing is considered part of mathematics.

Arithmetic (including U.S pre-algebra) should be made its own subject, beginning and ending in elementary school, with no mention of mathematics being made (since the students are not doing any).

Synthetic and Analytic geometry, trigonometry and general precalculus should all be put under the heading "Computer Science", since that is exactly where the domain of application for these things begins and ends. By getting students to write a program to draw a circle we address their lack of motivation and sense of perceived uselessness of the material.

Calculus belongs to Physics, these subjects should be studied together as old classics for the same reason that literature is studied.

English should be correctly labeled "Literature and Creative Fiction", and rhetoric (as a means of persuasion) should be ridiculed.

All mention of critical thinking, expository writing (clarity of communication) and reasoning(logic) belongs to mathematics. Mathematical Calculation should take on the generalized meaning of writing out ideas clearly and carefully so as to infer conclusions.

In my opinion these things have not happened because:

1) The English department has too much power, and they don't want to lose their monopoly on "writing" (which belongs to math at least as much as does arithmetic).

2) The computer science department enjoys not being associated with algebra/precalculus, because they receive more funding and more students. Very few freshmen going into computer science realize they will be doing more of what they think of as math (precalculus) then will be the freshmen going into mathematics.

 

[matt grime]

Synthetic and Analytic geometry, trigonometry and general precalculus should all be put under the heading "Computer Science", since that is exactly where the domain of application for these things begins and ends.

You're either joking, or, well, being deliberately provocative for no good reason, right?

Calculus belongs to Physics, these subjects should be studied together as old classics for the same reason that literature is studied.

Same comment applies.

 

[Math Is Hard]

1) The English department has too much power, and they don't want to lose their monopoly on "writing" (which belongs to math at least as much as does arithmetic).

You'd better keep it down... English professors could be reading this right now and reporting back to their departments. Make too much noise, and they'll send some of the boys around to take care of you.

 

[HallsofIvy]

I'm going to take a deep breath, a strong drink, and not say a word!

I've already gotten into trouble for repeatedly pointing out that mathematics is not physics!

 

[Crosson]

It seems that some of what I said is objectionable, and I would like you all to make your objections clear, so that I can address them. I stand by everything I said, I am not trolling (but I am proud of that I drove Ivy to drink :).

I've already gotten into trouble for repeatedly pointing out that mathematics is not physics!

Does something in my post suggest that I think mathematics is physics?

In fact, I advocate seperating the two properly, that is by including the method of basic physics (freshman calculus) under the label "physics", so as not to confuse it with mathematics, the science of theorems.

I think that finding an antiderivative relates more closely, in concept and appplication, to finding the force/charge/mass etc. than to showing that a function is integrable if it is bounded and continuous. The former is arithmetic (studied for application), the latter is mathematics (studied for aesthetics).

Edit: Matt - I guess I could have said "direct domain of application", but I still think what I said was true and implore to share your reasons for disagreeing.

 

[DaveC426913]

The problems with mathematics education cannot be fixed until the general public stops using these terms synonymously. Today the news (Good Morning America - ABC) reported an autistic savant who, although he is mentally disabled in many ways, could "solve complex mathematics".

Actually, I was going to say the problems with mathematics cannot be fixed until people stop looking to Good Morning America for their education.

Wait. They don't.

 

[matt grime]

You have said that you only wish for mathematics with application to the real world to be taught, and only in the context of its application to the real world. If you truly believe in what you said, that geometry should be taught only in computer science, because that is its only application, then you're missing the point of mathematics, not to mention seemingly being ignorant of some of its uses. Calculus (whatever you mean by it) certainly does not belong 'to physics', either as a subject in its own right, or as a subject to study for its own merits.

You complain that people are stereotyping mathematics unfairly as mere arithmetic, then go and make the same mistake yourself.

 

[JasonRox]

You have said that you only wish for mathematics with application to the real world to be taught, and only in the context of its application to the real world. If you truly believe in what you said, that geometry should be taught only in computer science, because that is its only application, then you're missing the point of mathematics, not to mention seemingly being ignorant of some of its uses. Calculus (whatever you mean by it) certainly does not belong 'to physics', either as a subject in its own right, or as a subject to study for its own merits.

You complain that people are stereotyping mathematics unfairly as mere arithmetic, then go and make the same mistake yourself.

Well said, and I'm quoting it to be said again.

 

[Crosson]

Ah, a typical forum misread.

In my post when I say "no mention of mathematics", I am referring to (in the sentence) elementary school. Obviously, if we are to separate mathematics and arithmetic, then we much teach arithmetic without making mention of mathematics! The kids in elementary school aren't ready to think they are doing math! Then they don't respect it!

Later I bring up mathematics as the science of theorems, teaching students to think, write and reason clearly as well as to read other's writings with depth. This subject is central (as English is now) to all students. This kind of (true) mathematics should be required and the "applications" math track be relegated to the Accounting/CS/Physics electives.

 

[yenchin]

Ah, a typical forum misread.

In my post when I say "no mention of mathematics", I am referring to (in the sentence) elementary school. Obviously, if we are to separate mathematics and arithmetic, then we much teach arithmetic without making mention of mathematics! The kids in elementary school aren't ready to think they are doing math! Then they don't respect it!

I kinda agree with this.

Later I bring up mathematics as the science of theorems, teaching students to think, write and reason clearly as well as to read other's writings with depth. This subject is central (as English is now) to all students. This kind of (true) mathematics should be required and the "applications" math track be relegated to the Accounting/CS/Physics electives.

But, does that mean Applied Mathematics is not mathematics and should be thrown out of math departments worldwide? I think we need to strike the balance between pure and applied, and make it clear to the students that they need to have a fair amount of exposure to both to get a good appreciation of mathematics. Over concentration in either extreme is not a good idea.

I dread math books with tonnes of applications and calculations with formulas pop up from nowhere without proper proofs or at least the sketch of the ideas behind the theorems. But that is not to say that we completely ignore teaching applications to students and let the other department do the job.

 

[Crosson]

But that is not to say that we completely ignore teaching applications to students and let the other department do the job.

Take a slice of a calculus class: a cs major, a bio major, an econ major, a phys major and a math major.

The problem comes when we force these people to study calculus from each other's intended realm of application. For the science majors, this leads to a dislike of math, and for the math majors it leads to a dislike of applications.

That makes the case for not studying applications in the math department (i.e. students elect to take the applications they are interested in at the departments they are interested in). But I propose something stronger: that calculus (of real functions of a single real variable) no more belongs to the math department than to any of the others! They should all teach their own brand of calculus tailored to be relevant an interesting to their students, then people wouldn't dislike math so much.

Now the guards will call me away for taking on the calculus cash cow

 

[FunkyDwarf]

ill say what feynman said: physics is to maths what sex is to masturbation :)

 

[murshid_islam]

but feynman was a physicist. he is supposed to be biased towards physics. :)

 

[Gib Z]

Richard Feynman was an excellent physicist, and if he wasn't he would have been an excellent Number Theorist. One of the most profound people to ever live. But I would reverse that quote.

 

[HallsofIvy]

It seems that some of what I said is objectionable, and I would like you all to make your objections clear, so that I can address them. I stand by everything I said, I am not trolling (but I am proud of that I drove Ivy to drink :).

Does something in my post suggest that I think mathematics is physics?

You said "Calculus belongs to Physics". Are you unaware that calculus is used in Biology, Economics, Meteorology, etc., etc.?

 

[Crosson]

You said "Calculus belongs to Physics". Are you unaware that calculus is used in Biology, Economics, Meteorology, etc., etc.?

Here is what I said three posts above this:

Take a slice of a calculus class: a cs major, a bio major, an econ major, a phys major and a math major.

And then I argue that calculus is no more a part of mathematics (mathematicians use it as a topic about which to prove theorems) then it (calculus) is a part of physics, bio, econ, meteorology, geology etc.

 

[Doodle Bob]

And then I argue that calculus is no more a part of mathematics (mathematicians use it as a topic about which to prove theorems) then it (calculus) is a part of physics, bio, econ, meteorology, geology etc.

That's right! Also, writing shouldn't be taught by an English professor, because so many other people besides English professors write.

History should not be taught by historians, because history involves so many non-historians.

Phys. Ed. shouldn't be taught by athletes because so many non-athletes like to do sports.

Biology shouldn't be taught by biologists, because so many non-biologists use biology (on a daily basis, no less).

 

[morphism]

And then I argue that calculus is no more a part of mathematics (mathematicians use it as a topic about which to prove theorems) then it (calculus) is a part of physics, bio, econ, meteorology, geology etc.

I think it is you who has no idea what mathematics is.

 

[Hootenanny]

That's right! Also, writing shouldn't be taught by an English professor, because so many other people besides English professors write.

History should not be taught by historians, because history involves so many non-historians.

Phys. Ed. shouldn't be taught by athletes because so many non-athletes like to do sports.

Biology shouldn't be taught by biologists, because so many non-biologists use biology (on a daily basis, no less).

I love this post, it gets my nomination for post of the year!

 

[matt grime]

Take a slice of a calculus class

IN AN AMERICAN UNIVERSITY:

a cs major, a bio major, an econ major, a phys major and a math major.

Don't judge the rest of us because of your education system.

 

[matt grime]

Here is what I said three posts above this:



And then I argue that calculus is no more a part of mathematics (mathematicians use it as a topic about which to prove theorems) then it (calculus) is a part of physics, bio, econ, meteorology, geology etc.

You realize you're contradicting yourself, right?

 

[FunkyDwarf]

Calculus is part of maths end of story. Maths is a tool for most things ie a means to an end not an end itself, however people do use it as an end, ie proofs and theorems that at the moment have no practical application.

 

[Crosson]

We all agree that there is a vast amount of good mathematics that has been created that does not have applications (pure math).

Question: How do we justify indoctrinating intelligent hardworking people into the field of pure mathematics?

I think it is because the skill of mathematical writing and reading are amazing skills that can be applied by anyone to improve communications and reasoning in general. I emphasize the communication aspect of math, it teaches us to read slowly and carefully (a respect no one on this forum grants me) and to give instructions or reports more precisely.

This is what I am saying the math curriculum should be: a focus on the reading and writing of math, not the concepts (which are chosen for applicability). Sure concepts could be taught i.e. arithmetic, algebra, calculus but these are not good reasons motivating students to study pure mathematics.

Question: Why isn't Arithmetic its own separate subject?

Answer: Because Arithmetic is used for examples and studied indirectly in many branches of pure mathematics, and studied directly in Number Theory.

Aren't these aspects of Arithmetic so removed from grade-school computational drills that we are justified in not including this subject under the header Math but instead "Arithmetic": its own subject with a well defined beginning and end. This would eliminate "what do we use this for?" questions because we are not confusing the students to think that the computational drills they are doing are anymore related to Math (claimed by a few to be interesting, currently a mystery to most) then would be their facility in drawing letters and diagrams.

 

[matt grime]

Q: how do you justify using the word indoctrinating?

 

[Chris Hillman]

Confrontation as a rhetorical technique?

We all agree that there is a vast amount of good mathematics that has been created that does not have applications (pure math).

Yet. (Or that you yourself happen to know about.)

Question: How do we justify indoctrinating intelligent hardworking people into the field of pure mathematics?

I think it is because the skill of mathematical writing and reading are amazing skills that can be applied by anyone to improve communications and reasoning in general. I emphasize the communication aspect of math, it teaches us to read slowly and carefully (a respect no one on this forum grants me) and to give instructions or reports more precisely.

This is what I am saying the math curriculum should be: a focus on the reading and writing of math, not the concepts (which are chosen for applicability). Sure concepts could be taught i.e. arithmetic, algebra, calculus but these are not good reasons motivating students to study pure mathematics.

So, were you just absurdly overstating your real case in order to get attention? If your real point was that calculus (I'd add linear algebra) should be taught to EveryMan as part of the standard intellectual toolkit, as an essential amplification of previous reading and writing skills, then you'd probably find greater acceptance here of your thesis.

It seems to me that a more logical (and otherwise reasonable!) conclusion from your last two paragraphs quoted above would be that more students should be required to take calculus, and that writing skills should be emphasized more in calculus classes, to help students understand why they are being inducted into the mysteries of analysis. If so, when I was a graduate student at UW, there was quite a bit of interest in the Math Department and "client departments" in exploring this and many other "curricular reform" suggestions. Some of us TAs, on our own initiative, experimented with including writing assignments in our calculus sections.

 

[Crosson]

Q: how do you justify using the word indoctrinating?

I'm Always chasing after students to read their definitions:

indoctrinate: To instruct in a body of doctrine or principles.

If your real point was that calculus (I'd add linear algebra) should be taught to EveryMan as part of the standard intellectual toolkit, as an essential amplification of previous reading and writing skills, then you'd probably find greater acceptance here of your thesis.

This is part of what I am saying, but I really want to emphasize:

Math class consists of reading and writing mathematical material. Think of the abundance of structures in elementary combinatorics/probability that have (simple enough) self-contained definitions and theorems. The focus is not on the content, but on the reasoning and writing, this is made clear to the students.

Classes like calculus/arithmetic are not labeled as math, they are labeled "Calculus" and "Arithmetic" and students take them for all the reasons that they currently do. This includes math students, because after a certain level their math classes begin to discuss theorems which have (computational/drills/conceptual) calculus as a prerequisite.

 

[HallsofIvy]

I think I am finally beginning to grasp the whole point of this! Basically, Crosson is opposed to the whole idea of "Liberal Arts Education". He is saying "don't require me to learn a lot of things that are not necessary for the job I want to do- just teach me how to do the job." In particular, do not require that he learn the ideas behind the mathematical tools necessary for physics, just give him the tools. And certainly, no physicist should be required to read Shakespeare! (Hence the polemic against the power of the English Department to force him to do so.)

In my humble opinion, a very shortsighted view. Much of the mathematics that would taught in such a physics course was not used in physics until a physicist who knew some abstract mathematics recognized that it could be applied. In any case, the purpose of education (as opposed to "training") is to produce an educated person- one who knows about other people, other societies, and has good understanding of his and other cultures, not just how to do his job.

 

[arildno]

Besides, I would say that arithmetics IS part of mathematics, and the study of (to us) trivial logical relations involved in, say, arithmetical algorithms, should be the starting point of deeper mathematical studies.

One of the main fallacies in early maths education is not the teaching of arithmetics, but the way in which that subject is taught.

 

[Crosson]

Halls Of Ivy -- No, that's not what I meant, but thank you for taking the time to respond. I appreciate all of you, and I take your comments as writing criticism; teaching me how to communicate with mathematicians who are in "casual mode".

One of the main fallacies in early maths education is not the teaching of arithmetics, but the way in which that subject is taught.

As a sample of what I am talking about:

Math Class: students are asked to write descriptions of the algorithms that they use to do arithmetic. The focus is on explaining the process clearly, not "practicing" the process.

Arithmetic Class: This is where students do the drills that are necessary for basic computational arithmetic life skills.

In general, I am adcocating that we isolate the "drills" of Arithmetic/Calculus/Algebra as separate subjects from math. Math should be about thinking, and the drills reinforce the opposite. The drills teach skills that are useful:

1) For applications.

2) As material about which to prove theorems.

I suspect it is reason (2) that makes most of you forum members say that these drills belong to mathematics, but I see (2) as a special case of (1) i.e. applying the drills to the subject of theorem-proving.

 

[DaveC426913]

In any case, the purpose of education (as opposed to "training") is to produce an educated person- one who knows about other people, other societies, and has good understanding of his and other cultures, not just how to do his job.

Is this not the difference between university and college?

In university you learn how to learn; you learn principles, academia, foundations, theory, giving you a knowledge base they will open many doors.

In college you learn a trade that will open a few doors.

(Maybe just here in Canada.)

 

[DaveC426913]

In general, I am adcocating that we isolate the "drills" of Arithmetic/Calculus/Algebra as separate subjects from math. Math should be about thinking, and the drills reinforce the opposite. The drills teach skills that are useful:

I think that, while this may seem logical in theory, in practice, it doesn't make sense.

Centuries of education demonstrate that those who grasp arithmetic intuitively will be more likely to grasp higher mathematical concepts intuitively. And that those who don't, don't.

Regardless of how you think they should be grouped, I believe that the grouping happens naturally, and the education system follows their lead.

 

[Crosson]

Centuries of education demonstrate that those who grasp arithmetic intuitively will be more likely to grasp higher mathematical concepts intuitively. And that those who don't, don't.

Based on the personal experience of at least myself, a friend, and Albert Einstein, this is totally wrong and disgusting!

Oh yes but you said "more likely", so you feel safe in ignoring exceptions. Unfortunately this is not acceptable when it comes to such a glorious subject, participation in which needs to be more encouraged!

 

[Crosson]

Regardless of how you think they should be grouped, I believe that the grouping happens naturally, and the education system follows their lead.

The educational system is a dinosaur! Computers have changed mathematics forever and those centuries of experience matter less and less with each passing day.

 

[DaveC426913]

Based on the personal experience of at least myself, a friend, and Albert Einstein, this is totally wrong and disgusting!

Based on an anecdotal sampling of three (3), you are comfortable labelling it as "disgusting".

OK, so much for a rational discussion...

Oh yes but you said "more likely", so you feel safe in ignoring exceptions. Unfortunately this is not acceptable when it comes to such a glorious subject, participation in which needs to be more encouraged!

I'm not ignoring exceptions, I'm acknowledging the catering is to the majority. Your solution is no better, it just divides it up differently, likewise leaving its own exceptions ignored.

The only way that's going to be fixed is via an overhaul in education that accounts for the exceptions as well as the majority.

But that's a whole different ball of wax.

 

[matt grime]

Unfortunately this is not acceptable when it comes to such a glorious subject, participation in which needs to be more encouraged!

And your suggestion is to teach mathematics to begin with purely by its applications (or more accurately, by what you think are its applications), thus implying that maths is only useful when applied. If that becomes ingrained over the first 11 years of schooling what is the chance that anyone would want to do mathematics at university?

 

[matt grime]

The educational system is a dinosaur! Computers have changed mathematics forever and those centuries of experience matter less and less with each passing day.

Really? How many examples can you cite? Perhaps the four colour theorem, and some useful conjecture indicators in number theory... Computer drawing packages have certainly helped with real 3-d algebraic geometry, I suppose. Certainly computer packages are useful for checking large (numbers of) examples, but I'm not even aware of a successful proof checker for a computer.

 

[arildno]

What do you suggest, Crosson?
Increase the number of Arithmetics+Maths hours to 10 per week, rather than 5 "Maths" hours per week?

 

[Crosson]

Really? How many examples can you cite?

Take for example Euler or Gauss, lagrange, laplace etc. Among their truly mathematical accomplishments were also their feats of numerical computation No mathematicians does this in todays computer age, and I consider this to be a big difference.

Also, note that the entire field of nonlinear dynamics and fractals was created because of the invention of modern computers.

And your suggestion is to teach mathematics to begin with purely by its applications (or more accurately, by what you think are its applications), thus implying that maths is only useful when applied.

You are totally misunderstanding what I am saying. No one is saying "Math should only be applications", I am saying that we should preserve the beauty of math by teaching the subject (theorem-proving) seperately from courses that involve "drill skills" i.e. Arithmetic. In a sense, I am saying the exact opposite of what you accuse me of saying.

Based on an anecdotal sampling of three (3), you are comfortable labelling it as "disgusting".

Yes, I think that every minority reserves equal representation, in idealistic discussions such as those on 'Physics Forums'. Ideallistically, this is something that America stands for, and it is the most beautiful aspect of this country. Don't bore me with what is practical, we can all see the obvious. Still, in a discussion of the way things should be, I think it is disgusting to exclude people unnecessarily.

I'm not ignoring exceptions, I'm acknowledging the catering is to the majority. Your solution is no better, it just divides it up differently, likewise leaving its own exceptions ignored.

Please, let these minorities speak up so that we can design the system to include them. The difference between your statement and mine is that you were explicitly excluding people who are bad at arithmetic, and personally that disgusts me because arithmetic drills are so irrelevant to higher math that the exclusion of even one hopeful mathematician on this basis is at least a personal tragedy (for that person's life) but also, I think, a tragedy for the entire mathematics community.

 

[arildno]

I wonder what sort of maths you've been doing. If you've done any error estimates of some shape, you wouldn't be able to do many of these without the arithmetical skills as an elementary, hardly thought about, but extensively needed skill.

 

[Crosson]

I wonder what sort of maths you've been doing. If you've done any error estimates of some shape, you wouldn't be able to do many of these without the arithmetical skills as an elementary, hardly thought about, but extensively needed skill.

Yes, arithmetic is necessary for doing error analysis. So is writing (english), but one of these is commonly included in mathematics and the other is not, so this criterion ("gets used") is not a convincing reason to include arithmetic as part of mathematics, unless we would also admit that writing is part of mathematics.

 

[morphism]

Arithmetic is part of mathematics. End of story.

 

[symbolipoint]

Crosson,

Would you settle for Mathematics instruction concentrating
on Arithmetic including number sense, units, simple Geometry,
consumer-quality graphs, word problems, until about 6th grade;
and then push for "Pre-Algebra" and Introductory Algebra, plus
review of the earlier concepts in junior high schoo and early high
school; and then Geometry, Intermediate Algebra, Trigonometry,
and maybe Calculus or Statistics for college preparatory students
in high school? Would you also prefer each or most of these to
be their own dedicated distrinct courses? That entire arrangement
should help you to feel better.

Most basic consumers should at least understand the basics of
algebra(introductory). The average consumer still should know
how to make sense of things as found by shape, collections of data,
numeric, and logic.

 

[matt grime]

Yes, arithmetic is necessary for doing error analysis. So is writing (english), but one of these is commonly included in mathematics and the other is not, so this criterion ("gets used") is not a convincing reason to include arithmetic as part of mathematics, unless we would also admit that writing is part of mathematics.

Writing clear and precise English is far more important than an ability to do arithmetic in mathematics.

 

[matt grime]

Take for example Euler or Gauss, lagrange, laplace etc. Among their truly mathematical accomplishments were also their feats of numerical computation No mathematicians does this in todays computer age, and I consider this to be a big difference.

You don't half make far reaching, and inflexible assertions that some of us think are indefensible.

Also, note that the entire field of nonlinear dynamics and fractals was created because of the invention of modern computers.

not really true. It is perfectly possible to do non-linear dynamics without any recourse to seeing a computer.

Sure, computers make some phenomena easier to observe, to calculate, to measure, but are you claiming that they seriously advanced the cause of mathematics more than the mathematicians who worked out the theory?

You seem to think that the only mathematics there is is applied.

 

[Crosson]

Writing clear and precise English is far more important than an ability to do arithmetic in mathematics.

Wonderful!

Increase the number of Arithmetics+Maths hours to 10 per week, rather than 5 "Maths" hours per week?

Now we are thinking along the same lines, but of course we cannot simply increase the load by five hours. I suggest that all students learn arithmetic (including pre-algebra i.e. arithmetic with an unknown). Due to increased use of calculators and information technology (and corresponding decrease in the importance of pen-paper computation) this should take only part of their schooling careers.

After that, all students are required to take math, in the sense of a writing subject where students practice reading and writing theorems (about arbitrary, but interesting, topics). In addition to this they can opt to take classes like analytic geometry/precalculus/calculus if they are pursuing careers in math or science, but all students are required to take mathematical communication/reasoning/critical_reading courses, to serve counterpoint to their "English" classes of fiction/rhetoric/literature.

 

[Math Is Hard]

People in the "ivory tower" of mathematics may not have an understanding of the true needs of the general population. Those in the ivory tower of English may suffer the same misunderstanding.

What we, the unwashed masses, need from our math classes is to learn basic accounting and basic logic to understand our bank balances and the mumbo jumbo we receive back from our insurance claims. As for English studies, most people need basic reading comprehension but not the skills to analyze great works of literature in depth.

Mathematics and English are both rich and beautiful subjects for those who wish to specialize, but we can't expect everyone to feel the need to delve deeply into either one. Most people are just looking for basic toolsets to manage their everyday lives.

 

[Crosson]

Symbolipoint, your compromise sounds an awful lot like the way things currently are (at least here in America)!

I really agree with you, Math_Is_Hard. Let's give the people what they want: basic skills that are useful throughout many areas of life. The ability to decode mumbo jumbo, to solve problems rationally, to make suggestions that are consistent and well-defined etc.

not really true. It is perfectly possible to do non-linear dynamics without any recourse to seeing a computer.

Okay, here is a problem for you to solve by hand: Given the following dynamical system, describe the geometry of the underlying attractor:

dx/dt = 10(y-x)
dy/dt = 28 x - y - xz
dz/dt = xy - 2.666z

...

You may think that my question is unmathematical, as many traditionalist do, but those who work in the field (N.D. & Chaos) think that the genuinely interesting answer to this question expands what ought to be considered mathematics.

The bottom line is this: no mathematicians prior to the computer age would have assigned any imprtance to this exact set of equations. The first clue appeared when Lorenz (1964) integrated the equations numerically and got total nonsense.

After re-writing the code (with punch cards) to create a routine which emphasized accuracy, Lorenz obtained something resembling an approximation to a (long since gauranteed to exist) solution to the above system. Apparently these equations "magnify errors".

People had been integrating linear equations on machines for a while, and no one had seen anything like this. Efforts to treat this exponential error magnification using mathematics (mostly analysis) led to the notion of lyaponov exponents. Now we can prove analytic results of this nature for various dynamical systems, so a lot of mathematical understanding was spawned by this numerical experimentation.

The counter argument could be "its possible to create of concepts like lyaponov exponents or correlation dimension without ever experimenting with computers" but this is highly suspect. Even after the discovery of error magnification in Lorenz's equation's, it took nearly a decade for computer graphics to get good enough to suggest that dynamical systems such as Lorenz's have "strange attractors" with non-integer dimension.

 

[Pathway]

Is this not the difference between university and college?

In university you learn how to learn; you learn principles, academia, foundations, theory, giving you a knowledge base they will open many doors.

In college you learn a trade that will open a few doors.

(Maybe just here in Canada.)

In America you might have a university, within which are contained several colleges. You might have the College of Engineering, the College of Letters and Science, and the Law School--or you could have what Harvard does: within Harvard University is Harvard College, where all undergraduates go. College does not imply a lack of prestige or an inferiority of education in the United States.

 

[CRGreathouse]

Is this not the difference between university and college?

In university you learn how to learn; you learn principles, academia, foundations, theory, giving you a knowledge base they will open many doors.

In college you learn a trade that will open a few doors.

(Maybe just here in Canada.)

In the US the first is called "college" (or tertiary education), in which you go to a university or college (the former is large, and may have subdivisions called colleges). The latter is called "trade school" or "vocational school".

 

[Alkatran]

So... this post is all about one guy wanting to radically changing how math is taught and expecting it to go well?



The large majority of people don't need to know how to prove theorems or why division works. We give them a little push in this direction, but that's all. There's no point in doubling the amount of time spent on math if it just doubles how much people forget.