Physics Education: A Different Method

[Raze]

I believe the way we currently teach physics should be changed. Sure, it seems to be working, but I can't help but feel that physics students would be much better off if they had better mathematical skills before they began really learning physics.


For a backdrop as to how it is done now, typically an incoming undergraduate student's plan of education will follow something like this (leaving out all courses not related to major, including chemistry/humanities, etc):



Year 1:

Semester 1:

Calculus 1
Introduction to Physics 1 (Newton's laws, statics/dynamics)

Semester 2

Calculus 2
Introduction to Physics 2 (thermodynamics/electromagnetism)

Year 2:

Semester 1

Mathematical Methods of Physicists 1 (covers multivariable calculus/linear algebra, ordinary differential equations, etc)
Introduction to Modern Physics 1 (special relativity, quantum mechanics)

Semester 2

Mathematical Methods of Physicists 2 (covers calculus of variations, tensors, partial differential equations, etc)
Introduction to Modern Physics 2 (mostly quantum mechanics)


Year 3:

Semester 1

Electricity and Magnetism I (including relativity/modern representation)
Classical Mechanics (including statistical physics)
Some elective related to emphasis of major (medical physics, astronomy, etc)

Semester 2

Electricity and Magnetism II (including relativity/modern representation)
Thermodynamics
Some elective related to emphasis of major


Year 4:

Semester 1
Quantum Mechanics I
Preparation for undergrad research
Scientific writing class
Elective study related to field of interest (General Relativity, astronomy, whatever)

Semester 2

Quantum Mechanics II
Research + research presentation
Electives related to field of interest.



* QM and E and M could be swapped.



Now, here is my motivation for asking these questions:

What is the purpose of any of the introductory classes? It's almost like you're learning the same thing twice, although the second time is FAR more rigorous. Now, I get that repetition isn't a bad thing, but in my humble opinion, things would be much easier if the intro physics courses were just replaced by math courses.

For example, do you REALLY need knowledge of calculus to learn Linear Algebra? I do not think so. So, if it were my choice, I'd teach Linear Algebra in place of Introduction to Physics I. And so on.


Here is what I would prefer to see in a typical physics education:



Year 1:

Semester 1:

Calculus I
Linear Algebra

Semester 2

Calculus II
Differential Equations I (use what you learn in Calc II in ODE as you learn integration methods)

Year 2:

Semester 1

Mathematical Methods of Physicists 1 (would be a partial math review)
Differential Equations II (Partial differential equations)

Semester 2

Mathematical Methods of Physicists 2 (again, part of this would be a math review)
A statistical math course of some sort







Year 3 and 4 would be the same as before.

Year 3:

Semester 1

Electricity and Magnetism I (including relativity/modern representation)
Classical Mechanics (including statistical physics)
Some elective related to emphasis of major (medical physics, astronomy, etc)

Semester 2

Electricity and Magnetism II (including relativity/modern representation)
Thermodynamics
Some elective related to emphasis of major


Year 4:

Semester 1
Quantum Mechanics I
Preparation for undergrad research
Scientific writing class
Elective study related to field of interest (General Relativity, astronomy, whatever)

Semester 2

Quantum Mechanics II
Research + research presentation
Electives related to field of interest.






Anyway, last year there were times during my Mathematical Methods of Physics class that I felt a bit overwhelmed at the pace and difficulty of the math. I was unprepared for some of it. I have wondered if maybe I had had a more rigorous math background things would have went a lot smoother. Hence this thread, lol.

[drkitten]

What is the purpose of any of the introductory classes? It's almost like you're learning the same thing twice, although the second time is FAR more rigorous. Now, I get that repetition isn't a bad thing, but in my humble opinion, things would be much easier if the intro physics courses were just replaced by math courses.

For example, do you REALLY need knowledge of calculus to learn Linear Algebra? I do not think so.

According to the mathematicians with whom I've discussed that question at length, "yes, you do." More accurately, it's not that you need the calculus per se, but you need the "mathematical maturity" to be able to handle the rigors of actual algebra (which bears little resemblance to the stuff you took under that name in high school).

From a mathematical perspective, it's not just about understanding the chain rule and integration by parts, but about understanding the difference between a theorem and a lemma, between a converse and a contrapositive, about the different between "if" and "iff" -- about understanding the actual mathematics.

Calculus is a brilliant introduction to that, because the mathematics is deep but sparse. There are really only about six theorems you need to master, but they're quite major ones, so you can spend lots of time exploring all their implications from the perspective of mathematical theory.



Anyway, last year there were times during my Mathematical Methods of Physics class that I felt a bit overwhelmed at the pace and difficulty of the math. I was unprepared for some of it. I have wondered if maybe I had had a more rigorous math background things would have went a lot smoother.

... and that's basically why they want to make sure that you understand the mathematics behind calculus (where it's fairly accessible) before they throw the mathematics behind linear algebra at you (where it is not).

Sure, any fool can learn to do the calculations. But doing the calculations isn't the same as understanding the math -- and physicists, in particular, tend to need to understand the math. (It's those pesky engineers who can just get away with crunching the numbers....)

[Raze]

According to the mathematicians with whom I've discussed that question at length, "yes, you do." More accurately, it's not that you need the calculus per se, but you need the "mathematical maturity" to be able to handle the rigors of actual algebra (which bears little resemblance to the stuff you took under that name in high school).

From a mathematical perspective, it's just about understanding the chain rule and integration by parts, but about understanding the difference between a theorem and a lemma, between a converse and a contrapositive, about the different between "if" and "iff" -- about understanding the actual mathematics.

Calculus is a brilliant introduction to that, because the mathematics is deep but sparse. There are really only about six theorems you need to master, but they're quite major ones, so you can spend lots of time exploring all their implications from the perspective of mathematical theory.




... and that's basically why they want to make sure that you understand the mathematics behind calculus (where it's fairly accessible) before they throw the mathematics behind linear algebra at you (where it is not).

Sure, any fool can learn to do the calculations. But doing the calculations isn't the same as understanding the math -- and physicists, in particular, tend to need to understand the math. (It's those pesky engineers who can just get away with crunching the numbers....)


Okay, maybe you're right. It COULD be that Linear Algebra seemed so easy to me BECAUSE I had already had decent mathematical experience in Calculus.


But still, looking at the material itself, the use of matrices, etc, it really doesn't seem that difficult. But again, I may be too biased due to the fact that I had already had several math courses before I took it.

ETA- of course, I am certainly NOT referring to more advance Linear Algebra classes, where it really gets into the mathematical theory of it. From what some professors have told me, the next step after the intro to Linear Algebra class is a jump from basic math to advanced math.

[lightfire22000]

Okay, maybe you're right. It COULD be that Linear Algebra seemed so easy to me BECAUSE I had already had decent mathematical experience in Calculus.


But still, looking at the material itself, the use of matrices, etc, it really doesn't seem that difficult. But again, I may be too biased due to the fact that I had already had several math courses before I took it.

I know of kids who took Linear Algebra before Calculus and they did fine. Many took Linear Algebra before Vector Calculus. With a firm Algebra background, one can derive much of Linear Algebra's basics like say,..., Kramer's Rule.

Calculus revolves around an understanding of the limit. Calculus is algebra with limits generally.

[drkitten]

I know of kids who took Linear Algebra before Calculus and they did fine.

Yes, yes, we know --- in your imaginary universe, no formal education is necessary.

Grownups are talking now, lightfire. Mummy will tuck you into bed in a few minutes.


Calculus revolves around an understanding of the limit. Calculus is algebra with limits generally.

Only if "algebra" means "high-school algebra" (i.e. manipulating equations in the field of the real numbers).

You've just proven my point for me, I'm afraid. The reason that Calculus is so often taught before linear algebra is becuase most calculus students don't actually understand what "algebra" means to an actual algebraist. Linear algebra is a convenient stepping stone precisely because to do it properly, one is no longer manipulating "equations" but matrices or other abstract objects, and one is doing it in an abstract ring (although it can become a field if one meets certain criteria).

For many (most?) students, this will be their first exposure to performing "mathematics" on things that aren't numbers, and it will typically hit them hard. Especially since a lot of them are cocky young know-it-all's who believe that skill at equation manipulation translates to actual mathematical talent, and they're astongished to learn that it isn't necessarily the case. (Those who do have talent generally stay in math or physics; the others tend to drop into engineering or some of the other hard sciences like chemistry.)

Indeed, this idea of doing math without numbers is probably the single hardest conceptual step in mathematical education -- and the foundations of abstract reasoning laid in calculus are crucial to make it. We can argue about the wisdom of using calculus per se as the course to develop mathematical maturity (I, personally, favor using symbolic logic, instead -- esp. as that translates much more clearly into algebra), but again, the physicists, chemists, and engineers need calculus for their calculations more than they need abstract logic, so calculus is a useful 100-level service course.

[drkitten]

Okay, maybe you're right. It COULD be that Linear Algebra seemed so easy to me BECAUSE I had already had decent mathematical experience in Calculus.

Or because you already had a talent for abstract reasoning. You want to get an argument going at a math department picnic, ask whether math is a "skill" (that can be taught) or a "talent" (that can only be given by God). I tend to be on the "skill" side of the fence myself, but I recognize that the other opinion exists and is even widespread.


But still, looking at the material itself, the use of matrices, etc, it really doesn't seem that difficult. But again, I may be too biased due to the fact that I had already had several math courses before I took it.

Well, for many (most?) students, this is the first time they're going to be doing "math" on anything except numbers. If I tell a fifth grader that I want him to multiply a color by a sound, he'll look at me like I'm bonkers -- when you do "multiplication," you do it on numbers.

And all of a sudden in linear algebra (at least if you teach it right), you're doing "math" on all sorts of abstract things that just don't make sense. You're not working on numbers. And all of the rules that you know about working with numbers are no longer true. All of a sudden you're working in a universe where AB is not the same thing as BA, and where the "inverse" of something depends on the direction it comes in from.

All routine to a physicist, of course. But it's a hell of a culture shock to a physics student.


ETA- of course, I am certainly NOT referring to more advance Linear Algebra classes, where it really gets into the mathematical theory of it. From what some professors have told me, the next step after the intro to Linear Algebra class is a jump from basic math to advanced math.

Maybe you simply didn't have a very good Linear Algebra class -- I'm not sure where you (or your professors) draw the line between basic and advanced. In my experience, Linear Algebra is one of the common spots where that line is crossed (either Linear Algebra or Differential Equations, depending on whether you want to keep going in the discrete or continuous domain). And, yes, those are the classes at which all our math majors wash out.

Of course, we have a rather unusual program (which I've criticized elsewhere): we teach Diff Eq and Lin Alg as a single course, which means you get a lightweight introduction to both and a mastery of neither. I think that this shortchanges our students for exactly the reason you might expect; it doesn't give them the abstract reasoning skills. But for some reasons, the math department is under no obligation to restructure its courses just to please me....

[Raze]

Or because you already had a talent for abstract reasoning. You want to get an argument going at a math department picnic, ask whether math is a "skill" (that can be taught) or a "talent" (that can only be given by God). I tend to be on the "skill" side of the fence myself, but I recognize that the other opinion exists and is even widespread.

I think it can be both. I mean, it is obviously a skill learned, but there are those with certain talents. Those who find abstract reasoning easier.



Well, for many (most?) students, this is the first time they're going to be doing "math" on anything except numbers. If I tell a fifth grader that I want him to multiply a color by a sound, he'll look at me like I'm bonkers -- when you do "multiplication," you do it on numbers.

I don't know. My introduction to doing math on non-numbers was Intro to Physics. But my introduction to math that didn't follow the "elementary math" things like commutative multiplication was linear algebra, and my intro to operators was linear algebra + Intro to Modern physics (in quantum mechanics).



And all of a sudden in linear algebra (at least if you teach it right), you're doing "math" on all sorts of abstract things that just don't make sense. You're not working on numbers. And all of the rules that you know about working with numbers are no longer true. All of a sudden you're working in a universe where AB is not the same thing as BA, and where the "inverse" of something depends on the direction it comes in from.

All routine to a physicist, of course. But it's a hell of a culture shock to a physics student.


Yes, these things were odd, but I didn't find them difficult because they were just based on more "rules." The hard stuff was the proofs. And when we really started juggling several matrices at once.






Maybe you simply didn't have a very good Linear Algebra class -- I'm not sure where you (or your professors) draw the line between basic and advanced. In my experience, Linear Algebra is one of the common spots where that line is crossed (either Linear Algebra or Differential Equations, depending on whether you want to keep going in the discrete or continuous domain). And, yes, those are the classes at which all our math majors wash out.

It's possible, but in my class we only went up to the first 6 chapters of our book, so maybe we didn't cover everything you would in your University's LA class. Then again, I was also learning some of the more important things of Linear Algebra in my Mathematical Methods of Physics class, so I suppose I did a few more homework problems in it than your run of the mill Linear Algebra student.

Anyway, I'm not saying it was "easy," exactly. It started out really easy, and then the difficulty of the course really ballooned. But if a student does his/her homework it shouldn't be too difficult to get a decent grade (whether or not you understand the mathematics, as you say, is a different question).

I will concede this though: When I took DE II (Partial DE's), my instructor made the class FAR more rigorous than anything I had yet seen, including his use of several seemingly unrelated mathematical concepts, including linear algebra. His emphasis on proofs and solid mathematical fundamentals makes me wonder if I had had HIS linear algebra class, maybe I would think the class is much more difficult.

Of course, we have a rather unusual program (which I've criticized elsewhere): we teach Diff Eq and Lin Alg as a single course, which means you get a lightweight introduction to both and a mastery of neither. I think that this shortchanges our students for exactly the reason you might expect; it doesn't give them the abstract reasoning skills. But for some reasons, the math department is under no obligation to restructure its courses just to please me....

That's the thing I'm complaining about in physics education at my school. Basically, physics students are not required to take ANY math course outside of Calculus I and II. So, we get 14 chapters, each a different math subject, thrown at us in two semesters in our Mathematical Physics class, and really, no real mastery is gained (a "jack of all trades" kind of thing).

I do like that course, because it is very challenging and gives you plenty of practice over a wide range of material, but again, some of the time it felt a bit overwhelming. Then again, maybe that's the point. To get your feet wet.

[SezMe]

From a mathematical perspective, it's not just about understanding the chain rule and integration by parts, but about understanding the difference between a theorem and a lemma, between a converse and a contrapositive, about the different between "if" and "iff" -- about understanding the actual mathematics.
I think a very important word got left out.

[drkitten]

I think a very important word got left out.

Yes, of course. Fixed and thank you.

[drkitten]

It's possible, but in my class we only went up to the first 6 chapters of our book, so maybe we didn't cover everything you would in your University's LA class. Then again, I was also learning some of the more important things of Linear Algebra in my Mathematical Methods of Physics class, so I suppose I did a few more homework problems in it than your run of the mill Linear Algebra student.

Sounds to me like a lightweight Lin Alg class, then -- and the reason you were having difficulty with Mathematical Methods was because you had gotten shortchanged. That's my initial guess, then.

At my undergraduate university, Lin Alg was (IIRC) a second-semester freshman class, taken concurrently with Calc II and required of all hard science majors... but taught by the math department to the standards expected of courses for majors.


Anyway, I'm not saying it was "easy," exactly. It started out really easy, and then the difficulty of the course really ballooned. But if a student does his/her homework it shouldn't be too difficult to get a decent grade (whether or not you understand the mathematics, as you say, is a different question).

Yes, it's not about grades. Frankly, it's not about homework, either. Homework just tests whether you can do the calculations (for the most part). Again, this is part of the transition from basic to advanced math -- are you solving problem sets, or are you doing proofs and conceptual arguments?

As it happens, I have my old Lin Alg book on the shelf. It's got two sets of exercises at the end of each section: "Exercises" and "Theoretical Exercises." The "Exercises" are standard high school algebra on matrices : " in exercises 15 through 18, solve the linear system with the given augmented matrix." The "Theoretical Exercises" include things like "Show that X is row equivalent to I_2 if and only if ad -bc != 0" or "Show that the values of \lambda for which the homogenous system [blah] has a nontrivial solution satisfy the equation [bla bla]."

The second type of exercise is what turns it into actual, rigorous, mathematics. Of course, you COULD teach the class without doing any of the second sort of exercises at all, and I suspect that if I taught it as a service class for the chemistry department, I would be asked to. I also suspect that the math faculty would crucify me if I did.

[lightfire22000]

You've just proven my point for me, I'm afraid. The reason that Calculus is so often taught before linear algebra is becuase most calculus students don't actually understand what "algebra" means to an actual algebraist. Linear algebra is a convenient stepping stone precisely because to do it properly, one is no longer manipulating "equations" but matrices or other abstract objects, and one is doing it in an abstract ring (although it can become a field if one meets certain criteria).

For many (most?) students, this will be their first exposure to performing "mathematics" on things that aren't numbers, and it will typically hit them hard. Especially since a lot of them are cocky young know-it-all's who believe that skill at equation manipulation translates to actual mathematical talent, and they're astongished to learn that it isn't necessarily the case. (Those who do have talent generally stay in math or physics; the others tend to drop into engineering or some of the other hard sciences like chemistry.)





Uhh...No. You're not even remotely close and appear to be deliberately misleading people. Since engineering is the purposeful use of science, and mathematics is the language of science, "actual mathematical talent" translates to talent in engineering. You're clearly just jealous of talented engineers and/or chemists.

Higher "algebras" come from the same principles of quantity that concern "High school" algebra. They're the same field. The concept of a ring comes from the concept of an integer. I don't see how one cannot conceptualize a ring before conceptualizing an integer. If you had any understanding of mathematics and its abstract logic you'd know this. However, you have no regard for reductionism and that is why I bet you don't produce anything like many ivory tower parasites.

[lightfire22000]

The second type of exercise is what turns it into actual, rigorous, mathematics. Of course, you COULD teach the class without doing any of the second sort of exercises at all, and I suspect that if I taught it as a service class for the chemistry department, I would be asked to. I also suspect that the math faculty would crucify me if I did.

Oooh, the department would crucify me for teaching math. Oh my God, what a disaster! You might want to switch departments if it'll get on your case that much for teaching math as a purely useful thing. After reading what you write, I'm starting to think it's pretty easy to get a Ph.D. in Mathematics.

As if there is a formal divide between the rudimentary and rigorous. Come on, don't be so full of yourself. As long as they teach how the math was derived in the first place, they have to teach the logic behind the math.

[drkitten]

Oooh, the department would crucify me for teaching math.

No, they'd crucify me for teaching mathematics that badly.

Now pipe down. Grownups are talking.

[drkitten]

Uhh...No. You're not even remotely close and appear to be deliberately misleading people.

Well, you'd know all about deliberately misleading people.

I've told you three times now that grownups are talking.

[drkitten]

Higher "algebras" come from the same principles of quantity that concern "High school" algebra. They're the same field.

That's right. The field of integers modulo 5 is exactly the same as the field of real numbers, which is exactly the same as the field of quaternions.

... in autodidactic mathematics.

For people who actually have learned the material, they're entirely different and require a substantial conceptual leap; simply understanding how to work with integers doesn't give you a particular good handle on complex numbers or on modular polynomials.

Which is why the "algebra" taught in high school is so different than the stuff taught at the junior level of college.


I don't see how one cannot conceptualize a ring before conceptualizing an integer.

Most high school mathematicians haven't "conceptualized" integers at all. And that's exactly the problem, and the reason they need to be introduced to these ideas.

[Hampster]

The underlying concept around Calculus is infinity- a rather difficult concept for a lot of people.
The underlying concept around linear algebra is abstractions- another difficult concept.

Some student may be better in one than the other, and others may be equally good or bad with these concepts. I have seen a lot of matrix manipulation and Cramers' rule in High School algebra, but not all schools teach this. It is possible to teach some of these before calculus.

In regards to physics, all I need to say is that algebra-based physics needs to be banned by the highest level of government. It is as anti-science as you can possibly get- forcing students to memorize equations and follow lab directions with no concept of what is really going on. This has turned more people away from science than anything else I can think of.

--Dave

[Gaspode]

Let's keep it civil please.

[drkitten]

The underlying concept around Calculus is infinity- a rather difficult concept for a lot of people.
The underlying concept around linear algebra is abstractions- another difficult concept.

Some student may be better in one than the other, and others may be equally good or bad with these concepts. I have seen a lot of matrix manipulation and Cramers' rule in High School algebra, but not all schools teach this.

Again, you're confusing symbol manipulation for mathematics. Lightfire did the same thing:


As long as they teach how the math was derived in the first place, they have to teach the logic behind the math.

This, of course, assumes that they do teach how the math was derived in the first place, an assumption that is not always followed. A much more blatant example of that is in the typical way statistics are taught to/by a psychology department, usually and pejoratively called "cookbook stats" because students are simply taught to follow a recipe ("How to make a t-test") without any understanding of where the concept of t-test comes from, the assumptions underlying it, or an understanding of when it's applicable -- which is why you will frequently see stats paper that use almost pathologically inappropriate statistics.

You can do the same thing with both calculus and linear algebra; most students of calculus cannot (re)derive the chain rule or the technique of integration by parts, or even explain how it works; they've simply memorized a formula. Similarly, merely being able to apply Cramer's rule does not mean you actually know any linear algebra if you've just been taught a formula to apply.

This, of course, is true at all levels of applied mathematics. I think I learned the formula that the distance fallen equals 16 t^2 feet when I was fourteen; I was given the more general formula 1/2 a t^2 at sixteen. I knew that distance equals rate times time at age 8. It wasn't until I was taught calculus that someone showed me that all these formulas were related, at which point I could re-derive 16 t^2 for myself -- and at that point, of course, I can actually go out and measure the acceleration of gravity instead of just copying it from a book.

If all you want to do is solve problems (systems of equations), you can just memorize Cramer's rule. But that's not what mathematicians regard as "linear algebra," and for a good reason.



In regards to physics, all I need to say is that algebra-based physics needs to be banned by the highest level of government. It is as anti-science as you can possibly get- forcing students to memorize equations and follow lab directions with no concept of what is really going on.

Yup. But just throwing around phrases like "Cramer's rule" or "matrix manipulation" is the equivalent of algebra-based physics. You learn to memorize equations (the solution to x is [bla]) and follow lab directions to get the answer to the problem at hand, with no concept of what is really going on.

[lightfire22000]

That's right. The field of integers modulo 5 is exactly the same as the field of real numbers, which is exactly the same as the field of quaternions.

... in autodidactic mathematics.

For people who actually have learned the material, they're entirely different and require a substantial conceptual leap; simply understanding how to work with integers doesn't give you a particular good handle on complex numbers or on modular polynomials.

Which is why the "algebra" taught in high school is so different than the stuff taught at the junior level of college.




Most high school mathematicians haven't "conceptualized" integers at all. And that's exactly the problem, and the reason they need to be introduced to these ideas.

More lies and libel. I'm beginning to doubt that you even hold an honest degree. If not, you should attempt to do something worthwhile rather than try to corrupt people's minds on a web forum.

The fact is that I never suggested that "The field of integers modulo 5 is exactly the same as the field of real numbers". Indeed, high school and college algebra are founded on the same logical principles if you reduce them both enough.

You don't have a remote understanding of mathematics. Understandings of complex numbers come from understanding of integers. I'd wager that a failure to understand that demonstrates your overall academic failure.

[lightfire22000]

Again, you're confusing symbol manipulation for mathematics. Lightfire did the same thing:



This, of course, assumes that they do teach how the math was derived in the first place, an assumption that is not always followed. A much more blatant example of that is in the typical way statistics are taught to/by a psychology department, usually and pejoratively called "cookbook stats" because students are simply taught to follow a recipe ("How to make a t-test") without any understanding of where the concept of t-test comes from, the assumptions underlying it, or an understanding of when it's applicable -- which is why you will frequently see stats paper that use almost pathologically inappropriate statistics.

You can do the same thing with both calculus and linear algebra; most students of calculus cannot (re)derive the chain rule or the technique of integration by parts, or even explain how it works; they've simply memorized a formula. Similarly, merely being able to apply Cramer's rule does not mean you actually know any linear algebra if you've just been taught a formula to apply.

This, of course, is true at all levels of applied mathematics. I think I learned the formula that the distance fallen equals 16 t^2 feet when I was fourteen; I was given the more general formula 1/2 a t^2 at sixteen. I knew that distance equals rate times time at age 8. It wasn't until I was taught calculus that someone showed me that all these formulas were related, at which point I could re-derive 16 t^2 for myself -- and at that point, of course, I can actually go out and measure the acceleration of gravity instead of just copying it from a book.

If all you want to do is solve problems (systems of equations), you can just memorize Cramer's rule. But that's not what mathematicians regard as "linear algebra," and for a good reason.




Yup. But just throwing around phrases like "Cramer's rule" or "matrix manipulation" is the equivalent of algebra-based physics. You learn to memorize equations (the solution to x is [bla]) and follow lab directions to get the answer to the problem at hand, with no concept of what is really going on.

This is irrelevant at best. Academic institutions are the number one cause of the prevailing lack of conceptual understanding. With basic logic, you can derive Cramer's rule. A matrix does not differ from an integer in true nature. Only in a rudimentary mind and a degenerate conception of an integer, do they truly differ. The basic principle is quantity in both cases.

There is no confusion of symbol manipulation and mathematics.

You're simply turning mathematics into a religion, forbidding people to investigate into its workings.

[lightfire22000]

You can do the same thing with both calculus and linear algebra; most students of calculus cannot (re)derive the chain rule or the technique of integration by parts, or even explain how it works; they've simply memorized a formula. Similarly, merely being able to apply Cramer's rule does not mean you actually know any linear algebra if you've just been taught a formula to apply.

No, it doesn't. That is precisely why institutionalized education is inherently flawed.


This, of course, is true at all levels of applied mathematics. I think I learned the formula that the distance fallen equals 16 t^2 feet when I was fourteen; I was given the more general formula 1/2 a t^2 at sixteen. I knew that distance equals rate times time at age 8. It wasn't until I was taught calculus that someone showed me that all these formulas were related, at which point I could re-derive 16 t^2 for myself -- and at that point, of course, I can actually go out and measure the acceleration of gravity instead of just copying it from a book.

That is your own fault and the fault of your teachers. With a proper understanding of quantity, ratios, and "high school algebra", you should be able to derive all of these equations. You do not need formal calculus to derive them since you should be able to conceptualize the concept of the limit and approximate it with discrete mathematics.

[drkitten]

That is your own fault and the fault of your teachers. With a proper understanding of quantity, ratios, and "high school algebra", you should be able to derive all of these equations. You do not need formal calculus to derive them since you should be able to conceptualize the concept of the limit and approximate it with discrete mathematics.

Except for the fact that the "concept of the limit" is not part of high school algebra. Sometimes it's taught as part of a "pre-calculus" class, sometimes it's simply folded into the first few weeks of the calculus class itself. High school algebra students cannot be expected to have the necessary skills to "conceptualize the concept of the limit."

What you're saying is that if you've never been taught calculus, you should be able to re-derive it yourself from first principles. Since mere words will not suffice, have an image instead:

http://pix.motivatedphotos.com/2009/6/15/633806685662365650-jesusfacepalm.jpg

[Wudang]

A matrix does not differ from an integer in true nature. Only in a rudimentary mind and a degenerate conception of an integer, do they truly differ. The basic principle is quantity in both cases.

Are you sure you don't want to reconsider this statement? :dig:

[lightfire22000]

Are you sure you don't want to reconsider this statement? :dig:

It's a question of sets and subsets. An integer is really a 1x1 matrix.

[lightfire22000]

Except for the fact that the "concept of the limit" is not part of high school algebra. Sometimes it's taught as part of a "pre-calculus" class, sometimes it's simply folded into the first few weeks of the calculus class itself. High school algebra students cannot be expected to have the necessary skills to "conceptualize the concept of the limit."

What you're saying is that if you've never been taught calculus, you should be able to re-derive it yourself from first principles. Since mere words will not suffice, have an image instead:

http://pix.motivatedphotos.com/2009/6/15/633806685662365650-jesusfacepalm.jpg

You should learn the skills necessary to re-derive it, yes. I had the "concept of the limit" taught in my Algebra II class Freshman year and from that, I could at least understand what was going on in Calculus once I understood the notation.

[Wudang]

It's a question of sets and subsets. An integer is really a 1x1 matrix.

Okay you made me laugh.

[drkitten]

It's a question of sets and subsets. An integer is really a 1x1 matrix.



The defense rests.

[lightfire22000]

It is a 1x1 matrix. Try to dispute that. The subset 'integers' does indeed fall under the larger set of all matrices. Purely indisputable.

drkitten, you are just nitpicking over semantics rather than demonstrating mathematical concepts.
Show me something now. Educate me. Provide some mathematical insight.

[Raze]

It is a 1x1 matrix. Try to dispute that. The subset 'integers' does indeed fall under the larger set of all matrices. Purely indisputable.

drkitten, you are just nitpicking over semantics rather than demonstrating mathematical concepts.
Show me something now. Educate me. Provide some mathematical insight.
lol whoops misread your post... disregard this.. .



But what do you people think about the idea of giving physics students a greater math background before actually teaching physics?

[lightfire22000]

lol whoops misread your post... disregard this.. .



But what do you people think about the idea of giving physics students a greater math background before actually teaching physics?

I think it is a great idea! It really allows one to have a whole lot of extra insight into equations. I've always found it odd that charge is a discrete quantity yet they describe with formulas involving continuous functions; I am by no means a physicist.

[mijopaalmc]

It is a 1x1 matrix. Try to dispute that. The subset 'integers' does indeed fall under the larger set of all matrices. Purely indisputable.

drkitten, you are just nitpicking over semantics rather than demonstrating mathematical concepts.
Show me something now. Educate me. Provide some mathematical insight.

The set of all matrices with addition and multiplication defined in their usual way for matrices is a ring, and a non-commutative ring at that. The set of integers is not a ring. Therefore, integers are not just 1x1 matrices.

[Fredrik]

I believe the way we currently teach physics should be changed. Sure, it seems to be working, but I can't help but feel that physics students would be much better off if they had better mathematical skills before they began really learning physics.
[...]
What is the purpose of any of the introductory classes? It's almost like you're learning the same thing twice, although the second time is FAR more rigorous. Now, I get that repetition isn't a bad thing, but in my humble opinion, things would be much easier if the intro physics courses were just replaced by math courses.

For example, do you REALLY need knowledge of calculus to learn Linear Algebra? I do not think so. So, if it were my choice, I'd teach Linear Algebra in place of Introduction to Physics I. And so on.
[...]
Anyway, last year there were times during my Mathematical Methods of Physics class that I felt a bit overwhelmed at the pace and difficulty of the math. I was unprepared for some of it. I have wondered if maybe I had had a more rigorous math background things would have went a lot smoother. Hence this thread, lol.
I completely agree with these things. There was a lot of repetition in my eductation too. E.g. we took "thermodynamics", and the next year "statistical mechanics", and the feeling I got was that the second one was a "let's do it again, but do it right this time" class. Same thing with quantum mechanics, and classical electrodynamics.

It made some sense in the case of quantum mechanics, or at least it would have if we had used a better book for the first class, and a better teacher for the second one. I think it's natural to start with a (short) class that focuses on "wave mechanics" (i.e. wavefunctions, the Schrödinger equation and so on), and make the next class deal with Hilbert spaces and stuff. I think the ideal literature for a set of 3-4 QM classes would be Griffiths, Isham and Ballentine, in that order. (Forget Sakurai. Ballentine is better).

In the case of electrodynamics, I think it's the advanced class that should be dropped (Jackson). I mean, it's mostly an excercise in solving really difficult boundary value problems. It doesn't give you a much deeper understanding of the physics. It would be much more useful to have an advanced analysis class instead, e.g. one based on Rudin's "Principle's of mathematical analysis", at least for students who intend to continue with theoretical stuff. If you haven't studied something like that, it's going to be insanely hard to study integration theory and functional analysis later, and you will have to if you're going to get into mathematical physics. If you're going to be an experimentalist, that time is probably better spent on a course that teaches experimental methods, or computer programming.

We studed linear algebra and calculus at the same time, and everyone agreed that linear algebra was much, much, much easier than calculus. (If we had been talking about how difficult it was to pass the exams, I would have had to repeat the word "much" at least five more times I think). It makes no sense to me that some people are saying that you need to study calculus before linear algebra.

I would like to see a good class on special relativity, one that takes a logical rather than historical approach, explains the philosophical aspects right, and prepares the student adequately for both GR (by teaching some basic differential geometry, not including curvature, but including e.g. a proof that the isometry group of Minkowski spacetime is the Poincaré group) and relativistic QM (including the group-theoretic definition of a particle in relativistic QM (see Weinberg, chapter 2)). But I don't think there's a book that covers those things well enough. I have actually been thinking about writing one myself, but I probably won't, at least not for the next two years or so. I still need to learn some of the things I think should be included in such a book, and even if I write the perfect book, it's still possible that no one will change the curriculum to include such a class.

[NewtonTrino]

There seems to be a lot of oddness in this thread. I took classes on calculus (including the concept of the limit and differential equations), complex numbers, finite mathematics, linear algebra etc. in high school. Do these things really not get taught in high school in the US? I grew up in Canada...

[drkitten]

There seems to be a lot of oddness in this thread. I took classes on calculus (including the concept of the limit and differential equations), complex numbers, finite mathematics, linear algebra etc. in high school. Do these things really not get taught in high school in the US? I grew up in Canada...

No, they don't.

Here, for example, are the NCTM (National Council of Teachers of Mathematics) standards (http://standards.nctm.org/document/appendix/alg.htm) for 9-12th grade algebra:



• generalize patterns using explicitly defined and recursively defined functions;
• understand relations and functions and select, convert flexibly among, and use various representations for them;
• analyze functions of one variable by investigating rates of change, intercepts, zeros, asymptotes, and local and global behavior;
• understand and perform transformations such as arithmetically combining, composing, and inverting commonly used functions, using technology to perform such operations on more-complicated symbolic expressions;
• understand and compare the properties of classes of functions, including exponential, polynomial, rational, logarithmic, and periodic functions;
• interpret representations of functions of two variables

Pathetic, isn't it? If you want to see some official state standards, check out South Carolina's (http://ed.sc.gov/agency/Standards-and-Learning/Academic-Standards/old/cso/standards/math/). Complex numbers get mentioned once in the algebra section -- finite mathematics and linear algebra not at all.

[lightfire22000]

The set of all matrices with addition and multiplication defined in their usual way for matrices is a ring, and a non-commutative ring at that. The set of integers is not a ring. Therefore, integers are not just 1x1 matrices.

That's faulty logic. The set of all integers is a ring!!!

[drkitten]

That's faulty logic. The set of all integers is a ring!!!

The set of integers is a ring.
The set of 1x1 matrices is a ring.
Therefore, the set of integers is the same as the set of 1x1 matrices.

... and you acuse US of faulty logic.

I think I've already used the Jesus facepalm on this thread. Does anyone know where I can find an Aristotle facepalm?

[mijopaalmc]

That's faulty logic. The set of all integers is a ring!!!

While the set of integers is a ring (I erroneously stated that it wasn't), it is not isomorhpic to the set of all matrices, because multiplication is commutative on the set of integers, while multiplication is not commutative (nor even necessarily defined) on the set of all matrices.

[drkitten]

While the set of integers is a ring (I erroneously stated that it wasn't), it is not isomorhpic to the set of all matrices, because multiplication is commutative on the set of integers, while multiplication is not commutative (nor even necessarily defined) on the set of all matrices.

Even if the set of all integers were isomorphic to the set of all matrices, it still wouldn't mean they're the same. Isomorphism only holds with regard to a particular property or set of properties; when you add other properties into them mix, you may break the isomorphism.

As an example, the set of integers is well understood to be isomorphic (in size) to both the set of natural numbers and the set of rational numbers : they're all "countably infinite." This means there is a function f() that uniquely converts integers into natural numbers and back, and another that uniquely converts integers into rational numbers and back (and of course, f-inverse*g() will uniquely convert natural numbers into rational numbers).

Does this mean these sets are all the same? Of course not. In particular, the rules of algebra are widly different for these three sets. Natural numbers are a group, integers are a ring, and rational numbers are a field. How is this possible? The "natural" interpretation of operations differs radically among these. If you say that a+b=c in the integers, you have no confidence that f(a)+f(b)=f(c) or that g(a)+g(b)=g(c).

Similarly, the comparison operation > breaks; in the natural numbers, there exists a number 0 such that every number > 0; no such number exists in the integers.

So, is the set of all 1x1 matrices isomorphic to the integers? No, not at all. I can have a 1x1 matrix of real numbers, which means there are uncountably many such matrices -- and only countably many integers. So in this case, the isomorphism fails to exist, at a very basic level.

But let's pretend that lightfire actually understood the material well enough to specify that he was talking about a 1x1 matrix of integers. Well, this is isomorphic in size to the set of integers, yes. But it's also isomorphic to the set of rational numbers. So if this isomorphism is sufficient to make these things the same, we get the rather odd property that 1x1 matrices are both a field and a ring that is not a field. Throw natural numbers into the mix and it becomes a group but not a ring as well -- throw in the free group on two generators and the set of all matrices becomes a non-abelian group as well.

Which is impressive, since all rings must be abelian.

Essentially, lightfire's argument is gibberish from one end to another.

Which suggests that perhaps high school isn't the place to study group theory.

[drkitten]

I completely agree with these things. There was a lot of repetition in my eductation too. E.g. we took "thermodynamics", and the next year "statistical mechanics", and the feeling I got was that the second one was a "let's do it again, but do it right this time" class. Same thing with quantum mechanics, and classical electrodynamics.

There are several reasons for that.

One is simply that students are not as bright as we wish they were, and they simply need to have things repeated several times.

Another is that many people do not need to "do it right"; if you're talking about thermodynamics, for example, a biologist can get by with a worse understanding than a chemist, who in turn can get by with a worse understanding than a physicist. So a lot of times, the introductory class is turned into a service class, one that will teach the chemist what she needs to know and provide an introduction to the physicist who can learn the correct stuff later.

This isn't confined to the sciences, either. What I learned about the thirty-years war in Western Civilization was largely supplanted by what I learned about it in European History.



We studed linear algebra and calculus at the same time, and everyone agreed that linear algebra was much, much, much easier than calculus. (If we had been talking about how difficult it was to pass the exams, I would have had to repeat the word "much" at least five more times I think). It makes no sense to me that some people are saying that you need to study calculus before linear algebra.

Again, sounds to me like a fairly lightweight linear algebra class. If you don't teach any actual algebra in a linear algebra class, you can make it as simple as you like. But the less you teach there, the harder you make the next course up....

[lightfire22000]

Even if the set of all integers were isomorphic to the set of all matrices, it still wouldn't mean they're the same. Isomorphism only holds with regard to a particular property or set of properties; when you add other properties into them mix, you may break the isomorphism.

As an example, the set of integers is well understood to be isomorphic (in size) to both the set of natural numbers and the set of rational numbers : they're all "countably infinite." This means there is a function f() that uniquely converts integers into natural numbers and back, and another that uniquely converts integers into rational numbers and back (and of course, f-inverse*g() will uniquely convert natural numbers into rational numbers).

Does this mean these sets are all the same? Of course not. In particular, the rules of algebra are widly different for these three sets. Natural numbers are a group, integers are a ring, and rational numbers are a field. How is this possible? The "natural" interpretation of operations differs radically among these. If you say that a+b=c in the integers, you have no confidence that f(a)+f(b)=f(c) or that g(a)+g(b)=g(c).

Similarly, the comparison operation > breaks; in the natural numbers, there exists a number 0 such that every number > 0; no such number exists in the integers.

So, is the set of all 1x1 matrices isomorphic to the integers? No, not at all. I can have a 1x1 matrix of real numbers, which means there are uncountably many such matrices -- and only countably many integers. So in this case, the isomorphism fails to exist, at a very basic level.

But let's pretend that lightfire actually understood the material well enough to specify that he was talking about a 1x1 matrix of integers. Well, this is isomorphic in size to the set of integers, yes. But it's also isomorphic to the set of rational numbers. So if this isomorphism is sufficient to make these things the same, we get the rather odd property that 1x1 matrices are both a field and a ring that is not a field. Throw natural numbers into the mix and it becomes a group but not a ring as well -- throw in the free group on two generators and the set of all matrices becomes a non-abelian group as well.

Which is impressive, since all rings must be abelian.

Essentially, lightfire's argument is gibberish from one end to another.

Which suggests that perhaps high school isn't the place to study group theory.

Nope. You're just stating completely irrelevant facts with seemingly bizarre intentions and grossly over-complicating the matter, at best. At worst, you're making huge mistakes. The steps that you performed are in fact logically consistent because rings CAN BE NON-ABELIAN under certain circumstances. Apparently you've never heard of non-commutative rings.

An integer satisfies all criteria for a one by one matrix simply because it is impossible to conceptualize an integer without conceptualizing a one by one matrix. Just ask the makers of MATLAB. Write an integer on a board without writing a one by one matrix. An integer is a scalar and a scalar is a 1x1 matrix. Hence, an integer is a 1x1 matrix. QED

Secondly, it's irrelevant because you still fail to address the point at hand. Elaborate mathematics is the application of logic to more rudimentary mathematics. If rudimentary mathematics and logic are taught, there is no need to isolate the "higher" level math from the lower level.

[lightfire22000]

The set of integers is a ring.
The set of 1x1 matrices is a ring.
Therefore, the set of integers is the same as the set of 1x1 matrices.

... and you acuse US of faulty logic.

I think I've already used the Jesus facepalm on this thread. Does anyone know where I can find an Aristotle facepalm?

No, I accuse you of flat out libel. I never posted any fallacious deduction of the sort.

[lightfire22000]

Essentially, lightfire's argument is gibberish from one end to another.

Which suggests that perhaps high school isn't the place to study group theory.

Gibberish, I wish. If my argument equaled Geber, I'd consider myself quite an intellectual. I have to say that if I could be as prolific as Geber, I'd be quite satisfied with my scientific career.

[mijopaalmc]

Nope. You're just stating completely irrelevant facts with seemingly bizarre intentions and grossly over-complicating the matter, at best. At worst, you're making huge mistakes. The steps that you performed are in fact logically consistent because rings CAN BE NON-ABELIAN under certain circumstances. Apparently you've never heard of non-commutative rings.

An integer satisfies all criteria for a one by one matrix simply because it is impossible to conceptualize an integer without conceptualizing a one by one matrix. Just ask the makers of MATLAB. Write an integer on a board without writing a one by one matrix. An integer is a scalar and a scalar is a 1x1 matrix. Hence, an integer is a 1x1 matrix. QED

Secondly, it's irrelevant because you still fail to address the point at hand. Elaborate mathematics is the application of logic to more rudimentary mathematics. If rudimentary mathematics and logic are taught, there is no need to isolate the "higher" level math from the lower level.

If you are so sure that a scalar is just a 1x1 matrix, multiply the 1x1 matrix (a) with the vector (b,c) and the 1x1 matrix (a) with (b,c)T. Then multiply the scalar a (with out the parentheses) by (b,c) and the scalar a (without the parentheses) by (b,c)T.

What are the results and code you used to obtain them.?

[Fredrik]

There seems to be a lot of oddness in this thread. I took classes on calculus (including the concept of the limit and differential equations), complex numbers, finite mathematics, linear algebra etc. in high school. Do these things really not get taught in high school in the US? I grew up in Canada...
I'm Swedish. We studied limits, derivatives, integrals, some differential equations, complex numbers, but no linear algebra. I don't think I had ever seen a matrix before the university, but I was familiar with vectors in R3. The calculus class at the university was much harder mainly because we had to know how to prove the relevant theorems, but also because the problems we were supposed to be able to solve were much harder.


Again, sounds to me like a fairly lightweight linear algebra class.

It was. It included vector spaces, the relationship between linear operators and matrices, eigenvalues and eigenvectors and so on, but it was still easy. The exam was a joke. You could pass it even if you had only read the first 90 pages in Howard Anton's linear algebra book, so you didn't even need to know what a vector space is. In other words, you could pass the class without knowing what linear algebra is.

The set of integers is a ring.
The set of 1x1 matrices is a ring.
Therefore, the set of integers is the same as the set of 1x1 matrices.

... and you acuse US of faulty logic.

He didn't actually say that. He was speaking loosely and said that an integer is a 1x1 matrix. I don't have any problem with that statement. It suggests that integers can be viewed as a subset of 1x1 matrices, nothing more, and it's obvious what subset that is, so it's really pointless to argue against it.

[ingoa]

My dear....
I did not read all the thread.
I did physics at a German university. Proofs and all the stuff...
All exercises. You know that there IS a solution.

Then
Diploma thesis (Master): you encouter an equation. You need a solution. Is there any? You don't know. It took me six months to come up with a solution. That´s mathematics!

Mathematics is easy as presented in course work. You know there is a solution. HOW you present it is quite unimportant.

[Mr.D]

Going all the way back to the OP, I suspect that one of the main reasons the typical US university 4 year science/engineering curriculum starts with two semesters of calculus is that of practicality. It's simply much simpler to funnel 95% your incoming freshman science/engineering majors into baseline Calc 101/102 classes.

You've got many incoming students with a large variation in the quality and depth of math and science backgrounds and you have to somehow "homogenize" and advance their math skillsets enough to begin to take 200 level science classes and meaningfully choose (or change!) majors.

While I can see some advantages to your idea for students who go into college as physics majors, it would seem to be a significant disadvantage to anyone who chose to select, switch into OR OUT OF physics after two or three semesters.

[lightfire22000]

If you are so sure that a scalar is just a 1x1 matrix, multiply the 1x1 matrix (a) with the vector (b,c) and the 1x1 matrix (a) with (b,c)T. Then multiply the scalar a (with out the parentheses) by (b,c) and the scalar a (without the parentheses) by (b,c)T.

What are the results and code you used to obtain them.?

You make a nice point. I don't know why you choose to say it like that, but it's a good point and I didn't catch it before. Since the number of columns of the first matrix must equal the number of rows of the second matrix in order to execute matrix multiplication, one should not be able to perform the operation [6]*[5;9], where ';' indicates a new column. You're right.


Here you go!
----------------------------------------------------------
Your MATLAB license will expire in 57 days.
Please contact your system administrator or
The MathWorks to renew this license.
----------------------------------------------------------
>> a=[6]

a =

6

>> a*(2,3)
??? a*(2,3)
|
Error: Expression or statement is incorrect--possibly unbalanced (, {, or
[.

>> a*[2,3]

ans =

12 18

>> [2,3]'

ans =

2
3

>> a*[2;3]

ans =

12
18

>> 6*[2;3]

ans =

12
18

>> 6*[2 3]

ans =

12 18

>> [6]*[2 3]

ans =

12 18

>> [6]*[2;3]

ans =

12
18

[The Sopwith Turtle]

To go back to the OP, the introductory courses are there to keep people interested in physics. If I did two years of only math method courses, then I would have been rather bored with the major before I got to the third year, and I was a relatively motivated student.

Also, math method courses for physicists are much harder if you're not doing the related physics work side-by-side. The examples keep it engaging. I took my math methods course in my last semester (I went to a liberal arts school, where there weren't enough physics faculty or majors to run every physics course every year) and I did fine up to that point. The two streams are so complementary that it's almost necessary to cross them.

[lightfire22000]

It appears that MATLAB makes exception for matrix multiplication when it's a scalar. It one by one matrices as scalars even though they don't behave as such. It really comes down to one's definition of matrix multiplication and semantics.

There are a lot of ways to introduce advanced topics to beginning students. They do that quite a bit. In fact, they do it way too much in my opinion.

[Raze]

Going all the way back to the OP, I suspect that one of the main reasons the typical US university 4 year science/engineering curriculum starts with two semesters of calculus is that of practicality. It's simply much simpler to funnel 95% your incoming freshman science/engineering majors into baseline Calc 101/102 classes.

You've got many incoming students with a large variation in the quality and depth of math and science backgrounds and you have to somehow "homogenize" and advance their math skillsets enough to begin to take 200 level science classes and meaningfully choose (or change!) majors.

While I can see some advantages to your idea for students who go into college as physics majors, it would seem to be a significant disadvantage to anyone who chose to select, switch into OR OUT OF physics after two or three semesters.

Huh. That is a really good point. I had not considered that.



ETA- as far as the "ease" of linear algebra, I suspect it could be because those classes are usually taught in a sort of "practicality" light instead of a purely mathematical one, perhaps for the same reason that Mr.D just outlined (students taking Intro to Linear Algebra are from nearly every scientific and math discipline in the university).

[quixotecoyote]

No, they don't.

Here, for example, are the NCTM (National Council of Teachers of Mathematics) standards (http://standards.nctm.org/document/appendix/alg.htm) for 9-12th grade algebra:




Pathetic, isn't it? If you want to see some official state standards, check out South Carolina's (http://ed.sc.gov/agency/Standards-and-Learning/Academic-Standards/old/cso/standards/math/). Complex numbers get mentioned once in the algebra section -- finite mathematics and linear algebra not at all.


Math in America does suck. We had trigonometry and algebra in high school and that was it. A few select students got into the single calc course, but I didn't have the pull.

If you try to talk to me about anything math related more complicated than factoring equations I will give you a blank look. It's not that I want to be mathematically illiterate, it's just that I don't have the instant knack for picking up math concepts and there's little motivation to spend the serious study time when the only area they overlap in social science is doing statistical studies, where I admit drkitten could probably slap me about with a trout over.

[lightfire22000]

Math in America does suck. We had trigonometry and algebra in high school and that was it. A few select students got into the single calc course, but I didn't have the pull.


That's the problem with the government system of education. Fundamentals are not taught and the higher math courses are only open to a few people. Anyone should be allowed to learn and receive credit for it.


...And although a scalar is not a 1x1 matrix, any arithmetic operation concerning exclusively scalars can be viewed as concerning exclusively 1x1 matrices.

[drkitten]

...And although a scalar is not a 1x1 matrix, any arithmetic operation concerning exclusively scalars can be viewed as concerning exclusively 1x1 matrices.

.... and similarly, any concept concerning clams can be viewed as concerning styrofoam packing, or shredded wheat, or peanut butter, if you don't mind the fact that you're getting the concept wrong in an effort to force-fit it into a rather stupid and inappropriate framework.

I mean, yes, you could do that. But I'd not eat at your seafood restaurant, and I'd not pass your students when I got them in a later class that required them to understand matrices.

[MattC]

That's the problem with the government system of education. Fundamentals are not taught and the higher math courses are only open to a few people. Anyone should be allowed to learn and receive credit for it.

I'm not entirely sure if you meant that to be some sort of assault on restricting courses, but I for one have no intention of allowing students into my class who have not successfully completed the prerequisities. I'd actually like more prereqs as opposed to less.

~ Matt

[drkitten]

I'm not entirely sure if you meant that to be some sort of assault on restricting courses, but I for one have no intention of allowing students into my class who have not successfully completed the prerequisities. I'd actually like more prereqs as opposed to less.


Lightfire has another perfectly cromulent thread in which he started arguing that formal education stifled autodidacts and geniuses like him by forcing them to learn the basics before they swept on to grant fallacious theories and then it degenerated into mother-mentioning.

[lightfire22000]

.... and similarly, any concept concerning clams can be viewed as concerning styrofoam packing, or shredded wheat, or peanut butter, if you don't mind the fact that you're getting the concept wrong in an effort to force-fit it into a rather stupid and inappropriate framework.

I mean, yes, you could do that. But I'd not eat at your seafood restaurant, and I'd not pass your students when I got them in a later class that required them to understand matrices.

So you'd fail the entire company of Mathworks? Their product, MATLAB, has saved lives and helped society a lot, but I guess its designers can't pass your class.

Besides, it's a false analogy. People change definitions of terms due to changing parameters all of the time! The word "mass" has had many definitions. Before gravity was documented and mathematically formulated, mass existed as a concept. However, gravity changed the way people define mass. They explained gravity in terms of inertial mass and then explained mass in terms of gravity. Sometimes, people have to get the concept wrong first in order to get the concept right. Without applying concepts, you have not way of determining whether the concept is right or not.

Also, matrix operations were decided by convention. Matrix multiplication is an arbitrary operation, not solely derived from scalar multiplication. Mathematicians made it up. Scalar multiplication solely requires the thinking, "If I have x groups of y units, how many total units do I have?" Matrix multiplication does not solely ask, "If I have AxB groups of CxD units how many units do I have?" Indeed, it cannot ask that because one cannot claim that an orange is made up of a group of apples. Clams are different from styrofoam naturally, so your analogy is ridiculous.

[lightfire22000]

I'm not entirely sure if you meant that to be some sort of assault on restricting courses, but I for one have no intention of allowing students into my class who have not successfully completed the prerequisities. I'd actually like more prereqs as opposed to less.

~ Matt

Yeah, that's an assault on restricting courses. Where would you stop Matt?
Students are restricted from prerequisites. They're restricted from the prerequisites for prerequisites. People have a right to knowledge and education should be tailored to that right.

[lightfire22000]

Lightfire has another perfectly cromulent thread in which he started arguing that formal education stifled autodidacts and geniuses like him by forcing them to learn the basics before they swept on to grant fallacious theories and then it degenerated into mother-mentioning.

Never do I claim to be a genius. I don't even claim to be a complete autodidact. Plus, you've got it backwards kitten. Formal institutions tend to forget the basics and force abstractions and fallacious theories on people. Self-learning forces one to learn the basics.

"Cromulent"? I've heard that word in only one place before and it was a Simpsons' episode that I haven't seen in ages where Lisa finds out that Jebediah Springfield was a pirate. Miss Hoover says it if I'm not mistaken. I'm pretty sure they made it up. If you believe in making up words that aren't formal, what do you have against informal education?

Informal education would solve Matt's problem and instill strong mathematical fundamentals and replicate Matt's curriculum.

[drkitten]

So you'd fail the entire company of Mathworks?

No, but I'd probably fail anyone who only learned algebra from MATLAB.

Their product, MATLAB, has saved lives and helped society a lot, but I guess its designers can't pass your class.

No, it's designers are actually fairly explicit about the fact that they know they're violating conventions. Read the design documents.



Also, matrix operations were decided by convention.

That's right. All of mathematics was decided by convention, and there's nothing particularly magical about matrix mathematics in that regard. A large part of mathematics education is learning the conventions so that you've got the right tool to hand when you need it.

You start blurring the conventions, and you're not learning the mathematics. Which is why I'd fail your students.


Clams are different from styrofoam naturally, so your analogy is ridiculous.

And matrices are different from scalars as well. Of course, they share similarities, but clams and styrofoam are both primarily made of carbon and hydrogen, so that doesn't help much.

The whole point of a matrix algebra class is to learn how matrices and scalars differ -- so saying "oh, but they're the same" is not just unhelpful, but actively harmful to the learning process.

More epic fail from the world's greatest autodidact.

[lightfire22000]

kitten,
you just don't get it.




That's right. All of mathematics was decided by convention, and there's nothing particularly magical about matrix mathematics in that regard. A large part of mathematics education is learning the conventions so that you've got the right tool to hand when you need it.

You start blurring the conventions, and you're not learning the mathematics. Which is why I'd fail your students.


Not true. Some of it is by convention, some of it is not by convention. If someone gave matrix multiplication a different name and called a different operation "matrix multiplication" they would still learn math. However, they wouldn't learn the same conventions. Mathematicians existed all over the world in ancient times and many used different conventions.





And matrices are different from scalars as well. Of course, they share similarities, but clams and styrofoam are both primarily made of carbon and hydrogen, so that doesn't help much.

The whole point of a matrix algebra class is to learn how matrices and scalars differ -- so saying "oh, but they're the same" is not just unhelpful, but actively harmful to the learning process.

More epic fail from the world's greatest autodidact.

A matrix is an array of scalars. A clam is not an array of styrofoam.

Matrix algebra was invented for practical purposes. The concept of a "matrix" exists to solve real life problems. It is not the other way around. Therefore, the point of a matrix algebra class is not to learn how matrices and scalars differ. That's absurd. The point of learning how matrices and scalars differ is to learn matrix algebra for use in daily life.

[MattC]

Yeah, that's an assault on restricting courses. Where would you stop Matt?
Students are restricted from prerequisites. They're restricted from the prerequisites for prerequisites. People have a right to knowledge and education should be tailored to that right.

I disagree with you entirely on your last point and this will, undoubtedly, be the source of our future disagreements. Someone has a right to the opportunity for knowledge.

The problem with saying it the way you have is that it presupposes that "knowledge" is some sort of readily packaged good that can be shoved into a box and handed off to someone freely. An odd mixture of laziness mixed with hedonism would suggest that "Here, take your knowledge-in-a-box and go!" is a better option, I must say that I frankly respect the students more than that. The presumption that knowledge is some sort of singular entity that can be essentially panhanded on a street corner is frankly a bit stunning, if not vaguely reminiscent of ancient Greece's philosophers.

To apply a small example, drkitten and I are completely different people with vastly different specialties. He, for example, spends a great deal of time in the economics forum - the only thing I've ever gotten out of that forum is a strong sense of confusion and inadequacy in the subject matter (drkitten's willingness to answer my very basic questions about economics and economic policy issues is something I'm very grateful for). I occasionally interject in the social issues forum whenever something related to my subject matter (criminology) comes up. The knowledge that we have is completely different and has very little interrelatedness (though we have seen some economists tackling criminological issues recently, but that's a different issue). Incorporating both of our vastly different specialties into the heading of "knowledge" does not do the students any favors.

I'm not entirely sure how to proceed from here. Is your argument against the necessity for prerequisites in general or the sheer amount of them required?

~ Matt

[drkitten]

I'm not entirely sure how to proceed from here. Is your argument against the necessity for prerequisites in general or the sheer amount of them required?

His thread is available here (http://forums.randi.org/showthread.php?t=144583). It's something like twenty pages long and filled with more nonsense than a warehouse full of Lewis Carroll reprints. Please, for the love of all that is holy, do not allow him to re-spam this thread with his drivel.....

[MattC]

His thread is available here (http://forums.randi.org/showthread.php?t=144583). It's something like twenty pages long and filled with more nonsense than a warehouse full of Lewis Carroll reprints. Please, for the love of all that is holy, do not allow him to re-spam this thread with his drivel.....

I confess the situation I'm in is very much akin to someone who's just found out that his treasured system for winning on the ponies has been proven inaccurate - you really don't want to abandon it because you've put a good deal of work into it, but at the same time the numbers don't lie.

I don't necessarily want to agree with you, but I'm forced to admit that the numbers suggest you're right.

~ Matt

[lightfire22000]

I'm not entirely sure how to proceed from here. Is your argument against the necessity for prerequisites in general or the sheer amount of them required?

~ Matt

My argument is against the necessity for formalized classroom prerequisites. People can have prerequisite knowledge from a course with a different title or from another school. They should be able to take placement tests to test out of classes. They usually are able to do so for course level purposes, but not usually for credit.

I confess the situation I'm in is very much akin to someone who's just found out that his treasured system for winning on the ponies has been proven inaccurate - you really don't want to abandon it because you've put a good deal of work into it, but at the same time the numbers don't lie.
~ Matt
To what numbers are you referring?

[MattC]

My argument is against the necessity for formalized classroom prerequisites. People can have prerequisite knowledge from a course with a different title or from another school. They should be able to take placement tests to test out of classes. They usually are able to do so for course level purposes, but not usually for credit.

Can have, yes. Their ability to prove that they do is something else entirely, which seems vaguely reminiscent of a conversation you had some time ago with drkitten and myself. I don't know anything about the matter of not getting credit for a course one successfully tests out of.

~ Matt

[lightfire22000]

Can have, yes. Their ability to prove that they do is something else entirely, which seems vaguely reminiscent of a conversation you had some time ago with drkitten and myself. I don't know anything about the matter of not getting credit for a course one successfully tests out of.

~ Matt

Sorry if I began to hijack the thread when I started discussing credit, but I think the rest is relevant. Their ability to prove that they have the prerequisites is simply in their ability to handle the coursework. You can usually tell if one has the prerequisite skills or not from a couple of tests and the student can usually tell after the first week of class. That's happened to me before and I dropped down a level. Then, I was fine.

Why is this relevant? It is relevant because it's an effective way for measuring the success of Raze's alternative curriculum.

[MattC]

Sorry if I began to hijack the thread when I started discussing credit, but I think the rest is relevant. Their ability to prove that they have the prerequisites is simply in their ability to handle the coursework. You can usually tell if one has the prerequisite skills or not from a couple of tests and the student can usually tell after the first week of class. That's happened to me before and I dropped down a level. Then, I was fine.

I'm glad to hear this worked out for you. I do not grant the seeming idea that because it worked for you, it will work for everyone else as well. Bear in mind that if I the instructor have to figure out what each individual student does and does not know about the subject matter they want to learn, we're never going to cover the actual material of the class. I as an instructor cannot be expected to fill in every bit of prerequisite knowledge needed to get to my class. A prerequisite is a very effective way to ensure that everyone is at least on a semi-level playing field.

Why is this relevant? It is relevant because it's an effective way for measuring the success of Raze's alternative curriculum.

He's not posting an alternative curriculum so much as an alternative class scheduling so far as I read the original post. It has a fair bit of merit to it as well.

~ Matt

[fuelair]

Even if the set of all integers were isomorphic to the set of all matrices, it still wouldn't mean they're the same. Isomorphism only holds with regard to a particular property or set of properties; when you add other properties into them mix, you may break the isomorphism.

As an example, the set of integers is well understood to be isomorphic (in size) to both the set of natural numbers and the set of rational numbers : they're all "countably infinite." This means there is a function f() that uniquely converts integers into natural numbers and back, and another that uniquely converts integers into rational numbers and back (and of course, f-inverse*g() will uniquely convert natural numbers into rational numbers).

Does this mean these sets are all the same? Of course not. In particular, the rules of algebra are widly different for these three sets. Natural numbers are a group, integers are a ring, and rational numbers are a field. How is this possible? The "natural" interpretation of operations differs radically among these. If you say that a+b=c in the integers, you have no confidence that f(a)+f(b)=f(c) or that g(a)+g(b)=g(c).

Similarly, the comparison operation > breaks; in the natural numbers, there exists a number 0 such that every number > 0; no such number exists in the integers.

So, is the set of all 1x1 matrices isomorphic to the integers? No, not at all. I can have a 1x1 matrix of real numbers, which means there are uncountably many such matrices -- and only countably many integers. So in this case, the isomorphism fails to exist, at a very basic level.

But let's pretend that lightfire actually understood the material well enough to specify that he was talking about a 1x1 matrix of integers. Well, this is isomorphic in size to the set of integers, yes. But it's also isomorphic to the set of rational numbers. So if this isomorphism is sufficient to make these things the same, we get the rather odd property that 1x1 matrices are both a field and a ring that is not a field. Throw natural numbers into the mix and it becomes a group but not a ring as well -- throw in the free group on two generators and the set of all matrices becomes a non-abelian group as well.

Which is impressive, since all rings must be abelian.

Essentially, lightfire's argument is gibberish from one end to another.

Which suggests that perhaps high school isn't the place to study group theory.Though group therapy might be a possibility.

[ZirconBlue]

Math in America does suck. We had trigonometry and algebra in high school and that was it. A few select students got into the single calc course, but I didn't have the pull.

I guess I was lucky. I went to a county high school in Kentucky and got a lot more math. I was on the "Advanced" track listed below.


Grade|Advanced|Typical|"Remedial"

8th|Algebra I|8th Grade Math|8th Grade Math
9th|Geometry|Algebra I|Pre-Algebra
10th|Algebra II|Geometry|Algebra I
11th|Pre-Calculus*|Algebra II|Geometry
12th**|A.P. Calculus|Pre-Calculus*|Algebra II


*"Pre-Calculus" was, I think, actually called "Trig. and Analytical Geometry".
** Some diplomas (we had 4 different diplomas) only required 3 years of math, so some students would not take the 12th grade math class.

[drkitten]

I guess I was lucky. I went to a county high school in Kentucky and got a lot more math. I was on the "Advanced" track listed below.

... which would still, I'm afraid, barely qualify as college prep mathematics in most of Europe.

The IB syllabus (http://www.cis.edu.hk/sec/math/ib/IBH.htm) (see also here (http://www.google.com/url?sa=t&source=web&ct=res&cd=2&url=http%3A%2F%2Fwww.education.umd.edu%2Fmathed%2F conference%2Fvbook%2Fmath.hl.08.pdf&ei=ybaNSoNIk5IxkrHdrwo&rct=j&q=international+baccalaureate+mathematics+HL+sylla bus&usg=AFQjCNHzog1wce2ln_XEdXbWOqDhOXugcA)) calls for students to be able to do vector algebra, solve first order differential equations, discuss the axioms of an Abelian group, calculate variance and confidence intervals from a group of statistical data, develop Taylor and Maclaurin series, and show that congruence mod p yields a set of equivalence classes over the natural numbers.

[Raze]

... which would still, I'm afraid, barely qualify as college prep mathematics in most of Europe.

The IB syllabus (http://www.cis.edu.hk/sec/math/ib/IBH.htm) (see also here (http://www.google.com/url?sa=t&source=web&ct=res&cd=2&url=http%3A%2F%2Fwww.education.umd.edu%2Fmathed%2F conference%2Fvbook%2Fmath.hl.08.pdf&ei=ybaNSoNIk5IxkrHdrwo&rct=j&q=international+baccalaureate+mathematics+HL+sylla bus&usg=AFQjCNHzog1wce2ln_XEdXbWOqDhOXugcA)) calls for students to be able to do vector algebra, solve first order differential equations, discuss the axioms of an Abelian group, calculate variance and confidence intervals from a group of statistical data, develop Taylor and Maclaurin series, and show that congruence mod p yields a set of equivalence classes over the natural numbers.

How many years are students in Europe undergraduates? (usually) How long are they usually in graduate school?

Usually here our undergraduates get 4 years of post secondary education, then anywhere from 2 to 6 in graduate school, depending on whether they are perusing a Masters or a PhD. How does it work in Europe?


The reason I ask is because I find it hard to believe that the top experts in a field over in Europe have significantly different knowledge of the field as experts in the Americas do. Although, it's possible, right?

[drkitten]

How many years are students in Europe undergraduates? (usually) How long are they usually in graduate school?

It varies with the school, program, and country. 3-4 years for a BA or equivalent is typical; Ph.D. programs can run anywhere from 3 years to infinity. And, yes, it's usually faster to get a Ph.D. in Europe (esp. in the UK) than in the States.


The reason I ask is because I find it hard to believe that the top experts in a field over in Europe have significantly different knowledge of the field as experts in the Americas do. Although, it's possible, right?

It's not only probable, but probable. Getting an American Ph.D. is typically a part-time job; you are expected to work about half-time either as a research assistant for another professor or (more commonly) as a teaching assistant, lecturing in calculus or composition or "nuts and sluts" or something. As a result, most American Ph.D.s are much better teachers coming out than their European counterparts, which gets reflected in the job market (it's harder to come out of a pure-research Oxbridge Ph.D. and get a professorship than it is to come out of Berkeley or Duke.)

The other main difference -- and this happens at secondary school level as well as the undergraduate level -- is that student in Europe specialize more. I have colleagues in the English faculty who have notentered a mathematics classroom since they were twelve; I have colleagues in physics who have not read an assigned work of fiction since they were fourteen. The idea of "breadth requirements" is almost unheard of -- if you're studying chemistry, why should you be required to take a course in history? By contrast, I think 40% of the courses my students are required to take are "core" courses required college- or university- wide; every math major and every history major has to take a course in philosophy and another in art, while every art major needs to take courses in history, math, and philosophy,.... you get the idea.

The effect is that European education is deep and narrow; US education is by comparison broad and shallow, which is why it takes something like four more years in total to achieve "expertise." The flip side is that most American mathematicians are more familiar with Shakespeare and Hemingway than their European counterparts.

[Raze]

It varies with the school, program, and country. 3-4 years for a BA or equivalent is typical; Ph.D. programs can run anywhere from 3 years to infinity. And, yes, it's usually faster to get a Ph.D. in Europe (esp. in the UK) than in the States.



It's not only probable, but probable. Getting an American Ph.D. is typically a part-time job; you are expected to work about half-time either as a research assistant for another professor or (more commonly) as a teaching assistant, lecturing in calculus or composition or "nuts and sluts" or something. As a result, most American Ph.D.s are much better teachers coming out than their European counterparts, which gets reflected in the job market (it's harder to come out of a pure-research Oxbridge Ph.D. and get a professorship than it is to come out of Berkeley or Duke.)

The other main difference -- and this happens at secondary school level as well as the undergraduate level -- is that student in Europe specialize more. I have colleagues in the English faculty who have notentered a mathematics classroom since they were twelve; I have colleagues in physics who have not read an assigned work of fiction since they were fourteen. The idea of "breadth requirements" is almost unheard of -- if you're studying chemistry, why should you be required to take a course in history? By contrast, I think 40% of the courses my students are required to take are "core" courses required college- or university- wide; every math major and every history major has to take a course in philosophy and another in art, while every art major needs to take courses in history, math, and philosophy,.... you get the idea.

The effect is that European education is deep and narrow; US education is by comparison broad and shallow, which is why it takes something like four more years in total to achieve "expertise." The flip side is that most American mathematicians are more familiar with Shakespeare and Hemingway than their European counterparts.

Not to be a Judas to the philosophy classes I have loved so much, but your description of European course of study has me a bit envious. So much wasted time...

[QED]

I'll start studying physics in Germany next month. The BA takes 3 years, the MSc takes 2 year, but it can of course take longer depending on the person (if you want to take additional courses for example). I completely agree with drkitten's assessment that "[...] European education is deep and narrow; US education is by comparison broad and shallow, which is why it takes something like four more years in total to achieve "expertise."

But I like it that way, I don't study physics so that I have to take history classes. :)

I may be a bit biased, but I think the extent to which subjects are covered in secondary school here in Germany is greater than in the US. That may as well reduce the need for a "breadth requirement" to some degree as well as the fact that we have a 13th grade*. Currently efforts are under way to reduce the number of grades to 12, which I find rather ill-conceived since the curriculum didn't change much. It's a much greater burden on the students.

On a side note: Most of the universities in Germany offer a preparatory course in maths for new physics students. In case of my university the preparatory course lasts 2 weeks, Monday till Friday from 9am to 6pm. It covers everything learned in school, plus topics that will be important for later analysis classes and which may have been left out in school. Complex numbers, Taylor series, set theory etc.


*However, only the first half of it is dedicated to learning new material, the latter half mostly consists of reviewing things that have already been covered and to prepare for the final examination. If time is still left at the end, additional material can be covered which may or may not be part of the official curriculum.

[Molinaro]

For many (most?) students, this will be their first exposure to performing "mathematics" on things that aren't numbers, and it will typically hit them hard.

I experienced that when I took a course titled "Non-Euclidean Geometry" as part of my Bachelor of Technology in Applied Computer Science.

I remember on the 1st test in that course seeing a question that said: "You are standing between the rails of a train track on level ground. The rails are 2m apart. How far away do they appear to meet?

My first thought was, "I don't recall any 2 variable equations being taught, such as distance = f(width)."

My 2nd thought was, hmmm.. I need to pay closer attention in this course!

The question may as well have been, "Since my cat has short hair, how many cups of coffee will a mouse drink for breakfast?"

[sol invictus]

I believe the way we currently teach physics should be changed. Sure, it seems to be working, but I can't help but feel that physics students would be much better off if they had better mathematical skills before they began really learning physics.

The way I see it, there are two good options for physicists:

1) learn math (including calculus) rigorously and well, from real mathematicians that care about teaching, in a curriculum that emphasizes mathematical reasoning over practical calculations.

2) learn math by using it for physics.

Then there are two bad options:

3) learn calculus from underpaid overworked grad students that can't speak English, don't like teaching, and really don't care about calc 1.

4) learn math in a "math methods for physicists class"

--

1) is great, but nearly impossible at large universities considering the demand for intro math teaching.

2) is uncommon, but I think - organized properly - it could work very well.

3) is the typical path, and it's a waste of time. Students come out of 3 semesters of calculus understanding nothing.

4) is hard because it comes after 3), and because it isn't 2).

[sol invictus]

The effect is that European education is deep and narrow; US education is by comparison broad and shallow, which is why it takes something like four more years in total to achieve "expertise." The flip side is that most American mathematicians are more familiar with Shakespeare and Hemingway than their European counterparts.

US Ph.D. level education in physics is at a level well above anything else in the world. Which is probably a good thing, because US students, crippled by terrible primary and high schools and handicapped by slower undergrad training in their field, take a while to catch up to their European and Asian counterparts.

In the end more good science gets done in the US than in Europe, I think. But that may change.

[drkitten]

US Ph.D. level education in physics is at a level well above anything else in the world.

I'm not sure I would accept that -- certainly my friends at Cambridge wouldn't agree. But it's certainly comparable; no one's going to brush off an MIT Ph.D. on the grounds that he's undertrained.

Which is probably a good thing, because US students, crippled by terrible primary and high schools and handicapped by slower undergrad training in their field, take a while to catch up to their European and Asian counterparts.

In the end more good science gets done in the US than in Europe, I think. But that may change.

... which raises the question about whether or not the "slower" training is actually a good thing. US scientists, for example, are typically much better writers and lecturers than their European counterparts, even among fluent English speakers. One reason that US science is better is because the Americans who do it tend to be so much better at expressing themselves -- they make the science both more accessible and more saleable.

And, yes, that's part of American Ph.D. training that you don't see very much in Europe. How to give a decent research talk or to write a decent paper.

[sol invictus]

I'm not sure I would accept that -- certainly my friends at Cambridge wouldn't agree. But it's certainly comparable; no one's going to brush off an MIT Ph.D. on the grounds that he's undertrained.

I agree, and I didn't express myself very well. There are some universities in the rest of the world, Cambridge among them, which are at the level of the best in the US. But I think there are as many top tier places in the US as there are in the rest of the world put together.

Do you feel the same way for math?


... which raises the question about whether or not the "slower" training is actually a good thing. US scientists, for example, are typically much better writers and lecturers than their European counterparts, even among fluent English speakers. One reason that US science is better is because the Americans who do it tend to be so much better at expressing themselves -- they make the science both more accessible and more saleable.

It may also be that a broader, slower education is an advantage, specifically that it makes it easier to think creatively. But there are so many confounding factors it's very hard to trust any such conclusion.