Math Elitists: A Controversial Opinion

[micromass]

Simple not all students want or can do proofs in a calculus class.

Since when do we care about what the student wants?? If we would follow what the average student wants, then there would be no school.
School should teach the students what is important and the school should challenge the students as much as possible.

By the same analogy, we shouldn't teach evolution, because not all students want to see it. Is this what you want??

I'm not suggesting outlawing proofs but the class should be about problem solving with applications.

You are really ignoring my posts here. Proofs ARE about problem solving. I do problem solving more when doing proofs then I do while doing applications.

I already listed some like finding the equations of motion of a particle, or even optimizing a complex shape like a sphere within a cube. These are all problems which have practical applications which the student will take with them. Anyone who wants to go into engineering or science probably has a idea of what pure math is and there is a reason they didn't choose that path.

I think very little people have an idea of what pure math is about. Only by seeing proofs and such things can we introduce to students what pure math is. Only by doing experiments, we can introduce what physics is.

Just take a look at the difference between a honors calculus test and a non honors test. The honors test will have question about the nature if a function can be integrated or not, while the non honors will have the student integrating a complicated function. The latter is more useful to students who want to go into sciences while the former is more useful to people who want to go into math.

Remind me how integrating a complicated function is useful again?? I mean, we have computers and software that do these kind of things right now... By your reasoning, we could just eliminate everything from calculus and just teach students how to use software.

I will restate my calculus BC class which I am very proud of had proofs and because of those proofs I still remember everything from 2 years ago but it also helped with my physics even more because the problems which applied in nature. I'm all about giving a choices there is room for a proof based class and there is room for a applications based class.

We shouldn't split up classes like that. A class should be BOTH theoretic and application based! A good class will contain both in a balanced way.
You make it sound like it is OR proofs OR applications. But we can have both! And both ways will reinforce each other! Proofs will help with applications and vice versa. Why not present the topic in various ways, instead of just focussing on the applications??

Again, I ask. Should we give experiments in physics class?? Why aren't we better of to just let the student take F=ma on faith and let the students make exercises on that?? Surely those experiments will be useless to future engineers?

 

[Kindayr]

Simple not all students want or can do proofs in a calculus class. I'm not suggesting outlawing proofs but the class should be about problem solving with applications. I already listed some like finding the equations of motion of a particle, or even optimizing a complex shape like a sphere within a cube. These are all problems which have practical applications which the student will take with them. Anyone who wants to go into engineering or science probably has a idea of what pure math is and there is a reason they didn't choose that path. Just take a look at the difference between a honors calculus test and a non honors test. The honors test will have question about the nature if a function can be integrated or not, while the non honors will have the student integrating a complicated function. The latter is more useful to students who want to go into sciences while the former is more useful to people who want to go into math. I will restate my calculus BC class which I am very proud of had proofs and because of those proofs I still remember everything from 2 years ago but it also helped with my physics even more because the problems which applied in nature. I'm all about giving a choices there is room for a proof based class and there is room for a applications based class.

I disagree with your claim that 'scientists and engineers' know what pure math is, and choose not to do it. None of my friends, including all of those in science and engineering (as well as my grandfather who has a BASc in Civil Engineering and a PhD in Surveying Science), know what I do in my pure math classes. None of them know what a proof is. They all suspect me to be doing an applied mathematician or physicist's work in applying math to the world, as opposed to the actuality of doing mathematics for the sake of mathematics.

Again, this is a personal experience, but I feel your claim is entirely without evidence. I highly doubt engineers and scientists huff, puff, and scoff at pure math because they know exactly what it is and what its work entails.

Again, I ask. Should we give experiments in physics class?? Why aren't we better of to just let the student take F=ma on faith and let the students make exercises on that?? Surely those experiments will be useless to future engineers?

I agree completely. I think the best example is in chemistry. To the OP: Should we only do experiments in chemistry and not teach the kids in high school the theory behind the reactions? Should we just give them the ingredients, show them what to do with them, and push them out into the real world hoping they took enough out of it? Of course not! The same goes for Calculus. You need a happy balance between application and theory.

 

[xdrgnh]

Since when do we care about what the student wants?? If we would follow what the average student wants, then there would be no school.
School should teach the students what is important and the school should challenge the students as much as possible.

By the same analogy, we shouldn't teach evolution, because not all students want to see it. Is this what you want??



You are really ignoring my posts here. Proofs ARE about problem solving. I do problem solving more when doing proofs then I do while doing applications.



I think very little people have an idea of what pure math is about. Only by seeing proofs and such things can we introduce to students what pure math is. Only by doing experiments, we can introduce what physics is.



Remind me how integrating a complicated function is useful again?? I mean, we have computers and software that do these kind of things right now... By your reasoning, we could just eliminate everything from calculus and just teach students how to use software.



We shouldn't split up classes like that. A class should be BOTH theoretic and application based! A good class will contain both in a balanced way.
You make it sound like it is OR proofs OR applications. But we can have both! And both ways will reinforce each other! Proofs will help with applications and vice versa. Why not present the topic in various ways, instead of just focussing on the applications??

Again, I ask. Should we give experiments in physics class?? Why aren't we better of to just let the student take F=ma on faith and let the students make exercises on that?? Surely those experiments will be useless to future engineers?

I'd like to think that if a student already knows what they want to be and on there own register for a calculus class they are responsible enough to know what they need. Most students don't need proof based math during high school or the 1st year of college unless they are going into math. Do you think students would also be forced to take extra classes that they don't want or need in there lives which just adds extra stress plus take away from classes they want or need for there major. Proofs are problem solving but they are many kinds of problems. If the emphasis is on proofs then students who don't intend to go into math will question the purpose of it. No one likes taking classes that seems to have no relevance to there interests. In most American colleges the classes are split up, you have a theoretical path and the applied path and it's working pretty well in college. Usu sally in physics you have your lecture then your lab class which is separate. If you try to make it both proofs and applications and you want a high quality class then that's just to much for a high school student or 1st year student. Try to make that work with a 90 minute class and see what the results are. It is in a way one way or the other that's why almost every college has a honors class and a non honors one because trying to do two in one doesn't work and is to demanding for anyone who wants to keep there sanity. In physics the intergalactic up to a point sort of needs to be made by hand, for example trying to find the gravitational force of a ball that is being pulled on by a disk is a complicated integral that needs to be made by hand before plugged into a computer system. By practicing integration students learn how to set up integral which is needed for all levels of physics. They don't need a proof of why certain function can't be integrated in terms of elementary functions at that level.

 

[micromass]

I'd like to think that if a student already knows what they want to be and on there own register for a calculus class they are responsible enough to know what they need. Most students don't need proof based math during high school or the 1st year of college unless they are going into math. Do you think students would also be forced to take extra classes that they don't want or need in there lives which just adds extra stress plus take away from classes they want or need for there major.

Colleges are not only there to teach you about your major. They should also give you a broad impression of what a certain field is about and what the methodology is that is used in that field.

Again, why give experiments in physics classes?? They are useless, no?? (I'm repeating myself, but that's because you fail to address the main point)

Proofs are problem solving but they are many kinds of problems.

Indeed, and we should present all kinds of problem solving. Thus proofs as well.

If the emphasis is on proofs then students who don't intend to go into math will question the purpose of it. No one likes taking classes that seems to have no relevance to there interests.

It's not because some people don't like the class, that we shouldn't teach it. Again, not all elementary school students like reading, but does that mean we shouldn't teach it?

In most American colleges the classes are split up, you have a theoretical path and the applied path and it's working pretty well in college. Usu sally in physics you have your lecture then your lab class which is separate. If you try to make it both proofs and applications and you want a high quality class then that's just to much for a high school student or 1st year student. Try to make that work with a 90 minute class and see what the results are. It is in a way one way or the other that's why almost every college has a honors class and a non honors one because trying to do two in one doesn't work and is to demanding for anyone who wants to keep there sanity.

It's not too demanding at all! In Belgium, everybody who follows calculus will see proofs. And they will see a mixed class that is both proof-based and application based. And the students do just fine! So it's not impossible for Belgian students. So why can't American students handle it?? Are they dumber than European students, is it that what you're claiming?

In physics the intergalactic up to a point sort of needs to be made by hand, for example trying to find the gravitational force of a ball that is being pulled on by a disk is a complicated integral that needs to be made by hand before plugged into a computer system. By practicing integration students learn how to set up integral which is needed for all levels of physics. They don't need a proof of why certain function can't be integrated in terms of elementary functions at that level.

OK, so we should just teach students how to set up an integral and then let a computer calculate the rest. Why actually teach substitution method and partial integration?? Who actually needs these things?? Can't we just teach students how to set up an integral? Isn't that enough.

You are trying to dumb down the college classes, and this is a very dangerous trend.

 

[Hurkyl]

By practicing integration students learn how to set up integral which is needed for all levels of physics. They don't need a proof of why certain function can't be integrated in terms of elementary functions at that level.

I hope you have a better example of what you mean; this one is terrible. The proof technique involved is not elementary calculus. (I think it's differential Galois theory; I'm not entirely sure what class, if any, one would encounter such a theory when in school)


The proofs you see in an elementary calculus course are generally ones that demonstrate the things you're supposed to be learning in elementary calculus -- i.e. how to use calculus to solve problems.

 

[thegreenlaser]

And on what do you base this vilification? I find it very hard to believe that people treat your work unfairly to feel superior. It is more likely that they do so because they actually find it trivial. Your random assigning of subconscious motivational factors to people who "trivialize" the work of others is much more rude than that which they do.

There's a reason that I said "you have to be careful to differentiate between people who are simply being realistic about what level of education you've reached in a certain subject area and people who are trying to put you down to make themselves feel smart."

I never said that superiority is always the reason that people put down the work of others. In fact, I'm pretty sure I said that most people who say something is easy compared to what they do aren't this way. All I was saying is that there are elitists out there: people who feel that they are inherently superior to everyone else; but you have to be careful to figure out if someone is actually an elitist or if they just honestly think your work is simple and don't necessarily view you as a lesser person because of it. I don't know how that's a random assignment of subconscious motivational factors. In fact, my point was that people shouldn't make assumptions like that. I apologize if that was unclear.

 

[xdrgnh]

Colleges are not only there to teach you about your major. They should also give you a broad impression of what a certain field is about and what the methodology is that is used in that field.

Again, why give experiments in physics classes?? They are useless, no?? (I'm repeating myself, but that's because you fail to address the main point)



Indeed, and we should present all kinds of problem solving. Thus proofs as well.



It's not because some people don't like the class, that we shouldn't teach it. Again, not all elementary school students like reading, but does that mean we shouldn't teach it?



It's not too demanding at all! In Belgium, everybody who follows calculus will see proofs. And they will see a mixed class that is both proof-based and application based. And the students do just fine! So it's not impossible for Belgian students. So why can't American students handle it?? Are they dumber than European students, is it that what you're claiming?




OK, so we should just teach students how to set up an integral and then let a computer calculate the rest. Why actually teach substitution method and partial integration?? Who actually needs these things?? Can't we just teach students how to set up an integral? Isn't that enough.

You are trying to dumb down the college classes, and this is a very dangerous trend.

Comparing career seeking high students and college student to elementary school students, I don't even know where to start at that. You're right we should present all kinds of problem solving even proofs but proof problem solving shouldn't be the main focus for a class filled with applied science students. In America we have separate classes in college for applied and theory and we have the best college system in the world. You're not always going to have a computer available will you and techniques that are used to integrate function can be used in other areas of math and physics that a computer can't do. If you took a physics class which I'm sure you did, then you would know how hard it is using a single integral to finder the gravitational attraction of a sphere, those skills are necessary for other things in physics proofs aren't necessary.

 

[micromass]

we have the best college system in the world.

Lol! Aren't we a bit elitist here??
I would certainly not consider american colleges to be the best of the world...

 

[xdrgnh]

I hope you have a better example of what you mean; this one is terrible. The proof technique involved is not elementary calculus. (I think it's differential Galois theory; I'm not entirely sure what class, if any, one would encounter such a theory when in school)


The proofs you see in an elementary calculus course are generally ones that demonstrate the things you're supposed to be learning in elementary calculus -- i.e. how to use calculus to solve problems.

That probably was a bad example I just came up with that off the top of my head. Just look at the questions on this thread https://www.physicsforums.com/showthread.php?t=312799&page=9. Those are the typical questions a scientist or a engineer doesn't need to know at that level yet unless they want to go into math.

 

[xdrgnh]

Well world news and other statistical organizations think so there is a reason why so many foreign students come here to study. In our colleges including MIT and Harvard the classes are separate because both address totally different needs. http://www.usnews.com/education/worlds-best-universities-rankings/top-400-universities-in-the-world If you look most of the top schools are American or British not a single Belgium school is on the list. We do though have a crappy high school system which is why a lot of our students are foreign and not domestic.

 

[micromass]

That probably was a bad example I just came up with that off the top of my head. Just look at the questions on this thread https://www.physicsforums.com/showthread.php?t=312799&page=9. Those are the typical questions a scientist or a engineer doesn't need to know at that level yet unless they want to go into math.

Let me look at the first file I come across.

The first question is a definition of integrability. So you say that people don't need to know the definition of an integrable function?? Really?

Questions (2) is calculating integrals without the fundamental theorem of calculus. Be sure that every engineer needs to be able to do this.

Questions (3)-(7) are really easy questions on calculating integrals. I don't see why an engineer won't need to knowthis

Only the last three questions are proof questions. Which I do think everybody should be able to do if they know the theory. Understanding calculus is very different from being able to make stupid calculations...

 

[micromass]

Well world news and other statistical organizations think so there is a reason why so many foreign students come here to study. In our colleges including MIT and Harvard the classes are separate because both address totally different needs. http://www.usnews.com/education/worlds-best-universities-rankings/top-400-universities-in-the-world If you look most of the top schools are American or British not a single Belgium school is on the list. We do though have a crappy high school system which is why a lot of our students are foreign and not domestic.

OK, that only shows that SOME colleges are extremely good. That doesn't mean that the entire college system is good.
Secondly, I would like to see the criteria on how they ranked these schools.

I really wouldn't know why the US colleges are so much better than European colleges. Please tell me what is so much better about US schools??

 

[xdrgnh]

Those are pretty standard questions actually but check the PDF questions Mathwong posted and you'll see what I am talking about.

 

[micromass]

Those are pretty standard questions actually but check the PDF questions Mathwong posted and you'll see what I am talking about.

These are the questions by Mathwonk... I find them pretty standard. Can you give me an exact question that you're complaining about?

 

[micromass]

OK, I looked at how the rankings of the universities was obtained. Look at this site:

http://www.usnews.com/education/wor.../21/worlds-best-universities-the-methodology-

Well, to be honest, the 6 criteria they used isn't really indicative on the teaching quality of the university. I mean, what does "citations per faculty member" has to do with how good the university is at teaching??

This survey lists the most WELL-KNOWN or FAMOUS universities. It doesn't necessarily list the BEST universities for students...

 

[Bourbaki1123]

Mathematics is formal logic,

As someone with a reasonable amount of knowledge in the major areas and history of mathematical logic, I can tell you that this is incorrect. It is neither historically true nor presently true.

Formal logic was developed tangentially, and although mathematics can be described by logical systems, it is not the be all end all of mathematics.

therefore to assert something you must prove it.

Generally correct, but in the real world there is a place in mathematics for conjecture, and even proofs based on the assumption that something is false (such as P = NP).

In addition to this, mathematics cannot be captured by a single formal logical system unless perhaps you allow for certain as-yet-to-be-fully-specified fuzziness (like with the human mind, which appears to be somewhat statistical in nature).

Also, many mathematical ideas and intuitions precede proof; proof simply helps to ensure that the intuition is consistent by tying it down to things that are simple, "obvious" and well-tested (often first order predicate calculus, but other logics can and have been used, and mathematical results have successfully been embedded in them).

If you look at the reverse mathematics programme (which aims to embed certain areas of pre-set theoretic mathematics in the simplest subsystems of second order arithmetic possible), it becomes exceedingly clear that logic provides a rich tool set for exploring mathematical ideas rather than providing a magical Platonic foundation for all of mathematics. There is no such foundation; at least, there is no such foundation that we know of.

 

[xdrgnh]

OK, that only shows that SOME colleges are extremely good. That doesn't mean that the entire college system is good.
Secondly, I would like to see the criteria on how they ranked these schools.

I really wouldn't know why the US colleges are so much better than European colleges. Please tell me what is so much better about US schools??

Regardless if you believe it or not, schools like Harvard and MIT are better in math and science then anything in Belgium and they separate the classes. But if I had to guess our college system is better then Europe's because we just put so much money into it.

 

[xdrgnh]

I can't see the questions because I am using a 11 year old computer on dial up right now but I imagined they would be pretty tough considering he is talking about lipschitz continuous function in calc II. Lipschitz continuous in calc II alone proves my point

 

[micromass]

I can't see the questions because I am using a 11 year old computer on dial up right now but I imagined they would be pretty tough considering he is talking about lipschitz continuous function in calc II. Lipschitz continuous in calc II alone proves my point

No it doesn't. Why wouldn't physicists and engineers have to use Lipschitz continuity??Lipschitz continuity is the easiest form of continuity out there, so the question probably won't be too hard.

For the record, I have seen engineering articles on image processing that uses Lipschitz continuity and even Holder continuity. I don't see why it shouldn't be useful to know...

 

[thegreenlaser]

we have the best college system in the world.

Regardless if you believe it or not, schools like Harvard and MIT are better in math and science then anything in Belgium

Wait, you started a thread because you're upset that people are being elitist about the way math should be taught?

 

[Kevin_Axion]

OK, that only shows that SOME colleges are extremely good. That doesn't mean that the entire college system is good.
Secondly, I would like to see the criteria on how they ranked these schools.

I really wouldn't know why the US colleges are so much better than European colleges. Please tell me what is so much better about US schools??

I actually read an article recently that showed that Canadian and British universities are better than American universities in general when the population and other factors are taken into account.

 

[xdrgnh]

Easiest form of continuity is that you can draw the function with a single stroke of a pen. Lipschitz continuity is something you would never see in a standard calculus class in college, probably in a honors class but not a regular one. These more abstract idea scientist and engineers need but not in a calc 1 and 2 class it's very extreme. I had to look it up to know what it meant because in all of the math and college lectures I watched I never saw it. Of coarse the easiest form isn't good enough for any college level class, there are other conditions that should be mentioned also.

 

[Hurkyl]

That probably was a bad example I just came up with that off the top of my head. Just look at the questions on this thread https://www.physicsforums.com/showthread.php?t=312799&page=9. Those are the typical questions a scientist or a engineer doesn't need to know at that level yet unless they want to go into math.

Really? Looking at the first pdf I saw, I see:

1. A question of knowing words and basic ideas
2. Some computations
3. A cookbook calculation
4. A calculation
5. A calculation that is straightforward, but tests a students ability to break a problem into parts and tackle them one at a time
6. A calculation
7. Some calculations
8. A combination of pattern recognition and an ability to apply convergence tests
9. A calculation
10. A test of a student's ability to work with a function defined via integrals
11. A test of a student's ability to work with a function defined via series
12. A calculation (but done in two different ways)
13. A test of problem solving skills (decide on your own how to check a fact, then use your method to actually check something)

Which of these are things are you proposing that an engineer or a scientist doesn't need to know?

 

[xdrgnh]

I actually read an article recently that showed that Canadian and British universities are better than American universities in general when the population and other factors are taken into account.

If you factor in America's terrible secondary education then yes you could argue that Canadians schools and British schools are better.

 

[micromass]

Easiest form of continuity is that you can draw the function with a single stroke of a pen. Lipschitz continuity is something you would never see in a standard calculus class in college, probably in a honors class but not a regular one. These more abstract idea scientist and engineers need but not in a calc 1 and 2 class it's very extreme. I had to look it up to know what it meant because in all of the math and college lectures I watched I never saw it. Of coarse the easiest form isn't good enough for any college level class, there are other conditions that should be mentioned also.

Well, mathwonk said that it IS a honours course test, so yeah.
But still, I don't see why people won't need Lipschitz continuity.

And really "you can draw the function with a single stroke of a pen". Is this really what you want the students to have as the definition of continuity? This is silly.

 

[xdrgnh]

Yes he did and he also said that his objection to AP calculus is because it doesn't substitute a honors class. The math elitists want the honors to become the standard for all students to take. I don't want students to have that definition but in truth a student could get away with that definition till they take more abstract classes. Something like Lipschitz continuity which is from my understanding that the absolute value of the slope of the secant line of a function can't exceed a certain number is very technical and isn't necessary for a class meant for scientists and engineers. This is why in America we separate the classes.

 

[Hurkyl]

Lipschitz continuity ... These more abstract idea

It is not abstract at all. It is merely something that an elementary calculus class can usually deal with in an ad-hoc manner (e.g. cases where we can simply invoke the mean value theorem).

Easiest form of continuity is that you can draw the function with a single stroke of a pen.

Now this is something that is actually quite abstract (assuming you meant it seriously and weren't just being glib).

Because you have to imagine that your pen is drawing a mathematical function. Oh, and that we're supposed to imagine the pen marking "continuously" on the paper. Oh, and by "stroke" you didn't really mean one stroke, but that many strokes can be taken as long as the pen keeps pressed against the paper. Oh, and that you're imagining a person can sketch out arbitrarily fine detail. And...

If you seriously tried to teach that as a definition, you're going to be faced with a lot of people who really don't get the difference between "continuous" and "smooth".

 

[atyy]

How about Ramanujan's formulas - were his divinations mathematics - or only his proofs?

I guess more generally, can the formulation of a conjecture be considered mathematics?

 

[xdrgnh]

It is not abstract at all. It is merely something that an elementary calculus class can usually deal with in an ad-hoc manner (e.g. cases where we can simply invoke the mean value theorem).



Now this is something that is actually quite abstract (assuming you meant it seriously and weren't just being glib).

Because you have to imagine that your pen is drawing a mathematical function. Oh, and that we're supposed to imagine the pen marking "continuously" on the paper. Oh, and by "stroke" you didn't really mean one stroke, but that many strokes can be taken as long as the pen keeps pressed against the paper. Oh, and that you're imagining a person can sketch out arbitrarily fine detail. And...

If you seriously tried to teach that as a definition, you're going to be faced with a lot of people who really don't get the difference between "continuous" and "smooth".

An ideal pen that doesn't leave any gaps, I'm not trying to be a teacher here but I'm sure you guys get the idea. If you are talking about the proper definition of a smooth function then no standard calculus class would talk about that either because it's not necessary. The purpose of a calc 1 and 2 for scientist and engineers is to show them the tools they will be using for there 1st and 2nd year classes, bringing in topic from analysis only gets in the way and makes the class harder.

 

[Fredrik]

I guess more generally, can the formulation of a conjecture be considered mathematics?

Of course. It's an essential part of the process that ends with the theorem being proved. I think that without non-rigorous arguments, mathematicians wouldn't be able to guess what statements they should try to prove. I'd be pretty surprised if the first person who proved the chain rule didn't do something like this first:

It follows immediately from the definition of the derivative that when h is small, f(x+h)\approx f(x)+hf'(x). Let's just use this formula twice, once on g and then once on f. f(g(x+h))\approx f\big(g(x)+hg'(x)\big)\approx f(g(x))+hg'(x)f'(g(x)) This implies that \begin{align}(f\circ g)'(x) &\approx \frac{f(g(x+h))-f(g(x))}{h}\approx \frac{f(g(x))+hg'(x)f'(g(x))-f(g(x))}{h}\\ &\approx f'(g(x))g'(x).\end{align} What's missing here is of course a proof that the error in this approximation really goes to zero when h goes to zero.

But still, I don't see why people won't need Lipschitz continuity.

I think that's actually a good example of something that only a very small number of people ever need. I only recall seeing it in an existence and uniqueness theorem for differential equations. Experimental physicists and engineers would be satisfied knowing that someone has proved the theorem. (Actually, most of them wouldn't even care about that). And a lot of people would be satisfied with a proof of a stronger version of the theorem, where the function is assumed to be continuous.