Math Elitists: A Controversial Opinion

[deRham]

Mathematics is formal logic, therefore to assert something you must prove it.

But formal logic is formal logic! Mathematics is conducted in the language of formal logic, sure.

Also, who said AP Calculus students don't make assertions and proofs? The question is how far you verify the details. You can go down to axiomatic set theory every time, but few do that.

Once you prove some theorems, actually a lot of proofs will not be far from what AP Calculus offers.

Now if you're in favor of forcing students to write in proper sentences, yes, if I taught AP Calculus ever, I'd do that. I would probably emphasize more details than is typical, but a clearly communicated explanation accounting for everything and acknowledging the assumptions is a proof!

 

[clope023]

Mathematics is formal logic, therefore to assert something you must prove it. And mathematics apply to many fields, I wouldn't like something to be used on so many fields with no logic behind it. That doesn't mean doing proofs is the best way to learn mathematics, I think that just knowing what the proof uses, its general idea and why that concept was needed in the first place many times is enough to have a solid understanding of the concept.

I study electrical engineering and the underlying concepts behind the math being used (usually calculus and differential equations) is not needed at all to do the engineering. Of course you would have to understand what a limit is, a derivative, and a differential equation is to understand the laplace transform and other such techniques like taking the maximums and minimums of functions but eventually it becomes second nature. I doubt most of my engineering classmates know many proofs but they can do the engineering because they have a minimum required understanding of the math so they can do the engineering, it's a trade off. I know some complex analysis, linear algebra, Fourier analysis, pde's, statistics and it's made the math in my engineering classes easier but I can't say it's helped my circuit designs.

 

[xdrgnh]

I enjoy there being those individuals who push for the rigor in mathematics; elitist or not I find that having others who have a grand knowledge of a mathematical subject provides motivation to TRY and achieve the same. In fact, it was only after seeing many of micromass's posts on this site regarding proofs in analysis, topology, and differential geometry that I was motivated to go beyond the level of rigor provided in physics texts that utilize the aforementioned subjects. When someone, like micromass, can present material in such a rigorous manner it just makes you want to be able to do the same (and it allows for more intelligent/precise conversations) and since we are on the subject of academics I don't see anything wrong with this =D. Cheers!

But that's the thing not everyone needs or wants rigor in their math. If you guys want to say that without lots of rigor and proofs math isn't math that is fine and you have a point. But for most people they use math as nothing but a tool. Rigorous math is very interesting and beautiful however to understand that beauty is very difficult and is something the individual has to be self motivated to do. Not all math students who sign up for a math class want that or need that. Forcing rigor into the class room is no different then forcing creationism in the classroom.

 

[deRham]

But that's the thing not everyone needs or wants rigor in their math.

I want to keep harping on this point - I agree with the spirit of what you're saying, but I want to also say that rigor is being misunderstood by many. The idea is to justify what you say.

That does NOT mean it has to be from the bare definitions. For instance, once you learn L'Hopital's rule, using it to show a limit is something or the other is perfectly valid.

The only difference is that some people want to give more details, and they are the ones who usually stand as "rigor-freaks" ... yet often they overestimate how much they are waving their hands, relative to someone who, say, does logic research.

I think rigor by my definition, namely being precise with what one means, is good. But giving "all" the details is often something implied by most people using the term, and that is what I find unnecessary.

 

[Vanadium 50]

Everyone is entitled to their opinion. It might convince more people if you gave new, cogent arguments, rather than just repeating inflammatory rhetoric.

 

[Tosh5457]

But formal logic is formal logic! Mathematics is conducted in the language of formal logic, sure.

Well, actually mathematics uses formal logic. Formal logic isn't a language.

But that's the thing not everyone needs or wants rigor in their math.

Sure, if you just want to apply maths you don't need to know the proofs or even what the proof uses. But then you won't know as much mathematics as the people who know how to relate the concepts (which is what proofs do), and knowing mathematics on that level can be really helpful even in applying mathematics.

 

[xdrgnh]

I'm not giving a rhetoric I'm giving facts, not everyone needs rigorous and proof based math, especially at the high school or freshmen college level. Forcing proof based math into the classroom is the same as forcing creationism. Of coarse it will probably never happen because this country prefers to dumb down the curriculum rather then make it more challenging which is if you ask me even worse then over emphasizing specialist concepts. By over emphasizing proofs time is taken away from problem solving and applications, something that most math students need. If you look at a honors calculus coarse the problems are more theory based then the ones in a non honors class. For those who don't want rigor they go into the non honors and for those who want to go into pure math they got into the honors one. Both types of math students have choices, what these math nazi's suggest is to remove choice and give everyone a proof based math curriculum.

 

[920118]

But that's the thing not everyone needs or wants rigor in their math. If you guys want to say that without lots of rigor and proofs math isn't math that is fine and you have a point. But for most people they use math as nothing but a tool. Rigorous math is very interesting and beautiful however to understand that beauty is very difficult and is something the individual has to be self motivated to do. Not all math students who sign up for a math class want that or need that. Forcing rigor into the class room is no different then forcing creationism in the classroom.

So you are saying that forcing rigor in a discipline, the theoretical side of which has always been aknowledged for its rigor, is like forcing people to learn about creationism? That just might be the worst analogy I have ever encountered.

It is likely that the "nazis" which you speak of by convention take "math" and "pure math" to coextend, which they do. Your problem seems to be that you want "doing math" to mean that you calculate by applying certain methods and theorems from the field of mathematics. To call pure math "math" and applied math "applied math" is custom.

 

[Kindayr]

I'm not giving a rhetoric I'm giving facts, not everyone needs rigorous and proof based math, especially at the high school or freshmen college level. Forcing proof based math into the classroom is the same as forcing creationism. Of coarse it will probably never happen because this country prefers to dumb down the curriculum rather then make it more challenging which is if you ask me even worse then over emphasizing specialist concepts. By over emphasizing proofs time is taken away from problem solving and applications, something that most math students need. If you look at a honors calculus coarse the problems are more theory based then the ones in a non honors class. For those who don't want rigor they go into the non honors and for those who want to go into pure math they got into the honors one. Both types of math students have choices, what these math nazi's suggest is to remove choice and give everyone a proof based math curriculum.

Okay first of all, stop calling us Nazis. It's offensive, especially when we aren't attacking you in any way, shape or form.

Secondly, do not compare rigour in mathematics to teaching creationism in the classroom. If anything, the analogy would be more likely comparable to introducing evolution to the classroom: teaching something that is actually relevant to the topic at hand. We aren't introducing faith into mathematics education, as your 'creationism' analogy implies.

Lastly, you claim that proof and rigour takes away from problem solving. I think its quite the opposite. When you teach the roots of a subject, it allows the students to understand where the concepts come from. I don't have a degree in education so this is only my opinion. However, I do know that I prefer to get a general sense of a subject before going into direct details as this helps me learn more efficiently.

Further, we're not asking for a jump to the most rigourous of teaching in mathematics, where we slap them on the hand for not explaining themselves axiomatically (as your use of Nazi implies). We ask for a general increase in the doing of actual mathematics in earlier years of a students life.

 

[xdrgnh]

Okay first of all, stop calling us nazis. Its offensive, especially when we aren't attacking you in any way shape or form.

Secondly, don't compare this to teaching creationism in the classroom. If anything, the analogy would be more likely comparable to introducing evolution to the classroom, as rigour and proof are real and applicable to the subject at hand. This isn't the introduction of faith to science.

Lastly, you claim that proof and rigour takes away from problem solving. I think its quite the opposite. When you teach the roots of a subject, it allows the students to understand where the concepts at hand come from. I don't have a degree in education, so this is my opinion. But I do know that I prefer to get a general sense of a subject before going into direct details helps me learn more efficiently. Though I cannot speak for anyone else.

Further, we're not asking for a jump to the most rigourous of teaching in mathematics, where we slap them on the hand for not explaining themselves axiomatically (as your use of Nazi implies). We ask for a general increase in doing of actual mathematics in earlier years of a students life.

When the emphasis of a class is proofs then it does take away from problem solving that could be used by engineers and scientists. Let me tell you about my AP calc BC class, I had a great math teacher for that class. When he introduced the topics he first proved them using appropriate mathematics for a 12th grade class, after the proof we all understood where it came from and why it was what it was. Afterward he gave us problems like finding the equation of motion of a falling object subject to air resistance or he would give something like finding the rate of change at which cars go through a intersections. What some radicals propose is that the emphasis should be on proofs and that rather giving problems like finding the equation of motion of a particle, the problems should be theory based and students should do many proofs on there own in a intro class. People who want to use math as a tool don't need that kind of math. Proofs even at a intro level are very difficult for students who have no interest in pure math.

It is like forcing creationism because it gets rid of choice in the end and like creationist these elitists think what they are doing is infallible. But I would like to say sorry if I offended you by using the term Nazi, I was trying to be parallel to grammar Nazis but I understand the weight of the word sorry. Doing proof based math in school was tried already in the USA and it failed during the 60s, topics like set and group theory were introduced in elementary school to foster greater understanding of numbers and it failed.

 

[deRham]

Well, actually mathematics uses formal logic. Formal logic isn't a language.

It might as well be. Of course there's such a thing formally defined in the study of logic as a language, a theory, a model, etc. But I guess in common speech, when we say "language," the logic is implied, and it's quite similar in the case of mathematics.

But aside from that, what you said is what I mean/agree with - mathematics uses formal logic. It isn't quite formal logic itself. You can say mathematics IS set theory, but in truth, it just uses it.

 

[deRham]

When the emphasis of a class is proofs then it does take away from problem solving that could be used by engineers and scientists.

And that's one step too far. You should read my latest posts. Rigor is not the same as giving all the details - it's about acknowledging what can and can't be assumed, and being precise. And frankly, without proper communication, there can be a weakness in understanding.

What some radicals propose is that the emphasis should be on proofs and that rather giving problems like finding the equation of motion of a particle, the problems

I don't know who they are, but most mathematicians I've talked to don't suggest any such thing. I'd be interested in those people's reasoning.

I dislike the harping on the word "proofs" though. A proof, as far as most math classes are concerned, is a precise explanation. That's all. And that's a good thing to expect. Jumping into esoteric theory is not necessary, of course.

"Proof" can of course mean something else to a logician, in terms of "deductive systems" or whatever. But the type of proof expected can vary greatly.

If you're suggesting emphasizing blind calculation, that's certainly bad. Calculation with the reasoning clearly stated is still, in a manner of speaking, a proof. Even very advanced math classes perform calculations.

 

[xdrgnh]

And that's one step too far. You should read my latest posts. Rigor is not the same as giving all the details - it's about acknowledging what can and can't be assumed, and being precise. And frankly, without proper communication, there can be a weakness in understanding.



I don't know who they are, but most mathematicians I've talked to don't suggest any such thing. I'd be interested in those people's reasoning.




I dislike the harping on the word "proofs" though. A proof, as far as most math classes are concerned, is a precise explanation. That's all. And that's a good thing to expect. Jumping into esoteric theory is not necessary, of course.

"Proof" can of course mean something else to a logician, in terms of "deductive systems" or whatever. But the type of proof expected can vary greatly.

If you're suggesting emphasizing blind calculation, that's certainly bad. Calculation with the reasoning clearly stated is still, in a manner of speaking, a proof. Even very advanced math classes perform calculations.

The people I am talking about those suggest in short that 1st year calculus should be like a light intro analysis class just look at any honors calculus class in college, most of the problems ask about the theory, not a actual calculation. Students who want to be engineers and scientists at first need to know how to do the calculations very well because that is what they will be doing for the first 2 or 3 years of college before the math becomes more abstract. There should be proofs or mathematical explanations but the emphasis should be applied problem solving in a standard 1st year college level class.

 

[Kindayr]

The people I am talking about those suggest in short that 1st year calculus should be like a light intro analysis class just look at any honors calculus class in college, most of the problems ask about the theory, not a actual calculation. Students who want to be engineers and scientists at first need to know how to do the calculations very well because that is what they will be doing for the first 2 or 3 years of college before the math becomes more abstract. There should be proofs or mathematical explanations but the emphasis should be applied problem solving in a standard 1st year college level class.

The problem roots from the fact that students are forced to rush through calculus in a very short amount of time. Like I've said in the "Calculus" thread, in Ontario we get 2.5 months of an introduction of calculus in our final year of high school, before going into university. A complete focus of application in university is just as hurtful to a student's education in mathematics than a rigourous analysis course in first year is. Without having knowledge or intuition of where the methods they are using come from, a lot of students fail to understand even the application of those methods.

In high school we were expected to understand a limit without explanation. This lead students to have the wrong intuition that the limit of any function is like the limit of a continuous function: that is, you just plug in the value to find the limit. So when we got to more complicated functions, their intuition failed them and they did poorly because they weren't taught what a limit really is. I'm not asking for a harsh and ridiculous notion of overwhelming epsilon-delta proofs at the first introduction of calculus (where they have no idea what a proof even is). Just a general sense of what a limit ACTUALLY is.

The state of the education of mathematics is ridiculous, and there needs to be some balance between rigour and application. For me, the solution would to introduce the ideas of reasoning your arguments at a younger age. This would allow teachers to introduce a little more theory, so students know what their doing and why it works in that way. Then all the time in the world could be used on application of those ideas.

But I'm just a 3rd year math student, so what do I know.

 

[deRham]

^ Yes it's unnecessary to make it like an intro to analysis class. However, emphasizing clear communication and understanding of the mathematics is a must.

You must be fair - not only engineers, but mathematicians too take the first course. It should involve clear reasoning, understanding the theory sufficiently to use it either in mathematics or other fields.

This means that one needn't know the proof of every theorem taught in standard calculus, but should be able to use them effectively, and with clear communication, both to solve intrinsically mathematical problems and inherently application-based problems, as I think at that basic level, even a pure mathematician should be exposed to the applications (it increases basic intuition).

 

[deRham]

I'm not asking for a harsh and ridiculous notion of overwhelming epsilon-delta proofs at the first introduction of calculus (where they have no idea what a proof even is). Just a general sense of what a limit ACTUALLY is.

This is gold, I think.

 

[Dembadon]

I'm not giving a rhetoric I'm giving facts, not everyone needs rigorous and proof based math, especially at the high school or freshmen college level. Forcing proof based math into the classroom is the same as forcing creationism. Of coarse it will probably never happen because this country prefers to dumb down the curriculum rather then make it more challenging which is if you ask me even worse then over emphasizing specialist concepts. By over emphasizing proofs time is taken away from problem solving and applications, something that most math students need. If you look at a honors calculus coarse the problems are more theory based then the ones in a non honors class. For those who don't want rigor they go into the non honors and for those who want to go into pure math they got into the honors one. Both types of math students have choices, what these math nazi's suggest is to remove choice and give everyone a proof based math curriculum.

Facts should be accompanied by data, not hyperbole. As others have suggested, it would help your arguments if you substantiated your claims. This is especially important, and blatantly obvious, when you're speaking to scientifically minded people.

 

[Kindayr]

This is gold, I think.

I guess I did contradict myself a bit there. Obviously its due to the fact that I wasn't introduced to logical reasoning until university. Just saying.

Joking aside, I just meant for them to have some sort of notion of what a limit actually is, not a rigourous introduction. I mean personally I loved learning epsilon-delta proofs. But I know its not necessary for someone just learning the methods of calculus. However, the concept of a limit is dire to understanding the methods of calculus. I see a difference, I hope everyone else does too.

 

[Fredrik]

Forcing proof based math into the classroom is the same as forcing creationism.

Are you seriously suggesting that forcing students to learn what mathematics really is and how to communicate mathematical results, is the same as forcing students to study brainwashed delusions that have been disproved by evidence as if they are facts?

 

[deRham]

Are you seriously suggesting that forcing students to learn what mathematics really is and how to communicate mathematical results, is the same as forcing students to study brainwashed delusions that have been disproved by evidence as if they are facts?

First, you can't disprove something using evidence. Evidence supports intuition, but proves nothing. Proofs are relative to some axioms.

Second, "what mathematics really is" happens to be a deeper question than most high schoolers will get to anyway. What we can do is enforce that they acknowledge their assumptions and explain their reasoning thoroughly, where thoroughness is measured by those assumptions all being clearly incorporated into the argument.

I agree the thing about creationism is confrontational and unnecessary. But answering confrontation with confrontation gets one nowhere, and in fact, I do believe there's a point beneath that rhetoric.

 

[micromass]

Hmm, I think the problem here is that we're falling into extremes here.
Sure, there are "math nazi's" out there who want to put in a ridiculous amount of rigor in high school courses, and sure, this is unnecessary. Seeing Dedekind cuts before analysis is rigorous, but it is certainly too much. (epsilon-delta's is something which can be given in high school however, it works in my country, so it could work everywhere).

However, the OP is the other extreme, I think. He is an "application nazi". That is, he just wants the people to be able to apply calculus, and not worry about the theory at all. This attitude is as bad as being a math nazi.

People who study calculus must study mathematics how it really is. I don't want to focus on the theory, but at least we should give proofs and show that this is all logically grounded and can be made extremely rigorous (if we want so). Not presenting the theory involved is actually lying to the students.

Of course we should give applications. In fact, the main focus should be in applications. But theory and proofs are necessary. I don't want a student come out of calculus say that he took it all on faith. If that happens, then the education failed.

The same thing happens with everything really. If I take a physics course, then it should be made clear to me that everything we do is verified by experiments. Fine, we can give physics without even mentioning the word experiments. Just show the fundamental laws and make exercises. But that would be lying. When teaching physics, the teacher should show the student the main methodology of physics: experiments. And when teaching math, we should give the methodolody of math: proofs.

OK, engineers will find the proofs useless in their later carreer. But at least we should give them some sort of "general culture". So that they don't graduate without knowing what math is really about. An engineer who can't do proofs isn't a bad engineer, but I still think that in that case, the education failed to show the engineer what math is really about.

Teaching a subject shouldn't just be giving all the techniques necessary. It should also try to give a broad picture of the field. If I TA courses, I always try to do this. Students don't always appreciate it, but I wouldn't feel good if I just let them make mind-numbing calculations...

Also, I find the comparison between creationism and proofs of an extreme bad taste. I think it's quite sad if you can't see the difference between these two examples...

 

[xdrgnh]

I agree that calculus isn't being taught well in my schools and that creates problems when students move on to higher level math classes. However the radicals see the fault in the curriculum and say it isn't rigorous enough. However if you look at the printed curriculum it's the same as any standard calculus sequence in college. I went to Brooklyn Technical High School a elite math and science school in new york. We understood limits very well, we used mathematical manipulation of stuff like (sin(5x)/x) as the limit goes to zero to show that is equals 5 and stuff like that. The curriculum's are good but in most lower tier schools the teacher don't teach it. By introducing more rigor in the end the tests will be harder, but the student wouldn't learn much more because they wouldn't be taught it by lower tier schools. This would result in more failure rates just like what happened in the 60s with the new math.

 

[xdrgnh]

Hmm, I think the problem here is that we're falling into extremes here.
Sure, there are "math nazi's" out there who want to put in a ridiculous amount of rigor in high school courses, and sure, this is unnecessary. Seeing Dedekind cuts before analysis is rigorous, but it is certainly too much. (epsilon-delta's is something which can be given in high school however, it works in my country, so it could work everywhere).

However, the OP is the other extreme, I think. He is an "application nazi". That is, he just wants the people to be able to apply calculus, and not worry about the theory at all. This attitude is as bad as being a math nazi.

People who study calculus must study mathematics how it really is. I don't want to focus on the theory, but at least we should give proofs and show that this is all logically grounded and can be made extremely rigorous (if we want so). Not presenting the theory involved is actually lying to the students.

Of course we should give applications. In fact, the main focus should be in applications. But theory and proofs are necessary. I don't want a student come out of calculus say that he took it all on faith. If that happens, then the education failed.

The same thing happens with everything really. If I take a physics course, then it should be made clear to me that everything we do is verified by experiments. Fine, we can give physics without even mentioning the word experiments. Just show the fundamental laws and make exercises. But that would be lying. When teaching physics, the teacher should show the student the main methodology of physics: experiments. And when teaching math, we should give the methodolody of math: proofs.

OK, engineers will find the proofs useless in their later carreer. But at least we should give them some sort of "general culture". So that they don't graduate without knowing what math is really about. An engineer who can't do proofs isn't a bad engineer, but I still think that in that case, the education failed to show the engineer what math is really about.

Teaching a subject shouldn't just be giving all the techniques necessary. It should also try to give a broad picture of the field. If I TA courses, I always try to do this. Students don't always appreciate it, but I wouldn't feel good if I just let them make mind-numbing calculations...

Also, I find the comparison between creationism and proofs of an extreme bad taste. I think it's quite sad if you can't see the difference between these two examples...

Now by calling me a application nazi just mud slinging, I haven't called anyone on this thread a math nazi. I'm just saying the emphasis should be on problem solving not proofs, there needs to be proofs in classroom so that students know where it all comes from so they will remember it. There's a reason why colleges have a standard calculus class and a honors one, the honors one is more based on theory while the standard is based off of problem solving.

 

[micromass]

Now by calling me a application nazi just mud slinging, I haven't called anyone on this thread a math nazi.

I'm not trying to mud sling here. I just continue on the path you're going. You always have extremes. I am the math nazi extreme, you are the application nazi extreme. I don't mean to insult everybody with this terminology, I'm just saying how I view the situation. Being a nazi is (in this context) not a bad thing. We need both math nazis and application nazis to balance each other out.


I'm just saying the emphasis should be on problem solving not proofs, there needs to be proofs in classroom so that students know where it all comes from so they will remember it. There's a reason why colleges have a standard calculus class and a honors one, the honors one is more based on theory while the standard is based off of problem solving.[/QUOTE]

Uumm, since when are proofs not about problem solving?? With proofs you achieve that
- the student thinks about the underlying principles
- the student solves abstract problems and thinks abstractly
- the student presents his findings in a clear and logical way

You can't possibly be against that, can you??

Calculus should be about more then just mind-numbing calculating derivatives. Calculus should also teach students the methodology of mathematics and that math is founded on underlying axioms.

So you also think that in physics classes we should just teach that F=ma holds always without mentioning experiments?? Should we just teach that formula and solve problems with it, without even mentioning where the formula comes from??

Do you also think that in biology we should teach evolution as a fact without presenting the evidence or underlying principles??

Do you think that in chemistry we should just let students calculate the reactions without even seeing them in real life??

School has a responsibility. And that responsibility is not only learning people to work with stuff, but it also involves teaching the scientific method and teaching where things come from. In the same manner, proofs should be taught.

 

[xdrgnh]

I'm not trying to mud sling here. I just continue on the path you're going. You always have extremes. I am the math nazi extreme, you are the application nazi extreme. I don't mean to insult everybody with this terminology, I'm just saying how I view the situation. Being a nazi is (in this context) not a bad thing. We need both math nazis and application nazis to balance each other out.


I'm just saying the emphasis should be on problem solving not proofs, there needs to be proofs in classroom so that students know where it all comes from so they will remember it. There's a reason why colleges have a standard calculus class and a honors one, the honors one is more based on theory while the standard is based off of problem solving.

Uumm, since when are proofs not about problem solving?? With proofs you achieve that
- the student thinks about the underlying principles
- the student solves abstract problems and thinks abstractly
- the student presents his findings in a clear and logical way

You can't possibly be against that, can you??

Calculus should be about more then just mind-numbing calculating derivatives. Calculus should also teach students the methodology of mathematics and that math is founded on underlying axioms.

So you also think that in physics classes we should just teach that F=ma holds always without mentioning experiments?? Should we just teach that formula and solve problems with it, without even mentioning where the formula comes from??

Do you also think that in biology we should teach evolution as a fact without presenting the evidence or underlying principles??

Do you think that in chemistry we should just let students calculate the reactions without even seeing them in real life??

School has a responsibility. And that responsibility is not only learning people to work with stuff, but it also involves teaching the scientific method and teaching where things come from. In the same manner, proofs should be taught.[/QUOTE]

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.

 

[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 adress 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 adress 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.