Math Elitists: A Controversial Opinion

[micromass]

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.

Indeed, and the tools are easiest to learn with a good deal of theory. How can one ever understand calculus without understanding the theory?? We don't want people to become like brainless calculating monkeys do we?

 

[xdrgnh]

I guess this is the difference between the Belgium system and the American system, in the American one the classes are separate while in Belgium they are together. Again I'm not saying take out the theory I am just saying don't make the main emphasis theory for a non honors calculus class.

 

[micromass]

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.

OK, fine, lipgarbagez continuity isn't the most useful concept around (although it's just saying that derivatives, if they exist, are bounded...). But I still think it's useful enough to include in a calculus class. It gets you thinking about the slope and about how fast the function changes.

I think it's quite ok for a calculus class to use such a advanced definitions. That way, students learn to deal with abstract things.

 

[micromass]

I guess this is the difference between the Belgium system and the American system, in the American one the classes are separate while in Belgium they are together. Again I'm not saying take out the theory I am just saying don't make the main emphasis theory for a non honors calculus class.

You keep saying that, but you always fail to say why you think that.
Obviously, applications should get attention, but there are ways to teach both applications and theory. If it can happen here in my country, it can happen everywhere. There's no real reason to separate theory and applications, other then to dumb down the course...

 

[xdrgnh]

I'm not a educator so I don't know why to do that. All I know is that the top schools in the world Harvard, MIT Cambridge have a theoretical path and a applied path in there math and physics studies. The reason is so that not all engineering and science students need or want that kind of rigor in there math class. It works very well in America if you look at every statistic for colleges.

 

[micromass]

I'm not a educator so I don't know why to do that. All I know is that the top schools in the world Harvard, MIT Cambridge have a theoretical path and a applied path in there math and physics studies. The reason is so that not all engineering and science students need or want that kind of rigor in there math class. It works very well in America if you look at every statistic for colleges.

Well, yeah, of course you should have a theoretical path and an applied path. But that doesn't mean that we don't have to confront the applied people with proofs. And that also doesn't mean that we don't have to confront the pure people with applications. Both are very much needed.

 

[thegreenlaser]

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 understand how the second sentence at all follows from the first. Since when was AP calculus designed to be a standard course that all students take? AP courses are supposed to be for honors students, that's why they're called Advanced Placement.

 

[xdrgnh]

Well, yeah, of course you should have a theoretical path and an applied path. But that doesn't mean that we don't have to confront the applied people with proofs. And that also doesn't mean that we don't have to confront the pure people with applications. Both are very much needed.

I'm not saying don't confront them with proofs I'm saying the main emphasis shouldn't be on proofs. Do they have applied math or physics classes in Belgium, in those classes you see very little proofs and it's mostly applied work. Proofs are needed to understand the material but shouldn't be the main emphasis especially when the people in that class just want to know the math so they can build a bridge or airplane.

 

[micromass]

I'm not saying don't confront them with proofs I'm saying the main emphasis shouldn't be on proofs. Do they have applied math or physics classes in Belgium, in those classes you see very little proofs and it's mostly applied work. Proofs are needed to understand the material but shouldn't be the main emphasis especially when the people in that class just want to know the math so they can build a bridge or airplane.

Again: teaching proofs is not mutual exclusive with teaching applications.

 

[xdrgnh]

I never said that, I just said it shouldn't be the main emphasis of a non honors math class.

 

[micromass]

I never said that, I just said it shouldn't be the main emphasis of a non honors math class.

You can have multiple main emphasises...

 

[xdrgnh]

Doesn't work like that in real life, you say it works in Belgium but in America where it is different our system works better. It's not a bad way of doing things but seperating the classes has just proved to be better.

 

[micromass]

Doesn't work like that in real life, you say it works in Belgium but in America where it is different our system works better. It's not a bad way of doing things but seperating the classes has just proved to be better.

Please provide factual evidence for this claim. (and no I won't accept that site that you linked earlier, for reasons I already explained).
Please provide evidence that Belgian engineers are worse than american engineers.

 

[xdrgnh]

Worse is a immature word really to describe two engineers that went to very reputable universities. You can't really compare them because they are both trained differently. The only proof I can show you is more statistics and if you don't accept that then I don't know much else. I can say though that America has more Nobel prize winners then Belgium.

 

[micromass]

Worse is a immature word really to describe two engineers that went to very reputable universities. You can't really compare them because they are both trained differently.
The only proof I can show you is more statistics and if you don't accept that then I don't know much else.

I accept statistics. But I might not accept the criteria which decide the quality of universities. Number of publictions is not a criteria on which we can decide that one university is better, for example.
So, find me a study with decent criteria.

I can say though that America has more Nobel prize winners then Belgium.

Yeah... Do I really need to answer this? Compare the population number...

 

[thegreenlaser]

Worse is a immature word really to describe two engineers that went to very reputable universities. You can't really compare them because they are both trained differently. The only proof I can show you is more statistics and if you don't accept that then I don't know much else. I can say though that America has more Nobel prize winners then Belgium.

Why are you calling him immature for using the word 'worse'? You were the one who said American universities are better, but apparently it's immature for him to ask why Belgium is worse? It's a perfectly fair question, given your pretty much baseless statement.

By the way, per capita there's actually very little difference between Belgium and the US as far as Nobel Prizes go.

 

[xdrgnh]

I accept statistics. But I might not accept the criteria which decide the quality of universities. Number of publictions is not a criteria on which we can decide that one university is better, for example.
So, find me a study with decent criteria.



Yeah... Do I really need to answer this? Compare the population number...

This will make you sad but Belgium only has two in science one in chemistry and one in medicine. So yah... that shows the difference between Belgium education and American. Also why do more foreign student study in America then in Belgium? The reason is because globally American schools have a better reputation.

 

[xdrgnh]

Why are you calling him immature for using the word 'worse'? You were the one who said American universities are better, but apparently it's immature for him to ask why Belgium is worse? It's a perfectly fair question, given your pretty much baseless statement.

By the way, per capita there's actually very little difference between Belgium and the US as far as Nobel Prizes go.

I called him immature really?

 

[thegreenlaser]

I called him immature really?

Yes, really.

 

[xdrgnh]

Somebody is either trolling or using the strawman argument, I said the word is I made no comment on him.

 

[micromass]

This will make you sad but Belgium only has two in science one in chemistry and one in medicine. So yah... that shows the difference between Belgium education and American. Also why do more foreign student study in America then in Belgium? The reason is because globally American schools have a better reputation.

http://www.nationmaster.com/graph/peo_nob_pri_lau_percap-nobel-prize-laureates-per-capita

Anyway, you start this thread saying that we shouldn't be elitist about things. And now it seems that you're the elitist one who claim that american schools are soo much better. Anyway, my link shows that Iceland and Sweden are the best Nobel prize countries. So maybe we should ask them how they do their education?

 

[micromass]

Somebody is either trolling or using the strawman argument, I said the word is I made no comment on him.

Well:

Worse is a immature word really

That's indirectly calling me immature... He has a valid point.

 

[thegreenlaser]

Somebody is either trolling or using the strawman argument, I said the word is I made no comment on him.

Oh come, now... "Worse is a immature word really to describe two engineers that went to very reputable universities."

Either you were calling him immature, or you were saying that comparing engineers from Belgium and the US is immature. The problem with the second option is that he was just restating what you said, which would mean you were calling yourself immature. That's why I assumed the first option.

 

[micromass]

I also found this striking:

http://www.nationmaster.com/graph/edu_mat_lit-education-mathematical-literacy

 

[Pyrrhus]

This thread is ridiculous. Anyone that is willing to go into a quantitative field (this includes physics, engineering fields, and other sciences) SHOULD NOT be afraid of furthering his or hers knowledge in mathematics. This is especially important if such an individual is thinking of career in research.

The learning of mathematics necessarily must be a progression, and this learning should not only be related to theoretical knowledge (Definitions, axioms, proofs), but also applied knowledge such as (mathematical methods used in various fields, and discussion of their existing models). An example of this progression is when you first learn statistical theory. You encounter Set theoretic options, and then it moves to calculus-based concepts. This is fine for a first course, where every random variable, and distributions you meet are "nice" (as in Riemann Integrable), and many advanced topics are relegated to advanced courses. A second course should be more close to the measure-theoretic instruction of the same previous familiar concepts in a new light (e.g. Riemann Integral vs. Lebesque Integral), and also new concepts such as characteristic functions.

The importance of applications of mathematics is everywhere. Examples include the Euler-Bernoulli Beam Theory (Engineering), Consumer Choice (Economics), Classical Mechanics (Physics), Cells Motility (Biology), and many other fields.

I believe the idea of cutting important elements in the instruction of mathematics is a dangerous notion. For two main reasons: It increase the already existing inequality due to heterogeneity (Students enrolling to the honors courses will be more prepared than those enrolling to the OP suggested courses); and it leads students terribly unprepared to resolve problems outside of the university environment (it may seem perfectly fine to fail a couple of problems in college, but how are you going to let your employer know that you cannot do the work you are expected?, because you don't know the math or even worse use it incorrectly). It is true that computer software have abstracted the most used methods, but still the human intuition and ingenuity is required to make sense of the results, and also to develop alternatives when the most used methods are no longer applicable.The bottom line is Mathematics is pervasive and entrenched in quantitative fields. Thus, students must either enjoy, accept (i.e. you should not necessarily become a mathematician, if you don't want to) or choose a non-quantitative field.

 

[thegreenlaser]

...

Some very good points. Knowing how to follow a cookie-cutter problem solving method is great... until you hit a problem you've never had before, and you find that the cookie cutter method doesn't work. Then, if you know your theory, you have a much better chance of being able to figure out exactly what part of the cookie-cutter method doesn't work and adjust the method appropriately. If you're an expert at using the cookie-cutter, but you have no idea why it works, then when you hit problems like that you're pretty much stuck explaining to your boss that you can't figure out how to solve the problem.

 

[deRham]

This is getting ridiculous. I haven't looked at this thread for some time and ...

Doesn't work like that in real life, you say it works in Belgium but in America where it is different our system works better. It's not a bad way of doing things but seperating the classes has just proved to be better.

To the one who began this thread: no, American schools are not in general superior to those who educate students with a mixture of proofs and applications. Who the heck said that? Arguably the best undergrad schools are in Europe. France, the UK, etc are filled with absolutely top class, insane mathematics.

I've written many posts supporting aspects of what you're suggesting, but I don't understand this sudden justification via the supposed superiority of US schools.

Bottom line is that people need to be able to communicate their reasoning clearly in engineering too. Epsilon-delta stuff can be minimized (really, I don't see the mania about this, because even in rigorous mathematics, you develop a lot of calculus after you just set up the basics using epsilon-delta type stuff). In practice, you only prove a few things using the basic principles.

The clamoring to use L'Hopital's rule is not just from the engineer, it's from the mathematician too - who the heck wants to compute out limits using first principles only, except as an illustrative exercise? We prove theorems for a reason - so they reduce our burden later, and illustrate the depth of the theory.

Blind calculation is not what engineering is about either. You just don't really care about proving the existence of the Riemann integral. Proving that it has certain properties can be a useful tool, however, because that involves manipulating the basic properties, and that actually CAN be useful in engineering derivations.

I think there's a great medium between a real analysis course and a totally computational calculus course, and that's what is needed.

I do NOT favor a totally computational course, because it leaves out the meaning of things you're tossing around. The meaning is important, though all the details are not crucial.

 

[Bourbaki1123]

This will make you sad but Belgium only has two in science one in chemistry and one in medicine. So yah... that shows the difference between Belgium education and American. Also why do more foreign student study in America then in Belgium? The reason is because globally American schools have a better reputation.

Totally illogical, and factually incorrect. Belgium has four in medicine, try to at least get the number right if you're going to argue the point.

Now: First correct for all prize winners who emigrated to the USA in their late teens or later, then all of those who attended private school (since we're presumably talking about public education). Next, take that number and divide it by 30 (USA has 30 times as many people overall, so you have to normalize). That is your number to compare to 5. I can tell you straight off the bat that even before correcting for the first two parameters, you come up with about 10 people.

Top American undergraduate institutions (incidentally, given the whining about AP courses I thought we were talking about high school) have a great reputation, and they also often teach rigorous calculus. Furthermore, I can say with a reasonable amount of confidence that the people who went on to get a Nobel were usually not the ones who were complaining about having to do epsilon-delta proofs.

 

[Hurkyl]

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.

I do get the idea -- but that's because I've already learned what it means to be continuous, so I can mentally substitute the elements of your analogy with things that are factually correct.

Someone who has not yet understood "continuous", on the other hand, cannot do so.


Even worse, people can be misled into thinking they understand "continuous" by hearing such a 'definition', when in fact they are pretty much incapable of actually using the notion for any purpose.

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.

Now this, boggled by. (and that's even if I ignore your change of focus)

"Piecewise smooth" (or even "piecewise analytic") is not an esoteric topic from real analysis -- it is the kind of function that most people use almost exclusively. Not only is it needed for second semester calculus (e.g. Taylor series), but your first and second year physics classes will be assuming you can take derivatives with impunity (except possibly at a handful of points), and possibly even using infinite series to express things.

Quite frankly, it is "continuous" that is the esoteric topic from analysis / topology. I believe the main reason the notion is introduced in elementary calculus is simply because people don't know of a better way to introduce the calculus of piecewise smooth functions.

 

[Manchot]

Electrical engineering grad student here. I've had a fair amount of Analysis and personally proved the fundamentals of undergraduate calculus starting from ZFC, but I can sympathize with the OP's point-of-view. If you do experimental work, rigor must eventually be sacrificed if you want to get anything done, because there simply aren't enough hours in the day to prove everything from first principles. I spent my first year of grad school holding myself to standards of rigor that were too high, and my research suffered as a result.

Having said that, I never really felt comfortable dealing with real numbers before I constructed them myself, and was even somewhat uncomfortable doing epsilon-delta proofs because I didn't know what could really be justified. I think there's a middle ground that differs from person-to-person, depending on their field and interests. IMO, my undergraduate department didn't place enough emphasis on rigor, and I would have benefited from a required introductory Analysis course.