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- Jan 17, 2013

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What do you think guys ?

- Thread starter ZaidAlyafey
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- Thread starter
- #1

- Jan 17, 2013

- 1,667

What do you think guys ?

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- Jan 26, 2012

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Well, I think it's extremely important to recognize that the vast majority of calculus students are engineers. They are not interested in proofs, and they likely never will be. I've found that engineers have little tricks and strategems to keep themselves out of any real serious theoretical trouble.

As for being able to construct proofs, students

As for constructing the real line before derivatives and integrals, I would say that the logical approach to teaching mathematics has some distinct disadvantages (over, say, a more intuitive approach): 1. Students come away from mathematics courses thinking that mathematicians think in definition-theorem-proof (DTP) format, and that this is the way mathematics is done. That is by no means the way mathematics is done. Mathematics requires imagination, intuition, trial-and-error, and lots of other ways of thinking than DTP format. Mind you, DTP is the standard in rigor, and must be taught, and it is part of the mathematician's toolbox. But it would be a significant mistake to think that's all there was to it. 2. It doesn't take into account where the students are. Most students do not naturally think that way, nor do they tend to learn best that way (with some exceptions, no doubt). If you are not speaking the same language as the students, you might as well be teaching to an empty room.

I was taught calculus initially using a more logical DTP format, and I can tell you: I found $\delta-\epsilon$ proofs extraordinarily difficult. The entire concept of the limit I found difficult. I had to take Calculus I and II three times, plus Multi-Variable Calculus, plus Complex Analysis, and then finally in Real Analysis senior year, I understood limits. Remember that these subtle concepts eluded Newton, Leibniz, and a number of other mathematicians up until the time of Cauchy, Weierstrass, and Riemann. And we're expecting students immediately to grasp these things in an introduction to calculus? Believe me, I'd like to think that the logical approach could work. I just have grave doubts.

- May 31, 2013

- 119

extraordinaryWell, I think it's extremely important to recognize that the vast majority of calculus students are engineers. They are not interested in proofs, and they likely never will be. I've found that engineers have little tricks and strategems to keep themselves out of any real serious theoretical trouble.

As for being able to construct proofs, studentsshouldbe able to do that out of high school.

As for constructing the real line before derivatives and integrals, I would say that the logical approach to teaching mathematics has some distinct disadvantages (over, say, a more intuitive approach): 1. Students come away from mathematics courses thinking that mathematicians think in definition-theorem-proof (DTP) format, and that this is the way mathematics is done. That is by no means the way mathematics is done. Mathematics requires imagination, intuition, trial-and-error, and lots of other ways of thinking than DTP format. Mind you, DTP is the standard in rigor, and must be taught, and it is part of the mathematician's toolbox. But it would be a significant mistake to think that's all there was to it. 2. It doesn't take into account where the students are. Most students do not naturally think that way, nor do they tend to learn best that way (with some exceptions, no doubt). If you are not speaking the same language as the students, you might as well be teaching to an empty room.

I was taught calculus initially using a more logical DTP format, and I can tell you: I found $\delta-\epsilon$ proofs extraordinarily difficult. The entire concept of the limit I found difficult. I had to take Calculus I and II three times, plus Multi-Variable Calculus, plus Complex Analysis, and then finally in Real Analysis senior year, I understood limits. Remember that these subtle concepts eluded Newton, Leibniz, and a number of other mathematicians up until the time of Cauchy, Weierstrass, and Riemann. And we're expecting students immediately to grasp these things in an introduction to calculus? Believe me, I'd like to think that the logical approach could work. I just have grave doubts.

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- Jan 17, 2013

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I very much liked the DTP format , especially in complex analysis. I don't know whether this suits other students or not because it is a self-study. I usually don't get convinced until I see the proof of a theorem and be able to understand it . Taking it for granted doesn't work for me . The pressure will surely be on instructors to introduce the idea the best so the majority of students will be able to digest the concept . It isn't necessary that they stick to the standard way , though.

When I first studied the epsilon and delta definition it made no sense for me . Because it was introduced as a step-by-step solution not focusing on the underlying concept. But , when I look at these things as general concept just like the limit point of an open set as a convergent sub-sequence , I more and more admire the concept .

I think you are right because you are not only looking at it as a student but also as an instructor.

- Feb 29, 2012

- 342

This is going to be rather long.

You have to consider that you probably have a very different background from those usually enrolled in calculus classes and certainly a very different interest. Engaging students is

What do you think guys ?

Let us not forget something else: it is not uncommon for us to work on further analysis unless something goes wrong first. Matter of fact, it so happened continuously in mathematics that people went on with things as they were until reaching the limits of the present theory, prompting further analysis of the concepts involved and what needed clarification.

You may disagree with me, but it seems that more often than not trying to be too careful on your problem analysis becomes a hindrance instead of productive.

How important is it to construct the real line? Do you think that using the axioms of the real line as an ordered field does not yield a firm foundation for other concepts? Nevertheless, I believe it only sheds light on those concepts once you've already mastered them. How can you appreciate abstraction and refinement if not aware of the difficulties inherent in the theory before they happened?

I agree, but the key word here isWell, I think it's extremely important to recognize that the vast majority of calculus students are engineers. They are not interested in proofs, and they likely never will be. I've found that engineers have little tricks and strategems to keep themselves out of any real serious theoretical trouble.

As for being able to construct proofs, studentsshouldbe able to do that out of high school.

As for constructing the real line before derivatives and integrals, I would say that the logical approach to teaching mathematics has some distinct disadvantages (over, say, a more intuitive approach): 1. Students come away from mathematics courses thinking that mathematicians think in definition-theorem-proof (DTP) format, and that this is the way mathematics is done. That is by no means the way mathematics is done. Mathematics requires imagination, intuition, trial-and-error, and lots of other ways of thinking than DTP format. Mind you, DTP is the standard in rigor, and must be taught, and it is part of the mathematician's toolbox. But it would be a significant mistake to think that's all there was to it. 2. It doesn't take into account where the students are. Most students do not naturally think that way, nor do they tend to learn best that way (with some exceptions, no doubt). If you are not speaking the same language as the students, you might as well be teaching to an empty room.

I was taught calculus initially using a more logical DTP format, and I can tell you: I found $\delta-\epsilon$ proofs extraordinarily difficult. The entire concept of the limit I found difficult. I had to take Calculus I and II three times, plus Multi-Variable Calculus, plus Complex Analysis, and then finally in Real Analysis senior year, I understood limits. Remember that these subtle concepts eluded Newton, Leibniz, and a number of other mathematicians up until the time of Cauchy, Weierstrass, and Riemann. And we're expecting students immediately to grasp these things in an introduction to calculus? Believe me, I'd like to think that the logical approach could work. I just have grave doubts.

Regarding the DTP format, as Gian-Carlo Rota says in his fabulous book Indiscrete Thoughts:

I do not believe I am exaggerating when I claim that everyone struggles with books in such format, lacking the motivation, explanations, insights or perspectives needed for the subject under study. One must not confuse the manner ofThe axiomatic method of presentation of mathematics has reached a pinnacle of fanaticism in our time. A piece of written mathematics cannot be understood and appreciated without additional strenuous effort. Clarity has been sacrificed to such hibboleths as consistency of notation, brevity of argument and the contrived linearity of inferential reasoning.Some mathematicians will go as far as to pretend that mathematics is the axiomatic method, neither more nor less.

The DTP format is worthwhile when you are already used to it and have some idea of where you are going. You apparently did not wander into complex analysis blindly, but had a goal to strive for. Would you not be convinced of Fermat's Last Theorem by trying out many cases or looking at the proofs of many simple ones? I am certain you are going to have some trouble just trying to understand the prerequisites of Wiles's proof. Case in point: Jordan curve theorem. Do you need to see the proof to be certain of the validity of the statement?I very much liked the DTP format , especially in complex analysis. I don't know whether this suits other students or not because it is a self-study. I usually don't get convinced until I see the proof of a theorem and be able to understand it . Taking it for granted doesn't work for me . The pressure will surely be on instructors to introduce the idea the best so the majority of students will be able to digest the concept . It isn't necessary that they stick to the standard way , though.

When I first studied the epsilon and delta definition it made no sense for me . Because it was introduced as a step-by-step solution not focusing on the underlying concept. But , when I look at these things as general concept just like the limit point of an open set as a convergent sub-sequence , I more and more admire the concept .

I think you are right because you are not only looking at it as a student but also as an instructor.

It is easy to say that you did not understand the epsilon-delta definition with the benefit of hindsight and maturity. I did not get it as I first studied calculus and even with real analysis I struggled a bit. Today, I see it is not a big deal as I thought. However, I know how much time I needed before it sunk in. John Von Neumann said:

Even as I study something, I do not in general enjoy dry books. I prefer the ones that show connections with other areas and interrelates concepts (this explains my recent interest in russian mathematics). Not many go that route, partially because requires great effort to present the "big picture" and commonly authors are more interested in covering ground, possibly intending for the book to serve both as textbook and reference for researchers. I firmly believe everybody gets benefited with production of texts focused on the reader.In mathematics you don't understand things. You just get used to them.

Whew, that's it for now.