Analysis vs Calculus: Learn limits,integration,proofs for Math Majors

[alyafey22]

For a person who is majoring in any field it is quite important to learn limits , integration , differentiation and sequences . I'll say we discussed all these notions as parts of the Calculus book in the Freshman year . But , I didn't like the way it's introduced maybe because I am more like into analysis more than pure computations . We only focused on the computational parts , rarely did we discuss any proofs of any theorems . I think this is quite understandable for a person who is not majoring in mathematics , but I am quite opposed to this idea for those who are searching a career in Mathematics. It is quite important to first construct the real line from scratch before even talking about integration and limits . Discussing limits as a part of analysis teaches you how to work a little bit more carefully before taking any further progress . When you study calculus you become careless about why it should work and you focus on how I should work the problem. That's why some students may face problems constructing formal proofs because they are not used to .

What do you think guys ?

 

[Ackbach]

Re: Calculus or analysis

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 should be 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.

 

[mathworker]

Re: Calculus or analysis

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 should be 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.

extraordinary(Bow)

 

[alyafey22]

Re: Calculus or analysis

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.

 

[Fantini]

Re: Calculus or analysis

This is going to be rather long. (Sweating)

For a person who is majoring in any field it is quite important to learn limits , integration , differentiation and sequences . I'll say we discussed all these notions as parts of the Calculus book in the Freshman year . But , I didn't like the way it's introduced maybe because I am more like into analysis more than pure computations . We only focused on the computational parts , rarely did we discuss any proofs of any theorems . I think this is quite understandable for a person who is not majoring in mathematics , but I am quite opposed to this idea for those who are searching a career in Mathematics. It is quite important to first construct the real line from scratch before even talking about integration and limits . Discussing limits as a part of analysis teaches you how to work a little bit more carefully before taking any further progress . When you study calculus you become careless about why it should work and you focus on how I should work the problem. That's why some students may face problems constructing formal proofs because they are not used to .

What do you think guys ?

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 hard. Even if you are capable of such achievement, there is another important question: should those computations be neglected? I say not. In mathematics it is very easy for us to lose sight of what we are doing in an immense sea of abstraction. I believe that firm knowledge rests upon being computationally able and theoretically sound.

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?

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 should be 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.

I agree, but the key word here is should. Most, even if exposed to proofs and moderately forced to attempt them, will likely not realize the importance nor the need for them. This creates a block right from the start.

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

The 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.

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 of presentation of mathematics with the way it is understood, thought and practiced. Else, why bother with all that fluff text? Just write out all the definitions, theorems and proofs and get done with it.

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.

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?

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:

In mathematics you don't understand things. You just get used to them.

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. Difficulty does not breed excellence, particularly if needlessly complicated.

Whew, that's it for now. (Whew)