Timeline for Assessing effectiveness of (epsilon, delta) definitions [closed]
Current License: CC BY-SA 3.0
54 events
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| Jul 26, 2019 at 16:05 | comment | added | benblumsmith | I have posted an answer arguing that the disagreement is illusory at matheducators.SE matheducators.stackexchange.com/a/16857/140 | |
| Jul 26, 2019 at 15:21 | comment | added | benblumsmith | @MikhailKatz - Putting aside the question of representativeness, I do not understand how you are reading Bishop as asserting that epsilon, delta is common sense before one masters the technique. The quote given in the OP is preceded by "Although it seems to be futile..." and followed with "They do not believe me." It seems clear even Bishop does not think students take to the definition easily or naturally. | |
| Dec 6, 2017 at 10:48 | history | edited | Mikhail Katz | CC BY-SA 3.0 |
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| Dec 5, 2017 at 9:58 | history | edited | Mikhail Katz | CC BY-SA 3.0 |
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| Dec 4, 2017 at 18:17 | review | Reopen votes | |||
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| Dec 4, 2017 at 12:15 | history | edited | Mikhail Katz | CC BY-SA 3.0 |
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| Dec 4, 2017 at 10:00 | history | edited | Mikhail Katz | CC BY-SA 3.0 |
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| Dec 3, 2017 at 11:49 | history | closed |
Joseph Van Name Ben McKay Jan-Christoph Schlage-Puchta David Handelman András Bátkai |
Not suitable for this site | |
| Nov 29, 2017 at 18:18 | review | Close votes | |||
| Dec 3, 2017 at 11:49 | |||||
| Jul 17, 2017 at 3:14 | answer | added | Pablo Lessa | timeline score: 6 | |
| Jul 16, 2017 at 22:22 | answer | added | Aryeh Kontorovich | timeline score: 8 | |
| Jul 16, 2017 at 16:44 | comment | added | Deane Yang | It's pretty clear that the key difficulty lies in the use of two quantifiers. I think when we teach calculus, we usually do not verify that students have a clear understanding of each quantifier alone. So when we string two of them together, they are totally lost. What students need is a strong grasp of the deductive logic involved. Unfortunately, I do not know how to teach this effectively. | |
| Jul 16, 2017 at 9:13 | history | edited | Mikhail Katz | CC BY-SA 3.0 |
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| Jul 11, 2017 at 22:51 | comment | added | Steven Landsburg | @MikhailKatz: yes, that's a fair point. | |
| Jul 11, 2017 at 9:21 | comment | added | Mikhail Katz | @StevenLandsburg, I agree with you that this material serves as a useful filter to weed out the weak students. However I think the material remains challenging even based on Keisler or related approaches using infinitesimals, so the role of a useful filter remains and is somewhat irrelevant to this question. | |
| Jul 10, 2017 at 20:52 | review | Close votes | |||
| Jul 11, 2017 at 7:15 | |||||
| Jul 10, 2017 at 18:19 | history | reopened |
Mikhail Katz user21574 R.P. paul garrett Paul Taylor |
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| Jul 10, 2017 at 15:06 | review | Reopen votes | |||
| Jul 10, 2017 at 17:43 | |||||
| Jul 10, 2017 at 14:57 | history | edited | Mikhail Katz | CC BY-SA 3.0 |
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| Feb 20, 2016 at 21:18 | review | Reopen votes | |||
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| May 1, 2014 at 17:54 | comment | added | Brian Rushton | @katz You should consider re-asking this question at matheducators.stackexchange.com | |
| Mar 8, 2014 at 3:04 | review | Reopen votes | |||
| Mar 8, 2014 at 10:23 | |||||
| Mar 5, 2014 at 15:05 | history | edited | Mikhail Katz | CC BY-SA 3.0 |
provided clarification requested by a fellow editor
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| Mar 4, 2014 at 16:04 | comment | added | François G. Dorais | The point is that the reason why Bishop calls the epsilon-delta definition "common sense" is because of his constructivist agenda. It is misleading to use this quote without this context and it is inappropriate to cite him as representative of mathematicians in general. | |
| Mar 4, 2014 at 15:41 | comment | added | Mikhail Katz | @François, I agree with your comments on Russell. My point here was to respond to a request by a fellow editor to provide citations from mathematicians, which I did by citing Kleinfeld, Margaret (who happens to quote Russell, but this is less relevant). I don't think Bishop's thinking about "common sense" is atypical in the math community. Can you cite an education specialist that thinks that epsilon, delta are "common sense" or "simple" (that is, before one masters the technique)? | |
| Mar 4, 2014 at 14:45 | comment | added | François G. Dorais | Russell and Bishop are both lacking much context. Since they both attempted to provide some kind of rigorous foundation for mathematics, it's natural that they would talk this way about a rigorous definition. It is also clear that they are not representative of mathematicians in general. Additionally, I doubt Russell ever taught anything one would recognize as calculus. Bishop did teach calculus but that is not the point of view he is representing in his review of Keisler's book. | |
| Mar 4, 2014 at 13:51 | history | edited | Mikhail Katz | CC BY-SA 3.0 |
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| Mar 3, 2014 at 20:00 | comment | added | user44143 | The first two sentences of the quote from Kleinfeld are a good example. The reference to Russell and the quote from Bishop both show the response of mathematicians rather than the response of mathematics students. | |
| Mar 3, 2014 at 18:57 | comment | added | Steven Gubkin | @katz I do not think that there is a difference between the two communities. These examples are cherry picked. I am sure that if you polled mathematicians with the question "Is it easy for most freshman students to understand $\epsilon-\delta$ definition of a limit", you would find that overwhelmingly they would say no. They might not have given a great deal of thought as to why, but they would say no. In my responses above I am merely pointing out that it becomes less difficult if you devote a significant amount of time to it, and this time is justified by the necessity of error analysis. | |
| Mar 3, 2014 at 18:03 | comment | added | Mikhail Katz | @StevenGubkin, my question was not to determine whether these definitions are difficult or easy, but to point out a difference in perception in the two communities. I provided some examples to illustrate the point. I certainly agree with you that epsilon, delta definitions are an essential part of the subject and should not be avoided (I can't see how anybody can disagree with that!). | |
| Mar 3, 2014 at 17:52 | comment | added | Steven Gubkin | It is also an opportunity to give students the exposure to these difficult modes of thought. Just because something is hard is not an excuse to avoid it! Things which are valuable are often difficult. | |
| Mar 3, 2014 at 17:50 | comment | added | Steven Gubkin | @katz: I am not saying that $\epsilon-\delta$ analysis is easy. It takes special training to understand statements with that many quantifiers. My point is that it is that $\epsilon-\delta$ should not be taught simply as a way to "back up" intuition with "rigorous proofs", but that in fact it is very important to develop an understanding of error analysis. As a practical skill, this is probably more important to the engineer or scientist than manipulating mathematical symbols formally: symbolic calculators can integrate or differentiate a lot better than humans can. | |
| Mar 3, 2014 at 17:41 | comment | added | Mikhail Katz | @Paul, I provided some examples. | |
| Mar 3, 2014 at 17:40 | history | edited | Mikhail Katz | CC BY-SA 3.0 |
provided examples requested by fellow editors
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| Feb 28, 2014 at 13:16 | comment | added | Yemon Choi | I agree whole-heartedly with @PaulSiegel and Neil Strickland, and find this yet another example of the OP extrapolating anecdote or personal experience into universal truth. Are we taking the USA as representative of all mathematics education here? | |
| Feb 28, 2014 at 13:05 | comment | added | Paul Siegel | A mathoverflow comment together with five upvotes does not convince me of the existence of a consensus in the mathematical community, particularly when my personal experience is that most mathematicians recognize students' difficulties in real analysis. I would consider a vote to re-open the question if you could justify the phrase "...upbeat assessment common in the mathematics community..." as well as you justified "...somber assessments common in the education community..." Otherwise it is hard to imagine an answer based on something other than personal opinion. | |
| Feb 28, 2014 at 11:17 | review | Reopen votes | |||
| Feb 28, 2014 at 13:56 | |||||
| Feb 28, 2014 at 11:00 | history | edited | Mikhail Katz | CC BY-SA 3.0 |
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| Feb 27, 2014 at 21:59 | history | closed |
Gerald Edgar Boris Bukh Stefan Kohl♦ Ben Webster♦ |
Opinion-based | |
| Feb 27, 2014 at 19:31 | comment | added | Mikhail Katz | @NeilStrickland, a good example of such an "upbeat assessment" is the comment by Steven Gubkin above. You just have to "properly motivate" it and the difficulties magically "disappear". Gubkin (and many others who have expressed similar sentiments) may well be right about this. However, the question still remains why there is such a different perception in the education community and the mathematical community. Could it be that in the education community they just have not caught on to the fact that definitions have to be "properly motivated"? I note that Gubkin's comment got 5 uparrows so far | |
| Feb 27, 2014 at 19:10 | review | Close votes | |||
| Feb 27, 2014 at 21:59 | |||||
| Feb 27, 2014 at 18:53 | comment | added | Gerald Edgar | All answers (so far) are opinions, leading to the suggestion to close the question... | |
| Feb 27, 2014 at 17:58 | answer | added | user44143 | timeline score: 9 | |
| Feb 27, 2014 at 17:38 | comment | added | Neil Strickland | @katz: what is this "upbeat assessment" of which you speak? I don't think I have heard many people deny that average students find rigorous analysis hard, and that current teaching methods do not succeed in making many of them understand it. People may think that rigorous analysis is valuable despite this, and that certain other proposed teaching methods are no better, but those are different questions. | |
| Feb 27, 2014 at 16:34 | answer | added | Lev Borisov | timeline score: 5 | |
| Feb 27, 2014 at 13:55 | answer | added | Jake Shreffler | timeline score: 7 | |
| Feb 20, 2014 at 18:29 | comment | added | Steven Gubkin | I think many of the difficulties with $\epsilon-\delta$ or $\epsilon-N$ analysis disappear when they are properly motivated. How many terms do you need in the Riemann sum to get an approximation with a maximum error of $0.001$? This is question is not esoteric at all: it is essential if you are going to numerically integrate anything in the real world. | |
| Feb 20, 2014 at 18:12 | comment | added | Mikhail Katz | @Steve, then you appear to agree with the education community about the difficulty of the material. If so the question remains to understand why the mathematics community often views it differently (a number of recent discussions at MO testify to this). | |
| Feb 20, 2014 at 17:52 | comment | added | Steven Landsburg | @katz: Obviously this is not the place for a continued discussion, but for the record I was entirely serious and I am baffled as to why you thought otherwise. | |
| Feb 20, 2014 at 17:13 | comment | added | Mikhail Katz | @BenMcKay, please, I don't want this closed as "subjective and argumentative". I think Steve's comment was tongue-in-cheek. I am sure he is a fine pedagogue. | |
| Feb 20, 2014 at 17:12 | comment | added | Ben McKay | @StevenLandsburg: don't ask me for a teaching reference letter. | |
| Feb 20, 2014 at 16:53 | comment | added | arsmath | Katz I feel like many of your questions are thinly disguised attempts to push an agenda about infinitesimals, which I don't think is really in the spirit of the site. | |
| Feb 20, 2014 at 16:13 | comment | added | Steven Landsburg | The fact that this circle of ideas can be difficult and discouraging is one of its good features, insofar as it aids in the early identification and weeding-out of students who don't have an aptitude for math. | |
| Feb 20, 2014 at 16:04 | history | asked | Mikhail Katz | CC BY-SA 3.0 |