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mathwonk

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I agree with you 100% about the constructive proofs being more persuasive. It seems there are 2 ways of doing calculus rigorously, either assuming the reals are represented by infinite decimals, or just assuming they form an ordered field with the least upper bound property. This second approach is much more abstract but is apparently favored by the experts as being cleaner and slicker for giving proofs. We ordinary people however who like to see real numbers concretely are more persuaded by the actual process of constructing a solution to our problem. The contrast is between exhibiting a concrete reasonably familiar model for the real numbers, as opposed to just stating axioms they should satisfy, without convincing us these axioms are reasonable. I.e. the abstract approach does not even consider the question of whether the reals actually exist. The tedious method of Dedekind cuts to construct them (as in Rudin) is also quite painful in my opinion. This is the contrast betwen representing a number c either in terms of the decimal expansion for c, or in terms of the entire (rational) part of the real axis lying to the left of c!

The most popular books for rigorous calculus, Spivak and Apostol, (as well as the less available but excellent books by Lang, Analysis I, and by Kitchen, Calculus of one variable), do things the abstract axiomatic way, and so did my intro college course, but the remarkable book by Courant does things using infinite decimals. Fortunately Courant was recommended reading for my course. To be sure Courant quickly ramps up and uses infinite decimals to prove an abstract property, the principle of the point of accumulation (every infinite subset of a closed bounded set must accumulate, or bunch up, about at least one point). Even this principle however is more concrete to me than a bare axiom, especially since it can be easily proved by subdividing intervals as in the proofs above.

It seems we are mostly given two types of presentations nowadays, either no proof at all of basic facts about limits in calculus, or abstract proofs based on axioms for the reals. I think many more people could be introduced to rigorous calculus if the constructive infinite decimal approach were more common.

I wonder what the common core approach to calculus is?

Edit: Well I just read the calculus chapter of the 2013 California standards for high school math. The first sentence is :

"When taught in high school, calculus should be presented with the same level of depth and rigor as are entry-level college and university calculus courses."

I think this is neither entirely meaningful, nor convincing. I.e. entry level (i.e. freshman) calculus courses throughout the country vary from completely intuitive ones to math 55 at Harvard, the "hardest course in America", apparently taken mainly by people who have already been Olympiad fellows. Moreover the whole advantage to taking a hard subject early is to have an easier, slower, introduction to it that will help later when a more rigorous version is encountered. In spite of the language in this chapter, my experience is that high schools tend to spend roughly twice as much time on an intro calc course as does a college course, even a non honors one. So do they mean "at the same depth", but twice as slowly?

The description given suggests the real model for the CA course is the AP test. To be fair, it also matches closely what is usually offered to non honors students in most non elite colleges, namely a course that states but does not prove the fundamental theorems underlying the calculus, the intermediate value and extreme value theorems, which we have just seen can be rigorously proved in a way that even high schoolers could easily grasp.

One thing which is taken for granted, but seems to me questionable, is the idea that a calculus course can be given at the same level as a college course when the high school teacher giving it has usually nowhere near the same training in math as a college teacher. I.e. even in top ranked private high schools of my acquaintance, calculus could be taught by anyone who had taken calculus in college, whereas in college it is often taught either by a graduate PhD student or, frequently, a professor with mathematical research experience. A quick look at my vita shows over 40 first year calculus courses taught in the 30 years after receiving the PhD, and it is incomplete.

The most popular books for rigorous calculus, Spivak and Apostol, (as well as the less available but excellent books by Lang, Analysis I, and by Kitchen, Calculus of one variable), do things the abstract axiomatic way, and so did my intro college course, but the remarkable book by Courant does things using infinite decimals. Fortunately Courant was recommended reading for my course. To be sure Courant quickly ramps up and uses infinite decimals to prove an abstract property, the principle of the point of accumulation (every infinite subset of a closed bounded set must accumulate, or bunch up, about at least one point). Even this principle however is more concrete to me than a bare axiom, especially since it can be easily proved by subdividing intervals as in the proofs above.

It seems we are mostly given two types of presentations nowadays, either no proof at all of basic facts about limits in calculus, or abstract proofs based on axioms for the reals. I think many more people could be introduced to rigorous calculus if the constructive infinite decimal approach were more common.

I wonder what the common core approach to calculus is?

Edit: Well I just read the calculus chapter of the 2013 California standards for high school math. The first sentence is :

"When taught in high school, calculus should be presented with the same level of depth and rigor as are entry-level college and university calculus courses."

I think this is neither entirely meaningful, nor convincing. I.e. entry level (i.e. freshman) calculus courses throughout the country vary from completely intuitive ones to math 55 at Harvard, the "hardest course in America", apparently taken mainly by people who have already been Olympiad fellows. Moreover the whole advantage to taking a hard subject early is to have an easier, slower, introduction to it that will help later when a more rigorous version is encountered. In spite of the language in this chapter, my experience is that high schools tend to spend roughly twice as much time on an intro calc course as does a college course, even a non honors one. So do they mean "at the same depth", but twice as slowly?

The description given suggests the real model for the CA course is the AP test. To be fair, it also matches closely what is usually offered to non honors students in most non elite colleges, namely a course that states but does not prove the fundamental theorems underlying the calculus, the intermediate value and extreme value theorems, which we have just seen can be rigorously proved in a way that even high schoolers could easily grasp.

One thing which is taken for granted, but seems to me questionable, is the idea that a calculus course can be given at the same level as a college course when the high school teacher giving it has usually nowhere near the same training in math as a college teacher. I.e. even in top ranked private high schools of my acquaintance, calculus could be taught by anyone who had taken calculus in college, whereas in college it is often taught either by a graduate PhD student or, frequently, a professor with mathematical research experience. A quick look at my vita shows over 40 first year calculus courses taught in the 30 years after receiving the PhD, and it is incomplete.

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