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In some cases the historical development of a mathematical subject/tool is not straightforward. Mathematicians define a particular notion and work in an accepted direction. After a while they come across some problems/complexities and refine/redefine their methods/tools/notions in a completely new way and so on.

Texts are not based on historical development of the mathematical subjects necessarily because the recent refined approaches are more regular and well-designed but these approaches don't reflect the original motivations and essential problems of the field properly. These roots are very useful for introducing subjects to students in a "natural" way.

Question. Which one should I choose when there is a contradiction between historical and official development processes of a particular subject? What are the advantages and disadvantages of each method? Is it possible to have an effective combination of these two methods?

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There is always a significant difference between the historical development of a subject and the logical, mathematical reasoning taught in the classroom. The historical development involved lots of twists and turns, dead-ends and loops, occurring possibly over centuries. It is rarely practical to teach a topic (to first-time learners) in a historically valid fashion. Take advantage of the hindsight and use limits in calculus, for example. (If you are driving to the west coast for the first time, take the interstate!)

At the same time, the precise sterile logic of the topic hides the excitement of mathematical exploration. So if you get a chance, digress for a short time to describe the history (Newton's fluxions and fluents!) and show that it was not straightforward. (Pull off the interstate and drive a few miles on route 66 before going back onto the interstate.)

I have little historical vignettes I stick into my lectures to try and add a human side to the topic. But the topic itself should be taught as logically as possible, so that the logic is reasonable and understandable. (Cantor's first proof on the cardinality of the reals is probably NOT what you want to teach to newbies learning cardinality! But you can say something about the impact Cantor's work had on both Cantor and others in the late nineteenth century.)

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  • $\begingroup$ That is exactly what I want to do! You described a practical way for combining two historical and logical methods for using their advantages and avoiding their disadvantages. $\endgroup$
    – user230
    Mar 25 '14 at 16:28
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The "official" development (with modern notation, notions, ...) is for a reason. It simplifies understanding, makes relationships that were noticed much later aparent, generalizes results that were very narrow, and in general eases the path for the casual wanderer.

Yes, it does violence on my sense of historical accuracy. It is often linked with scientific "biographies" that are wide off the mark, misrepresenting the real struggle and triumphs.

But in final summary, I'd stick with the "official story," just reminding students that this is not the real history, but is heavily edited.

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