Mathematical sign-posts

When you think about mathematics, what are your markers?   How do you organize in your mind the development of mathematical reasoning and ideas?  How do you integrate your historical, social, and personal perspectives?   Day-to-day, as a teacher or as a researcher, you may have very clear practical and narrow goals: to effectively convey important and useful knowledge and methodology,  or to reduce an open-ended problem to a well-defined and solvable setting.   But all this is informed by a larger view of mathematics and its essential interest and importance.  What are the landmarks that guide you, and how did they come to be a part of your landscape?   What do you try to pass on to your students?   How do you pass these on?

I look forward to your comments!

In the meantime, here is my featured book for this post.

Featured Book of the Day

1Mathematical Omnibus: Thirty Lectures on Classic Mathematics

By Dmitry Fuchs: University of California, Davis, CA,
Serge Tabachnikov: Pennsylvania State University, University Park, PA

(Paraphrasing the Bookstore.) The book consists of thirty lectures on diverse topics, covering a broad area of the mathematical landscape. The common thread in the selected subjects is their illustration of the unity and beauty of mathematics. A special feature of the book is an abundance of drawings (more than four hundred), artwork by an award-winning artist, and about a hundred portraits of mathematicians. Almost every lecture contains surprises for even the seasoned researcher.


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3 Responses to Mathematical sign-posts

  1. Avatar schremmer says:

    This is just a general comment about the blog: Mathematics is described by researchers for their own purpose. Expounders of mathematics on the other hand have not realized that most learners cannot see mathematics that way and thus have failed to recast mathematics for the purpose of having it be learned by “just plain folks.” As Hestenes once said, one ought to “reconstruct course content.”

    • Avatar Eriko Hironaka says:

      Thank you for your comment. I am very interested in alternative ways of teaching, and possibly alternative kinds of textbooks. Do you have examples of texts that do reach “plain folks” effectively?

      • Avatar schremmer says:

        One should distinguish “ways of teaching” from ways of “reconstructing contents”. For a short example of the latter, in his “A very short introduction to mathematics”, Gowers doesn’t mention real numbers, as I recall not even once. Instead, he deals thoroughly with decimal numbers. For adult beginners, this makes all the difference.
        Do you know any introductory textbook that deviates from the “standard” treatment other than by atomizing it?
        (Other than mine of course, to be found at!)

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