Numbers and geometry

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Professor John Stillwell writes, in the preface to his book Numbers and Geometry (New York: Springer-Verlag, 1998): "What should every aspiring mathematician know? The answer for most of the 20th century has been: calculus. . . . Mathematics today is . . . much more than calculus; and the calculus now taught is, sadly, much less than it used to be. Little by little, calculus has been deprived of the algebra, geometry, and logic it needs to sustain it, until many institutions have had to put it on high-tech life-support systems. A subject struggling to survive is hardly a good introduction to the vigor of real mathematics. ". . . . In the current situation, we need to revive not only calculus, but also algebra, geometry, and the whole idea that mathematics is a rigorous, cumulative discipline in which each mathematician stands on the shoul…

…ery influential on my thinking about math education. A link that furthered my process of pondering how to teach mathematics better was Richard Askey's review of the book Knowing and Teaching Elementary Mathematics by Liping Ma. http://www.aft.org/pubs-reports/american_educator/fall99/ame... Another review of that excellent book http://www.ams.org/notices/199908/rev-howe.pdf is also food for thought. Professor John Stillwell writes, in the preface to his book Numbers and Geometry (New York: Springer-Verlag, 1998): "What should every aspiring mathematician know? The answer for most of the 20th century has been: calculus. . . . Mathematics today is . . . much more than calculus; and the calculus now taught is, sadly, much less than it used to be. Little by little, calculus has been deprived of the algebra, geometry, and logic it needs to sus…

Professor John Stillwell writes, in the preface to his book Numbers and Geometry (New York: Springer-Verlag, 1998): "What should every aspiring mathematician know? The answer for most of the 20th century has been: calculus. . . . Mathematics today is . . . much more than calculus; and the calculus now taught is, sadly, much less than it used to be. Little by little, calculus has been deprived of the algebra, geometry, and logic it needs to sustain it, until many institutions have had to put it on high-tech life-support systems. A subject struggling to survive is hardly a good introduction to the vigor of real mathematics. ". . . . In the current situation, we need to revive not only calculus, but also algebra, geometry, and the whole idea that mathematics is a rigorous, cumulative discipline in which each mathematician stands on the shoul…

…om that thread: http://www.springer.com/physics/book/978-0-306-45036-5 http://www.amazon.com/Mathematics-Short-Introduction-Timothy... I agree with the recommendation of An Introduction to Mathematical Reasoning in this thread. Another participant has already recommended my favorite for background reading, Concepts of Modern Mathematics by Ian Stewart. http://www.amazon.com/Concepts-Modern-Mathematics-Ian-Stewar... Get that right away. Sawyer's A Mathematician's Delight is surely also good (I've read other books by Sawyer). http://www.amazon.co.uk/gp/product/0486462404/ Read those for background as you get my favorite overviews of mathematics: Basic Mathematics by Serge Lang and Numbers and Geometry by Joseph Stillwell. http://www.amazon.com/Basic-Mathematics-Serge-Lang/dp/038796... (Basic Mathematics is mostly high school level math, wit…

A suggestion for an addition (or replacement) for the list: http://www.amazon.com/Numbers-Geometry-John-Stillwell/dp/038... Numbers and Geometry by John Stillwell is very readable and full of very interesting math problems, covering a lot of topics that are accessible to anyone with a good high school education but often aren't covered even by an undergraduate math degree.

Stillwell is wrong, or at least not doing his homework, Have you read his book Numbers and Geometry, or any of Stillwell's other books? I actually think he takes the approach you advocate, but you haven't given me quite enough examples of the approach you think best for me to be entirely sure.

…ath professor says about the delightful aspects of mathematics to learn besides calculus as it is now taught. Professor John Stillwell writes, in the preface to his book Numbers and Geometry (New York: Springer Verlag, 1998): "What should every aspiring mathematician know? The answer for most of the 20th century has been: calculus. . . . Mathematics today is . . . much more than calculus; and the calculus now taught is, sadly, much less than it used to be. Little by little, calculus has been deprived of the algebra, geometry, and logic it needs to sustain it, until many institutions have had to put it on high-tech life-support systems. A subject struggling to survive is hardly a good introduction to the vigor of real mathematics. ". . . . In the current situation, we need to revive not only calculus, but also algebra, geometry, and the wh…

Some favorites about mathematics: Concepts of Modern Mathematics by Ian Stewart http://www.amazon.com/Concepts-Modern-Mathematics-Ian-Stewar... Numbers and Geometry by John Stillwell. http://www.amazon.com/Numbers-Geometry-John-Stillwell/dp/038... The Pleasures of Counting by T. W. Körner http://www.amazon.com/Pleasures-Counting-T-W-K%C3%B6rner/dp/... Mathematics: A Very Short Introduction by Timothy Gowers http://www.amazon.com/Mathematics-Short-Introduction-Timothy...

…discrete, countable things, that is numbers, with reasoning about shapes, that is geometry. An especially good book about both aspects of mathematics is John Stillwell's Numbers and Geometry. http://www.amazon.com/Numbers-Geometry-John-Stillwell/dp/038... I'll quote a bit of Stillwell's preface to give the flavor of the book: "What should every aspiring mathematician know? The answer for most of the 20th century has been: calculus. . . . Mathematics today is . . . much more than calculus; and the calculus now taught is, sadly, much less than it used to be. Little by little, calculus has been deprived of the algebra, geometry, and logic it needs to sustain it, until many institutions have had to put it on high-tech life-support systems. A subject struggling to survive is hardly a good introduction to the vigor of real mathematics. ". . . .…