Mathematica U. and AP Calculus

The Wolfram Company which makes the premier mathematical computing software program "Mathematica" has branched out into all different directions, but one of their core areas of interest is teaching mathematics to all levels of students. They have apparently established a 'University' where online study can be found. Some of their latest work involves advanced placement Calculus.

And from their Facebook presentation we get:

[20180919]

Proofs from the Book

Paul Erdos used to remark that a particularly elegant of beautiful proof of a mathematical theorem was one for the "Book." This reference is to a hypothetical book Erdos figured God was keeping with only the very best proofs. Oh, if one could only be so lucky as to read this book! As things worked out two mathematicians took his book concept to heart and, after careful consultation with Erdos, decided to give it their best shot. The following link interviews the two authors and gives an interesting background to their book, "Proofs from the Book", 5th edition. As mentioned at the end off the interview there may be a sixth edition available at some point.


Proofs from the Book, authors interview.

[20180322]

More to the point:

A conjecture both deep and profound
Is whether a circle is round.
In a paper of Erdős
Written in Kurdish
A counterexample is found.

Simon's Course in Modern Analysis

Professor Barry Simon of Caltech has a series of books on modern analysis. As a prelude to this set he has written an ~130 page 'companion guide' which is available free from the Am. Math. Society at this link. It's a good read for anyone irrespective of level of math skills. Please avail yourself of this free publication! Give yourself a Valentine's Day present.

Now, right out of the gate, Prof. Simon tells us:

"Analysis is the infinitesimal calculus writ large. Calculus as taught to most high school students and college freshmen is the subject as it existed about 1750—I’ve no doubt that Euler could have gotten a perfect score on the Calculus BC advanced placement exam. Even “rigorous” calculus courses that talk about ε-δ proofs and the intermediate value theorem only bring the subject up to about 1890 after the impact of Cauchy and Weierstrass on real variable calculus was felt.

"This volume [vol 1] can be thought of as the infinitesimal calculus of the twentieth century. From that point of view, the key chapters are Chapter 4, which covers measure theory—the consummate integral calculus—and the first part of Chapter 6 on distribution theory—the ultimate differential calculus.
"But from another point of view, this volume is about the triumph of abstraction. Abstraction is such a central part of modern mathematics that one forgets that it wasn’t until Frechet’s 1906 thesis that sets of points with no a priori underlying structure (not assumed points in or functions on Rn) are considered and given a structure a posteriori (Frechet first defined abstract metric spaces). And after its success in analysis, abstraction took over significant parts of algebra, geometry, topology, and logic."

[20180214]

Math under the Microscope

This blog give an interesting overview of teaching methods and instructional goals.

[20171119]

This is probably the most general description of mathematics ever written down.

Ref. QED The Strange Theory of Light and Matter, p. 63, Richard Feynman

[161229]

Prof. H.-H. Wu, Cal

Prof. H-H. Wu at Cal-Berkeley has contributed a large number of original papers about how to teach grade school mathematics. From the Link:

"The following are papers on mathematics education, together with a few videos at the end. The papers are separated into two groups. The first group consists of papers that are directly related to the Common Core Mathematics Standards, and the second group contains the papers on mathematics education in general. Within each group, the papers are listed more or less in chronological order."
[20170113]

Number Sense

This is where we need to go! 

From the link:

"If you are connected with the world of K-12 mathematics education, it’s highly unlikely that a day will go by without you uttering, writing, hearing, or reading the term “number sense”. In contrast everyone else on the planet would be hard pressed to describe what it is. Though entering the term into Google will return close to 38 million hits, it has yet to enter the world’s collective consciousness. Stanford mathematician Keith Devlin explains what it is."

[20170113]

And further this is how to get there, from the inventor of the computer program Mathematica.
[20170115]