https://www.coursera.org/course/maththink
Introduction to Mathematical Thinking
Dr. Keith Devlin
Stanford University

This course (and the textbook with the same name) looks interesting and does not overlap heavily with the usual algebra-through-calculus sequence. Course reviews (generally favorable) are at http://coursetalk.org/coursera/introduction-to-mathematical-thinking?sort=newest . The prerequisites are moderate:

"High school mathematics. Specific requirements are familiarity with elementary symbolic algebra, the concept of a number system (in particular, the characteristics of, and distinctions between, the natural numbers, the integers, the rational numbers, and the real numbers), and some elementary set theory (including inequalities and intervals of the real line)."

Devlin has written several books on math for popular audiences. I'll buy a few for my eldest son, such as "The Language of Mathematics: Making the Invisible Visible", and if he likes them, I will mention the course to him.

Thanks, cool find, seems like it would make a great enrichment course that enhances rather than muddies the standard secondary school throughline (though, personally, I'm more inclined to think this topic set would be more useful to more people than geometry or algebra 2.) I may see if DS8 wants to try it. I also noticed in their related links the course Creative Problem Solving which also seems potentially useful.

I'm actually taking this myself right now for fun. Peeking is allowed! Anyone who wants to see what it's like can enroll to check out the videos and assignments even though the course has already started.

I sent you both a PM.

Devlin's background is in mathematical logic, so it's natural that he'd choose it as a focal point, but it's unrepresentative —  very few mathematicians do research in mathematical logic. You might be interested in the great mathematician Henri Poincare's essay "Intuition and Logic in Mathematics" http://www-history.mcs.st-and.ac.uk/Extras/Poincare_Intuition.html

I don't know of good video courses that give an accessible introduction to more representative mathematical thinking, but some books that do are

• Groups and Symmetry: A Guide to Discovering Mathematics and Knots and Surfaces: A Guide to Discovering Mathematics by David Farmer and Theodore Stanford.
• The Knot Book by Colin Adams.
• The Shape of Space by Jeffrey Weeks.
• Yearning for the Impossible: The Surprising Truths of Mathematics by John Stillwell.
• Elementary Number Theory by Jones and Jones.
• Galois' Theory Of Algebraic Equations by Jean-Pierre Tignol
• Integer Partitions by George E. Andrews and Kimmo Eriksson

each of which require no more than high school algebra.

I have no basis to value your term "unrepresentative." It seems to imply that the popularity of fields of mathematical research is relevant to the selection of mathematical education topics.

@Zen Scanner — Mathematicians' activity taken as a whole reflects an aggregated consensus concerning the aesthetic value and depth of different mathematical topics. Tastes vary, but one can expect that on average students will get the most out of what knowledgable people find most interesting.

Oh well.

I disagree with Jonah on this. It might make sense to avoid topics which few educated people find interesting, but beyond that, it makes no sense to choose fields of study by their research popularity (as opposed, say, to their appeal to the potential student, their relevance to possible careers, or the availability of good materials).

Even if it did make sense to do that, dismissing mathematical logic because relatively few mathematicians work in that field would be silly. Much research in mathematical logic is now done in computer science departments (albeit often by people whose PhDs are in maths), because of the importance of logic as foundation of computer science. Many mathematicians are relatively ignorant of logic, because for historical reasons (and because there are few mathematics professors pushing for courses in it, because they have skipped to CS departments!) it is often neglected in undergraduate mathematics syllabuses. Neither of these facts makes mathematical logic any less suitable for study by a mathematical youth, however.

I'd go even further-- to suggest that actually-- for quite young students, it quite possibly makes MORE sense to pursue those areas of inquiry which are less mainstream.

The reason? Because it slows down the inevitable progression into post-secondary (and post-graduate) topics in the subject.

Since opportunities at that level come with certain strings attached (an expectation of maturity that will allow a student to function as an autonomous adult on a college campus, for starters-- or to be escorted continuously by a parent at all times); it makes sense to explore WIDELY in an effort to delay that as long as seems feasible.

I'm not talking about depriving a child of learning-- but in gently steering that learning where possible so that the maturity gap isn't quite so wide. Vector calculus with 20yo classmates.... 14yo? Or 16yo?

KWIM?

@ ColinsMum — I agree with the criteria that you list for the suitability of choices of enrichment topics. My thought was that more mainstream math meets them better than mathematical logic. But you're right about mathematical logic showing up more in computer science – this is an important point that hadn't occurred to me, thanks.

@ HowlerKarma — I agree with your general point.

• For the most part, the material in the books that I listed doesn't show up in the standard undergraduate curriculum, and many mathematicians aren't exposed to it until graduate school, and also falls under the "exploring widely" heading (e.g. many mathematicians don't know knot theory).
• Being exposed to mainstream math helps students decide whether or not to pursue math further.
• Most children who are interested in math won't go on to math graduate school.
• Deep mathematical subjects take a long time to absorb, often including returning to them multiple times before one assimilates them, so it's not necessarily bad to study them before taking courses on them.
• For those children who will go on to math graduate school and who have already mastered the relevant material before graduate school, such that they don't want to take the coursework, acceleration won't be problematic from the point of view of age, and graduate schools are more flexible about students skipping courses than high schools (for example).

I purchased The Language of Mathematics: Making the Invisible Visible and Introduction to Mathematical Thinking. I also added Mathematics for the Nonmathematician (for myself). I will see what my ds thinks of them, he will give them a chance I am sure.
I love all the suggestions, thanks.