Emphasis on meaningfulness and sense-making as central to mathematical development. Memorization is fragile and recovery is difficult (as in forgot my password or lock cutter at my gym for forgotten combinations) but reasoning from basic meaning back to some fact is possible.
#MTBoS #TeachMath

I am currently working with my co-author to finish up our textbook using complex number topics to prepare students for success in upper-level math courses. It mixes proving with investigating and gives students a nice taste of beautiful, accessible yet deep mathematics.

I was lucky to have been a grad student at Courant when he was director. Never took a class from him but he was part of my oral exam for candidacy (as was Fritz John), which was quite an experience. He gave a minilecture during my exam! I learned so much from his works and his personal example.

I refer to this as "math hacking". What could give an answer of ln(x) ... guess x ln(x) and then fix it. We should call the general idea in books "integration by (missing) parts" since we never have two product rule terms in sight. Use this later to approximate ln(n!) as sum, useful for series.

Will you follow up with Newton polynomial interpolation? Stage is set.

depends on grade level, but zometools are really fun. Not sure if a basic 3-D printer fits that budget either.

Sophs with what background? It's fun to do hyperbolic and spherical geometry, with Escher prints as some motivation. Circle/limit prints map into upper half plane setting, where math is easier. Soph/early junior math majors/minors have loved it. They find the Mobius subgroup generators for UHP.

I draw a right triangle with letters a, b, and c in the usual spots. We either have a hypotenuse relabeled as x or one side is x. Side case splits depending on whether we have c or have a. So the basic forms come from one picture with label choices. Final exam now. Easy to give advice b4 grading.

Besides those mentioned, if this is for secondary Ss, any book in the Anneli Lax series (MAA press). Many classics. Strogatz book on calculus. Several from my grad school friend Mark Levi, particularly those from Princeton U. press. Math circle titles too. More advanced: Tabachnikov and Fuchs.