Promoting intuition (for undergraduate students): visual thinking, geometic approaches, etc. in the classroom

Question

Note: This question is ment to extend the scope of some related questions of mine. I would appreciate very much any suggestion to improve the way the question is posed.

---

I would like to ask what is -- according to your experience as students beforehand and educators then -- a sensible approach to develop undergraduate students' mathematical intuition (*) (that is visual thinking, etc.).

As the question is very broad, collections of references to research papers in the field of mathematical education and to fitting textbooks are very welcome ad accepted as helpful answers (at least by me).

(*) in general, but in particular as regards the following fields:

• Analysis;
• Mathematical/theoretical physics/ dynamical systems;
• Linear Algebra;
• Geometry;
• Abstract Algebra;
• Number Theory;
• Probability/Combinatorics.

Answer 1

For Linear Algebra, try this classic MAA volume: Resources for Teaching Linear Algebra. It gives a lot of examples of different ways teachers have worked to make linear algebra more concrete, applications oriented, visual, technology-based, etc.

http://eric.ed.gov/?id=ED407244

For Vector Calculus, I particularly liked the problems and curricula from the Vector Calculus Bridge Project. They apparently have disabled the download link, though, and you'll have to email them to get a copy of their labs and instructor's guide.

http://math.oregonstate.edu/bridge/

Answer 2

I've been using origami and pop-up card design as vehicles to teach geometry (and algebra and combinatorics and ...). Here is one example; pardon the self promotion:

"From Pop-Up Cards to Coffee-Cup Caustics: The Knight's Visor," 2012. (arXiv link).

---

               
               

---