Learn Math With Own Example Space

G. Polya / Paul Halmos advocate getting math students to construct not just one but classes of examples to:

1. Extend & enrich own Example Spaces;

2. Develop full appreciation of concepts, definitions, techniques that they are taught.

[Polya, Halmos, Feynman]: they collect and build a personal ‘repertoire’ of “Examples Space” (include counter-examples) for each abstract math idea, which they can relate to a concrete object.

Examples:

Group abelian = (Z,+)

Ring = Z

Principal Ideal = nZ

Equivalence Relation = mod (n)

Cosets = {3Z, 1+3Z, 2+3Z}

…