See: Math Big Picture
Understand mathematics by describing it as an activity.
Consider math as the most general discipline for figuring things out, for relating questions and answers, in the spirit of games of understanding. Look for the role of duality and deviation from it.
Study the most basic math, what is taught in grade school. Study geometries as theories.
Given a theory, as in geometry, collect and study the questions that arise.
What kinds of mathematical activity are there, how are they related and what do they involve?
A math result is an Alexandrian pattern which brings together:
Counting: characterizing a set.
Note: the action becomes an object in its own right. For example, an object is characterized by whether it is an instance of itself or is not. The multiset of instances is then characterized by distinguishing a set of distinct instances. A set is characterized by counting it. But the count is not a set but a list. In this way, algebra unfolds to yield counting.
Study polymorphism of principles of basic math
Math should derive from logic, from perfect duality, and variation from it. The relation between logic and metalogic is given by our interpretation of a variable with reference to one of four levels. This establishes an analogy between logic, set theory, category theory and...? See logic.
List basic principles of math: