Apart from books on heuristics by George Polya.
When trying to engage with and understand mathematical concepts and when applying abstract mathematical concepts to model "continuum" or real world problems.
I find it fruitful to engage in Socratic questioning based on the Socratic method and principles of critical thinking such as:
Analyzing thought (questioning the components of my thinking, for example questioning goals and purpose, questioning assumptions)
Assessing thought (questioning standards of my thinking, for example questioning clarity, accuracy, precision, breadth and depth)
I understand the only way to "learn mathematics is to do mathematics", but if this is approached unthinkingly, the learning in my experience tends to be rather superficial.
Since the quality of our thinking is driven by questions, "doing mathematics" should be approached in the spirit of Socrates who acknowledged “The only true wisdom is in knowing you know nothing”.
I suppose if meaning is context dependent, most of Socrates questions were rigorously focused on definitions. Not sure if Godel's incompleteness theorem regarding logical inconsistency is somehow related to this?
The importance of asking essential questions was fundamentally important to Georg Cantor given that his 1867 Doctoral thesis was entitled "In mathematics the art of proposing a question must be held of higher value than solving it."
Does anyone know of any textbooks which focus on the relationship between the Socratic method and deep mathematical thinking?