Thinking Mathematics or Thinking Physics for math is a hypothetical book that teaches math in the style of Thinking Physics.
actually, is there a Thinking Mathematics? (maybe AOPS books are it?) it feels like math would work less well for some reason, but i am having trouble articulating why. maybe because you need to introduce a lot of terminology and concepts into someone's ontology first before you can even ask the question? probability/combinatorics is like one of the few branches where you don't need to do this (and this might be why the AoPS book is so enjoyable).
UPDATE: i think the thing that makes Thinking Physics possible is that you can pose the question to anyone, in a way that is understandable to them, without first having to lecture them about a bunch of definitions or what a mathematical proof is or whatever.
There are a few books like Creative Mathematics by H.S. Wall and 'the essence of mathematics through elementary problems' that might be trying to do this, but i have not looked at either of them in depth.