I started reading a book and I'm only in chapter one but three things popped out at me.
1. Mathematics is patterns
2. Mathematicians work in collaboration with each other rather than in isolation.
3. Computation begins with the question and finds the answer. Mathematics begins with the answer and finds the question.
This last one I found quite interesting because as part of the warm-up this past year I posed some answers and had the students work towards the question. My warm-ups are on the smartboard, ready and waiting till the students arrive and class starts. They do the warm-up while I take roll, etc.
Often I have said something like you have three different numbers and two operations to get -35. I let the students write the problem. The more advanced might do -6 x 5 + -5 while the less advanced might to -40 + 5 x 1. I find this type of warm-up gets them thinking and helps them practice their order of operations in a fun way. I've had students tell others they messed up on the order of their math.
Most of my students prefer to work together. This coming year, I am going to request that they have to ask at least three other people before they ask me for help when they are working together. They have a culture where you must do it correctly the first time so they don't like working independently.