#1. A true mathematician is a pattern-seeker. The majority of math concerns patterns in numbers, but a skilled mathematician finds himself well equipped to deal in logic and philosophy as well - because it is not numbers, intrinsically, but patterns, that concern him. The better you are at noticing, conceptualizing, defining and explaining patterns, the better you are at mathematics.
#2. A true mathematician relishes challenge. Never work out on paper what you think you might be able to do in your head. Take ten seconds to think about it. If you aren’t sure, then check it by working it out (when you’re being graded, always work it out). Likewise, only use the calculator to save time! Never use it as a magic conch! If you can’t explain to yourself or someone else how the calculator got the answer it did, you are doing divinations, not mathematics!