Mathematics: The Art of Being Careful

Lately I have been reviewing some of the old mathematics that I learned (and subsequently forgot) in decades gone by.  I'm going chapter-by-chapter in the Schaums series, right now appropriately at a "beginner's" level for me: pre-calculus.  I have been making a daily routine of doing problem(s) from the book and am now on the fourth chapter.  The problems in the Schaums series are explained and solved, so should I have difficulty in one of them, I can read the solution and return to it, (re)learning in the process.  But besides becoming reacquainted with math, I have also found that I derive the same degree of pleasure from successfully solving a problem that I get from doing sudoku, kakuro, cryptograms, Jumbles, or other puzzles.  And in all of these, there is one overriding principle: in order to solve any of them, I have to be very careful in the steps I take!

As a matter of fact, I might just argue that mathematics in itself is the art of being careful. Back when I was in mathematics classes at school, careless mistakes in my assignments counted against me, but the teacher also usually gave "partial credit" if it appeared that I had grasped the general principles of the concepts involved.  Looking back on it, I think that was a mistake.  One cannot truly develop a skill in anything, especially mathematics, without making accuracy and mistake avoidance an absolute priority.  Carefulness must become an ingrained habit.  On the other hand, whizzing through problems and dismissing sections of instruction because they seem intuitively obvious are not only counterproductive: they are in themselves examples of carelessness!