Mathematics is about problems, and problems must be made the focus of a student's mathematical life. Painful and creatively frustrating as it may be, students and their teachers should at all times be engaged in the process - having ideas, not having ideas, discovering patterns, making conjectures, constructing examples and counterexamples, devising arguments, and critiquing each other's work.
Paul LockhartMathematicians enjoy thinking about the simplest possible things, and the simplest possible things are imaginary.
Paul LockhartDoing mathematics should always mean finding patterns and crafting beautiful and meaningful explanations.
Paul Lockhart[Math] curriculum is obsessed with jargon and nomenclature seemingly for no other purpose than to provide teachers with something to test the students on.
Paul LockhartMental acuity of any kind comes from solving problems yourself, not from being told how to solve them.
Paul LockhartThe thing I want you especially to understand is this feeling of divine revelation. I feel that this structure was "out there" all along I just couldn't see it. And now I can! This is really what keeps me in the math game- the chance that I might glimpse some kind of secret underlying truth, some sort of message from the gods.
Paul Lockhart