Mathematics is the art of explanation. If you deny students the opportunity to engage in this activity-- to pose their own problems, to make their own conjectures and discoveries, to be wrong, to be creatively frustrated, to have an inspiration, and to cobble together their own explanations and proofs-- you deny them mathematics itself.
Paul LockhartThere is nothing as dreamy and poetic, nothing as radical, subversive, and psychedelic, as mathematics.
Paul LockhartNo mathematician in the world would bother making these senseless distinctions: 2 1/2 is a "mixed number " while 5/2 is an "improper fraction." They're EQUAL for crying out loud. They are the exact same numbers and have the exact same properties. Who uses such words outside of fourth grade?
Paul LockhartThe mathematical question is "Why?" It's always why. And the only way we know how to answer such questions is to come up, from scratch, with these narrative arguments that explain it. So what I want to do with this book is open up this world of mathematical reality, the creatures that we build there, the questions that we ask there, the ways in which we poke and prod (known as problems), and how we can possibly craft these elegant reason-poems.
Paul LockhartThere is nothing as dreamy and poetic, nothing as radical, subversive, and psychedelic, as mathematics. It is every bit as mind blowing as cosmology or physics (mathematicians conceived of black holes long before astronomers actually found any), and allows more freedom of expression than poetry, art, or music (which depends heavily on properties of the physical universe). Mathematics is the purest of the arts, as well as the most misunderstood.
Paul Lockhart