As one reads mathematics, one needs to have an active mind, asking questions, forming mental connections between the current topic and other ideas from other contexts, so as to develop a sense of the structure, not just familiarity with a particular tour through the structure.
William ThurstonYou don't see what you're seeing until you see it but when you do see it, it lets you see many other things.
William ThurstonMathematics is not about numbers, equations, computations, or algorithms: it is about understanding.
William ThurstonExperience has shown repeatedly that a mathematical theory with a rich internal structure generally turns out to have significant implications for the understanding of the real world, often in ways no one could have envisioned before the theory was developed.
William ThurstonWe humans have a wide range of abilities that help us perceive and analyze mathematical content. We perceive abstract notions not just through seeing but also by hearing, by feeling, by our sense of body motion and position. Our geometric and spatial skills are highly trainable, just as in other high-performance activities. In mathematics we can use the modules of our minds in flexible ways - even metaphorically. A whole-mind approach to mathematical thinking is vastly more effective than the common approach that manipulates only symbols.
William Thurston