Again, we note that expert teachers exist at all levels, from those expert elementary teachers described by Ma (2010) as having "profound understanding of fundamental mathematics" to university mathematicians like George Pólya.
This knowledge in general has been referred to as "mathematical knowledge for teaching" or MKT (Ball et al. 2005; Ball et al. 2008; Hill and Ball 2004); the expert version of MKT was called "Profound Understanding of Fundamental Mathematics" (PUFM) by Ma (2010).
I do not understand how these claims can be justified, but personally speaking, I feel most creative during moments of profound melancholy – in mathematics and during writing.
Behind this unexciting observation, however, lies a profound fact of mathematics: that the ratio of circumference to diameter is the same for one circle as for another.
For "profound" work in mathematics and for elucidating how the skin senses pain, respectively, Jean Bourgain of the Institute for Advanced Study in Princeton, New Jersey, and David Julius of the University of California, San Francisco, have become instant millionaires by each winning a Shaw Prize.
By the way, the techniques of string theory have found application in other branches of physics such as quark-gluon plasmas, superconductivity, fluid mechanics and quantum information theory (as well as having a profound influence on pure mathematics).
In the century from 1832 to Noether's death in 1935, the field of mathematics—specifically algebra underwent a profound revolution, whose reverberations are still being felt.
Moreover, the Renaissance enthusiasm for mathematics had resulted in a profound interest in rational principles, necessary propositions, and innate ideas.
The evolution of calculation tools available for the learning of mathematics has been quick and profound.
If such an easier route were to turn up, profound knock-on effects would ripple through mathematics because it would mean all the other hard problems could be simplified as well.
