On this Quora space, we often receive questions from intelligent people asking why they have trouble with mathematics. Some believe that there is a “math gene” that they are missing. Certainly, research at the frontiers of mathematics requires a very high level of intelligence where neural efficiency, short term memory, and pattern recognition come into play–qualities possessed by only the small number of gifted people. But even at these levels, the greatest minds struggle with mathematics because it requires, in addition to these special neural qualities, a prolonged focus on the abstract thought. John von Neumann, one of the intellectual giants of the 20th century observed, “Young man, in mathematics, you don’t understand things, you just get used to them.” Einstein, whose field equation was described as “one of the greatest achievements of human thought” admitted to a struggling student, “Do not worry about your difficulties in mathematics. I can assure you mine are still greater.” When attempting to express his field equations in terms of tensors, Einstein had to seek help from a former classmate Marcel Grossmann.
Indeed, doing mathematics at the level presented in the movie Good Will Hunting requires a special talent. However, I believe, based on decades of learning and teaching university level mathematics, that virtually all highly intelligent people are capable of learning mathematics at the undergraduate level. But, as von Neumann asserted, it takes unrelenting effort gained through applying its techniques to solving a myriad of problems until the deep ideas become internalized. Have you ever observed that when working on a problem over a period of time and coming up empty, you decided to abandon the effort; then, on returning to the problem at a later time, the solution came to you almost immediately? Physicist Henri Poincaré described how several of his discoveries occurred when he let a problem incubate in his subconscious and returned to it at a later time. This incubation process is vital in assimilating mathematics.
The key to mastering mathematics at the undergraduate level is to persist relentlessly until your mind has internalized your learning, restructuring itself so that what once seemed difficult soon becomes familiar and natural. Someone once observed, “a brain expanded by the study of mathematics, never shrinks to its original size.”
