Understanding Mathematics

Question on a LinkedIn discussion group:

Which way do you learn and understand Mathematics in general? -> From the “forest” to the “tree” (-> first the Big Picture and then going into the details) or vice versa (-> seeing the Big Picture after learning and understanding some details) and explain why.

My answer:

In learning mathematics, it definitely starts from a tree to a forest. I think only professional research mathematicians, or exceptional teachers of mathematics, can really think from a “forest” to a “tree”. I recall a recent conversation with my former MSc and PhD student who now works, in a senior position, in software development for a serious, and widely known, Internet company. I asked her: “Does it help you that you have a PhD in mathematics”. Her response: “When they have a project too big to handle, they come to me and I cut it in smaller more manageable parts”. This is thinking from top down. It remains a rare skill, hard to teach, and hard to learn. And the issue is of critical importance not only for mathematics, but for other walks of life, too. Just a few weeks ago I had a conversation about that with a lecturer of architecture (and she was also a very successful practicing architect).