Physicists often remark that no one actually understands quantum mechanics (and those who say they do are lying), but they use it because it consistently works.
Similarly, polymath John von Neumann once famously said that , “…in mathematics you don’t understand things, you just get used to them.”
And in a similar vein David Wells quotes applied mathematician, Oliver Heaviside, thusly:
“The prevalent idea of mathematical works is that you must understand the reason why first, before you proceed to practise. That is fudge and fiddlesticks. I know mathematical processes that I have used with success for a very long time, of which neither I nor anyone else understands the scholastic logic. I have grown into them, and so understood them that way.”
It seems to me that a lot of the emphasis these days from professional mathematicians, as well as in Common Core’s approach, is for students to develop a much deeper understanding of mathematical logic and connections first (and foremost), and for rote processes to follow thereafter. A change in perspective or outlook perhaps??? (or maybe math education has simply always been a mixture of both, in a sort of chicken-and-egg fashion).
Anyway, I’ve spent this week offering up a few snippets (with one more coming Sunday) from Wells’ wonderful 20-year-old volume “The Penguin Book of Curious and Interesting Mathematics.” It is one of the most delightful math reads I’ve stumbled upon in quite awhile, with no particular order (that I can detect) to its succinct, highly-varied contents. Some of the best bits in it are stories/narratives/anecdotes, too long to quote verbatim, about specific famous mathematicians (I especially found the background on Stanislav Ulam fascinating, for example). If you can find a copy I highly recommend it.