Digging Deep or Digging Hard

Had an interesting conversation today about learning math, though it can be easily applied to anything that requires learning and practice.The whole notion of figuring out how to work a problem. Not just get the answer, but doing all the work yourself. So, what's the purpose -- and when does it get in the way?It boils down to the "show your work" mentality that comes with a lot of math. Before you can use a calculator you need to know how to do it manually. Before a computer: manually. Before anything: manually.I understand it to a point.But it's a strange bit of nit-picky annoyance in the current world.Now, I am a proponent of knowing why you are doing something, but if you can't derive the solution yourself does it make you not right?

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From an engineering perspective, do you care if the engineer knew the math or that they knew what they were doing and the bridge didn't fall down? Sure, it's cool if they can work the math by themselves... but in the end everyone will trust the finite element analysis run on the computer.Likewise, isn't the important part knowing that you need to take an integral of something and knowing why you need it. The fact that Mathematica can give you an answer that you trust is good enough for me. It frees you, the human who has a limited time on this planet, to move on to bigger and better things.Most times the result is matters, not how you got it. (Of course if what you're working on is extending the state of the art then much of this argument tends to fall by the wayside)

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This, in many ways, is somewhat ironic. This is coming from someone that wants to dive in to the guts of damn near everything.But is is really?I tend to stop digging when it starts to get to rote work. Once I have the main understanding of the principals and how they are applied I tend to get bored. I'm sure it can be spun that I lack the perseverance to do the hard parts.I do the hard parts once or twice to know what the hell is going on. Then I delegate.Do you need to know how a car works before you get a driver's license?

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This gets back to teaching kids math. What matters? Should we hide the calculator? Should we keep them in the dark?It's an interesting question. Which side does more good? For how long?