Exactly, understanding the intentions and history behind a concept is key to achieving comprehension. I've had to teach myself nearly all of the more advanced mathematical concepts I know, and I'm finally starting to reach a point where I feel I have a nice approach toward achieving an understanding:
- Learn the definition
- Learn the motivations and history
- Peruse a few examples
- Try to map the above to a brief synopsis that explains the concept in intuitive terms that relate to your own life. Rely on pictures.
Finally, I find it helpful sometimes to try and "deduce" identities from "first principles". e.g. assume I didn't know that n^0 = 1. How might I reach that conclusion? If I have an understanding of what exponentiation means I should be able to come up with a few different propositions (these could even be relatively informal) that make such a conclusion make sense.
My childhood was rife with mathematics teachers that focused more on rote memorization of identities instead of careful explanation of definitions and development of "intuition". There's pretty much no better way to ensure you'll produce students that dislike and suck at math for the rest of their lives than proceeding by mind-numbing rote memorization.
I think that's a great approach. If possible, you can add another step: Teaching it to someone else.
>There's pretty much no better way to ensure you'll produce students that dislike and suck at math for the rest of their lives than proceeding by mind-numbing rote memorization.
- Learn the definition - Learn the motivations and history - Peruse a few examples - Try to map the above to a brief synopsis that explains the concept in intuitive terms that relate to your own life. Rely on pictures.
Finally, I find it helpful sometimes to try and "deduce" identities from "first principles". e.g. assume I didn't know that n^0 = 1. How might I reach that conclusion? If I have an understanding of what exponentiation means I should be able to come up with a few different propositions (these could even be relatively informal) that make such a conclusion make sense.
My childhood was rife with mathematics teachers that focused more on rote memorization of identities instead of careful explanation of definitions and development of "intuition". There's pretty much no better way to ensure you'll produce students that dislike and suck at math for the rest of their lives than proceeding by mind-numbing rote memorization.