> memorized the equation so I could carry it around with me in my head and play with it. If m and a were big numbers, what did that do to f when I pushed it through the equation? If f was big and a was small, what did that do to m? How did the units match on each side? Playing with the equation was like conjugating a verb. I was beginning to intuit that the sparse outlines of the equation were like a metaphorical poem, with all sorts of beautiful symbolic representations embedded within it.
I cannot emphasize how useful this approach is. What's more, I am very surprised by how many students lack the basic ability to sit and "play around" with math concepts.
I used to tutor algebra, trig, and pre-calc. I was surprised by how few students had the ability to look at something and break it into pieces. If you gave them piece A which they knew, piece B which they knew, and put them together, the result was something new the student couldn't understand and they'd sit and wait for an explanation.
This was sad because this is largely what math learning is. Given pieces you do understand, you put together bigger pieces and then build an understanding of those bigger pieces.
I have always advocated something similar to what what the author said here:
> What I had done in learning Russian was to emphasize not just understanding of the language, but fluency. Fluency of something whole like a language requires a kind of familiarity that only repeated and varied interaction with the parts can develop.
I love this point. I am borderline convinced it it is near impossible to learn math concepts more than one or two steps beyond the point that you are fluent in. Beyond that, it's just memorization, guessing, and poor heuristics that just get you to skate past the test and do little or nothing for understanding or retention.
The bigger picture here is that the author has learned how to learn. It amazes me how significant a divide there is between people who know how to learn and people who don't. Forget IQ or test scores, I think knowing how to learn is the biggest indicator of how far someone will go in life. So perhaps most of all I love the fact that they began this journey relatively late (most students going through a similar math path would have done in their late teens or very early 20s what the author was doing in their late 20s).
I cannot emphasize how useful this approach is. What's more, I am very surprised by how many students lack the basic ability to sit and "play around" with math concepts.
I used to tutor algebra, trig, and pre-calc. I was surprised by how few students had the ability to look at something and break it into pieces. If you gave them piece A which they knew, piece B which they knew, and put them together, the result was something new the student couldn't understand and they'd sit and wait for an explanation.
This was sad because this is largely what math learning is. Given pieces you do understand, you put together bigger pieces and then build an understanding of those bigger pieces.
I have always advocated something similar to what what the author said here:
> What I had done in learning Russian was to emphasize not just understanding of the language, but fluency. Fluency of something whole like a language requires a kind of familiarity that only repeated and varied interaction with the parts can develop.
I love this point. I am borderline convinced it it is near impossible to learn math concepts more than one or two steps beyond the point that you are fluent in. Beyond that, it's just memorization, guessing, and poor heuristics that just get you to skate past the test and do little or nothing for understanding or retention.
The bigger picture here is that the author has learned how to learn. It amazes me how significant a divide there is between people who know how to learn and people who don't. Forget IQ or test scores, I think knowing how to learn is the biggest indicator of how far someone will go in life. So perhaps most of all I love the fact that they began this journey relatively late (most students going through a similar math path would have done in their late teens or very early 20s what the author was doing in their late 20s).