Anecdote: I’m not especially talented at math compared to the Lesswrong average, I’ve done math up to diffEq and linear algebra, but it’s not my main interest.
I realize that the level of abstraction involved in the sort of math I am familiar with isn’t very high in the first place, and perhaps if the level of abstraction were higher I’d change my tune. But I’ve found that the best way for me to learn math is to continually stare at the definitions and the most general theorems until they make sense, and then work through the examples. The examples are a useful check to make sure I understand, but they are never the mechanism by which I understand—unless I’m dealing with something completely unfamiliar, in which case examples do help.
I’ve had a few professors who lead with a few examples, and then lay down the basic definitions and general theorems. This style of teaching causes me to become completely and utterly lost...my mind keeps sounding alarm bells every time something that doesn’t make complete sense appears, and this prevents me from just going ahead and processing it anyway. Or, I’ll just instinctively ignore the example itself entirely, and instead allocate my attention to trying to glean an underlying principle within the example.
is there something to be learned?
I think yes, from both sides.
From the “example” learning style, I need to learn the value of being able to work through a problem procedurally, even if you don’t completely understand absolutely everything that is necessary to derive what you are doing...you can always figure out the more abstract stuff later. I’ve performed poorly in some of the more applied math classes, as well as organic chemistry, as a result of this mental block in the past—O-chem, especially as traditionally taught, seems like a subject where you need to master the example-based learning style, and I wasted an embarrassing amount of time trying to learn the theory hoping that it would eventually allow me to derive things without ever working through examples.
I think the value of the “abstract” learning style is the more intense focus on first principles, and the ability to differentiate whether one really understands something, or is just going through the motions which get the right answer—perhaps related to a refusal to “guess the teacher’s password” even when doing so would actually lead to a correct and useful answer.
I don’t think these two styles are necessarily dichotomous—some people seem to be able to do both.
Anecdote: I’m not especially talented at math compared to the Lesswrong average, I’ve done math up to diffEq and linear algebra, but it’s not my main interest.
I realize that the level of abstraction involved in the sort of math I am familiar with isn’t very high in the first place, and perhaps if the level of abstraction were higher I’d change my tune. But I’ve found that the best way for me to learn math is to continually stare at the definitions and the most general theorems until they make sense, and then work through the examples. The examples are a useful check to make sure I understand, but they are never the mechanism by which I understand—unless I’m dealing with something completely unfamiliar, in which case examples do help.
I’ve had a few professors who lead with a few examples, and then lay down the basic definitions and general theorems. This style of teaching causes me to become completely and utterly lost...my mind keeps sounding alarm bells every time something that doesn’t make complete sense appears, and this prevents me from just going ahead and processing it anyway. Or, I’ll just instinctively ignore the example itself entirely, and instead allocate my attention to trying to glean an underlying principle within the example.
I think yes, from both sides.
From the “example” learning style, I need to learn the value of being able to work through a problem procedurally, even if you don’t completely understand absolutely everything that is necessary to derive what you are doing...you can always figure out the more abstract stuff later. I’ve performed poorly in some of the more applied math classes, as well as organic chemistry, as a result of this mental block in the past—O-chem, especially as traditionally taught, seems like a subject where you need to master the example-based learning style, and I wasted an embarrassing amount of time trying to learn the theory hoping that it would eventually allow me to derive things without ever working through examples.
I think the value of the “abstract” learning style is the more intense focus on first principles, and the ability to differentiate whether one really understands something, or is just going through the motions which get the right answer—perhaps related to a refusal to “guess the teacher’s password” even when doing so would actually lead to a correct and useful answer.
I don’t think these two styles are necessarily dichotomous—some people seem to be able to do both.