This question doesn’t seem worthy of it’s own Discussion post, and seems to fit reasonably well here.
Whenever I try to learn math, I try to know why such a thing is a certain way, making sure I’m not just carrying a feeling of being correct, or even a vague feeling for what the answer is without there being an explicit sequence of ideas that convey that answer. But if I do get such a sequence of ideas, I never know if I’m finished, so I try to assume that I’m not finished.
For example, I was watching some of Salman Khan’s introductory videos on probability theory two days ago. The first few videos were very basic, using coin flips to explain where the fractions representing probability came from.
It occurred to me that I wasn’t really thinking about the video. I just knew that a probability of 1⁄4 means that there’s a 25% chance of something happening, and was more waiting for the video to end so I could go on to something new. I asked myself “Wait, can I take that fractional representation of probability and connect it directly to what probability actually is, or am I just playing Chinese room with numbers?”
I thought about it for a minute, and decided that the denominator represented the quantity of “runs” needed for the desired “results”, the quantity of which was represented by the numerator. A probability of 5⁄9 means that if the situation is run 9 times, the result which is assigned a probability of 5⁄9 will occur 5 times.
Which is all well and good, but I still felt confused. How do I know when I’m finished thinking about this? My only arbiter is a subjective feeling of comprehension, and comprehension can occur after knowing how to use a method, as well as knowing why a method works.
This question doesn’t seem worthy of it’s own Discussion post, and seems to fit reasonably well here.
Whenever I try to learn math, I try to know why such a thing is a certain way, making sure I’m not just carrying a feeling of being correct, or even a vague feeling for what the answer is without there being an explicit sequence of ideas that convey that answer. But if I do get such a sequence of ideas, I never know if I’m finished, so I try to assume that I’m not finished.
For example, I was watching some of Salman Khan’s introductory videos on probability theory two days ago. The first few videos were very basic, using coin flips to explain where the fractions representing probability came from.
It occurred to me that I wasn’t really thinking about the video. I just knew that a probability of 1⁄4 means that there’s a 25% chance of something happening, and was more waiting for the video to end so I could go on to something new. I asked myself “Wait, can I take that fractional representation of probability and connect it directly to what probability actually is, or am I just playing Chinese room with numbers?”
I thought about it for a minute, and decided that the denominator represented the quantity of “runs” needed for the desired “results”, the quantity of which was represented by the numerator. A probability of 5⁄9 means that if the situation is run 9 times, the result which is assigned a probability of 5⁄9 will occur 5 times.
Which is all well and good, but I still felt confused. How do I know when I’m finished thinking about this? My only arbiter is a subjective feeling of comprehension, and comprehension can occur after knowing how to use a method, as well as knowing why a method works.