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Understanding why has always been my primary way to learn things. Do you have any suggestions for books or courses I could dig into to better understand this ap

Understanding why has always been my primary way to learn things. Do you have any suggestions for books or courses I could dig into to better understand this approach. My oldest who is 8 is quite good at math and on the math team and his school does do a lot to teach them but I want to help him how to think. In almost every other field I can but I am not good enough at math to understand which "games" to play.

ColinWright 9y ago
Everyone has a balance of these things, but in my experience there are two extrema to approaching maths. Some people are happy to be solving the puzzles, and simply enjoy the thrill of the chase. Others insist on knowing why they should be learning this, and what it's good for. Both are valid, but if there is A and B and you're naturally good at A, the natural tendency is to ignore or avoid B. But that then limits your box of tricks, and makes you one-dimensional. By all means take advantage of what you're good at, but work hard on the other aspects to enrich your abilities. Also, they cross-fertilise, so you get more than you might expect. With regards your oldest, it's worth trying a bunch of stuff to see if you can work out what they prefer. Then do both, but be prepared for having to work harder on one aspect than the other. Your oldest might like puzzles for their own sake, or might insist on knowing what it's for. Explore both. Find maths in strange places, work through puzzles just for their own sake. The critical key is to be doing stuff all the time. Caveat: I don't have children, I have no formal training, but I do a lot of outreach, enrichment, and enhancement.
lsc 9y ago
I personally find Serge Lang's "Basic Mathematics" to be super useful in explaining the "why" of a particular formula in pre-college level math, as most books at my current level just seem to say "memorize this" He has an explanation of why x^0 is 1 and not 0 that doesn't involve calculus, for instance.
- yorwba 9y ago
  What would be an explanation that involves calculus? The only one I know is that x^(a+b) = x^a * x^b holds for positive a and b, and to uphold that pattern, it's necessary that x^0 = 1, x^-1 = 1/x and so on. For me, all of that was covered long before calculus.
  - sethammons 9y ago
    I actually referred to a different pattern to help knowing what x^0 is above. Not from calc, but just a different way than yours. If you start with x^3, x^2, x^1, x^0, x^-1, x^-2, and expand the forms, you get xxx, xx, x, ... where the pattern is each step divides by x. So x/x = 1 is the next, and 1/x is the next, and 1/xx is the next, ...
  - lsc 9y ago
    Interesting. When I took high school math as a kid, there weren't many explanations, (and for some reason, I wasn't up for looking them up/figuring them out myself- weird, 'cause I totally was up for that sort of thing in some other classes) - just x^(a+b) = x^a * x^b and x^0=1 -- memorize these facts, with the emphasis on memorization rather than on the connection between those facts. Most of what I remember is a lot of guess and check and factorization; something I automated using the rom basic on my Tandy model 100. These were the mid '90s, and looking back with a less resistant mind, I think they really were trying to get us to develop a mathematical intuition, and that if I had read the math text book the way I read the history or English lit textbooks (they were full of stories!) I would have done just fine. But I still can't read math books the way I read literature, and I haven't figured out why yet. I know my first decade of reading, I skipped a lot, like if I didn't understand something, I moved on and filled out the blanks from what I did understand as best I could. (These days when I'm reading literature I get all excited when I see a word or phrase I don't understand, and I immediately stop and research. (the world is awesome for that today. Once I was blocked on my way home, waiting for a family of skunks to move off of the trail. I was introduced to rimbaud by miller, so I bought and downloaded a bunch of his stuff to my kindle. By the time I was done reading, it was dark and the skunks started to approach me, so I took off (without incident) but I wish I could do that with math, and I'm not quite sure why I cant. I mean, reading things I half understand, and halfway getting something out of it.) the calculus explanation I heard from a friend that I was talking about this about was extrapolating from x^.5 x^.4, etc, etc, and as the exponent approaches 0 the answer approaches 1 for all non zero values of x - but this was maybe two years back and the friend in question was educated overseas, and I still am not educated enough to use calculus notation to write that down (or even really to understand if I understood him correctly.) so it could be garbled, or only tradition on the continent, or something like that.

ColinWright 9y ago

Everyone has a balance of these things, but in my experience there are two extrema to approaching maths. Some people are happy to be solving the puzzles, and simply enjoy the thrill of the chase. Others insist on knowing why they should be learning this, and what it's good for. Both are valid, but if there is A and B and you're naturally good at A, the natural tendency is to ignore or avoid B. But that then limits your box of tricks, and makes you one-dimensional. By all means take advantage of what you're good at, but work hard on the other aspects to enrich your abilities. Also, they cross-fertilise, so you get more than you might expect. With regards your oldest, it's worth trying a bunch of stuff to see if you can work out what they prefer. Then do both, but be prepared for having to work harder on one aspect than the other. Your oldest might like puzzles for their own sake, or might insist on knowing what it's for. Explore both. Find maths in strange places, work through puzzles just for their own sake. The critical key is to be doing stuff all the time. Caveat: I don't have children, I have no formal training, but I do a lot of outreach, enrichment, and enhancement.

lsc 9y ago

I personally find Serge Lang's "Basic Mathematics" to be super useful in explaining the "why" of a particular formula in pre-college level math, as most books at my current level just seem to say "memorize this" He has an explanation of why x^0 is 1 and not 0 that doesn't involve calculus, for instance.

yorwba 9y ago
What would be an explanation that involves calculus? The only one I know is that x^(a+b) = x^a * x^b holds for positive a and b, and to uphold that pattern, it's necessary that x^0 = 1, x^-1 = 1/x and so on. For me, all of that was covered long before calculus.
- sethammons 9y ago
  I actually referred to a different pattern to help knowing what x^0 is above. Not from calc, but just a different way than yours. If you start with x^3, x^2, x^1, x^0, x^-1, x^-2, and expand the forms, you get xxx, xx, x, ... where the pattern is each step divides by x. So x/x = 1 is the next, and 1/x is the next, and 1/xx is the next, ...
- lsc 9y ago
  Interesting. When I took high school math as a kid, there weren't many explanations, (and for some reason, I wasn't up for looking them up/figuring them out myself- weird, 'cause I totally was up for that sort of thing in some other classes) - just x^(a+b) = x^a * x^b and x^0=1 -- memorize these facts, with the emphasis on memorization rather than on the connection between those facts. Most of what I remember is a lot of guess and check and factorization; something I automated using the rom basic on my Tandy model 100. These were the mid '90s, and looking back with a less resistant mind, I think they really were trying to get us to develop a mathematical intuition, and that if I had read the math text book the way I read the history or English lit textbooks (they were full of stories!) I would have done just fine. But I still can't read math books the way I read literature, and I haven't figured out why yet. I know my first decade of reading, I skipped a lot, like if I didn't understand something, I moved on and filled out the blanks from what I did understand as best I could. (These days when I'm reading literature I get all excited when I see a word or phrase I don't understand, and I immediately stop and research. (the world is awesome for that today. Once I was blocked on my way home, waiting for a family of skunks to move off of the trail. I was introduced to rimbaud by miller, so I bought and downloaded a bunch of his stuff to my kindle. By the time I was done reading, it was dark and the skunks started to approach me, so I took off (without incident) but I wish I could do that with math, and I'm not quite sure why I cant. I mean, reading things I half understand, and halfway getting something out of it.) the calculus explanation I heard from a friend that I was talking about this about was extrapolating from x^.5 x^.4, etc, etc, and as the exponent approaches 0 the answer approaches 1 for all non zero values of x - but this was maybe two years back and the friend in question was educated overseas, and I still am not educated enough to use calculus notation to write that down (or even really to understand if I understood him correctly.) so it could be garbled, or only tradition on the continent, or something like that.