she sounds very wise. i had trouble even waking up for class. i had a 9am honors calculus class tuesday - thursday - saturday. oh boyy..and laundry, ouch, i still remember the time my roommate added his red socks to my whites. i also identify with the fun of being in the hearts card game every night. the main problem was that as a nerd, i had never before been in an environment where my peers were my classmates. In high school it was " what did you get on the test?..,,..,I hate you!" In college everyone had the props to hold their own on a test. and my scores did not intimidate anyone.

this might seem naive, but remembering my college days makes me think of something i seldom hear discussed that relates to college preparation. namely a child that goes away to college and lives in a dorm has to be able to get himself up early in the morning, get ready for the day, and make it to class on time, without mom or dad helping. Then they have to make sure they do all their assignments with no nagging, and get them in on time. I.e. they are suddenly treated like an adult in terms of responsibility. i was totally lacking in this department and I missed so much class, and slacked off on so many projects, i actually failed out. fortunately after a year off working, i went back and finished successfully. do people today realize they must let up on the reins a bit in high school to give practice in maturity? Probably you all do, and kids today seem more mature anyway. But this one thing can matter more to success than all the AP preparation and advanced curricula in the world. I remember in my day students missed so many meals from sleeping in, meals that mom and dad had paid for in tuiton, that there was a famous sandwich shop, Elsie's, that practically coined money selling sandwiches at night. Of course Elsie's food was better than Harvard's, and cheaper. Here is Elsie's obit and a remembrance: https://www.ccgfuneralhome.com/obit/elsie-j.-baumann

to TerriM: yes I did. They offered me a larger scholarship than vandy which seemed stingy by comparison, so I felt dissed by vandy and went north. what a shock when i got out of the subway in busy harvard square! i had never been to the city. Other impressions i had were receiving the paperwork with questions like " have you always wanted to go to harvard?" my mental response was : "who do you people think you are?" I.e. in 1960 most students in the south did not wake up every day hoping to get into a school in "the north". and here's to Ollie! of course my affection for Harvard grew when my brother started walking around with t-shirts that said "Harvard, the Vanderbilt of the north."

reminds me of my college application process in the pleistocene era. i lived in nashville and my mom wanted me to stay nearby so i applied only to vanderbilt, literally across the street from my high school. in spring of senior year my math teacher asked where i was going and i told her my plan. she asked if i had applied anywhere else and i said no, so she asked if she could inquire on my behalf. a few weeks later i got a summons from the softball field to the principal's office where a phone call was waiting that accepted me to Harvard, (without my having applied). My question was literally "where is harvard?" I think i had not even heard of cal tech.

suggested advice from an old man: you are definitely gifted. now go out and find your gift, and then use it.

you are asking about math for a 2 year odl. may i suggest you yourself surely know more than enough math to challenge her/him. just make stuff up, and interact with your brilliant child. the plus side is reinforced interaction with you and pleasure of that relationship. you don't need professional help teaching a 2 year old. go for it!

this is the accelerated learner board, but that seems to have many meanings. One is to go through routine level work but sooner than normal. another would be to do higher level work that requires more thinking and deeper understanding. if the latter is desired, then saxon is never indicated, in my opinion (I am a retired college math professor who has worked with bright youngsters, including my own). I firmly believe saxon is indicated only for those who are struggling and need a very focused rote approach without challenging problems, aiming at minimum computational competency. It helped one of my children superficially, who had attention and short term memory problems, but ultimately let him down in terms of interest, depth, and creative challenge. His school which adopted it in hopes of raising standardized test scores, abandoned it at last because they realized the children "didn't understand anything" (quote from the lead teacher). I recommend Harold Jacobs' books, as well as the somewhat duller but substantive AOPS books, for those who can enjoy them.

"where the difference between problem solving and just having conceptual understanding. Is one without the other okay?" no.

For kids who do not have the kind of high end intellectual stimulation of AL kids, "hands on" is by far the most effective way to connect. In fact also for me, who was an AL learner as a child, I have since then met so many subjects I did not grok, that I would have to admit hands on usually works best for me too. As a refinement, start with hands on, and be ready to upgrade in abstraction as soon as the student seems bored and ready for it. I.e. no one approach works for everyone, only attentive consideration of the student's response does.

To me a rational response is just to upgrade their curriculum until they are challenged. But in reality I think it helps you, to learn that/whether your kid is considered "gifted", as it may guide your search for appropriate materials. We had a kid we were kind of ignoring since he seemed pretty average, (so we were not as alert as you are to your child's potential). After testing revealed that he was considered "gifted" we took more seriously the challenge of educating him. So the test was more for our benefit. I.e. it helped us understand the abilities of a kid who was talented but just very laid back. Of course the reverse side of this might be that if he had been found "not gifted", we might have continued to shortchange him. So maybe this sort of labeling is not so reliable, as every kid deserves as much challenge as she/he can benefit from. So i guess I think testing can give useful information, but I would try not to interpret it as a binary decision, i.e. there are more categories than gifted/not gifted. I.e. your children obviously can absorb more of something, so indeed please feel liberated to offer it to them.

I am biased, as a mathematician, but I agree with suggestions above. It is probably not math that is unchallenging but saxon math. One thing to watch for however as you crank up the level, is that a child who has never seen any real math may balk at first when the math gets hard, as it may challenge the child's confidence, so it is important to remind her/him that essentially everyone thinks (real) math is challenging, but that challenge is worth it. Good luck, and good for you for noticing that something needs enhancing with this child. Here is a little example of a simple sounding but apparently very hard math problem: a positive integer ≥ 2 is "prime" if it has no integer factors greater than 1 and smaller than itself. Then all such prime numbers greater than 2 are odd, so the sum of any two of them is an even number greater than 4. (The first few primes are 2,3,5,7,11,13,17, 19, 23,29,31,37, 41, 43,47,...) What about the reverse? Is every even number greater than 4 the sum of two odd primes? It sort of looks as if it is so, since 6 = 3+3, 8 = 5+3, 10 = 7+3 = 5+5, 12 = 7+5, 14 = 11+3 = 7+7, 16 = 11+5 = 13+3, 18 = 11+7 = 13+5, 20 = 13+7 (also 17+3)..... Even a bright grammar school child may appreciate this question, but no one alive knows if this is always true. In general I think number puzzles offer a fun way to keep a clever child engaged with math. E.g. the fact that if a number is divisible by 9 then so is the sum of its digits. E.g. 81 and 8+1 are both divisible by 9; 135 and 1+3+5 are both divisible by 9,..... In fact this works backwards too, so since 1+3+5 is divisible by 9 it follows that not only 135 but also 531 and 351 and 153 and 315 and 513 must all be divisible by 9 too! The child may have fun checking this by dividing. Or maybe the way to present it would be as a question. Ask the child to take any number that is divisible by 9 and rearrange the digits and see if it still is divisible by 9. Ask if that seems always true. Try it with other numbers, are there other divisors where it still works?

or to reprise epi's explanation, -X is defined as the thing that "cancels" X, i.e. such that X + (-X) = 0. But if -X is the thing that cancels X, then -(-X) is the thing that cancels -X, namely X. I.e. since by definition (-X) + (-(-X)) =0, adding X to both sides, gives (-(-X)) = X.

The key idea as stated before is that X-Y is the same as X + (-Y). In fact this is a definition of subtraction. Then A-B = C means that C+B = A. so A - (-B) = C means that C + (-B) = C-B = A, which means that C = A+B. I.e. A - (-B) = A +B. But I myself like informal explanations such as your example that taking away a debt means adding funds. But the above sequence is the logical derivation.

here is an example of linear mathematics: suppose you can make a batch of 2 dozen cookies using 2 cups of flour, one cup of sugar, and one egg. how much of those ingredients do you need to make 4 dozen cookies? you guessed it: 4 cups flour, 2 cups sugar and 2 eggs. but suppose that investing $1,000 at 7% annual interest for 10 years returns $2,000 and you want to know how much it returns at 14%. I doubt if it is $4,000, but you could check me on this. e.g 14 times the rate, or close to 100% annual interest, would almost double yearly, so return close to 2^10 ≈ $1,000,000! thats why people like warren buffet's 20% return. unfortunately the price of those shares in berkshire hathaway have always been out of reach. they went over the time i watched them from 3.000 to maybe 300,000. this is non linear math. on the other hand investing twice as much money at the same rate does seem to yield twice the return, i.e. it is linear in the investment amount, but not in the interest rate. technically, (1+r)^10. P is linear in P but not in r. of course (1+r)^n. P is not linear in n either, so tell your kids, if they can't find a better rate, then start investing earlier, to make n larger. i.e. if it doubles after 10 years, it multiplies by 2^5 = 32 after 50 years.

anybody can structure a course so that it uses any other course, but intrinsically, linear algebra does not use calculus at all. linear algebra is about linear functions, while calculus is about non linear functions (much harder), and in fact is about approximating non linear functions by linear functions, so linear algebra should logically be a prerequisite for calculus, not the other way around. In fact it is taught that way nowadays at most good schools, (like Harvard). there are lots of free linear algebra books on the web, at all levels of difficulty, and essentially none of them really use calculus. I have written three or four of them myself. the most sophisticated math they use is usually polynomials in one variable. Many books do not use even that. let me explain briefly why calculus may come up in a linear algebra class. An operation F is linear if F(x+y) = F(x) + F(y) for all inputs x,y, and also F(cx) = c.F(x) for all scalars c and all inputs x. In fact differentiation obeys these laws, i.e. if D is differentiation, and f and g are functions, then D(f+g) = D(f) + D(g) and D(cf) = cD(f). So differentiation is an example of a linear operation and hence may be mentioned in a linear algebra course. So if you learn calculus first, and linear algebra afterwards, they may point out that you could have understood calculus better by using concepts of linear algebra. But sophisticated courses teach linear algebra first and then use it to teach calculus. Because non honors courses usually teach in a historical order instead of a logical order, one often sees courses offered in the traditional but illogical ordering: calculus, differential equations, linear algebra; instead of linear algebra, calculus, differential equations. But honors courses at top schools do it right. here is the syllabus for the first half of (super honors) advanced calc at harvard: http://www.math.harvard.edu/~ctm/home/text/class/harvard/55a/08/html/syl.html and here is the syllabus for thje second half: http://www.math.harvard.edu/~ctm/home/text/class/harvard/55b/10/html/syl.html notice the 1st half is all algebra and the second half is all analysis (calculus). Mind you i do not recommend taking this particular course since the emphasis at harvard seems to be on making it really hard, but that is because of the speed, not the ordering. Take it somewhere where they use the same ordering but you go slow enough to learn it well. here is a link to a free linear algebra book, that i have not lookd at lately but i remember as being very easy. http://joshua.smcvt.edu/linearalgebra/ well i just looked at it and maybe it is not my best recommendation, i prefer the book by paul shields, because it works in 2 and three dimensions and this is better for beginners than n dimensions. here is shields' book: (which i also have not looked at lately but recall from when my wife took the course from it. shields is from stanford.) https://www.amazon.com/Elementary-Linear-Algebra-Paul-Shields/dp/0879011211