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Everything posted by mathwonk

  1. 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.
  2. 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
  3. 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."
  4. 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.
  5. suggested advice from an old man: you are definitely gifted. now go out and find your gift, and then use it.
  6. 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!
  7. 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.
  8. "where the difference between problem solving and just having conceptual understanding. Is one without the other okay?" no.
  9. 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.
  10. 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.
  11. 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?
  12. 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.
  13. 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.
  14. 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.
  15. 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
  16. i loved jacobs's elementary algebra for the cartoons as much as the explanations. the whole approach seemed more fun than any other book i had (or have) seen. my son worked through about half of it in maybe 7th grade, and then won the state mathcounts contest with a perfect paper. so he seemed to get a lot out of it. to give perspective, the next year or so, he did no such independent work at all and finished much lower in the regional math contest.
  17. as for modular artihmetic, you are just setting multiples of some fixed integer, the modulus, equal to zero, so the result is the remainder after dividing by the modulus. this sort of thing was in old days called "casting out", i.e. "casting out nines" meant setting multiples of 9 equal to zero and seeing what was left. this is easy since in base 10, every power of 10 gives 1 when nines are cast out, so 638 becoms 6+3+8 = 9+8 = 8, after casting out nines. casting out 10's is the easiest since it just leaves the ones term, i.e. 638 is equal to 8 "modulo 10". you can cast out multiples of polynomials too, and then a polynomial p(X) is equivalent modulo X-a to what ever is the remainder after division by X-a. Of course this means multiples of X-a are set equal to zero, in particular X-a is set equal to zero, so X is set equal to a, which means of course the result, i.e. the remainder, is just p(a).
  18. i used to be afraid of polynomial division until i realized it is just like integer division only easier. i.e. an integer is essentially a polynomial with x set equal to 10, only there is no carrying needed for polynomials, hence your first guess in long division always works! e.g. when dividing 649 by 38, our first guess of 2 (actually 20), does not work because of carrying, but if we divide 6X^2 + 4X + 9 by 3X + 8, then the first guess of 2X does work! I think I myself was a college professor teaching this stuff by the time i noticed this, but probably some of your kids could figure this out on their own. (when i speak of "carrying" or not, of course i mean that with integers, a large enough number of ones changes the number of tens, and enough tens changes the number of hundreds, but with polynomials, no number of ones changes the number of X's and no number of X's changes the number of X^2's.)
  19. Some elementary schools in our experience tested for IQ very early, but did not tell a student his score until perhaps in high school when it was thought it might motivate him to do better in an area where he was weak. Still even being told early on just that the score was relatively high seemed to have a negative effect. On the other hand, when in a reading group most cannot read the advanced words another child can read, how do you keep that kid feeling he is comparable in ability to the others? It is obvious that something is different. So some method of keeping a level head and a good attitude must be found, in spite of at least implicit information on high (or low) scores getting out. In gifted classes I have taught, kids who acted as if they thought the others were relatively dull, was a harder problem than the kids who felt they were not as smart as the others. I.e. self esteem and confidence seemed a little easier to teach than maturity and respect. You can boost a shy kid a lot by praising one success, but convincing a narcissistic person to respect others can be a long job.
  20. This man had a big impact on my view of calculus teaching. His book Calculus is a wonderful resource for any high level introduction to the topic, and has been so used for decades at many places. Ironically, the advance of popularity of accelerated high school presentation of math ("AP" programs), including calculus for most students has largely killed off the outstanding college programs using his book. The problem is that high school students are usually not offered the subject at so high a level, but then when they enter college they do not want to begin over at the beginning. So for example at Stanford and Harvard, students are not offered the chance to be introduced to this high level of beginning calculus, they must have alteady had it, and there are few chances to get it in high school. I.e. when my son went there Stanford offered only volume 2 of Apostol, not volume 1. University of Georgia kept the tradition going until very recently of offering a comparable course from Spivak's similarly high level Calculus, but I am told it tooi was recently abandoned for lack of an appropriate incoming audience. This makes almost the only remaining audience for his fine book the home school crowd. When I retired I donated my copies of his books to the undergraduate student math library at UGA, as well as copies of Spivak, and Courant. I still miss them from time to time. For those still seeking such courses, they existed until recently still at University of Chicago (math 16000, Spivak based) and MIT ("calculus with theory") from Apostol). A few years ago when teaching Euclid and Archimedes to brilliant 10 year old, I discovered that (by using "Cavalieri's" principle) Archimedes had found the volume not only of a sphere but also of a "bicylinder" , the region between two perpendicular intersecting cylinders of the same radius, and realized how to generalize the calculation he had made from the sphere to that case, a much easier argument than the one using calculus that we teach in college. Then I found a current article by Apostol generalizing these two solids to a whole range of similar ones. I have forgotten it now but it involved solids of a certain degree of convexity for which the calculation of surface area followed directly from that of volume. E.g. for a sphere the solid is viewed as a cone with vertex at the center, hence the volume is 1/3 the product of the surface area and the radius (the "height" of the "cone"). It was very pleasing to learn that such a man as Apostol also found interest in relatively elementary geometry deriving from that of the great Greeks, and even added to it. I didn't know then he was of Greek descent. I recommend anything he has written for seriously mathy types. http://mamikon.com/USArticles/CircumSolids.pdf http://www.maa.org/press/books/new-horizons-in-geometry
  21. Just 2 cents. I have been reading a book of popular anthropology written by a college friend of mine (and now a retired professor of anthropology) and learned that so called "IQ testing" was originally developed by Alfred Binet as a means of detecting areas of deficiency in students that could and should be corrected, i.e. to help them learn more effectively. It was only later in the US that Lewis Terman at Stanford introduced the idea of using it to measure some kind of intrinsic intellectual ability, and nowadays "giftedness". The idea of being labeled gifted or ungifted based on a test seems potentially harmful, not to say unscientific, but the possibility of learning how a student's education could be enhanced seems helpful. I would take this into account myself.
  22. I taught a class for elementary teachers that used this book, and I think it is excellent as a guide to teaching all the elementary math topics. You can go probably at any speed you want through various subjects. but since it is expensive i recommend looking at it in a math ed library first to see if you like it. http://www.amazon.com/Mathematics-Elementary-Teachers-Activities-4th/dp/0321825721/ref=sr_1_1?s=books&ie=UTF8&qid=1461458470&sr=1-1&keywords=sybilla+beckmann i don't know if it has enough drill for a standalone text for the child but a gifted child could well benefit from it at least as a supplement, as well as the parent. I recommend it because everyone here is aspiring to be the best teachers we can be, so it seems prudent to have a book like this as well as standard subject texts. even though i am a professional mathematician, this book helped me with my arithmetic operational skills. you may laugh at this, since i am a researcher on complex algebraic geometry of curves and abelian varieties, but i always kind of struggled with borrowing and paying back in subtraction. maybe that doesn't make sense, but that's what i call it. the book explains it more correctly as "regrouping", and I really liked that concept. i think i understand multiplication better. my students all thought of it as repeated addition, but i see it as forming a rectangle in geometry. i.e. the product of say 5 times 6 is the area of a rectangle with sides 5 and 6, i.e. the number of squares in that rectangle. that way multiplication makes sense also for any two numbers, not necessarily integers, say pi and sqrt(2), i.e. it's just the area of the rectangle of sides pi and sqrt(2), but you may not agree. of course you may be thinking of how to carry out multiplication with decimals, rather than how to define it as a concept. pardon me i do rattle on discussing teaching math. here is a copy of beckmann's book for one cent! http://www.amazon.com/Mathematics-Elementary-Teachers-Sybilla-Beckmann/dp/0201725878/ref=la_B001ITXCCA_1_5?s=books&ie=UTF8&qid=1461809590&sr=1-5
  23. i thought better of this post. please forgive me. the idea was not to stress out about tests, since their goal is to place kids appropriately. so it is potentially a useful result to score in the middle, since that helps us choose where to send them.
  24. i thought you meant does intuition slow down with advanced age, as i worry mine is declining in my 70's. i don't think it slows down much with kids under 30! that uneven rocket ship analogy followed by slow stuff seems very familiar to me. probably my intuition is slowing down partly because of my lack of exposure to math activity. i think intuition follows thoughtful exploration. some people think intuition slows down when people learn too much. one of my profs* suggested we try to prove things ourselves "before you have filled your head with too many of other people's ideas". this is probably not a problem for the first several decades. *here is a nice interview with that prof: Raoul Bott: http://www.ams.org/publications/journals/all/fea-bott.pdf
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