well i myself had been put off Euclid my whole life by various things that are said about it, and by the formality of the first few pages when I once had picked it up. What opened it up for me, at age 60+, was the book by Robin Hartshorne, Geometry: Euclid and beyond. He starts off with a little introduction, and then has exercises, the first being something like: write down whatever you remember from high school. Then he asks you to read the first so many propositions in any translation of Euclid you like. I recommend the Green Lion edition, http://www.amazon.com/Euclids-Elements-Euclid/dp/1888009195/ref=sr_1_1?ie=UTF8&qid=1421876344&sr=8-1&keywords=euclid The fact he does not include Euclid's text in his book, nor any substitute for it, but assumes you are reading it on your own alongside, forced me to actually open Euclid. But it is still potentially off putting to actually plunge into the book. I think I recommended my students to ignore the definitions and axioms and postulates and just start reading the theorems, to see if you can understand them. I guess you could read the postulates without much trouble, it's the definitions that may be confusing. Or you could look at them as long as you don't let not understanding some of them scare you off. "A straight line is a line that lies evenly with the points on itself." What does that mean? Fortunately it does not matter. Or you might start with Hartshorne's section 2, on constructions, and afterwards try the exercises at the end of that section. In section 3 Hartshorne discusses Euclid's axiomatic method. I myself wrote some rough notes when I taught from Euclid to youngsters at epsilon camp, available free on my uga math webpage. http://www.math.uga.edu/~roy/camp2011/10.pdf There was also a companion series of afternoon classes on constructions for which there are no notes, so the students knew more than just what is in the rough class notes, which are largely concerned with examining the proofs in Euclid. Hartshorne also has a nice essay on teaching from Euclid, comparing it to other more modern approaches: http://www.ams.org/notices/200004/fea-hartshorne.pdf

Oh of course! I meant geometry was pretty much the first math subject, but someone might still prepare for it with some other activity. I see. Also one does not need in reading Euclid to pay so much attention to the axioms. One can treat much of it as a sequence of fun constructions with compass and straight edge. Forgive my ignorance. I am not used to so much compartmentalization of mathematical knowledge, to me its all just one huge body of natural phenomena, beautifully interrelated. Sort of like my wife the pediatrician's surprise to learn that food manufacturers have separated various prepared foods into stages at which they should be given to the baby, pre vegetables, vegetables, ... she just fed ours whatever we were eating.

i don't mean this flippantly, but since the first math book pretty much in history is Euclid, historically there is no such thing as pre - geometry. I.e., I recommend over pre - geometry, rather "pre modern" geometry, i.e. Euclid. really. just a suggestion, feel free to ignore.

sadly, the only copies i see for sale are priced at over $400. I will not pay that. I would rather seek it in a university library.

@Arcadia; I have not read it, but learned of it too at the same time as of Liu Hui, who perhaps commented on it. I am newly interested in seeing it after this discussion, since I benefited so much from reading Euclid recently. The man who introduced me to both Chinese authors, David Mumford (Fields medalist, former Harvard professor and personal acquaintance), used that book and its similarity to Euclid, to argue that mathematics is a universal truth because in all cultures the same phenomena are observed. So I would be interested to see how the nine chapters develop geometry.

here is another problem that interested my student, from a contest, that he asked me how to solve: start from a circle of radius 1, and inscribe a square around it. Then another circle around that square, then a regular octagon around that circle, another circle, then a regular 16 sided polygon,... and continue, alternating circles with regular polygons of 2^n sides. what is the limiting value of the radii of the circles? (answer π/2). After I solved this, I was asked by the contest maker if my method would also solve the case where each succeeding polygon has, not twice as many sides, but only one more side than the previous one. I could not see how. Then my student suggested doing the case where each succeeding polygon has 3 times the number of sides as the previous one, a much better next case scenario. I made progress on this one but did not settle it. Still his quick insight into finding the next natural question impressed me. Can anyone solve this? Forgive me if I am just indulging myself here, outside the realm of interest of the thread. But it is so interesting it sparks these comments.

here is a nice example of problem finding by one of my 10 year old scholars. he was interested in fibonacci numbers (1,1,2,3,5,8,13,21,.....) where each number after the second is the sum of the two previous ones. He observed that it seemed as if, denoting by f(n) the n th fibonacci number, then f(k) divides f(n) whenever k divides n. I had never noticed this and it did not seem obvious, but he had checked it for maybe all k up to 6. Together we proved it, using an idea of his. It turns out to be a well known property of fibonacci numbers, proved early in maybe a grad class on them, or maybe any number theory class, but i was very impressed at his perspicacity. After reading up a bit on these numbers, we were also able to show that the the gcd of f(n) and f(k) equals f(gcd(k,n)). that was our best day in calculus class, i.e. the day when he asked "can we discuss something besides calculus today?". To me this shows more clearly than anything else the fact that this boy can be, and really already is, a mathematician. It is often true that a senior mathematician, equipped with more technical tools, can prove something he has his attention drawn to. But we all admire, even envy, the imagination and insight of the young person with a fresh eye, who sees the truth in nature even before having the power to prove it. Without this ongoing process of new discovery, mathematical research would stop. Of course many senior mathematicians also have fine insight and can look into the future past what their skills can yet reveal, to suggest avenues of research for the next generation.

The more imaginative and insightful students do generate their own problems, but in math this is not at all always the case. It is very easy in math to pose a problem that one will not be able to solve as a student. so the advisor can help focus and mitigate the difficulty of the project proposed by a naive student. Experimental sciences are also different since one can do experiments and explore, generating both positive and negative results, and then comment on them and write them up. In pure math. negative results cannot be published. you either find a new truth or you wind up after months of work with nothing at all. E.g. one may be interested to know whether every even integer > 2 is or is not the sum of exactly two prime numbers. Since however this has been studied for many decades without success, it is unsuitable for a thesis. The advisor might however suggest a modification, such as trying to prove that the ratio of the number of even numbers less than X, divided by the number of positive integers less than X which are sums of 2 primes, approaches 1 as X --> infinity. this also may or may not be trivial, as i have not thought about it, but i would guess one could attack it with the famous "prime number theorem". Or, a perfect number is one whose smaller factors add up to the number, such as 6 = 1+2+3, or 28 = 1+2+4+7+14. It is widely believed that no odd perfect numbers exist, but if a student proposed to work on this, his advisor might say that a suitable PhD thesis might be just to prove that if one did exist, it would have a large number of prime factors. someone of my acquaintance generated such a problem for himself as a student. Often a student just knows he/she wants to work on a topic say in homotopy theory, or singularity theory, or moduli, and the advisor may then suggest several more narrow areas, and try to help the student discern those which interest her/him especially, finally arriving together at a specific problem. But many students just ask for a problem to solve and the advisor provides one. If the student solves it too easily, the advisor may provide a more challenging one until he/she feels he has the one that teaches the student as much as possible in the given amount of time. One friend of mine actually entered grad school with a "thesis worthy" problem already identified and already solved, and merely wrote it up after a requisite amount of time, along with other discoveries made in the meantime. I myself needed a lot of help identifying a suitable problem, but once I became familiar with it, made significant progress, and helped generate new methods that, in the hands of more talented workers, led to resolving previously unsolved questions of wider interest.

this reminds me of the importance i once had pointed out to me, of "problem finders" in mathematics. some of us just try to solve problems given to us by others, but who decides what problems are interesting? most PhD students depend upon their advisors to provide a problem, which is both interesting and solvable, but some students have noticed problems on their own with which they are fascinated. I suspect problem finders are generally held in even higher regard than solvers, but that regard is usually established by their first solving an important problem, and afterwards posing one that they have not solved. this latter contribution allows others to play in the game as well, and every mathematician would prefer to be given a problem to solve rather than just listen to someone explain how he solved one. Similarly, a good teacher is not one who just explains clearly how to solve a problem, but who presents an interesting problem to his students and inspires them to attack it. Learning math is not about just acquiring information and technique, but about acquiring a way of thinking, a habit of mind. Sometimes we meet students who do not like books that present math as a sequence of problems to be solved, or discoveries to be made, but merely want a simple statement of ":what do i do next?" These students are missing the main benefit of a mathematics education, and it is our challenge to initiate them gently into the wider world of thought somehow. In my own case, I think I shrank from such exploratory presentations for a long time , out of fear of not getting "the right answer". This is also why we seek books with such answers in the back. If we can somehow convey or at least suggest that the search, or journey, is part of the game, and try to remove the compulsion to get the answer, we may help. I.e. the benefit is in the struggle. It may help also to encourage students to pose questions in math whose answers they are merely curious about. As long as I only wanted to always know the answer, I was afraid to raise too many such questions. [edit: see Barb's following post and the end note: “I am not a teacher, but an awakener.†― Robert Frost]

here's an interesting related post. Of the 150 "greatest" mathematicians listed here, born by 1930, only 3 are Asians, and by far most are Europeans. Ignorance? I myself was over 60 when I first learned of Liu Hui. What would a more up to date list have? There may also be some bias in the choices since only 2 are women, (#26 is Emma Noether. This reminds me of the refusal to grant her membership to the Gottingen math faculty based on her gender, to which David Hilbert responded that the faculty was a scientific body, and "not a bath house!") http://fabpedigree.com/james/mathmen.htm here is a list of the 56 fields medalists (somewhat similar to Nobel prize in math) since 1936, of whom about 6 have Asian sounding names. Thus in the past 2000 years Asian names on these 2 lists are up from 2% to over 10%. http://en.wikipedia.org/wiki/Fields_Medal

I found the comments above on sounds in languages interesting as well, and the possible link with mental functions. There is an ancient teaching in Indian tradition on the link between sounds of different frequencies and different locations in the physical body, or "chakras", which are thought of as centers of energy. The higher the location of the point in the body, the higher is the frequency of the sound said to resonate with it. E.g. bass notes resonate theoretically with the lowest, or 1st chakra, (the base of the spine) associated traditionally with survival. The 4th chakra is one most people may actually have experience with associating to a note. This is the "heart" chakra, associated traditionally with compassion, and resonating with the sound "ah" which occurs in many ballads of love and worship. Vocalize the sound of "ah- men", and you can perhaps feel the vibrations in your chest from the first syllable, (and in your head at the end when the humming sound of "nnnn" occurs). Higher pitched sounds are thought to resonate with the (5th) throat chakra associated with teaching, the (6th) forehead or "3rd eye" associated with objective insight, and the highest ones with the top of the head, or 7th chakra (???). Interestingly, this would suggest that higher pitched female voices are more likely to vibrate the "higher" chakras. For full disclosure, my impression is that western science does not confirm the theories of chakras and their associations. I confess to once hearing a music professor remark that he had no idea why "high" notes are so called as he knew no connection between altitude and pitch, and almost raising my hand. (His comment seems odd though since higher pitch is associated with higher frequency, or did he mean why is a larger number called a "higher" one?) As to connections between language use and mental function, my most recent AARP bulletin had an article stating that people who speak two languages throughout life are less disposed to alzheimer's, (although learning a second language as an adult apparently does not help). On still another aspect discussed here, parental involvement, I used to notice that some of the best athletes in basketball, especially ones from "underrepresented groups", such as Bobby Hurley, Mike Dunleavy, Pete Maravich, had fathers who were basketball coaches. In the same way, highly accomplished academicians and mathematicians may often benefit from having parents who are teachers. E.g. everyone on this forum is a teacher, and this presumably especially benefits your children.

This is a very interesting thread, and the personal experiences make it extremely believable. I think all the facets of the problem mentioned here are valid, and I cannot discern one reason for the phenomenon being examined. I am led to add a few remarks on the US cultural aspect of motivation to become a professional mathematician, especially a college math professor. I.e. in the US, being a math professor is not a position that is highly respected generally (although of course it is in some circles) or highly paid. This may make it more attractive to recent immigrants, who may come from a culture where the opposite is true. I suspect bright and/or competitive women, and men also, in the US, are more likely to obtain high paying and respected jobs in business, law, medicine, entertainment, technology,..., than in pure math. A quick count of the full math professors on the UGA website showed roughly 10 out of 19 were foreign born, and of the 2 distinguished research professors, one was foreign born and the other, an American, had an asian surname. Of the 3 recent female full professors who come to mind, 2 were foreign born. As a college professor, I saw years of attempts by the US government to increase participation in math by various "underrepresented groups", especially racial minorities (except asians), and females. In some cases, as a department responding to these pressures on favored hiring or graduate acceptance, we chose European or middle eastern females and males from Africa. Then the pressure shifted to a stress on promoting participation from US citizens, since that is where the real deficit was and still is. There is a multi million dollar NSF program called VIGRE, which is open only to US citizens or permanent residents. This aspect of the program is not even mentioned prominently on some university web pages describing the program only by its acronym: "vertical integration of research and education(?)". http://www.stat.ncsu.edu/programs/grad/aid.html http://www.math.uchicago.edu/~may/VIGRE/ When we introduced this program at UGA, we found some of the entering students benefiting from it were not ready for beginning graduate work. There was during this time some tension created by the availability of especially generous funding for which the qualification was restricted to American students, when in many instances the most mathematically qualified students were foreigners. The program has apparently since been terminated, and replaced by another federally funded one.

I'm not sure of the definition of "Asian" in these cases. Of the kids I know in this highly gifted group, several seem to have one Asian parent and one Caucasian parent. Thus although they might look Asian in a photo, they are perhaps genetically, and culturally(?), as Caucasian as they are Asian. And of course they are all Americans.

i agree $40 is a lot less than i expected and i would also buy it for that; and my kids are grown, so it would be for me.

By the way, there is a superb summer program in Park City Utah, and overseen by the Institute for Advanced Study in Princeton. It is not really suitable for very young kids, but extremely beneficial to a select group of older ones, college age and up. This is worth knowing about I think, for its future value to some of the people here. The program is so advanced that as a professional mathematician and Professor, I learned a lot from the lectures aimed at graduate and even undergraduate students. This is really a good program in math. https://pcmi.ias.edu/program-index/2015 One big benefit is the inclusive nature, with a diverse group of participants. I enioyed also interacting with the high school teachers there. Once I met Irving Adler there and had lunch with him. He was an academic who was so principled and brave that when obliged to sign a loyalty oath as a condition of employment at his institution in the 1950's he refused and earned a living writing popular books on math and science. (I found a better article than his obituary.) http://en.wikipedia.org/wiki/Irving_Adler

next years epsilon camp, i.e. summer 2015, will be in st louis at wASHINGTON UNIVERSITY ST LOUIS. http://www.epsiloncamp.org/#apply But you probably all know this. I wish it had been here in seattle again so i could vist but maybe it will benefit some of you to be more centrally located.

Although there are obviously many fixed formal math techniques, and these are what is taught in school and books, math is also an open ended world of new and related problems and these continually call for new ideas and insights, which spring from intuition. So the further one goes in math the more important intuition becomes. Math research is solving a problem no one else has solved before, and the reason is often that existing methods were not quite adequate, i.e. the solution calls for some new insight. The way to prepare for the day when a problem is encountered that does not yield to the known methods is to pursue ones own way, via ones own intuition, in solving more elementary problems. One of our profs at college, the great Raoul Bott, said we should try to do the problems he assigned by our own methods "before your heads get so full of other people's ideas, you are no longer able to generate your own". So learning to do math is more than just learning the existing methods that past mathematicians have discovered for doing those parts of it they have mastered. But when we do succeed in solving a problem by our own methods, we need to have the discipline to clarify our intuition, to reveal why it works, both so others can understand us, and so we can apply the same principle again. So intuition gives rise to methods, but only if it is made precise and clear. And since we may not always have a bright insight, we need to possess a toolbox of tried and true methods as well.

I'm not sure what your child's goal is, but if it is merely to advance in depth of math understanding, independent study from a good theoretical calculus book might suit, such as Spivak, or Apostol. As you may know, Spivak is one variable calc done from scratch, but at a deeper level than AP calculus.

I am myself conflicted on when one should insist on a teaching technique the student rebels against, even if one is sure of its value, but I do recall that the best teachers I have known, would never tell a student the answer.

I was puzzled that some of those Davidson verbal cutoff scores are higher than I would have thought realistic, based on my experience taking the SAT in about 1959, and my kids taking it in the late 1980's and early 1990's. Then I read this, which says the verbal scores were raised by 70-80 points in April 1995: http://www.wsj.com/news/articles/SB124397818883378713?mg=reno64-wsj&url=http%3A%2F%2Fonline.wsj.com%2Farticle%2FSB124397818883378713.html So bear that in mind if comparing with scores obtained 20+ years ago. I also see now that Davidson program is for profoundly gifted. So it all depends on your definition of gifted. Having taught college for years to students who thought they were doing well with math SAT scores around 500, I think any middle schoolchild with scores over 500 or especially 600 is pretty special. Still thinking of the SAT test from 20+ years back, I seem to recall they removed the one section that was most sensitive to measuring what to me is a key indicator of "intelligence", namely the analogies questions. Those were the questions that actually required thinking and reasoning. I don't know if the SAT measures giftedness or intelligence, but it does measure preparedness. To avoid discouraging a kid, I would suggest purchasing an actual book of old SAT tests and taking them before the real thing. In my day, there was very little in the way of formal preparation, but I think those of us who were on the math team for instance, and who thus spent a lot of time taking similar tests had a huge advantage. But in any case there is now no need, and I think little reason, to take it without knowing in advance roughly how you will score. I.e. taking the test is not a lark, but a number with consequences, so it benefits one to try to control that number. Somewhat related if peripheral, is there real money available for high scoring kids today? In the 1960's the merit scholarship was based on SAT scores and paid very generous college stipends, more than 100% of Harvard tuition for example. Maybe we could advocate for wealthy companies to resuscitate these scholarship programs.

i am puzzled. did you say both of you are math phd's, and yet you are asking for advice on math education?

in regard to dumb hiring decisions, as a personnel person i can remark that it is extremely difficult for us to discern the quality of a potential hire. we just don't know enough to recognize the best person, so we fall back on stuff like, where did they go to school, and where did they publish, and how many pages did they publish in how many years. otherwise we have to spend hundreds of hours actually reading their works, and even then we don't know enough to assess it. so most such stupid decisions are made because we couldn't tell the difference between the better person from the small school and the weaker person from the famous school. but eventually the difference becomes evident. in an attempt to do justice to the hundreds of applications we received for one job, i once spent over 30 hours reading applications one weekend, without food or sleep, but after a while you get tired, and your children need their father. i hope we hired the right person, but many fine applicants probably went un detected.

UNC is an excellent school in many areas I think, an example of an outstanding university in a region (my home region of the south) not known for its educational advantages. Kudoes should go, I have heard, to a governor some years ago, who set an admirable standard of funding education in NC, including high school level opportunities.

here are some more "it depends who you ask" examples. Vanderbilt is ranked something like #46 by US news, and UGA and UVA tie at #52, But the American Math Society ranks UVA in its Group I, and ranks both Vandy and UGA in Group II. Wait a minute, there seem to be 2 schools called NYU, and the more famous one is ranked in the top 10. That makes more sense.

those rankings of math depts may be open to various interpretations as well. E.g. as mentioned, GaTech ranks #28, UGA ranks #52, and there are at least two math professors at GaTech, one of whom received the PhD from UGA, and the other transferred to GaTech from the UGA faculty, (one was my colleague, the other my student in graduate algebra). NYU ranks #73 I think, but has several (to me) more famous math faculty, such as Bogomolov, Cheeger, Gromov, than either of the other two places named. (oops, correction below.) I am sure many other such possibly unexpected phenomena exist.