Congrats to you and your daughter.I like girls, think they all are cute, sweet, pretty, darling, adorable, precious, should never be told “no”, etc. and hate to see them suffer. And too many of them have suffered. At times I’ve suffered and hate to think that a girl would ever suffer. Maybe that’s my being ‘protective’; Mom said that if I’d had a sister, then I wouldn’t have had such a high opinion of girls!I hope you and your daughter don’t suffer.I know if I had to do anything that ‘girls do’, then I’d likely fail and if successful would no doubt have worked so hard at something I just was not built to do I would suffer. So, if a girl tries to do ‘boy things’, then maybe she will suffer, but I can’t say for sure because to me girls are like something from Venus for someone from Mars.For some aspects of ‘girl thinking’, math is cruel, unforgiving stuff. Math has it’s own way of doing things independent of and oblivious to what any human feels.Math totally saved my tail feathers: In the early grades, I was so socially awkward the teachers apparently wished I’d go out with the trash paper. But in the ninth grade I discovered math and that I could do it, and then no teacher on this planet, no matter what the heck she thought about me, could refuse to give me credit.In college, once I got out of the required ‘humanities’ courses, I took only math and physics, and the math was nothing but fun and games all day; I got “Honors in Math”, a good GRE math score, etc. Early in my career in mostly DoD work around DC, math saved me again. My current project has some original math at the core, and the math prerequisites are beyond all but a tiny fraction of full professors of computer science. I know the core math is solid because I did it with theorems and proofs (essentially all math taken at all seriously by the math community is, when polished and presented, just theorems and proofs).So, if can do math, then it can be powerful stuff. How powerful? My view is that it is the main key, with no alternative in sight, to most of the powerful future of computing, first by humans and later by ‘real AI’.So, I like math. But, then, I’ve been able to do it.If your daughter can do math and likes it, good. I wish her well. If she has to strain to do well in math, say to achieve some recent ‘feminist’ goals, then later she may, did I mention, suffer?But even “good” has to be qualified: Even if your daughter likes math and is good at it, so far the world of math, say, from trigonometry and solid geometry in high school on, is mostly a ‘man’s’ subject. In some colleges, there is a second track for ‘teacher training’, for K-12 math teachers to be, and there the books are easier and there are more women. Else math is a severe discipline, and the people the best at it now are men with high natural talent and uncompromising, iron determination.Unless the world of math changes a lot by the time your daughter gets to college, what she will see in solid math courses are essentially 100% men. But, there, she can exploit the same situation I did: If she can prove the theorems, then no one can deny her. There is some opportunity here since some of the exercises are difficult enough that likely at most one student in the class will get it. At least in ugrad school, a student who gets a few such exercises will be “honored above all other” students.Starting now, it should be easy for her to race ahead if both of you want. Maybe a person ends up knowing the material the same if they learn it after the course or before the course, but the second is much more impressive during the course! So, get her started with high school algebra. Prerequisites? Essentially none. So, get her several highly recommended books and let her dig in — for each ‘lesson’, read the material and, then, work the exercises. Let her have several books and concentrate on the one she likes best but have the others available if they are helpful occasionally (no book is perfect or is the best at everything it does; if something is not at all clear, then try another book). If know a good college or university math prof for occasional consultation for you and/or her, then better. No, a K-12 math teacher won’t qualify! If for the books you can get the teacher’s manual with the answers to the exercises, terrific! Let her learn how to work her way through a math book this way, that is, essentially independently; if you wish, give her praise when she is successful and help where she struggles.Then, sure, do the same for plane geometry (gorgeous stuff; like eating caramel popcorn — can’t stop), second year algebra, trigonometry, solid geometry, a little analytic geometry (the conic sections, amazing stuff), and college calculus. Same way: For each subject, have several of the best recommended books, pick a favorite and make occasional use of the rest if wish, study the text, work the exercises, check results with the teacher’s manual, and have a good math prof for occasional consultation.Then continue with linear algebra; work through 2-3 highly recommended books; and there end with the classic Halmos, ‘Finite Dimensional Vector Spaces’ (a crown jewel of civilization; a finite dimensional introduction to Hilbert space theory; from the knee of von Neumann when he was at the Institute for Advanced Study). Continue with Rudin, ‘Principles of Mathematical Analysis’ (calculus done carefully, plus a lot). For the exterior algebra, use Cartan’s book now available in English (high end approach to some of the crucial math of general relativity). Optionally take a pass through Spivak, ‘Calculus on Manifolds’ (there is a typo in Spivak I can help her with if she gets that far!). If do well, then have done well with the Harvard Math 55, sometimes called the most difficult ugrad math course in the country since at least at one time Halmos, Rudin, and Spivak were the standard texts. Next take a good pass through ‘abstract algebra’ — get some really good facility with set theory and then groups, rings (do some coding theory in high end electronic engineering), some number theory (right, for cryptology if nothing else), fields, vector spaces (a little more general than saw in linear algebra), maybe some category theory, maybe some group representations (e.g., part of how to identify molecules). Take a pass through ordinary differential equations (e.g., Coddington, elegant, polished, balanced); touch on partial differential equations (warning: can get lost here, so don’t; the level of precision will likely lower than in, say, Halmos or Rudin and, otherwise, way too difficult to read, and a lot of ‘engineering intuition’ is justified).Now can knock off quite a lot of serious mathematical physics as footnotes or small exercises. Would be right at the head of the class in anything about ‘machine learning’ or ‘big data’ if can stand the horribly low quality of that stuff!Take a first college course in probability and statistics (don’t take it very seriously; do that later).If wish, take a pass through optimization as in old ‘operations research’ and now part of computer science, machine learning, much of now classic mathematical economics, and wherever else it can be used now or in the future (there stand to be some uses) — linear, network linear, non-linear, dynamic.Then, right, heck, take the math GRE. If do well, then pick a university with a good math department, e.g., Princeton, get the materials they recommend for their Ph.D. qualifying exams, and dig in. Last time I read what Princeton said about that, no courses were offered for preparation for those exams, and students were expected to prepare on their own! So, your daughter will be fully ready to dig in. Likely she will have to dig into measure theory and functional analysis as in Royden, ‘Real Analysis’ and the first half of Rudin, ‘Real and Complex Analysis’ — gorgeous stuff; in places beyond belief. ‘Measure theory’? The grown up version of freshman calculus.With measure theory and functional analysis, have what is now something of a special opportunity so far not heavily pursued by US pure math departments: Take a pass through a few of the high end texts on ‘graduate probability’. The main authors are M. Loeve (the whole two volumes are a bit much), J. Neveu (succinct and just brilliant), L. Breiman (super nice to read), K. Chung. Then take a pass through stochastic processes, say, Cinlar’s little introduction (it has a lot of implicit monotone class arguments and, thus, often can’t be taken too seriously by a student on a first pass), and then, say, Karatzas and Shreve and/or Chung and Williams (Williams is a woman!). Then are set to do some high end work in some parts of mathematical physics, electronic engineering, and machine learning (the top end of that sad field). E.g., can do power spectral estimation on the data from the 3 degree Kelvin background radiation! Also will have an excellent start on high end versions of math for Wall Street.Now, pick some problems, pure or applied, in math, physics, engineering, computer science, the social sciences, etc., do some research to get solutions, largely just as math exercises, publish a few papers, and show up at a university for a year and get a Ph.D., possibly with the published papers as the dissertation. Social science? Maybe find a good problem in ‘social media’ and/or virality!A ‘good problem’ in academic research? First, a problem can solve where the solution will be ‘new, correct, and significant”. Second, for business, the solution will be powerful, valuable intellectual property can convert to software and run hidden in a server farm!Or, if want to learn more about how to do research from some real masters, then take some graduate courses (at Princeton supposedly they will all be introductions to research) and do some supervised research. From such on-campus work can get some of the ‘values’ and ‘socialization’ available there, if want such.Then get on with life. “Look, Ma, few or no classes!”, tuition, dorm life, drunken frat parties, etc.The description above is essentially what I did, that is, what worked for me, largely independent study, although I was dragged down by the classes. Tough to study when the teacher is exercising their vocal chords! In grad school, far too much of the week was spent driving to/from the university and sitting in classes, thus, leaving too little time for the crucial studying.For how to do research in applied math, I essentially discovered that on my own. My view is that the exercises in Halmos and Rudin are good starts. A famous recipe for rabbit stew starts out, “First catch a rabbit.”. Well, my recipe for good applied math is, “First get an application.”, that is, a good problem to solve. For more, have a lot of intuition, wild guesses, and evaluating wild guesses with lots of intuition; the careful theorems and proofs are mostly or all just at the end — no one ever told me this; maybe you first saw it here! Some really good math profs may occasionally have some more good ‘meta’ lessons to share. Note: I’m not such a “really good math prof”!Some advice: Don’t regard the books as perfect. While the level of quality of the best books is astoundingly high, there may not be a perfect book yet. So, each book can have some errors, obscure passages, poor explanations, out of place or way too hard exercises, wrong answers in the back of the book, a section that is way too hard considering the level of the rest of the book, etc. So, don’t get stuck. Don’t spend two weeks on a single hard exercise. If study well and still something is obscure, then look at alternate sources, go on and maybe come back later, use Stack Overflow, talk to a good math prof, etc. It is common to assume that need to understand every point, topic, theorem, proof, exercise or may be missing something crucial; instead, nearly never will be, and even if so then can patch it up later. So, don’t let some such chuckholes in the road cause giving up on the whole effort.Three big topics to skip over quickly, i.e., don’t get stuck, (1) the axiom of choice, (2) the Navier-Stokes equations, and (3) the question of P versus NP. Take one of those problems seriously and can go deep under water and never come up again.Most of what I have outlined here is ‘pure’ math and is by far the most highly polished, exact, and reliably correct part of civilization; so, if move into applications in science, engineering, technology, etc., then be prepared to face a lot of really sloppy work!In nearly all high end work in science, engineering, technology, and computer science, the workers are seriously handicapped by not having enough in math and, thus, struggle terribly. With what I’ve outlined here, your daughter will find a lot of such work as easy as shooting fish in a barrel.Before having your daughter go very far along what I have outlined here, get some second opinions!Generally, be careful, be very careful.Most important of all, don’t let her suffer. The girl who would get damaged is your daughter.In the end, you might prefer that she concentrate on finding a good husband and then giving you some good grand children instead of a stack of research papers.I’m failing to see why a career in math is better than being a good wife and mother.

She’s kicking ass at math…just got 100% on the state practice exam. She’s actually good with people (albeit charmingly sassy). Spelling…not her forte. {She can’t be bothered.} You’re probably right, I need to cut this off at the pass with full skirts, rainbows and unicorns.

You would think so but when they removed all the books in the library to make way for a paperless learning centre and didn’t give parents or teachers adequate learning support the shininess soon rubbed off. If is fantastic in some ways but there are massive inefficiencies just trying to use the technology. I would caution any school to spend as much time on teachers and parents as they do the kids to make the project a feasible reality a lot faster. Additionally, I fundraise to build libraries in developing countries and I sure as hell didn’t think I would have to be lobbying for a real library with books at our school.

You’re quite right.

Sorry, missed the write up and it referring its interaction with TV. The watch interacting/directing other devices will be big thing.