I've seen many posts from people here looking for similar resources, so I thought I'd announce that I've just released a new book aimed at gifted math students in grades 3-5. It's called Hard Math for Elementary School.

In my day job I'm a professor of economics at MIT, but I'm also the parent of three very talented daughters. When we couldn't find other good options I've coached middle school math teams and held informal classes for elementary and high school students.

The book and accompanying workbook are based on what I did with my elementary school group. It's aimed at gifted students in grades 3-5. The idea is give kids an alternative to moving ahead to do math with a higher grade.

Instead, I try to cover grade-level math in greater depth and cover other things that they'll never see in school. I ask much harder questions to try to build problem solving skills and try to make math fun.

I envision the book as something kids can do with the parents at home or talk their teacher into letting them do at school. (I've found that this is easier than trying to get other accommodations made in reluctant schools.) The book is designed to be very teacher/parent friendly -- the text spends more time than other books explaining the math because there are lots of things many teachers won't really know and I also sell an answer key at a very low price.

For now the book's only for sale on Amazon. If anyone has any questions about the book or anything else I'm happy to answer them.

I like the book - think "AOPS, but different in several respects". (I quickly looked on amazon which shows much of the book.)

Among many good examples:
- p. 26: "put in some addition and subtraction to make round numbers"
- p. 207: "which is bigger: 87/437 or 85/435?" ("funny subtraction").

The author's description above is correct: "Hard Math for Middle School: IMLEM Plus Edition" by the same author also looks good.

(The "middle school" book is probably more competition-oriented, while the "elementary school" book is more like "a must know, solid elementary math".)

Thanks for posting about this book; it looks interesting. Plus (I guess more for the middle school book) nice to know about what kids study for math competitions if you're in an area where there isn't much of that available.

Actually it occurred to me that it would be great to hear what math training you recommend for kids who are gifted in math and thinking about what they might do with it when they grow up--is it just the regular courses that kids need and they will do fine (e.g., algebra, geometry, calculus in high school and then a math or economics major in college??) or are there other areas (for example, coding/programming) that are very helpful in economics or other math careers? I am not mathy myself and my DH is a statistician, so I would like to learn more about what our very mathy DD might be able to do with it and what she might need. Perhaps there is another book or a professional society that provides some guidance in this area?

Dbat, undergrad economics only required calculus here. Most of the work through grad school involves basically the first year university series of calculus, stats (flowing into econometrics), and linear algebra. Programming is usually done in economic/stats software packages like Stata, Eviews, SAS, etc, but knowledge of the basic mechanics of programming going into university would be a considerable advantage.

Feel free to PM me if ever you have any economics questions, as economics was my first love.

The older Middle School book is more competition-oriented. It is similar to AOPS books. Apart from the organization/prose that's optimized for MA math teams, I think of the biggest difference is that my book is more designed to be read and you could skip doing the problems whereas for AOPS books you really have to do the problems as you go along.

The Elementary School book is similar in philosophy -- you read it and it explains things -- but the prose is all about the joys of doing hard math. Unless kids are also extremely advanced verbally it's probably more a read-with-your-parents book. For kids who don't love reading and/or are impatient it can also be a text for the parent -- the kid just works on the workbook worksheets and the parent who has read the text can explain things when they're stuck.

Mostly I think kids just need to learn math up through Calculus BC in high school because they will take the other things they need in college. I'm not a fan of the AP Stats curriculum and also don't see a whole lot of value to learning multivariable calculus while in HS. Many students who go on to graduate school in economics major (or double major) in math as an undergraduate.

If kids are able to do more in middle school/high school I do think it's valuable for kids to get the deeper knowledge of high school math that you get from doing contests like the AMCs. And while this is appropriate for fewer kids, I also think it is also great if kids can get some exposure to abstract math and proofs while it high school.

I am a big fan of learning to program in HS. Kids won't learn the particular languages that economists use, but the skills they learn will translate. When I'm hiring research assistants I care more about programming skills than anything else. You can obviously put this off and learn programming in college, but it can be intimidating to go into a college programming class if you don't have prior experience and a lot of kids end up never learning.

The charter DS will be attending this coming year is touted to be especially beneficial for those who are either behind or else ahead academically.

The principal/teacher was speaking to us at an orientation and she said she was teaching Calculus to an eighth grader.

What do you suggest as a way to market this to charter schools?

Where I live there's no requirement that schools accommodate gifted students, so it's all about trying to persuade the school that what you want is a good idea and not too hard for them.

My first step is always trying to establish credibility so they know I'm not just a pushy parent. I know some people do this with testing. I've taken the approach of letting them get to see my kids in class and spending a lot of time in the school volunteering.

During a school year I start with the classroom teacher. Teachers are happy to have easy problems, so if you explain what you think the needs are, and show them materials that you think solve the problem, they can be happy with this. It's easier to get an accommodation that your child can work on their own during math time than to get them to actively teach something, but I'm always happy if I improve on the status quo. If your class has kids who could form a math group and work on the book together that's obviously better. Having a group of parents supporting each other is also more powerful.

If you're trying to set something up in advance for next year, you do have to go through the principal. Your principal anecdote sounds promising. If she's personally a math teacher, then that's even better and encouraging her read to through the book could be a good method of approach.

Others here probably have more experience than I do.

As the mother of a 7 year-old daughter with exceptional mathematical aptitude, I generally feel saddened and concerned when I see "hard" or "advanced" math books with only boys or principally boys on the cover. You have the appreciation of at least one mom for placing a girl on the cover of your books. Thank you!

gellison--thanks very much for your posts, and the advice. We are trying to get DD into a school that we hope will challenge her in math but are trying also to line up other options in case that does not happen for whatever reason, so it is helpful to hear what others have done.

Mom2277--in addition to the "Hard Math" books, have you seen the Danica McKellar math book series? I guess they're supposed to sound a little like teen girls' gossip magazines in some ways, but DD has enjoyed them and they're mainly about math with some girly stuff thrown in to make it less threatening(??). Which I suppose might be objectionable in some ways but overall it seems like a good thing, especially as just one part of what girls might read about math.

I have always had a lot of girls on my math teams. As a social scientist I have to say that it's probably just a coincidence plus the fact that I have daughters. But it could also be that my more reading-oriented approach is a good fit for many girls.

If it's not too much bragging, you can find a story about some of my girls at
http://mathprizeforgirlscommunity.blogspot.com/2012/08/newton-north-math-prize-shout-out.html

gellison--that is awesome; thanks for posting that! Congratulations to you and your daughters. It's wonderful that you took the time to do all that for the team, and that they did so well.

I have been looking into this kind of team a little, but it seems like opportunities here (a mid-sized, Southeastern city) are pretty limited. Which is kind of surprising to me given the number of high-tech, mathy businesses and also universities not too far away. Maybe I need to look harder to find the local groups, which I will do, but I am also wondering whether there are any such or similar teams that are not geographically limited. After all, in the age of the internet and Khan Academy, it seems like maybe somebody would be running a group via Google Hangouts or something similar?? I will keep searching. Not that our own DD is very into competitions at this point, but it would be nice to have more options. And I'm sure there are other girls in a similar situation who might be even more interested in math clubs.

WOW!!!!! You must be excruciatingly, if not unbearably, proud of your daughters.

Thank you for sharing the link about them. My daughter beamed, seeing the picture of "real" girls who had won the Math Prize for Girls, and reading of the team's growth from middle school to their unparalleled caliber in high school. I think those stories of growth are so important, regardless of how gifted someone may be.

Thank you for allowing my daughter and many other children to benefit from the work you did for your daughter and other local children by publishing your worksheets and books. You are to be lauded for your work with your children and their schools. With your professional demands, I cannot imagine anyone faulting you if you acted no further. A heartfelt thank you for doing the additional work to ensure that children like my little girl who do not live in an area with robust resources could benefit from your formidable mathematical mind as an elementary and middle school student.

Dbat, thank you for the suggestion to look at the Danica McKellar books. I'll check them out.

We face the same issue. We live in the southeast somewhat close to a mid-size city, and I haven't been able to find math circles, etc. in which they could participate, even when older. I've thought of starting a math circle, but their age and ability are so mismatched that I cannot really envision it working. My children have zero tolerance of something too easy. (Obviously, we homeschool.)

gellison, thank you for posting this. Wish it had been available a couple of years ago! Due to his triple acceleration, DS is beyond the elementary book, but I ended up buying your middle school book from Amazon.

One of the comments/reviews referenced a high school version of your book, but I could not find a link to it. Is that book out of print or did the reviewer mean to reference someone else's book.

I think that reviewer is probably just mistaken. I do have some handwritten lecture notes from when I've taught high school students about writing proofs, but have only given copies to a few kids and none sound like the one she mentions.

If kids are interested in high school contest-style math I'd recommend the AOPS books.

Thanks for all of the great insights-- I realize this is probably a touch off-topic for the thread, but--


Would you mind elaborating a bit?

Because of the lack of any direct instruction at all (long story-- cyberschool), we've had to take a strange path through high school mathematics. As a PhD scientist, statistics is an area where I feel far more comfortable than I do with teaching calculus to my DD. She is not probably 'mathy' the way some kids are; that is, gravitating to theoretical exploration. But she certainly enjoyed AP Physics B after she slogged through algebra-based classical mechanics, and loved geometry and algebra II. She doesn't like economics, even though she could see the appeal of the math in it (again, this is a longish story).


What things should I be on the look-out for as I coach my DD through AP stats? (Understanding that I've taught the material to undergraduate biology and chemistry majors, and a few graduate students in pharmacology.)

Any recommendations for improving understanding while working with that curriculum? Or is it just fatally flawed in your estimation?

One thing that I noticed with AP Physics B (aside from the clunky approach mandated by algebra) was that the syllabus was very prep-focused, VERY broad, and not very mastery-oriented. This wound up working out fine for my PG DD, who was more than able to work to a mastery level at that pace, we just had to offer explanations and add a supplemental text that was more conceptual than the example-heavy Giancoli that the course used. She found that she really enjoyed the modern physics semester.

Anyway. I'm just curious here and wanting to pick your brain a bit.

Our copy arrived last night, and I plan to share it with DD9 this weekend.

Howlerkarma, my lack of enthusiasm for AP Stats is related to the mastery issue.

I think statistics is a beautifully logical and consistent subject as well as an important one. But the right way to understand it is to start by learning probability, then learn the random variable concept, and then think of statistics as functions of random variables. If you do this everything makes a great deal of sense.

In my daughter's AP Statistics class they went in the opposite order. They spent the first few months talking about statistics and then did probability later in the year. When you do it this way, it seems like it becomes an exercise in learning that there are a large number of formulas you'd apply in different situations, but doesn't form a coherent whole.

In our department we teach statistics using Degroot and Schervish which is good, but I know it requires calculus and perhaps also some linear algebra. I don't know if there are AP-level textbooks that go in the more mathematically appealing order. (Like physics I also think statistics is better done using calculus.)

One other thing I would recommend if you are doing AP Stats is to also work with some statistical programming package. Our daughter's class did a lot of things with a graphing calculator which I thought was insane. No one really doing any statistical analysis would start by typing a dataset into a calculator, so I see no point in learning how to do that.


ElizabethN, I hope she likes it. For most 9-year-olds I'd start with Chapter 1 to build comfort. For some advanced and/or impatient kids Chapter 2 can will be a better place to start to let them know there will be things that are more novel.

I appreciate the insight. Sounds like I'll have the same-- exact-- feelings about this class as you have. I agree wholeheartedly with everything you've stated about starting with probability and building from there, because the math just falls effortlessly out of it. Well, as noted-- from the calculus approach it does. I also agree re: statistical analysis programs and calculator magic. GREAT tip to seek out a stats package for the PC.



Again, I appreciate the insight and the tip on a text with a superior approach. Thank you very much.

Thanks! I'll report back how it goes. My experience is that sometimes she dives in full-bore, and sometimes she refuses to look at something for a while and then "sneaks" it when I'm not around. She just finished 4th grade math (she's in a gifted program that works one year ahead), so I think she's probably right in the sweet spot for it.

My DS7 has just finished grade 5 maths (home/virtual schooling), and he has also self-taught himself most of prealgebra by playing around online. But most of what he has seen is "routine schoolwork", and he has not seen much in the way of "challenging problems".

Do you have any suggestions for books or other resources he could use?

Any ideas for getting ready for AMC8 in a year or two?


Also a question about MIT. In some countries it is possible to do an undergraduate degree purely in STEM subjects, or even purely in mathematics, whereas in the USA you usually have to spend a significant amount of time on non-STEM subjects. What is the situation at MIT? Can you focus more on STEM subjects compared to other American institutions, or do you have to do the same kind of "liberal arts" degree that most American institutions have?

My Elementary School book is aimed at this kind of need -- covering some of the finer points of 3rd-5th grade math and asking much harder problems. Parts might be too elementary, but there are very few 7-year-olds who'd have this problem with the whole book.

Among books of elementary school challenge problems, George Lechner's books are probably my favorite. The only drawback is that they're fairly expensive. I'd also recommend any of Martin Gardner's books. Their emphasis is more on puzzles than curricular math, but they're stimulating and there are Dover editions of some. you can get for around $5. Some people also like Edward Zaccaro's "Challenge Math" books. They're more curricular and a bit less challenging.

There are several good books oriented toward Mathcounts preparation. The AMC 8 is less demanding than Mathcounts, but the material is similar so any of these books would be good preparation. "Art of Problem Solving: Volume 1" is a classic. Jason Batterson's "Competition Math for Middle School" is also very good. I, of course, also like my middle school book.

There are also lots of good free online resources at this level. The AOPS website has a resources section where people have posted most old AMC problems. Their "Alcumus" is a nice online tool that asks questions and adapts to what kids know. They also have a set of "Mathcounts Mini" video lessons. My book is organized around old problems from "IMLEM". The IMLEM website (www.imlem.org) has a link to (free!) past contests which provide hundreds of problems with detailed solutions.

MIT is more STEM focused than most American colleges, but you still spend time on non-STEM subjects as well. It's more complex, but roughly the requirements for an MIT degree are: (1) Take 8 core STEM classes (2 math, 2 physics, 1 chemistry, 1 bio, 2 others); (2) Take 8 courses in humanities, arts, and social sciences; and (3) Take a number of courses (12-16) in your major area. The typical student ends up taking 35-40 courses in total. But for a math major, for example, this would probably be at most 50% math by the time you do the other requirements (and most students also do other things just because they're interested).

Thankyou very much for the information.

Looking at the contents for the "Elementary" book, I agree that there are topics not covered in my son's courses. And it looks like the "middle school" book also is more broad than the regular curriculum.

I notice the "Elementary" book has an associated Workbook and Answer Key. Is the same true for the "middle school" book?

I haven't ever gotten around to writing a workbook for the middle school book.

In part this is because the IMLEM website has 500 pages worth of past contest problems with detailed solutions arranged according to the chapter numbers in my book. It's a great resource -- probably better than anything I would have written. And it's completely free!

See http://www.imlem.org/meetdoc/index.htm

For example, to find problems on the first topic covered in my book (Meet 1 - Geometry - Angle Measures) you can click on any of the number 1's you see and then scroll down in the PDF file until to come to Category 2 - Geometry. There will be a sheet with three problems arranged in difficulty order: the first is (relatively) easy, the second is medium, and the third is (sometimes very) hard. Each page is followed by a sheet giving the answers with explanations.

I can speak to this, having graduated from MIT after transferring from UC Berkeley. With the AP courses that I had taken, I could have graduated from Berkeley taking one (upper-division) humanities course the whole time that I was there (which I did take during my one year there). When I transferred to MIT, because they did not give AP credit for many of my classes (and I had 5's on every test except French, where I had a 4), and because they had more humanities requirements, I had to take three or four more, and also more science distribution requirements than I would have had at Berkeley. That said, humanities courses at MIT do tend to have a more science-y feel to them, and they do tend to be taught with the assumption that this is a fun extra, not your core interest.

Glenn, thanks for the pm. I just saw it today and responded.

We got these books and DS7 is enjoying them. Even though he's finished (routine) K-5 Math courses, there are still several topics in the "Hard Math for Elementary School" book that are interesting and new for him.

Thanks for the notice about the new books, and thanks for the interesting comments in this thread. Several of the comments mention the issue of "problems" versus "exercises," and I should mention a FAQ about that point,

http://www.epsiloncamp.org/ProblemsversusExercises.php

which I first prepared for my local mathematics classes (I use the Art of Problem Solving textbook on Prealgebra

http://www.artofproblemsolving.com/Store/viewitem.php?item=prealgebra

in classes including third, fourth, and fifth graders) and is now hosted on the Epsilon Camp website. I'd be delighted to hear any commments any of you have about the FAQ link, and I have put the books kindly mentioned here into my Amazon wish list for likely future purchase for my own family, and maybe for my local classes too.

Just bought the books and love them so far. gellison- I sent you a pm with a question.

I purchased the middle school (IMLEM) book for DS although we are not in Massachusetts. DS is a rising 5th grader who will be studying Algebra in the fall and is able to go through this book independently. It is definitely deeper and more difficult than your typical middle school curriculum. I would recommend this book as a supplement for other strong math students.