I come to you by way of your computational complexity blog. I get that there is some really good stuff there, but frankly don’t understand about 99% of it. What I do understand is that math can be fascinating and that knowing how to ask the right questions can fill a lifetime with wonder. I am now homeschooling one of my children. He has a good grasp of basic math of the elementary and junior high level, but I just don’t have the experience to pose the questions that don’t have clear answers and that can keep a young mind engaged on a deeper level.

I am writing in the hope that you can share with me, or even poll your readers on, the questions that inspired you and them to further your own investigations and advancement in the worlds of math. Ideally, I would be interested in creating some structure that I could share with the homeschooling community so that an unsophisticated mathematical mind could enjoy this journey with its homeschooled child.

You might try some of the books I enjoyed as a child, Flatland, Gödel, Escher, Bach and the puzzles of Raymond Smullyan.

Everybody's story is different. While I did well in math growing up, I didn't really consider taking a career in a math-related field until college when I discovered my need to know why. But what did fascinate me as a child was process, for example what happens to my letter from when the mailman picked it up until it got delivered. Once I started having access to computers as a teen I started thinking about what they could and couldn't do. For example, I remember asking myself whether our school's computer with its fixed amount of memory could eventually print all the digits of π (no it can't). Not until late in my college career did I find out there was a whole field of study devoted to the mathematical understanding of the possibilities and limits of computation. And I was hooked.

Gödel, Escher, Bach is a very special book about Gödel's uncompleteness theorem, not everybody will like the lots of logic it contains. I would suggest The Music of the Primes: a really fascinating account of the human side of Number Theory…

ReplyDeleteI was a perfectly capable math up until my first go-round as an undergraduate. Then it all seemed so horribly unmotivated, and so consequently was I. I went into math related fields because I was 'supposed to'. It ended badly.

ReplyDeleteIn my second go-round, I had the good fortune of taking quantum mechanics from a very smart guy, and that is when I got my first real taste of math as a rich and expressive medium. So, for me it wasn't love at first sight, but it grew on me like moss.

Many math-loving people got their start reading Martin Gardner.

ReplyDeleteThe Art of Problem Solving books make me wish I had learned from them when I was younger http://www.artofproblemsolving.com/

Like Lance, I had no idea what real math was like or that I could do it until partway through undergrad. The key for me was not "big" inspiring questions but rather learning that I could solve "small" ones with just a little flash of insight, and it was only a lot of solving little questions that gave me the confidence to pose my own. This is not something that I would have figured out how to do on my own, or even if my parents had handed me a book or two (though that might have helped).

Your child may enjoy learning to program, in which case something like http://projecteuler.net/ might be fun (but a bit much for junior high math).

For the young student who is just starting out in math, I can't recommend two books highly enough: "The I Hate Mathematics! Book" and its somewhat more advanced sequel "Math for Smarty-Pants".

ReplyDeleteI

devouredthese books in early elementary school. They are full of zany comics, fun activities/experiments/puzzles/magic tricks with surprising mathematical depth, and great concepts that they don't teach in school: Mobius strips, perfect numbers, "huge" numbers (e.g., googol and googolplex), the Josephus problem, etc.The book also poses problems, some of which end up being very deep. (Some have even been open for centuries!) I remember thinking about whether all perfect numbers are even, and that led to all kinds of independent explorations and discoveries in number theory. I can't praise these books highly enough.

You didn't say how old your son was. =)

ReplyDeleteLet me second the Martin Gardner recommendation. And he has so many books that one can go years without exhausting them all.

You may also want to look at math competitions, which start as young as 6th grade (or younger?). Whether your child actually "competes" or not, you can download the exams from the web to give him something challenging to work on.

If he's interested in fundamentals of computers you can try "Elements of Computing Systems..." by Nisan. Surely other can recommend for TCS-focused books.

You read Godel, Escher, Bach as a child? Umm. I read it as an undergrad and I still thought it was heavy reading at the time.

ReplyDeleteArt of Problem Solving is a website that promotes a number of books for kids who are really into math. Their tagline is "Is math class too easy for you? Looking for a greater challenge? You've come to the right place."

>> I remember asking myself whether our school's computer with its fixed amount of memory could eventually print all the digits of π (no it can't).

ReplyDeleteWhy not? I always thought it could. Given sufficient time it could print *any* digit of Pi you want :)

1) (ME) In 4th grade I wanted to find out how many seconds were in a century so I figured it out

ReplyDelete(I think I didn't count leap years--- oh well.)

I doubt this is that interesting to anyone nowadays with computers so prevalent.

2) (ME still) 9th grade--- the fact that you could PROVE the quadratic formula I thought was fascinating, and I decided RIGHT THEN to become a math major to find out why you could not solve a quintic equation.

(I doubt my thoughts were quite so clear as I remember them, but that was my general sense.)

3) For YOUR kid, I AGREE

with prior posters-- I also read Martin Gardner

in high school and thought that was great.

4) For more recent books--I recently wrote a BOOK REVIEW of several

books of the type you may be looking for. Its

at

www.cs.umd.edu/~gasarch/bookrev/bookrev.html

5) Here are some math things that have amused my great nieces and nephews (ages 9-11).

a) NIM games: here is one,

there is more information on the web.

There are n stones on the table. Each player can remove 1,2 or 3 stones.

Whoever removes the last stone wins. For which values of n does player I wins.

(could be more concrete at first- have 10 stones on the table)

b) FIND THE NUMBER- I am

thinking of a number between 1 and 100. You can

ask questions about it.

Try to find it in as few questions as usual.

c) Look at the sums of odd number

1

1+3

1+3+5

etc. Try go guess what the

pattern is.

d) Same for sums of evens.

bill gasarch

I grew up in Africa and was home schooled from grade 8 to 12. I was an average student in Mathematics in grade 6 and 7 (and I had a very lazy math teacher, who hardly got up from his chair!) My interest in Math was sparked by a British grade 10 mathematics textbook called "O-Level Additional Mathematics" by Michael Browne. It was a delightful textbook. At University, I was lucky to have professors who were excellent Mathematics teachers and kept my interest in Mathematics

ReplyDeletePhysics was (and is) my main motivation for math, though there is a lot of neat pure math as well.

ReplyDeleteIn this post by Scott Aaronson,he links to and discusses a paper by Paul Lockhart, of which I have only read the first few pages, and it is more about what is wrong with math than how to do it right (I think). Still, like all good science, it can be great to know all the things that don't work.

I loved A.K.Dewdney's The Planiverse . Like Flatland, it looks at life in a two dimensional universe, but goes way beyond Flatland in terms of exploring what engineering, physics, biology, and chemistry would look like in a two dimensional world. How would you play volleyball in a two dimensional world? What sorts of vehicles can you build? How does a two dimensional body with a digestive track avoid falling apart?

ReplyDeleteFlatland was written as a social satire and, even with that in mind, can be hard to stomach at times. Parents whose kids read Flatland will need spend significant time explaining the satire, as opposed to the mathematics, including the extreme sexism and social hierarchy of the Flatland society. The satire has its value, but is not necessarily what readers expect when they pick up a book that is famous now almost entirely for its mathematical content.

The thing that got me into math was a love of puzzles. I always kind of liked math somewhat but the Junior Waterloo Math Contest problems for grades 9-11 (and the MAA contest) really hooked me - multiple choice problems on a varied range of combinatorics, simple algebra, geometry, and logic. I would work on copies of old contests during my other classes.

ReplyDeleteThe funny thing is that though these math contests got me into the field, I now have little patience for doing math puzzles of the sort where someone else already knows the answer.

Physics was my first love as a kid. I was fascinated by the possibility of answering many big questions (life, universe and all that) through the scientific approach. At the time, I knew very little 'real math' and I was mainly reacting to the excellent popular writings of Gamow, Feynman, etc.

ReplyDeleteAs I have grown up and looked at some real math, I find that one of the most fantastic things about it is its utility in the description and analysis of phenomena that would otherwise seem too vague. Perhaps it is not a bad idea to introduce children to this idea (in conjunction with the notion of pure math as the ultimate form of entertainment).

You might want to see if there are any "math circles" nearby or otherwise any small learning groups for math (often related to training for math contests). There is a book on the topic of math circles, written by some people who started some, which is pretty interesting from the perspective of teaching, and the philosophy of exploring math, called Out of The Labyrinth.

ReplyDeleteAlso, try letting your child pick the material if you haven't already. When I was much younger I found a surprising amount of value in going to high school/local/university libraries and book stores, and just browsing the math section.

Books with lots of information in them, along with lots of fun problems to explore, might be good. Two stand out in my mind:

ReplyDeleteMathematical Journeys, by Schumer

Excursions in Calculus, by Young

I'm not particularly good at judging what level reader these (or any other) texts are appropriate for. But I like both of these.

No one has mentioned science fiction (in the classic SF sense). The short stories of Cordwainer Smith had a big influence on me ... these stories characterically combine complexity theory, game theory, evolutionary biology, politics, and morality, in the context of wonderful narratives.

ReplyDeleteTwo stories that come to mind are

Alpha Ralpha BoulevardandThe Burning of the Brain(hope I remembered the titles correctly!)I second the Art of Problem Solving recommendation, and I think that the forum is a specially good source, in the sense that it is easy to find there very diverse problems, in terms of both breadth and depth.

ReplyDeleteI don't find GEB to be a good book to learn about math, and I don't like most of its references, although maybe I might have liked them when I was younger, and, well, of course different people can have different experiences about it.

Many thanks to Chris. I had totally forgotten about "The I Hate Mathematics! Book". I think it was the only math book in my elementary school's library. I definitely remember learning about Moebius strips and googleplexes from it!

ReplyDeleteI'd like to recommend Conway and Guy's

ReplyDeleteThe Book of Numbers. It's not as entertaining as GEB, and requires a fair amount of patience to think and work through some of the problems. But I think it is rewarding as it gives multiple approaches to answering questions about why some results are true. For example, as to the question what is the pattern of 1, 1+3, 1+3+5, ..., they give both algebraic and geometric formulations of the problem.I am not a theorist, but I sometimes write about my (often frustrating) experiences with theory in the "problem solving" and "learning" memories of my LJ.