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The Art of Problem Solving is an online self-teaching maths website with a really strong focus on curriculum quality. So if the kid likes maths, you can let them learn more of it!

Here's one discussion of 9 year olds using the sites successfully.

The Art of Problem Solving is an online self-teaching maths website with a really strong focus on curriculum quality. So if the kid likes maths, you can let them learn more of it!

Here's one discussion of 9 year olds using the sites successfully.

Amen to this.

I am a "homeschooling" parent. This does NOT mean my children are taught solely by myself and/or my wife, and it does NOT mean they are taught solely at home. It DOES mean that we have personally selected and combined a number of different educational opportunties for them. These include (but are not limited to):

Enrolling in college coursework while still in high school. Example: Harvard Math 23b. The majority of students in this class are admitted Harvard freshmen, but it is also available in an open enrollment capacity through Extension for anyone of any age willing to pay tuition. I like that peer group for "socialization" a whole lot better than the kids at my local public high school.

Hiring the chair of the language department at a local private high school to come to our home to provide personalized one-on-one instruction in classical Greek and Latin.

Hiring multiple music teachers for piano, guitar, theory, and composition.

Participation in team sports at the local health club.

Engaging a flight instructor for our son to earn a private pilot's rating.

Successfully completing qualifying flights for TARC

The Internet (Obviously). Taking advantage of online educational programs such as AOPS and edX and Open Courseware

Stocking our home with thousands of quality print books and plenty of subscriptions to lots of quality print journals (e.g. Economist, Nature, Lapham's Quarterly, IEEE publications, etc.)

Buying a whole bunch of the Great Courses

Joining CTY

Plenty of socratic dialogue with Mom & Dad. And plenty of unstructured time.

Flexibility to travel (including abroad) during the school year.

Concrete advice for OP: First, read The Underground History of American Education. Make of it what you will --- just include it (or criticisms of it) as a data point. Next, decide if any your local school choices (either public or private) are awesome. Do they approach the quality of Exeter or Boston Latin or Bronx Science? Understand the concept of a feeder school and that this concept can start at the elementary level. Got great public or private school options you like and can afford? Go for it. Not so much? Then go ahead and homeschool kindergarten. I guarantee you that your drop-out wife is capable of teaching your child to read and anything else they are supposed to learn in kindergarten. I guarantee you that unless you are completely negligent that your child will (if you choose) be able to enter first grade after a year of homeschooling and do fine. And I guarantee you that after a year you will be in a much better position to understand if more homeschooling is the right choice.

There are some good math texts out there. Art of Problem Solving http://www.artofproblemsolving.com/ has written a number of math text books. We homeschool, and after looking at *many* curricula chose AoPS for the spine of our daughter's math education. AoPS isn't aimed at homeschoolers -- it is aimed at kids in regular school who are not well served there. The texts are excellent, IMO. The online lecture meets once per week in a chatroom -- no video, no audio, but it *does* support direct entry of LaTeX. My daughter has thrived on it -- she is 12 years old and preparing for the AP Calc this spring (now you know why we homeschool).

The AoPS texts are available at a reasonable cost to anyone and I highly recommend them. The online classes move at a very rapid pace -- they are not for everybody. But you can take the text books at your own pace and get a great math education.

Another good set of online math lectures are put out by ThinkWell http://www.thinkwell.com/ -- the math lectures by Dr. Ed. Burger are outstanding. Burger is the math teacher you wish you had. These are not live, ThinkWell uses recorded videos. Very good and very reasonable priced.

Online classes are changing the world. Clayton Christensen (author of Innovator's Dilemma) wrote a book "Disrupting Class" about how online classes are causing a classic disruptive innovation in the world of education. I recommend it. If the world follows his time line, soon schools as we know them will go the way of the dinosaurs. About time.

Totally agree with you about LaTeX. TeX (on which LaTeX is based) was done back in the days of punch cards -- it was the only game in town for typesetting mathematics papers on a computer. TeX is a really an amazing accomplishment, when you think about it. Score another one for Knuth. But.... it is about as far from WYSIWYG as can be imagined, with all the good and bad brought on by that circumstance.

LaTeX can be learned with effort. The learning curve is nasty, but you get very nice math typesetting as a result. My daughter does on line math classes from Art of Problem Solving. http://www.artofproblemsolving.com/ The lecture classroom is a chatroom that supports LaTeX. So it's pretty wild to see a bunch of middle-school and high-school kids blasting LaTeX into the chat. LaTeX isn't dead -- but only math nerds are fluent.

I mean, there's two ways you can go. There's high school level problem solving texts and then there's college level math.

If they're looking for interesting high school level math problems, then I'd look into Art of Problem Solving series (the two texts at the bottom). There's also Problem Solving Strategies by Engel, which is also an excellent book. If they're looking into some accessible "higher level" (meaning, proof-based) math, I'd suggest number theory. Niven has an excellent treatment of introductory number theory.

As far as introductory college level math goes, you could always hand them a multivariable calculus, linear algebra or differential equations book. As far as recommendations go, Axler has a pretty good treatment of linear algebra if you already have some background, Apostol's series on calculus (vol. 1 and vol. 2) has a more theoretical approach to single and multi variable calculus (it's more advanced than high school calculus, but not quite an undergraduate analysis text), and Birkhoff/Rota's differential equations book has an advanced approach to differential equations probably not for high schoolers. If you don't think they'll be scared by rigor, you could always refer them to Little Rudin, which is pretty much the standard for an undergraduate real analysis class. If you're looking for an analysis book that isn't ridiculously overpriced, I've heard about this one but I've never actually read it. For a pretty readable treatment of algebra (a.k.a. modern algebra or abstract algebra), see Artin. For a more theoretical approach (less recommended for high schoolers I guess), see MacLane/Birkhoff.

http://www.artofproblemsolving.com/

The essence of understanding math is being able to use it to solve problems. Math problems are like chess problems: they both have a start state and an end state and a solution consists of a sequence of legal moves. Routine problems are easy, like mate-in-1 or a simple application of a single mathematical rule. Non-routine problems require you to think a few moves ahead, but if you can't do that, you don't really understand the moves/material.

It's important to become proficient at non-routine applications of basic material before moving on to more advanced material like calculus. As the author of The Calculus Trap writes: Rather than learning more and more tools, students are better off learning how to take tools they have and apply them to complex problems.

To this end, I recommend The Art of Problem Solving Volume 1: the basics & The Art of Problem Solving Volume 2: and beyond. They are the best math textbooks I have ever seen. The intuitive explanations really sink in, so no memorization is required. But the key is that each section is followed by a bunch of non-routine problems from middle-school and high-school math contests like MATHCOUNTS and AMC. These are a fun way to make the material second nature, and besides, it's pretty motivating to know that a bunch of middle- or high-school kids solved the problem you're struggling with. (I want a shirt that says I'm as good as a middle schooler on the front, and on the back says MATHCOUNTS.)

After studying the first few chapters of Volume 1, you will be able to solve problems such as these:

• The formula N = 8 * 10^8 * x-3/2 gives, for a certain group, the number of individuals whose income exceeds x dollars. What is the smallest possible value of the lowest income of the wealthiest 800 individuals? (AHSME 1960)
• Find Sqrt[53 - 8 Sqrt[15]]. (MATHCOUNTS 1990)
• If for three distinct positive numbers x, y, and z: y/(x-z) = (x+y)/z = x/y, then find the numerical value of x/y. (AHSME 1992)
• For each of n = 84 and n = 88, find the smallest integer multiple of n whose base 10 representation consists entirely of 6's and 7's. (USAMTS 1)

This post is based in part upon similar posts of mine at Reddit and MathNotations.

The essence of understanding math is being able to use it to solve problems. Math problems are like chess problems: they both have a start state and an end state and a solution consists of a sequence of legal moves. Routine problems are easy, like mate-in-1 or a simple application of a single mathematical rule. Non-routine problems require you to think a few moves ahead, but if you can't do that, you don't really understand the moves/material.

It's important to become proficient at non-routine applications of basic material before moving on to more advanced material like calculus. As the author of The Calculus Trap writes: Rather than learning more and more tools, students are better off learning how to take tools they have and apply them to complex problems.

To this end, I recommend The Art of Problem Solving Volume 1: the basics & The Art of Problem Solving Volume 2: and beyond. They are the best math textbooks I have ever seen. The intuitive explanations really sink in, so no memorization is required. But the key is that each section is followed by a bunch of non-routine problems from middle-school and high-school math contests like MATHCOUNTS and AMC. These are a fun way to make the material second nature, and besides, it's pretty motivating to know that a bunch of middle- or high-school kids solved the problem you're struggling with. (I want a shirt that says I'm as good as a middle schooler on the front, and on the back says MATHCOUNTS.)

After studying the first few chapters of Volume 1, you will be able to solve problems such as these:

• The formula N = 8 * 10^8 * x-3/2 gives, for a certain group, the number of individuals whose income exceeds x dollars. What is the smallest possible value of the lowest income of the wealthiest 800 individuals? (AHSME 1960)
• Find Sqrt[53 - 8 Sqrt[15]]. (MATHCOUNTS 1990)
• If for three distinct positive numbers x, y, and z: y/(x-z) = (x+y)/z = x/y, then find the numerical value of x/y. (AHSME 1992)
• For each of n = 84 and n = 88, find the smallest integer multiple of n whose base 10 representation consists entirely of 6's and 7's. (USAMTS 1)

This post is based in part upon similar posts of mine at Reddit and MathNotations.

The essence of understanding math is being able to use it to solve problems. Math problems are like chess problems: they both have a start state and an end state and a solution consists of a sequence of legal moves. Routine problems are easy, like mate-in-1 or a simple application of a single mathematical rule. Non-routine problems require you to think a few moves ahead, but if you can't do that, you don't really understand the moves/material.

It's important to become proficient at non-routine applications of basic material before moving on to more advanced material like calculus. As the author of The Calculus Trap writes: Rather than learning more and more tools, students are better off learning how to take tools they have and apply them to complex problems.

To this end, I recommend The Art of Problem Solving Volume 1: the basics & The Art of Problem Solving Volume 2: and beyond. They are the best math textbooks I have ever seen. The intuitive explanations really sink in, so no memorization is required. But the key is that each section is followed by a bunch of non-routine problems from middle-school and high-school math contests like MATHCOUNTS and AMC. These are a fun way to make the material second nature, and besides, it's pretty motivating to know that a bunch of middle- or high-school kids solved the problem you're struggling with. (I want a shirt that says I'm as good as a middle schooler on the front, and on the back says MATHCOUNTS.)

After studying the first few chapters of Volume 1, you will be able to solve problems such as these:

• The formula N = 8 * 10^8 * x-3/2 gives, for a certain group, the number of individuals whose income exceeds x dollars. What is the smallest possible value of the lowest income of the wealthiest 800 individuals? (AHSME 1960)
• Find Sqrt[53 - 8 Sqrt[15]]. (MATHCOUNTS 1990)
• If for three distinct positive numbers x, y, and z: y/(x-z) = (x+y)/z = x/y, then find the numerical value of x/y. (AHSME 1992)
• For each of n = 84 and n = 88, find the smallest integer multiple of n whose base 10 representation consists entirely of 6's and 7's. (USAMTS 1)

This post is based in part upon similar posts of mine at Reddit and MathNotations.

I am from the opposite end of things, someone who did math competitions from elementary through undergrad and who misses having them in graduate school. That said, The Art of Problem Solving books might work for you. They are intended to help students prepare for middle and high school math competitions, have solution manuals, and are $73 for both books and their solution manuals. There is also a new strictly algebra book available. My main reason for recommending this is that the whole point of most math competitions and these books is to teach you problem solving techniques, you will learn algebra, geometry, trig, etc, but also learn more of how to apply them to more interesting/applicable problems. http://www.artofproblemsolving.com/

If you want to see it, my science fair paper may be found at http://www.artofproblemsolving.com/Resources/Paper s/FracBase.pdf.

Keep that up every year as mandatory courses along with the other mandatory courses, and there won't be educational problems.

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