When we were kids, growing up in different states, Bill and I both enjoyed solving math puzzles and creating math challenges for ourselves. Yet we saw the kids around us struggling with math, claiming it was too hard and that they hated it.

I didn't believe this. How could anyone hate math? And how could math be "hard" when it was so logical?

Bill went to M.I.T., where he earned a degree in math; I was a math major at Boston College before I transferred to R.P.I. and switched to engineering. We eventually met , got married, and had kids, who we homeschooled. All of our children have gone on to do well in math. Most of them have already won, or placed in the top three, of at least one local and or national math competition.

This sounds like annoying bragging, doesn't it? Bear with me. I'm not saying that our kids are all genetic math geniuses. In fact, if they were in school, at least three would have been diagnosed with ADD, and one with dyslexia. We haven't had a tremendous amount of time to spend drilling them on math, either. Thanks to my duties at PRACTICAL HOMESCHOOLING, I actually have less time to spend on such things than the average homeschool mom.

What I do have is my dad the college professor, who taught me to be analytical about how I learned things from a very early age. I remember how he got me excited about math, what helped me learn the quickest, and what was a massive waste of time. I also had the tremendous experience of him teaching me six grades of math (from grade 2 through grade 7) in the summer after first grade. So even before I began homeschooling, I knew math could be taught much more quickly and effortlessly than almost anyone believes today. For what I am about to share with you, I am indebted to Dr. Stuart Martin of Boston College, my father.

**Competition & Rewards**

Let's start at the beginning, with motivation. Competition and rewards, as much as some people like to knock them, were a big factor in both Bill and me getting turned on to math. My dad offered me, at the age of five, a penny for every page I completed in a set of math workbooks he bought me. After I had very quickly earned a buck, he declined to issue any further remuneration, but by then I had learned (1) I could do this! and (2) math was fun and profitable!

In Bill's case, his third-grade teacher had her class copy out four pages of math problems each week. Every Wednesday, they would have a timed math bee, filling in the problems. The first person to get them all right was the winner. A clickable ball-point pen (then costing around 5¢) was the prize. Bill only won about once every five weeks, but that was enough to motivate him to start thinking about math as an exciting game.

The idea is to make a child's first math experience a thrill. Some variations: do math problems with M&Ms, and let them eat the answers they get right. Let them do math with pennies, and keep the change if they get the problem right. You won't want to keep doing this for months at a time, as the kids will either get fat or rich, but as my dad found, it isn't necessary to keep up these rewards once the children start enjoying math for its own sake.

The Barnum Software Quarter Mile program brings just this surge of excitement. Not only are kids drilling the math facts, they are competing against their own previous best speeds. This can be taken to a competitive level if you sign up for one of their International Math Tournaments. Our kids all enjoyed competing very much, and it certainly gave them incentive to practice! For a noncomputerized alternative, the Providence Project Calculadder timed math drill sheets also "rev" kids up and give them that sense of progress.

The great beauty of math drill software is that you, the devoted parent, don't have to be there for the child to improve his skills. But if you prefer to use traditional flashcard, here's a tip: let the child hold the cards he gets right, then run through the deck again. Every time he gets a card, the number of facts to drill narrows down to those that need the most work; and for some reason, holding the cards make the child feel like a winner. He will count the cards he has gotten right so far (more math practice!) and calculate how many he has to go (yet more practice!), while feeling he has already succeeded at least in part.

**What Counts?**

Most of us have 10 fingers and 10 toes to count. But that gets old pretty fast. Try counting backwards, like a rocket lift-off countdown: "10, 9, 8, 7, 6, 5, 4, 3, 2, 1, BLAST OFF!" Little kids love it! Try learning to count in several foreign languages. This is a great way to introduce the concept of other languages, and to help kids keep their ability to make the sounds of languages other than our own. It also teaches them that "2" is "2," no matter what it's called.

I have a dress with 20 buttons. This made it easy to practice counting with teeny-tiny ones on my lap. Try counting flowers on the wallpaper, leaves on a branch, cans on the shelf . . . whatever is right there in front of you. Remember to make a big excited fuss when the little one gets it right!

**Math Manipulatives**

For the math facts, I still think money makes the best math manipulatives. Coins and bills aren't rods you can lay side by side, or cubes you can snap together. For that very reason, they encourage abstract math thinking skills. A dime doesn't look like it's worth more than a nickel; it just is worth more than a nickel. You have to assign a value of 10 to the dime and 5 to the nickel. This is prealgebra and advanced math at its most basic; assigning values and making calculations based on those values.

A word more about the value of abstract thinking. The National Council of Teachers of Mathematics has been for some time promoting the use of hands-on math manipulatives designed to help children think about math concretely. "This five-rod is as long as five one-rods. I can see with my eyes how many one-rods it takes to make a rod of the same length as one five-rod." This is not the way math manipulatives have worked throughout history. The sheepherder who made a knot in his string for every ten sheep, or the abacus user who moved a bead to the top of his abacus to indicate "five," were using the knot or the bead to indicate abstract value. There is nothing "ten-ish" about a knot, or "five-ish" about a bead, any more than there is anything "ten-ish" about a dime or "five-ish" about a nickel. People had to think abstractly to handle the knots and beads. They couldn't just count up a number of one-rods to get the answer.

A dime does not resemble ten pennies in the least. A ten-rod resembles ten one-rods; maybe not in its color, but in having the exact same overall length and shape. Therefore, I encourage you to use rods, cubes, and even cuter manipulatives such as the Delta Fast Food math items reviewed elsewhere in this issue, to demonstrate math principles, but not to work math problems. Rather, try using coins, with all their different values. Your kids will learn the most essential math skill of all, needed in all math from algebra on up - how to solve problems where you can't simply count your way to the answer. At the moment, it will be a nickel standing for five pennies; later, it will be x standing for a number; still later, it will be x standing for an entire statement including other variables; later yet, it will be f(x) standing for a series, or a number set, or the area under a curve, and so on. All of this is really easy if you understand the concept of abstract value. But I guarantee you that a child who has always been able to stick manipulatives end-to-end or in a square to solve a problem will be totally lost when he runs into negative numbers, irrational numbers, or imaginary numbers. Don't get stuck in the concrete, when you could teach your children abstract math thinking with your pocket change.

One more point about money: even kids who struggle with numbers catch on very quickly when those numbers have two decimal places and dollar signs. I have yet to meet a kid who can't learn to shop!

**Girls & Math**

Research shows that American girls do noticeably worse than American boys on standardized math tests. But not homeschooled girls! Our own daughter Sarah recently scored a perfect 80 out of 80 on the math portion of the PSAT. (I admit it, that was bragging!)

The better performance of homeschooled girls probably has to do with (1) using math more in their everyday lives and (2) not being surrounded by boys who not-so-subtly prefer cute airheads to smart young misses. As long as the parents expect their daughter to do well in math, there should be nothing in the homeschool environment to hold her back.

One area that most girls do need extra help with, even at home: spatial skills. If I ask you to visualize a shape in your head, then rotate it various ways, most guys can do this with ease, but most gals can't. Two resources I have found that are a great help in improving spatial skills are:

F The Factory (Mac, Win, Apple, and DOS versions available; $89.95) and Factory Deluxe (Mac or Win CD; $89.95) software from Sunburst (1-800-321-7511). Or (for Mac only, $129.95) the Spatial Sense CD-ROM, which includes Factory, Super Factory, and Building Perspective. Check them out on www.sunburst.com.

F The most excellent D.I.M.E. Blocks ($17) and associated 3-D Build-Up books (three available for $7 each) from Timberdoodle (1-360-426-0672). This forms an entire mini-course in spatial skills, as you rotate and fit the odd-shaped blocks together. Oodles of fun, too!

**Puzzles & Construction**

Puzzles and construction toys are another great way to develop your spatial skills. Bill's mom was, and is, an avid puzzler, who also bought him Tinkertoys and Erector sets. Oddly enough, my parents also gave me tons of puzzles, and the very same building sets Bill grew up with.

Today, "Erector" sets are called "Meccano," and products such as Duplo and Lego bricks are mainstays of many homeschools. While Lego-like products are great for creativity, they only fit together in a few limited ways. For more advanced spatial and engineering skills, an up-to-date construction series such as Fishertechnik (available from Timberdoodle: 1-360-426-0672) or the motorized Robotix kits (available from Home Life: 1-800-346-6322) is what you need.

**A Word About Words**

One fact of mathematical life that nobody talks about is the nomenclature: the jargon mathematicians use. Getting a grip on the lingo as early as possible is a way for your child to "pre-digest" some of what he'll be encountering in later math courses. We were only kidding about the Baby's First Calculus course in our Homeschool Admirer parody section in this issue, because babies can't learn calculus. However, young kids can learn calculus terminology. So go ahead: "integrate" these math words into your daily life and help your children "function" better as you push them to their "limits"! (Note: the words in quotes are mathematical terms. I don't really think kids should be pushed anywhere unless they're in a baby carriage or wheelchair.)

A great book I just found that can help you with this is G is for Googol, from Tricycle Press pictured on the first page of this article ($15.95, 1-800-841-2665). This "alphabet" book is not for teaching little kiddies to read. Instead, this 57-page, oversized, fully illustrated hardcover is devoted to explaining advanced math to the young. Starting with "A is for abacus," in which we see an abacus, learn its history, and see how it works, each letter stands for one or more important topics beloved of mathematicians. I'm not talking about wimpy stuff like addition or fractions, either. Cast your ovoid oculars on this list:

Abacus, Binary, Cubit, Diamond, Equilateral, Exponent, Fibonacci, Googol, Googolplex, Hundred, "If," Jupiter, Königsberg, Light-year, Möbius Strip, Nature, Obtuse, Probability, Quantity and Quality, Rhombicosidodecahedron, Symmetry, Tesselate, Unit, Venn Diagram, "When are we ever going to use this stuff, anyway?," X, Y-axis, Zillion.

Each concept is explained in easy-reading detail, with full-color drawings and diagrams. As a bonus, for each letter you'll also find a list of other math concepts starting with the letter in question. So A is for Abacus (the main entry), and also for acute, algebra, angle, art, architecture, area, asymmetry, average, and axis. No definitions are provided for the bonus list, but any parent with a smidgen of mathematical training can use this list for inspiration for further explanations and explorations.

Three last tips: (1) math puzzle books, available at bookstores and teacher's stores: (2) checkers, and (3) chess. All are great, fun ways to improve math thinking skills. Let's do it!