I invented the method. We use algebraic operators for all arithmetic problems to break down the problem into easier sub-problems. Engineers use these methods in their heads. We use Saxon only as a source of problems and for drill work. It's archaic.
In our system, all equations for word problems have units, set operations, what we call "truth statements" (such as 1 dozen oranges = 12 oranges), and calculated identities (1 dozen oranges/12 oranges = 1). "We can always multiply any number by one." No hand waving allowed. This rigorous method makes word problems a snap. There is no "borrowing" in subtraction operations. We add another equation that equals zero to adjust the subtrahend. "We can always add zero, can't we?" Any time they apply a principle, axiom, or theorem they are to cite it as if the problem was a proof.
Rigor. It's that simple. Rigorous simple problems makes complex problems easier to do later. This is the necessary preparation that will make chemistry and physics far simpler to learn.
Also be sure to expose them to analytic subjects outside straight "math" such as symbolic logic, computer programming, etc., and when they do calculus make sure they are learning what it means and what it is for, not just how to solve the problems.
Symbolic logic will be taught after the first exposure to calculus (so that we can teach physics) and after an extended period of diagramming sentences. My hope is to teach them to convert sentence structure to mathematics. My goal is that they could do legal analysis with symbolic logic. (That is a skill that would make them truly dangerous.)
I am an engineer. I won't expose them to computer programming until they routinely see science in the world and have mastered paper else paper may too easily bore them. For now, the computer is a word processor. I do not want them thinking that the computer is the world.
I agree, but my husband builds networks for a living and he is insistent that our children will learn at his knee, which will be difficult to avoid since he works from home a lot, and studies constantly. I want my children to be able to interact with intellectual resources from any source or medium. He shares your interest in creating "dangerous" children, hehehe, and he does make good points to the effect that for them to truly be such in the coming era they will need to understand the fundamentals of computing and not be end users only. My dad is a programmer, and one thing I have been really surprised to learn from my husband is that the kind of network/systems/hardware stuff he does is actually more fundamental than (most) programming. When his programmer friends are hanging out here, it is obvious that they know much less than he does about what makes the machines tick. What kind of engineering do you do?
I have to admit that I'm not quite comprehending your method, I think I would need to see a full example laid out. I mean I think I get the basic idea, but I'm not quite sure how it would work on a kid who didn't already know arithmetic. You should really publish this, even if only on the web. There are a lot of people out here who want to teach our kids to really understand math, not just by rote, from the start, but are at a loss for methods. I looked at a Saxon book once, it looked just like my high school math books, only more systematic. But not deeper.
Textbooks? I like Copi's.