LMom,

It sounds as if you and your spouse may have me at least tied as math geeks � and I do in fact mean that as a compliment.

You wrote:
>You are years ahead of us and you have much more insight.

Well� my kids are a bit older than yours, and we�ve been homeschooling longer, so I�ve had more time to make mistakes! As to how much insight I�ve gained from those mistakes�

You wrote:
>I hear you on more than one resource. How does one manage that? I fully understand where you are coming from, but I can see myself overdoing it and trying to do way too much�

Well, the short answer is that I think the approach you are taking so far with the �Fred� books is basically the right approach. My attitude is that I am entitled to ruthlessly �strip-mine� workbooks and textbooks: i.e., use whatever I think serves my kids� best interests whenever and however we think is appropriate and ignore anything that we find does not serve our purposes.

I think the central principle is to have your kid do as much work as he needs to do in order to master something� and no more. The goal is understanding and mastery, not work for the sake of work.

The workbooks/textbooks exist for us � we are not servants of the books.

Parents, especially when we�re homeschooling, should be able to figure out what the kid knows � if he needs more problems from another book in order to master the stuff, have him do them. If not, don�t.

I�m having our kids do almost all the problems in Singapore Math because SP is not known for busywork problems and the kids seem to find them challenging. Since Fred Fractions repeats much of what we did in SP, I�m more lax there; we�ll probably be more thorough on Fred Algebra.

Reading through Fred Fractions with your kid as you did and working with him on problems makes good sense even if it is not exactly the procedure the author had in mind.

My kids are just getting to the age when I myself was reading good kids� books about math (such as Adler�s �Giant Golden Book of Mathematics� that I mentioned earlier), and I have not quite figured out how to get them to read them as I did � should I assign reading, just make the books available or what?

I do think one of the most important things is to try to talk with the kids about math, including stuff that you know may be over their heads, and see what happens. I talked with them while they were still learning simple arithmetic about the sum of the angles of a triangle being 180 and how this only occurred in flat space and was connected to the existence of a unique parallel. We looked at triangles on a sphere and they could easily see that it does not work on a sphere. I�m doubtful that they really grasped the connection to the existence of a unique parallel (although I certainly understand it better now after trying to explain it to them!).

When we learned the commutative laws for arithmetic in first grade, I pointed out how rotations did not always commute so that commutative laws don�t hold for everything. This is very easy to show with a couple of identical cracker boxes: rotate one 90 degrees around the vertical axis and then 90 degrees around a horizontal axis. Do the same operations in reverse with the second box. The non-commutativity is obvious. The kids still remember this from a couple years ago.

I�ve talked with them about simple ideas having to do with vectors, which they do seem to get. I�ve also had them do simple calculations with a rotation matrix: this is easy to do once you know the basics of fractions. It gave them some practice with fractions and with graph paper, though of course they still do not really understand matrices.

Similarly, we�ve talked about infinity � they understand some of it, but not all of it. Kids do seem interested in the idea of infinity, though.

You know of E. D. Hirsch�s espousal of developmentally *inappropriate* teaching? Hirsch argues that instead of spoon-feeding them each tiny next logical step, kids deserve to hear about the stuff that is actually interesting. Black holes and supernovae are more interesting than pulleys and levers (we�ve read and talked about all of those) and infinity is more interesting than dividing fractions (we�ve worked more seriously on fractions, but we have discussed infinity).

It would of course be unfair to test even profoundly gifted grade-school kids on the mathematical theory of black holes or on Cantor�s theory of transfinite numbers, whereas, it�s fair to expect them to prove some knowledge about levers and about dividing fractions. But that does not mean you can�t talk about stuff that you expect they will be interested in, even though you know they will not fully grasp it.

After all, as adults, we would be really irritated if the news media only told us about things that we were expected to internalize so fully that we could later pass a test on it!

Anyway, it sounds as if you are taking a similar approach to what I am advocating and trying to do myself. I think the important point is to not be afraid of trying out stuff even though it may turn out to be beyond your kid�s understanding (or your ability to explain). Try it out and see what happens. That, after all, is what we adults do with each other. If the child ends up not understanding, well, he will have a shot again when he is a little older.

I hope it�s clear that I�m not advocating in any way skimping on the core material that really must be mastered in grade school � the four arithmetic operations for whole numbers and for the various sorts of fractions and some basic ideas of measurement. But the nice thing about dealing with bright kids is that you can cover that more rapidly than the public schools do, and still have some fun exploring other things.

Sorry for having written a treatise rather than a reply again, but I hope this clarifies what I�ve been trying to do with our kids in math.

All the best,

Dave