About the power of knowing facts--I agree that children benefit from knowing facts AT SOME POINT--but facts with NO number sense are pretty much meaningless. The son on the trip to Mexico had terrific number sense. I taught a child who attended Kumon school (Yes, I wrote about this several months ago, so if you know how this turns out, just skip on!) She worked on basic facts constantly and could answer basic arithmetic problems--stated exactly like 8 times 7--with great speed and accuracy. One day, though, we were looking at a collection of creatures. There were 4 head shapes and 8 other variables for each shape. We had drawn 4 columns and put 8 rows under them showing all the types. I asked how many of these creatures all together were there? My Kumon child responded 90. I asked her why 90 and she said, well, she liked 90 and that was probably the answer. Other children used a variety of strategies--two rows of four is 8, four rows of 8 is 16 and two 16s is 32 etc. etc. At the end of the day the mom of my facts expert was visiting and I told her that the child had not been able to come up with a meaningful answer to the problem--And it was 4 x 8 (I said). The little girl who had been standing nearby sai, Oh! 4 x 8 is 32 and gave a big smile, quite justifiably proud of herself for knkowing the fact. She knew the fact, but she sure didn't know the numbers!
We had problems like this all year--she could do the drills beautifully but rarely had answers to simple problems like the one above (which she could have even counted to get the answer) which made much sense. I hope I helped her develop some sense over the year, but it was almost impossible to get her to budge from thinking that math is a series of facts and rules. She told me later in the year (after her dad had spent a long morning with us when we'd done some terrific calculations froma chart we'd developed based on data we'd collected) that her dad really felt that workbook math was the best, anyway.
So, my point is--knowing your facts is great, but number sense is far, far more important! What kids are missing when they have to figure out simple arithmetic is not knowledge of facts, it's understanding of number. (I do believe that learning facts can be helpful, but are not anywhere near enough!) Cindy