These are the thoughts that guide me as I go about re-teaching math to my son, N.
When I undertook to teach him math last year, I found that there were big holes in his understanding of basic operations. It was clear that he had missed important steps and concepts in his elementary school math training. This probably springs from two sources. One is that he had so much difficulty following verbal instructions that he simply missed much of what was being "taught." (My very wise professor in Special Education Assessment says that if a student hasn't learned something, the first hypothesis ought to be that he was not taught it. Covering material for a student is not the same thing as teaching that student). The second problem with math instruction in public schools is that, in their frantic urgency to improve standardized test scores, districts tend to change curricular programs very often. Program changes in the middle of someone's education tend to create gaps in learning because each program builds on previous instruction in different ways.
Aside: In my opinion, this problem extends far beyond N. and far beyond math. I believe that one reason our schools are failing is that we have taught our kids that filling in the blank or the bubble for the "right" answer in all areas is far more important than actually thinking something through.
But enough! Back to the main story.
We have developed the following principles for re-teaching N. math.
But the ultimate goal is not to be fast! It is to appreciate numbers and enjoy applying math to real-world problems in order to get good results and satisfying answers to the big questions.