I noticed, while teaching Robert to divide decimals, that he made quite a few mistakes. It was clear that he lost some of the previously acquired skills.
He made most, if not all, errors while dividing large numbers by 7, 8, or 9. I went back to dividing two digit numbers with reminder. I soon found out that Robert had difficulties with these and similar problems: 61:8, 60:8, 62:8 but he was fine with those problems 64:8, 65:8, 67:8.
I noticed a pattern. When the dividend was a digit, two or three below 64, Robert had problems. When dividend was slightly larger than 64, Robert didn’t make mistakes. This pattern repeated itself with other dividends and divisors.
At first, I just wanted to do reteaching using the old approach. Upon hearing direction, “Help yourself”, Robert wrote the multiples of eight (the divisor): 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, placed 61 (60 or 62) between 56 and 64 , found a quotient of 7, and finished dividing without problems. I believe that repeating this strategy many times would at some point lead to an improvement.
Still, I wanted Robert to grasp the idea behind choosing 7 or 8 as a quotient . I designed other worksheets. In the center of the page I wrote in big numbers, 64:8. I drew a line through the center of the page (but not through the division). Above the line I wrote all the divisions with dividends larger than 64 (65, 66, 67, 68, 69), below the line I wrote the ones with dividends smaller than 64 (60, 61, 62, 63).
I made similar pages for different problems.
Why I did that? What was the difference?
Since Robert could find most of the quotients and reminders, I did not want to lose too much time by reteaching using the old method. I believed (this is all domain of beliefs not knowledge yet) that the new approach would help Robert relearn quickly.
I knew, however, that this method could be helpful only if Robert develops better understanding of numbers. On the other hand, Robert’s understanding of numbers could be greatly improved by mastering this approach.
krymarh 