Identifying Errors, Diagnosing Problems, Designing Intervention

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.

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