Robert memorized multiplication facts during his 2005/2006 year in Collaborative program. I cannot describe how Robert learned multiplication facts because my role was only marginal. It, probably, helped that in 2004, Robert knew already how to count by five, two, and ten or that I practiced with him counting by other numbers. I made worksheets to practice changing repetitive addition into multiplication and worksheets in which a new multiplication fact was scattered among the facts that Robert already mastered. The Collaborative Program, however, led the way. I think that the worksheets the teachers used relied a lot on patterns. I recall a worksheet that introduced multiplying nine by consecutive numbers by emphasizing a pattern of results: increasing ten’s digit and simultaneously decreasing one’s digit. I am not sure if at that time Robert and I worked on those lessons from Saxon Math that presented multiplication as an array of objects. Maybe not yet. I recall vividly, that Robert learned quickly and almost effortlessly as it was always the case when the school assumed responsibility for teaching. Unfortunately, not much later, Robert was forced to leave this program.

For the next four months I was teaching Robert at home much more intensively than before. We practiced multiplication facts and family of facts as a way to almost mechanically tie multiplication to division. (That is what we had done before when Robert had been learning to memorize subtraction facts). That allowed Robert to memorize division facts relatively quickly. The problem arose when reminders had to be taken into account. It is one thing to remember that 32 :8=4 and another to remember what is 34, 35, or 38 divided by 8.

We started by learning to divide by 2. On top of each page, I wrote the multiples of two on a number line in large print and the odd numbers in small. Robert had to place each dividend on the number line and then look to the left to find the closest multiple. For 17, he pointed to 16, for 9, he found 8 and so on. In the next step, we skipped number line, but Robert still had to write all the multiples of two with empty spaces between them and locate the odd number somewhere between the multiples. For 11, for instance, he had to point to empty space between 10 and 12, then move his finger to 10, and proceed from here.

Finally, when Robert practiced division by two, he was prompted to help himself by listing the multiples of 2 in order but only when he was unsure or confused.

We repeated these steps for all the divisors from 3 to 9. Of course while learning to divide by 3 with a reminder, Robert also practiced, on a separate page, dividing by 2. Then I mixed two kinds of problems on the same page.

I have to say, however, that we didn’t go past dividing by more than 12.

Learning to divide with a reminder helps with long division and with changing improper fractions to mixed numbers. It is clear that it would be beneficial to practice dividing by divisors larger than 12. That, however, is still a problem, and when we have to do that, I just ask Robert to find a few multiples of the divisor (by multiplying). Cumbersome process that still requires hand over hand (prompt over prompt) guidance. We do that only rarely, when Robert wants to finish the problem which is a little over his (or mine) head.

The algorithm of dividing by numbers larger than 12 depends on ability to round numbers up or down. This is a very hard skill to teach, and Robert doesn’t have it.

For school this is NOT a functional skills, so instead Robert is practicing counting values of nickles: 5cents, 10 cents, 15 cents…

Didn’t he do that 10 years ago?

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