The immediate result was that I became very slow at doing math assignments. I had always been moderately slow doing worksheets. Teachers had commented earlier to my parents that though all the questions attempted were usually correct, I seldom finished my assignments. When multiplication entered the picture, my pace became glacial.
After a while, though, I started to pick up some tricks that helped greatly. Since doubling a number in my head was easy and tripling wasn't too bad,
I soon started factoring multipliers where possible and then multiplying by the factors. Thus four became double double and six became triple double. Since five and 10 were also easy to multiply by, that covered everything but seven. For seven and for larger numbers, I often relied on the distributive property though I had no idea that's what I was doing. 11 Times a number is the same as 10 times a number plus the original number. 19 times a number is equal to 20 times the number minus the original number.
This created an entirely new "problem" or at least extra step when I did my math homework. Whenever I had to multiply two numbers together I had to decide which forms of the numbers to use. Eight could be 2×2×2 or 4×2 or 2×4. Nine could be 3×3 or 10-1. Some forms made the problems easier than others but, amazingly, no matter which form I used the answer was always the same.
In the short term, my self imposed pedagogical experiment could not be called the success. My work was slow and my test performance dropped. Fortunately I had supportive parents and exceptionally understanding and flexible teachers and administrators.
I say "fortunately" because in the end things did work out well. The turnaround started with fractions. Ideas like simplifying and finding the lowest common denominators came naturally because of the groundwork that had been laid earlier. With algebra, the effect was even more pronounced.
This was partially because of the specific concepts I had been using, but the more important factor was the general approach to problem solving which I have been forced to take. Questions that would have been trivial had I done what I was told to do, instead required a great deal of thought. I had to think about all the different ways I could find to state the problem, then I had to decide which approach would be the easiest and quickest. In the long run, it turned out that this method was a pretty good way of approaching most mathematics.
Just to be clear, I am not saying that we should stop teaching multiplication tables. I'm not even saying that I would not have been better off had I simply done what I was supposed to do.
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