Recently, Jacob and I have been working on this times tables. They aren't doing them in school yet, but we try to spend some time learning beyond the curriculum. The nice thing with math is that Jacob enjoys it.
I always did well in school, but I really don't think I learned to effectively study until law school. In most law school classes, your grade for the semester is determined by a single final exam. Law school exam questions typically lay out a fact pattern and require you to identify the issues and relevant points of law. So while memorizing the cases and relevant law is not sufficient to pass an exam, it is necessary to know the relevant law.
Law students almost universally create an outline of the course to study for the exam. The outlines are a condensed version of the law and cases. The students study from their outlines, but are not allowed to take them to exams (with a few exceptions). Some students outlines were very detailed and grammatically correct(my friend Karl Sanders for example). I took a much more Spartan approach. My outlines would only have a few bullet points about the cases, often containing abbreviations and partially completed sentences.
Unlike some outlines which were borrowed by other students, my outlines were useless to anyone but myself. They worked so well for me for a few reasons. First, by the time I was done making the outline, I had a pretty good grasp of the case. My bullet points gave me enough information so that I could recall the important parts of the case. Secondly, I would effectively memorize my outline by reviewing it over and over. I could not efficiently memorize a two paragraph explanation of Marbury v. Madison, but a few bullet points were manageable.
The way I memorized my outline was to break it down into sections. For instance, if there were twenty cases relevant for a course, I would start out by reviewing the first five. Once I could recite the important part of those cases, I would add five more. I would continue this pattern until I had memorized everything. As I added more material though, I would continue to repeat the previous material as well. By the time I got to the last case, I may have repeated the first case a dozen or more times. It was a fairly effective strategy.
I'm trying to use that repetition to help Jacob. We might review three times one through three times five. Once he has that down, we'll add in three times six and three times seven. He complains sometimes about having to repeat it, but it seems to be fairly effective. He knows his 0's, 1's, 2's, 3's, 4's, 5's, 6's, 9's, and 10's.
One of the great things about teaching is that you can learn something yourself. I had heard that there was an easy trick to doing the nine tables, but since I had already memorized them quite a long time ago, I never really tried to figure out the shortcut. While I was teaching Jacob, I took some time and figured it out. Here is what I've found.
For 1 through 10
Take the number that you are multiplying nine with, and subtract one from it. That number will go in the ten position. Subtract the number you put in the ten position from nine and that goes in the one position. An example shows this more easily.
9 x 6
Subtract 1 from 6 to get 5. Put 5 in the ten position. Subtract 5 from 9 to get 4, which you put in the ones position. This gives you 54.
9 x 9
Subtract 1 from 9 to get 8. Put 8 in the ten position. Subtract 8 from 9 to get 1, which you put in the ones position. This gives you 81.
I don't have a pattern for 11, but its easy to multiply.
From 12 to 20
This works largely the same as 1 - 10, but you subtract 2 from the number you are multiplying with nine.
9 x 13 = 117
13 - 2 =11 (place in the ten's and hundreds position)
9 - 1 - 1 = 7 (place in the ones)
I've only done the 1-10 with Jacob, and it has really helped him pick up on the nine table. Now, if only I can come up with a way to motivate him to write.
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