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We can prove that the answers are correct, but how did we find them? I answer this question in the Appendix. Not only do we find three answers, we also can learn that there are exactly three. We also would learn that if we took 1/4 of 4 (or even 1/4 of 3, for that matter), there would be four answers. If you are familiar with middle school arithmetic you know that when you take the square root of a number you get two answers, and for the fourth root you get four answers.
These mathematical concepts can be understood by third graders, as I have discovered first hand by explaining it to them. Yet, I have seen teacher training films on algebra that teach this information on modulo arithmetic completely incorrectly.
The explanation that zero is nothing boxes in a child's and teacher's true comprehension of the basics of mathematics. A different explanation gives us more consistent answers and allows us to be creative in applying that knowledge. There are supplementary texts that fill children's heads with mathematical nonsense as they build upon the notations of zero being nothing and zero being a place holder. In the number 02.0 the zeros are not place holders. Those who understand number structure know that we have:
02.0 = 0 x 10 + 2 x 1 + 0 x 1/10 = 2.
2 = 2 x 1 = 2.
And the zero in the number 202 is as much of a place holder as is the one in the number 212, because if we dropped either, the number becomes 22, which is neither 202 or 212.
In my experience, many of the standard texts, new and old, as well as many supplemental texts have several such incorrect or misleading lessons.
Teachers have recognized that there is more to teaching math than just the illustrating the concepts. You have to get the child's interest, you must build up the child's confidence, and you must teach the child how to study. Much attention has been focused on building up the child's interest, but building confidence and teaching the child how to study are left as an art for highly qualified teachers. Games and computers have become the traditional ways of capturing a child's interest. These approaches would be good if they were coupled with the proper teaching of math. Unfortunately, the ones with which I am familiar do not teach math properly. Their limited success may be in part due to an innate ability of bright children to appreciate math beyond what they are taught. For example, some children can figure out that to add nine to a number you add ten and count back one, whereas many teachers are unaware of this technique. Because I have conducted workshops for teachers I know some of the strengths and weakness of teachers. Because I understand math I found that it was easy to discover techniques for building the confidence of children and to teach them to study.
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