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One may see a weakness in how mathematics is being taught by following the example below. We are told that zero in our number system is special because it has a value of nothing and it serves as a place holder. Even though this statement is true, it is not properly understood by children. In fact, the best way of understanding the (in)effectiveness of math education is to ask children (or adults) to explain what they have learned about zero.
When asked how much is 0 + 5, people will correctly respond "5." When asked why, they will tell you that zero equals nothing. This puzzles me, because if you add 5 hours to 12 on the clock, the answer will be 5 o'clock--and surely 12 is not nothing. People then say that 12 is like nothing, and leave it at that. Well, 12 on a clock is like zero in that they both come before 1. Our focus now is on why 1 is so important. One reason is that 1 is used to generate all the other numbers.
Anecdote has it that when Bill Cosby told his mother that he had learned that 1 + 1 = 2, his mother complimented him. Then young Bill asked, "Mom, what is 2?" This is no laughing matter; 2 is the definition of 1 + 1. Similarly, 3 is the definition of 1 + 1 + 1. These simple definitions along with the associative and equivalency concepts lead us to discover addition. Each number is obtained by adding one to the previous number. Besides learning how to add one to a number, we learn how to sequence the numbers: 1, 2, 3, 4, 5, etc.
If one were to ask what number comes before one, we would say that it must have the property that it + 1 = 1. We have learned to call that number "zero." 0 + 1 = 1. Logic then leads us to say that the number before one has a value of zero, and that the number before zero has the property of -1. We would discover by analogy that 11 (on a clock) has the property of -1, both numbers being two positions before 1. The concept of zero comes about from how we define and order our numbers. As far as zero being a place holder, every number is a place holder.
If we were not bothered by this third grade math we might say that this is interesting, but ask what difference does it make? Incorrect or incomplete knowledge keeps us from understanding and from making discoveries. On the clock what is 1/3 of 3? Most people will be satisfied with answering "1." However, there are three answers: 1, 5, and 9! To check those results we have:
1 + 1 + 1 = 3
5 + 5 + 5 = 15 = 15 - 12 = 3 on the clock.
9 + 9 + 9 = 27 = 27 - 24 = 3 on the clock.
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