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One of the problems faced by middleschool-aged children in the study of mathematics finds its root in the properties of binary operations when confronting various mathematical systems. By the time these children begin the study of the properties of each of the four basic operations, they have already done extensive work with the operation itself. That is, children have learned to add, subtract, multiply, and divide long before the properties of these operations are introduced formally. By the time formal study of the operations and their properties is begun, the students are often no longer able to distinguish the property under consideration from the operation itself. Thus it becomes quite difficult to explain to some children exactly what is meant by the words “closure, associative, commutative, identity, and inverse.” Even though a teacher may succeed in fixing these concepts in the minds of his students, he must then face the problem of convincing each student, in a meaningful way, that every operation that is associative is not necessarily commutative, and vice versa. This is not an easy task, because the student is generally aware that addition and multiplication in the set of natural numbers have both properties, whereas subtraction and division in the same set have neither; in fact they do not even possess the closure property!