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• #### Index 441-454

Students who are proficient with arithmetic are generally assumed to have facility with the arithmetic operations of addition and multiplication and their inverses (subtraction and division), with computations written in a horizontal form, and with the equivalence of numerical expressions. These notions, however, are not always as well cultivated in elementary school mathematics as they should be if they are to serve as a basis for algebraic reasoning.

Students emerging from six or seven years of elementary school mathematics are ordinarily aware of the close relationship between addition and subtraction. After all, they check subtraction written vertically by adding the answer (the difference) to the number above it (the subtrahend) to see if it gives the number in the top line of the subtraction (the minuend). But they seem less comfortable with moving among the written forms of this relationship—for example, from an addition statement written horizontally to its equivalent subtraction (e.g., writing 35+42=77 as 35=77–42). Thus, these students seem somewhat bewildered when asked in initial algebra instruction to express, say, x+42=77 as x=77–42. The same confusion over the written notation for the inverse relationship between addition and subtraction is seen in the errors students make in solving equations53 when they judge, say, x+37=150 to be equivalent to x=37+150 and x+37=150 to be equivalent to x+37–10=150+10.