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.

Solving equations and simplifying expressions require the ability to reason about operations as expressions of quantitative relationships rather than just procedures. Researchers have found that sixth graders lack adequate experience in developing this ability. Students were asked to judge the equivalence (without computing the totals) of three-term arithmetic expressions with a subtraction and an addition operation;54 for example, 685–492+947, 947+492–685, 947–685+492, and 947–492+685. The typical answer was that you needed to calculate to decide whether the expressions were equivalent. Similar results were found in another study55 when students of the same age were presented with the task of stating the value of □ in the expression (235+□)+(679–122)=235+679. Findings such as these illustrate that traditionally instructed students who are proficient with numbers need to shift from thinking about “finding the answer” to thinking about the “numerical relationships” underlying the calculations they perform and the nature of the methods they use.

Traditionally instructed students who are proficient with numbers need to shift from thinking about “finding the answer” to thinking about the “numerical relationships” underlying the calculations they perform and the nature of the methods they use.

Students’ experience with equivalence in earlier grades is often restricted to their study of equivalent fractions. For example, is equivalent to which is equivalent to and so on. But this equivalence is one of numbers, not of operations or expressions. There are few opportunities in the present



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