Every algebraic manipulation involves an anticipatory element, a sense of the direction in which you want to be going and of what the desired expression will look like once you get there.6 The development of this sense of anticipation provides an alternative to the “blind” manipulation that is so often performed by beginning algebra students.7 Research suggests, however, that such anticipatory thinking is not acquired without effort. Even students with extensive algebra experience can make poor strategic decisions that leave them “going round in circles” because they cannot seem to “see” the right thing in algebraic expressions.8

The transformational aspects of algebra have traditionally been emphasized in U.S. textbooks, which have tended to pay more attention to the rules to be followed in manipulating symbolic expressions and equations than to the concepts that support those rules or give meaning to the expressions or equations being manipulated. Although few experimental comparisons have been conducted, research has shown that rule-based instructional approaches that do not give students opportunities to create meaning for the rules or to learn when to use them can lead to forgetting,9 unsystematic errors,10 reliance on visual clues,11 and poor strategic decisions.12 For example, experienced algebra students were found to choose inappropriate strategies when deciding what to do next in the simplification of an algebraic expression and would often end up with an expression that was more difficult to deal with, even though they had performed legal transformations.13 Beginning algebra students were found to be quite haphazard in their approach; they might simplify 4(6x–3y)+5x as 4(6x–3y–5x) on one occasion and do something else on another.14 When the consecutive numbers problem was given to 113 high school students who had studied algebra, only 8 worked the problem correctly.15 The rest made a variety of errors, including substituting a few values for x to show the sum’s “oddness,” using different letters for each number (x and y), representing the consecutive numbers as 1x and 2x, and setting the expression x+(x+1) equal to a fixed odd number and then solving for x. In one of the few experimental studies of rule-based instruction, students who were taught an estimate-and-test sense-making strategy performed better in solving systems of equalities and inequalities than students taught rule-based equation solving.16

Data from the National Assessment of Educational Progress (NAEP) further reveal the shortcomings of traditional school algebra. For example, one of the NAEP tasks from the second mathematics assessment involved completing the table shown in Box 8–2. Most of the students with one or two years of algebra could recognize the pattern—adding 7—from the given nu-

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