Teachers' and researchers' beliefs about the development of algebraic reasoning

Mathematics teachers and educational researchers ordered arithmetic and alg ebra problems according to their predicted problem-solving difficulty for s tudents. Predictions deviated systematically from algebra students' perform ances but closely matched a view implicit in textbooks. Analysis of student s' problem-solving strategies indicates specific ways that students' algebr aic reasoning differs from that predicted by most teachers and researchers in the sample and portrayed in common textbooks. The Symbol Precedence Mode l of development of algebraic reasoning, in which symbolic problem solving precedes verbal problem solving and arithmetic skills strictly precede alge braic skills, was contrasted with the Verbal Precedence Model of developmen t, which provided a better quantitative fit of students' performance data. Implications of the findings for student and reacher cognition and for alge bra instruction are discussed.