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.