What Kinds of Numbers Do Students Assign to Literal Symbols? Aspects of the Transition from Arithmetic to Algebra

Three experiments used multiple methods-open-ended assessments, multiple-choice questionnaires, and interviews-to investigate the hypothesis that the development of students' understanding of the concept of real variable in algebra may be influenced in fundamental ways by their initial concept of number, which seems to be organized around the notion of natural number. In the first two experiments 91 secondary school students (ranging in age from 12.5 to 14.5 years) were asked to indicate numbers that could or could not be used to substitute literal symbols in algebraic expressions. The results showed that there was a strong tendency on the part of the students to interpret literal symbols to stand for natural numbers and a related tendency to consider the phenomenal sign of the algebraic expressions as their "real" sign. Similar findings were obtained in a third, individual interview study, conducted with tenth grade students. The results were interpreted to support the interpretation that there is a systematic natural number bias on students' substitutions of literal symbols in algebra.