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

Abstract

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.

ACKNOWLEDGMENTS

We would like to thank Xenia Vamvakoussi, Lyn English, and the anonymous reviewers who provided useful suggestions resulting in significant improvements of this article.

Funding for the research was obtained through the program PYTHAGORAS-EPEAΕK II, with 75% contribution from European Social Funds and 25% contribution from Greek Funds.

Notes

¹By “variable” we mean “real variable,” that is a variable whose range is a subset of the real numbers. Also, the term “concept” is used to refer to the normative, socially shared, and culturally accepted concept, while the term “conception” is used in this paper to refer to individuals' subjective constructions of the mathematical concept (see also Leinhardt, Zaslavsky, & Stein, 1990; Sfard, 1991). The term “initial concept” is also used to refer to what is assumed to be the initial, psychological, concept of number, which we argue is close to the mathematical notion of natural number.

²With a few exceptions such as that a denominator cannot equal zero.

³The response that zero cannot be substituted for “g” or “b” in the case of “1/g” and “a/b” respectively was also considered as correct.

⁴Only one of the students tried to solve the first two inequalities formally but she did not succeed and accepted the interviewers' suggestion to try with numbers.