Students' understanding of algebraic notation:

TLDR: Evidence is presented that some common misinterpretations can be explained by considering factors more accessible than cognitive level to diagnosis and possible remediation, and that difficulties in learning to use algebraic notation have several origins.

Abstract: Research studies have found that the majority of students up to age 15 seem unable to interpret algebraic letters as generalised numbers or even as specific unknowns. Instead, they ignore the letters, replace them with numerical values, or regard them as shorthand names. The principal explanation given in the literature has been a general link to levels of cognitive development. In this paper we present evidence for specific origins of misinterpretation that have been overlooked in the literature, and which may or may not be associated with cognitive level. These origins are: intuitive assumptions and pragmatic reasoning about a new notation, analogies with familiar symbol systems, interference from new learning in mathematics, and the effects of misleading teaching materials. Recognition of these origins of misunderstanding is necessary for improving the teaching of algebra. The Concepts in Secondary Mathematics and Science (CSMS) research project (Hart, 1981) provided evidence linking students' levels of under- standing of algebraic letters to Piagetian stages of cognitive development and to IQ scores. It was concluded that most of the 13 to 15-year-olds tested were unable to cope with items that required interpreting letters as generalised numbers or even as specific unknowns. In the many years since the CSMS project, it has been widely accepted that cognitive level is a suf- ficient explanation for the way in which algebraic notation is interpreted. If cognitive level is viewed as a barrier to the construction of certain concepts, it explains why students cannot do certain algebraic tasks. However, it does not explain why they misinterpret the notation in different ways and why they make certain errors. In this paper we take this next step. We show that some common misinterpretations can be explained by considering factors more accessible than cognitive level to diagnosis and possible remediation. We present evidence that difficulties in learning to use algebraic notation have several origins, including: