Developing an understanding of algebraic symbols

The major objective of this research project was to investigate the difficulties that beginning algebra students experience in developing an understanding of the meaning and use of algebraic symbols. Learning problems identified by relevant research projects during the previous two decades provided a starting point, and items used in these projects for written tests or interviews were valuable in the formation of a new test instrument. By incorporating aspects investigated by several other researchers, a broad-based approach was employed to extend their work of applying psychological understandings of cognition to the learning processes involved in early algebra. Investigations examined interrelationships between measures previously studied in separate projects. Data were collected for analysis from a sample of 208 Year 7 secondary school students as they began their study of algebra in the form of generalized arithmetic. Methods of data collection were repeated written tests, interviews and lesson observations. To locate the responses of the beginning Year 7 students in the learning continuum about algebraic symbols as numerical variables, research data were also collected from another 309 Years 7 to 12 students. Scales were established for measuring and reporting on the patterns of thinking revealed by the students' responses. The pool of research information about the learning of algebra was expanded by the frequency data for individual items and for scaled groups of items. Comparisons and contrasts with findings of earlier researchers were reported where possible. Hierarchies of difficulty, as proposed by previous researchers for distinguishing levels of understanding of algebraic symbols, were tested for their applicability to the student sample and to see if they reflected any identifiable learning sequences. The most difficult challenge for students beginning their study of the algebra of generalized arithmetic was found to be the attainment of an understanding of algebraic symbols as representing numerical variables. Some Year 7 students made little progress towards this goal during the seven months of the study. The tendency to regard symbols as standing for objects or people was one focus of attention. Evidence supported the view that the level of achievement on the algebraic tasks presented is related to the degree of progress towards understanding algebraic symbols as numerical variables. Empirical data were shown to agree with psychological reasons for arranging some of the tasks into hierarchical orders of difficulty and/or into sequential orders of learning. There was some elucidation of the key steps in learning which distinguish students likely to progress in algebra.