The understanding of generalised arithmetic (algebra) by secondary school children
A class test, Algebra, was developed as part of the research effort of the mathematics wing of the programme "Concepts in Secondary school Mathematics and Science". The test was developed in the light of interviews with 27 children and written trials involving 11 classes. In the summer of 1976, the final version of the test was then given to representative samples of 1128 2nd year, 961 3rd year, and 731 4th year children (13, 14, and 15 year olds) in English secondary schools (together with an unrepresentative sample of 103 5th year children). The test was designed to investigate the different ways in which children interpret the letters in generalised arithmetic: six ways were identified, which were called Letter Evaluated, Letter Not Used, Letter as Object, Letter as Specific Unknown, Letter as Generalised Number, and Letter as Variable. Subsequently selected items from the test were classified into four "levels of understanding". The items were selected by using statistical methods ('spider diagrams' and factor analysis) based on the correlation coefficient phi. Other coefficients were also investigated, as was a method based on characteristic curves and Guttman scalogram analysis. Performance on the test was compared with performance on other CSMS mathematics tests and also with a Piagetian class task devised by the science wing of CSMS. The latter comparison was used to classify the items on the test into Piagetian substages.