An attempt to understand students' understanding of basic algebra

This paper reports the results obtained with a group of 24 14-year-old students when presented with a set of algebra tasks by the Leeds Modelling System, LMS. These same students were given a comparable paper-and-pencil test and detailed interviews some four months later. The latter studies uncovered several kinds of student misunderstandings that LMS had not detected. Some students had profound misunderstandings of algebraic notation: Others used strategies such as substituting numbers for variables until the equation balanced. Additionally, it appears that the student errors fall into several distinct classes: namely, manipulative, parsing, clerical, and “random.” LMS and its rule database have been enhanced as the result of this experiment, and LMS is now able to diagnose the majority of the errors encountered in this experiment. Finally, the paper gives a process-oriented explanation for student errors, and re-examines related work in cognitive modelling in the light of the types of student errors reported in this experiment. Misgeneralization is a mechanism suggested to explain some of the mal-rules noted in this study.     

The cognitive structure of social categories

Support for the prototype theory of categorization was found in a study of the structure of social categories. Though occupational terms such as DOCTOR are socially defined, they do not have the classical structure their clear definitional origins would predict. Conceptions of social categories are richer and more complex than those of physical object categories and subjects agree upon them. Comparison of various instructions for eliciting attributes of categories showed that whether subjects are asked to define a term, give characteristics, or describe ways they recognize members of categories, the attributes they list contribute to a prototype structure. These data provide evidence against the view that prototype structure is relevant only to an identification procedure and not to the core of concepts, as has been suggested.