Correlation between Student Performance in Linear Algebra and Categories of a Taxonomy

This paper concerns a study of the performance of students in a recent linear algebra examination. We investigated differences in performance in tasks requiring understanding of the concepts with those that required only the use of routine procedures and factual recall. Central to this study was the use of a taxonomy, based on Bloom's Taxonomy, for characterising assessment tasks, which we have described in previous publications. The full taxonomy has 8 categories, which fall into 3 broad groups. The first group (A) encompasses tasks which could be successfully done using a surface learning approach, while the other two (B and C) require a deeper learning approach for their successful completion. Tasks on the examination paper were put into one of the three groups and comparisons were made concerning the performance of individual students in each of these areas. There are several interesting areas to investigate. The first is to identify those students whose performance in group A was markedly different to their performance on groups B and C. There is considerable disquiet amongst mathematics lecturers at tertiary level as to the routine algebraic skills of incoming students and of students studying mathematics at university (see for example the ICMI Study into the Teaching and Learning of Mathematics at University Level, 2001). There is a conjecture that students who have poor technical skills are not able to succeed in university mathematics. The contrapositive conjecture that good technical skills (such as algebraic dexterity) are necessary for success in university mathematics is often taken for granted. The taxonomy allows us to test this hypothesis as we can compare performance in group A tasks (routine) with performance in higher level B and C tasks. We have also investigated whether or not the data supports any systematic effect of differences in sex or language background in the performance on the three groups. The sample contained a large cohort of students with who had a home language other than English. We tested the hypothesis that such students would have difficulty with the conceptual aspects of the course, since these normally require greater language facility. This proved not to be the case.