The role of example-generation tasks in students' understanding of linear algebra

There is little doubt that linear algebra is a fundamentally important course in undergraduate studies. It is required for students majoring not only in mathematics, but also in engineering, physics, and economics, to name just a few. However, to date, research on students’ understanding of linear algebra is rather slim. This study is a contribution to the ongoing research in undergraduate mathematics education, focusing on linear algebra. It is guided by the belief that better understanding of students’ difficulties leads to improved instructional methods. The questions posed in this study are: What is students’ understanding of the key concepts of linear algebra? What difficulties do students experience when engaged in these tasks? What can example-generation tasks reveal about students’ understanding of linear algebra? Are these tasks effective and useful as a data collection tool for research in mathematics education? This study identifies some of the difficulties experienced by students with learning several key concepts of linear algebra: vectors and vector spaces, linear dependence and independence, linear transformations, and basis, and also isolates some possible obstacles to such learning. In addition, this study introduces learner-generated examples as a pedagogical tool that helps learners partly overcome these obstacles. There are several contributions of this study to the field of undergraduate mathematics education. Firstly, focusing on specific mathematical content, it provides a finer and deeper analysis of students’ understanding of linear algebra. Secondly, focusing on methodology, it introduces an effective data collection tool to investigate students’ learning of mathematical concepts. Thirdly, focusing on pedagogy, it enhances the teaching of linear algebra by developing a set of example-generation tasks that are a valuable addition to the undergraduate mathematics education. The tasks I have designed can be used in assignments, tutorials, and other educational settings serve not only as an assessment tool but also as an instructional tool that provides learners with an opportunity to engage in mathematical activity.