Three interpretations of the matrix equation Ax=b

Over the past years we have been involved in a series of studies researching the teaching and learning of linear algebra at the undergraduate level in the United States. Through work with our research colleagues and our own teaching of linear algebra we have come to reflect on the nature of the cognitive demands that an introductory undergraduate linear algebra course places on students. Many of the struggles students face are related to a central set of ideas. In this article, we present a framework that offers insight into student thinking related to these central ideas. This framework can serve as a tool for researchers, curriculum designers and instructors to deconstruct their expert knowledge in order to anticipate challenges that students may face. The framework also functions as a diagnostic tool that can be helpful in making sense of the unexpected and seemingly idiosyncratic ways in which students often blend ideas, particularly as they begin learning linear algebra.