A Tale of Two Courses; Teaching and Learning Undergraduate Abstract Algebra

The abstract algebra course is an important point in the education of undergraduate mathematics majors and secondary mathematics teachers. Abstract algebra teachers have multiple goals for student learning, and the literature suggests that students have difficulty meeting these goals. Advisory reports have called for a move away from lecture toward investigation-based class sessions as a means of improving student understanding. Thus, it is appropriate to understand what is happening in the current teaching and associated learning of abstract algebra. The present study examined teaching and learning in two abstract algebra classrooms, one consciously using a lecture-based (i.e., deduction-theory-proof, or DTP) mode of instruction and the other an investigative approach. Instructional data was collected in classroom observations, and multiple written instruments and a set of interviews were used to evaluate student learning. Each instructor hoped students would develop a deep and connected knowledge base and attempted to create classroom environments where students were constantly engaged as a means of doing so. In the lecture class, writing proofs was the central activity of class meetings; nearly every class period included at least one proof. In the investigative class, the processes of computing and searching for patterns in various structures were emphasized. At the end of the semester, students demonstrated mixed levels of proficiency. Generally, students did well on items that were relatively familiar, and poorly when the content or context was unfamiliar. In the DTP course, two students demonstrated significant proficiency with analytical argument; the remainder demonstrated mixed proficiency with proof and very little proficiency with other content. The students in the investigative class all seemed to develop similar levels of proficiency with the content, and demonstrated more willingness to explore unknown structures. This study may prompt discussions about the relative importance of developing proof-proficiency, students' ability to formulate and investigate hypotheses, developing students' content knowledge, and students' ability to operate in and analyze novel structures

When we Grade Students’ Proofs, Do They Understand our Feedback?

Instructors often write feedback on students’ proofs even if there is no expectation for the students to revise and resubmit the work. It is not known, however, what students do with that feedback or if they understand the professor’s intentions. To this end, we asked eight advanced mathematics undergraduates to respond to professor comments on four written proofs by interpreting and implementing the comments. We analyzed the student’s responses using the categories of corrective feedback for language acquisition, viewing the language of mathematical proof as a register of academic English

Students\u27 sense-making frames in mathematics lectures

The goal of this study is to describe the various ways students make sense of mathematics lectures. Here, sense-making refers to a process by which people construct personal meanings for phenomena they experience. This study introduces the idea of a sense-making frame and describes three different types of frames: content-, communication-, and situating-oriented. We found that students in an abstract algebra class regularly engaged in sense-making during lectures on equivalence relations, and this sense-making influenced their note-taking practices. We discuss the relationship between the choice of frame, the students\u27 sense-making practices, and the potential missed opportunities for learning from the lecture. These results show the importance of understanding the ways students make sense of aspects of mathematics lectures and how their sense-making practices influence what they might learn from the lecture. © 2013 Elsevier Inc

Is Grading Papers an Effective Teaching Practice?

College teachers devote a great deal of effort to grading students’ papers. We do this not simply to assign grades, but for the more important purpose of giving students feedback that will help them improve their work in the future. We should ask, then, whether grading papers is an effective teaching practice; that is, to what extent do students benefit from the comments we write on their papers? We will report on an examination of this question in the context of mathematics and then allow time for the audience to discuss this question in the context of other disciplines