Search

In many advanced mathematics courses, comprehending theorems and proofs is an essential activity for both students and mathematicians. Such activity requires readers to draw on relevant meanings for the concepts involved; however, the ways that concept meaning may shape comprehension activity is currently undertheorized. In this paper, we share a study of student activity as they work to comprehend the First Isomorphism Theorem and its proof. We analyze, using an onto-semiotic lens, the ways that students' meanings for quotient group both support and constrain their comprehension activity. Furthermore, we suggest that the relationship between understanding concepts and proof comprehension can be reflexive: understanding of concepts not only influences comprehension activity, but engaging with theorems and proofs can serve to support students in generating more sophisticated understanding of the concepts involved.

The metaphors that students form and encounter have been shown to exert a powerful influence on how they think about mathematics. In this paper, we explore the linguistic metaphors about learning and doing mathematics that were prevalent in 11 advanced mathematics lectures. We present four metaphor clusters that were common in the corpus that we analyzed: Learning Mathematics is a Journey, Doing Mathematics is Work, Mathematics is Discovery, and Presenting Mathematics is a Story. We conclude the paper by discussing the implications of the metaphors that mathematicians use and students hear in their mathematics courses.

This paper presents the findings from a survey used to investigate how mathematicians perceive the genre of mathematical proof writing at the undergraduate level. Mathematicians were asked whether various proof excerpts highlighted in four partial proofs were unconventional in each one of three pedagogical contexts: undergraduate mathematics textbooks, what instructors write on the blackboard in undergraduate mathematics courses, and how students write mathematics in these courses. There are four main findings. First, there are some potential breaches of mathematical language that participants found unconventional regardless of the context in which they occur. Second, there are some differences in how mathematicians perceived the linguistic conventions in blackboard proofs and student-produced proofs. Third, textbook authors are expected to adhere to stricter writing norms than mathematics instructors and undergraduate students when writing proofs. Fourth, there were some potential breaches of mathematical language that the literature suggests were unconventional, which were not evaluated as unconventional by the mathematicians. We argue that this diversity of expectations regarding the language of mathematical proof writing in undergraduate classrooms, together with the potential disconnect between those expectations and the types of proofs that students see presented in those classrooms, could make it difficult for students to become proficient in this important mathematical discourse.

This study examined the genre of undergraduate mathematical proof writing by asking mathematicians and undergraduate students to read 7 partial proofs and identify and discuss uses of mathematical language that were out of the ordinary with respect to what they considered conventional mathematical proof writing. Three main themes emerged: First, mathematicians believed that mathematical language should obey the conventions of academic language, whereas students were either unaware of these conventions or unaware that these conventions applied to proof writing. Second, students did not fully understand the nuances involved in how mathematicians introduce objects in proofs. Third, mathematicians focused on the context of the proof to decide how formal a proof should be, whereas students did not seem to be aware of the importance of this factor.

This article presents an inquiry-oriented lesson for teaching Lagrange's theorem in abstract algebra. This lesson was developed and refined as part of a larger grant project focused on how to "Orchestrate Discussions Around Proof" (ODAP, the name of the project). The lesson components were developed and refined with attention to how well they supported active and broad student participation. By guiding student exploration of a few key example groups and structuring the exploration to identify recurrent structure, students formulate key lemmas that they can combine to prove Lagrange's theorem. [ABSTRACT FROM AUTHOR]

In this article, we describe and illustrate the process by which we developed and validated short, multiple-choice, reliable tests to assess undergraduate students' comprehension of three mathematical proofs. We discuss the purpose for each stage and how it benefited the design of our instruments. We also suggest ways in which this process could be employed by other researchers to develop and validate their own reliable proof comprehension tests.

We describe a case study in which we investigate the effectiveness of a lecture in advanced mathematics. We first videorecorded a lecture delivered by an experienced professor who had a reputation for being an outstanding instructor. Using video recall, we then interviewed the professor to determine the ideas that he intended to convey and how he tried to convey these ideas in this lecture. We also interviewed 6 students to see what they understood from this lecture. The students did not comprehend the ideas that the professor cited as central to his lecture. Based on our analyses, we propose 2 factors to account for why students did not understand these ideas.

In mathematics education, researchers commonly infer students' standards of conviction from the justifications that they produce. Specifically, if students justify a mathematical statement with an empirical justification, researchers often infer that example-based justifications provide the students with certainty that a general mathematical statement is true. In this article, we present a theoretical framework for interpreting individuals' proof-related behavior that challenges the aforementioned interpretations. Adapting constructs from expectancy value theory, we argue that whether an individual will seek a deductive proof or settle for an empirical justification depends on several factors, including the value they place on knowing the veracity of the mathematical statement being considered, the cost in terms of time and effort in searching for a proof, and their perceived likelihood of success of being able to find a proof. We demonstrate that mathematicians consider value, cost, and effort in deciding what statements they will try to prove, so it would not be irrational or unmathematical for students to make the same considerations. We illustrate the explanatory power of our framework by studying the justification behavior of 11 preservice and in-service secondary mathematics teachers in a problem-solving course. Although these individuals frequently justified mathematical statements empirically, these individuals were aware of the limitations of empirical justifications and they usually did not obtain certainty from these justifications. The notions of value, cost, and likelihood of success could explain why they settled for empirical justifications and ceased seeking proofs. [ABSTRACT FROM AUTHOR]