The Expected Value of a Random Variable: Semiotic and Lexical Ambiguities

In calculus-based statistics courses, the expected value of a random variable (EVORV) is discussed in relation to underlying mathematical notions. This study examines students’ understanding of the mathematical notions of EVORV in connection with its semiotic and lexical representations. It also assesses students’ computational competency revolving around EVORV. We collected qualitative data via surveys and interviews from eight students enrolled in a calculus-based university statistics course. The results suggest that while the students in general had the computational accuracy to correctly calculate EVORV, they struggled to understand the notion, and in particular to make sense of the term “random” in “random variable” and the symbol E() in the mathematical context. The study provides a basis for understanding potential challenges to students’ learning of EVORV and other related statistics topics and how such challenges may emerge from the semiotic and lexical ambiguities inherent in terms and symbols used in statistics

Divergent definitions of inquiry-based learning in undergraduate mathematics

Inquiry-based learning is becoming more important and widely practiced in undergraduate mathematics education. As a result, research about inquiry-based learning is similarly becoming more common, including questions of the efficacy of such methods. Yet, thus far, there has been little effort on the part of practitioners or researchers to come to a description of the range(s) of practice that can or should be understood as inquiry-based learning. As a result, studies, comparisons and critiques can be dismissed as not using the appropriate definition, without adjudicating the quality of the evidence or implications for research and teaching. Through a large-scale literature review and surveying of experts in the community, this study begins the conversation about possible areas of agreement that would allow for a constituent definition of inquiry-based learning and allow for differentiation with non-inquiry pedagogical practicesAccepted manuscrip

Teacher Questioning and Invitations to Participate in Advanced Mathematics Lectures

In this study, we were interested in exploring the extent to which advanced mathematics lecturers provide students opportunities to play a role in considering or generating course content. To do this, we examined the questioning practices of 11 lecturers who taught advanced mathematics courses at the university level. Because we are unaware of other studies examining advanced mathematics lecturers’ questioning, we first analyzed the data using an open coding scheme to categorize the types of content lecturers solicited and the opportunities they provided students to participate in generating course content. In a second round of analysis, we examined the extent to which lecturers provide students opportunities to generate mathematical contributions and to engage in reasoning researchers have identified as important in advanced mathematics. Our findings highlight that although lecturers asked many questions, lecturers did not provide substantial opportunities for students to participate in generating mathematical content and reasoning. Additionally, we provide several examples of lecturers providing students some opportunities to generate important contributions. We conclude by providing implications and areas for future research

Mathematics lectures as narratives: insights from network graph methodology

Although lecture is the traditional method of university mathematics instruction, there has been little empirical research that describes the general structure of lectures. In this paper, we adapt ideas from narrative analysis and apply them to an upper-level mathematics lecture. We develop a framework that enables us to conceptualize the lecture as consisting of collections of narratives to identify connections between the narratives and to use the narrative structure to identify key features of the lecture. In particular, we use the idea of framing and embedded narratives to identify central ideas in the lecture and to understand how examples, diagrams, and smaller claims contribute to the development of those ideas. Additionally, we create a graph structure for the lecture using the repetition of mathematical ideas across embedded narratives, and we employ graph-theoretic analysis to support our identification of framing narratives and to highlight particular roles of embedded narratives

Characterizing instructor gestures in a lecture in a proof-based mathematics class

Researchers have increasingly focused on how gestures in mathematics aid in thinking and communication. This paper builds on Arzarello’s (2006) idea of a semiotic bundle and several frameworks for describing individual gestures and applies these ideas to a case study of an instructor’s gestures in an undergraduate abstract algebra class. We describe the role that the semiotic bundle plays in shaping the potential meanings of gestures; the ways gestural sets create complex relationships between gestures; and the role played by polysemy and abstraction. These results highlight the complex ways in which mathematical meanings—both specific and general—are expressed in gesture, and to highlight the integrated nature of elements of the semiotic bundle

Student Understanding of Symbols in Introductory Statistics Courses

This study explores student understanding of the symbolic representation system in statistics. Furthermore, it attempts to describe the relation between student understanding of the symbolic system and statistical concepts that students develop as the result of an introductory undergraduate statistics course. The theory, drawn from the notion of semantic function that links representations and concepts, seeks to expand the range of representations considered in exploring students’ statistical proficiencies. Results suggest that students experience considerable difficulty in making correct associations between symbols and concepts; that they describe the relationship as seemingly arbitrary; and that they are unlikely to understand statistics as quantities that can vary. Finally, this study describes students’ need for robust knowledge of preliminary concepts in order to understand the construct of a sampling distribution

P-37 When we grade proofs, do our students understand what we’re saying?

The ability to write clear, correct proofs is a central goal of the curriculum for undergraduate mathematics majors. In an earlier study Moore (under review) investigated the proof-grading practices of four mathematics professors and showed that these professors devote much time and effort to reading students’ written proofs and marking the papers with corrections and suggestions for improvement. To learn how students interpret and make use of such feedback, we interviewed eight advanced mathematics undergraduates and asked them to respond to professor comments on three or four written proofs. The participants were asked to interpret and justify each comment and then write a revised version of each proof. Using the theoretical frameworks of communities of practice and legitimate peripheral participation, we analyzed the interviews and written data, compared the students’ interpretations of the comments to expert consensus, and identified patterns and commonalities in their responses and actions. A noteworthy finding was that even though students were able to identify and correctly implement the professor’s recommended changes, they sometimes misinterpreted the professor’s intentions

Student Interpretations of Written Comments on Graded Proofs

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

How mathematicians assign homework problems in abstract algebra courses

Embargo of 24 months from article publicationWhile many aspects of the teaching and learning of advanced mathematics have been explored, the role, construction, and values of homework have been virtually ignored. This report draws on task-based interviews with six mathematicians to explore the relationship between an instructor’s learning goals and factors considered when selecting homework problems. All participants viewed homework as critical to student learning, and the majority of the participants’ claims focused on either the mathematics or how the problem would help students learn; no instructor gave primacy to evaluative reasons for homework. We highlight six themes used by participants to evaluate and select items for inclusion in homework. They are (1) knowing and recalling ax- ioms and definitions, (2) developing an arsenal of examples, (3) developing new problem ap- proaches, (4) remediating misconceptions, (5) making connections to prior and future material, and (6) valuing reading notes or text