RUME 14 (Volume 1)
2011
Portland, Oregon
Number of pages: 266
Number of papers: 20
- Stacy Brown
- Sean Larsen
- Karen Marrongelle
- Michael Oehrtman
2011
Portland, Oregon
Number of pages: 266
Number of papers: 20
In advanced undergraduate mathematics, students are expected to make sense of abstract definitions of mathematical concepts, create conjectures about those concepts, write proofs and exhibit counterexamples of these abstract concepts. In all of these actions, students may draw upon a rich store of examples in order to make meaningful progress. We have drawn on the concept of an example space (Watson & Mason, 2008) for a particular concept. We adapted it and defined the concepts of example neighborhood, methods of example construction, and the functions of examples to create a methodology for studying the teaching of proof-based courses. We demonstrate our method via a case study from an undergraduate abstract algebra course.
Over the past 15 years, mathematics departments have begun to incorporate online homework systems in mathematics courses, touting benefits for students and instructors alike. However, the impact of web-based homework systems on student engagement, learning, and perception are poorly understood. This preliminary study seeks to add to this body of research by comparing the performance and experience of students taking an undergraduate finite mathematics course in a traditional paper/pencil homework format to that of students completing the same course, with the same instructor, using an online homework format. While statistically significant results comparing the two sections on learning outcomes were few, descriptive analysis yields consistent trends suggesting that student learning may be enhanced through online homework participation. However, despite potential positive impacts on learning, students in the online homework section were significantly less likely than their traditional homework counterparts to rate their course as “excellent” on end-of-semester course evaluation forms.
Despite best efforts, hundreds of thousands of students are not succeeding in postsecondary general education mathematics courses each year. Using data from 11,970 enrollments in College Algebra, Foundations of Mathematics, and Elementary Calculus from fall 2007 to spring 2010 at the University of Memphis, we compare the impact of the Memphis Mathematics Method (MMM), a blended learning instructional model, to the traditional lecture teaching method on student performance and retention. Our results show the MMM was positive and significant for raising success rates particularly in Elementary Calculus. The results also show the MMM as a potential vehicle for closing the achievement gap between Black and White students.
This study traces how one real analysis student was able to develop a property-based rather than exemplar-based notion of sequence convergence by first instantiating conditions on the parameters of sequence convergence in a metaphorical domain. The analysis uses the radical constructivism framework of cognitive development (von Glasersfeld, 1995). Once he had encapsulated the conditions that related the parameters within the formal definition, he was able to accommodate the conditions into a mathematical schema in which he could reason flexibly about convergent sequences in ways compatible with standard formal practice. I observe his language use and the various lines of reasoning that his concept of the sequence convergence definition did or did not support over the course of a month to identify changes in his convergence schema.
Calculus is an important tool for building mathematical models of the world around us and is thus used in a variety of disciplines, such as physics and engineering. These disciplines rely on calculus courses to provide the mathematical foundation needed for success in their discipline courses. Unfortunately, many students leave calculus with an exceptionally primitive understanding and are ill-prepared for discipline courses. This study has begun to work with presumed experts (undergraduate mathematics and other discipline faculty members) to develop a small number of prototype tasks that will elicit, document, and measure students’ understanding of a few calculus concepts the faculty participants believe to be essential to successful academic pursuits within their respective disciplines. This paper discusses the data and analysis from the first round of interviews. Implications of these findings for calculus curriculum are presented.
This paper provides a method for analyzing undergraduate teaching of proof-based courses based on Toulmin’s model of argumentation. The paper presents a case-study of one instructor’s presentation of proofs and shows that she was inconsistent in the amount of detail that she included in her written proofs. The paper then describes how that analysis can be used as a predictor of subsequent student proof-writing performance. Second, student work is analyzed using Toulmin’s (1969) model for argumentation. The data shows that the students had adopted the modeled proof-writing behavior of the instructor in terms of the type and quantity of elements of argumentation. The method of analysis was developed via research in a lecture- based abstract algebra class, but has applications to any lecture-based proof-intensive course. This method provides one way to link classroom teaching activities to student performance that forces instructors to assume more responsibility for their students’ demonstrated end-of-course performance.
In a widely cited paper, Leron (1983) proposed presenting proofs in a novel format that he called “structured proofs” and suggested that this format improves comprehension. In a qualitative study, we found that participants had difficulty with this format for several reasons, including a lack of familiarity with the format and the requirement on the reader to jump around between different sections of the proof. In a larger quantitative study, we found that students reading a structured proof were better than students reading a linear proof at identifying a good summary of the proof, but performed slightly (but not statistically reliably) worse on questions concerning justifications within the proof, transferring the ideas from the proof to another setting, and illustrating the ideas of the proof using examples.
Students demonstrate a variety of behaviors while solving problems involving limits. However, no model currently exists to address what students do understand about limits while accounting for these varied behaviors. This study presents one possible model. This paper proposes that student behavior while solving limit problems may be interpreted by strands that reflect the student’s method for solving a problem involving limits, the student’s justification for the solution, and the applicability of the student’s method and justification within the context of the problem. This paper concludes with the presentation of a model which demonstrates how these strands connect to each other and a proposal of future directions for its refinement.
The study of sociomathematical norms initiated by Yackel and Cobb (1996) has become a popular way to make sense of the complexity of mathematical activity in the classroom. In this study we explore the role authority plays in the negotiation and legitimization of sociomathematical norms. We found that sociomathematical norms in this setting were introduced with hierarchal authorities whether the expectation was introduced implicitly or explicitly. We also found, when sociomathematical norms were introduced by a student, mathematical authority played a strong role in the negotiation process including interpretation, and legitimization.
We report on our work to build an applied theory for intercultural competence development for mathematics teaching and learning in secondary and tertiary settings. We use research in social anthropology and communications to investigate the nature of intercultural competence development for mathematics instruction among in-service secondary mathematics teachers and college faculty participating in a university-based mathematics teacher professional development program. We present results from quantitative and qualitative inquiry into the intercultural orientations of individuals and some groups (teachers, teacher-leaders, university faculty and graduate students) and offer details on the development of case stories for use in the professional development of mathematics university teacher educators, in-service teacher leaders, and secondary school teachers.
The article reviews efforts to develop an observation protocol to assess the pedagogical content knowledge (PCK) that middle and high school teachers may develop and demonstrate in the classroom over time as part of their participation in a master’s program for secondary mathematics teachers. We observed each of 16 teachers in real time using the instrument, before involvement in the project and again during the second semester of participation. Aspects of the protocol measure four critical components of PCK including curricular content, discourse, anticipatory, and implementation knowledge. We present a quantitative analysis of the observations and discuss various challenges faced in the instrument development and its relation to similar protocols used by others previously.
In this article we aim to understand what it looks like when community college instructors, with little to no experience with inquiry-oriented curriculum, implement inquiry-based curriculum for the first time. To approach this question we focused our attention on a community college instructor’s first implementation of an inquiry-oriented task. In total we identified six moves used during the implementation of this task. These moves are (1) zooming out, (2) real world examples, (3) counter-examples, (4) selective restating (5) referring to definitions, and (6) sequencing of student sharing. By identifying these teacher moves, it is indicated that mathematics instructors, even ones who primarily engage in teacher-center teaching, have techniques that they can draw on as they enact inquiry-oriented curriculum materials. Identifying such techniques can serve as a starting point for understanding how to support college-level teachers in changing their teaching practices.
The purpose of this paper is to share preliminary results from a pilot study on mathematical definitions. Interviews with university mathematicians were designed to gain insight into mathematicians' processes for developing understanding of new definitions. We asked the participants to talk about what helps them understand a new definition and how they support students’ understanding of definitions. We also observed them while they engaged in a definition task. Analysis revealed a noticeable difference in the emphasis on examples between what the participants described that they do and what they actually did while working on the definition task. We hypothesize that mathematicians’ processes for making sense of a definition necessarily involve considering the definition’s usefulness within a particular mathematical setting. Furthermore, these data indicate that mathematicians see examples as a multi-faceted, but not comprehensive, tool for understanding definitions.
We present a genetic decomposition using APOS Theory, about the way in which students can construct the concepts of spanning set and span in Linear Algebra. We also present empirical data coming from interviews made with 11 university students who had completed a course on Analytic Geometry which included an introduction to Linear Algebra. We report on our observations and suggest possible modifications to our initial theoretical analysis.
A configuration for analyzing vector representations based on multiple representations, semiotic representation, cognitive development, and mathematical conceptualization, to serve as a new unifying framework for studying undergraduate student approaches and difficulties in understanding and use of vectors is proposed. Using this configuration, the study will explore five important transitions: physics to mathematics, arithmetic to algebraic, analytic to synthetic, geometric to symbolic, concrete to abstract, and corresponding student difficulties along epistemological and ontological axes. As a part of validation of the framework, a mini-study on undergraduate students’ approaches and difficulties in understanding and use of vectors is introduced, and we see how useful this new framework is to describe and analyze student approaches and difficulties in understanding and use of vectors.
At last year’s conference, we presented a qualitative study providing insight into what mathematicians believe makes a good proof for pedagogical purposes based on eight mathematicians’ revisions of two proofs (see Lai & Weber, 2010). In this paper, we empirically test four hypotheses generated from last year’s study. This year’s study provides quantitative support for the claims that mathematicians believe (1) adding an introductory sentence stating the goals of the proof improves its pedagogical quality, (2) formatting key equations in a proof to emphasize their importance improves their pedagogical quality, and (3) unnecessary statements in a proof lowers its pedagogical quality.
This report presents the findings of an exploratory study into the perceptions held by students regarding the use of criterion-referenced assessment in an undergraduate differential equations class. Students in the class were largely unaware of the concept of criterion referencing and of the various interpretations that this concept has among mathematics educators. Our primary goal was to investigate whether explicitly presenting assessment criteria to students was useful to them and guided them in responding to assessment tasks. Quantitative data and qualitative feedback from students indicates that while students found the criteria easy to understand and useful in informing them as to how they would be graded, the manner in which they actually approached the assessment activity was not altered as a result of the use of explicitly communicated grading criteria.
In recent years, researchers have given attention to the new mathematics graduate student as a mathematics instructor. In contrast, this study explores the academic side of the transition to graduate school in mathematics—the struggles students face, the expectations they must meet, and the strategies they use to deal with this new chapter in their academic experience. I will identify several resulting themes—Isolation vs. Community, Academic Relationships, Role of the Department, and Realizations of Self—from semi-structured interviews with mathematics graduate students designed to explore multiple aspects of the academic transition to graduate school. I will also use the social theory of legitimate peripheral participation (Herzig, 2002; Lave & Wenger, 1991) to discuss potential implications for graduate students.
The purpose of this research was to gain insights into how calculus students might come to understand the formal definitions of sequence, series, and pointwise convergence. In this paper we discuss how one pair of students constructed a formal ε-N definition of series convergence following their prior reinvention of the formal definition of convergence for sequences. Their prior reinvention experience with sequences supported them to construct a series convergence definition and unpack its meaning. We then detail how their reinvention of a formal definition of series convergence aided them in the reinvention of pointwise convergence in the context of Taylor series. Focusing on particular x-values and describing the details of series convergence on vertical number lines helped students to transition to a definition of pointwise convergence. We claim that the instructional guidance provided to the students during the teaching experiment successfully supported them in meaningful reinvention of these definitions.
We compare the effect of incorporating inquiry-based sessions versus traditional lecture sessions, and a blend of the two approaches, in an elementary algebra course in which the pedagogy consistent among treatments is computer-assisted instruction. Our research hypothesis is that inquiry-based sessions benefit students significantly in terms of mathematical content knowledge, problem-solving, and communications. All students receive the same computer- assisted instruction component. Students are randomly assigned for the semester to one of three treatments (two inquiry-based meetings, two lecture meeting, or one of each, weekly). Measures, including pre- and post-tests with both open-ended and objective items, are described. Statistically significant differences have previously been observed in similar quasi-experimental studies of multiple sections of finite mathematics (Fall, 2008) and elementary algebra (Fall, 2009) with two treatments. Undergraduates, including many pre-service elementary teachers, who do not place into a credit-bearing mathematics course take this developmental algebra course.