RUME 14 (Volume 2)
2011
Portland, Oregon
Number of pages: 273
Number of papers: 19
- Stacy Brown
- Sean Larsen
- Karen Marrongelle
- Michael Oehrtman
2011
Portland, Oregon
Number of pages: 273
Number of papers: 19
At a research university near the east coast, researchers have restructured a College Algebra course by formatting the course into two large lectures a week, an active recitation size laboratory class once a week, and an extra day devoted to active group work called Supplemental Practice (SP). SP was added as an extra day of class where the SP leader has students to work in groups on a worksheet of examples and problems, based off of worked example research, that were covered in the previous week’s class material. Two sections of the course were randomly chosen to be the experimental group and the other section was the control group. The experimental group was given the SP worksheets and the control group was given a question and answer session. The experimental group significantly outperformed the control on a variety of components in the course, particularly when prior knowledge is factored in.
This descriptive study investigates the ways in which proofs are presented in upper division proof-based undergraduate mathematics courses at a large comprehensive research university in the Midwest. To pursue this inquiry, four faculty members who were teaching such courses were interviewed and three of the four participated in video-taped observations periodically throughout the course of the semester. Interview data were used to construct a framework with which to analyze the observation data. The observation data were analyzed to determine the level of engagement that the professor seemed to expect of the students, and to identify some proof presentation strategies that were used.
Over the past half-century, one strand of problem solving research has investigated problem solvers’ behaviors during various phases of the problem solving process. A second strand of problem solving research has explored the mental actions, reasoning abilities, and understandings involved in using specific concepts to solve novel problems. These two strands of research provide valuable contributions to mathematics education, but they have remained as separate strands of inquiry within the research literature. This paper reports on the results of a study that investigated three students’ problem solving behaviors in the context of their confronting novel problems when learning central ideas of trigonometry. The study’s findings illustrate specific mental processes that enabled or hindered the students in engaging in productive problem solving behaviors. Particularly, the study describes how a student’s ability to engage in quantitative reasoning shaped their movement through the problem solving phases.
We propose viewing the ways in which people use symbols and drawings as having an intrinsic physicality. When perceived as an extension of gesture-making, symbol use can give us insight into how symbol users experience the mathematics they’re considering. To illustrate this, we apply this lens to examine selections from a video-recorded interview of a mathematician regarding one of his published papers. This results in several phenomenological characterizations of the mathematician’s embodied symbol use that, in turn, offer potential insight into many instances of symbol use beyond this single case study.
Little research exists on the ways in which students may develop an understanding of formal limit definitions. We conducted a study to i) generate insights into how students might leverage their intuitive understandings of sequence convergence to construct a formal definition and ii) assess the extent to which a previously established approximation scheme may support students in constructing their definition. Our research is rooted in the theory of Realistic Mathematics Education and employed the methodology of guided reinvention in a teaching experiment. In three 90-minute sessions, two students, neither of whom had previously seen a formal definition of sequence convergence, constructed a rigorous definition using formal mathematical notation and quantification equivalent to the conventional definition. The students’ use of an approximation scheme and concrete examples were both central to their progress, and each portion of their definition emerged in response to overcoming specific cognitive challenges.
This report describes a case study in an undergraduate elementary linear algebra class about the relationship between students’ understanding of span and linear independence and their intuition and language use. The study participants were seven students with a range of understanding levels. Findings indicated that students were more likely to successfully solve problems and explain concepts involving span and linear independence if they had low levels of self-evident intuitions and were better able to communicate their thinking in writing. Examples of students’ interfering intuitions regarding span and linear independence are described. The report also includes possible instructional implications.
A group of mathematicians and mathematics educators are collaborating in the fine- grained examination of selected ‘slices’ of video recordings of lectures drawing on Schoenfeld’s ROG framework of teaching-in-context. We seek to examine ways in which this model can be extended to examine university lecturing. In the process we have identified a number of lecturer behaviours. There are times when, in what appears to be an internal dialogue, lecturing decisions are driven by the mathematician within the lecturer despite the pre-stated intentions of the lecturer to be a teacher. We analyse three scenarios: in the first the teacher prevails, in the second the mathematician, and in the third the mathematician appears to prompt the teacher to be more rigorous. We analyse these behaviours against the lecturers’ ROGs, both stated and inferred. The value of the approach used as a professional development model is considered.
Given the rise in distance delivered graduate programs, educators continue to seek ways to improve teaching and learning in online environments. In particular, the need for high quality K-12 teachers requires superior teacher education programs that model good instructional practice, especially in mathematics. This paper explores the challenges and opportunities presented to the instructor of an online mathematics education course designed for inservice mathematics teachers. The mixed-methods study utilized data from class observations and survey data from participants to capture perceptions of the course. Results of these data are presented and used with the instructor’s reflections to make specific recommendations for improving the course and to offer insight to others using distance learning technology to teach graduate mathematics education courses.
Eleven exemplary high school mathematics teachers were interviewed to investigate their views on mathematical knowledge for teaching. Teachers took part in a one-hour interview and discussed a written lesson plan. Teachers believed the following aspects of mathematical knowledge for teaching to be important: (a) building mathematical ideas using prior mathematics, (b) a range of examples that illustrate a mathematical concept, (c) appropriate applications of a concept, (d) connections between mathematical ideas within and beyond the high school curriculum, and (e) various approaches to problem solving. Teachers also discussed the development of their mathematical knowledge for teaching, which they believed came from their teaching experience and personal experiences in addition to coursework.
In this paper, we investigate interpersonal difficulties that student teachers and cooperating teachers experience during the teaching internship by exploring the tension between one high school mathematics student teacher and his cooperating teacher. We identified seven causes of this tension, which included different ideas about what mathematics should be taught and how it should be taught and a strained personal relationship. We compare these findings to results from interviews with 6 other student teachers and 8 of their mentors. These results suggest that (a) cooperating teachers may offer less freedom than they realize, (b) mathematics educators and cooperating teachers may have very different goals for student teaching, (c) cooperating teachers may hold unrealistic expectations about the student teachers prior to their student- teaching experience, and (d) personal relationships can greatly impact the overall student- teaching experience.
This research explored principles in designing an instructional intervention to promote students’ reflective thinking about multiple quantifications. We suggested the Mayan activity and examined to what extent it promoted students’ reflective thinking about the independence ofε from N in the ε-N definition of the limit of a sequence. After the Mayan activity, students recognized the problem caused by describing ε as dependent on N and hence understood the significance of the order between them. Students also developed proper reasoning and tested their hypotheses by using the Mayan stonecutter story. These results indicate that the Mayan activity provides an example logically compatible with describing ε as dependent on N, tractable, and transferable.
Although some researchers argue that diagrams can aid undergraduates’ proof constructions, most undergraduates have difficulty translating a visual argument to a formal one. The processes by which undergraduates construct proofs based on visual arguments are poorly understood. We investigate this issue by presenting eight mathematicians with a mathematical task that invites the construction of a diagram and examine how they used this diagram to produce a formal proof. The main findings from the paper were that it was not trivial for mathematicians to translate an intuitive argument into a formal proof and mathematicians used diagrams for multiple purposes, including noticing mathematical properties, verifying logical deductions, representing ideas or assertions, and suggesting proof approaches.
Often university mathematics departments teach some formal logic early in a transition-to- proof course in preparation for teaching undergraduate students to construct proofs. Logic, in some form, does seem to play a crucial role in constructing proofs. Yet, this study of forty-two student-constructed proofs of theorems about sets, functions, real analysis, abstract algebra, and topology, found that only a very small part of those proofs involved logic beyond common sense reasoning. Where is the logic? How much of it is just common sense? Does proving involve forms of deductive reasoning that are logic-like, but are not immediately derivable from predicate or propositional calculus? Also, can the needed logic be taught in context while teaching proof-construction instead of first teaching it in an abstract, disembodied way? Through a theoretical framework emerging from a chunk-by-chunk analysis of student- constructed proofs and from task-based interviews with students, I try to shed light on these questions.
This study explores how students read from an online mathematics textbook. The particular textbook that we are exploring is Precalculus: Pathways to Calculus, which was developed at Arizona State University as part of a redesigned precalculus course that focuses on developing students’ ability to reason conceptually about functions and quantity. We are interested in understanding the way students read their mathematical textbooks so that research informed activities can be developed and incorporated into online textbooks to increase comprehension and retention. In order to investigate authentic student reading habits as closely as possible, we used nonintrusive screen capture software to measure activities such as scrolling, latency, and browsing, as students completed their regular reading assignments in a study hall setting. Other data sources include brief surveys, assessments and interviews. Interventions include reading instruction and embedded activities with feedback and sequences of hints that are intended to promote deeper engagement with the text.
This research project aimed to explore the van Hiele levels of undergraduate mathematics majors, most of whom were prospective mathematics teachers. The participants were taking a proof-intensive Euclidean geometry course that relied on technology, The Geometer’s Sketchpad (KCP Technologies, 2006), to help them make and prove conjectures. Data was collected from classes in consecutive years, the first with 21 participants, the second with 24 participants. Data collection included a pre- and a posttest of participants’ van Hiele levels. Data analysis suggests similar results for both sets of participants in that the course had greater influence on the van Hiele levels of female participants. Results also suggest that the van Hiele test instrument used for this study operated well with university students.
Quantitative reasoning combined with gestures, visual representations, or mental images has been at the center of much research in the field of mathematics education. In this research we extend these studies to include the arithmetic of complex numbers and the analysis of complex valued functions. Our data consists of videotaped interviews with experts, including mathematicians, physicists, and graduate students. Microethnography and phenomenological methods were used to analyze and interpret the data. In this case study we synthesize how one mathematician, Ricardo, employs geometric representations, gestures, metaphor, diagrams, and models to describe his understanding of complex variables topics. Linear transformations were fundamental in his connections between analytical and geometrical representations. He routinely characterized complex linear functions as linear transformations on R2 with added symmetries. These findings may serve as a foundation for creating teaching experiments to help students develop geometrical representations of the mathematics behind complex variables.
In this paper, we examine the strategies that two successful undergraduate students used to comprehend six mathematical proofs. These students spent nearly as much time studying the theorem as they did studying the proof, both by reformulating the theorem and trying to understand why it was true prior to reading the proof. When reading the proof, these participants attended to proof frameworks, partitioned the proof, and considered specific examples. How these ideas might be used to improve students’ proof comprehension is discussed.
Students are expected to apply the mathematics learned in their mathematics courses to concepts and problems in physics. Little empirical research has investigated how readily students are able to transfer their mathematical knowledge and skills from their mathematics classes to other courses. In physics education research, few studies have distinguished between difficulties students have with physics concepts and those with either the mathematics concepts, application of those concepts, or the representations used to connect the math and the physics. We report on empirical studies of student conceptual difficulties with (single-variable) integration on mathematics questions that are analogous to canonical questions in thermodynamics.
We report on the process by which we extended a local instruction theory for number sense development from the whole-number domain to the rational-number domain. Students involved in an earlier teaching experiment developed improved number sense, particularly in the form of flexible mental computation. The previous research was informed by a conjectured local instruction theory and informed the refinement and elaboration of that local instruction theory. The present study concerns a recent classroom teaching experiment in which envisioned learning routes that were developed in the context of whole-number mental computation and estimation were applied to reasoning about fraction size. In this way, the application of the local instruction theory was extended from whole-number sense to rational-number sense.