RUME 14 (Volume 4)
2011
Portland, Oregon
Number of pages: 237
Number of papers: 53
- Stacy Brown
- Sean Larsen
- Karen Marrongelle
- Michael Oehrtman
2011
Portland, Oregon
Number of pages: 237
Number of papers: 53
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This study addressed the following research question: To what extent are K-8 pre- service teachers' personal mathematics teacher efficacy beliefs aligned with their content knowledge for teaching mathematics? 18 K-8 pre-service teachers enrolled in a teacher preparation mathematics content course completed semi-structured interviews and follow-up written assessments in which efficacy beliefs and content knowledge regarding specific mathematical teaching scenarios were assessed. Preliminary analyses indicate that the efficacy beliefs of pre-service teachers with low content knowledge vary according to the nature of the teaching scenario. Consequently, the extent to which teacher efficacy beliefs and knowledge are aligned for these pre-service teachers depends on the mathematics involved. Keywords pre-service teachers, teacher efficacy beliefs, mathematical content knowledge Teacher efficacy beliefs are considered an important topic of study, in part because of the apparent positive correlations between teacher efficacy beliefs and a variety of desirable outcomes including student achievement (Ghaith & Yaghi, 1997; Riggs & Enochs, 1990; Ross, 1992). While there is an increasing body of literature on teacher efficacy beliefs, few researchers have examined the potential links between pre-service teachers' mathematics teacher efficacy beliefs and mathematical content knowledge. In fact, investigating the extent to which pre- service teachers' teacher efficacy beliefs are aligned with their mathematical content knowledge att
The Effects of Online Homework in a University Finite Mathematics Course Preliminary Report M. Axtell and E. Curran University of St. Thomas, St. Paul, MN 55105
Over the past decade, a growing trend has been to center instruction around student learning, rather than teacher performance. Such instruction often elicits student parti pation through various classroom activities: answering questions; solving problems; having students do board work; working on collaborative tasks in groups or pairs; making and testing conjectures; presenting ideas, proofs, and solutions; and debating. Through the and other classroom activities, students are expected to engage in the learning process, participate in mathematical thinking, and contribute to classroom discourse.
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number of possible ways a set of requirements can be satis.ed. Students are often taught to use combinatorial formulas, such as permutation or combination formulas, to solve such problems.
It seems clear that students' activity while working with definitions differs from that of mathematicians. The constructs of concept definition and concept image have served to support analyses of both mathematicians' and students' work with definitions (c.f. Edwards & Ward, 2004; Tall & Vinner, 1981). As part of an ongoing study, we chose to look closely at how mathematicians make sense of definitions in hopes of informing the ways in which we interpret students' activity and support their understanding of definitions. We conducted interviews with mathematicians in an attempt to reveal their process when making sense of definitions. A striking observation relates to the role of examples. We will share a preliminary analysis of these interviews and engage the audience in reflecting on the ideas. KEY WORDS: mathematical definitions, advanced mathematical thinking, mathematicians' practice, examples How do we come to understand mathematical definitions? Is the process different for students than it is for mathematicians? What can be learned from the practice of mathematicians that could support students' learning? In their chapter on advanced mathematical thinking, Harel, Selden, & Selden (2006) identified mathematical definitions as one area of focus when comparing the activity of students with the practice of mathematicians. The constructs of concept definition and concept image have served to support analyses of both mathematicians' and students' work with definitions (c.f. Edwards & Ward, 2004; Tall & Vinner, 1981). Our current research attempts to bring such ideas together into explanatory models for mathematicians' and students' activity. In this presentation we will focus on mathematicians and how their ability to build adequate concept images might develop. Mathematicians encounter definitions in their work in a variety of ways. There are definitions included in courses they teach, definitions proposed by other mathematicians, and perhaps even new definitions created in the course of their own research work. In instructional settings, mathematicians must decide how to present definitions to students. In the context of current mathematical work, mathematicians must judge the clarity and appropriateness of stated definitions. In preparing to share proposed definitions, mathematicians must also consider presentation, clarity, and usefulness. Each of these settings requires some level of making sense of a given definition within a mathematical setting. We set out to create an interview context in which aspects of this activity were brought out and thus became accessible for analysis. The interviews provided opportunities for the mathematicians to articulate their perspectives on making sense of definitions and to participate in definition-related tasks (Watson & Mason, 2005). In the interviews, participants were first asked to describe how they make sense of new mathematical definitions and to provide a recent example of doing this, if possible. The second interview question asked participants to share how they support students' work with definitions. Participants were then asked to engage in an example-generation activity, and finally were given a formal definition from an unfamiliar context and asked to share how they would go about 441
Tools in the mathematics classroom are often not given the credence or the attention they warrant. Considering Vygotsky's view of mediation, tools may play a larger role in mathematics then originally thought. This preliminary report presents a framework for attempting to identify the implications of tools in student learning. Using Pickering analytic framework (1995) distinguishing individual, disciplinary and material agencies, 1 am interested in how material agency takes form in the interaction of students with tools. While teaching an education class of pre-service mathematics teachers I will analyze their interactions with a Dynamic Geometric software, specifically Geometer's Sketchpad. In the process of solving a problem I will analyze students' engagement with the tool in terms of the different types of agencies, based on their spoken words and their actions in using the program. Key words: agency, disciplinary agency, material agency, mediation, dynamic geometry software, Geometer's Sketchpad Introduction A tension has always existed between the advocates of mathematics as being more of a mental discipline and, both academics and pedagogues, who consider the physical role of objects, materials or machines playing a formative role in the learning of mathematics. While both sides recognize that tools play their role in the practice of mathematics, the mental mathematicians may consider that tools or machines play a small role to either simplify a calculation to arrive at a particular theorem or merely serve as a vessel that serves the sole purpose of "getting" to the mathematics. This attitude is not so much explicitly stated as it is practiced. Whether stemming from Plato's vision of mathematics as a separate, distinct and pure discipline, that is accessible solely through contemplation (Tarnas, p. 6), mathematical production acts often state no reference to materials or tools
Large numbers of college students study probability and statistics, but research indicates many are not learning with understanding. The concept of probability distribution undergirds development of conceptual connections between probability and statistics and a principled understanding of statistical inference. Using a control-treatment design, this study employed differing technology-based lab assignments and investigated the impact of instruction aimed at fostering development of stochastic reasoning on students' understanding of probability distribution. Participants were approximately 200 undergraduate students enrolled in a lecture/recitation, calculus-based, introductory probability and statistics course. This preliminary research report will discuss the framework used to develop the stochastic lab materials and preliminary results of an assessment of students' understandings. Key words: Probability distribution, stochastic reasoning, technology-based instruction, instructional intervention. Statement of research issue: Large numbers of university students study probability and statistics (Moore & Cobb, 2000), but research indicates that many of these students exhibit difficulties in learning and applying probabilistic and statistical concepts (Garfield & Ben-Zvi, 2007; Shaughnessy, 1992, 2007). Inappropriate reasoning in probability and statistics is widespread and persistent across all age levels. After probability instruction, many post-calculus students demonstrate merely instrumental understanding (Skemp, 1976) and present notions about probability that are not aligned with formal probabilistic concepts (Barragues, Guisasola, & Morais, 2007). This study draws on constructivist and situated learning perspectives and assumes understandings are built through learning experiences, which are impacted by the learner, teachers, and the instructional material. The study assumes that: (1) teaching impacts learning and can facilitate learning with understanding; (2) effective teaching elicits students' pre-existing understandings and builds on that understanding; (3) effective teaching helps students develop deep knowledge connections in the context of a conceptual frame for the content domain (Bransford, Brown, & Cocking, 2000). This research was designed to evaluate the effectiveness of an instructional intervention that builds on students' initial understandings of probability and statistics and facilitates student understanding of content within a connected conceptual framework. The study seeks to measure and describe individual understandings of probability distribution. The concept of probability distribution is a powerful springboard for the development of stochastic reasoning as it may facilitate making deep conceptual connections around probabilistic understandings related to variability, independence, sample space, and distribution (Liu & Thompson, 2007). Principled knowledge (Spillane, 2000) refers to an understanding of the ideas and concepts that support mathematical procedures. Principled knowledge of probability
In this study, we observed first semester calculus students solving related rates problems in a peer- led collaborative learning environment. The development of a robust mental model has been shown. to be a critical part of the solution process for such problems. We are interested in determining whether the collaborative learning environment promotes the development of such a mental model. Through our observations, we were able to determine the amount of time students spent engaging with the diagrams they drew to model the problem situation. Our analysis strove to also determine the quality of the student interactions with their diagrams. This analysis provided insights about the mental models with which the students were working. Engaging students with complex, non- routine problems resulted in the students spending more time developing robust mental models.
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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 seeks to identify the fundamental calculus concepts necessary for successful academic pursuits outside the undergraduate mathematics classroom, describe appropriate understanding of these concepts, and collect tasks that elicit, document, and measure this understanding. Data were collected through a series of interviews with select undergraduate mathematics and other discipline faculty members. The data were used to build descriptions of and frameworks for understanding the calculus concepts and generate the pool of tasks. Implications of these findings for calculus curriculum are presented.
Traditionally, research on technology in mathematics education focuses on interac tions between the user and the technology, but little is known is about how technology can facili- tate interaction among students. In this preliminary report we will explore how students use iPads while negotiating mathematical meaning in a community of learners. We are currently studying the use of iPads in an introductory business calculus course. We will report on classroom observations and a series of small-group interviews in which students explore the concepts of local and global extrema. Our preliminary results are that the portability of Pads and the intuitive applications have allowed students to easily incorporate the iPad into their collaborations.
In advanced undergraduate mathematics, students are expected to make sense of abstract definitions of mathematical concepts, to create conjectures about those concepts, and to write proofs and exhibit counter-examples of these abstract concepts. In all of these actions, students must be able to draw upon a rich store of examples in order to make meaningful progress. We have created a methodology to evaluate what students might learn from a particular course by describing and analyzing the enacted example space (Mason & Watson, 2008) for a particular concept. This method will both give a means to create testable hypotheses about individual student learning as well as provide a way to compare disparate pedagogical treatments of the same content. Here, we describe and assess the enacted example space by studying the teaching of abstract algebra.
Technology is a cornerstone for NCTM and is agreed to be beneficial, but the level of effectiveness is still very vague. This research questions exactly how effective is technology in the mathematics classroom, and what are the definitive benefits. Afier studying over 300 articles, technology has proven to be beneficial in five ways: providing instantaneous visual feedback, creating student-centered learning environments, providing multiple representations of similar concepts, combining learning environments for generalizations, and retracing previous steps for self-assessment. The most frequently discussed topic was multiple representations, usually in the form of CAS and dynamic geometry systems. The research shows that providing multiple representations allows students with varying levels of intelligence to better understand tricky and abstract concepts.
Tests in underergduate mathematics courses are generally high stakes, and yet have low reliability The current study aims to increase the reliability of such exams by studying the qualities of test items that determine the ability of the item to contribute, to the information of the test. Using q three parameter item response theory model, 695 items contained in'25 different tesis for 5 different first-year undergraduate mathematics courses have been gnalyzed to determine the abjlity of each item 10 contribute to the test's reliability. During the conference presentation, the speakers with m the participants regarding the types of qualities of thesé items that 'may contribiyfe to their information inex. These qualities may include cognitiv mathematical content, linguistic, or other descriptions.
Preliminary results of research into the effectiveness of an innovative on-line mathematics review and practice tool (www.mathessentials.ca) will be reported (data collection completion in Dec. 2010). The goal of the web-site is to provide students with the opportunity to review and practice developmental 'math skills (fractions, percents, etc.), thus filling in gaps in their knowledge. The development of the web-site begat the development of an innovative evaluation model, which can be used to evaluate online educational technologies. Key to the model is not simply evaluating improvement with pre/post test scores, or with anecdotal reports, but through tracking built into the site, which has the potential to provide a multidimensional view of improvement, usage and engagement (usability score). We believe that the web-site itself (support of student success) and the evaluation model ('gold standard' for evaluation of educational technologies) have implications for both teaching and research
We apply a Vygotskian perspective on the interplay between spontaneous and scientific concepts to identify and characterize calculus students' idiosyncratic use of examples in the process of trying to formulate a rigorous definition for convergence of a sequence. Our data is drawn from a larger teaching experiment, but analyzed for this study to address questions of the origins, nature, and implications of students' nonstandard ways of reasoning. We observed two students interpreting a damped oscillating sequence as divergent, drawing from considerations from an initial, intuitively-framed definition, but remaining persistent and consistent over the duration of multiple sessions. We also trace some of the implications of their idiosyncratic reasoning for their reasoning and ultimately for their definition of convergence. We conclude by posing several questions about the nature of such example use in terms of our Vygotskian perspective.
We report on our work to build an applied theory for intercultural competence development for mathematics teaching and learning in secondary and tertiary settings. Based on social anthropology and communications research, we 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 subgroups (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.
While many studies have focused on student knowledge of function, few studies have focused on composition. This report describes a curriculum analysis of the treatment of composition in the secondary (algebra, geometry, algebra 2, precalculus) and early college (precalculus, calculus) mathematics curriculum. In this study composition is conceptualized as a sequence of functions and as a binary operation on functions. The curriculum analysis utilizes a framework of conceptual, procedural, and conventional knowledge elements as well as representations and types of functions. Preliminary data will be presented during the session and a discussion will center on conceptual, procedural, and conventional knowledge elements for composition.
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Teachers implementing inquiry-oriented, discourse-promoting tasks can face a number of challenges (Speer & Wagner, 2009; Ball, 1993). In this study we will examine the challenges faced by two community college instructors as they implement such a task in a "transition to proof" course. In this task students initially use their informal ideas of symmetry to develop a criteria to quantify the symmetry of six figures (see Larsen & Bartlo, 2009), these criteria are then formalized into definitions for symmetry and equivalent symmetries. During this task a number of conflicts arise, and to resolve these conflicts the students engage in rich mathematical discourse. While this task and ensuing discourse offer opportunities for learning mathematics, they also offer significant challenges for effective implementation. We aim to identifying these challenges and the ways in which these challenges were navigated as the class worked towards formal definitions of symmetry and equivalent symmetries. While working on a project aimed to develop a community college "transition to proof" course, bases on an inquiry-oriented abstract algebra curriculum, we began to wonder what sort of challenges the community college instructors would face as they navigated the curriculum. In order to begin looking at this question we decided to focus our attention on an inquiry-oriented task in which the students reinvent and define the concepts of symmetry and equivalent symmetries. In this task students are initially given six shapes (see figure below) and are asked to arrange the figures from least to most symmetric. The students work on this task individually and then in small groups prior to a whole class discussion. The groups share how they ordered the figures and how they came to that decision. The students are then asked to determine a way to quantify the symmetry of each figure and, using their quantification criteria, the groups rank the figures and present both their criteria and their ranking to the whole class. Following these presentations the groups work to develop both a definition of what a symmetry is and what makes two symmetries equivalent (see Larsen & Bartlo, 2009). bis Bae your an Fig. 1 Symmetry Task Launch
In undergraduate mathematics classrooms where instructors are beginning to focus on more student-centered instruction, teachers' moves foster mathematical discourse among students and teachers as a way to further the mathematics. While some are studying these teacher moves in K-12 classrooms, there seems to be little research focusing on this in the university classroom. We define a pedagogical content move to be a discursive or inscriptive act by an instructor that is purposely used to promote or further the mathematical agenda in the classroom (Lee, Keene, Lee, Holstein, Early& Ely, 2009). In an earlier paper, we presented several of these moves as identified in our data collection and analysis. In this proposal, we further this research by answering the question:
Understanding and Overcoming Difficulties with Building Mathematical Models in Engineering: Using Visua n to Aide in Opt 'ion Courses. Rachael Kenney, Nelson Uhan, Ji Soo Yi, Sung-Hee Kim, Mohan Gopaladesikan, Aiman Shamsul and Amit Hundia
A configuration of vector representations based on multiple represen- tation, cognitive development, and mathematical conceptualization, to serve as a new unifying framework for studying undergraduate student approaches and difficulties in understanding and using of vectors is proposed. Using this configuration, the study will explore 5 impor- tant 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 s undergraduate students' approaches and difficulties in unders and using of vectors with both quantitative and qualitative methods will be introduced, and we will see how useful this new framework is to analyze student approaches and difficulties in understanding and using of vectors.
I propose the use of systemic functional linguistics (SFL) as a tool to better understand how mathematical ideas are conveyed through multiple semiotic resources. To demonstrate the tools that SFL offers, mathematical symbols and written language in college beginning algebra textbooks will be examined. I argue that using SFL to research how mathematical content is communicated to undergraduate students can expose important nuances that may otherwise go unnoticed. Key Words: Beginning Algebra, Language and Mathematics, Mathematical Symbolism, Systemic Functional Linguistics, Textbooks How well do college beginning algebra textbooks integrate mathematical symbolism and language? To answer this question I will use systemic functional linguistics (SFL) to link the mathematical symbolism to language. To make this connection, two topics will be focused on: the use of the hyphen, as both an operator for subtraction and modifier for the opposite; and the simplification of algebraic expressions. These topics were chosen not only because many students in these courses struggle with them, but because their simplicity can reveal how SFL can be used as an aid for researcher to tease out subtle distinctions in a subject matter that is so clear in their minds that they might otherwise be overlooked. This research is extended from the work of Kay O'Halloran (i.e. 2000; 2005) which looks at the multisemiotic nature of mathematics through the systemic functional linguistic perspective. My research differs from much of the linguistic research in mathematics education (i.e. Herbel- Eisenmann, 2007; Mesa & Change, 2010; Wagner & Herbal-Eisenmann, 2008) in that it looks at the linguistic nature of the mathematical symbols alongside the use of language with a focus on content. The choice to examine college beginning algebra textbooks comes from of the lack of research in teaching and learning in college development mathematics (Stigler, Givvin, Thompson, 2010) despite the need, and the potential role of the textbook. Developmental mathematics The number of college students needing developmental mathematics is larger than many realize. More than one out of five students entering college are required to take a developmental mathematics course and in two-year public institutions more than one out of every three students needs to take at least one developmental mathematics course (NCES, 2003). Developmental college mathematics courses include arithmetic, beginning algebra, and intermediate algebra, and are labeled developmental or remedial because it is expected that students would have acquired this knowledge in high school or earlier. Compounding this issue, the a majority of students (70%) taking developmental mathematics courses need more than one attempt to pass these courses (Attewell, et al., 2006).
This study seeks to contribute to research on the teaching and learning of combinatorics at the undergraduate level. In particular, the authors draw upon a distinction characterized in combinatorial texts between set-oriented and process-oriented definitions of basic counting principles. The aim of the study is to situate the dichotomy of set-oriented versus process-oriented thinking within the domain-specific combinatorial problem-solving activity of students. The authors interviewed post-secondary students as they solved counting problems and examined alternative solutions. Data was analyzed using grounded theory, and a number of preliminary themes were developed. The primary theme reported in this study is that students showed a strong tendency to utilize set-oriented thinking during the problem-solving phase that Carlson & Bloom (2005) refer to as checking, especially when they engaged in the evaluation of alternative solutions.
There is an abundance of recommendations and articles that extol the virtues of writing in the mathematics classroom. The National Council of Teachers of Mathematics encouraged mathematical communication in its 1989 Standards for School Mathematics and again in its update of the Standards (2000). The Mathematical Association of America (2004) underscored the need for developing communication skills in mathematics and was joined by a host of other countries that encouraged writing (Ntenza, 2006). Yet, with twenty years of advocacy by researchers and policy-makers, very few students have experience with mathematical writing when they come to college (Borasi & Rose, 1989; Ntenza, 2006, Pugalee, 2004).
In recent years, researchers have given much 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. This talk will look at preliminary results and analysis from a qualitative study designed to explore these aspects of the transition to graduate school in mathematics from a post-positivist perspective. In order to explore the transition as fully as possible, interview data from a varied sample of graduate students and faculty members at one university are being incorporated to gain multiple perspectives on the transition experience. Potential implications for graduate recruitment, retention, and program protocols in mathematics will be discussed.
One direction taken by course reform over the past few years has been the development of sophisticated computer-assisted instruction. This approach has been applied to large-enrollment service courses in mathematics, including algebra. Elementary algebra is typically taken by under-graduate students who do not place into a credit-bearing course. Traditionally, the goal of such a developmental algebra course has been to enhance students' "algebra skills," for example, dealing procedurally with rational numbers and expressions. Higher-order thinking may be largely absent. Alternately, one might focus on developing quantitative reasoning and communications skills, rather than, or in addition to, training to acquire a set of specific algebraic skills (Wiggins, 1989; Blais, 1988). Our position is that incorporating an inquiry-based component, either together with, or in place of, a didactic component, into a computer-a instructional environment may enhance student learning. Two previous studies in the literature bear this out (Mayer, 2009, 2010).
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In this study we compare teaching approaches of 14 community college mathematics instructors with their classroom questioning and their classroom non-mathematical discursive interactions. The teaching approaches were drawn from interviews and the application of an analytical framework derived from the higher education literature. The questioning and the non- mathematical discursive interactions were characterized using transcripts of classroom observations and the application of an analytical framework derived from the mathematics education and higher education literature. From the interviews, we found a wide range of espoused teaching approaches, although the majority of instructors favored instructor-centered approaches. From the observations, we found that these instructors ask a large amount of questions, a sizable proportion of which generate opportunities for students to engage with authentic mathematical knowledge. Also, we found that these espoused teaching approaches are related to observed non-mathematical discursive interactions.
Representations of teaching can be seen not only as cases of practice but also as probes on the rationality that practitioners use as they teach (Herbst & Chazan, 2003). Herbst and Chazan have developed a new kind of representation of teaching-animations of classroom scenarios, deliberately designed to probe some of the unspoken norms of classroom practice. Herbst and Miyakawa (2008) provided some details of how those animations are produced to be prototypes of models of instructional situations: Instructional situations are identified and modeled by hypothesizing the norms or tacit responsibilities of classroom participants in a situation, then scenarios are created that fulfill some of those norms but breach others; finally those scenarios are prototyped in a cartoon animation. Herbst, Nachlieli, and Chazan (in press) have shown how such animations can elicit data that informs about the rationality of teaching.
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For this preliminary research report, I have two goals: a) to present initial findings from the pilot study, and b) to use feedback from the session to design a more robust follow-up study.
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 KOG 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. Introduction In contrast with the manner in which a school teacher's Knowledge, Orientation and Goals (KOGs) determine their decision making, we present evidence that for mathematicians this decision making is additionally complicated by an inner argument between the lecturer-as-mathematician and the lecturer-as-teacher. Are there conflicting orientations and goals active in the decision moment? The way in which the decisions play out is thus a function not only of the lecturer's knowledge of mathematics but of the way they work mathematically. Research base This paper reports on an aspect of a project that explores how Schoenfeld's KOGs may be used to direct lecturers' attention to aspects of their decision-making in the lecture theatre as a professional development activity. The project is informed by research concerning how a teacher's knowledge, orientations or beliefs and goals impact on their teaching practice (Schoenfeld, 2007; Ball, Bass & Hill, 2004; Shulman, 1986; Speer, Smith & Horvath, 2010; Torner, Tolka, Rosken & Sriraman, 2010) However, these are studies of teacher practice in primary and secondary schools and similar work at the college level is 'virtually non-existent' (Speer et al, 2010, p 99). The project is also designed to build on the effectiveness of communities of practice (Lave & Wenger, 1999) and a culture of enquiring conversation (Rowland, 2000) for professional development. The project is described in more detail in Barton, Oates, Paterson and Thomas (to be published). Structure of project and data collection A group of four mathematicians and four mathematics educators are collaborating in the fine-grained examination and discussion of lecturer actions in video recordings of lectures (Kazemi, Franke, Lampert, 2009; Prushiek, McCarty, & Mcintyre, 2001). The theoretical approach draws on Schoenfeld's theory of teaching-in-context (Schoenfeld, 2002). The data for each lecture consists of videotape, an observer record, and a written lecturer-KOG (a statement by the lecturer of the knowledge used, orientation held, and goals, both specific goals intended for the lecture and more 4a7T
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algebra, and topology, found that only 1.7% of proof lines 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 line-by-line analysis of proofs and task-based interviews with students, | try to shed light on these questions.
Many would agree that reading is critical for gaining understanding within a discipline, and that students will not reap the full benefits of their studies if they skim through (or worse yet, ignore) their reading assignments. Even in quantitative disciplines such as mathematics, teachers may assign readings from the textbook with the intent of having students come to class more prepared and giving them exposure to more material than can be taught in the time allotted to class meetings. However, few teachers would be so naive as to believe that students actually read the text, and often complain about the unpreparedness of the students for instruction. On their part, students complain about how hard it is to read mathematics textbooks, perhaps because they lack appropriate reading strategies that might remedy the situation. Indeed, even first-year undergraduate who are good general readers do not read mathematics textbooks well (Shepherd, Selden & Selden, 2009).
Calculus appeared from the real world application, has a real world context, and is fundamentally a dynamic conception; this is why the framework of Realistic Mathematics Education (RME) should be the most efficient approach to teaching and learning calculus. The current study is devoted to investigation of the computer simulated bodily path optimization calculus. I adapted the conception of "tacit intuitive model' for the particular calculus task of path optimizations. My hypothesis is that tacit mental modeling takes place with the allocentric frame of reference. I designed a paradigm in the Second Life virtual environment which allows simulating the navigational task of path optimization with two different mediums and with voluntary choice between allocentric/egocentric views. The reinventing the calculus problem of path optimization from the virtual navigation and its mathematizing would give a powerful intuitive link between the everyday real world problem and its symbolic arithmetic.
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 report we extend these studies to include complex numbers and complex variables. We provide a construct analysis for the teaching and learning of complex variables, which includes a description of existing frameworks that hypothesize about how students can best comprehend the arithmetic operations of complex numbers. In order to test these conjectures, we interviewed mathematicians, physicists, and electrical engineers to explore how they perceive complex variables content. Through phenomenolgogical and microethnography analysis methods we found how these experts integrate perceptuo-motor activity and metaphors into their descriptions.
In this study we use APOS theory to propose a genetic decomposition for the concept of spanning set in Linear Algebra. We give examples of interviews that were conducted with a group of university students who were taking an analytic geometry course and their analysis in relation to our genetic decomposition. We also comment on the nature of difficulties that students experience in constructing this notion. One of the results that are obtained in this research that is in line with previous results reported in the literature is the difficulty in distinguishing a spanning set from a basis. Another aspect is that students have varying levels of difficulty when working with different types of vector spaces. As was expected, the concept of linear combination plays a very important role in the understanding of the notion of spanning.
Free, open, online, help forums are found on public websites and allow students to post queries from their course assignments that can be responded to asynchronously by anonymous others. Several of these forums are tailored to helping students with mathematics assignments from various courses, and Calculus, in particular, is a heavily trafficked area. Students use the forums when they have reached an impasse, either in constructing or understanding a solution to an exercise that they have encountered, or to seek verification of their own reasoning. The queries posted by students include both computational tasks as well as proof constructions. In this project, we examine threads on limit proofs for single-variable functions from two popular online forums. Our goal is twofold: to characterize the help students are receiving as they wrestle with using the formal definition of limit, and to compare the construction of proof to other tasks in online forums.
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Aaron Wangberg, Winona State University Nicole Engelke, California State University Fullerton Gulden Karakok, UMERC (Umea Mathematics Education Research Centre), Umea University
, Proof is a dominant means of conveying mathematics to undergraduates in their advanced mathematics courses, yet research suggests that students learn little from the proofs they read and find proofs to be confusing and pointless. In this presentation, we examine the behavior of two successful mathematics majors as they studied six proofs to identify productive proof comprehensive strategies. Prior to reading a proof, these students would attempt to understand the theorem by rephrasing and trying to determine why it was true. While reading a proof, these students would partition the proof into sections, attend to the proof framework being employed, and illustrate confusing aspects of the proof with examples. Implications and limitations of this study will be discu:
Student Understanding of Integration in the Context and Notation of Thermodynamics: Concepts, Representations, and Transfer Preliminary Research Report
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Think alouds are a research tool originally developed by cognitive psychologists for the purpose of studying how people solve problems. The basic idea being that if a subject can be trained to think out aloud while completing a certain task then the introspections can be analyzed and may provide insights into misunderstandings as well as higher thinking. This talk is a preliminary report of a think aloud conducted with calculus students to understand their difficulties with work problems in integral calculus.
is not keeping pace with demand (Liu et al., 2008; National Research Council, 2002). According to Ingersoll and Perda (2009), the problem is more than a number game. Part of the problem resides in the fact that teachers are not happy with the profession once they are out in the fields, causing the number of teaching leaving the profession to be greater than the number of teachers entering the profession. This is especially the case in low-income areas where they are 77% more likely to be taught by out of field teachers compared to students from high socioeconomic backgrounds (Ingersoll, 2003). Attrition is also a compounding factor as recent research reveals 20 to 30% of teachers have left the profession within the first five years (Darling-Hammond, 2001).
This study is a pilot to a larger design research project that aims to explore an alternative approach to teaching a Calculus I course. Central to this approach is the introduction of the integral first, utilizing a non-standard definition, but which is equivalent to the standard definition. This is immediately followed by the introduction of derivative. This approach allows methods of derivation and integration, which are analogs of one another to be introduced in close succession, allowing the relationships between these methods to be a major theme of the course. The alternative definition of integral is the focus of this study. I present preliminary results of a teaching experiment that explores how students develop an understanding of this alternative definition of integral and how these understandings relate to prerequisite notions, such as area and arithmetic mean.