PROCEEDINGS OF THE 16TH ANNUAL CONFERENCE ON RESEARCH IN UNDERGRADUATE MATHEMATICS EDUCATION (Vol 2)
2013
Denver, Colorado
Number of pages: 707
Number of papers: 112
- STACY BROWN
- GULDEN KARAKOK
- KYEONG HAH ROH
- MICHAEL OEHRTMAN
2013
Denver, Colorado
Number of pages: 707
Number of papers: 112
2013
Denver, Colorado
This study explores in-service high school mathematics teachers’ conception of various forms of a complex number and the ways that they transition between different representations’ (algebraic and geometric) of these forms. Data were collected from three high school mathematics teachers via a ninety- minute interview after they completed professional development on complex numbers. Results indicate that these teachers do not necessarily objectify exponential form of complex numbers and only conceptualized it at the operational level. On the other hand, two teachers were very comfortable with Cartesian form and showed process/object duality by translating between different representations of this form. It appeared that our participants’ ability to develop a dual conception of complex numbers was bound by their conceptualization of the various forms, which in turn was hindered by their representations of each form.
2013
Denver, Colorado
In this report we present the commonly identified error patterns and students’ misconceptions about vectors, vector operations, orthogonality, and linear combinations. Twenty three freshmen students participated in this study. The participants were non-mathematics majors pursuing liberal arts degrees. The main research question was: What misconceptions about vector algebra were still prevalent after the students completed a freshmen-level linear algebra course? We used qualitative data in the form of artifacts and students’ work samples to identify, classify, and describe students’ mathematical errors. Seventy four percent of students in this study were unable to correctly solve a task involving vectors and vector operations. Two types of errors were commonly identified across the sample: a lack of students’ understanding about vector operations and projections, and a lack of understanding (or distinction) between vectors and scalars. Final results and conclusions include research suggestions and practitioner-based implications for teaching linear algebra in high school and college.
2013
Denver, Colorado
Mathematical knowledge for teaching (MKT) is essential for effective teaching of elementary mathematics. Given the importance of MKT, MKT and conceptions of teaching effectiveness should not develop independently. The purpose of this study was to examine whether and how K-8 pre-service teachers’ MKT and personal mathematics teacher efficacy beliefs are related. Results indicated overconfidence in teaching ability was prevalent, with the majority of participants exhibiting a strong sense of personal mathematics teacher efficacy but low levels of MKT. Pre-service teachers with high levels of MKT, however, reported a more accurate assessment of their teaching effectiveness. Results also indicated that examining pre-service teachers’ self-evaluations of MKT is helpful for understanding pre-service teachers’ personal mathematics teacher efficacy beliefs. Moreover, the results of this study point to the inadequacies of existing measures of teacher efficacy beliefs that do not parse out differences in efficacy beliefs according to a number of contextual factors.
2013
Denver, Colorado
In this study, we open up discussions regarding one of the unexplored aspects of mathematical sophistication, the inductive work of conjecturing. We consider the following questions: What does conjecturing entail? How do the conjectures of experts and novices differ? What characteristics, behaviors, practices, and viewpoints distinguish novice from expert conjecturers? and What activities enable individuals to make conjectures? To answer these questions, we conducted a qualitative research study of eight participants at various levels of mathematical maturity. Answers to our research questions will begin to provide an understanding about what helps students develop the ability to make mathematical conjectures and what characteristics of tasks and topics may effectively elicit such behaviors, informing curriculum development, assessment, and instruction.
2013
Denver, Colorado
The purpose of this study is to investigate a relationship between mathematical content knowledge and pedagogical knowledge of content and students (Hill, Ball, & Shilling, 2008), in the context of algebra. As participants in a paired teaching experiment, mathematics education doctoral students revealed their understandings of commutativity and associativity (cf. Larsen, 2010). Although the participants’ knowledge of children’s initial understandings of algebra and familiarity with mathematics education literature influenced their own mathematics reasoning, the difficulties they encountered were similar to those of undergraduates without such pedagogical content knowledge.
2013
Denver, Colorado
This quantitative study compared the implementation of a problem-based curriculum in precalculus and a modular-style implementation of traditional curriculum in precalculus to the historical instructional methods at a western Tier 2 public university. The goal of the study was to determine if either alternative approach improved student performance in precalculus and better prepared students for success in a calculus sequence. The study used quantitative data collection and analysis. Results indicate students who experienced the problem-based curriculum should be better prepared to learn calculus but mixed results in terms of retention and success in calculus.
2013
Denver, Colorado
This paper examines mathematics majors' evaluations of indirect proofs and of the compound statements used in this form of proof. Responses to survey items with a cohort of 23 students and six 1-hour clinical interviews, indicate that the students who could successfully evaluate indirect arguments and who could successfully recognize logically equivalent statements, tended to use partially unpacked (Selden & Selden, 1995) versions of the statements and the proofs and, in so doing, demonstrated a productive use of the symbolic proof scheme, whereas both successful and unsuccessful students tended to use a proof framework (Selden & Selden, 1995) for indirect proofs. Moreover, successful students' approaches are suggestive of activities, which are rarely found in introductory proof texts, yet may benefit novice proof writers.
2013
Denver, Colorado
In this study, seventeen math education majors completed a test on fractions and quotient. From this group, one above-average calculus student was selected to participate in a six-lesson teaching experiment. The major question investigated was “what constrains and affords the development of the productive meanings for division and fractions articulated by Thompson and Saldanha (2003)?” The student’s thinking was described using Steffe and Olive’s (2010) models of fractional knowledge. The report focuses on the student’s part-whole meaning for fractions and her difficulty assimilating instruction on partitive meanings for quotient. Her part-whole meaning for fractions led to the resilient belief that any partition of a length of size m must result in m, unit size pieces. It was non-trivial to develop the basic meanings underlying the concept of rate of change, even with a future math teacher who passed calculus.
2013
Denver, Colorado
There is little mixed methods research into the patterns of course taking, performance, and persistence among mathematics majors, in general, and among secondary mathematics majors, in particular. Drawing from a sample of 42,825 mathematics enrollment records at two universities over a six-year period, this study presents quantitative summaries of mathematics majors' performance and persistence in undergraduate mathematics courses alongside qualitative themes from interviews of nine secondary mathematics majors at one of the universities. Implications include potential strategies for mathematics programs and faculty to support the success of mathematics majors in undergraduate mathematics coursework, with special emphasis on prospective secondary mathematics teachers.
2013
Denver, Colorado
At small liberal arts colleges, a single calculus sequence must successfully accommodate students from various majors, such as mathematics, biology, chemistry, and economics. This qualitative case study considers mathematics professors' perspectives about the required nature of calculus in various disciplines, attempts to identify how calculus instructors teach with the aim of preparing students to apply calculus knowledge in their future coursework, and how the disciplinary focus of their students affects professors' design and teaching of calculus courses. Framed using aspects of teaching and learning shown to promote transfer of knowledge, results suggest that the professors teach for understanding and allow in-class processing time, but could improve their emphasis on applying calculus in non-mathematics disciplines. This study contributes to the growing body of undergraduate mathematics education research intended to document undergraduate teaching practices.
2013
Denver, Colorado
Little is known about how students learn the basic ideas of ring theory. While the literature addressing student learning of group theory is certainly relevant, the concept of zero-divisor in particular is one for which group theory has no analog. In order to better understand how students come to understand zero-divisors, this talk will present results from a study that investigated how students can capitalize on their intuitive notions of solving equations to reinvent the definitions of ring, integral domain, and field. In particular, the emergence and progressive formalization of the concept of zero-divisor at various stages of the reinvention process will be detailed and discussed.
2013
Denver, Colorado
Recent reform movements, including the Mathematical Association of Amer- ica’s Curriculum Foundations Project, have sought to make calculus courses more coherent with other fields (Hughes Hallet, 2000). Despite this and other interdisciplinary reform efforts, post-calculus instructors still express extreme dissatisfaction with their students’ mathemat- ical preparation (Ferguson, 2012). We do not downplay the importance of the lamentations of other disciplines, but instead suggest that there also exist coherence issues within math- ematics. Our hypothesis is that there is a mismatch between how students are expected to know calculus at the end of a course and how students are required to use calculus in future courses. Specifically, we examine calculus knowledge for differential equations.
2013
Denver, Colorado
In this paper, we describe an inquiry-oriented method of using metaphors to support students’ development of conventional logical reasoning in advanced mathematics. Our model of instruction was developed to describe commonalities observed in the practice of two inquiry-oriented real analysis instructors. We present the model via a general thought experiment and one representative case study of a students’ metaphorical reasoning. Part of the success of the instructional method relates to its ability to help students reason about, assess, and communicate about the logical structure of mathematical activity. In the case presented, this entailed a students’ shift from using properties to describe examples to using examples to relate various properties. The metaphor thus imbued key example sequences with meta-theoretical significance. We introduce the term “wedge” to describe such examples that distinguish oft-conflated properties. We also present our analytical criteria for empirically verifying the specific influence of the metaphorical aspect of instruction.
2013
Denver, Colorado
Previous reports show that not only are too few students pursuing Science, Technology, Engineering, or Mathematics (STEM) fields, but also many who originally intend to pursue these fields leave after their experiences in introductory STEM courses. Based on data gathered in a national survey, we will present an analysis of 5381 STEM intending students enrolled in introductory Calculus in Fall 2010, 12.5% of whom switched out of a STEM trajectory after their experience in Calculus I. When asked why these students no longer intended to continue taking Calculus (an indicator of continuing their pursuit of a STEM major), 31.4% cited their negative experience in Calculus I as a contributing factor. We analyze student and their instructor survey responses on various aspects of their classroom experience in Calculus I to better understand what aspects of this experience contributed to their persistence.
2013
Denver, Colorado
This study was designed to document the development of teachers’ ways of thinking about quantitative reasoning, one of the standards for mathematical practice in the Common Core State Standards. Using a Models and Modeling Perspective, the authors designed a model- eliciting activity (MEA) that was implemented in a graduate mathematics education course focusing on quantitative reasoning. Teachers were asked to create a quantitative reasoning task for their students, which they subsequently revised three times in the course after receiving instructor, peer, and student feedback. The MEA documented the development of the teachers’ models of quantitative reasoning, and this report details one group of three teachers’ development over the course. Findings include an overall model of teachers’ development that is both generalizable and sharable for other researchers and teacher educators.
2013
Denver, Colorado
The Models and Modeling Perspectives (MMP) has evolved out of research that began 26 years ago. MMP research uses Model Eliciting Activities (MEAs) to elicit students’ models of mathematical concepts. In this study MMP were used as conceptual framework to investigate the nature of undergraduate students’ models of curve fitting. Participants of this study were prospective mathematics teachers enrolled in an undergraduate mathematics problem solving course. Videotapes of the MEA session, class observation notes, and anecdotes from class discussions served as the sources of data for this study. Iterative videotape analyses as described in Lesh and Lehrer (2003) were used to analyze the videotapes of the participants working on the MEA. Results of this study discuss the nature of students’ models of the concept of curve fitting and add to the introductory undergraduate statistics education research by investigating the learning of the topic curve fitting.
2013
Denver, Colorado
This study investigated the ways in which college mathematics teachers might encourage the development of student reasoning through critiquing activities. In particular, we focused on identifying situations in which the instructional interventions were implemented to encourage the critiquing of arguments and in which students explained another’s reasoning. Data for the study come from two teaching experiments – one from the domain of combinatorics and the other from real analysis. Through open coding of the data, Devil’s Advocate and Peer Interpretations emerged as effective interventions for the creation of sources of perturbation for the students and for assisting in the resolution of a state of disequilibrium. These two interventions differ in design and in the type of reasoning students evaluate, but they both provoke students to further develop their reasoning, and therefore their understanding. We discuss the implications of these interventions for both research and teaching practice.
2013
Denver, Colorado
This case study explored how a student could use Venn diagrams to explain his reasoning while solving counting problems. An undergraduate with no formal experience with combinatorics participated in nine teaching sessions during which he was encouraged to explain his reasoning using visual representations. Open coding was used to identify the representations he used and the ways of thinking in which he engaged. Venn diagrams were introduced as part of an alternate solution written by a prior student. Following this introduction, the student in this study often chose to use Venn diagrams to explain his reasoning and stated that he was envisioning them. They were a powerful model for him as they helped him visualize the sets of elements he was counting and to recognize over counting. Though they were originally introduced to express additive reasoning, he also used them to represent his multiplicative reasoning.
2013
Denver, Colorado
The undergraduate mathematical preparation of elementary teachers often occurs through mathematics departments. This study looks at the issue of teacher education as a site for collective work between mathematicians and mathematics educators. It works on a dual agenda of understanding what is involved, on the one hand, in using instructional support materials, and on the other hand, in creating usable instructional support materials. We analyzed three mathematicians' reviews of materials intended to teach mathematical knowledge for teaching to prospective elementary teachers. The analysis suggests that the mathematical issues most salient to these mathematicians concerned the coherence of the mathematical curriculum to be taught, the use and choice of mathematical representations, and what features of mathematical objects to make explicit. We discuss implications of this observation for development of materials for the mathematical preparation of teachers.
2013
Denver, Colorado
In this study we draw on the constructs of eagerness, flexibility and willingness to characterize the necessary disposition for critical thinking required in learning statistics in addition to specific content knowledge (Ennis, 1989). We investigated the challenges that students who are highly successful in mathematics might have in doing statistics and found that while a student might have an inquisitive disposition and good proficiency with the foundational mathematical concepts such as functions and function transformations, that same student might struggle in statistics. Even concepts that are seemingly related to their mathematical counterparts, such as what is a variable when considering population and sample, may cause problems as the statistical sense is distinct enough from the mathematical sense. We suggest that such students may experience greater than usual affective problems in a statistics class and may, therefore, give up easier and earlier than students who have been less successful mathematically.
2013
Denver, Colorado
Twelve participants were asked to decode a proof of Fermat’s Little Theorem and present it in a form of a script for a dialogue between two characters of their choice. Our analysis of these scripts focuses on issues that the participants identified as ‘problematic’ in the proof and on how these issues were addressed. Affordances and limitations of this dialogic method of presenting proofs are exposed, by means of analyzing how the students’ correct, partial or incorrect understanding of the elements of the proof are reflected in the dialogues. The difficulties identified by the participants are discussed in relation to past research on undergraduate students’ difficulties in proving and in understanding number theory concepts.
2013
Denver, Colorado
Researchers have found that students’ beliefs about mathematics impact the way in which they learn and approach mathematics in general. The purpose of this study is to categorize college students’ various conceptions concerning mathematics as a discipline. Results from this study were used to create a preliminary framework for categorizing student conceptions. The results of this study indicate that the conceptions are numerous and range greatly in complexity. The results also suggest the need for further study to qualify the various student conceptions and the roles they play in students' understanding of and approach to performing mathematics.
2013
Denver, Colorado
One challenge of teaching content courses for prospective teachers is organizing instruction in ways that represent the discipline with integrity while serving the needs of future teachers—for example, choosing math problems that provide a logical development of a topic while also addressing mathematical knowledge for teaching. This paper examines the work entailed in structuring in-class work in mathematics courses for teachers. It argues that practices of teaching that are mathematical—such as representing ideas, grounding reasoning in mathematical observations available to the class, using definitions, or using mathematical language—can be used to negotiate mathematical and pedagogical aims, and therefore can be used to organize instruction of mathematical knowledge for teaching while simultaneously developing a disciplinary understanding.
2013
Denver, Colorado
Combinatorial enumeration has a variety of important applications, but there is much evidence indicating that students struggle with solving counting problems. In this paper, the use of the problem-solving heuristic of solving smaller, similar problems is tied to students’ facility with sets of outcomes. Drawing upon student data from clinical interviews in which post-secondary students solved counting problems, evidence is given for how numerical reduction of parameters can allow for a more concrete grasp of outcomes. The case is made that the strategy is particularly useful within the area of combinatorics, and avenues for further research are discussed.
2013
Denver, Colorado
Examples play a critical role in mathematical practice, particularly in the exploration of conjectures and in the subsequent development of proofs. Although proof has been an object of extensive study, the role that examples play in the process of exploring and proving conjectures has not received the same attention. In this paper, results are presented from interviews conducted with six mathematicians. In these interviews, the mathematicians explored and attempted to prove several mathematical conjectures and also reflected on their use of examples in their own mathematical practice. Their responses served to refine a framework for example- related activity and shed light on the ways that examples arise in mathematicians’ work. Illustrative excerpts from the interviews are shared, and five themes that emerged from the interviews are presented. Educational implications of the results are also discussed.
2013
Denver, Colorado
At a research university near the east coast, a College Algebra class has been restructured into two large lectures a week, an active recitation size laboratory once a week, and an extra day of SP. SP was added as an extra day of class where the SP leader has students 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 in light of the number of days students attended SP.
2013
Denver, Colorado
An extensive body of research exists on students’ function concept in the context of graphing in the Cartesian coordinate system (CCS). In contrast, research on student thinking in the context of the polar coordinate system (PCS) is sparse. In this report, we discuss the findings of a teaching experiment that sought to characterize two undergraduate students’ thinking when graphing in the PCS. As the study progressed, the students’ capacity to engage in covariational reasoning emerged as critical for their ability to graph relationships in the PCS. Additionally, such reasoning enabled the students to understand graphs in the CCS and PCS as representative of the same relationship despite differences in appearance. Collectively, our findings illustrate the importance of covariational reasoning for conceiving graphs as relationships between quantities’ values and that graphing in the PCS might create one opportunity to promote such reasoning when combined with graphing in the CCS.
2013
Denver, Colorado
In this report we propose an alternate account of mathematical reification as compared to Sfard’s (1991) description, which is characterized as an “instantaneous quantum leap”, a mental process, and a static structure. Our perspective is based on two in-service teachers’ exploration of the function f (z) = eaz , using Geometer’s Sketchpad. Using microethnographic analysis techniques we found that the long road to beginning to reify the function entailed interplay between body-generated motion and object self-motion, kinesthetic continuity between different sides of the “same” thing, cultural and emotional background of life with things-to-be, and categorical intuitions. Our results suggest that perceptuomotor activities involving technology may serve as an instrument in facilitating reification of abstract mathematical objects such as complex-valued functions.
2013
Denver, Colorado
Little is known about preservice elementary teachers’ mathematical knowledge for teaching number theory concepts, especially greatest common factor. As part of a larger case study investigating preservice elementary teachers’ understanding of topics in number theory, both content knowledge and pedagogical content knowledge (Shulman, 1986), a theoretical model for how preservice elementary teachers understand GCF story problems was developed. An emergent perspective (Cobb & Yackel, 1996) was used to collect and analyze data in the form of field notes, student coursework, and responses to task-based one-on-one interviews. The model resulted from six participants’ responses to three sets of interview tasks where participants discussed concrete, visual, and story problem representations of GCF. In addition to discussing the model and relevant empirical evidence, I suggest language with which to discuss GCF representations.
2013
Denver, Colorado
The Polar Coordinate System (PCS) arises in a multitude of contexts in undergraduate mathematics. Yet, there is a limited body of research investigating students’ understandings of the PCS. In this report, we discuss findings from a teaching experiment concerned with exploring four pre-service teachers’ developing understandings of the PCS. We illustrate ways students’ meanings for angle measure influenced their construction of the PCS. Specifically, students with a stronger understanding of radian angle measure more fluently constructed the PCS than their counterparts. Also, we found that various aspects of the students’ understandings of the Cartesian coordinate system (CCS) became problematic as they transitioned to the PCS. For instance, mathematical differences between the polar pole and Cartesian origin presented the students difficulties. Collectively, our findings highlight important understandings that can support or prevent students from developing a robust conception of the PCS.
2013
Denver, Colorado
We report results from a guided reinvention of the definition of sequence convergence conducted in three second-semester calculus classes. This report contributes to the growing body of research on how students come to understand and reason with formal limit definitions, focusing on the emergence of students’ understanding of the epsilon quantity, conceived in terms of error bounds. Using Sfard's framework of the condensation of processes to entities, we mapped the possible conceptual trajectories followed by the students in the study. In this report, we detail our map, these trajectories and students’ reasoning about other aspects of the formal definition, and the influence of reasoning about approximations and error analyses in students’ progression.
2013
Denver, Colorado
Previous studies in proof reading have identified several strategies to improve students’ proof comprehension. In this study, we report on what we have learned from the first two iterations of a design experiment in which students were taught to apply these strategies. When coupled with modest teacher support, students made significant learning gains. Particular success was found by encouraging students to consider how they would prove the theorem, to use examples to understand assertions in the proof, and to examine definitions before reading the proof. We also discuss some of the difficulties we encountered in teaching students to implement these reading strategies, and the steps we took to attempt to resolve these difficulties.
2013
Denver, Colorado
Over the past two decades education researchers have demonstrated that various types of knowledge, including pedagogical content knowledge, influence teachers' instructional practices and their students’ learning opportunities. Findings suggest that by engaging in the work of teaching, teachers acquire knowledge of how students think, but we have not yet captured this learning as it occurs. We examined whether novice instructors can develop such knowledge via the activities of attending to student work and we identified mechanisms by which such knowledge development occurs. Data come from interviews with physics graduate teaching assistants as they examined and discussed students' written work on problems involving rates of change. During those discussions, some instructors appear to develop new knowledge–either about students’ thinking or about the content—and others did not. We compare and contrast three cases representing a range of outcomes and identify factors that enabled some instructors to build new knowledge.
2013
Denver, Colorado
This paper provides a framework for analyzing and explaining successes and failures when working with summation notation. Cognitively, the task of interpreting a given summation-notation expression differs significantly from the task of expressing a long- hand sum using summation notation. As such, we offer separate cognitive models that 1) outline the mental steps necessary to carry out each of these types of tasks and 2) provide a framework for explaining why certain types of errors are made.
2013
Denver, Colorado
High school mathematics teachers must have coherent systems of mathematical meanings to teach mathematical ideas well. One hundred five teachers were given a battery of items to discern meanings they held in with respect to quantities, variables, functions, and structure. This paper reports findings on a sample of items that, by themselves, should alert college mathematics professors that foundational understandings they assume students have in advanced mathematics courses likely are commonly missing.
2013
Denver, Colorado
This study discusses various theoretical perspectives on abstract concept formation. Students’ reasoning about abstract objects is described based on proposition that abstraction is a shift from abstract to concrete. Existing literature suggested a theoretical framework for the study. The framework describes process of abstraction through its elements: assembling, theoretical generalization into abstract entity, and articulation. The elements of the theoretical framework are identified from students’ interpretations of and manipulations with elementary abstract algebra concepts including the concepts of binary operation, identity and inverse element, group, subgroup. To accomplish this, students participating in the abstract algebra class were observed during one semester. Analysis of interviews and written artifacts revealed different aspects of students’ reasoning about abstract objects. Discussion of the analysis allowed formulating characteristics of processes of abstraction and generalization. The study offers theoretical assumptions on students reasoning about abstract objects. The assumptions, therefore, provide implications for instructions and future research.
2013
Denver, Colorado
The purpose of this study is to investigate students’ concept images of span and linear (in)dependence and to utilize the mathematical activities of defining, example generating, problem solving, proving, and relating to provide insight into these concept images. The data under consideration are portions of individual interviews with linear algebra students. Grounded analysis revealed a wide range of student conceptions of span and/or linear (in)dependence. The authors organized these conceptions into four categories: travel, geometric, vector algebraic, and matrix algebraic. To further illuminate participants’ conceptions of span and linear (in)dependence, the authors developed a framework to classify the participants’ engagement into five types of mathematical activity: defining, proving, relating, example generating, and problem solving. This framework could prove useful in providing finer-grained analyses of students’ conceptions and the potential value and/or limitations of such conceptions in certain contexts.
2013
Denver, Colorado
Researchers have documented deficiencies in prospective elementary teachers’ mathematics content knowledge. The construct of number sense captures the qualities of mathematical reasoning that mathematics teacher educators wish for elementary teachers. However, few researchers have documented interventions that were successful in improving the number sense of prospective elementary teachers. The field is in need of theory concerning prospective elementary teachers’ number sense development. I report findings from a study of number sense development in the setting of a mathematics content course for prospective elementary teachers. Drawing on analyses of collective activity and individuals’ number sense development, I highlight findings that illuminate our understanding of the phenomenon of prospective elementary teachers’ number sense development.
2013
Denver, Colorado
The purpose of this paper is to examine descriptive axiomatizing as a classroom mathematical activity. More specifically, if given the opportunity, how do students select axioms and how might their intellectual needs influence these decisions? These two case studies of axiomatizing address these questions and elaborate on how students engage in this practice within a classroom setting. The results of this research suggest that while students may at first be resistant to axiomatizing, this mathematical activity also affords them opportunities to create meaning for new mathematical content and for the axiomatic method itself.
2013
Denver, Colorado
As part of a larger study of student understanding of concepts in linear algebra, we interviewed 10 university linear algebra students as to their conceptions of functions from high school algebra and linear transformation from their study of linear algebra. Analysis of these data led to a classification of student responses into properties, computations and a series of five interrelated clusters of metaphorical expressions. In this paper, we use this classification to analyze students’ written and verbal responses to questions regarding one-to-one in the context of function from high school algebra and in the context of linear transformation from their study of linear algebra. We found that students’ ability to construe sameness across the two contexts is related to their reliance on properties versus metaphors. We conjecture that this phenomenon is likely to occur for other mathematical constructs as well.
2013
Denver, Colorado
The goal of our study was to characterize the processes and to identify the ways in which different kinds of expertise (mathematics vs. mathematics education) unfolded in the planning and teaching of an undergraduate course on Mathematical Proof and Proving (MPP), which was co-taught by a professor of mathematics and a professor of mathematics education. The content of the course consisted of topics that were supposed to be familiar to the students, i.e., high school level algebra, geometry, and basic number theory. In particular, we looked at a case study describing the design and implementation of a particular task in order to help understand an instance in which each professor’s expertise contributed to the course and complemented the other. The findings indicate that by co-teaching and constantly reflecting on their thinking and teaching, the instructors became aware of the added value of working together and the unique contribution each one had.
2013
Denver, Colorado
A group of prospective mathematics teachers was asked to imagine a conversation with a student centered on a particular proof regarding the derivative of even functions and produce a script of this imagined dialogue. These scripts provided insights into the script-writers’ mathematical knowledge, as well as insights into what they perceive as potential difficulties for their students, and by extension difficulties they may have had themselves when learning the concepts. The paper focuses on the script-writers’ understandings of derivative and of even/odd functions.
2013
Denver, Colorado
In this paper, we examine 26 instances in which mathematics majors attempted to write a proof based on an informal explanation. In each of these instances, we represent students’ informal explanations using Toulmin’s (1958) scheme, we use Stylianides’ (2007) conception of proof to identify what one would need to accomplish to transform the informal explanation into a proof. We then compare this to the actions that the participant took in attempting to make this transformation. The results of our study are categories of actions that led students to successfully construct valid proofs and actions that may have hindered proof construction.
2013
Denver, Colorado
Mathematicians distinguish two modes of their practice – problem solving and theory building. While problem solving has a robust presence in the mathematics curriculum, it is less clear whether theory building does, or should, have such a place. I will report on a curricular design to support a kind of simulation of mathematical theory building. It is based on the notion of a “common structure problem set” (CSPS). This is a small set of mathematical problems with a two-part assignment: I. Solve the problems; and II. Find, articulate, and demonstrate a mathematical structure common to all of them. Some examples will be presented and analyzed. Relations of this construct to earlier ideas in the literature will be presented, in particular to the notion of isomorphic problems, and cognitive transfer. Designing effective instructional enactments of a CSPS is still very much in an experimental stage, and feedback about this would be welcome.
2013
Denver, Colorado
Sjogren (2010) suggested that formal proof could be understood as an explication (Carnap, 1950) of informal proof. Explication describes the supplanting of an intuitive or unscientific concept by a scientific or formal concept. I clarify and extend Sjogren’s claim by applying Carnap’s criteria for explication (similarity, exactness, and fruitfulness) to definitions, theorems, axioms, and proofs. I synthesize a range of proof-oriented research constructs into one overarching framework for representing and analyzing students’ proving activity. I also explain how the analytical framework is useful for understanding student difficulties by outlining some results from an undergraduate, neutral axiomatic geometry course. I argue that mathematical contexts like geometry in which students have strong spatial and experiential intuitions may require successful semantic style reasoning. This demands that students’ construct rich ties between different representation systems (verbal, symbolic, logical, imagistic) justifying explication as a reasonable analytical lens for this and similar proof-oriented courses.
2013
Denver, Colorado
In the past decade, educators and statisticians have made new suggestions for teaching undergraduate statistics. In light of these new recommendations it is important to (re)evaluate how individuals come to understand statistical concepts and how such research should impact curricular efforts. One concept that plays a major role in introductory statistics is hypothesis testing and the computation of the test statistic to draw conclusions in a hypothesis test. This proposal presents a theoretical approach through the development of a hypothetical learning trajectory of hypothesis testing by utilizing sampling distributions as the building block to understand statistical inference. In addition, this proposal presents how this hypothetical learning trajectory may support the development of research-based curricula that foster an understanding of the test statistic and its role in hypothesis testing.
2013
Denver, Colorado
This paper puts forth a new theoretical perspective for students’ understanding of sampling. The Action, Process, Object, and Schema Theory for Sampling serves as a potential bridge between Saldanha’s and Thompson’s Multiplicative Conception of Sampling and APOS Theory. This theoretical perspective provides one potential way to describe the development of a student’s conception of sampling. Additionally this perspective differs from most other perspectives in that it does not focus on the sample size the student uses or the sampling method, but rather how the student understands sampling in terms of a sampling distribution.
2013
Denver, Colorado
The accepted framing of pedagogical content knowledge (PCK) as mathematical knowledge for teaching has centered on the question: What mathematical reasoning, insight, understanding, and skills are required for a person to teach mathematics? Many have worked to address this question, particularly among K-8 teachers. What about teachers with broader mathematics knowledge (e.g., from algebra to proof-based understandings of topics in advanced mathematics)? There is a need for examples and theory in the context of teachers with greater mathematical preparation and older students with varied and complex experiences in learning mathematics. This theory development piece offers background and examples for an extended theory of PCK as the interplay among conceptually-rich mathematical understandings, experience of teaching, and multiple culturally-mediated classroom interactions.
2013
Denver, Colorado
The primary goal of this work is to articulate a theoretical foundation based on Realistic Mathematics Education (RME) that can support the analysis of student learning, both individual and collective, by documenting changes in local activity. To do so, I will build on previous work on the analytic implications of the Emergent Perspective, specifically Rasmussen and Stephan’s (2008) analytic approach to documenting the establishment of classroom mathematical practices. The Emergent Perspective is broadly consistent with RME, but the existing analytic methods related to the Emergent Perspective fail to draw on the theoretical constructs provided by RME. For instance, current analytic methods fail to draw on the RME Emergent Models heuristic to inform the analysis of the development of mathematical practices related to models of/for student mathematical activity. Here I will be explicitly considering the roles that RME constructs could play in analytic processes consistent with the Emergent Perspective.
2013
Denver, Colorado
Model analysis is a quantitative research method used in physics education research; we have adapted this method to study proof schemes in a transition-to-proof course. Model analysis accounts for the fact that students may hold more than one idea or conception at a time, and may use different ideas and concepts in response to different situations. Model analysis is uniquely suited to study students' proof schemes, as students often hold multiple, sometimes conflicting proof schemes, which they may use at different times. Model analysis treats each student’s complete set of responses as a data point, rather than treating each individual response as a separate data point. Thus, model analysis can capture information on the self-consistency of a student’s responses. We collected data from a transition-to-proof course at the beginning and the end of the Fall 2012 semester. We find that model analysis offers significant insights not offered by traditional analysis.
2013
Denver, Colorado
Perceived mathematical authority plays an important role in how students engage in mathematical interactions, and ultimately how they learn mathematics. This paper elaborates the concept of mathematical authority (Engle, 2011) by introducing two concepts: scope and relationality. This elaborated view is applied to a number of peer-interactions in a specialized peer-assessment context. In this context, self-perceived authority influenced the way feedback was framed (as either questions or assertions).
2013
Denver, Colorado
The formal definition of a limit, or the epsilon delta definition is a critical topic in calculus for mathematics majors’ development and the first chance for students to engage with formal mathematics. This report is a microgenetic study of one student understanding of the formal definition focusing on a particularly important relationship between epsilon and delta. diSessa’s Knowledge in Pieces and Knowledge Analysis provide frameworks to explore in detail the structure of students’ prior knowledge and their role in learning the topic. The study documents the progression of the student’s claims about the dependence between delta and epsilon and explores relevant knowledge resources.
2013
Denver, Colorado
The formal definition of a limit, or the epsilon delta definition is a critical topic in calculus for mathematics majors’ development and the first chance for students to engage with formal mathematics. Research has documented that the formal definition is a roadblock for most students but has de-emphasized the productive role of their prior knowledge and sense making processes. This study investigates the range of knowledge resources included in calculus students’ prior knowledge about the relationship between delta and epsilon within the definition. diSessa’s Knowledge in Pieces provides a framework to explore in detail the structure of students’ prior knowledge and their role in learning the topic.
2013
Denver, Colorado
In this work, we examine students' ways of thinking when presented with a novel linear algebra problem. We have hypothesized that in order to succeed in linear algebra, students must employ and coordinate three modes of thinking, which we call computational, abstract, and geometric. This study examines the solution strategies that undergraduate honors linear algebra students employ to solve the problem, the variety of productive and reflective ways in which the computational mode of thinking is used, and the ways in which they coordinate the computational mode of thinking with other modes.
2013
Denver, Colorado
This study is a part of ongoing research on development of Mathematical Knowledge for Teaching (MKT) in mathematical content courses. Reflective practice represents a central theme in teacher education. The purpose of this reported study was to understand the role of guided reflections on mathematical tasks in a college geometry course. We were also interested in understanding how guided reflections on mathematical tasks would effect teachers’ development of MKT. Our research data consist of participants’ reflections, teaching scenarios, and pre-post test results. In this study we developed a workable framework for data analysis. Audience discussion will address questions related to the proposed analysis framework and development of MKT in college mathematics courses.
2013
Denver, Colorado
Effective mathematics teachers are able to make connections between mathematical content and pedagogy in their professional practice. One of the most readily prescribed approaches for facilitating teachers’ ability to make such connections is through the development of collaborations between mathematicians and mathematics educators in venues related to teacher professional development. Most prior research related to collaborative endeavors between these two groups has focused on the products, rather than the process, of collaboration. In this preliminary research report, I present the results of an interpretative phenomenological case study that investigated the team-teaching experiences of a mathematician and a mathematics educator within the context of an undergraduate mathematics teacher preparation program. I present extracts from interviews that highlight the instructors’ perceptions related to crossing the boundaries of their professional communities of practice, and engage participants in discussion about relevant “boundary crossing” in their own institutional contexts.
2013
Denver, Colorado
This article describes the results obtained from a diagnostic instrument to establish the difficulties in understanding and using variables that engineering students have at the moment of their entrance to a public Mexican university that does not examine the candidates prior to admittance. This work is part of a study that analyses the possible impact that failing to understand the uses of variables may have in understanding systems of linear equations.
2013
Denver, Colorado
Despite the importance of intuition and analysis in proof tasks, students have various difficulties with both types of reasoning. Such difficulties may be attributed to insufficient intuition, logical reasoning skills, or concept images. However, dual-process theory asserts that intuition can form inaccurate or incomplete representations of tasks based on systematic errors before analysis can respond. Thus, students' difficulties may be attributed to systematic intuitive errors rather than inadequate intuitive or analytical reasoning. In this study, I conducted task-based interviews with four undergraduate and one graduate mathematics major in which they completed prove-or-disprove tasks. In this paper, I discuss the systematic intuitive errors committed by these students on a monotonicity task. These errors led all five students to believe incorrectly that the statement in the task was true. Furthermore, each student engaged in correct mathematical reasoning guided by their incorrect intuitive representations.
2013
Denver, Colorado
This paper describes preliminary results of a study aimed at examining the effects of working in cooperative groups on acquisition and development of proof skills. Particular attention will be paid to the varying tendencies of students to switch proof methods (direct, induction, contradiction, etc) based on their level of proof expertise. Namely, as students progress from novice to expert provers, they tend to change proof methods more frequently until they reach the final stages of development (Hart 1994).
2013
Denver, Colorado
Previous research has reported an absence of a theoretical thinking component in college-level Calculus courses. While valid arguments can be made for or against the necessity of incorporating such a component, our belief is that students who wish to engage in theoretical thinking should be given the chance to do so. Our goal is to determine whether instructors can provide their students with opportunities to engage in theoretical thinking, despite constraints they often face such as time, course content, and assessment material. This report presents a preliminary analysis of a study in which we presented students in a Calculus class with tasks intended to provoke theoretical thinking. Using Sierpinska et al.’s (2002) model for theoretical thinking we show that students who engaged in these optional tasks where in fact engaging in theoretical thinking. We conclude that, despite the institutional constraints, incorporating such a component is indeed feasible.
2013
Denver, Colorado
Undergraduate mathematics programs must prepare teachers for the challenges of teaching statistical thinking as advocated in standards documents and statistics education literature. This preliminary report presents initial results from a study of pre-service secondary mathematics teachers at the end of their undergraduate educations. Although nearly all had completed a required upper-division statistics course, most were challenged by two tasks which required a critical analysis of the use of statistics in newspaper articles. Some patterns emerged in the incorrect answers, including a tendency to focus on potential sampling issues which were not relevant to the tasks. The session will explore the nature and sources of these difficulties with statistical thinking and statistical communication and it will explore the implications for undergraduate mathematics and statistics teacher preparation.
2013
Denver, Colorado
This research focuses on mental challenges that students face and how they resolve these challenges while transitioning from intuitive reasoning to constructing a more formal mathematical structure of Riemann sum while modeling “real life” contexts. A pair of Calculus I students who had just received instruction on definite integral defined using Riemann sums and illustrated as area under the curve participated in multiple interview sessions. They were given contextual problems related to Riemann sums but were not informed of this relationship. Our intent was to observe students’ transitioning from model of to model for reasoning while modeling these problem situations. Results indicate that students conceived of five major conceptions during their first task and their reasoning from the first task that became a model for reasoning about their next task. In this paper we detail those conceptions and their reasoning that became model for reasoning on the second task.
2013
Denver, Colorado
An initial investigation into students’ understanding of Eigentheory using semi-structured interviews was conducted with students at the end of a first-semester course in quantum mechanics. Many physics faculty would expect students to have mastery of basic matrix multiplication after a course in Linear Algebra, and especially so after fairly extensive use of matrices in quantum mechanics in the context of Ising model spin problems. Using a previously published interview protocol by Henderson et al, student reasoning patterns were investigated to probe to what extent there reasoning patterns were similar to those identified among Linear Algebra students. Reasoning patterns appeared quite consistent with previous work; that is, students used superficial algebraic cancellation, and demonstrated difficulty interpreting their result even when they arrived at a correct solution. The interview protocol was modified slightly to probe whether or not students felt the tasks they were engaging in were mathematical or physics-related. Additional questions were added at the end of the protocol about how these concepts were used in their quantum mechanics course. Students were somewhat successful relating them to Hamiltonians and energy eigenvalues, but couldn’t articulate the type of physical situations where they might be useful.
2013
Denver, Colorado
Although there is much research showing that proof serves more than just a verification function in mathematics, there is little research documenting which functions of proof undergraduate students understand. Additionally, research suggests that students have difficulty in determining the validity of a given proof. This study examines the effects of a teaching intervention called collaborative revision on student beliefs regarding proof and on student proof validation skills. Student assessment data was collected and interviews were conducted with students in the treatment course and in a comparison course. At the end of this study, we will produce a categorization of the proof functions that students appreciate, as well as a determination of the value of the teaching intervention on students’ abilities to correctly classify proofs as valid or invalid.
2013
Denver, Colorado
In this study we use the tradition of semiotics to motivate an exploration of the knowledge of, and facility with, the symbol system of statistics that students bring to university. We collected a sample of incoming mathematics majors in their first semester of study, prior to taking any statistics coursework, and engaged each in a task-based interview using a think-aloud protocol with questions designed to assess their fluency with basic concepts and symbols of statistics. Our findings include that students find symbols arbitrary and difficult to associate with the concepts. Second, that generally, no matter the amount of statistics that students took in high school, including Advanced Placement courses, they have relatively little recall of topics. Most can calculate the mean, median and mode, but they generally remember little beyond that. Finally, students have difficulty connecting practices or procedures to meaning.
2013
Denver, Colorado
Students who graduate with an engineering or science degree using applied mathematics are expected to synthesize concepts from calculus to solve problems. First semester calculus students attempting to understand the derivative as a rate of change encounter difficulties. Specifically, the challenges arise while making the decision to apply an average rate of change or an instantaneous rate of change (Zandieh, 2000) to the problem. This paper discusses how students view the derivative in an applied mathematical setting and investigates how the concept of time and other related quantities contribute to the development of a solution.
2013
Denver, Colorado
A models & modeling theoretical perspective has been suggested to supersede both con- structvism (Lesh, Doerr, Carmona, & Hjalmarson, 2003) and problem-solving paradigms (Zawojewski, 2007). This is important to the RUME community because many of our foun- dational works are rooted in theoretical and methodological perspectives derived from these paradigms. In the case of problem solving and modeling, it is important to re-view the problem-solving research settings with a mathematical modeling lens. Without this glance backwards we cannot connect new ideas to old knowledge and we should not supplant a theory without ensuring the next can account for existing observations. The objective of this paper is to revisit a well-known problem setting to explore alternative interpretations of students’ mathematical work.
2013
Denver, Colorado
One of the challenges of teaching introductory calculus is the large variance in student backgrounds. Formative assessment can be used to target which students need help, but little is known about why formative assessment is effective with adult learners. The purpose of this qualitative study was to investigate which functions of formative assessment as described by Black & William’s 2009 framework help students progress through their Zone of Proximal Development. By regularly collecting information from low-stakes opportunities for students to demonstrate their current understanding, instructors were able to target subsequent class discussion on critical scaffolding for student growth. The formative assessments also enabled students to evaluate their own progress and ask clarifying questions and, provided students who would not ordinarily ask questions during class opportunities for legitimate peripheral participation.
2013
Denver, Colorado
Data was collected from three sections of an introductory proofs course that was taught from an inquiry-based perspective. Inquiry-based learning (IBL) gives authority to students and allows them to present to their peers, rather than having the instructor be the focus of the class and authority on proof. Data from the final exams of 68 students was analyzed with a focus on proof structure. Proofs chosen to analyze included concepts considered “prior knowledge”, as well as problems that required new concepts from class. This research utilizes an adaptation of Toulmin’s method for argumentation analysis. Our goal was to compare the proof structures generated by these students to previous research also applying some form of Toulmin’s scheme to mathematical proof. There was significant variety of proof structures, which could be a result of the IBL atmosphere.
2013
Denver, Colorado
Previous researchers have examined students’ understanding of derivative and their difficulties in solving applied problems and/or their difficulties in applying the basic knowledge of derivative in different contexts. There has not been much research approaching students’ ways of thinking about derivative through the lens of applied questions. In this research, first I categorized the students’ way of thinking about the basic concept of derivative by running a survey of questions addressing the different ways of thinking about derivative based on the existing research works. While analyzing these surveys, I used grounded theory and added more ways of thinking about derivative. I specially noticed very incomplete ways of thinking about derivative as described below. Since my goal was looking at the students’ ways of thinking about derivative through the lenses of applied questions, I also piloted my applied questions survey with 51 multivariable calculus students. I noticed a lot of students struggling with defining variables (the initial translation as described below) and if they could define the variable, a lot of them struggled on applying their ways of thinking about derivative into solving the applied problem. These difficulties are great venues to study their ways of thinking about derivative using their struggle in the applied questions. This is a summary of my initial works on this ongoing research, the goal of which is to shed new insights into students’ solving of applied problems.
2013
Denver, Colorado
Researchers have documented difficulties that elementary school students have in understanding volume. Despite its importance in higher mathematics, we know little about college students’ volume understanding volume. This study investigated calculus students’ understanding of volume. Clinical interview transcripts and written responses to volume problems were analyzed. One finding is that some calculus students, when asked to find volume, find surface area instead and others blend volume and surface area ideas. We categorize students’ formulae according to their volume and surface area elements. Clinical interviews were used to investigate why students might find surface area when asked for volume. We found that some students believe adding the areas of an object’s faces measures three-dimensional space. Findings from interviews also revealed that understanding volume as an array of cubes is connected to successfully solving volume problems. This finding and others are compared to those for elementary school students. Implications for calculus teaching and learning are discussed.
2013
Denver, Colorado
Shifting demographics show America rapidly diversifying, yet research indicates that an alarming number of diverse students continue to struggle to meet learning outcomes of collegiate mathematics curriculum. Consequently, recruitment and retention of diverse students in STEM majors is a pervasive issue. Using a sociocultural perspective, this study examined the effect of two pedagogical strategies (traditional instruction and cooperative learning) in a diverse College Algebra course on enhancing students’ mathematics self-efficacy. Particular attention was paid to investigating the role student discourse and interaction play in facilitating learning, improving conceptual understanding, and empowering students to engage in future self-initiated communal learning. The goal is to develop an effective classroom model that cultivates advancement in content knowledge and enculturation into the STEM community, culminating in a higher retention rate of diverse students in STEM. Preliminary data analysis suggests that a hybrid model encompassing both traditional instruction and cooperative learning successfully enhances students’ self-efficacy.
2013
Denver, Colorado
Although students have reported on their study habits outside of the classroom, there has been little verification of their actual behavior during group study sessions. This project observed undergraduate students studying together outside of the classroom setting in order to determine what study groups formed and what structures described these groups. Elements of social network analysis were employed to identify the groups that students formed. Transcripts of the study sessions were coded and frequency counts were established for each type of student interaction in order to characterize the roles students assumed while studying. This paper discusses the process of identifying the study groups and sets the groundwork for sharing the student roles. One main finding of this work is that the presence of a student recognized as an authority or facilitator of the group impacts the type of conversations that occur in the group setting.
2013
Denver, Colorado
Gestures are profoundly integrated into communication. This study focuses on the impact that gestures have in a mathematical setting, specifically in an undergraduate calculus workshop. We identify two types of gesture – dynamic and static – and note a strong correlation between these movements and diagrams produced. Gesture is a primary means for students to communicate their ideas to each other, giving them a quick way to share thoughts of relative motion, relationships, size, shape, and other characteristics of the problem. Dynamic and static gestures are part of the students’ thinking, affecting how they view the problem, sway group thinking, and the construction of their diagrams.
2013
Denver, Colorado
The past 20 years has produced a growing body of research on instructional technologies in mathematics classrooms, as well as a small, but growing, body of work on the impact of technology in introductory statistics classrooms (e.g., Chance, Ben-Zvi, Garfield, & Medina, 2007; Chance, delMas & Garfield, 2004). However, much of the work touting the benefits of using technology in the statistics classroom has not been empirically based (see Mills, 2002) and Shaughnessy (2007) has argued that there is a lack of research investigating the use of technology in statistics classrooms. Specifically, research is needed that explores how technology changes the way students think about statistics and the ways technology can be used to enable students to construct models to solve statistical problems. There are two fundamental reasons why such research is badly needed: (1) statistical literacy and computer literacy are vital skills in an information age; and (2) new dynamic educational technologies carry much promise for supporting student learning, but without knowledge of how these technologies impact student thinking they fall short of their potential. This proposal focuses on student challenges interpreting a single trial of a statistical experiment.
2013
Denver, Colorado
This study sought to describe the tools and reasoning techniques used by mathematicians to construct and write proofs. Task-based clinical interviews were conducted with 3 research mathematicians in varying research fields. The tasks were upper-undergraduate and lower- graduate level proofs from linear algebra, basic analysis, and abstract algebra. Data were coded based on a framework constructed from Dewey’s theory of inquiry and the characterizations of conceptual insight and technical handle. Preliminary results indicate the task of discovering a conceptual insight that can potentially lead to a proof can be problematic, and there are distinct moments in the construction process when the problem changes from “why should this be true?” to “how can I prove that?”
2013
Denver, Colorado
Like several other research groups, we have been investigating multiple measures for capturing change in middle and high school teachers’ mathematical pedagogical content knowledge (PCK). This article reports on results among 14 teachers (of 16 enrolled) who have completed a distance-delivered master’s program in mathematics education. The degree program seeks to develop content proficiency, cultural competence, and pedagogical expertise for teaching mathematics. Analysis included pre- and post-program data from classroom observations and written PCK assessment. Results indicate significant changes in curricular content knowledge on the observation instrument and significant changes in discourse knowledge on both the observation instrument and the written assessment. Path analyses suggest teacher discourse knowledge as measured by the written assessments is significantly related to discourse knowledge as measured by the post-program observation.
2013
Denver, Colorado
Self-inquiry is the process of posing questions to oneself while solving a problem. The self- inquiry of thirteen undergraduate mathematics students was explored via structured interviews requiring the solution of both mathematical and non-mathematical problems. The students were asked to verbalize any thought or question that arose while they attempted to solve a mathematical problem and its nonmathematical logical equivalent. The thirteen students were volunteers who had each taken at least four upper division proof-based mathematics courses. Using transcripts of the interviews, a coding scheme for questions posed was developed and all questions were coded. While data analysis of the posed questions is ongoing, initial analysis suggests that the “good” mathematics students focus more questions on legitimizing their work and fewer questions on specification of the problem-solving task. Additionally, the self-inquiry of “fast” problem solvers mirrored that of the strong students with even less focus on specification questions.
2013
Denver, Colorado
Many first year college students struggle with foundational mathematics skills even after one semester of mathematics. JUMP math, a systematized program of teaching mathematics, claims that its approach, though initially designed for K-8, can strengthen skills at the foundations college math level as well. Students in sixteen sections of Foundations Mathematics at a college in Canada were randomly assigned to be taught with either the JUMP math approach or a typical teaching approach. Students were measure before and after on their competence (Wechsler test of Numerical Operations) and attitudes (Mathematics Attitudes Inventory) to identify any improvements. Results showed that students in JUMP classes had modest, but consistently higher improvements in competence when compared to students in non- JUMP classes, even after controlling for potential confounding variables, while improvements in Math Attitudes showed no differences.
2013
Denver, Colorado
This study looks at the interplay between the concept image and concept definition when students are given a task that requires direct application of the definition of continuity of a function at a point. Data was collected from 37 first year university students. It was found that different students apply the definition to different levels, which varied from formal deductions (based on the application of the definition) to intuitive responses (based on rather loose and incomplete notions in their concept image).
2013
Denver, Colorado
This article documents a new way of conceptualizing vectors in college geometry. The complexity and subtlety of the construct of vectors highlight the need for a new framework that permits a layered view of the construct of vectors. The framework comprises three layers of progressive refinements: a layer that describes a global distinction between physical vectors and mathematical vectors, a layer that recounts the difference between the representational perspective and the cognitive perspective, and a layer that identifies ontological and epistemological obstacles in terms of transitions towards abstraction. Data was gathered from four empirical studies with ninety-eight total students to find evidence of the three major transition points in the new framework: physical to mathematical coming from the first layer, geometric to symbolic and analytic to synthetic from the second layer, and the prevalence of the analytic approach over the synthetic approach while developing abstraction enlightened by the third layer.
2013
Denver, Colorado
The CSPCC (Characteristics of Successful Programs in College Calculus) project is a large empirical study investigating mainstream Calculus 1 to identify the factors that contribute to success, to understand how these factors are leveraged within highly successful programs. Phase 1 of CSPCC entailed large-scale surveys of a stratified random sample of college Calculus 1 classes across the United States. Phase 2 involves explanatory case study research into programs that are successful in leveraging the factors identified in Phase 1. Here we report preliminary findings from a pilot case study that was conducted at a private liberal arts university. We briefly describe the battery of interviews conducted at the pilot site and discuss some of the themes that have emerged from our initial analyses of the interview data.
2013
Denver, Colorado
Some mathematics educators and mathematicians have suggested that students should base their proofs on informal reasoning (Garuti et al. 1998). However, the ways in which students implement informal representations are not well understood. In this study, we investigate informal representations made by undergraduates during proof construction. Their use of informal representations will be compared to mathematicians’ use of informal representations as described in Alcock (2004) and Samkoff et al. (2012). Further, an analysis of different types of informal representations will investigate the necessity to treat these different representations more carefully in the future.
2013
Denver, Colorado
The author is writing an electronic “book” to support the teaching of calculus to business students with the assumption that they will use a spreadsheet as their main computational engine. With the change in technology, it is appropriate to rethink the content of the course, as a different technology makes different tasks accessible. This study looks at what the content of the course should be. It compares the official learning objects of the course, with de facto learning objectives obtained by analyzing final exams from 20 sections of the course, and with the results of a survey of the faculty of the business school, the client discipline. It is intended that this preliminary study will establish a baseline that can be used to evaluate the effectiveness of the new approach to the course.
2013
Denver, Colorado
This report considers student proof construction in small groups within an inquiry-orientated abstract algebra classroom. During an initial analysis, several cases emerged where students used familiar knowledge from another mathematical domain to provide informal intuition. I will report on two episodes in order to illustrate how this intuition could potentially aid or hinder the construction of a valid proof.
2013
Denver, Colorado
We have developed a short (16 question) basic skills test for use in our institution's transition-to-proof course that assesses basic skills required to succeed in such a course. Using this test in our core introductory proof course, we have found that students are generally deficient in a number of skills assumed by instructors. In addition, using this test as a pre/post-test we have found that in this course students are learning some concepts well, but that learning gains on other concepts are much below desired levels. Finally, administration of the test to students in a higher level course has allowed us to assess retention of these skills. At this preliminary stage these skills appear to be retained into higher-level proof courses, but more data collection is needed, as well as a more extensive instrument to assess proof skills, rather than simply basic logic and comprehension.
2013
Denver, Colorado
We summarize the preliminary results of a study of conceptual understanding of mathematics by pre-service secondary school math teachers. Our research involves the statistical analysis of data from an actual mathematics Praxis II licensure exam, which was administered nationwide. Through a quantitative, item by item analysis, using a classification of these test items by conceptual difficulty, we obtain insight into the conceptual issues that pre-service teachers have great difficulty with. Our preliminary results show a significant gap between computational and abstract mathematical processes. This in turn, affects the ability of pre- service teachers to be fluent in the domains of both subject and pedagogical content knowledge.
2013
Denver, Colorado
Studies have shown that students have difficulty with the concept of limit, especially when reasoning about formal limit definitions. We conducted a five-day teaching experiment (TE) in a second semester calculus classroom in which students were asked to reinvent a formal sequence convergence definition. Oehrtman et al. (2011) detailed how pairs of students reinvented sequence convergence definitions but did not attempt the same instructional heuristic in the classroom. Our analysis focused on the instructor prompts and the TE students' subsequent group discussion through their use of key words and visuals in revising their definition. An interview with the instructor was conducted to investigate his intention of using specific prompts and his thinking about the TE group's choice of words and visuals. In our preliminary analysis, we found that the roles of the instructor were extended beyond those roles previously reported as roles for facilitators with pairs of students.
2013
Denver, Colorado
Starting in 2009 we have implemented a Team Based Learning (TBL) model of delivery in two mathematics courses and one mathematics education course involving a total of 295 students. Qualitative data from evaluations, observations and interviews is used to begin to answer four questions raised by the Seldens (2001) regarding teaching mathematics at tertiary levels. Our analysis indicates that students say that TBL creates an environment in which they are active, have productive arguments and discussions and benefit from immediate feedback. There is scant evidence of any group being disadvantaged by this model of delivery.
2013
Denver, Colorado
This study started with a thorough analysis of student work on problems involving related rates of change in a first-year differential calculus course at a large, research- focused university. In two sections of the course, students' written solutions to geometric related rates problems were coded and analyzed, and students' learning was tracked throughout the term. Three months after the end of term, "think-aloud'' interviews were conducted with some of the students who completed the course. The interviews and some of the written assessments were structured based on the classification of key steps in solving related rates proposed by Martin (2000). Our preliminary findings revealed a widespread, persistent use of algorithmic procedures to generate a solution, observed in both the treatment of the physical and geometric problem, and the approach to the differentiation, and raised the question of whether traditional exam questions are a true measure of students' understanding of related rates.
2013
Denver, Colorado
The “one-minute paper” (Stead, 2005) is a technique for facilitating communication between students and the teacher and promoting reflection. In this paper we focus on the types of questions students ask and how they may be related to success. We present preliminary results from an introductory university-level calculus course, indicating that the nature of questions asked by more successful and less successful students are different, suggesting that the types of reflections that students engage in may have a significant impact on the efficacy of such an intervention.
2013
Denver, Colorado
This study investigates how two professors and pre-service secondary mathematics teachers engage in argumentation and proof in two courses. One course under investigation is a geometry course; the second is a methods of teaching mathematics course. The research also studies the how professors and pre-service teachers construct arguments and proofs. Examining the classroom discourse to understand how it may impact argumentation practices is another aspect of the research. Case study and grounded theory approaches are used to guide the data collection and analysis. Some data collected include interviews with the two professors and pre-service teachers and observations of the two courses and the pre-service teachers’ classrooms during their student teaching. Data analysis so far indicates the geometry professor engages students in argumentation and proof in multiple ways.
2013
Denver, Colorado
A link between proving and problem solving has been well established in the literature (Furinghetti & Morselli, 2009; Weber, 2005). In this paper, I discuss similarities and differences between proving and problem solving by using the Multidimensional Problem-Solving Framework created by Carlson and Bloom (2005) on Livescribe pen data from a study of proving (Savic, 2012). I focus on two participants’ proving processes: Dr. G, a topologist, and L, a mathematics graduate student. Many similarities were revealed by using the Carlson and Bloom framework, but also some differences distinguish the proving process from the problem- solving process. In addition, there were noticeable differences between the proving of the mathematician and the graduate student. This study may influence a proving-process framework that can encompass both the problem-solving aspect of proving and the differences found.
2013
Denver, Colorado
As a first step in studying students’ spatial reasoning ability, preference, and their impact on performance in second semester calculus, I ran a pilot study to develop interview tasks and a coding scheme for analyzing the interviews. Four videotaped interviews were conducted with each of the five participants and the video was coded for graphical reasoning. I will discuss my coding scheme and share some preliminary results. I hypothesize that the coding scheme may help identify a student’s preference and ability for spatial reasoning.
2013
Denver, Colorado
This preliminary report describes the second stage of data collection and analysis in a larger study that examines students’ written and verbal language when studying basic theorems in a first-semester calculus course. We examine students’ difficulties with understanding and using mathematical language and notation in both formal written work and informal verbal descriptions. Not surprisingly, the students in our study rarely use formal mathematical language without being prompted to do so. One surprising result was that while many students do understand the mathematical notation in the theorems, and can illustrate this graphically when prompted, they still do not use this notation when providing their own written (or verbal) description of a theorem. Preliminary results suggest that our biggest obstacle as teachers is not in getting our students to understand the notation, but instead lies in convincing our students of the power that comes from this notation in describing a concept, thus encouraging our students to use this notation in their own written work.
2013
Denver, Colorado
The world of a mathematician, with all its creativity and precision is fascinating to most people. This study is an account of collaboration between mathematicians and mathematics educators. In order to examine a mathematician’s daily activities, we have primarily employed Schoenfeld’s goal-orientated decision making theory to identify his Resources, Orientations and Goals (ROGs) in teaching an abstract algebra class. Our preliminary results report on a healthy and positive atmosphere where all involved freely express their views on mathematics and pedagogy.
2013
Denver, Colorado
Research in science education, particularly physics education, indicates that students in Interactively-Engaged classrooms are more successful on tests of basic conceptual knowledge. Despite this, undergraduate mathematics courses are dominated by lectures in which students take a passive role. Given the value of such tests in assessing students' conceptual knowledge, the method for measuring such change is largely unexplored. In our study, students were given one such inventory, the Calculus Concept Inventory, in introductory Calculus classes as a pretest and posttest. We address issues of how gains might be measured on this instrument using two techniques, and the implications of using each of these measures.
2013
Denver, Colorado
Past research has shown that students struggle when applying the definite integral concept, and these difficulties stem from incomplete understanding of the integral’s underlying structure. This study aims to provide insight into the construction of effective mental structures for integrals by examining experts’ solutions to volume problems. Seven mathematics faculty members from a large, public university solved three volume problems (two routine, one novel) in videotaped interview sessions. Preliminary analysis shows that the experts have a rich understanding of definite integrals, and the few instances of errors seemed to be a result of inattention as opposed to a deficit in understanding. Their problem-solving process was highly structured and detailed. The experts’ visual representations varied from sparse and static to fully 3-dimensional and dynamic. We hope to use this and past student data to construct a framework for analyzing student understanding of integral volume problems.
2013
Denver, Colorado
In this small comparative study, we explore the impact of “flipping” the instructional delivery of content in an undergraduate Calculus III course. Two instructors collaborated to determine daily content and lecture notes; one instructor altered the instructional delivery of the content (not the content itself), utilizing videos to communicate procedural course content to students out-of-class, with time in-class spent on conceptual activities and homework problems. Student performance on tests for both classes will be compared to determine any significant differences in achievement related to “flipping” the instructional delivery.
2013
Denver, Colorado
This paper describes a model of the understandings of two first-semester calculus students, Brian and Neil, as they participated in a teaching experiment focused on exploring ways of thinking about rate of change of two-variable functions. I describe the students’ construction of directional derivative as they attempted to generalize their understanding of one-variable rate of change functions, and characterize the importance of quantitative and covariational reasoning in this generalization.
2013
Denver, Colorado
In this talk, we describe the development of the ways of thinking of 25 vector calculus students over the course of one term. In particular, we characterize the generalizations that students made within and across interviews. We focus on the construction of the semi-structured pre and post interviews, trace the construction of explanatory constructs about student thinking that emerged from those interviews, and describe how those constructs fit within the broader literature on student thinking in advanced calculus. We conclude by exploring implications for future research and practical applications for educators.
2013
Denver, Colorado
It is important for us to develop an understanding of how students make sense of lectures, particularly in upper-level mathematics classes, where the focus is on definitions, proofs and examples. To do this, we observed abstract algebra lectures, collected students’ notes, and interviewed students about their experiences while re-reading their notes and re-watching the lectures. Our preliminary findings indicate that students attempt to make sense of components of the lecture on multiple levels, which influences their understanding of the mathematical aspects of the lecture.
2013
Denver, Colorado
Math content courses aim to develop mathematical reasoning and communication skills in future teachers. Instructors often assign problems requiring in-depth written explanations to develop these skills. However, when a student’s conception is incorrect, does written feedback from the instructor create the cognitive dissonance necessary to effect realignment of the student’s understanding? These conceptions may be mathematical (“what is a fraction?”) or meta-mathematical (“what constitutes a justification?”). Assigning problem revisions theoretically creates space for cognitive dissonance by having students rethink their solutions. I investigate a revision assignment in a course for future teachers to understand the nature of students’ revisions and the possible impetuses for these revisions. In particular, I find preliminary evidence that students’ revisions demonstrate changes in their language, mathematics, and use of examples and representations. Further, students’ adoption of new representations in their solutions are largely due to observing peers’ presentations rather than to instructor feedback.
2013
Denver, Colorado
Many mathematics students experience proof-based classes primarily through lectures, although there is little research describing what students actually learn from such classroom experiences. Here we outline a framework, drawing on the idea of the implied observer, to describe lecture content; and apply the framework to a portion of a lecture in an abstract algebra class. Student notes and interviews are used to investigate the implications of this description on students' opportunities to learn from proof-based lectures. Our preliminary findings detail the behaviors, codes, and competencies that an algebra lecture requires. We then compare those with how students behave in response to the same lecture with respect to sense- making and note-taking, and thereby how they approach opportunities to learn.
2013
Denver, Colorado
Previous studies have shown that the normative solutions of the Pin-Pong Ball Conundrum and the Pin-Pong Ball Variation are difficult to understand even for learners with advanced mathematical background such as doctoral students in mathematics. This study examines whether this difficulty is due to the way they are set in everyday life experiences. Some variations of the Pin-Pong Ball Conundrum and the Pin-Pong Ball Variation and their abstract versions set in the set theoretic language without any reference to everyday life experiences were given to a doctoral student in mathematics. Data collected suggest that the abstract versions can help learners see beyond the metaphorical language of the paradoxes. The main contribution of this study is revealing the possible negative effect of the metaphorical language of the paradoxes of infinity on the understanding of the learner.
2013
Denver, Colorado
This preliminary study explores how students’ perceptions of the equality of trigonometric expressions evolve during the process of verifying trigonometric identities (VTI). If students already view the purported equality as being true, VTI may not offer much in the way of learning experiences for students. Using a semiotic perspective to analyze student work, this study attempts to describe the evolution of students’ perceptions of identities upon application of VTI, focusing on the components of the student’s VTI process that contribute to the evolution. Initial analyses of interviews conducted while students proved identities indicate that students are not fully convinced that the identities are initially true. However, successful VTI, signaled, for example, by the use of an idiosyncratic equality construction, endows equality on not only the purported identity but on ancillary equality statements generated as part of the VTI process.
2013
Denver, Colorado
Students’ difficulties with relating the graphs of functions to the graphs of their derivatives have been well documented in the literature. Here I present a Geometer’s Sketchpad based applet, which was used as part of a technologically enriched Calculus I course. Individual interviews with students conducted after this in-class activity show evidence of varied and powerful student problem solving strategies that emerged after participation in the activity.
2013
Denver, Colorado
We present how we are using tablets with an open-source online homework system to collect students’ written work to calculus problems. The new whiteboard feature captures all student written work in real time. An annotation tool has also been incorporated into the system. Through this tool, we are examining how students solve chain rule problems and what actions they take to correct their mistakes. At this poster, we will allow users to try out the annotation tool and provide results of how we have used it to date.
2013
Denver, Colorado
This poster illustrates and describes student thinking when responding to applied problems to relate two quantities that cannot be directly related by a single formula. Students who understood the meaning of the directive to define one quantity in terms of another, and who also conceptualized variables as representing varying values that a quantity can assume were successful in constructing a meaningful formula to relate the values of two quantities that cannot be directly related by a single formula. Students who failed to construct meaningful formulas during their solution process either held the view that a variable is an unknown value to be solved for or could not meaningfully interpret the directive of the problem statement.
2013
Denver, Colorado
This study evaluated program changes that required university supervisors to attend professional development and to use an observation protocol that reflected high quality mathematics pedagogy. The study used both qualitative and quantitative data to analyze the impact of the professional development. Qualitative data consisted of background information, observations, and interviews. Quantitative data included teacher candidates’ performance on the Reformed Observation Teaching Protocol (RTOP) and belief scores from the Mathematics Beliefs Instrument (MBI). This study was designed to fill in a gap in the literature to investigate the role university supervisors play in changing teacher candidates’ beliefs about mathematics and their instructional practice. By examining the effects of professional development, this study provided research about the type of support university supervisors need to challenge teacher candidates’ beliefs about mathematics.
2013
Denver, Colorado
The past few years have seen a substantial rise in the use and interest in a teaching and learning paradigm most commonly known as the flipped classroom. It is called the flipped class model because the whole classroom/homework paradigm is "flipped". In its simplest terms, what used to be classwork (the lecture) is done at home via teacher-created videos and what used to be homework (assigned problems) is now done in class.
2013
Denver, Colorado
Productive groupwork requires “group-worthy” problems (cf. Complex Instruction; Featherstone et al., 2011), which are nontrivial, require multiple competencies, and often have multiple solution paths (see also “problem aesthetic”; Schoenfeld, 1991). In contrast, groupwork with standard tasks often degenerates into one student “teaching” the other, because the tasks do not support collaborative learning. Short of full-blown collaborative groupwork, partner work is often a productive practice. In this poster, I discuss a particular type of partner work (peer- assessment) and the affordances required of tasks to make the activity productive. I dub these problems “peer-worthy.” Peer-worthy problems should satisfy a number of the following criteria; they: (1) are nontrivial, (2) have multiple solution paths, (3) require students to generate examples, and (4) involve explanation. In short, peer-worthy problems require students to generate mathematics in problem situations that allow for many different productive pathways, allowing students to deepen their understanding by making connections.