PROCEEDINGS OF THE 15TH ANNUAL CONFERENCE ON RESEARCH IN UNDERGRADUATE MATHEMATICS EDUCATION (Vol 1)
2012
Portland, Oregon
Number of pages: 531
Number of papers: 39
- STACY BROWN
- SEAN LARSEN
- KAREN MARRONGELLE
- MICHAEL OEHRTMAN
2012
Portland, Oregon
Number of pages: 531
Number of papers: 39
2012
Portland, Oregon
Combinatorial topics are prevalent in undergraduate curricula, and research indicates that students face difficulties when solving counting problems. The literature has not sufficiently addressed students' ways of thinking about combinatorial concepts at a level that enables researchers to understand how students conceptualize counting problems. In this paper, a model of students’ combinatorial thinking is presented that emphasizes relationships between formulas/expressions, counting processes, and sets of outcomes; additionally, the model is used to frame several examples of students’ reasoning about counting problems. The model serves as a conceptual analysis of students' thinking and activity related to counting, providing language to describe and explain aspects of students' counting activity. In this way, the model has practical implications, both for researchers (providing a lens through which to examine data on combinatorics education) and for teachers (providing an aid to instructional design based on student thinking).
2012
Portland, Oregon
While the unit circle is a central concept of trigonometry, students’ and teachers’ understandings of trigonometric functions typically lack connections to the unit circle. In the present work, we discuss a teaching experiment involving two pre-service secondary teachers that sought to characterize and produce shifts in their unit circle notions. Initially, both students experienced difficulty when given a circle that did not have a stated radius of one. The students relied on memorized procedures, including “unit-cancellation,” to relate the unit circle to given circles. In an attempt to foster more robust connections between novel circle contexts and the unit circle, we implemented tasks designed to foster thinking about a circle’s radius as a unit of measure. We report on the students’ progress during these tasks.
2012
Portland, Oregon
We document the evolving meanings that preservice elementary teachers ascribed to the sociomathematical norms of a mathematics class designed to foster mathematical sophistication. Specifically, we explore the developing meanings students gave to: a) their instructor’s request for general solutions to problems; b) classroom norms concerning problem solving behaviors; and c) their instructor’s expectation for mathematical justification. Finally, we document changes in student mathematical sophistication during the course of a semester, and illuminate the reflexive relationship between their mathematical sophistication and their interpretations of these classroom sociomathematical norms.
2012
Portland, Oregon
In this report we examine linear algebra students’ conceptions of inverse and invertibility. In the course of examining data from semi-structured clinical interviews with 10 undergraduate students in a linear algebra class, we noted that all the students said the result of composition of a function and its inverse is 1. We propose that this may stem from the several meanings of the word “inverse” or the influence of notation from linear algebra. In addition, we examined how students attempted to reconcile their initial incorrect predictions with their later computational results, and found that students who succeeded in this reconciliation used what we termed “do-nothing function” ideas. This analysis highlights several implications for classroom practice, including a possible method to help students develop object conceptions of function, as well as the need to pay more explicit attention to often- backgrounded notational issues.
2012
Portland, Oregon
One direction taken by course reform over the past few years has been the use of computer- assisted instruction, often applied to large-enrollment service courses, and justified in part by cost-effectiveness. Elementary algebra is typically taken by undergraduate students who do not place into a credit course. The goal of such a developmental algebra course has been to enhance students' “algebra skills,” for example, dealing procedurally with rational expressions. Higher-order thinking may be largely absent from such an approach. Our motivating question is “What approach maximizes the student’s chance to succeed in subsequent courses?” In view of our theoretical perspective that an inquiry-based approach enhances learning, a subsidiary question is “Is it effective to blend a focus on skills development (through computer-assisted instruction)with a focus on problem-solving (through cooperative group learning)?” Results of the analysis suggest that effectiveness is a matter of what student outcomes are valued, balanced against cost-effectiveness.
2012
Portland, Oregon
Counting problems ask students to compute the number of ways a certain set of requirements can be satisfied, and they are important in such mathematical subjects as probability, combinatorics, and abstract algebra, among others. Students are often taught to solve counting problems by looking for specific clues to help categorize the problems and identify solution strategies. In this study, we investigate how the wording of certain counting problems, specifically whether or not “order matters,” affects students' solution strategies. In particular, we gave students questions involving explicit statements as to whether or not order matters, some of which were intentionally misleading, and questions that do not contain such an explicit statement. Data was collected in the form of written responses and student interviews. The results show that many students do, in fact, rely heavily on such explicit statements about whether order matters, even when such statements are misleading.
2012
Portland, Oregon
This study uses reader-oriented theory and the analysis of example spaces to understand abstract algebra textbooks. Textbooks can lay the foundation for a course and greatly influence student understanding of the material. Multiple undergraduate abstract algebra texts were studied to investigate potential audiences of the books, the level of detail in explanations, examples, and proofs, and the overall material included in the book. Conclusions were drawn regarding some discrepancies between the intended reader and the implied reader and the appropriateness and differences among example spaces.
2012
Portland, Oregon
This paper will take a close look at the construction of a graphical image for reasoning with approximation in the context of Taylor series. In particular, it is a comprehensive case study of the genesis and evolution of an image created by one student, who draws extensively on other images and knowledge from calculus and physics to supplement gaps in his understanding of Taylor series and reason with Taylor series approximation tasks. His process resulted in a graphical image that was leveraged to build knowledge and reason with the situation, even while lacking key considerations that are central to an understanding of Taylor series. In this paper, we speak not only to considerations of a student’s understanding of this particular content. This work also provides a detailed examination of the processes of constructing a graphical image used for problem solving, for which it was necessary for the student to obtain and utilize evidence to amend that graphical image.
2012
Portland, Oregon
For more than a decade, capstone courses have been recommended as a way for pre-service secondary mathematics teachers to connect the mathematics they learn in college to the mathematics they will teach in their own classrooms. Yet little is known about the extent and nature of the implementation of these courses in the United States. This paper presents findings from a 2011 survey of U.S. colleges and universities that investigated whether and how capstone courses for pre-service secondary mathematics teachers have been implemented.
2012
Portland, Oregon
We have implemented a classroom experiment similar to a recent study in Physics (Deslauriers, Schelew, & Wieman, 2011): each of two sections of the same Calculus 1 course at a research-focused university were subject to an “intervention” week where a less- experienced instructor encouraged a much higher level of student engagement by design; we employed a modified quasi-experiment structure for our methods comparison with a Calculus 1 student population and with further steps to improve validity. Our instructional choices encouraged active learning (answering “clicker” questions, small-group discussions, worksheets) during a significant amount of class time, building on assigned pre-class tasks. The lesson content and analysis of the assessments were informed by existing research on student learning of mathematics, in particular the APOS framework. We report improved student performance, on conceptual items in particular, in the higher engagement section in both cases.
2012
Portland, Oregon
This is a first attempt to describe how students might develop a statistical symbol sense and what such a symbol sense entails. The paper first presents a genetic decomposition for a symbolic understanding of the arithmetic mean, the standard deviation and the standard error of the sample means of a sampling distribution by drawing on Sfard’s (1991) process-object duality. There is currently little research in which to ground a genetic decomposition and, as a result, the one presented here draws primarily upon the authors’ experience teaching statistics. It needs extensive testing and revision, but it is meant to serve as a starting point for future investigations into students’ development of understanding of statistical symbols. The paper ends by describing some important attributes of a symbol sense in statistics based upon Sfard’s (1991) framework and Arcavi’s (1994) description of a symbol sense in mathematics.
2012
Portland, Oregon
The literature is replete with evidence of student difficulty in abstract algebra. In response, innovative approaches for teaching group theory have been developed, yet no corresponding methods exist for ring theory. In an effort to simultaneously fill this void and build upon Larsen’s (2009) guided reinvention efforts in group theory, I conducted a study to investigate how students might be able to reinvent fundamental notions from introductory ring theory. Rooted in the theory of Realistic Mathematics Education, this paper reports on a teaching experiment conducted in nine sessions (up to 120 minutes each) with two students, neither of whom had prior exposure to abstract algebra. Using the construct of an emergent model, I show how these students formalized their intuitive understandings of linear equation solving and used them to reinvent the definitions of ring, integral domain, and field. In particular, the milestones of the reinvention process are identified and explicated.
2012
Portland, Oregon
Mathematics teachers at all levels are called to promote gender equity in their classrooms. During a college course on mathematics and gender, future K-12 teachers indicated their intentions to foster gender equity in their own classrooms. To investigate whether, and how, this resolve for equity persisted and influenced their own classroom practice, we present case study data of four former students from this course. Using a grounded approach (Glaser, 1992) to analyze classroom observations and semi-structured interviews, we report how closely the former students’ current descriptions of an equitable classroom align with their classroom practice, and with NCTM’s call for equity. We find that these teachers’ self- assessment of their success in achieving equitable classrooms appears to be accurate. We also highlight the learning experiences they feel most contributed to their views and practice regarding equity and equitable teaching. The results suggest possible implications for mathematics teacher preparation programs.
2012
Portland, Oregon
In this report we detail linear algebra students’ interpretations of linear transformations. Data for this analysis comes from mid semester, semi-structured problem solving interviews with 13 undergraduate students in linear algebra. We identified two main strategies used by students: 1) students used structural reasoning with entries of the matrix, columns of the matrix, and orientation of the shape and 2) students used operational reasoning through matrix and vector multiplication. We examine the patterns that emerged from student strategies, and discuss possible explanations for these patterns.
2012
Portland, Oregon
Calculus is an important tool for building mathematical models of the world around us and is thus used in a variety of disciplines, such as physics and engineering. These disciplines rely on calculus courses to provide the mathematical foundation needed for success in their courses. Unfortunately, due to the basal conceptions of what it means to understand calculus, many students leave their calculus course(s) with an understanding misaligned with what is needed in the follow-on discipline courses and are thus ill-prepared. By working with presumed experts (undergraduate mathematics and other discipline faculty members) to develop a small number of prototype tasks that elicit, document, and measure students’ understanding of a few calculus concepts they believe are essential to successful academic pursuits within their respective disciplines, this study documents how the faculty participants’ underlying conceptions about understanding changed and converged. Implications for calculus instruction and curriculum are mentioned.
2012
Portland, Oregon
There is a need to explain the relationship between teaching (classroom activities) and the resulting student learning, especially in advanced mathematics classes. This study represents a first attempt to describe the opportunity to learn present in an abstract algebra lecture, as an exemplar of advanced mathematics. Based on Weinberg and Wiesner’s (2010) work on the implied reader of a mathematics textbook, we describe the implied observer of a lecture as a bundle of codes, competencies and behaviors that are needed to make a meaningful interpretation of the lecture. We also use the framework to analyze an abstract algebra class and describe needed codes, competencies and behaviors of a student of that class. Finally, we close the paper by discussing both theoretical and methodological questions that remain and how those questions give rise to disagreements about how to interpret a component of the lecture by expert observers.
2012
Portland, Oregon
Students’ proof abilities were explored in the context of an inquiry-based learning (IBL) approach to teaching an introductory proofs course. IBL is a teaching method that focuses on student discussion and exploration in contrast to lecture-based instruction. Data was collected from three sections of an introductory proofs course, which included 70 students total. Data collection included a portfolio from each student, consisting of their work on every proof assigned throughout the course, as well as each student’s final exam. Contrary to previously published research related to courses taught in a more traditional lecture-based setting, this data analysis suggests that students developed an understanding of how to correctly use definitions and assumptions within the context of their proofs. Results also suggest that within the IBL setting, students generally organized their proofs in an efficient, thoughtful, and logical manner.
2012
Portland, Oregon
This paper aims to address students’ ways of thinking about the sets of elements being counted in enumerative combinatorics problems, known as solution sets. Fourteen undergraduates with no formal experience with combinatorics participated in individual task-based interviews in spring 2011. Open coding was used to identify students’ ways of thinking about solution sets. One category of ways of thinking which emerged from the data analysis involves holding an item constant and cycling through possible items for the remaining spots in order to generate all elements of the solution set. This category is known as Odometer thinking and two ways of thinking from this category, Standard Odometer and Wacky Odometer, are presented here. The conjectured Generalized Odometer way of thinking, which involves holding an array of items constant, is introduced as an extension of Wacky Odometer thinking.
2012
Portland, Oregon
We report on our work to build an interculturally aware theory for pedagogical content knowledge (PCK) in the context of teacher leadership. The effort is based on existing and continuing work on developing pre- and in-service teacher classroom PCK and intercultural competence. The RUME session focused on two discussion topics. Discussion Item 1: How do we identify and capture evidence of what might be called “teacher leader pedagogical content knowledge” in interculturally aware ways? Discussion Item 2: What question formats (for written assessments, surveys, interviews) might be productive for eliciting information from teacher leaders about their awareness of and attention to the intercultural aspects of mathematics instruction? ...of mathematics itself?...of teacher leadership?
2012
Portland, Oregon
This work aims to establish a new theoretical construct, mathematical activity for teaching – the mathematical work teachers engage in while teaching. Given this new construct, it is possible to investigate relationships and patterns of interaction between teacher activity and that of their students. By analyzing the classroom video data of mathematicians implementing an inquiry- oriented abstract algebra curriculum I was able to identify four patterns of interaction between mathematical activity for teaching, pedagogical activity, and student mathematical activity. My analysis shows a variety of ways in which teachers’ mathematical and pedagogical activity may interact– with some episodes illustrating ways in which these two forms of activity may be somewhat disjoint and other episodes illustrating ways in which these two forms of activity may be tightly integrated.
2012
Portland, Oregon
This report is based on work completed within an ongoing project to develop a calculus course which serves as the foundation for the mathematical education of STEM-focused elementary teachers at a large southeastern university. In the process of designing and implementing the course materials, several research-based activities have been developed, tested and refined. In this paper we discuss how we used a design research approach to create and implement an activity that introduces the concept of limit of a sequence using popular characters from Sesame Street. We report on the first two design cycles in the ongoing design of this activity and discuss the modifications made in both the broad learning goals and the activity drafts.
2012
Portland, Oregon
This study explores student understanding of the symbolic representation system in statistics. Furthermore it attempts to describe the relation between student understanding of the symbolic system and statistical concepts that students develop as the result of an introductory undergraduate statistics course. The theory, drawn from the notion of semantic function that links representations and concepts seeks to expand the range of representations considered in exploring students’ statistical proficiencies. Results suggest that students experience considerable difficulty in making correct associations between symbols and concepts; that they describe the relationship as seemingly arbitrary and that they are unlikely to understand statistics as quantities that can vary. Finally, this study describes students’ need for robust knowledge of preliminary concepts in order to understand the construct of a sampling distribution.
2012
Portland, Oregon
This study is a teaching experiment investigating the effect of reading assignments in Calculus II on student performance. Students from a test section and a control section of Calculus II taught during a summer semester were compared. Both sections used traditional lecture methods, the same on-line homework assignments, and common exams. In one section the students completed additional reading assignments with open-ended questions and in-class quizzes evaluating reading comprehension. The study compares student performance on the common exam covering series convergence and the level of writing fluency in student’s written arguments on this exam. In addition, four interviews from comparable students, two in each section, were conducted to investigate the ways in which they read, comprehend, and create a series convergence argument.
2012
Portland, Oregon
The purpose of this study was to gain insight into how exposure to hands-on and computer resampling methods affected a statistically naïve student’s emergent understandings of statistical inference. In this study, simulation design activities provided a vehicle for engaging a student with the core ideas of hypothesis testing. The results highlight challenges the student experienced in coordinating the components of the logic into a coherent scheme of ideas and sheds light on aspects of engagement which need to be emphasized in order to resolve the inherent conceptual difficulties associated with reasoning that invokes a modus tollens-like argument. Moreover, I report on a heuristic the student used to make his inferential decisions—one that does not produce correct inferences. I’ve termed this the “similarity heuristic” because of a specific similarity relationship the student would look for and then use as a method for rejecting or not rejecting the hypothesis being tested.
2012
Portland, Oregon
In this paper we present a case study of an individual student who consistently used semantic reasoning to write proofs in calculus but infrequently used semantic reasoning to write proofs in linear algebra. We argue that the differences in these reasoning styles can be partially attributed to this student’s familiarity with the content, the teaching styles of the professors who taught him, and the time he was given to complete the tasks. These results suggest that there are factors that have been ignored in previous research, including domain, instruction, and methodological constraints, that researchers should consider when ascribing to students a proving style.
2012
Portland, Oregon
This study combines interview data and observation data to investigate the teaching practices of mathematics faculty members when teaching upper-division proof-based undergraduate mathematics courses. Four case studies of faculty members at a large research institution who were teaching in different mathematics content areas are used to construct a model describing the ways in which examples are used to motivate and support proof presentations in class.
2012
Portland, Oregon
This study explores features of university calculus students' discourses on the derivative using a communicational approach to cognition. The data was collected from a survey and interviews in three calculus classes at a public Midwestern university. During the interview, 12 students explained their solution processes on the survey problems. The analysis of interviews focuses on students' descriptions about the derivative and the relationships between a function, the derivative function, and the derivative at a point. The results show that their descriptions were closely related to how they think about the derivative as a number and as a function. A common description of the derivative as a tangent line, which is a point-specific object but also a function defined on an interval, was identified. This description was closely related to their use of the word, "derivative" for both "the derivative function" and "the derivative at a point."
2012
Portland, Oregon
In this study, seven mathematicians and seven undergraduates were asked to read and summarize mathematical proofs that they read to investigate which ideas they consider to be important in a proof. Mathematicians’ ideas consisted of a) the overarching goals of the proof, b) the ideas they found novel or unfamiliar, c) theorems or facts used in the proof, d) encapsulations of inferences as applications to general methods, e) diagrams, or f) cues for reconstructing the proof. Students did not mention the goals, important theorems or facts, or the methods as being important, but some focused on whether the proof was indirect or direct. We present a model for accounting for many of the different types of ideas found important by mathematicians.
2012
Portland, Oregon
This paper reports what six mathematicians did when they came to impasses while constructing proofs on an unfamiliar topic, from a set of notes, alone, and with unlimited time. Detailed information is given on two of the mathematicians. By an impasse, I mean a period of time during the proving process when a prover feels or recognizes that his or her argument has not been progressing and that he or she has no new ideas. What matters is not the length of time but its significance to the prover and his or her awareness thereof. I point out two kinds of actions these mathematicians took to recover from their impasses: one kind relates directly to the ongoing argument, while the other kind consists of doing something unrelated, either mathematical or non-mathematical. Data were collected using technology and a new technique being developed to capture individuals’ autonomous proof constructions in real-time.
2012
Portland, Oregon
To understand the mathematical transition students make between secondary school and the university requires an in-depth look at the mathematical topics students learn at the time of this transition and the contextual, institutional changes that simultaneously occur. This report explores how linear algebra students at both the secondary school and university in Germany understand vectors and linear independence and dependence in the course of video-recorded, think-aloud problem-solving interviews. Analysis of these interviews indicates not only differences in mathematical content and sophistication between secondary school and university students, but also in students’ disposition, particularly towards new mathematical experiences. A look at more informal data about the various institutional environments, the Gymnasium and the University, provides a potential reason for these differences. This report concludes with a discussion on how to create a blended analysis of these individual understandings and dispositions and their relationship with the institutional context as a better means of understanding the transition to university-level mathematics.
2012
Portland, Oregon
As students progress through the college mathematics curriculum, enter graduate school and eventually become practicing mathematicians, reading mathematics textbooks and journal articles appears to comes easier and these readers appear to gain quite a bit from reading mathematics. This preliminary study was designed to help us begin to understand how more advanced readers of mathematics read for understanding. Three faculty members and three graduate students participated in this study and read from a first year graduate textbook in an area of mathematics unfamiliar to each of them. The reading methods of the faculty level mathematicians were all quite similar and were markedly different from all the students the researcher has encountered so far, including the more advanced students in this study. A proposed Mathematics Reading Framework is given based on this study and years of observations of first-year undergraduate students reading their mathematics textbooks.
2012
Portland, Oregon
Using a theoretical perspective of embodied cognition, we explored how six experts integrated metaphors to reason and communicate about arithmetic and analytic complex variables concepts. We found that experts who displayed evidence of reification of a complex variables concept or had a need to use a concept imparted their sense of understanding through enacted metaphors. These metaphors were often invented or reinterpreted, based on personal experiences and created to convey nuances of the experts’ understanding to students. The experts appeared conscientious of using metaphors relevant to their own students. This research may support practitioners’ efforts to create opportunities for students to create or reinterpret experts’ metaphors into personally meaningful metaphors that both capture important mathematical concepts accurately and align with their own understandings, experiences, and culture. Further research may investigate how technology may serve as a tool for such an endeavor.
2012
Portland, Oregon
Despite the consensus among mathematics educators that prior knowledge is essential to student success, calculus instructors vary widely in their assessment of prior knowledge errors found on student assignments and exams. This phenomenological study of five calculus instructors at a large research institution investigated the influence that instructor belief systems have on the consistency of grading across instructors. The results showed that the intricacies of instructor sensible systems play a vital role in the assessment of student errors.
2012
Portland, Oregon
Historically grounded in Oliver Byrne's reworking of Euclid's Elements, and based on a student- generated proof, we investigate the use of coloring to enhance geometry proofs. Charlotte Knight, an undergraduate mathematics major enrolled in a modern geometry course, regularly employed coloring techniques as a tool in her proof-writing. We met for a single semi-structured, task-based interview to discuss Charlotte’s use of coloring in her organization and understanding of geometry proofs. Results indicate that Charlotte’s use of diagrams is closely related to her construction of a proof. In particular, her use of color serves several purposes: (1) as an organizational tool to connect her diagrams to the content of her proofs, (2) to enhance her understanding of the proof she is writing, and (3) to illustrate relationships within her diagrams and proofs. We believe this small study has particularly interesting pedagogical implications at the post-secondary level as well as for K-12 mathematics instruction.
2012
Portland, Oregon
In this paper we develop the notion of a hypothetical collective progression (HCP). We offer this construct as an alternative to the construct of hypothetical learning trajectory in order to (a) foreground the mathematical development of the collective rather than that of individuals, and (b) highlight the integral role of the teacher within this development. We offer an abbreviated example of an HCP from introductory linear algebra based on the “Italicizing N” task sequence, in which students work to generate and combine matrices that correspond to geometric transformations specified within the problem context. In particular, we describe the ways in which the HCP supports students in developing and extending local “matrix acting on a vector” views of matrix multiplication (focused on individual mappings of input vectors to output vectors) to more global views in which matrices are conceptualized in terms of how they transform a space in a coordinated way.
2012
Portland, Oregon
In an influential article, Rowland (2001) suggested “generic proofs” might improve students understanding and appreciation of the proofs that they study. In this paper, we present a qualitative and quantitative study exploring how well students understand generic proofs. In a qualitative study, we found students generally have positive opinions about generic proofs, believing generic proofs can be a useful tool for improving understanding. In a larger quantitative study, we conducted a randomized experiment where we assessed how well undergraduates understood the same proof when presented traditionally and generically. Those who read the proof generically performed somewhat (although not statistically reliably) better on questions applying the ideas of the proof to specific examples but statistically reliably worse on other types of assessment questions.
2012
Portland, Oregon
In this paper, we investigate how and why mathematicians read the published proofs of their colleagues. Based on qualitative interviews with nine mathematicians, we posit that mathematicians understand proofs in three ways: as cultural artifacts with a social history, as a sequence of inferences, and as the application of methods. Each type of understanding is based, at least in part, on non-deductive evidence. A survey with 118 mathematicians confirms the generality of these findings. We conclude by arguing that more comprehensive frameworks for how mathematicians gain conviction are needed.
2012
Portland, Oregon
I report on the classroom mathematical practices that developed in a mathematics content course for prospective elementary teachers. The course focused on number and operations and was intended to promote number sense development. Instruction was guided by a local instruction theory for number sense development, which has been described previously. The present report focuses on the classroom mathematical practices that emerged and became established in the class during a recent teaching experiment. The actual learning route identified informs elaboration and refinement of the local instruction theory and sheds light on prospective teachers’ number sense development.
2012
Portland, Oregon
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 results led to a classification of student responses into properties, computations and a series of five interrelated clusters of metaphorical expressions. We see this classification as providing richness and nuance to existing literature on students’ conceptions of function. In addition, we are finding these categories helpful in describing the compatibilities and distinctions in student understanding of function and linear transformation.