Abstracts - Contributed Reports

Title: Making the familiar strange: An analysis of language in postsecondary calculus textbooks then and now

Veda Abu-Bakare

Simon Fraser University

Abstract: Three calculus textbooks covering a span of about 40 years were examined to determine whether and how the language used has changed given the reform movement and the impetus to make mathematics accessible to all. Placed in a discourse analytic framework using Halliday’s (1978) theory of functional components –ideational, interpersonal and textual, and using the exposition of the concept of a function as a unit of comparison, the study showed that language is an integral indicator of the author’s view of mathematics and an important factor for textbook adoption in the pursuit of student success.

Keywords: discourse analysis, calculus textbooks, language of mathematical discourse

The Effectiveness of Blended Instruction in Postsecondary General Education

Mathematics Courses

Anna Bargagliotti, Fernanda Botelho, Jim Gleason, John Haddock, Alistair Windsor

University of Memphis

Abstract: Despite best efforts, hundreds of thousands of students are not succeeding in postsecondary general education mathematics courses each year. Low student success rates in these courses are pervasive, and it is well documented that the nation needs to improve student success and retention in general mathematics.

    Using data from 11,970 enrollments in College Algebra, Foundations of Mathematics, and Elementary Calculus from fall 2007 to spring 2010 at the University of Memphis, we compare the impact of the Memphis Mathematics Method (MMM), a blended learning instructional model, to the traditional lecture teaching method on student performance and retention.

    Our results show the MMM was positive and significant for raising success rates particularly in Elementary Calculus. In addition, the results show the MMM as a potential vehicle for closing the achievement gap between Black and White students in such courses.

Keywords: Calculus, general education mathematics, classroom research, teaching experiment

Obstacles to Teacher Education for Future Teachers of Post-Secondary Mathematics

Mary Beisiegel

Harvard Graduate School of Education

Abstract: The purpose of this study was to uncover issues and difficulties that come into play as mathematics graduate students develop their views of their roles as university teachers of mathematics. Over a six-month period conversations were held with mathematics graduate students exploring their experiences and perspectives of mathematics teaching. Using hermeneutic inquiry and thematic analysis, the conversations were analyzed and interpreted with attention to themes and experiences that had the potential to influence the graduate students’ ideas about and approaches to teaching. Using Lave and Wenger’s notion of legitimate peripheral participation, themes that are explored in this paper are the replication of mathematics teaching practice and identity, and resulting feelings of resignation. It is hoped that this research will contribute to the understanding of teaching and learning in post-secondary mathematics as well as provide guidance in structuring post-secondary teacher education in mathematics.

Keywords: post-secondary, mathematics graduate students, community of practice, teacher identity

Designing and Implementing a Limit Diagnostic Tool

Timothy Boester

Wright State University

Abstract: The purpose of this study is to create and utilize a tool that evaluates students’ comprehension of the logical structure and implications of the formal definition of limit. This study continues the trajectory of recent limit research involving classroom-based interventions that reveal student metaphors and conceptions (Boester, 2010; Oehrtman, 2009; Roh, 2008, 2010). The diagnostic tool, based on seven concepts embedded in the formal definition, uses a set of delta/epsilon diagrams that students must explain, either accepting them as correct, or augmenting them to make them correct. The assessment was used after giving students in a conceptually-based calculus class a problem meant to introduce the logical structure of the formal definition. While students did not spontaneously show many of the concepts based on the problem alone, an interview protocol following the assessment prompted the students to rethink the implications of the problem, thus promoting the missing concepts.

Keywords: calculus, limit, assessment, conceptual decomposition

Assessing Active Learning Strategies in Teaching Equivalence Relations

Jim Brandt

Southern Utah University

Abstract: In this study, students in transition-to-proof courses were introduced to equivalence relations either using a traditional classroom lecture or using small group learning activities. Students’ understanding of equivalence relations were then assessed using task-based interviews aimed at assessing concept image, concept definition, as well as concept usage in terms of writing proofs. The students involved in small group activities made stronger connections to partitions and were more successful in writing proofs. In addition, the concrete learning activities gave many participants a strong prototypical example that aided in encapsulating the essential features of an equivalence relation.

Key words: transition to proof, classroom teaching experiment, concept definition

Surveying Mathematics Departments to Identify Characteristics of Successful Programs in College Calculus

Marilyn Carlson

Arizona State University

Chris Rasmussen

San Diego State University

David Bressoud

Macalester College

Michael Pearson

The Mathematical Association of America

Sally Jacob

Scottsdale Community College

Jess Ellis & Eric Weber

San Diego State University

Abstract: This report describes results of a process to develop a suite of instruments to assess calculus instruction at universities and colleges across the nation. The instruments were developed as part of a national study of the Mathematical Association of America to identify characteristics of successful programs in college calculus. The report will focus on the development of the student pre- post-surveys to illustrate the development process. We describe the theoretical framework consisting of a broad taxonomy of primary variables that guided the development of these surveys. We then describe the research we conducted to develop and validate specific survey items to assure that they assess the intended taxonomy variables and are consistently interpreted. This report contributes knowledge of the primary variables that affect success in calculus and insights into the processes involved in survey design and validation.

Keywords: calculus, survey design, instrument validation

Translating Definitions Between Registers as a Classroom Mathematical Practice

Paul Dawkins

Northern Illinois University

Abstract: Many have noted that mathematical definitions constitute a duality between a category of objects and the definition that delineates that category (Alcock & Simpson, 2002; Edwards & Ward, 2008; Mariotti & Fischbein, 1997; Tall & Vinner, 1981). Prior research has readily identified conflict between these two elements of students’ conceptions, but reliable mechanisms for explaining and resolving such conflicts are still forthcoming. The present study observed a real analysis classroom in which the duality was embodied and addressed directly in class dialogue and activities. Particularly, three linguistic registers (metaphorical, common, and symbolic) arose to express different aspects of the definitions themselves (conceptual and formal). Translation across these registers provided a mechanism by which some students were able to segue their concept image and concept definitions successfully. Some students corrected errors in their concept image as a result of this practice.

Keywords: mathematical defining, real analysis, translating definitions, harmonisation, classroom communication

The Role of Conjecturing in Developing Skepticism: Reinventing the Dirichlet Function

Brian Fisher

Pepperdine University

Abstract: The study presented in this research report was born out of the desire to develop pathways for students from informal to formal modes of thinking. The data from this report stems from a series of small group interviews using a process of guided reinvention incorporating frequent student conjectures in order reinvent the definitions of limit and continuity. During these interviews, students used the practice of skepticism in order to suspend judgment on various mathematical statements. In the process of exploring a developed conjecture, the students’ suspension of judgment allowed them to alter their initial beliefs about the nature of continuity and their interactions with functions.

Keywords: Conjecturing, Skepticism, Calculus, Continuity, Dirichlet Function

Toulmin Analysis: A Tool for Analyzing Teaching and Predicting Student Performance in Proof-Based Classes

Tim Fukawa-Connelly

University of New Hampshire

Abstract: This paper provides a method for analyzing undergraduate teaching of proof- based courses based on Toulmin’s model of argumentation. The paper then describes how that analysis can be used as a predictor of subsequent student proof-writing performance and shows that the predictions are reasonable approximations of students’ subsequent proof-writing. The method of analysis was developed via research in a lecture-based abstract algebra class, it has application, to any lecture-based, proof- intensive course. This method provides one possible way to directly link classroom teaching activities to subsequent student performance that would force instructors to assume more responsibility for their students’ demonstrated end-of-course performance.

Keywords: proof, Toulmin analysis, abstract algebra, classroom research

A Multi-Strand Model for Student Comprehension of the Limit Concept

Gillian Galle

University of New Hampshire

Abstract: In analyzing interview transcripts to assess student understanding of limits for first year calculus students, the application of the 7 Step Genetic Decomposition created by Cottrill, et. al. (1996) indicated that the interviewed students possessed no higher than a 3rd step understanding. Despite an inability to clearly articulate their understanding in terms of the expected lexicon, several students were able to create valid examples and counterexamples while justifying their answers. This suggests that these students possessed a better understanding of the limit concept than they were able to articulate. Thus, this study concludes that there exists additional criterion that should be taken into account in order to accurately diagnose student understanding of the limit concept. In particular a model for student understanding of limits should contain strands reflecting the student’s method for solving a problem involving limits, the student’s justification for the solution, and the applicability of the student’s method and justification within the context of the problem.

Keywords: limits, student understanding, calculus, interview methodology

Authority in the Negotiation of Sociomathematical Norms

Hope Gerson and Elizabeth Bateman

Brigham Young University

Abstract: The study of sociomathematical norms initiated by Yackel and Cobb (1996) has become a popular way to make sense of the complexity of mathematical activity in the classroom. Levenson, Tirosh, and Tsamir (2009) found that teachers and students did not share the same interpretations of teacher-initiated sociomathematical norms. In this study we explore the role authority plays in the negotiation and legitimization of student-initiated soiciomathematical norms. We found that mathematical authority legitimized through mathematical argument and justification played a major role in the negotiation of sociomathematical norms in an inquiry based, university honors calculus II course. We suggest creating an environment where students rather than teachers are encouraged to initiate and negotiate sociomathematical norms will lead to better agreement on the expectations. We also believe that if teachers introduce a sociomathematical norm, they should be aware of the potentially obstructive role their non- mathematical authorities may play in the negotiation process.

Keywords: authority, calculus, inquiry, sociomathematical norms

Student Understanding of Eigenvectors in a DGE: Analysing Shifts of Attention and Instrumental Genesis

Shiva Gol Tabaghi

Simon Fraser University

Abstract: This study examines the potentialities of the theory of instrumental genesis and shifts of attention in analysing students’ evolving understanding as they interacted with a dynamic geometry representation of eigenvectors and eigenvalues. Although the former theory provides a framework to analyse students’ interactions with tools and transformation of tools into instruments, it makes an assumption about the role of instrument in cognitive development. According to Verillon and Rabardel (1995), the founders of the theory, the role of instrument in cognitive development is a sensitive point. I thus explore the complementary use of the theory of instrumental genesis with the theory of shifts of attention to enable an analysis of students’ cognitive development in a digital technology environment.

Keywords: Technology, linear algebra, instrument and attention

University Students’ Understanding of Function is Still a Problem!

Zahra Gooya & Mehdi Javadi Shahid

Beheshti University, Iran

Abstract: A research study was designed using the conceptual model consisting two cells of concept images and concept definition developed by Vinner (1983) and has been used by many researchers since then, to investigate students’ understanding of different concepts of calculus. A related literature review made us believe that students’ understanding of function as one of the pillar of calculus is still problematic. 53 first year university students participated in this study that its purpose was to shed more light into the students’ understanding of function in terms of their concept images and concept definitions. The study showed that the most common concept images of function among the students were having a rule, and using a machine as a metaphor for a function. The study also indicated that a concept image of having a rule for each function acted as an obstacle for students to understand the concept definition of function.

Key words: Conceptual Model, Concept Image, Concept Definition, Function, Calculus.

The Limit Notation: What is it a Representation of?

Beste Güçler

University of Massachusetts Dartmouth

Abstract: Student difficulties with the notion of limit are well-documented by research. These studies suggest that students mainly realize limits through dynamic motion, which can hinder further realizations of the concept. Some studies mention the overemphasis on the dynamic aspects of limits in classrooms but research on the teaching of limits is quite scarce. This work investigates the development of discourse on limits in a beginning-level undergraduate calculus classroom with a focus on the limit notation and uses a communicational approach to learning, a framework developed by Sfard (2008). The study explores how the limit notation is utilized by an instructor and his students and compares the realizations of limit in their discourse. The findings indicate that the shifts in the instructor's word use when talking about the notation supported students' realizations of limit as a process despite the frequency with which the instructor talked about limit as a number in his discourse.

Keywords: teaching of calculus, limits, the limit notation, discourse analysis

Student Outcomes from Inquiry-Based College Mathematics Courses: Benefits of IBL for Students from Under-Served Groups

Marja-Liisa Hassi, Marina Kogan, Sandra Laursen

University of Colorado at Boulder

Abstract: Our large, mixed-methods study examines cognitive and affective outcomes of inquiry-based learning (IBL) in a variety of undergraduate mathematics courses at four universities. Student outcomes are measured by pre/post-survey items, self-reported gains and historical transcript data. Students in IBL courses report higher cognitive and affective gains than do non-IBL students. IBL students also report increase in motivation and interest, whereas non-IBL students’ motivation drops after mathematics courses. The historical transcript data also shows IBL students’ higher interest compared to their non-IBL peers. These benefits of IBL instruction are especially important for women and low achieving students, who are often under-served by the traditional college mathematics courses. Our findings suggest that IBL instructional methods support positive learning outcomes in various groups of students, including those under-served and under-supported by the traditional college mathematics courses.

Keywords: inquiry-based learning, mixed methods, learning outcomes, undergraduate students

On exemplification of probability zero events

Simin Chavoshi Jolfaee

Simon Fraser University

Abstract: In this study the example space of pre-service secondary teachers on probability zero events is examined. Different aspects of such events as perceived by the respondents are discussed and their perception of impossible events versus improbable is studied. The examples are categorised in terms of the type of sample space and once again categorised in terms of how do they fit the classic definition of probability. The role of measure theory to approach probability is briefly looked at via the examples. Meanwhile the participants’ understanding of “more complicated” is explored and different ways they add complexity to their examples are analysed.

Keywords: example space, classic probability, impossible events.

Improving the Quality of Proofs for Pedagogical Purposes: A Quantitative Study

Yvonne Lai

University of Michigan

Juan-Pablo Mejia Ramos &

Keith Weber

Rutgers University

Abstract: In university mathematics courses a primary means of conveying mathematical information is by mathematical proof. A common suggestion to realize learning goals related to proof is to increase the quality of the proofs that we present to students. The goal of this paper is to investigate evidence related to the question: What changes to a proof do mathematicians believe will improve the quality of a proof for pedagogical purposes?

    We present quantitative findings that corroborate hypotheses generated by a qualitative study of features of proofs that mathematicians find pedagogically valuable. Our work examines hypotheses related to typesetting, brevity, and the framework of a proof. One of our findings suggests that there is not a consensus among mathematicians what level of justification is desirable or necessary for the purposes of teaching undergraduates, though there may be common themes in the warrants they give for the level chosen.

Key words: proof evaluation, mathematicians, proof revision, quantitative study.

Putting Research to Work: Web‐Based Instructor Support Materials for an Inquiry Oriented Abstract Algebra Curriculum

Sean Larsen, Estrella Johnson & Travis Scholl

Portland State University

Abstract: For several years we have been researching students’ and instructors’ experiences with an inquiry‐oriented group theory curriculum. This research has resulted in a number of insights; include findings that may be significant only for instructors and students engaged with this specific curriculum, as well as findings that appear to have broader significance. We are putting this research to work as we develop web‐based instructor support materials to accompany the curriculum. These materials include 1) information about the rationale for each task/sequence, 2) insights about student thinking related to the task/sequence, and 3) discussion of task/sequence implementation considerations. In this presentation we will share some of our findings (both general and specific) and illustrate how we have incorporated these findings into the web‐based instructor support materials in the form of text, video‐clips, and images culled from our research efforts.

Keywords: abstract algebra, curriculum, teaching, student thinking

Students’ Modeling of Linear Systems: The Car Rental Problem

Christine Larson

Vanderbilt University

Michelle Zandieh

Arizona State University

Abstract: In this talk, we characterize the nature of students’ thinking about real-world problem situations that mathematicians might choose to reason about using ideas from linear algebra such as eigen theory, matrix equations, and/or systems of linear equations. We documented students working in groups on the “Car Rental Problem,” a task that our research team specifically designed to elicit students’ thinking about problem contexts that might be modeled in the aforementioned ways. We will describe the models students create to reason in this problem context, illustrating the variety in the final solutions of four different groups of linear algebra students and discuss the trends that appeared across the four groups as they worked toward their solution. Our analysis follows Lesh & Kelly’s (2000) multi-tiered approach, and will focus on the mathematical topic areas drawn upon, the inscriptions created, and the quantitative reasoning that the students engaged in as they worked toward a solution.

Key Words: Linear Algebra, Modeling, Student Thinking

Navigating the Straits: Critical Instructional Decisions in Inquiry-Based College Mathematics Classes

Sandra Laursen, Marja-Liisa Hassi, and Anne-Barrie Hunter

University of Colorado at Boulder

Abstract: Inquiry-based learning (IBL) approaches engage college mathematics students in analyzing and solving problems and inventing and testing mathematical ideas for themselves. But to effectively apply IBL teaching methods, instructors must make good decisions both in planning their syllabus, assignments, and assessment before the term begins, and in the moment, as they monitor classroom progress, manage interpersonal dynamics, and decide what to do when things do not go as planned. Using interview data from 40 IBL instructors at four campuses, including graduate teaching assistants and faculty at a range of experience levels, we identify critical instructional decisions that can affect the success of IBL classes. We describe why these decisions are more salient in IBL classrooms than in those using lecture-based methods, and we examine patterns in instructors’ ability to identify these issues for themselves and suggest appropriately nuanced solutions to common IBL classroom dilemmas.

Student perceptions of an explicitly criterion referenced assessment activity in a differential equations class

Dann G. Mallet and Jennifer Flegg

Queensland University of Technology

Abstract: This report presents the findings of a study into the perceptions held by students regarding the use of criterion referenced assessment in an undergraduate differential equations class. Students in the class were largely unaware of the concept of criterion referencing and of the various interpretations that this concept has among mathematics educators. Our primary goal was to investigate whether explicitly presenting assessment criteria to students was useful to them and guided them in responding to assessment tasks. The data and feedback from students indicates that while students found the criteria easy to understand and useful in informing them as to how they would be graded, it did not alter the way the actually approached the assessment activity.

Keywords: differential equations, assessment experiment, criterion referenced assessment

Reaching out to the Horizon: Teachers’ use of Advanced Mathematical Knowledge

Ami Mamolo & Rina Zazkis

Simon Fraser University

Abstract: This paper explores teachers’ use of advanced mathematical knowledge (AMK) – that is, the knowledge acquired during undergraduate university or college mathematics courses. In particular, our interest is in the use of advanced mathematical knowledge as an instantiation of knowledge at the mathematical horizon (KMH). With this tie to undergraduate mathematics education, we re-conceptualize the notion of knowledge at the mathematical horizon and illustrate its value with excerpts from instructional situations.

Keywords: advanced mathematical knowledge; horizon knowledge; group theory; calculus

Students’ Reinvention of Formal Definitions of Series and Pointwise Convergence

Jason Martin

Arizona State University

Michael Oehrtman

University of Northern Colorado

Kyeong Hah Roh

Arizona State University

Craig Swinyard

University of Portland

Catherine Hart-Weber

Arizona State University

Abstract: The purpose of this research was to gain insights into how calculus students might come to understand the formal definitions of sequence, series, and pointwise convergence. In this paper we discuss how one pair of students constructed a formal ε-N definition of series convergence following their prior reinvention of the formal definition of convergence for sequences. Their prior reinvention experience with sequences supported them to construct a series convergence definition and unpack its meaning. We then detail how their reinvention of a formal definition of series convergence aided them in the reinvention of pointwise convergence in the context of Taylor series. Focusing on particular x-values and describing the details of series convergence on vertical number lines helped students to transition to a definition of pointwise convergence. We claim that the instructional guidance provided to the students during the teaching experiment successfully supported them in meaningful reinvention of these definitions.

Keywords: Reinvention of Definitions, Series Convergence, Pointwise Convergence, Taylor Series

Exploring Collaborative Concept Mapping In Calculus

David Meel

Bowling Green State University

Abstract: For the past 25 years, concept mapping has been considered primarily a solitary assessment instrument where individuals build an external illustration representative of some notion of held concept images. This study explored the role of concept mapping to collaborative settings and what discourse is generated as Calculus students engage with their individual concept maps to construct a map representative of the group's collective perceptions of calculus concepts. By using adjacency matrices to explore the structure of the concept maps, the study compared individual maps against one another, against aggregated maps and finally against the collaborative concept maps. In particular, the study identified differences in structure and emphasis across the students' maps and identified different discourse models generated by various methodologies employed to generate the collaborative maps. These observations were triangulated with student utterances during the collaborative concept mapping activity and reflections on both the individual and collaborative concept mapping activities.

Keywords: Concept mapping, Collaborative, Calculus, Concept Image

An Analysis of Examples in College Algebra Textbooks for Community Colleges: Opportunities for Student Learning

Vilma Mesa, Heejoo Suh, Tyler Blake, & Tim Whittmore

University of Michigan

Abstract: We analyzed 348 examples from sections on graphing, logarithmic and exponential functions in seven College Algebra textbooks used in community colleges to have a deeper understanding of instructional materials available for students by disclosing what textbooks are offering. We analyzed (1) their cognitive demand, (2) the strategies available to control solutions, (3) the types of responses, and (4) the use of representations. We found that 10% of examples in these textbooks were at a high-level of cognitive demand and that strategies to control solutions were not frequently offered. About 50% of examples expected answer only. Symbols and numbers were the most common representations in the statement and solution, respectively. Given that students rely on examples when they meet difficulties doing homework, shortcomings highlight the need to modify the examples or supplement from outside sources if we want the textbooks to be more useful to students.

Keywords: college algebra, community colleges, textbook analysis, opportunity to learn.

Promoting Success in College Algebra by Using Worked Examples in Weekly Active Group Work Sessions

David Miller and Matthew Schraeder

West Virginia University

Abstract: At a research university near the east coast, researchers have restructured a College Algebra course by formatting the course into two large lectures a week, an active recitation size laboratory class once a week, and an extra day devoted to active group work called Supplemental Practice (SP). SP was added as an extra day of class where the SP leader has students to work in groups on a worksheet of examples and problems, based off of worked example research, that were covered in the previous week’s class material. Two sections of the course was 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 a question and answer session. The experimental group significantly outperformed the control on a variety of components in the course, especially when SP attendance was factored into the analysis.

Keywords: College Algebra, Cognitive Science, Worked Examples, Large Lecture Supplemental Sessions

Differences in Beliefs and Teaching Practices between International and U.S. Domestic Mathematics Teaching Assistants

Minsu Kim

The University of Oklahoma

Abstract: International Mathematics Teaching Assistants (MTAs) and U.S. domestic MTAs are an indispensable part of mathematics departments regarding teaching a substantial portion of undergraduate students. Because MTAs’ beliefs are significant to their pedagogical methods, this study examines the contrast between international and U.S. domestic MTAs’ beliefs and teaching practices. This research aims to answer the following questions: 1) What are the differences in beliefs and teaching practices between international and U.S. domestic MTAs? and 2) How are MTAs’ different teaching practices shaped by their beliefs? The goals of this study are to help understand international and U.S. domestic MTAs’ different approaches to education. The results indicate significant differences between the two groups centered on how they taught students to understand definitions and problems and how they motivated students to learn mathematics. The findings also describe MTAs’ beliefs in relationship with their teaching practices.

Keywords: U.S. domestic mathematics teaching assistants (MTAs), international mathematics teaching assistants (MTAs), beliefs and teaching practices

Relationships between Quantitative Reasoning and Students’ Problem Solving Behaviors

Kevin C. Moore

University of Georgia

Abstract: This presentation reports on the results of a study into precalculus students’ reasoning when solving novel problems. The study intended to identify students’ mental actions that support or hinder their ability to provide meaningful and correct solutions, while also characterizing the role of quantitative reasoning in the students’ solutions. Analysis of clinical interviews with each student revealed that a student’s propensity to reason about quantities and a problem’s context significantly influenced his or her problem solving approach. Students who spent a significant amount of time orienting to a problem by identifying quantities and relationships between quantities leveraged the resulting mental images throughout their problem solving activity. Contrary to this, students who focused on recalling procedures and performing calculations spent little time reasoning about a problem’s context and encountered difficulty providing meaningful and correct solutions. These findings offer insights into the relationship between students’ reasoning and their problem solving behaviors.

Keywords: Precalculus, Problem Solving, Student Reasoning, Quantitative Reasoning

The Physicality of Symbol-Use: Projecting Horizons and Traversing Improvisational Paths Across Inscriptions and Notations

Ricardo Nemirovsky

San Diego State University

Michael Smith

San Diego State University & University of California San Diego

Abstract: The way people use symbols and drawings has an intrinsic physicality. Viewed as an extension of gesture-making, symbol-use can give us insight into how symbol-users experience the mathematics at hand. Using a theoretical framework of embodied cognition, we explore this matter by conducting a phenomenological analysis of a 2- minute selection from an interview with a topologist about one of his published papers. We propose an interpretation of the mathematician’s symbol-use in terms of two related constructs: realms of possibility in what the mathematician perceives as available to him and paths within and between these realms. Both of these are projected onto the writing surface and embodied through gestures, speech, eye gaze, and many other means. We explore the origins and relevance of these in our presentation.

Keywords: Embodied cognition, phenomenology, gesture, mathematicians

From Intuition to Rigor: Calculus Students’ Reinvention of the Definition of Sequence Convergence

Michael Oehrtman

University of Northern Colorado

Craig Swinyard

University of Portland

Jason Martin, Catherine Hart-Weber & Kyeong Hah Roh

Arizona State University

Abstract: Little research exists on the ways in which students may develop an understanding of formal limit definitions. We conducted a study to i) generate insights into how students might leverage their intuitive understandings of sequence convergence to construct a formal definition and ii) assess the extent to which a previously established approximation scheme may support students in constructing their definition. Our research is rooted in the theory of Realistic Mathematics Education and employed the methodology of guided reinvention in a teaching experiment. In three 90-minute sessions, two students, neither of whom had previously seen a formal definition of sequence convergence, constructed a rigorous definition using formal mathematical notation and quantification nearly identical to the conventional definition. The students’ use of an approximation scheme and concrete examples were both central to their progress, and each portion of their definition emerged in response to overcoming specific cognitive challenges.

Keywords: Limits, Definition, Guided Reinvention, Approximation, Examples

How Intuition and Language Use Relate to Students’ Understanding of Span and Linear Independence

Frieda Parker

University of Northern Colorado

Abstract: This report describes a case study in an undergraduate elementary linear algebra class about the relationship between students’ understanding of span and linear independence and their intuition and language use. The study participants were seven students with a range of understanding levels. The purpose of the research was to explore the relationship between students’ “natural” thinking and their conceptual development of formal mathematics and the role of language in this conceptual development. Findings indicate that students with low indicators of intuition and stronger language skills developed better understanding of span and linear independence. The report includes possible instructional implications.

Keywords: Intuition, Language use, Linear algebra, Linear independence, Span

The Impact of Technology on a Graduate Mathematics Education Course

Robert A. Powers, David M. Glassmeyer, & Heng-Yu Ku

University of Northern Colorado

Given the rise in distance delivered graduate programs, educators continue to seek ways to improve teaching and learning in an online environment. In particular, the need for high quality K-12 teachers requires superior teacher-education programs that model good instructional practice, especially in mathematics. In this article, the instructor of a mathematics education course describes the opportunities and difficulties he encountered in designing and implementing an online course for inservice mathematics teachers. In addition to anecdotal evidence from class observations, researchers collected survey data from participants. Results of these data are presented and used with the instructor’s reflections to make specific recommendations for improving the course and to offer insight to others using distance-learning technology to teach graduate mathematics education courses.

Keywords: online professional development, mathematics teacher education, teaching geometry

Using College Lesson Study to introduce discovery and real-life problem solving skills

Kirthi Premadasa

University of Wisconsin-Marathon County

Kavita Bhatia

University of Wisconsin-Marshfield/Wood County

Abstract: Four College Lesson Studies conducted at different locations with students of different levels are discussed, to show how careful planning of the research lesson by several teachers can provide students with opportunities which are missed in a routine lesson due to time and planning constraints. In two of the examples, students "discover" the precise definition of a limit and the integral formula for the arc length, showcasing the opportunity to enhance discovery skills. The other two examples provide students with a "hands-on" opportunity to explore significant real world applications, such as making predictions about the SARS epidemic and finding the optimal payment plan for credit card debt. The classes range from College Algebra to advanced calculus and the lesson studies were done in the US and South Asia.

Keywords: Calculus, Precalculus, Lesson study

Student Teacher and Cooperating Teacher Tensions in a High School Mathematics Teacher Internship: The Case of Luis and Sheri

Kathryn Rhoads, Aron Samkoff, & Keith Weber

Rutgers University

Abstract: We investigate difficulties that student teachers and cooperating teachers experience during the student internship experience by exploring the tension between one high school mathematics student teacher and his cooperating teacher. The student teacher and the cooperating teacher each took part in an individual interview. A follow-up interview with the student teacher, written artifacts, and an interview with the university supervisor were also analyzed. We identified seven causes of tension, which included different ideas about what mathematics should be taught and how it should be taught and a strained personal relationship. These results suggest that (a) cooperating teachers may offer less freedom than they realize, (b) mathematics educators and cooperating teachers may have very different goals for student teaching, (c) cooperating teachers may hold unrealistic expectations about the student teachers prior to their student-teaching experience, and (d) personal relationships can greatly impact the overall student-teaching experience.

Keywords: Preservice secondary teachers Teaching internship Case study

Promoting Students’ Reflective Thinking of Multiple Quantifications via the Mayan Activity

Kyeong Hah Roh

Arizona State University

Yong Hah Lee

Ewha Womans University

Abstract: The aim of this presentation is to introduce the Mayan activity as an instructional intervention and to examine how the Mayan activity promotes students’ reflective thinking of multiple quantifications in the context of the limit of a sequence. The students initially experienced difficulties due to the lack of understanding of the meaning of the order of variables in the definition of convergence. However, such difficulties experienced were resolved as they engaged in the Mayan activity. The students also came to understand that the independence of the variable ε from the variable N is determined by the order of these variables in the definition. The results indicate the Mayan activity plays a crucial role an instructional intervention in understanding why the order of variables matters in the context of limit.

Keywords: Quantification, Reflective Thinking, Proof Evaluation, Convergent Sequence, Cauchy Sequence

How Mathematicians Use Diagrams to Construct Proofs

Aron Samkoff,

Rutgers University

Yvonne Lai

University of Michigan

Keith Weber

Rutgers University

Abstract: Although some researchers argue that diagrams can aid undergraduates’ proof constructions, most undergraduates have difficulty translating a visual argument to a formal one. The processes by which undergraduates construct proofs based on visual arguments are poorly understood. We investigate this issue by presenting ten mathematicians with a mathematical task that invites the construction of a diagram and examine how they used this diagram to produce a formal proof. The talk focuses on the extent to which mathematicians based their proofs on a diagram, the ways in which they used the diagram, and the skills and strategies they used to translate an intuitive argument into a formal one. We observed that mathematicians used diagrams to notice mathematical properties, to verify logical deductions, and to justify assertions. However their use of diagrams relied on sophisticated proving strategies and a range of logical skills, such as the ability to strategically reformulate logical statements.

Keywords: Mathematicians Informal diagrams Proof construction

Exploring the van Hiele Levels of Prospective Mathematics Teachers

Carole Simard

Cal Poly, San Luis Obispo

Todd A. Grundmeier

Cal Poly, San Luis Obispo

Abstract: This research project aimed to assess the influence of an inquiry-oriented, technology- based, proof-intensive geometry course on the van Hiele levels of prospective mathematics teachers. Data was collected in an upper division geometry course taught from an inquiry- oriented perspective. The course relied on technology (The Geometer’s Sketchpad) to help students make and prove conjectures. Data was collected from classes in consecutive years, the first with twenty-one participants and the second with twenty-four participants. Most participants were prospective secondary mathematics teachers. Data collection included a pre- and a posttest of participants’ van Hiele levels. Data analysis suggests similar results for both sets of participants in that the course had greater influence on the van Hiele levels of female participants. Results also suggest that the van Hiele test instrument used for this study operated well with university students.

Keywords: Geometry, van Hiele levels, teacher preparation, secondary

The effect of statistical coursework on preservice secondary teacher understanding of, and efficacies and attitudes toward, statistics learning: The case of Betty

Stephen M. Lancaster

California State University Fullerton

Abstract: In this qualitative study, I describe the characteristics of Betty, a senior undergraduate mathematics education major, who has never completed a full course in statistics either in high school or at the undergraduate level. To investigate and to provide perspective on Betty’s characteristics, Betty and five of her cohorts were administered a series of surveys, which measured statistical understanding, self‐efficacy, and attitudes. The participants also developed and presented a statistics lesson and were then interviewed to investigate their confidence in the implementation of their statistics lesson. The results indicate that the lack of a complete statistics course may have contributed to Betty’s confidence in her ability to learn statistics, her low confidence levels in her ability to currently perform statistical analyses, her weak understanding of statistical concepts, he unwillingness to take risks on statistical knowledge surveys, and her lack of confidence in her lesson plan.

 

Keywords: statistics, preservice secondary teachers, efficacy, attitudes

Classroom Activity with Vectors and Vector Equations: Integrating Informal and Formal Ways of Symbolizing Rn

George Sweeney

San Diego State University

Abstract: Instructional design based upon realistic problems and scenarios allow students to examine the mathematics from a variety of mathematical positions, and create meaning that integrates geometric, algebraic, and formal linear algebra. However, a potential consequence of researching student work on complex activities in difficult mathematics is that classroom mathematical activity from this perspective requires examining how meaning for mathematical objects gets generated over time as a process of collective action and negotiation. In this talk, I will answer two questions: What are the activities that students engage in as they learn to symbolize vector spaces in Rn using realistic situations? And, what is the process by which the classroom community developed these activities? Answering these questions can provide teachers ways being responsive to student needs and thinking as they lead their classrooms in symbolizing vectors and vector equations.

Keywords: Linear Algebra, Symbolizing, Sociocultural Perspectives

Changing Mathematical Sophistication in Introductory College Mathematics Courses

Jennifer E. Szydlik, Eric Kuennen, John Beam, Jason K. Belnap, & Amy Parrott

University of Wisconsin Oshkosh

Abstract: The Mathematical Sophistication Instrument (MSI) measures the extent to which students’ mathematical values and ways of knowing are aligned with those of the mathematical community based on eight interwoven categories: patterns, conjectures, definitions, examples and models, relationships, arguments, language, and notation. In this paper, we present the results of a study designed to explore whether students’ scores on the MSI improved during their introductory college mathematics courses. A large sample of five sections of a first course for elementary education majors, five sections of College Algebra, and seven sections of mathematics for liberal arts majors completed the instrument both at the start and end of the spring 2009 term. Results showed that students in courses where instructors used inquiry-based pedagogies scored markedly higher on the instrument at the end of the semester than at the start. In courses where instructors used traditional pedagogies, only slight changes in scores were observed.

Keywords: nature of mathematics, inquiry-based pedagogy, preservice teachers, mathematical acculturation

Using interactions with children and/or artifacts of children’s mathematical reasoning to promote the development of preservice teachers’ subject matter knowledge

Eva Thanheiser, Krista Heim, & Briana Mills

Portland State University

Abstract: We examine the use of artifacts of children’s mathematical thinking to develop preservice elementary teachers’ (PSTs’) subject matter knowledge. Using three design principles: (a) connecting to previous knowledge, (b) experiencing sense making, and (c) connecting various types of knowledge; artifacts are chosen to specifically address PSTs’ conceptions and allow them to experience sense making and move beyond their reliance on meaningless algorithms. Results show that tasks including such artifacts can be used (a) to assist teachers in developing subject matter knowledge, (b) allow PSTs to move beyond their reliance on the algorithms, and (c) serve as a gateway to allow PSTs to open up to listening to other students’ thinking. However, these changes are not easy. Conceptual difficulties are discussed.

Keywords: Preservice elementary teachers, subject matter knowledge, artifacts of children’s mathematical thinking

Diary of a Mad Grad Student: Folding Autoethnography and Narrative Inquiry into Mathematics Education

Allison F. Toney

University of North Carolina Wilmington

Abstract: Taking the perspective that one cannot reflect on teaching and learning mathematics without reflecting on their thoughts, emotions, and feelings as a mathematics learner and teacher, I illustrate the use of autoethnography and narrative inquiry in mathematics education research. I used autoethnographic and narrative approaches in my research to investigate the nature of the graduate school related experiences of women in collegiate mathematics education doctoral programs. The research relied on interview data with 9 women at 3 universities. Each woman had an advanced degree in mathematics and chose to move into a collegiate mathematics education doctoral program housed in a mathematics department. As a co-participant, I was one of the 9 women. Starting with interviews and personal journals, I developed individual vignettes for each participant. Then, I synthesized the vignettes into a diary of a single fictional woman in a mathematics education doctoral program.

Keywords: Graduate experiences, mathematics education graduate student, women in mathematics education, autoethnography, narrative inquiry.

Using The Emergent Model Heuristic to Describe the Evolution of Student Reasoning regarding Span and Linear Independence

Megan Wawro

San Diego State University

Michelle Zandieh

Arizona State University

George Sweeney

San Diego State University

Christine Larson

Vanderbilt University

Chris Rasmussen

San Diego State University

Abstract: A prominent problem in the teaching and learning of undergraduate mathematics is how to build on students’ current ways of reasoning to develop more generalizable and abstract ways of reasoning. A promising aspect of linear algebra is that it presents instructional designers with an array of applications from which to motivate the development of mathematical ideas. The purpose of this talk is to report on student reasoning as they reinvented the concepts of span and linear independence. The reinvention of these concepts was guided by an innovative instructional sequence known as the Magic Carpet Ride problem, whose creation was framed by the emergent models heuristic (Gravemeijer, 1999). During our talk we will: explain how this instructional sequence differs from a popular “systems of equations first” approach, present the instructional sequence via the framing of the emergent models heuristic; and provide samples of students’ sophisticated thinking and reasoning.

Keywords: Linear algebra, Student Reasoning, Realistic Mathematics Education, Inquiry-Oriented Instruction

Individual and Collective Analysis of the Genesis of Student Reasoning Regarding the Invertible Matrix Theorem in Linear Algebra

Megan Wawro

San Diego State University &

University of California, San Diego

Abstract: I present research regarding the development of mathematical meaning in an introductory linear algebra class. In particular, I present analysis regarding how students– both individually and collectively–reasoned about the Invertible Matrix Theorem over the course of a semester. To do so, I coordinate the analytical tools of adjacency matrices and Toulmin’s (1969) model of argumentation at given instances as well as over time. Synthesis and elaboration of these analyses was facilitated by microgenetic and ontogenetic analyses (Saxe, 2002). The cross-comparison of results from the two analytical tools, adjacency matrices and Toulmin’s model, reveals rich descriptions of the content and structure of arguments offered by both individuals and the collective. Finally, a coordination of both the microgenetic and ontogenetic progressions illuminates the strengths and limitations of utilizing both analytical tools in parallel on the given data set. These and other results, as well as the methodological approach, will be discussed in the presentation.

Keywords: linear algebra, individual and collective, genetic analysis, Toulmin scheme, adjacency matrices.

The Role of Quantitative Reasoning in Representing Functions of Two-Variables

Eric Weber

Arizona State University

Abstract: This paper describes two second-semester calculus students’ understandings of functions of two variables in a teaching experiment that focused on thinking about function as the simultaneous variation of quantities. Analysis of the data, using an open, axial coding scheme suggested that the students thought about functions’ graphs in two distinct ways. The students’ actions, responses, and construction of graphs revealed that one student thought about graphs as a malleable wire and another student considered graphs as covariation of quantities. The subsequent sections review the importance of understanding students’ ways of thinking about functions of more than one variable, the methodology used to collect and analyze data. I conclude by discussing the implications of thinking about graphical representation of functions either as shapes or as covariation of quantities.

Keywords: Two-Variable Functions, Covariation, Quantitative Reasoning, Representation

Questions or comments to the website should be directed to Jason Dolor.

Last Updated January 31, 2011