Abstracts - Preliminary Reports

Students’ Logical Reasoning in Undergraduate Mathematics Courses

Homer W. Austin

Salisbury University

Abstract: This preliminary report describes research results from a pilot study conducted at Salisbury University on undergraduate mathematics/computer science students’ understandings of logical inference. The study was guided by a theoretical framework derived from APOS theory and Balacheff’s theory. The results from the pilot study are crucial for the implementation of a proposed study to be conducted during the 2011-1012 academic year at Salisbury University.The main purpose of the study is to describe students understandings of logical inference in ways that will have implications for providing better instruction in their undergraduate mathematics courses.

Keywords: student understandings of proof, logical inference, transition to proof

Examining Personal Teacher Efficacy Beliefs and Specialized Content Knowledge of Pre-service Teachers in Mathematical Contexts

Jathan Austin

University of Delaware

Abstract: This study addressed the following research question: To what extent are K-8 pre- service teachers’ personal mathematics teacher efficacy beliefs aligned with their content knowledge for teaching mathematics? 18 K-8 pre-service teachers enrolled in a teacher preparation mathematics content course completed semi-structured interviews and follow-up written assessments in which efficacy beliefs and content knowledge regarding specific mathematical teaching scenarios were assessed. Preliminary analyses indicate that the efficacy beliefs of pre-service teachers with low content knowledge vary according to the nature of the teaching scenario. Consequently, the extent to which teacher efficacy beliefs and knowledge are aligned for these pre-service teachers depends on the mathematics involved.

Keywords: pre-service teachers, teacher efficacy beliefs, mathematical content knowledge

The Effects of Online Homework in a University Finite Mathematics Course

Mike Axtell & Erin Curran

University of St. Thomas

Abstract: Over the past 15 years, mathematics departments have begun to incorporate online homework systems in mathematics courses. Several studies of online homework systems have shown them to be as effective as traditional homework, while others have shown them to be less effective for certain audiences. Our study seeks to add to the body of research examining the effectiveness of web-based homework systems by examining the performance of students in two Finite Mathematics classes. This study will compare the individual final exam items and overall exam performance of the students in the web-based and traditional homework sections. Additionally, the study will examine and compare the types of questions that traditional and web-based homework students tend to get correct (or incorrect) in order to gain insight into the depth of learning that may be promoted using either homework system.

Keywords: online homework, learning outcomes, effective practice

Building Knowledge within Classroom Mathematics Discussions

Jason K. Belnap

The University of Wisconsin—Oshkosh

Abstract: The growing emphasis on student-centered instruction has generated a va- riety of instructional forms. The presence of alternate pedagogies does not always indicate quality instruction or guarantee quality student involvement. Ascertaining this requires deeper questions about the nature and extent of both instructional tasks and student contributions to the discourse. Previously, I used a framework developed by Belnap and Withers, to de- termine a conversations composition, the nature and extent of participants contributions, and key discussion characteristics in the context of a profes- sional development program (Belnap & Withers, 2010; Belnap, 2010). This study represents an attempt to adapt this framework to classroom dis- cussions to answer these questions: How are learners contributing to the discussion? What is the nature of those contributions? What role are they playing in the discussion? and What significance and impact do their con- tributions have on the developing content?

Using Video to Inform Pedagogical Practices of Female Mathematics Teachers

Tetyana Berezovski

Saint Joseph’s University

Teri Sosa

Saint Joseph’s University

Abstract: This paper reports a study investigating the use of videotaping for professional development of female mathematics educators. Participants in the study were two elementary and two secondary mathematics teachers who videotaped a self-selected mathematics lesson. Using criteria defined in Alba Thompson’s study of mathematics teaching, participants identified and explored desirable pedagogical practices. Participants then used this critical understanding of desirable pedagogical practices to reflectively analyze their videotaped lesson. Researchers added additional reflections based on their observations.

    Using videotapes in conjunction with lesson study adds new opportunities for mathematics educators to reflect and refine their practice. In this paper, we analyze the reflective writings of participants, identifying specific issues of pedagogical practice made visible by analysis of video. We also consider the commonalities between research participants. While not generalizable, our results provide insight into the aspects of classroom pedagogy that female teachers value and can be considered when designing learning environments for pre-service teachers.

Keywords: innovative methodology, teacher education, reflective practice, video recording

Using Think Alouds to Remove Bottlenecks in Mathematics

Kavita Bhatia

University of Wisconsin Marshfield/Wood County

Kirthi Premadasa

University of Wisconsin Marathon County

Abstract: Think alouds are a research tool originally developed by cognitive psychologists for the purpose of studying how people solve problems. The basic idea being that if a subject can be trained to think out aloud while completing a certain task then the introspections can be analyzed and may provide insights into misunderstandings as well as higher thinking. This talk is a preliminary report of a think aloud conducted with calculus students to understand their difficulties with work problems in integral calculus.

Keywords: Calculus, cognitive science, classroom research, think alouds

Tracking and Influencing Concepts of Proof

David E. Brown

Utah State University

Abstract: Anecdotal remarks and somewhat quantifiable data lead us to believe there are moments in a student’s Mathematical development which can lead to or indicate a change in the perception of, or comfort for, proofs. We attempt to identify and record these moments in the contrasting situations of a first-semester calculus course and a third-year course in Discrete Mathematics. In this prelim- inary research report, we attempt to track changes in understandings of, roles of, reasons for, and comfort levels with proof via video-taped interviews which are ethnographic in spirit – interviews unfold with minor direction on our part except when comments which we think are interesting are encountered. The interviews begin by asking the students to comment on strategically chosen results which are proved in class or solved in a homework assignment; for example, the “Product Rule” in the calculus course and the “cocktail party problem” in the Discrete Math class. Briefly, the strategy behind choosing the proofs for interview fodder is couched in our idea of what the proofs or problems represent: ways to verify a claim which is “known” to be true – all of the stu- dents in the calculus class are familiar with the “Product Rule” – or as a way to deal with a large number of cases at once – an exhaustive case-by-case consideration of the cocktail party problem would require examining 215 = 32, 768 circumstances.

Keywords: Student conceptions of proofs, student respect for proofs, dis- course analysis, teaching proofs.

An Investigation of Students’ Proof Preferences: The Case of Indirect Proofs

Stacy Brown

Pitzer College

Abstract: This paper reports findings from an exploratory study regarding undergraduate natural sciences students’ proof preferences, as they relate to indirect proof. While many agree that students dislike indirect proofs and fail to find them convincing, quantitative studies of students’ proof preferences have not been conducted. The purpose of this study is to build on the existing qualitative research base and to determine if the identified preferences and conviction levels can be established as general tendencies among undergraduates. Specifically, the aim of the study is to explore two common claims: (1) students experience a lack of conviction when presented with indirect proofs; and (2) students prefer direct and causal arguments, as opposed to indirect arguments. The purpose of this preliminary report is to share findings from the proof preference pilot study.

COUNTING PROBLEM STRATEGIES OF PRESERVICE AND INSERVICE TEACHERS

Todd CadwalladerOlsker, Scott Annin, & Nicole Engelke

California State University, Fullerton

Abstract: “Counting problems” are a class of problems in which the solver is asked to determine the number of possible ways a set of requirements can be satisfied. Students are often taught to use combinatorial formulas, such as permutation or combination formulas, to solve such problems. However, it is common for students to incorrectly apply such formulas. Heuristics, such as “look- ing for whether or not order matters,” can be unhelpful or misleading. We will discuss an ongoing analysis of preservice and inservice secondary and community-college level teachers’ responses to six counting problems in order to determine the strategy or formula used in attempting to solve the problem. We are particularly interested in whether or not an explicit statement about order “mattering” helps or hinders the participants’ ability to choose an appropriate strategy.

Keywords: combinatorics, counting problems, preservice teachers, inservice teachers

How Do Mathematicians Make Sense of Definitions?

Laurie Cavey, M. Kinzel,T. Kinzel,  K. Rohrig & S. Walen

Boise State University

Abstract: It seems clear that students’ activity while working with definitions differs from that of mathematicians. The constructs of concept definition and concept image have served to support analyses of both mathematicians’ and students’ work with definitions (c.f. Edwards & Ward, 2004; Tall & Vinner, 1981). As part of an ongoing study, we chose to look closely at how mathematicians make sense of definitions in hopes of informing the ways in which we interpret students’ activity and support their understanding of definitions. We conducted interviews with mathematicians in an attempt to reveal their process when making sense of definitions. A striking observation relates to the role of examples. We will share a preliminary analysis of these interviews and engage the audience in reflecting on the ideas.

Keywords: mathematical definitions, advanced mathematical thinking, mathematicians’ practice, examples

Material Agency: questioning both its role and meditational significance in mathematics learning.

Sean Chorney

Simon Fraser University

Abstract: Tools in the mathematics classroom are often not given the credence or the attention they warrant. Considering Vygotsky’s view of mediation, tools may play a larger role in mathematics then originally thought. This preliminary report presents a framework for attempting to identify the implications of tools in student learning. Using Pickering’s analytic framework (1995) distinguishing individual, disciplinary and material agencies, I am interested in how material agency takes form in the interaction of students with tools. While teaching an education class of pre-service mathematics teachers I will analyze their interactions with a Dynamic Geometric software, specifically Geometer’s Sketchpad. In the process of solving a problem I will analyze students’ engagement with the tool in terms of the different types of agencies, based on their spoken words and their actions in using the program.

Keywords: agency, disciplinary agency, material agency, mediation, dynamic geometry software, Geometer’s Sketchpad

The Impact of Instruction Designed to Support Development of Stochastic Understanding of Probability Distribution

Darcy L. Conant

University of Maryland, College Park

Abstract: Large numbers of college students study probability and statistics, but research indicates many are not learning with understanding. The concept of probability distribution undergirds development of conceptual connections between probability and statistics and a principled understanding of statistical inference. Using a control-treatment design, this study employed differing technology-based lab assignments and investigated the impact of instruction aimed at fostering development of stochastic reasoning on students’ understanding of probability distribution. Participants were approximately 200 undergraduate students enrolled in a lecture/recitation, calculus-based, introductory probability and statistics course. This preliminary research report will discuss the framework used to develop the stochastic lab materials and preliminary results of an assessment of students’ understandings.

Keywords: Probability distribution, stochastic reasoning, technology-based instruction, instructional intervention.

Supplemental Instruction and Related Rates Problems

Nicole Engelke, California State University, Fullerton Todd CadwalladerOlsker, California State University, Fullerton

Abstract: In this study, we observed first semester calculus students solving related rates problems in a peer- led collaborative learning environment. The development of a robust mental model has been shown to be a critical part of the solution process for such problems. We are interested in determining whether the collaborative learning environment promotes the development of such a mental model. Through our observations, we were able to determine the amount of time students spent engaging with the diagrams they drew to model the problem situation. Our analysis strove to also determine the quality of the student interactions with their diagrams. This analysis provided insights about the mental models with which the students were working. Engaging students with complex, non- routine problems resulted in the students spending more time developing robust mental models.

Keywords: Calculus, related rates, mental model, collaborative learning

Exploring student’s spontaneous and scientific concepts in understanding solution to linear single differential equations

Arlene M Evangelista

School of Human Evolution and Social Change, Arizona State University

Abstract: In this study, we use the zone of proximal development to characterize students’ spontaneous and scientific concepts of rate of change, rate proportional to amount, exponential function and long-term behavior of solutions for a system of one and two linear autonomous differential equations. Our focus on the dynamics of the differential equation systems is to investigate how these spontaneous and scientific concepts are incorporated from a system one linear differential equation into a larger system of two linear differential equations. We use and adapt previously used instructional activities from an inquiry-oriented differential equation course to help us gather our data by doing semi-structured interviews with five students. We present only preliminary findings on student’s thinking of solutions mainly for single differential equations, with some insights of student thinking of solutions on a system of two differential equations.

Keywords: Differential equations, solutions, rate of change, zone of proximal development

Concepts Fundamental to an Applicable Understanding of Calculus

Contributed Research Report Leann Ferguson and Richard Lesh

Indiana University, Bloomington

Abstract: Calculus is an important tool for building mathematical models of the world around us and is thus used in a variety of disciplines, such as physics and engineering. These disciplines rely on calculus courses to provide the mathematical foundation needed for success in their discipline courses. Unfortunately, many students leave calculus with an exceptionally primitive understanding and are ill-prepared for discipline courses. This study seeks to identify the fundamental calculus concepts necessary for successful academic pursuits outside the undergraduate mathematics classroom, describe appropriate understanding of these concepts, and collect tasks that elicit, document, and measure this understanding. Data were collected through a series of interviews with select undergraduate mathematics and other discipline faculty members. The data were used to build descriptions of and frameworks for understanding the calculus concepts and generate the pool of tasks. Implications of these findings for calculus curriculum are presented.

Keywords: Calculus, understanding, design research

Evaluating Mathematical Quality of Instruction in Advanced Mathematics Courses By Examining the Enacted Example Space

Tim Fukawa-Connelly & Charlene Newton

University of New Hampshire

Abstract: In advanced undergraduate mathematics, students are expected to make sense of abstract definitions of mathematical concepts, to create conjectures about those concepts, and to write proofs and exhibit counter-examples of these abstract concepts. In all of these actions, students must be able to draw upon a rich store of examples in order to make meaningful progress.

    We have created a methodology to evaluate what students might learn from a particular course by describing and analyzing the enacted example space (Mason & Watson, 2008) for a particular concept. This method will both give a means to create testable hypotheses about individual student learning as well as provide a way to compare disparate pedagogical treatments of the same content. Here, we describe and assess the enacted example space by studying the teaching of abstract algebra.

Keywords: example spaces, classroom research, teaching, evaluation, mathematical quality of instruction

A Proposal for Further Research into Students’ Transference of Trigonometry Concepts to Applications in Physics

Gillian Galle Mathematics and Statistics

University of New Hampshire

Abstract: Many universities offer an algebra based physics course for undergraduates. Research has shown that students in these courses encounter difficulty in transferring their knowledge of trigonometry to applications in physics. This paper proposes a possible research study to identify the obstacles encountered by students in an algebra-based physics course as they learn about simple harmonic motion.

Keywords: Student understanding, transfer of knowledge, trigonometry, simple harmonic motion, physics

Title: Determining Mathematical Item Characteristics Corresponding With Item Response Theory Item Information Curves

Jim Gleason, Calli Holaway & Andrew Hamric

University of Alabama

Abstract: Tests in undergraduate mathematics courses are generally high stakes, and yet have low reliability. The current study aims to increase the reliability of such exams by studying the qualities of test items that determine the ability of the item to contribute to the information of the test. Using a three parameter item response theory model, 695 items contained in 25 different tests for 5 different first-year undergraduate mathematics courses have been analyzed to determine the ability of each item to contribute to the corresponding test’s reliability. During the conference presentation, the speakers with solicit input from the participants regarding the types of qualities of these items that may contribute to their information index. These qualities may include cognitive, mathematical content, linguistic, or other descriptions.

Keywords: Assessment, test writing, item response theory

Assessing the Effectiveness of an On-line Math Review and Practice Tool in Foundational Mathematics.

Tara Gula & Julie Gaudet

George Brown College

Mina Singh

York University

Abstract: 150 words Preliminary results of research into the effectiveness of an innovative on-line mathematics review and practice tool (www.mathessentials.ca) will be reported (data collection completion in Dec. 2010). The goal of the web-site is to provide students with the opportunity to review and practice developmental math skills (fractions, percents, etc.), thus filling in gaps in their knowledge. The development of the web-site begat the development of an innovative evaluation model, which can be used to evaluate online educational technologies. Key to the model is not simply evaluating improvement with pre/post test scores, or with anecdotal reports, but through tracking built into the site, which has the potential to provide a multidimensional view of improvement, usage and engagement (usability score). We believe that the web-site itself (support of student success) and the evaluation model (‘gold standard’ for evaluation of educational technologies) have implications for both teaching and research.

Keywords: online practice, developmental math, educational technology, introductory statistics

Horizontal and Vertical Concept Transitions

May Hamdan

Lebanese American University

Abstract: Transfer of concepts, ideas and procedures learned in mathematics to a new and unanticipated situation or domain is one of the biggest challenges for teachers to communicate and for students to learn because it involves high cognitive skills. This study is an attempt to find ways for driving students to generalize and expand mathematical results from one domain to another in a natural way, and to promote that mathematics is not a collection of isolated facts by providing meaningful ways for students to construct, explain, describe, manipulate or predict patterns and regularities associated with a given system of theorems and mathematical behavior. One would wish there were a universal genetic decomposition for generalization and for the abstraction of properties from a given structure and applying it to a new domain. In this study I plan to focus on particular cases of generalizations in calculus and distinguish between two different types of examples.

Keywords: genetic decomposition, abstraction, generalization, Calculus, RME

The Nature and Effect of Idiosyncratic Examples in Student Reasoning about Limits of Sequences

Catherine Hart-Weber

Arizona State University

Michael Oehrtman

University of Northern Colorado

Jason Martin

Arizona State University

Craig Swinyard

University of Portland

Kyeong Hah Roh

Arizona State University

Abstract: We apply a Vygotskian perspective on the interplay between spontaneous and scientific concepts to identify and characterize calculus students’ idiosyncratic use of examples in the process of trying to formulate a rigorous definition for convergence of a sequence. Our data is drawn from a larger teaching experiment, but analyzed for this study to address questions of the origins, nature, and implications of students’ nonstandard ways of reasoning. We observed two students interpreting a damped oscillating sequence as divergent, drawing from considerations from an initial, intuitively-framed definition, but remaining persistent and consistent over the duration of multiple sessions. We also trace some of the implications of their idiosyncratic reasoning for their reasoning and ultimately for their definition of convergence. We conclude by posing several questions about the nature of such example use in terms of our Vygotskian perspective.

Keywords: Limits, Definition, Examples, Spontaneous and Scientific Concepts

Transitioning from Cultural Diversity to Intercultural Competence in Mathematics Instruction

Shandy Hauk

WestEd & University of Northern Colorado

Nissa Yestness

University of Northern Colorado

Jodie Novak

University of Northern Colorado

Abstract: We report on our work to build an applied theory for intercultural competence development for mathematics teaching and learning in secondary and tertiary settings. Based on social anthropology and communications research, we investigate the nature of intercultural competence development for mathematics instruction among in-service secondary mathematics teachers and college faculty participating in a university-based mathematics teacher professional development program. We present results from quantitative and qualitative inquiry into the intercultural orientations of individuals and subgroups (teachers, teacher-leaders, university faculty and graduate students) and offer details on the development of case stories for use in the professional development of mathematics university teacher educators, in-service teacher leaders, and secondary school teachers.

Keywords: secondary teacher preparation, cultural competence, intercultural development, cultural diversity

The Treatment of Composition the Secondary and Early College Mathematics Curriculum

Aladar Horvath

Michigan State University

Abstract: While many studies have focused on student knowledge of function, few studies have focused on composition. This report describes a curriculum analysis of the treatment of composition in the secondary (algebra, geometry, algebra 2, precalculus) and early college (precalculus, calculus) mathematics curriculum. In this study composition is conceptualized as a sequence of functions and as a binary operation on functions. The curriculum analysis utilizes a framework of conceptual, procedural, and conventional knowledge elements as well as representations and types of functions. Preliminary data will be presented during the session and a discussion will center on conceptual, procedural, and conventional knowledge elements for composition.

Keywords: composition, curriculum analysis, conceptual and procedural knowledge, representations

What Do We See? Real Time Assessment of Middle and Secondary Mathematics Teachers’ Pedagogical Content Knowledge and Sociomathematical Norms

Billy Jackson

Saint Xavier University

Lisa Rice

University of Wyoming

Kristin Noblet

University of Northern Colorado

Abstract. The article reviews efforts to develop an observation protocol to assess the pedagogical content knowledge (PCK) and sociomathematical norms (SMN) that middle and high school teachers may develop over time as part of their participation in a master’s program for secondary mathematics teachers. We observed each of 16 teachers in real time using the instrument, before involvement in the project and again after one year. Aspects of the protocol measure four critical components of PCK including curricular content, discourse, anticipatory, and implementation knowledge as well as some sociomathematical classroom norms. We present preliminary quantitative and qualitative analysis of the observations and discuss various challenges faced in the instrument development and its relation to similar protocols used by others previously.

Keywords: Pedagogical content knowledge, sociomathematical norms, inter-rater reliability, teaching moves

Navigating the Implementation of an Inquiry-Oriented Task in a Community College

Estrella Johnson, Carolyn McCaffery, & Krista Heim

Portland State University

Abstract: Teachers implementing inquiry-oriented, discourse-promoting tasks can face a number of challenges (Speer & Wagner, 2009; Ball, 1993). In this study we will examine the challenges faced by two community college instructors as they implement such a task in a “transition to proof” course. In this task students initially use their informal ideas of symmetry to develop a criteria to quantify the symmetry of six figures (see Larsen & Bartlo, 2009), these criteria are then formalized into definitions for symmetry and equivalent symmetries. During this task a number of conflicts arise, and to resolve these conflicts the students engage in rich mathematical discourse. While this task and ensuing discourse offer opportunities for learning mathematics, they also offer significant challenges for effective implementation. We aim to identifying these challenges and the ways in which these challenges were navigated as the class worked towards formal definitions of symmetry and equivalent symmetries.

Keywords: teaching, symmetry, community college, mathematical discourse

Linking Instructor Moves to Classroom Discourse and Student Learning in Differential Equations Classrooms

Karen Allen Keene

North Carolina State University

J. Todd Lee

Elon University

Hollylynne Lee

North Carolina State University

Abstract: This presentation provides a preliminary analysis of how teacher moves in an undergraduate classroom can be specifically linked to student learning about one overarching mathematical topic: parametric curves. Preliminary analysis of one teacher and classroom using an inquiry oriented discursive move framework and grounded theory supports the hypothesis that a teacher’s mathematics and his pedagogical choices provide focus for student discourse and learning about parametric curves. The authors found that the teacher’s moves motivated by his own lateral and vertical curriculum knowledge, desire to deepen students currently held knowledge, and promotion of the students’ abilities to think like mathematicians and develop mathematical habits of mind links to student learning of the parametric equations and graphs as seen through discourse and student work. Finally, the research offers ideas about how university professors can be more aware of their choices of pedagogy to influence learning about large mathematical ideas.

Keywords: differential equations, discourse, teaching moves, parametric equations

Understanding and Overcoming Difficulties with Building Mathematical Models in Engineering: Using Visualization to Aid in Optimization Courses

Rachael Kenney, Nelson Uhan, Ji Soo Yi, Sung-Hee Kim, Mohan Gopaladesikan, Aiman Shamsul, and Amit Hundia

Purdue University

Abstract: In an optimization course, many students find modeling – the process of translating a verbal description of a decision making problem into a valid mathematical optimization model – difficult to learn. To identify the types of mistakes and difficulties experienced by engineering students, we examined various textbooks to create a taxonomy of the types of problems encountered in these courses, and analyzed student performance on modeling questions given on past exams and quizzes to create a taxonomy of the types of mistakes typically made. In our analysis, we observed students often made errors that indicate that they did not have a sound conceptual understanding of the word problem models and the variables and symbols involved. Based on this research, we have designed a preliminary web-based visualization tool using node- link diagrams that aims to help students to gain a better conceptual understanding of modeling problems and formulate valid optimization models.

Keywords: Mathematical Modeling, Engineering, Technology, Visualization

A Systemic Functional Linguistics Analysis of Mathematical Symbolism and Language in Beginning Algebra Textbooks

Elaine Lande

University of Michigan

Abstract: I propose the use of systemic functional linguistics (SFL) as a tool to better understand how mathematical ideas are conveyed through multiple semiotic resources. To demonstrate the tools that SFL offers, mathematical symbols and written language in college beginning algebra textbooks will be examined. I argue that using SFL to research how mathematical content is communicated to undergraduate students can expose important nuances that may otherwise go unnoticed.

Keywords: Beginning Algebra, Language and Mathematics, Mathematical Symbolism, Systemic Functional Linguistics, Textbooks

Student Use of Set-Oriented Thinking in Combinatorial Problem Solving

Elise Lockwood & Steve Strand

Portland State University

Abstract: This study seeks to contribute to research on the teaching and learning of combinatorics at the undergraduate level. In particular, the authors draw upon a distinction characterized in combinatorial texts between set-oriented and process-oriented definitions of basic counting principles. The aim of the study is to situate the dichotomy of set-oriented versus process-oriented thinking within the domain- specific combinatorial problem-solving activity of students. The authors interviewed post-secondary students as they solved counting problems and examined alternative solutions. Data was analyzed using grounded theory, and a number of preliminary themes were developed. The primary theme reported in this study is that students showed a strong tendency to utilize set-oriented thinking during the problem-solving phase that Carlson & Bloom (2005) refer to as “checking,” especially when they engaged in the evaluation of alternative solutions.

Keywords: combinatorics, counting, problem-solving, grounded theory

How Do iPads Facilitate Social Interaction in the Classroom?

Brian Fisher and Timothy Lucas

Pepperdine University

Abstract: Traditionally, research on technology in mathematics education focuses on interac- tions between the user and the technology, but little is known is about how technology can facili- tate interaction among students. In this preliminary report we will explore how students use iPads while negotiating mathematical meaning in a community of learners. We are currently studying the use of iPads in an introductory business calculus course. We will report on classroom obervations and a series of small-group interviews in which students explore the concepts of local and global extrema. Our preliminary results are that the portability of iPads and the intuitive applications have allowed students to easily incorporate the iPad into their collaborations.

Keywords:business calculus, social constructivism, classroom technology, iPad

An Exploration of the Transition to Graduate School in Mathematics

Sarah L. Marsh

University of Oklahoma

Abstract: In recent years, researchers have given much attention to the new mathematics graduate student as a mathematics instructor. In contrast, this study explores the academic side of the transition to graduate school in mathematics—the struggles students face, the expectations they must meet, and the strategies they use to deal with this new chapter in their academic experience. This talk will look at preliminary results and analysis from a qualitative study designed to explore these aspects of the transition to graduate school in mathematics from a post-positivist perspective. In order to explore the transition as fully as possible, interview data from a varied sample of graduate students and faculty members at one university are being incorporated to gain multiple perspectives on the transition experience. Potential implications for graduate recruitment, retention, and program protocols in mathematics will be discussed.

Keywords: graduate students, academic transition, semi-structured interview, case study

Inquiry and Didactic Instruction in a Computer-Assisted Context: a Quasi-Experimental Study

John Mayer, Rachel Cochran, Jason Fulmore, Thomas Ingram, Laura Stansell & William Bond

University of Alabama at Birmingham

Abstract: We compare the effect of incorporating inquiry-based sessions versus traditional lecture sessions, and a blend of the two approaches, in an elementary algebra course in which the pedagogy consistent among treatments is computer-assisted instruction. Our research hypothesis is that inquiry-based sessions benefit students significantly in terms of mathematical content knowledge, problem-solving, and communications. All students receive the same computer- assisted instruction component. Students are randomly assigned for the semester to one of three treatments (two inquiry-based meetings, two lecture meeting, or one of each, weekly). Measures, including pre- and post-tests with both open-ended and objective items, are described. Statistically significant differences have previously been observed in similar quasi-experimental studies of multiple sections of finite mathematics (Fall, 2008) and elementary algebra (Fall, 2009) with two treatments. Undergraduates, including many pre-service elementary teachers, who do not place into a credit-bearing mathematics course take this developmental algebra course.

Keywords: Elementary algebra, teaching experiment, computer-assisted instruction, inquiry- based instruction, didactic instruction.

Do Leron’s structured proofs improve proof comprehension?

Juan Pablo Mejia-Ramos

Rutgers University

Evan Fuller

Montclair State University

Keith Weber, Aron Samkoff, Kathryn Rhoads, Dhun Doongaji, & Kristen Lew

Rutgers University

Abstract: In undergraduate mathematics courses, proofs are regularly employed to convey mathematics to students. However, research has shown that students find proofs to be difficult to comprehend. Some mathematicians and mathematics educators attribute this confusion to the formal and linear style in which proofs are generally written. To address this difficulty, Leron (1983) suggested an alternative format for presenting proofs, named structured proofs, designed to enable students to perceive the main ideas of the proof without getting lost in its logical details. However, we are not aware of any empirical evidence that such format actually helps students comprehend proofs. In this presentation we report preliminary results of a study that employs a recent model of proof comprehension to assess the extent to which Leron’s format help students comprehend proofs.

Keywords: proof comprehension, structured proofs, proof reading.

Teaching Approaches of Community College Mathematics Faculty: Do Teaching Conceptions and Approaches Relate to Classroom Practices?

Vilma Mesa &Sergio Celis

University of Michigan

Abstract: In this study we compare teaching approaches of 14 community college mathematics instructors with their classroom questioning and their classroom non-mathematical discursive interactions. The teaching approaches were drawn from interviews and the application of an analytical framework derived from the higher education literature. The questioning and the non- mathematical discursive interactions were characterized using transcripts of classroom observations and the application of an analytical framework derived from the mathematics education and higher education literature. From the interviews, we found a wide range of espoused teaching approaches, although the majority of instructors favored instructor-centered approaches. From the observations, we found that these instructors ask a large amount of questions, a sizable proportion of which generate opportunities for students to engage with authentic mathematical knowledge. Also, we found that these espoused teaching approaches are related to observed non-mathematical discursive interactions.

Keywords: classroom research, community college, mathematics teaching

Using Animations of Teaching to Probe the Didactical Contract in Community College Mathematics

Vilma Mesa & Patricio Herbst

University of Michigan

Abstract: In this presentation we want to share with participants prototypes of animations that have been developed as part of a larger project that investigates mathematics instruction in community colleges. The animations have been developed to study the norms of the didactical contract that regulate classroom activity in trigonometry classrooms. We describe the design process that led to generate the raw material for the animations focusing on an instructional situation that we call “finding the values of trigonometric functions” and specifically on a case of this situation that occurs as instructors and students solve examples on the board. Participants will engage in discussing how using the animations can generate data to test hypothesis about the contract that is being probed.

Mathematicians’ Pedagogical Thoughts and Practices in Proof Presentation

Melissa Mills

Oklahoma State University

Abstract: Little is known about how mathematicians present proofs in undergraduate courses. This descriptive study uses ethnographic methods to explore proof presentations at a large comprehensive research university in the Midwest. We will investigate three research questions: What pedagogical moves do mathematics faculty members make when presenting proofs in a traditional undergraduate classroom? What do mathematics faculty members contemplate as they plan lectures that include proof presentations? To what degree and in what ways do faculty members engage students when presenting proofs? To pursue these questions, four faculty members who were teaching proof-based mathematics courses were interviewed and 6-7 observations of each classroom were conducted throughout the course of the semester. The data were analyzed to identify some of the pedagogical content tools that were used, to develop an observation instrument, and to understand how mathematicians think about the pedagogy of proof presentation.

Keywords: proof presentation, pedagogical content tools, teaching proof, ethnographic methods

Where is the Logic in Proofs?

Milos Savic

New Mexico State University

Abstract: Often university mathematics departments teach some formal logic early in a transition- to-proof course in preparation for teaching undergraduate students to construct proofs. Logic, in some form, does seem to play a crucial role in constructing proofs. Yet, this study of 43 student- constructed proofs of theorems about sets, functions, real analysis, abstract algebra, and topology, found that only 1.7% of proof lines involved logic beyond common sense reasoning. Where is the logic? How much of it is just common sense? Does proving involve forms of deductive reasoning that are logic-like, but are not immediately derivable from predicate or propositional calculus? Also, can the needed logic be taught in context while teaching proof- construction instead of first teaching it in an abstract, disembodied way? Through a theoretical framework emerging from a line-by-line analysis of proofs and task-based interviews with students, I try to shed light on these questions.

Keywords: Logic, transition-to-proof courses, analysis of proofs, task-based interviews

Mathematics Faculty’s Efforts to Improve the Teaching of Undergraduate Mathematics

Susana Miller

University of Delaware

Abstract: In recent years, much attention has been given to the pre-service preparation and professional development of mathematics teachers at the elementary, middle, and high school levels. Researchers have concluded that strong content knowledge is not enough to insure effective teaching. Yet, many colleges require little to no professional development for their mathematics faculty. Without supports similar to those provided to K-12 teachers, how do college mathematics faculty members develop and improve their teaching of undergraduate mathematics? A department-wide survey and follow-up interviews were used to investigate if and how the mathematics faculty at one research university have acquired and honed skills for teaching undergraduate mathematics. Preliminary analyses of this data will be presented, and feedback for future directions will be solicited. Understanding if and how mathematics faculty currently seek supports for improving their teaching can inform the design of future professional development programs for college mathematics faculty.

Keywords: professional development, undergraduate mathematics instruction, teaching resources, mixed methods research

Student Approaches and Difficulties in Understanding and Using of Vectors

Oh Hoon Kwon

Michigan State University

Abstract: A configuration of vector representations based on multiple represen- tation, cognitive development, and mathematical conceptualization, to serve as a new unifying framework for studying undergraduate student approaches and difficulties in understanding and using of vectors is proposed. Using this configuration, the study will explore 5 impor- tant transitions, ‘physics to mathematics’, ‘arithmetic to algebraic’, ‘analytic to synthetic’, ‘geometric to symbolic’, ‘concrete to abstract’, and corresponding student difficulties along epistemological and ontological axes. As a part of validation of the framework, a study on undergraduate students’ approaches and difficulties in understanding and using of vectors with both quantitative and qualitative methods will be introduced, and we will see how useful this new framework is to analyze student approaches and difficulties in understanding and using of vectors.

Keywords: Vector, Representation, Vector Representation, Undergraduate Mathematics Education

Geometric Constructions to Activate Inductive and Deductive Thinking Among Secondary Teachers

Eric Pandiscio

University of Maine

Abstract: In a pilot study, the goal was to show that students in an inquiry-oriented, construction-based experience dealing with Euclidean geometry topics can gain in their ability to write deductive proofs. A learning environment was created that involved extensive work with constructions using traditional compass and straightedge techniques as well as with dynamic geometry software. A major piece of the work was a rigorous program of “deconstructions” whereby participants gave written and oral validations of each construction. A pre-test/post test consisting of formal, written proofs served as one assessment instrument. Preliminary data show promise for an increase in the proficiency on such tasks, indicating a potential mechanism for enhancing deductive reasoning.

Keywords: Geometry, Secondary Teachers, Deductive Proof

The Internal Disciplinarian: Who is in Control?

Judy Paterson, Claire Postlethwaite, & Mike Thomas

Auckland University

Abstract: A group of mathematicians and mathematics educators are collaborating in the fine- grained examination of selected ‘slices’ of video recordings of lectures drawing on Schoenfeld’s KOG framework of teaching-in-context. We seek to examine ways in which this model can be extended to examine university lecturing. In the process we have identified a number of lecturer behaviours There are times when, in what appears to be an internal dialogue, lecturing decisions are driven by the mathematician within the lecturer despite the pre-stated intentions of the lecturer to be a teacher.

Keywords: Professional development, lecture research, decisions

Mathematical Knowledge for Teaching: Exemplary High School Teachers’ Views

Kathryn Rhoads

Rutgers University

Abstract: Eleven exemplary high school mathematics teachers were interviewed to investigate their views on mathematical knowledge for teaching. Teachers took part in a one-hour interview and discussed a written lesson plan. Results indicated that these teachers believed the following aspects of mathematical knowledge for teaching to be important: (a) making connections between mathematical ideas in the high school curriculum and beyond, (b) recognizing key examples that illustrate a mathematical concept, (c) knowing appropriate applications of a concept, (d) recognizing several approaches to problem-solving for a particular concept, and (e) understanding various representations of a concept. Teachers also discussed the development of their mathematical knowledge for teaching, which they believed came from their teaching experience and personal experiences rather than formal coursework. These results point to suggestions for areas of focus in undergraduate mathematics teacher education.

Keywords: Mathematical knowledge for teaching High school mathematics teachers Interview study

Analysis of Undergraduate Students’ Cognitive Processes When Writing Proofs about Inequalities

Kyeong Hah Roh & Aviva Halani

Arizona State University

Abstract: The purpose of this presentation is to discuss undergraduate students’ cognitive processes when they attempt to write proofs about inequalities involving absolute values. We employ the theory of conceptual blending to analyze the cognitive process behind the students’ final proof of inequalities. Two undergraduate students from transition-to-proof courses participated in the study. Although the instruction about inequalities was given graphically, the students recruited algebraic ideas mainly when they attempted to construct a proof for the inequality. We illustrate how students apply the algebraic ideas and proving structures for their mental activity in their proving activity.

Keywords: proof construction, inequalities, absolute values, conceptual blending

The van Hiele Theory Through the Discursive Lens: Prospective Teachers’ Geometric Discourses

Sasha Wang

Michigan State University

Abstract: This project investigates changes in prospective elementary and middle school teachers’ van Hieles levels, and in their geometric discourses, on classifying, defining and constructing proofs with geometric figures, resulting from their participation in a university geometry course. The project uses the van Hiele Geometry Test from the Cognitive Development and Achievement in Secondary School Geometry (CDASSG) project, in a pretest and posttest, to predict prospective teachers’ van Hiele levels (Usiskin, 1982), and also uses Sfard’s (2008) framework to analyze these same prospective teachers’ geometric discourses based on in-depth individual interviews. Additionally, the project produces a translation of van Hiele levels into a detailed model that describes students’ levels of geometric thinking in discursive terms. The discussion will focus on studying college students’ reasoning and methods of proof regarding

geometric figures in Euclidean geometry.

Keywords: prospective teachers, Euclidean geometry, mathematical discourse, the van Hiele Theory

Reading Online Mathematics Textbooks

Mary Shepard & Carla van de Sande

Arizona State University

Abstract: This study explores how students read from an online mathematics textbook. The particular textbook that we are exploring is Precalculus: Pathways to Calculus, which was developed at Arizona State University as part of a redesigned precalculus course that focuses on developing students’ ability to reason conceptually about functions and quantity. We are interested in understanding the way students read their mathematical textbooks so that research- informed activities can be developed and incorporated into online textbooks to increase comprehension and retention. In order to investigate authentic student reading habits as closely as possible, we used nonintrusive screen capture software to measure activities such as scrolling, latency, and browsing, as students complete their regular reading assignments in a study hall setting. Other data sources include brief surveys, assessments and interviews. Interventions include reading instruction and embedded activities with feedback and sequences of hints that are intended to promote deeper engagement with the text.

Keywords: online textbooks; precalculus; reading; textbooks

Calculus from a virtual navigation problem

Olga Shipulina

Simon Fraser University

Abstract: Calculus appeared from the real world application, has a real world context, and is fundamentally a dynamic conception; this is why the framework of Realistic Mathematics Education (RME) should be the most efficient approach to teaching and learning calculus. The current study is devoted to investigation of the computer simulated bodily path optimization calculus. I adapted the conception of ‘tacit intuitive model’ for the particular calculus task of path optimizations. My hypothesis is that tacit mental modeling takes place with the allocentric frame of reference. I designed a paradigm in the Second Life virtual environment which allows simulating the navigational task of path optimization with two different mediums and with voluntary choice between allocentric/egocentric views. The reinventing the calculus problem of path optimization from the virtual navigation and its mathematizing would give a powerful intuitive link between the everyday real world problem and its symbolic arithmetic.

Keywords: calculus, virtual navigation, egocentric/allocentric view, tacit intuitive model, Realistic Mathematics Education

Construct Analysis of Complex Variables: Hypotheses and Historical Perspectives

Hortensia Soto-Johnson Michael Oehrtman

University of Northern Colorado

Abstract: Quantitative reasoning combined with gestures, visual representations, or mental images has been at the center of much research in the field of mathematics education. In this report we extend these studies to include complex numbers and complex variables. We provide a construct analysis for the teaching and learning of complex variables, which includes a description of existing frameworks that hypothesize about how students can best comprehend the arithmetic operations of complex numbers. In order to test these conjectures, we interviewed mathematicians, physicists, and electrical engineers to explore how they perceive complex variables content. Through phenomenolgogical and microethnography analysis methods we found how these experts integrate perceptuo-motor activity and metaphors into their descriptions.

Keywords: Complex variables, Operational components, Perceptuo-motor activity, Structural components

Conceptual Writing and Its Impact on Performance and Attitude

Elizabeth J. Malloy, Virginia (Lyn) Stallings, Frances Van Dyke

American University

Abstract: In a small study, the authors found that writers improved more than non-writers numerically on a post test; but the difference was not significant overall except in the case of the lower level mathematics class. Furthermore, the authors found that within the writers group: 1) females had more negative attitudes about communicating mathematically than males and, 2) students who were the most diligent in their writing about concepts had significantly more negative attitudes about their ability to do mathematics which seemed to correspond with the adage, “The more I learn, the less I know,” The previous study used a complex writing heuristic and, as a result, the authors believe that more focused writing is key to conceptual understanding. They propose to conduct a larger study using a visual assessment skills instrument that contains concept questions that are not directly related to the course.

Keywords: conceptual understanding, writing to learn mathematics, visual skills assessment, attitudes toward writing in mathematics, attitudes toward mathematics

Spanning set: an analysis of mental constructions of undergraduate students

María Trigueros, Asuman Oktaç, Darly Kú

Instituto Tecnológico Autonoma de Mexico

Abstract: In this study we use APOS theory to propose a genetic decomposition for the concept of spanning set in Linear Algebra. We give examples of interviews that were conducted with a group of university students who were taking an analytic geometry course and their analysis in relation to our genetic decomposition. We also comment on the nature of difficulties that students experience in constructing this notion. One of the results that are obtained in this research that is in line with previous results reported in the literature is the difficulty in distinguishing a spanning set from a basis. Another aspect is that students have varying levels of difficulty when working with different types of vector spaces. As was expected, the concept of linear combination plays a very important role in the understanding of the notion of spanning.

Keywords: Spanning set, APOS Theory

Concrete Materials in Mathematics Education: Identifying “Concreteness” and Evaluating its Pedagogical Effectiveness

Dragan Trninic

University of California at Berkeley

Abstract: A growing body of research suggests cognitive difficulties associated with the use of concrete learning materials. I argue that this research program may benefit from a critical examination of its underlying assumptions. Thus, this report was motivated by a concern that extant approach to evaluating the pedagogical effectiveness of “concreteness” in education is by-and-large undertheorized, resulting suboptimal interpretation of reform- based philosophy and recommendations, ultimately to the detriment of students. I hope to open up a space for a discussion of a more nuanced conceptualization of both (1) “concreteness” as a concept and (2) the observed cognitive difficulties evident in classroom implementation of concrete materials.

Keywords: cognitive research, theoretical perspectives, concrete problems

Technologizing Math Education: The case of multiple representations

Tyler Gaspich &

Tetyana Berezovski

Saint Joseph's University

Abstract: Technology is a cornerstone for NCTM and has been accepted to be beneficial, but the level of effectiveness is still very vague. This research questions exactly how effective is technology in the mathematics classroom, and what are the definitive benefits. After studying over 300 articles, technology has proven to be beneficial in five ways: providing instantaneous visual feedback, creating student-centered learning environments, providing multiple representations of similar concepts, combining learning environments for generalizations, and retracing previous steps for self-assessment. The most frequently discussed topic was multiple representations, usually in the form of CAS and dynamic geometry systems. The research shows that providing multiple representations allows students with varying levels of intelligence to better understand tricky and abstract concepts.

Keywords: Technology, Multiple representations, Multiple intelligences, technology effectiveness, mathematics education

The Construction of Limit Proofs in Free, Open, Online, Help Forums

Carla Van de Sande & Kyeong Hah Roh

Arizona State University

Abstract: Free, open, online, help forums are found on public websites and allow students to post queries from their course assignments that can be responded to asynchronously by anonymous others. Several of these forums are tailored to helping students with mathematics assignments from various courses, and Calculus, in particular, is a heavily trafficked area. Students use the forums when they have reached an impasse, either in constructing or understanding a solution to an exercise that they have encountered, or to seek verification of their own reasoning. The queries posted by students include both computational tasks as well as proof constructions. In this project, we examine threads on limit proofs for single-variable functions from two popular online forums. Our goal is twofold: to characterize the help students are receiving as they wrestle with using the formal definition of limit, and to compare the construction of proof to other tasks in online forums.

 

Keywords: computer-mediated discourse; limits; online help; student understanding of proof

 

Function Composition and the Chain Rule in Calculus

Aaron Wangberg

Winona State University

Nicole Engelke

California State University, Fullerton

Gulden Karakok

UMERC

Umea University

Abstract: The chain rule is a calculus concept that causes difficulties for many students. While several studies focus on other aspects of calculus, there is little research that focuses specifically on the chain rule. To address this gap in the research, we are studying how students use and interpret the chain rule while working in an online homework environment. We are particularly interested in answering three questions: 1) What characterizes student’s understanding of composition of functions? 2) What characterizes student’s understanding of chain rule? and 3) To what extent do students’ understanding of composition of functions play a role in their understanding and ability to use chain rule in calculus?

Keywords: Calculus, precalculus, procedural knowledge, conceptual knowledge, technology

Effective Strategies That Successful Mathematics Majors Use to Read and Comprehend Proofs

Keith Weber & Aron Samkoff

Rutgers University

Abstract: Proof is a dominant means of conveying mathematics to undergraduates in their advanced mathematics courses, yet research suggests that students learn little from the proofs they read and find proofs to be confusing and pointless. In this presentation, we examine the behavior of two successful mathematics majors as they studied six proofs to identify productive proof comprehensive strategies. Prior to reading a proof, these students would attempt to understand the theorem by rephrasing and trying to determine why it was true. While reading a proof, these students would partition the proof into sections, attend to the proof framework being employed, and illustrate confusing aspects of the proof with examples. Implications and limitations of this study will be discussed.

Keywords: Proof, proof reading, proof comprehension.

Student Understanding of Integration in the Context and Notation of Thermodynamics: Concepts, Representations, and Transfer

Thomas Wemyss, Bajracharya Rabindra, John Thomson, & Joseph Wagner

University of Maine

Abstract: Students are expected to apply the mathematics learned in their mathematics courses to concepts and problems in physics. Little empirical research has investigated how readily students are able to “transfer” their mathematical knowledge and skills from their mathematics classes to other courses. In physics education research (PER), few studies have distinguished between difficulties students have with physics concepts and those with either the mathematics concepts, application of those concepts, or the representations used to connect the math and the physics. We report on empirical studies of student conceptual difficulties with (single-variable) integration on mathematics questions that are analogous to canonical questions in thermodynamics. We interpret our results considering the representations used as well as the lens of knowledge transfer, with attention to how students solve problems involving the same mathematical principles in the differing contexts of their physics and mathematics classes.

Keywords: Physics, integrals, conceptual understanding, representations, transfer

Extending a Local Instruction Theory for the Development of Number Sense to Rational Number

Ian Whitacre & Susan D. Nickerson

San Diego State University

Abstract: We report on results of the implementation of a local instruction theory for number sense development in a course for prospective elementary teachers. Students involved in an earlier teaching experiment developed improved number sense, particularly in the form of flexible mental computation. The previous research was informed by a conjectured local instruction theory and informed the refinement and elaboration of that local instruction theory. The present study concerns a recent iteration of the classroom teaching experiment, in which the local instruction theory guided instructional planning. In the recent iteration, the local instruction theory was extended from the whole-number portion of the course to the rational-number portion. Envisioned learning routes that were developed in the context of mental computation and estimation were applied to reasoning about fraction size. In this way, the application of the local instruction theory was extended from whole-number sense to rational-number sense.

Keywords: Local instruction theory, number sense, prospective teachers, rational number

Redefining Integral: Preparing for a New Approach to Undergraduate Calculus

Dov Zazkis

San Diego State University

Abstract: This study is a pilot to a larger design research project that aims to explore an alternative approach to teaching a Calculus I course. Central to this approach is the introduction of the integral first, utilizing a non-standard definition, but which is equivalent to the standard definition. This is immediately followed by the introduction of derivative. This approach allows methods of derivation and integration, which are analogs of one another to be introduced in close succession, allowing the relationships between these methods to be a major theme of the course. The alternative definition of integral is the focus of this study. I present preliminary results of a teaching experiment that explores how students develop an understanding of this alternative definition of integral and how these understandings relate to prerequisite notions, such as area and arithmetic mean.

Keywords: Calculus, arithmetic mean, new methodology, teaching experiment.

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

Last Updated January 31, 2011