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Saturday

5:25 – 6:20 pm

Grand Foyer

Poster Session

Impact of advanced mathematical knowledge on the teaching and learning of secondary mathematics

Eileen Murray, Debasmita Basu, Matthew Wright

This poster presents preliminary data from an exploratory study that aims to advance our understanding of the nature of mathematics offered to prospective mathematics teachers by looking at mathematical connections. In this study, we investigate how in-service and pre-service middle school teachers make connections between tertiary and secondary mathematics as well as if and how the understanding of connections influences teachersÕ thoughts about teaching and learning mathematics.

Analyzing classroom developments of language and notation for interpreting matrices as linear transformations.

Ruby Quea, Christine Andrews-Larson

As part of a larger study of students reasoning in linear algebra, this research analyzes how students make sense of language and notation introduced by instructors when learning matrices as linear transformations. This paper examines the implementation of an inquiry-oriented instruction that consists of students generating, composing, and inverting matrices in the context of increasing the height and leaning a letter ÒNÓ placed on a 2-dimensional Cartesian coordinate system (Wawro et. al., 2012). I analyzed two classroom implementations and noted how instructors introduced and formalized mathematical language and notation in the context of this particular instructional sequence, and then related that to the ways that language and notation were subsequently taken up by students. This work was conducted in order to enable me to build theory about the relationship between student learning and the ways in which language and notation are introduced.

What do students attend to when first graphing in R3?

Allison Dorko

This poster considers what students attend to as they first encounter R3 coordinate axes and are asked to graph functions with free variables. Graphs are critical representations, yet students struggle with graphing functions of more than one variable. Because prior work has revealed that studentsÕ conceptions of multivariable graph are often related to their conceptions about single variable functions, we used an actor-oriented transfer perspective to identify what students see as similar between graphing functions with free variables in R2 and R3. We considered what students attended to mathematically, and found that they focused on equidistance, parallelism, and coordinate points.

Support for mathematicians' teaching reform in an online working group for inquiry oriented differential equations

Nicholas Fortune

There is more need for research on how mathematicians can alter their teaching style to a reform approach (Speer, Smith, & Horvath, 2010), especially if they have always been teaching the same way (Speer & Wagner, 2009; Wagner, Speer, & Rossa, 2007). One particular area that needs more work is investigations of support structures for mathematicians hoping to reform their teaching practice. This poster focuses on supports designed to aid in the reform of teaching practice and specifically discusses the Teaching Inquiry-oriented Mathematics: External Supports (TIMES) project and one online working group (OWG) used as a mode of support in the project. Results indicate that facets of the OWG are successful support structures for mathematicians who desire to align their practice to an inquiry oriented (IO) approach to undergraduate differential equations (Rasmussen & Kwon, 2007; Rasmussen, 2003).

StudentsÕ understanding of mathematics in the context of chemical kinetics

Kinsey Bain, Alena Moon and Marcy Towns

This work explores general chemistry studentsÕ use of mathematical reasoning to solve quantitative chemical kinetics problems. Personal constructs, a variation of constructivism, provides the theoretical underpinning for this work, asserting that students engage in a continuous process of constructing and modifying their mental models according to new experiences. The study aimed to answer the following research question: How do non-major students in a second-semester general chemistry course and a physical chemistry course use mathematics to solve kinetics problems involving rate laws? To answer this question, semi-structured interviews using a think-aloud protocol were conducted. A blended processing framework, which targets how problem solvers draw from different knowledge domains, was used to interpret studentsÕ problem solving. Preliminary findings describe instances in which students blend their knowledge to solve kinetics problems.

A comparative study of calculus I at a large research university

Xiangming Wu, Jessica Deshler, Marcela Mera Trujillo, Eddie Fuller and Marjorie Darrah

In this report, we describe the results of analyzing data collected from 502 Calculus I students at a large research university in the U.S. Students were enrolled in one of five different versions of Calculus I offered at the university. We are interested in (i) whether the different characteristics of each version of the course affect studentsÕ attitudes toward mathematics and (ii) how each course might affect studentsÕ intentions of pursuing science, technology, engineering and mathematics (STEM) degrees. We examine data related to these two issues gathered from students via surveys during one semester in Calculus I.

Active learning in undergraduate precalculus and single-variable calculus

Naneh Apakarian and Dana Kirin

The study presented here examines the active learning strategies currently in place in the Precalculus through single variable calculus sequence. While many lament the lack of active learning in undergraduate mathematics, our work reveals the reality behind that feeling. Results from a national survey of mathematics departments allow us to report the proportion of courses in the mainstream sequence utilizing active learning strategies, what those strategies are, and how those strategies are being implemented.

A Proposed Framework for Tracking Professional Development Through GTAÕs

Hayley Milbourne and Susan Nickerson

There are several different models of graduate teaching assistant programs in mathematics departments across the nation (Ellis, 2015). One particular public university has recently reformatted their calculus program toward a peer-mentor model and this is the first year of implementation. In the peer-mentor model, there is a lead TA who serves as a support for the other TAs in the program. Because of this, the professional development in which the TAs are engaged is formally directed by both faculty and a peer. We are interested in discussing a framework known as the Vygotsky Space as a methodology for tracking the appropriation and sharing of pedagogical practices among those responsible for calculus instruction.

How Calculus students at successful programs talk about their instructors

Annie Bergman and Dana Kirin

The CSPCC (Characteristics of Successful Programs in College Calculus) project was a 5-year study focused on Calculus I instruction at colleges and universities across the United States with overarching goals of identifying the factors that contribute to successful programs. In this poster, we draw from student focus group interview data collected from schools that were identified by the CSPCC project as being successful. The analyses we will present in this poster will characterize the ways in which calculus students talk about their instructors in an attempt to understand how their perceptions shape their experience.

Investigating university students difficulties with algebra

Sepideh Stewart and Stacy Reeder

Algebra is frequently referred to as the ÒgatewayÓ course for high school mathematics. Even among those who complete high school Algebra courses, many struggle with more advanced mathematics and are frequently underprepared for college level mathematics. For many years, college instructors have viewed the final problem solving steps in their respective disciplines as Òjust AlgebraÓ, but in reality, a weak foundation in Algebra maybe the cause of failure for many college students. The purpose of this project is to identify common algebraic errors students make in college level mathematics courses that plague their ability to succeed in higher level mathematics courses. The identification of these common errors will aid in the creation of a model for intervention.

A qualitative study of the ways students and faculty in the biological sciences think about and use the definite integral

William Hall

In this poster, I share my methods and pilot interview results concerning a qualitative study of the ways undergraduate students and faculty from the biological sciences think about and use the definite integral. In this research, I utilize task-based interviews including five applied calculus tasks in order to explore how students and faculty think about area, accumulation, and the definite integral. Early results from pilot interviews helped me revise the interview protocols and indicate that student reasoning may be affected by experience and context. In presenting this poster, I hope to gain feedback from the community on my research methodology and potential analytical strategies.

Root of Misconceptions – the Incorporation of Mathematical Ideas in History

Kuo-Liang Chang

The evolution of a mathematical concept in history has been the process of merging different ideas to form a more rich, general, and rigorous concept. Ironically, students, when learning such well-developed concepts, have similar difficulties and make the same misconceptions again and again. To illustrate, despite the well-developed and defined concept of real numbers, many students still have difficulties in comparing fractions or doing basic operations on irrational numbers. In this poster, the incorporation of different ideas to form a general and rigorous mathematical concept in history is examined. StudentsÕ struggles and misconceptions in learning the concepts are investigated from the perspective of the incorporation process. Finally, a model for differentiating and validating the variations of a general mathematical concept is suggested for resolving learning difficulties and misconceptions.

Student Experiences in a Problem-Centered Developmental Mathematics Class

Martha Makowski

Given the large numbers of students who enroll each year in developmental mathematics classes, community colleges have started creating developmental classes that both engage students in meaningful mathematics and provide them with an efficient pathway to the college-level curriculum. This study examines how community college students enrolled in a problem-based developmental algebra class experience the curriculum and instruction. Eight students from a target classroom were interviewed about their experiences in the class, focusing on how they experienced group work, problem solving, and the role of the teacher. Surveys were also used to measure their attitudes towards mathematics along several dimensions at the beginning and end of the semester. Students who liked group work were energized by the opportunity to work with people while learning and valued the multiple opportunities group work provided for them to check their work. Few students saw group work as an opportunity to explore conceptual ideas. The results suggest that implementations of such classes could benefit from structured discussion about group work norms and explicit discussion about the purpose of solving problems.

A collaborative effort for improving calculus through better assessment practices

Justin Heavilin, Hodson Kyle and Brynja Kohler

Like many institutions across the country, Utah State UniversityÕs Department of Mathematics and Statistics has embarked on an effort to improve the calculus sequence with the following objectives: (1) improve our studentsÕ comprehension and application of key topics, (2) retain/recruit more students into STEM majors, and (3) provide more consistency across sections. After initial planning and preparation in the 2014-15 academic year, new practices were ready for implementation. In the fall of 2015, teams of instructors worked from common guided course notes, and met weekly to discuss instruction and develop common assessments. This poster displays the methodology of test design and item analysis we employed in the Calculus 2 course. While our team is only at the beginning stages of this work, the methods for creating reliable and relevant measures of student learning hold promise for achieving the goals of our reform.

Exploring student understanding of the negative sign in introductory physics contexts

Suzanne Brahmia and Andrew Boudreaux

Recent studies in physics education research demonstrate that although physics students are generally successful executing mathematical procedures, they struggle with the use of mathematical concepts for sense making. In this poster we investigate student reasoning about negative numbers in contexts commonly encountered in calculus-based introductory physics. We describe a large-scale study (N > 900) involving two introductory physics courses: calculus-based mechanics and calculus-based electricity and magnetism (E&M). We present data from six assessment items (3 in mechanics and 3 in E&M) that probe student understanding of negative numbers in physics contexts. Our results reveal that even mathematically well-prepared students struggle with the way that we symbolize in physics, and that the varied uses of the negative sign in physics can present an obstacle to understanding that persists throughout the introductory sequence.

Classroom observation, instructor interview, and instructor self-report as tools in determining fidelity of implementation for an intervention

Shandy Hauk, Katie Salguero and Joyce Kaser

A web-based activity and testing system (WATS) has features such as adaptive problem sets, instructional videos, and data-driven tools for instructors to use to monitor and scaffold student learning. Central to WATS adoption and use are questions about the implementation process: What constitutes ÒgoodÓ implementation and how far from ÒgoodÓ is good enough? Here we report on a study about implementation that is part of a state-wide randomized controlled trial examining student learning in community college algebra when a particular WATS suite of tools is used. Discussion questions for conference participants dig into the challenges and opportunities in researching fidelity of implementation in the community college context, particularly the role of instructional practice as a contextual component of the research.

Separating issues in the learning of algebra from mathematical problem solving

R. Cavender Campbell, Kathryn Rhoads and James A. Mendoza Epperson

StudentsÕ difficulty in learning school algebra has motivated a plethora of research on knowledge and skills needed for success in algebra and subsequent undergraduate mathematics courses. However, in gateway mathematics courses for science, technology, engineering, and mathematics majors, student success rates remain low. One reason for this may be to the lack of understanding of thresholds in student mathematical problem solving (MPS) practices necessary for success in later courses. Building from our synthesis of the literature in MPS, we developed Likert scale items to assess undergraduate studentsÕ MPS. We used this emerging assessment and individual, task-based interviews to better understand studentsÕ MPS. Preliminary results suggest that studentsÕ issues in algebra do not prohibit them from using their typical problem solving methods. Thus, the assessment items reflect studentsÕ MPS, regardless of possible misconceptions in algebra, and provide a mechanism for examining MPS capacity separate from procedural and conceptual issues in algebra.

Communicative artifacts of proof: Transitions from ascertaining to persuading

David Plaxco and Milos Savic

With this poster, we wish to highlight an important aspect of the proving process. Specifically, we revisit Harel and SowderÕs (1998, 2007) proof schemes to extend the authorsÕ constructs of ascertaining and persuading. With this discussion, we reflect on the original theoretical framework in light of more recent research in the field and draw focus to a critical aspect of the proving process in which the prover generates the communicative artifacts of proof (CAP) critical to shifts between ascertaining to persuading. We also discuss possible ways in which an attention to the psychological and social activities involved in the development of the CAP might inform research and instruction.

On the variety of the multiplication principleÕs presentation in college texts

Zackery Reed and Elise Lockwood

The Multiplication Principle is one of the most foundational principles of counting. Unlike foundational concepts in other fields, where there is uniformity in presentation across text and instruction, we have found that there is much variety in the presentation of the Multiplication Principle. This poster highlights the multiple aspects of this variety, specifically those with implications for the combinatorial research and education community. Such topics include the statement types, language and representation of statements, and mathematical implications.

Assessing studentsÕ understanding of eigenvectors and eigenvalues in linear algebra

Kevin Watson, Megan Wawro, and Michelle Zandieh

Many concepts within Linear Algebra are extremely useful in STEM fields; in particular are the concepts of eigenvector and eigenvalue. Through examining the body of research on student reasoning in linear algebra and our own understanding of eigenvectors and eigenvalues, we are developing preliminary ideas about a framework for eigentheory. Based on these preliminary ideas, we are also creating an assessment tool that will test studentsÕ understanding of eigentheory. This poster will present our preliminary framework, and examples of the multiple-choice- extended questions we have created to assess student understanding.

6:30 – 9:00 pm

Grand Ballroom Salons 2-4

Using adjacency matrices to analyze a proposed linear algebra assessment

Hayley Milbourne, Katherine Czeranko, Chris Rasmussen and Michelle Zandieh

An assessment of student learning of major topics in linear algebra is currently being created as part of a larger study on inquiry-oriented linear algebra. This includes both the assessment instrument and a way to understand the results. The assessment instrument is modeled off of the Colorado Upper-division Electrostatics (CUE) diagnostic (Wilcox & Pollock, 2013). There are two parts to each question: a multiple-choice part and an explanation part. In the explanation part, the student is given a list of possible explanations and is asked to select all that could justify their original choice. This type of assessment provides information on the connections made by students. However, analyzing the results is not straightforward. We propose the use of adjacency matrices, as developed by Selinski, Rasmussen, Wawro, & Zandieh (2014), to analyze the connections that students demonstrate.