Abstracts of Presentations

RR 1 Type of Talk: Research Report, Author: Rani Satyam

Talk Title: Affect Graphing: Inviting Students to Draw A Graph of their Emotions

Keywords: Affect, emotions, proving, methodological tools

Abstract: Affect (e.g., beliefs, attitudes, emotions) plays a crucial role in mathematics learning, but reliance on verbal and written responses (from surveys, interviews, etc.) can limit students’ expression of their affective states. As a complement to existing methods that rely on verbal reports, we explore how graphing can be used to study affect during mathematical experiences. I present some studies that used graphing to represent, stimulate recall, and reflect on affect. In each, students were asked to draw their perception of an affective construct, such as confidence or intensity of emotion, against time. The affordances of graphing include reduced dependence on verbal data, temporal ordering of participants’ recollections, explicit representation of change over time, and the creation of objects (the graph) for discussion. These studies show that well-structured graphing can productively supplement existing methods for studying affect in mathematics education, as a different medium through which students can communicate their experience.

----------------------

RR 2 Type of Talk: Research Report, Author: Pablo Duran Oliva

Talk Title: Intersubjectivity and Participatory Equity in Active Learning Calculus Classrooms

Keywords: Active learning, Participatory Equity, Intersubjectivity

Abstract: Evidence of the effectiveness of active learning (AL) classrooms in STEM includes increases in students’ achievement, attitudes, and persistence in college (Freeman et al. 2010; Ellis et al. 2011; Duran et al., 2022). In terms of classroom equity, however, results are mixed. Some studies reported that AL contributed to vanishing gaps in achievement for minoritized students (Theobald et al., 2020), while other studies found that these gaps, on the contrary, increased after AL implementations (Reinholz et al., 2022). These conflicting results revealed the need to better understand the elements contributing to more equitable implementations of AL. In this talk, we will present preliminary findings of discourse analysis on student interactions in Calculus classrooms where active learning is in use. Our results contribute to a better understanding of the interface between intersubjective and participatory equity. We will conclude with a brief discussion of the relevant components in instruction that could contribute to leveraging the engagement of all students in mathematics discussions in active learning classrooms. 

----------------------

RR3 Type of Talk: Research Report, Author: Zachary Coverstone

Talk Title: What Problem-Based Learning Helped Me Learn about Student Understanding of Infinite Series Convergence

Keywords: calculus, infinite series, active learning

Abstract: In 2021, I designed and implemented a three-week curriculum to help students understand infinite series convergence to fit into a traditional calculus sequence. The design of these materials is patterned after the Park City Mathematics Institute’s (PCMI) Teacher Leadership Program (TLP), where participants worked on problems in teams with little direct instruction. The problems for each day during the TLP build on each other, using common contexts interwoven throughout the workshop. One reason for using a problem set-based curriculum design in the calculus curriculum was to encourage students to make connections between infinite series concepts.

Following the first semester of implementation, I analyzed student work samples for themes about what students understand about infinite series convergence. This analysis led me to improve the curricular materials through two subsequent semesters, the most recent of which is currently in progress.

In this talk, I plan to discuss the overarching ideas that this unit address, some of the common contexts that were chosen, and the thematic take-aways from student work samples. I then want to consider some ideas for future study with respect to student development of concept images about infinite series convergence.

----------------------

RR 4 Type of Talk: Research Report, Author: Vladislav Kokushkin

Talk Title: Conceptual Chunking and Mathematical Proofs

Keywords: Proofs, Chunking, Cognition

Abstract: Conceptual chunking is one of the driving mechanisms that allows learners to meaningfully engage in a cognitively demanding task. Although it is a well-researched psychological phenomenon, the nature of chunking in the context of mathematical proofs has remained unclear. In this report, I discuss how certain aspects of conceptual chunking turn out to be problematic when undergraduate students produce proofs of algebraic conjectures. 

----------------------

Special Topic, Author: Elise Lockwood

Talk Title: RUME-Related Funding Opportunities at the National Science Foundation

Keywords:  Funding opportunities, NSF, research projects

Abstract: As Program Officers in the Division of Undergraduate Education at the National Science Foundation (NSF), we want to share information about funding opportunities at the NSF that may be relevant for the RUME community. In this session, we will provide a brief overview of programs at the NSF that may be of interest to members of the RUME community. We will also share some broad tips for writing successful proposals, discuss some common myths about the funding process, and allow plenty of time for Q and A.

----------------------

IR/PR 1 Type of Talk: Ideas for Research Studies/Preliminary Research Reports, Authors: Modiu Olaguro

Talk Title: Probability as Attribution: A Sensible Model

Keywords: Probabilistic reasoning, attribution theory, sensible systems, alternative conceptions

Abstract: The study draws on the notion of students’ lived experience to propose a model of probabilistic reasoning. Contrary to what is considered normative probabilistic reasoning (Konold, 1989), I adapted the attributional model of motivation (Weiner, 1994) to normalize students’ informal and subjective reasoning in chance situations, arguing that the existing developmental framework of probabilistic reasoning (Jones et al., 1997) that regard informal or subjective reasoning as a misconception, naïve-thinking, and a deviation from the norm thrives on a deficit perspective that alienates students’ lived experiences from their mathematical realities. Adapting the sensible systems framework (Leatham, 2006), the study hypothesizes students as “sensible beings”, ascribing the use of heuristics and other subjectivities (representativeness, luck, God, etc.) as a way of resolving “perturbations” within the probabilistic system containing also the classical and frequentist clusters—the hallmark of (formal) school probability. Within this framing, probabilistic reasoning ties to locus (internal/external), stability, and controllability (Weiner, 1994).

----------------------

IR/PR 2 Type of Talk: Ideas for Research Studies/Preliminary Research Reports, Author: Marcus Wolfe

Talk Title: The Types of Gestures Used to Connect Mathematics to Video Games

Keywords: Fractions, Gestures, Video Games 

Abstract: Approximately 30% of undergraduate students are enrolled in remedial mathematics courses (Ngo, 2019). However, remediation is often unsuccessful for students in these courses (Bahr 2008, Lundeberg 2018). One reason for this challenge is students’ poor fraction skills, which have long term effects on students success in future math classes . Therefore, helping students through remediation requires developing their understanding of fractions. Previous studies have shown that there is value in using video games to address and develop students STEM knowledge (Gaydos & Squire, 2012).Students in recent studies began to make connections between video games designed for learning and math concepts if prompted (Williams-Pierce, 2016). With this in mind, I conducted a study for students to develop their fraction knowledge using video games. Students in this study played 4 video game levels in order to evoke 4 actions iteration, splitting, dis-embedding and partitioning that are found within fraction scheme literature, in order to develop their fraction knowledge. It was designed such that students upon reflecting on the level would develop fraction knowledge through connecting their prior knowledge to the gameplay. Data on their understanding of fractions was collected through observing their gestures inside and outside of the game. This study was done with 4 participants who were enrolled in a summer developmental math program that will stands as a possible barrier to entry into the four-year institution that it is attached to. This talk will highlight emerging findings to the research, including how students gesture when describing a game they played constructed via grounding metaphor for mathematical knowledge. 

----------------------

IR/PR 3 Type of Talk: Ideas for Research Studies/Preliminary Research Reports, Authors: Cheryl Vallejo

Talk Title: Development of Undergraduate Mathematics Student Perceived Caring Instrument

Keywords: mathematics education, instrument development, caring

Abstract: Amy Hackenberg (2005) created a mathematical framework for her dissertation in the early 2000s from the theoretical framework of schema to create student learning models she calls Mathematical Caring Relations (MCRs) based off Nell Noddings (1984, 2002) notion of caring relations between teachers and students. During an exploration of literature (Hackenberg, 2005, 2010a, 2010b; Noddings 2012a, 2012b), I discovered there was an opportunity to extend Amy Hackenberg’s MCRs to include perceptions of caring from students, specifically related to the energy exchanged during an interaction in the classroom. Hackenberg’s approach to MCRs is through second-order learning models, an interpretation the researcher makes of the student’s thinking while solving a mathematics problem. Therefore, Hackenberg’s research includes only the researcher’s interpretation of the energy exchange and caring relations during a student interview. The students are not asked about whether they felt cared for or cared for the instructor. Nor were the students asked how their interaction with the instructor influenced their understanding of the mathematics.

In addition to Noddings and Hackenberg, Jones (2009, 2017, 2018) includes a caring component of his MUSIC Model of Motivation. The C in MUSIC represents Caring. Jones defines caring as a belief “that others in the learning environment care about their learning and them as a person” (Jones, 2018, p. 9). I am interested in expanding Dr. Brett Jones's Caring Scale of the College Student version of the MUSIC Model of Motivation. I plan on expanding each item of the six-item instrument. I will use the expanded survey to gather data from undergraduate students on how they perceive care from their mathematics teachers and the influence of the perception on the students’ attitudes and motivation towards mathematics.

This instrument I plan on creating will be a noncognitive construct, students’ beliefs and attitudes towards perceived care. Jones (2009, 2017) Caring Scale of the MUSIC Model is the starting point of the perception of care necessary to create a model supportive of students in developmental undergraduate mathematics. Care is not a direct relationship. Feeling secure is an indirect modality of care. It has been reported that college freshman express feeling cared for and more validated when they feel secure in a school setting (Bergin & Bergin, 2009). Feeling secure in a classroom setting generally generated by attachment security to the instructor. Bergin and Bergin (2009) define attachment as “a deep and enduring affectionate bond that connects one person to another across time and space” (p. 142). A secure attachment gives a person unrestrictive permission to explore their environment and has been shown to influence academic achievement. For attachment to be effective in schools, instructors “must connect and care” (p. 150) for the student, creating a bond.

----------------------

IR/PR 4 Type of Talk: Ideas for Research Studies/Preliminary Research Reports, Author: Katie Bjorkman

Talk Title: The Mathematics of Transfer: Who are the Gatekeepers?

Keywords: Community College, Equity, Institutional Structures

Abstract: Community college students represent a large proportion of students in the US and underrepresented groups are overrepresented among community college students. Many of these students intend to transfer to a 4-year institution and complete a 4-year degree with community college providing a low-cost entry into higher education, yet students lose an average of 43% of earned credits upon transfer. Mathematics courses, in particular, tend to act as gatekeeper courses with strict course sequences so that failing to transfer a mathematics course may have disproportionate impact. Richard Bland College is a two-year school with more than 40 transfer partners including public and private 4-year colleges within Virginia and surrounding states. The purpose of this proposed research is to determine:

(1) what individuals at transfer-accepting institutions determine if/how a mathematics course transfers and

(2) what criteria they use to determine if/how a mathematics course transfers

within the context of the roughly 40 transfer partners and 11 college-credit mathematics courses of Richard Bland College.

The intent of this presentation is to elicit ideas for how best to structure the study to maximize response rates and their validity and to refine the research questions asked.

----------------------

RR 5 Type of Talk: Research Report, Author: Praveen Chhikara and Rochelle Gutiérrez

Talk Title: "Understanding mathematics professors' instructional goals related to guided reinvention and creation" 

Keywords: Realistic Mathematics Education, Rehumanizing Mathematics, Guided Reinvention, Creation

Abstract: Guided reinvention (Realistic Mathematics Education) provides instructional sequences from “realistic” situations to facilitate students to reinvent “higher” level formal concepts. (Gravemeijer, 2008; Van den Heuvel-Panhuizen, & Drijvers, 2020). The interactivity principle of realistic mathematics education emphasizes the collaborative work of students but in groups, students can develop multiple understandings, some of which might not align with standard knowledge. An example of non-standard knowledge is an alternative axiomatic model, which might be sense-making to students. Thus, we consider the creation dimension of Rehumanizing Mathematics that acknowledges students’ conceptions and diverse forms of expressing mathematics (Gutiérrez, 2018). Juxtaposing guided reinvention with creation can help each other’s use. Guided reinvention is a pedagogical practice that offers concrete guidelines for teachers, whereas creation does not. Creation, on the other hand, is a theoretical tool that honors and supports space for teachers and learners to develop alternative, multiple understanding, and even non-standard knowledge, whereas guided reinvention does not encourage it. Thus, both concepts can help expand each other.

To obtain empirical evidence, our ongoing qualitative case study of a single mathematics professor in India explores his teaching goals that can be related to guided reinvention and creation. The research participant was one of ten participants of a professional development program that we organized. The program centered on guided reinvention and creation for university mathematics professors. One out of three interviews was focused on guided reinvention, another interview centered on creation, and the final interview centered on both concepts. The last interview consisted primarily of “member checking,” for which we included our first two interview interpretations in the questions to the participant. The questions also intended to identify the resistance to guided reinvention and creation.

During the data analysis, guided reinvention and creation are providing us a lens to understand the participant’s perceptions of his role to facilitate students for formal concepts and students’ alternative conceptions (Fujii, 2020), conjecture making, and collaborative learning. Preliminary findings suggest that the professor finds it essential to provide realistic scenarios to students but thinks it becomes difficult to find such scenarios at “higher” abstraction levels. We identified that, for this professor, it was important to alert his students about informal writing so that they can avoid “errors,” but doing so might prevent guided reinvention or creation. The findings suggest guided reinvention and creation can be developed into an integrated framework, which can be used in future studies.

----------------------

RR 5 Type of Talk: Research Report, Authors: Andrew Richman, Heather Ortiz, Ahsan Chowdhury, and Eric Henry

Talk Title: Multiple pathways and corequisite supports in early undergraduate mathematics: Supporting change in Arkansas

Keywords: mathematics pathways, corequisite support, developmental mathematics, gateway mathematics

Abstract: Traditionally, early undergraduate mathematics has consisted of a single pathway designed to prepare students for calculus. For students unprepared for college-level work, this pathway begins with a series of precollege-level algebra classes. Recent evidence suggests critical limitations of this structure. Students required to enter coursework at the precollege level, many of whom are historically marginalized or from low-income communities, often leave before acquiring college credit (Barnett & Reddy, 2017; Ganga et al., 2018). Further, non-STEM intending students find that it is not an effective preparation for their fields of study and eventual professions (Bickerstaff et al., 2018; Ganga & Mazzariello, 2018).

In response, a call to action has emerged (AMATYC, 2021; Saxe & Braddy, 2015). This call insists on replacing stand-alone prerequisite developmental mathematics classes with corequisite classes taken concurrently with introductory college-level classes. It further challenges institutions of higher education (IHEs) to develop multiple mathematics pathways, such as quantitative reasoning and statistical reasoning, aligned to programs of study. While these changes have taken hold in many IHEs across the nation, others have struggled to implement them. National efforts have emerged to support states and IHEs undertaking these changes (e.g., Launch Years Initiative | UT Dana Center, 2023). These efforts are accumulating knowledge about critical factors that support and inhibit these changes.

Disseminating these lessons is a necessary step for broad-based reform. This presentation contributes to this dissemination. It reports on a three-year initiative by The Charles A. Dana Center at The University of Texas at Austin to support statewide implementation and scale of mathematics pathways and corequisite supports in Arkansas. This initiative represented a multi-dimensional approach to implementation and scale that emphasized state, regional, and campus-level technical assistance support and engagement across broad stakeholder groups. The Dana Center facilitated multi-institution convenings, targeted workshops, and regional coordinators to work directly with institutions to support holistic change. Ten interviews and focus groups were conducted with faculty and administrators from fourteen two- and four-year IHEs to explore the most influential state- and campus-level factors that supported and inhibited change. We will present these factors and examine the Dana Center’s theory of scale (Ortiz & Cook, 2019) to determine the extent to which the new data is explained by this theory, and, where appropriate, introduce other relevant theories of organizational change that fill gaps in this explanation. Investigating the Dana Center’s theory of change and statewide reform efforts in Arkansas will deepen our collective understanding of student success reform and change factors that can support and inhibit math pathways and corequisite work.