Invited Paper Session Abstracts - Implications for Practice: Applying Education Research to our Shared Disciplinary Work

This paper describes ways of thinking students employ when they choose to use calculations or produce algebraic expressions to respond to mathematical tasks and their expectations regarding the meanings of what they produce. My findings suggest that students’ reasoning in symbolization activity is often guided by perceptual features of tasks, such as the numbers explicitly given in prompts and key words students identify. I describe the construct “emergent symbol meaning” as a potentially productive way of thinking in symbolization activity based on synthesizing prior work and the results of this study. I close by making connections between similar work in analyzing students’ ways of thinking about graphs and discussing how my work contributes to the common instructional goal of promoting connections between representations.

Many transition to proof courses and early proof courses include sections on logic. Logic is often taught using abstract syntax (truth tables or Venn diagrams). We have developed and researched some alternative teaching approaches in which students compare mathematical statements to learn about logical structures. We find this helps students understand the questions that logic answers as they encounter the standard logical tools. Furthermore, we have learned much about how untrained undergraduate students read mathematical language and find that they find it easier to make sense of quantified statements (“for all integers x, x is even or x is odd”) than they do non-quantified statements (“15 is even or 15 is odd”). Since so many of the statements of interest in mathematics are quantified, we argue that approaches to logic instruction may benefit from studying connectives (or, and, if then) using quantified mathematical statements and attending to the underlying set structure.

In recent years, there has a been a push for undergraduate mathematics classrooms to move away from purely lecture to a model where students are more actively engaged in their own learning. Such a transition is hardly a trivial task and requires robust instructional supports. Our recent work endeavors to adapt research-based supports from the K-12 level to the undergraduate abstract algebra classroom. In this report, we share preliminary results from a design-based research project directly aimed at adapting best practices to this new setting. We share several illustrations of how particular teaching routines (author, year; Teachers Development Group, 2013) can productively unfold in a proof-based setting.

Our research project, the Calculus Videos Project (https://calcvids.org/), focuses on designing and evaluating the effectiveness of instructional videos for introductory calculus. We draw on research about student thinking and learning in the design of our materials. The structure and imagery in our videos is based in ideas of quantitative and covariational reasoning—specifically, we focus on coordinating amounts of change of quantities and representing these quantities in systematic ways. We also draw on the idea of intellectual need to structure the video watching process—specifically, we design intellectual need-provoking tasks for each concept and ask students to attempt these tasks prior to watching the videos.

Despite mounting recommendations to adopt more evidence-based methods in STEM classes, change at scale is hard to achieve, even for instructors who desire to make such changes. Studies indicate that successful institutional change initiatives all tend to share common features: they provide ongoing support, align with the views participants, and account for the existing landscape of institutional values. In this work, we modify an existing model of change theory by adding a new dimension, peer observation, in order to foster community, collaboration, and teacher reflection. We share preliminary outcomes from the REFLECT project (Redesigning Education For Learning through Evidence and Collaborative Teaching), which aims to help STEM instructors sustain the adoption of new pedagogies by engaging them in reflective peer observation. A tangible product of this work is a new classroom observation protocol designed for formative assessment of evidence-based educational practices. Early testing of the protocol has shown positive faculty outcomes with regard to personal reflection.

Introductory mathematics courses, specifically calculus and precalculus, function as pathways into STEM majors, making student success in calculus particularly impactful. However, the quality of instruction in calculus and precalculus courses has been found to negatively impact persistence in STEM majors, which can be especially detrimental for Black and Latinx students managing stereotypes of ability and feelings of isolation in mathematical spaces. As part of a larger study, COURAGE (Challenging, Operationalizing, and Understanding Racialized and Gendered Events) in Undergraduate Mathematics, this presentation characterizes forms of benevolence in calculus and pre-calculus instructors’ perceptions of classroom practices, which contrasts Black and Latinx students’ perceptions of the same practices as potentially racialized and gendered. Our report details how instructors perceived themselves as lacking agency to reform calculus instruction in ways that supported the student retention in the calculus sequence. However, Black and Latinx students identified multiple ways that instructors could exercise their classroom authority to mitigate gatekeeping effects of instruction with racialized-gendered impacts.