Understanding instructor support of student prior knowledge in introductory calculus
Date
2024
Authors
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Journal ISSN
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Publisher
University of Delaware
Abstract
Introductory calculus is mandatory for postsecondary science, technology, engineering, and mathematics (STEM) majors (Bressoud, 2015); however, many students do not persist through introductory calculus to further STEM study (Rasmussen et al., 2019). One reason for this may be that students struggle with the prior mathematical knowledge required for success (Sadler & Sonnert, 2018), creating a barrier to course completion (Bressoud et al., 2013; Chen, 2013). One way to help students succeed in introductory calculus classrooms may therefore be to support their prior knowledge instructionally. Instructional support of student prior knowledge can be defined as instruction that helps students make connections between current and previous content and engage students prior knowledge in some fashion during instruction (Tobias, 1976). Research suggests that instructional support of student prior knowledge occurs in three ways: Explicit Connections, Writing and Discourse, and Reinforcement of Prerequisite Skills (Campbell et al., 2017; Chappell & Killpatrick, 2003; Miller et al., 2006; Swanson & Collins, 2018). In the context of introductory calculus, however, there is limited research on whether and how instructors use practices aligned with these constructs. Understanding instructional practices that support prior knowledge, their frequency across calculus classrooms, and the extent to which they intersect with practices that support conceptual understanding is an essential first step in taking an incremental approach to improving instruction in ways that support students’ learning and success (e.g., Litke, 2020; Star, 2016). Thus, to better understand current practice, this dissertation aims to understand instructional support of student prior knowledge in postsecondary introductory calculus courses. ☐ To do this, I employ a mixed-methods approach. I first use surveys (n=142) to understand instructional practices reported by introductory calculus instructors across postsecondary institutions. Next, I conducted interviews (n=6) to investigate how instructors describe their instruction to support student prior knowledge. Survey analysis suggests that instructors report engaging more often in practices that support student prior knowledge and less often in providing opportunities for students to engage in activities that could support understanding of required prior knowledge. Additionally, I find that instructors describe reinforcing prerequisite skills when asked how student prior knowledge affects their instruction, in addition to reporting that they engage in practices that utilize explicit connections or discourse and writing when asked about their instruction more generally. This finding could suggest that instructors engage in practices that have the potential to support student prior knowledge without utilizing these practices to do so. Finally, I find four distinct clusters of instructors when analyzing participant responses regarding the frequency of instructional support of student prior knowledge and conceptually-oriented instruction. ☐ The analysis of instructor interviews from each cluster shows differences in their descriptions of instructional support of student prior knowledge and the use of activities that may support student understanding of prior knowledge. Despite differences, all instructors described instructional practices aligned with their cluster description. Interview participants agreed on the critical prior knowledge necessary for success in introductory calculus. Additionally, they spoke both about the practices that they currently enact as well as practices they aspire to enact to support student prior knowledge. For those practices they aspire to, they cited institutional impediments (e.g., class size, classroom space) as reasons they were unable to successfully use them. Finally, my interview participants described both more conceptually-oriented and procedurally-oriented instruction when answering questions about how they support student prior knowledge. This indicates that instructors with various stances toward the course and various instruction modes can engage in instructional support of student prior knowledge. ☐ In my dissertation, I present a theory of action that considers what we know about the skills students need to succeed in introductory calculus and the instructional practices we know to support student understanding to move toward instructional improvement. The next step in my theory of action is to understand what instructors are currently doing in this classroom so that we can suggest instructional improvement based on common current practices. This study begins that work. I also suggest that because introductory calculus is a linchpin course in students’ mathematical understanding (Usiskin, 1999), we must work to incorporate practices that both create conditions for student learning and support student prior knowledge. By finding and exploiting the intersection of these types of practices, we can support student prior knowledge without sacrificing calculus content or course time.
Description
Keywords
Calculus teaching, Instructional practices, Postsecondary education, Cluster description