Abstract:

Since 2012, U.S. student enrollment in post-secondary courses declined steadily, yet enrollment in distance-learning courses increased each year. Currently over 33% of post-secondary students take at least one such course each semester and almost 16% are exclusively distance-learning students (U.S. Department of Education, 2013-2018). Much of the research on online learning currently focuses on student engagement and affect in online settings, characteristics of successful online students, and comparisons between student performance in similar courses delivered online vs. face-to-face. We argue that what is missing in the research literature, and an area critical for improving student success in online mathematics courses, is a focus on the quality of mathematical meanings students develop while engaged in online learning. Are students constructing deep, flexible, and coherent meanings for mathematical ideas proven necessary for success in later courses? What aspects of the online curriculum support or hinder students’ construction of these important meanings?

The context for our research is an online precalculus course created based on our work developing, testing, and refining face-to-face Precalculus materials designed to improve retention rates and students’ success in Calculus. Our face-to-face curriculum is a student-centered problem-solving approach with careful scaffolding to support students in building robust meanings for important ideas such as rate of change, variation and variables, proportional reasoning, and connections between representations of mathematical ideas. Students using our face-to-face materials in 11 U.S. universities have the highest gains and mean score on the Precalculus Concept Assessment (PCA) (Carlson, Oehrtman, & Engelke, 2010) compared to students using other curriculum materials. The PCA is a validated instrument assessing the conceptual foundations for Calculus and can predict success in Calculus courses.

This paper outlines our design process and principles for creating the online course. Lessons are written so that students take an active role in the development of important meanings instead of just practicing techniques demonstrated in text or on a video. Lessons are also designed to promote students’ quantitative reasoning skills (Thompson, 1990, 2012) (their ability to recognize attributes of objects so that those attributes might take on measurements and be related to each other), covariational reasoning skills (Carlson, Jacobs, Coe, Larsen, & Hsu, 2002; Saldanha & Thompson, 1998; Thompson & Carlson, 2017) (their ability to unite two varying quantities and track how they change together), and to encourage their development of emergent symbol meaning (O’Bryan, 2018, 2019) (their belief that operations they perform, the expressions they write, and the orders of operations they indicate have quantitative significance). We share promising quantitative data over four semesters of piloting (including less than 10% of students withdrawing from the course and 86% of students completing the course earning a C or better) and shifts on student affective characteristics (such as confidence and persistence). We conclude by discussing the next steps for our research, including tracking students’ success in future courses, designing new iterations of the course, and looking more closely at how shifts in students’ beliefs about mathematics connect to shifts in other affective categories.

Keywords:

Online learning, calculus preparation, curriculum design.