Promoting Students' Ability to Think Conceptually in Calculus

Abstract

An overview is given of three conceptual lessons that can be incorporated into any first-semester calculus class. These lessons were developed to help promote calculus students' ability to think conceptually, in particular with regard to the role that infinity plays in the subject. A theoretical basis for the value of these lessons is provided, and both indirect and direct evidence, that demonstrates that these lessons were successful in beginning to achieve their goal, is described. Ideas for how to incorporate these lessons into the standard calculus curriculum are presented as well as thoughts on how to further improve the lessons' ability to promote students' conceptual understanding.

Bridging the Gap Between the Derivatives And Graph Sketching in Calculus

“…Researchers revealed that students faced difficulties in graphical interpretation of the derivative which occurred on a straight line or complicated curves (Planinic, Ivanjek, Susac, & Milin-Sipus, 2013), computing the slope of a tangent line from the graphs (Dominguez et al, 2017), lack of visualization ability on the conceptual image of functions (Zerr, 2010), cognitively combining with the interval information of the derivatives and critical point(s) on graph (Borji et al, 2018;Hayfa & Ballout, 2015;Orhun, 2012). As a result, students experiencing these difficulties have problems to sketch the graph of a given (polynomial or rational) function even though they may be able to solve for its derivative and identify its direction, relative extrema, curvature, inflection points and/or asymptotes.…”

“…Moreover, this unfavourable result was also reported by Hung, Huang, and Hwang (2014) in which the learning achievements of the e-learning group and the traditional instruction group did not have a significant difference. Furthermore, sketching a graph of rational function involves mastering the concept infinity via limits (both infinite limits and limits at infinity), which is another confusing abstract concept for students (Zerr, 2010).…”

“…Typical first-level calculus course focuses on the fundamental theorem of calculus and covers topics like limits, rules of derivative and integrations, and their applications. Most students are doing well for the topics like limits, and rules of derivative and integrations that are predominantly solvable procedurally; even if they fail to grasp the conceptual knowledge well (Zerr, 2010). Nonetheless, many of them have a hard time coping with questions related to the applications of these concepts.…”

“…There appears a positive and long-lasting impact of designing an embodied learning environment (Duijzer, Heuvel-Panhuizen, Veldhuis, Doorman, & Leseman, 2019). When it comes to the teaching and learning of calculus, designing problem-based learning that focuses on the infinite concept that promotes capability of the students to think calculus conceptually (Zerr, 2010); and the importance of balanced emphasis among the algebraic, numeric, and graphic representations of functions under investigation (Çetin, 2009) is the emphasis for a meaningful learning process.…”

“…The tool designed is named Graph Puzzle (GP) (Liew, Chen, Tuh, Ling & Ahmad Bakri, 2019). GP is a type of board-game tool innovated for an embodied learning environment with a balanced emphasis among the algebraic, numeric and graphic presentations of functions and their derivatives so that capability of the students to think calculus conceptually could be promoted (Duijzer et al, 2019;Hong & Thomas, 2015;Zerr, 2010;Çetin, 2009). On top of this, the power of technology is also exploited to expand the capacity of the amount of content that could be included in it.…”