Promoting Prospective Mathematics Teachers' Understanding of Derivative across Different Real-life Contexts

Abstract

This study investigates prospective secondary school mathematics teachers' developing conceptions about the meaning of derivative (as rate of change) while working on a series of tasks involving different non-motion real-life contexts. The study was conducted by employing a case study method. The participants of the study were 25 prospective secondary school mathematics teachers enrolled in the mathematics teaching methods course. A series of contextual tasks involving different real-life situations about derivative was designed and implemented in the form of problem-based learning as a regular part of the course. Pre and post questionnaires, written group solutions to the tasks, individual reflection papers, and video-recorded online classroom discussions were the data sources. A constant comparative analysis method was employed in analysing the data. The results showed that prospective teachers have difficulties in interpreting the meaning of derivative in different real-life contexts. While they were frequently using “amount of change” and “slope” interpretations, they started to use “rate of change” interpretation. However, the classroom discussions showed that most of the prospective teachers were using the “rate of change” expression as a memorized term without engaging in its contextual meaning. We made some inferences about the teaching and learning of derivative and also the professional development needs for teachers.

Slope conception as rate of change starter kit: Malaysian pre-service secondary mathematics teachers’ subject matter knowledge in rate of change

“…These misinterpretations portray the inconsistency of both participants regarding the occurrence of variables or quantities involved in rate of change within the situated context. This problem is similar to past studies (e.g., Byerley & Thompson, 2017;Kertil & Dede, 2022;Mkhatshwa & Doerr, 2018), where teachers and students often confused "amount" with "amount of change". In one study, a student named Brian interpreted rate of change as an amount of quantity rather than the changes between quantities.…”

“…There is a lack of thorough research focusing on preservice mathematics teachers' knowledge of rate of change. Existing studies (e.g., Dane et al, 2016;Fothergill, 2011;Kertil, 2014;Kertil & Dede, 2022) among pre-service mathematics teachers primarily focus on broader calculus topics such as derivatives, limits, or integrals. These studies have consistently shown weak knowledge of derivatives conception and a lack of awareness of its connection to rate of change.…”

“…However, most students only know that 𝑓(𝑥) is different from 𝑑𝑦 𝑑𝑥 due to the differentiation process (derivatives), indicating a procedural understanding of how to obtain 𝑑𝑦 𝑑𝑥 rather than understanding the conceptual contexts of 𝑑𝑦 𝑑𝑥 (Hashemi et al, 2014;Makonye & Luneta, 2014;Ubuz, 2007;Weber & Thompson, 2014). Common mistakes reported in past studies show that students tend to verbalize 𝑑𝑦 𝑑𝑥 as amount of quantity, despite knowing that it represents a rate of change, as a result of memorizing derivatives as a rate of change (Byerley & Thompson, 2017;Kertil & Dede, 2022;Mkhatshwa & Doerr, 2018). This further suggests that students do not conceive 𝑓(𝑥 + ℎ) and 𝑓(𝑥) as quantities associated with specific inputs (Weber et al, 2012), leading to their inability to understand 𝑓(𝑥 + ℎ) − 𝑓(𝑥) as the quantity change relative to the change in input (Weber et al, 2012).…”