Relating Flexibility in Concept Image and Understanding of Limits and Derivatives

Mathematics educators accept that if students demonstrate procedural knowledge - how to get to an answer in mathematics - it does not necessarily imply that these students understand why the procedure works (e.g. Tall, 1992; Rittle-Johnson & Alibali, 1999; Hiebert, 2013). In contrast, if students have conceptual knowledge, we consider they do understand these principles. Conceptual knowledge can be described as "Explicit or implicit understanding of the principles that govern a domain and of the interrelations between pieces of knowledge in a domain" (Rittle-Johnson & Alibali, 1999, p. 175). One of the ways that students demonstrate conceptual knowledge is by procedural flexibility, where students have "knowledge of multiple ways to solve problems and when to solve them" Rittle-Johnson & Star, 2007). Tall and Vinner (1981) defined conceptual understanding of ;the necessary procedures as the concept image, the "total cognitive structure that is associated with the concept" including ''all the mental pictures and associated properties and processes" that are "built up over the years through experiences of all kinds, changing as the individual meets new stimuli and matures" (p. 152).