How can Students Generalize the Chain Rule? The Roles of Abduction in Mathematical Modeling

Abstract

The purpose of this study is to design a modeling task to facilitate students' inquiries into the chain rule in calculus and to analyze the results after implementation of the task. In this study, we take a modeling approach to the teaching and learning of the chain rule by facilitating the generalization of students' models and modeling activities. We assumed abductive reasoning to be one of the key factors which can support the generalization of students' models and modeling activities. We believe that analogical reasoning and diagrammatic reasoning are key factors in fostering students' use of abduction. As a result, we determined that the students' models and modeling activities were generalized to the chain rule by their use of abductive reasoning, and the students found the chain rule to be a generalized rule for describing changes of various quantities.

How Can Mathematical Modeling Facilitate Mathematical Inquiries? Focusing on the Abductive Nature of Modeling

“…Building a model of a problem situation (or mathematizing B->C in Figure 1) involves searching mathematical rules to explain the observation results of a problem situation and interpreting a given situation using mathematical objects or procedures (Park & Lee, 2016). Given that, we can consider that abductive reasoning intervenes in building models.…”

How can Students Generalize Examples? Focusing on the Generalizing Geometric Properties

“…Most figures used in Euclidean geometry are both an icon and a diagram, as conventional rules apply to represent relationships among constituent elements (Hoffmann, 2004). An experiment on a diagram is the transformation of representations based on conventional rules; hence, the results of an experiment are assumed to have some degree of rationality (Park & Lee, 2016;Hoffmann, 2004). Accordingly, Otte (2006) took the role of diagrammatic reasoning into account when supporting the identification of hidden relationships by students in order to form new general properties.…”

Tendencias en la educación matemática 2022

“…En educación matemática las investigaciones sobre el razonamiento abductivo reconocen su importancia en la formulación de conjeturas, demostraciones y generalizaciones plausibles (Rivera y Becker, 2007). Estos estudios se han enfocado en los contextos de geometría (Baccaglini-Frank, 2019;Pedemonte, 2018;Pedemonte y Reid 2011), álgebra (Hidayah, Sa' dijah y Sudirman, 2020;Reid, 2003;Rivera, 2018;Rivera y Becker, 2007), cálculo (Park y Lee, 2016) y resolución de problemas (Cifarelli, 1997(Cifarelli, , 2016. Otras han combinado su estudio con otras formas del razonamiento matemático, como abductiva e inductiva (Rivera y Becker, 2007;Radford, 2008), abductiva y deductiva (Shodikin, 2017), y abductiva, inductiva, deductiva y analógica (Conner, et al , 2014;Reid y Knipping, 2010;Soler-Álvarez y Manrique, 2014;Arce y Conejo, 2019;Cervantes-Barraza, Ordóñez-Cuastumal y Carballo-Morales, 2020).…”