The opaque nature of generic examples: The structure of student teachers’ arguments in multiplicative reasoning

Kirsti Rø,Kristin Krogh Arnesen

doi:10.1016/j.jmathb.2019.100755

Kirsti Rø, Kristin Krogh Arnesen

Open Access

https://doi.org/10.1016/j.jmathb.2019.100755

Copy DOI

Abstract

The study aims to explore the structural aspects of generic examples, to get better insight into what makes them potentially opaque for learners. We have analyzed 27 written arguments, for which student teachers (grades 1–10) were asked to use a generic example to prove a given statement in multiplication. Using Toulmin’s framework, we developed five categories of arguments based on their structure: examples, empirical arguments, leap arguments, embedded arguments, and other arguments. Also, we conclude that none of the student teachers provided arguments that we recognize as complete generic examples. The results bring us to a discussion about features of generic examples making them difficult to come to grips with, having implications for how teacher educators can support student teachers’ learning to prove. From this, we propose a definition of generic examples that attends to the criteria suggested in previous research, yet, emphasizing their structural nature.

Highlights

Reasoning and proving are central aspects of mathematics as a discipline, and many researchers have argued that they should be a central part of school mathematics at all grades and in all topics (e.g., Balacheff, 1988; Ball, Hoyles, Jahnke, & Movshovitz-Hadar, 2002; Stylianides, 2007, 2008)
We show the number of arguments identified, we provide a description of the category’s characteristic features and give an example from the student teachers’ arguments, both in its written form and as a Toulmin scheme
It is remarkable that none of the student teachers provided arguments that we recognized as complete generic examples

Summary

Introduction

Reasoning and proving are central aspects of mathematics as a discipline, and many researchers have argued that they should be a central part of school mathematics at all grades and in all topics (e.g., Balacheff, 1988; Ball, Hoyles, Jahnke, & Movshovitz-Hadar, 2002; Stylianides, 2007, 2008). In exploring ways to teach proof, several studies have shown the crucial role that a teacher plays in helping students identifying the structure of a proof, presenting arguments, and distinguishing between correct and incorrect arguments (see e.g., Stylianides, 2007). Previous research has identified (student) teachers’ challenges when learning to prove in mathematics, there is still a problem finding ways to better support student teachers’ learning in terms of communicating clearly what makes an argument a mathematical proof. We focus on identifying the challenges stemming from the structural nature of proofs, rather than from the student teachers’ beliefs or capacities in mathematical reasoning and proving

Objectives

Methods

Results

Conclusion