Spontaneous Usage of Different Shortcuts Based on the Commutativity Principle

Robert Gaschler,Alexandra Eichler,Hilde Haider,Peter A Frensch,Bianca Vaterrodt

doi:10.1371/journal.pone.0074972

Robert Gaschler, Alexandra Eichler + Show 3 more

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https://doi.org/10.1371/journal.pone.0074972

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Abstract

Based on research on expertise a person can be said to possess integrated conceptual knowledge when she/he is able to spontaneously identify task relevant information in order to solve a problem efficiently. Despite the lack of instruction or explicit cueing, the person should be able to recognize which shortcut strategy can be applied – even when the task context differs from the one in which procedural knowledge about the shortcut was originally acquired. For mental arithmetic, first signs of such adaptive flexibility should develop already in primary school. The current study introduces a paper-and-pencil-based as well as an eyetracking-based approach to unobtrusively measure how students spot and apply (known) shortcut options in mental arithmetic. We investigated the development and the relation of the spontaneous use of two strategies derived from the mathematical concept of commutativity. Children from grade 2 to grade 7 and university students solved three-addends addition problems, which are rarely used in class. Some problems allowed the use of either of two commutativity-based shortcut strategies. Results suggest that from grade three onwards both of the shortcuts were used spontaneously and application of one shortcut correlated positively with application of the other. Rate of spontaneous usage was substantial but smaller than in an instructed variant. Eyetracking data suggested similar fixation patterns for spontaneous an instructed shortcut application. The data are consistent with the development of an integrated concept of the mathematical principle so that it can be spontaneously applied in different contexts and strategies.

Highlights

Given the role of self-guided learning and performance in the development of mathematical abilities and concepts, some recent studies have focused on spontaneous recognition of mathematical aspects of situations [1,2,3]
In the current study we introduce a paper and pencil-based as well as an eyetracking approach to unobtrusively assess spontaneous application of two shortcuts that are procedurally different but are both based on the concept of commutativity
Knowledge of mathematical principles is especially helpful if we recognize without instruction or direct cues when we can apply it for efficient processing of arithmetic problems [4]

Summary

Introduction

Given the role of self-guided learning and performance in the development of mathematical abilities and concepts, some recent studies have focused on spontaneous recognition of mathematical aspects of situations [1,2,3]. Verschaffel and colleagues [4] have called for helping students to become experts in flexibly selecting the most efficient strategy for the current task and social context This often, in the first place, involves recognizing that there is an alternative option for solving the current arithmetic problem. Rather than studying the acquisition of new concepts or strategies, our focus was on factors that determine whether or not a person recognizes and applies a shortcut option without being instructed to do so The advantage of such spontaneous shortcut application is that it might reveal different aspects of the quality of the person’s knowledge than those aspects that can be tapped by more direct testing

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Conclusion