Abstract

An increasing number of learning goals refer to the acquisition of cognitive skills that can be described as ‘resource-based,’ as they require the availability, coordination, and integration of multiple underlying resources such as skills and knowledge facets. However, research on the support of cognitive skills rarely takes this resource-based nature explicitly into account. This is mirrored in prior research on mathematical argumentation and proof skills: Although repeatedly highlighted as resource-based, for example relying on mathematical topic knowledge, methodological knowledge, mathematical strategic knowledge, and problem-solving skills, little evidence exists on how to support mathematical argumentation and proof skills based on its resources. To address this gap, a quasi-experimental intervention study with undergraduate mathematics students examined the effectiveness of different approaches to support both mathematical argumentation and proof skills and four of its resources. Based on the part-/whole-task debate from instructional design, two approaches were implemented during students’ work on proof construction tasks: (i) a sequential approach focusing and supporting each resource of mathematical argumentation and proof skills sequentially after each other and (ii) a concurrent approach focusing and supporting multiple resources concurrently. Empirical analyses show pronounced effects of both approaches regarding the resources underlying mathematical argumentation and proof skills. However, the effects of both approaches are mostly comparable, and only mathematical strategic knowledge benefits significantly more from the concurrent approach. Regarding mathematical argumentation and proof skills, short-term effects of both approaches are at best mixed and show differing effects based on prior attainment, possibly indicating an expertise reversal effect of the relatively short intervention. Data suggests that students with low prior attainment benefited most from the intervention, specifically from the concurrent approach. A supplementary qualitative analysis showcases how supporting multiple resources concurrently alongside mathematical argumentation and proof skills can lead to a synergistic integration of these during proof construction and can be beneficial yet demanding for students. Although results require further empirical underpinning, both approaches appear promising to support the resources underlying mathematical argumentation and proof skills and likely also show positive long-term effects on mathematical argumentation and proof skills, especially for initially weaker students.

Highlights

  • Today, educators in formal and informal learning settings deal with increasingly complex skills as learning goals, such as argumentation or complex problem solving (e.g., National Research Council, 2012; Osborne, 2013; Greiff et al, 2014), which require the availability, coordination, and integration of multiple underlying cognitive resources.Research from educational psychology focusing on the support of complex skills has long been examining part- and whole-task approaches for learning (e.g., Naylor and Briggs, 1963; Anderson, 1968; Lim et al, 2009)

  • In a quasi-experimental study with university mathematics students, we investigated whether supporting each of the four resources sequentially one after the other or supporting the resources concurrently in the context of mathematical proofs yields learning gains on the resources as well as on overall mathematical argumentation and proof skills

  • No indications for violations against normal distribution or equality of variances were found for resources and mathematical argumentation and proof skills

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Summary

Introduction

Educators in formal and informal learning settings deal with increasingly complex skills as learning goals, such as argumentation or complex problem solving (e.g., National Research Council, 2012; Osborne, 2013; Greiff et al, 2014), which require the availability, coordination, and integration of multiple underlying cognitive resources.Research from educational psychology focusing on the support of complex skills has long been examining part- and whole-task approaches for learning (e.g., Naylor and Briggs, 1963; Anderson, 1968; Lim et al, 2009). Respective research has focused on different parts of larger, complex tasks, which can be decomposed into a number of discrete subtasks, and how those can be learned and transferred to the overall task. It did not focus on different (dispositional) resources possibly required for a specific skill. The second approach may overwhelm students with the resourcebased cognitive skill and its underlying resources all at once, yet allows an integrated learning of the resources in authentic settings that support the integration of the resources and already trains their concurrent application within mathematical argumentation and proof tasks

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