Academic journal article The Mathematics Enthusiast

Reasoning-and-Proving within Ireland's Reform-Oriented National Syllabi

Academic journal article The Mathematics Enthusiast

Reasoning-and-Proving within Ireland's Reform-Oriented National Syllabi

Article excerpt

Introduction

Mathematicians have argued that proof is the material with which mathematical structures are constructed (Schoenfeld, 2009). Proof is also becoming instantiated as an important component through which one learns school mathematics (Common Core State Standards Initiative [CCSSI], 2010; Epp, 1998; Hanna, 2000; Martin et al., 2009; National Council of Teachers of Mathematics [NCTM], 2000). Due to the acknowledgement of proof as important in the practice and learning of mathematics researchers are beginning to analyze this practice (Hanna & de Bruyn, 1999) or practices related to it such as reasoning-andproving (RP) (Davis, Smith, Roy, & Bilgic, 2013; Stylianides, 2009) in textbooks. While analyses of standards at the state level in the United States for reasoning (Kim & Kasmer, 2006) or for conjecturing and proving (Porter, McMaken, Hwang, and Yang, 2011) have been conducted, we know little about the standards in other countries with regard to proof or its related actions of pattern identification or conjecture formulation. National standards play an important role in shaping classroom practices in the United States (Cogan, Schmidt, & Houang, 2013), Ireland (National Council for Curriculum and Assessment [NCCA], 2012), and other countries (Eurydice, 2011). The curriculum documents at the center of this study are two national syllabi designed to describe the learning expectations for students ages 1218 studying mathematics in Ireland. These frameworks have been recently developed to drive a nation-wide reform of the Irish secondary mathematics system. This study introduces readers to a framework and methodology for examining RP in national curriculum documents and addresses the dearth of research of this type by enumerating the nature of RP within these two documents. The analysis of these documents for RP expands our knowledge of the nature of this important process in national curriculum documents and adds to our understanding of the potential effectiveness of RP in this reform. More broadly, this paper makes suggestions for how RP can be more interwoven into curriculum frameworks.

Background

Centrality of Proof-Related Constructs in Mathematics and Mathematics Education

Mathematicians have pointed out that the act of constructing proofs is essential to the practice of mathematics (Ross, 1998) or as Schoenfeld (2009) has stated, "If problem-solving is the 'heart of mathematics', then proof is its soul (p. xii). National curriculum documents in the United States emphasize the centrality of proof in the learning of mathematics. Specifically, the Principles and Standards for School Mathematics (PSSM) (NCTM, 2000), which has driven reform in the United States for over a decade breaks down the instruction of mathematics into five content areas and five processes, one of which is reasoning and proof. The Common Core State Standards for Mathematics (CCSSM) (CCSSI, 2010) begins their document with eight standards for mathematical practice, which they argue should be present as students engage in the learning of mathematics. The third standard advocates for students' construction of mathematical arguments or proof as well as the critiquing of arguments constructed by others. Other countries have also emphasized the importance of proof in the instruction of mathematics. For example, each of the syllabi documents produced by the NCCA in Ireland break mathematics content down into five different mathematics content strands. At the end of each of these content strands is a topic with the title: synthesis and problem solving skills. This topic includes the identification of patterns, formulation of conjectures, and explanation/justification of assertions. These three actions comprise the related processes of reasoning-and-proving as defined by Stylianides (2009).

Ireland's Secondary Educational System

The secondary educational system in Ireland consists of three components. The first component is called the junior cycle and lasts three years. …

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