Mathematical reasoning competency and digital tools

In recent decades, among others, two areas of focus have emerged in mathematics education research. The first emphasizes mastering mathematics as competencies, proficiency, or literacy, as opposed to merely knowing mathematical facts and skills (Niss, 2016). The second area explores the use of digital technologies in teaching and learning mathematics (Artigue & Trouche, 2021).  The Danish competency framework, known as the KOM-framework, outlines eight mathematical competencies that describe activities related to doing and dealing with mathematics (Niss & Højgaard, 2019). Among the eight competencies, mathematical reasoning competency, which involves analyzing or producing arguments to justify mathematical claims, is the focus of this project (Niss & Højgaard, 2019). The KOM-framework, along with the use of digital tools, is featured in the Danish mathematics curricula across primary, lower secondary, and upper secondary education (Danmarks Evalueringsinstitut [EVA] 2009). In addition, digital mathematics tools, such as dynamic geometry environments and computer algebra systems, increasingly integrate functionalities from both geometry and algebra (Freiman, 2014; Sutherland & Rojano, 2014), offering new possibilities and complexities that often surpass the understanding of laypersons. This study reported in this dissertation has both practical and theoretical aims. Practically, it investigates how the integration of digital tools in geometry and algebra can support students’ reasoning processes in lower secondary mathematics education and emphasizes enabling students to exercise their mathematical reasoning competency rather than developing the competency as such. Theoretically, the study seeks to promote sustainable theoretical development by linking the KOM-framework with international mathematics education research. This is achieved by adopting a networking perspective (Prediger, Bikner-Ahsbahs, et al., 2008) on theory development.  Design research (Bakker, 2018; Cobb et al., 2003; Gravemeijer & Prediger, 2019; McKenney & Reeves, 2014). as the methodological framework for this project, guiding the collection and analysis of data, as well as the establishment of learning situations that incorporate mathematical reasoning competency, digital technologies, and the variable as a generalized number. A microworlds (Hoyles, 1993) of variable points along with task sequences were developed, aiming for students to exercise mathematical reasoning competency by investigating basic algebraic expressions and their structural implications in the dynamic behavior of variable points.  In addition to the KOM-framework, the project employs other theoretical perspectives, such as the instrumental approach to mathematics education (IAME) (Guin & Trouche, 1998) and the elaborated notion of a scheme (Vergnaud, 1998b) in relation to the scheme-technique duality (Drijvers et al., 2013), and Toulmin’s (2003) argumentation model. These perspectives are used for analyzing, describing, and explaining empirical data, particularly regarding students’ use of digital tools and their mathematical reasoning competency.  The dissertation comprises six research papers and an accompanying kappa that provides the theoretical background, methodology, additional analysis, and results. Paper 1 is a literature review that identifies potential tools within GeoGebra, a dynamic geometry and algebra environment, for student justifications of the variable as a generalized number, which informed the task design. Papers 2 through 5 analyze empirical data from students working with different tasks and the development of an analytical tool for instrumented justification. Paper 2 introduces the initial connection between the KOM and IAME frameworks, interpreted through Toulmin’s model, leading to the creation of an analytical tool. Paper 3 refines this tool, defines instrumented justification, and describes students’ justification processes using artifacts. Paper 4 focuses on the goal of student tool use within a task that is further developed, examining its potential and challenges for exercising reasoning competency. Paper 5 emphasizes the scheme aspect of the analytical tool, analyzing Vergnaud’s (1998) scheme components and elaborating on how students’ conceptual knowledge integrates into their instrumented justification processes. Paper 6 addresses the notion of foreground and background theory within the networking of theories perspective.  The dissertation contributes three design principles for task design that support students’ exercise of reasoning competency, a microworld for exploring and justifying the dynamic behavior of variable points, and associated tasks. It also elaborates on reasoning competency in students’ instrumented justification processes and the scheme-technique duality and provides suggestions for supporting these processes in the classroom and through task design. Additionally, it identifies a hybrid conception between continuous and discrete understandings of variables in predicting variable behavior within the dynamic geometry and algebra environment and suggests theoretical links between the KOM and IAME frameworks as potentials for further theoretical networking.