The analysis of the transplantability as a tool for the development of mathematical discourse

The topic of this dissertation is a study conducted in mathematics education. The aim is to investigate the learning opportunities offered by dynamic explorative objects (DEO) to deal with algebraic notions concerning equations for high schools’ students with a history of severe and persistent difficulties in mathematics. The investigation leans on previous findings on students’ difficulties and misconceptions in tackling the notions of unknown and variable, equal sign and algebraic expressions, that together are basic in the processes of equation solving. I frame the investigation within the Theory of Commognition (Sfard, 2008). Mathematics, in this perspective, is a special discourse about mathematical objects and learning mathematics is seen as becoming participant to such discourse. Thus, investigating teaching and learning processes in this framework means analysing students’ discourse according to four features that are words, visual mediators, routines and endorsed narratives. Two are the main research objectives. According to the will of looking at learning processes within digital environments, the first research objective is leaded by the aim of elaborating tools to guide the design of technology-enhanced activities. Indeed, the commognitive perspective offers refined tools to analyse students’ discourse but does not explicitly deal with the involvement of digital technologies within learning processes. Thus, starting from the instruments offered by the Commognition, the aim is to develop a theoretical integration to investigate learning processes in digital environments. The second research objective involves the analysis of learning processes with respect to algebra and, especially, equations solving. The study is conducted within three cycle of design-implementation-a posteriori analysis phases, according to the design base research (Cobb et al., 2003). It is developed through the design of a teaching sequence within DEOs to introduce and deal with the notions overmentioned and its implementation to look at students’ learning processes according to a discursive perspective. The data collection, carried out in two different implementation’s sessions involves 23 students of 10th grade classrooms from different high schools who recognize themselves as having a history of severe and persistent difficulties in mathematics. They have come, in the afternoon, to an out-of-school learning center once a week, for 5 or 6 weeks. Data available are audio-video recordings of the sessions, together with screen recordings of the tablet on which students interact with DEOs and accomplish the proposed task situation, and students written productions. The study leads to the integration in the Theory of Commognition of some theoretical constructs. Specifically, I expand upon the notions of metaphor and transplanting proposed by Sfard (2008) to develop the construct of analysis of the transplantability. Moreover, I elaborate the notion of bridging discourse and blended discourse. These theoretical tools guide the design of new DEOs and task situations, as well as the elaboration of teacher’s intervention. In addition, they offer new insight into the processes of learning algebra through digital environments, focusing on the treatment of indeterminate quantities and the relation of equality between algebraic expressions. This dissertation is structured as follows. In the first chapter I position our study within the literature concerning algebra and its teaching and learning processes, delving into the issues that such multifaceted topic carries along and pointing out the main students’ difficulties with respect to this mathematical content. Relying on the previous research on mathematics education and digital environments, I stress the potentialities of technology-enhanced activities I want to take advantage of. In chapter 2 I present the theoretical framework within which has been carried out this study, that is the Theory of Commognition. I discuss the position taken on by the commognitive perspective with respect to mathematics and its teaching and learning. In describing the theoretical tools offered by this theory to investigate the processes of teaching and learning, I put the light on some theoretical constructs I have needed to elaborate and refine in order to conduct the investigation. Thus, in conclusion of this chapter, I present the two research objectives that have guided this study. One is referred to the theoretical elaborations I aimed at developing, the other to the analysis of learning processes with respect to the algebraic discourse. In chapter 3 I introduce the methodology used to deal with my research objectives. Together with this, I present the timeline of the study and the evolution of our research objectives according to the progression of the investigation. In chapter 4 I propose a proposal to fulfil the first research objective. I present the notion of analysis of transplantability, as a theoretical construct to guide the design of digital environments and related task situations in terms of discursive aims. I propose a classification of discourses that can allow to clarify the aims of the elaborated design as well as to analyse some processes emerged during the experimentations. Specifically, I propose the construct of bridging discourse as the discourse about the relation between two different discourses, that in our case have been the discourse related to the DEO and the mathematical discourse. This specific construct has guided the introduction of new task situations in our teaching sequence, involving the notion of silent video. In chapter 5 I present and analyse the technology-enhanced activities I have developed within this study. Specifically, I use the theoretical developments introduced in Chapter 4 to provide the a priori analysis of each DEO involved in our teaching sequence, together with the task situations designed and the tutor’s intervention elaborated. In chapter 6 I provide information about the data collection, focusing on how data have been collected and analysed. The data analysis is proposed in chapter 7 and in chapter 8. In the first one I have collected the analysis of some episodes taken by the first cycle of implementation session, focusing on the features of emerged discourse and their relationship with the algebraic one. Chapter 8 is devoted to the analysis of episodes from the third cycle of implementation and is directed to the processes related to equations solving involving both DEOs and equations. The concluding chapter (9) is dedicated to the research objectives and the reached proposals to fulfil them, highlighting the theoretical contributions offered by this study and contextualizing our findings within the commognitive framework. Moreover, I discuss possible didactical implications of our findings, focusing on the reinvestment of our results in a classroom context. In conclusion, I describe possible directions for further research, that include the refinement of the theoretical elaborations according to their use for the design and elaboration of other DEOs, as well as the investigation about the role of the teacher in promoting the transition from discourses related to DEO to the mathematical discourse. In addition, considering the context in which our study has been conducted, and especially the students involved, I could use our findings as a starting point to characterize remedial interventions.