Idoneidad didáctica de la probabilidad en documentos normativos y materiales curriculares de Educación Secundaria. Implicaciones para la formación de profesores

Abstract

This doctoral thesis focuses on the treatment of probability in Secondary Education curriculum materials in Peru, the implications for the teaching and learning of this content. On one hand, given the crucial role these resources play in the teaching and learning of mathematics, the thesis seeks to address the need to evaluate their adequacy, as well as analyze their influence on the understanding of concepts and anticipate possible difficulties. On the other hand, it addresses the importance of designing and implementing formative actions with teacher trainees with the goal of fostering a critical use of curriculum materials supported by mathematical-didactic knowledge and competencies in the context of probability. The objectives guiding this study are: a) to analyze the representativeness and articulation of the meanings of probability in the normative documents and Secondary Education curriculum materials; b) to review and adapt criteria and indicators of didactic suitability, to generate a tool that allows the evaluation of the study of probability in these documents, considering not only the epistemic dimension, but also the cognitive-affective and instructional; c) to apply this tool to evaluate normative documents and student workbooks on probability in Secondary Education; d) to plan and implement formative actions in which prospective teachers can become familiar with the use of the didactic suitability guide to assess the relevance and potential of these resources for their teaching management. To address the established objectives, firstly, a systematic review of the literature is conducted on three essential and intertwined elements of mathematics education: the didactic, probability, and teacher training (Chapter 1). Given the research interest, the theoretical framework employed is the Onto-semiotic Approach (OSA) to Mathematical Knowledge and Instruction (Godino et al., 2007; 2019). This framework facilitates the theoretical and methodological tools with which to address the research problem: the notion of pragmatic meaning (Batanero, 2005; Godino et al., 2007), the theory of didactic suitability (Godino, 2013) and the model of mathematical-didactic knowledge and competencies of the teacher (Godino et al., 2017). These tools are described in Chapter 2 of this dissertation. The study adopts a predominantly qualitative approach. To develop the didactic suitability assessment guides, content analysis (Cohen et al., 2011) is applied to normative documents, research on the teaching and learning of probability, and research on the analysis of curriculum materials (norms and textbooks, essentially). In the part of the research dedicated to teacher training, the approach typical of design-based research is followed, according to the proposal of didactic engineering of the OSA (Godino et al., 2014). This methodology detailed in Chapter 2, comprises four distinct phases that guide the development of the research: preliminary study, design, implementation, and retrospective evaluation or analysis. In the preliminary phase, the meanings of probability in the current official curriculum materials, such as curricular guidelines and workbooks corresponding to the sixth cycle (first and second grade of Secondary Education), are explored and the didactic suitability assessment guide that we will use in the following phases is specified. In the development of this guideline, which we intend to be a tool for a global assessment of the material, we consider in addition to the epistemic and ecological aspect, the cognitive, affective, interactional and mediational facets that had not been considered in previous research on probability curriculum materials. The aim is to have a holistic view of the relevance of the instructional processes foreseen or intended through these resources. In the design phase, formative interventions with prospective Peruvian Secondary Education teachers are planned. The a priori didactic analysis of the materials used in the workshops (analysis of practices, objects, and processes in Chapter 3; analysis of the didactic suitability of the curricular program and the workbooks on probability in Chapter 4) is carried out based on the instruments developed in the previous phase (Chapter 3). The implementation and evaluation phase involves observing the interactions between prospective teachers and with the didactic resources. In particular, the a priori analysis facilitates the confrontation of the observed results in the participants' productions with those anticipated by expert analysis. The results of the analysis of practices, objects, and processes in the curriculum materials (Chapter 3) reveal an emphasis on the classical meaning of probability to the detriment of intuitive and frequential meanings, which may lead to a biased teaching of probability. Based on this analysis of meanings, the application of the didactic suitability assessment guide to the materials (Chapter 4) allows us to identify shortcomings in the materials associated with different facets, as well as to propose possible improvements. There is an absence of essential concepts or imprecise use of terms and expressions in the definitions, for example, of sample space or event. The conditions for using Laplace's rule are not clarified, or it is applied inappropriately when the situation does not allow it because the elementary events are not equiprobable. The problem situations associated with the frequential meaning ignore experimentation or simulation and are reduced to the calculation of relative frequency (percentages) from graphs or tables that collect absolute frequencies. From a cognitive point of view, little attention is paid to prior knowledge. In the workbooks, disparities are observed with the curricular program, lack of articulation between classical, frequential, and intuitive meanings, scarcity of experimental and simulation situations, and lack of group tasks to foster interaction among students, among other limitations. These results should be considered by teachers using curriculum materials and normative frameworks. Distinguishing the meanings and identifying the objects involved in mathematical practices, although a challenge for prospective teachers, improves their ability to analyse the potential of the tasks they propose to students and anticipate possible conflicts. The results of the formative actions with prospective teachers, detailed in Chapters 5, 6, and 7, show how, as is common in previous research in other educational contexts, there is deficient common knowledge in probability in prospective teachers. This limitation may motivate their difficulties in differentiating mathematical practices, identifying mathematical objects, and recognizing the meanings of probability involved. Prospective teachers also show limitations in assessing indicators of suitability and difficulties in making reasoned judgments about curriculum materials. These limitations could be due to the lack of specific training and the short time they had to familiarize themselves with the suitability indicators. However, it is observed that having the indicators helps to recognize the shortcomings of the material in such an assessment. The research suggests that, to improve these results, it is necessary to dedicate more space to reflection on a greater variety of problem situations that allow achieving an adequate level of competence in the analysis of meanings and ontosemiotic analysis of mathematical practices. To develop their capacity for assessment and management of materials, it is necessary to reinforce the didactic-mathematical knowledge in probability from the epistemic point of view (practices, objects, and processes characteristic of the different meanings of probability and how they relate), cognitive (factors that influence the complexity of probability situations and biases), as well as in the other facets, where there is a confused idea of the affective aspects, what autonomous student work entails, or the importance of adopting the curriculum in the materials to ensure a progression of learning without gaps.