Challenging inequity in mathematics education by making pedagogy more visible to learners

This paper reports on initial findings from the Visible Maths Pedagogy research project, a collaboration between an academic researcher and two teacher researchers (the paper’s authors). The aim of the project was to explore the effects of making pedagogy more visible on students’ success in school mathematics. We adopted a Participatory Action Research methodology to plan and evaluate five strategies used alongside ‘progressive’ teaching approaches to make the teacher’s pedagogical rationale more visible to learners. Our findings show that students, particularly those from disadvantaged backgrounds, were initially prone to misinterpret the intentions of the teacher. However, the five strategies helped students gain a greater appreciation of the teacher’s pedagogical rationale and how to respond appropriately. We discuss the implications of these findings for enabling all students to access the benefits of progressive teaching approaches and for opening up to scrutiny what it means to be a successful learner of mathematics

Understanding the impact of school´s ethnic composition on Mathematics results of students with immigrant origin in primary school

Scientific interest concerning the impact of school’s compositional effect on student performance has intensified throughout the current century, as the subject has been vastly explored by researchers in various national contexts (Agirdag, Van Houtte & Van Avermaet, 2012; Jensen & Rasmussen, 2011; Van der Slik, Driessen & De Bot, 2006; Schnepf, 2007; Goldsmith, 2003, Lleras, 2008). These effects have usually been studied taking student outcomes as measured by standardised tests on different school subjects and levels of education. In Portugal we conducted a research using an extensive database (16 269 students and 417 schools) with the results of fourth grade students on Portuguese Language national standardised tests in the Lisbon Metropolitan Area (2009-10). We intent to reveal what are the main factors behind students results in public schools giving special attention to the effect of ethnicity, considered both at the student level (its national origin) and at the school level (the schools ethnic composition). A multilevel analysis was developed involving both individual level variables and school level variables to answer the following research questions: Does the schools’ ethnic composition effect on Portuguese Language scores stand when students’ gender, social and ethnic origins are taken into account? Does this effect stand when the schools’ social composition is taken into account? How does the schools’ ethnic composition moderate the relation between having/not having an immigrant background and students’ Portuguese Language scores, when the socio-economic status (SES) of both students and schools is controlled?info:eu-repo/semantics/publishedVersio

Visible mathematics pedagogy: A model for transforming classroom practice

This paper focuses on the development of a model of research and professional development which aims to bring about transformations in classroom practice in situations that have previously proved resistant to change. We explore reasons why conventional models have failed to address one such situation, the continuing predominance of teacher-centred pedagogies in mathematics classrooms. We draw on findings from the Visible Mathematics Pedagogy research project to highlight how a critical model of participatory action research can be refined to enhance its potential to bring about changes in classroom practice. We report on research tools and processes that were developed, distinct from those commonly used in research, including the organisation of research team meetings around participatory principles, the active involvement of teachers in designing and employing data collection tools, and in generating protocols associated with video-stimulated reflection. We demonstrate how these research tools and processes enhanced collaboration and teacher agency, the trustworthiness of the research findings and teachers’ critical reflection on existing practice. We argue that our refined model of participatory action research can inform and support teachers and researchers wishing to transform classroom practice, especially in situations analogous to many mathematics classrooms, in which conventional models have had limited impact

The effect of school´s ethnic composition on Mathematics results of students with immigrant origin in primary school

Scientific interest in the impact of the school composition effect on student performance has been reflected in the literature for several decades (since Coleman, 1966) and has recently intensified. In Portugal, sociological research in this field has been scarce, in particular research based on extensive methodologies. Taking 1st Cycle students and their Mathematics results as reference, the present study sought to contribute to this field. The objective was to identify the main effect and the moderating effect of school composition (socioeconomic and ethnic) on students' results, controlling the effect of individual and school variables, whose effects on school outcomes are already known, such as socioeconomic status (SES), gender, school trajectory, and school size. We particularly sought to understand whether school composition affects the students’ results, and whether the effects vary with students’ national origins. The research was supported by an extensive database containing information on 23,143 students at 522 public schools in the Lisbon Metropolitan Area. A multilevel analysis was developed, considering student and school levels. The dependent variable was the Mathematics results of fourth-grade students in the 2015 National Attainment Test. We concluded that the effect of school’s social composition is more important to student performance than that of school’s ethnic composition and also that the students from PALOPs (Portuguese-speaking African Countries) benefit more than any others when they attend schools with a higher average SES.info:eu-repo/semantics/submittedVersio

Progressive pedagogies made visible: Implications for equitable mathematics teaching

This paper makes a significant contribution to contemporary debates over the direction curriculum reforms should take. It challenges claims that progressive pedagogies can exclude disadvantaged learners from gaining access to powerful knowledge and argues that greater attention needs to be given to learner agency and subject didactics. It reports on the findings from the Visible Maths Pedagogy research project, which aimed to develop and evaluate strategies for making progressive pedagogies more visible to mathematics learners. Evidence collected from student surveys and interviews suggests that these novel strategies were successful in heightening students’ appreciation of the teacher’s pedagogic rationale for employing progressive teaching approaches. They appeared to have a positive impact on students’ mathematical engagement and awareness of how to achieve success in the secondary school mathematics classroom, particularly for those from disadvantaged backgrounds. The findings highlight the potential of progressive pedagogies made visible for establishing an alternative didactic situation based on socio-mathematical norms associated with ‘sense-making’, rather than ‘answer-getting’, which can help develop students’ individual and collective agency. We argue that such a didactic situation offers a pathway towards a more equitable mathematics curriculum, that enables wider access to powerful knowledge, and which forms an integral part of a school curriculum designed to address the environmental, economic and social challenges currently faced by our global society

On digit frequencies in beta-expansions

We study the sets DF(β) of digit frequencies of β-expansions of numbers in [0,1]. We show that DF(β) is a compact convex set with countably many extreme points which varies continuously with β; that there is a full measure collection of non-trivial closed intervals on each of which DF(β) mode locks to a constant polytope with rational vertices; and that the generic digit frequency set has infinitely many extreme points, accumulating on a single non-rational extreme point whose components are rationally independent

Symbol ratio minimax sequences in the lexicographic order

Consider the space of sequences of letters ordered lexicographically. We study the set of all maximal sequences for which the asymptotic proportions of the letters are prescribed, where a sequence is said to be maximal if it is at least as great as all of its tails. The infimum of is called the -infimax sequence, or the -minimax sequence if the infimum is a minimum. We give an algorithm which yields all infimax sequences, and show that the infimax is not a minimax if and only if it is the -infimax for every in a simplex of dimension 1 or greater. These results have applications to the theory of rotation sets of beta-shifts and torus homeomorphisms