This chapter looks at an in-depth application of meaning equivalence reusable learning objects (MERLO) to mathematics education and teacher professional development. The study has been conducted during professional development courses for in-service teachers and is focused on mathematics teachers' praxeologies, namely their didactical techniques and theoretical aspects embraced to accomplish a task. Specifically, the task given to the teachers consists in designing MERLO items to be used in their classrooms, working in groups or individually, after having been trained by researchers in mathematical education. The chapter presents two case studies with data, one dealing with secondary school teachers in Italy and one concerning primary teachers in Australia. One of the main aims of the study is the analysis of the praxeologies of these teachers when they are engaged in designing MERLO items during professional development programs. The chapter demonstrates, with these examples, the generalizability potential of MERLO items and that they can be used in different cultural and institutional ecosystems.
TopIntroduction
A common and recognized world wide challenge is to improve the efficacy and efficiency of teaching, learning and assessing. Teaching the right things in the right way is an important objective for both teachers and education research.
In recent years many studies in different disciplines, including mathematics education, have been carried out with the aim of providing teachers, students and educational systems with new didactical and methodological tools. A collection of research contributions appeared in the Proceedings of the International Commission on Mathematical Instruction, a book fully dedicated to the design task in mathematics education (Margolinas, 2013).
MERLO (Meaning Equivalence Reusable Learning Objects) is an innovative methodological-didactical tool originally introduced by Shafrir and Etkind, and later developed by others, that uses alternative, multi-semiotic, representations in different sign systems to focus the attention of learners on meanings. MERLO allows teachers to assess the depth of understanding of contents, and to enhance students’ conceptual thinking (Etkind, Kenett & Shafrir, 2010).
Classroom activities using MERLO can be performed in every subject matter and at different grade levels (e.g. see contributions in other chapters of this book; Etkind, Kenett & Shafrir, 2016; Arzarello, Robutti & Carante, 2015; Robutti et al., in press).
In this chapter, we present the use of MERLO in mathematics education, with a double perspective:
According to the Meta-Didactical Transposition framework (Arzarello et al., 2014), the main aim of this study is to analyse the praxologies of teachers involved in professional development courses that make use of MERLO and engage participants in designing tasks for their students. In order to find similarities and differences between the teachers’ praxeologies in designing MERLO for their students, we present two different methodologies: i) the Italian one, which involves mathematics teachers of secondary school, and ii) the Australian one, which involves primary school teachers. In both cases, teachers collaborate together and/or with researchers in mathematics education. As demonstrated in recent studies, working with colleagues and with academics is an important aspect of teachers’ professional development (Robutti et al., 2016). The chapter focuses on the design of MERLO items for mathematics teaching, learning and assessment, as part of professional development program.
TopTeachers’ Praxeologies
The purpose of studying teachers involved in a professional development is pursued in this study according to the frame of Meta-Didactical Transposition (Arzarello et al., 2014), which offers an interpretative model of teachers’ praxeologies. A praxeology is introduced by Chevallard (1999) in his Anthropological Theory of Didactics, to indicate four interrelated components: task, technique, technology and theory. The given task, and the corresponding technique used to solve the task, are the practical counterpart of the praxeology (the praxis), while the technology (in the sense of justification) and the theory are the theoretical counterpart that validates the use of that technique (the logos). A mathematical praxeology is made of a task (for example, to find the equation of the tangent to the graph of a generic function f) that the students have to solve, the employed technique and the justification for using it, all within a specific mathematical theory.
At the same time, the teacher’s questions and actions used to build such a mathematical praxeology with students constitute a didactical praxeology. What may occur is:
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The teacher introduces students to a type of task (task);
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The teacher manages how to organize such an approach (technique);
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The teacher knows why he/she has to organize it like that (technology);
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The teacher justifies why she/he knows that he/she has to organize it like that (theory).