The objective of this paper is to study students' difficulties when they have to ascribe the same meaning to different representations of the same mathematical object. We address two theoretical tools that are at the core of Radford's cultural semiotic and Godino's onto-semiotic approaches: objectification and the semiotic function. The analysis of a teaching experiment involving high school students working on the tangent, shows how students' difficulties in ascribing sense to different representations of a common mathematical object can be traced back to the kind of objectification processes and semiotic functions they are able to establish.

The COVID-19 crisis has strongly affected the school system. In Italy, at-distance forms of didactics have been activated, changing the physiognomy of schools in terms of social interaction, practices and the identity of the individuals. In this paper, we address the issue of how teachers are facing the crisis: our focus is on assessment, as a key variable catalyzing personal history; beliefs; the interface between students; teachers and the school system. We study teachers’ beliefs as part of their identities and assessment as a fundamental variable of beliefs. A qualitative content analysis of the open-ended answers to an online questionnaire is carried out to understand the main characteristics associated with assessment by teachers and the obstacles to overcome in the context of long distance learning (LDL). The data show that teachers did not identify valid assessment methods for LDL during the lockdown, especially due to the lack of control over the students. A misconception emerges concerning the definition of formative assessment together with a new awareness of the possibilities offered by digital technologies regarding the individualization of didactics. This study helps to understand which teachers’ beliefs are related to assessment are and how they are shaped.

In this article, we present the first results of the project OPEN-MATH. The research project aims at acknowledging the need for learning environments with differentiation strategies for all. We developed a model for inclusive mathematics learning, based on the Theory of Objectification and a broad idea of differentiation realized through Open Learning. It poses an interdisciplinary research issue that requires the collaboration of two sub-disciplines pertaining to the area of educational studies: Inclusive Education and Mathematics Education. The results we present here are related to the dialogue between theory and practice, whose outcome is a teaching and learning model for inclusion in mathematics. The construction of the teaching and learning model moves along two complementary paths: (1) concerning the theoretical point of view, we implemented connecting theory strategies to network Open Education and the Theory of Objectification; (2) concerning the methodological point of view, we implemented Educational Design Research. The new teaching–learning model is the result of theoretical and methodological validation in real contexts according to an interdisciplinary approach. This study shows the strengths of interdisciplinary research for the pursuit of inclusive mathematics and high standards of learning.

We shift the view of a special needs student away from the acknowledged view, that is as a student who requires interventions to restore a currently expected functioning behaviour, introducing a new paradigm to frame special needs students’ learning of mathematics. We use the theory of objectification and the new paradigm to look at (and characterize) students’ learning experiences in mathematics as generalizing reflexive mediated activity. In particular, from this perspective, we present examples of shifts to higher levels of generalization of a student with mathematical learning difficulties working with Mak-Trace, a Logo-like educational software for the iPad.Formas de generalización en estudiantes con dificultades de aprendizaje en matemáticasEn este artículo introducimos un nuevo paradigma para enmarcar el aprendizaje de las matemáticas de alumnos con necesidades especiales. Consideramos una visión de los estudiantes con necesidades especiales diferente a la comúnmente aceptada que los considera como estudiantes que requieren intervención para reestablecer el comportamiento actualmente esperado. Utilizamos la teoría de la objetivización y ese nuevo paradigma para observar (y caracterizar) las experiencias de aprendizaje de las matemáticas entendido como actividad reflexiva y mediada de generalización. En particular, desde esta perspectiva proponemos ejemplos de acceso a niveles superiores de generalización de un estudiante con dificultades de aprendizaje de las matemáticas que utiliza Mak-Trace, un software didáctico para iPad parecido a Logo.Handle: http://hdl.handle.net/10481/34993WOS-ESCINº de citas en WOS (2017): 1 (Citas de 2º orden, 1)

The issue of students’ difficulties in processing operations with percentages has been addressed in several international research studies from a qualitative perspective. In this study, we analyze students’ difficulties on this topic, focusing on the transition from middle school to high school with a mixed methods research design. We focus on students’ responses in a specific task belonging to the Italian large-scale assessment analyzed through the Rasch model, and we deepen the task analysis thanks to interviews, which enlightened image schemas and metaphors underlying students’ reasoning. From the qualitative point of view, the Rasch model shows that students’ difficulties in dealing with percentages is a macrophenomenon that involves the higher levels of competences. From the qualitative point of view, the metaphoric approach outlines the image schemas that foster the correct conceptualization of percentage and those that hinder their correct learning and can be one of the possible causes of the emerging aforementioned macrophenomenon.