Mereological foundations of point-free geometry via multi-valued logic

We suggest possible approaches to point-free geometry based on multi-valued logic. The idea is to assume as primitives the notion of a region together with suitable vague predicates whose meaning is geometrical in nature, e.g. ‘close’, ‘small’, ‘contained’. Accordingly, some first-order multi-valued theories are proposed. We show that, given a multi-valued model of one of these theories, by a suitable definition of point and distance we can construct a metrical space in a natural way. Taking into account that interesting metrical approaches to geometry exist, this looks to be promising for a point-free foundation of the notion of space. We hope also that this way to face point-free geometry provides a tool to illustrate the passage from a naïve and ‘qualitative’ approach to geometry to the ‘quantitative’ approach of advanced science

CRUCIAL EVENTS IN PRE-SERVICE PRIMARY TEACHERS’ MATHEMATICAL EXPERIENCE

Relationship with mathematics is a crucial variable in the professional development of pre-service primary teachers: it can largely affect pre-service teachers’ reactions to educational prompts and also their future didactical choices in classroom. Many researches showed that pre-service primary teachers have often developed a negative relationship with mathematics during their mathematical experience as students. This paper adopts a narrative methodology to investigate about the origins and development of these relationships. Specifically, it investigates how pre-service primary teachers tell the events that they recognize as crucial for the development of their relationship with mathematics. Results indicate the relevance of success/failure experiences, and the key-role of the teacher therein

On prospective teachers' evaluations of freshman students' written arguments

n this paper we present a qualitative study in which we analyse how a group of prospective mathematics teachers evaluates written arguments produced by freshman students solving a problem involving graphs. The prospective teachers were attending a course of Mathematics Education for the second year of a master degree in Mathematics. In particular we focus on how prospective teachers in their evaluations look at mathematical content, language, and argumentative structure. The outcomes show, on the one hand, a certain attention to some linguistic aspects, on the other hand, a greater attention to the content than to the argumentative structure of the texts. We suggest that prospective teachers' models for linguistic education are often the standard ones, based on conformity to grammar or style rather than on adequacy with respect to goals

Mathematical induction at the tertiary level: looking behind appearences

The relevance of inductive proofs in Mathematics is beyond question and the research in Mathematics Education has widely documented the students’ difficulties in under- standing and applying mathematical induction, both at secondary school level and at university level. In this paper, we present a qualitative study involving third year Mathematics degree students aimed at investigating the solidity/fragility of mathe- matical induction comprehension. The results highlight that mathematical induction is a very hard topic also in this context, in which are involved mathematical competent students. We argue the need to design non-standard activities able to get the miscon- ceptions emerge, in order to support a deep understanding of the topic

Turning moments: the crossroads of the prospective secondary teachers’ attitude towards math

Relationship with mathematics strongly affects teachers’ practices. Specialist mathematics teachers’ positive relationship is often taken for granted, although recent studies suggest that this may not be the case. In this paper we present a narrative research aimed at investigating which events do prospective secondary teachers recognise as crucial in the development of their relationship with mathematics. The analysis reveals that experiences of success/failure in mathematics, teacher’s opinion, teacher’s charisma, and experiences of helping someone with maths are frequent factors influencing these events, impacting on prospective teachers' emotions, perceived competences, and view of mathematics

Turning moments: the crossroads of the prospective secondary teachers' attitude towards math

Relationship with mathematics strongly affects teachers’ practices. Specialist mathematics teachers’ positive relationship is often taken for granted, although recent studies suggest that this may not be the case. In this paper we present a narrative research aimed at investigating which events do prospective secondary teachers recognise as crucial in the development of their relationship with mathematics. The analysis reveals that experiences of success/failure in mathematics, teacher’s opinion, teacher’s charisma, and experiences of helping someone with maths are frequent factors influencing these events, impacting on prospective teachers' emotions, perceived competences, and view of mathematics