In this article, grounded in the concept of interpretative knowledge (IK), which is well known in the literature, we introduce and discuss the construct of semiotic interpretative knowledge (SIK). This theoretical tool unfolds the interpretation of conceptual knowledge into a broader construct that intertwines the use of semiotic representations with the manifold aspects of mathematical learning. In the first part of the article, we first introduce Duval’s semio-cognitive approach, in which semiotics is the element characterizing the specific cognitive functioning of mathematics. On the basis of some classic examples from the relevant literature, we then show the necessity of introducing the semiotic component into IK to expand the interpretive power of the teacher. In the second part of the article, through the analysis of six episodes involving students with a specific learning disorder (SLD) engaged in mathematical activities, we show how the development of the teacher’s SIK emerges as a necessary condition for the implementation of inclusive teaching practices. This permits us to face the research question focused on how SIK allows us to understand the student’s behavior in a special needs educational context and provide effective feedback. In this sense, SIK is shown to be an intrinsic prerequisite of inclusive teaching, thus extending the concept of IK to the components of pedagogical content knowledge (PCK).

Background: While it is usually taken for granted that logic taught in the mathematics classroom should consist of elements of classical propositional or first-order predicate logic, the situation may differ when referring to students’ discursive productions. Objectives: The paper aims to highlight how classical logic cannot grasp some epistemic aspects, such as evolution over time, uncertainty, and quantification on blurred domains, because it is specifically tailored to capture the set-theoretic language and to validate, rather than to consider epistemic aspects. The aim is to show that adopting classical and non-classical lenses might lead to different results in analysis. Design: Nyaya pragmatic and empiricist logic, with Peircean non-standard quantification, both linked by the concept of free logic, are used as theoretical lenses in analysing two paradigmatic examples of classroom argumentation. Setting and Participants: excerpts from a set of data collected by prof. Paolo Boero from the University of Genoa during research activities in a secondary school mathematical class. Methodology: The examples are discussed by adopting a hermeneutic approach. Results: The analysis shows that different logical lenses can lead to varying interpretations of students’ behaviour in argumentation and presenting proof in mathematics and that the adopted non-classical lenses expand the range of possible explanations of students’ behaviour. Conclusion: In mathematics education research, the need to consider an epistemic dimension in the analysis of classroom argumentation and proof production leads to the necessity to consider and combine logical tools in a way specific to the discipline, which might differ from those usually required in mathematics.

In Mathematics Education (ME), research dealing with topic-specific (TS) issues (e.g., what levels of development exist in learning fractions) produces usually local results and is considered less fashionable and attractive for innovative research projects than research dealing with context-specific (CS) issues that have more general and abstract scopes (e.g., what is mathematical instruction? what is the field of ME?) and produces middle-range or grand theories. TS- and CS-research run along separate tracks with little or no crossover, at least from the beginning of the socio-political-turn in ME, but connecting them could help to single out hidden variables in CS-research. This paper shows that TS-research creates specific mathematical objects that allow us to reduce the distance between these two lines of research. Mathematical objects specific to ME research are shown to be both a technical link between the two lines of research because they allow topic-specificity to access more abstract and general realms of research, as well as factor at stake when aspects related to the social, political, and ethical implications of the ontological creativity of TS-research are discussed in a critical postmodern approach. Discussing its impact on textbooks, teacher-training, teaching practices, further TS-research-practices, as well as on the researcher’s epistemological empowerment and on the self-referentiality of ME research, TS-research moves from the periphery to the heart of CS-research.

Questo studio esplora le concezioni di studenti universitari riguardo alla dimostrazione matematica, focalizzandosi su un gruppo di 70 studenti iscritti a corsi di laurea triennale e magistrale in Matematica presso un’università italiana. Attraverso la raccolta e l’analisi di produzioni testuali personali (TEP), la ricerca indaga gli aspetti che determinano le concezioni degli studenti e come queste si modificano nel tempo, influenzate dal percorso accademico seguito. I risultati evidenziano una varietà di concezioni, raggruppate in categorie deduttive, desunte dalla letteratura, e induttive, ricavate dai dati, che riflettono sia le funzioni della dimostrazione, sia aspetti legati alle modalità di apprendimento, sia questioni legate alla struttura logica della dimostrazione. Lo studio offre una mappatura dettagliata delle concezioni sulla dimostrazione emerse durante la ricerca, facendo riferimento a una agenda di codifica appositamente predisposta per lo studio.

In this theoretical paper, an epistemic logical model for analysis of students’ argumentation and proof processes is presented. The model is conceived as a methodological tool addressed to the researcher in mathematics education that aims to shed light on the relations between argumentation and proof, highlighting the continuities and discontinuities within and between them. It reconciles the epistemic logic approach, which takes into account the exploratory phases of a statement, linked to argumentative processes, and the deductive logic approach, which takes into account the phases linked to proof in a classical sense. The model is based on Vergnaud’s concepts- and theorems-in-action, on Duval’s distinction between the epistemic and logical value of verbalised propositions, and on elements of Oostra’s intuitionistic existential graphs, a kind of graphical topological logic rooted in Peircean thought, adapted to mathematics education research by considering also shifts in the classical existential graphs. After exposing the theoretical grounding the model is based on, some examples taken from the literature are examined to exemplify how it works.