The use of technology in teaching-learning processes has become an issue for education policy makers at a national and international level, that involves important financial and human resources - e.g. the Italian government has recently launched an important program for the digitalization of the entire education system (MIUR, 2015). In order to analyse the cognitive, pedagogical and social impact of technology in schools and devise an effective didactical engineering, we require solid theoretical perspectives.

A key issue in education is the investigation of e-Learning dynamics and tools (Ninoriya et al, 2011; Noesgaard & Ørngreen, 2011; Sufeng & Runjuan, 2013; Harasim, 2012). Disregarding specific classifications we will use the term virtual classroom as a reference to something that allows teachers and students to work and collaborate using a digital platform.

The research  provides, within an activity theory perspective, a tentative framework in order to investigate mathematics teaching-learning processes of the same class when the physical and virtual classrooms are entangled.

Mathematics education considers the teaching-learning processes within the complexity of Chevallard’s didactic triangle (Chevallard, 1987, 1992) formed by three “vertexes”: student, knowledge, teacher (Brousseau, 1987; D’Amore 2001). A real classroom, at a social level, is formed by N pupils and one mathematics teacher, therefore the triangle represents the pole student in a single vertex, as if the N pupils - each one with their cognitive style and personal history - collapsed in an ideal pupil that does not exist. Instead we should rather think of a “star” formed by N triangles with the common side knowledge-teacher. A new set of sides arises, the social relationship between the A1 , A2 ,....AN  pupils each one positioned at the vertex of their triangle.  

To face the complexity of the social-cultural activity in the mathematics classroom, we turn to Radford’s (2008, 2016) Theory of Knowledge Objectification (TKO): a cultural-historical-activity perspective for mathematical thinking and learning (Roth & Radford, 2011). TKO conceives learning as a process of objectification: a social reflexive activity, mediated by semiotic means of objectification (SMO) (Radford, 2003), that actively endows the cultural meaning with the student’s personal meaning deriving from their embodied experience.

Also the vertex “knowledge” has to be conceived as several vertices since the ideal mathematical entity is stratified in levels of generality (Radford, 2006).

The educational issue is to foster the encounter between each student and a multi-layered object of knowledge.

The didactical system can be imagined as set of planes each containing a student with their personal meaning, that has to intersect a multi-layered cultural meaning. The affordance of the mathematical object on the part of the student depends on the SMO he is exposed to. The educational issue is who and how orchestrates the mediation that binds the plane of the student with the affordances of the mathematical object .

There are basically two learning environments:

• the traditional physical space, containing material and symbolic SMO, where impermanent activities are synchronic, in a fixed time lag and in the same space;

• the virtual one - with a broad range of SMO available in the web - where space is dematerialized, both open and private, shared and limitless; activities are both synchronous, asynchronous and diachronic, and carried out in open and closed time frames. In the virtual space-time environment, dematerialized SMO have a double affordance: they allow both disembodied and embodied activities. Furthermore the virtual space is a social-historical community due to the possibility of archiving the teaching-learning activities.

Within this framework, our research  question is:

What characterization of SMO and modes of social activity emerge when the same class works in entangled virtual and physical classrooms?

The research presents a longitudinal case study focussing on a specific mathematical topic faced both within the physical and virtual classroom. It is carried out in a secondary school in Bologna (Italy) which offers a cross-curriculum that requires the learning of humanities, languages, science and mathematics. In particular, the research involves a class with a language curriculum formed by 21 students (one with special needs), novices in the use of a virtual classroom, with good learning standards in mathematics and high social interaction. With the TKO lens we focus on: - the characterization of the SMO that mediate activity in the virtual environment interwoven with the SMO used in the physical classroom: embodied, disembodied, iconic, symbolic, space-visual, kinaesthetic etc. - the modes and the tones of social interaction emerging in the virtual classroom that influence the original personal meaning students develop in the physical classroom, analyzing the subsequent affordances of the mathematical object. The research develops along the following steps: 1-The researchers and the teacher introduce Google Classroom as a management tool for virtual classroom and provide the class with its basic functionalities. 2-The teacher, in the physical classroom, presents the mathematical topic according to his/her traditional teaching methodology. 3-The teaching-learning process intertwines the physical classroom with the virtual one. The teacher asks explicitly the students to carry out in Google Classroom the following assignments: - share their difficulties with others and each student can help his peers overcoming a learning hindrance. The teacher can play the role of a mediator for the students who are reluctant in exposing in first person directly; - share their learning conquests; - share their observation, doubts, questions, solutions, curiosities regarding the mathematical topic or any assignment of the teacher; - share both their personal material (notes, schemes, summaries...) and any other material related to the mathematical topic collected around the web (videos, web pages, softwares,...). The virtual classroom becomes a new social cultural space monitored by their teacher and the researchers. 4-The researchers administer a questionnaire to the students. The questions consist both of multiple choice and open questions and its aim is to scrutinize the students personal beliefs, interpretation, emotion and perception regarding the use of Google Classroom in their meaning-making process of the specific mathematical topic. 5-The research data is collected from: - the Stream of Google Classroom; - the archive of materials and communications shared between students and the teacher; - the answers to the questionnaire.

We expect the following answer to our research question. - Interaction of the virtual and the physical classroom allows students to share their personal artifacts that give rise to new forms of semiosis, mediating a new communal personal meaning that pushes towards the interpersonal cultural ones. - The radical change is that the SMO are subsumed in a digital form and integrated in the social-didactic system of the virtual classroom. We assist to a new form of embodied reflexive mediated activity, in the virtual space-time of Google Classroom. For example, our preliminary observation brings evidence of a student who, concerned about the difficulties of her classmates, spontaneously shares a photo of her notes, Google Classroom's Stream only after she has enriched them with the appropriate symbols, icons, and schemes. The student’s personal artifacts become common SMO that mediate the social reflexive activity of the whole class both in the virtual space of the Google Classroom and in the physical classroom. This suggest that a scenario where physical and virtual classroom are in a harmonic coexistence, allows the students to experience deeper forms of subjectification that enhance self-confidence, self-efficacy and the sense of belonging to a larger community. This behavior can be crucial in characterizing a new mode of being and knowing mathematics (Radford, 2008). The new social space can give a contribution to the rise of the communal self, an individual with unerasable ethical concerns and in search for intersubjectivity: “Instead of the idea of the self-regulated Enlightened individual [...], the theory of knowledge objectification suggests the idea of a communitarian self, one busy with learning how to live in the community that is the classroom, learning how to interact with others, to opening oneself up to understanding other voices and other consciousnesses, in brief, being-with-others” (Radford, 2008, p. 229).

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