In recent years, within the field of mathematics education there has been a growing interest for inclusive approaches that, in line with the international documents claiming for inclusive education (OECD, 2008; UNESCO; 2017), consider the uniqueness of every student a crucial aspect of effective learning processes (Faragher, Hill, & Clarke, 2016; Santi, & Baccaglini-Frank, 2015; Sullivan, 2015). 

Inclusive education is here defined in broad terms, as a process aimed at granting joint presence, and meaningful learning and social outcomes for all students (Ainscow, 2016; Slee, 2018), considering their individual and social characteristics. On the classroom level, this conceptualisation is strongly connected with the idea of differentiation (Tomlinson, 2014):  the creation of learning environments that value diversity in student population and respond to individual needs in terms of interests, readiness and learning profiles. The aim of differentiation is to create learning communities within each individual finds his own space and has meaningful opportunities to learn. Open learning settings (Demo, 2016), like station learning, where students are free to choose among different options in relation to their talents, needs or interests, represent one of the main resources to put the principle of differentiation into practice.  

In the field of mathematical learning, the Theory of Objectification (TO, Radford, 2008) offers an interesting perspective to address this broad idea of inclusion. According to TO, knowledge emerges as fixed patterns of individuals’ mathematical activity mediated by cultural-historical artefacts, which can bear a material or an ideal form of existence. Learning in TO is theorised as the process of objectification through which the learner gradually becomes aware of historically constituted cultural meanings as well as shared forms of reasoning and action. While learning, the individual co-produces himself as a subject and positions himself with respect to the object of knowledge: this process is called subjectification (Radford, 2020). In fact, subjectification processes describe how individuals find their own positioning in the objectification, it is fundamental to consider it in order to grant access to all students to mathematical activities and becomes the crucial link between TO and inclusion. Moreover, the important role attributed to artifacts, understood as semiotic means of objectification, seems suitable to promote educational differentiation in mathematical practice. The mediation of the artifact in the processes of objectification and subjectification allows the students to experience a mathematical object in ways that are consistent with his specific way of learning, working on different levels.  

On this background, the Open-Math project, funded by the Free University of Bozen, is conceived as an interdisciplinary work for inclusive mathematics education that intertwines theories and methods from the two fields of inclusive education and Mathematics education. In line with the approach of educational design research (McKenney, & Reeves, 2019), the project has two main goals. First, it aims to develop a conceptual framework of inclusive mathematics learning and ensure the integration on a theoretical and methodological level of the Theory of Objectification, of the broad idea of differentiation for inclusive education and of Open Learning. Second, the expected outcome is a model of lesson plan implemented and refined though the pilot use in one class 

In particular, in this work we are going to present our analysis instruments: the coding system, result of the collaboration between our two fields, that allowed us to establish the inclusiveness of a task, and to improve our design in its various cycle of implementation.

The project is an educational design research (McKenney, & Reeves, 2019) conducted with a class of grade 7 between October 2020 and May 2021, during the mathematics lessons. The class is composed by 17 students, the mathematics and the support teachers. We defined a lesson plan, the Open Activity Theory Lesson Plan (OATLP), starting from Radford’s lesson plan (2015) and adding a specific focus on differentiation. In particular, the original activity lesson plan consists of three main phases (presentation by the teacher, small group work and whole class discussion) whereas in the OATLP, before the group work, students are asked to work on differentiated station activities considering their earning differences. The introduction of this phase aims at facilitating everybody’s access to groupwork, mediating the mathematical knowledge at play in various ways. After whole class discussion, another moment of differentiation is planned for generalization. The class involved in the project works with the OATLP once a month in school year 20/21, covering different topics in mathematics. Groupwork and whole class discussion are videotaped, and students’ protocol are collected. Coherently with the definition of inclusion, that aims at granting significant learning and socialisation processes for all different students, six students selected based on their different competencies in mathematics and learning styles are observed in group interactions. Furthermore, the six students and the mathematics teacher are interviewed after every cycle. Interdisciplinary categories were used in order to analyse the collected data. Learning processes are described on the basis of categories referred to TO and to socialization processes. The analysis instruments are the following: 1)Theory of objectification: a)Level of generalization: Factual generalization; Contextual generalization; Symbolic generalization (Radford, 2006). b)Description of the use of the artifact (Radford, 2008) 2)Inclusive education: a)Participation. Scale adapted from Ianes (2020): Pupil's deliberate marginalisation; physical absence; Physical presence without interaction; Activation of first level of communication skills; Activation of second level of communication skills; Activation of communication and leadership skills. b)Self-determination scale: Autonomy, Self-regulation, Empowerment, Self-realisation (Whemeyer et al., 2003). The analysis is conducted by two researchers separately. Using a triangulation perspective to guarantee reliability, according to the inclusive perspective and to mathematics education, the two analysis are then compared and discussed among the researchers.

The presentation will offer insights in the two main outputs of the research project. First, the complete OATLP will be presented and an example of the implementation of it with reference to the faced mathematics topics, the pythagorean theorem, will be provided. Data analysis will show how the lesson plan was reviewed according to students’ interviews and to the analysis of their protocols. For example, it became clear that there was a need to increase the time attached to each task, to make explicit how to deal with the request for help from classmates or from the teachers, and to design mathematics tasks for the stations more targeted on specific learning objectives in mathematics but differentiated along the different learning styles. Second, the analysis instruments will be presented and their application on some collected data will be shown. Some preliminary results that describe learning and socialisation processes activated by the OATLP for the six monitored students will be discussed. Contextually, some characteristics of the subjectification processes acted by the students will be highlighted. To conclude, the innovative value of the first results of the Open Math project is the insight into an interdisciplinary work that has invested on the creation of a joint new perspective both on the level of intervention in the inclusive mathematic classroom and on the level of research instruments used to describe students’ learning and participation processes. In this sense, they can contribute to the discussion about the theoretical and concrete challenges of the interdisciplinary work between inclusive and mathematics education.

Ainscow, M. 2016. “Diversity and Equity: A Global Education Challenge.” New Zealand Journal of Educational Studies 51 (2): 143–155. Demo, H. (2016). Didattica Aperta e Inclusione. Trento: Erickson. Faragher, R., Hill, J., & Clarke, B. (2016). Inclusive Practices in Mathematics Education. In K. Makar (Ed.), Research in Mathematics Education in Australasia 2012–2015 (pp. 119-141). Singapore: Springer. Ianes D., Cramerotti S., Fogarolo F. (2020) Il nuovo pei in prospettiva bio-psico-sociale ed ecologica, Erickson, Trento McKenney, S. E., & Reeves, T. C. (2019). Conducting educational research design. Routledge. OECD (2008). Ten steps to equity in education. https://www.oecd.org/education/school/39989494.pdf Last access 19/01/2021 Radford, L., & Peirce, C. S. (2006). Algebraic thinking and the generalization of patterns: A semiotic perspective. In Proceedings of the 28th conference of the international group for the psychology of mathematics education, (Vol. 1, pp. 2-21). Radford, L. (2008). The ethics of being and knowing: Towards a cultural theory of learning. In L. Radford, G. Schubring & F. Seeger (Eds.), Semiotics in mathematics education: epistemology, history, classroom, and culture (pp. 215-234). Rotterdam: Sense Publishers Radford, L. (2015). Methodological Aspects of the Theory of Objectification. Perspectivas da Educação Matemática, 8(18), 547-567. Radford, L. (2020). Play and the Production of Subjectivities in Preschool. Mathematics Education in the Early Years (pp. 43-60). Springer, Cham. Santi, G., Baccaglini-Frank, A. (2015). Forms of generalization in students experiencing mathematical learning difficulties, PNA, 9(3), 217-243. Slee, R. 2018. “Defining the Scope of Inclusive Education.” Paper Commissioned for the 2020 Global Education Monitoring Report, Inclusion and Education. Sullivan, P. (2015). Maximising opportunities in mathematics for all students: Addressing within-school and within-class differences. In A. Bishop, H. Tan, & T. N. Barkatsas (Eds.), Diversity in mathematics education—towards inclusive practices (pp. 239–253). Cham: Springer publishing. Tomlinson, C. (2014). The differentiated classroom. Alexandria: ASCD Wehmeyer, M. L., Abery, B. H., Mithaug, D. E., & Stancliffe, R. J. (2003). Theory in self-determination: Foundations for educational practice. Charles C Thomas Publisher. UNESCO (2017). A Guide for Ensuring Inclusion and Equity in Education. Paris: UNESCO. https://unesdoc.unesco.org/ark:/48223/pf0000248254 Last access 19/01/2021