Preparing prospective mathematics teachers to become teachers who recognize andrespond to students’ mathematical needs is challenging. In this study, we use the construct of critical incident as a tool to support prospective mathematics teachers’ reflection on their authentic fieldwork activities, notice students’ thinking, and link it to the complexity of mathematics teaching. Particularly, we aim to explore the characteristics and evolution of prospective mathematics teachers’ noticing of students’ mathematical thinking when critical incidents trigger reflective discussions. Critical incidents are moments in which students’ mathematical thinking becomes apparent and can provide teachers with opportunities to delve more deeply into the mathematics discussed in the lesson. In the study, twenty-two prospective mathematics teachers participated in fieldwork activities that included observing and teaching secondary school classrooms. The prospective teachers identified critical incidents from their observations and teaching, which were the foci for reflective discussion in university sessions. By characterizing the prospective teachers’ reflective talk in these discussions, we demonstrate the discussion’s evolution.In it, participants questioned learning and teaching mathematics and suggested alternate explanations. This characterization also shows that using critical incidents in the university discussions enabled the prospective teachers to link students’ thinking with the teacher’s teaching practices while supporting their reflection using classroom evidence. We emphasize the importance of descriptive talk in the discussion, which allows for deepening the prospective teachers’ reflections. Further, we explore the teacher educator’s contributions in those discussions, showing that the teacher educator mainly maintained the reflective talk by contextualizing the critical incidents and pressing the participants to explain further issues they raised in the discussions. Implications for mathematics teacher education are discussed.

Teachers face many issues when trying to integrate digital resources (DR) into mathematics classes. This article applies an identity-based perspective to understand teachers’ roles in the practices that evolve in such classes. We focus on Victor, a Greek mathematics teacher, who, when viewed from levels beyond the classroom, experienced to become a reform-oriented teacher and a designer of DR. We explore how these experiences of professional identity shapes and is shaped by his work with DR at the classroom level. We show how Victor’s identity changed from ‘being a Mathematics teacher who struggles with inquiry-based teaching’ to ‘becoming a mathematics Teacher who uses DR to support inquiry-based learning’ and outline what fuelled these changes. Our results suggest the importance of connecting an identity perspective to classroom interactions and mathematics.

We study the interactions over 22 years between one mathematics teacher and his resources for teaching, especially digital ones, with a dual focus on the teacher’s documentational and identity trajectories and professional development. We combined a theoretical framework on teachers’ work with resources  —documentational approach to didactics (DAD)— with a framework based on social practice theory—patterns of participation (PoP). The DAD analysis provided rich descriptions of which digital resources the teacher interacted with and of the transformative evolution of these interactions over time. The PoP analysis offered explanations of how and why the teacher transformed his interactions with the resources through foregrounding affective and contextual factors as significant for the teacher’s formation of identity. We conclude that the combination of the two frameworks provided deeper and complementary insights into the teacher’s long-term professional development with digital resources, and that such networking is needed to develop balanced understandings of teachers’ long-term interactions with digital resources.

In this paper we use learning trajectories to study 11th grade students’ conceptualization of function as a covariational relationship between two quantities while they engaged in modeling tasks to support their experimentation and conceptualizations. Pairs of students used digital tools that offer integrated representations of functions while working on an instructional sequence of modeling tasks in their mathematics classrooms. The analysis shows students’ progressive conceptualization of functional relationships starting from quantitative and covariational relationships using learning trajectories. The findings indicate the potential of upper secondary students to conceptualize function as a covariational relationship involving the rate of change, as well as the role of the available tools and the role of models and their connections in students’ conceptualizations.

This study explores teacher educators’ (TEs’) activity as they support mathematics and science teacher collaboration in co-designing and jointly implementing tasks. We view TEs’ activity through the lens of Activity Theory and expansive learning and draw evidence from data generated within the mascil project that linked mathematics and science teaching with workplace situations through inquiry-based teaching. We focus on five TEs’ actions and goals, use data from their professional development sessions with teachers and from the TEs’ interactions during their own meetings, and highlight the illuminating case of one teacher educator. We trace evidence indicating paths of actions followed by each Teacher Educator and look for indications of their professional learning. Our analysis reveals generic and content-focused actions. All TEs faced different kinds of contradictions and had difficulties handling them. In terms of professional learning, all TEs adapted their prior teacher education practices and appreciated the critical role of epistemological differences between the two disciplines.

The reported study is situated within the process of developing a reform-oriented national mathematics curriculum for compulsory education in Greece by a design team that involved teachers, academic researchers, and policy-makers. From an activity theory perspective, we identify the activity systems of mathematics teaching, research in mathematics education, and educational policy interacting in the design process. We focus on the contradictions between the three activity systems and how these were dealt with. We based our analysis on email exchanges during the curriculum design, field notes from whole-team sessions, and interviews with key persons. Our results highlight that the emerging contradictions primarily concerned the teaching and research activity systems. Members of the team who acted as brokers between the different activity systems and facilitated their interaction played an important role in overcoming the contradictions.

In this paper, we explore the process of becoming a teacher educator in the pedagogical use of digital tools in mathematics teaching. The study took place in the context of an in-service program during the trainees’ engagement in their practicum fieldwork activities including the process observation–reflection–design–implementation–reflection. We explored the features of this context that facilitated the trainees’ transition from the level of trainee educator to the level of teacher educator as well as the nature of the trainees’ documentation work for teachers. The results showed that observation of other teacher educators’ teaching in conjunction with reflection during the program’s respective sessions facilitated the trainees’ transition to the professional level. The identified operational invariants underlying the trainees’ designs concerned the focus of their observation in teacher education classrooms, the importance they attributed to the constraints and opportunities provided by the wider educational context and epistemological issues regarding the teaching and learning of mathematics with technology. The analysis of trainees’ designs revealed three kinds of documents (‘‘explanatory,’’ ‘‘instructive’’ and ‘‘facilitative’’) and corresponding roles of trainees during the implementation. These documents targeted different aspects of TPACK depending on the trainees’ conceptualizations of teachers’ roles either ‘‘as students’’ or ‘‘of students.’’

The general goal of this paper is to explore the potential of computer environments for the teaching and learning of functions. To address this, different theoretical frameworks and corresponding research traditions are available. In this study, we aim to network different frameworks by following a ‘double analysis’ method to analyse two empirical studies based on the use of computational environments offering integrated geometrical and algebraic representations. The studies took place in different national and didactic contexts and constitute cases of Constructionism and Theory of Didactical Situations. The analysis indicates that ‘double analysis’ resulted in a deepened and more balanced understanding about knowledge emerging from empirical studies as regards the nature of learning situations for functions with computers and the process of conceptualisation of functions by students. Main issues around the potential of computer environments for the teaching and learning of functions concern the use of integrated representations of functions linking geometry and algebra, the need to address epistemological and cognitive aspects of the constructed knowledge and the critical role of teachers in the design and evolution of students’ activity. We also reflect on how the networking of theories influences theoretical advancement and the followed research approaches.

In this paper we aim to contribute to the process of networking between theoretical frames in mathematics education by means of forging connections between Constructionism and Instrumental Theory to discuss a design for instrumentalisation. We specifically focus on instrumentalisation, i.e. the ways in which students make changes to digital artifacts and generate meanings in reference to these, as something which will not inevitably happen during activity with digital media. We discuss the issue of designing artifacts and corresponding activities in order to facilitate an instrumentalisation process which will be rich in the generation of mathematical meanings. We report findings from research aimed at shedding light on the meanings of angle in 3D space generated by 13 year olds students while using a specially designed Turtle Geometry microworld. The analysis indicates that connections between the two theories on the issue of designing for instrumentalisation enhances our efficiency to explore the instrumental genesis in technology-rich environments.

The present article reports a study concerning the analysis of 19 activity plans (we call them ‘scenarios’) developed by mathematics teacher educators-in-training for the pedagogical use of digital tools. The development of these scenarios took place during their training program and it was designed as an activity for increasing reflection, for expressing creative pedagogical ideas and for an active engagement in the design of curricula enriched with the use of technology. Our analysis shows that the trainee teacher educators deconstructed and reconstructed respective parts of the formal curriculum regarding the mathematical concepts they chose to embody in their scenarios.

This paper reports on a design experiment conducted to explore the construction of meanings by 17 year old students, emerging from their interpretations and uses of algebraic like formalism. The students worked collaboratively in groups of two or three, using MoPiX, a constructionist computational environment with which they could create concrete entities in the form of models by using equations and animate them to link the equations’ formalism to the produced visual representation. Our aim was to further study the ways in which the use of formalism in constructionist environments can create contexts for the emerging of mathematical meanings. Some illustrative examples of two groups of students’ work indicate the potential of the activities and tools for expressing and reflecting on the mathematical nature of the available formalism. We particularly focused on the students’ engagement in reification processes, i.e. making sense of structural aspects of equations, involved in conceptualising them as objects that underlie the behaviour of the respective models

The paper describes a study of the contexts of six teams, expert in research and development of digital media for learning mathematics, who cross-experimented in classrooms with the use of each other’s artefacts. Contextual issues regarding the designed tasks and technologies, the socio-systemic milieu and the ways in which the researchers worked with the teachers were in focus. We analysed the ways in which a set of mutually constructed and negotiated questions aiming to illuminate otherwise tacit contextual issues operated as boundary objects amongst the teams. We discuss the need to develop special tools such as these boundary objects in order to elicit issues of context and the ways they may affect the production of theory.

This paper reports on a case-study design experiment in the domain of fraction as number-measure. We designed and implemented a set of exploratory tasks concerning comparison and ordering of fractions as well as operations with fractions. Two groups of 12-year-old students worked collaboratively using paper and pencil as well as a specially designed microworld which combines graphical and symbolic notation of fractions represented as points on the number line. We used the students’ interactions with the available representations as a window into their conceptual understanding and struggles in making sense of fraction asnumber-measure. We report on the features of the available representations from an epistemological point of view, on the design of activities aiming at creating meaningful problem contexts for fractions as well as on the meanings generated by the students by some illustrative examples of their work indicating the potential of the activities and tools for expressing and reflecting on the mathematical nature of fraction as number-measure.

Enlarging-shrinking geometrical figures by 13 year-olds is studied during the implementation of proportional geometric tasks in the classroom. Students worked in groups of two using ‘Turtleworlds’, a piece of geometrical construction software which combines symbolic notation, through a programming language, with dynamic manipulation of geometrical objects by dragging on sliders representing variable values. In this paper we study the students’ normalising activity, as they use this kind of dynamic manipulation to modify ‘buggy’ geometrical figures while developing meanings for ratio and proportion. We describe students’ normative actions in terms of four distinct Dynamic Manipulation Schemes (Reconnaissance, Correlation, Testing, Verification). We discuss the potential of dragging for mathematical insight in this particular computational environment, as well as the purposeful nature of the task which sets up possibilities for students to appreciate the utility of proportional relationships.

Στο άρθρο αυτό παρουσιάζονται ερευνητικά αποτελέσματα που αφορούν τη χρήση ειδικού υπολογιστικού εργαλείου δυναμικού χειρισμού γεωμετρικών μεγεθών στο πλαίσιο του πειραματισμού 13χρονων μαθητών για την αυξομείωση γεωμετρικών κατασκευών με βάση σχέσεις αναλογίας μεταξύ μεταβλητών μεγεθών. Τα παιδιά εργάστηκαν σε ομάδες στο εργαστήριο υπολογιστών του σχολείου τους χρησιμοποιώντας ειδικά σχεδιασμένα υπολογιστικά εργαλεία συμβολικής και γραφικής αναπαράστασης των μεταβλητών μεγεθών, που παράλληλα μπορούσαν να τα χειριστούν ελέγχοντας με δυναμικό τρόπο την αριθμητική μεταβολή τους. Η ανάλυση εστιάζεται στα σχήματα που αναδύθηκαν κατά τη χρήση του εργαλείου δυναμικού χειρισμού αλλά και στις μεταξύ τους διασυνδέσεις. Στα ευρήματα καταγράφεται η αξιοποίηση του δυναμικού χειρισμού ως πλαισίου αναγνώρισης και έκφρασης συσχετίσεων μεταξύ των μεταβλητών μεγεθών μιας γεωμετρικής κατασκευής με στόχο την αυξομείωσή της στο πλαίσιο κατάλληλα σχεδιασμένων δραστηριοτήτων.

In the context of the Kaleidoscope Network of Excellence, six European research teams developed a methodology for integrating their research approaches. In this paper, we present the methodology based on a cross-experimentation, showing how it gave insight to the understanding of each team’s research and on the relationship between theoretical frameworks and experimental research.