Students’ Evolving Meaning About Tangent Line with the Mediation of a Dynamic Geometry Environment and an Instructional Example Space

In this paper I report a lengthy episode from a teaching experiment in which fifteen Year 12 Greek students negotiated their definitions of tangent line to a function graph. The experiment was designed for the purpose of introducing students to the notion of derivative and to the general case of tangent to a function graph. Its design was based on previous research results on students’ perspectives on tangency, especially in their transition from Geometry to Analysis. In this experiment an instructional example space of functions was used in an electronic environment utilising Dynamic Geometry software with Function Grapher tools. Following the Vygotskian approach according to which students’ knowledge develops in specific social and cultural contexts, students’ construction of the meaning of tangent line was observed in the classroom throughout the experiment. The analysis of the classroom data collected during the experiment focused on the evolution of students’ personal meanings about tangent line of function graph in relation to: the electronic environment; the pre-prepared as well as spontaneous examples; students’ engagement in classroom discussion; and, the role of researcher as a teacher. The analysis indicated that the evolution of students’ meanings towards a more sophisticated understanding of tangency was not linear. Also it was interrelated with the evolution of the meaning they had about the inscriptions in the electronic environment; the instructional example space; the classroom discussion; and, the role of the teacher

Teachers and didacticians: key stakeholders in the processes of developing mathematics teaching

This paper sets the scene for a special issue of ZDM-The International Journal on Mathematics Education-by tracing key elements of the fields of teacher and didactician/teacher-educator learning related to the development of opportunities for learners of mathematics in classrooms. It starts from the perspective that joint activity of these two groups (teachers and didacticians), in creation of classroom mathematics, leads to learning for both. We trace development through key areas of research, looking at forms of knowledge of teachers and didacticians in mathematics; ways in which teachers or didacticians in mathematics develop their professional knowledge and skill; and the use of theoretical perspectives relating to studying these areas of development. Reflective practice emerges as a principal goal for effective development and is linked to teachers' and didacticians' engagement with inquiry and research. While neither reflection nor inquiry are developmental panaceas, we see collaborative critical inquiry between teachers and didacticians emerging as a significant force for teaching development. We include a summary of the papers of the special issue which offer a state of the art perspective on developmental practice. © 2014 FIZ Karlsruhe

Mathematics teaching development as a human practice: identifying and drawing the threads

This article was published in the journal, ZDM Mathematics Education [© FIZ Karlsruhe] and the definitive version is available at: http://dx.doi.org/10.1007/s11858-012-0437-7The didactic triangle links mathematics, teachers and students in a consideration of teaching– learning interactions in mathematics classrooms. This paper focuses on teachers and teaching in the development of fruitful learning experiences for students with mathematics. It recognises primarily that teachers are humans with personal characteristics, subject to a range of influences through the communities of which they are a part, and considers aspects of teachers’ personhood, identity and agency in designing teaching for the benefit of their students. Teaching is seen as a developmental process in which inquiry plays a central role, both in doing mathematics in the classroom and in exploring teaching practice. The teacher-as-inquirer in collaboration with outsider researchers leads to growth of knowledge in teaching through development of identity and agency for both groups. The inclusion of the outsider researcher brings an additional node into the didactic triangle

Il Laboratorio delle Macchine Matematiche: dalla tradizione a un progetto regionale di formazione degli insegnanti della scuola secondaria

Il Laboratorio delle Macchine Matematiche (MMLab) dell\u2019Universit\ue0 di Modena e Reggio Emilia \ue8 un centro di ricerca sull\u2019insegnamento e l\u2019apprendimento della matematica con l\u2019uso di strumenti, che opera anche come aula didattica decentrata per attivit\ue0 laboratoriali rivolte a classi di studenti di scuola secondaria. Da novembre del 2008 \ue8 stata avviata una collaborazione tra il Laboratorio delle Macchine Matematiche e la regione Emilia Romagna per la realizzazione di un progetto biennale (denominato \u201cLaboratorio delle Macchine Matematiche per l\u2019Emilia Romagna\u201d) che ha come obiettivo la messa a punto di un modello operativo di diffusione su scala regionale di una metodologia di attivit\ue0 di laboratorio di matematica che segua le indicazioni proposte dalla Commissione UMI-CIIM nel progetto curricolare Matematica per il cittadino . Questo progetto \ue8 la prima azione del progetto regionale Scienze e Tecnologie in Emilia-Romagna promosso dall\u2019Assessorato Scuola, Formazione Professionale, Universit\ue0, Lavoro, Pari Opportunit\ue0 e realizzato in partenariato con l\u2019Ufficio Scolastico Regionale e l\u2019Agenzia Nazionale per l\u2019Autonomia Scolastica ANSAS (ex IRRE ER). Nella prima parte di questo capitolo sono presentate le diverse prospettive con cui si interpreta l\u2019idea di laboratorio di matematica e le ricerche su cui si fondano le attivit\ue0 del Laboratorio delle Macchine Matematiche di Modena. Queste attivit\ue0 sono descritte nella seconda parte in cui, partendo dalle attivit\ue0 di laboratorio con le macchine matematiche condotti ormai da anni nel laboratorio di Modena, \ue8 presentata una sintesi delle pi\uf9 recenti attivit\ue0 legate al progetto regionale

The use of concrete artefacts in Geometry Teacher Education for secondary schools

This presentation deals with the use of some concrete geometry artefacts (called Mathematical Machine) for the purpose of drawing curves and realizing geometric transformations within the MMLab-ER project developed by UNIMORE. In the ancient Greece, at the time of Euclid, some concrete artefacts (such as the straightedge and compass) were used in both practical and theoretical geometry. Other artefacts were known in the ancient age and were considered again by the most important European mathematicians as from the 16th century. This presentation reports today\u2019s use of working copies of those instruments (complementary to dynamic geometry system) in secondary school teacher education and development for the purpose of realizing laboratory activities in their own classroom

L’educazione geometrica attraverso l’uso di strumenti: un esperimento didattico

Abstract. In questo lavoro ci si propone di presentare un esperimento didattico a lungo termine sull’introduzione in una classe V elementare di strumenti e modelli per la prospettiva. La fase di cui ci occuperemo, temporalmente iniziale rispetto l’intero esperimento, riguarda l’uso di particolari artefatti culturali, che hanno permesso la realizzazione della trasposizione nella classe del costrutto teorico della mediazione semiotica