Drawing On A Didactical Analysis To Redesign Professional Tasks For Middle And High School Mathematics Teachers
Author(s):
Javier Diez-Palomar (presenting / submitting) Joaquim Gimenez (presenting) Yuly Vanegas Vicenç Font
Conference:
ECER 2014
Network:
24. Mathematics Education Research
Format:
Paper

Session Information

24 SES 02 A, Issues in Mathematics Teacher Education (Part 1)

Paper Session: to be continued in 24 SES 03

Time:
2014-09-02
15:15-16:45
Room:
B113 Sala de Aulas
Chair:
Javier Diez-Palomar

Contribution

In this paper we analyse how the didactical analysis of specific professional tasks in mathematics may led mathematics teachers’ trainers to redesign specific mathematics tasks to better support prospective teachers to develop their mathematical and educative competences. We aim to encourage future teachers of mathematics to improve their professional practice by equipping them with didactical tools based on a rigorous theoretical foundation (Zavlaswski & Sullivan, 2011). For this reason we discuss in this paper the results of a research process framed in the “Mathematics Teacher Training Program” [Master Program] addressed to future teachers for middle and high schools, provided at the University of Barcelona. We expect to discuss tools for a descriptive and explanatory analysis aiming to answer “what happens in the classroom and why?”

Our general intention in such a program is led future teachers to develop the [professional] ability to (re)design sequences of suitable tasks, as well as to make them able to re-design their own designs of school tasks. In our study we call ‘professional task’ those tasks that we propose to future teachers in order to encourage them doing didactic analysis and developing their didactical analysis competencies. We understand ‘competence’ the ability for designing, applying and evaluating sequences of learning by means of didactic analysis techniques and quality criteria. It is also assumed that someone may reflect and improve their competence in terms of the analysis of mathematical classrooms, in order to make best use of the opportunities for being a teacher as teacher enquirer (Mason & Johnston-Wilder, 2004).

We want to focus on some immediate effects over the Program. We found them when analysing prospective teachers’ thoughts emerging from their feedback with the researchers; and also emerging from our analysis of some impacts of the program itself. Such above mentioned development, it is stated when future teachers incorporate and use tools for the description, explanation and process valuation of mathematical school teacher/learning practices.

In this paper we discuss data emerging out from a teaching research project based on an inquiry and reflective practicing framework in which we design and implement diverse teacher training cycles as teaching experiments (Tzur, Sullivan & Zaslavsky, 2008) for developing transversal competencies such as “citizenship”, “digital competency”, “didactical analysis”, among others.

The development of the cycles was based from the very beginning of the research process in six big types of professional tasks: (a) analysis of practices, objects and mathematical processes; (b) analysis of didactic interactions, conflicts and norms; (c) evaluation of tasks and classroom episodes using criteria of didactic suitability or quality; (d) design and implementation of a lesson in their [prospective teachers] period of internship; (e) analysis and valuation of the suitability of the didactic implemented unit; (f) improvement of their lessons designs (for future implementation).

The analysis and description of the mathematical activity is conducted using the theoretical constructs proposed by the ‘Ontosemiotic’ approach (Font, Planas y Godino, 2010). According to this perspective, the mathematical activity plays a central role and it is modelled in terms of systems of operative and discursive practices. From these practices the different types of related mathematical objects emerge building cognitive or epistemic configurations among them. Problem-situations support and contextualize the mathematical activity; languages (symbols, notations, and graphics) may serve as tools for action; arguments justify the procedures and propositions embedded within the concepts. According to this approach, the objects appearing in mathematical practices might be considered from a dual dimension. 

Method

This proposal was conducted in the frame of the Master's Final Project (MFP). In this paper we draw on data collected concerning the first two types of tasks (a - b). Data was collected from video recorded observations, sorting sheets produced by teacher trainers, students’ reflections at the end of the workshops and documentation housed in the Moodle platform (slides, reading material, tasks and the students’ responses to them, and questionnaires and the students’ responses to them). The samples were 3 groups of 24-26 and 25 prospective teachers. This amount of teachers includes almost the totality of students recruited in the Teacher Program in the University. During the first year, future teachers did many naïf comments regarding the first tasks (a-b). We conjectured that protocols were static. During the next year we decided to use more videos and transcripts. Prospective teachers designed and implemented tasks (type b), with protocols showing constructs as cognitive and semiotic conflicts, epistemic obstacles, types of norms, patterns of models of management, interaction analysis, and so on. After that, they analysed a lesson focused on equations applying suitability criteria (task type c). During the second year, we decided to implement some tasks type (a), by emphasizing the analysis of processes; and tasks type (b) by using new video sources. In the new tasks (type a) we proposed the observation of three short ways of introducing perpendicular bisector with 12-13 years old students, by observing three different teachers. The main purpose was to present a discussion about the different practices, objects and mathematics processes and to introduce a reflection associated to how each of these classes contributes to introduce different kind of epistemic configurations and objects associated to three different definitions. After the second year of experience observing the analysis realized by the future teachers, some difficulties still appeared. As a consequence, the changes proposed for the third year were the following: (1) to join the categories for epistemic suitability from OSA with categories from the quality for mathematics instruction given by Hill (2010). In such way, it was introduced new criteria for valuing mathematical quality (mathematical richness, coherence, errors, etc.); (2) to select new case studies from previous years with more wide and complex explanations than the case studies used previous years. The aim was to connect echoes and voices to produce more consistent arguments (Garuti & Boero, 2002) to justify mathematical quality of didactical sequences.

Expected Outcomes

After the observations collected during the first year, we found that some future teachers had difficulties to relate didactical analysis to epistemic ideas. For instance, Student 5 claimed: “When I did the didactic unit I didn’t contextualize enough the exercises. Now, I think it’s important to use activities proposed in the article: ‘Algebra for all Junior High School students’. In these kinds of sentences, we expected to talk more about the specific iterative algebraic approach as an explicit content in the evidence presented by the student. However, this student preferred to focus on general aspects not connected to the teaching activity. That lack of a deepest (in epistemological and cognitive terms) understanding of the mathematical task was fully discussed during the whole research process, drawing on the tasks presented to the students (future teachers). At the end of the third year we found the students becoming more aware of the important role of theory (in the mathematics educational field) when presenting their didactical analysis, as well as when discussing their redesigns. In fact, some students affirmed that by doing the master’s degree work, “we had been developed our competence of didactical analysis”. More concrete: we found that suitability criteria used for redesigning the tasks (considered as teaching experiments and corresponding case studies) has anticipatory purposes as hypothetical trajectories, but also helps to improve didactic training trajectories. We recognized the final master degree as the starting point for developing research competency for future teachers. It gives opportunities for students to learn and recognize problems connected to their professional context. Drawing on our theoretical lenses, we intend to see didactical analysis beyond the banality, considering classroom situation as an integral but dynamic system evolving in time, promoting autonomous mathematical thinking and independent validation of its results as future teacher (Laborde, Perrin-Glorian, Sierpinska, 2005).

References

Font, V., Planas, N. & Godino, J. D. (2010). Modelo para el análisis didáctico en educación matemática [A model for didactic analysis in mathematics education]. Infancia y Aprendizaje, 33(1), 89-105. Garuti,R & Boero,P (2002) Interiorisation of forms of argumentation: A case study. In A.D. Cockburn & E. Nardi (eds) Proceedings of 26th PME Conference, (Vol. 2, pp. 408-415). Norwich UK University of West Amglia. Laborde, C., Perrin-Glorian, M.J.; Sierpinska, A. (2005). Beyond the Apparent Banality of the Mathematics Classroom (1-12). Netherlands: Springer. Mason, J.; & Johnston-Wilder, S. (2004). Designing and Using Mathematical Tasks. Tarquin: London. Tzur, R., Sullivan, P., & Zaslavsky, O. (2008). Examining teachers' use of (non-routine) mathematical tasks in classrooms from three complementary perspectives: Teacher, teacher educator, researcher. In O. Figueras & A. Sepúlveda (Eds.), Proceedings of the Joint Meeting of the 32nd Conference of the PME, and PME-NA (Vol. 1, pp. 133-137). México: PME. Zaslavsky, O., & Sullivan, P. (Eds.). (2011). Constructing knowledge for teaching: Secondary mathematics tasks to enhance prospective and practicing teacher learning. New-York: Springer.

Author Information

Javier Diez-Palomar (presenting / submitting)
University of Barcelona
Mathematics and Sciences Education
Barcelona
Joaquim Gimenez (presenting)
University of Barcelona, Spain
Yuly Vanegas
University of Barcelona, Spain
Vicenç Font
University of Barcelona, Spain

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