Analysis of algebraic reasoning and its different levels in primary

Proposal of a Workshop to develop in the 13th ICME-13 on the theme:
Analysis of algebraic reasoning and its different levels in
primary and secondary education
Organizers:
Juan D. Godino, University of Granada (Spain), [email protected]
Miguel R. Wilhelmi, Public University of Navarra (Spain)
Teresa Neto, University of Aveiro (Portugal)
1. Background and expertise of the organizers
Dr. Juan D. Godino is Professor of Didactic of Mathematics at the University of
Granada (Spain), principal researcher of the Project, “Assessment and development of
mathematical and didactical competencies of prospective primary education teachers”
(Ministry of Economy and Competitiveness, Spain).
Dr. Miguel R. Wilhelmi is member of the mentioned project and faculty member of the
Public University of Navarra (Spain)
Dr.Teresa Neto is also member of the project and faculty member of Aveiro University
(Portugal).
In the workshop some consolidated results of the project, specifically on the recognition
of algebraization levels of school mathematics activity and the connections between
algebraic thinking in primary and secondary education will be implemented.
Some publications related with the workshop theme are below included:
Aké, L., Godino, J. D., Gonzato, M. & Wilhelmi, M. R. (2013). Proto-algebraic levels
of mathematical thinking. In Lindmeier, A. M. & Heinze, A. (Eds.). Proceedings of
the 37th Conference of the International Group for the Psychology of Mathematics
Education, Vol. 2, pp. 1-8. Kiel, Germany: PME.
Godino, J. D. Aké, L., Gonzato, M. & Wilhelmi, M. R. (2014). Niveles de algebrización
de la actividad matemática escolar. Implicaciones para la formación de maestros.
Enseñanza de las Ciencias, 32.1, 199-219.
Godino, J. D., Neto, T., Wilhelmi, M. R., Aké, L., Etchegaray, S. & Lasa, A. (2015).
Levels of algebraic reasoning in primary and secondary education. CERME 9,
TWG 3: Algebraic Thinking.
Godino, J. D., Castro, W., Aké, L. & Wilhelmi, M. D. (2012). Naturaleza del
razonamiento algebraico elemental. Boletim de Educação Matemática - BOLEMA,
26 (42B), 483-511.
Godino, J. D., Neto, T., Wilhelmi, M. R., Aké, L., Etchegaray, S. & Lasa, A. (2015)
Niveles de algebrización de las prácticas matemáticas escolares. Articulación de las
perspectivas ontosemiótica y antropológica. Avances de Investigación en
Educación Matemática (in press).
Godino, J. D., Aké, L. P., Contreras, A., Díaz, C., Estepa, A., Blanco, T. F., Lacasta, E.,
Lasa, A., Neto, T., Oliveras, M. L. & Wilhelmi, M. R (2015). Diseño de un
cuestionario para evaluar conocimientos didáctico - matemáticos sobre
razonamiento algebraico elemental. Enseñanza de las Ciencias, 33.1, 127 - 150.
Godino, J. D., Wilhelmi, M. R, Neto, T., Blanco, T. F., Contreras, A., Díaz-Batanero,
C., Estepa, A. & Lasa, A. (2015). Assessing prospective primary school teachers’
didactic-mathematical knowledge on elementary algebraic reasoning. Revista de
Educación (in press).
2. Description of the topic theme including aims and rationale for the topic
An important objective in various curricular guidelines (e. g., NCTM, 2000) is the
enhancement of algebraic reasoning since the first educational levels. This objective
implies that we assume a new view of school algebra as being not limited to handling
algebraic expressions (Aké, Godino, Gonzato & Wilhelmi, 2013; Godino, Ake, Gonzato
& Wilhelmi, 2014). The effective implementation of this new conception of school
algebra poses a challenge for the training of mathematics teachers, because few current
training programs include the development of such new vision.
The objective of the workshop is to implement some practical activities aimed at
recognizing the main features of School Algebraic Reasoning (SAR), which can be used
to train teachers to promote algebraic thinking in primary and secondary education. The
wider view of school algebra that will be presented and discussed takes into account the
processes of generalization, symbolization, as well as structural and functional
modeling and analytical calculation. It also creates a meaningful link between algebraic
thinking in primary and secondary education.
After discussing two main references about SAR, the attendees will be asked to solve a
set of tasks with the aim of identifying the algebraic objects and processes put at stake,
and recognizing the different algebraization levels involved in the mathematical activity
carried out when solving these tasks.
3. Key questions and issues for participants in the Workshop
The workshop aims to achieve the following objectives:
a) Reflecting on the characteristics of Algebraic Reasoning in Primary and Secondary
Education and deepening the participants’ knowledge of them.
b) Recognizing algebraic reasoning levels required to solve primary and secondary
education mathematical activities.
c) Designing and analyzing tasks whose solution involves changes in the algebrization
levels.
4. Description of the structure
The workshop aim for a deeper interaction with the specific topic emphasizing activities
of the participants. It will be organized in two modules of 45 minutes each:
Module 1:
1. Introducing the workshop and discussing the Proto-Algebraic Reasoning for
Elementary Education model. Previous reading:
Aké, L., Godino, J. D., Gonzato, M. & Wilhelmi, M. R. (2013). Proto-algebraic levels
of mathematical thinking. In Lindmeier, A. M. & Heinze, A. (Eds.). Proceedings of
the 37th Conference of the International Group for the Psychology of Mathematics
Education, Vol. 2, pp. 1-8. Kiel, Germany: PME.
2. Practical Activity 1: Working in teams to perform the following tasks:
i) Solving a selected set of tasks on School Algebraic Reasoning (Part I).
ii) Assigning levels of algebraic thinking to different solutions given to the tasks by
taking into account the previously identified algebraic objects and processes.
iii) Enunciating related tasks whose solution involves changes in the levels of
algebrization.
3. Presentation and discussion of results.
Module 2:
1. Discussing the model of Algebraic Reasoning for Secondary Education. Previous
reading:
Godino, J. D., Neto, T., Wilhelmi, M. R., Aké, L., Etchegaray, S. & Lasa, A. (2015).
Levels of algebraic reasoning in primary and secondary education. CERME 9,
TWG 3: Algebraic Thinking.
2. Practical Activity 2: Working in teams to perform the following tasks:
i) Solving a selected set of tasks on School Algebraic Reasoning (Part II).
ii) Assigning levels of algebraic thinking to different solutions given to the tasks,
taking into account the previously identified algebraic objects and processes.
iii) Enunciating related tasks whose solution involves changes in the levels of
algebrization.
3. Presentation and discussion of results.