Abstract
We present the ArAl project, conceived as an integrated system of teacher education and classroom innovation, aiming at renewing the teaching of arithmetic in an early algebra perspective, guiding pupils towards the discovery of letters to express generalities. We focus on some theoretical key points (KP) and on the main language constructs (LC). Through excerpts of class-discussions, we show the incidence of KP and LC on the progressive construction and refinement of pupils’ early algebraic thinking . Finally we discuss the difficulties met by teachers at the K–8 levels when they reflect upon their knowledge, beliefs, attitudes, and ways of relating with the pupils.
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Notes
- 1.
‘Devolution’ is a term introduced by Brousseau (1997) in his Theory of Didactical Situations in Mathematics (Didactique des Mathématiques 1970–1990). It indicates a process between the teacher and her students where she, in presenting a problematic situation, brings them to assume the responsibility to deal with it. The devolution is fulfilled when the students actually accept the uncertainty implied in this assumption and they take on the commitment.
- 2.
For typing questions, here we have written “=?=”. In our teaching experiments, the teachers put the question mark on top of the equal sign to stress that they are in front of a hypothetical equality; the pupils then have to express the reasons that support or refute it.
- 3.
The ArAl Units (at the moment there are 12 of them)—supported by the theoretical frame and the glossary—can be seen as models of teaching pathways for arithmetic in an algebraic perspective. They are structured in such a way as to make the teaching process transparent in relation to the problem situation being examined (methodological choices, activated class dynamics, key elements of the process, extensions, potential behavior of pupils, and difficulties they may encounter).
- 4.
The ArAl Unit 11 is devoted to the construction of this property.
- 5.
This term unifies a set of teacher’s characteristics, such as knowledge, beliefs, orientations, goals, and ways of being inside the classroom.
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Malara, N.A., Navarra, G. (2018). New Words and Concepts for Early Algebra Teaching: Sharing with Teachers Epistemological Issues in Early Algebra to Develop Students’ Early Algebraic Thinking. In: Kieran, C. (eds) Teaching and Learning Algebraic Thinking with 5- to 12-Year-Olds. ICME-13 Monographs. Springer, Cham. https://doi.org/10.1007/978-3-319-68351-5_3
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