The benefit of the metaphor “contract” for understanding of concept formation in mathematics
Treść / Zawartość
The paper reports on a design and research project conducted in a secondary school on the Indonesian island Sumba. The aim of this project is to foster students’ understanding in learning mathematics and, consequently, to educate them better for their career opportunities. The essential idea is thereby to use the introduction of negative numbers and arithmetic operations on integers as a backgroundmotivating the learners to a new cognitive behavior, especially to critical thinking, monitoring and reflection. The new approach to teaching and learning uses the metaphor contract as a representation of a specific way of thinking when dealing with systems of concepts. Thereby, the systemic thinking in the sense of Sierpinska (Sierpinska et al., 2002; Sierpinska, 2005) and axiomatic thinking in the sense of Freudenthal (1963; 1968, p. 7) are meant. To establish this metaphor in students’ minds, a learning environment (LE) for the introduction of negative numbers was designed. Since the context used in this learning environment relates to banking operations with assets and debts in a fictitious bank, the learners conceive this context as a kind of game and not as a part of their everyday experience. Their activities are performed in a thought experiment and not as hands-on activities with concrete objects. The problem in designing the LE was how to find access to students’ intuitive knowledge related to integers and to reconstruct the intuitively understood way of dealing with these numbers and not how to find a simple way to make arithmetic calculations quickly. The emphasis is placed thereby on the process of reconstruction and precise representation of the intuitive knowledge up to a contractual (axiomatic) definition of addition, subtraction and the inverse of a number in the realm of integers. The access is created through the use of the metaphor “contract”. In the LE, the learners have to conclude a contract with the bank in order to achieve that the bank performs banking operations in a way which can be accepted by the students, based on their intuitive knowledge on debts and assets.Based on students’ written works, the paper presents some effects and explains the benefits of using the metaphor “contract”. The aim of this qualitative analysis of students’ solutions is not to provide the reader with a quantitative explanation of the teaching success, but to document and explain the intended understanding, in particular, the axiomatic view and some aspects of the systemic thinking as they are indicated in the effects of the teaching and learning process documented in this paper. It can be stated that the metaphor “contract” enables the learners to experience mathematics as human activity (Freudenthal, 1991, p. 14) – in particular as an “activity of discovering and organizing in an interplay of content and form” (ibid., p. 15) – and to use this experience in order to establish new cognitive habitus in learning mathematics.