Research approaches
A - Realistic Mathematics Education
Realistic Mathematics Education (RME) is an approach to maths education developed in The Netherlands by the maths educators of the Freudenthal Institute. It is proposed here that the teaching and learning of maths should be connected to reality, stay close to children’s experience and be relevant to society, in order to be of human value. Maths lessons according to Freudenthal should give students the ‘guided’ opportunity to ‘re-invent’ maths by doing it; the focal point should not be on maths as a closed system but on the activity, on the process of mathematization.
The features of RME include the following.
- Use of realistic situations to develop maths
- Well researched activities that encourage students to move from informal to formal representations
- Less emphasis on algorithms, more on making sense
- Use of 'guided reinvention'
- Progress towards formal ideas seen as a long-term process
B - Constructivist
The lessons here can, in places, appear very structured and teacher-led. However, they are designed to help students explore and contemplate mathematical ideas. The intention is that students are given the opportunity to reveal their thinking, to value their ideas, and to explore, critique and develop them through discussion and activities guided by the teacher. In the process, it is hoped that the teacher will get a richer understanding of their students’ thinking and of the complex nature of multiplicative reasoning. The lessons are predicated on the belief that multiplicative reasoning is not learnt in a ‘linear’, step by step, level by level way, but that it comprises a complex network of ideas that is constructed, strengthened, extended and modified over a long period of time. These few lessons can only provide snapshots of some of these ideas but it is hoped they will stimulate the teacher to revisit and take them further with their students.
C - Enquiry/cognitive conflict
A collaborative enquiry-based approach to learning where students are engaged in cognitive conflict has been shown to promote long-term learning (see for example Birks (1987) ‘Reflections: a Diagnostic Teaching Experiment’, Cobb (1988) ‘The tension between theories of learning and instruction in Mathematics Education’, Onslow (1986) ‘Overcoming conceptual obstacles concerning rates: Design and Implementation of a diagnostic Teaching Unit’, and Swan (1983) ‘Teaching Decimal Place Value – a comparative study of ‘conflict’ and ‘positive only’ approaches’). Students become aware of the inconsistencies in their own conceptions and this awakens a curiosity and desire to seek resolution through discussion. A final whole class discussion allows students to share their different understandings and provides an opportunity for generalisation and extending what has been learned.