To develop the ability of teacher students to reason mathematically

The aim of the project The project aims at developing the ability of teacher students to reason mathematically and to understand pupils' reasoning in mathematics. The problem in focus For a teacher to be able to talk to pupils in a way that suits the learning of the individual pupil she has a need to be capable of expressing herself in many different ways and to give arguments and to reason in such a way that the pupil understands. This aim is rarely reached by the ways teacher courses are structured today. A stronger focus on mathematical reasoning and communication is necessary. What is the ability to reason mathematically? Mathematics as a science is characterised among other things by the method to work with deductions and proofs which means that starting from certain given conditions you can with full certainty prove propositions and theorems. To be able to follow and develop such ways of thinking and arguing you have to develop your own ability to reason mathematically. In the school curriculum deductions and proofs have been given new importance. It is necessary for the teacher to have developed her own ability to reason mathematically in order to be able to work with the pupils to reach the aims of the curriculum in this area. The teacher must in conversations with the pupils be flexible and be able to understand different ways of thinking and reasoning. This makes it necessary for the teacher to be able to reason, to follow multiple ways of thinking, and to compare them and to value how viable they are. What do we mean by ability to reason mathematically? A good ability to reason mathematically means that you can reformulate questions and propositions in different ways, make and test conjectures, reject them or verify them, formulate counterexamples, specialise, generalise, draw conclusions, find alternative ways when you are stuck, decide if a solution is reasonable and judge the validity of arguments, describe, explain, and convince others about your arguments and to prove your claims. Mason, Burton and Stacey (1982) have demonstrated how to encourage, develop and foster the processes mentioned above, which seem to come naturally to mathematicians. They suggest a method of working that is highly practical by starting from exemplary questions and problems and involving the learner in the discussions. In this way a deeper awareness of the nature of mathematical thinking and reasoning can grow. Methods We want to take a departure in research results from mathematics education and offer the student teachers opportunities to develop their own ability to reason mathematically in a more conscious way. We want to develop the mathematics parts of the courses. Instead of starting in a traditional way by presenting the theory and then continuing with problem solving exercising the application of the theory, we want to start with open or exploratory problem solving that give the students an experiential background for the introduction of theoretical concepts. The work will be done in small groups and the focus on reporting the work will be on presenting their findings and on convincing others about the reasonableness of the conclusions they have drawn. To work with different pieces of mathematical argument, both from former teacher students and pupils, will be another type of task. Alongside they will work with a selected number of more conventional mathematical problems, but be stimulated to continue to use the group as a resource, discussing different difficulties and explaining to each other different ways of solving the problems. Comparing and valuing the viability of alternative solutions is vital here. We will construct open problems and exercises that foster reasoning and try them out in group-work with student teachers. After evaluation and reconstruction if necessary the problems will be used in future mathematics teacher training courses