PROBLEM-BASED LEARNING AND MATHEMATICS

Starting from the assumption that problem-based learning is an instructional approach suitable for transmitting

a real understanding of mathematics, we have to regret that such a methodology is not common in real teaching practice at school, as teachers usually rely on self perpetuating “traditional” methods. These  include:

• training courses for pre-service and in-service teachers;

• organizing problem-based mathematical laboratories in school (both in primary school and at a higher level);

• designing interactive exhibitions;

• organizing web-based mathematical game contests;

• publishing books and magazines.

It is my believe that the role of Research is crucial in carrying on synergical actions both on the teachers and on the pupils in order to improve the effectiveness of the teaching of mathematics.

Problem-based learning (PBL) is a constructivist learner-centred instructional approach based on the analysis, resolution and discussion of a given problem. It can be applied to any subject, indeed it is especially useful for the teaching of mathematics . For a neat definition of PBL we refer to according to which PBL “is an instructional (and curricular) learner-centered approach that empowers learners to conduct research, integrate theory and practice, and apply knowledge and skills to develop a viable solution to a defined problem”. Typically a PBL session follows these steps :

• pupils are given a problem;

• they discuss the problem and/or work on the problem in small groups, collecting information useful to solve the problem;

• all the pupils gather together to compare findings and/or discuss conclusions; new problems could arise from this discussion, in this case

• pupils go back to work on the new problems, and the cycle starts again.

In spite of researches documenting the effectiveness of PBL, we have to regret that such a methodology is not common in real teaching practice at school, as teachers usually rely on self perpetuating “traditional” methods . Research should thus focus on finding ways to support teachers practice in schools.

When designing refresher courses for teachers, it is important to focus not only on disciplinary contents, but also on methodological aspects. That is, on one side one has to deal with the peculiar nature of mathematics, keeping in mind that ordinary people usually have a conflicting relation with mathematics. This bad attitudes is often due to wrong beliefs about the subject, which is seen as a dull set of rules to be learned by heart. Thus teachers should give their students the opportunity to see mathematics as a living subject, that is as an exploratory, dynamic, and evolving science. Teachers should teach their students that the short answer is not what they really need, and that things must be looked at in depth. And this is the place where methodology steps in: simply put, if mathematics is an exploratory subject, then it must be taught in an exploratory way. The socio-constructivist perspective enforce this point of view, and doing mathematics in a PBL learner centred environment is a chance to give students a real understanding of the subject. Once again, the basic idea is that you do not appreciate and learn mathematics if you do not do mathematics.

Moreover, once accepted that PBL is an effective way to teach mathematics, one has to act on teachers’ beliefs in order to have them actually use PBL in their teaching practice. According to my experience, the most effective way to have teacher willing to use a PBL methodology is to have they experience themselves the joy of mathematical discovery, hence the courses for in-service teachers now adopt PBL. By undergoing PBL, teachers acquire beliefs in favour of  a PBL methodology as they see that in this way they learn some good maths . The main criticism concerns the fact that PBL is time consuming, and it is thus important to show teachers that the time is well spent as pupils get a real understanding of the subject, and gain at the same time many collateral competencies . A further objection coming from the teacher is that such activities rely on “good” problems and in-service teachers do not feel to be competent enough to create suitable ones. Researchers can provide teachers with ready to use material. There is one more aspects that needs to be stressed in order to use PBL teachers must be trained as their role is crucial. Once more the personal experience is the best ground on which they can build an effective teaching practice.