Teaching Mathematical Problem-Solving with the Brain in Mind: How Can Opening a Closed Problem Help?

By Ambrus, András | CEPS Journal : Center for Educational Policy Studies Journal, April 1, 2014 | Go to article overview

Teaching Mathematical Problem-Solving with the Brain in Mind: How Can Opening a Closed Problem Help?


Ambrus, András, CEPS Journal : Center for Educational Policy Studies Journal


Introduction

For a long time in the Hungarian teaching practices of mathematics, the scientific aspect of mathematics dominated, while the psychological, ped- agogical, social, biological aspects were mostly neglected. It is not surprising that in Hungarian mathematics curricula and in the mathematics textbooks (including in lower grades) there is a chapter with the title 'Sets. Logic', which cannot be found in other European and American curricula and mathematics textbooks. In short, in Hungarian mathematics teaching, the symbolic, abstract and verbal aspects are dominant.

Regarding mathematical problems and tasks, the so-called closed prob- lems are predominantly used. I can characterise our mathematics teaching by quoting the opinion of Laurinda Brown, who, after numerous visits to Hungar- ian secondary schools, summarised her experiences in the following way: 'You in Hungary are teaching mathematics; we in England children'.

Another main characteristic of Hungarian mathematics teaching is the fostering of talented pupils, which is in the centre of mathematics teaching. Hungary is a small country from which many world famous mathematicians come. The idea is that such a small country must honour its talent, because they can contribute in a great manner to the development of our country. A direct consequence of focusing mainly on fostering talented students is that 90% of the students suffer from this situation. Teaching not only the rather talented but also average students, I slowly started to seek some possibilities to help the aver- age pupils. Many books and articles have been published recently on the topic of learning with a specific focus on how the brain works, which can be also ap- plied for mathematics education. Based on my studies, I started to change my mathematics teaching style.

In this article, I will report about my experience with a selected problem, which was formulated and used in Hungarian mathematics teaching in a closed form. However, seeing the immense difficulties my students had, I opened the problem. I have fifty years mathematics teaching experience; nevertheless, I think sometimes it may be appealing to watch and listen to other experts. In terms of research method, we may classify this study as a case study, but I will refer not only to the analysed experience with my students attending my math- ematics group study sessions but also to my former class teaching experience. The question I wanted to study is whether it is possible to that more students solve the problem individually or with a small amount of help.

Some theoretical considerations

I use the term 'open problem' for problems in which at least one of the three basic notions (initial state, transformation (solution) steps, and goal state) is not exactly determined. We speak about 'open-ended problems' if the goal state is not determined. In this sense, many Hungarian textbook and task col- lection tasks are open problems because, although their starting situation and goal situation are explicitly given, finding the solution path is a quite difficult task. Our closed problem was transformed into an open-ended problem.

How can the teacher think about mathematics?

Mathematicians view their subject from any one of three different per- spectives: Platonist, Formalist, or Intuitionist.

Platonist: Mathematics exists in an abstract plane. Objects of mathemat- ics are as real as everyday life. Mathematics reality exists outside the human mind. Mathematicians' function is to discover or observe mathematical objects.

Formalist: Mathematics is a game in which one manipulates symbols in ac- cordance with precise formal rules. Mathematical objects have no relation to reali- ty and are solely a set of symbols that satisfy the axioms and theorems of geometry.

Intuitionist: Mathematical objects are solely constructions of the human mind. Mathematics does not exist in the real world, but only in the brain of the mathematician who invents it. …

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