Mathematical thinking: foundations – development – education
Niepubliczna Szkoła Podstawowa Edukacji Matematycznej “Edu&MATH”
(Edu&MATH Private Elementary School for Mathematical Education)
January 24th–26th 2019
Kampus Wielicki (Wieliczka Campus) , ul. Marszałka Józefa Piłsudskiego 105,
32-020 Wieliczka, Poland
Subject – inspiration
It is difficult to define mathematics. Depending on the philosophical system followed, one might say that mathematics is a way to describe, or to give structure to, the reality we live in, including a method to model processes which we observe. The special role of mathematics stems from its effectiveness. The effectiveness depends on, on the one hand, close ties between mathematics and the human cognitive system, and, on the other, a highly precise language, inbuilt in mathematics. No other branch of knowledge requires so strict a discipline of thought and such precision of statement, as mathematics does.
The Conference is addressed to:
A strong feedback exists between mathematical thinking, mathematical imagination and mathematical intuition, as well as mathematical knowledge based thereon on the one hand, and mathematics education on the other. For mathematics education to be effective, it should start early and be consistent, taking into consideration a pupil’s abilities and knowledge. Another important factor is proper preparation of teachers in terms of their mathematical knowledge and teaching methods mastered. Mathematical abilities and skills are often mistakenly identified with computational proficiency. It is quite frequently that we hear the following common opinion on a child’s abilities: He/she is much talented in mathematics, because he/she proficiently operates numbers, and large ones. It should be expressly emphasised that there is no direct connection between computational proficiency and mathematical abilities. Persons are known who were able to mentally do extremely complex arithmetical operations, but never revealed mathematical abilities. And the other way round, outstanding mathematicians are known who computed slowly and rather reluctantly. Mathematical abilities should rather be seen as predisposition to a specific kind of thinking, which we call mathematical thinking.
It is by far more important to instil in pupils the mathematical way of thinking (based on logically correct foundations) and mathematical imagination, than to train them in arithmetical technicalities. Accordingly, while teaching mathematics, a special emphasis should be put on the development of such thinking and not on computational proficiency. Pupils’ difficulty with solving word problems, so often observed, is just the difficulty with building a mathematical micro-model corresponding to the situation described in a given word problem. The ability to independently build a correct micro-model is conditional upon a pupil’s having appropriately developed mathematical imagination.
Education directed towards the development of mathematical thinking requires taking dedicated steps. At various education levels these steps take various forms. From modelling elementary mathematical objects with actual physical objects at the pre- and early-school levels, to problem formulating at the level of university mathematical majors.
Under the current educational model, mathematics does not represent a consistent entirety. The ways it is treated in the early-school education and in higher grades do differ. Nor is teaching of mathematics organised consistently in higher grades, with the stress the proficiency in computation and solving standard problems. It should expressly be emphasised that there is just one mathematics, and it is only teaching methods that need to be adapted to pupils’ perception abilities. To instil in pupils the mathematical way of thinking and mathematical imagination is much more important than for pupils to acquire technical mathematical skills.
Undertaking research into the subjects outlined above and formulating constructive conclusions require multi-party cooperation of specialists representing various disciplines, including: mathematicians, mathematics education specialists, mathematics teachers, logicians, philosophers, psychologists, cognitive and pedagogy scientists.
- scholars – mathematicians, logicians, philosophers, psychologists and cognitive scientists – who conduct research into the foundations and development of mathematical thinking in relations to both individual development in correlation with endo- and exogenesis, and social environment and considerations;
- mathematicians who conduct research into mathematics education from both the theoretical and practical (mathematics teaching methods) perspectives;
- mathematics teachers (at any educational level) who record their own observations concerning teaching of mathematics;
- anyone interested in the Conference subject matter.
- Identification of problems and exchange of views and opinions on mathematics education, including in particular in relation to the concepts of the foundations and development of mathematical thinking.
- Establishment of working contacts for exchanging opinions and developing new teaching methods.
- Publication of post-Conference materials and undertaking various projects initiated at the Conference, including publications concerning teaching syllabi/curricula and methods.