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SUBSTANTIAL LEARNING ENVIRONMENTS IN GEOMETRY

Examples of substantial learning environments that stimulate individualization and enable natural
differentiation which is one of powerful motivating factors.


Guided by: A. Hošpesová, F. Roubíček, M. Tichá
It is a generally accepted fact that there are differences in how children learn, that they need different length of time to master the same subject matter under the same conditions with pupils of the same age and with the same teacher. This makes considerable demands on teachers' professional competences. One of the crucial competences is the ability to cope with heterogeneity and to foster favourable conditions for development of educational prerequisites of all their pupils. That is why an increasing number of educators and teachers try to apply the principles of individualization in teaching. Educators looked for and proposed various methods of differentiation which would respect the differences between pupils and their individualities, and whose objective is optimum pupils' motivation and development.
In the working seminar Substantial Learning Environments in Geometry we will discuss questions related to the creation and discovery of substantial learning environments (SLE) and the potential they bear for general improvement of education. The concrete examples of SLE will be shown and characterized (mathematical content, source of notions and solving methods...). We will also demonstrate
  • what potential these SLEs have for individualization and natural differentiation as one of the forms of inner differentiation,
  • power of motivation of natural differentiation.
The work will start with video recordings and transcripts of short episodes from mathematics lessons realised in the "geometrical" learning environments named Mosaics, Way (work with maps and plans), and Room (work with plans of room equipment).
Mosaics: During the teaching experiment, the pupils were working with different types of mosaics; they for example assembled shapes that they fancied, assembled shapes with a given number of pieces of the mosaic/with given particular pieces of mosaic.
Way: The environment Way works with an idealized street plan. Pupils became familiar with the map, with the structure of a description of a way, with interpretation of instructions, with criteria that a description should meet. After that we focused on three types of problems: (a) drawing a verbally described way in the map, (b) verbal description of a mapped way, (c) pupils' individual choice of a way and its description.
Room: Two basic activities were used: (a) modelling of 3D space and objects in 2D, (b) arrangement of objects in space. This environment opens space especially to development of modelling, of estimating, of work with scales, of the concepts of transformations (reflection, translation, rotation), of filling up the space.

We suppose discussion especially on following problems and questions:
  • Stimuli to natural differentiation in mentioned activities (learning environments).
  • Expectations concerning benefit of inclusion of work within mentioned environments. Are realistic?
  • Benefits of organizing the tasks that enable natural differentiation into mathematics lessons.
  • Further geometrical environments that would be possible to mark as SLEs, in frame of which it would be possible to pose tasks that enable natural differentiation.
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