Jean Piaget

For reasearch purposes, Piaget pursued a separate understanding of topological space, projective space and Euclidean Space. It is obvious, of course, that there is actually only one space and that these are three interrelated way by which human intelligence attempts to grasp the patterns and connections in that one space. As our eyes dart around the world about us, visual perception by itself gives us little basis for understanding how that world works. Mature understanding evolves when operationally guided manipulation of the objects of the world and coordination of that manipulation with vision have resulted in knowledge of topological relationships, have provided a child with the capacity to imagine any of the infinite possible perspectives, and have resulted in the recognition of Euclidean relationships. When all three of these can be related to each other to achieve a connected understanding, the a child is prepared to exercise well developed spatial ability. (D-TSC 41f)

Piagets verschiedenen Arten der räumlichen Beziehungen:

• topologische Beziehungen - kategoriale Relationen: präoperationale Phase ( bis 7 Jahre)
• projektive Beziehungen - Ordnungsrelationen: konkret-operationale Phase (7-11 Jahre)
• euklidische Beziehungen - Distanzrealtionen: konkret - operationale Phase (7-11 Jahre)

Topologische Beziehungen

Topological relationships involve such things as proximity, separation, order, enclosure, and continuity. These are the simplest of spatial relationships, but the basis for the development of all others. (Dixon-TSC 30)

Nähe

• Nähe (proximity)

Proximity is the observation that two things occur near each other. (D-TSC 30)

Trennung

• Trennung (separation)

Separation is the observation that although two things may be in close proximity, they are disctinct, separate things. (D-TSC 30)

Ordnung

• Ordnung (order)

Order means that a sequence is observed, i.e. three or more separate objects remain in some consistent order. Essential to the full comprehension of order is the ability to reverse order and to see that this reversal represents the same ordered relationship seen from the opposite direction. (D-TSC 30)

Einschluss

• Einschluss (enclosure)

Enclosure is the observation that something is surrounded by other things. (D-TSC 30f)

Kontinuität

• Kontinuität (continuity)

Continuity is the observation that although a number of objects may be focused on as distinct from each other, they all occur together in certain continuous relationships, so that they make up some whole entity. (D-TSC 31)

Projektive Beziehungen

The full development of projective skills isn't reached until children can create in their mind a fairly accurate image of how objects, even though they have seen the objects from only one position. ... In other words, it involves the imagination and coordination in your mind of the infinite possiblities of lines of sight. (D-TSC 34)

• Projektiver Raum: Gerade durch Steichhölzer aufstellen, die durch Anfangs- und Endpunkt vorgegeben ist.
• Drei-Berge-Versuch: Die Blickwinkel einer Puppe, die die drei Berge aus verschiedenen Blickwinkeln stellvertretend sieht, sollen nachgebildet werden.

Euklidische Beziehungen

Euclidean understanding of space involves knowing when a few rather special properties occur and when these properties are preserved in the structure of objects. (D-TSC 39)