Geometry is a fundamental part of mathematical learning. Since ancient time the study of geometry was considered as one of the most important subjects in school. In the arcade of the famous school of Athens, where Plato taught, it was written that entry was not permitted to people who did not know geometry. In the Renaissance period, geometry was part of the 'quadrivium', which was considered a needed work preparatory for a serious study of philosophy. Nevertheless, despite geometry is one of the main areas of mathematical learning, the cognitive processes underlying geometry-related academic achievement have not been studied in detail.
The present dissertation has three important aims. First, to investigate the relationship between various aspects of geometry and visuospatial working memory (VSWM). Second, to investigate whether the children with nonverbal learning disabilities (NLD) symptoms present difficulties in various aspects of geometry. Third, to investigate the relationship between various aspect of geometry, working memory (WM) and intelligence (g).
In the second chapter, a general overview of the relationship between geometry, WM and g is provided. Since geometry concerns the study of the space, it requires a particular involvement of spatial abilities. Thus, WM, and in particular VSWM should be crucially involved. In addition, solving geometrical problems requires to reason and to find out a solution among various alternatives. Thus, g should be crucially involved in solving geometrical problems.
In the third chapter, the relationship between academic achievement in geometry, intuitive geometry (i.e., a part of geometry which seems to be independent from the culture), and VSWM will be examined. Two studies will be presented.
In the first study, the involvement of VSWM in intuitive geometry and in school performance in geometry at secondary school was tested. A total of 166 pupils were administered: (1) six VSWM tasks, comprising simple storage and complex span tasks; (2) the intuitive geometry task devised by Dehaene, Izard, Pica, and Spelke (2006), which distinguishes between core, presumably independent form the culture, and culturally-mediated principles of geometry; and (3) a task measuring academic achievement in geometry.
In the second study, VSWM and intuitive geometry were examined in two groups aged 1113; one with children displaying symptoms of NLD, and the other, a control group without learning disabilities. The two groups were matched for general verbal abilities, age, gender, and socioeconomic level. The children were presented with simple storage and complex-span tasks involving VSWM and with the intuitive geometry task devised by Dehaene and colleagues (Dehaene et al., 2006).
In the fourth chapter, we report a study on the relationship between geometry, WM, and intelligence aimed to find out the model of WM which provided the best fit to the data and to examine the strength of the relations between WM and intelligence (part I) and the relationship between geometry (intuitive geometry and geometrical achievement), WM and g testing several models (part II).
In the last chapter a general overview of the important theoretical and applied implications of the three studies will be discussed. The limits of the present dissertation and possible future researches will also be outline
