Completion of the mathematical knowledge for teaching theory with figural concepts

Da bi bili učenci uspešni in da bi razvili dobro znanje, ni potreben zgolj njihov kognitivni potencial, ampak mora tudi njihov učitelj imeti ustrezna znanja, da tega potenciala ne zatre, ampak ga razvije v znanje. V tej doktorski disertaciji obravnavamo prav matematično znanje, potrebno za poučevanje na področju geometrije. V teoretičnem delu med seboj povežemo več za poučevanje in posebej za poučevanje geometrije pomembnih teorij. Podrobneje predstavimo Fischbeinovo teorijo konceptov s podobo, van Hielejevo teorijo skupaj z njenimi izboljšavami ter teorijo in model matematičnega znanja za poučevanje po Ball, Thames in Phelps. Izhajajoč iz analize omenjenih teorij in še nekaterih teorij razvoja konceptov ter s prepletom vseh, dopolnimo model matematičnega znanja za poučevanje še za (do zdaj nekoliko zapostavljeno) področje geometrije. Z raziskavo, ki jo predstavljamo v empiričnem delu, smo želeli dobiti vpogled v vlogo konceptualne in vizualne komponente konceptov s podobo pri študentih (bodočih učiteljih) razrednega pouka. Zanimalo nas je, ali na vključevanje konceptualne komponente pri reševanju šolskih geometrijskih nalog vplivata stopnja zamejenosti in topološka dimenzija vključenih konceptov. Izkazalo se je, da ima pri reševanju ključno vlogo vizualna komponenta, konceptualna komponenta je upoštevana predvsem pri rutinskih nalogah. Vključenost konceptualne komponente ni povezana s stopnjo zamejenosti ali topološko dimenzijo. Še posebej nas je zanimal problem ločevanja med likom in krivuljo, ki ga omejuje. To je pomemben problem, saj vpliva tudi na razvoj konceptov s področja merjenja (dolžine, ploščine in prostornine). Obravnave tega problema v literaturi predhodno še nismo zasledili. Izkazalo se je, da ima tudi tu vodilno vlogo vizualna komponenta in da izključitev konceptualne komponente vodi v neločevanje med likom in krivuljo. Na osnovi združitve spoznanj različnih teorij s področja poučevanja geometrije in rezultatov naše empirične raziskave oblikujemo smernice za izobraževanje učiteljev in poučevanje učencevIt is not only pupils’ cognitive potential that is necessary for them to be successful and develop good knowledgetheir teachers must also possess suitable knowledge not to suppress pupils’ potential but to develop it into knowledge. This doctor’s thesis discusses mathematical knowledge necessary for teaching geometry. Several teaching theories and theories particularly important for teaching geometry are combined in the theoretical section. The Fischbein theory of figural concepts, the van Hiele theory with improvements and the Ball, Thames and Phelps’ theory and model of mathematical knowledge for teaching are presented in more detail. On the basis of the analyses of the aforementioned theories, several other concepts development theories and their intertwining, the model of mathematical knowledge for teaching was further complemented in the field of geometry (so far a somewhat neglected field). The research presented in the empirical section provides an insight in the roles of conceptual and visual components of figural concepts of students (pre-service teachers) of primary education. We were interested whether inclusion of the conceptual component when solving geometrical tasks is influenced by the degree of limitedness and the topological dimension of included concepts. It was established that the visual component plays the key role in solving tasks, and the conceptual component is observed particularly in routine tasks. The inclusion of the conceptual component is not related to the degree of limitedness or the topological dimension. We were further interested in the problem of distinguishing between a flat shape and a curve that limits it. This is a significant problem as it also affects the development of concepts in the field of measures (length, area and volume). The discussion of this problem has not been detected in literature yet. It was established that the visual component has a leading role in this field as well, and that the exclusion of the conceptual component hinders the distinction between a flat shape and a curve. On the basis of combining the findings from different theories in the field of teaching geometry and the results of our empirical research, we formed guidelines for the education of teachers and the teaching of pupils