This thesis is focused on mathematical definitions and concepts, from cognitive and didactical points of view, in the field of plane geometry, solid geometry and equations. Starting from research questions formulated in some classic theoretical framework, I set up a questionnaire, which has been proposed to 145 students attending the fourth year of liceo scientifico (grade 12), 24 students attending the fourth year of liceo classico (grade 12) and 149 grade 8 students. I based the study on the theoretical constructs of Concept Image and Concept Definition proposed by Tall and Vinner. I investigate the students Concept Image and, in particular, if it was tightly coherent with their Concept Definition. In this regard, as for geometry, I refer to the researches in didactics of mathematics done by R. Hershkowitz and N.C. Presmeg with respect to the prototypes phenomenon and on the Fischbeins theory of Figural Concepts. The well-known and long-debated issue of figure classification in plane geometry, i.e. inclusive classification vs. exclusive classification, helped the analysis of the definitions given by pupils for the main plane figures. With respect to the equations I attempted to cross two different theoretical frameworks, in order to analyse students behaviour by means of two distinct lens: the constructs of Concept Image and Concept Definition, and the distinction between Conceptual Knowledge and Procedural Knowledge operated by J. Hiebert. The results of this analysis confirm the ones of previous researches in the field of plane geometry and show a huge gap between the level of conceptual and procedural knowledge with respect to the meanings and the solving process of equations.
Questo lavoro di tesi si concentra su alcuni aspetti cognitivi e didattici relativi a definizioni e concetti nellambito della geometria piana e solida e delle equazioni. Lindagine si avvale di un questionario, costruito a partire da domande di ricerca formulate in alcuni quadri teorici classici della ricerca in didattica della matematica, che è stato somministrato a 145 studenti frequentanti la quarta Liceo Scientifico, 24 studenti frequentanti la quarta Liceo Classico e a 149 studenti frequentanti la classe terza delle scuole medie. Sulla base del quadro teorico proposto da Tall e Vinner sui costrutti di Concept Image e Concept Definition, ho indagato la Concept Image degli studenti e la sua coerenza con la Concept Definition. A tal fine, nellambito della geometria ho fatto riferimento alle ricerche effettuate in didattica della matematica sul fenomeno dei prototipi, in particolare da R. Hershkowitz e N.C. Presmeg, e alla Teoria dei Concetti Figurali di E. Fischbein. Per svolgere unanalisi delle definizioni fornite dagli studenti per le principali figure piane ho inoltre richiamato la spinosa ed annosa questione della loro classificazione, distinguendo tra classificazione inclusiva ed esclusiva. Per quanto riguarda le equazioni sono stati incrociati due diversi quadri teorici. In altri termini, il comportamento messo in atto dagli studenti è stato descritto e interpretato mediante due diverse lenti, i costrutti di Concept Image e Concept Definition, e la distinzione tra Conoscenza Concettuale e Conoscenza Procedurale presentata da J. Hiebert. I risultati di questa analisi confermano quelli degli studi precedenti relativamente alla geometria e mostrano una enorme discrepanza tra il livello di conoscenza concettuale e quello procedurale relativamente ai significati e alla risoluzioni di equazioni.
Definizioni e concetti in matematica: analisi cognitiva e didattica nella scuola secondaria di primo e di secondo grado.
GUARGUAGLINI, YLENIA
2018/2019
Abstract
This thesis is focused on mathematical definitions and concepts, from cognitive and didactical points of view, in the field of plane geometry, solid geometry and equations. Starting from research questions formulated in some classic theoretical framework, I set up a questionnaire, which has been proposed to 145 students attending the fourth year of liceo scientifico (grade 12), 24 students attending the fourth year of liceo classico (grade 12) and 149 grade 8 students. I based the study on the theoretical constructs of Concept Image and Concept Definition proposed by Tall and Vinner. I investigate the students Concept Image and, in particular, if it was tightly coherent with their Concept Definition. In this regard, as for geometry, I refer to the researches in didactics of mathematics done by R. Hershkowitz and N.C. Presmeg with respect to the prototypes phenomenon and on the Fischbeins theory of Figural Concepts. The well-known and long-debated issue of figure classification in plane geometry, i.e. inclusive classification vs. exclusive classification, helped the analysis of the definitions given by pupils for the main plane figures. With respect to the equations I attempted to cross two different theoretical frameworks, in order to analyse students behaviour by means of two distinct lens: the constructs of Concept Image and Concept Definition, and the distinction between Conceptual Knowledge and Procedural Knowledge operated by J. Hiebert. The results of this analysis confirm the ones of previous researches in the field of plane geometry and show a huge gap between the level of conceptual and procedural knowledge with respect to the meanings and the solving process of equations.È consentito all'utente scaricare e condividere i documenti disponibili a testo pieno in UNITESI UNIPV nel rispetto della licenza Creative Commons del tipo CC BY NC ND.
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https://hdl.handle.net/20.500.14239/18925