Research in mathematics education has highlighted the learning opportunities offered by the use of open-ended problems in geometry, in terms of supporting cognitive processes of conjecture generation and argumentation. In order to contribute to this research topic, the study aims to propose and describe different formulations of open-ended geometric problems in Euclidean geometry, and to investigate how the implementation of these different formulations could support different processes. In particular, the study focuses on different solvers behaviors with respect to the production of a final statement (i.e., a possible solution to a given problem) and the structure of the solvers’ argumentations. The study was conducted with two groups of ninth grade students, in a high school in northern of Italy. During four sessions, students were asked to solve four geometric problems belonging to different categories as they were coded for this study. The solvers’ productions and argumentations were qualitatively analyzed using Toulmin’s model (i.e., a structural model that describes the relationship between the elements that constitute an argument). Special attention has been paid to the extent to which the solvers’ domain of the Theory of Euclidean geometry has influenced the solutions. The data analysis shows how different categories have induced different behaviors during the exploration phase, which in turn contribute to form different solvers’ strategies both in reaching a final statement and in constructing an argumentation. These differences are evident in how solvers analyze the figure to solve the given problem, which corresponds to different ways of providing an argumentation for a particular statement. Moreover, this study allows us to observe qualitative differences in the solvers’ productions, in terms of different formulations of conditional statements and geometric knowledge that was mobilized by the solvers. Finally, the study supports the formulation of specific hypotheses for educational implications and highlights perspectives for future research.
La ricerca in Didattica della Matematica ha messo in luce le opportunità, in termini di supporto a processi di congettura e argomentazione, offerte dall’utilizzo di problemi geometrici proposti in forma aperta. Allo scopo di contribuire a tale tema di ricerca, lo studio intende proporre e descrivere diverse formulazioni di problemi geometrici ed esplorare l’ipotesi secondo la quale esse possono indurre diversi comportamenti nei solutori rispetto al modo di formulare l’enunciato, che costituisce una delle possibili soluzioni a tali problemi, e alla struttura delle argomentazioni proposte. Lo studio è stato condotto in due classi prime di scuola secondaria di II grado. Durante quattro incontri è stato chiesto agli studenti di risolvere quattro diversi problemi geometrici, appartenenti alle diverse categorie codificate per questo studio. Le produzioni e le argomentazioni dei solutori sono state analizzate qualitativamente utilizzando il modello di Toulmin, un modello strutturale che descrive il rapporto tra i diversi elementi che compongono una argomentazione. Particolare attenzione è stata posta al dominio della teoria della Geometria Euclidea manifestato dei solutori, per descrivere in che modo esso influenza le soluzioni proposte. L’analisi dei dati mostra come le diverse categorie di problemi inducano comportamenti diversi nella fase di esplorazione, che a loro volta contribuiscono a formare strategie diverse sia nella costruzione dell’enunciato, sia nell’argomentazione. Tali differenze sono riscontrabili nel diverso modo di analizzare la figura per risolvere il problema, a cui corrispondono diversi modi di argomentare gli enunciati proposti come possibili soluzioni. Lo studio ha consentito anche di osservare differenze nella formulazione degli enunciati, in termini di condizionalità, in relazione alle diverse categorie di problema nonché nella conoscenza geometrica mobilitata dagli studenti. Alla luce delle analisi condotte, lo studio consente di avanzare ipotesi didattiche e proposte per ricerche future.
Formulazioni aperte di problemi in ambito geometrico: uno studio esplorativo in due classi di scuola secondaria di II grado
IZZO, ILARIA
2022/2023
Abstract
Research in mathematics education has highlighted the learning opportunities offered by the use of open-ended problems in geometry, in terms of supporting cognitive processes of conjecture generation and argumentation. In order to contribute to this research topic, the study aims to propose and describe different formulations of open-ended geometric problems in Euclidean geometry, and to investigate how the implementation of these different formulations could support different processes. In particular, the study focuses on different solvers behaviors with respect to the production of a final statement (i.e., a possible solution to a given problem) and the structure of the solvers’ argumentations. The study was conducted with two groups of ninth grade students, in a high school in northern of Italy. During four sessions, students were asked to solve four geometric problems belonging to different categories as they were coded for this study. The solvers’ productions and argumentations were qualitatively analyzed using Toulmin’s model (i.e., a structural model that describes the relationship between the elements that constitute an argument). Special attention has been paid to the extent to which the solvers’ domain of the Theory of Euclidean geometry has influenced the solutions. The data analysis shows how different categories have induced different behaviors during the exploration phase, which in turn contribute to form different solvers’ strategies both in reaching a final statement and in constructing an argumentation. These differences are evident in how solvers analyze the figure to solve the given problem, which corresponds to different ways of providing an argumentation for a particular statement. Moreover, this study allows us to observe qualitative differences in the solvers’ productions, in terms of different formulations of conditional statements and geometric knowledge that was mobilized by the solvers. Finally, the study supports the formulation of specific hypotheses for educational implications and highlights perspectives for future research.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/20.500.14239/17218