This master's thesis presents a research study in mathematics education that investigates processes of argumentation and proof as they are undertaken and performed by students. Exploiting data collected within the context of Rally Matematico Transalpino (i.e., a contest between classes, promoting collaborative resolution of mathematical problems) some argumentative texts were analyzed. The analysis focused on a geometric problem, intended for 8th, 9th, and 10th grade students; the problem was chosen for its capability to stimulate rich and varied explorations. Through the use of Balacheff's classification, the study analyzes the students’ productions based on four types of proof, namely naive empiricism, crucial experiment, generic example, and mental experiment. The aim is to operationalize the theoretical constructs of depersonalization, detemporalization, and decontextualization by making explicit descriptive criteria to analyze students’ written productions. Therefore, specific analytical schemes were constructed to classify the types of proof and evaluate the three features. These features are the ones identified by Balacheff as crucial for the transition from pragmatic proofs to intellectual proofs. The application of these schemes allowed us to evaluate how these features affect each other and the nature of the proofs, focusing on the changes between different school grades, in particular within the transition from one cycle to another. The analysis of students' argumentations highlighted, at each grade level, a prevalence of pragmatic proofs; however, there is a tendency to produce more intellectual proofs in higher grades. In particular, as the grade level increases, the proofs appear more decontextualized, suggesting a greater generality and conceptualization of the involved knowledge. Finally, directions for future research were outlined in order to make the classification more general and applicable to different situations. These new lines of research emerge from the results of this study and its limitations, which are essentially related to the specific context of the present study and the nature of the data.
Questa tesi di laurea magistrale presenta un lavoro di ricerca in Didattica della Matematica che indaga i processi argomentativi e dimostrativi messi in atto dagli studenti. Utilizzando dati provenienti dall'archivio del Rally Matematico Transalpino —un confronto tra classi che promuove la risoluzione collaborativa di problemi matematici— si sono analizzate le argomentazioni di gruppi di studenti di scuola secondaria di grado 8, 9 e 10, focalizzandosi su un quesito geometrico scelto per la sua capacità di stimolare esplorazioni ricche e variegate. Attraverso l'utilizzo della classificazione dei tipi di prove proposta da Balacheff, il lavoro analizza le argomentazioni prodotte dagli allievi in base a quattro tipologie (empirismo naive, esperimento cruciale, esempio generico ed esperimento mentale) con lo scopo di rendere operativi i costrutti teorici di depersonalizzazione, detemporalizzazione e decontestualizzazione attraverso l’esplicitazione di criteri descrittivi per analizzare le argomentazioni degli studenti. Quindi, sono stati costruiti specifici schemi analitici per la classificazione in base ai quattro tipi di prove e per la valutazione delle tre caratteristiche identificate da Balacheff come cruciali per la transizione da prove pragmatiche a prove intellettuali. Questo ha consentito di valutare come tali caratteristiche coesistano e quali effetti abbiano sulle prove prodotte dagli studenti, osservando in particolare come queste cambino al variare del grado di scolarizzazione, nel passaggio tra il primo e il secondo ciclo di istruzione. L'analisi delle argomentazioni degli studenti ha evidenziato, in ogni grado scolare, una predominanza di prove di tipo pragmatico; si nota però una tendenza all'evoluzione verso prove più intellettuali nei gradi scolastici superiori. In particolare, all’aumentare del grado scolare le prove risultano più decontestualizzazione, suggerendo una maggiore generalità e concettualizzazione delle conoscenze coinvolte. Infine, è stato possibile delineare direzioni per future ricerche al fine di rendere la classificazione più generale ed applicabile a situazioni differenti, a partire dai risultati di questo studio e dalle sue limitazioni legate essenzialmente al contesto specifico della ricerca e alla natura dei dati analizzati.
Tipologie di prove nella scuola secondaria: analisi di alcuni testi argomentativi nel contesto del Rally Matematico Transalpino
VIDULIS, CHIARA
2022/2023
Abstract
This master's thesis presents a research study in mathematics education that investigates processes of argumentation and proof as they are undertaken and performed by students. Exploiting data collected within the context of Rally Matematico Transalpino (i.e., a contest between classes, promoting collaborative resolution of mathematical problems) some argumentative texts were analyzed. The analysis focused on a geometric problem, intended for 8th, 9th, and 10th grade students; the problem was chosen for its capability to stimulate rich and varied explorations. Through the use of Balacheff's classification, the study analyzes the students’ productions based on four types of proof, namely naive empiricism, crucial experiment, generic example, and mental experiment. The aim is to operationalize the theoretical constructs of depersonalization, detemporalization, and decontextualization by making explicit descriptive criteria to analyze students’ written productions. Therefore, specific analytical schemes were constructed to classify the types of proof and evaluate the three features. These features are the ones identified by Balacheff as crucial for the transition from pragmatic proofs to intellectual proofs. The application of these schemes allowed us to evaluate how these features affect each other and the nature of the proofs, focusing on the changes between different school grades, in particular within the transition from one cycle to another. The analysis of students' argumentations highlighted, at each grade level, a prevalence of pragmatic proofs; however, there is a tendency to produce more intellectual proofs in higher grades. In particular, as the grade level increases, the proofs appear more decontextualized, suggesting a greater generality and conceptualization of the involved knowledge. Finally, directions for future research were outlined in order to make the classification more general and applicable to different situations. These new lines of research emerge from the results of this study and its limitations, which are essentially related to the specific context of the present study and the nature of the data.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/20.500.14239/17406