The study presented in this thesis investigates the Proof by Mathematical Induction as a research problem in mathematics education. Mathematical induction (MI) is characterized by a considerable complexity, both from a historical-epistemological and a cognitive-didactic point of view. On the one hand, in fact, MI has a fundamental and foundational role in modern mathematics, a role that has been reached after a long process of historical development. On the other hand, research in mathematics education is unanimous in highlighting how MI is extremely problematic from a didactic point of view, with difficulties of different and various nature that are recorded transversely from novice to expert students. The purpose of this research is to shed some light on this complexity, by observing and analysing it from different standpoints. In line with this research objective, a conceptual framework is structured, obtained as a combination of different perspectives, in order to get a multi-faced insight into the research problem. First of all, this study takes into consideration a historical-epistemological perspective. In particular, an analysis of the historical genesis of the proof by induction is conducted, focusing on a series of traces of proofs by induction that historiographic research has identified. Subsequently, starting from this historical-epistemological analysis, other different theoretical perspectives are considered, each with a different focus in relation to the study of MI from a cognitive and educational point of view. Preliminarily, the theoretical standpoint on ‘Argumentation and Proof’ adopted in this thesis is presented. This standpoint, which emerges within the studies on Cognitive Unity allows to extend the focus of this study not only to the formal proofs by MI, but also to those informal argumentations related to MI, the recursive argumentations, as defined in the thesis. Another perspective considered is the APOS Theory, adopted in this study to investigate students’ interiorization of crucial processes for the construction of the Schema of MI (i.e., in APOS terms, the Genetic Decomposition of MI). In addition to this, the theoretical framework of Intuitions according to Fischbein, is considered, with the aim of investigating on the students’ intuitive acceptance of the proof by MI. Finally, a multimodal semiotic perspective, with the construct of the Semiotic Bundle, is also considered in order to identify and analyse an extended variety of signs (speech, gestures, inscriptions) produced and used by students as a lens to observe their processes related to MI during problem solving activities. Within this conceptual framework, the research questions of the study are then formulated, each in line with one of the adopted perspectives. Besides the research question related to the historical-epistemological analysis (R.Q.1), the others address, respectively, students’ interiorization of the Explain Induction Process, one of the crucial processes involved in the MI Schema, in APOS terms (R.Q.2), students’ intuitive acceptance of the proving scheme by MI (R.Q.3), and students’ use and production of multimodal signs in the generation and construction of recursive argumentations and proofs by MI during problem solving (R.Q.4). To investigate these research questions, two different empirical studies are conducted. The first one is based on an online survey with some close-ended and open-ended questions. It involves a total of 307 participants including undergraduate and master’s students from various academic courses from different Italian universities. The second empirical study involves some task-based interviews with experienced (Doctoral) and less experienced (second or third year undergraduate) students in mathematics and physics. The data collected are video and audio recordings, interview transcripts, and solvers’ written inscriptions. Analyses of the data collected in the two empirical studies are then presented and discussed, highlighting some interesting results in relation to the research objective. On the one hand, these results include the identification of some problematic aspects for students in relation to the Explain Induction process, in particular the construction of chains of logical inferences (Modus ponens and Modus tollens), or in relation to the intuitive acceptance of MI (e.g. in accepting the validity of the implication involved in the inductive step as independent from knowing the truth value of antecedent and consequent). On the other hand, these results also include the identification of two categories of signs (named Linking and Iteration signs) and the construction of an interpretative tool (the distinction between ground and meta-level signs) that allowed to identify, observe, and analyse some effective processes of students involved in the construction of recursive argumentations and proofs by MI during problem solving. Lastly, these results are discussed in order to provide an answer to the research questions. The findings of this thesis are then contextualised within the existing literature, highlighting how some of them confirm and extend previous studies on MI, and how others offer an original contribution in relation to the research problem. Finally, this thesis concludes by presenting possible research directions that this study offers and some didactical implications that arise from it.

Proving by mathematical induction: an analysis from history and epistemology to cognition / Bernardo Nannini. - (2022).

Proving by mathematical induction: an analysis from history and epistemology to cognition

Bernardo Nannini
2022

Abstract

The study presented in this thesis investigates the Proof by Mathematical Induction as a research problem in mathematics education. Mathematical induction (MI) is characterized by a considerable complexity, both from a historical-epistemological and a cognitive-didactic point of view. On the one hand, in fact, MI has a fundamental and foundational role in modern mathematics, a role that has been reached after a long process of historical development. On the other hand, research in mathematics education is unanimous in highlighting how MI is extremely problematic from a didactic point of view, with difficulties of different and various nature that are recorded transversely from novice to expert students. The purpose of this research is to shed some light on this complexity, by observing and analysing it from different standpoints. In line with this research objective, a conceptual framework is structured, obtained as a combination of different perspectives, in order to get a multi-faced insight into the research problem. First of all, this study takes into consideration a historical-epistemological perspective. In particular, an analysis of the historical genesis of the proof by induction is conducted, focusing on a series of traces of proofs by induction that historiographic research has identified. Subsequently, starting from this historical-epistemological analysis, other different theoretical perspectives are considered, each with a different focus in relation to the study of MI from a cognitive and educational point of view. Preliminarily, the theoretical standpoint on ‘Argumentation and Proof’ adopted in this thesis is presented. This standpoint, which emerges within the studies on Cognitive Unity allows to extend the focus of this study not only to the formal proofs by MI, but also to those informal argumentations related to MI, the recursive argumentations, as defined in the thesis. Another perspective considered is the APOS Theory, adopted in this study to investigate students’ interiorization of crucial processes for the construction of the Schema of MI (i.e., in APOS terms, the Genetic Decomposition of MI). In addition to this, the theoretical framework of Intuitions according to Fischbein, is considered, with the aim of investigating on the students’ intuitive acceptance of the proof by MI. Finally, a multimodal semiotic perspective, with the construct of the Semiotic Bundle, is also considered in order to identify and analyse an extended variety of signs (speech, gestures, inscriptions) produced and used by students as a lens to observe their processes related to MI during problem solving activities. Within this conceptual framework, the research questions of the study are then formulated, each in line with one of the adopted perspectives. Besides the research question related to the historical-epistemological analysis (R.Q.1), the others address, respectively, students’ interiorization of the Explain Induction Process, one of the crucial processes involved in the MI Schema, in APOS terms (R.Q.2), students’ intuitive acceptance of the proving scheme by MI (R.Q.3), and students’ use and production of multimodal signs in the generation and construction of recursive argumentations and proofs by MI during problem solving (R.Q.4). To investigate these research questions, two different empirical studies are conducted. The first one is based on an online survey with some close-ended and open-ended questions. It involves a total of 307 participants including undergraduate and master’s students from various academic courses from different Italian universities. The second empirical study involves some task-based interviews with experienced (Doctoral) and less experienced (second or third year undergraduate) students in mathematics and physics. The data collected are video and audio recordings, interview transcripts, and solvers’ written inscriptions. Analyses of the data collected in the two empirical studies are then presented and discussed, highlighting some interesting results in relation to the research objective. On the one hand, these results include the identification of some problematic aspects for students in relation to the Explain Induction process, in particular the construction of chains of logical inferences (Modus ponens and Modus tollens), or in relation to the intuitive acceptance of MI (e.g. in accepting the validity of the implication involved in the inductive step as independent from knowing the truth value of antecedent and consequent). On the other hand, these results also include the identification of two categories of signs (named Linking and Iteration signs) and the construction of an interpretative tool (the distinction between ground and meta-level signs) that allowed to identify, observe, and analyse some effective processes of students involved in the construction of recursive argumentations and proofs by MI during problem solving. Lastly, these results are discussed in order to provide an answer to the research questions. The findings of this thesis are then contextualised within the existing literature, highlighting how some of them confirm and extend previous studies on MI, and how others offer an original contribution in relation to the research problem. Finally, this thesis concludes by presenting possible research directions that this study offers and some didactical implications that arise from it.
2022
Samuele Antonini, Veronica Gavagna
ITALIA
Bernardo Nannini
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Utilizza questo identificatore per citare o creare un link a questa risorsa: https://hdl.handle.net/2158/1270844
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