This research project aims to investigate covariation understood not only as the ability to visualize two or more magnitudes while co-varying simultaneously (Thompson & Carlson, 2017), but in a broader epistemological sense, as the ability to grasp relationships of invariance between two quantities. The need to better characterize more complex forms of reasoning performed by students in mathematical modelling activities, led us to introduce second-order covariation, a form of covariation that consists in describing relations in which not only variables are involved but also parameters (Arzarello, 2019). These enable to represent families of relationships between variables that is classes of real phenomena characterized, from a mathematical standpoint, through parameters, which determine the specificities of the mathematical model. The discussion of this theme arises not only from research needs in the field of Mathematics Education, i.e., the existence of a theoretical framework only partially useful to describe the covariational reasonings of students, but above all by its relevance in terms of teaching practices. There is a wide literature showing that in mathematical modelling situations the ability to reason covariationally is essential because it allows to visualize the invariant relationships that exist between quantities involved in dynamic situations (Thompson, 2011). The indications for teaching mathematics in high schools (MIUR, 2010) underline the relevance of introducing mathematical modelling as a representation of classes of real phenomena. However, despite the acknowledged relevance of covariation for the learning of numerous mathematical concepts, in the National Indications, as well as in most textbooks, references to this approach are generally absent. The teachers themselves have little knowledge of covariation and therefore struggle to introduce it into their teaching practices. The data analyzed in this project come from three didactical experiments developed in some classes of a scientific high school and whose aim is the mathematical description of some real situations: specifically, the motion of a ball along an inclined plane and the relationship between temperature and humidity described in the so-called psychrometric diagram. Using appropriate technological tools, the students were guided in deriving a mathematical formula that described such phenomena and in recognizing the different role played by variables and parameters in the writing and reading of different registers of mathematical representations. Students' reasoning processes and the evolution of the different semiotic aspects (spoken, gestural, representational) involved in the teaching-learning processes were analyzed; as well the support of technology and the role of the teacher in enhancing covariational reasoning through appropriate adaptive teaching strategies, were considered. This study led us not only to the elaboration of a broader theoretical framework, which consistently includes second-order covariation, but also to hypothesize the existence of a third-order covariation. In addition, some research studies complementary to the main one described above, allowed us to explore the theme of assessment of covariation as a form of conceptual understanding and to elaborate a mathematical interpretation of the covariation construct using category theory (MacLane, 1978) and the cognitive mechanisms of conceptual blending (Fauconnier & Turner, 2002).

Questo progetto di ricerca si propone di indagare la covariazione intesa non solo come capacità di visualizzare due o più grandezze mentre co-variano simultaneamente (Thompson & Carlson, 2017), ma in un più ampio senso epistemologico, come capacità di cogliere relazioni di invarianza tra due grandezze. L’esigenza di caratterizzare meglio forme di ragionamento più complesse messe in atto da studenti in attività di modellizzazione matematica, ci ha portato a introdurre la covariazione al secondo ordine, una forma di covariazione che consiste nel descrivere le relazioni in cui sono coinvolte non solo variabili ma anche parametri (Arzarello, 2019). Questi ultimi consentono di rappresentare famiglie di relazioni tra variabili cioè classi di fenomeni reali caratterizzati, da un punto di vista matematico, da parametri che determinano le specificità del modello matematico. La trattazione di questo tema nasce non solo da esigenze a livello di ricerca nel settore della Didattica della Matematica, ovvero l’esistenza di un quadro teorico solo parzialmente utile a descrivere i ragionamenti covariazionali degli studenti, ma soprattutto da una sua rilevanza a livello di pratiche didattiche. Infatti, esiste un’ampia letteratura che mostra come in situazioni di modellizzazione matematica sia essenziale la capacità di ragionare in modo covariazionale poiché essa consente di visualizzare le relazioni invarianti che sussistono tra grandezze fisiche coinvolte in situazioni dinamiche (Thompson, 2011). Le indicazioni per l’insegnamento della matematica nei licei (MIUR, 2010) sottolineano l’importanza dell’introduzione alla modellizzazione matematica intesa come rappresentazione di classi di fenomeni reali eppure, nonostante la riconosciuta importanza della covariazione per l’apprendimento di numerosi concetti matematici, nelle Indicazioni Nazionali così come nella maggior parte dei libri di testo i riferimenti a questo approccio sono generalmente assenti. Gli insegnanti stessi hanno poche conoscenze in merito alla covariazione e quindi faticano a introdurla nelle loro pratiche didattiche. I dati analizzati in questo progetto provengono da tre sperimentazioni didattiche condotte in alcune classi di un liceo scientifico e aventi come obiettivo la descrizione matematica di alcune situazioni reali quali, nello specifico, il moto di una pallina lungo un piano inclinato e la relazione tra temperatura e umidità descritta nel cosiddetto diagramma psicrometrico. Attraverso l’utilizzo di opportuni strumenti tecnologici, gli studenti sono stati guidati nel ricavare una formula matematica che descrivesse tali fenomeni e nel riconoscere il differente ruolo svolto da variabili e parametri nella scrittura e lettura di diversi registri di rappresentazione matematica. Sono stati analizzati i processi di ragionamento degli studenti, l’evoluzione dei diversi aspetti semiotici (parlato, gestualità, rappresentazioni) coinvolti nei processi di insegnamento-apprendimento, il supporto della tecnologia e il ruolo dell’insegnante nel favorire il ragionamento covariazionale adottando adeguate strategie didattiche adattive. Questo studio ci ha portato non solo all’elaborazione di un più ampio quadro teorico che includesse in modo coerente la covariazione al secondo ordine, ma anche a ipotizzare l’esistenza di un terzo ordine di covariazione. Inoltre, alcuni studi di ricerca complementari a quello principale finora descritto, ci hanno permesso di esplorare il tema della valutazione della covariazione intesa come forma di apprendimento concettuale e ad elaborare un’interpretazione matematica del costrutto covariazione usando la teoria delle categorie (MacLane, 1978) e i meccanismi cognitivi del blending concettuale (Fauconnier & Turner, 2002).

Covariazione al secondo ordine: un’analisi dei ragionamenti degli studenti e degli interventi dell’insegnante in situazioni di modellizzazione di fenomeni reali / Sara Bagossi , 2022 Feb 24. 34. ciclo, Anno Accademico 2020/2021.

### Covariazione al secondo ordine: un’analisi dei ragionamenti degli studenti e degli interventi dell’insegnante in situazioni di modellizzazione di fenomeni reali

#### Abstract

This research project aims to investigate covariation understood not only as the ability to visualize two or more magnitudes while co-varying simultaneously (Thompson & Carlson, 2017), but in a broader epistemological sense, as the ability to grasp relationships of invariance between two quantities. The need to better characterize more complex forms of reasoning performed by students in mathematical modelling activities, led us to introduce second-order covariation, a form of covariation that consists in describing relations in which not only variables are involved but also parameters (Arzarello, 2019). These enable to represent families of relationships between variables that is classes of real phenomena characterized, from a mathematical standpoint, through parameters, which determine the specificities of the mathematical model. The discussion of this theme arises not only from research needs in the field of Mathematics Education, i.e., the existence of a theoretical framework only partially useful to describe the covariational reasonings of students, but above all by its relevance in terms of teaching practices. There is a wide literature showing that in mathematical modelling situations the ability to reason covariationally is essential because it allows to visualize the invariant relationships that exist between quantities involved in dynamic situations (Thompson, 2011). The indications for teaching mathematics in high schools (MIUR, 2010) underline the relevance of introducing mathematical modelling as a representation of classes of real phenomena. However, despite the acknowledged relevance of covariation for the learning of numerous mathematical concepts, in the National Indications, as well as in most textbooks, references to this approach are generally absent. The teachers themselves have little knowledge of covariation and therefore struggle to introduce it into their teaching practices. The data analyzed in this project come from three didactical experiments developed in some classes of a scientific high school and whose aim is the mathematical description of some real situations: specifically, the motion of a ball along an inclined plane and the relationship between temperature and humidity described in the so-called psychrometric diagram. Using appropriate technological tools, the students were guided in deriving a mathematical formula that described such phenomena and in recognizing the different role played by variables and parameters in the writing and reading of different registers of mathematical representations. Students' reasoning processes and the evolution of the different semiotic aspects (spoken, gestural, representational) involved in the teaching-learning processes were analyzed; as well the support of technology and the role of the teacher in enhancing covariational reasoning through appropriate adaptive teaching strategies, were considered. This study led us not only to the elaboration of a broader theoretical framework, which consistently includes second-order covariation, but also to hypothesize the existence of a third-order covariation. In addition, some research studies complementary to the main one described above, allowed us to explore the theme of assessment of covariation as a form of conceptual understanding and to elaborate a mathematical interpretation of the covariation construct using category theory (MacLane, 1978) and the cognitive mechanisms of conceptual blending (Fauconnier & Turner, 2002).
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Second-order covariation: an analysis of students’ reasonings and teacher’s interventions when modelling real phenomena
24-feb-2022
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Descrizione: Tesi definitiva Bagossi Sara
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11380/1265204`