caterina cumino | Politecnico di Torino (original) (raw)
Papers by caterina cumino
We present and analyze an example of activity aiming at introducing graphic language and standard... more We present and analyze an example of activity aiming at introducing graphic language and standards to students of the first year bachelor's in architecture. Our example concerns the 2D representation of an architectural object, an essential competency in the architect's profession, and specifically deals with roofing systems made up of vaults generated by cylinders and their intersections. It is a first attempt to create activities that involve all students regardless of their different levels of geometric understanding and of familiarity with graphic language; in particular, this example exploits the introduction of physical or virtual models to support the mathematical thinking of students in completing the task and takes place during regular lesson times, without modifying or adding mathematical subject contents. Research literature on mathematical modeling, in particular on the so-called prescriptive one, provides us tools to frame and implement our study case; it also allows presenting mathematical modeling as a goal in an extra-mathematical educational context.
We present a set of guided tours and educational workshops, where the halls of a magnificent baro... more We present a set of guided tours and educational workshops, where the halls of a magnificent baroque royal residence become classrooms and visitors are led to discover architectural shapes and to understand their geometry with the use of various origami modeling activities, rigorously supported by mathematics (descriptive geometry and basics in analytic and differential geometry).
Education and new developments, Jun 21, 2019
The role of mathematical visualization in problem solving process and the incidence of material t... more The role of mathematical visualization in problem solving process and the incidence of material tools on apprehension of mathematical concepts have been subjects of intensive research for a long time. In the present paper we analyze the use of a manipulative in solving a space Geometry problem, providing a critical account of its theoretical framework and its effectiveness in various teaching contexts. Previous researches (Leikin, Stylianou, & Silver, 2005) examined students' use of visual representations, across various ages and mathematical education levels, proposing a challenging task to the study participants: describe the net of a truncated right cylinder. Given the visual nature of the problem, any solution strategy required a translation between 3D and 2D representations, regardless of the more or less advanced mathematical knowledge. In the behavior of the successful visualizers, similarities were identified, such as selection of what needs to be visualized and transfer of visual images in a "symbol system", see Nemirovsky (1994); while both young and older low-achieving students reported difficulties in isolating object's relevant characteristics. About the same problem, a physical manipulative was designed (Cumino, Spreafico, & Zich, 2017) in a dissemination context where Geometry was a tool for understanding architectural shapes, under the need of quickly and correctly communicating the solution, avoiding mathematical formalizations: It shows a right cylinder truncated by oblique planes of various inclination and allows to obtain its net by a simple unwrap. Currently, this manipulative is being tested at the university level, on first year students of the bachelor program in Architecture and at the middle school level, on second year students of a Technology course; in both teaching contexts, different for age and levels of mathematical knowledge, it is proving useful in making students able to elaborate individual solving strategies, to set up the problem correctly and to improve its translation in symbolic language for the holders of advanced mathematical tools, speeding up teachers' interventions.
ICERI proceedings, Nov 1, 2017
Rendiconti del Seminario Matematico della Università di Padova, 1977
L'accès aux archives de la revue « Rendiconti del Seminario Matematico della Università di Padova... more L'accès aux archives de la revue « Rendiconti del Seminario Matematico della Università di Padova » (http://rendiconti.math.unipd.it/) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/
Annali di Matematica Pura ed Applicata, Dec 1, 1986
In this paper, using the theory o] henselian rings and in particular the notion o] heuselian and ... more In this paper, using the theory o] henselian rings and in particular the notion o] heuselian and strict henselian couples, we study the behaviour o] analytic branches o] an a]]ine algebraic variety at a point x, when x varies along a singular subvariety ; moreover we construct an algebraic and canonical procedure, which parametrizes the branches.
HAL (Le Centre pour la Communication Scientifique Directe), Sep 12, 2020
We present an interdisciplinary activity, directed to students of the first year Bachelor's in Ar... more We present an interdisciplinary activity, directed to students of the first year Bachelor's in Architecture, about the visualization of mathematical objects through physical and digital models, with specific regard to surfaces generated by intersections of cylinders. We focus on the geometrical structure of barrel vaults with lunettes and propose the use of various kind of models to improve the accessibility of interdisciplinary elements and to translate specialistic knowledge, building a common language for students with different backgrounds. The work is a result of a broader research project, dedicated to enhancing the relationship between Mathematics and Architecture through Geometry and Representation.
Il contributo analizza il Catalog Mathematischer Modelle, edito da Schilling nel 1903, nella sua ... more Il contributo analizza il Catalog Mathematischer Modelle, edito da Schilling nel 1903, nella sua duplice accezione di catalogo: di modelli matematici e di immagini dei modelli stessi. Il lavoro è un primo esito della lettura critica di una fonte grafico/testuale/analitica avviata con un approccio interdisciplinare, che indaga i modelli fisici come linguaggio geometrico, tattile e visivo comune all'Architettura e alla Matematica. Il catalogo rappresentò, per la vendita di modelli matematici, una tipologia di sistematizzazione nuova rispetto ad analoghi esempi editi nel ventennio precedente, proponendo una presentazione delle informazioni per famiglia e non solo per innovazione del prodotto da immettere sul mercato, mediando quindi il suo ruolo da catalogo di vendita a riferimento per la scelta degli ausili visivi per la didattica della Matematica. Il contesto socio-scientifico e l'analisi condotta, sostanzialmente visiva, non solo degli strumenti scelti per la rappresentazione, ma anche delle dinamiche editoriali, hanno portato a leggere criticamente il Catalogo come medium di comunicazione di risultati matematici con un risvolto visuale sul Pensiero Visivo nella materia di appartenenza, indagandone quindi le connessioni tra immagini scritte e parole disegnate, con uno sguardo verso l'oggetto reale, tra collezioni fisiche e digitali. Parole chiave Comunicazione della Geometria, modelli fisici, oggetti matematici, geometria delle superfici algebriche, Brill-Schilling.
DOAJ (DOAJ: Directory of Open Access Journals), Sep 1, 2000
We study the behaviour of analytic branches of a projective variety with respect to hypersurface ... more We study the behaviour of analytic branches of a projective variety with respect to hypersurface sections and we give conditions under which their number and their orders are preserved. CORE Metadata, citation and similar papers at core.ac.uk
arXiv (Cornell University), Jan 24, 2007
ABSTRACT Some results about the analytic branches of an algebraic affine variety along a singular... more ABSTRACT Some results about the analytic branches of an algebraic affine variety along a singular subvariety are proved, using the theory of henselian rings. Precisely, let X=Spec(A), with A a noetherian domain, and Y a closed irreducible subvariety of X corresponding to the prime p of A. The first result is that the global branches of X along Y, which by definition are the minimal primes of the henselization of the couple (A,p), correspond to the connected components of p -1 (Y), where p:X ' →X is the normalization morphism. Moreover, there is an open subset U of X such that there is a natural canonical correspondence between the global branches of U along U∩Y and the branches of X at the generic point y of Y. A similar result is then proved for the geometric global branches of X along Y, i.e. the minimal primes of the strict henselization of the couple (A,p), replacing the branches of X in y with the geometric branches of X in y. Furthermore it is shown how to reconstruct the local rings of the branches at each point of a dense open subset of Y knowing the branches at the generic point y, under some conditions for the behaviour of X along Y. This result is finally extended to the general case, passing to a suitable étale covering of X. For closely related results proved with completely different topological techniques see the paper by G. Tedeschi, Boll. Unione Mat. Ital., VI. Ser., D, Algebra Geom. 4, No.1, 17-27 (1985; see the preceding review)].
Journal of Pure and Applied Algebra, Jun 1, 2011
We collect a few results about jets of line bundles on curves and Wronskians, with a special emph... more We collect a few results about jets of line bundles on curves and Wronskians, with a special emphasis to those arising from the canonical involution of a hyperelliptic curve.
Lecture notes in networks and systems, 2023
Publistampa Edizioni, 2018
Education and New Developments, 2019
The role of mathematical visualization in problem solving process and the incidence of material t... more The role of mathematical visualization in problem solving process and the incidence of material tools on apprehension of mathematical concepts have been subjects of intensive research for a long time. In the present paper we analyze the use of a manipulative in solving a space Geometry problem, providing a critical account of its theoretical framework and its effectiveness in various teaching contexts. Previous researches (Leikin, Stylianou, & Silver, 2005) examined students' use of visual representations, across various ages and mathematical education levels, proposing a challenging task to the study participants: describe the net of a truncated right cylinder. Given the visual nature of the problem, any solution strategy required a translation between 3D and 2D representations, regardless of the more or less advanced mathematical knowledge. In the behavior of the successful visualizers, similarities were identified, such as selection of what needs to be visualized and transfer of visual images in a "symbol system", see Nemirovsky (1994); while both young and older low-achieving students reported difficulties in isolating object's relevant characteristics. About the same problem, a physical manipulative was designed (Cumino, Spreafico, & Zich, 2017) in a dissemination context where Geometry was a tool for understanding architectural shapes, under the need of quickly and correctly communicating the solution, avoiding mathematical formalizations: It shows a right cylinder truncated by oblique planes of various inclination and allows to obtain its net by a simple unwrap. Currently, this manipulative is being tested at the university level, on first year students of the bachelor program in Architecture and at the middle school level, on second year students of a Technology course; in both teaching contexts, different for age and levels of mathematical knowledge, it is proving useful in making students able to elaborate individual solving strategies, to set up the problem correctly and to improve its translation in symbolic language for the holders of advanced mathematical tools, speeding up teachers' interventions.
We present and analyze an example of activity aiming at introducing graphic language and standard... more We present and analyze an example of activity aiming at introducing graphic language and standards to students of the first year bachelor's in architecture. Our example concerns the 2D representation of an architectural object, an essential competency in the architect's profession, and specifically deals with roofing systems made up of vaults generated by cylinders and their intersections. It is a first attempt to create activities that involve all students regardless of their different levels of geometric understanding and of familiarity with graphic language; in particular, this example exploits the introduction of physical or virtual models to support the mathematical thinking of students in completing the task and takes place during regular lesson times, without modifying or adding mathematical subject contents. Research literature on mathematical modeling, in particular on the so-called prescriptive one, provides us tools to frame and implement our study case; it also allows presenting mathematical modeling as a goal in an extra-mathematical educational context.
We present a set of guided tours and educational workshops, where the halls of a magnificent baro... more We present a set of guided tours and educational workshops, where the halls of a magnificent baroque royal residence become classrooms and visitors are led to discover architectural shapes and to understand their geometry with the use of various origami modeling activities, rigorously supported by mathematics (descriptive geometry and basics in analytic and differential geometry).
Education and new developments, Jun 21, 2019
The role of mathematical visualization in problem solving process and the incidence of material t... more The role of mathematical visualization in problem solving process and the incidence of material tools on apprehension of mathematical concepts have been subjects of intensive research for a long time. In the present paper we analyze the use of a manipulative in solving a space Geometry problem, providing a critical account of its theoretical framework and its effectiveness in various teaching contexts. Previous researches (Leikin, Stylianou, & Silver, 2005) examined students' use of visual representations, across various ages and mathematical education levels, proposing a challenging task to the study participants: describe the net of a truncated right cylinder. Given the visual nature of the problem, any solution strategy required a translation between 3D and 2D representations, regardless of the more or less advanced mathematical knowledge. In the behavior of the successful visualizers, similarities were identified, such as selection of what needs to be visualized and transfer of visual images in a "symbol system", see Nemirovsky (1994); while both young and older low-achieving students reported difficulties in isolating object's relevant characteristics. About the same problem, a physical manipulative was designed (Cumino, Spreafico, & Zich, 2017) in a dissemination context where Geometry was a tool for understanding architectural shapes, under the need of quickly and correctly communicating the solution, avoiding mathematical formalizations: It shows a right cylinder truncated by oblique planes of various inclination and allows to obtain its net by a simple unwrap. Currently, this manipulative is being tested at the university level, on first year students of the bachelor program in Architecture and at the middle school level, on second year students of a Technology course; in both teaching contexts, different for age and levels of mathematical knowledge, it is proving useful in making students able to elaborate individual solving strategies, to set up the problem correctly and to improve its translation in symbolic language for the holders of advanced mathematical tools, speeding up teachers' interventions.
ICERI proceedings, Nov 1, 2017
Rendiconti del Seminario Matematico della Università di Padova, 1977
L'accès aux archives de la revue « Rendiconti del Seminario Matematico della Università di Padova... more L'accès aux archives de la revue « Rendiconti del Seminario Matematico della Università di Padova » (http://rendiconti.math.unipd.it/) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/
Annali di Matematica Pura ed Applicata, Dec 1, 1986
In this paper, using the theory o] henselian rings and in particular the notion o] heuselian and ... more In this paper, using the theory o] henselian rings and in particular the notion o] heuselian and strict henselian couples, we study the behaviour o] analytic branches o] an a]]ine algebraic variety at a point x, when x varies along a singular subvariety ; moreover we construct an algebraic and canonical procedure, which parametrizes the branches.
HAL (Le Centre pour la Communication Scientifique Directe), Sep 12, 2020
We present an interdisciplinary activity, directed to students of the first year Bachelor's in Ar... more We present an interdisciplinary activity, directed to students of the first year Bachelor's in Architecture, about the visualization of mathematical objects through physical and digital models, with specific regard to surfaces generated by intersections of cylinders. We focus on the geometrical structure of barrel vaults with lunettes and propose the use of various kind of models to improve the accessibility of interdisciplinary elements and to translate specialistic knowledge, building a common language for students with different backgrounds. The work is a result of a broader research project, dedicated to enhancing the relationship between Mathematics and Architecture through Geometry and Representation.
Il contributo analizza il Catalog Mathematischer Modelle, edito da Schilling nel 1903, nella sua ... more Il contributo analizza il Catalog Mathematischer Modelle, edito da Schilling nel 1903, nella sua duplice accezione di catalogo: di modelli matematici e di immagini dei modelli stessi. Il lavoro è un primo esito della lettura critica di una fonte grafico/testuale/analitica avviata con un approccio interdisciplinare, che indaga i modelli fisici come linguaggio geometrico, tattile e visivo comune all'Architettura e alla Matematica. Il catalogo rappresentò, per la vendita di modelli matematici, una tipologia di sistematizzazione nuova rispetto ad analoghi esempi editi nel ventennio precedente, proponendo una presentazione delle informazioni per famiglia e non solo per innovazione del prodotto da immettere sul mercato, mediando quindi il suo ruolo da catalogo di vendita a riferimento per la scelta degli ausili visivi per la didattica della Matematica. Il contesto socio-scientifico e l'analisi condotta, sostanzialmente visiva, non solo degli strumenti scelti per la rappresentazione, ma anche delle dinamiche editoriali, hanno portato a leggere criticamente il Catalogo come medium di comunicazione di risultati matematici con un risvolto visuale sul Pensiero Visivo nella materia di appartenenza, indagandone quindi le connessioni tra immagini scritte e parole disegnate, con uno sguardo verso l'oggetto reale, tra collezioni fisiche e digitali. Parole chiave Comunicazione della Geometria, modelli fisici, oggetti matematici, geometria delle superfici algebriche, Brill-Schilling.
DOAJ (DOAJ: Directory of Open Access Journals), Sep 1, 2000
We study the behaviour of analytic branches of a projective variety with respect to hypersurface ... more We study the behaviour of analytic branches of a projective variety with respect to hypersurface sections and we give conditions under which their number and their orders are preserved. CORE Metadata, citation and similar papers at core.ac.uk
arXiv (Cornell University), Jan 24, 2007
ABSTRACT Some results about the analytic branches of an algebraic affine variety along a singular... more ABSTRACT Some results about the analytic branches of an algebraic affine variety along a singular subvariety are proved, using the theory of henselian rings. Precisely, let X=Spec(A), with A a noetherian domain, and Y a closed irreducible subvariety of X corresponding to the prime p of A. The first result is that the global branches of X along Y, which by definition are the minimal primes of the henselization of the couple (A,p), correspond to the connected components of p -1 (Y), where p:X ' →X is the normalization morphism. Moreover, there is an open subset U of X such that there is a natural canonical correspondence between the global branches of U along U∩Y and the branches of X at the generic point y of Y. A similar result is then proved for the geometric global branches of X along Y, i.e. the minimal primes of the strict henselization of the couple (A,p), replacing the branches of X in y with the geometric branches of X in y. Furthermore it is shown how to reconstruct the local rings of the branches at each point of a dense open subset of Y knowing the branches at the generic point y, under some conditions for the behaviour of X along Y. This result is finally extended to the general case, passing to a suitable étale covering of X. For closely related results proved with completely different topological techniques see the paper by G. Tedeschi, Boll. Unione Mat. Ital., VI. Ser., D, Algebra Geom. 4, No.1, 17-27 (1985; see the preceding review)].
Journal of Pure and Applied Algebra, Jun 1, 2011
We collect a few results about jets of line bundles on curves and Wronskians, with a special emph... more We collect a few results about jets of line bundles on curves and Wronskians, with a special emphasis to those arising from the canonical involution of a hyperelliptic curve.
Lecture notes in networks and systems, 2023
Publistampa Edizioni, 2018
Education and New Developments, 2019
The role of mathematical visualization in problem solving process and the incidence of material t... more The role of mathematical visualization in problem solving process and the incidence of material tools on apprehension of mathematical concepts have been subjects of intensive research for a long time. In the present paper we analyze the use of a manipulative in solving a space Geometry problem, providing a critical account of its theoretical framework and its effectiveness in various teaching contexts. Previous researches (Leikin, Stylianou, & Silver, 2005) examined students' use of visual representations, across various ages and mathematical education levels, proposing a challenging task to the study participants: describe the net of a truncated right cylinder. Given the visual nature of the problem, any solution strategy required a translation between 3D and 2D representations, regardless of the more or less advanced mathematical knowledge. In the behavior of the successful visualizers, similarities were identified, such as selection of what needs to be visualized and transfer of visual images in a "symbol system", see Nemirovsky (1994); while both young and older low-achieving students reported difficulties in isolating object's relevant characteristics. About the same problem, a physical manipulative was designed (Cumino, Spreafico, & Zich, 2017) in a dissemination context where Geometry was a tool for understanding architectural shapes, under the need of quickly and correctly communicating the solution, avoiding mathematical formalizations: It shows a right cylinder truncated by oblique planes of various inclination and allows to obtain its net by a simple unwrap. Currently, this manipulative is being tested at the university level, on first year students of the bachelor program in Architecture and at the middle school level, on second year students of a Technology course; in both teaching contexts, different for age and levels of mathematical knowledge, it is proving useful in making students able to elaborate individual solving strategies, to set up the problem correctly and to improve its translation in symbolic language for the holders of advanced mathematical tools, speeding up teachers' interventions.