Experimental Approaches to Theoretical Thinking: Artefacts and Proofs

Arzarello, Ferdinando; Bartolini, Maria Giuseppina; Leung, Allen; Mariotti, Maria Alessandra; Stevenson, Ian

doi:10.1007/978-94-007-2129-6_5

Cited by 33 publications

(17 citation statements)

References 27 publications

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“…Arzarello et al (2012) examine how this tension "regulates the actions of students who are asked to solve mathematical problems by first making explorations with technological tools, then formulating suitable conjectures and finally proving them" (p. 98) This was illustrated through student-centred analyses using four theoretical constructs: almost-empiricism and experimental mathematics, abductive versus deductive activities in mathematics learning, cognitive unity between arguments and proofs, and negation from a mathematical and cognitive point of view. In the context of exploring geometrical problems in DGEs leading to explanation and proof, the authors discussed the MD scheme and a DGE proof by contradiction to illustrate the complex interplays between inductive, abductive and deductive reasoning in the transition between empirical and theoretical proof perspectives.…”

Section: Explanation and Proof In Dynamic Geometry Environmentsmentioning

confidence: 99%

“…The visual-discursive duality becomes even more epistemological and cognitive relevant by the tension created between approximated measurements in DGE and the actually Euclidean measurement. This is the almost-empiricism that Arzarello et al (2012) referred to where in this case DGE can be regarded as an experimental laboratory with conceptual mathematical objects as objects of investigation. Olivero and Robutti (2007) proposed different categories of DGE measuring modalities (e.g., exploring, guiding, perceptual and proof) to study the shift between the spatio-graphic field and the theoretical field.…”

Section: Explanation and Proof In Dynamic Geometry Environmentsmentioning

confidence: 99%

“…The experience of a student using an exploratory artifact of a Turtle geometry model of the hyperbolic surface was discussed in Arzarello et al (2012). The students; experience was indicative of the value of a digital artefact to offer learners a more concrete and visual situation that can supplement and support traditional proofs, which often appear abstract and lack visual intuition.…”

Section: Non-euclidean and Other Geometriesmentioning

confidence: 99%

See 2 more Smart Citations

Recent research on geometry education: an ICME-13 survey team report

Sinclair

Bartolini

Villiers

et al. 2016

ZDM Mathematics Education

113

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This survey on the theme of Geometry Education (including new technologies) focuses chiefly on the time span since 2008. Based on our review of the research literature published during this time span (in refereed journal articles, conference proceedings and edited books), we have jointly identified seven major threads of contributions that span from the early years of learning (pre-school and primary school) through to post-compulsory education and to the issue of mathematics teacher education for geometry. These threads are as follows: developments and trends in the use of theories; advances in the understanding of visuo spatial reasoning; the use and role of diagrams and gestures; advances in the understanding of the role of digital technologies; advances in the understanding of the teaching and learning of definitions; advances in the understanding of the teaching and learning of the proving process; and, moving beyond traditional Euclidean approaches. Within each theme, we identify relevant research and also offer commentary on future directions.

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Section: Explanation and Proof In Dynamic Geometry Environmentsmentioning

confidence: 99%

Section: Explanation and Proof In Dynamic Geometry Environmentsmentioning

confidence: 99%

Section: Non-euclidean and Other Geometriesmentioning

confidence: 99%

See 1 more Smart Citation

Recent research on geometry education: an ICME-13 survey team report

Sinclair

Bartolini

Villiers

et al. 2016

ZDM Mathematics Education

113

View full text Add to dashboard Cite

show abstract

“…Guven and Baki (2010) theorised van Hiele levels of understanding for spherical geometry similar to 2D, which appeared to be reasonably confirmed by a Guttman scalogram analysis, though future studies would be useful. There have been studies focused on a Turtle geometry model of the hyperbolic surface (Arzarello, Bartolini Bussi, Leung, Mariotti, & Stevenson, 2012) and on topological surfaces (the Mobius strip, the torus and the Klein bottle) using a DGE (Hawkins & Sinclair, 2008). In contrast, there has been little or no research on the teaching and learning of fractals over the past 10 years.…”

Section: Moves Beyond Traditional Euclidean Approaches To Geometrymentioning

confidence: 99%

Geometry Education, Including the Use of New Technologies: A Survey of Recent Research

Sinclair

Bartolini

Villiers

et al. 2017

Proceedings of the 13th International Congress on Mathematical Education

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This is a summary report of the ICME-13 survey on the theme of recent research in geometry education. Based on an analysis of the research literature published since 2008, the survey focuses on seven major research threads. These are the use of theories in geometry education research, the nature of visuospatial reasoning, the role of diagrams and gestures, the role of digital technologies, the teaching and learning of definitions, the teaching and learning of the proving process, and moves beyond traditional Euclidean approaches. Within each theme, there is commentary on promising future directions for research.

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“…Cabri Géomètre, Geogebra, Sketchpad, etc.) can support forms of cognitive continuity, at least in the case of elementary geometry (Arzarello et al 2012: see the discussion about abductions in the final section).…”

mentioning

confidence: 99%

Compendium for Early Career Researchers in Mathematics Education

Kaiser¹,

Presmeg²

2019

ICME-13 Monographs

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The paper analyses a basic gap, highlighted by most of the literature concerning the teaching of proofs, namely, the distance between students' argumentative and proving processes. The analysis is developed from both epistemological and cognitive standpoints: it critiques the Toulmin model of reasoning and introduces a new model, the Logic of Inquiry of Hintikka, more suitable for bridging this gap. An example of didactical activity within Dynamic Geometry Environments is sketched in order to present a concrete illustration of this approach and to show the pedagogical effectiveness of the model. Keywords Proof Á Logic of inquiry Á Argumentation Á Dynamic geometry environments 10.1 Introduction In their wonderful book Anschauliche Geometrie, Hilbert and Cohn-Vossen (1932) wrote 1 : In mathematics, as in all scientific research, we find two tendencies: the tendency to abstraction-it seeks to work out (herauszuarbeiten) the logical points of view from the manifold material and bring this into systematic connection-and the other tendency, that

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Experimental Approaches to Theoretical Thinking: Artefacts and Proofs

Cited by 33 publications

References 27 publications

Recent research on geometry education: an ICME-13 survey team report

Recent research on geometry education: an ICME-13 survey team report

Geometry Education, Including the Use of New Technologies: A Survey of Recent Research

Compendium for Early Career Researchers in Mathematics Education

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