The width of a proof

This paper’s aim is to discuss the concept of width of a proof put forward by Timothy Gowers. It explains what this concept means and attempts to show how it relates to other concepts discussed in the existing literature on proof and proving. It also explores how the concept of width of a proof might be used productively in the mathematics curriculum and how it might fit with the various perspectives on learning to prove

La amplitud de una demostración

This document was originally published as G. Hanna (2013). The width of proof. In B. Ubuz , Ç. Haser, & M. A. Mariotti (Eds.), Proceeding of the 8th Congress of European Research in Mathematics Education (pp. 146-155). Antalya, Turkey: Middle East Technical University.This paper’s aim is to discuss the concept of width of a proof put forward by Timothy Gowers. It explains what this concept means and attempts to show how it relates to other concepts discussed in the existing literature on proof and proving. It also explores how the concept of width of a proof might be used productively in the mathematics curriculum and how it might fit with the various perspectives on learning to prove.El objetivo de este artículo es discutir el concepto de amplitud de una demostración presentado por Timothy Gowers. Se explica el significado de este concepto y se trata de mostrar cómo se relaciona con otros conceptos discutidos en la literatura existente sobre prueba y demostraciones. También se explora cómo el concepto de amplitud de una demostración podría utilizarse productivamente en el currículo de matemáticas y cómo podría encajar con las diferentes perspectivas sobre el aprendizaje de la demostración

As Thurston says? : On using quotations from famous mathematicians to make points about philosophy and education

© 2020 Springer-Verlag. The final publication is available at Springer via https://doi.org/10.1007/s11858-020-01154-w.It is commonplace in the educational literature on mathematical practice to argue for a general conclusion from isolated quotations from famous mathematicians. In this paper, we supply a critique of this mode of inference. We review empirical results that show the diversity and instability of mathematicians’ opinions on mathematical practice. Next, we compare mathematicians’ diverse and conflicting testimony on the nature and purpose of proof. We lay especial emphasis on the diverse responses mathematicians give to the challenges that digital technologies present to older conceptions of mathematical practice. We examine the career of one much cited and anthologised paper, WP Thurston’s ‘On Proof and Progress in Mathematics’ (1994). This paper has been multiply anthologised and cited hundreds of times in educational and philosophical argument. We contrast this paper with the views of other, equally distinguished mathematicians whose use of digital technology in mathematics paints a very different picture of mathematical practice. The interesting question is not whether mathematicians disagree—they are human so of course they do. The question is how homogenous is their mathematical practice. If there are deep differences in practice between mathematicians, then it makes little sense to use isolated quotations as indicators of how mathematics is uniformly or usually done. The paper ends with reflections on the usefulness of quotations from research mathematicians for mathematical education.Peer reviewe

Human-Machine Collaboration in the Teaching of Proof

This paper argues that interactive theorem provers (ITPs) could play an important role in fostering students’ appreciation and understanding of proof and of mathematics in general. It shows that the ITP Lean has three features that mitigate existing difficulties in teaching and learning mathematical proof. One is that it requires students to identify a proof strategy at the start. The second is that it gives students instant feedback while allowing them to explore with maximum autonomy. The third is that elementary formal logic finds a natural place in the activity of creating proofs. The challenge in using Lean is that students have to learn its command language, in addition to mathematics course content and elementary logic

The role of networks to overcome large-scale challenges in tomography : the non-clinical tomography users research network

Our ability to visualize and quantify the internal structures of objects via computed tomography (CT) has fundamentally transformed science. As tomographic tools have become more broadly accessible, researchers across diverse disciplines have embraced the ability to investigate the 3D structure-function relationships of an enormous array of items. Whether studying organismal biology, animal models for human health, iterative manufacturing techniques, experimental medical devices, engineering structures, geological and planetary samples, prehistoric artifacts, or fossilized organisms, computed tomography has led to extensive methodological and basic sciences advances and is now a core element in science, technology, engineering, and mathematics (STEM) research and outreach toolkits. Tomorrow's scientific progress is built upon today's innovations. In our data-rich world, this requires access not only to publications but also to supporting data. Reliance on proprietary technologies, combined with the varied objectives of diverse research groups, has resulted in a fragmented tomography-imaging landscape, one that is functional at the individual lab level yet lacks the standardization needed to support efficient and equitable exchange and reuse of data. Developing standards and pipelines for the creation of new and future data, which can also be applied to existing datasets is a challenge that becomes increasingly difficult as the amount and diversity of legacy data grows. Global networks of CT users have proved an effective approach to addressing this kind of multifaceted challenge across a range of fields. Here we describe ongoing efforts to address barriers to recently proposed FAIR (Findability, Accessibility, Interoperability, Reuse) and open science principles by assembling interested parties from research and education communities, industry, publishers, and data repositories to approach these issues jointly in a focused, efficient, and practical way. By outlining the benefits of networks, generally, and drawing on examples from efforts by the Non-Clinical Tomography Users Research Network (NoCTURN), specifically, we illustrate how standardization of data and metadata for reuse can foster interdisciplinary collaborations and create new opportunities for future-looking, large-scale data initiatives

Proof and proving in mathematics education : the 19th ICMI study

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Resumen tomado de la publicaciónEl objetivo de este artículo es discutir el concepto de amplitud de una demostración presentado por Timothy Gowers. Se explica el significado de este concepto y se trata de mostrar cómo se relaciona con otros conceptos discutidos en la literatura existente sobre prueba y demostraciones. También se explora cómo el concepto de amplitud de una demostración podría utilizarse productivamente en el currículo de matemáticas y cómo podría encajar con las diferentes perspectivas sobre el aprendizaje de la demostración.ES

La amplitud de una demostración

El objetivo de este artículo es discutir el concepto de amplitud de una demostración presentado por Timothy Gowers. Se explica el significado de este concepto y se trata de mostrar cómo se relaciona con otros conceptos discutidos en la literatura existente sobre prueba y demostraciones. También se explora cómo el concepto de amplitud de una demostración podría utilizarse productivamente en el currículo de matemáticas y cómo podría encajar con las diferentes perspectivas sobre el aprendizaje de la demostración.This paper’s aim is to discuss the concept of width of a proof put forward by Timothy Gowers. It explains what this concept means and attempts to show how it relates to other concepts discussed in the existing literature on proof and proving. It also explores how the concept of width of a proof might be used productively in the mathematics curriculum and how it might fit with the various perspectives on learning to prove