Proof in classroom social practice

How does the construction of proof relate to the social practice developed in the mathematics classroom? This report addresses the role of diagrams in order to focus the complementarity of participation and reification in the process of constructing a proof and negotiating its meaning. The discussion is based on the analysis of the mathematical practice developed by a group of four 9th grade students and is inspired by the social theory of learning

Reasoning and proof in classroom (9th grade)

This paper is research oriented and pretends to contribute toward giving empirical evidence about how students develop their reasoning and how they achieved to a proof construction in school context. Its main theme is epistemology. It describes the way in which four students in 9th Grade explored a task related with the discovery of symmetry axes in various geometric figures. The proof constructed by students had essentially an explaining function and it was related with the symmetry axes of regular polygons. The teacher’s role in meaning negotiation of the proof and its need is described through illustrative episodes. The paper presents part of a study which purpose is to analyse the nature of mathematical proof in classroom, its role and the nature of the relationship between the construction of a proof and the social interactions. Assuming a social perspective, attention is focussed on the social construction of knowledge and on the structuring resources that shape mathematical experience. The study’s methodology has an interpretative nature. One outcome of the study discussed here is that students develop first a practical understanding with no awareness of the reasons founding mathematical statements and after a theoretical one leading them to a proof elaboration.

Producing an explanatory proof in a 9th grade classroom

Este artigo apresenta parte de um estudo fundamentado na problemática da demonstração na matemática escolar. Descreve o modo como quatro alunos do 9.º ano exploraram uma tarefa relacionada com a descoberta de eixos de simetria em várias figuras geométricas. A demonstração, que os mesmos construíram, teve essencialmente uma função explicativa. O papel da professora na negociação do significado de demonstração e da sua necessidade é igualmente analisado. Os alunos desenvolvem primeiro uma compreensão prática sem consciência das razões que fundamentam as afirmações matemáticas e só depois uma compreensão teórica que os conduz à construção de uma demonstração.

Topics in geometry, analysis and inverse problems

The thesis consists of three independent parts. Part I: Polynomial amoebas We study the amoeba of a polynomial, as de ned by Gelfand, Kapranov and Zelevinsky. A central role in the treatment is played by a certain convex function which is linear in each complement component of the amoeba, which we call the Ronkin function. This function is used in two di erent ways. First, we use it to construct a polyhedral complex, which we call a spine, approximating the amoeba. Second, the Monge-Ampere measure of the Ronkin function has interesting properties which we explore. This measure can be used to derive an upper bound on the area of an amoeba in two dimensions. We also obtain results on the number of complement components of an amoeba, and consider possible extensions of the theory to varieties of codimension higher than 1. Part II: Differential equations in the complex plane We consider polynomials in one complex variable arising as eigenfunctions of certain differential operators, and obtain results on the distribution of their zeros. We show that in the limit when the degree of the polynomial approaches innity, its zeros are distributed according to a certain probability measure. This measure has its support on the union of nitely many curve segments, and can be characterized by a simple condition on its Cauchy transform. Part III: Radon transforms and tomography This part is concerned with different weighted Radon transforms in two dimensions, in particular the problem of inverting such transforms. We obtain stability results of this inverse problem for rather general classes of weights, including weights of attenuation type with data acquisition limited to a 180 degrees range of angles. We also derive an inversion formula for the exponential Radon transform, with the same restriction on the angle.

The Nature of Scientific Proof in the Age of Simulations. Is numerical mimicry a third way of establishing truth?

Is numerical mimicry a third way of establishing truth? Kevin Heng received his M.S. and Ph.D. in astrophysics from the Joint Institute for Laboratory Astrophysics (JILA) and the University of Colorado at Boulder. He joined the Institute for Advanced Study in Princeton from 2007 to 2010, first as a Member and later as the Frank & Peggy Taplin Member. From 2010 to 2012 he was a Zwicky Prize Fellow at ETH Z¨urich (the Swiss Federal Institute of Technology). In 2013, he joined the Center for Space and Habitability (CSH) at the University of Bern, Switzerland, as a tenure-track assistant professor, where he leads the Exoplanets and Exoclimes Group. He has worked on, and maintains, a broad range of interests in astrophysics: shocks, extrasolar asteroid belts, planet formation, fluid dynamics, brown dwarfs and exoplanets. He coordinates the Exoclimes Simulation Platform (ESP), an open-source set of theoretical tools designed for studying the basic physics and chemistry of exoplanetary atmospheres and climates (www.exoclime.org). He is involved in the CHEOPS (Characterizing Exoplanet Satellite) space telescope, a mission approved by the European Space Agency (ESA) and led by Switzerland. He spends a fair amount of time humbly learning the lessons gleaned from studying the Earth and Solar System planets, as related to him by atmospheric, climate and planetary scientists. He received a Sigma Xi Grant-in-Aid of Research in 2006

The Standard of Proof in EC Merger Control-Conclusions from the Sony BMG Saga. IES WORKING PAPER 4/2008

One of the most important developments in EC competition policy during 2006 was the Court of First Instance’s (CFI) Impala v. Commission judgment annulling the European Commission’s approval of the merger between the music units of Sony and Bertelsmann. It harshly criticized the Commission’s Decision because it found that the evidence relied on was not capable of substantiating the conclusion. This was the first time that a merger decision was annulled for not meeting the requisite legal standard for authorizing the merger. Consequently, the CFI raised fundamental questions about the standard of proof incumbent on the Commission in its merger review procedures. On July 10, 2008, the European Court of Justice overturned Impala, yet it did not resolve the fundamental question underlying the judicial review of the Sony BMG Decision; does the Commission have the necessary resources and expertise to meet the Community Court’s standard of proof? This paper addresses the wider implications of the Sony BMG saga for the Commission’s future handling of complex merger investigations. It argues that the Commission may have set itself an impossible precedent in the second approval of the merger. While the Commission has made a substantial attempt to meet the high standard of proof imposed by the Community Courts, it is doubtful that it will be able to jump the fence again in a similar fashion under normal procedural circumstances.

Operative masonry; or, A theoretical and practical treatise of building; containing a scientific account of stones, clays, bricks, mortars, cements, &c.; a description of their component parts, with the manner of preparing and using them. The fundamental rules in geometry, on masonry and stone-cutting, with their application to practice. Illustrated with forty copper-plate engravings.

Republished in 1846 with title: Practical masonry: or A theoretical and operative treatise of building ...

Some extremal problems in geometry and analysis /

Includes bibliographies.

A plain statement of the trusts and recent administration of Lady Hewley's charities, as now in proof in the suit of the attorney-general, v. Shore, esq., and others; with remarks on efforts now making to effect "a total disconnexion between church and state":

Mode of access: Internet.

A school course in geometry, including the elements of trigonometry and mensuration and an introduction to the methods of co-ordinate geometry,

Mode of access: Internet.

Practical exercises in geometry / by W. D. Eggar ...

Mode of access: Internet.

Lessons in geometry. For the use of beginners.

Mode of access: Internet.