Investigating mathematics and learning to teach mathematics

This paper deals with an idea that plays an increasing role in teaching and in teacher education—investigating as a powerful paradigm of knowledge construction. Investigations may be carried out both in learning mathematics and in learning how to teach mathematics at preservice and inservice levels. I look into investigations in mathematics and in the mathematics curriculum, pointing out some issues that teachers face proposing them in the classroom. Then, I discuss teacher education and professional development, stressing the value of investigations about practice as a means of developing knowledge. I conclude with examples of work done by preservice and inservice teachers and by teams of teachers and researchers focusing on pupils’ investigative work in mathematics classes that illustrate the educational value of this activity and discuss the roles of the teacher.Este artigo baseia-se numa ideia que desempenha um papel crescente no ensino e na formação de professores – investigar constitui um paradigma poderoso de construção do conhecimento. Tanto podem ser realizadas investigações no ensino da Matemática como na formação inicial e contínua do professor de Matemática. Assim, analiso o papel das investigações em Matemática e no currículo de Matemática, apontando algumas questões que os professores enfrentam quando as propõem na sala de aula. De seguida, discuto a formação de professores e o desenvolvimento profissional, dando ênfase ao valor das investigações sobre a prática como meio de desenvolver novo conhecimento. Concluo com exemplos de trabalho realizado por professores em formação inicial e contínua e por equipas de professores e investigadores que se centram no trabalho investigativo dos alunos realizado nas aulas de Matemática, exemplos esses que ilustram o valor educacional desta actividade e permitem discutir os papéis do professor

Introducing Formalism in Economics: The Growth Model of John von Neumann

The objective is to interpret John von Neumann's growth model as a decisive step of the forthcoming formalist revolution of the 1950s in economics. This model gave rise to an impressive variety of comments about its classical or neoclassical underpinnings. We go beyond this traditional criterion and interpret rather this model as the manifestation of von Neumann's involvement in the formalist programme of mathematician David Hilbert. We discuss the impact of Kurt Gödel’s discoveries on this programme. We show that the growth model reflects the pragmatic turn of the formalist programme after Gödel and proposes the extension of modern axiomatisation to economics..Von Neumann, Growth model, Formalist revolution, Mathematical formalism, Axiomatics

Revisiting Major Discoveries in Linguistic Geometry with Mosaic Reasoning

AbstractIn discovering the nature of the Primary Language of the human brain introduced by J. von Neumann, two components have been investigated. The first component is Linguistic Geometry (LG), the algorithm of optimizing war-fighting strategies. LG's universal applicability in various domains and its power in generating human-like strategies suggested that LG should be a component of the Primary Language. The second component is the algorithm of inventing new algorithms. This paper researches the role of mosaic reasoning, a component of the Algorithm of Discovery, in obtaining the key results in LG such as the algorithms for generating trajectories and zones

Pluto and the Platypus: An Odd Ball and an Odd Duck — On Classificatory Norms

Some astronomers believe that we have discovered that Pluto is not a planet. I contest this assessment. Recent discoveries of trans-Neptunian Pluto-sized objects do not require that we exclude Pluto from the planets. But the obvious alternative, that classificatory revision is a matter of arbitrary choice, is also unpalatable. I argue that this classificatory controversy — which I compare to the controversy about the classification of the platypus — illustrates how our classificatory practices are laden with normative commitments of a distinctive kind. I argue that the “norm-ladenness” of classification has philosophically significant ramifications for how we think about scientific disputes and debates in the metaphysics of classification such as the monism/pluralism debate

Presume It Not: True Causes in the Search for the Basis of Heredity

Kyle Stanford has recently given substance to the problem of unconceived alternatives, which challenges the reliability of inference to the best explanation (IBE) in remote domains of nature. Conjoined with the view that IBE is the central inferential tool at our disposal in investigating these domains, the problem of unconceived alternatives leads to scientific anti-realism. We argue that, at least within the biological community, scientists are now and have long been aware of the dangers of IBE. We re-analyze the nineteenth-century study of inheritance and development (Stanford’s case study) and extend it into the twentieth century, focusing in particular on both classical Mendelian genetics and the studies by Avery et al. on the chemical nature of the hereditary substance. Our extended case studies show the preference of the biological community for a different methodological standard: the vera causa ideal, which requires that purported causes be shown on non-explanatory grounds to exist and be competent to produce their effects. On this basis, we defend a prospective realism about the biological sciences

2006 SQ372: A Likely Long-Period Comet from the Inner Oort Cloud

We report the discovery of a minor planet (2006 SQ372) on an orbit with a perihelion of 24 AU and a semimajor axis of 796 AU. Dynamical simulations show that this is a transient orbit and is unstable on a timescale of 200 Myrs. Falling near the upper semimajor axis range of the scattered disk and the lower semimajor axis range of the Oort Cloud, previous membership in either class is possible. By modeling the production of similar orbits from the Oort Cloud as well as from the scattered disk, we find that the Oort Cloud produces 16 times as many objects on SQ372-like orbits as the scattered disk. Given this result, we believe this to be the most distant long-period comet ever discovered. Furthermore, our simulation results also indicate that 2000 OO67 has had a similar dynamical history. Unaffected by the "Jupiter-Saturn Barrier," these two objects are most likely long-period comets from the inner Oort Cloud

ASIME 2018 White Paper. In-Space Utilisation of Asteroids: Asteroid Composition -- Answers to Questions from the Asteroid Miners

In keeping with the Luxembourg government's initiative to support the future use of space resources, ASIME 2018 was held in Belval, Luxembourg on April 16-17, 2018. The goal of ASIME 2018: Asteroid Intersections with Mine Engineering, was to focus on asteroid composition for advancing the asteroid in-space resource utilisation domain. What do we know about asteroid composition from remote-sensing observations? What are the potential caveats in the interpretation of Earth-based spectral observations? What are the next steps to improve our knowledge on asteroid composition by means of ground-based and space-based observations and asteroid rendez-vous and sample return missions? How can asteroid mining companies use this knowledge? ASIME 2018 was a two-day workshop of almost 70 scientists and engineers in the context of the engineering needs of space missions with in-space asteroid utilisation. The 21 Questions from the asteroid mining companies were sorted into the four asteroid science themes: 1) Potential Targets, 2) Asteroid-Meteorite Links, 3) In-Situ Measurements and 4) Laboratory Measurements. The Answers to those Questions were provided by the scientists with their conference presentations and collected by A. Graps or edited directly into an open-access collaborative Google document or inserted by A. Graps using additional reference materials. During the ASIME 2018, first day and second day Wrap-Ups, the answers to the questions were discussed further. New readers to the asteroid mining topic may find the Conversation boxes and the Mission Design discussions especially interesting.Comment: Outcome from the ASIME 2018: Asteroid Intersections with Mine Engineering, Luxembourg. April 16-17, 2018. 65 Pages. arXiv admin note: substantial text overlap with arXiv:1612.0070