ZDM Mathematics Education, 2016

This article is a commentary on the mathematical working space (MWS) approach and draws on the articles contained in this ZDM issue. The article is divided into three parts. In the first part I discuss the place of the MWS approach among the French theories of didactique des mathématiques. In the second part I outline what I think are the central ideas of the MWS approach. I conclude the article with a sketch of what seems to me to be its accomplishments and challenges, focusing mainly on the epistemolog-ical and cognitive stances that the MWS approach conveys in order to elucidate the manner in which this approach theoretically assumes that things are known and learned. Keywords Cognition · Epistemology · Activity Theory · Conceptions of the student · Collective versus individual learning

2013

In this report we propose an alternate account of mathematical reification as compared to Sfard’s (1991) description, which is characterized as an “instantaneous quantum leap”, a mental process, and a static structure. Our perspective is based on two in-service teachers’ exploration of the function f (z) = e , using Geometer’s Sketchpad. Using microethnographic analysis techniques we found that the long road to beginning to reify the function entailed interplay between body-generated motion and object self-motion, kinesthetic continuity between different sides of the “same” thing, cultural and emotional background of life with things-to-be, and categorical intuitions. Our results suggest that perceptuomotor activities involving technology may serve as an instrument in facilitating reification of abstract mathematical objects such as complex-valued functions.

Mathematics and mathematical metaphors are important because they help to make heterogeneous processes – including those of globalization – seem unified. Like the theology they have replaced, they have the effect of putting human beings in contact with the transcendent. The argument, then, is: although mathematical shapes are historically acquired and learned, they are subsequently naturalized, apprehended, and widely used in enacting the real world. How does this work? To answer this question we need to note that mathematical referents are powerfully located and enacted in texts. In science, and especially in mathematical texts, a `figure' or an inscription is combined with a legend. The referent – the figure – is right there in the text. So mathematics endlessly creates new referents, and a few are converted into representations of the real world. This happens because they are embedded in metrology, the art of measuring. Metrology strengthens the capacity of mathematics to convince. It helps to frame what is to be taken into account or forgotten, made present or absent. So the argument is that mathematically stable shapes (theorems), mobilized as metaphors in diverse real-world situations and supported by metrology, become invincible ontological tools. Mathematics is important in contemporary reality partly because of its metaphorical plasticity. On the one hand, it is taken to be certain, determined, exact, and decidable. On the other hand, in the first half of the 20th century, mathematicians concluded that it is impossible to determine the logical consistency of any reasonably large mathematical system. For instance, the logical consistency of arithmetic can be neither proved nor disproved. At the beginning of the 21st century, uncertainty is recognized in mathematics in two more ways. First, it is argued that any shape (or `order') will always be present in a space (a `universe') that is sufficiently large. This means that the appearance of disorder in the world can be understood as a matter of scale. Second, it is argued that the radical indeterminacy (that is, the nondecidability or uncalculability) of most `objects' that populate mathematical spaces means that they cannot be caught in any kind of regular network (`order'). In what follows I will work these arguments through by considering first the enactment of mathematics, and what one might think of as the practical transcendence of mathematical shapes or theorems. Then I will explore its character as an ontological tool. I will next consider the mathematics of order and scale, on the one hand, and uncalculability, on the other. I will conclude with a few observations about the potentially subversive character of these 20th-century forms of mathematics to the global and the local.

Mathematical Thinking and Learning, 2019

in: Vera Bühlmann, Ludger Hovestadt (eds.): Symbolizing Existence - Metalithikum III (Birkhäuser, Vienna 2016)

of Basel. Her work focuses on the double-articulation of semiotics and (mathematical) communication, especially on how an algebraic understanding of code and programming languages enable us to consider computability within a general literacy of architectonic articulations. She has edited several books in this fi eld, has published many essays and is author of Die Nachricht, ein Medium : Annäherungen an Herkünfte und Topoi städtischer Architektonik (ambra, 2014). www. monasandnomos.org 158 VI 1 a spectrograph 163-ii the spectrometer 163 -iii the generic 164 · grammatizing symbolic domains 165 · an abstract object's integrity: political subjectivization 167 · beyond urban comfort, in a state of expulsion 173 · generic as an adverb, the liveliness of nature 174 · bodies-to-think-in live in algebraic universality 176 -iv characterizations of the generic 178 · characterization on a grammatical level 178 · the man without qualities (robert musil) 179 · the city without identity (rem koolhaas) 179 -v falling in love with the in-sinuousness proper to an economy of entropy 181 · primary abundance 181-vi the master 184 · toward an informationbased architectonics 184 · within the generic city: master, yet in "whose" house? 187 -vii characterizations of the master 189 · attracted by the volatility of a flirtation between the philosophical stances of "critical rationalism" and "speculative realism" 189 · cosmic untendedness, prosaicness in verse 193 · cosmo-politics, or putting to work a symbolist meter 199 · cosmo-literacy, or the alphabetization of the nature of number 200viii acquiring a body-to-think-in 203 · the most common representation of the nature of numbers … 203 · … and how it got into trouble still not resolved today 204 · algebraic operations, or how the nature of numbers can be brought to work 205 -ix masterpieces, and why there are so few of them 208

THEORIA. An International Journal for Theory, History and Foundations of Science

In the first part of the paper, previous work about embodied mathematics and the practice of topology will be presented. According to the proposed view, in order to become experts, topologists have to learn how to use manipulative imagination: representations are cognitive tools whose functioning depends from pre-existing cognitive abilities and from specific training. In the second part of the paper, the notion of imagination as “make-believe” is discussed to give an account of cognitive tools in mathematics as props; to better specify the claim, the notion of “affordance” is explored in its possible extension from concrete objects to representations.

2017

In line with the emerging field of philosophy of mathematical practice, this book pushes the philosophy of mathematics away from questions about the reality and truth of mathematical entities and statements and toward a focus on what mathematicians actually do—and how that evolves and changes over time. How do new mathematical entities come to be? What internal, natural, cognitive, and social constraints shape mathematical cultures? How do mathematical signs form and reform their meanings? How can we model the cognitive processes at play in mathematical evolution? And how does mathematics tie together ideas, reality, and applications? Roi Wagner uniquely combines philosophical, historical, and cognitive studies to paint a fully rounded image of mathematics not as an absolute ideal but as a human endeavor that takes shape in specific social and institutional contexts. The book builds on ancient, medieval, and modern case studies to confront philosophical reconstructions and cutting-edge cognitive theories. It focuses on the contingent semiotic and interpretive dimensions of mathematical practice, rather than on mathematics’ claim to universal or fundamental truths, in order to explore not only what mathematics is, but also what it could be. Along the way, Wagner challenges conventional views that mathematical signs represent fixed, ideal entities; that mathematical cognition is a rigid transfer of inferences between formal domains; and that mathematics’ exceptional consensus is due to the subject’s underlying reality. The result is a revisionist account of mathematical philosophy that will interest mathematicians, philosophers, and historians of science alike.

2003

The core of this paper consists of a reflective account of the means a solution of a particular problem was obtained. The problem solving literature now includes a number of books that commend such reflective activity, and sest out particular frameworks encompassing heuristics for arriving at solutions. The account in this paper is largely descriptive in the sense that it does not claim to offer a heuristic menu for others to use. It does, however, contrast successful deductive and inductive approaches to solving the same problem, and is intended to encourage students and other problem solvers to maintain reflective awareness of the possibilities available to them as they work on a problem.

Educational Studies in Mathematics, 1997

In this paper, we explore the relationship between learners' actions, visualisations and the means by which these are articulated. We describe a microworld, Mathsticks, designed to help students construct mathematical meanings by forging links between the rhythms of their actions and the visual and corresponding symbolic representations they developed. Through a case study of two students interacting with Mathsticks, we illustrate a view of mathematics learning which places at its core the medium of expression, and the building of connections between different mathematisations rather than ascending to hierarchies of decontextualisation.

This book presents a new approach to the epistemology of mathematics by viewing mathematics as a human activity whose knowledge is intimately linked with practice. The crucial idea of a continuum is used to provide an account of the development of mathematical knowledge that reflects the actual experience of doing math and makes sense of the perceived objectivity of mathematical results.