Embodied Semiotic Activities and Their Role in the Construction of Mathematical Meaning of Motion Graphs

This paper examines the relation between bodily actions, artifact-mediated activities, and semiotic processes that students experience while producing and interpreting graphs of two-dimensional motion in the plane. We designed a technology-based setting that enabled students to engage in embodied semiotic activities and experience two modes of interaction: 2D freehand motion and 2D synthesized motion, designed by the composition of single variable function graphs. Our theoretical framework combines two perspectives: the embodied approach to the nature of mathematical thinking and the Vygotskian notion of semiotic mediation. The article describes in detail the actions, gestures, graph drawings, and verbal discourse of one pair of high school students and analyzes the social semiotic processes they experienced. Our analysis shows how the computerized artifacts and the students’ gestures served as means of semiotic mediation. Specifically, they supported the interpretation and the production of motion graphs; they mediated the transition between an individual’s meaning of mathematical signs and culturally accepted mathematical meaning; and they enable linking bodily actions with formal signs.     

Reflections on learning and cognition

The occasion of my 9th ICME—the first being in Berkeley in 1980, the most recent being in Seoul in 2012—provides an opportunity for reflecting on changes in the field over more than 30 years. “Learning and cognition” have a very different meaning now than they did in 1980. I argue that in various ways, the papers in this volume (derived from the ICME 12 Topic Study Group on Learning and Cognition) represent a significant evolution of the field—with mathematical sense making being a central conception, and with the evolution of the very notions of learning and cognition to include embodied, sociocultural, and historical perspectives. In this volume one sees a focus on classroom activities as they engender aspects of sense making, framed in ways that were not even part of the discourse on learning in 1980; one also sees a widely varied set of research methods for addressing such issues. I reflect on the state of the art, and then discuss some possibly productive directions with regard to the characterization and support of mathematically productive classrooms.     

The catwalk task: Reflections and synthesis: Part 1

In this article I recount my experiences with a series of encounters with the catwalk task and reflect on the professional growth that these opportunities afforded. First, I reflect on my own mathematical work on the catwalk task, including my efforts to fit various algebraic models to the data. Second, I reflect on my experiences working with a group of high school students on the catwalk task and my interpretations of their mathematical thinking. Finally, I reflect on the entire experience with the catwalk problem, as a mathematics learner, as a teacher, and as a professional.     

Perception and Virtue Reliabilism

In some recent work, Ernest Sosa rejects the “perceptual model” of rational intuition, according to which intuitive beliefs (e.g., that 2 + 2 = 4 2 + 2 = 4 ) are justified by standing in the appropriate relation to a nondoxastic intellectual experience (a seeming-true, or the like), in much the way that perceptual beliefs are often held to be justified by an appropriate relation to nondoxastic sense experiential states. By extending some of Sosa’s arguments and adding a few of my own, I argue that Sosa is right to reject the perceptual model of intuition, and that we should reject the “perceptual model” of perception as well. Rational intuition and perception should both receive a virtue theoretic (e.g., reliabilist) account, rather than an evidentialist one. To this end, I explicitly argue against the Grounds Principle, which holds that all justified beliefs must be based on some adequate reason, or ground.     

Réflexion entre deux diffusions conjuguées

We observed, in a previous work, that Brownian motion reflected on an independent time-reversed Brownian motion is again Brownian motion. We present the generalisation of this result to pairs of conjugate diffusions (which are also dual, in the sense of Siegmund). To cite this article: F. Soucaliuc, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 1119–1124.     

Hume’s Argument for the Ontological Independence of Simple Properties

In this paper, I will reconstruct Hume's argument for the ontological (in the sense of rigid existential) independence of simple properties in A Treatise of Human Nature, Book 1 (1739). According to my reconstruction, the main premises of the argument are the real distinctness of every perception of a simple property, Hume's Separability Principle and his Conceivability Principle. In my view, Hume grounds the real distinctness of every perception of a simple property in his atomistic theory of sense perception and his Copy Principle. I will also show why Hume's argument should be seen as relevant nowadays. David Lewis and his followers in metaphysics continue Hume's line of thinking in this respect, which is opposed by power ontologists (Brian Ellis, Stephen Mumford), for example.     

The role of mathematics in educational systems

In this paper, I will explain my views on education, particularly mathematics education, stressing my critique on the socio-cultural dimensions involved in mathematics education. The state of the world and ethical behavior play an important role in my critique. In exposing my views, the importance of my relationship with Hans-Georg Steiner will be clearly stated. In my memory, our discussions on these issues were very influential to my thinking. The importance of the support given by Hans-Georg to the development of mathematics education worldwide will be underlined. Both in the position of the chairman of the International Program Committee of ICME 3 and in the early years of the IDM in Bielefeld, Hans-Georg provided the material and intellectual space for the strengthening of Didactics of Mathematics as a research field. Following these preliminary remarks, I will briefly present the Program Ethnomathematics and explain how the program fits into what I call the modern trivium of literacy, matheracy and technoracy, which is a curriculum alternative for contemporary education.     

On the sources of my book Moderne algebra

In December 1971, Garrett Birkhoff asked me to give my view on the main sources for my book. I wrote him a seven-page letter with two supplements. He intended to publish an edited version of my letter, with some commentary of his own, but in the course of our correspondence it turned out that both versions were unsatisfactory. I shall now present an extended record, explaining more fully how I came to write the book and what was the general situation in algebra at that time.     

You want them to remember? Then make it memorable! Means for enhancing operations research education

In this article I summarize the main points I made in the keynote presentation of the same title I gave at the EURO XXIV conference in Lisbon, Portugal in July of 2010. Each of these points deals in some way with making communications between an operations research professional (academic or practitioner) and a student, client, subordinate, supervisor, or colleague more effective. Furthermore, each point is directly related to some realization (or epiphany) that I have had with regard to communication since I began teaching ORMS in 1984. It is noteworthy that these communications share a common objective; we are trying to facilitate learning. Since I have spent most of my career in academia, my primary emphasis is on communication with students (particularly those enrolled in introductory ORMS courses). However, I have also spent a great deal of time working on operations research problems outside of academia, either as an employee in private industry or as an operations research consultant to corporations and not-for-profit organizations, and I hope as a consequence my discussion is also relevant to those working in the practice of ORMS.     

Computational Literacy and “The Big Picture” Concerning Computers in Mathematics Education

This article develops some ideas concerning the “big picture” of how using computers might fundamentally change learning, with an emphasis on mathematics (and, more generally, STEM education). I develop the big-picture model of computation as a new literacy in some detail and with concrete examples of sixth grade students learning the mathematics of motion. The principles that define computational literacy also serve as an analytical framework to examine competitive big pictures, and I use them to consider the plausibility, power, and limitations of other important contemporary trends in computationally centered education, notably computational thinking and coding as a social movement. While both of these trends have much to recommend them, my analysis uncovers some implausible assumptions and counterproductive elements of those trends. I close my essay with some more practical and action-oriented advice to mathematics educators on how best to orient to the long-term trajectory (big picture) of improving mathematics education with computation.     

‘Seeing is believing’ and ‘believing is seeing’

The principal concern of my paper is a distinction between two ways of appreciating works of art, characterised here in terms of the phrases ‘seeing is believing’ and ‘believing is seeing’. I examine this distinction in the light of an epistemological requirement at times at least grounded in what David Davies, in his Art as Performance, refers to as the ‘common sense theory of art appreciation’ in order to assess exactly what aspect of the philosophical approach generally known as aesthetic empiricism his account commits him to reject. I argue that the ‘experiential requirement’, if only conceived in a slightly broader way than is usual, might very well have an important role to play not only in the appropriate appreciation of works that do not demonstrate the need for such a requirement (primarily works of late modern and conceptual art), but also in the ontological account Davies himself favours.     

Towards an embodied,cultural, and material conception of mathematics cognition

In this paper I sketch an embodied, cultural, and material conception of cognition and discuss some of the implications for mathematics education. This approach, which I term sensuous cognition, rests on a cultural and historical dialectical materialist understanding of the senses, sensation, and the material and conceptual worlds. Sensation and matter are considered to be the substrate of mind, and of all psychic activity (cognitive, affective, volitional, etc.). I argue that human cognition can only be understood as a culturally and historically constituted multimodal sentient form of creatively responding, acting, feeling, transforming, and making sense of the world. To illustrate these ideas I briefly refer to a classroom episode involving 7- to 8-year-old students dealing with pattern generalization.     

Necessary and sufficient conditions for finiteness of lifetime

We give a geometric characterization for the finiteness of conditioned Brownian motion for a general class of simply connected domains, extending previous results and exhibit some new examples of domains with infinite area and finite lifetime.I would like to thank Professor Rodrigo Bañuelos, my academic advisor, for his help and guidance on this paper which is part of my Ph.D. thesis.     

Analyzing Telephone Network Data

Abstract

As our reliance on computer networks grows, it becomes increasingly important that we understand the behavior of the traffic that they carry. Because of the speeds and sizes involved, working with traffic collected from a computer network usually means working with large datasets. This article describes our experience with traffic collected from the common channel signaling (CCS) network. I will briefly describe the data and how they are collected and discuss similarities with and differences from other large datasets. Next, I will describe our analysis tools and outline the reasons that they have been found useful. Finally, I will discuss the challenges facing us as we strive for a better understanding of more data from faster networks. Although my emphasis in this article is on the CCS network, it has been my experience that both the problems and the solutions generalize to data from other networks.     

Naïve Realism,Adverbialism and Perceptual Error

My paper has three parts. First I will outline the act/object theory of perceptual experience and its commitments to (a) a relational view of experience and (b) a view of phenomenal character according to which it is constituted by the character of the objects of experience. I present the traditional adverbial response to this, in which experience is not to be understood as a relation to some object, but as a way of sensing. In the second part I argue that acceptance of (a) is independent of acceptance of (b). I then present a modified adverbialism that presents experience as relational in nature but whose character is nevertheless to be explained in terms of the way in which one senses an object. Finally, I will offer an explanation of how a naïve realist about experience can adopt this modified adverbialism and in so doing accommodate the possibility of perceptual error.     

Reasoning about relationships between quantities to reorganize inverse function meanings: The case of Arya

Researchers have argued high school students, college students, pre-service teachers, and in-service teachers do not construct productive inverse function meanings. In this report, I first summarize the literature examining students’ and teachers’ inverse function meanings. I then provide my theoretical perspective, including my use of the terms understanding and meaning and my operationalization of productive inverse function meanings. I describe a conceptual analysis of ways students may reorganize their limited inverse function meanings into productive meanings via reasoning about relationships between covarying quantities. I then present one pre-service teacher’s activity in a semester long teaching experiment to characterize how her quantitative, covariational, and bidirectional reasoning supported her in reorganizing her limited inverse function meanings into more productive meanings. I describe how this reorganization required her to reconstruct her meanings for various related mathematical ideas. I conclude with research and pedagogical implications stemming from this work and directions for future research.     

Spinoza on Having a False Idea

Naturalism pervades Spinoza’s doctrines of The Ethics, but the contours of it often bewilder us. In this light, I consider the account of falsity, or having a false idea, as presented by Spinoza in Proposition thirty_five of the Second Part, its demonstration, and the subsequent note. Based on my interpretation I argue for the claim that his account has coherence and makes sense. Further, I examine the significance of what Spinoza says about falsity for comprehension of his philosophy overall, especially as regards its contrasts with the philosophy of Descartes.     

An epistemic model of an agent who does not reflect on reasoning processes

This paper introduces an epistemic model of a boundedly rational agent under the two assumptions that (i) the agent’s reasoning process is in accordance with the model but (ii) the agent does not reflect on these reasoning processes. For such a concept of bounded rationality a semantic interpretation by the possible world semantics of the Kripke (1963) type is no longer available because the definition of knowledge in these possible world semantics implies that the agent knows all valid statements of the model. The key to my alternative semantic approach is the extension of the method of truth tables, first introduced for the propositional logic by Wittgenstein (1922), to an epistemic logic so that I can determine the truth value of epistemic statements for all relevant truth conditions. In my syntactic approach I define an epistemic logic–consisting of the classical calculus of propositional logic plus two knowledge axioms–that does not include the inference rule of necessitation, which claims that an agent knows all theorems of the logic. As my main formal result I derive a determination theorem linking my semantic with my syntactic approach. The difference between my approach and existing knowledge models is illustrated in a game-theoretic application concerning the epistemic justification of iterative solution concepts.     

Hume, the Occult, and the Substance of the School

I have not been able to locate any critique of Hume on substance by a Schoolman, at least in English, dating from Hume's period or shortly thereafter. I have, therefore, constructed my own critique as an exercise in ??post facto history??. This is what a late eighteenth-century/early nineteenth-century Scholastic could, would, and should have said in response to Hume's attack on substance should they have been minded to do so. That no one did is somewhat mysterious. My critique is precisely in the language of the period, using solely the conceptual resources available to a Schoolman at that time. The arguments, however, are as sound now as they were then, and in this sense the paper performs a dual role??contributing to the defence of substance contra Hume, and filling, albeit two hundred?years or so too late, a gap in the historical record.