We argue that pedagogic authority relies on love, which is misunderstood if seen as a matter of actions and subjects. Love is based not on finite subjects and objects existing in Euclidean space and linear time, but, rather, on the non-finite ontology, space and time of relations. Loving authority is a matter of calling and vocation, arising from the spontaneous and simultaneous call-and-response of a lively relation. We make this argument through a reading of Buber’s I–You relation and Murdoch’ s (...) account of the responsiveness of ascetic discipline. In presenting this analysis, we draw upon a case study from a research project on Australian teachers and students. (shrink)

Who were the five strangest mathematicians in history? What are the ten most interesting numbers? Jam-packed with thought-provoking mathematical mysteries, puzzles, and games, Wonders of Numbers will enchant even the most left-brained of readers. Hosted by the quirky Dr. Googol--who resides on a remote island and occasionally collaborates with Clifford Pickover--Wonders of Numbers focuses on creativity and the delight of discovery. Here is a potpourri of common and unusual number theory problems of varying difficulty--each presented in brief chapters that (...) convey to readers the essence of the problem rather than its extraneous history. Peppered throughout with illustrations that clarify the problems, Wonders of Numbers also includes fascinating "math gossip." How would we use numbers to communicate with aliens? Check out Chapter 30. Did you know that there is a Numerical Obsessive-Compulsive Disorder? You'll find it in Chapter 45. From the beautiful formula of India's most famous mathematician to the Leviathan number so big it makes a trillion look small, Dr. Googol's witty and straightforward approach to numbers will entice students, educators, and scientists alike to pick up a pencil and work a problem. (shrink)

This volume presents the proceedings from the Eleventh Brazilian Logic Conference on Mathematical Logic held by the Brazilian Logic Society in Salvador, Bahia, Brazil. The conference and the volume are dedicated to the memory of professor Mario Tourasse Teixeira, an educator and researcher who contributed to the formation of several generations of Brazilian logicians. Contributions were made from leading Brazilian logicians and their Latin-American and European colleagues. All papers were selected by a careful refereeing processs and were revised and updated (...) by their authors for publication in this volume. There are three sections: Advances in Logic, Advances in Theoretical Computer Science, and Advances in Philosophical Logic. Well-known specialists present original research on several aspects of model theory, proof theory, algebraic logic, category theory, connections between logic and computer science, and topics of philosophical logic of current interest. Topics interweave proof-theoretical, semantical, foundational, and philosophical aspects with algorithmic and algebraic views, offering lively high-level research results. (shrink)

Research In Mathematics Education In Australasia 2016–2019 (sometimes shortened to RiMEA 2016–2019) is a timely and important edited collection where contributors provide critical reviews on mathem...

We analyze the cognitive strategies underlying performance in the Number task, a Math game that requires both arithmetic fluency and mathematical creativity. In this game all elements in a set of numbers have to be used precisely once to create a target number with basic arithmetic operations. We argue that some instances of this game are NP complete, by showing its relation to the well-known Partition problem. We propose heuristics based on the distinction in forward and backward reasoning. The Number (...) Game is part of Math Garden, a popular online educational platform for practicing and monitoring math skills using innovations in computerized adaptive testing. These educational games generate enormous amounts of rich data on children's cognitive development. We found converging evidence for the use of forward proximity heuristics in the data of Math Garden, consisting of more than 20 million answers to 1,700 items. Item difficulties and the structure of correct answers were analyzed. (shrink)

ABSTRACT In this paper we discuss the potential of digital games to create meaningful educational experiences that contribute to the learning of ethics in higher education Science, Technology, Engineering and Mathematics degrees. We describe the design of a new digital ethics game with a focus on the challenges we encountered when applying existing theoretical frameworks for educational games and propose ways to address these challenges. We contend that existing design frameworks fail to account for the ‘wickedness’ (...) of ethical problems – i.e. their inconclusive, complex, and sometimes inherently contradictory nature – as they are centred around consequentiality and consistent game-system feedback to players’ actions. Drawing from a Deweyan account of the ‘educative experience’ we seek to contribute to a domain-adequate theory of transformational experience and transformational play in the context of educational ethics game design. (shrink)

We analyze the cognitive strategies underlying performance in the Number task, a Math game that requires both arithmetic fluency and mathematical creativity. In this game all elements in a set of numbers have to be used precisely once to create a target number with basic arithmetic operations. We argue that some instances of this game are NP complete, by showing its relation to the well-known Partition problem. We propose heuristics based on the distinction in forward and backward reasoning. The Number (...) Game is part of Math Garden, a popular online educational platform for practicing and monitoring math skills using innovations in computerized adaptive testing. These educational games generate enormous amounts of rich data on children's cognitive development. We found converging evidence for the use of forward proximity heuristics in the data of Math Garden, consisting of more than 20 million answers to 1,700 items. Item difficulties and the structure of correct answers were analyzed. (shrink)

The article is devoted to the applicability of Wittgenstein’s following the rule in the context of his philosophy of mathematics to real mathematical practice. It is noted that in «Philosophical Investigations» and «Remarks on the Foundations of Mathematics» Wittgenstein resorted to the analysis of rather elementary mathematical concepts, accompanied also by the inherent ambiguity and ambiguity of his presentation. In particular, against this background, his radical conventionalism, the substitution of logical necessity with the «form of life» of the (...) community, as well as the inadequacy of the representation of arithmetic rules by a language game are criticized. It is shown that the reconstruction of the Wittgenstein concept of understanding based on the Fregian division of meaning and referent goes beyond the conceptual framework of Wittgenstein language games. (shrink)

This paper surveys central justifications and approaches adopted by educators interested in incorporating history of mathematics into mathematics teaching and learning. This interest itself has historical roots and different historical manifestations; these roots are examined as well in the paper. The paper also asks what it means for history of mathematics to be treated as genuine historical knowledge rather than a tool for teaching other kinds of mathematical knowledge. If, however, history of mathematics is not subordinated (...) to the ideas and methods at the heart of the usual mathematical curriculum – algebraic equations, functions, derivatives, analytic geometry – if it becomes an essential subject for mathematics, then the attempt to find a place for history of mathematics requires refining our understanding of the nature of mathematics education itself. Thus, the paper asks not only how can history of mathematics be incorporated into mathematics education but also how the idea of mathematics education may need to be adjusted to accommodate history. (shrink)

Educational interventions are necessary to develop mathematical competence at early ages and prevent widespread mathematics learning failure in the education system as indicated by the results of European reports. Numerous studies agree that domain-specific predictors related to mathematics are symbolic and non-symbolic magnitude comparison, as well as, number line estimation. The goal of this study was to design 4 digital learning app games to train specific cognitive bases of mathematical learning in order to create resources and (...) promote the use of these technologies in the educational community and to promote effective scientific transfer and increase the research visibility. This study involved 193 preschoolers aged 57–79 months. A quasi-experimental design was carried out with 3 groups created after scores were obtained in a standardised mathematical competence assessment test, i.e., low-performance group, high-performance group, and control group. The results show that training with the 4 digital learning app games focusing on magnitude, subitizing, number facts, and estimation tasks improved the numerical skills of the experimental groups, compared to the control group. The implications of the study were, on the one hand, provided verified technological tools for teaching early mathematical competence. On the other hand, this study supports other studies on the importance of cognitive precursors in mathematics performance. (shrink)

The fifteenth Journal for Research in Mathematics Education monograph had its origins in a conference titled An Interdisciplinary Conference on Assessment in K--12 Mathematics: Collaborations Between Mathematics Education and Psychometrics, which was held in 2011 in Atlanta, Georgia. The basis for the conference was the renaissance in the field of psychometrics in which an increasing variety of psychometric models are becoming available through advances in computer hardware and software. This is opening new avenues for studying (...) the mathematical knowledge of teachers and students. The overarching purpose of the monograph is to guide further interdisciplinary collaborations between mathematics education researchers and psychometricians by examining theoretical and conceptual issues that have arisen in recent efforts to apply contemporary psychometric models to mathematics education research. Specifically, its chapters are intended to (a) illustrate for mathematics education researchers the two main categories of psychometric models-models that locate individuals along continua and models that place individuals into discrete groups, as well as hybrids of these approaches; (b) provide examples that apply these different categories of psychometric models to mathematics education research; (c) illustrate how researchers have selected different psychometric models on the basis of the researchers' goals and (d) demonstrate issues related to item development. With these goals in mind, the monograph will enhance an awareness among mathematics education researchers that it is increasingly possible to select from a variety of model options when pursuing particular research goals and that choosing among models involves trade-offs. (shrink)

Doctoral programs at U.S. universities play a critical role in the development of human resources both in the United States and abroad. This volume reports the results of an extensive study of U.S. research-doctorate programs in five broad fields: physical sciences and mathematics, engineering, social and behavioral sciences, biological sciences, and the humanities. Research-Doctorate Programs in the United States documents changes that have taken place in the size, structure, and quality of doctoral education since the widely used 1982 (...) editions. This update provides selected information on nearly 4,000 doctoral programs in 41 subdisciplines at 274 doctorate-granting institutions. This volume also reports the results of the National Survey of Graduate Faculty, which polled a sample of faculty for their views on the scholarly quality of program faculty and the effectiveness of doctoral programs in preparing research scholars/scientists. This much-anticipated update of such an essential reference will be useful to education administrators, university faculty, and students seeking authoritative information on doctoral programs. (shrink)

Open peer commentary on the article “Examining the Role of Re-Presentation in Mathematical Problem Solving: An Application of Ernst von Glasersfeld’s Conceptual Analysis” by Victor V. Cifarelli & Volkan Sevim. Upshot: Cifarelli and Sevim’s analysis of Marie’s problem solving activity raises two questions for me. The first regards what Marie is reflectively abstracting: the use of the generic phrase her solution activity finesses a largely unarticulated disagreement in the mathematics education community about what the nature of actions are (...) in Piaget’s theory. The second question involves the implications that radical constructivist research in mathematics has for day-to-day mathematics instruction. (shrink)

Mathematics achievement in different education systems around the world is assessed periodically in the Programme for International Student Assessment. PISA is deemed to yield robust...

Most scientific domains have guidelines concerning ethics in research. These guidelines are designed to enable researchers to conduct good research while avoiding potential harm to research participants. This paper examines the bases for ethical codes in research and discusses how these codes are operationalized in research conducted in the field of mathematics education.

How can we deal with the diversity of theories in mathematics education? This was the main question that led the authors of this book to found the Networking Theories Group. Starting from the shared assumption that the existence of different theories is a resource for mathematics education research, the authors have explored the possibilities of interactions between theories, such as contrasting, coordinating, and locally integrating them. The book explains and illustrates what it means to network theories; (...) it presents networking as a challenging but fruitful research practice and shows how the Group dealt with this challenge considering five theoretical approaches, namely the approach of Action, Production, and Communication (APC), the Theory of Didactical Situations (TDS), the Anthropological Theory of the Didactic (ATD), the approach of Abstraction in Context (AiC), and the Theory of Interest-Dense Situations (IDS). A synthetic presentation of each theory and their connections shows how the activity of networking generates questions at the theoretical, methodological and practical levels and how the work on these questions leads to both theoretical and practical progress. The core of the book consists of four new networking case studies which illustrate what exactly can be gained by this approach and what kind of difficulties might arise. (shrink)

How can we deal with the diversity of theories in mathematics education? This was the main question that led the authors of this book to found the Networking Theories Group. Starting from the shared assumption that the existence of different theories is a resource for mathematics education research, the authors have explored the possibilities of interactions between theories, such as contrasting, coordinating, and locally integrating them. The book explains and illustrates what it means to network theories; (...) it presents networking as a challenging but fruitful research practice and shows how the Group dealt with this challenge considering five theoretical approaches, namely the approach of Action, Production, and Communication (APC), the Theory of Didactical Situations (TDS), the Anthropological Theory of the Didactic (ATD), the approach of Abstraction in Context (AiC), and the Theory of Interest-Dense Situations (IDS). A synthetic presentation of each theory and their connections shows how the activity of networking generates questions at the theoretical, methodological and practical levels and how the work on these questions leads to both theoretical and practical progress. The core of the book consists of four new networking case studies which illustrate what exactly can be gained by this approach and what kind of difficulties might arise. (shrink)

Generally, research with Black youth/population in mathematics education have largely focused on their mathematics experiences in the context of United States with a special focus on issues of race/racism, equity, and social justice. While these studies are important and have informed our understanding of the schooling experiences of Black youth, there is a dearth of research with specificity on the mathematics learning experiences of Black African immigrant youth. In this paper, using duoethnography, we analyze our previous (...) projects with Black African immigrant youth/families to argue for more nuances in the theoretical and methodological lenses when engaging with this population. (shrink)

The purpose of this paper is to critically analyse and discuss the views of constructivism, on the teaching and learning of mathematics. I provide a background to the learning of mathematics as constructing and reconstructing knowledge in the form of new conceptual networks; the nature, role and possibilities of constructivism as a learning theoretical framework in Mathematics Education. I look at the major criticisms and conclude that it passes the test of a learning theoretical framework but (...) there is still a gap between theory and mathematics classroom practice. (shrink)

Current formal mathematics, being divorced from the empirical, is entirely a social construct, so that mathematical theorems are no more secure than the cultural belief in two-valued logic, incorrectly regarded as universal. Computer technology, by enhancing the ability to calculate, has put pressure on this social construct, since proof-oriented formal mathematics is awkward for computation, while computational mathematics is regarded as epistemo-logically insecure. Historically, a similar epistemological fissure between computational/practical Indian mathematics and formal/spiritual Western mathematics (...) persisted for centuries, during a dialogue of civilizations, when texts on "algorismus" and "infinitesimal" calculus were imported into Europe, enhancing the ability to calculate. It is argued here that this epistemological tension should be resolved by accepting mathematics as empirically based and fallible, and by revising accordingly the mathematics syllabus outlined by Plato. (shrink)

Current formal mathematics, being divorced from the empirical, is entirely a social construct, so that mathematical theorems are no more secure than the cultural belief in two-valued logic, incorrectly regarded as universal. Computer technology, by enhancing the ability to calculate, has put pressure on this social construct, since proof-oriented formal mathematics is awkward for computation, while computational mathematics is regarded as epistemo-logically insecure. Historically, a similar epistemological fissure between computational/practical Indian mathematics and formal/spiritual Western mathematics (...) persisted for centuries, during a dialogue of civilizations, when texts on "algorismus" and "infinitesimal" calculus were imported into Europe, enhancing the ability to calculate. It is argued here that this epistemological tension should be resolved by accepting mathematics as empirically based and fallible, and by revising accordingly the mathematics syllabus outlined by Plato. (shrink)

Journal of Philosophy of Education, EarlyView.

In lieu of an abstract, here is a brief excerpt of the content:A Wittgensteinian Approach to Discerning the Meaning of Works of Art in the Practice of Critical and Contextual Studies in Secondary Art EducationLeslie Cunliffe (bio)In order to get clear about aesthetic words you have to describe ways of living.Wittgenstein, Lectures and Conversations on Aesthetics, Psychology and Religious Belief1Language is a labyrinth of paths. You approach from one side and know your way about; you approach the same place from (...) another side and no longer know your way about.Wittgenstein, Philosophical Investigations2IntroductionThis article explores a Wittgensteinian approach for dealing with the meaning of works of art in the practice of critical and contextual studies in art education. The article is aimed at developing a methodology for eleven to nineteen year olds, known in the UK as the secondary phase of education, but the approach could be generalized to other educational phases.Henry Finch3 interprets four references to meaning found in Ludwig Wittgenstein's philosophy of language as a quadrant that combines the horizontal axis of meaning as physiognomies (the experiential and visual) and meaning as constructions (concepts, uses, science, and mathematics), with a vertical axis of meaning as imperatives (rule following, obligations, and law) and meaning as custom (ceremonies, myths, and ritual).4 The quadrant represents Finch's understanding of Wittgenstein's views about how language serves a different but complementary role in the way human beings engage [End Page 65] with meaning in the world and their place in it.5 Such meanings are embedded in social practices, including first-person responses to a physiognomy, although these start off as biologically primed reactions.Michael Luntley argues that Wittgenstein's philosophy of language and the related remarks on perception should be understood as setting conditions for the possibilities of judgment in "seeing the world aright, in taking the right attitude to the world"6 rather than simply mapping the possibilities of meaning, as the former underscores the role of will7 in the way meaning is actively witnessed and ethically processed by an agent. Having the right view is still only a view, but for Wittgenstein this amounted to an immense ethical and philosophical project to see things accurately and in correct relationship to each other. Such perspectival stances result from an agent's capacity to selectively engage with meaning, with a more unitary understanding resulting from criss-crossing the same territory from different points of view. Wittgenstein's four references to meaning can be thought of as forming four different points on a compass that direct the "self-as-will engaged with that which is independent of will "8 in a quest to see things aright. By extension, such an approach could be used in art education to enable students to develop the dispositions to see works of art aright.Marcia Eaton describes the tension between epistemic and interpretative approaches to the meaning of works of art in this way: "In which direction should one go? In other words, how can judgments which seem so personal (relative, subjective, etc.) also play a role in interpersonal (absolute, objective, etc.) language games?"9 Michael Podro gives a similar description of the apparent tension between a context-bound analysis of works of art and their irreducible and autonomous qualities as like "constantly treading a tightrope."10 Ralph Smith highlights the difference "between having knowledge about art and personally experiencing its presence and power."11Ben Tilghman12 advocates a Wittgensteinian way of going beyond these twofold approaches for understanding and interpreting art by focusing on the role that ponderable and imponderable evidence play in achieving the right stance when attempting to condense the meaning of works of art. Following Wittgenstein, Tilghman argues that a judgment based on imponderable evidence can often be supported by ponderable evidence, a process that is always linked to the relative experience, knowledge, and wisdom of the person making the judgment. Wittgenstein's four references to meaning could be thought of as points on a compass for directing the will in fittingness when endeavoring to understand the role and relationship of ponderable and imponderable evidence in any attempt to see works of art aright. But before these ideas can be... (shrink)

In the mathematics education research literature, there is a growing body of scholarship on how mathematicians practice their craft. The purpose of this chapter is to survey some of this literature and explain how it can contribute to the philosophy of mathematical practice. We first describe how mathematics educators use empirical methodologies to investigate the behaviors of mathematicians and argue that findings from these studies can inform the philosophy of mathematical practice. We then illustrate this by summarizing (...) research on mathematicians’ understandings and mathematicians’ proof reading. (shrink)

The wide range of semiotic possibility, through networks of motor communications, reveal processes for decision-making with playful meaning. We describe a physical education experience according to a sequence based on five motor games, corresponding to five networks of motor communications, with the purpose of revealing the signs to interpret a fully comprehension of playful communication. A total of 180 high school students were part of this pedagogic experience. Events were obtained through the systematic observation of three game conducts, (...) to be compared through the five games and thus offer indicators of motor complexity regarding decoding. Through the game sequencing studied, three blocks with different networks of motor communications were shown: their exclusivity and stability, their instability, and ambivalence, in addition to signs based on relationships of opposition, reidentification, and relativization. We propose that these signs should be transferred to a significant pedagogy with curricular interest for the physical education class. (shrink)

In this paper I present a picture of the social status of the Antwerp arithmetic teachers. This requires first a closer look at the Antwerp schoolmaster in general. Because no fewer than two-thirds of them were also reckoning masters most of the conclusions about teachers are also valid for reckoning masters. The consequences of the sack of Antwerp in 1585 provides us with a wealth of information about the social position of its inhabitants. The material about reckoning teachers is presented. (...) Especially important, in this period of religious turmoil and civil war, is the information concerning the religious convictions of reckoning masters. Finally, a closer look is given to the contents of the arithmetic books and the introduction of arabic numerals. (shrink)