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...

In this article, modern standards of early years mathematics education are criticized and a proposal for change is presented. Today's early years mathematics education standards rest on a view of mathematics that became obsolete already at the end of the 19th century while the spirit of children's mathematics is precisely the spirit of modern mathematics. The proposal for change is not a return to the “new mathematics” movement, but something different.

This book offers an up-to-date overview of the research on philosophy of mathematics education, one of the most important and relevant areas of theory. The contributions analyse, question, challenge, and critique the claims of mathematics education practice, policy, theory and research, offering ways forward for new and better solutions. The book poses basic questions, including: What are our aims of teaching and learning mathematics? What is mathematics anyway? How is mathematics related to society (...) in the 21st century? How do students learn mathematics? What have we learnt about mathematics teaching? Applied philosophy can help to answer these and other fundamental questions, and only through an in-depth analysis can the practice of the teaching and learning of mathematics be improved. The book addresses important themes, such as critical mathematics education, the traditional role of mathematics in schools during the current unprecedented political, social, and environmental crises, and the way in which the teaching and learning of mathematics can better serve social justice and make the world a better place for the future. (shrink)

The field of mathematics education has been fashioned by a diversity of theoretical and philosophical perspectives. The purpose of this study is to add to this field an analysis of the philosophical position of critical realism. To achieve this objective, the study addresses the following questions: what does critical realism have to offer mathematics education? How may critical realism underlabour for this discipline? In addressing these questions, the study provides an overview of the (...) basic theories and the possible weak points of and arguments against critical realism and realism in general. It then draws upon the notion of a dialectical phenomenology to provide a sequence of Achilles' heel critiques of some of the perspectives that constitute broadly the theoretical landscape of mathematics education research. This critique proceeds with an analysis of didactically oriented empiricism, hermeneutics, pragmatism, postmodernism, traditionally recognized forms of constructivism, traditionally recognized theories of activity, and ethnomathematics. The last section summarizes the implications of philosophical underlabouring for a more beneficial science of mathematics education. (shrink)

Indonesia has recently faced problems in various aspects of life. The results of a social media survey in Indonesia in early 2021 that the biggest threat to the Pancasila ideology is communism and other western ideologies. Communism has a dark history in the life of the Indonesian people. It shows the problem of thinking and philosophical views of the Indonesian people. This research is textbook research that aims to analyze philosophy books, namely mathematics education philosophy textbooks written with (...) a Western cultural background by Paul Ernest. This book was chosen as the object of analysis because this book is the essential reference textbook for the philosophy of mathematics education in universities providing prospective mathematics teachers in Indonesia. The research results show that this book contributes to developing Western thoughts that are contrary to Pancasila. It needs to eliminate the most critical thing in this book. It is the atheistic and individualistic philosophy underlying the ideology of mathematics education because this is not in line with the theistic collectivistic Pancasila. Therefore, this research recommends developing a textbook on the philosophy of mathematics education. It should be under the Pancasila ideology or written based on the Pancasila ideology. (shrink)

The traditional way of presenting mathematical knowledge is logical deduction, which implies a monolithic structure with topics in a strict hierarchical relationship. Despite many recent developments and methodical inventions in mathematics education, many curricula are still close in spirit to this hierarchical structure. However, this organisation of mathematical ideas may not be the most conducive way for learning mathematics. In this paper, we suggest that flattening curricula by developing self-contained micro topics and by providing multiple entry points (...) to knowledge by making the dependency graph of notions and subfields as sparse as possible could improve the effectiveness of teaching mathematics. We argue that a less strictly hierarchical schedule in mathematics education can decrease mathematics anxiety and can prevent students from ‘losing the thread’ somewhere in the process. This proposal implies a radical re-evaluation of standard teaching methods. As such, it parallels philosophical deconstruction. We provide two examples of how the micro topics can be implemented and consider some possible criticisms of the method. A full-scale and instantaneous change in curricula is neither feasible nor desirable. Here, we aim to change the prevalent attitude of educators by starting a conversation about the flat curriculum alternative. (shrink)

This book discusses both the philosophy of mathematics and of mathematics education. The first part is a critique of existing approaches and a new philosophy of mathematics. Chapters include: (1) "A Critique of Absolutist Philosophies of Mathematics," (2) "The Philosophy of Mathematics Reconceptualized," (3) "Social Constructivism as a Philosophy of Mathematics," (4) "Social Constructivism and Subjective Knowledge," and (5) "The Parallels of Social Constructivism." The second part of the book explores the philosophy of (...) mathematics education and shows that many aspects of mathematics education rest on underlying, usually implicit, philosophical assumptions. Making these assumptions explicit puts a critical tool into the hands of teachers and researchers. Chapters in this section are: (6) "Aims and Ideologies of Mathematics Education," (7) "Groups with Utilitarian Ideologies," (8) "Groups with Purist Ideologies," (9) "The Social Change Ideology of the Public Educators," (10) "Critical Review of Cockcroft and the National Curriculum," (11) "Hierarchy in Mathematics, Learning, Ability and Society," (12) "Mathematics, Values and Equal Opportunities," and (13) "Investigation, Problem Solving and Pedagogy." Each chapter contains notes. (Contains 654 references.) (MKR). (shrink)

This editorial, introducing the Journal for Theoretical & Marginal Mathematics Education, is historically situated in a moment when the field of mathematics education research is on the precipice of acknowledging that the old world is dying. That is to say, the way research has been done before is no longer adequate for operating within the White, Colonial, cis-hetero Patriarchal, Abled Capitalist dystopia we find ourselves. This inadequacy is, however, not a reason for despair but instead for (...) celebration. Instead of directing our resources towards reifying existing norms and structures, towards adopting exhausted methods and theories—as is the way of the old world—we can instead foster a ‘new aesthetics’ of mathematics education research, wherein we remake the very material that constitutes mathematics education research itself. In doing so, we engage with the politics of the margins of mathematics education research. We discuss the critical, philosophical, and psychoanalytic as three potential, but non-exhaustive, departures from mathematics education research as it has been. Each of these marginal approaches, in their own way, points to the ways that presuming a fixed mathematics education research enacts exclusionary politics and precludes change. Let this be our manifesto, written from the margins, for a more root-grasping, more self-reflective, and freer mathematics education research. (shrink)

It is a rather safe statement to claim that the social dimensions of the scientific process are accepted in a fair share of studies in the philosophy of science. It is a somewhat safe statement to claim that the social dimensions are now seen as an essential element in the understanding of what human cognition is and how it functions. But it would be a rather unsafe statement to claim that the social is fully accepted in the philosophy of (...) class='Hi'>mathematics. And we are not quite sure what kind of statement it is to claim that the social dimensions in theories of mathematics education are becoming more prominent, compared to the psychological dimensions. In our contribution we will focus, after a brief presentation of the above claims, on this particular domain to understand the successes and failures of the development of theories of mathematics education that focus on the social and not primarily on the psychological. (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)

The goal of this paper is to highlight the fact that the Philosophy for Children Approach can be used to stimulate pupil’s reflection within the framework of school subjects such as mathematics. First we situate P4C within the field of socio-constructivist epistemology. Then, P4C as adapted to mathematics is introduced. Finally, we describe an experiment linked to five types of exchanges, manifested between the beginning and the end of a school year while the pupils were learning to philosophize (...) about mathematics. In the discussion, emphasis is placed on the educational interventions of the teachers who facilitate the P4CM workshops. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Philosophy of Music Education Review 11.2 (2003) 141-156 [Access article in PDF] Music Education as Critical PracticeA Naturalist View Lauri Väkevä University Of Oulu, Finland I This essay defends naturalism as a framework for philosophy of music education. I have three general reasons for supporting naturalism. First, by taking naturalism seriously we can keep our philosophies up-to-date with scientific inquiry. Second, naturalism can emancipate us (...) from transcendental pseudo-questions and instead help us to deal with the empirical problems of everyday life. Third, with naturalism, we can free ourselves from a priori theories of morals and aesthetics and thus view the active part that normative and aesthetic concerns play in human praxis. Even if I endorse naturalism, I do not subscribe to scientism. 1 Rather, I submit to a pragmatist view that stresses the role of philosophy (and science) in dealing with the questions of our life practices. Of the recent perspectives in the philosophy of music education my approach comes closest to the critical praxialism discussed in the MayDay group. 2All in all, I find a lot in common with the praxialist discussion (within or outside the MayDay group) and the kind of pragmatism that follows John Dewey's mature philosophy. 3 However, it seems to me that the naturalist bent of Deweyan pragmatism has not been thoroughly analyzed in our field. 4 One reason for this [End Page 141] might be the bad report that 'hard' or reductive naturalism has been given in connection with the experimental paradigm of music education research. 5 I want to argue that the critical potential of Deweyan pragmatism should not (and cannot) be separated from a naturalist approach. Taken together, critical and naturalist views can contribute to any philosophy, especially in fields that have a long tradition of leaning on otherworldly explanations.In what follows, I begin by outlining (in broad brushstrokes) the historical position of pragmatism and its status in present day philosophy. Next, I focus on the implications of Deweyan naturalist pragmatism and the possibility of maintaining a critical pragmatist attitude with naturalist presumptions in a pedagogical context. In the final part of the essay, I outline an interpretative framework for a critical pragmatism of music education that takes advantage of the naturalizing tendencies in contemporary philosophy without losing its grasp on the things that matter musically. II For a while, pragmatism had a bad name in academic philosophy. This was especially true after the linguistic turn in the first half of the last century. 6 By making the linguistic turn, many philosophers hoped to disassociate themselves from the speculative tradition(s) once and for all. Logical analysts also made it more difficult for philosophers to ignore the findings of scientists. Thus they paved the way for contemporary reductive naturalism that takes science as the measure of what can be known. 7However, analytic philosophers were not alone in their attempt to provide philosophy with scientific rigor. Classical pragmatists, too, had integrated a naturalist penchant in their output. Embracing naturalism with an open-ended experimental spirit, they wanted to focus on "the problems of men" as the chief subject matter of philosophy. 8 To many analytic philosophers, this seemed like a return to armchair speculation with an ingredient of sentimental ethics thrown in for good measure. Worse still, it looked as though the classical pragmatists were giving in to the spirit of the corporate West: speaking of the truth as "cash-value" surely seemed like a blasphemy to intellectuals who sought from formal logic and mathematics a purity of mind to clear up the messiness of everyday experience. 9Logical analysis also had its shortcomings. While many mainstream philosophers turned to logical analysis for clarity and certainty, at the same time they submitted to Cartesian-Kantian mind-body dualism with its problematic epistemology. Thus, they kept supporting the tensions between the rationalist and empiricist or analytic and synthetic domains of knowledge that haunted modernist neo-Kantian arguments.These distinctions turned out to be easy targets for postanalytic critics. 10 [End Page 142]Classical pragmatists anticipated this... (shrink)

This essay addresses a critical epistemology of mathematics as an investigation into the epistemic limitations of mathematical thinking. After arguing for the relevance of a critical epistemology of mathematics, I discuss assumptions underlying standard arithmetic and assumptions underlying standard logic as examples for such epistemic limitations of mathematical thinking. Looking into the work of philosophically inte­res­ted scholars in mathematics education such as Alan Bishop and Ole Skovsmose, I discuss some early insights for a (...) class='Hi'>critical epistemology of mathematics. I conclude that these insights can only be the beginning, that we are yet far away from a proper under­standing of the epistemic limitations of mathematics, and that more research is needed. (shrink)

In the field of mathematics education, one of the main questions remaining under debate is whether students’ development of mathematical reasoning and problem-solving is aided more by solving tasks with given instructions or by solving them without instructions. It has been argued, that providing little or no instruction for a mathematical task generates a mathematical struggle, which can facilitate learning. This view in contrast, tasks in which routine procedures can be applied can lead to mechanical repetition with little (...) or no conceptual understanding. This study contrasts Creative Mathematical Reasoning (CMR), in which students must construct the mathematical method, with Algorithmic Reasoning (AR), in which predetermined methods and procedures on how to solve the task are given. Moreover, measures of fluid intelligence and working memory capacity are included in the analyses alongside the students’ math tracks. The results show that practicing with CMR tasks was superior to practicing with AR tasks in terms of students’ performance onpracticed test tasksandtransfer test tasks. Cognitive proficiency was shown to have an effect on students’ learning for both CMR and AR learning conditions. However, math tracks (advanced versus a more basic level) showed no significant effect. It is argued that going beyond step-by-step textbook solutions is essential and that students need to be presented with mathematical activities involving a struggle. In the CMR approach, students must focus on the relevant information in order to solve the task, and the characteristics of CMR tasks can guide students to the structural features that are critical for aiding comprehension. (shrink)

The theory of knowledge objectification, initially presented and developed by Luis Radford, has gained some traction in the field of mathematics education. As with any developing theory, its presentation contains statements that may contradict its stated intents; and these problems are exacerbated in its uptake into the work of other scholars. The purpose of this study is to articulate a Spinozist-Marxian approach, in which the objectification exists not in things—semiotic means that mediate interactions—but as real relation between people. (...) As a consequence, the concept of mediation is unnecessary and can be abandoned. A concrete classroom example from Radford’s own studies is used to exemplify and develop pertinent issues. In particular, the societal nature of the ideal—a synecdoche of relations between objects that reflect relations between people—should be added and the notion of mediation no longer is required. (shrink)

Imre Lakatos' philosophical and scientific papers are published here in two volumes. Volume I brings together his very influential but scattered papers on the philosophy of the physical sciences, and includes one important unpublished essay on the effect of Newton's scientific achievement. Volume 2 presents his work on the philosophy of mathematics (much of it unpublished), together with some critical essays on contemporary philosophers of science and some famous polemical writings on political and educational issues.

Values, moral values and democratic values are attracting the attention of education researchers in general and mathematics education researchers in particular. Little research has studied pre-service teachers’ perceptions of values in the classroom, their perceptions of the relationship between the different variables of values in the classroom, as well as their relationship with the democratic society. The present research attempts to do so. Twenty-two graduate pre-service teachers who participated in ‘New trends in mathematics education’ course (...) discussed how to cultivated values in the mathematics classroom. Moreover, they answered survey questions related to the cultivation of values in this classroom. We used a combination of deductive and inductive content analysis to characterize the pre-service teachers’ texts. The research results indicate that the pre-service teachers perceived values as encouraging students’ activity in the mathematics classroom. In addition, the pre-service teachers perceived values as encouraging specific categories of values needed as skills for the citizen in a democratic society, as creativity, critical thinking and metacognition. (shrink)

This article sheds new light on Karl Marx’s theoretical legacy to promote educational philosophy and theory in science, technology, engineering, arts, and mathematics (STEAM) education. Accordingly, this article selects Marxian thought as an exemplar case to elaborate critical pedagogy and aesthetics and culminate with an overview of the Marxian approach to the theoretical underpinnings of STEAM education. Hence, the article critically reviews Marx’s original works and earlier Marxian scholars’ contributions to extending critical pedagogy and its (...) applications to the STEAM paradigm and informational and communication technology (ICT) development. This article critiques notable structural problems of STEAM education that produce power differentials such as educational gaps and digital divides in the current age of digital capitalism. Given the context, this article refines art and communication theories through the lens of his biography and writings. Additionally, this article incorporates the Marxian viewpoints about reproduction and aesthetic literacy into STEAM education. Overall, this article reviews the theoretical legacy of Marx and his essential critical and pedagogical theory, calling on scholars to pay close attention to re-orienting STEAM pedagogy as an emerging multidisciplinary field. (shrink)

Purpose: One of my goals in the paper is to investigate why realists reject radical constructivism (RC) as well as social constructivism (SC) out of hand. I shall do this by means of commenting on Peter Slezak’s critical paper, Radical Constructivism: Epistemology, Education and Dynamite. My other goal is to explore why realists condemn the use of RC and SC in science and mathematics education for no stated reason, again by means of commenting on Slezak’s paper. (...) Method: I restrict my comments to Slezak’s paper and leave it to the reader to judge which, if any, of the reasons that I advance for these two states of affairs are not specific to Slezak’s paper. Other readers might not agree with my interpretations of Slezak’s paper, including Slezak himself, but I offer them after having worked with von Glasersfeld in interdisciplinary research in mathematics education for over 25 years. Findings: My findings are that Slezak: (1) rejects RC and SC on the basis of unjustified criticisms, (2) does not explore basic tenets of RC nor of SC beyond the unjustified criticisms, (3) rejects how SC and RC have been used in science and mathematics education, based at least in part on the unjustified criticisms, (3) dislikes how SC has been used in science and mathematics education, a dislike that fuels his rejection of any constructivism, and (4) doesn’t explore how RC has been used in scientific investigations in mathematics education. On the basis of these findings, I conclude that how epistemological models of knowing might be used in science or mathematics education would be better left to the educators who use them in interdisciplinary work. (shrink)

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)

Joseph Agassi is known primarily among fellow academics as an exemplary historian and philosopher of science; an ardent critic and disciple of Karl Popper; a critical admirer of the work of Michael Polanyi; and a Socratic fly with the “sting of a bee” for all those who wear the intellectual fashions of the day. To most of Agassi’s students he is known primarily as an exemplary model of the Socratic teacher. The question of most urgency for educators today who (...) care about the intellectual development of students is: How do we make ready our educational institutions for more Socratic teachers? The philosophical or theoretical question is: Why do we want Socratic teachers? In outline, of the many of Agassi’s educational essays selected for this book, Agassi answers those questions: authoritarianism (or anti-democracy) blocks the democratic reform of educational institutions where Socratic teachers and students could find a safe haven; and, Socratic teaching is the main anti-dote to authoritarianism. The removal of authoritarianism from education also removes the hazard that education has become to students; to their happiness, creativity, and dignity as autonomous individuals. See Less Copyright Year: 2014 E-Book (PDF) Availability: Published ISBN: 978-94-6209-626-4 Publication date: 01 Jan 2014 Hardback Availability: Published ISBN: 978-94-6209-624-0 Publication date: 01 Jan 2014 Paperback Availability: Published ISBN: 978-94-6209-623-3 Publication date: 01 Jan 2014 Login via Institution Prices from (excl. shipping): -/- $56.00 -/- ​ Available formats Add to Cart View PDF Flyer Get Permissions. (shrink)

Along with critical thinking, collaboration and communication, creativity is considered a crucial skill to prepare students for uncertain societal challenges and future jobs in the twenty-first century. Therefore, it is not enough to just encourage creativity in education, but it is also important to assess it, because assessing creativity helps to recognise and understand students' creative abilities. In this paper, we focus on mathematical creativity and link it to general definitions of creativity. We thoroughly investigate and analyse methods (...) that foster mathematical creativity and enable its assessment, highlighting innovations that result from the application of these methods. The methods we consider focus on approaches to problem posing and problem solving, including open-ended tasks. The results show that these methods not only enhance students' mathematical creativity, but also contribute to more connected mathematical knowledge and improve problem-solving skills. (shrink)

First published in 1991, _In Praise of Cognitive Emotions_ comprises fourteen of Scheffler's most recent essays - all of which challenge contemporary notions of education and rationality. While defending the ideal of rationality, he insists that rationality not be identified with a mental faculty or a mechanism of inference but taken rather as the capactity to grasp principles and purposes and to evaluate them in the light of relevant reasons. Examining a broad range of issues - from computers in (...) school to math education, from metaphor to morality - these essays are unified by Scheffler's conviction of the primacy of critical thought in education. Scheffler is especially concerned to promote a broad interpretation of rationality to counteract the narrowing of vision accompanying the technological revolution now sweeping education. Addressing three specific areas of curriculum, the work offers a critique of computer applications to education, develops a notion of strategic rationality in understanding mathematical reasoning, and, contrary to prevalent notions of moral education, connects reason with care, thus emphasizing the intimate connection between emotion and reason and challenging the dominant perception of the two as oppositional. (shrink)

Globalisation is a contested concept and therefore refers to contradictory social processes. Additionally, immigration is established alongside the dualities of globalisation and ghettoisation. This paper is theoretically driven by doctoral research data and aims to present concerns and possibilities for critical mathematics education regarding immigrants in the context of the challenges posed by globalisation and ghettoisation in a world impacted by socially and racially structured injustices. The following topics will be addressed: (1) Firstly, I will clarify what (...) this paper refers to regarding globalisation and ghettoisation and discuss how these concepts relate to mathematics education and immigrant students. (2) Next, I will present the participants' reports in the voices of mathematics teachers from schools with immigrant students in São Paulo, Brazil. (3) Following this, I will engage in a theoretical discussion on "meeting amongst differences" in mathematics classes with immigrant students. (4) Finally, I will explore, through Pedagogical Imagination, the hypothetical project "Ghettoisation—Is Racism a Factor?" as a possibility, considering the reality of immigrant students in developing spaces for meeting to overcome the distances created by ghettoisation. (shrink)

An Archaeology of Educational Evaluation: Epistemological Spaces and Political Paradoxes outlines the epistemology of the theories and models that are currently employed to evaluate educational systems, education policy, educational professionals and students learning. It discusses how those theories and models find their epistemological conditions of possibility in a specific set of conceptual transferences from mathematics and statistics, political economy, biology and the study of language. The book critically engages with the epistemic dimension of contemporary educational evaluation and is (...) of theoretical and methodological interest. It uses Foucauldian archaeology as a problematizing method of inquiry within the wider framework of governmentality studies. It goes beyond a mere critique of the contemporary obsession for evaluation and attempts to replace it with the opening of a free space where the search for a mode of being, acting and thinking in education is not over-determined by the tyranny of improvement. This book will appeal to academics, researchers and post-graduate students in the fields of educational philosophy, education policy and social science. (shrink)

Imre Lakatos' philosophical and scientific papers are published here in two volumes. Volume I brings together his very influential but scattered papers on the philosophy of the physical sciences, and includes one important unpublished essay on the effect of Newton's scientific achievement. Volume 2 presents his work on the philosophy of mathematics, together with some critical essays on contemporary philosophers of science and some famous polemical writings on political and educational issues.

This paper explores the nature of the language used when teaching mathematics to young children. It proposes that an important part of the teaching of a mathematical concept is the introduction of specific terminology. Children may need to be taught new meanings for already familiar words. The timing of these introductions to new words or meanings is critical to their understanding of the concepts being taught. It will be argued that there are two aspects of the children's learning (...) that need to be considered. First, their understanding of the concept being introduced, and secondly, their learning the appropriate word to describe that concept. By assessing children's understanding of new mathematical concepts through their own use of the terminology, the teacher can then negotiate new meanings with them through practical experiences, introducing new word meanings only when the concepts have been understood. (shrink)

The critique of philosophical foundations of neoclassical economics is significant, because of its hegemony on economic education and research programs in Iran and worldwide academies. Due to an epistemological fallacy, methodological individualism plays a prominent role in the philosophy of economic; since the ontological aspects of economy are reduced to methodological considerations. Accordingly, critique of methodological individualism is regarded as the main entry for philosophical analysis of neoclassical economics. This article aims to analyze and appraise the methodological individualism from (...) critical realism’s point of view. Critical realism is taken as a stand for criticizing methodological individualism, because critical realism and neoclassical economics as a positive approach to economics, both share a realistic worldview. The critical realism’ critique of methodological individualism is bases on analyzing the concepts of formalism, syllogism, social atomism, isolation and reduction. From critical realism’ point of view because of unrealistic postulations of methodological individualism, neoclassical economics is turned to an abstract and deductive science such as mathematics with a low potentiality of explanation. (shrink)

One of the criticisms of standard teaching practices is that they support merely ‘ritual’ as opposed to ‘principled’ knowledge, that is, knowledge which is procedural rather than being founded on principled explanation. This paper addresses issues and assumptions in current debate concerning the nature of mathematical knowledge, focusing on the ritual/principle distinction. Taking a discussion of centralism in logic and mathematics as its start-point, it seeks to resolve these issues through an examination of mathematics as a community of (...) practice and the teacher's role as epistemological authority in inducting pupils into such practices. (shrink)

Abstract In the present article, we provide a critical overview of the emerging field of ‘neuroeducation’ also frequently referred to as ‘mind, brain and education’ or ‘educational neuroscience’. We describe the growing energy behind linking education and neuroscience in an effort to improve learning and instruction. We explore reasons behind such drives for interdisciplinary research. Reviewing some of the key advances in neuroscientific studies that have come to bear on neuroeducation, we discuss recent evidence on the brain (...) circuits underlying reading, mathematical abilities as well as the potential to use neuroscience to design training programs of neurocognitive functions, such as working memory, that are expected to have effects on overall brain function. Throughout this review we describe how such research can enrich our understanding of the acquisition of academic skills. Furthermore, we discuss the potential for modern brain imaging methods to serve as diagnostic tools as well as measures of the effects of educational interventions. Throughout this discussion, we draw attention to limitations of the available evidence and propose future avenues for research. We also discuss the challenges that face this growing discipline. Specifically, we draw attention to unrealistic expectations for the immediate impact of neuroscience on education, methodological difficulties, and lack of interdisciplinary training, which results in poor communication between educators and neuroscientists. We point out that there should be bi-directional and reciprocal interactions between both disciplines of neuroscience and education, in which research originating from each of these traditions is considered to be compelling in its own right. While there are many obstacles that lie in the way of a productive field of neuroeducation, we contend that there is much reason to be optimistic and that the groundwork has been laid to advance this field in earnest. Content Type Journal Article Category Original Paper Pages 1-13 DOI 10.1007/s12152-011-9119-3 Authors Daniel Ansari, Numerical Cognition Laboratory, Department of Psychology, The University of Western Ontario, Westminster Hall, London, ON N6A 3K7, Canada Bert De Smedt, Parenting and Special Education Research Group, Katholieke Universiteit Leuven, Leuven, Belgium Roland H. Grabner, Institute for Behavioral Sciences, Swiss Federal Institute of Technology (ETH) Zurich, Zurich, Switzerland Journal Neuroethics Online ISSN 1874-5504 Print ISSN 1874-5490. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Method and Mathematics:Peter Ramus's Histories of the SciencesRobert GouldingPeter Ramus (1515–72) was, at first sight, the least likely person to write an influential history of mathematics. For one thing, he was clearly no great mathematician himself. His sympathetic biographer Nicholas Nancel related that Ramus would spend the mornings being coached in mathematics by a team of experts he had assembled, and in the afternoon would lecture (...) on the very same subjects.1 But there can be no doubt that Ramus held mathematics in particular esteem and even played a crucial role in linking philosophical discourse to mathematics and in promoting the use of quadrivial reasoning in the study of the natural world.2The origins for his enthusiasm for mathematics are to be found in his account of the nature of the arts and critique of the curriculum of the universities. From his earliest career, Ramus was a logician and remained one in all his works, whether writing on mathematics or Virgil, and he conceived [End Page 63] of all the arts, especially mathematics, as unchanging structures of necessarily true propositions—not prima facie a position which lends itself to notions of historical change and development.3 His career as a historian of mathematics, I will argue, was directed by problems that arose in his theoretical understanding of mathematics as he began to immerse himself in the sciences of the ancient world.Ramus and the Reform of DialecticRamus set out his fundamental logical positions in his very first printed work, the Dialecticae institutiones (Education in Dialectic) of 1543, a contribution to the ongoing humanist attack on scholastic logic, in which Ramus rehearsed many of the commonplace criticisms of the university dialectic. Humanists complained that logic no longer concerned itself with real human reasoning; instead, it had become a discipline studied for its own sake, using its own incomprehensible jargon, and was of no practical interest. The new humanist dialectics of Lorenzo Valla and Rudolph Agricola attempted to teach the kind of practical reasoning useful for composing a speech or letter, and they borrowed extensively from rhetorical works of Cicero and Quintilian to develop a highly rhetoricized logic. Questions about the formal validity of arguments were of little interest; what mattered was whether the arguments were persuasive.4In his Dialecticae institutiones, Ramus took this humanist reformulation of dialectic a step further. Dialectic—or, in fact, any art or science—consisted of nature, doctrine, and exercise. The natural workings of the mind formed the most significant element. Doctrine was nothing more than a record of natural reasoning; in importance it paled next to nature and practice.5 The logic of the universities bore no resemblance to the true, natural dialectic, as he argued at length in the companion volume to the Institutiones, the innocuously titled but exuberantly offensive Aristotelicae animadversiones [End Page 64] (Observations on Aristotle).6 The question, then, remained: how can we gain access to this "natural reasoning"?Ramus's answer was surprising. He asks us to find a group of men—not scholars, but completely uneducated vineyard workers. Question them about the coming year: the fertility of the soil, the quality and quantity of the crop. "And then," he writes "from their minds as from a mirror an image of nature will be reflected."7 In the reasoned replies of these uneducated men, says Ramus, we discover every part of logic needed for any purpose, whether everyday discourse or composition of poetry: the invention of arguments, the assessment of their truth and their proper and orderly presentation. Other humanists had praised man's natural logical faculties, elevating them over artificial technique, but Ramus was the first to look beyond the walls of the university and the writings of the ancients to find natural dialectic at work.If even the uneducated have some possession of the arts, then the arts taught at the university should do no more than clear away the misleading junk in the mind, and allow its natural clarity to shine through.8 Logic should be easy to learn—not because this is a pedagogic ideal, but from the very nature of the art. And if logic as it is... (shrink)

A volume in the series The Montana Mathematics Enthusiast: Monograph Series in Mathematics Education. Series Editor: Bharath Sriraman, The University of Montana The title of the book is Critique as Uncertainty. Thus Ole Skovsmose sees uncertainty as an important feature of any critical approach. He does not assume the existence of any blue prints for social and political improvements, nor that certain theoretical structures can provide solid foundations for a critical activities. For him critique is (...) an open and uncertain activity. This also applies to critical mathematics education. Critique as Uncertainty includes papers Ole Skovsmose already has published as well as some newly written chapters. The book addresses issues about: landscapes of investigations, students' foregrounds, mathematics education and democracy, mathematics and power. Finally it expresses concerns of a critical mathematics education. (shrink)

Harvey Siegel has argued that successful critical thinking requires both a critical mindset and expertise in evaluating reasons. In this article, I focus on the latter, the business of accurately appraising competing reasons in the practical domain. More specifically, I critically examine the limitations inherent in two commonly used epistemic frameworks: the balance account and reasons-for-and-against. I argue neither is suitable for scenarios requiring especially nuanced appraisals or complex metrics. This is because they primarily deal with somewhat crude (...) approximations and/or binary choices. I will propose a new framework to address this concern. I dub this the Stress-Testing-Critique-Support-Explanationist Account (STCSEA). STCSEA is engineered to handle complex practical epistemic situations, ones where determining what to do is not straightforward or immediately calculable. To fully understand the accurate and reliable determination of competing reasons’ strengths and weaknesses, educators require sophisticated tools. Such tools can help us fully to grasp the concept and mechanics of rational defeat—that being figuring out the strength of competing reasons. To do so, we must also formulate and continuously refine an operationalized defeasibility taxonomy.I proceed as follows. I begin by providing a brief overview of the critical thinking debate. I then survey John Pollock’s work on defeaters. This work is germane to the conversation yet remains something of a lacuna in the contemporary philosophy of education literature. After this, I discuss the balance account and reasons-for-and-against, specifically their evident limitations. I then make a case for the superiority of STCSEA when it comes to handling ‘hard cases’. These are cases where what we should, must, or may do necessitates a complex computation. Such computations often exceed the mathematical or epistemic resources associated with ‘balancing metaphors’ and ‘for and against’ determinations. (shrink)

In this article, we review approaches to modeling a connection between spatial and mathematical thinking across development. We critically evaluate the strengths and weaknesses of factor analyses, meta-analyses, and experimental literatures. We examine those studies that set out to describe the nature and number of spatial and mathematical skills and specific connections between these abilities, especially those that included children as participants. We also find evidence of strong spatial-mathematical connections and transfer from spatial interventions to mathematical understanding. Finally, we map (...) out the kinds of studies that could enhance our understanding of the mechanisms by which spatial and mathematical processing are connected and the principles by which mathematical outcomes could be enhanced through spatial training in educational settings. (shrink)

Recently within the critical accounting literature Funnell (1998) has argued that accounting was implicated in the Holocaust. This charge is primarily related to the technical, mathematical nature of accounting and its ability to dehumanise individuals. Broadbent (1998, see also DeMoss and McCann, 1997) has also contended that "accounting logic" excludes emotion. She suggests that a more emancipatory form of accounting could be possible if emotion were given a voice and allowed to be heard within accounting discourse (see also Kjonstad (...) and Wilmott, 1995). This paper contends that emotion should be introduced into accounting education and in particular emotional commitment to other individuals should be encouraged. It is suggested that one way to do this may be through business ethics education. It is also suggested that increasing ethical commitment to other individuals may go some way towards combating the tendency for accountancy to dehumanise other people. While there have been specific studies of ethics and accounting education there has, as yet, been little open debate about what the objectives of accounting ethics education should be or the specific techniques that could be used to meet the desired aims. This paper contends that accountancy has become dangerously dehumanised and that one of the most important objectives for any business ethics education must be to develop an empathy with "the other". The paper studies the developments within the medical, legal and engineering profession in order to suggest some specific methods which could be employed in order to re-humanise accountancy and develop a sense of moral commitment towards other individuals. (shrink)

There has been little overt discussion of the experimental philosophy of logic or mathematics. So it may be tempting to assume that application of the methods of experimental philosophy to these areas is impractical or unavailing. This assumption is undercut by three trends in recent research: a renewed interest in historical antecedents of experimental philosophy in philosophical logic; a “practice turn” in the philosophies of mathematics and logic; and philosophical interest in a substantial body of work in adjacent (...) disciplines, such as the psychology of reasoning and mathematics education. This introduction offers a snapshot of each trend and addresses how they intersect with some of the standard criticisms of experimental philosophy. It also briefly summarizes the specific contribution of the other chapters of this book. (shrink)

Using educational robots to integrate computational thinking with cross-disciplinary content has gone beyond Science, Technology, Engineering, and Mathematics, to include foreign-language learning and further cross-context target-language acquisition. Such integration must not solely emphasise CT problem-solving skills. Rather, it must provide students with interactive learning to support their target-language interaction while reducing potential TL anxiety. This study aimed to validate the effects of the proposed method of pair programming along with question-and-response interaction in a board-game activity on young learners’ CT (...) skills and TL learning across contexts. Two Grade 6 classes, one with 16 students who were studying Chinese as a Second Language and the other with 16 students who were studying English as a Foreign Language, participated in the activity. A series of instruments on achievement assessment, questionnaires on CT skills and TL anxiety, and sequential learning behaviour analysis were used to critically examine the results. The main conclusion is that the EFL group showed better social skills of cooperation on CT and lower TL learning anxiety, while the CSL group demonstrated better problem-solving skills in CT, but presented more behaviours of trial-and-error loops. Results not only contribute suggestions for cross-disciplinary learning but also provide support for cross-context instruction beyond educational coursework. (shrink)

This dissertation concerns Plato's theory of education and the problem of how one can actually acquire knowledge of the Forms. Plato's theory of education aims to make one a good person, which requires knowledge of the Form of the Good. Yet, how exactly one would acquire such knowledge has remained a mystery. Various models of learning are presented by Plato: elenctic refutation ; hypothesis; recollection; the mathematical, dialectical, and political studies of the Republic's curriculum; and diairesis to name (...) just those that are explicit. I have argued here that as rich as these learning strategies are we need to augment our examination of Socratic pedagogy with a full exegesis of Platonic epistemology. ;In order finally to appreciate how they, Socratic education and Platonic epistemology, are meaningfully inseparable, I have explored the following questions: what kind of teacher Socrates is and the way in which he struggles against the sophistic methods of teaching; why Plato has Socrates include ten years of mathematical study in the ideal philosophical curriculum and how we can take the Republic's paradoxical curriculum seriously; how the philosopher utilizes the mathematical training in coming to know a Platonic Form; what role opposites play in Plato's metaphysical imagination; and what kind of person would one be if one actually knew the Form of the Good. I have shown that if one were to partake of a Socratic education, to pursue the requisite mathematical training, and to arrive finally at knowledge of the Platonic Forms, then one would be the sort of person that Socrates has in mind when he articulates the principles of Socratic intellectualism. ;What has been novel about this approach is its demonstration of how, despite the fact that Socratic learning and Platonic epistemology have separately been the object of perennial scholarly attention, the two taken genuinely together have been in need of fresh critical engagement. This inquiry reveals the metaphysical constraints on knowledge and in so doing reconciles Plato's theory of knowledge with Socrates' style of educating potential philosophers. (shrink)

In this article, Kurt Stemhagen reconstructs mathematics education in light of Dewey's democratic theory and his ideas about mathematics and mathematics education. The resulting democratic philosophy and pedagogy of mathematics education emphasizes agency and the connections between mathematics and students' social experiences. Stemhagen considers questions about the disconnect between constructivist reformers and critical mathematics educators, and he positions Dewey's ideas as a way to draw on the best of both to (...) create an active and more democratic school math experience. (shrink)

The present work presents partial results of our research, entitled: “This also works: the valuation of alternative paths to conventional algorithms for solving arithmetic problems in the classroom”. The research aims to understand how an arithmetic teaching practice in the early years of elementary school that emphasizes alternative algorithms, as opposed to conventional ones, contributes to mathematics education in a more dialogic and critical perspective. The theoretical support used is Critical Mathematics Education. It is (...) a qualitative research in which we will investigate our own pedagogical practice as an educator of children in the Elementary School, through instruments such as participant observation, field notes, conversation circles, children's productions and didactic material. The data will be analyzed with a lens of the mentioned theories, in search of units or categories. It is expected that the research brings some contributions of a teaching practice based on valuing alternative algorithms. Currently, the literature review and the design of the theoretical and methodological framework for the research have been completed, and some data have already been collected and analyzed. For this study, we will bring 1) an overview of the research, with elements of the theoretical and methodological framework of the research and 2) the literature review carried out, with the categories found. (shrink)

The article presents the theoretical construction of the relationship between mathematics learning and philosophy for/with children. The selection of a documentary corpus allowed delimiting the object of study. Then, thematic and analytical cards were elaborated to interpret the main statements, in which divergences and convergences between philosophy for/with children and the learning of mathematics were evidenced. The text is organized in six sections: Mathematical thinking as experience; Philosophy for/with children: a space for thinking; Reasoning, argumentation and logical thinking: (...) a matter of philosophy for/with children and mathematics; the role of the question in mathematical thinking; the community of inquiry: a matter of mathematics; and, finally, from concept to mathematical thinking. The article gives an account of the reflection on the need to link the learning of mathematics with the exercise of philosophizing in order to democratize mathematics. This leads to the formation of critical, creative and ethical thinking, fundamental in the learning processes of mathematics, which becomes a pedagogical experience and a didactic bet from critical mathematics education. In conclusion, the interaction between these two disciplines transforms pedagogical practices and school contexts. The classroom becomes a space for inquiry, wonder, creation, reasoning, modeling, argumentation, training and individual and collective transformation. The learning of mathematics encompasses the need to emancipate the subjects in order to solve social injustices from the contexts, to promulgate equality and democracy through the learning of mathematics. (shrink)

The increased distribution of fake news on internet and social media raises concerns for democratic processes. Sometimes, argumentation in deceptive information is built on numbers, which gives reason to include mathematics when working with fake news in education. In this paper, I suggest a tool to facilitate students’ critical thinking related to numbers, or other mathematical representations, presented in the media. It may not be straight forward, or even possible, to judge the validity of presented numbers, or (...) whether numbers are used with the intention to deceive. Complex topics are associated with uncertainty, which implies that numbers may be questionable without hidden intentions, and that evaluating a number’s relevance may be quite challenging. The developed tool consists of a set of questions to help reflecting on the validity of numbers, which again is developed into categories reflecting degrees of validity and whether the mathematical representation has a deceptive role. The categories are illustrated with examples from the media and from a classroom situation in teacher education to indicate how the tool can help raise critical questions. The developed categories are based on academic literature on fake news, typographies of uncertainty and on critical mathematics education. (shrink)

This article argues that children represent one vanguard of an emergent shift in Western subjectivity, and that adult-child dialogue, especially in the context of schooling, is a key locus for the epistemological change that implies. Following Herbert Marcuse's invocation of a ‘new sensibility’, the author argues that the evolutionary phenomenon of neoteny—the long formative period of human childhood and the pedomorphic character of humans across the life cycle—makes of the adult-collective of school a primary site for the reconstruction of belief. (...) After exploring child-adult dialogue more broadly as a form of dialectical interaction between what Dewey called ‘impulse’ and ‘habit’, three key dimensions of dialogic schooling are identified, all of which are grounded in a fourth: the form of dialogical group discourse called community of philosophical inquiry (CPI), which is based on the problematisation and reconstruction of concepts through critical argumentation. As a discourse-model, CPI grounds practice in all of the dialogic school's emergent curricular spaces, whether science, mathematics, literature, art, or philosophy. Second, it opens a functional space for shared decision-making and collaborative governance, making of school an exemplary model of direct democracy. Finally, CPI as a site for critical interrogation of concepts encountered in the curriculum (e.g. ‘alive’, ‘justice’, ‘system’, ‘biosphere’) and as a site for democratic governance leads naturally to expression in activist projects that model an emergent ‘new reality principle’ through concrete solutions to practical problems on local and global levels. (shrink)

Purpose: This study aimed to explore the relationship between mathematic achievement and programming self-efficacy, and adopt a mediation model to verify the mediating role of creativity on the relationship between mathematic achievement and programming self-efficacy.Methods: A total of 950 upper-secondary school students were surveyed using their math test scores, the Kirton Adaption-Innovation and the Programmed Self-Efficacy Scale. SPSS-26 was used for descriptive statistical analysis and correlation analysis of related variables. The PROCESS plugin was used to test the mediating effect of (...) creativity.Results: Mathematic achievement has a positive effect on programming self-efficacy, mathematic achievement is positively related to creativity, and creativity also has a positive influence on programming self-efficacy. Creativity has a mediating effect on the relationship between mathematic achievement and programming self-efficacy.Conclusion: The results revealed that mathematic achievement affected programming self-efficacy directly and also indirectly through creativity. This provided certain ideas for the development of programming education for teenagers. Since students’ mathematics learning and creativity are related to programming learning, it is necessary to pay attention to the integration of the disciplines of programming education and mathematics. Further, the cultivation of innovative thinking is also critical to facilitate programming learning. (shrink)

The vision of integrating artificial intelligence in education is part of an ongoing push for harnessing digital solutions to improve teaching and learning. Drawing from Jasanoff and Hasse, this paper deliberates on how sociotechnical imaginaries are interrelated to the implications of new technologies, such as AI, in education. Complicating Hasses’s call for the development of Socratic ignorance to consider our predispositions about new technologies and open new prospects of thought, this paper revisits postphenomenology and Feenberg’s critical constructivist (...) theories. While embracing the notion of Socratic ignorance, this paper stresses the importance of developing a nuanced understanding of technology that realizes its lack of neutrality and supports the creation of a deeper understanding of how knowledge is produced, deployed, and interpreted in the digital age. Thus, this paper argues that an amalgam of Hasse’s call for advancing Socratic ignorance combined with postphenomenology and critical constructivism can support students in developing a critical understanding of technology and opening new landscapes of imaginaries. (shrink)

Like other science, technology, engineering, and mathematics fields, ecological research needs ethics. Given the rapid pace of technological developments and social change, it is important for scientists to have the vocabulary and critical-thinking skills necessary to identify, analyze, and communicate the ethical issues generated by the research and practices within their fields of specialization. The goal of introducing ethics education for ecological researchers would be to promote a discipline in which scientists are willing and able to engage (...) in ethical questions and problem solving, even if they do so inadequately at first. Practicing ecologists ought to be able to identify and critically evaluate the ethical dimensions of their field studies because ecologists are at the forefront of important interfaces between humans and other-than-human organisms and natural systems. They are among the first to identify the impact of anthropogenic changes to the environments. Rapidly changing local and global environments mean that ecologists will be on the front line of any efforts to create a sustainable lifestyle for humans on this planet. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Kant’s Critique of the Power of Judgment: Critical EssaysTed KinnamanPaul Guyer, editor. Kant’s Critique of the Power of Judgment: Critical Essays. Lanham, MD: Rowman & Littlefield, 2003. Pp. xxiii + 253. Cloth, $75.95. Paper, $27.95.The volume under review is a collection of essays on a wide range of topics concerning Kant's Critique of the Power of Judgment. All the papers included here have been published previously, (...) although many of them in places where they might not be easily accessible to English-speaking philosophers. While some of the articles will be familiar to Kant scholars, this collection should be useful to specialists and valuable to students first delving into the third Critique.One essay in the volume, Guyer's own piece on "Kant's Principles of Reflecting Judgment," will be of particular interest to Kant scholars. Guyer's thesis here is that the account of reflecting judgment Kant offers in the Critique of the Power of Judgment is both continuous with the account of regulative principles of reason in the Critique of Pure Reason and the basis of the unity of the third Critique as a whole. He supports this by demonstrating how each of the sorts of judgment discussed in the work can be seen as an application of the broader account of reflecting judgment in general. Guyer identifies five such applications, and shows that each of them has three features central to regulative principles: they set a goal of constructing a system within a certain specific range of cognitions; they require the suitability of objects for attainment of the goal in question, and thus are transcendental [End Page 499] and not merely logical principles; and they supply heuristic advice for the investigation of nature. Thus every form of judgment can with justice be considered as an instance of the general activity of reflecting, that is, finding a universal for a given particular.Several of the essays included here concern the most prominent issues relating to the Analytic of the Beautiful. Nick Zangwill's piece deals with the disinterestedness of aesthetic judgment, while Christopher Janaway's "Kant's Aesthetics and the Empty Cognitive Stock" debunks what he sees as misguided readings of the role of concepts in judgments of taste. In "The Idealism of Purposiveness," an excerpt from his 1993 book Kantian Aesthetics Pursued (Edinburgh, 1993), Anthony Savile criticizes what he calls the "established reading" of Kant's claim that beautiful objects exhibit "purposiveness without purpose." On this reading, to say that a thing is purposive without purpose is to say that it appears to be designed, but to no particular purpose. Savile argues instead that we should take purposiveness without purpose to indicate that an object is suitable for a function, but not as a result of conscious design. Articles by Eva Schaper (from her well-known book on Kant's aesthetics) and Malcolm Budd discuss the distinction between free and dependent beauty and the account of the sublime, respectively.Some of the other selections in the volume are interesting for their focus on somewhat neglected aspects of the third Critique. One such aspect is the role of the imagination in Kant's account of aesthetic appreciation, and by extension in cognition more generally. In "Imaginative Freedom and the German Enlightenment," Jane Kneller argues that appreciation of the role of imaginative freedom in Kant's account of aesthetic judgment helps to connect Kant to the Enlightenment rationalist tradition that saw the value of art in its ability to educate and improve those who appreciate it. This education cannot, of course, be of the most straightforward nature, since Kant emphatically argues that to judge a thing beautiful is altogether different from judging it morally good. Instead, Kneller argues, our awareness of our own imaginative freedom provides us with evidence of our ability to realize the "highest good" on earth, evidence otherwise denied us in Kant's philosophy. Also addressing the theme of imagination is Donald Crawford's "Kant's Theory of Creative Imagination," in which he points out striking parallels between Kant's account of the imagination's role in mathematics and in the creation of fine art.Another aspect of Kant... (shrink)