Innovations in Evidence and Proof' brings together leading scholars and law teachers from the US, Australia, Canada, South Africa, and the UK to explore the latest developments in evidence scholarship.--Résumé de l'éditeur.

Collectively these essays inform educators and researchers at different grade levels about the teaching and learning of proof at each level and, thus, help ...

This book's activities highlight the important cycle of exploration, conjecture, and justification in all five mathematical strands. Students recognize patterns and make conjectures, learn the value of a counterexample, explore the strengths and weaknesses of visual proofs, discover the power of algebraic representations, and learn that theoretical approaches can substantiate empirical results. The supplemental CD-ROM features interactive electronic activities, master copies of activity pages for students, and additional readings for teachers. --publisher description.

Based on extensive research conducted by the authors, Reasoning and Sense Making in the Mathematics Classroom, Pre-K-Grade 2, is designed to help classroom teachers understand, monitor, and guide the development of students' reasoning and sense making about core ideas in elementary school mathematics. It describes and illustrates the nature of these skills using classroom vignettes and actual student work in conjunction with instructional tasks and learning progressions to show how reasoning and sense making develop and how instruction can support students (...) in that development. Students who can make sense of mathematical ideas can apply those ideas in problem solving, even in unfamiliar situations, and can use them as a foundation for future learning. Without them, students are reduced to rote learning, often experiencing frustration and failure. But what do reasoning and sense making during learning and teaching look like? Each chapter of Reasoning and Sense Making in the Mathematics Classroom, Pre-K-Grade 2 explores a different topic that young children encounter in mathematics, demonstrating with actual student work and classroom dialogue how their mathematical knowledge and reasoning ability move through "levels of sophistication" or learning progressions: After opening with a discussion of the nature of reasoning and sense making and their critical importance in developing mathematical thinking, chapter 1 examines how young students attempt to make sense of the concepts of place value and length measurement. Chapter 2 focuses on how early childhood instruction can engage students in mathematical reasoning while helping them construct a rich sense of number and operations. Chapter 3 identifies core algebraic ideas and shows how students can engage with these ideas in ways that not only deepen their understanding of arithmetic but also lays the foundation for the future study of algebra. Children's reasoning and sense making as they decompose and compose geometric shapes--including reasoning about area--is examined in chapter 4. The use of learning progressions to understand students' reasoning and to guide their sense making with appropriate teaching is also discussed. Not just a theoretical discussion, the book also provides specific suggestions for related instructional activities for each topic. Supplementary online resources can be accessed at NCTM's More4U website. Reasoning and Sense Making in the Mathematics Classroom, Pre-K-Grade 2 will be a valuable and practical addition to your professional library. (shrink)

Upshot: This response focuses on the following issues, which summarize the points made by the commentaries: (i) further reflection on and details of the methodological framework that was applied to studying the proposed design of our innovation course, (ii) the issue of generalizability of the findings for teaching innovation (in this context the question of generic or transferable skills will become central), and (iii) finally, more precise explanation of what we mean by “learning from the future as it emerges.”.

Structural proof theory is a branch of logic that studies the general structure and properties of logical and mathematical proofs. This book is both a concise introduction to the central results and methods of structural proof theory, and a work of research that will be of interest to specialists. The book is designed to be used by students of philosophy, mathematics and computer science. The book contains a wealth of results on proof-theoretical systems, including extensions (...) of such systems from logic to mathematics, and on the connection between the two main forms of structural proof theory - natural deduction and sequent calculus. The authors emphasize the computational content of logical results. A special feature of the volume is a computerized system for developing proofs interactively, downloadable from the web and regularly updated. (shrink)

Proof theory began in the 1920s as a part of Hilbert’s program, which aimed to secure the foundations of mathematics by modeling infinitary mathematics with formal axiomatic systems and proving those systems consistent using restricted, finitary means. The program thus viewed mathematics as a system of reasoning with precise linguistic norms, governed by rules that can be described and studied in concrete terms. Today such a viewpoint has applications in mathematics, computer science, and the philosophy of mathematics.

Apologies. The purpose of the following talk is to give an overview of the present state of aims, methods and results in Pure Proof Theory. Shortage of time forces me to concentrate on my very personal views. This entails that I will emphasize the work which I know best, i.e., work that has been done in the triangle Stanford, Munich and Münster. I am of course well aware that there are as important results coming from outside this triangle (...) and I apologize for not displaying these results as well.Moreover the audience should be aware that in some points I have to oversimplify matters. Those who complain about that are invited to consult the original papers.1.1. General. Proof theory startedwithHilbert's Programme which aimed at a finitistic consistency proof for mathematics.By Gödel's Theorems, however, we know that we can neither formalize all mathematics nor even prove the consistency of formalized fragments by finitistic means. Inspite of this fact I want to give some reasons why I consider proof theory in the style of Gentzen's work still as an important and exciting field of Mathematical Logic. I will not go into applications of Gentzen's cut-elimination technique to computer science problems—this may be considered as applied proof theory—but want to concentrate on metamathematical problems and results. In this sense I am talking about Pure Proof Theory.Mathematicians are interested in structures. There is only one way to find the theorems of a structure. Start with an axiom system for the structure and deduce the theorems logically. These axiom systems are the objects of proof-theoretical research. Studying axiom systems there is a series of more or less obvious questions. (shrink)

Winner of the 1999 Best Book in Ecological and Transformational Politics presented by the American Political Science Association's Section on Ecological and Transformational Politics The discipline of political science has reached a crossroads. The frequency with which terms such as "post-liberal," "post-modern," "post-patriarchical," "post-materialist," and "post-structural" are used in contemporary political discourse testifies to the pervasive conviction that an era has ended. Similarly, phrases such as "new world order," "new paradigm," "new age," and "third wave" convey the widely-shared expectation that (...) what lies ahead politically will be qualitatively unlike what has gone before. Transformational Politics argues that traditional political science is failing to identify and address fundamental political phenomena of our time and proposes an alternative value-based political science that not only studies phenomena, but also uses knowledge to promote democracy, sustainability, and social conscience. Part I of the book defines transformational political theory as an emerging paradigm and draws on a wide array of theories—empowerment, feminist, democratic, communitarian, chaos, quantum, conflict resolution, self-actualization. Part II examines how a transformational perspective guides the study of politics in both research and teaching. Part III offers guidance about how to practice the theory and apply the study with a concern for creating a better world. (shrink)

The ability to reason and think in a logical manner forms the basis of learning for most mathematics, computer science, philosophy and logic students. Based on the author's teaching notes at the University of Maryland and aimed at a broad audience, this text covers the fundamental topics in classical logic in an extremely clear, thorough and accurate style that is accessible to all the above. Covering propositional logic, first-order logic, and second-order logic, as well as proof theory, computability (...) theory, and model theory, the text also contains numerous carefully graded exercises and is ideal for a first or refresher course. (shrink)

This chapter focuses on the development of Gerhard Gentzen's structural proof theory and its connections with intuitionism. The latter is important in proof theory for several reasons. First, the methods of Hilbert's old proof theory were limited to the “finitistic” ones. These methods proved to be insufficient, and they were extended by infinitistic principles that were still intuitionistically meaningful. It is a general tendency in proof theory to try to use weak principles. (...) A second reason for the importance of intuitionism for proof theory is that the proof-theoretical study of intuitionistic logic has become a prominent part of logic, with applications in computer science. (shrink)

Occasional Papers in Cassirer and Cultural-Theory Studies presented at the University of Glasgow's Centre for Intercultural Studies. Given the growing disenchantment, on all sides, with the 'high theory' of the 1970s and 1980s, and with the dominant master-trope of literary and cultural reflexion of the 1980s and 1990s, the extended metaphor or 'allegory', this volume offers a timely re-examination of what, according to Goethe, is a deeper mode of understanding the symbol. Via the life-long preoccupation of Ernst Cassirer (...) with the problems of 'symbolic form', as he christened it, the papers collected here try to come to terms with the thinking of Goethe and Schiller on the symbol, and on related issues. Taken together, they attempt to elucidate the filiation of German classicism down through the nineteenth century to the present, in the belief that some of Cassirer's ideas have fed, often unacknowledged, into the mainstream of contemporary cultural theory, and that the rigour of his thought can help clear up much of the confusion in that 'theory'. (shrink)

This book is a specialized monograph on the development of the mathematical and computational metatheory of reductive logic and proof-search, areas of logic that are becoming important in computer science. A systematic foundational text on these emerging topics, it includes proof-theoretic, semantic/model-theoretic and algorithmic aspects. The scope ranges from the conceptual background to reductive logic, through its mathematical metatheory, to its modern applications in the computational sciences. Suitable for researchers and graduate students in mathematical, computational and philosophical logic, (...) and in theoretical computer science and artificial intelligence, this is the latest in the prestigous world-renowned Oxford Logic Guides, which contains Michael Dummet's Elements of intuitionism (2nd Edition), Dov M. Gabbay, Mark A. Reynolds, and Marcelo Finger's Temporal Logic Mathematical Foundations and Computational Aspects, J. M. Dunn and G. Hardegree's Algebraic Methods in Philosophical Logic, H. Rott's Change, Choice and Inference: A Study of Belief Revision and Nonmonotonic Reasoning, and P. T. Johnstone's Sketches of an Elephant: A Topos Theory Compendium: Volumes 1 and 2. (shrink)

Fuzzy logics are many-valued logics that are well suited to reasoning in the context of vagueness. They provide the basis for the wider field of Fuzzy Logic, encompassing diverse areas such as fuzzy control, fuzzy databases, and fuzzy mathematics. This book provides an accessible and up-to-date introduction to this fast-growing and increasingly popular area. It focuses in particular on the development and applications of "proof-theoretic" presentations of fuzzy logics; the result of more than ten years of intensive work by (...) researchers in the area, including the authors. In addition to providing alternative elegant presentations of fuzzy logics, proof-theoretic methods are useful for addressing theoretical problems and developing efficient deduction and decision algorithms. Proof-theoretic presentations also place fuzzy logics in the broader landscape of non-classical logics, revealing deep relations with other logics studied in Computer Science, Mathematics, and Philosophy. The book builds methodically from the semantic origins of fuzzy logics to proof-theoretic presentations such as Hilbert and Gentzen systems, introducing both theoretical and practical applications of these presentations. (shrink)

Apologies. The purpose of the following talk is to give an overview of the present state of aims, methods and results in Pure Proof Theory. Shortage of time forces me to concentrate on my very personal views. This entails that I will emphasize the work which I know best, i.e., work that has been done in the triangle Stanford, Munich and Münster. I am of course well aware that there are as important results coming from outside this triangle (...) and I apologize for not displaying these results as well.Moreover the audience should be aware that in some points I have to oversimplify matters. Those who complain about that are invited to consult the original papers.1.1. General. Proof theory startedwithHilbert's Programme which aimed at a finitistic consistency proof for mathematics.By Gödel's Theorems, however, we know that we can neither formalize all mathematics nor even prove the consistency of formalized fragments by finitistic means. Inspite of this fact I want to give some reasons why I consider proof theory in the style of Gentzen's work still as an important and exciting field of Mathematical Logic. I will not go into applications of Gentzen's cut-elimination technique to computer science problems—this may be considered as applied proof theory—but want to concentrate on metamathematical problems and results. In this sense I am talking about Pure Proof Theory.Mathematicians are interested in structures. There is only one way to find the theorems of a structure. Start with an axiom system for the structure and deduce the theorems logically. These axiom systems are the objects of proof-theoretical research. Studying axiom systems there is a series of more or less obvious questions. (shrink)

Transition to Real Analysis with Proof provides undergraduate students with an introduction to analysis including an introduction to proof. The text combines the topics covered in a transition course to lead into a first course on analysis. This combined approach allows instructors to teach a single course where two were offered. The text opens with an introduction to basic logic and set theory, setting students up to succeed in the study of analysis. Each section is followed (...) by graduated exercises that both guide and challenge students. The author includes examples and illustrations that appeal to the visual side of analysis. The accessible structure of the book makes it an ideal refence for later years of study or professional work. (shrink)

Chapter 1 constitutes an introduction to Gentzen calculi from two perspectives, logical and philosophical. It introduces the notion of generalisations of Gentzen sequent calculus and the discussion on properties that characterize good inferential systems. Among the variety of Gentzen-style sequent calculi, I divide them in two groups: syntactic and semantic generalisations. In the context of such a discussion, the inferentialist philosophy of the meaning of logical constants is introduced, and some potential objections – mainly concerning the choice of working with (...) semantic generalizations – are addressed. Finally, I’ll introduce the case studies that I’ll be dealing with in part II. Chapter 2 is concerned with the origins and development of Jaśkowski’s discussive logic. The main idea of this chapter is to systematize the various stages of the development of discussive logic related researches from two different angles, i.e., its connections to modal logics and its proof theory, by highlighting virtues and vices. Chapter 3 focuses on the Gentzen-style proof theory of discussive logic, by providing a characterization of it in terms of labelled sequent calculi. Chapter 4 deals with the Gentzen-style proof theory of relevant logics, again by employing the methodology of labelled sequent calculi. This time, instead of working with a single logic, I’ll deal with a whole family of them. More precisely, I’ll study in terms of proof systems those relevant logics that can be characterised, at the semantic level, by reduced Routley-Meyer models, i.e., relational structures with a ternary relation between states and a unique base element. Chapter 5 investigates the proof theory of a modal expansion of intuitionistic propositional logic obtained by adding an ‘actuality’ operator to the connectives. This logic was introduced also using Gentzen sequents. Unfortunately, the original proof system is not cut-free. This chapter shows how to solve this problem by moving to hypersequents. Chapter 6 concludes the investigations and discusses the future of the research presented throughout the dissertation. (shrink)

Accession Number: ATLA0001712127; Hosting Book Page Citation: p 187-206.; Physical Description: table ; Language(s): English; General Note: Bibliography: p 204-206.; Issued by ATLA: 20130825; Publication Type: Essay.

This book is a specialized monograph on the development of the mathematical and computational metatheory of reductive logic and proof-search, areas of logic that are becoming important in computer science. A systematic foundational text on these emerging topics, it includes proof-theoretic, semantic/model-theoretic and algorithmic aspects. The scope ranges from the conceptual background to reductive logic, through its mathematical metatheory, to its modern applications in the computational sciences. Suitable for researchers and graduate students in mathematical, computational and philosophical logic, (...) and in theoretical computer science and artificial intelligence, this is the latest in the prestigous world-renowned Oxford Logic Guides, which contains Michael Dummet's Elements of intuitionism, Dov M. Gabbay, Mark A. Reynolds, and Marcelo Finger's Temporal Logic Mathematical Foundations and Computational Aspects, J. M. Dunn and G. Hardegree's Algebraic Methods in Philosophical Logic, H. Rott's Change, Choice and Inference: A Study of Belief Revision and Nonmonotonic Reasoning, and P. T. Johnstone's Sketches of an Elephant: A Topos Theory Compendium: Volumes 1 and 2. (shrink)

Proof Theory of Modal Logic is devoted to a thorough study of proof systems for modal logics, that is, logics of necessity, possibility, knowledge, belief, time, computations etc. It contains many new technical results and presentations of novel proof procedures. The volume is of immense importance for the interdisciplinary fields of logic, knowledge representation, and automated deduction.

Linear logic is a new logic which was recently developed by Girard in order to provide a logical basis for the study of parallelism. It is described and investigated in Gi]. Girard's presentation of his logic is not so standard. In this paper we shall provide more standard proof systems and semantics. We shall also extend part of Girard's results by investigating the consequence relations associated with Linear Logic and by proving corresponding str ong completeness theorems. Finally, we (...) shall investigate the relation between Linear Logic and previously known systems, especially Relevance logics. (shrink)

In China, the improvement of the learner performance is critical a challenge for the teaching staff and the management in intercultural communication class. Indeed, the administration of the Chinese schools is failed to provide effective learning to the students with innovative methods. The objective of this study was to identify the role of college students' learning performance, teaching skills, and teaching innovation in intercultural communication class. This study is based on the quantitative data collected on a five-point Likert (...) scale from the target respondents who were the students of different colleges and 700 questionnaires distributed for it. The study concludes that there is a significant relationship among abstract conceptualization, active participation, and reflective observation in students' learning performance. Furthermore, this study highlights that the mediating role of teaching innovation is critical for students' learning performance. This study contributes to the literature with a significant theoretical framework. Finally, this study provides significant theoretical implications and practical implications which are key game-changers for improving the performance of the students in the intercultural communication class. (shrink)

The current paper studies the formal properties of the Global Reflection Principle, to wit the assertion “All theorems of$\mathrm {Th}$are true,” where$\mathrm {Th}$is a theory in the language of arithmetic and the truth predicate satisfies the usual Tarskian inductive conditions for formulae in the language of arithmetic. We fix the gap in Kotlarski’s proof from [15], showing that the Global Reflection Principle for Peano Arithmetic is provable in the theory of compositional truth with bounded induction only ($\mathrm (...) {CT}_0$). Furthermore, we extend the above result showing that$\Sigma _1$-uniform reflection over a theory of uniform Tarski biconditionals ($\mathrm {UTB}^-$) is provable in$\mathrm {CT}_0$, thus answering the question of Beklemishev and Pakhomov [2]. Finally, we introduce the notion of a prolongable satisfaction class and use it to study the structure of models of$\mathrm {CT}_0$. In particular, we provide a new model-theoretical characterization of theories of finite iterations of uniform reflection and present a new proof characterizing the arithmetical consequences of$\mathrm {CT}_0$. (shrink)

Goal Directed Proof Theory presents a uniform and coherent methodology for automated deduction in non-classical logics, the relevance of which to computer science is now widely acknowledged. The methodology is based on goal-directed provability. It is a generalization of the logic programming style of deduction, and it is particularly favourable for proof search. The methodology is applied for the first time in a uniform way to a wide range of non-classical systems, covering intuitionistic, intermediate, modal and substructural (...) logics. The book can also be used as an introduction to these logical systems form a procedural perspective. Readership: Computer scientists, mathematicians and philosophers, and anyone interested in the automation of reasoning based on non-classical logics. The book is suitable for self study, its only prerequisite being some elementary knowledge of logic and proof theory. (shrink)

Theory of mind is a key consideration in autism spectrum conditions and is frequently associated with social, emotional, behavioural and mental health difficulties. The latest practical workbook in the Talkabout series, this book is designed to support those for whom theory of mind does not come naturally. It teaches strategies that can be used to identify others' thoughts and feelings based on their behaviour, as well as how to adapt behaviour in order to competently manage social situations and (...) have positive interactions. With fully illustrated activities covering topics such as thoughts, feelings and actions, knowledge and beliefs, and respect, the programme outlined in this book can be used with children and young people to develop and confidently implement an awareness of theory of mind. Key features include: Assessments, targets, lesson plans and over ninety activities to support theory of mind Structured activities which progress from simple concepts to more complex skills Opportunities for skills to be practiced and recapped Fully photocopiable and downloadable resources Packed full of flexible activities to suit different levels and ages, this is a vital resource for educators and therapists looking to support children and young people with poor theory of mind as they develop the skills necessary to create positive interactions. (shrink)

Education for Total Liberation is a collection of essays from leaders in the field of critical animal pedagogy (CAP). CAP emerges from activist educators teaching critical animal studies and is rooted in critical theory as well as the animal advocacy movement. Critical animal studies (CAS) argues for an interdisciplinary approach to understanding our relationships with nonhuman animals. CAS challenges two specific fields of theory: (1) animal studies, rooted in vivisection and testing on animals in the hard sciences and (...) (2) human-animal studies, which reinforces a socially constructed binary between humans and animals and adopts abstract theoretical approaches. In contrast, CAS takes a progressive and committed approach to scholarship and sees the exploitation of nonhuman animals as interrelated with oppression of humans based on class, gender, race, ability, sexuality, age, and citizenship. CAS promotes the liberation of all animals and challenges all systems of domination. Education for Total Liberation is appropriate for undergraduate and graduate level readers (and beyond) who wish to learn from examples of radical pedagogical projects shaped by CAS and critical pedagogy. Contributing to this collection are Anne C. Bell, Anita de Melo, Carolyn Drew, Amber E. George, Karin Gunnarsson Dinker, Sinem Ketenci, John Lupinacci, Anthony J. Nocella II, Sean Parson, Helena Pedersen, Ian Purdy, Constance L. Russell, J.L. Schatz, Meneka Repka, William E. Shanahan III, and Richard J, White. (shrink)

This paper provides a proof-theoretic study of quantified non-normal modal logics. It introduces labelled sequent calculi based on neighbourhood semantics for the first-order extension, with both varying and constant domains, of monotone NNML, and studies the role of the Barcan formulas in these calculi. It will be shown that the calculi introduced have good structural properties: invertibility of the rules, height-preserving admissibility of weakening and contraction and syntactic cut elimination. It will also be shown that each of the (...) calculi introduced is sound and complete with respect to the appropriate class of neighbourhood frames. In particular, the completeness proof constructs a formal derivation for derivable sequents and a countermodel for non-derivable ones, and gives a semantic proof of the admissibility of cut. (shrink)

Our first aim is to make the study of informal notions of proof plausible. Put differently, since the raison d'étre of anything like existing proof theory seems to rest on such notions, the aim is nothing else but to make a case for proof theory; ...

I argue that if we allow that Moore’s Method, which involves taking an ordinary knowledge claim to support a substantive metaphysical conclusion, can be used to support Moore’s proof an external world, then we should accept that Moore’s Method can be used to support a variety of incompatible metaphysical conclusions. I shall refer to this as “the problem of many proofs”. The problem of many proofs, I claim, stems from the theory-ladenness of perception. I shall argue further that (...) this plethora of proofs for incompatible positions leads to a darker form of skepticism, one which maintains that each of the dogmatic views is probably false. We will conclude by considering various ways a Moorean might respond to these difficulties. (shrink)

In a classical 1976 paper, Solovay proved the arithmetical completeness of the modal logic GL; provability of a formula in GL coincides with provability of its arithmetical interpretations of it in Peano Arithmetic. In that paper, he also provided an axiomatic system GLS and proved arithmetical completeness for GLS; provability of a formula in GLS coincides with truth of its arithmetical interpretations in the standard model of arithmetic. Proof theory for GL has been studied intensively up to the (...) present day. However, it might sound somewhat strange that no proof theory for GLS was ever developed nor even suggested thus far, except for the axiomatic system offered by Solovay. In this paper, we develop a proof theory for GLS based on the sequent calculus method. We provide a sequent calculus for GLS and prove the cut-elimination and some standard consequences of it: the interpolation and definability theorems. As another consequence of cut-elimination, we also prove the equivalence of GL and GLS with respect to a special form of formulas which we call Gödel sentences, using a purely proof-theoretical method. (shrink)

In this paper, we show the equivalence between the provability of a proof system of basic hybrid logic and that of translated formulas of the classical predicate logic with equality and explicit substitution by a purely proof–theoretic method. Then we show the equivalence of two groups of proof systems of hybrid logic: the group of labelled deduction systems and the group of modal logic-based systems.

The human species is more reliant on cultural adaptation than any other species, but it is unclear how observational learning can give rise to the faithful transmission of cultural adaptations. One possibility is that teaching facilitates accurate social transmission by narrowing the range of inferences that learners make. However, there is wide disagreement about how to define teaching, and how to interpret the empirical evidence for teaching across cultures and species. In this article I argue that disputes about the nature (...) and prevalence of teaching across human societies and nonhuman animals are based on a number of deep-rooted theoretical differences between fields, as well as on important differences in how teaching is defined. To reconcile these disparate bodies of research, I review the three major approaches to the study of teaching – mentalistic, culture-based, and functionalist – and outline the research questions about teaching that each addresses. I then argue for a new, integrated framework that differentiates between teaching types according to the specific adaptive problems that each type solves, and apply this framework to restructure current empirical evidence on teaching in humans and nonhuman animals. This integrative framework generates novel insights, with broad implications for the study of the evolution of teaching, including the roles of cognitive constraints and cooperative dilemmas in how and when teaching evolves. Finally, I propose an explanation for why some types of teaching are uniquely human, and discuss new directions for research motivated by this framework. (shrink)

Paul Cohen’s method of forcing, together with Saul Kripke’s related semantics for modal and intuitionistic logic, has had profound effects on a number of branches of mathematical logic, from set theory and model theory to constructive and categorical logic. Here, I argue that forcing also has a place in traditional Hilbert-style proof theory, where the goal is to formalize portions of ordinary mathematics in restricted axiomatic theories, and study those theories in constructive or syntactic terms. (...) I will discuss the aspects of forcing that are useful in this respect, and some sample applications. The latter include ways of obtaining conservation results for classical and intuitionistic theories, interpreting classical theories in constructive ones, and constructivizing model-theoretic arguments. (shrink)

There are many algorithm texts that provide lots of well-polished code and proofs of correctness. Instead, this book presents insights, notations, and analogies to help the novice describe and think about algorithms like an expert. By looking at both the big picture and easy step-by-step methods for developing algorithms, the author helps students avoid the common pitfalls. He stresses paradigms such as loop invariants and recursion to unify a huge range of algorithms into a few meta-algorithms. Part of the goal (...) is to teach the students to think abstractly. Without getting bogged with formal proofs, the book fosters a deeper understanding of how and why each algorithm works. These insights are presented in a slow and clear manner accessible to second- or third-year students of computer science, preparing them to find their own innovative ways to solve problems. (shrink)

Learning and Teaching in the Early Years provides a comprehensive, contemporary and practical introduction to early childhood teaching in Australia. A strong focus on the links between theory, policy and practice firmly aligns this text with the Early Years Learning Framework. Written for students of early childhood programs, this book covers learning and development, as well as professional practice in teaching children from birth to eight years. In recognition of the evolving role of educators, topic areas include learning, teaching, (...) working with families, leading, advocating and researching. Each chapter contains learning objectives, key terms and reflection points. Detailed case studies document the intersection between research, policy and practice, enhancing pre-service and practicing educators' appreciation of how a policy-aligned approach reinforces learning and development in the early years. This text draws on the latest research to present children's learning as a dynamic and active process requiring specific, intentional teaching behaviours. (shrink)

Learning and Teaching in Early Childhood: Pedagogies of Inquiry and Relationships is an introduction for early childhood educators beginning their studies. Reflecting the fact that there is no single correct approach to the challenges of teaching, this book explores teaching through two lenses: teaching as inquiry and teaching as relating. The first part of the book focuses on inquiry, covering early childhood learning environments, learning theories, play pedagogies, approaches to teaching and learning, documentation and assessment, and the policy, curriculum and (...) regulatory requirements in Australia. The second part explores relationships in early childhood contexts and covers topics such as fostering meaningful and respectful relationships with children, and working with families, staff and the wider community. Written by well-respected academics in the field, Learning and Teaching in Early Childhood is a vital resource for those entering the early childhood education and care profession. (shrink)

This volume covers a wide range of topics in the most recent debates in the philosophy of mathematics, and is dedicated to how semantic, epistemological, ontological and logical issues interact in the attempt to give a satisfactory picture of mathematical knowledge. The essays collected here explore the semantic and epistemic problems raised by different kinds of mathematical objects, by their characterization in terms of axiomatic theories, and by the objectivity of both pure and applied mathematics. They investigate controversial aspects of (...) contemporary theories such as neo-logicist abstractionism, structuralism, or multiversism about sets, by discussing different conceptions of mathematical realism and rival relativistic views on the mathematical universe. They consider fundamental philosophical notions such as set, cardinal number, truth, ground, finiteness and infinity, examining how their informal conceptions can best be captured in formal theories. The philosophy of mathematics is an extremely lively field of inquiry, with extensive reaches in disciplines such as logic and philosophy of logic, semantics, ontology, epistemology, cognitive sciences, as well as history and philosophy of mathematics and science. By bringing together well-known scholars and younger researchers, the essays in this collection – prompted by the meetings of the Italian Network for the Philosophy of Mathematics (FilMat) – show how much valuable research is currently being pursued in this area, and how many roads ahead are still open for promising solutions to long-standing philosophical concerns. Promoted by the Italian Network for the Philosophy of Mathematics – FilMat. (shrink)

Making Sense of Education provides a contemporary introduction to the key issues in educational philosophy and theory. Exploring recent developments as well as important ideas from the twentieth century, this book aims to make philosophy of education relevant to everyday practice for teachers and student teachers, as well as those studying education as an academic subject.

No longer confined to traditional institutions devoted to Talmudic studies, havruta work, or the practice of students studying materials in pairs, has become a relatively widespread phenomenon across denominational and educational settings of Jewish learning. However, until now there has been little discussion of what havruta text study entails and how it might be conceptualized and taught. This book breaks new ground from two perspectives: by offering a model of havruta text study situated in broader theories of interpretation (...) and learning, and by treating havruta text study as composed of textual, interpersonal and intra-personal practices which can be taught and learned. We lay out the conceptual foundations of our approach and provide examples of their pedagogical implementation for the teaching of havruta text study. Included are illustrative lesson plans, teachers' notes and students’ reflections, exercises for students, and other instructional materials for teaching core concepts and practices. (shrink)

Historical case studies of scientific concepts are a useful medium for showing how scientific ideas originate and how they change over time. They are thus a useful tool for conveying knowledge about the nature of science. This paper focuses on the concepts of heat and temperature and discusses some issues related to choosing the content for a historical case study which incorporates not only nature of science perspectives but understandings related to what we know about the teaching and learning (...) of these concepts. The case study is designed for first-year university chemistry students as an introduction to their study of thermodynamics. The paper includes a general chemistry textbook analysis of the heat and temperature concepts and a discussion of the caloric theory of heat, thermometry, and a brief survey of how the energy concept transformed our understanding of heat and temperature. (shrink)

This book is a guide to understanding and applying the essential, heretofore elusive notion of context in language study and pedagogy: Éva Illés offers a new, critical, systematic theoretical framework, then applies that framework to practical interactions and issues in communicative language teaching rooted in English as a Lingua Franca. By linking theory and practice for research and teaching around the world, this book brings a new awareness of how context can be conceptualized and related to language pedagogy (...) to advanced students, teachers, teacher educators, and researchers of language teaching, applied linguistics, and pragmatics. (shrink)

Feminist theory is a central strand of cultural studies. This book explores the history of feminist cultural studies from the early work of Mary Wollstonecraft, Charlotte Perkins Gilman, Virginia Woolf, Simone de Beauvoir, through the 1970s Women's Liberation Movement. It also provides a comprehensive introduction to the contemporary key approaches, theories and debates of feminist theory within cultural studies, offering a major re-mapping of the field. It will be an essential text for students taking courses within both cultural (...) studies and women's studies departments. (shrink)