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Trivalent Discrete Surfaces and Carbon Structures (SpringerBriefs in the Mathematics of Materials #5)
by Hisashi NaitoThis book discusses discrete geometric analysis, especially topological crystallography and discrete surface theory for trivalent discrete surfaces. Topological crystallography, based on graph theory, provides the most symmetric structure among given combinatorial structures by using the variational principle, and it can reproduce crystal structures existing in nature. In this regard, the topological crystallography founded by Kotani and Sunada is explained by using many examples. Carbon structures such as fullerenes are considered as trivalent discrete surfaces from the viewpoint of discrete geometric analysis. Discrete surface theories usually have been considered discretization of smooth surfaces. Here, consideration is given to discrete surfaces modeled by crystal/molecular structures, which are essentially discrete objects.
The Troika of Adult Learners, Lifelong Learning, and Mathematics
by Katherine Safford-Ramus Pradeep Kumar Misra Terry MaguireThis book presents a synopsis of six emerging themes in adult mathematics/numeracy and a critical discussion of recent developments in terms of policies, provisions, and the emerging challenges, paradoxes and tensions. It also offers an extensive review of the literature adult mathematics education. Why do adults want to learn mathematics? Did they enjoy mathematics at school so much that they want to continue? NO! Most of these adults have to learn mathematics because it is part of a formal qualification they need, because their job demands the ability to apply mathematics, or because they need basic numeracy in their daily lives. Lastly, the authors discuss five potential strategies to promote lifelong learning of mathematics among adult learners.
Tropical and Logarithmic Methods in Enumerative Geometry (Oberwolfach Seminars #52)
by Renzo Cavalieri Hannah Markwig Dhruv RanganathanThis book is based on the lectures given at the Oberwolfach Seminar held in Fall 2021. Logarithmic Gromov-Witten theory lies at the heart of modern approaches to mirror symmetry, but also opens up a number of new directions in enumerative geometry of a more classical flavour. Tropical geometry forms the calculus through which calculations in this subject are carried out. These notes cover the foundational aspects of this tropical calculus, geometric aspects of the degeneration formula for Gromov-Witten invariants, and the practical nuances of working with and enumerating tropical curves. Readers will get an assisted entry route to the subject, focusing on examples and explicit calculations.
Tropical Circuit Complexity: Limits of Pure Dynamic Programming (SpringerBriefs in Mathematics)
by Stasys JuknaThis book presents an enticing introduction to tropical circuits and their use as a rigorous mathematical model for dynamic programming (DP), which is one of the most fundamental algorithmic paradigms for solving combinatorial, discrete optimization problems. In DP, an optimization problem is broken up into smaller subproblems that are solved recursively. Many classical DP algorithms are pure in that they only use the basic (min,+) or (max,+) operations in their recursion equations. In tropical circuits, these operations are used as gates. Thanks to the rigorous combinatorial nature of tropical circuits, elements from the Boolean and arithmetic circuit complexity can be used to obtain lower bounds for tropical circuits, which play a crucial role in understanding the limitations and capabilities of these computational models. This book aims to offer a toolbox for proving lower bounds on the size of tropical circuits.In this work, the reader will find lower-bound ideas and methods that have emerged in the last few years, with detailed proofs. Largely self-contained, this book is meant to be approachable by graduate students in mathematics and computer science with a special interest in circuit complexity.
Tropical Intraseasonal Variability and the Stochastic Skeleton Method (Mathematics of Planet Earth)
by Andrew J. Majda Samuel N. Stechmann Shengqian Chen H. Reed Ogrosky Sulian ThualIn this text, modern applied mathematics and physical insight are used to construct the simplest and first nonlinear dynamical model for the Madden-Julian oscillation (MJO), i.e. the stochastic skeleton model. This model captures the fundamental features of the MJO and offers a theoretical prediction of its structure, leading to new detailed methods to identify it in observational data. The text contributes to understanding and predicting intraseasonal variability, which remains a challenging task in contemporary climate, atmospheric, and oceanic science. In the tropics, the Madden-Julian oscillation (MJO) is the dominant component of intraseasonal variability. One of the strengths of this text is demonstrating how a blend of modern applied mathematical tools, including linear and nonlinear partial differential equations (PDEs), simple stochastic modeling, and numerical algorithms, have been used in conjunction with physical insight to create the model. These tools are also applied in developing several extensions of the model in order to capture additional features of the MJO, including its refined vertical structure and its interactions with the extratropics. This book is of interest to graduate students, postdocs, and senior researchers in pure and applied mathematics, physics, engineering, and climate, atmospheric, and oceanic science interested in turbulent dynamical systems as well as other complex systems.
Trotter-Kato Approximations of Stochastic Differential Equations in Infinite Dimensions and Applications
by T. E. GovindanThis is the first comprehensive book on Trotter-Kato approximations of stochastic differential equations (SDEs) in infinite dimensions and applications. This research monograph brings together the varied literature on this topic since 1985 when such a study was initiated. The author provides a clear and systematic introduction to the theory of Trotter-Kato approximations of SDEs and also presents its applications to practical topics such as stochastic stability and stochastic optimal control. The theory assimilated here is developed slowly and methodically in digestive pieces.The book begins with a motivational chapter introducing several different models that highlight the importance of the theory on abstract SDEs that will be considered in the subsequent chapters. The author next introduces the necessary mathematical background and then leads the reader into the main discussion of the monograph, namely, the Trotter-Kato approximations of many classes of SDEs in Hilbert spaces, Trotter-Kato approximations of SDEs in UMD Banach spaces and some of their applications. Most of the results presented in the main chapters appear for the first time in a book form. The monograph also contains many illustrative examples on stochastic partial differential equations and one in finance as an application of the Trotter-Kato formula. The key steps are included in all proofs which will help the reader to get a real insight into the theory of Trotter-Kato approximations and its use. This book is intended for researchers and graduate students in mathematics specializing in probability theory. It will also be useful to numerical analysts, engineers, physicists and practitioners who are interested in applying the theory of stochastic evolution equations. Since the approach is based mainly in semigroup theory, it is accessible to a wider audience including non-specialists in stochastic processes.
Trotter-Kato Product Formulæ (Operator Theory: Advances and Applications #296)
by Valentin A. Zagrebnov Hagen Neidhardt Takashi IchinoseThe book captures a fascinating snapshot of the current state of results about the operator-norm convergent Trotter-Kato Product Formulæ on Hilbert and Banach spaces. It also includes results on the operator-norm convergent product formulæ for solution operators of the non-autonomous Cauchy problems as well as similar results on the unitary and Zeno product formulæ.After the Sophus Lie product formula for matrices was established in 1875, it was generalised to Hilbert and Banach spaces for convergence in the strong operator topology by H. Trotter (1959) and then in an extended form by T. Kato (1978). In 1993 Dzh. L. Rogava discovered that convergence of the Trotter product formula takes place in the operator-norm topology. The latter is the main subject of this book, which is dedicated essentially to the operator-norm convergent Trotter-Kato Product Formulæ on Hilbert and Banach spaces, but also to related results on the time-dependent, unitary and Zeno product formulæ. The book yields a detailed up-to-date introduction into the subject that will appeal to any reader with a basic knowledge of functional analysis and operator theory. It also provides references to the rich literature and historical remarks.
The Trouble with Maths: A practical guide to helping learners with numeracy difficulties
by Steve ChinnNow in third edition, with updates to reflect developments in our understanding of learning difficulties in maths, this award-winning text provides vital insights into the often confusing world of numeracy. By looking at learning difficulties in maths and dyscalculia from several perspectives, including the vocabulary and language of maths, thinking styles and the demands of individual procedures, this book provides a complete overview of the most frequently occurring problems associated with maths teaching and learning. Drawing on tried-and-tested methods based on research and Steve Chinn’s years of classroom experience, it provides an authoritative yet accessible one-stop classroom resource. Combining advice, guidance and practical activities, this user-friendly guide will help you to: develop flexible thinking styles use alternative strategies to replace an over-reliance on rote learning for pupils trying to access basic facts understand the implications of underlying skills, such as working memory, on learning implement effective pre-emptive measures before demotivation sets in recognise the manifestations of maths anxiety and tackle affective domain problems find approaches to solve word problems select appropriate materials and visual images to enhance understanding. With useful features such as checklists for the evaluation of books and a comprehensive overview of resources, this book will equip you with essential skills to help you tackle your pupils’ maths difficulties and improve standards. This book will be useful for all teachers, classroom assistants, learning support assistants and parents.
The Trouble with Maths: A Practical Guide to Helping Learners with Numeracy Difficulties
by Steve ChinnNow in its fourth edition, with updates to reflect developments in our understanding of learning difficulties in maths, this award-winning text provides vital, pragmatic insights into the often-confusing world of numeracy. By looking at learning difficulties in maths and dyscalculia from several perspectives, for example, the vocabulary and language of maths, cognitive style and the demands of individual procedures, this book provides a complete overview of the most frequently occurring problems associated with maths teaching and learning. Drawing on tried-and-tested methods based on research and Steve Chinn’s decades of classroom experience, it provides an authoritative yet accessible one-stop classroom resource. Combining advice, guidance and practical activities, this user-friendly guide will help you to: develop flexible cognitive styles use alternative strategies to replace an over-reliance on rote-learning for pupils trying to access basic facts understand the implications of underlying skills, such as working memory, on learning implement effective pre-emptive measures before demotivation sets in recognise the manifestations of maths anxiety and tackle affective domain problems find approaches to solve word problems select appropriate materials and visual images to enhance understanding. With useful features such as checklists for the evaluation of books and an overview of resources, this book will equip you with essential skills to help you tackle your pupils’ maths difficulties and improve standards for all learners. This book will be useful for all teachers, classroom assistants, learning support assistants and parents.
Trucks and Diggers One to Ten Counting
by CaterpillarKids love trucks and diggers, and with the help of full color photographs featuring rugged Caterpillar machines, these budding little contractors will love learning counting, too!
Trust and Credit in Organizations and Institutions: As Viewed from the Evolution of Cooperation (Theoretical Biology)
by Mayuko NakamaruThis book shows that evolutionary game theory can unravel how mutual cooperation, trust, and credit in a group emerge in organizations and institutions. Some organizations and institutions, such as insurance unions, credit unions, and banks, originated from very simple mutual-aid groups. Members in these early-stage mutual-aid groups help each other, making rules to promote cooperation, and suppressing free riders. Then, they come to “trust” not only each other but also the group they belong to, itself. The division of labor occurs when the society comes to have diversity and complexity in a larger group, and the division of labor also requires mutual cooperation and trust among different social roles. In a larger group, people cannot directly interact with each other, and the reputation of unknown people helps other decide who is a trustworthy person. However, if gossip spreads untruths about a reputation, trust and cooperation are destroyed. Therefore, how to suppress untrue gossip is also important for trust and cooperation in a larger group. If trustworthiness and credibility can be established, these groups are successfully sustainable. Some develop and evolve and then mature into larger organizations and institutions. Finally, these organizations and institutions become what they are now. Therefore, not only cooperation but also trust and credit are keys to understanding these organizations and institutions.The evolution of cooperation, a topic of research in evolutionary ecology and evolutionary game theory, can be applied to understanding how to make institutions and organizations sustainable, trustworthy, and credible. It provides us with the idea that evolutionary game theory is a good mathematical tool to analyze trust and credit. This kind of research can be applied to current hot topics such as microfinance and the sustainable use of ecosystems.
Trust in Numbers: The Pursuit of Objectivity in Science and Public Life
by Theodore M. PorterA foundational work on historical and social studies of quantificationWhat accounts for the prestige of quantitative methods? The usual answer is that quantification is desirable in social investigation as a result of its successes in science. Trust in Numbers questions whether such success in the study of stars, molecules, or cells should be an attractive model for research on human societies, and examines why the natural sciences are highly quantitative in the first place. Theodore Porter argues that a better understanding of the attractions of quantification in business, government, and social research brings a fresh perspective to its role in psychology, physics, and medicine. Quantitative rigor is not inherent in science but arises from political and social pressures, and objectivity derives its impetus from cultural contexts. In a new preface, the author sheds light on the current infatuation with quantitative methods, particularly at the intersection of science and bureaucracy.
Trust in Numbers: The Pursuit of Objectivity in Science and Public Life
by Theodore M. PorterThis investigation of the overwhelming appeal of quantification in the modern world discusses the development of cultural meanings of objectivity over two centuries. How are we to account for the current prestige and power of quantitative methods? The usual answer is that quantification is seen as desirable in social and economic investigation as a result of its successes in the study of nature. Theodore Porter is not content with this. Why should the kind of success achieved in the study of stars, molecules, or cells be an attractive model for research on human societies? he asks. And, indeed, how should we understand the pervasiveness of quantification in the sciences of nature? In his view, we should look in the reverse direction: comprehending the attractions of quantification in business, government, and social research will teach us something new about its role in psychology, physics, and medicine. Drawing on a wide range of examples from the laboratory and from the worlds of accounting, insurance, cost-benefit analysis, and civil engineering, Porter shows that it is "exactly wrong" to interpret the drive for quantitative rigor as inherent somehow in the activity of science except where political and social pressures force compromise. Instead, quantification grows from attempts to develop a strategy of impersonality in response to pressures from outside. Objectivity derives its impetus from cultural contexts, quantification becoming most important where elites are weak, where private negotiation is suspect, and where trust is in short supply.
Trustworthy Artificial Intelligence for Healthcare: Second International Workshop, TAI4H 2024, Jeju, South Korea, August 4, 2024, Proceedings (Lecture Notes in Computer Science #14812)
by Daguang Xu Hao Chen Yuyin Zhou Varut Vince VardhanabhutiThis book constitutes the proceedings of Second International Workshop on Trustworthy Artificial Intelligence for Healthcare, TAI4H 2024, held in Jeju, South Korea, in August 2024, in conjunction with the International Joint Conference on Artificial Intelligence, IJCAI 2024. The 13 full papers included in this book were carefully reviewed and selected from 21 submissions. They focus on trustworthy artificial intelligence, healthcare, generalization, explainability, fairness, privacy, multi-modal fusion, foundation models.
Trustworthy Communications and Complete Genealogies: Unifying Ancestries for a Genealogical History of the Modern World (Synthesis Lectures on Information Concepts, Retrieval, and Services)
by Reagan W. MooreThis book explains how all persons with Western European ancestry are related through a Unifying Ancestry. The author presents the knowledge metrics needed to derive the best representation for a Unifying Ancestry. The book uses example lineages to historically notable persons including relationships to U.S. Presidents, royal families, actors, and singers in order to demonstrate these points. This third edition is based upon expanded data, applying the algorithms to validate the Unifying Ancestry to a 348,844-person Research Genealogy. The author successfully demonstrates that a Unifying Ancestry with 38,533 persons is sufficient to identify familial relationships between Western Europeans.
Trustworthy Cyber-Physical Systems Engineering (Chapman & Hall/CRC Computer and Information Science Series #36)
by Alexander Romanovsky Fuyuki IshikawaFrom the Foreword "Getting CPS dependability right is essential to forming a solid foundation for a world that increasingly depends on such systems. This book represents the cutting edge of what we know about rigorous ways to ensure that our CPS designs are trustworthy. I recommend it to anyone who wants to get a deep look at these concepts that will form a cornerstone for future CPS designs." --Phil Koopman, Carnegie Mellon University, Pittsburgh, Pennsylvania, USA Trustworthy Cyber-Physical Systems Engineering provides practitioners and researchers with a comprehensive introduction to the area of trustworthy Cyber Physical Systems (CPS) engineering. Topics in this book cover questions such as What does having a trustworthy CPS actually mean for something as pervasive as a global-scale CPS? How does CPS trustworthiness map onto existing knowledge, and where do we need to know more? How can we mathematically prove timeliness, correctness, and other essential properties for systems that may be adaptive and even self-healing? How can we better represent the physical reality underlying real-world numeric quantities in the computing system? How can we establish, reason about, and ensure trust between CPS components that are designed, installed, maintained, and operated by different organizations, and which may never have really been intended to work together? Featuring contributions from leading international experts, the book contains sixteen self-contained chapters that analyze the challenges in developing trustworthy CPS, and identify important issues in developing engineering methods for CPS. The book addresses various issues contributing to trustworthiness complemented by contributions on TCSP roadmapping, taxonomy, and standardization, as well as experience in deploying advanced system engineering methods in industry. Specific approaches to ensuring trustworthiness, namely, proof and refinement, are covered, as well as engineering methods for dealing with hybrid aspects.
Trustworthy Global Computing: 8th International Symposium, TGC 2013, Buenos Aires, Argentina, August 30-31, 2013, Revised Selected Papers (Lecture Notes in Computer Science #8358)
by Martín Abadi and Alberto Lluch LafuenteThis book constitutes the thoroughly refereed post-conference proceedings of the 8th International Symposium on Trustworthy Global Computing, TGC 2013, held in Buenos Aires, Argentina, in August 2013. The 15 revised full papers presented together with 3 invited talks were carefully reviewed and selected from 29 submissions. The papers cover a wide range of topics in the area of global computing and safe and reliable computation. They are organized in topical sections on security, π-calculus, information flow, models, specifications and proofs and quantitative analysis.
Truth and Falsehood
by Heinrich Wansing Yaroslav ShramkoThe book presents a thoroughly elaborated logical theory of generalized truth-values understood as subsets of some established set of (basic) truth values. After elucidating the importance of the very notion of a truth value in logic and philosophy, we examine some possible ways of generalizing this notion. The useful four-valued logic of first-degree entailment by Nuel Belnap and the notion of a bilattice (a lattice of truth values with two ordering relations) constitute the basis for further generalizations. By doing so we elaborate the idea of a multilattice, and most notably, a trilattice of truth values - a specific algebraic structure with information ordering and two distinct logical orderings, one for truth and another for falsity. Each logical order not only induces its own logical vocabulary, but determines also its own entailment relation. We consider both semantic and syntactic ways of formalizing these relations and construct various logical calculi.
Truth Decay: An Initial Exploration of the Diminishing Role of Facts and Analysis in American Public Life
by Kavanagh Michael D. RichPolitical and civil discourse in the United States is characterized by “Truth Decay,” defined as increasing disagreement about facts, a blurring of the line between opinion and fact, an increase in the relative volume of opinion compared with fact, and lowered trust in formerly respected sources of factual information. This report explores the causes and wide-ranging consequences of Truth Decay and proposes strategies for further action.
The Truth Detective: How to make sense of a world that doesn't add up
by Tim Harford'Tim Harford is peerless at making sense of a complicated world and our place within it. This is a book that all children should read' - Matthew Syed-> Did you know that a toy spaceship can teach you about inflation?-> Or that a pooping cow can show you how to invest your pocket money?-> And that even the greatest detectives have been fooled by fake news and dancing fairies?The world is often full of bamboozling headlines and numbers that don't add up. But don't panic. Within these pages you will transform into a Truth Detective, hunting down the truth about the world around you. You will meet heroic truth detectives, such as Florence Nightingale who started a revolution with a pie chart. You will encounter dastardly villains who have tried to trip us up with dodgy data and misinformation. And you will learn how being smart and savvy with numbers, will help you be smart and savvy about everything else in life too.So grab your detective cap, pick up your magnifying glass and start seeing the world like never before.A must read for curious kids from presenter of BBC Radio 4's "More or Less", Tim Harford and with wonderful illustrations to enjoy by Ollie Mann.
The Truth Detective: How to make sense of a world that doesn't add up
by Tim Harford'Tim Harford is peerless at making sense of a complicated world and our place within it. This is a book that all children should read' - Matthew Syed-> Did you know that a toy spaceship can teach you about inflation?-> Or that a pooping cow can show you how to invest your pocket money?-> And that even the greatest detectives have been fooled by fake news and dancing fairies?The world is often full of bamboozling headlines and numbers that don't add up. But don't panic. Within these pages you will transform into a Truth Detective, hunting down the truth about the world around you. You will meet heroic truth detectives, such as Florence Nightingale who started a revolution with a pie chart. You will encounter dastardly villains who have tried to trip us up with dodgy data and misinformation. And you will learn how being smart and savvy with numbers, will help you be smart and savvy about everything else in life too.So grab your detective cap, pick up your magnifying glass and start seeing the world like never before.A must read for curious kids from presenter of BBC Radio 4's "More or Less", Tim Harford and with wonderful illustrations to enjoy by Ollie Mann.
Truth, Existence and Explanation: FilMat 2016 Studies in the Philosophy of Mathematics (Boston Studies in the Philosophy and History of Science #334)
by Mario Piazza Gabriele PulciniThis book contains more than 15 essays that explore issues in truth, existence, and explanation. It features cutting-edge research in the philosophy of mathematics and logic.Renowned philosophers, mathematicians, and younger scholars provide an insightful contribution to the lively debate in this interdisciplinary field of inquiry. The essays look at realism vs. anti-realism as well as inflationary vs. deflationary theories of truth. The contributors also consider mathematical fictionalism, structuralism, the nature and role of axioms, constructive existence, and generality. In addition, coverage also looks at the explanatory role of mathematics and the philosophical relevance of mathematical explanation.The book will appeal to a broad mathematical and philosophical audience. It contains work from FilMat, the Italian Network for the Philosophy of Mathematics. These papers collected here were also presented at their second international conference, held at the University of Chieti-Pescara, May 2016.
Truth or Beauty
by David OrrellFor millennia, scientists and philosophers have strived to show that the universe is governed by a few simple principles. These principles are not derived from science. They do not come from looking through telescopes or carefully examining the results generated by particle colliders. Rather,they are based on aesthetic laws and concepts such as symmetry, beauty, and unity. Scientist and author David Orrell considers how aesthetics have influenced the models we create in hopes of explaining our universe. His book begins with a look at early scientific thinkers, from the ancient Greeks to Galileo. The ancients constructed a concept of the world based on musical harmony;later thinkers overturned this concept, but replaced it with a program, based on Newton's "rational mechanics," to reduce the universe to a few simple equations. Orrell then turns to the scientific program of the twentieth century, culminating in supersymmetric string theory, which was againinfluenced by deep aesthetic principles. In a final section of the book, Orrell broadens his discussion to other fields of research, including economics, architecture, and health. Recent history has shown us what happens when financiers rely on a model of economics that resembles what a good theory"should look like" rather than the messy reality of human interaction. Ideas of mathematical elegance have inspired, entranced, and "sometimes misled" thinkers in their desire to find the laws that govern our universe. Orrell evaluates these aesthetic principles as a means of understanding the structure of the universe - let alone messy human society - and questionswhether they reflect an accurate way to understand our world. Truth or Beauty comes at an interesting period, when the multi-billion-dollar Large Hadron Collider in Switzerland has been designed to test the existence of exotic phenomena such as supersymmetric particles. Will these too turn out to be nothing more than a beautiful illusion?
Truth-Seeking by Abduction (Synthese Library #400)
by Ilkka NiiniluotoThis book examines the philosophical conception of abductive reasoning as developed by Charles S. Peirce, the founder of American pragmatism. It explores the historical and systematic connections of Peirce's original ideas and debates about their interpretations. Abduction is understood in a broad sense which covers the discovery and pursuit of hypotheses and inference to the best explanation. The analysis presents fresh insights into this notion of reasoning, which derives from effects to causes or from surprising observations to explanatory theories.The author outlines some logical and AI approaches to abduction as well as studies various kinds of inverse problems in astronomy, physics, medicine, biology, and human sciences to provide examples of retroductions and abductions. The discussion covers also everyday examples with the implication of this notion in detective stories, one of Peirce’s own favorite themes.The author uses Bayesian probabilities to argue that explanatory abduction is a method of confirmation. He uses his own account of truth approximation to reformulate abduction as inference which leads to the truthlikeness of its conclusion. This allows a powerful abductive defense of scientific realism. This up-to-date survey and defense of the Peircean view of abduction may very well help researchers, students, and philosophers better understand the logic of truth-seeking.
Try It! Even More Math Problems for All
by Jerry KaplanThis is not your typical math book.Try It! Even More Math Problems for All is the third of three collections of offbeat, open-ended problems designed to make even the most math-averse student excited about working through these challenging yet accessible problems.The Hints and Solutions section guides you to probe, suggest, and encourage students to explore even their most unusual insights on the way to solving these 25 new, illustrated problems of varying difficulty. As a result, you will be able to motivate your students to think creatively on their own and to engage in teamwork. And when students solve a problem, you will see and hear their accomplishments.Perfect for any math classroom, club, after school activity, or coaching session, Try It! celebrates not only the destination, but the journey, giving students a chance to think differently, and, above all, have fun!Can’t get enough? Volume 1 in the series, Try It! Math Problems for All, and Volume 2, Try It! More Math Problems for All, are also available at Routledge.com.Optional Student Workbook PacksIn addition to this teachers' guide, companion student workbooks are available in packs of ten. The student workbooks feature ample room for student responses and notes, make reviewing and providing feedback on student work easy, provide students with a quick reference to use during discussions, and they save time — there is no need to reproduce student handouts.