Developing Essential Understanding of Rational Numbers for Teaching Mathematics in Grades 3/5


Book Description

What is the relationship between fractions and rational numbers? Can you explain why the product of two fractions between 0 and 1 is less than either factor? How are rational numbers related to irrational numbers, which your students will study in later grades? How much do you know… and how much do you need to know? Helping your upper elementary school students develop a robust understanding of rational numbers requires that you understand this mathematics deeply. But what does that mean? This book focuses on essential knowledge for teachers about rational numbers. It is organised around four big ideas, supported by multiple smaller, interconnected ideas-essential understandings. Taking you beyond a simple introduction to rational numbers, the book will broaden and deepen your mathematical understanding of one of the most challenging topics for students and teachers. It will help you engage your students, anticipate their perplexities, avoid pitfalls and dispel misconceptions. You will also learn to develop appropriate tasks, techniques and tools for assessing students’ understanding of the topic. Focus on the ideas that you need to understand thoroughly to teach confidently.




Understanding and Explanation


Book Description

Understanding and Explanation clarifies the "explanation versus understanding" debate that has become central to the philosophy of the social sciences.




Faithfully Seeking Understanding


Book Description

The selections are taken from chapters of Kuhn's books, including his Catholic Dogmatics, and from various articles that appeared in the Tübingen quarterly, Die theologische Quartalschrift. They are arranged according to the following topics: faith and reason; revelation and the personhood of God; responses to Protestant liberalism; theological topics, including grace and freedom, theology and the university; and Catholic fundamental theology. Though diverse in subject, when read together the essays yield a complete theological vision that derives from a carefully constructed theological foundation.




Understanding Abortion


Book Description

Books on abortion (other than collections of readings) typically express and defend a particular position. This book gives both sides, as evenly and objectively as possible; it gets to the heart of each position, the core idea which animates it. It then leaves the reader to make up his or her own mind. It is an introduction to the issue, not only to the basic positions on the issue. Despite being brief, it contains careful analyses and discussions of many topics often not found at all in other works. The treatment is thorough and detailed, but succinct. Understanding Abortion: From Mixed Feelings to Rational Thought is aimed at all people who want a better understanding of what the two sides on this issue are really saying, and what reasons they give for their position. Many people assume that this issue is an interminable one, with "no clear answers"; a purely emotional debate that cannot be addressed by the use of reasoned arguments. The book shows that this is not the case.




The Psychology and Law of Criminal Justice Processes


Book Description

Psychological science now reveals much about the law's response to crime. This is the first text to bridge both fields as it presents psychological research and theory relevant to each phase of criminal justice processes. The materials are divided into three parts that follow a comprehensive introduction. The introduction analyses the major legal themes and values that guide criminal justice processes and points to the many psychological issues they raise. Part I examines how the legal system investigates and apprehends criminal suspects. Topics range from the identification, searching and seizing to the questioning of suspects. Part II focuses on how the legal system establishes guilt. To do so, it centres on the process of bargaining and pleading cases, assembling juries, providing expert witnesses, and considering defendants' mental states. Part III focuses on the disposition of cases. Namely, that part highlights the process of sentencing defendants, predicting criminal tendencies, treating and controlling offenders, and determining eligibility for such extreme punishments as the death penalty. The format seeks to give readers a feeling for the entire criminal justice process and for the role psychological science has and can play in it.










Rational Causation


Book Description

We explain what people think and do by citing their reasons, but how do such explanations work, and what do they tell us about the nature of reality? Contemporary efforts to address these questions are often motivated by the worry that our ordinary conception of rationality contains a kernel of supernaturalism—a ghostly presence that meditates on sensory messages and orchestrates behavior on the basis of its ethereal calculations. In shunning this otherworldly conception, contemporary philosophers have focused on the project of “naturalizing” the mind, viewing it as a kind of machine that converts sensory input and bodily impulse into thought and action. Eric Marcus rejects this choice between physicalism and supernaturalism as false and defends a third way. He argues that philosophers have failed to take seriously the idea that rational explanations postulate a distinctive sort of causation—rational causation. Rational explanations do not reveal the same sorts of causal connections that explanations in the natural sciences do. Rather, rational causation draws on the theoretical and practical inferential abilities of human beings. Marcus defends this position against a wide array of physicalist arguments that have captivated philosophers of mind for decades. Along the way he provides novel views on, for example, the difference between rational and nonrational animals and the distinction between states and events.




Essays on Ethics, Social Behaviour, and Scientific Explanation


Book Description

When John Harsanyi came to Stanford University as a candidate for the Ph.D., I asked him why he was bothering, since it was most un likely that he had anything to learn from us. He was already a known scho lar; in addition to some papers in economics, the first two papers in this vol ume had already been published and had dazzled me by their originality and their combination of philosophical insight and technical competence. However, I am very glad I did not discourage him; whether he learned any thing worthwhile I don't know, but we all learned much from him on the foundations of the theory of games and specifically on the outcome of bar gaining. The central focus of Harsanyi's work has continued to be in the theory of games, but especially on the foundations and conceptual problems. The theory of games, properly understood, is a very broad approach to social interaction based on individually rational behavior, and it connects closely with fundamental methodological and substantive issues in social science and in ethics. An indication of the range of Harsanyi's interest in game the ory can be found in the first paper of Part B -though in fact his owncontri butions are much broader-and in the second paper the applications to the methodology of social science. The remaining papers in that section show more specifically the richness of game theory in specific applications.




Learning Forensic Assessment


Book Description

Major developments in the field since the publication of Learning Forensic Assessment are integrated in this revised edition, including revised editions of the DSM-5, HCR-20 scale, and child custody guidelines. This textbook is designed for graduate students learning forensic assessment and psychologists coming to forensic practice later in their careers. It is organized around five broad areas: Professional and Practice Issues, Adult Forensic Assessment, Juvenile Forensic Assessment, Civil Forensic Assessment, and Communicating Your Findings. Each chapter begins with a strong teaching and learning foundation. The latter part of each chapter is assessment specific, covering available assessment measures and approaches to assessment. The authors go well beyond simple descriptions of assessment measures and provide a conceptual discussion of the evaluation process that helps the reader understand how assessment measures fit into the overall evaluation process. The evaluation component is geared toward assessing the important aspects of the construct as laid out in the early part of each chapter. Each chapter then concludes with a case example to illustrate the measures and techniques described.