This fourth volume in the series of yearbooks by the Association of Mathematics Educators in Singapore entitled Reasoning, Communication and Connections in Mathematics is unique in that it focuses on a single theme in mathematics education. The objective is to encourage teachers and researchers to advance reasoning, communication and connections in mathematics classrooms. Several renowned international researchers in the field have published their work in this volume. The fifteen chapters of the book illustrate evidence-based practices that school teachers and researchers can experiment with in their own classrooms to bring about meaningful learning outcomes. Three major themes: mathematical tasks, classroom discourse, and connectivity within and beyond mathematics, shape the ideas underpinning reasoning, communication and connections in these chapters. The book makes a significant contribution towards mathematical processes essential for learners of mathematics. It is a good resource for mathematics educators and research students.

These are the proceedings of the 8th European Conference on Symbolic and Quantitative Approaches to Reasoning with Uncertainty, ECSQARU 2005, held in Barcelona (Spain), July 6–8, 2005. The ECSQARU conferences are biennial and have become a major forum for advances in the theory and practice of r- soning under uncertainty. The ?rst ECSQARU conference was held in Marseille (1991), and after in Granada (1993), Fribourg (1995), Bonn (1997), London (1999), Toulouse (2001) and Aalborg (2003). The papers gathered in this volume were selected out of 130 submissions, after a strict review process by the members of the Program Committee, to be presented at ECSQARU 2005. In addition, the conference included invited lectures by three outstanding researchers in the area, Seraf ́ ?n Moral (Imprecise Probabilities), Rudolf Kruse (Graphical Models in Planning) and J ́ erˆ ome Lang (Social Choice). Moreover, the application of uncertainty models to real-world problems was addressed at ECSQARU 2005 by a special session devoted to s- cessful industrial applications, organized by Rudolf Kruse. Both invited lectures and papers of the special session contribute to this volume. On the whole, the programme of the conference provided a broad, rich and up-to-date perspective of the current high-level research in the area which is re?ected in the contents of this volume. IwouldliketowarmlythankthemembersoftheProgramCommitteeandthe additional referees for their valuable work, the invited speakers and the invited session organizer.

We do not perceive the present as it is and in totality, nor do we infer the future from the present with any high degree of dependability, nor yet do we accurately know the consequences of our own actions. In addition, there is a fourth source of error to be taken into account, for we do not execute actions in the precise form in which they are imaged and willed. Frank H. Knight [R4.34, p. 202] The “degree” of certainty of confidence felt in the conclusion after it is reached cannot be ignored, for it is of the greatest practical signi- cance. The action which follows upon an opinion depends as much upon the amount of confidence in that opinion as it does upon fav- ableness of the opinion itself. The ultimate logic, or psychology, of these deliberations is obscure, a part of the scientifically unfathomable mystery of life and mind. Frank H. Knight [R4.34, p. 226-227] With some inaccuracy, description of uncertain consequences can be classified into two categories, those which use exclusively the language of probability distributions and those which call for some other principle, either to replace or supplement.

This book provides professional development leaders and teachers with a framework for integrating authentic real-world performance tasks into science, technology, engineering, and mathematics (STEM) classrooms. We incorporate elements of problem-based learning to engage students around grand challenges in energy and environment, place-based leaning to motivate students by relating the problem to their community, and Understanding by Design to ensure that understanding key concepts in STEM is the outcome. Our framework has as a basic tenet interdisciplinary STEM approaches to studying real-world problems. We invited professional learning communities of science and mathematics teachers to bring multiple lenses to the study of these problems, including the sciences of biology, chemistry, earth systems and physics, technology through data collection tools and computational science modeling approaches, engineering design around how to collect data, and mathematics through quantitative reasoning. Our goal was to have teachers create opportunities for their students to engage in real-world problems impacting their place; problems that could be related to STEM grand challenges demonstrating the importance and utility of STEM. We want to broaden the participation of students in STEM, which both increases the future STEM workforce, providing our next generation of scientists, technologists, engineers, and mathematicians, as well as producing a STEM literate citizenry that can make informed decisions about grand challenges that will be facing their generation. While we provide a specifi c example of an interdisciplinary STEM module, we hope to do more than provide a single fish. Rather we hope to teach you how to fish so you can create modules that will excite your students.

This book constitutes the refereed proceedings of the 11th European Conference on Symbolic and Quantitative Approaches to Reasoning with Uncertainty, ECSQARU 2011, held in Belfast, UK, in June/July 2011. The 60 revised full papers presented together with 3 invited talks were carefully reviewed and selected from 108 submissions. The papers are organized in topical sections on argumentation; Bayesian networks and causal networks; belief functions; belief revision and inconsistency handling; classification and clustering; default reasoning and logics for reasoning under uncertainty; foundations of reasoning and decision making under uncertainty; fuzzy sets and fuzzy logic; implementation and applications of uncertain systems; possibility theory and possibilistic logic; and uncertainty in databases.