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)

La parution récente en français du livre de M. Denton : “Evolution. Une théorie en crise” , qui traite des theories explicatives actuelles de l'évolution, nous amine à rappeler les caracteres généraux des grandes theories biologiques et à présenter une critique sommaire du livre en question.La science West pas une simple accumulation de connaissances. Le scientifique ne doit pas se contenter de decrire et de mesurer des faits. Son but eat d'essayer de les relier et de construire des théories qui (...) tentent de nous faire comprendre le “reel”, objet de ses recherches.Cependant, la nature du matériel biologique — c'est-à-dire les espèces vivantes et les catégories qu'on y a reconnues, classe ou ordre par exemple — est si complexe qua la biologie actuelle est fort loin encore Wavoir réalisé une synthèse qui ressemble, méme très approximativement, à la classification de Mendéleiev pour les éléments chimiques.Malgré cette complexité, les biologques ont réussi à construire la th6orie du fait de l'evolution et ils proposent aujourd'hui une théorie explicative de cette évolution qui, bien qua critiquée, est incontestablement bien étayee. Nous allons essayer de définir la structure des grandes théories biologiques; ensuite nous analyserons sommairement quelques-unes des critiques qui sont adressees aux theories de l'évolution.Considering the publication in French version of Michael Denton's book, Evolution : A Theory in crisis , in which he critizes the synthetical theory of the biological evolution, we thought it useful to point out the main errors of argumentation contained in that book by first referring to the characteristics of general biological theories.In biology, there exist elementary theories - such as that dealing with the extinction of giant Reptiles - and general theories - such as the chromosomic theory of heredity and the explanatory theory of evolution. The latter generally consist of a group of elementary theories imbricated so that they form a whole like a jig-saw puzzle. An example is provided by the synthetic theory of evolution that can be defined in 20 propositions divided into two categories : the propositions relating to the naturalists' observations - which correspond to descriptive concepts - and those relating to the data of the population genetics - which correspond to explanatory concepts . The general scheme of the theory is formed by a network of relations between the descriptive concepts and the explanatory concepts. There also exist other theories in general biology supported by thousands of minor facts which only together can be of proof of validity. These theories are quite reliable and cannot be questioned by facts still to be explained. Questioning can only happen if they are confronted to well-established data in nearly equal numbers.In his work, Denton does not deny the phenomenon of biological evolution, but he criticizes the syntheticians' ideas by emphasizing that no great evolutive changes can be accounted for by the complex mechanism of random mutations and selection. He goes back to a typological concept of the zoological classification in some way similar to that of Vialleton's. However, Denton agrees with the phenomenon of the biological evolution and accepts the existence of "saltations" among the great zoological groups. The arguments used by Denton merely show that he is not conversant with the complexity of sciences particularly in comparative anatomy and zoology, the teaching of which is being somehow neglected today.By giving as an example the structure of the aortic arches in Amphibians and by a summary of the variations of those structures and the evolution of the blood-circulation in Vertebrates, we proved that the typological argument is of no value. As a matter of fact, a study of the now living beings clearly shows the gradual transition - e.g. a transition by a very large number of feasible stages - from the Lower Vertebrates to the Higher Vertebrates. In particular, there have been several attempts at double circulation and there still exist many species in which the blood of the double circulation is mixed. It seems therefore obvious that the double circulation has been achieved, stage by stage, with many intermediate schemes; we are inclined, therefore, to conclude that this succession of small transformations was produced by a huge mechanism of selected small mutations.Our present-day biological knowledge includes a number of examples large enough to consider the “argument of the impossible” used by Denton is due to inadequate fossil data and not to any mysterious mechanism such as macromutations. Let us say again that the typological structures seem to occur only when the data of comparative anatomy and zoology are over-simplified. (shrink)