Samarium Barbier reactions of alpha-iodomethyloxazoles and thiazoles with aliphatic aldehydes

[reaction: see text] The reductive coupling of substituted alpha-iodomethyloxazoles and thiazoles with aliphatic aldehydes under Barbier conditions provides an effective method for the direct incorporation of intact heterocyclic systems.     

A Dialectic Analysis of Generativity: Issues of Network-SupportedDesign in Mathematics and Science

New theoretical, methodological, and design frameworks for engaging classroom learning are supported by the highly interactive and group-centered capabilities of a new generation of classroom-based networks. In our analyses, networked teaching and learning are organized relative to a dialectic of (a) seeing mathematical and scientific structures as fully situated in sociocultural contexts and (b) seeing mathematics as a way of structuring our understanding of and design for group-situated teaching and learning. An engagement with this dialectic is intended to open up new possibilities for understanding the relations between content and social activity in classrooms. Features are presented for what we call generative design in terms of the respective “sides” of the dialectic. Our approach to generative design centers on the notion that classrooms have multiple agents, interacting at various levels of participation, and looks to make the best possible use of the plurality of emergent ideas found in classrooms. We close with an examination of how this dialectic framework also can support constructive critique of both sides of the dialectic in terms of content and pedagogy.     

Understanding Qualitative Calculus: A Structural Synthesis of Learning Research

More than a decade of research and innovation in using computer-based graphing and simulation environments has encouraged many of us in the research community to believe important dimensions of calculus-related reasoning can be successfully understood by young learners. This paper attempts to address what kinds of calculus-related insights seem to typify this early form of calculus reasoning. The phrase “qualitative calculus” is introduced to frame the analysis of this “other” calculus. The learning of qualitative calculus is the focus of the synthesis. The central claim is that qualitative calculus is a cognitive structure in its own right and that qualitative calculus develops or evolves in ways that seem to fit with important general features of Piaget's analyses of the development of operational thought. In particular, the intensification of rate and two kinds of reversibility between what are called “how much” (amount) and “how fast” (rate) quantities are what interactively, and collectively,characterize and help to define understanding qualitative calculus. Although sharing a family resemblance with traditional expectations of what it might mean to learn calculus, qualitative calculus does not build from ratio- or proportion-based ideas of slope as they are typically associated with defining rate. The paper does close, however, with a discussion of how understanding qualitative calculus can support and link to the rate-related literature of slope, ratio and proportion. Additionally, curricular connections and implications are discussed throughout to help illustrate and explore the significance of learning qualitative calculus. This revised version was published online in July 2006 with corrections to the Cover Date.     

Chemoselective amination of 5-bromo-2-chloro-3-fluoropyridine

The chemoselective functionalization of 5-bromo-2-chloro-3-fluoropyridine (1c) is described. Catalytic amination conditions (Pd2dba3, Xantphos, base) afford exclusively the bromide substitution product (2) for both secondary amines and primary anilines. A reversal in chemoselectivity is observed under neat conditions in the absence of palladium catalysis, with substitution at the 2-chloro position preferred to generate 3. Last, selective substitution of the 3-fluoro group is achieved under SNAr conditions to afford the dihalo adduct (4).     

On a new stopping rule for stochastic approximation

Summary A new stopping rule for the Robbins-Monro process, based on an F-statistic criterion is proposed and its asymptotic behavior established. On the basis of evidence obtained through experimental sampling, the procedure seems to work well over a wide variety of situations. A two-stage procedure, coupling the new rule with an earlier one proposed by Sielken [1973] is recommended for practical use.