Abstract. We describe the derived functor DRep V (A) of the affine representation scheme Rep V (A) parametrizing the representations of an associative k-algebra A on a finite-dimensional vector space V . We construct the characteristic maps Tr V (A)n : HCn (

Effective mathematics teachers have a large body of professional knowledge, which is largely undocumented and shared by teachers working in a given country's education system. The volume and cultural nature of this knowledge make it particularly challenging to share curricula and instructional methods between countries. Thus, approaches based on knowledge engineering-designing a software system by interviewing human experts to extract their knowledge and heuristics-are particularly promising for cross-cultural curriculum implementations. Reasoning Mind's Genie 2 system demonstrates the viability of such an approach, bringing elements of Russian mathematics education (known for its effectiveness) to the United States. Genie 2 has been adopted on a large scale, with around 67,000 United States students participating in the 2012-2013 school year. Previously published work (some of it in peer reviewed articles and some in non-peer-reviewed independent evaluations) has associated Genie 2 with high student and teacher acceptance, increases in test scores relative to "business as usual" conditions, high time on task, and a positive affective profile. Here, we describe for the first time the design, function, and use of the Genie 2 system. Based on this work, we suggest general principles-which collectively represent a proposed methodology-for the design of intelligent tutoring systems intended for cross-cultural transfer of curriculum and instructional methods.

This chapter discusses the interplay between instruction modeling and applied learning science in developing blended learning programs. Far from being two alternative methods, these two approaches complement one another. We discuss in depth the Edifice program; this program systematically applies Mayer’s cognitive theory of multimedia learning (CTML), which is perhaps the most developed and empirically supported theory guiding the design of blended learning interfaces.

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Blended learning, which is instruction that combines online with face-to-face components, is becoming increasingly popular in American schools. However, few first-hand accounts have been provided by program creators concerning the design, use, and experiences of widely used blended learning programs. This book fills this gap. Written by a leading program designer, it describes the creation and implementation of the blended learning programs of Reasoning Mind, a non-profit organization which has served hundreds of thousands of elementary and middle school students. Besides general insights into blended learning design and implementation, the book offers a detailed discussion of instruction modeling, a blended learning design approach used by Reasoning Mind. A wide range of methods can be used in designing instructional technology programs: some rely on research in cognitive psychology, others on gaming, and still others on modern statistical methods such as “big data.” By contrast, instruction modeling relies on the careful analysis of existing instructional traditions. The idea of instruction modeling is to study high-quality offline instruction and use blended learning to recreate it on a larger scale. The aim is to give students equivalent educational experiences to those of children in the world’s best classrooms. This book describes the instruction modeling technique in detail and makes the case for its broader use. The book will be relevant to anyone interested in the practical design and evolution of blended learning, including researchers, instructional designers, teachers, and students of education.