Continued Progress: Promising Evidence on Personalized Learning

The findings are grouped into four sections. The first section on student achievement finds that there were positive effects on student mathematics and reading performance and that the lowest-performing students made substantial gains relative to their peers. The second section on implementation and the perceptions of stakeholders finds that adoption of personalized learning practices varied considerably. Personalized learning practices that are direct extensions of current practice were more common, but implementation of some of the more challenging personalized learning strategies was less common. The third section relates implementation features to outcomes and identifies three elements of personalized learning that were being implemented in tandem in the schools with the largest achievement effects. Finally, the fourth section compares teachers' and students' survey responses to a national sample and finds some differences, such as teachers' greater use of practices that support competency-based learning and greater use of technology for personalization in the schools in this study with implementation data

Groups and semigroups with a one-counter word problem

We prove that a finitely generated semigroup whose word problem is a one-counter language has a linear growth function. This provides us with a very strong restriction on the structure of such a semigroup, which, in particular, yields an elementary proof of a result of Herbst, that a group with a one-counter word problem is virtually cyclic. We prove also that the word problem of a group is an intersection of finitely many one-counter languages if and only if the group is virtually abelian

Competing orders, non-linear sigma models, and topological terms in quantum magnets

A number of examples have demonstrated the failure of the Landau-Ginzburg-Wilson(LGW) paradigm in describing the competing phases and phase transitions of two dimensional quantum magnets. In this paper we argue that such magnets possess field theoretic descriptions in terms of their slow fluctuating orders provided certain topological terms are included in the action. These topological terms may thus be viewed as what goes wrong within the conventional LGW thinking. The field theoretic descriptions we develop are possible alternates to the popular gauge theories of such non-LGW behavior. Examples that are studied include weakly coupled quasi-one dimensional spin chains, deconfined critical points in fully two dimensional magnets, and two component massless $QED_3$. A prominent role is played by an anisotropic O(4) non-linear sigma model in three space-time dimensions with a topological theta term. Some properties of this model are discussed. We suggest that similar sigma model descriptions might exist for fermionic algebraic spin liquid phases.Comment: 11 pages, 1 figur