Next-to-MHV Yang-Mills kinematic algebra

Kinematic numerators of Yang-Mills scattering amplitudes possess a rich Lie algebraic structure that suggest the existence of a hidden infinite-dimensional kinematic algebra. Explicitly realizing such a kinematic algebra is a longstanding open problem that only has had partial success for simple helicity sectors. In past work, we introduced a framework using tensor currents and fusion rules to generate BCJ numerators of a special subsector of NMHV amplitudes in Yang-Mills theory. Here we enlarge the scope and explicitly realize a kinematic algebra for all NMHV amplitudes. Master numerators are obtained directly from the algebraic rules and through commutators and kinematic Jacobi identities other numerators can be generated. Inspecting the output of the algebra, we conjecture a closed-form expression for the master BCJ numerator up to any multiplicity. We also introduce a new method, based on group algebra of the permutation group, to solve for the generalized gauge freedom of BCJ numerators. It uses the recently introduced binary BCJ relations to provide a complete set of NMHV kinematic numerators that consist of pure gauge.

Ringel Duals of Brauer Algebras via Super Groups

We prove that the Brauer algebra, for all parameters for which it is quasi-hereditary, is Ringel dual to a category of representations of the orthosymplectic super group. As a consequence we obtain new and algebraic proofs for some results on the fundamental theorems of invariant theory for this super group over the complex numbers and also extend them to some cases in positive characteristic. Our methods also apply to the walled Brauer algebra in which case we obtain a duality with the general linear super group, with similar applications.

The Unintended Consequences of an Algebra-for-All Policy on High-Skill Students

In 1997, Chicago implemented a policy that required algebra for all ninth-grade students, eliminating all remedial coursework. This policy increased opportunities to take algebra for low-skill students who had previously enrolled in remedial math. However, little is known about how schools respond to the policy in terms of organizing math classrooms to accommodate curricular changes. The policy unintentionally affected high-skill students who were not targeted by the policy—those who would enroll in algebra in its absence. Using an interrupted time-series design combined with within-cohort comparisons, this study shows that schools created more mixed-ability classrooms when eliminating remedial math classes, and peer skill levels declined for high-skill students. Consequently, their test scores also declined.

An Examination of Algebra for All through Historic Context and Statewide Assessment Data

Since 2003, California has enacted a policy through its education accountability system that encourages schools and districts to place all 8th grade students into algebra courses and therefore, be tested in algebra in the statewide assessment program. Ten years later, there are a great many more 8th graders taking algebra now. However, there are also many students repeating algebra, instead of going on taking higher level mathematics tests. This article aims to provide the historic context of this policy, previous and recent studies on 8th grade algebra, and our study based on the California Standardized Testing and Reporting (STAR) data. We analyzed 8th grade algebra test-taking and the following years� higher level mathematics test-taking to examine the college preparation course taking pipeline. Our longitudinal study compared two groups of students� performance on 9th grade algebra between those who previously scored below proficient on algebra at 8th grade and those who scored proficient or above on general mathematics at 8th grade. Further, another longitudinal study linked 7th grade mathematics sub-scores to 8th grade algebra achievement. The results show that �algebra for all� policy increased the number of students taking algebra at 8th grade and subsequently, taking higher level mathematics tests. However, the pipeline of the college preparation course taking has a significant leak because the number of students taking higher level mathematics decreased dramatically after algebra. Longitudinal study shows that students who pass the general mathematics test at 8th grade have a 69% greater chance to pass the algebra test at 9th grade compared to their peers who failed the algebra test at 8th grade. We also find that the sub-score rational numbers is a strong predictor of 8th grade algebra achievement. Alternatives to help all students achieve in mathematics learning are also discussed in addition to recommendations for future research.

Book Review: Mathematics Educators Respond to Kaput's “Algebra Problem”: A Review of Algebra in the Early Grades

In the last few decades, mathematics education in the United States has seen a perfect storm with respect to the teaching and learning of algebra—one that is difficult for our colleagues in other countries to fathom. As part of recent largescale education reform in the United States, the increasingly widely perceived need for greater mathematical literacy and the desire to make access to college more equitable in the society have led to promotion of “algebra for all” and the codification of this desire in high-stakes accountability measures (e.g., as illustrated by Achieve's American Diploma Project (ADP, 2004)). Algebra in the Early Grades, an edited volume by a group of mathematics education researchers, is in important ways a response of mathematics educators to these developments.