Graphs, Matrices, and the GraphBLAS: Seven Good Reasons

The analysis of graphs has become increasingly important to a wide range of applications. Graph analysis presents a number of unique challenges in the areas of (1) software complexity, (2) data complexity, (3) security, (4) mathematical complexity, (5) theoretical analysis, (6) serial performance, and (7) parallel performance. Implementing graph algorithms using matrix-based approaches provides a number of promising solutions to these challenges. The GraphBLAS standard (istc- bigdata.org/GraphBlas) is being developed to bring the potential of matrix based graph algorithms to the broadest possible audience. The GraphBLAS mathematically defines a core set of matrix-based graph operations that can be used to implement a wide class of graph algorithms in a wide range of programming environments. This paper provides an introduction to the GraphBLAS and describes how the GraphBLAS can be used to address many of the challenges associated with analysis of graphs.Comment: 10 pages; International Conference on Computational Science workshop on the Applications of Matrix Computational Methods in the Analysis of Modern Dat

Alternative High School Math Pathways in Massachusetts: Developing an On-Ramp to Minimize College Remediation in Mathematics

Of the Massachusetts graduates from the Class of 2005 who enrolled in public colleges, an appalling 29 percent enrolled in a developmental (remedial) math course during the fall semester. Nationally, 63 percent of college students who remediate in mathematics do not earn a 2- or 4-year degree. At a time when a college degree is one of the critical components of one's ability to afford a home and support a family, that such high rates of Massachusetts' high school graduates require remediation in math is cause for alarm - and action. The Rennie Center for Education Research and Policy has produced a policy brief that proposes a new pathway in high school mathematics aimed at eliminating the need for college remediation in math.The policy brief, entitled Alternative High School Math Pathways in Massachusetts: Developing an On-Ramp to Minimize College Remediation in Mathematics, proposes a plan designed to significantly reduce, and ultimately, eliminate the number of students who require college remediation in mathematics.Rather than the traditional progression of math courses (Algebra I, Geometry, Algebra II, Calculus), we propose three new math courses at the middle and high school levels - including a new fourth year math course titled: Topics in Applied Mathematics for College Preparation that would provide an alternative to Pre-calculus/Calculus for students pursuing non-math related majors. We recommend that Massachusetts policymakers and school and district leaders should take the following steps toward establishing to a well-aligned, effective system that ensures all students are ready for college-level mathematics:Ensure mastery of arithmetic by the end of seventh grade;Focus on mastery and application of algebraic concepts;Offer the ACCUPLACER(R) test to high school juniors;Provide guidance based on the Elementary Algebra ACCUPLACER(R) score; andEncourage all students to take mathematics during their first college semester

Making Upper Division Mathematics Courses Relevant for Pre-Service Teachers

This article addresses the disconnect that in-service and pre-service secondary school teachers feel between the material presented in upper division mathematics courses and high school classroom practice. Two examples are given from an abstract algebra course in which this problem is addressed

Technology Solutions for Developmental Math: An Overview of Current and Emerging Practices

Reviews current practices in and strategies for incorporating innovative technology into the teaching of remedial math at the college level. Outlines challenges, emerging trends, and ways to combine technology with new concepts of instructional strategy

GHOST: Building blocks for high performance sparse linear algebra on heterogeneous systems

While many of the architectural details of future exascale-class high performance computer systems are still a matter of intense research, there appears to be a general consensus that they will be strongly heterogeneous, featuring "standard" as well as "accelerated" resources. Today, such resources are available as multicore processors, graphics processing units (GPUs), and other accelerators such as the Intel Xeon Phi. Any software infrastructure that claims usefulness for such environments must be able to meet their inherent challenges: massive multi-level parallelism, topology, asynchronicity, and abstraction. The "General, Hybrid, and Optimized Sparse Toolkit" (GHOST) is a collection of building blocks that targets algorithms dealing with sparse matrix representations on current and future large-scale systems. It implements the "MPI+X" paradigm, has a pure C interface, and provides hybrid-parallel numerical kernels, intelligent resource management, and truly heterogeneous parallelism for multicore CPUs, Nvidia GPUs, and the Intel Xeon Phi. We describe the details of its design with respect to the challenges posed by modern heterogeneous supercomputers and recent algorithmic developments. Implementation details which are indispensable for achieving high efficiency are pointed out and their necessity is justified by performance measurements or predictions based on performance models. The library code and several applications are available as open source. We also provide instructions on how to make use of GHOST in existing software packages, together with a case study which demonstrates the applicability and performance of GHOST as a component within a larger software stack.Comment: 32 pages, 11 figure