An Advanced Mathematics Program for Middle School Teachers

The Conference Board of the Mathematical Sciences (CBMS), National Council of Teachers of Mathematics, and other organizations recommend twenty-one credits of mathematics coursework for prospective middle school teachers, beginning with a foundation based on mathematics for the elementary school curriculum, and followed by advanced courses directly addressing middle school mathematics. Three simultaneous factors—the emergence of the Interdisciplinary Liberal Studies Program at James Madison University, the release of CBMS guidelines, and a statewide focus on a critical shortage of qualified middle school teachers—provided an immediate audience for new upper-division courses built around the guidelines in probability/statistics, algebra, geometry, and calculus/analysis. We will discuss our experience with course planning and adaptation of other programs

Course, Counselor, and Teacher Gaps: Addressing the College Readiness Challenge in High Poverty High Schools

This paper presents a new analysis of education data on high schools in the 100 largest school districts that highlights the role of inadequate K-12 preparation as a barrier to postsecondary success for students who live in poverty. In particular, the analysis highlights stark differences in the quality of college preparation that high school students receive based on their schools' concentration of poverty. The paper compares characteristics of high-poverty high schools (more than 75 percent of students eligible for free or reduced lunch) to mid-high poverty (50-75 percent eligible), mid-low poverty (25-50 percent eligible), and low-poverty high schools (fewer than 25 percent of students eligible). Key findings include: Less-experienced and less-qualified teachers.College prep courses less likely to be offered.More schools without counselors

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

The yoga of commutators

In the present paper we discuss some recent versions of localisation methods for calculations in the groups of points of algebraic-like and classical-like groups. Namely, we describe relative localisation, universal localisation, and enhanced versions of localisation-completion. Apart from the general strategic description of these methods, we state some typical technical results of the conjugation calculus and the commutator calculus. Also, we state several recent results obtained therewith, such as relative standard commutator formulae, bounded width of commutators, with respect to the elementary generators, and nilpotent filtrations of congruence subgroups. Overall, this shows that localisation methods can be much more efficient, than expected

Hilbert's "Verunglueckter Beweis," the first epsilon theorem, and consistency proofs

In the 1920s, Ackermann and von Neumann, in pursuit of Hilbert's Programme, were working on consistency proofs for arithmetical systems. One proposed method of giving such proofs is Hilbert's epsilon-substitution method. There was, however, a second approach which was not reflected in the publications of the Hilbert school in the 1920s, and which is a direct precursor of Hilbert's first epsilon theorem and a certain 'general consistency result' due to Bernays. An analysis of the form of this so-called 'failed proof' sheds further light on an interpretation of Hilbert's Programme as an instrumentalist enterprise with the aim of showing that whenever a `real' proposition can be proved by 'ideal' means, it can also be proved by 'real', finitary means.Comment: 18 pages, final versio