Jim Gleason

On the index of invariant subspaces in spaces of analytic functions of several complex variables

Abstract Let be the open unit ball in ℂ d , d ≧ 1, and Hd 2 be the space of analytic functions on determined by the reproducing kernel (1 − 〈 z, λ 〉)−1. This reproducing kernel Hilbert space serves a universal role in the model theory for d -contractions, i.e. tuples T = (T 1,…,Td ) of commuting operators on a Hilbert space such that ||T 1 x 1 + ⋯ + Td x d ||2 ≦ ||x 1||2 + ⋯ + ||xd ||2 for all x 1, … ,xd ∈ . If is a separable Hilbert space then we write Hd 2( ) ≅ Hd 2 ⊗ for the space of -valued Hd 2 functions and we use Mz = (,…, ) to denote the tuple of multiplication by the coordinate functions. We consider Mz -invariant subspaces ℳ ⊆Hd 2( ). The fiber dimension of ℳ  is defined to be . We show that if ℳ  has finite positive fiber dimension m, then the essential Taylor spectrum of Mz |ℳ , σe (Mz |ℳ ), equals ∂ plus possibly a subset of the zero set of a nonzero bounded analytic function on and ind(Mz − λ) |ℳ = (−1) dm for every λ ∈ \σe (Mz |ℳ ). As a corollary we prove that if T = (T 1,…,Td ) is a pure d-contraction of finite rank, then σe (T ) ∩ is contained in the zero set of a nonzero bounded analytic function and (−1) d ind(T − λ) = κ (T ) for all λ ∈ \σe (T ). Here κ(T ) denotes Arveson’s curvature invariant. We will also show that for d > 1 there are such d-contractions with σe (T ) ∩ ≠ ∅. These results answer a question of Arveson, [William Arveson, The Dirac operator of a commuting d-tuple, J. Funct. Anal. 189(1) (2002), 53–79]. We also prove related results for the Hardy and Bergman spaces of the unit ball and unit polydisc of ℂ d .

Developing and Validating a Reliable TPACK Instrument for Secondary Mathematics Preservice Teachers

Abstract Within the realm of teaching middle and high school mathematics, the ability to teach mathematics effectively using various forms of technology is now more important than ever, as expressed by the recent development of the Common Core State Standards for Mathematical Practice. This article presents the development process and the results from 15 institutions and more than 300 surveys completed by secondary mathematics preservice teachers. The results suggest that technological, pedagogical, and content knowledge; technology knowledge; content knowledge; and pedagogical knowledge constructs are valid and reliable, whereas pedagogical content knowledge, technological content knowledge, and technological pedagogical knowledge domains remain difficult for preservice teachers to separate and self-report.

Using Technology-Assisted Instruction and Assessment to Reduce the Effect of Class Size on Student Outcomes in Undergraduate Mathematics Courses

The implementation of online texts, videos, homework, and tests has changed the process of instruction in introductory college mathematics courses. With this change, more of the students’ learning takes place outside of the traditional college classroom and in places such as tutoring centers and dorm rooms. A combination of chi-square tests—for independence with unordered categorical data—and Mann-Whitney two-sample rank-sum tests—for continuous data and ordered categorical data—were used to analyze student outcomes generated from College Algebra and Applied Calculus courses with class sizes ranging from 37 to 129, with common syllabi, homework, quizzes, and tests. These tests showed that medium classes (30–55 students) had little to no benefit over large classes (110–130 students) in student learning and student achievement, with large classes having small to medium positive-effect sizes over medium classes in the area of student satisfaction. The only area in which the small classes had a small positive effect was in the area of student engagement.

Mathematics Classroom Observation Protocol for Practices (MCOP2): A validation study

ABSTRACT This article reports the robust validation process of the Mathematics Classroom Observation Protocol for Practices (MCOP2). The development of the instrument took into consideration both direct and dialogic instruction encompassing classroom interactions for the development of conceptual understanding, specifically examining teacher facilitation and student engagement. Instrument validation involved feedback from 164 external experts for content validity, interrater and internal reliability analyses, and structure analyses via scree plot analysis and exploratory factor analysis. Taken collectively, these results indicate that the MCOP2 measures the degree to which actions of teachers and students in mathematics classrooms align with practices recommended by national organizations and initiatives using a two-factor structure of teacher facilitation and student engagement.

Item efficiency: an item response theory parameter with applications for improving the reliability of mathematics assessment

ABSTRACT Many large universities, community colleges and some smaller four-year colleges are turning to hybrid or online instruction for remedial and entry level mathematics courses, often assessed using online exams in a proctored computer lab environment. Faculty face the task of choosing questions from a publishers text bank with very little, if any, background in test theory and design. We present a new item parameter, item efficiency, that is calculated from the results of an item response theory analysis of a comprehensive college algebra final examination and show that this new parameter may be used to identify items better suited for similar comprehensive final assessments. Further, by relating Item Efficiency to classical test theory item statistics, we propose guidelines that can be used to identify suitable items prior to testing with little or no background in psychometric theory.

Identification, Development and Implementation of Nanoscience Activities for Alabama K-12

We report on a pair of MSP (Mathematics & Science Partnership) START pilot projects designed to identify nanoscience experiments that will fit within the Alabama course of study for use in Alabama K-12 classrooms. As part of the first project we are testing the development, refinement and evaluation of an activity already partly developed. The form of this activity has had input from a focus group of RETs who were tasked to provide input into the activity and how it can be matched to components of the Alabama Course of Study. This activity consists of using sparks generated by abrasion of misch metal by sand paper of different grit size. Different grit sizes produce metal particles of different sizes, resulting in sparks of different size and length. If done in a dry box no sparks are produced and the powder left is not pyrophoric, demonstrating that high surface area, heat and oxygen are all required to produce sparks. SEM characterization of the powder allows the particle sizes to be determined, giving the correlation between size, grit size and spark track length. The activity was tested on groups of middle school science campers at McWane Science Center, and after evaluation, further modified to increase student interest and impact. The activity was then tested on grades 6-8 in a middle school classroom by a graduate student/undergraduate student team.