Given a continuous linear operator T ∈ L(X) defined on a separable F -space X, we will show that T satisfies the Hypercyclicity Criterion if and only if for any strictly increasing sequence of positive integers {n k } k such that sup k {n k+1 −n k } < ∞, the sequence {T n k } k is hypercyclic. In contrast we will also prove that, for any hypercyclic vector x ∈ X of T , there exists a strictly increasing sequence {n k } k such that sup k {n k+1 − n k } = 2 and {T n k x} k is somewhere dense, but not dense in X. That is, T and {T n k } k do not share the same hypercyclic vectors. (2000). Primary 47A16 ; Secondary 37D45, 46A04.
Mathematics Subject Classification
Previous investigations of the ability of content experts and test developers to estimate item difficulty have, for the most part, produced disappointing results. These investigations were based on a noncomparative method of independently rating the difficulty of items. In this article, we argue that, by eliciting comparative judgments of difficulty, judges can more accurately estimate item difficulties. In this study, judges from different backgrounds rank ordered the difficulty of SAT® mathematics items in sets of 7 items. Results showed that judges are reasonably successful in rank ordering several items in terms of difficulty, with little variability across judges and content areas. Simulations of a possible implementation of comparative judgments for difficulty estimation show that it is possible to achieve high correlations between true and estimated difficulties with relatively few comparisons. Implications of these results for the test development process are discussed.
In 2017, the mathematics assessments that are part of the National Assessment of Educational Progress (NAEP) program underwent a transformation shifting the administration from paper-and-pencil formats to digitally-based assessments (DBA). This shift introduced new interactive item types that bring rich process data and tremendous opportunities to study the cognitive and behavioral processes that underlie test-takers’ performances in ways that are not otherwise possible with the response data alone. In this exploratory study, we investigated the problem-solving processes and strategies applied by the nation’s fourth and eighth graders by analyzing the process data collected during their interactions with two technology-enhanced drag-and-drop items (one item for each grade) included in the first digital operational administration of the NAEP’s mathematics assessments. Results from this research revealed how test-takers who achieved different levels of accuracy on the items engaged in various cognitive and metacognitive processes (e.g., in terms of their time allocation, answer change behaviors, and problem-solving strategies), providing insights into the common mathematical misconceptions that fourth- and eighth-grade students held and the steps where they may have struggled during their solution process. Implications of the findings for educational assessment design and limitations of this research are also discussed.
In order to facilitate the interpretation of test scores from the TOEIC® Bridge as a measure of English language proficiency, one form of the test was administered to more than 6000 test takers in three South American c ountries — Colombia, Chile and Ecuador. The appropriateness of the TOEIC Bridge test as a measure of English language skill was evaluated using several approaches, such as factor analysis and computation of correla tions between the TOEIC Bridge scores and other measures like local English test scores, student self-assessment scores, and teacher ratings. The evidence collected was generally consistent with the interpretation of TOEIC Bridge scores as indicators of English language competencies for the students examined.
The validity of TOEIC Bridge™ scores as a measure of English language skill was examined from the standpoint of a unified concept of test validity. In this study, more than 6,000 test takers in 3 Latin American countries (Chile, Colombia, and Ecuador) took 1 form of the TOEIC Bridge test, and their scores were compared to additional information about the students (teacher judgments, self‐assessments, and performance on other academic achievement measures). The evidence collected was generally quite consistent with the interpretation of TOEIC Bridge scores as indicators of the English language competencies for the students examined.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.