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"Mathematical Terminology."
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Primary teachers notice the impact of language on children's mathematical reasoning
Mathematical reasoning is now featured in the mathematics curriculum documents of many nations, but this necessitates changes to teaching practice and hence a need for professional learning. The development of children's mathematical reasoning requires appropriate encouragement and feedback from their teacher who can only do this if they recognise mathematical reasoning in children's actions and words. As part of a larger study, we explored whether observation of educators conducting mathematics lessons can develop teachers' sensitivity in noticing children's reasoning and consideration of how to support reasoning. In the Mathematical Reasoning Professional Learning Research Program, demonstration lessons were conducted in Australian and Canadian primary classrooms. Data sources included post-lesson group discussions. Observation of demonstration lessons and engagement in post-lesson discussions proved to be effective vehicles for developing a professional eye for noticing children's individual and whole-class reasoning. In particular, the teachers noticed that children struggled to employ mathematical language to communicate their reasoning and viewed limitations in language as a major barrier to increasing the use of mathematical reasoning in their classrooms. Given the focus of teachers' noticing of the limitations in some types of mathematical language, it seems that targeted support is required for teachers to facilitate classroom discourse for reasoning. [Author abstract]
Journal Article
Mathematical competencies revisited
This article deals with what it means to possess competence in mathematics. It takes its point of departure in the fact that the notions of mathematical competence and mathematical competencies have gained a foothold as well as momentum in mathematics education research, development and practice throughout the last two decades. The Danish so-called KOM Project (KOM: Competencies and the Learning of Mathematics), the report from which was published in 2002, has played an instrumental role in that development. Since then, a host of new developments has taken place, and we—as the authors of the original report—have felt the need to take stock of this development and revisit the conceptualisation of the basic notions in order to provide an updated version of the original conceptual framework and terminology. Whilst the fundamentals of this framework have been preserved in this article, the version presented here in addition to an up-to-date terminology offers greater clarity and sharpness and richer explanations than found in the original.
Journal Article
The many faces of working memory and short-term storage
The topic of working memory (WM) is ubiquitous in research on cognitive psychology and on individual differences. According to one definition, it is a small amount of information kept in a temporary state of heightened accessibility; it is used in most types of communication and problem solving. Short-term storage has been defined as the passive (i.e., non-attention-based, nonstrategic) component of WM or, alternatively, as a passive store separate from an attention-based WM. Here I note that much confusion has been created by the use by various investigators of many, subtly different definitions of WM and short-term storage. The definitions are sometimes made explicit and sometimes implied. As I explain, the different definitions may have stemmed from the use of a wide variety of techniques to explore WM, along with differences in theoretical orientation. By delineating nine previously used definitions of WM and explaining how additional ones may emerge from combinations of these nine, I hope to improve scientific discourse on WM. The potential advantages of clarity about definitions of WM and short-term storage are illustrated with respect to several ongoing research controversies.
Journal Article
Quantitative imaging biomarkers: A review of statistical methods for technical performance assessment
Technological developments and greater rigor in the quantitative measurement of biological features in medical images have given rise to an increased interest in using quantitative imaging biomarkers to measure changes in these features. Critical to the performance of a quantitative imaging biomarker in preclinical or clinical settings are three primary metrology areas of interest: measurement linearity and bias, repeatability, and the ability to consistently reproduce equivalent results when conditions change, as would be expected in any clinical trial. Unfortunately, performance studies to date differ greatly in designs, analysis method, and metrics used to assess a quantitative imaging biomarker for clinical use. It is therefore difficult or not possible to integrate results from different studies or to use reported results to design studies. The Radiological Society of North America and the Quantitative Imaging Biomarker Alliance with technical, radiological, and statistical experts developed a set of technical performance analysis methods, metrics, and study designs that provide terminology, metrics, and methods consistent with widely accepted metrological standards. This document provides a consistent framework for the conduct and evaluation of quantitative imaging biomarker performance studies so that results from multiple studies can be compared, contrasted, or combined.
Journal Article
Clinical prediction models: diagnosis versus prognosis
Clinical prediction models play an increasingly important role in contemporary clinical care, by informing healthcare professionals, patients and their relatives about outcome risks, with the aim to facilitate (shared) medical decision making and improve health outcomes. Diagnostic prediction models aim to calculate an individual's risk that a disease is already present, whilst prognostic prediction models aim to calculate the risk of particular heath states occurring in the future. This article serves as a primer for diagnostic and prognostic clinical prediction models, by discussing the basic terminology, some of the inherent challenges, and the need for validation of predictive performance and the evaluation of impact of these models in clinical care.
Journal Article
The emerging science of quantitative imaging biomarkers terminology and definitions for scientific studies and regulatory submissions
The development and implementation of quantitative imaging biomarkers has been hampered by the inconsistent and often incorrect use of terminology related to these markers. Sponsored by the Radiological Society of North America, an interdisciplinary group of radiologists, statisticians, physicists, and other researchers worked to develop a comprehensive terminology to serve as a foundation for quantitative imaging biomarker claims. Where possible, this working group adapted existing definitions derived from national or international standards bodies rather than invent new definitions for these terms. This terminology also serves as a foundation for the design of studies that evaluate the technical performance of quantitative imaging biomarkers and for studies of algorithms that generate the quantitative imaging biomarkers from clinical scans. This paper provides examples of research studies and quantitative imaging biomarker claims that use terminology consistent with these definitions as well as examples of the rampant confusion in this emerging field. We provide recommendations for appropriate use of quantitative imaging biomarker terminological concepts. It is hoped that this document will assist researchers and regulatory reviewers who examine quantitative imaging biomarkers and will also inform regulatory guidance. More consistent and correct use of terminology could advance regulatory science, improve clinical research, and provide better care for patients who undergo imaging studies.
Journal Article
Cyclification of Orbifolds
Inertia orbifolds homotopy-quotiented by rotation of
geometric
loops play a fundamental role not only in ordinary cyclic cohomology, but more recently in constructions of equivariant Tate-elliptic cohomology and generally of transchromatic characters on generalized cohomology theories. Nevertheless, existing discussion of such
cyclified stacks
has been relying on ad-hoc component presentations with intransparent and unverified stacky homotopy type. Following our previous formulation of transgression of cohomological charges (“double-dimensional reduction”), we explain how cyclification of
∞
-stacks is a fundamental and elementary base-change construction over moduli stacks in cohesive higher topos theory (cohesive homotopy-type theory). We prove that Ganter/Huan’s extended inertia groupoid used to define equivariant quasi-elliptic cohomology is indeed a model for this intrinsically defined cyclification of orbifolds, and we show that cyclification implements transgression in group cohomology in general, and hence in particular the transgression of degree-4 twists of equivariant Tate-elliptic cohomology to degree-3 twists of orbifold K-theory on the cyclified orbifold. As an application, we show that the universal shifted integral 4-class of equivariant 4-Cohomotopy theory on ADE-orbifolds induces the Platonic 4-twist of ADE-equivariant Tate-elliptic cohomology, and we close by explaining how this should relate to elliptic M5-brane genera, under our previously formulated
Hypothesis H
.
Journal Article
Standardizing Terminology and Definitions of Medication Adherence and Persistence in Research Employing Electronic Databases
Objective: To propose a unifying set of definitions for prescription adherence research utilizing electronic health record prescribing databases, prescription dispensing databases, and pharmacy claims databases and to provide a conceptual framework to operationalize these definitions consistently across studies. Methods: We reviewed recent literature to identify definitions in electronic database studies of prescription-filling patterns for chronic oral medications. We then develop a conceptual model and propose standardized terminology and definitions to describe prescription-filling behavior from electronic databases. Results: The conceptual model we propose defines 2 separate constructs: medication adherence and persistence. We define primary and secondary adherence as distinct subtypes of adherence. Metrics for estimating secondary adherence are discussed and critiqued, including a newer metric (New Prescription Medication Gap measure) that enables estimation of both primary and secondary adherence. Discussion: Terminology currently used in prescription adherence research employing electronic databases lacks consistency. We propose a. clear, consistent, broadly applicable conceptual model and terminology for such studies. The model and definitions facilitate research utilizing electronic medication prescribing, dispensing, and/or claims databases and encompasses the entire continuum of prescription-filling behavior. Conclusion: Employing conceptually clear and consistent terminology to define medication adherence and persistence will facilitate future comparative effectiveness research and meta-analytic studies that utilize electronic prescription and dispensing records.
Journal Article
Tame distillation and desingularization by p-alterations
We strengthen Gabber's l'-alteration theorem by avoiding all primes invertible on a scheme. In particular, we prove that any scheme X of finite type over a quasi-excellent threefold can be desingularized by a char(X)-alteration, i.e., an alteration whose order is only divisible by primes noninvertible on X. The main new ingredient in the proof is a tame distillation theorem asserting that, after enlarging, any alteration of X can be split into a composition of a tame Galois alteration and a char(X)-alteration. The proof of the distillation theorem is based on the following tameness theorem that we deduce from a theorem of M. Pank: if a valued field k of residue characteristic p has no nontrivial p-extensions, then any algebraic extension l/k is tame.
Journal Article