Connecting numbers to discrete quantification: A step in the child’s construction of integer concepts

The present study asks when young children understand that number words quantify over sets of discrete individuals. For this study, 2- to 4-year-old children were asked to extend the number word five or six either to a cup containing discrete objects (e.g., blocks) or to a cup containing a continuous substance (e.g., water). In Experiment 1, only children who knew the exact meanings of the words one, two and three extended higher number words (five or six) to sets of discrete objects. In Experiment 2, children who only knew the exact meaning of one extended higher number words to discrete objects under the right conditions (i.e., when the problem was first presented with the number words one and two). These results show that children have some understanding that number words pertain to discrete quantification from very early on, but that this knowledge becomes more robust as children learn the exact, cardinal meanings of individual number words

A picture of eight turtles: the child’s understanding of cardinality and numerosity

An essential part of understanding number words (e.g., eight) is understanding that all number words refer to the dimension of experience we call numerosity. Knowledge of this general principle may be separable from knowledge of individual number word meanings. That is, children may learn the meanings of at least a few individual number words before realizing that all number words refer to numerosity. Alternatively, knowledge of this general principle may form relatively early and proceed to guide and constrain the acquisition of individual number word meanings. The current article describes two experiments in which 116 children (2½- to 4-year-olds) were given a Word Extension task as well as a standard Give-N task. Results show that only children who understood the cardinality principle of counting successfully extended number words from one set to another based on numerosity—with evidence that a developing understanding of this concept emerges as children approach the cardinality principle induction. These findings support the view that children do not use a broad understanding of number words to initially connect number words to numerosity but rather make this connection around the time that they figure out the cardinality principle of counting

Learning to represent exact numbers

This article focuses on how young children acquire concepts for exact, cardinal numbers (e.g., three, seven, two hundred, etc.). I believe that exact numbers are a conceptual structure that was invented by people, and that most children acquire gradually, over a period of months or years during early childhood. This article reviews studies that explore children’s number knowledge at various points during this acquisition process. Most of these studies were done in my own lab, and assume the theoretical framework proposed by Carey (2009). In this framework, the counting list (‘one,’ ‘two,’ ‘three,’ etc.) and the counting routine (i.e., reciting the list and pointing to objects, one at a time) form a placeholder structure. Over time, the placeholder structure is gradually filled in with meaning to become a conceptual structure that allows the child to represent exact numbers (e.g., There are 24 children in my class, so I need to bring 24 cupcakes for the party.) A number system is a socially shared, structured set of symbols that pose a learning challenge for children. But once children have acquired a number system, it allows them to represent information (i.e., large, exact cardinal values) that they had no way of representing before

On asymptotic approach to reliability improvement of multi-state systems with components quantitative and qualitative redundancy : series and parallel systems

The paper is composed of two parts, in this part after introducing the multi-state and the asymptotic approaches to system reliability evaluation the multi-state homogeneous series and parallel systems with reserve components are defined and their multi-state limit reliability functions are determined. In order to improve of the reliability of these systems the following methods are used: (i) a warm duplication of components, (ii) a cold duplication of components, (iii) a mixed duplication of components, (iv) improving the reliability of components by reducing their failure rate. Next, the effects of the systems’ reliability different improvements are compared

Reliability and risk function improvement of bulk cargo transportation system

In the paper the basic notions of the ageing multistate systems reliability analysis are introduced. The multistate system reliability functions are defined and the mean values of the multistate system lifetimes in the reliability state subsets and in the particular reliability states are determined. The notion of the multistate system risk function and the moment of the system exceeding the critical reliability state are introduced. Further, in the developed reliability models, it is assumed that the system’s components have the multistate Weibull reliability functions with various parameters in their different reliability state subsets. Under this assumption, the proposed multistate system reliability models are applied in maritime transport to the reliability analysis of a bulk cargo transportation system and its reliability function, moreover other main characteristics are determined

On asymptotic approach to reliability improvement of multi-state systems with components quantitative and qualitative redundancy : „m out of n” systems

The paper is composed of two parts, in this part the multi-state homogeneous „m out of n” systems with reserve components are defined and their multi-state limit reliability functions are determined. In order to improve of the reliability of these systems the following methods are used: (i) a warm duplication of components, (ii) a cold duplication of components, (iii) a mixed duplication of components, (iv) improving the reliability of components by reducing their failure rate. Next, the effects of the systems’ reliability different improvements are compared

Models of reliability and availability improvement of series and parallel systems related to their operation processes

Integrated general models of approximate approaches of complex multi-state series and parallel systems, linking their reliability and availability improvement models and their operation processes models caused changing reliability and safety structures and components reliability characteristics in different operations states, are constructed. These joint models are applied to determining improved reliability and availability characteristics of the considered multi-state series and parallel systems related to their varying in time operation processes. The conditional reliability characteristics of the multi-state systems with hot, cold single reservation of component and the conditional reliability characteristics of the multi-state systems with reduced rate of departure by a factor of system components are defined

General approach to Baltic electric cable critical infrastructure network operation process Modelling

In the paper, the critical infrastructure operation process is defined and its main parameters are fixed. Next, a general model of operation process of critical infrastructure network is defined and its parameters are described. A special case of the general when its component critical infrastructures are independent model is considered and applied to the Baltic Electric Cable Critical Infrastructure Network