Some Mathematical Expressions Probability and Statisties

1 Probability1.1 Some definitions (1)Outcome A result of an experiment is called an outcome. (2)Sample space The set of possible outcomes of an experiment is the sample space for the experiment. (3) Fair experiment If a sample space has n possi-     

Some Mathematical Expressions of Probability and Statistics

1.1 Some definitions (1)Outcome A result of an experiment is called an outcome. (2)Sample space The set of possible outcomes of an experiment is the sample space for the experiment.     

Some Mathematical Expressions of Sets and Inequalities

SETS(1 )Aisasetconsistingofa,b,c,dande.(2 )Thesetofallnaturalnumbers (inte gers,rationalnumbers,realnumbers)isusuallycalledanaturalnumerical (integer,rationalnumber,realnumber)set,denotedbyN (Z ,Q ,R ) .(3 )“∈”“ ”Signsthatrepresentthemembershiprelationbetweenasetandaele ment.ForthesetNwhosenumbersareone ,two ,three,etc.FourisamemberofN (4∈N) .FourpointfiveisnotamemberofN(4.5 N) .(4)“ ”Asignthatrepresentstherelationbetweentwosets.A B ,AisasubsetofBorAispartofBorAisincludedinB…     

Mathematical Model of Weight in Evaluation System

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Developing Children''''s Gommunication Skills to Aid Mathematical Understanding

Mathematics teaching and learning has movedaway from a mechanical view of mathematics,to onewith an emphasis on problem solving,understandingand communicating mathematically with others.Inthe past children have been mainly taughtalgorithmic formulae and repeatedly practiced suchan approach,in tasks that have been removed frommeaningful contexts and with an expectation ofworking individually.They are now,however,expected to work with others to solve problems,totrial strategies,and to find effective strategies thatwork for them.     

Some Mathematical Expressions of Functions and its Properties

1　ThedefinitionoffunctionsLetAandBbetwonon emptysetsofnumbers.Ifbyagivencorrespondingrelationf ,wecanassigntoeveryelementxinAauniqueelementf(x)inB ,thenwecallf :A→Bisafunction ,denotedbyy =f(x) ,x∈A .Wherexiscalledtheindependentvariable,yisthedependentvariable.ThesetAisthedomainoffandBisthecodomainoff.Thenumberf(x) iscalledthevalueofthefunction fatxandtheset { f(x) |x∈A}istherangeoff .Thepointset { (x ,y) |y =f(x) ,x∈A}issaidtobethegraphoff ,where (x ,y)isapointinthecoordinatesystem…     

Study on the Mathematical Model of Hydraulic Jump Atomization

An equation of atomization quantity from energy dissipation by hydraulic jump was derived from the dimensional analysis. By applying Gauss diffusion equation, the spray diffusion rule in valley was studied under the condition of continuous linear source and random wind direction. By considering the spray-rain switching process, coagulation, condensation and evaporation of droplets, the air temperature, air relative humidity,spray density and the rainfall intensity in the lower reaches of the linear source were calculated. The 3-D numerical simulation fitted well with prototype monitoring. Finally, the prediction of atomization influence on environments for Xiangjiaba Hydropower Station was conducted.     

Mathematical Description of Acousto-Optic Tunable Filter Orthogonal Polarizing Method

The foremost acousto-optic tunable filter(AOTF) was proposed in 1967[1], but the appliedAOTF based on TeO2crystal was not put into practiceuntil 1974, when the non-linear optical design meth-od[2]was proposed. In recent years, AOTF has beenapplied widely as optical splitting component. And ithas attracted more and more attention for its meritssuch as rapid scanning speed, wide tuning range ofspectrum and large aperture of angle of incidence.AOTF is often used for spectrum measurement te…     

Outside-In – Inside-Out: Seventh-Grade Students' Mathematical Thought Processes

Building on the research of Vygotsky regarding the role of social interaction and the zone of proximal development (ZPD) in learning and development, this paper explored the relation between students' oral thought processes and written thought processes. It is argued that the practice of writing provides a context for a new learning zone: the zone of proximal practice (ZPP). In this new zone, students independently organize their thinking about mathematical concepts and ideas. An interpretative case study of seven middle grade students is presented to support this contention. The case study describes the strategies and procedures students employed while solving mathematical problems and documents students' oral and written thought processes through interview protocols and writing samples. The position that students' mathematical understanding is further developed through writing as a communicative tool, while taking advantage of mediated social practices, is discussed to make clear the rationale for introducing a new learning zone.This revised version was published online in September 2005 with corrections to the Cover Date.     

Analyzing Mathematical Tasks: A Catalyst for Change?

In this study we investigate a strategy for engaging high school mathematics teachers in an initial examination of their teaching in a way that is non-threatening and at the same time effectively supports the development of teachers’ pedagogical content knowledge [Shulman (1986). Educational Researcher, 15(2), 4–14]. Based on the work undertaken by the QUASAR project with middle school mathematics teachers, we engaged a group of seven high school mathematics teachers in learning about the Levels of Cognitive Demand, a set of criteria that can be used to examine mathematical tasks critically. Using qualitative methods of data collection and analysis, we sought to understand how focusing the teachers on critically examining mathematical tasks influenced their thinking about the nature of mathematical tasks as well as their choice of tasks to use in their classrooms. Our research indicates that the teachers showed growth in the ways that they consider tasks, and that some of the teachers changed their patterns of task choice. Further, this study provides a new research instrument for measuring teachers’ growth in pedagogical content knowledge. An earlier version of this paper was presented at the American Educational Research Association Annual Meeting, New Orleans, LA, April 2002.     

Young Children’s Mathematical Learning From Intelligent Characters

Children’s math learning (N = 217; M_age = 4.87 years; 63% European American, 96% college-educated families) from an intelligent character game was examined via social meaningfulness (parasocial relationships [PSRs]) and social contingency (parasocial interactions, e.g., math talk). In three studies (data collected in the DC area: 12/2015–10/2017), children’s parasocial relationships and math talk with the intelligent character predicted quicker, more accurate math responses during virtual game play. Children performed better on a math transfer task with physical objects when exposed to an embodied character (Study 2), and when the character used socially contingent replies, which was mediated by math talk (Study 3). Results suggest that children’s parasocial relationships and parasocial interactions with intelligent characters provide new frontiers for 21st century learning.     

Mathematical Analysis of Energy Efficiency in IEEE 802.11 DCF

The standardized IEEE ,802. II distributed coordination function （ DCF） provides a contention-based distributed channel access mechanism for mobile stations to share the wireless medium. However, when stations are mobile or portable units, power consumption becomes a primary issue since terminals are usually battery driven. This paper proposes an analytical model that calculates the energy efficiency of both the basic and the RTS/CTS access mechanisms of the IEEE 802. II protocol. The model is validated with simulation results using NS-2 simulation package. The effects of the network size, the average packet length, the initial contention window and maximum backoff stages on the energy efficiency of both access mechanisms are also investigated. Results show that the basic scheme has low energy efficiency at large packet length and large network size, and depends strongly on the number of stations and the backoff procedure parameters. Conversely, the RTS/CTS mechanism provides higher energy efficiency when the network size is large, and is more robust to variations in the backoff procedure parameters.     

Teachers’ Attitudes Towards Their Pupils’ Mathematical Errors

This article is mainly concerned with the didactical and the epistemological approach towards pupils’ errors in mathematics. The findings of an investigation into a representative sample of Cypriot teachers’ attitudes of errors in mathematics are presented. Although teachers draw on behaviourism as a way of understanding pupils’ errors, a considerable number of teachers seem to be influenced by both cognitive science error theory and the obstacle theory. Inferential statistical analysis revealed that teachers’ attitudes of errors are mainly associated with the attendance of a specialised INSET course in mathematics. Teachers who attended the INSET course in mathematics are generally more influenced by the cognitive science theory and the obstacle theory. Implications for the development of policy on teacher training are discussed. It is argued that with appropriate training teachers will be able to adopt a model of interpreting errors in mathematics based on the didactical and the epistemological approaches.     

Gender Differences in Lunar‐related Scientific and Mathematical Understandings

This paper reports an examination on gender differences in lunar phases understanding of 123 students (70 females and 53 males). Middle‐level students interacted with the Moon through observations, sketching, journalling, two‐dimensional and three‐dimensional modelling, and classroom discussions. These lunar lessons were adapted from the Realistic Explorations in Astronomical Learning (REAL) curriculum. Students’ conceptual understandings were measured through analysis of pre‐test and post‐test results on a Lunar Phases Concept Inventory (LPCI) and a Geometric Spatial Assessment (GSA). The LPCI was used to assess conceptual learning of eight science and four mathematics domains. The GSA was used to assess learning of the same four mathematical domains; however, the GSA test items were not posed within a lunar context. Results showed both male and female groups to make significant gains in understanding on the overall LPCI test scores as well as significant gains on five of the eight science domains and on three of the four mathematics domains. The males scored significantly higher than the females on the science domain, phase—Sun/Earth/Moon positions, and on the mathematics domain geometric spatial visualisation. GSA results found both male and female groups achieving a significant increase in their test scores on the overall GSA. Females made significant gains on the GSA mathematics domains, periodic patterns and cardinal directions, while males made significant gains on only the periodic patterns domain. Findings suggest that both scientific and mathematical understandings can be significantly improved for both sexes through the use of spatially focused, inquiry‐oriented curriculum such as REAL.     

Mathematical Subject Classification在数学文献信息分类中的应用

美国《数学评论》的分类体系《数学主题分类表》(Mathematical Subject Classification)，是国际数学界影响最大使用最广的数学专业分类表。数学工作者可以使用美国数学会在网站上提供的Mathematical Subject Classification为自己的数学论文标引分类号，也可从该分类途径检索网上的数学信息。     

Mathematical Manipulative Models: In Defense of “Beanbag Biology”

Mathematical manipulative models have had a long history of influence in biological research and in secondary school education, but they are frequently neglected in undergraduate biology education. By linking mathematical manipulative models in a four-step process—1) use of physical manipulatives, 2) interactive exploration of computer simulations, 3) derivation of mathematical relationships from core principles, and 4) analysis of real data sets—we demonstrate a process that we have shared in biological faculty development workshops led by staff from the BioQUEST Curriculum Consortium over the past 24 yr. We built this approach based upon a broad survey of literature in mathematical educational research that has convincingly demonstrated the utility of multiple models that involve physical, kinesthetic learning to actual data and interactive simulations. Two projects that use this approach are introduced: The Biological Excel Simulations and Tools in Exploratory, Experiential Mathematics (ESTEEM) Project (http://bioquest.org/esteem) and Numerical Undergraduate Mathematical Biology Education (NUMB3R5 COUNT; http://bioquest.org/numberscount). Examples here emphasize genetics, ecology, population biology, photosynthesis, cancer, and epidemiology. Mathematical manipulative models help learners break through prior fears to develop an appreciation for how mathematical reasoning informs problem solving, inference, and precise communication in biology and enhance the diversity of quantitative biology education.     

Mathematical Model and Simulation of Cutting Force in Plunge Milling Process

Plunge milling,also calledZ-axis milling,is one ofthe most effective machining methods in high speed metalremoval machining.Asfor curvedsurface of difficult-to-ma-chine material ,slot-cut working and machining withlongercutter of suspendinglength,the mach…     

Adults’ Use of Mathematics and Its Influence on Mathematical Competence

The Programme for the International Assessment of Adult Competencies (PIAAC) has recently drawn additional attention to “mathematical literacy” as an important influential factor for individuals’ life chances. High levels of mathematical literacy have thereby been linked to using mathematics in daily and working life frequently. In this paper, based on the data from Germany, we focus on the construct “use of mathematics” in two ways: First, we analyze in depth how it can be utilized to describe different groups of adults. Second, we investigate its role as predictor of mathematical competence and mediator of other relevant background variables. Results show that three groups of adults can be distinguished that use mathematics differently in daily and working life. However, the construct can sensibly be described as unidimensional. In a path model, “use of mathematics” turns out to be the strongest predictor of mathematical competence. In addition, it mediates effects of the mathematical requirements of the job, duration of education, and gender.