What Is a Sigma-Field

There are many ideas from set theory that undergird probability. One such idea is that of a sigma-field. A sigma-field refers to the collection of subsets of a sample space that we should use in order to establish a mathematically formal definition of probability. The sets in the sigma-field constitute the events from our sample space. Definition The definition of a sigma-field requires that we have a sample space S along with a collection of subsets of S. This collection of subsets is a sigma-field if the following conditions are met: If the subset A is in the sigma-field, then so is its complement AC.If An   are countably infinitely many subsets from the sigma-field, then both the intersection and union of all of these sets is also in the sigma-field. Implications The definition implies that two particular sets are a part of every sigma-field. Since both A and AC are in the sigma-field, so is the intersection. This intersection is the empty set. Therefore the empty set is part of every sigma-field. The sample space S must also be part of the sigma-field. The reason for this is that the union of A and AC must be in the sigma-field. This union is the sample spaceS. Reasoning There are a couple of reasons why this particular collection of sets is useful. First, we will consider why both the set and its complement should be elements of the sigma-algebra. The complement in set theory is equivalent to negation. The elements in the complement of A are the elements in the universal set that are not elements of A. In this way, we ensure that if an event is part of the sample space, then that event not occurring is also considered an event in the sample space. We also want the union and intersection of a collection of sets to be in the sigma-algebra because unions are useful to model the word â€Å"or.† The event that A or B occurs is represented by the union of A and B. Similarly, we use the intersection to represent the word â€Å"and.† The event that A and B occurs is represented by the intersection of the sets A and B. It is impossible to physically intersect an infinite number of sets. However, we can think of doing this as a limit of finite processes. This is why we also include the intersection and union of countably many subsets. For many infinite sample spaces, we would need to form infinite unions and intersections. Related Ideas A concept that is related to a sigma-field is called a field of subsets. A field of subsets does not require that countably infinite unions and intersection be part of it. Instead, we only need to contain finite unions and intersections in a field of subsets.

Analysis Of The Color Purple - 1393 Words

Even though today s society tries to be just, that does not always happen. Learning the past injustice that has happened can help structure the way todays world is but history can repeat itself. Rape, racism and many more acts of hate are still not dealt with just like back in the 1930 s where both of these novels took place. Both of these novels conversant one of the most egregious injustices; rape. Rape has always been erroneous. In The Color Purple the novel begins with Celie describing being raped by her father and no one knowing about it. She elucidate s what was said and done to her â€Å"You gonna do what your mammy wouldn t. When that hurt, I cry. He start to choke me, saying you better shut up and get used to it. But I don t†¦show more content†¦Celie being violated growing up does not enjoy intercourse even with her husband. She said â€Å"Mr.__ clam on top of me, do his business, in ten minutes us both asleep (Walker,p.66).† Similarly, many females had t here own views on men but mass women had a negative view. Miss Maudie stated her view by this comment â€Å"all men lie, that all Negroes are basically immoral beings, that all Negro men are not to be trusted around our women (Lee, p.273).† Tom is a prime example that there was injustice to Negros as well during this time. Coloured men had stereotypes and unfair accusations against them from the white race. Tom was not a liar, nor immoral . Not all men were evil and caustic. Celie growing up around atrocious men, has the same salient view as Miss Maudie on men. Although there were many treatments of blacks that can be considered unjust, racism is the most recognizable and alive oppression. This was exceedingly frequent in two novels. Racism is inevitable, it has always been around. Racism has got better over time but it has not constantly been like this. Both novels substantiate this and bring enlightenment back to how racism was in the 1930 s. The caucasian race was superio r during this time and more powerful. The caucasians knew they were superior, they had set views on the negro s. Not all the whites preformed colossal acts against the negro s, but even minor acts were to make the negro s feel degraded. AShow MoreRelatedAnalysis Of The Color Purple 1043 Words   |  5 Pagesessay, I am going to analyze the intersectionality of oppression in Alice Walkers novel, The Color Purple. I am going to show how the political categories of race, sexuality and gender play a role throughout. I am also going to discuss Walker’s own term, â€Å"Womanism† and how that plays throughout the story. I will be focusing on the main character Celie, as well as other characters to help me demonstrate my analysis effectively. Celie, the main character, starts out the book writing letters to god, a godRead MoreLiterary Analysis Of The Color Purple1074 Words   |  5 PagesIn Search of Our Mothers’ Gardens. Yet Ms. Walker’s most critically acclaimed novel remains The Color Purple. The Color Purple tells the complex tale African American women, their brutal living conditions, everyday abuse, and their instinct to survive. The Color Purple was an immediate success due to its simple writing style, the intricate storyline, and compelling characters. In 1983 The Color Purple was recognized for these very reasons and graciously awarded The Pulitzer Prize For Fiction. EveryRead MoreThe Color Purple Character Analysis1362 Words   |  6 Pagesâ€Å"The Transformation of Celie in The Color Purple† The 1900’s – a time when blacks were segregated from whites and women were seen as inferior to men. Alice Walkers character Celie, from The Color Purple, was ugly and terrified as a young girl. Though many trials and tribulations, she would become strong and independent. In addition, Celie built up a resistance to the hurt and suffering that she painfully endured from her stepfather and husband. She eventually allowed herself to not take simpleRead MoreAnalysis Of Alice Walkers The Color Purple2079 Words   |  9 Pages In Alice Walkers The Color Purple, she explores the thin grey line that stands between survival and living. Through her protagonist, Celie, she examines the dramatic shifts of empowerment; focusing on the young black girl in the 1850’s. Walker introduces the reader to the protagonist, Celie, through a series of letters. In these letters the reader finds Celie amidst her mother’s death. The author chooses to address her letters to God, giving Celie a greater willpower to survive. Celie’s upbringingRead MoreLiterary Analysis: The Color Purple Essay1388 Words   |  6 Pagesmost prevalent themes in historical types of these kinds of literature is racism. In America specifically, African Americans endured racism heavily, especially in the South, and did not gain equal rights until the 1960s. In her renowned book The Color Purple, Alice Walker narrates the journey of an African American woman, Celie Johnson (Harris), who experiences racism, sexism, and enduring hardships throughout the course of her life; nonetheless, through the help of friends and family, she is ableRead MoreAnalysis Of The Color Purple By Alice Walker1173 Words   |  5 Pagestheir heads!† â€Å"Touch the spinning wheel!† â€Å"Did I say anything about purple dragons?† We all recognize the quotes of Disney’s infamous female villains. In these, and all other Disney movies, there is usually a female protagonists and antagonists who reign supreme to their male counterparts. But what happens in reality? Do women have that same authority over their lives? Alice Walker sheds light on this debate in her novel The Color Purple. Celie, the main character, grows up in a predominantly, male-ruledRead MoreAnalysis Of The Color Purple By Alice Walker1444 Words   |  6 PagesMasamoto, Lani Ms. Britton Eng. 4 AP/Per. 3 28 August 2014 Literary Review 1. The Color Purple by Alice Walker; The color purple is commonly used to symbolize royalty and a deep connection spiritually. Celie, the main character in the novel, writes many of her very personal letters and thoughts that she has never shared with to anyone else to God. She deeply connects spiritually with God. She knows that, unlike everyone else present around her in society, God will neither discriminate or judgeRead MoreAnalysis Of The Color Purple By Alice Walker776 Words   |  4 PagesMany of her books including, The Color Purple displays countless examples of these themes. Because these themes can be so sensitive and traumatizing in life, they should not be produced in books therefore exhibiting why the book should be banned. Furthermore, Alice Walkers delicate topics become popular throughout her novel, specifically, the idea of homosexuality. In a critical essay written about Walkers life and novels, researcher Lambert says, The Color Purple is purely adventitious: men areRead MoreFinding Your Voice: an Analysis of the Color Purple1864 Words   |  8 PagesFinding Your Voice: An Analysis of The Color Purple â€Å"Who do you think you is? he say... Look at you. You black, you pore, you ugly, you a woman. Goddam, he say, you nothing at all.† (187) Alice Walker, the author of The Color Purple, focuses on the struggles of a poor and uneducated African American girl, who is verbally, physically and sexually abused by several men in her life. She feels worthless and becomes completely submissive. Her only way to express her feelings is through privateRead MoreAn Analysis Of Jim Crow s The Color Purple1972 Words   |  8 PagesQinying Sun Ms. Hughes English 10, 5th period 2016/5/18 To mutely comply is to survive, to firmly defy is to live: The Impact of Jim Crow in The Color Purple Coward is how people refer to someone who lacks the courage to face a plight; but what if its hardness is impregnable for a single person, though the ambition to revolt never dies? At the end of the 19th century, Jim Crow was defined as a system that imposed racial segregation and discrimination on African Americans. Under the Jim Crow

Technology in Education Free Essays

Software Support Tools According to Rogers (2008), â€Å"the landscape of the classroom is being transformed, with technology becoming a driving force† (p. 92). In today’s classroom there are numerous software support tools available to enhance learning and cater to the needs of all learners. We will write a custom essay sample on Technology in Education or any similar topic only for you Order Now Roblyer and Doering (2010) presented six categories of software support tools. They listed these as: material generators, data collection and analysis, graphics, planning and organizing, research and reference, and tools for the content areas. Material generators are software tools that are used for making instructional materials such as posters, brochures, quizzes, tests, rubrics, puzzles, worksheets, games, awards, and IEPs for special education students. Many teachers regard these tools as essential to the day-to-day operation of the classroom. Wellert (2008) stated that the use of games allow teachers to directly cater to the learning style needs of the visual and tactile learners, and that games can expedite the learning process when constructed with the different learning styles in mind. Data collection and analysis software tools allow teachers to collect information about their students and closely monitor their performance and progress throughout the year. According to Roblyer and Doering (2010), the electronic grade book is preferred by many teachers because of its special feature to analyse data from tests and print reports based on the information given. Student information systems (SIS) help teachers keep account of students’ data such as attendance and test scores. Roblyer and Doering also referred to computerized testing systems which offer benefits such as immediate feedback of results. Additionally, it allows students to take tests on screens or scanned sheets. According to the authors, these simplify test scheduling because everyone does not take the test at the same time. Graphic tools consist of draw/paint programs, image editing tools, charting/graphing tools, and clip art animation, sound, video, and font collections (Roblyer Doering, 2010). These allow teachers and students to create, illustrate, modify, and combine drawings, illustrations, clip art, pictures, charts and graphs. They also give students the opportunity to create their own designs and show them on paper or screen. Additionally, teachers and students can insert these into documents and reports to give a more professional look. Catchings and MacGregor (as cited in Roblyer Doering, 2010) believed that these programs aid in the development of students’ visual-verbal literacy and creativity. Planning and organizing software tools entails outlining and concept mapping, software, lesson planners, and scheduling and time management tools. Outlining tools assist students in outlining their story ideas and help them to get their stories started, while the other tools help teachers organize their time and prepare for instruction. Research and reference tools such as electronic encyclopedias, atlases, and dictionaries help facilitate research. Electronic atlases and maps are useful in the classroom because students get to see and learn information being taught, and are especially helpful because they are interactive. Wellert (2008) stated that students need to have active participation and involvement in lessons that include technology. Furthermore, student engagement is vital to ensure motivation during the learning process. Finally, the last category of software support tools put forward by Roblyer and Doering (2010) is content area tools. These support teaching and learning in the different content areas. Multimedia Multimedia refers to multiple media or a combination of media. Media can be still graphics, and photographs, sound, motion video, animation, and or text items along with products used to communicate information in multiple ways. Multimedia impacts the classroom in various ways. For example it increases motivation as it offers a variety of options for the user and encourages the learner to be proactive. For students who might not be strong in written expression but have visual aptitude, multimedia offers flexible learning modes. It also fosters the development of creative and critical thinking skills as it opens up creative avenues for students to apply higher thinking skills.. For those who struggle with organizing and presenting information, multimedia offers improved writing process as students are equipped to create characters and linking dialogue to it rather than simply creating a lengthy story. Creativity is encouraged by this process. The use of multimedia also decreases the time it takes to present lengthy presentations as information is presented in chunks as oppose to monotonous reports. According to Mayer and Moreno, 1998; Moreno and Mayer, 2002 (as sited in Roblyer and Doering, 2010) learners exhibited greater comprehension and retention of learned materials when pictures were accompanied by spoken words rather than by written words. They observed that the â€Å"split attention† effect was consistent with a model of working memory that had separate visual and auditory channels. As oppose to chalk and talk which rarely engage students, the use of multimedia encourages interaction and positive immediate feedback. Students are prompted when answers are incorrect giving them the option to correct their answers. Multimedia also provides easy navigation with clearly defined procedures. It is learner controlled for those who are less experienced as well as those who are advanced in technology. Color is also used to bring certain elements to the learner’s attention. Utility provides assortment of functionalities and features necessary to accomplish their instructional tasks, while usability deals with the effectiveness, efficiency and satisfaction of the user. Aesthetics on the other hand, represents elements of design which enhance or heightens the learner’s experience as oppose to just satisfying the pedagogical or instructional objectives. In addition (Lavie Tractinsky, 2004; Wilson, 2005), research on emotion suggests that aesthetically pleasing objects positively influence our emotions which in turn help students to actively process material. Commercial multimedia products include instructional software such as tutorials, drills, simulations, instruction and problem solving tools. It also includes interactive storybooks and eBooks for toddler’s right up to adults. There are digital libraries, Google books, listening to books online, Grimm Fairy Tales as well as Clifford Interactive Storybooks. In addition to these, these products include reference materials such as Encarta (Encyclopedia), Almanacs, Atlases, newspapers and newsletters, collection of development resources like Clip art, Audio clips, Video clips and Animations, all of which can be used for learning. When producing power points and graphics, fancy fonts and animation can all be incorporated. Multimedia tools also provide editing systems which allows the polishing of the end product. The question about how to effectively integrate support tools, multimedia, and hypermedia into teaching and learning for the highest potential relative advantage, this question can be answered by simply choosing the right tools to meet specific needs of the audience being taught. It is no longer debatable whether or not these tools are relevant, but which tools are relevant for what and who is being taught. Hypermedia According to (Roblyer and Doering, 2010), Hypermedia is also known as â€Å"linked media†. It is a system in which information stored in various media is connected, often through the internet. Most multimedia products are also hypermedia products. There exist a synonymous connection of their uses and alternative purpose. The ability to get information using multimedia tools, from one media element to another, makes them hypermedia. There are six kinds of hypermedia formats, such as: commercial hypermedia software, presentation software, video production and editing systems, hypermedia design and development software, virtual reality environments and Web 2. 0. Hypermedia has been impacting the educational system in many ways. Students are able to attend classes in the comfort of their home. Teachers are able to peruse through a wide variety of activities to effectively and efficiently deliver lessons. Hypermedia is geared at providing multiple channels by which students are able to learn and process information. It has been stated by researchers that students who learn through verbal and visual means are better able to solve problems in comparison to those who learn only through verbal means. In using hypermedia in the classroom, it saves time. Students and teachers are able to search through multiple types of information in a short length of time, with little or no cost. It can be stressful and frustrating for those who are not fully equipped with the skills of navigating. Users can also use hypermedia to add, modify or share information. The software is designed to be easy to use and understand. The programs available can incorporate video and audio of CD’s and DVD’s on the web. This creates a merger of both concepts which can realize learning at its maximum relative advantage. In considering Gardner’s Theory of Multiple Intelligences, we can integrate support tools multimedia and hypermedia into teaching and learning. This can be done in various ways. Software programs that supports tutorials, drill and practice, simulation, instructional games and problem solving can all be incorporated using an interactive approach. Students as well as teachers are motivated to learn and present information. According to Edyburn (as cited in Roblyer Doering, 2010), the accessibility capabilities of special needs students are often challenging. Special programs are provided along with software for these students, but in order to use the programs, one must be especially skilled. Roblyer and Doering (2010), further stated that in order for hypermedia to be used effectively, some hardware are needed in order for the software to function, such as: computer with keyboard, mouse and monitor, digital cameras, scanners, video digitizers, camcorders and other video output, microphones and audio speakers. Financial resources are necessary for this system to take place. Adobe Flash Reader has become the standard for creating interactive web based animation and other software tools. Flexible learning modes exist but teachers must carefully select programs that promote higher order thinking, which are few in the case of reading and language. The response based methods are critical and students are able to use their creativity to access information. Students can also make the distinction between various texts and think critically about which is appropriate and applicable. Teachers are able to assess students based on the outcome of assigned projects while challenging them to learn independently at times. Students use the Web 2. 0, and presentation software such as Microsoft Power point Presentation to deliver work the way they understand. Hypermedia helps students to generate new ideas, raise the bar for creativity and independence and thus provide a, variety of sources which students can use to understand and interpret information, thus reaching their highest potential relative advantage. Factors Influencing the Effective Integration of Technology Gorder (2008) stated that integrating technology is not about having technology available in the classroom, but more about how teachers use technology as a tool for learning. To effectively and successfully integrate support tools, multimedia and hypermedia into teaching and learning for the highest potential relative advantage, certain factors need to be considered. Franklin (2008) outlined four factors that influence the integration of technology: availability and access to technology, teacher preparation and training, good leadership, and time. According to the National Center for Education Statistics (as cited in Franklin, 2008), the availability of classroom technology correlates to teacher use of computers. This means that when more computers are available in the classroom, teachers tend to incorporate more activities involving technology. While the average elementary classroom in the United States has two computers (NCES, as cited in Franklin, 2008), classrooms in Belize are yet to experience that luxury. Access to computers in Belizean elementary schools is very limited, often through a computer lab. Oftentimes many of these computers are outdated, or not working. The lack of hardware and software tools do very little to promote the use and integration of technology in education. Teacher preparation and training is also an important factor in the effective integration of technology. Franklin (2008) indicated that teachers who feel better prepared to use technology are more likely to have their students use technology than teachers who feel unprepared. With the advent of numerous new technological advances, more experienced teachers are feeling unprepared to face the technological world. The NCES (as cited in Franklin, 2008) reported that only one third of elementary teachers in the United States felt they were well prepared to use technology in the classroom. Gorder (2008) referred to this as a change from educating students for the industrial age, to one of educating students for the technological age. This change will require teachers to become students again, seeking assistance from fellow teachers as well as from their own students. In light of this change, Ash (2007) felt that for teachers to implement technology effectively in the classroom there needs to be some form of professional development. Ash reported that in order to meet the professional development needs of teachers, some school districts have encouraged teachers to form on-line learning communities with their colleagues where they can seek technology advice, share resources and tips. Additionally, Ash suggested having a technology specialist on staff to assist teachers with new technology as well as to provide training for teachers in the form of after school training and workshops. Pitler (2006) said that most importantly teachers need to be provided with ongoing technology professional development. Ash (2007) believed strong leadership is required to effectively integrate technology into teaching and learning and considered the principal as the key factor to ensuring the successful integration of technology into the school curriculum. Hope and Stakenas (as cited by Gosmire Grady, 2007) suggested three primary roles for the principal as technology leader: role model, instructional leader, and visionary. They stated that principals themselves need to have a good working knowledge of technology tools and be able to use these for their own administrative and managerial duties. When principals have a good working knowledge of technology they are more committed to its importance in teaching and learning and tend to impart this to their teachers (Franklin, 2008). Furthermore, principals are the instructional leaders of the school and are primarily responsible for overseeing technology integration. As a result, principals need to have a good knowledge of technology hardware and software to be able to acquire those that best fit the needs of their school. Hope and Stakenas emphasized that the most important thing is for principals to have a vision for technology in their school. They need to have a clear understanding of how technology will be used by teachers and students to better facilitate teaching and learning. Franklin concluded that principals who provide strong leadership â€Å"help create a climate in which experimentation with technology is looked on with favour and given encouragement† (p. 55). Finally, the fourth factor that influences the integration of technology is time. Franklin (2008) believed that for technology integration to be effective it is necessary to provide the time for teachers to learn, practice, and plan how they will integrate technology into the curriculum. She stated that this might, at times, require teachers to be released from their regular classroom duties in order to facilitate this experience. Apart from the time teachers need to learn and plan for technology, students also need time to use technology. Franklin indicated that time needs to be placed in the daily schedule when students will be using technology. Schacter and Fagnano (1999) also suggested several practices to aid in the successful implementation of technology in the classroom. These include: (a) getting students involved in discussions and debates where they will be required to work with a team to evaluate their work, as well as the work of others, (b) allowing students to analyse situations and think independently through self reflection and thought, (c) getting students to design programs that encourage critical thinking, judgement, and personal involvement, (d) using project-based learning activities, (e) engaging students in contextualized and meaningful projects, and (f) teaching students how to use computers to design projects. The authors emphasized that the effective use of technology increases students’ learning, understanding, achievement and motivation as well as develops their critical thinking and problem solving skills. Conclusion Hence, it is clear that in order to integrate technology into teaching and learning for the highest potential relative advantage, it is not just about having the technology and being knowledgeable about its use and benefits, but also about having a vision and plan of how technology will be integrated into the school curriculum and providing the necessary training for teachers. Pitler (2006) stated that all the new technologies in the world will not impact student achievement if the school does not have a clear vision of how and why they will use the technology. It is important to understand that the acquisition of technology alone will not improve the quality of instruction. Day (2010) emphasizes that â€Å"the fundamental determinants of instructional quality have always been the course content, the teachers, the learning activities in which the students are engaged and the students themselves† (p. 49). How to cite Technology in Education, Essay examples Technology in Education Free Essays Nowadays, the technology has become essential in people’s daily life because it makes human beings’ lives more colorful. In the same way, technology also changed traditional education in many ways. For example, technology makes education more efficiency, more interesting, and more flexible. We will write a custom essay sample on Technology in Education or any similar topic only for you Order Now However, it also has negative effects in education. Technology makes education more efficiency. Students use smartphones in their daily lives, which is providing some ways to let student have access to internet, and therefore making things such as checking their email in palm of their hand. Technology make education more interesting. For example, â€Å"prezi† have the same functions as power point, but it looks more interesting because it has more animations to show you ideas when you make presentation. Which makes a presentation more visually appealing. Now prezi is used at some of university, for instance,Oregon state university and Dwight school and also elsewhere in primary education and higher education. Prezi can achieve that a group present a same presentation, but allow students present their knowledge in different learning styles. (http://en. wikipedia. rg/wiki/Prezi) Technology also make education more flexible. Not everyone has the schedule that fits to enroll a traditional school to take a class. Some people will take online classes, which gives them access to get degree they want. And more and more people are earning their degrees online, by doing this, they can further their education, and are open to more opportunities as well. Although, technology can be convenient for people with special needs, it also have the negative effects in education. For example, technology makes it easy, fast, and convenient to type or write an essay. The downfall to this is that student’s handwriting skills are reduced, due to the fact that students rely on technology too much. And other negative points for technology in education is easy to make students distracting when they use technology products (smart phone or laptop) on class. Students cannot put 100% attention on teacher, they will spent too much time on twitter or facebook. In conclusion, education become more efficiency, interesting, and flexibly, however, there also are negative effect because of technology. But, there are more benefits than limit, so technology push forward education develop quickly. How to cite Technology in Education, Essays