Proof :: Math Education Papers

Proof Proof. What is it and why does this simple term cause such a stir among mathematics educators and mathematics students? If you were to ask a young child to prove a mathematical fact, they would be happy to show you many examples of how it works. This does not constitute a proof but it is a step in the right direction. If you were to ask a high school student or first year college student to do a proof, you will most likely be met with groans and feelings of disgust. Students at this age have probably encountered proof in a geometry class where they were expected to follow a strict format without much freedom to express proofs on their own. However, if you were to ask a mathematician about proof they would begin to tell you about how beautiful proof in mathematics can be. Proof has always been a topic of interest for me. In high school geometry and my first year of college, I too did not understand proof. I felt like many other students, frustrated by the fact that we were asked to prove theorems that the book had already told us were true. It was as though the instructor was playing magical games on the chalkboard and all of the sudden we had a proof. However, as time progressed, I began to see the beauty of proof. Then, mathematical induction introduced me to the power of proof. In this paper I hope to address the concept of proof, how it relates to understanding and the implications for mathematics education. BACKGROUND In the 1950’s and 60’s proof played a significant role in mathematics education. Then in 1989, the National Council of Teachers of Mathematics (NCTM) deemphasized proof and replaced it with reasoning. Following this, mathematics educators began to see that students had difficulty with proof because they had little contact with it. In response, NCTM in the 2000 standards, elevated proof to a standard, emphasizing that it should be part of all students mathematical experiences (Knuth). Schoenfeld states proof is inseparable from mathematics. It is essential in communicating, doing, and recording mathematics (153). Throughout most of the history of mathematics education, proof has been more of a topic of study instead of a way to understand mathematics (Knuth 73). In addition, proof has only been limited to the college bound student or the student enrolled in geometry.

Business Statistical Analysis Scenario Essay

A GMC manager recently noticed that there is a production difference between the early shift and the late shift. The manager would like to determine why there is a difference in production between the shifts and asked for research on the issue. The research team came up with the following research question: Is there a reason for the different production levels between the day and evening shifts. Our team has come up with the following two null hypotheses: H0 – There is a significant difference in employee productivity between shifts due to worker age. Ha – There is no significant difference in employee productivity between shifts due to worker age. In order to accomplish this, we need to find the average number of errors along with the standard deviation. By doing so, we can set up a confidence interval to see if the late shift is truly doing better when it comes to quality. If they are, we can make the nomination that the manager has the late shift supervisors provide guidance to the early shift supervisors. There are various statistical reports that outsource the relationship between the independent variable(s) and the dependent variable. Here we will realize how GMC will translate the organization problem into a statistical problem, provide a solution to the problem statistically, and then translate the statistical solution into an actionable solution for the company. Literature Review General Motors realized that to stay competitive a number of changes would have to be made starting with production efforts. Based on economic  uncertainty and gas price volatility, adjustments were made to accommodate the needs of the population interested in purchasing vehicles during times of uncertainty. After the fall of the three leading automotive giants, coming back would have to be well thought-out and cost beneficial to the consumer. GM got to work. Production in most plants was one shift, Monday – Friday, 8 hours per-day. At one facility adding a third shift let the plant produce more than 26,000 vehicles within three months, more than it had previously done when it produce 20,000 vehicles. Before incorporating a third shift the plant had approximately 3,300 workers who averaged fewer than 14,000 vehicles per month (Funk, 2011). In another plant that focused on the production of only one vehicle, this plant employed eight hundred production workers who were a one- shift, eight hour Monday-Friday production facility. Another GM plant that makes the Chevrolet Tahoe and Suburban, GMC Yukon and Cadillac Escalade add a third shift and 800 new jobs was created, the assembly plant employed approximately 2,500 hourly and salaried employees and operates two production shifts. In 2011, the plant produced nearly 270,000 vehicles. â€Å"Demand for full-size SUVs remains stable.† The third shift enable the Assembly plant to meet market demand for the current generation of full-size SUVs and provide relief for employees who have worked extensive overtime since the fourth quarter of 2009. The installation of new tooling and equipment required to build GM’s new line of SUVs limited vehicle production in 2013. A third shift will provide the plant needed production capacity during this transition time (Funk, 2012). There was a time when working a straight day-shift was enough to meet the production demands for General Motors products, however with the versatility and the types of vehicle’s being produced third shift help to meet and exceed demand and provide relief for employees working extensive overtime. Many years, companies have pursued to advance worker output and labor settings. One approach is substitute toil schedules, which comprise flextime, work division, and a compacted workweek. Sleep-related aids are mainly supportive for the salaried period; insufficient sums of nap reduce work output and general well-being. Statistics shows consistent amounts of sleep Improved performance and alertness in the workplace. Therefore, we propose creating work-sleep equilibrium, comparable to work-life stability, as a standard for planning and refining work plans. Prolonging operational times outside nine periods per day did not effect in equal growths in GMC manufacturing. Production increased as working hours increased. According to the survey 1977 Quality of Employment, the impact of unusual toil plans on workers’ personal time has resulted with: Fewer periods in household parts.  Advanced plans of particular forms of struggle among family life besides work. One of the amendments is the decision by GMC to implement twelve-hour work days for all Union employees. This decision did not do well with the Union because most were used to working 8-hour days with weekends off. GMC decision is based on new lines and the demand to cope up with production forecasts. This forecast demands the organization to switch to a 24/7 hour operation. Union employees disagree, as they have a feeling that working 12-hour days will lower their morale. They view the alternative schedule as time that they will be missing from their family. While there has been deep research on the subject of scheduled overtime on the productivity of construction labor, there is little research available that looks at the efficiency, or lack of efficiency that is associated with work accomplished on the second shift. In 2008, Hanna, Chang, Sullivan and Lackey in the Journal of Construction Engineering and Management, looked at why and how shift work impacts worker productivity, and they attempted to find a relationship between the length of shift work and resulting efficiency. Their objective was to quantify the effects of shift work on labor productivity. Their results showed the potential of shift work can be both beneficial and detrimental to productivity. Small amounts of well-organized shift work can perform very effectively in response to schedule compression (p.203), but the study also showed that prolonged use of a second shift can lead to a negative impact on efficiency and productivity. The author addresses the attitudes of workers as it relates to the question of productivity of the employee’s in the U.S. automobile industry. Norsworthy and Zabala (1985), attempt to explain the effect on direct labor efficiency (a measure of labor productivity) and product quality of industrial relations performance, as measured by grievances, disciplinary actions, absenteeism, contract demands, negotiations length, and the climate of industrial relations. Their finding state grievance rates are associated with low productivity of production workers which results in high unit costs of production. The study also â€Å"offers strong exhibit that the benefit of improving workers behavior can be enormous† (p.557). The line between labors policies and workers can lead to improved worker attitudes and a rise of productivity. Shift work is the way of life for those who work within the manufacturing industry. Depending on what type of schedule workers on required to be on can have an effect on employee risk or costs to the company. In a report done by Circadian, who specialize in workforce solutions, having those that will be doing the labor work for that manufacturer involved in the decision making with respect to what type shift schedule they are on can go a long ways in having happier, more productive workforce. A resounding message throughout the reports was that one of mostly commonly seen things that affect such things as errors during particular shifts was fatigue. Research methodology Efficiency is an integral requirement in the workplace as it encompasses proper input, output, and high returns for the company (Anderson, 2013). An analysis conducted at GMC, a manufacturing firm, has revealed some inconsistency in the levels of productivity resulting from the activities of its workforce. The early shift workers or employees have posted better output and results more than the late shifts. Two main hypotheses have been supplemented to explain this anomaly. First, the disparity has been linked to machine failure but this has been ruled out due to the routine upgrades and checks, which are usually done on the machines. Secondly, employees in the late night shift may be under-performing as they are making a higher quality product, which takes more time. Management believes there are discrepancies in the quality and quantity of work being produced. Our team was brought in to conduct data analysis to solve and/or assist with the decisions to be made. After listening to their concern the following two variables were developed: Population and Size Collectively, the early and late night shift is composed of 385 employees with twenty in each shift. The population selected for this research activity will be the members of the workforce involved in the early and late night shifts of the company. Target Population and Justification The target populations are the employees who are involved in both the late and early shifts. Their involvement in the research process is based on the fact that they are involved in the activity or area where a problem has been identified in GMC. The equal representation of the employees in the research’s sample size is meant to ensure uniformity in the findings and the recommendations channeled or communicated to the company’s management (Anderson, 2013). Sampling Method The sampling will be conducted through observation and a survey actualized through the administration of a 5-question questionnaire. One research officer will be expected to carefully observe the activities of workers who are part of the early shift while another one will observe the same number of employees in the late night shift. The observation process will be conducted randomly over a 2-week period, with emphasis on the motivation levels of the employees, their input and subsequent output. To supplement the observation process, questionnaires will then be given to employees involved in the analysis process. The questionnaire will contain 5 questions which are listed below: 1. Age: 20-25 26-35 36-45 46-55 >55 2. Sex: F M 3. How many years have you worked nights? 0-5 6-10 11-15 16-20 >20 4. Do you feel management is available to assist with your needs? Yes No 5. Are all the tools and supplies necessary for your jobs accessible? Yes No The two sampling techniques will be pivotal in establishing the problems, which are leading or contributing to the company’s low levels of productivity. Random Sampling Technique used and Justification The research will use the simple random sampling technique to obtain people who are supposed to be involved in the study. The rationale behind using the random sampling method is the fact that it gives all the members of the population an equal opportunity to be involved in the research process. The use of this random sampling technique leads to the elimination of bias in the research process thus enhancing the credibility and integrity of the study’s findings and subsequent recommendations (Oliver, 2010). Protection of Human Subjects The 384 employees involved in the research or study process will be protected by ensuring the anonymity of their responses, comments, and conduct on the company’s premises. Where necessary, alphabetical numbers will be assigned to the various subjects involved in the research process. This will prevent any cases of victimization or witch-hunt, which may result from the remarks communicated by the employees. Data Collection As mentioned earlier, the data will be collected through the observation and survey methods. The survey will be conducted using a 5-question questionnaire as shown above. Data or information collected from the analysis process will be recorded into the soft copy format and then stored in the researchers’ Drop Box account. This is an online information storage service, which is preferred for its top-notch security measures (May, 2011). The secure storage of the data will make certain that it cannot be tampered with to influence the recommendations that are meant to be obtained from the findings. According to Anderson (2013), password protection where cloud or online storage is involved is integral, Drop Box provides this option to all its users. Only authorized individuals will be allowed or able to access the data stored in this online service. Challenges Faced It was difficult and time consuming to reach all the selected employees as some ware available only during night shift. The researchers had to work at night. Descriptive Statistics Interpretation Employees Early and Late Shift 384 employees from the early and late shift were randomly selected. The sample size was 192 employees being selected from each shift. Random sampling was conducted to afford all of the 384 employees an opportunity of being selected and to eliminate bias. Their ages ranged between 20 and 55 years, with variations of plus or minus 19 years. One half or more of the employees were 37.5 years of age or older. The middle half of those selected fell between 28 and 47 years. The most frequent age was 37 years. Their ages were considered to help gage motivation, maturity, and productivity of the employees on both shifts. Interpretation of the results Employees with the least experience (0-5 years) work night shift with those with much experience work (over 20 years) day shift. From the results it’s clear that experience is not affecting productivity. Notably, employees with working experience of 11-15 years’ work day shift. This implies that most night shift workers are youth while those for day shift are aged. This is supported by the number of years night shift employees have spent in the company (Less than 5). Results and Findings Age Almost seventy-eight percent of the workers were between the ages of 26-45. 15.8% were between 20-25, 36.4% were from 26-35, 31.2% were from 36-45, 14% were from 46-55 and only 2.6% were over 55 years of age. Sex Just over twenty-five percent (25.2%) of workers sampled were women and the remaining 74.8% were men. We did not tie worker gender to which shift the worker was employed. Years of Experience The majority of workers selected had either less than 5 years of experience (36.4%) or between 11-15 years of experience (28.6%). Of the 140 workers with less than 5 years of experience, 30% worked day shift and 70% worked night shift. Of the 110 with 11-15 years of experience, 70% worked day shift and 30% worked night shift. See Appendix B for further details. Management Availability and Resources Almost ninety percent (89.9) of workers sampled felt that management was available to assist with any problems and 80% felt they had the required tools to perform their job successfully. Conclusion Team B’s research was not able to show a specific correlation to productivity based on age. There is however, a difference in productivity between the two shifts. Our research appears to show that the day shift, with a larger percentage (65.1%) is more productive and that the decrease in production at night could be related to less experience. Recommendations The management should mix the young employees with the aging ones in both the day and night shift. This does not necessarily need to be a permanent change in shifts but by moving some of the more experienced workers around they can aid in the mentorship of the lesser experienced workers. Doing this can speed the development of those with less experience in order to strengthen the respective shift if management decides alter the shifting in the future when significant ground has been made with evening out the productivity of both shifts. Area for further study Further study should be done to tie and evaluate the impact of age on productivity. Management could also look at better working conditions as an indicator of productivity. Lastly, look age and experience of the employees across the two shifts and how to better mix the levels of both. References Anderson, N. G. (2013). Practical Process Research & Development (Revised ed.). San Diego: Academic Press. Basu, C. (n.d.). Examples of independent variables in business. . Retrieved August 1, 2014, from http://www.scribd.com/doc/141597585/Scienc Boudreau, N. S., & McClave, J. T. (2011). Student’s Solutions Manual, Statistics for Business, 11th edition, Boston, MA: Prentice Hall. Davis, W., & Aguirre, A. (2009). Shift Scheduling and Employee Involvement: The Key to Successful Schedules. Retrieved August 7, 2014. Lankford, W. M. (1998, June 21). Changing Schedules: A Case for Alternative Schedules of Work. Career Development international, 3.4, 161-163. Retrieved October 8, 2013, from Miller Library. May, T. (2011). Social Research Issues, Methods and Process (4th ed.). Maidenhead, Berkshire, England: McGraw Hill, Open University Press. Oliver, P. (2010). Understanding the Research Process. Los Angeles: SAGE. Staines, G. L., & Pleck, J. H. (1984). Nonstandard Work Schedules and Family Life. Journal of Applied Psychology, 69(3), 515-523. doi:10.1037/0021-9010.69.3.515. Takahashi, M. (2012, March 13). Prioritizing Sleep for Healthy Work Schedules. Physiological Anthropology. Retrieved October 8, 2013, from Miller Library. Taylor, E. (n.d.). Dependent and independent variables Retrieved August 1, 2014, from http://de.cyclopaedia.net/wiki/Dependent_and_independent_variables. Hanna, A.S., Chang, C., Sullivan, K.T. and Lackney, J. A. (2008, March). Impact of Shift Work on Labor Productivity Contractor. Journal of Construction Management, 134(3), 197-204. Norsworthy, J. and Zabala, C. (1985). Worker Attitudes, Worker Behavior, and Productivity in The U.S. Automobile Industry, 1959-1976. Industrial & La bor Relations Review, 38(4), 544-557. Robert Schoenenberger, 2011, General Motors Sets Overtime Shift for Lordstown to Meet Growing Chevrolet Cruze demand. Retrieved from http://www.cleveland.com/business/index.ssf/2011/08/general_motors_sets_overtime_s.html GM News. 2012, GM to Add Third Shift, 800 Jobs at Arlington Assembly, http://media.gm.com/media/us/en/gm/news.detail.html/content/Pages/news/us/en/2012/Jun/0622_arlington.html

Borderline Personality Case Study Maria - 1402 Words

Case study: Maria The patient Maria offers a case study in typical borderline personality symptoms. Maria has manifested a marked instability in the relationships in her personal life. Borderline patients often express over-idealization or demonization of the individuals in their lives. Their personalities are often histrionic, and they tend to see both people and events in black and white terms, either as bad or good. It is typical for a BPD (borderline personality disorder) patient to have suffered from sexual abuse or emotional abuse in childhood, which is one reason that it is hypothesized that so many have obsessive real or imagined fears of abandonment. However, in the case of Maria, intense emotions alternated with feelings of depersonalization (which often bring upon cutting episodes, substance abuse, or binge eating) may be evident from birth, even in the absence of sexual abuse. In an effort at self-stabilization, some use physical pain which has been demonstrated to reduce emotional arousal (Dingfelder 2004:46). Abandonment fears are often enacted in the therapeutic relationship, and can make it very difficult to treat borderline patients. For example, a therapist is often confronted with a borderline patient who will threaten suicide if the therapist does not talk to him or her immediately outside of office hours, as the patient repeatedly demands proof of her affections. BPD provides many challenges to practitioners. They frequently seek out help, butShow MoreRelatedConcept Analysis Essay1230 Words   |  5 PagesConcept Analysis Assignment Grand Canyon University Theoretical Foundations for Nursing Roles and Practice NUR 502 Ms. Maria Mendez June 13, 2013 Concept Analysis Assignment Today’s nursing profession encompasses a variety of specialties and disciplines that demonstrate a wide range of phenomena. 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