Reflective Assessment on Mathematics and Calculus

Reflective Assessment on Mathematics and Calculus Relearning the calculus, relating it to real-life Mela Aziza Background I have loved doing mathematics since I was in elementary school. However, this feeling changed a little bit when I was at secondary school. My mathematics teacher asked me to memorise many formulas and principles related to advanced topics without knowing when I can use those in my real-life. I thought that an advanced topic was really hard to learn because it was commonly abstract concept. Consequently, a student like me would find difficulties how to make it concrete and connect it to the real world. In addition, my mathematics teacher only encouraged us to study mathematics hard in order to achieve high scores in examinations. She rarely explained about the application of mathematics in our daily life. This situation made me less enjoyed learning mathematics. For example, while I was learning calculus that I assumed as an advanced topic, I did not know when I can use it in my life so that I was not motivated to learn it. At the time, I guessed calculus was useless. Calculus was ju st about patterns, formulas, and calculations without knowing why I needed to learn it. Therefore, this experience has been inspiring me in how I should teach my students in the future. I hoped to explain and show my students about how powerful and useful mathematics can be. Unfortunately, it was really hard to find the connection between mathematics and daily activities, especially for the calculus. My students were questioning when they could use calculus in their life. I became confused and could not give the appropriate answer because I have not known the application of calculus that was relevant to my students life. I taught calculus using the similar method to my previous mathematics teacher, solving any kind of calculus questions from my own textbooks using the formulas or rules. However, I am interested in exploring and developing the usefulness of calculus in daily life because I want to establish answers for my own previous question, when I can use it. Hence, when getting the chance to take the developing subject knowledge course, I was excited to focus on some calculus questions using real-life contexts. Solving calculus problems I started my independent learning by solving the max box problem given by my personal tutor (see Appendix A). This problem about the paper which has side a, then I was instructed to make a box by cutting a square with side x from each of the four corners. I have to find the value of x so that I can make the biggest box. I tried to find the x value for creating the biggest box by doing some algebraic equations and finally, I obtained the pattern for finding the x value. Finding out the answer gave me an opportunity to relate it to the concept of differentiation. It was a new thing for me and when I searched on the internet, found it was popular in teaching and learning mathematics related to the calculus topic. However, I did not know why I found Indonesian mathematics teachers rarely used this practical question while teaching the concept of differentiation. Next, I moved to how to introduce the first principle of differentiation, f'(x), from function f(x). I started by drawing a graph of the function, then formulated gradient of two adjacent points using the gradient of a straight line and limit concept (see Appendix B). Finally, I found that the first derivative equals with the gradients of a point from the function. Then, I tried similar calculations for some different functions, and finally, I established the pattern of the first derivative. While doing this, I was thinking which I should teach first, gradient or differentiation, in order to make students understand where the first derivative comes. Furthermore, a noticeable point for me by solving this problem, I was aware that as a teacher I can teach mathematics through using algorithmic/algebraic/analytic/calculating, visual (image/graph), and inductive (pattern) thinking. For example, when finding the maximum value of the function, I acquired the same answer by using two differe nt methods, graphing and calculating. In addition, I explored how to draw the graph of the first derivatives of different functions by using gradient concept (see Appendix C). I drew both common and uncommon functions. I felt those were interesting and challenging because I could create the graph of the first and the second derivative just by looking at the graph of the original function. However, when I want to find the first derivative function, I have to calculate using an algebraic method. Although I could not get directly what the function of the first derivative f(x) through drawing, I could differentiate when the function reached maximum value, (when f (x) f (x) > 0), and neither maximum nor minimum value (when f (x) = 0), for instance, f(x)= x3-6x2+12x-5 having an inflexion point (see Figure 1). I also tried to find the gradient of uncommon functions such as an absolute function (f(x)=|x|) by plotting the graph manually and checking it using software GSP (The Geometers Sketchpad), then I found that there was a point on the |x|function that cannot be differentiated (non-differentiable point) that was when x = 0, but for other points, those were differentiable (see Figure 2). Furthermore, I explored six common mistakes (Cipra, 2013) that students made in doing calculus related to how they solve some routine problems and understand a concept of finding the area of function by integral concept (see Appendix D). The students mostly just calculated the area using formula without drawing the function so that occasionally they found a negative area. The area will be never negative. The students should know that the area above x-axis will be positive because y-axis values are always positive while the area below x-axis will be negative because of y-axis negative values (Stewart, 2016). Hence, students have to multiply the area of function below x-axis with negative (-) in favour of becoming a positive area. Reflection During this course, I relearned calculus concept by solving some problems. I felt back a sense of doing mathematics when solving the problems both routine and real-life problems. This sense made me excited to find the solutions for every problem that I faced. I became aware that abstract concepts cannot be separated from calculus. Although routine problems are commonly abstract, students will be able to learn the importance of symbol concepts in calculus through solving these problems. I also tried to connect calculus by solving some real-life problems which use real-life contexts and can be imagined as daily experiences (Gravemeijer Doorman, 1999), for instance, the max box problem that can be connected to a manufacturer. After doing some real-life problems, I agree that these problems should be taught in the classroom (Gainsburg, 2008). Teachers are able to use these problems to enhance students motivation and to develop reasoning as well as problem-solving skills of students in l earning mathematics (Karakoà § AlacacÄ ±, 2015). Therefore, the teachers will be able to make mathematics become more meaningful for their students through real-life problems. On the other hand, I think not all real-life problems are practicable for students because the problems do not relate to their life directly. I have done some problems from some websites and a textbook of calculus (SMP, 1973), but not all problems were relevant to a real context and could be solved. I encountered there was a problem when some facts are abandoned in order to make students understand the question easily. A problem which is relevant to one students life may not be relevant for others. Therefore, teachers should check the effectiveness of the problems by asking students first (Burkhardt, 1981), and then they will notice the good problems that can be used in the future. In addition, calculus is advanced knowledge for most students because they find it difficult to concretise so that occasionally it should remain abstract (Wilensky, 1991). Furthermore, teachers need to consider the time when they give the students real-life problems. They cannot give them these problems fo r every meeting because they also should provide opportunities to students for learning all calculus concepts, both concrete and abstract. Thus, most teachers assumed the nature of mathematics topic and the time may become limitations for connecting it to the real-world (Karakoà § AlacacÄ ±, 2015). Teachers can motivate students to think inductively in learning mathematics. They may involve students to find the first derivative pattern by using the gradient of a straight line and limit concept. They should not give a pattern f'(xn) =nxn-1 directly to students when introducing differentiation, but they ask students to establish the first derivative pattern by their own self. In addition, I found that teachers are able to use a slope of zero (f'(x)=0) for figuring out what is the maximum or minimum value of the function quickly. However, teachers also have to ask students to check the graph or the second derivative of the function to find the exact category of the x value (maximum, minimum, or inflexion point). Hence, as a mathematics teacher, I should deem some factors before deciding an effective teaching method that encourages my students to understand calculus concepts easily. I assumed that using technology can make sense of calculus for students. I considered using GSP while teaching to draw a graph of the function and to look closer whether the function can be differentiated for every point. Furthermore, I think that mathematics teachers may be able to explore any kind of calculus questions on websites such ashttps://www.math.ucdavis.edu andhttp://www.dqime.uni-dortmund.de which I assert as resources for finding real-life mathematics problems using the English language. However, teachers who come from non-native-English-speaking countries should be careful in understanding the meaning of the problems because there was a specific English term of mathematics that sounds unfamiliar or synonymous. For instance, I was confused to distinguish between two words that felt to be synonyms like capacity and volume. I firstly thought that those two words had similar meaning, however, capacity related to how much liquid held while volume related to how many material s needed (solid) in the container. Teachers also may adapt examples of the calculus projects and the application of calculus videos that are provided on the internet. Personally, I obtained the new perspective by watching some videos showing activities that teachers did like creating a group project related to the application of calculus. However, teachers should consider about the time because doing a project or watching a video will be time-consuming. Teachers require looking at why students made the mistakes and analysed what they should do to prevent similar mistakes among students. Students solve calculus problems using algorithms involving symbol concepts but they commonly forget to crosscheck the use of the symbol in their works. As a result, they made mistakes in using symbols which are shown in Appendix D. Furthermore, visual thinking is an important skill to abandon mistakes in finding solutions for calculus problems especially to find the area of the function. It happened because they did not draw the graph of the function. Meanwhile, students only will be able to identify the position of the graph either above or below the x-axis when they look the graph directly. Hence, teachers should be more aware that algebraic, symbol, and drawing the graph or visualisation are crucial concepts in learning calculus. Implementation in Indonesia One of the reasons why I wanted to explore the usefulness of mathematics and how to teach it in the classroom is the aim of teaching mathematics in Indonesia. Indonesia has adopted RME (Realistic Mathematics Education) from the Netherlands, and then known as PMRI (Pendidikan Matematika Realistik Indonesia) which correlates to teaching mathematics in real contexts and emphasises the application of mathematics (Sembiring, 2008). However, RME does not mean teachers have to involve the students in real activities but create a meaningful learning activity so that students can imagine it like they do reality (Van den Drijvers, 2014). Even though some previous researchers found that the implementation of PMRI in Indonesia had positive effects on students mathematics achievement (Armanto, 2002; Fauzan, 2002), Indonesia has not made relevant PMRI curriculum materials (Sembiring, 2008). Therefore, Indonesia still needs to develop some resources related to the implementation of PMRI. In addition, Indonesian mathematics teachers ability itself will be a difficulty in implementing teaching mathematics in real contexts. Although one of their concerns is connecting mathematics to the real world in order to encourage students to deal with their daily life problems (Zamroni, 2000), some of them are only able to teach instrumental understanding (Skemp, 1976) in the classroom so that students learn calculus as formulas without realising how they use it. Students just follow teachers instruction; memorising formulas, understanding the examples, and then solving the exercises. Undeniably, students own negative perspectives on mathematics, including the calculus, are due to this fact. Thus, teachers should find ways to improve these students perspectives in order to enhance their understanding and achievement in mathematics. Mathematics teachers can develop realistically applied mathematics in the classroom through the collection of realistic problems (Burkhardt, 1981) that provide an opportunity for students to apply their mathematical skills. Personally, there are some real-life problems that Indonesian teachers can use such as Max box. I am curious what will happen when I and other teachers use this problem before introducing calculus to the students, maybe, we will recognise kinds of methods from the students that we have never imagined before. Furthermore, Indonesian mathematics teachers should explore resources on the internet and use software like GSP in order to stimulate students sense of learning calculus. However, they may encounter further difficulty in using GSP or e-based learning method because not all of them can operate it and not every school has technological equipment as well as internet connection. Another point that Indonesian mathematics teachers should deem is students common mist akes in learning calculus. Teachers should be aware that students have to check their own work to find the mistakes because if they check by themselves, they likely will not repeat the same mistake. Teachers also have to check their students mistakes to analyse the reasons, then reviewing and correcting the misconceptions that student have from the mistakes. Conclusion Despite the fact that it is common that students feel calculus is difficult to be understood, solved and applied, I think there will be some solutions that teachers can do such as giving both realistic and unrealistic problems, using software, and watching application of calculus on videos. Besides these ways being likely to motivate and encourage students to learn calculus, these ways also can stimulate students using it in their real-life. However, teachers have to consider the practical problems for students and keep giving some routine problems to look closer what some misconceptions or mistakes that they made in doing calculus.

Censorship :: essays research papers

Censorship by definition is the suppression of words, images or ideas that are â€Å"offensive†. It occurs when certain people succeed in imposing their personal or moral values on others. The debate over censorship deals mainly with the first amendment and whether it is constitutional for a group of people to decide what is right for other people. It has exploded within recent years with the advent of the Internet. Many different kinds of people can and are considered censors. Parents, teachers, administrators and employers who forbid others to speak in certain ways can be censors. In fact, the word censor actually comes from ancient Rome, which referred to someone whose job it was to oversee morals and conduct. Before I get into censorship on the Internet, here are other cases of censorship: Perhaps the first known case of censorship occurred in 1873 with the passing of the Comstock law by Congress. The law was advocated by Anthony Comstock, head of the Society for the Suppression of Vice (Explain vice). The law forbade the mailing of anything, in his opinion, lewd, obscene or indecent. In his life burned 120 tons of books and art including the works of Chaucer- Caterbury Tales, Oscar Wilde, Ernest Hemmingway, Eugene O’Neill and John Steinbeck. Censorship primarily occurs in the arts, especially music. The PMRC or Parents Music Resource Center finally after about a decade of pushing forced the RIAA or Recording Industry of America to start use labeling systems and logos such as â€Å"Parental Advisory – Explicit Lyrics†. However, the putting of the labels on the records was left entirely to the record companies who neglected the sticker on many deserving albums. In 1992, Washington became the first state to pass a law forbidding the purchase of cdâ⠂¬â„¢s with certain logos if you’re under 18, this law was quickly deemed unconstitutional. Some performers have been banned from performing certain songs in certain places, for example Ozzy Osbourne was sued thrice after three teens committed suicide and their parents blamed his lyrics in â€Å"Suicide Solution†, Ice-T was heavily protested for his song Cop Killer in the early 90’s which promoted violence specifically on police. Madonna was arrested in Toronto after she ignored warnings of her potential arrest and went on with her normal act of acting out masturbation on stage. The debate over music most definitely flared up again after the shootings at Columbine in which Marilyn Manson was once again accused of causing horrible crimes as many people continue their war with him and his music.

Pet Wonderland Services (PWS) Essay

Introduction to the organization Pet Wonderland Services (PWS) is a new business in pets industry. PWS first shop was at Shah Alam section 13. Although there are several pets shop in Shah Alam, PWS wil brings new services to the animal lovers communities as. Vision To serve the best quality, friendly and effective service and fulfill customer satisfaction. Mission To be one of the top pet shop that brings the most high quality and effective services in the Malaysia. â€Å"Your pets is our responsible too† Objective The main objective of this service is to ease community that they would not have to bring their pet to our shop but we will go to their house and give the services that they wanted to. In our services, we only provide the best pets product and we will consult with the owner about their pets first so that we can use an appropriate an suitable pet’s product to their pets. We want to deliver a quality services so that customer will believe in us. We are not just gaining profit, we are also gaining trust to our customer and love to the animal.

Comparison of AvantGarde Formal Art work essays

Comparison of AvantGarde Formal Art work essays A Comparison of Formal and Avant-Garde Artwork Modern art is a unique creation all its own, and since its beginnings there have been two very distinct groups present. They are the Formalists and the Avant-Garde. The Formalist group believes in the literal representation of the art work. They value the form used, whether it be how the colors are stressed or the techniques used, over the idea behind the art work. The Avant-Garde artists on the other hand are more interested in the creation of art in order to challenge what exactly art is and can be. They have tried to break down the boundaries of what I selected Piet Modrian for as the artist I looked at regarding Formalism. His work seemed to change over a period of time. His Woods done in 1910 showed a typical painting of the forest. By typical I mean that the depiction of the forest was done to look like a forest on the canvas. The next piece of Tree done in 1911 showed a more abstract form of trees while you could still detect on a lot of movement within the painting. The shapes and negative space between the lines is what became more important in his work. He then created Composition with Trees in 1914. This piece as his previous ones was even more abstract and was honing in on the form of the painting. Later he created a piece known only as Composition, removing the trees and woods all together from the title. He was trying to create the essence of nature using rectangles and simplifying lines to their primary essence. He also used more primary colors versus his previous works done in black and white. This idea that Mondrian was using while creating his works of art was one of the ideas that Clement Greenberg was dealing with in his works. Greenberg said that the actual painting over the form is what makes the art work abstract.1 Greenberg believed that on...

When The Legends Die Essays - English-language Films, Free Essays

When The Legends Die Essays - English-language Films, Free Essays When The Legends Die When The Legends Die Novel/Movie Contrast Joe McNew June 8, 2000 The book, When the Legends Die, was pretty good. Then some idiot had to go and make a movie out of it. In contrasting the two, I found alot more differences than similarities. The first similarity came when Blue Elk came and took Thomas Black Bull to the reservation. Tom had trouble with the kids as in the book. Brother Bear was chained up and eventually Tom took him back to the woods and left him just as Tom did in part I of the book. Of course, Red Dillion was a jerk in both the movie and the book, although the book illustrated a better image in the readers mind. There were quite a few things the two had in common but the differences outnumber the similarities. I am not going to attempt to represent every difference but I wil give a few examples. For instance, we first see Tom, or any characters for that matter, after George and his mom dies. The movie represents Tom as older, during his rodeo days. In the book, during Red and Toms 2 fight, he only knocks him down but in the movie he beats him up really bad. He goes home with the nurse but in the book he does nothing of that sort. So far he hasnt gone for the bear yet but were not finished yet. The book had alot going for it and I thought that it was much better than the movie. The movie was a horrible experience and is probably the only time I wished my teacher wouldnt have shown it. Usually, I like the movie better than the book but this was the exception. Whoever made this movie should be charged with murder (of the book). The movie missed alot of the key points. The movie was interesting and did illustrate some things better than the book. It did an excellent job of show Toms transitiion from rags to riches. In addition, the movie moved alittle faster which made it more exciting. I believe that it did a better job when Tom actually ditched Red and got into the big tournament. Both had their strong points. The book was better...way better. This is a good example of why a movie shouldnt be made from a book or at least a person may want to consider skipping the movie if they really liked the book. Although, the movie was better in some areas overall I would definitely recommend the book over the movie. In conclusion, movie = F book = A.

The Story of Baseball Player Michael When He Gets Injured and Loses Essay

The Story of Baseball Player Michael When He Gets Injured and Loses Sight - Essay Example Despite all Michael's fame and fortune, Michael feels incredibly lonely at times. His father, Phil, is virtually never around home because he spends a large part of the year traveling around the world as a symphony orchestra conductor. Up until now, Phil has been able to make up for his long absences with expensive gifts from the four ends of the earth. On the other hand, Michael’s mother, Sarah, is completely the opposite type of character; she is very weak physically as a result of a recent stroke. As such, Sarah cannot travel as much as Phil does. Despite her limitations, Sarah is a strong-willed woman who tries her best to keep Michael in check. Because of their close proximity to each other, Michael and his mother have a close yet sometimes fractured relationship. To make up for her lack of mobility, Sarah has made use of Julia, who works as a servant around the family home. Life for Michael couldn’t be better when tragedy strikes – he suffers a facial injur y during a baseball game that could result in the loss of sight in both or one of his eyes. At first, Michael does not know what to think. Ever since he was a young child, his goal was to become a professional in one of the bigger leagues. If he cannot recover from this injury, his lifelong dream may be in jeopardy. After flying to a specialist eye hospital many hours away, the doctor examines Michael and then informs him of some depressing news – Michael has suffered massive internal bleeding that has the potential to threaten his long-term eyesight. However, there is one small ray of hope that Michael clings onto for dear life. It might be possible for him to have an operation to regain his eyesight, but only after his internal bleeding is cured. This is a process that may take up to two years and is a long shot at best.  

Culture, Service and Product Essay Example | Topics and Well Written Essays - 2000 words

Culture, Service and Product - Essay Example These barriers have to be addressed so that entering a new market would be successful. This paper comes in two parts. First part discusses glocalization issues, and the second part talks about case of Disneyland as it entered glocalization in HongKong and the soon to be opened Shanghai Disney. Conceptual framework of the study analyzes importance of glocalization, gives its meaning, and how it is applied by Disney in its business practice in setting up its resorts to an environment totally different from its origin. These comes as are positive challenges to the this icon of entertainment. In this study, I borrowed the concept of Luigi and Vineran (n.d.) that proposed: â€Å"in order to be successful globally, managers must act locally in the different markets they chose to enter.† Glocalization is coined word from globalization and glocalization which is used describe a product or service that is developed and distributed globally, but is also fashioned to accommodate the user or consumer in a local market†. (Investopedia. n.d.) Glocalization meant differently as to each region because of differences cultures, language, religion, traditions and laws. Luigi, D. and S. Vineran (n.d.) in their study put forward a definition that states: â€Å"providing a global offer (brand, idea, product, idea, or service, etc.) while taking local issues into account†. A global product / service that can face competition from both local and international brands in a better way because it meets certain local needs or preferences at lower costs due the global edge of the company. What key areas does a company need to consider when moving a product/service developed in one country to a different country? How should a company assess how a product or service is being received in a different country? Global marketing has introduced may changes on business approaches, such as the way how a company transacts business to