# Reflective Analysis on Mathematics and Calculus

Relearning the calculus, relating it to real-life

Mela Aziza

Background

I have enjoyed doing mathematics since I used to be in elementary university. However, this sense changed a bit after i was at secondary college. My mathematics teacher asked me to memorise many formulas and rules related to advanced topics without knowing when I can use those in my real-life. I thought an advanced topic was really hard to learn since it was commonly abstract notion. Consequently, students like me would find difficulties how to make it concrete and hook up it to real life. Furthermore, my mathematics teacher only motivated us to review mathematics hard in order to attain high results in examinations. She hardly ever explained about the application of mathematics inside our daily life. This situation made me less enjoyed learning mathematics. For instance, while I was learning calculus which i assumed as a sophisticated topic, I did not know when I can use it in my own life so that we was not motivated to learn it. At that time, I guessed calculus was inadequate. Calculus was just about habits, formulas, and calculations without knowing why I needed to learn it. Therefore, this experience has been uplifting me in how I should train my students in the foreseeable future.

I hoped to describe and show my students about how powerful and useful mathematics can be. Unfortunately, it really was difficult to find the connection between mathematics and day to day activities, specifically for the calculus. My students were questioning when they could use calculus in their life. I became puzzled and may not supply the appropriate answer because I have not known the application of calculus that was highly relevant to my students' life. I taught calculus using the similar method to my previous mathematics teacher, fixing almost any calculus questions from my very own books using the formulas or rules. However, I am interested in exploring and growing the effectiveness of calculus in lifestyle because I want to set up answers for my own past question, "when I could use it". Hence, when getting the chance to take the growing subject knowledge course, I had been excited to concentrate on some calculus questions using real-life contexts.

Solving calculus problems

I began my independent learning by handling the max package problem given by my personal teacher (see Appendix A). This issue about the newspaper which has part a, then I was instructed to make a box by trimming a rectangular with side x from each of the four corners. I must find the value of x so that I can make the largest box. I tried to get the x value for creating the biggest pack by doing some algebraic equations and lastly, I obtained the style for locating the x value. Learning the answer gave me an opportunity to relate it to the idea of differentiation. It was a new thing for me and when I looked on the internet, found it was popular in coaching and learning mathematics related to the calculus subject. However, I did so not know why I found Indonesian mathematics professors hardly ever used this practical question while educating the idea of differentiation.

Next, I transferred to how to add the first rule of differentiation, f'(x), from function f(x). I started by sketching a graph of the function, then designed gradient of two adjacent items using the gradient of an straight series and limit strategy (see Appendix B). Finally, I came across that the first derivative equals with the gradients of a spot from the function. Then, I attempted similar calculations for a few different functions, and lastly, I set up the style of the first derivative. While accomplishing this, I was considering which I should coach first, gradient or differentiation, to make students understand where the first derivative comes. Furthermore, a obvious point for me by solving this problem, I was aware that as a professor I can teach mathematics through using algorithmic/algebraic/analytic/calculating, visual (image/graph), and inductive (structure) thinking. For example, when locating the maximum value of the function, I attained the same answer by using two different methods, graphing and calculating.

In addition, I explored how to get the graph of the first derivatives of different functions by using gradient idea (see Appendix C). I drew both common and uncommon functions. I felt those were interesting and challenging because I possibly could create the graph of the first and the second derivative just by considering the graph of the original function. However, when I want to find the first derivative function, I must analyze using an algebraic method. Although I could not get immediately what the function of the first derivative f'(x) through pulling, I could distinguish when the function come to maximum value, (when f" (x) < 0), bare minimum value (when f" (x) > 0), and neither maximum nor minimum amount value (when f" (x) = 0), for illustration, f(x)= x3-6x2+12x-5 having an inflexion point (see Number 1).

I also tried to find the gradient of unusual functions such as a complete function (f(x)=|x|) by plotting the graph by hand and checking out it using software GSP (The Geometer's Sketchpad), i quickly found that there was a point on the |x|function that can't be differentiated (non-differentiable point) that was when x = 0, but for other things, those were differentiable (see Physique 2).

Furthermore, I explored six common flaws (Cipra, 2013) that students made in doing calculus related to the way they solve some routine problems and understand an idea of finding the section of function by essential theory (see Appendix D). The students mostly just calculated the region using formula without sketching the function so that occasionally they found a poor area. The region will be never negative. The students should know that the region above x-axis will be positive because y-axis worth are always positive as the area below x-axis will be negative because of y-axis negative ideals (Stewart, 2016). Hence, students have to increase the region of function below x-axis with negative (-) in favour of becoming a positive area.

Reflection

During this program, I relearned calculus notion by solving some problems. I sensed back a sense of doing mathematics when dealing with the problems both regular and real-life problems. This sense made me thrilled to get the solutions for each and every problem that I faced. I became aware that abstract concepts cannot be segregated from calculus. Although regular problems are commonly abstract, students will be able to learn the value of symbol ideas in calculus through fixing these problems. I also tried out to hook up calculus by fixing some real-life problems designed to use real-life contexts and can be imagined as daily experiences (Gravemeijer & Doorman, 1999), for illustration, the max box problem that can be linked to a company. After doing some real-life problems, I agree that these problems should be trained in the school room (Gainsburg, 2008). Teachers have the ability to use these problems to improve students' motivation also to develop reasoning as well as problem-solving skills of students in learning mathematics (Karako & Alacac±, 2015). Therefore, the instructors can make mathematics are more meaningful for their students through real-life problems.

On the other palm, I think not absolutely all real-life problems are practicable for students because the problems do not relate with their life straight. 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 problems when some facts are abandoned in order to make students understand the question easily. An issue which is pertinent to one student's life might not exactly be relevant for others. Therefore, teachers should check the potency of the issues by asking students first (Burkhardt, 1981), and then they will notice the good problems that can be utilized in the foreseeable future. Furthermore, calculus is advanced knowledge for most students because they find it hard to concretise so that occasionally it should continue to be abstract (Wilensky, 1991). Furthermore, teachers need to consider the time when they give the students real-life problems. They can not give them these problems for 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 subject matter and enough time may become restrictions allowing you to connect it to the real-world (Karako & Alacac±, 2015).

Teachers can motivate students to believe inductively in learning mathematics. They may involve students to find the first derivative routine by using the gradient of your straight range and limit idea. They should not give a design f'(xn) =nxn-1 directly to students when presenting differentiation, but they ask students to establish the first derivative pattern by their own self. In addition, I found that teachers have the ability to use a slope of zero (f'(x)=0) for determining what is the maximum or minimal value of the function quickly. However, professors also have to ask students to check on the graph or the second derivative of the function to find the exact category of the x value (maximum, minimum amount, or inflexion point). Hence, as a mathematics educator, I should deem some factors before deciding an effective coaching method that encourages my students to understand calculus concepts easily.

I assumed that using technology can seem sensible of calculus for students. I considered using GSP while educating to get a graph of the function also to look closer whether the function can be differentiated for each and every point. Furthermore, I feel that mathematics teachers might be able to explore any type 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, educators who result from non-native-English-speaking countries should be careful in understanding this is of the issues because there is a specific British term of mathematics that looks new or synonymous. For example, I was baffled to tell apart between two words that felt to be synonyms like capacity and quantity. I first of all thought that those two words possessed similar meaning, however, capacity related to how much water held while volume level related to how many materials needed (sturdy) in the pot. Educators also may modify types of the calculus tasks and the application of calculus videos that are given on the internet. In my opinion, I obtained the new point of view by watching some videos showing activities that instructors have like creating a group job related to the application of calculus. However, educators should consider about the time because doing a project or observing a training video will be time-consuming.

Teachers require taking a look at why students made the errors and analysed what they must do to prevent similar faults among students. Students solve calculus problems using algorithms including symbol concepts however they commonly ignore to crosscheck the utilization of the symbol in their works. Because of this, they made mistakes in using symbols which can be shown in Appendix D. Furthermore, visual thinking is an important skill to abandon mistakes in finding alternatives for calculus problems especially to find the area of the function. It just happened because they did not get the graph of the function. On the other hand, students only can identify the positioning of the graph either above or below the x-axis when they look the graph immediately. Hence, teachers should be more aware that algebraic, image, and attracting the graph or visualisation are crucial principles in learning calculus.

Implementation in Indonesia

One of the reasons why I needed to explore the effectiveness of mathematics and exactly how to instruct it in the class room is the aim of educating mathematics in Indonesia. Indonesia has adopted RME (Realistic Mathematics Education) from holland, and then known as PMRI (Pendidikan Matematika Realistik Indonesia) which correlates to instructing mathematics in real contexts and emphasises the use of mathematics (Sembiring, 2008). However, RME will not mean educators have to entail the students in real activities but produce a important learning activity so that students can see right now it like they actually reality (Vehicle den & Drijvers, 2014). Despite the fact that some previous research workers found that the implementation of PMRI in Indonesia experienced results on students' mathematics success (Armanto, 2002; Fauzan, 2002), Indonesia hasn't made relevant PMRI curriculum materials (Sembiring, 2008). Therefore, Indonesia still must develop some resources related to the implementation of PMRI.

In addition, Indonesian mathematics teacher's ability itself will be a difficulty in implementing coaching mathematics in real contexts. Although one of their concerns is hooking up mathematics to the real world to be able to encourage students to cope with their lifestyle problems (Zamroni, 2000), some of them are only able to instruct instrumental understanding (Skemp, 1976) in the class room so that students learn calculus as formulas without realising the way they utilize it. Students just follow instructors' instruction; memorising formulas, understanding the illustrations, and then fixing the exercises. Undeniably, students own negative perspectives on mathematics, like the calculus, are because of this fact. Thus, teachers should find ways to improve these students' perspectives to be able to improve their understanding and achievements in mathematics.

Mathematics teachers can form realistically applied mathematics in the classroom through the collection of practical problems (Burkhardt, 1981) that provide a chance for students to use their mathematical skills. Individually, there are some real-life issues that Indonesian teachers can use such as Max container. I am curious exactly what will happen as i and other teachers utilize this problem before introducing calculus to the students, maybe, we will recognise kinds of methods from the students that people have never dreamed before. Furthermore, Indonesian mathematics teachers should explore resources on the internet and use software like GSP in order to encourage students' sense of learning calculus. However, they may face further difficulty in using GSP or e-based learning method because not all of them can operate it and not every school has technical equipment as well as internet connection. Another point that Indonesian mathematics teachers should deem is students' common blunders in learning calculus. Instructors 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 duplicate the same mistake. Teachers also have to check their students' flaws to analyse the reason why, then researching and fixing the myths that college student have from the faults.

Conclusion

Despite the actual fact that it's common that students feel calculus is difficult to be known, resolved and applied, I believe there will be some alternatives that teachers can do such as giving both reasonable and unrealistic problems, using software, and watching request of calculus on videos. Besides these ways being likely to inspire and encourage students to learn calculus, these ways can also stimulate students using it in their real-life. However, teachers have to consider the practical problems for students and keep presenting some boring problems to look deeper what some myths or mistakes that they manufactured in doing calculus.

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