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 just 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 different 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) < 0), minimum value (when 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).
Figure 1. An inflexion point on the graph of f(x)= x3-6x2+12x-5
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 Geometer’s 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).
Figure 2. The graph off(x)=|x|
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 learning mathematics (Karakoç & Alacacı, 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 student’s 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 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 topic and the time may become limitations for connecting it to the real-world (Karakoç & Alacacı, 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 as and 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 materials 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 teacher’s 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 mistakes 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.