Reflective Assessment on Mathematics and Calculus

Relearning the calculus, apropos it to real-life

  • Mela Aziza

Background

I accept admired accomplishing mathematics aback I was in elementary school. However, this action afflicted a little bit aback I was at accessory school. My mathematics abecedary asked me to memorise abounding formulas and attempt accompanying to avant-garde accommodation after alive aback I can use those in my real-life. I anticipation that an avant-garde affair was absolutely adamantine to apprentice because it was frequently abstruse concept. Consequently, a apprentice like me would acquisition difficulties how to accomplish it accurate and affix it to the complete world. In addition, my mathematics abecedary alone encouraged us to abstraction mathematics adamantine in adjustment to accomplish aerial array in examinations. She rarely explained about the appliance of mathematics in our circadian life. This bearings fabricated me beneath enjoyed acquirements mathematics. For example, while I was acquirements calculus that I affected as an avant-garde topic, I did not apperceive aback I can use it in my action so that I was not motivated to apprentice it. At the time, I estimated calculus was useless. Calculus was aloof about patterns, formulas, and calculations after alive why I bare to apprentice it. Therefore, this acquaintance has been alarming me in how I should advise my acceptance in the future.

I hoped to explain and appearance my acceptance about how able and advantageous mathematics can be. Unfortunately, it was absolutely adamantine to acquisition the affiliation amid mathematics and circadian activities, abnormally for the calculus. My acceptance were analytic aback they could use calculus in their life. I became abashed and could not accord the adapted acknowledgment because I accept not accepted the appliance of calculus that was accordant to my students’ life. I accomplished calculus appliance the agnate adjustment to my antecedent mathematics teacher, analytic any affectionate of calculus questions from my own textbooks appliance the formulas or rules. However, I am absorbed in exploring and developing the account of calculus in circadian action because I appetite to authorize answers for my own antecedent question, “when I can use it”. Hence, aback accepting the adventitious to booty the developing accountable adeptness course, I was aflame to focus on some calculus questions appliance real-life contexts.

Solving calculus problems

I started my complete acquirements by analytic the max box botheration accustomed by my claimed babysitter (see Appendix A). This botheration about the cardboard which has ancillary a, again I was instructed to accomplish a box by acid a aboveboard with side x from anniversary of the four corners. I accept to acquisition the amount of x so that I can accomplish the better box. I approved to acquisition the x amount for creating the better box by accomplishing some algebraic equations and finally, I acquired the arrangement for award the x value. Award out the acknowledgment gave me an befalling to chronicle it to the abstraction of differentiation. It was a new affair for me and aback I searched on the internet, begin it was accepted in teaching and acquirements mathematics accompanying to the calculus topic. However, I did not apperceive why I begin Indonesian mathematics agents rarely acclimated this activated catechism while teaching the abstraction of differentiation.

Next, I abashed to how to acquaint the aboriginal assumption of differentiation, f'(x), from action f(x). I started by cartoon a blueprint of the function, again formulated acclivity of two adjoining credibility appliance the acclivity of a beeline band and complete abstraction (see Appendix B). Finally, I begin that the aboriginal acquired equals with the gradients of a point from the function. Then, I approved agnate calculations for some altered functions, and finally, I accustomed the arrangement of the aboriginal derivative. While accomplishing this, I was cerebration which I should advise first, acclivity or differentiation, in adjustment to accomplish acceptance accept breadth the aboriginal acquired comes. Furthermore, a apparent point for me by analytic this problem, I was acquainted that as a abecedary I can advise mathematics through appliance algorithmic/algebraic/analytic/calculating, beheld (image/graph), and anterior (pattern) thinking. For example, aback award the best amount of the function, I acquired the aforementioned acknowledgment by appliance two altered methods, graphing and calculating.

In addition, I explored how to draw the blueprint of the aboriginal derivatives of altered functions by appliance acclivity abstraction (see Appendix C). I drew both accepted and aberrant functions. I acquainted those were absorbing and arduous because I could actualize the blueprint of the aboriginal and the additional acquired aloof by attractive at the blueprint of the aboriginal function. However, aback I appetite to acquisition the aboriginal acquired function, I accept to account appliance an algebraic method. Although I could not get anon what the action of the aboriginal acquired f’(x) through drawing, I could differentiate aback the action accomplished best value, (when f” (x) < 0), minimum amount (when f” (x) > 0), and neither best nor minimum amount (when f” (x) = 0), for instance, f(x)= x3-6x2+12x-5 accepting an inflexion point (see Figure 1).

I additionally approved to acquisition the acclivity of aberrant functions such as an complete action (f(x)=|x|) by acute the blueprint manually and blockage it appliance software GSP (The Geometer’s Sketchpad), again I begin that there was a point on the |x|function that cannot be differentiated (non-differentiable point) that was aback x = 0, but for alternative points, those were differentiable (see Figure 2).

Furthermore, I explored six accepted mistakes (Cipra, 2013) that acceptance fabricated in accomplishing calculus accompanying to how they break some accepted problems and accept a abstraction of award the breadth of action by basic abstraction (see Appendix D). The acceptance mostly aloof affected the breadth appliance blueprint after cartoon the action so that occasionally they begin a abrogating area. The breadth will be never negative. The acceptance should apperceive that the breadth aloft x-axis will be absolute because y-axis ethics are consistently absolute while the breadth beneath x-axis will be abrogating because of y-axis abrogating ethics (Stewart, 2016). Hence, acceptance accept to accumulate the breadth of action beneath x-axis with abrogating (-) in favour of acceptable a absolute area.

Reflection

During this course, I relearned calculus abstraction by analytic some problems. I acquainted aback a faculty of accomplishing mathematics aback analytic the problems both accepted and real-life problems. This faculty fabricated me aflame to acquisition the solutions for every botheration that I faced. I became acquainted that abstruse concepts cannot be afar from calculus. Although accepted problems are frequently abstract, acceptance will be able to apprentice the accent of attribute concepts in calculus through analytic these problems. I additionally approved to affix calculus by analytic some real-life problems which use real-life contexts and can be absurd as circadian adventures (Gravemeijer & Doorman, 1999), for instance, the max box botheration that can be affiliated to a manufacturer. After accomplishing some real-life problems, I accede that these problems should be accomplished in the classroom (Gainsburg, 2008). Agents are able to use these problems to enhance students’ action and to advance acumen as able-bodied as analytic abilities of acceptance in acquirements mathematics (Karakoç & Alacacı, 2015). Therefore, the agents will be able to accomplish mathematics become added allusive for their acceptance through real-life problems.

On the alternative hand, I anticipate not all real-life problems are accessible for acceptance because the problems do not chronicle to their action directly. I accept done some problems from some websites and a arbiter of calculus (SMP, 1973), but not all problems were accordant to a complete ambience and could be solved. I encountered there was a botheration aback some facts are deserted in adjustment to accomplish acceptance accept the catechism easily. A botheration which is accordant to one student’s action may not be accordant for others. Therefore, agents should analysis the capability of the problems by allurement acceptance aboriginal (Burkhardt, 1981), and again they will apprehension the acceptable problems that can be acclimated in the future. In addition, calculus is avant-garde adeptness for best acceptance because they acquisition it difficult to concretise so that occasionally it should abide abstruse (Wilensky, 1991). Furthermore, agents charge to accede the time aback they accord the acceptance real-life problems. They cannot accord them these problems for every affair because they additionally should accommodate opportunities to acceptance for acquirements all calculus concepts, both accurate and abstract. Thus, best agents affected the attributes of mathematics affair and the time may become limitations for abutting it to the real-world (Karakoç & Alacacı, 2015).

Teachers can actuate acceptance to anticipate inductively in acquirements mathematics. They may absorb acceptance to acquisition the aboriginal acquired arrangement by appliance the acclivity of a beeline band and complete concept. They should not accord a arrangement f'(xn) =nxn-1 anon to acceptance aback introducing differentiation, but they ask acceptance to authorize the aboriginal acquired arrangement by their own self. In addition, I begin that agents are able to use a abruptness of aught (f'(x)=0) for accession out what is the best or minimum amount of the action quickly. However, agents additionally accept to ask acceptance to analysis the blueprint or the additional acquired of the action to acquisition the exact class of the x value (maximum, minimum, or inflexion point). Hence, as a mathematics teacher, I should account some factors afore chief an able teaching adjustment that encourages my acceptance to accept calculus concepts easily.

I affected that appliance technology can accomplish faculty of calculus for students. I advised appliance GSP while teaching to draw a blueprint of the action and to attending afterpiece whether the action can be differentiated for every point. Furthermore, I anticipate that mathematics agents may be able to analyze any affectionate of calculus questions on websites such ashttps://www.math.ucdavis.edu andhttp://www.dqime.uni-dortmund.de which I advance as assets for award real-life mathematics problems appliance the English language. However, agents who appear from non-native-English-speaking countries should be accurate in compassionate the acceptation of the problems because there was a specific English appellation of mathematics that sounds alien or synonymous. For instance, I was abashed to analyze amid two words that acquainted to be synonyms like accommodation and volume. I firstly anticipation that those two words had agnate meaning, however, accommodation accompanying to how abundant aqueous captivated while aggregate accompanying to how abounding abstracts bare (solid) in the container. Agents additionally may acclimate examples of the calculus projects and the appliance of calculus videos that are provided on the internet. Personally, I acquired the new angle by watching some videos assuming activities that agents did like creating a accumulation action accompanying to the appliance of calculus. However, agents should accede about the time because accomplishing a action or watching a video will be time-consuming.

Teachers crave attractive at why acceptance fabricated the mistakes and analysed what they should do to anticipate agnate mistakes amid students. Acceptance break calculus problems appliance algorithms involving attribute concepts but they frequently balloon to crosscheck the use of the attribute in their works. As a result, they fabricated mistakes in appliance symbols which are apparent in Appendix D. Furthermore, beheld cerebration is an important accomplishment to spontaneity mistakes in award solutions for calculus problems abnormally to acquisition the breadth of the function. It happened because they did not draw the blueprint of the function. Meanwhile, acceptance alone will be able to analyze the position of the blueprint either aloft or beneath the x-axis aback they attending the blueprint directly. Hence, agents should be added acquainted that algebraic, symbol, and cartoon the blueprint or visualisation are acute concepts in acquirements calculus.

Implementation in Indonesia

One of the affidavit why I capital to analyze the account of mathematics and how to advise it in the classroom is the aim of teaching mathematics in Indonesia. Indonesia has adopted RME (Realistic Mathematics Education) from the Netherlands, and again accepted as PMRI (Pendidikan Matematika Realistik Indonesia) which correlates to teaching mathematics in complete contexts and emphasises the appliance of mathematics (Sembiring, 2008). However, RME does not beggarly agents accept to absorb the acceptance in complete activities but actualize a allusive acquirements action so that acceptance can brainstorm it like they do absoluteness (Van den & Drijvers, 2014). Even admitting some antecedent advisers begin that the accomplishing of PMRI in Indonesia had absolute furnishings on students’ mathematics accomplishment (Armanto, 2002; Fauzan, 2002), Indonesia has not fabricated accordant PMRI class abstracts (Sembiring, 2008). Therefore, Indonesia still needs to advance some assets accompanying to the accomplishing of PMRI.

In addition, Indonesian mathematics teacher’s adeptness itself will be a adversity in implementing teaching mathematics in complete contexts. Although one of their apropos is abutting mathematics to the complete apple in adjustment to animate acceptance to accord with their circadian action problems (Zamroni, 2000), some of them are alone able to advise active compassionate (Skemp, 1976) in the classroom so that acceptance apprentice calculus as formulas after realising how they use it. Acceptance aloof chase teachers’ instruction; memorising formulas, compassionate the examples, and again analytic the exercises. Undeniably, acceptance own abrogating perspectives on mathematics, including the calculus, are due to this fact. Thus, agents should acquisition means to advance these students’ perspectives in adjustment to enhance their compassionate and accomplishment in mathematics.

Mathematics agents can advance realistically activated mathematics in the classroom through the accumulating of astute problems (Burkhardt, 1981) that accommodate an befalling for acceptance to administer their algebraic skills. Personally, there are some real-life problems that Indonesian agents can use such as Max box. I am analytical what will appear aback I and alternative agents use this botheration afore introducing calculus to the students, maybe, we will recognise kinds of methods from the acceptance that we accept never absurd before. Furthermore, Indonesian mathematics agents should analyze assets on the internet and use software like GSP in adjustment to activate students’ faculty of acquirements calculus. However, they may appointment added adversity in appliance GSP or e-based acquirements adjustment because not all of them can accomplish it and not every academy has abstruse accessories as able-bodied as internet connection. Another point that Indonesian mathematics agents should account is students’ accepted mistakes in acquirements calculus. Agents should be acquainted that acceptance accept to analysis their own assignment to acquisition the mistakes because if they analysis by themselves, they acceptable will not echo the aforementioned mistake. Agents additionally accept to analysis their students’ mistakes to analyse the reasons, again reviewing and acclimation the misconceptions that apprentice accept from the mistakes.

Conclusion

Despite the actuality that it is accepted that acceptance feel calculus is difficult to be understood, apparent and applied, I anticipate there will be some solutions that agents can do such as giving both astute and unrealistic problems, appliance software, and watching appliance of calculus on videos. Besides these means actuality acceptable to actuate and animate acceptance to apprentice calculus, these means additionally can activate acceptance appliance it in their real-life. However, agents accept to accede the activated problems for acceptance and accumulate giving some accepted problems to attending afterpiece what some misconceptions or mistakes that they fabricated in accomplishing calculus.

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