Pada perkuliahan Psikologi Belajar Matematika yang disampaikan oleh Dr. Marsigit pada 16 November 2009, kami belajar bersama mengenai hakekat siswa belajar matematika dan hakekat matematika sekolah.
HAKEKAT SISWA BELAJAR MATEMATIKA
1.Motivasi (Motivation)
Istilah “motivasi” sama artinya dengan “apersepsi” dalam bidang pendidikan dan “kesiapan (readiness)” dalam bidang psikologi. Motivasi siswa dalam kegiatan belajar matematika dapat diawali dengan rasa senang yang diciptakan melalui kesan pertemuan pertama. Itulah mengapa ada iklan suatu produk yang menjadi terkenal karena slogannya, “kesan pertama begitu menggoda...”, yang ternyata ada benarnya. Agar pada pertemuan/ proses pembelajaran selanjutnya dapat dengan mudah terjalin komunikasi yang efektif maka langkah awal yang sebaiknya dilakukan adalah memberikan kesan yang mendalam pada pertemuan/ proses pembelajaran pertama. Persiapkan sebaik mungkin kemudian realisasikan seoptimal mungkin.
Motivasi ini juga dapat diciptakan dengan sikap guru yang bersedia menampung aspirasi siswa. Bagaimanapun siswa merupakan pihak yang semestinya terlibat secara aktif dalam proses pembelajaran sehingga sudah menjadi keharusan bagi guru agar mampu melayani kebutuhan siswa-siswanya. Dengan kata lain, guru tidak sekedar menyampaikan ilmu pengetahuan kepada anak didiknya tetapi guru juga dituntut untuk memainkan peran yang bertujuan untuk mengembangkan potensi anak didiknya secara optimal. Guru hendaknya dapat menyediakan fasilitas yang memungkinkan anak didiknya dapat belajar secara optimal, misalnya dengan memberikan dukungan sehingga anak didik memiliki motivasi yang tinggi dalam belajar.
Motivasi atau apersepsi merupakan persiapan yang dilakukan dalam upaya meminimalisir trauma. Tentu saja persiapan ini tidak dilakukan dengan seketika. Persiapan ini membutuhkan proses. Sebagai contoh, orang-orang di dunia Timur melakukannya dengan memulai persiapan dari luar diri mereka. Terkadang, mereka membutuhkan suatu slogan sebagai motivasi. Misalnya di Jawa terdapat filsafat mengenai perjalanan hidup manusia yang dimulai dari mijil, kemudian dilanjutkan dengan sinom, maskumambang, asmarandhana, dandang gula, gambuh, pangkur, megatruh, dan yang terakhir pucung.
Dalam dunia pendidikan, pemberian motivasi dapat tercermin dalam gambaran sederhana mengenai suatu proses pembelajaran berikut.
10 menit istirahat (cerita, games, dan sebagainya)
30 menit materi II
10 menit penutup (pesan, berdoa)
Pembelajaran tersebut diawali dengan pembuka yang terdiri dari say hello guru kepada anak didiknya, pengkondisian, kemudian berdoa. Hal ini dimaksudkan agar anak didik lebih siap menerima dan mengikuti materi pelajaran.
2.Individu (unique)
Meminjam kalimat Dr. Marsigit bahwa “nilaimu adalah keunikanmu, keunikanmu adalah karena orang lain ada”. Hal ini menunjukkan bahwa setiap individu itu unik dan tidak ada orang yang persis sama di dunia ini. Oleh karena itu, guru harus dapat menyadari kenyataan ini dengan cara menghargai perbedaan di antara anak-anak didiknya. Menghargai perbedaan atau dengan kata lain menghargai keunikan setiap individu dapat dilakukan melalui proses identify, yaitu mengenal keunikan anak didik misalnya melalui LKS dan portofolio. Cara lain yang dapat dilakukan untuk memunculkan pemikiran bahwa “kamu (setiap individu) itu unik” adalah dengan memberikan kesempatan yang sama kepada masing-masing individu untuk mengeluarkan pemikirannya. Selain itu, guru juga diharapkan dapat menanamkan sikap saling menghargai setiap perbedaan di antara anak didiknya, misalnya dalam menghadapi perbedaan pendapat. Jika ada di antara mereka yang tidak setuju dengan suatu pendapat maka usahakan agar mereka tidak setuju hanya terhadap pendapat tersebut, bukan terhadap individu yang mengeluarkan pendapat. Dengan kata lain, “disagree with the opinion, not the person”.
Maka ijinkan saya menuliskan sebait kalimat berikut. Semoga dapat kita jadikan renungan bersama.
Kamu adalah sesuatu yang lain daripada yang lain
Tak ada seorang pun yang menyerupaimu dalam catatan sejarah kehidupan ini
Belum pernah ada seorang pun yang diciptakan sama dengan kamu,
dan tidak akan pernah ada orang yang serupa denganmu di kemudian hari
Karenanya, jangan memaksakan diri untuk berbuat latah
dan meniru-niru kepribadian orang lain
Tetaplah berpijak dan berjalan pada kondisi dan karaktermu sendiri
Hiduplah sebagaimana kamu diciptakan
Tuntunlah dirimu dengan wahyu Illahi,
tapi jangan melupakan kondisimu dan membunuh kemerdekaanmu sendiri
(Diadaptasi dari “La Tahzan: Jangan Bersedih!” karya Dr. ‘Aidh al-Qarni)
3.Kerjasama (Cooperation)
Kerjasama menjadi sangat penting dalam proses pembelajaran. Kerjasama yang baik dan efektif dapat menentukan keberhasilan pembelajaran. Bentuk kerjasama ini berupa kerjasama antara sesama guru sebagai pendidik, kerjasama guru sebagai pendidik dengan anak didiknya, juga kerjasama di antara sesama peserta didik. Kemudian muncul pemikiran, di jaman sekarang ini masih bisakah kerjasama dipertahankan di tengah persaingan dan kompetisi yang semakin ketat? Hal itu kembali ke diri kita masing-masing dalam merefleksikan berbagai pemikiran, namun dalam kondisi apapun usahakan agar pikiran dan tindakan kita tetap terkontrol.
Dalam kondisi real, kerjasama tetap diperlukan bahkan menempati posisi yang penting dalam melakukan suatu usaha. Meskipun pada kenyataannya persaingan selalu menuntut adanya kompetisi namun usahakan agar kompetisi yang kita lakukan bukan bertujuan untuk mengalahkan, tapi untuk meningkatkan prestasi.
4.Kontekstual (Contextual)
Kontekstual berhubungan dengan konteks yang sifatnya terhubung dan dalam kondisi siap. Konteks dapat terjadi di dalam maupun di luar kelas. Jika lama tidak digunakan, maka seseorang dapat kehilangan konteks. Misalnya, jika seseorang yang terbiasa hidup di lingkungan yang modern dengan segala fasilitas dan kemudahan mengakses teknologi dan informasi, kemudian karena suatu hal dia tersesat di padang pasir yang luas dan sepi tanpa seorang pun yang dapat ditemuinya maka semakin lama dia hidup dia akan kehilangan konteksnya yang dulu. Dia jadi lupa bagaimana mengoperasikan komputer, misalnya, atau bahkan handphone sekalipun. Hal ini juga bisa dialami oleh seorang siswa dalam belajar. Jika dia mengulang secara teratur pelajaran yang telah dia dapatkan maka informasi tentang pelajaran tersebut akan masuk ke dalam memori jangka panjang yang artinya jika sewaktu-waktu dia mebutuhkannya akan dengan mudah mengingatnya. Sebaliknya, jika dia merasa tidak membutuhkan informasi maka dia cenderung akan lupa dan semakin lama konteks yang telah dia punyai mengenai informasi tersebut juga akan hilang.
HAKEKAT MATEMATIKA SEKOLAH
1.Pola (Pattern)/ Hubungan (Relationship)
Menurut Oxford Learner’s Pocket Dictionary, pola (pattern) berarti sesuatu yang terjadiatau yang telah dilakukan secara teratur, sedangkan hubungan (relationship) berarti sesuatu (cara) dimana dua orang, negara, dan sebagainya bertindak ke arah atau berhubungan satu sama lain. Jika di dalam proses pembelajaran matematika sekolah tidak terdapat pola atau hubungan maka seharusnya dipertanyakan. Pola dan hubungan penting peranannya dalam pembelajaran matematika yaitu untuk meningkatkan pemahaman terhadap materi yang dipelajari. Semakin banyak pola atau hubungan yang terjadi maka semakin mudah pula suatu materi dapat dipelajari. Hal ini tergambar dalam matematika realistik berikut.
Matematika sekolah merupakan batas antara matematika vertikal dan matematika horisontal. Contoh real untuk menunjukkan sesuatu itu yang dinamakan kubus, dalam matematika vertikal benda yang diguanakan untuk merepresentasikannya disebut “model kubus”, dan selanjutnya ini dijadikan sebagai contoh. Namun, dalam matematika horisontal, benda tersebut dinamakan “kubus” karena digunakan sebagai bukti.
2.Pemecahan Masalah (Problem Solving)
Masalah (problem) adalah sesuatu yang sulit untuk dimengerti. Demikian juga dalam belajar matematika, siswa berkesulitan belajar merupakan salah satu masalah. Untuk itu diperlukan cara untuk menyelesaikan masalah-masalah yang muncul dalam proses pembelajaran matematika di sekolah.
3.Investigasi (Investigation)
Dalam Oxford Learner’s Pocket Dictionary dijelaskan bahwa investigasi (investigation) adalah menguji fakta-fakta di sekitarnya dengan tujuan untuk menemukan kebenaran. Contoh investigasi dalam pembelajaran matematika misalnya membuktikan suatu teorema dengan definisi-definisi yang diketahui atau yang berhubungan. Contoh lainnya yaitu membuktikan suatu hipotesis menggunakan teknik analisis data yang sesuai.
4.Komunikasi (Communication)
Komunikasi menjadi sangat penting dalam suatu proses pembelajaran. Tanpa komunikasi yang terjalin dengan baik di antara komponen-komponen yang terlibat di dalamnya maka proses pembelajaran menjadi kurang atau bahkan tidak efektif. Tanpa komunikasi, kemungkinan yang akan terjadi antara lain adanya efek traumatis yang disebabkan oleh hilangnya konsep (miss concept), hilangnya pemahaman (miss understanding), dan hilangnya interpretasi (miss interpret).
ON THE PSYCHOLOGY OF TEACHING LEARNING MATHEMATICS:
Applying Mathematical Thinking Related to Mathematical Contents in Learning
the Arithmetics Sequence and Series of the 9th Grade Student of Junior High School
Ratna Ayu Setiyo Siwi
Yogyakarta State University
The psychology of teaching learning mathematics especially mathematical contents guided this small research. The researcher tried to uncover the student attempts in applying mathematical thinking related to the mathematical contents in learning arithmetics sequence and series of the 9th grade student of Junior High School. The results of the research described that the student attempts to finding relation between variables, express relationship as formulas, and read the meaning of formulas. Mathematical thinking related to the mathematical contents can be compared to mathematical skills as well as estimation or deep mathematical knowledge.
INTRODUCTION
In the National Education Ministerial Decree No. 19 year 2007 about Education Management Standard by Base and Middle Education Unit, quality of study in the school/ Islamic school is developed by purpose so that educative participant reach patterned thinking and free of thinking causing can execute intellectual activity which in the form of thinking, argument, question, study, find, and predict. In other word, the educative participant expected can proceed the informations into knowledges and then using the knowledges to solving problems. According to Kazuyoshi Okubo (2006), A characteristic of school mathematics is that the learning contents in each grade are based on what the children learn in previous year or earlier in the same year, which involves a new stage of learning. Especially to this research, in the National Education Ministerial Decree No. 23 year 2006 about SKL “Graduate Competence Standard” mentioned the competence standard of lesson group in science and technology (include mathematics) for Junior High School is developing logic, ability think and analysis of educative participant. It can be realized by applying mathematical thinking related to mathematical contents. By applying mathematical contents, a student can analized and solved problem such as ideas of fundamental formulas. Above all, the researcher have an opinion that learning arithmetics sequence and series of the 9th grade student of Junior High School needs application of mathematical thinking related to mathematical contents to increasing student’s ability in case to understanding formulas of this concept.
THEORETICAL FRAMEWORK
In his blog http://powermathematics.blogspot.com, Dr. Marsigit writed Niss’s opinion (in Ernest,1991) about the purpose of mathematics education. He said that it should be enable students to realize, understand, judge, utilize and sometimes also perform the application of mathematics in society, in particular to situations which are of significance to their private, social and professional lives. So, mathematics education has many important roles to solve problems, at present and future. According to Kazuyoshi Okubo (2006), mathematics education has two main purpose, both are following.
1.To enable children to make use what they acquire in the study of mathematics, understand phenomena in their daily life in a mathematical way, and examine and process phenomena through logical thinking.
2.To enable children to enjoy intellectual pleasure through learning mathematics creatively. In this purpose, for a change in approach to allow children to take part in more creative way, have them experience intellectual pleasure and make them feel that they are “creating” mathematics.
Moreover, his matter is also insisted by Shigeo Katagiri (2004) in Marsigit (2006) that mathematical activities can not just be pulled out of a hat, they need to be carefully chosen so that children form concepts, develop skills, learn facts and acquire strategies for investigating and solving problem.
Mathematical Thinking
According to Madihah Khalid (2006), mathematical thinking is the mathematical mode of thought that we use to solve any problem in our daily life including at school. It can be defined as applying mathematical techniques, concepts and processes, either explicitly or implicitly, in the solution of problems. It is, according to Shigeo Katagiri (2006), used during mathematical activities, and is therefore intimately related to the contents and methods of arithmetics and mathematics. According to him, mathematical thinking can be divided into three categories:
1.Mathematical attitudes
2.Mathematical thinking related to mathematical methods
3.Mathematical thinking related to mathematical contents
Mathematical attitudes is mathematical thinking which related to the attitudes, actions, and attempts done by students depend on how interested they are in the problem solving or the lesson. It is considered as the driving force behind the two later categories. Mathematical thinking related to mathematical methods is mathematical thinking which related to mathematical patterns. According to Mason, Burton, and Stacey (1982) in Stacey (2006), it is consisting of specializing (trying special cases, looking at examples), generalizing (looking for patterns and relationships), conjecturing (predicting relationships and results), and convincing (finding and communicating reason why something is true). Mathematical thinking related to mathematical contents is discussed in following sub chapter.
Mathematical Thinking Related to Mathematical Contents
In the Oxford Learner’s Pocket Dictionary, content (plural: contents) can be meant as what is contained in something. Therefore, mathematical contents is the objects (what is contained) in mathematics. According to Madihah Khalid (2006), it includes ideas of sets, units, expressions, operations, algorithms, approximation, fundamental properties and formulas. These can be compared to mathematical skills as well as estimation or deep mathematical knowledge.
According to Kazuyoshi Okubo (2006), A characteristic of school mathematics is that the learning contents in each grade are based on what the children learn in previous year or earlier in the same year, which involves a new stage of learning. Consequently, sufficiently understanding the contents and procedures of the subject is vital for the children to work on problem solving activies. The course of study guidelines mention four areas of learning contents for Junior High School, they are numbers and formulas, graphic figures, functions, and quantitative relations.
LEARNING CONTENTS ABOUT “NUMBERS AND FORMULAS”
Let’s assume that the students are going to learn contents about “numbers and formulas”. It is important to the students: (1) understand the meaning of formula and which formula should be used in a certain situasion, (2) understand the meaning of the procedure previously introduced, if the students learn such ways of learning as the basics then it serves as a great strength when they proceed to next step, (3) understand the manner in which they learned to understand it, (4) in that sense, the teacher sould clearly understand what the students need to learn and how this will be used in future. Therefore, the way to study mathematics is an important part of mathematical thinking.
Mathematical thinking related to mathematical contents is mathematical thinking related to finding relation between variables, express relationship as formulas, and read the meaning of formulas. This is a simple step to learning contents about formulas and then using the problem solving to examine mathematical thinking.
In his blog http://powermathematics.blogspot.com, Dr. Marsigit writed Ebbutt, S. and Straker, A. (1995) said that Mathematics can provide an important set of tools for problems in the main, on paper and in real situations. Students of all ages can develop the skills and processes of problem solving and can initiate their own mathematical problems. Hence, the teacher may help the students learn mathematics by: (1) providing an interesting and stimulating environment in which mathematical problems are likely to occur, (2) suggesting problems themselves and helping students discover and invent their own, (3) helping students to identify what information they need to solve a problem and how to obtain it, (4) encouraging the students to reason logically, to be consistent, to works systematically and to develop recording system, (5) making sure that the students develop and can use mathematical skills and knowledge necessary for solving problems, and (6) helping them to know how and when to use different mathematical tools. In this matter, according to Madihah Khalid (2006), the students are encouraged to use thinking skills and problem solving strategies during mathematics lessons and not just learn mathematical skills and concepts from listening to the teacher. It is featured that if mathematical thinking is not emphasized, the students would end up learning mathematics by rote memorization, without understanding and without the ability to think integently.
According to Kazuyoshi Okubo (2006), a major objective of mathematics education at present is to foster the ability to solve problems. Then, according to Polya in Sri Subarinah (2006), there are four stages be needed in problem solving:
1.The stage of understanding the problem
2.The stage of devising a plan for solution
3.The stage of carrying out the plan for solution
4.The stage of examining the solution
Polya’s four stages seem to have been derived focusing on problem solving as an activity of individuals (Kazuyoshi Okubo, 2006). It is suitable to this research that applying mathematical thinking related to the mathematical contents in learning arithmetics sequence and series of the 9th grade student of Junior High School, in which the subject of research is only one student. The understanding of mathematical thinking related to mathematical contents about “number and formulas” that the student acquire in each stage based on Polya’s four stages (adapted from Kazuyoshi Okubo, 2006) are following.
1.The Stage of Understanding the Problem
To understanding the problem or question, the student must consider the phenomenon mathematically. In order to establish a mathematical formula necessary to answer the question, the student must understand the following:
a.The mathematical conditions related to the question
b.What must be found
c.Remembering previously learned related questions
d.Establishing a formula by selecting division as a procedure, which was previously learned
The teacher should extend guidance to help them develop the ability to practice using a mathematical way of thinking such as that mentioned above. The teacher should also use ingenuity to present a concrete phenomenon and a concrete situation related to the question shown to them.
2.The Stage of Devising a Plan for Solution
The ability to translate such a problem into a formula depends on how much they understand the of this fundamental formula or other formulas from past lessons. Then, they can finding relationship between formulas. When the student try to find the answer to a question, it is important to have an estimate of the answer and the ability to estimate which procedure should be use to find the answer. Having an estimated answer is important in the sense that they have their own viewpoint and respond using their own way of thinking.
3.The Stage of Carrying Out the Plan for Solution
It is expected that the student can find the correct procedure for finding formula. Choosing the correct procedure means to have the ability to foresee the process that leads to the result. By considering which procedure should be used to find the result before taking the steps to solve the problem, one can reasonably and efficiently find the solution. Then, the student doing the selected procedure that the student assumed correct.
4.The Stage of Examining the Solution
As described by previous stage, each student can think using his/ her own way of thinking. There are various procedures for finding the solution. The student can deepen their understanding by exchanging their opinions among themselves, finding similarities and differences and correlating them. It is important to have them examine and understand which procedure is more intelligible and easier to understand, or have them find a more general means of solving the problem, if necessary (which procedure would seem to be most suitable for future development). When one ascertains that one’s way of thinking is correct, or one shows that it is correct and conveys it to others, it is important to think logically, that is, inductively, analogously and deductively.
RESEARCH METHOD
The main aim of the research is uncovering the student attempts in applying mathematical thinking related to the mathematical contents in learning arithmetics sequence and series of the 9th grade student of Junior High School. The subject of research is only one student, she named herself “Chako”. Because of in the National Education Ministerial Decree No. 19 year 2007 was mentioned that one of education management standard by base and middle education unit is the educative participant expected can proceed the informations into knowledges and then using the knowledges to solving problems, then one matter to indicating student’s success in studying mathematics is the student has ability to finding mathematics concepts, definitions, patterns, and relationships so that can use knowledges in the process of problem solving. The instrument used for collecting data consists of observation and interview. Observation done participatively. Interview done directly, it means the researcher collects the data from the student directly. Because of the topic about arithmetics sequence and series has not studied yet, the design of the research included: preparation, implementation, and reflection.
LESSON PLAN
Day and Date: Sunday, December 20th 2009
Time: 08.30-10.00
Subject: Chako
Grade: 9th of Junior High School
Competency Standard: Understanding the arithmetics sequence and series and using both in
the process of problem solving
Basic Competencies: 1. Determining the nth term of arithmetics sequence
2. Determining the sum of the first n terms of arithmetics series
3. Solving problem related to the arithmetics sequence and series
Teaching Scenario: 1. Apperception
2. Developing concepts
3. Reflection
4. Conclusion and closing
PART 1
Aim: Determining the nth term of arithmetics sequence
1.Introduction: Topic Comprehension
a.The teacher writes some numbers pattern on the whiteboard in which all of them described the arithmetics sequence.
b.The teacher lets the student observes given numbers pattern, what is she thinking about them?
c.The teacher informs that each number set is called sequence number and each number of sequence is called sequence term. nth sequence term is symbolized by Un. Then, the teacher lets the student defines sequence number and sequence term, and symbolized each number of given sequence number.
d.The teacher informs in each sequence number, the differences between each two term in series is always constant number, it is called balance (beda, Indonesia) and it is symbolized by b.
2.Activity 1: Understanding the Problem
a.The teacher lets the student defines the arithmetics sequence.
b.The teacher lets the student identify the terms of arithmetics sequence.
c.The teacher lets the student defines the concept of nth term of arithmetics sequence.
3.Activity 2: Devising a Plan for Solution
a.The student attempts define the terms of arithmetics sequence.
b.The student learned that the next term is sum of previous term and balance (b).
4.Activity 3: Carrying Out the Plan for Solution
The student learned that the nth term of arithmetics sequence is Un =Un-1 + b.
5.Activity 4: Examining the Solution
a.The student presented that Un = a + (n - 1) b
in which
Un is nth term
ais the first term
nis number of term
bis balance (beda, Indonesia)
b.The student needed to have clarification whether her formula was correct?
PART 2
Aim: Determining the sum of the first n terms of arithmetics series
1.Introduction: Topic Comprehension
a.The teacher informs that if U1, U2, U3, ..., Un are sequence number then the sum of them is called number series. It is symbolyzed by Sn. Then, the teacher lets the student defines number series and symbolyzed it.
b.The teacher lets the student defines Sn in which n = 1, 2, 3, ..., n in related to the terms of sequence number. Then, the teacher lets the student symbolyzed them.
2.Activity 1: Understanding the Problem
a.The teacher lets the student defines the arithmetics series (based on the previous informations).
b.The teacher informs the sum of the first n terms of arithmetics series (or symbolized by Sn) can be determined by sum of two arithmetics series.
c.The teacher lets the student defines this concept of sum of the first n terms of arithmetics series.
3.Activity 2: Devising a Plan for Solution
a.The student attempts define the sum of arithmetics series.
b.The student learned that the sum of the first n terms of arithmetics series (or symbolized by Sn) can be determined by sum of two arithmetics series.
4.Activity 3: Carrying Out the Plan for Solution
The student learned that the sum of the first n terms of arithmetics seriesis
Sn = n [2a + (n - 1) b] / 2.
5.Activity 4: Examining the Solution
a.The student presented that Sn = n [2a + (n - 1) b] / 2
in which
ais the first term
nis number of term
bis balance (beda, Indonesia)
b.The student needed to have clarification whether her formula was correct?
PART 3
Aim: Solving problem related to the arithmetics sequence and series
1.Problem/: The first term of arithmetics series is 3, the nth term is 41, and its balance is
Question3, find the sum of the first n terms of arithmetics series!
2.Activity 1: Understanding the Problem
a.The student attempts to understanding the relation between variables (the relations between the first term of arithmetics series, the nth term, and its balance) and what must be found (the sum of the first n terms of arithmetics series).
b.The student remembers previously learned related question
c.The student establishes a formula by selecting division as procedure which as previously learned.
3.Activity 2: Devising a Plan for Solution
a.The student learned that the sum of the first n terms of arithmetics series (or symbolized by Sn) can be determined by following formula:
Sn = n [2a + (n - 1) b] / 2
b.The student learned that the first term is U1 = a and the nth term is Un = a + (n - 1) b.
4.Activity 3: Carrying Out the Plan for Solution
a.The student learned that the nth term is Un = a + (n - 1) b, so that number of term is
n = (Un – a + b) / b
b.The student learned that
Sn = n [2a + (n - 1) b] / 2
So this formula can be expressed by
Sn = n [a + (a + (n - 1) b)] / 2
Therefore, because of U1 = a and Un = a + (n - 1) b, then
Sn = n (U1 + Un) / 2
According to this formulas, the student attempts to solving the problem by put U1 = a = 3, Un = 41, andb = 2 into the formulas.
n = (Un – a + b) / b = (41 – 3 + 2) / 2 = 20
and
Sn = n (U1 + Un) / 2 = 20 (3 + 41) / 2 = 440
So, the sum of the first n terms of arithmetics seriesis 440.
5.Activity 4: Examining the Solution
a.The student presented that the sum of the first n terms of arithmetics seriesis 440.
b.The student needed to have clarification whether her formula was correct?
ANALYSIS OF DATA
1.Topic Comprehension
Part 1: Determining the nth term of arithmetics sequence
It is aimed to informing studied topic because the topic about arithmetics sequence and series has not studied yet by the student. So, the student is coming to know about this topic.
The first, the teacher writes numbers pattern on the whiteboard:
(1)2, 4, 6, 8, ...
(2)1, 3, 5, 7, ...
(3)3, 6, 9, 12, ...
The student observed given numbers pattern. The teacher guides the student in order to understanding similarities between them. The student defined “the differences between two number in series of (1) is 2, the differences between two number in series of (2) is 2, and the differences between two number in series of (3) is 3”.
The second, the teacher informs the differences is called balance (beda, Indonesia) in which symbolized by b, and its value is always constant. The teacher guides the student understands each numbers pattern in (1), (2), and (3) are following a certain pattern. The teacher informs each number set is called sequence number and each number of sequence is called sequence term or usually called term. The nth sequence term is symbolized by Un. The student defined “sequence number is a numbers set in which each numbers are following a certain pattern” and “sequence term is each numbers in sequence number”. Therefore, the student defined “in sequence number of (1), U1 = 2, U2 = 4, U3 = 6, U4 = 8”.
Part 2: Determining the sum of the first n terms of arithmetics series
The teacher informs that if U1, U2, U3, ..., Un are sequence number then the sum of them is called number series. It is symbolyzed by Sn. Then, the student defined number series as U1 + U2 + U3 + ... + Un. the student determined Sn in which n = 1, 2, 3, ..., n. It means the student defines Sn as result generalized of S1 (sum of the first 1 term), S2 (sum of the first 2 terms), S3 (sum of the first 3 terms), ... . Therefore, Sn is sum of the first n terms. Then, symbolyzed them as follows: S1 = U1, S2 = U1 + U2, S3 = U1 + U2 + U3, ..., Sn = U1 + U2 + U3 + ... + Un. The teacher informs this is a arithmetics series, then the teacher lets the student defines it.
2.Understanding the Problem
Part 1: Determining the nth term of arithmetics sequence
After the student defined, the teacher informs its sequence is called the arithmetics sequence. According to the teacher information, the student defined arithmetics sequence based on related to balance as “a sequence number in which its differences between two term is constant”. The teacher guides based on this concept, sequence number of (1) can be written as follows.
The teacher lets the student defines the concept of nth term of arithmetics sequence. Previously, the teacher informs the nth term of arithmetics sequence (or symbolized by Un) can be determined by takes example by U1, U2, U3, ..., Un are the terms of arithmetics sequence with first term is a and balance is b. Understanding this concept, observe previous information about sequence number of (1).
Part 2: Determining the sum of the first n terms of arithmetics series
The student defined the arithmetics series as “ the sum of all terms of arithmetics series”.
Teacher informs the sum of the first n terms of arithmetics series (or symbolized by Sn) can be determined by sum of two arithmetics series. The teacher lets the student defines this concept of sum of the first n terms of arithmetics series.
3.Devising a Plan for Solution
Part 1: Determining the nth term of arithmetics sequence
After getting information from the teacher, the student defined in the sequence number of (1),U1 = a = 2 and b = U2 – U1 = U3 – U2 = U4 – U3 = 2. From here, the student generalized this concept into U1 = a and b = U2 – U1 = U3 – U2 = U4 – U3 = ... = Un – Un-1.
The student attempts define the terms of arithmetics sequence (U1, U2, U3, ..., Un) in a and b formulas. By using previous formula, the student expresses the relationship as formulas, such as because of b = U2 – U1 then U2 =U1 + b, because of b = U3 – U2 then U3 =U2 + b, therefore, because of b = Un – Un-1 then Un =Un-1 + b.
In other word, the student learned that the next term is sum of previous term and balance (b).
Part 2: Determining the sum of the first n terms of arithmetics series
The student attempts defined that Sn is the sum of the first n terms of arithmetics series, so that:
Sn= U1 + U2 + U3 + ... + Un-1 + Un
Because of
U1= a
U2= a + b
U3= a + 2b
Un-2=a + (n -3) b
Un-1=a + (n -2) b
Un= a + (n - 1) b
Therefore,
Sn = a + (a + b) + (a + 2b) + ... + [a + (n - 3) b] + [a + (n - 2) b] + [a + (n - 1) b]
4.Carrying Out the Plan for Solution
Part 1: Determining the nth term of arithmetics sequence
Student defined that:
The first term is U1= a= a(according to understanding the problem)
The second term is U2=U1 + b= a + b
The third term isU3=U2 + b= a + 2b
The fourth term isU4=U3 + b= a + 3b
Therefore, the nth term is Un =Un-1 + b. Then appear a question, what is the express Un in a and b formula because the express of Un-1 is known.
Then, based on the last formulas, the student defined that:
U1= a= a + (1 - 1) b
U2= a + b= a + (2 - 1) b
U3= a + 2b= a + (3 - 1) b
U4= a + 3b= a + (4 - 1) b
Therefore,
Un= a + (n - 1) b
Part 2: Determining the sum of the first n terms of arithmetics series
The student learned that the sum of the first n terms of arithmetics series (or symbolized by Sn) can be determined by sum of two arithmetics series, so that:
Sn = a + (a + b) + (a + 2b) + ... + [a + (n - 3) b] + [a + (n - 2) b] + [a + (n - 1) b]
Sn = a + (a + b) + (a + 2b) + ... + [a + (n - 3) b] + [a + (n - 2) b] + [a + (n - 1) b]
So, the sum of the first n terms of arithmetics series is Sn= n [2a + (n - 1) b] / 2.
5.Examining the Solution
Part 1: Determining the nth term of arithmetics sequence
The student presented her formula:
Un= a + (n - 1) b
in which
Unis nth term
ais the first term
nis number of term
bis balance (beda, Indonesia)
So, the nth term of arithmetics sequence is Un = a + (n - 1) b.
Part 2: Determining the sum of the first n terms of arithmetics series
Student presented that Sn= n [2a + (n - 1) b] / 2
DISCUSSION (STUDENT’S REFLECTION)
Researcher: How do you find the nth term of arithmetics sequence formula?
Student: By generalized the terms of this sequence.
Researcher: What do you mean?
Student: The first term is a and the balance is b in which b = U2 – U1 = U3 – U2 = U4 – U3 = ... = Un – Un-1, so we can conclused that the next term is sum of previous term and balance. Then, we can expressed them in a and b formulas. The last, by generalized them, we found the formula.
Researcher: Then, how about the sum of the first n terms of arithmetics series formula?
Student: It can be determined by sum of two arithmetics series.
Researcher : How?
Student: Instance, we have two arithmetics series. We can sum both, one series written from first term to last term but the other written from last term to first term. Then, we have a sum of two series. Therefore, the sum of one series is half of it.
Researcher: So, do you wish to say that the formula be found by expressing relationship between previously fundamental formulas?
Student: I think like that.
CONCLUSION
In this research, tried to uncover the student attempts in applying mathematical thinking related to the mathematical contents in learning arithmetics sequence and series of the 9th grade student of Junior High School. The results of the research described that the student attempts to finding relation between variables, express relationship as formulas, and read the meaning of formulas. In other word, the formula be found by expressing relationship between previously fundamental formulas. The research stated that mathematical thinking related to mathematical content can be known by indicating student’s success in studying mathematics.It means the student has ability to finding mathematics concepts, definitions, patterns, and relationships so that can use knowledges in the process of problem solving. The schema of teaching learning activities are following.
1.Topic Comprehension
a.It is aimed to informing studied topic because the topic about arithmetics sequence and series has not studied yet by the student.
b.Observed given some numbers pattern which described the arithmetics sequence.
c.Defined sequence number and sequence term, and symbolized each number of given sequence number.
d.Defined number series and symbolyzed it.
e.Defined Sn in which n = 1, 2, 3, ..., n in related to the terms of sequence number and symbolyzed them.
2.Understanding the Problem
a.Defined the arithmetics sequence.
b.Identified the terms of arithmetics sequence.
c.Defined the concept of nth term of arithmetics sequence.
d.Defines the arithmetics series.
e.Got notices to define the concept of sum of the first n terms of arithmetics series.
f.In problem solving related to the arithmetics sequence and series:
(1)attempted understand the relation between variables
(2)remembered previously learned related question
(3)established a formula by selecting division as procedure which as previously learned.
3.Devising a Plan for Solution
a.Defined the terms of arithmetics sequence.
b.Learned that the next term is sum of previous term and balace (b).
c.Defined the sum of arithmetics series.
d.Learned that the sum of the first n terms of arithmetics series (or symbolized by sn) can be determined by sum of two arithmetics series.
e.In problem solving related to the arithmetics sequence and series:
(1)learned that the sum of the first n terms of arithmetics series is Sn = n [2a + (n - 1) b] / 2
(2)learned that the first term is U1 = a and the nth term is Un = a + (n - 1) b.
4.Carrying Out the Plan for Solution
a.Learned that the nth term of arithmetics sequence is Un =Un-1 + b.
b.Learned that the sum of the first n terms of arithmetics seriesis Sn = n [2a + (n - 1) b] / 2
a.In problem solving related to the arithmetics sequence and series:
(1)learned that number of term is
n = (Un – a + b) / b
(2)learned that
Sn = n (U1 + Un) / 2
5.Examining the Solution
a.Presented that Un = a + (n - 1) b
b.Presented that Sn = n [2a + (n - 1) b] / 2
c.In problem solving related to the arithmetics sequence and series:
Presented that the sum of the first n terms of arithmetics seriesis 440
Note:
Problem/ question related to the arithmetics sequence and series:
The first term of arithmetics series is 3, the nth term is 41, and its balance is 3, find the sum of the first n terms of arithmetics series!
REFERENCES
Badan Standar Nasional Pendidikan. 2006. Peraturan Menteri Pendidikan Nasional No. 23 Tahun 2006 tentang Standar Kompetensi Lulusan (SKL) (The National Education Ministerial Decree No. 23 year 2006 about SKL “Graduate Competence Standard”).
Badan Standar Nasional Pendidikan. 2007. Peraturan Menteri Pendidikan Nasional No. 19 Tahun 2007 tentang Standar Pengelolaan Pendidikan oleh Satuan Pendidikan Dasar dan Menengah (The National Education Ministerial Decree No. 19 year 2007 about Education Management Standard by Base and Middle Education Unit).
Katagiri, Shigeo. 2006. Mathematical Thinking and How to Teach It, in Progress Report of The APEC Project, “Collaborative Studies on Innovations for Teaching and Learning Mathematics in Different Cultures (II) – Lesson Study Focusing on Mathematical Thinking – Center for Research on International Cooperation in Educational Development (CRICED) University Of Tsukuba.
Khalid, Madihah. 2006. Incorporating Mathematical Thinking in Addition and Substraction of Fraction: Real Issues and Challenges.
Marsigit, Mathilda S., Elly A. 2006. Lesson Study on Mathematical Thinking: Developing Mathematical Methods in Learning the Total Area of a Right Circular Cylinder and Sphere as well as the Volume of a Right Circular Cone of the Indonesian 8th Students.
Okubo, Kazuyoshi. 2006. Mathematical Thinking from the Perspectives of Problem Solving and Area of Learning Contents, in Progress Report of The APEC Project, “Collaborative Studies on Innovations for Teaching and Learning Mathematics in Different Cultures (II) – Lesson Study Focusing on Mathematical Thinking – Center for Research on International Cooperation in Educational Development (CRICED) University Of Tsukuba.
Oxford University. 2009. Oxford Learner’s Pocket Dictionary Third Edition. New York: Oxford University Press.
Stacey, K. 2006. What is Mathematical Thinking and Why is It Important?, in Progress Report of The APEC Project, “Collaborative Studies on Innovations for Teaching and Learning Mathematics in Different Cultures (II) – Lesson Study Focusing on Mathematical Thinking – Center for Research on International Cooperation in Educational Development (CRICED) University Of Tsukuba.
Subarinah, Sri. 2006. Inovasi Pembelajaran Matematika Sekolah Dasar. Departemen Pendidikan Nasional Direktorat Jenderal Pendidikan Tinggi.
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