Berpikir Kritis – Aplikasi Untuk Metodologi Pengajaran Matematika

Menurut Ioana Marcut (2005), berpikir kritis dapat dilihat sebagai yang memiliki dua komponen:

1. Seperangkat keterampilan untuk memproses dan mengembangkan informasi dan keyakinan;

2. Kebiasaan, berdasarkan komitmen intelektual, menggunakan keterampilan tersebut untuk menuntun perilaku

Demikian itu dipertentangkan dengan:

1. Semata-mata pemerolehan dan penyimpanan informasi itu sendiri, (karena hal ini melibatkan suatu cara tertentu dimana informasi dicari dan dicobakan);

2. Semata-mata kepemilikan seperangkat keterampilan, (karena hal ini melibatkan penggunaan berkelanjutan komponen tersebut);

3. Semata-mata penggunaan tentang keterampilan tersebut (“sebagai suatu latihan”) tanpa penerimaan hasil dari penggunaan keterampilan tersebut.

Referensi : Iona Marcut, 2005. Critical thinking – applied to the methodology of teaching mathematics. Educatia Matematica Vol. 1, Nr. 1 (2005), 57–66.

Perlu Dipikirkan Pengganti UN SD

JAKARTA, KOMPAS.com - Jika Ujian Nasional (UN) untuk jenjang sekolah dasar jadi dihapuskan, maka perlu instrumen yang tepat untuk menggantikannya. Direktur Pembinaan Sekolah Dasar (SD) Kementerian Pendidikan dan Kebudayaan Ibrahim Bafadal megnatakan, perlu ada pengganti evaluasi yang sesuai sehingga dapat menjadi acuan untuk melanjutkan ke jenjang Sekolah Menengah Pertama (SMP).

Namun, jika tak tak ada pula evaluasi yang dijadikan acuan, Ibrahim mengatakan, pihak SMP harus menggelar ujian tes masuk.

“Jadi jika nanti UN SD dihapus maka perlu ada seleksi SMP yang tak menggunakan UN,” kata Ibrahim di Jakarta, Jumat (8/3/2013).

Ibrahim sepakat dengan penghapusan UN untuk SD jika Kurikulum 2013 telah diterapkan. Pasalnya, UN sebagai evaluasi anak didik pada kurikulum baru tidak sesuai dengan metode pembelajaran tematik integratif yang menjiwai Kurikulum 2013.

“Tapi kami akan lihat penilaiannya nanti bagaimana. Jika memang harus diubah, kami siap saja,” ujar Ibrahim.

Namun, Ibrahim tak berkomentar lebih banyak mengenai rencana penghapusan UN untuk SD tersebut. Menurutnya, keputusan bukan menjadi kewenangannya.

“Kami hanya melaksanakannya. Kalau ada ya dilaksanakan. Tergantung pada pusat penilaian,” tandasnya.

Editor :
Caroline Damanik

Jenius Kian Langka?

Artikel ini dapat diperoleh dari tempointeraktif

“Ilmuwan jenius kini semakin langka,” kata Keith Simonton, seorang akademisi di University of California at Davis, AS—Simonton memakai istilah scientific genius. Saat ini, menurut Simonton, sains modern hanya menyisakan ruang kecil bagi orang-orang seperti Galileo (di masa lalu), yang menggunakan teleskop untuk mempelajari langit, atau Charles Darwin, yang mengajukan teori evolusi.

Dalam suratnya yang dimuat di jurnal Nature edisi Januari 2013, Simonton mengatakan, kemajuan masa depan dibangun di atas apa yang telah diketahui dan bukan mengubah fondasi pengetahuan. Bila saya tak keliru tafsir, ini berarti tidak akan ada perubahan mendasar seperti ketika Albert Einstein meruntuhkan fondasi fisika Newtonian. Pendeknya, tidak ada revolusi saintifik seperti yang disimpulkan oleh Thomas Kuhn, melainkan hanya sains normal semata.

Selama abad yang baru lewat, kata Simonton—yang menulis buku Origins of Genius, tidak tercipta disiplin yang benar-benar orisinal. Pendatang baru umumnya berupa hibrida dari disiplin yang sudah ada, contohnya astrofisika dari astronomi dan fisika serta biokimia dari biologi dan kimia. Semakin sukar pula bagi individu untuk membuat kontribusi yang mendasar (groundbreaking contributions), sebab karya yang sangat mutakhir seringkali dikerjakan oleh tim yang besar dan didanai dengan sangat baik.

Ini bukan yang pertama kali orang meramalkan bahwa hari-hari paling menggairahkan dalam sains segera berakhir. Pandangan Simonton itu mengingatkan saya kepada buku yang ditulis oleh John Horgan pada tahun 1997. Judulnya provokatif: The End of Science. Sewaktu terbit, banyak pihak menyambutnya dengan antusias, yang kontra pun tak kurang banyak. Tesis yang diusung Horgan kira-kira seperti ini: sains modern telah mencapai batas terdepannya. Bersamaan dengan itu, “kesempurnaan sains berbanding lurus dengan akhir petualangannya.”

Dari hasil studinya maupun wawancara dengan para pemuka ilmu pengetahuan modern di antaranya Steven Weinberg dalam fisika dan Thomas Kuhn yang menggegerkan dengan teorinya tentang revolusi sains, Horgan berpendapat bahwa tak ada lagi yang perlu dibuktikan, tak ada lagi yang perlu dicapai, dan tak ada lagi yang lebih menarik (dari temuan yang sudah ada).

Horgan menyimpulkan bahwa sains-sains terdepan telah mencapai batasnya. Tidak ada lagi pertanyaan-pertanyaan penting yang harus dijawab karena memang sudah terjawab. Benarkah kesimpulan Horgan? Tidak mudah untuk menjawabnya, namun sebelum datangnya teori relativitas dan mekanika kuantum, pada awal abad ke-20, sebagian ilmuwan sudah meramalkan bahwa semua penemuan besar sudah dilakukan. Penemuan berikutnya, menurut mereka, hanya memerinci penemuan besar. Terbukti kemudian, ramalan mereka meleset.

Karena itu, kemajuan di masa depan niscaya tetap tidak terduga. Bukan tidak mungkin, apa yang telah kita yakini sebagai kebenaran ternyata keliru dan harus dibongkar hingga fundamennya. Proyek Genome maupun proyek perburuan Partikel Tuhan memang mendukung apa yang diargumenkan oleh Simonton: riset sains modern kini melibatkan ratusan ilmuwan dan membutuhkan dukungan dana yang sangat besar. Namun, bukankah Einstein merevolusi sains hanya dengan berbekal kertas dan pensil? **

Meta-Cognitive Therapy More Effective for Adult ADHD Patients

Apr. 1, 2010 — Mount Sinai researchers have learned that meta-cognitive therapy (MCT), a method of skills
teaching by use of cognitive-behavioral principles, yielded significantly greater improvements in symptoms
of attention deficit hyperactivity disorder (ADHD) in adults than those that participate in supportive therapy.
The study is now published in the American Journal of Psychiatry.
Mary Solanto, Ph.D., Associate Professor in the Department of Psychiatry and Director of the Attention
Deficit/Hyperactivity Disorder Center at The Mount Sinai Medical Center examined the effectiveness of a
12-week meta-cognitive therapy group. The intervention was intended to enhance time management,
organizational, and planning skills/abilities in adults with ADHD.
“We observed adults with ADHD who were assigned randomly to receive either meta-cognitive therapy or a
support group,” said Dr. Solanto. “This is the first time we have demonstrated efficacy of a non-medication
treatment for adult ADHD in a study that compared the active treatment against a control group that was
equivalent in therapist time, attention, and support.”
The study observed 88 adults with rigorously diagnosed ADHD, who were selected following structured
diagnostic interviews and standardized questionnaires. Participants were randomly assigned to receive
meta-cognitive therapy or supportive psychotherapy in a group setting. Groups were equated for ADHD
medication use.
Participants were evaluated by an independent (blind) clinician using a standardized interview assessment of
core inattentive symptoms and a subset of symptoms related to time-management and organization. After 12
weeks, the MCT group members were significantly more improved than those in the support group. The
MCT group was also more improved on self-ratings and observer ratings of these symptoms.
Meta-cognitive therapy uses cognitive-behavioral principles and methods to teach skills and strategies in time
management, organization, and planning. Also targeted were depressed and anxious thoughts and ideas that
undermine effective self-management. The supportive therapy group matched the MCT group with respect to
the nonspecific aspects of treatment, such as providing support for the participants, while avoiding discussion
of time management, organization, and planning strategies.

Web address:

http://www.sciencedaily.com/releases/2010/03/

100330142437.htm

What is Mathematical Thinking and Whay is it Important?

WHAT IS MATHEMATICAL THINKING
AND WHY IS IT IMPORTANT?
By: Kaye Stacey
University of Melbourne, Australia
INTRODUCTION
This paper and the accompanying presentation has a simple message, that
mathematical thinking is important in three ways.
• Mathematical thinking is an important goal of schooling.
• Mathematical thinking is important as a way of learning mathematics.
• Mathematical thinking is important for teaching mathematics.
Mathematical thinking is a highly complex activity, and a great deal has been written and studied about it. Within this paper, I will give several examples of mathematicalthinking, and to demonstrate two pairs of processes through which mathematicalthinking very often proceeds:
• Specialising and Generalising
• Conjecturing and Convincing.
Being able to use mathematical thinking in solving problems is one of the most thefundamental goals of teaching mathematics, but it is also one of its most elusive goals.
It is an ultimate goal of teaching that students will be able to conduct mathematical investigations by themselves, and that they will be able to identify where the mathematics they have learned is applicable in real world situations. In the phrase of the mathematician Paul Halmos (1980), problem solving is “the heart of mathematics”. However, whilst teachers around the world have considerable successes with achieving this goal, especially with more able students, there is always a great need for improvement, so that more students get a deeper appreciation of what it means to think mathematically and to use mathematics to help in their daily and working lives.
MATHEMATICAL THINKING IS AN IMPORTANT GOAL OF SCHOOLING
The ability to think mathematically and to use mathematical thinking to solve problems is an important goal of schooling. In this respect, mathematical thinking will support science, technology, economic life and development in an economy. Increasingly, governments are recognising that economic well-being in a country is underpinned by strong levels of what has come to be called ‘mathematical literacy’(PISA, 2006) in the population. Mathematical literacy is a term popularised especially by the OECD’s PISA program of international assessments of 15 year old students. Mathematical literacy is the ability to use mathematics for everyday living, and for work, and for further study, and so the PISA assessments present students with problems set in realistic contexts. The framework used by PISA shows that mathematical literacy involves many components of mathematical thinking, including reasoning, modelling and making connections between ideas. It is clear then, that mathematical thinking is important in large measure because it equips students with the ability to use mathematics, and as such is an important outcome of schooling.
At the same time as emphasising mathematics because it is useful, schooling needs to give students a taste of the intellectual adventure that mathematics can be. Whilst the highest levels of mathematical endeavour will always be reserved for just a tiny minority, it would be wonderful if many students could have just a small taste of the spirit of discovery of mathematics as described in the quote below from Andrew Wiles, the mathematician who proved Fermat’s Last Theorem in 1994. This problem had been unsolved for 357 years.One enters the first room of the mansion and it’s dark. One stumbles around bumping into furniture, but gradually you learn where each piece of furniture is. Finally, after six months of so, you find the light switch, you turn it on, and suddenly it’s all illuminated.
You can see exactly where you were. Then you move into the next room and spend another six months in the dark. So each of these breakthroughs, while sometimes they’re momentary, sometimes over a period of a day or two, they are the culmination of, and couldn’t exist without, the many months of stumbling around in the dark that precede them. (Andrew Wiles, quoted by Singh, 1997, p236, 237) At the APEC meeting in Tokyo in January 2006, Jan de Lange spoke in detail about the use of mathematics to equip young people for life, so I will instead focus this paper on two other ways in which mathematical thinking is important.
WHAT IS MATHEMATICAL THINKING?
Since mathematical thinking is a process, it is probably best discussed through examples, but before looking at examples, I briefly examine some frameworks provided to illuminate mathematical thinking, going beyond the ideas of mathematical literacy. There are many different ‘windows’ through which the mathematical thinking can be viewed. The organising committee for this conference (APEC, 2006) has provided a substantial discussion on this point. Stacey (2005) gives a review of how mathematical thinking is treated in curriculum documents in Australia, Britain and USA. One well researched framework was provided by Schoenfeld (1985), who organised his work on mathematical problem solving under four headings: the resources of mathematical knowledge and skills that the student brings to the task, the heuristic strategies that that the student can use in solving problems, the monitoring and control that the student exerts on the problem solving process to guide it in productive directions, and the beliefs that the student holds about mathematics, which enable or disable problem solving attempts. McLeod (1992) has supplemented this view by expounding on the important of affect in mathematical problem solving.
In my own work, I have found it helpful for teachers to consider that solving problems with mathematics requires a wide range of skills and abilities, including:
• Deep mathematical knowledge
• General reasoning abilities
• Knowledge of heuristic strategies
• Helpful beliefs and attitudes (e.g. an expectation that maths will be useful)
• Personal attributes such as confidence, persistence and organisation
• Skills for communicating a solution.
Of these, the first three are most closely part of mathematical thinking.
In my book with John Mason and Leone Burton (Mason, Burton and Stacey, 1982), we provided a guide to the stages through which solving a mathematical problem is likely to pass (Entry, Attack, Review) and advice on improving problem solving performance by giving experience of heuristic strategies and on monitoring and controlling the problem solving process in a meta-cognitive way. We also identified four fundamental processes, in two pairs, and showed how thinking mathematically very often proceeds by alternating between them:
• specialising – trying special cases, looking at examples
• generalising – looking for patterns and relationships
• conjecturing – predicting relationships and results
• convincing – finding and communicating reasons why something is true.
I will illustrate these ideas in the two examples below. The first example examines the mathematical thinking of the problem solver, whilst the second examines the mathematical thinking of the teacher. The two problems are rather different – the second is within the mainstream curriculum, and the mathematical thinking is guided by the teacher in the classroom episode shown. The first problem is an open problem, selected because it is similar to open investigations that a teacher might choose to use, but I hope that its unusual presentation will let the audience feel some of the mystery
and magic of investigation afresh.
MATHEMATICAL THINKING IS IMPORTANT AS A WAY OF LEARNING MATHEMATICS
In this section, I will illustrate these four processes of mathematical thinking in the context of a problem that may be used to stimulate mathematical thinking about numbers or as an introduction to algebra. If students’ ability to think mathematically is an important outcome of schooling, then it is clear that mathematical thinking must feature prominently in lessons.Number puzzles and tricks are excellent for these purposes, and in the presentation I will use a number puzzle in a format of the Flash Mind Reader, created by Andy Naughton and published on the internet (HREF1). The Flash Mind Reader does not look like a number puzzle. Indeed its creator writes: We have been asked many times how the Mind Reader works, but will not publish that information on this website. All magicians […] do not give away how their effects work.The reason for this is that it spoils the fun for those who like to remain mystified and when you do find out how something works it’s always a bit of a let-down. If you are really keen to find out how it works we suggest that you apply your brain and try to work it out on paper or search further afield. (HREF1). As with many other number tricks, an audience member secretly chooses a number (and a symbol), a mathematical process is carried out, and the computer reveals the audience member’s choice. In this case, a number is chosen, the sum of the digits is subtracted from the number and a symbol corresponding to this number is found from a table. The computer then magically shows the right symbol. The Flash Mind Reader is too difficult to use in most elementary school classes, the target of this conference, but I have selected it so that my audience of mathematics education experts can experience afresh some of the magic and mystery of numbers. As the group works towards a solution, we have many opportunities to observe mathematical thinking in action. Through this process of shared problem solving as we investigate the Flash Mind Reader, I hope to make the following points about mathematical thinking. Firstly, when people first see the Flash Mind Reader, mathematical explanations are far from their minds. Some people propose that it really does read minds, and they may try to test their theory by not concentrating hard on the number that they choose. Others hypothesise that the program exerts some psychological power over the person’s choice of number. Others suggest it is only an optical illusion, resulting from staring at the screen. This illustrates that a key component of mathematical thinking is having a disposition to looking at the world in a mathematical way, and an attitude of seeking a logical explanation. As we seek to explain how the Flash Mind reader works, the fundamental processes of thinking mathematically will be evident. The most basic way of trying to understand a problem situation is to try the Flash Mind Reader several times, with different numbers and different types of numbers. This helps us understand the problem (in this case, what is to be explained) and to gather some information. This is a simple example of specialising, the first of the four processes of thinking mathematically processes. As we enter more deeply into the problem, specialising changes its character. First we may look at one number, noting that if 87 is the number, then the sum of its digits is 15 and 87 – 15 is 72.
Beginning to work systematically leads to evidence of a pattern:
87 8 + 7 = 15 87 – 15 = 72
86 8 + 6 = 14 86 – 14 = 72
85 8 + 5 = 13 85 – 13 = 72
84 8 + 4 = 12 84 – 12 = 72
and a cycle of experimentation (which numbers lead to 72?, what do other numbers lead to?) and generalising follows. Of course, at this stage it is important to note the value of working with the unclosed expressions such as 8+7 instead of the closed 15, because this reveals the general patterns and reasons so much better. Working with the unclosed expression to reveal
structure is an admirable feature of Japanese elementary education.
87 87 – 7 = 80 80 – 8 = 72
86 86 – 6 = 80 80 – 8 = 72
85 85 – 5 = 80 80 – 8 = 72
84 84 – 4 = 80 80 – 8 = 72
It is also worthwhile noting at this point, that although we are working with a specific example, the aim here is to see the general in the specific. This generalising may lead to a conjecture that the trick works because all starting numbers produce a multiple of 9 and all multiples of 9 have the same symbol. But this conjecture is not quite true and further examination of examples (more specialising) finally identifies the exceptions and leads to a convincing argument. In school, we aim for students to be able to use algebra to write a proof, but even before they have this skill, they can be produce convincing arguments. An orientation to justify and prove (at an appropriate level of formality) is important throughout school. If students are to become good mathematical thinkers, then mathematical thinking
needs to be a prominent part of their education. In addition, however, students who have an understanding of the components of mathematical thinking will be able to use these abilities independently to make sense of mathematics that they are learning. For example, if they do not understand what a question is asking, they should decide themselves to try an example (specialise) to see what happens, and if they are oriented to constructing convincing arguments, then they can learn from reasons rather than rules. Experiences like the exploration above, at an appropriate level build these dispositions.
MATHEMATICAL THINKING IS ESSENTIAL FOR TEACHING MATHEMATICS.
Mathematical thinking is not only important for solving mathematical problems and for learning mathematics. In this section, I will draw on an Australian classroom episode to discuss how mathematical thinking is essential for teaching mathematics. This episode is taken from data collected by Dr Helen Chick, of the University of Melbourne, for a research project on teachers’ pedagogical content knowledge. For other examples, see Chick, 2003; Chick & Baker, 2005, Chick, Baker, Pham & Cheng, 2006a; Chick, Pham & Baker, 2006b). Providing opportunities for students to learn about mathematical thinking requires considerable mathematical thinking on the part of teachers. The first announcement for this conference states that a teacher requires mathematical thinking for analysing subject matter (p. 4), planning lessons for a specified aim (p. 4) and anticipating students’ responses (p. 5).These are indeed key places where mathematical thinking is required. However, in this section, concentrate on the mathematical thinking that is needed on a minute by minute basis in the process of conducting a good mathematics lesson. Mathematical thinking is not just in planning lessons and curricula; it makes a difference to every minute of the lesson.The teacher in this classroom extract is in her fifth year of teaching. She stands out in Chick’s data as one of the teachers in the sample exhibiting the deepest pedagogical content knowledge (Shulman, 1986, 1987). Her pupils are aged about 11 years, and are in Grade 6. This lesson began by reviewing ideas of both area and perimeter. We will examine just the first 15 minutes.The teacher selected an open and reversed task to encourage investigation and mathematical thinking. Students had 1cm grid paper and were all asked to draw a rectangle with an area of 20 square cm. This task is open in the sense that there are multiple correct answers, and it is ‘reversed’ when it is contrasted to the more common task of being given a rectangle and finding its area. The teacher reminded students that area could be measured by the number of grid squares inside a shape. In terms of the processes of mathematical thinking, the teacher at this stage is ensuring that each student is specialising. They are each working on a special case, and coming to know it well, and this will provide an anchor for future discussions and generalisations. I make no claim that the teacher herself analyses this move in this way. As the teacher circulated around the room assisting and monitoring students, she came to a student who asked if he could draw a square instead of a rectangle. In the dialogue which follows, the teachers’ response highlighted the definition of a rectangle, and she encouraged the student to work from the definition to see that a square is indeed a rectangle.
S: Can I do a square?
T: Is a square a rectangle?
T: What’s a rectangle?
T: How do you get something to be a rectangle? What’s the definition of a rectangle?
S: Two parallel lines
T: Two sets of parallel lines … and …
S: Four right angles.
T: So is that [square] a rectangle?
S: Yes.
T: [Pause as teacher realises that student understands that the square is a rectangle, but
there is a measurement error] But has that got an area of 20?
S: [Thinks] Er, no.
T: [Nods and winks]
Other responses to this student would have closed down the opportunity to teach him about how definitions are used in mathematics. To the question “Can I do a square?”, she may have simply replied “No, I asked you to draw a rectangle” or she might have immediately focussed on the error that led the student to ask the question. Instead she saw the opportunity to develop his use of definitions. When the teacher realised that the student had asked about the square because he had made a measurement error, she judged that this was within the student’s own capability to correct, and so she simply indicated that he should check his work. In the next segment, a student showed his 4 x 5 rectangle on the overhead projector, and the teacher traced around it, confirmed its area is 20 square cm and showed that multiplying the length by the width can be used instead of counting the squares, which many students did. In this segment, the teacher demonstrated that reasoning is a key component of doing mathematics. She emphasised the mathematical connections between finding the number of squares covered by the rectangle by repeated addition (4 on the first row, 4 on the next, …) and by multiplication. In her classroom, the formula was not just a rule to be remembered, but it was to be understood. The development of the formula was a clear example of ‘seeing the general in a special case’. The formula was developed from the 4 x 5 rectangle in such a way that the generality of the argument was highlighted. The teacher paid further attention to generalisation and over-generalisation at this point, when a student commented: ‘That’s how you work out area – you do the length times the width’. The teacher seized on this opportunity to address students’ tendency to over-generalise, and teased out, through a short class discussion, that LxW only works for rectangles.
S1: That’s how you work out area — you do the length times the width.
T: When S said that’s how you find the area of a shape, is he completely correct?
S2: That’s what you do with a 2D shape.
T: Yes, for this kind of shape. What kind of shape would it not actually work for?
S3: Triangles.
S4: A circle.
T: [With further questioning, teases out that LxW only applies to rectangles]
In the next few minutes, the teacher highlighted the link between multiplication and area by asking students to make other rectangles with area 20 square centimetres. Previously all students had made 4 x 5 or 2 x 10, but after a few minutes, the class had found 20 x 1, 1 x 20, 10 x 2, 2 x 10, 4 x 5 and 5 x 4 and had identified all these side lengths as the factors of 20. Making links between different parts of the mathematics curriculum characterises her teaching. Then, in another act of generalisation, the teacher begins to move beyond whole numbers:
T: Are there any other numbers that are going to give an area of 20? [Pauses, as if
uncertain. There is no response from the students at first]
T: No? How do we know that there’s not?
S: You could put 40 by 0.5.
T: Ah! You’ve gone into decimals. If we go into decimals we’re going to have heaps,aren’t we?
After these first 15 minutes of the lesson, the students found rectangles with an area of 16 square centimetres and the teacher stressed the important problem solving strategy of working systematically. Later, in order to contrast the two concepts of area and perimeter, students found many shapes of area 12 square cm (not just rectangles) and determined their perimeters.
Even the first 15 minutes of this lesson show that considerable mathematical thinking on behalf of the teacher is necessary to provide a lesson that is rich in mathematical thinking for students. We see how she draws on her mathematical concepts, deeply understood, and on her knowledge of connections among concepts and the links between concepts and procedures. She also draws on important general mathematical principles such as
• working systematically
• specialising – generalising: learning from examples by looking for the
general in the particular
• convincing: the need for justification, explanation and connections
• the role of definitions in mathematics.
Chick’s work analyses teaching in terms of the knowledge possessed by the teachers. She tracks how teachers reveal various categories of pedagogical content knowledge (Shulman, 1986) in the course of teaching a lesson. In the analysis above, I viewed the lesson from the point of view of the process of thinking mathematically within the lesson rather than tracking the knowledge used. To draw an analogy, in researching a students’ solution to a mathematical problem, a researcher can note the mathematical content used, or the researcher can observe the process of solving the problem. Similarly, teaching can be analysed from the “knowledge’ point of view, or analysed from the process point of view. For those us who enjoy mathematical thinking, I believe it is productive to see teaching mathematics as another instance of solving problems with mathematics. This places the emphasis not on the static knowledge used in the lesson asabove but on a process account of teaching. In order to use mathematics to solve a problem in any area of application, whether it is about money or physics or sport or engineering, mathematics must be used in combination with understanding from the area of
application. In the case of teaching mathematics, the solver has to bring together expertise in both mathematics and in general pedagogy, and combine these two domains of knowledge together to solve the problem, whether it be to analyse subject matter, to create a plan for a good lesson, or on a minute-by-minute basis to respond to students in a mathematically productive way. If teachers are to encourage mathematical thinking in students, then they need to engage in mathematical thinking throughout the lesson themselves.
References
APEC –Tsukuba (Organising Committee) (2006) First announcement. InternationalConference on Innovative Teaching of Mathematics through Lesson Study. CRICED, University of Tsukuba.
Chick, H. L. (2003). ‘Pre-service teachers’ explanations of two mathematical concepts’Proceedings of the 2003 conference, Australian Association for Research in Education.From: http://www.aare.edu.au/03pap/chi03413.pdf
Chick, H.L. and Baker, M. (2005) ‘Teaching elementary probability: Not leaving it to chance’, in P.C. Clarkson, A. Downton, D. Gronn, M. Horne, A. McDonough, R. Pierce & A. Roche (eds.) Building Connections: Theory, Research and Practice. (Proceedings of the 28th annual conference of the Mathematics Education Research Group of Australasia), MERGA, Sydney, pp. 233-240.
Chick, H.L., Baker, M., Pham, T., and Cheng, H. (2006a) ‘Aspects of teachers’ pedagogical content knowledge for decimals’, in J. Novotná, H. Moraová, M. Krátká, & N.Stehlíková (eds.), Proc. 30th conference e International Group for the Psychology of Mathematics Education, PME, Prague, Vol. 2, pp. 297-304.
Chick, H.L., Pham, T., and Baker, M. (2006b) ‘Probing teachers’ pedagogical content knowledge: Lessons from the case of the subtraction algorithm’, in P. Grootenboer, R.Zevenbergen, & M. Chinnappan (eds.), Identities, Cultures and Learning Spaces (Proc.29th annual conference of Mathematics Education Research Group of Australasia),MERGA, Sydney, pp. 139-146.
Halmos, P. (1980). The heart of mathematics. American Mathematical Monthly, 87(7), 519– 524.HREF1 CyberGlass Design – The Flash Mind Reader. http://www.cyberglass.biz Accessed 28 November 2006.
Mason, J. Burton, L. and Stacey, K. (1982) Thinking Mathematically. London: Pearson.(Also available in translation in French, German, Spanish, Chinese, Thai (2007))
McLeod, D.B. (1992) Research on affect in mathematics education: a reconceptualisation.In D.A. Grouws, Ed., Handbook of research on mathematics teaching and learning, (pp.575–596).New York: MacMillan, New York.
PISA (Programme for International Student Assessment) (2006) Assessing Scientific,Reading and Mathematical Literacy. A Framework for PISA 2006. Paris: OECD.
Schoenfeld, A. (1985) Mathematical Problem Solving. Orlando: Academic Press.
Shulman, L.S. (1986) Those who understand: Knowledge growth in teaching, EducationalResearcher 15 (2), 4-14.
Shulman, L.S. (1987) Knowledge and teaching: Foundations of the new reform, HarvardEducational Review 57(1), 1-22.
Singh, S. (1997) Fermat’s Enigma, New York: Walter
Stacey, K. & Groves, S. (1985) Strategies for Problem Solving. Lesson Plans forDeveloping Mathematical Thinking. Melbourne: Objective Learning Materials.
Stacey, K. & Groves, S. (2001) Resolver Problemas: Estrategias. Madrid: Lisbon.
Stacey, Kaye (2005) The place of problem solving in contemporary mathematics curriculum documents. Journal of Mathematical Behavior 24, pp 341 – 350.

Konsep Limit dan Orang yang Menimbang Gula

Situasi merupakan bagian dari proses mengawali terbentuknya pengalaman dan pengetahuan. Sekarang, konteks dalam pembelajaran menjadi penting untuk mendukung kegiatan pembelajaran. Misalnya matematika, kegunaan konteks ternyata bukan hanya merangsang peserta didik untuk melakukan pendeteksian terhadap pengetahuan yang mereka miliki sebelumnya, tetapi juga dapat digunakan untuk membangun pengetahuan baru. Konsep yang dikonfigurasi akan membentuk suatu hasil elaborasi yang tidak jarang membuat peserta didik lebih memahami apa yang mereka hasilkan. Untuk itu, paham konstrukstivisme meletakkan dasar-dasar materi kontekstual dalam sintaks pembelajaran yang dikembangkannya.

Matematika memang tidak begitu mudah untuk ditemukan dalam kehidupan sehari dibandingkan dengan mata pelajaran lain. Mudah atau sulitnya terletak seberapa sering pengguna memaknai eksistensi matematika tersebut. Jadi, kalau mau untuk selalu memandang bahwa benar matematika merupakan ratu ilmu pengetahuan, berarti seluruh dasar yang menopang perkembangan ilmu pengetahuan pasti dikendalikan oleh matematika. Tapi, bagaimana matematika melakukannya?

Sebuah contoh, menarik menurut saya ketika ingin menentukan dalam 1 bungkus gula dengan berat 2 kg, seberapa tepat banyaknya butiran gula yang akan dituang ke dalam 1 bungkus gula tersebut. Tentunya ada galat yang akan mempengaruhi kinerja timbangan dan perhitungan si penimbang, sebab saat itu si penimbang akan menentukan takaran 2 kg dengan menggunakan konsep limit (mendekati). Yang mana yang menggunakan konsep limit, apakah timbangan atau gula pasirnya? Tentunya gula pasirnya, dan tidak heran kalau saat mengukurnya akan terjadi penaksiran tepat 2 kg atau kurang atau lebih malah, tetapi sebegitulah yang ada.

Kalau ada setuju dengan konteks di atas untuk merepresentasikan konsep limit digunakan dalam aktivitas manusia, dan konsep limit adalah salah satu materi dalam pelajaran matematika. Dapat disimpulkan, matematika ada bersama dengan aktivitas dan pengetahuan manusia. Silahkan memberikan tanggapan, jika ada! Semoga bermanfaat.

Bukan Hanya Siswa, Tetapi juga Guru

Sekarang, pemerintah, khususnya Kementerian Pendidikan dan Kebudayaan sudah merancang kurikulum baru, kurikulum 2013. Dengan adanya kurikulum baru ini, ada beberapa mata pelajaran terintegrasi dengan mata pelajaran lain. Misalnya, TIK akan terpadu kegiatan pembelajarannya dengan mata pelajaran lain. Hal ini tentulah menarik dan menantang bagi guru untuk dapat menerapkan kegiatan pembelajarannya dengan integrasi pembelajaran TIK di dalamnya. Untuk itu, bagi guru yang belum terbiasa menggunakan perangkat teknologi seperti komputer, laptop, proyektor, dan lainnya, harus sudah mengubah kebiasaan dan belajar mengenai perangkat-perangkat tersebut.

Tantangan selanjutnya adalah apakah pengintegrasian TIK dengan mata pelajaran menjadi solusi yang dibutuhkan pebelajar kita saat ini. Mungkin saja! Sejatinya, persoalan mental baik untuk siswa kita sendiri dan bahkan gurunya untuk menikmati pembelajaran yang digelutinya. Perlu diingat bahwa seberapa sering guru kita menempatkan hasil pengajaran kepada siswanya sebagai acuan untuk membuat sesuatu yang lebih baik. Begitu pula dengan siswa yang diajar, seberapa serius mereka mengikuti instruksi yang diberikan oleh guru agar diikuti dalam kehidupan sehari-hari.

Pelibatan teknologi dan komunikasi dalam kegiatan pembelajaran merupakan suatu transformasi dunia pendidikan yang mengagumkan. Seperti yang kita ketahui bahwa berbagai kemudahan disediakan dengan penggunaan teknologi tersebut dalam berbagai kegiatan, termasuk kegiatan pembelajaran di sekolah. Namun, tidak dipungkiri dari kemajuan ini kalau dibalik hal tersebut membutuhkan prasyarat kepada guru, misalnya, untuk melengkapi pengetahuan dan keterampilan dalam mengelola pembelajaran dan teknologi secara bersamaan. Belum lagi, jika mereka berada pada kondisi dan situasi yang terbatas sehingga keberlangsungan kegiatan pembelajaran masih bergantung pada beberapa faktor pendukung, misalnya pasokan arus listrik yang memadai, jumlah item teknologi yang memperhatikan jumlah pengguna, dan lain sebagainya.

Untuk hal ini, guru sebenarnya tidak harus berpangku tangan untuk menunggu kebijakan pemerintah yang terkadang lama hingga bahkan tidak terealisasi. Terpenting, kita melaksanakan kegiatan pembelajaran semampu yang dapat dilakukan. Berhasil atau tidak, sebaiknya pemerintah mengetahui bukan dari nilainya saja tetapi juga bagaimana para guru mengusahakan kepada  siswanya untuk mengikuti kegiatan pembelajaran dengan nilai tersebut. Nas

Peran Budaya dalam Pembelajaran Matematika

Pembelajaran matematika yang masih dianggap sebagian orang masih menjadi momok merupakan pendapat yang keliru. Banyak riset yang menunjukkan bahwa pembelajaran matematika realistik mampu mengubah hasil belajar matematika siswa, dan perubahan itu ditunjukkan dalam proses pembelajaran yang melibatkan guru dan siswa.
Mungkin saja, tetapi secara teoritis, Dewey dengan tinjauannya mengemukakan bahwa kebutuhan terhadap pengalaman belajar menjadi nutrisi tambahan untuk melengkapi kompetensi yang ingin dicapai, dengan pengalaman tersebut akan menjadi bahan pelajaran untuk berbagai proses pembelajaran selanjutnya. Perubahan kurikulum demi kurikulum yang dilakukan pemerintah, tentunya telah memberikan kontribusi meskipun itu sedikit. Berkaitan dengan adanya kurikulum sebagai panduan dalam menentukan orientasi pembelajaran di sekolah, saat ini, dikenal adanya kurikulum tingkat satuan pendidikan dimana penekanan kurikulum untuk memperhatikan standar kompetensi dan kompetensi dasar sebagai tolok ukur yang akan dicapai oleh tenaga pendidik untuk peserta didik.
Ketidaksiapan dapat saja menjadi faktor bagi peserta didik begitu juga pendidik dalam menyiapkan bahan pembelajaran yang mengacu pada standarisasi dalam KTSP.
Mungkin perlu diketahui, kemajemukan bangsa Indonesia misalnya dari segi aspek budaya. Menjadi potensi objektif untuk mendukung kegiatan pembelajaran, misalnya pembelajaran matematika. Alasan yang dipahami berkaitan dengan hal ini, di dalam budaya menurut Nasrullah (2011) terdapat aturan yang dengan hal tersebut mengandung unsur pembelajaran dapat dimanfaatkan untuk diberikan kepada siswa.

10 Alasan untuk Mengajar Kurikulum Terintegrasi

Saat ini guru perlu memperhatikan situasi dan kondisi pembelajaran yang dapat membantu mereka untuk dapat bekerja secara efektif dan efisien. Mungkin, sudah saatnya untuk menerapkan kurikulum terintegrasi. Berikut 10 alasan untuk menggunakan kurikulum seperti itu, diantaranya:

  1. Alokasi waktu pembelajaran yang tidak sejalan dengan materi pembelajaran yang begitu banyak. Tentunya, hal ini membuat guru harus berpikir ekstra untuk menyiapkan waktu seefektif mungkin agar materi yang ditargetkan sesuai jadwal yang ditentukan.
  2. Materi pelajaran yang terpisah membuat siswa tidak dapat mengembangkan keterampilannya dengan baik, bahkan sebagian menganggap ada mata pelajaran yang menurut mereka tidak berkaitan dengan kehidupan sehari-hari.
  3. Otak dapat bekerja dengan baik jika berkaitan.
  4. Perkembangan ilmu pengetahuan dan kehidupan tidak berlangsung secara terpisah. Hidup ini bukanlah karena matematika, sains, membaca, menulis, kajian sosial, atau cuti saja. Sebaliknya, kesemuanya merupakan perpaduan yang membangun dunia.
  5. Keterampilan pemecahan masalah dapat meningkat ketika semua pengetahuan kita dan berpikir tingkat tinggi dari seluruh bidang dalam kurikulum disentuh.
  6. Pustaka yang jelas dalam buku-buku yang ada menyediakan bantuan pendalaman yang autentik ke dalam seluruh subjek pembelajaran. Pustaka yang baik memberikan model untuk pemecahan masalah, hubungan sejawat, pengembangan karakter, dan bangunan keterampilan sehingga siswa terpikat dengan kesenangan petualangan bersama karakter realistik mereka yang terlibat bersama masalah seperti yang mereka miliki atau masalah (seperti perang) dimana mereka akan belajar kebenaran sejarah.
  7. Sekolah memperolehnya terbalik! Dalam dunia nyata anda diuji dengan suatu masalah dan selanjutnya harus merebut jawabannya, tetapi dalam sekolah tradisional anda diberikan jawaban dan diminta untuk meluapkannya.
  8. Interaksi kelompok dan bangunan tim yang melekat ke dalam suatu integrasi kurikulum bergantung pada penggunaan berbagai kekuatan dan keterampilan untuk menciptakan jembatan pemahaman.
  9. Skor tes standar anda akan mencapai puncaknya! Dengan menginspirasi siswa untuk berpikir, mencintai pembelajaran, dan menempatkan pembelajaran mereka untuk bekerja dengan cara autentik, anak-anak anda akan dilengkapi dengan apapun liku-liku dimana mereka mungkin terlempar berdasar tes standar dan dalam hidup!
  10. Siswa menyukai suatu integrasi kurikulum dan menikmati tantangannya.

Dikutip dari “Ten Reasons to Teach an Integrated Curriculum“, semoga bermanfaat!

Menjumlahkan Bilangan Bulat dan Pecahan

Siswa mengalami kesulitan dalam melakukan operasi penjumlahan terhadap bilangan bulat dan pecahan. Ketika soal yang diberikan tentang, “tentukan nilai dari 3 + ½ “ masih sebagian besar siswa mengalami kesulitan dalam menyelesaikannya. Bahkan, diantara mereka ada yang berpendapat bahwa hasilnya sama dengan 3/2. Tentu saja hal ini salah karena nilai dari 3/2 itu sama dengan 1,5 yang lebih kecil dari 3. Seharusnya hasil jumlah dari 3 + ½ harus lebih besar dari 3. Jadi, bagaimana solusi untuk mengatasi permasalahan pemahaman siswa dalam melakukan operasi terhadap bilangan-bilangan tersebut.

Suatu cara yang digunakan adalah dengan mengubah bentuk bilangan bulat tersebut menjadi bilangan pecahan juga. Dengan begitu, bilangan bulat 3 jika diubah ke dalam bentuk pecahan akan menjadi 6/2. Bilangan ini memiliki nilai yang sama dengan 3, tentunya. Ternyata, masih ada siswa yang mengalami kebingungan terhadap bilangan 6/2 (Darimana asalnya?). Cara yang digunakan untuk mengatasi kesulitan siswa adalah dengan mengarahkan pemahaman mereka untuk memahami kalau 6. 2/2 = 3 . Dengan cara ini, siswa mulai memahami cara yang dapat dilakukan untuk mengatasi permasalahan dalam menyelesaikan masalah matematik berkaitan operasi diantara bilangan bulat dan bilangan pecahan.

Semoga bermanfaat.

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