Oct 6, 2014

Imagination and Intuitive Knowledge of Mathematics




From LinkedIn:

Do you think that students find difficulties for the imagination in mathematics especially in geometry? If you think so,what's the technics used for solving this situation?

Salim Mchennec Professeur de l'enseignement secondaire qualifiant de Mathématiques chez Ministère de l'éducation nationale Top Contributor

Marsigit Dr MA
Lecturer at Yogyakarta State University
Imagination lies in Intuitions, however teaching activities frequently have not developed intuitions yet or even have eliminated them.

Mehtamatics = mathematics with a difference
I am inclined to agree with Dr Marsigit. But I do think that there is a distinction to be made between geometrical visualisation and mathematical problem-solving in multi-dimensional space. I certainly cannot visualise 5-dimensional space but mathematically, I am perfectly comfortable working with 15 or even 500 dimensional spaces.

I have found that some of my pupils have good 3-d imagination and others don't. There appears to be no systematic difference between them otherwise. At first I had thought that people who were also interested in or studying art may have a better understanding of perspective and projections - from higher to lower dimensions. But that has not been the case.

I regret that I have no answers to Salim's second question, though.

3D Innovator / Educator / Computer Industry Veteran
Yes they do, however using tools like 3d visualization to graphically depict numerical constructs provides a trifecta of visual, audio, and tactile sensory experience for learners. The millennials learn different lyrics and it is up to educators to adapt their teaching styles to accommodate these new approach in the process of educating.

Professeur de l'enseignement secondaire qualifiant de Mathématiques chez Ministère de l'éducation nationale
Top Contributor
Thank you for your contribution to this subject.I think intuition plays a crucial role in mathematical work, it can be regarded as a form of intelligence. As stated by Mr James we may use tracking software or geometrical forms .But beyond the 3rd dimension it becomes more complicated, arriving to the infinite dimension where all notions of ordinary geometry changes completely

Lecturer at Yogyakarta State University
Imagination lies in Intuitions, and intuitions emerge through experiences. Experiences composes of spirit, motivation, activity, interaction, knowledge, skill, and minor experiences. So imagination and intuitions work not only for youngster but also for elders. For elders, mathematics is both formal mathematics and intuitive mathematics; but for youngsters (early childhood) mathematics is mainly intuitive mathematics. Intuitive mathematics is mathematics that you understand but you need not to define. The youngster understand well about the concept of big, small, close, tall, high,...without needed to define them. They emerge from their interaction both at home (family), school, and society. Most teachers behave arrogantly by introducing formal mathematics to the youngsters. It is bad!.

Research Guide Mathematics JJTUniversity
I agree to you Dr. Experiance in Maths is another form of practice which we lag behind. it needs constant struggle and time which youngesters dont have. Mathematics itself is intuitive.What do you mean by formal. please explain

3D Innovator / Educator / Computer Industry Veteran
Hi, To clarify, I was not proposing we (necessarily) teach linear algwbra, actually use the visual depth of 3d as a teaching medium for introducing concepts and providing spacial representation to math topics.

Lecturer at Yogyakarta State University
@Dr Prabha: I mean Formal Mathematics is Pure Mathematics/Axiomatic Mathematics in which mathematicians always perceive as a body of knowledge/structure of truth/logic. While for the younger learner it is better mathematics to be introduced as School Mathematics, in which mainly mathematics can be perceive as activities to find the pattern and relationship and to communicate their findings. However, we need intuitive knowledge to accomplish both of them.

Research Guide Mathematics JJTUniversity
@ Dr. Understood pure maths is too tough even teachers also find difficult to adjust it but there are fast developments day to day it is difficult to cope up and maths has become too vast to handle at school level also. We have to bring down Axiomatic approach to school level. To develop intitutional approach you need to absorb yourself in the subject where is the time. can we have any solution.
@James maths is a subject requires personal touch. It is very abstract yes 3D will help as a teaching medium but teacher should be good enough to representthe concepts in a friendly way otherwise maths is too hostile.

Mehtamatics = mathematics with a difference
Dr Rastogi, what you say at the end is true of any subject. There is beauty in mathematics and there is tedium. A good teacher is one who can inspire pupils with the first and help pupils deal with the second. I was lucky: I had two such teachers - one when I was around 9 years old and the other when I was 18. I am now in pay-back mode: using what inspired me to inspire others.

Lecturer at Yogyakarta State University
@Dr.Prabha: There is a huge gap between the ideas of pure mathematicians (the "first") and school math educators (the "second") in perceiving how the younger may learn mathematics. Your idea " We have to bring down Axiomatic approach to school level", is no problem for you and for pure mathematicians; however, it can be very big problem, strange, odd and absurd in sense of younger psychology of learning. The "first" strive to force their own math concepts to the youngster; while the "second" strive to communicate mathematics with the younger based on studying and understanding about "the way of the younger learner think". So the "second" try to creatively build a bridge e.g. by creating School Math in order to approach the younger learner gently. If you and the "first" try to force your method and are not willing to uncover the younger psychology of learning, there will be no solution. However, please note that for many centuries your method of teaching math can be perceived as fail and producing most of the youngster dislike math. Anand Mehta is only a few example of the success student, but all youngster have their right to learn mathematics based on their backgrounds.

Department Head - Communications/Mathematics
How can students use their imagination in mathematics when they aren't taught the fundamentals of problem solving correctly? Once they master topics and generalized solutions to problems, then they can move on to creating their own problems and becoming more imaginative.

Research Guide Mathematics JJTUniversity
@ Dr Marsigit. I understand the problem but do we have the solution.To teach addition and subtraction at tender age is very difficult. unless they have to byheart it . this tender age is very difficult to handle. So you cant bring down more to lower level but pressure at higher level increases the problems. That stage you feel helpless. Maths is not only for mathematician but applicable every where wether you likt the subject or not. May be we should use calculators or Ipads for calculations and abstract concepts should be taught thoroughly in schools but teachers should be trained enough.
@ Anand I agree to you good teacher makes an impact on the life of students. I was not lucky .I had to struggle thats why I became maths teacher but problem is when you get more of the students sitting in the class without interest. Your real test is to make concept simple and reach to them but time is detterent. You have to finish a lot in limited time

3D Innovator / Educator / Computer Industry Veteran
Dr. Prabha Rastogi,
Agreed a good instructor is required. that said, using 3d to illustrate concepts, I have seen pretest to post test increases of up to 30% regardless of class type. that includes visual learners, ADHD, ASD, and both average and a students. isn't the same 30% of course but is repeatable and measurable. key is the development of teaching materials by master teachers, and not by programmers. "Content is king" in that arena. 3d tech gets a bad name from poor implementations. does work well with good production team behind it.

Research Guide Mathematics JJTUniversity
@James I have the knowledge of maths but not much idea of programming. I am associated with educational publications as a writer but want to do somthing on e learning so that I can reach to more people. You are right when you say Content is King. I have seen the effects of visual learning on kids.If concepts are made simple student friendly you can reach easily to students

Lecturer at Yogyakarta State University
Dr Prabha and others: In the sense of Psychology and Pedagogy of learning mathematics the worst thing happened when Pure Mathematician strive to deal with education/teaching. Pure Mathematicians are frequently soliloquy express their own subjective perception about their experience in dealing with mathematics as a science, e.g. mathematics is beautiful (Anand), Content is King (Nord), bring down axiomatic approach (Prabha), absorb mathematics (Anand), introducing concepts (Nord), finish a lot in a limited time (Prabha)...All you have produced of your own notions without considering what happened to the younger learner of math. Again, all of those are about you (ambition of the adults), and not about your younger generation. Unluckily, your younger generations have not equal chances and right to express their own world. So all your claims and judgements about good teaching practice of math are unfair and meaningless for them. To solve the problems, please ask them how really to learn mathematics. I challenge you all to make your younger generations to be your real teacher!.

Research Guide Mathematics JJTUniversity
Dr I accept that I am not much related to younger generation but I would like to understand the mind of younger generation because even grown up students have same biockage for maths. How you deal with students.At college level also students have same hate to maths how much you simplify the concept

Research Guide Mathematics JJTUniversity
Need your sugesstions

Lecturer at Yogyakarta State University
Dear Dr Prabha, it is not an utopia or fiction that we need to ask to young learners on the way they think (math). Too much to be able to be indicated about the younger knowledge/concept in which they understand well but they can not define, e.g. the concept of big, small, long, short, high, short, near, far away, few, many, one, two, some, sad, happy, love...etc. Those all I call as Intuitive Knowledge. In fact, it is not only younger learner who employ Intuitive Knowledge but also the adults and even the elders. So our live really consists full of Intuitive Knowledge. The characteristics of Intuitive Knowledge i.e. when we understand something but we do not remember when and how we did so. Intuitive Knowledge emerge from the Context of life (family, society, and schools) through Interaction/activities. So that why mathematics for younger learner should be defined as Social Activities (Paul Ernest, 1995) and Activities to Search the Pattern, Problem Solving, Investigation and Communication (Ebbut and Straker, 1994). From social activities there will emerge a differentiation pattern of concepts in such a way that we get our Categories of Knowledge (math) in our Mind (Aristoteles and Immanuel Kant). Those categories make us be ready to think all new analytic, a priori, a posteriori and synthetic knowledge. Mathematics is said to be a Science if it is Synthetical a Priori. Thank

25 comments:

  1. KHOIRUDIN
    14301244002
    S1 Pendidikan Matematika I 2014

    Imajinasi dan intuisi penting dimiliki manusia karena dasar dari kesuksesan adalah imajinasi dan intuisi. Kreatifitas tidak bisa dimiliki tanpa kita punya yang namanya imajinasi. Padahal imajinasi atau kreatifitas akan membantu kita untuk memudahkan menyelesaikan masalah.

    ReplyDelete
  2. KHOIRUDIN
    14301244002
    S1 Pendidikan Matematika I 2014

    Yang boleh bermimpi tidak hanya anak orang kaya. Semuanya berhak bermimpi. Berhak bermimpi menjadi sastrawan, insinyur, dokter, dan sebagainya. Dalam perwujudan mimpi inilah kekuatan imajinasi berperan. Imajinasi adalah energi. Energi yang kalau diolah terus-menerus akan mewujud dalam apa yang kita imajinasikan itu.

    ReplyDelete
  3. Suci Renita Sari
    14301241052
    S1 Pendidikan Matematika I 2014

    Pembelajaran geometri adalah pembelajaran yang terkait dengan bangun-bangun, baik bangun datar maupun bangun ruang. Pembelajaran geometri yang berhubungan dengan bangun-bangun masih membuat siswa merasa kesulitan dalam mengimasjinasikannya. Hal itu di karenakan bangun-bangun tersebut bersifat abstrak. Oleh karena itu, untuk mempelajari bangun-bangun geometri tersebut dibutuhkan bangun-bangun konkrit yang ada disekitar siswa, sehingga siswa dapat memahami bangun tersebut.

    ReplyDelete
  4. Hana' Aulia Dewi
    14301241054
    S1 Pendidikan Matematika A 2014

    Dalam belajar matematika, imajinasi dan intuisi memegang peranan yang penting. Dengan imajinasi, seseorang akan memperoleh gambaran awal dari konsep matematika yang bersifat abstrak. Kemudian, dengan intuisi, seseorang itu akan memperdalam gambaran awal konsep yang dipelajarinya hingga diperoleh konsep matematika yang sebenarnya..

    ReplyDelete
  5. Jeanete Nenabu
    PPS PMat D (15709251004)

    Dalam percakapan tersebut Prof Marsigit menyampaikan bahwa sebenar-benar hidup kita terdiri atas Pengetahuan Intuitif. Adapun karakteristik Pengetahuan intuitif yaitu ketika kita memahami sesuatu tapi kita tidak ingat kapan dan bagaimana kita melakukannya. Pengetahuan intuitif muncul dari Konteks kehidupan (keluarga, masyarakat, dan sekolah) melalui Interaksi / kegiatan. Oleh karena itu lah matematika untuk pelajar muda harus didefinisikan sebagai Kegiatan Sosial (Ernest, 1995).

    ReplyDelete
  6. Hana' Aulia Dewi
    14301241054
    S1 Pendidikan Matematika A 2014

    Ketika siswa belajar matematika di sekolah, proses belajar siswa dimulai dari mempelajari konsep dari benda konkret hingga mempelajari konsep yang abstrak. Karena objek matematika bersifat abstrak, maka selama belajar siswa memerlukan imajinasi agar gambaran objek itu dapat ditangkap oleh otak. Kemudian, setelah memperoleh gambaran awal, siswa dapat memperdalam konsep matematika itu dengan menggunakan intuisinya. Oleh karena itu, penting bagi guru untuk dapat membangkitkan imajinasi dan intuisi siswa selama proses pembelajaran.

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  7. Jeanete Nenabu
    PPS PMat D (15709251004)

    Menurut Ebbut dan Straker, (1994), Matematika juga harus dimaknai sebagai kegiatan penelusuran pola atau hubungan, problem solving, investigasi, dan komunikasi. Matematika akan menjadi Ilmu jika matematika dibangun di atas intuisi. Senada dengan pendapat tersebut, Kant menyampaikan bahwa matematika akan menjadi ilmu jika dibangun di atas intuisi dalam kerangka ruang dan waktu. Kant kemudian menyimpulkan bahwa "Matematika akan menjadi Ilmu jika dia bersifat SINTETIK A PRIORI". Jadi hakekat matematika sekolah itu lah yang selama ini digunakan sebagai alternatif agar matematika di sekolah tampak lebih bersahabat dan menyenangkan bagi siswa. Namun di balik semua itu, peran guru tetaplah penting dalam upaya membuat matematika menjadi sesuatu yang mudah dan menyenangkan di benak siswa. Maka tugas kita sebagai guru agar membuat siswa menyukai matematika.

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  8. Ainun Fidyana Syafitri
    14301244006
    S1 Pendidikan Matematika A 2014

    Mata pelajaran geometri dianggap sulit oleh sebagian siswa, hal ini dikarenakan dalam geometri dibutuhkan daya imajinasi siswa. Benda-benda geometri bukanlah benda nyata, dalam pembelajaran disajikan gambar model benda/bangun geometri. Hal inilah yang kiranya menjadikan siswa kesulitan dalam mengimajinasikannya sehingga kesulitan pula dalam pelajaran.

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  9. Angga Kristiyajati
    17709251001
    Pps UNY P.Mat A 2017

    Thank you very much, Mr. Prof. Marsigit.
    This is another inspiring script of debate. This debate between pure mathematics mindset and educational mathematics mindset. I do agree with Paul Ernest that Mathematics education should be have:
    a. Provision of a well structured environment and experience for mathematics learning;
    b. Counseling children in active and autonomous inquiry in mathematics;
    c. Attention to feelings, motivations, attitudes, and behavior from negative aspects.
    (Ernest, 2004)

    ReplyDelete
  10. Auliaul Fitrah Samsuddin
    17709251013
    PPs P.Mat A 2017

    Thank you for your sharing your brilliant idea through this script, sir. I cannot help but agreeing with you on we should separate between formal and educational mathematics. Eventhough both are mathematics, but the way we deliver them to children should be different with that to college students for example. I also noticed your concern related to spacial ability which needed intuition and imagination. Fortunately we have mathematics-based application which can ease us to deal with three or more dimension related problems.

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  11. Junianto
    PM C
    17709251065

    One of the problems in learning mathematics, especially in geometry is student cannot imagine the object because they can learn well if the object can be seen or can be hold by them. When teacher study together with student in the classroom, teacher must fasilitate student so it can help student to imagine the object of geometry. For example, teacher use object in daily life that can represent one of the object in geometry like cube, triangle, etc. After that, new students are able to imagine when one day do not use concrete objects. For example, when the test took place, students are able to imagine because at the time of learning students have been given examples several object in geometry.

    ReplyDelete
  12. Nama : Mirza Ibdaur Rozien
    NIM : 17709251064
    Kelas : Pascasarjana Pendidikan Matematika C

    BISMILLAHIRROHMANIRROHIM
    I’m not on one way with the statemen that students find difficulties for the imagination in mathematics especially in geometry. Why I say that? Actually they understand the geometry, but sometimes we as mathematics teacher just focus on what we think and what we understand about geometry. We as math teacher ignore students understanding about geometry. Actually, what they understand is some intuition. We can’t forbid them to develope their intuition in understanding about geometry. What we should do is enter to their understanding and bering them to the right way to understand geometry as slow as possible to indeep their understanding about geometry based on their understanding. So something that should be done is to change math teacher tecnics in teaching.
    TAMMA BIHAMDILLAH

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  13. Gamarina Isti R
    17790251036
    Pendidkan Matematika Kelas B (Pascasarjana)

    Geometri merupakan salah satu pelajaran yang dianggap sulit dipahami karena memiliki berbagai macam dimensi. Oleh karena itu materi ini akan lebih dipahami menggunakan intuisi, yang diperoleh dari pengalaman – pengalaman siswa melalui proses interaksi sosial, dieksplorasi, diinvestigasi dan siswa mencoba secara langsung. Mengajarkan geometri apabila hanya memberikan rumus saja tidak dapat membuat siswa paham tetapi hanya sebatas tahu. Oleh karena itu dengan pengalaman langsung dapat membuat siswa paham dan berguna untuk mengembangkan kemampuan menentukan strategi yang nantinya akan bermanfaat pada materi selanjutnya.

    ReplyDelete
  14. Nama : Habibullah
    NIM : 17709251030
    Kelas : PM B (S2)

    Assalamualaikum wr.wb

    Geometry is one of the hard enough material because it demands the ability of students in using their imagination to understand the various forms of images that sometimes form three dimensions. Students have two kinds of intelligence that is intuitive and reflective intelligence. Intuitive intelligence is a spontaneous intelligence whose thinking appears on the spot. while reflective intelligence is the intelligence that comes after going through the process of experience in learning. so the task of the teacher is how to understand the intelligence of students and communicate the intelligence into the subject matter one of which is on the material geometry.

    ReplyDelete
  15. Nama: Hendrawansyah
    NIM: 17701251030
    S2 PEP 2017 Kelas B
    Asslamualaikum wr wb.Excitedly, I look a deep conversation about imagination and intuitive of Mathematics.Sometimes, there are students having problem when they study of mathematics.I agree that intuitive has function in developing of knowledge like geometry.I cannot imagine the students complain when they face with aducation of it ,a part of them consider that mathematics is difficult and make them afraid. It is big problem. why the studenst feel as it?May be, they got only a little motivation from their teacher.Duty of teacher is how to built spirit and give more attention to them.Wassalamualaikum wr wb.

    ReplyDelete
  16. Wisniarti
    17709251037
    PM B Pascasarjana

    Terimakasi pak sebelumnya atas postingan yang sangat menarik ini. Saya akan menyampaikan pendapat saya dengan menggunakan bahasa saja. Saya sangat dengan awalan dari postingan ini yaitu mengajukan suatu pertanyaan yang bertujuan menggali pendapat mengenai suatu fenomena permasalahan yang memang sering ditemui pada siswa baik pada tingkat sekolah dasar hingga pada tingkat perguruan tinggi, yaitu materi geometri. Beberapa ahli dari berbagai bidang berpendapat mengenai kesulitan dalam memahami geometri, dan setiap pendapat tersebut memiliki keunggulan masing-masing. Kesulitan yang dialami siswa dalam geometri berdasarkan diskusi ini yaitu siswa mengalami kesulitan dalam membayangkan bentuk kongkrit dari objek geometri 3 dimensi. Selain siswa, guru juga sering mengalami kesulitan dalam menjelaskan dan menuntun siswa untuk dapat paham mengenai materi geometri 3 dimensi atau geometri ruang. Sehingga, salah satu solusi yang ditawarkan dalam diskusi ini adalah dengan menggunakan alat bantu komputasi.

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  17. Wisniarti
    17709251037
    PM B Pascasarjana

    Postingan ini juga membahas kemampuan yang dimiliki oleh siswa itu yang dominan adalah kemampuan matematika intuitif dan kemapuan imaginasi dalam mempelajari geometri. Kemampuan matematika intuitif itu yaitu siswa mengetahui sesuatu tanpa memperdulikan darimana dan bagaimana sesuatu itu terjadi dan muncul. Sedangkan kemampuan berimaginasi maksudnya, siswa dapat membayangkan objek geometri yang bersifat abstrak kemudian menggambarkannya dalam bentuk yang kongkrit.

    ReplyDelete
  18. Bulan nuri
    17709251028
    PPs PM B 2017

    Kemampuan intuitive, merupakan suatu kemampuan yang tidak hanya dimiliki oleh anak-anak saja, tetapi orang dewasapun memilikinya. Dalam melaksanakan proses belajar mengajar guru harus mampu melihat mana murid yang mempunyai kemampuan intuitive tinggi mana yang memiliki kemapuan reflektif tinggi. Sehingga guru tersebut dapat mengontrol proses belajar mengajar dengan baik. Terkait dengan geometri, menggunakan 3-d dalam proses pembelajaran geometri juga sangat membantu siswa dalam membangun imajinasi mereka, sehingga semuanya terasa lebih dekat dan nyata bagi siswa.

    Demikian, terimakasih.

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  19. Alfiramita Hertanti
    17709251008
    S2- Pendidikan Matematika kelas A 2017

    Assalamualaikum wr.wb
    Thank you for your posts Prof, this is very helpful for teachers in learning Gemoetry space. Indeed, this material requires a special strategy in the presentation because not all students have high spatial power, so the props that can support or enhance students to geometry for example by using 3d to describe the concept of geometry.

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  20. Muhammad Sabri
    17701251034
    S2 PEP B

    Dalam belajar geometri imajinasi diperlukan agar siswa dapat lebih mudah membayangkan bentuk dari benda yang sedang dipelajari atau dibahas sedangkan intuisi diperlukan ketika mulai membahas perhitungan matematis dari benda yang dibahas, baik luas, keliling, besar sudut, dan lain sebagainya. Jadi untuk mempelajari geometri imajinasi dan intuisi sama-sama diperlukan, kedua punya tugas masing untuk membantu siswa memahami materi-materi geometri.

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  21. Kholifatun Nur Rokhmah
    17709251011
    Pend. Matematika A 2017

    Saya setuju dengan statemen di atas karena saya sendiri mengalami kesulitan tersebut. Kemudian saat SMA guru saya membawakan media yaitu berupa bangun-bangun ruang yang terbuat dari kawat. Kami di minta untuk mengamati media tersebut. Setelah melihat media untuk selanjutnya saya lebih mudah membayangkan benda ruang tersebut saya mengerjakan soal. Saya rasa dengan media seperti itu sangat baik untuk digunakan dalam pembelajaran geometri. Kemudian semakin berkembangnya teknologi, kehadiran aplikasi seperti geogebra tentunya akan sangat membantu kita dalam menyampaikan materi geometri.

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  22. Hari Pratikno
    17709251032
    Pendidikan Matematika S2 (Kelas B)

    Fokus diskusi diatas berkaitan dengan matematika anak dan matematika dewasa. Matematika anak bersifat intuitif, maksudnya adalah mengerti sesuatu tapi tidak mengerti definisinya, hal ini bisa terjadi karena pengalaman anak sebelumnya di lingkungan sehari-hari. Pada siswa menengah biasanya sudah muncul definisi-definisi yang tercantum di setiap buku cetak. Sebenarnya definisi itu penting untuk membatasi apakah masuk konsep atau tidak. Namun guru harus kreatif memberikan definisi, yaitu definisi bisa diberikan ditengah atau di akhir pembelajaran setelah, jadi diawal guru bisa memberikan contoh-contoh real materi matematika secara kontekstual.

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  23. Sofi Saifiyah
    17701251033
    S2 PEP B

    Kesulitan siswa dalam mengimajinasikan matematika terutama dalam materi geometri. Setuju dengan pendapat yang Bapak sampaikan, bahwa imajinasi terletak pada intusi. Seperti yang telah dijelaskan pula oleh Pak Marsigit pada perkualihan pertemuan ke enam bahwa intuisi tidak hanya digunakan oleg anak-anak saja, melainkan orang dewasa pun perlu menggunakan intuisi asalkan sesuai dengan ruang dan waktunya. Maka penting bagi guru matematika untuk menggunakan intuisi dalam proses pembelajaran, artinya seorang guru perlu memberikan pembelajaran berdasarkan atau pengelamannya, karena pengalaman adalah seuatu yang konkrit dan nyata, sehingga siswa lebih mudah untuk berimajainasi. Jika seorang guru tidak mengembangkan intuisi untuk murid-muridnya atau bahkan menghilangkannya, jelas siswa akan mengalami kesulitan untuk mengimajinasikan matematika terutama dalam materi geometri.

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  24. Kartika Pramudita
    17701251021
    S2 PEP B

    Geometri adalah cabang matematika yang seringkali siswa sulit mempelajarinya. Kesulitan itu karena siswa sulit untuk membayangkan dan berimajinasi tentang geometri. Sehingga kemampuan untuk berimajinasi diperlukan dalam belajar matematika. Imajinasi berkaitan dengan intuisi. Setiap orang memiliki intuisi. Intuisi merupakan hal yang diperlukan untuk belajar matematika bagi anak-anak. Dengan intuisi anak akan lebih mudah memahami materi belajar matematika. Namun, terkadang dalam pembelajaran matematika tidak melibatkan intuisi dan lebih bahayanya lagi apabila pembelajaran justru menurunkan intuisi.

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  25. Latifah Pertamawati
    S2 PM B
    17709251026

    Bismillaah..
    When I was in (senior) high school, geometry is such a difficult concept for me, to be honest. It is the other way around for those who have surprisingly high spatial intelligence. They enjoy it, so.. much. They could easily imagine some geometrical objects without drawing them on the paper. However, those people who find difficulties in understanding some geometrical objects have to draw them first, in order to understand the underlying concepts. In my case, teacher's (wrong) perception of his/ her students' (real) mathematical abilities could cause students' failure on understanding the concept. It is sad to admit that there were always teachers who gave a definition of A, then directly gave task B to F or even to Y to the students. For a reserved student like me, it was not easy for asking questions thus I just let it go and let it go till in the end. I didn't understand fully what have been taught by the teacher. So, yes, in conclusion, we need to discover the concept (together). Especially on geometry, the concept can be discovered by applying some approach which facilitate students' intuition, not just giving definitions, formulas, or just.. laws? We need to give the students experiences, by showing them the objects and engaging students in learning process connected to their everyday lives.

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