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

13 comments:

  1. Atin Argianti
    18709251001
    PPs PM A 2018
    Berdasarkan percakapan dalam artkel tersebut, saya belajar bahwa imajinasi berasal dari intuisi, intuisi muncul karena pengalaman. Matematika yang berkaitan dengan kehidupan sehari-hari untuk orang dewasa dan anak-anak berbeda. Matematika untuk orang dewasa, matematika formal dan matematika intuitif dan untuk anak-anak, matematika terutama adalah matematika intuitif.

    ReplyDelete
  2. Rindang Maaris Aadzaar
    18709251024
    S2 Pendidikan Matematika 2018

    Assalamualaikum warahmatullahi wabarakatuh
    Dari percakapan diatas yang membahas tentang imajinasi dan pengetahuan intuitif, saya dapat mengambil kesimpulan bahwa kedua hal tersebut merupakan suatu kemampuan yang muncul dari dalam siswa. Hal ini cenderung personal dan tidak dapat diramalkan karena kemampuan intuitif adalah memahami pengetahuan tanpa melalui penalaran yang rasional dan kemampuan intelektualitas dari siswa.
    Wassalamualaikum warahmatullahi wabarakatuh

    ReplyDelete
  3. Dini Arrum Putri
    18709251003
    S2 P Math A 2018

    Intuisi dalam matematika itu penting terlebih dalam memecahkan masalah, intuisi muncul tanpa perlu harus berpikir dan bernalar. Dan siswa akan memiliki kemampuan intuisi seiiring pengalamannya dalam menyelesaikan soal atau memecahkan soal matematika , bagaimana dengan imajinasi? Imajinasi muncul ketika siswa memiliki kemampuan intuisinya keduanya berkaitan dan perlu dimiliki dalam proses pembelajaran matematika.

    ReplyDelete
  4. Amalia Nur Rachman
    18709251042
    S2 Pendidikan Matematika B UNY 2018

    Dalam pembelajaran matematika, siswa memerlukan pemahaman yang kuat akan konsep matematika. Pemahaman konsep dapat dicapai dengan banyak metode dan berbagai alat peraga maupun media pembelajaran untuk memudahkan siswa. Hal ini sangat siperlukan karena matematika merupakan ilmu yang bersifat abstrak dan memerlukan bentuk real untuk memahamkan pada anak anak. Pengetahuan yang mereka peroleh didapatkan dari pengalaman. Disinilah intuisi mempunyai peran yang lebih besar pada pembelajaran matematika anak. Dalam artikel di atas teknologi juga digunakan sebagai media pembelajaran kontekstual untuk menunjukkan konsep tersebut terhubung pada hal yang lebih nyata.

    ReplyDelete
  5. Aizza Zakkiyatul Fathin
    18709251014
    Pps Pendidikan Matematika A

    Dari dialog dalam postingan ini dapat disimpulkan bahwa imajinasi berada pada intuisi. Intuisi muncul dari pengalaman siswa. Pengalaman ini terdiri dari semangat, motivasi, aktivitas, interaksi, pengetahuan, keterampilan, dan pengalaman kecil. Jadi dalam pembelajaran matematika yang diberikan adalah matematika sekolah. Matematika sekolah memanfaatkan intuisi untuk membangun konsep. Saya setuju bahwa pembelajaran geometri untuk mengarahkan siswa mengimajinasikan 3d bisa menggunakan berbagai macam software agar siswa dapat secara visual melihat objek-objek 3d. Kemudian guru juga tetap mengajarkan siswa dengan tanpa software, setelah melihat langsung kemudian siswa dapat mengimajinasikan dalam otaknya.

    ReplyDelete
  6. Yoga Prasetya
    18709251011
    S2 Pendidikan Matematika UNY 2018 A
    Jika seorang siswa menemukan kesulitan dalam belajar matematika tentunya sebagai pelajar yang telah belajar matematika harus berusaha mengajarkannya dan calon pendidik juga harus sigap akan hal itu. Terkadang dalam proses pembelajaran seorang siswa memiliki tingkat kecerdasan yang berbeda, seperti misalnya siswa menemukan sebuah opini ataupun pertanyaan yang harus dijawab agar siswa tersebut mengerti. Sebagai calon pendidik yang sudah mempelajari konsep pembelajaran dan memiliki pengalaman dalam proses belajar, misalnya metode dan media pembelajaran sangat diperlukan untuk mengkover seluruh siswa dalam proses pembelajaran.

    ReplyDelete
  7. Elsa Apriska
    18709251005
    S2 PM A 2018

    Dalam elegi di atas disebutkkan bahwa imajinasi terletak pada intuisi, namun kegiatan mengajar masih belum mengembangakan intuisi bahkan menghilangkannya. Intuisi muncul dari pengalaman dan pengalaman terdiri dari semangat, motivasi, interaksi, pengetahuan, keterampilan dan pengalaman kecil. Jadi imanjinasi dan intuisi bekerja tidak hanya untuk anak muda tetapi juga untuk orang dewasa. Untuk orang dewasa matematika formal dan matematika intuitif, tetapi untuk anak-anak matematika intuitif.

    ReplyDelete
  8. Hasmiwati
    18709251023
    S2 Pend.Matematika B 2018

    Assalamualaikum Warahmatullahi Wabarakatuh.
    Memang, masih banyak siswa yang merasa kesulitan dalam membayangkan materi matematika, terutama geometri. Untuk mengatasi hal ini, diperlukan inovasi dan langkah tepat dari pendidik. Pendidik harus menggunakan media yang tepat dan maksimal. Penggunaan media ini akan sangat membantu siswa dalam mengimajinasikan pembelajaran yang sedang berlangsung.

    ReplyDelete
  9. Hasmiwati
    18709251023
    S2 Pend.Matematika B 2018

    Assalamualaikum Warahmatullahi Wabarakatuh.
    Banyak siswa merasa kesulitan untuk berimajinasi dalam materi geometri. Hal ini adalah tanggung jawab guru. Dalam mengajarkan materi geometri, guru harus memulai dari hal yang dapat dibayangkan siswa, hal yang dekat dengan kehidupan mereka, barulah mematematisasi secara horisontal dan vertikal. Pelajaran yng dimulai dari apa yang dekat dengan siswa akan terasa mudah dan menyenangkan bagi siswa itu sendiri, karena mereka mengeksplorasi apa yang ada dalam bayangan mereka, bukan berimajinasi ke hal-hal yang tak ada dalam bayangan mereka.

    ReplyDelete
  10. Hasmiwati
    18709251023
    S2 Pend.Matematika B 2018

    Assalamualaikum Warahmatullahi Wabarakatuh.
    Salah satu masalah yang sering di hadapi siswa ketika belajar geometri adalah kesulitan mereka dalam berimajinasi. Imajinasi sendiri terletak pada intuisi yang dapat muncul akibat adanya pengalaman. Sesuai dengan tahapan perkembangan anak, bahwa anak mengenal hal-hal yang konkret, maka agar imajinasi anak dapat berkembang dengan baik guru dapat memulai pembelajaran geometrinya dengan menunjukkan anak kepada hal-hal yang konkret terlebih dahulu. Sehingga mereka nantinya memiliki pengalaman lalu dapat berimajinasi.

    ReplyDelete
  11. Fabri Hidayatullah
    18709251028
    S2 Pendidikan Matematika B 2018

    Berdasarkan pengalaman dan juga data yang saya peroleh dari litbang, penguasaan materi siswa terhadap geometri lebih rendah dibandingkan dengan materi yang lain. Berdasarkan pengalaman saya, siswa cenderung mengalami kesulitan dalam geometri, terutama dalam kegiatan pemecahan masalah. Sementara menurut litbang, penguasaan materi pada bangun geometri secara nasional hanya mencapai 52,04%. Penguasaan materi ini lebih rendah jika dibandingkan dengan penguasaan materi yang lainnya, yaitu operasi bilangan 60,64%, operasi aljabar 57,28%, statistika dan peluang 60,78%. Mengingat pentingnya geometri bagi siswa, maka perlu adanya solusi untuk membantu siswa mudah dalam belajar geometri. Misalnya dengan membuat inovasi dalam pembelajaran, menciptakan pelajaran yang aktif, konstruktif, menyenangkan, dan realistis. Guru perlu menggunakan alat peraga agar dapat memudahkan siswa dalam belajar geometri. Yang tidak kalah penting, guru juga perlu merujuk teori dari I Kant bahwa pengetahuan ialah sintetik a priori. Maka guru perlu mengkonstruk pengetahuan siswa melalui intuisi yang dimiliki siswa.

    ReplyDelete
  12. Sintha Sih Dewanti
    18701261013
    PPs S3 PEP UNY

    Menanggapi pertanyaan “Do you think that students find difficulties for the imagination in mathematics especially in geometry?” Untuk mempelajari geometri, khususnya geometri ruang diperlukan kecerdasan spasial, yaitu kecerdasan yang mencakup kemampuan berpikir dalam gambar, serta kemampuan untuk menyerap, mengubah dan menciptakan kembali berbagai macam aspek dunia visual-spasial. Kecerdasan visual-spasial berkaitan dengan kemampuan menangkap warna , arah ,ruang secara akurat. Kecerdasan spasial sebagai sekumpulan kemampuan-kemampuan yang berhubungan dengan pemilihan, pemahaman dimana proyeksi visual , imajinasi mental pemahaman ruang, manipulasi imajinasi.

    ReplyDelete
  13. Cinta Adi Kusumadewi
    18709251059
    PPs Pendidikan Matematika C 2018

    Dalam proses pembelajaran matematika,pemahaman konsep pada siswa menjadi tolak ukur yang sangat penting. Oleh karena matematika merupakan ilmu yang bersifat abstrak dan memerlukan bentuk real/konkrit untuk memahamkan pada anak anak, maka pengetahuan dibentuk dari pengalaman. Hal inilah yang menjadi peran intuisi dalam proses pembelajaran, lebih khusus dalam memecahkan masalah matematika. Sehingga, ciptakanlah pengalaman yang baik saat pembelajaran berlangsung agar intuisi yang terbentuk juga baik untuk pembelajaran.

    ReplyDelete