Mar 9, 2013

Mathematics and Language 3




Edited by Marsigit
From Linked In


Doug Hainline:

Teachers who are themselves poor at mathematics will be poor teachers of mathematics. The best teachers of mathematics are confident about their mathematical abilities (with good reason), and enjoy mathematics.


They need not be mathematicians, but they should take pleasure in beautiful proofs and demonstrations, and in showing these to their students. They should be the kind of people who like to read popularized mathematics books, such as the kind the late Martin Gardner wrote, or that Ian Stewart or Keith Devlin write.

Of course, this is a counsel of perfection. We have to start where we are, and try to ensure, for the moment, that every child is exposed to at least one teacher like this in his or her first years of school, and that there is then some institutionalized path that children who become interested in mathematics can follow: Maths Clubs, for instance.

Of course, good teachers will know, and use, whatever mathematical abilities they discover latent in their students. But mathematics has to be taught. The idea that it is just lying, latent, in our pupils, is wrong (despite Socrates' famously, supposedly, demonstrating this the Meno).

Marsigit Dr MA :

@ Doug: Yes I agree that the teachers who are themselves poor at mathematics will be poor teachers of mathematics. Also I do agree that the best teachers of mathematics are confident about their mathematical abilities (with good reason), and enjoy mathematics.

By the way, I should be very careful with your term "beautiful proofs and demonstrations".

"Beautiful" proofs is your perception; but, your students may perceive that it is very "bad" proofs.

"Demonstrations" is your behave; but, what your students should behave?

"Showing the formula" is also your behave; and, it's a pity that your students just to look passively.

You seemed to force your life (math) to your students. From my perspective, it is dangerous for your students. So, again I have proved that the problems of math teaching are coming from the adults and not from the younger (students).

For me, the students need to learn mathematics; and not as your notion " But mathematics has to be taught". It is the students who NEED to learn math.

Your last notion indicate that your students are shadowed by your ambition to implant math to young generations.

For me, the younger students are free to learn; it is okay if they do not like math and they do not want to learn math. The problems, again, are not coming from the students but from the adults (teachers)

It is better if the students themselves (not the teacher) who claim beautiful math/formulas and demonstrate them to their mate or even to the teacher.

So, the problem is how the teacher are to facilitate their students in order to learn math happily.

David Reid:

Doug's point that school mathematics teachers need not be mathematicians is very realistic: most mathematics teachers never do any mathematics. Mathematicians rarely teach at the school level; most school mathematics teachers are only trained to apply mathematics, and after university they rarely do this beyond the level of school mathematics. However, loving mathematics is indeed a help, because a teacher who doesn't is likely to pass this dislike on to his/her students.

Marsigit has a point that beauty is in the eye of the beholder, and Doug's point that mathematical knowledge is not inborn is complementary to Marsigit's implicit point that the ability to find beauty in math is also not inborn. Alas, the world seems to be designing curricula which downplay or eliminate proof altogether, so that the student never gets much chance to judge whether he/she finds a proof beautiful. The reason is not only the tendency to teach what industry wants, but also that proof, when it was taught, has usually been taught in a way that led more to memorization that to thinking, especially with the school culture of learning for tests.

The dichotomy between teachers-teaching and students-learning is an artificial one: a student will not learn without being taught, and the teaching will be for naught if the students doesn't try to learn. There is effort needed on both sides. And it is an effort: we have to teach to the math-lovers, the math-indifferent, and the math-haters.

Marsigit is rather harsh on Doug, reading psychological tendencies between the lines that are not justified from the text: a rather strange practice for someone who is involved in mathematics, with its axiomatic method.

Marsigit accuses Doug of wishing to force mathematics on to students. This leads to the very thorny question as to how much mathematics, and which mathematics, is needed. Most of the students will not need more than a small percentage of the mathematics they need, whereas a small percentage of the students will need a large percentage of the mathematics. The problem is how to decide who will need what, and which mathematics to teach. Some school systems start splitting up the children according to ability and desire early in primary school, whereas most systems take the easy way out and teach everyone more or less the same mathematics. This is an unresolved problem, and I don't pretend to offer a solution here. But as long as the society requires the teaching/learning of more mathematics than most people need, then simply allowing the students to choose their own way of learning will not work, since most students will choose not to learn, due to a number of unfortunate factors. Therefore, although the word "facilitate" is now a very modish word in education, it is not clear what is really meant by it.
Doug Hainline :

 I'm not sure that I understand what Dr Marsigit is saying.

I have seen a fair amount of writing, among academics infected with the post-modernist disease, which seem at least congruent to some of the things Dr Marsigit seems to be advocating. I have to force myself to read these things.

What I believe is that there is an objective reality independent of human consciousness, that mathematical truths are absolute, that understanding them does not come easy to most people, but that it is worth understanding as much as you can. (Not so that your country's economy will grow faster, although that may be an outcome, but because expanding this sort of knowledge and ability is what it means to be human.)

I believe the truths of mathematics, and the skills involved in problem-solving and proofs, should be taught to all of our children, regardless of their wishes, until they become old enough to decide for themselves what they want to learn, which in our current culture is some time around 16 years of age.

How children learn the truths of mathematics and how they become, when they do, proficient mathematicians, is still an open question.
We as yet know little of how the brain works, and there are multiple ambiguities when it comes to discussing teaching and learning mathematics.

I favour Direct Instruction, including learning things by heart, as well as guided discovery and open-ended problems, collaborative learning, what have you. Whatever works, which may be different across cultures, and in any case has to be implemented by what are often very imperfect human agents and in social contexts which are not conducive to learning.

You can spend days reading about different approaches to teaching mathematics. I think many of these debates are sterile, and that the people who love to conduct them have little experience of actual classrooms.

The children we encounter there range from the highly intelligent, intellectually curious, across to those who are ... not. We have a duty to all of them.

I cannot see how any reasonable person would disagree with this.
David Reid :

I agree with the pedagogical side of Doug's last post. There are a couple of philosophical points about which his last line "I cannot see how any reasonable person would disagree with this " is unjustified. That is, he takes the side of Philosophical Materialism and the side of Classical Platonism. Whereas both of these sides have good arguments, so do their opposites, Philosophical Idealism and Neo-Formalism, respectively. (The "Neo-" is used to distinguish it from the Formalism of Hilbert's Program, which was discredited by the results of Kurt Gödel.)

Actually, most mathematicians and physicists take a golden mean between these extremes, since each extreme, while having their merits, also have their defaults. So, whatever your philosophical stance in these matters, it is simply not fair to say that your opponents are not reasonable. In any case, whichever your philosophical stance, it will not affect the pedagogical methods or aims below university level that Doug is discussing.
David Reid :

Dr. Marsigit, I am familiar with Prof. (emer.) Ernest's philosophy. On one side, any good logician (which Prof. Ernest studied for a while) or student of Model Theory clearly knows that mathematics rests upon the axioms chosen, that the axioms are chosen by human mathematicians, and that these mathematicians are a product of their genetic and social background. So far is obvious, and so I do not defend the straw man (Aunt Sally, if you are British) of naive mathematical absolutism that Prof. Ernest argues against.

On the other hand, the common misinterpretation of Prof. Ernest's social constructivism that mathematics depends entirely on social constructs is also flawed, since mathematics contains strong filters to restrict these choices. (Hence mathematics differs strongly from sociology.) In other words, given the infinite amount of mathematics possible, we choose part of it, but what is chosen would probably be found to be valid also by intelligent mathematicians from another planet. This is a far cry from the silliest social constructivism (not Prof. Ernest's, even though his theories easily give way to misinterpretation) which simply says that mathematics is invalid because it is a social construct.

In any case, the position that most practicing mathematicians adopt is somewhat in the middle between the naive absolutism and the extreme social constructivism which was so popular in the 1960's and 1970's. Unfortunately, it is not the more nuanced and realistic theories that are behind many of the theories of education which are practiced today because, as has been repeatedly pointed out, most teachers, as well as those who set the curricula or write the textbooks are not mathematicians. Nonetheless, most major theories -- social constructivism and others -- have yielded some good pedagogical practices (and, alas, a lot of rubbish as well), and one should take the best from each.

In general, the proof of the pudding is in the eating. Unfortunately, so far no one pedagogical theory has come up with a pudding worth eating, and so we will be depending on the ad hoc hybrids that characterize most education today -- including in those schools which give lip service to a singular concept but are then forced to alter these ideas in the face of the reality of the classroom.
Doug Hainline :

David: I agree absolutely with what you have said here, although I would be interested to learn more about the good pedagogical practices which have come from social constructivism. (I've recently been reading through back issues of Paul Ernest's journal, looking for useful work. I haven't found much yet, but I have not finished. I did come across an interesting piece on why we should teach logic along with mathematics, which, in the light of our discussion above, I thought made a lot of sense.)

I keep reading, from one side, that 'consructivism is a theory of learning, not of teaching'. I don't think it's even a theory, because it's not testable, but I have no objection to someone calling themselves a constructionist, if their kids learn maths.

I just want to see the evidence. I can well believe that some form of 'discovery-learning' by bright, motivated students under the guidance of a wise teacher, especially among older kids, can yield very good results. I think it's probably the best way to teach the last year or two of undergraduate maths (wasn't there a fellow named Moore at the University of Texas who did this -- his students were given proofs to do by themselves, as I recall, and became very good mathematicians as a result).

I doubt it works in inner-city classrooms among nine year olds. And I think there have been several meta-analyses of studies of 'constructivist' teaching vs 'direct instruction' which have shown that the former doesn't really get the results that the latter does.
Marsigit Dr MA :

@Reid: O yea great. I think Ernest's work is very great, because he produced something like a map of educational philosophy/ideology. I think you are very good reader of his works. Although, I still found that we have different interpretation of some of his points. I found that you still used a certain criteria from a certain dimension of life to judge the criteria from different dimension. It lead to a condition in which you seemed unfair in making some judgements.

In my point of view, your claimed that mathematics contains strong filters to restrict the choices based on the position of axiomatic mathematics. Again, in the case of younger learner, there are different world of life (learning math). There will be many mistakes if we judge them from the world of pure math. The young students do not learn the infinite amount of math possible. It is also not about the validity from other planet. The criteria for validity, truth, proof, construct, etc.are totally different. The children are the victims of naive absolutist who intervene much on the younger learner.
@Reid and Doug: I wish to argue the validity of the ground of pure math in developing school math curriculum and its textbooks. According to me, they are not accountable in doing so. I did not find any reason for pure math for their intervention in primary or secondary math teaching. It is clear that pure mathematicians will also teach pure mathematics to younger children. It is wrong. The younger learner must not learn pure mathematics. As Ebbutt and Straker (1995) suggested that the solution is that there should be School Mathematics i.e. mathematics that the student should learn.
David Reid :

For readability, I will make one post to answer Doug, and another to answer Marsigit. This one is addressed to Doug. You are referring to the late Prof. Robert Lee Moore, who developed a method which is still used in some universities in selected courses. A follower of the social constructivist school would have been proud to claim that Moore's method followed from social constructivism, but that would have been false; nor could it have, since there are important differences: the social constructivist would conclude that the students' final proofs would be valid proofs for that group, albeit perhaps not for another group.

That position, of course, would be rubbish, and this was not the position of Prof. Moore. However, there have been other spin-offs from social constructivism that are more worthwhile and not as naive. A good example is "Computer-supported collaborative learning" which is useful, for a couple of reasons. First, there is a trend in many Western secondary school classrooms for the students to think that the classroom is a chat room in which the teacher is just an annoying bit of noise that one can ignore most of the time.

However, these students are more willing to listen to one of their peers than the teacher. This is the "collaborative learning" part which is useful; in fact, I often use this trick: I get a student to say more or less the same that I would, but the students listen. In these cases I just insert necessary corrections or questions. Works like a charm. (Again, in secondary school. I haven't tried this in primary school.) Then, there is the computer part. Pupils above a certain age will more willingly listen to, and follow the instructions of, a computer than a human. (Some psychology student should do his doctoral thesis on this phenomenon.)

This is true even if the computer is just carrying a human's presentation. As well, then, an interactive computer lesson (once you figure out how to stop students from switching to Facebook) has other advantages, such as being able to have quick references. But apparently this method need not be reserved for undisciplined students: I have seen some research in South Korea (but I am afraid I cannot give you a reference at this moment) reporting some positive results from this method (although going under a different name, and practiced at university level). However, in South Korea the method was mixed with other methods, so the results were largely anecdotal. Nonetheless, I figure that this would have a place, not as the unique method of instruction, but as one element in a school mathematics program. That is, there is no reason not to mix this with the Direct Instruction method.
Finally, I fully agree that logic, when taught correctly (which it rarely is, in my view), should be taught at school level. Since we are in agreement, I will say no more in this post; however, I will comment a bit further on this in my post to Marsigit, since he does not agree.
David Reid :

To Dr. Marsigit: first, we must separate mathematics from mathematics education. As I pointed out in my post to Doug, Prof. Ernest's work has had some good spin-offs, even if indirect, for mathematics education. With respect to mathematics, his conclusions can be separated into two categories: the obvious ones about the limitations of mathematics, and the ones that are fuzzy enough to easily give rise to erroneous interpretations of the complex relationships between mathematics and society.

You refer to mathematics education, so I will say no more about his ideas on mathematics per se. Here, Prof. Ernest's main conclusion is that student's learning styles will depend on their respective social backgrounds, and that one must take this into consideration. I have taught in a large number of different cultures, and in each one I adapted my teaching style to the student's society's; even in a single school, my teaching styles for two different groups will usually differ. But this conclusion is not unique to the style of social constructivism espoused by the followers of Prof. Ernest.

Therefore Prof. Ernest's philosophy cannot always take credit for this conclusion. I understand what you mean when you say that the criteria for proof and truth are different in different environments. But you need to separate the ways of learning from mathematics per se. For mathematics per se, these criteria are not society-dependent, but the way a student to be convinced of something without formal proof is society-dependent.

The problem is that eventually the two concepts of truth and proof -- one subjective, one mathematical, need to eventually brought together. Students should eventually learn that their subjective ideas are not necessarily correct. I fully agree with you that one cannot start immediately in a topic in primary school with this synthesis, especially in primary school (and you seem to concentrate on primary education), as one needs to first appeal to their intuition. But even in primary school one can, little by little, impress upon the students that, whereas their intuitions are a useful starting point, and whereas one can develop this intuition quite a ways, one cannot always rely on intuition to take you where you want to go.

Mathematics partly uses intuition, but also partly uses methods which are no longer intuitive, nor which can be made to be so. It is especially important in secondary to get away from the pure subjective approach. This is also why pure mathematicians could be useful in organizing a curriculum -- both at primary and secondary level. An important aspect of mathematics which could be profitable for all students, regardless of their eventual professions, would be introducing concepts from logic, to help students think clearly. But not the formal notation of logic; rather the concepts. This can start in primary, and continue in secondary.

However, every time it has been introduced, it has been a fiasco, as most teachers below university level themselves do not have a clear idea of these concepts, and have made a mess of it. The solution has, alas, been to strip school mathematics of the very essence of mathematics, that of clear analysis. I am not sure what you consider pure mathematics, but at university level, pure mathematics includes Mathematical Logic, and some of the more useful and learnable aspects of Mathematical Logic would be more useful, if put into a form which students could understand and practice, than most of the formulas which are presently memorized by students.


23 comments:

  1. Agnes Teresa Panjaitan
    S2 Pendidikan Matematika A 2018
    18709251013

    Dalam tulisan ini, terdapat beberapa pandangan yang menjadi lanjutan dari tulisan sebelumnya, yaitu memahami matematika dalam dunia pendidikan. tentu saja kita ketahui bahwa matematika adalah ilmu pasti yang memiliki batasan-batasan mutlak dalam teorinya, namun kemutlakan dalam mempelajari matematika bagi siswa yang berstatus pemula dalam pembelajaran matematika (young learner) harus dipisahkan dari sudut pandang matematika murni. saya setuju bahwa matematika murni adalah teori yang belum sesuai untuk memasuki dunia dari siswa Sekolah Dasar dan Sekolah menengah pertama.

    ReplyDelete
    Replies
    1. Umi Arismawati
      18709251037
      S2 Pendidikan Matematika B 2018

      Saya setuju dengan saudari Agnes bahwa kemutlakan dalam mempelajari matematika bagi siswa yang berstatus pemula dalam pembelajaran matematika (young learner) harus dipisahkan dari sudut pandang matematika murni. Memang harus dibedakan antara matematika murni dan matematika sekolah. Matematika murni sangat identik dengan definisi teorema dan lain-lain. Sedangkan jika hal tersebut langsung diterapkan atau diberikan kepada siswa, maka siswa akan kesulitan dalam memahaminya. Untuk itu, guru perlu merancang pembelajaran yang dapat mengkonstruksi konsep matematika di benak siswa.

      Delete
  2. Dini Arrum Putri
    18709251003
    S2 P Math A 2018

    Matematika mempelajari ilmu yang pasti, ilmu yang bersifat konkret. Memang perlu adanya perhatian khusus pada seseorang yang baru mengenal matematika, itulah mengapa sejak sekolah dasar siswa sudah dibekali dengan pengetahuan matematikanya, mengenal konsep-konsep yang berhubungan dengan kehidupan sehari-hari, karena memang matematika tidak pernah jauh dari kehidupan sehari-hari manusia, sehingga tidak perlu adanya sangkut paut matematika murni pada siswa yang baru belajar matematika.

    ReplyDelete
  3. Nani Maryani
    18709251008
    S2 Pendidikan Matematika (A) 2018
    Assalamu'alaikum Wr.Wb

    Menurut Doug Hainline, seorang guru yang lemah dalam matematika akan menjadi guru matematika yang lemah. Guru matematika terbaik adalah guru yang percaya diri dengan kemamuan matematikanya dan guru yang menganggap matematika adalah hal yang menyenangkan, dengan kata lain guru tersebut mencintai matematika dalam hati dan pikirannya.

    Wassalamu'alaikum Wr.Wb

    ReplyDelete
  4. Janu Arlinwibowo
    18701261012
    PEP 2018

    matematika berisi filter yang kuat untuk membatasi pilihan berdasarkan posisi matematika aksiomatik. Sekali lagi, dalam kasus pelajar muda, ada dunia yang berbeda dari kehidupan (belajar matematika). Akan ada banyak kesalahan jika kita menilai mereka dari dunia matematika murni . Siswa adalah korban absolut naif yang campur tangan banyak pada peserta didik yang lebih muda.

    ReplyDelete
  5. Rindang Maaris Aadzaar
    18709251024
    S2 Pendidikan Matematika 2018

    Assalamualaikum warahmatullahi wabarakatuh
    Banyak guru yang tidak mengajar dengan sepenuh hati dan tidak memberikan pembelajaran terbaiknya kepada siswa. Biasanya faktor-faktor yang menyebabkannya adalah kurang percaya dirinya sebagai diri bahwa sebenarnya mampu menyampaikan pembelajaran dengan metode yang lebih baik lagi dari pada metode yang berpusat kepada guru seperti metode ceramah. Oleh karena itu sebagai guru yang baik harus bisa memotivasi dan menginspirasi siswanya sehingga siswanya bisa berpikir kreatif lagi dan bisa memecahkan masalah dengan cara berpikir lebih kritis lagi.
    Wassalamualaikum warahmatullahi wabarakatuh

    ReplyDelete
    Replies
    1. Umi Arismawati
      18709251037
      S2 Pendidikan Matematika B 2018

      Assalamu'alaikum, wr, wb.
      saya sangat setuju dengan pendapat saudari rindang bahwa sebagai guru yang baik harus bisa memotivasi dan menginspirasi siswanya sehingga siswanya bisa berpikir kreatif lagi dan bisa memecahkan masalah dengan cara berpikir lebih kritis lagi. Motivasi memang sangat kuat pengaruh dalam siswa mengikuti pembelajaran. Siswa yang motivasinya tinggi, dalam pembelajaran juga akan semangat dan aktif. Sedangkan siswa yang motivasinya rendah akan malas-malasan selama pembealajaran. Sehingga guru harus sering memberikan motivasi di dalam kelas agar siswa semangat. hal lain juga bisa dilakukan dengan mendesain pembelajaran yang bervariasi dan menarik bagi siswa.

      Delete
  6. Tiara Cendekiawaty
    18709251025
    S2 Pendidikan Matematika B 2018

    Seberapa mahir seorang guru dalam matematika tetapi apabila ia tidak dapat mengkomunikasikan matematika kepada siswa dengan benar maka transfer ilmu tidak akan berjalan dengan maksimal. Tidak ada larangan kepada siswa untuk tidak menyukai matematika dan tidak mau belajar matematika karena disinilah peran orang dewasa dibutuhkan yaitu bagimana menumbuhkan rasa suka siswa kepada matematika. Guru harus memfasilitasi siswa dalam belajar matematika dengan menyediakan/menyelenggarakan pembelajaran matematika yang menyenangkan. Dan yang paling penting adalah matematika murni belum sesuai/belum dapat diterapkan/diajarkan kepada siswa sekolah dasar dan menengah karena proses berpikirnya masih berpikir konkret.

    ReplyDelete
  7. Bayuk Nusantara Kr.J.T
    18701261006
    PEP S3


    Saya sepakat bahwa seorang guru matematika harus memiliki kepercayaan diri terhadap apa yang akan diajarkan. Ketika kitabtidak yakin dalam materi yang akan kita ajarkan, maka, kita akan sangat sulit sekali untuk menyampaikannya. Contohnya saja ketika saya tidak memahami materi, maka yang terjadi adalah siswa tidak mengerti apa yang saya ajarkan. Selain itu, kedua belah pihak, guru dan siswa harus menikmati proses pembelajaran matematika tanpa paksaan.

    ReplyDelete
  8. Septia Ayu Pratiwi
    18709251029
    S2 Pendidikan Matematika 2018

    Matematika mempunyai peranan yang penting dalam pembentukan kognitif anak. oleh sebab itu matematika diajarkan sejak memasuki sekolah dasar. Diperlukan perhatian khusus dalam mengomunikasikan matematika kepada anak, karena anak yang baru mengenal matematika tidak memiliki gambaran apapun tentang konsep matematika. Konsep matematika merupakan konsep yang abstrak sehingga perlu tahapan-tahapan untuk membelajarkannya.

    ReplyDelete
    Replies
    1. Umi Arismawati
      18709251037
      S2 Pendidikan Matematika B 2018

      Assalamu'alaikum, wr, wb.
      saya sangat setuju dengan pendapat saudari Septia bahwa Konsep matematika merupakan konsep yang abstrak sehingga perlu tahapan-tahapan untuk membelajarkannya. Dalam pembelajaran matematika jangan mengawalinya dengan sebuah definisi karena hal tersebut sangat abstrak dibenak siswa. Pembelajaran dapat dimulai dengan sesuatu yang mudah diterima siswa yaitu dengan memberi contoh soal dalam kehidupan sehari-hari.

      Delete
  9. Sintha Sih Dewanti
    18701261013
    PPs S3 PEP UNY

    Seorang guru yang memiliki kepercayaan diri akan dapat dengan mudah mengekspresikan segala potensinya secara penuh, dan ia tidak akan merasa ragu atau bahkan terkekang. Kepercayaan diri seorang guru akan dapat tergambar dalam setiap tingkah lakunya. Karena itu, untuk melihat kepercayaan diri seseorang maka dapat dilihat dari sikap dan penampilan perilakunya. Untuk dapat percaya diri dalam menyampaikan materi pembelajaran kepada siswanya, diperlukan kemampuan yang tinggi dan wawasan yang luas terkait dengan materi yang diajarkan. Hal ini sependapat dengan artikel ini yang menuliskan bahwa Guru yang miskin dalam matematika akan menjadi guru matematika yang miskin. Guru-guru matematika terbaik percaya diri tentang kemampuan matematika mereka (dengan alasan yang baik), dan menikmati matematika.

    ReplyDelete
  10. Rosi Anista
    18709251040
    S2 Pendidikan Matematika B

    Assalamualaikum wr wb
    Matematika harus diakui adalah sebuah pelajaran yang relatif sulit, namun apabila kita memahami prinsip-prinsipnya, maka kita akan bisa belajar matematika dengan baik.Matematika bukanlah sekedar pelajaran untuk menghitung sesuatu. Matematika adalah pelajaran yang melatih logika kita untuk problem solving. Berbeda dengan pelajaran lain, satu bab pelajaran matematika bisa meliputi berbagai macam masalah (soal) yang harus kita selesaikan. Caranya adalah dengan membiasakan otak kita untuk menyelesaikan soal-soal itu.

    ReplyDelete
  11. Umi Arismawati
    18709251037
    S2 Pendidikan Matematika B 2018

    Assalamu'alaikum, wr, wb.
    Saya sangat setuju dengan ungkapan bapak Marsigit yaitu “It is clear that pure mathematicians will also teach pure mathematics to younger children. It is wrong. The younger learner must not learn pure mathematics”. Memang harus dibedakan matematika murni dan matematika yang diajarkan disekolah. Matematika murni merupakan ilmu yang abstrak yang berisi tentang teorema definisi dan lain sebagai nya. Dalam taraf di sekolah, kognitif siswa tidak akan mampu atau sulit mempelajari ilmu abstrak tersebut. Seperti Ebbutt dan Straker (1995) mengemukakan bahwa solusinya adalah bahwa harus ada Matematika Sekolah yaitu matematika yang harus dipelajari siswa.

    ReplyDelete
  12. Umi Arismawati
    18709251037
    S2 Pendidikan Matematika B 2018

    Assalamu'alaikum, wr, wb.
    Membelajarkan matematika memang ada jenjangnya. Tidak bisa matematika murni yang abstrak diajarkan kepada siswa disekolah. Matematika murni untuk orang dewasa mungkin tidak masalah tetapi sangat bermasaalah jika diberikan kepada siswa. Dalam psikologi belajar matematika siswa memiliki tahap-tahap tersendiri dalam mempelajari matematika. Siswa biasanya belajar matematika lebih mudah dengan mengkaitkannya dengan kehidupan sehari-hari yang dia kenal. Sehingga siswa akan merasa matematika itu sangat dekat dan berguna untuk lingkungan sekitar. Untuk itu pembelajaran matematika kepada siswa akan lebih baik dikaitkan dengan sesuatu yang realistic daripada langsung kepada materi yang abstrak seperti definisi.

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

    Kebenaran matematika bersifat mutlak tergantung pada siapa yang akan mempelajarinya sesuai dengan tingkatan atau jenjang masing masing. Begitu pula dengan pendekatan yang dilakukan juga harus sesuai dengan tingkatan berfikir peserta didik. Matematika sekolah lebih sesuai diajarkan untuk tingkatan anak anak SD dan SMP. Sedangkan, matematika murni untuk level universitas itu sudah sangat sesuai karena pada tingkat ini peserta didik sudah dapat untuk memikirkan hal hal yang bersifat abstrak. Pengembangan rumus dan interpretasi matematika murni dapat di lakukan pada level universitas.

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  14. Fabri Hidayatullah
    18709251028
    S2 Pendidikan Matematika B 2018

    Siswa tidak akan mampu mengerti matematika murni. Jika dipaksakan hal tersebut hanya akan membuat siswa menganggap bahwa matematika adalah mata pelajaran yang sulit dan enggan bermatematika. Maka guru perlu menghadirkan matematika yang sesuai dengan tahap perkembangan siswa. Namun, masih terdapat banyak permasalahan dalam pelaksanaannya. Permasalahan tersebut tidak hanya terdapat pada implementasi matematika di sekolah. Implementasi tersebut tidak terlepas dari sistem pendidikan yang ada. Kenyataannya, yang menyusun kurikulum adalah para matematikawan murni, maka hal ini tentu memberikan dampak yang serius terhadap implementasi pembelajran matematika di sekolah.

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  15. Herlingga Putuwita Nanmumpuni
    18709251033
    S2 Pendidikan Matematika B 2018

    Matematika adalah cabang ilmu pengetahuan yang luas cakupannya. Sering kali kita mendengar istilah matematika murni dan matematika sekolah. Mengajar pembelajaran matematika memanglah tidak sesederhana yang dibayangkan. Bahkan ada pedapat pada bacaan diatas jikalau ingin menjadi guru matematika sekolah dasar maka ia tidak harus menjadi ahli matematika murni.

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  16. Anggoro Yugo Pamungkas
    18709251026
    S2 Pend.Matematika B 2018

    Assalamualaikum Warahmatullahi Wabarakatuh.
    Berdasarkan artikel diatas, terdapat beberapa pandangan yang menjadi lanjutan dari artikel sebelumnya, yaitu memahami matematika dalam dunia pendidikan. Kita ketahui bahwa matematika adalah ilmu pasti yang mempunyai batasan-batasan mutlak dalam teorinya, namun kemutlakan dalam mempelajari matematika bagi siswa yang berstatus pemula dalam pembelajaran matematika harus dipisahkan dari sudut pandang matematika murni. Dengan pernyataan diatas saya setuju bahwa matematika murni merupakan teori yang belum sesuai untuk siswa Sekolah Dasar dan siswa Sekolah Menengah Pertama, karena matematika murni sangat identik dengan definisi teorema dan lain-lain.

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  17. Nur Afni
    18709251027
    S2 Pendidikan Matematika B 2018

    Assalamualaikum warahmatullahi wabarakatuh.
    Meskipun seorang guru sudah memiliki ilmu terkait dengan matematika tetapi itu semua perlu didukung oleh komunikasi sehingga siswa dan guru terkoneksi dengan benar secara matematik. Guru adalah penerjemah matematika sehingga matematika yang diterjemahkan juga haruslah dengan menggunakan bahasa yang tepat dan mudah dipahami oleh siswa. terimakasih

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  18. Ngaenun Nangim
    19709251058
    S2 Pendidikan Matematika D 2019

    Realitanya, guru yang memiliki banyak ilmu saja belum tentu dapat mengantarkan siswa-siswanya pada kepandaian, apalagi guru yang tidak memiliki ilmu. Karenanya, meskipun sudah menjadi guru tidak serta merta fokusnya hanya mengajar, tetapi juga harus tetap belajar. Meningkatkan kualitas ilmu dan dimensi cara mengajarnya untuk yang lebih tinggi. Selain itu, sayapun sepakat jikalau guru profesional mengajar di SD. Hal ini karena SD merupakan tempat di mana pondasi awal keilmuwan formal dibentuk dan diberikan. Sehingga jika pondasi itu sudah kuat, maka membangun bangunan di atasnya pun dapat lebih kokoh.

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  19. Vera Yuli Erviana
    NIM 19706261005
    S3 Pendidikan Dasar 2019

    Assalamu’alaikum Wr. Wb.
    Matematika adalah ilmu yang konkret. Guru yang lemah dalam matematika akan menjadi guru yang lemah dalam mengajari peserta didik mengenai pembelajaran matematika. Guru yang lemah dalam matematika akan lemah dalam menyampaikan materi yang akan diajarkan. Penggunaan metode serta model pembelajaran tidak akan maksimal jika guru masih lemah dalam matematika. Oleh karena itu guru matematika harus kuat dalam hal matematika agar dapat mengajarkan konsep matematika kepada peserta didik.

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