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.


  1. Devi Anggriyani
    S2 PEP B 2016

    Siswa akhirnya harus belajar bahwa ide-ide subjektif mereka belum tentu benar. David mengatakan bahwa sedikit demi sedikit intuisi siswa adalah titik awal yang berguna, dan bahwa tidak bisa selalu mengandalkan intuisi untuk membawa kita pada yang diinginkan. Sebagian matematika menggunakan intuisi, tetapi sebagian ada juga yang menggunakan metode yang tidak lagi intuitif, atau yang dapat dibuat begitu. Namun, setiap kali diperkenalkan, selalu gagal karena kebanyakan guru di bawah tingkat universitas sendiri tidak memiliki gagasan yang jelas tentang konsep-konsep ini dan mengacaukan semuanya.

  2. Misnasanti
    PPs PMAT A 2016

    Dalam proses belajar mengajar, kebanyakan guru hanya menyuguhkan suatu bentuk hafalan kepada siswa. Siswa sendiri sudah bisa dianggap sebagai suatu mesin foto kopi yang diharuskan menghafal berlembar-lembar materi dan juga para siswa hanya di instruksikan mendengarkan guru mereka berbicara saja tanpa ikut terlibat dalam pembelajaran. Disini siswa tidak diajak untuk terlibat dan ikut berpikir tentang bagaimana caranya untuk mengembangkan sesuatu dari apa yang telah mereka pelajari. Dan guru juga cenderung memaksakan siswa untuk memahami materi yang telah guru jelaskan. Oleh karena itu, guru yang baik harus bijaksana dalam membimbing, mengarahkan, dan melayani siswanya. dengan begitu, ke depannya tidak ada lagi istilah paksaan dalam belajar matematika. Yang ada hanyalah siswa memiliki keinginan yang berasal dari diri mereka pribadi untuk senantiasa belajar matematika.

  3. Johanis Risambessy
    PPs PEP B 2016

    Seperti kata David pada artikel di atas bahwa sedikit demi sedikit intuisi siswa adalah titik awal yang berguna, dan bahwa tidak bisa selalu mengandalkan intuisi untuk membawa kita pada yang diinginkan. Sebagian matematika menggunakan intuisi, tetapi sebagian ada juga yang menggunakan metode yang tidak lagi intuitif, atau yang dapat dibuat begitu. Dalam pembelajaran matematika di sekolah, pendidik sering kali mengkomunikasikan matematika dalam bentuk hafalan, sehingga siswa sudah terbiasa dengan model seperti itu dan ketika harus memecahkan masalah dengan model yang berbeda, siswa merasa bingung. Oleh karena itu, guru harus dapat menggunakan intuisi yang tepat agar siswa dapat memahami setiap konsep dalam matematika.

  4. Mega Puspita Sari
    PPs Pendidikan Matematika
    Kelas B

    Dalam pembelajaran konstruktivisme, guru tidak lagi menduduki tempat sebagai pemberi ilmu. Tidak lagi sebagai satu-satunya sumber belajar. Namun guru lebih diposisikan sebagai fasiltator yang memfasilitasi siswa untuk dapat belajar dan mengkonstruksi pengetahuannya sendiri pembelajaran ini lebih menekankan bagaimana siswa belajar bukan bagaimana guru mengajar. Pembelajaran ini dapat diterapkan pada proses pembelajaran matematika, dimana dengan pembelajaran ini siswa dapat membangun konsep dan pengetahuannya.

  5. Fevi Rahmawati Suwanto
    PMat A / S2

    Menanggapi pendapat Doug Hainline, yang intinya menyebutkan bahwa guru yang miskin matematika akan menjadi guru matematika yang miskin. Guru matematika yang baik adalah guru yang percaya dengan kemampuan matematikanya dan menikmati matematika. Saya setuju dengan pendapat ini bahwa guru yang kurang akan pengetahuan matematika murni akan berbanding lurus dengan guru matematika yang buruk. Untuk dapat mengkomunikasikan pengetahuan yang dimiliki, kita harus memiliki bekal pengetahuan yang akan disampaikan. Apa yang disampaikan intinya sama, namun dengan cara yang berbeda, yang dimodifikasi sesuai dengan taraf berpikir responden menggunakan strategi yang kreatif dan inovatif.

  6. Kumala Kusuma Putri
    Pendidikan Matematika I 2013

    Assalamualaikum Wr. Wb.
    Yes, that is right that most of teacher think that all of their student have same ability to receive their knowledge. It is same with force their students to be understand with all of their explaination. But, I found some teacher who think that their students have different ability to receive their knowledge. I ever found teacher that know that her students have different ability. One day, one of her cleverest student kept asking to continue her explanation. But, she refused to continue her explanation because she knew that some of her students didn't understand yet. Then she told her cleverest student to wait and not to be selfish. She is wonderful teacher indeed. She know that her student have different ability. Teacher should often communicate with students to understand their students ability. Of course, mathematics teacher should know many knowledge about mathematics material, but best teacher is teacher who can communicate with their students, right? Knowledge without socialization is nothing, and otherwise. I think that is enough. Thank you.

    Wassalamualaikum Wr. Wb.

    PEP S2 B

    Guru sebagai ujung tombak dalam keberhasilan pembelajaran matematika tentunya mempunyai peranan penting dalam upaya mencerdaskan anak bangsa terutama dalam hal matematika harus mempunyai strategi yang pas dalam mengajarkan murid - murid nya, sehingga apa yang disampaikan oleh seorang guru dapat dimengerti oleh mereka. Guru yang baik di dalam mengajarkan matematika, sebaiknya guru jangan menggunakan metode ceramah yang akan membuat siswa menjadi jenuh dan tidak bersemangat, gunakan lah metode belajar yang bervariasi yang akan membuat siswa berpartisifasi aktif dalam pembelajaran. Selain itu suasana belajar harus fun atau menyenangkan sehingga siswa tidak terbebani dengan soal – soal matematika yang dianggap sulit tersebut. Namun dalam pembelajaran pun harus tetap fokus terhadap apa yang diajarkan supaya apa yang disampaikan menjadi prioritas utama dalam pembelajaran.

    PEP S2 B

    Seorang guru pun harus selalu bisa tersenyum kepada murid – muridnya, baik murid yang dianggap pintar atau sebalikya. Seorang guru khususnya guru matematika harus bisa memotivasi siswa yang kurang bisa memecahkan permasalahan matematika. Hindari kata-kata 3S dan 3T: sulit, sukar ,susah, tidak bisa, tidak mungkin, dan tidak mampu.Contohnya : ”Soal matematika ini begitu sulit, kamu pasti tidak dapat mengerjakannya” . Kata – kata tersebut hanya akan membuat para murid menjadi minder dan down dan akhirnya mereka malas untuk mengerjakan soal matematika tersebut. Lalu, bagaimana cara belajar matematika supaya murid – murid menjadi mengerti apa yang disampaikan oleh seorang guru? Ketika banyak siswa yang takut terhadap pelajaran matematika, atau terlihat bosan, seorang guru perlu melakukan segala sesuatu yang dapat membuat pelajaran matematika menarik. Perlu bagi seorang guru untuk melakukan segala sesuatu untuk menolong siswa agar merasa senang dengan pelajaran matematika. Langkah yang harus digapai adalah bagaimana siswa nyaman dalam pembelajaran, yang dalam hal ini siswa harus senang terlebih dahulu akan apa yang ia terima. Banyak cara yang dilakukan, diantaranya adalah dengan merubah simbol dalam matematika menjadi nyata seperti contoh ketika dalam suatu soal cerita tentang perhitungan uang atau jual beli, mengapa tidak kita menggunakan uang asli agar siswa tergambarkan jual beli barang tersebut.

    PEP S2 B

    Selain itu ada faktor internal dari seorang guru sehingga murid tidak menguasai matematika yaitu jika seorang guru tidak menyenangi terhadap suatu pelajaran, maka para siswanya juga tidak akan menyenangi pembelajarannya. Semakin banyak energi positif yang dimiliki seorang guru terhadap sebuah pelajaran, akan semakin menyenangkan pembelajarannya terutama dalam matematika. Selain itu supaya pembelajaran matematika tertuju pada sasaran yang diinginkan maka ketika seorang guru membuat perencanaan untuk pelajaran matematika, perlu mewujudkannya dengan kreatif, membentuk pelajaran matematika interaktif yang melibatkan para siswa dalam proses pembelajaran.

    PEP S2 B

    Jika memungkinkan, rencanakan aktifitas yang akan menjadikan siswa-siswa berdiri dan bergerak di dalam atau di sekitar kelas. Tentunya hal ini dapat merangsang keaktifan siswa di kelas sehingga belajar matematika itu menjadi menyenangkan. Ada beberapa tips untuk perencanaan (pembelajaran) matematika , diantaranya rencanakan permainan-permainan matematika jika memungkinkan dan rencanakan kerja kelompok yang memberikan kesempatan bagi siswa-siswa yang mampu untuk maju membantu siswa-siswa yang lambat belajarnya.

    PPS 2016PEP B
    I agree with Hainline that someone who is good at mathematics will be a good teacher of mathematics, but some one with unability to understand mathematical concepts will stay unable to be a good mathematical teacher. mathematics is not the particularity of some, but the property of everyone who devotes to leaning it . lazy people cannot be successful in mathematics because mathematics requires to work hard and hard. basing on what Marsigit said, the approaches used by the teacher gets evaluated as successful by observing the results of students in mathematics.

  12. Azwar Anwar
    Pendidikan Matematika S2 Kelas B 2016

    Guru memegang peranan yang sangat strategis terutama dalam membentuk karakter bangsa serta mengembangkan potensi siswa. Hal-hal yang harus diketahui oleh seorang guru dalam upaya mengembangkan peserta didik dalam memahami konsep matematika, mengoptimalkan perkembangan peserta didik. Pentingnya peranan guru dalam keberhasilan peserta didik maka hendaknya guru mampu beradaptasi dengan berbagai perkembangan yang ada dan meningkatkan kompetensinya sebab guru pada saat ini bukan saja sebagai pengajar tetapi juga sebagai pengelola proses belajar mengajar di kelas. Dengan begitu matematika yang sulit dapat mudah dipahami jika guru benar-benar mengajarkan konsepnya serta mampu berkomunikasi dengan baik terhadap siswanya.

  13. Fatya Azizah
    Pendidikan Matematika B PPS UNY 2016

    benar adanya pernataandavid yang menyatakan bahwa kebanyakan siswa dalam kelas lebih mendengarkan temannya dibandingkan guru yang mengajar, disinilah peran kolaboratif learning yang memanfaatkan pembelajaran dengan teman sebaya dimana siswa diminta untuk belajar dengan bekerjasama melengkapi pengetahuan satu sama lain sehingg adiharapkan kemampuan siswa dalam satu kelas bisa terdistribusi secara merata.

  14. Erni Anitasari
    S2 Pend. Matematika Kelas A

    Disini dikatakan bahwa guru yang baik adalah guru yang percaya akan kemampuannya. Apabila seorang guru matematika maka yang baik adalah yang yakin dengan kemampuan matematikanya. Tapi di sisi lain seorang guru tidak serta merta meluapakan kewajibannya untuk mengajarkan materi matematika kepada siswanya. Dan dalam proses mengajar kepada siswa tersebut tidak cukup seorang guru hanya dengan modal yakin dengan kemampuannya. Misalnya dalam mengajar materi matematika kita sebagai guru telah mempersiapkan materi dan penyampaian formula matematika yang menurut kita sudah bagus dan mudah dimengerti siswa akan tetapi itu hanyalah persepsi dari diri kita. Hal tersebut belum tentu terjadi di dalam persepsi para siswa. Karena yang menurut kita materi yang sudah kita kemas secara sederhana untuk disampaikan ke siswa belum tentu sederhana dimata siswa. Jadi, kita sebagai guru matematika harus memperhatikan jenjang sekolah agar kita dapat menyesuaikan model pembelajaran yang akan digunakan dalam mengajar.

  15. Rospala Hanisah Yukti Sari
    S2 Pendidikan Matematika Kelas A Tahun 2016

    Assalamu’alaikum warohmatullahi wabarokatuh.

    Dari percakapan di atas, bahwa pembelajaran matematika memang harus disesuaikan dengan taraf berfikir si anak didik. Jika dalam level universitas, matematika murni termasuk logika matematika dapat diajarkan, karena dalam level universitas, taraf berpikir mahasiswa sudah dalam tahap berpikir abstrak, sehingga hal tersebut tidak menjadi masalah.

    Namun, yang menjadi masalah ketika matematika murni termasuk logika matematika diajarkan di SD dan SMP, di mana siswa masih berpikir konkrit. Seharusnya, dalam jenjang tersebut siswa lebih ditanamkan kepada akidah islam dan membentuk kepribadian islam dengan menyatukan pola pikir (‘aqliyyah) dan pola sikap (nafsiyyah) yang sesuai dengan aturan islam. Karena pada jenjang tersebut, siswa masih memerlukan bimbingan dalam membentuk kepribadiannya.

    Sistem pendidikan kapitalis-sekuler telah merenggut fitrah dan potensi anak. dalam sistem pendidikan ini, membuat siswa ‘dipaksa’ untuk belajar matematika yang murni, maka hal itu tidak sesuai dengan taraf berpikirnya dan bahkan secara perlahan dapat ‘melumpuhkan’ kinerja otak si anak tersebut.

    Hal inilah yang patut diwaspadai bagi para pembuat kebijakan, kaum intelektual, dan masyarakat keseluruhan, agar sistem pendidikan kapitalis-sekuler segera diganti dengan sistem yang shahih yang berasal dari Allah SWT dengan penerapan syariat islam secara sempurna, dan jangan sampai kita membiarkan anak didik kita diasuh dengan sistem yang destruktif tersebut.

    Hanya dengan upaya yang sungguh-sungguh yang dibekali dengan keimanan dan keyakinan, serta pertolongan Allah sajalah masa ‘kegelapan’ ini akan diganti dengan ‘cahaya terang’ kehidupan yang berasal dari aturan Allah SWT .

    Wassalamu’alaikum warohmatullahi wabarokatuh.

  16. Nuha Fazlussalam
    s1 pendidikan matematika c 2013

    dari tulisan ini saya sangat setuju dengan pernytaan "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" baaimanas eseorang bisa mengajarkan sesuatu jika orang tersbut tidak bisa, tidak mampu dan tidak paham sesuat itu. jika tidak bsa demikaina, maka akan memnagshilkan bad teaching. terkait dengan menujukkan pe buktian matematika dan demosntrsi, untuk hal ini kita kemabilaka lagi pada etode terbaik yaitu student centered, apapun proses pembeajarannya, bahgimana kita bisa memberikan kesempatan kepada siswa untuk terlibat aktif dalam kegatan bpembelajaran

    PPs Pmat A 2016

    Dalam makalah aini, dia menulis gaya belajar merupakan suatu kombinasi dari bagaimana ia menyerap, dan kemudian mengatur serta mengolah informasi. Gaya belajar bukan hanya berupa aspek ketika menghadapi informasi, melihat, mendengar, menulis dan berkata tetapi juga aspek pemrosesan informasi sekunsial, analitik, global atau otak kiri- otak kanan, aspek lain adalah ketika merespon sesuatu atas lingkungan belajar (diserap secara abstrak dan konkret).
    Perlu disadari bahwa tidak semua orang punya gaya belajar yang sama. Walaupun bila mereka berada di sekolah atau bahkan duduk di kelas yang sama. Kemampuan seseorang untuk memahami dan menyerap pelajaran sudah pasti berbeda tingkatnya. Ada yang cepat, sedang dan ada pula yang sangat lambat. Karenanya, mereka seringkali harus menempuh cara berbeda untuk bisa memahami sebuah informasi atau pelajaran yang sama.
    Di lingkungan sekolah, sebagian siswa lebih suka guru mereka mengajar dengan cara menuliskan segalanya di papan tulis. Dengan begitu mereka bisa membaca, kemudian mencoba memahaminya. Sebagian siswa lain lebih suka guru mereka mengajar dengan cara menyampaikannya secara lisan dan mereka mendengarkan untuk bisa memahaminya. Sementara itu, ada siswa yang lebih suka membentuk kelompok kecil untuk mendiskusikan pertanyaan yang menyangkut pelajaran tersebut.
    Cara lain yang juga kerap disukai banyak siswa adalah model belajar yang menempatkan guru tak ubahnya seorang penceramah. Guru diharapkan bercerita panjang lebar tentang beragam teori dengan segudang ilustrasinya, sementara para siswa mendengarkan sambil menggambarkan isi ceramah itu dalam bentuk yang hanya mereka pahami sendiri.
    Setelah mengenal gaya belajar pada siswa, seorang guru menjadi tahu cara mengidentifikasi dan mengajar siswa yang memiliki berbagai macam gaya belajar dengan keunikannya masing-masing.

  18. Wan Denny Pramana Putra
    PPs Pendidikan Matematika A

    Dalam ruang lingkup pelajar muda, terdapat perbedaan yang sangat besar antara matematika murni dengan matematika sekolah. Jika matematika murni diterapkan ke sekolah maka akan terjadi banyak kesalahan dalam kita menilai mereka. Anak-anak adalah korban naif absolut dengan cara mengintervensi mereka jika mereka tetap dipaksakan mempelajari matematika murni pada bangku sekolah. Ebbut dan Straker (1955) mengemukakan bahwa solusinya adalah harus ada sekolah matematika tempat dimana anak-anak belajar matematika sesuai dengan dimensinya. Disinilah kelebihan pendidikan matematika.


    Kenyataan telah menunjukkan bahwa intelektual seorang anak berkembng secara kualitatif. Proses belajar mengajar akan efektif bila kemampuan berpikir anak diperhatika. Proses belajar mengajar dikatakan sukses apabila terjadi transfer belajar, yaitu materi pelaharan yang disajikan oleh guru dapat diserap ke dalam struktur kognitif siswa. Siswa dapat menguasai materi tersebut tidak hanya terbatas pada tahap ingatan tanga pengertian (meaningful learning).


    Kenyataan telah menunjukkan bahwa intelektual seorang anak berkembng secara kualitatif. Proses belajar mengajar akan efektif bila kemampuan berpikir anak diperhatika. Proses belajar mengajar dikatakan sukses apabila terjadi transfer belajar, yaitu materi pelaharan yang disajikan oleh guru dapat diserap ke dalam struktur kognitif siswa. Siswa dapat menguasai materi tersebut tidak hanya terbatas pada tahap ingatan tanpa pengertian atau pemahaman secara mendalam (meaningful learning).

  21. Erlinda Rahma Dewi
    S2 PPs Pendidikan Matematika A 2016

    Guru terbaik matematika yakin tentang kemampuan matematika mereka (dengan alasan yang baik), dan menikmati matematika. Mereka tidak perlu matematika, tetapi mereka harus mengambil kesenangan dalam bukti indah dan demonstrasi dalam menunjukkan ini untuk siswa mereka. Saya setuju dengan Doug. Mungkin apa yang dimaksud dengan pernyataannya adalah bukti indah dan demonstrasi adalah untuk siswa, tidak hanya untuk guru. Ya, saya bisa mengerti apa yang Prof Marsigit katakan bahwa bukti-bukti yang indah harus baik dari persepsi guru dan siswa. Tapi kadang-kadang, apa yang kita anggap benar-benar berarti tidak dapat diungkapkan dengan tepat oleh kata-kata.

  22. Rhomiy Handican
    PPs Pendidikan Matematika B 2016

    Matematika sebagian menggunakan intuisi, tetapi juga sebagian menggunakan metode yang tidak lagi intuitif, atau yang dapat dibuat menjadi begitu. Hal ini terutama penting dalam sekunder untuk menjauh dari pendekatan subjektif murni. Ini juga mengapa matematika murni dapat berguna dalam mengatur kurikulum - baik di tingkat primer dan sekunder. Sebuah aspek penting dari matematika yang bisa menguntungkan bagi semua siswa, terlepas dari profesi akhirnya mereka, akan memperkenalkan konsep dari logika, untuk membantu siswa berpikir jernih. Tapi tidak notasi formal logika; bukan konsep. Hal ini dapat dimulai pada primer, dan terus di sekunder.

  23. Arifta Nurjanah
    PPs P Mat B

    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.