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.


26 comments:

  1. Shelly Lubis
    17709251040
    S2 P.Mat B

    most students do prefer to memorize methods in studying mathematics in school or in solving problems given by the teacher. they may do so because they follow the example of the teacher, which only shows the steps of settlement. if so, then the math they have learned becomes useless and they will find it difficult to apply it.

    ReplyDelete
  2. Gamarina Isti R
    17709251036
    Pendidkan Matematika Kelas B (Pascasarjana)


    Dalam postingan ini membahas mengenai kepercayaan diri seorang guru dalam mengajar berdasarkan kompetensi yang mereka miliki sehingga tidak dierlukan lagi matematikawan. Hal yang sering menjadi persepsi dalam masyarakat adalah orang matematika murni lebih memiliki kemampuan lebih mumpuni dari pada orang yang belajar pendidikan matemtaika. Memang anggapan itu tidak bisa juga untuk disalahkan karena seorang matematikawan tentu saja lebih banyak waktu untuk belajar konsep matematika. Namun apabila ingin mengaplikasikan dalam pembelajaran tidak hanya cukup dengan kemampuan matematiika saja namun kemampuan matematika beserta cara guru mengajar. Sehingga saat guru ingin menjadikan matematikanya paham mengenai matematika maka guru juga harus meningkatkan profesi dan pedagogiknya.

    ReplyDelete
  3. Gina Sasmita Pratama
    17709251003
    S2 P.Mat A 2017

    As it is said in the above article that the world seems to be designing a curriculum that underestimates or omits evidence altogether, so that the student never gets much chance to judge whether he or she finds any beautiful evidence. The reason is not just a tendency to teach what industry wants, but it is usually taught in a way that leads to more memorization, especially with learning culture to learn for tests. In fact, students should learn and build their own knowledge. Therefore, the mathematics learning process should emphasize student activity.

    ReplyDelete
  4. Rahma Dewi Indrayanti
    17709251038
    PPS Pendidikan Matematika Kelas B

    Selama ini masih banyak guru yang menggunakan model ceramah. Dampak dari model pembelajaran yang seperti ini adalah membentuk sifat pasif pada siswa. Suasana yang membuat tertarik untuk belajar tidak dibiasakan sehingga siswa tidak mau belajar kecuali jika ada ujian atau penilaian. Apabila sikap ini terus-menerus dipupuk bukan tidak mungkin sikap ini akan dibawa sampai ia dewasa sehingga muncul karakter yang tidak mendukung sebagai generasi muda penerus bangsa yang harusnya memiliki jiwa pantang menyerah dan rasa ingin tahu yang tinggi. Mestinya sedikit demi sedikit hal ini harus kita ubah, salah satunya mungkin dengan menerapkan berbagai model pembelajaran yang variatif.

    ReplyDelete
  5. Irham Baskoro
    17709251004
    S2|Pendidikan Matematika A 2017|UNY

    Pembelajaran matematika itu seharusnya membangkitkan antusiasme tidak hanya paad guru yang mengajar melainkan juga siswa sebagai pebelajar. Seperti yang diungkapkan David Reid: “a student will not learn without being taught, and the teaching will be for naught if the students doesn't try to learn” atau dengan kata lain siswa tidak belajar kalau tidak diajari dan guru tidak bisa mengajar kalau siswa tidak mau belajar. Baik antara guru maupun siswa harus memiliki komitmen yang sama untuk membangun pengetahuan dalam pembelajaran matematika. Untuk itu rasa cinta terhadap matematika perlu dibangun dalam diri siswa agar siswa mau dan termotivasi untuk belajar matematika.

    ReplyDelete
  6. Mariana Ramelan
    17709251056
    S2 Pend. Matematika C 2017

    Guru yang baik ialah guru yang mampu menunjukkan/menggunakan berbagai sumber belajar untuk siswanya. Idealnya, tugas guru adalah menfasilitasi siswanya dalam kegiatan pembelajaran matematika. Selebihnya biarlah siswa yang menentukan sendiri apakah dia menyukai matematika atau tidak. Jika guru tetap bersikeras memaksa siswanya untuk menyukai matematika, hal tersebut malah membuat siswa menjadi tertekan. Sehingga dapat menghilangkan intuisi siswa itu dalam belajar matematika. Oleh karena itu, guru yang baik harus bijaksana dalam membimbing, mengarahkan, dan melayani siswanya.

    ReplyDelete
  7. Junianto
    PM C
    17709251065

    Guru professional sudah menguasai bagaimana pola pikir siswa dan bagaimana karakter mereka. Dengan demikian, guru dapat menyampaikan ilmu matematika dan mengajak siswa untuk belajar. Guru juga mengetahui bagaimana menjadi fasilitator dan mendidik siswa dalam belajar disamping menguasai materi yang akan di ajarkan. Selain itu, perlu juga dipahami bagaimana kondisi psikologi siswa dalam belajar, bagaimana alur berpikir siswa, apa saja kesulitan belajar siswa, dll.

    ReplyDelete
  8. Putri Solekhah
    17709251006
    S2 Pend. Matematika A

    Assalamu'alaikum wr wb,

    Selain kemampuan penguasaan materi yang baik serta cara mengajar yang bersemangat, persepsi atau sudut pandang guru dalam mengajar juga penting bagi terlaksananya pembelajaran matematika yang baik dan menyenangkan. Dikatakan bahwa guru sebaiknya menyajikan materi atau mengajarkan materi matematika dengan cara yang menyenangkan, tetapi guru juga harus tahu seperti apa ‘menyenangkan’ itu bagi siswa. Bisa jadi metode yang digunakan guru dan ia anggap menyenangkan justru dianggap siswa merupakan metode yang menyebalkan. Contohnya guru menggunakan pembelajaran berbasis diskusi dengan model bangun datar dalam pembelajaran mengenali macam-macam bangun datar. Guru menganggap itu menyenangkan bai siswa karena ia berfikir bahwa kegiatan diskusi dan tukar pikiran akan lebih menambah wawasan dan masukan dalam mempelajari suatu hal, serta menggunakan model nyata dapat membantu visualisasi siswa. Namun ternyata, siswa justru bingung jika disuruh diskusi karena dalma kelompoknya taka da satupun siswa yang paham sehingga mereka menjadi bingung berjama’ah. Serta mereka merasa menggunakan model dua dimensi itu membosankan karena dengan gambar pun mereka mampu memvisualisasikan macam-macam bangun datarnya.

    ReplyDelete
  9. Latifah Fitriasari
    17709251055
    PPs PM C

    Guru yang baik adalah guru yang benar benar paham mengenai matematika sehingga tidak menyesatkan para siswa. Sehingga memiliki kompetensi dalam karyanya. Guru yang baik adalah yang dapat menyampaikan matematike dengan baik pula kepada para siswanya, seperti berbicara mengenai hal-hal sperti sejarah tentang matematika, atau berbagai macam pengalaman lainnya, sehingga para siswa menjadi tertarik dan mulai mencintai matematika sebagai sebuah kebutuhan juga.

    ReplyDelete
  10. Muh Wildanul Firdaus
    17709251047
    Pendidikan matematika S2 kls C

    Matematika murni berbeda dengan matematika sekolah, apa yang siswa pelajari bukanlah matematika murni seperti yang dipelajari oleh para matematikawan. Pemikiran siswa mengenai matematika tidak akan sampai pada pemahaman matematika murni yang sangat abstrak. Oleh karena itu, matematika yang dipelajari siswa terutama siswa-siswa yang berada di sekolah dasar dan menengah bukanlah matematika murni, melainkan matematika sekolah yang dapat dipelajari sesuai dengan tingkat perkembangan mereka.

    ReplyDelete
  11. Arina Husna Zaini
    PEP S2 B
    17701251024
    Assalamualaikum Wr.Wb

    Bahasa dalam proses pembelajaran matematika harus memiliki pola dan bentuk yang uniq. Kekreativitasan guru ditaruhkan dalam hal ini. Ketika guru memasuki kelas untuk pembelajaran, maka hal yang baik bukanlah langsung masuk dalam materi inti, namun sebelumnya harus menggunakan bahasa yang tepat untuk mengungkapkan kembali materi sebelumnya yang berkaitan dengan materi inti. Hal ini biasa disebut dengan memberikan apersepsi kepada siswa agar intuisis siswa mulai terbangun sehingga apa yang akan dipelajari akan nyambung dan komunikatif dengan materi sebelumnya. Terima Kasih

    ReplyDelete
  12. Novita Ayu Dewanti
    17709251053
    S2 PMat C 2017

    Bismillah
    Bahasa matematika perlu disampaikan oleh guru dalam bahasa yang mudah diterima oleh siswa. Penyampaian ini dapat digunakan dengan berbagai macam metode. Guru disarankan tidak menggunakan metode ceramah saja dalam penyampaiannya. Guru dituntut untuk kreatif dalam penyampaiaannya.

    ReplyDelete
  13. Isoka Amanah Kurnia
    17709251051
    PPs Pendidikan Matematika 2017 Kelas C

    Education certainly also sees the psychological development of a learner as an important thing. Trying to teach the material to children will be great if it adjusts their psychological development level. In elementary school-age children whose psychology tends to play with the surrounding environment, then if applied learning refers to a pure mathematical approach would be very wrong. Because their psychology is not in line with it. Providing contextual teaching will give students a personal interest to learn and try to solve the problems given so as to make math lessons more fun.

    ReplyDelete
  14. Ulivia Isnawati Kusuma
    17709251015
    PPs Pend Mat A 2017

    Karakter siswa satu dengan sisswa lainnya itu berbeda-beda. Oleh karena itu pembelajaran sebaiknya memfasilitasi siswa dengan karatkter yang berbeda tersebut. Oleh karena itu, pembelajaran sebaiknya inovatif dan harus memuat akuntability, sustainability. Akuntability adalah sesuatu hal yang mencakup bagaimana menciptakan suatu kondisi yang kondusif selama proses pembelajaran. Sedangkan sustainability adalah suatu hal cara menjaga proses yang sesuai secara terus menerus

    ReplyDelete
  15. Nama: Dian Andarwati
    NIM: 17709251063
    Kelas: Pendidikan Matematika (S2) Kelas C

    Assalamu’alaikum. untuk mengubah paradigma pembelajaran guru dari pembelajaran tradisional ke pembelajaran inovative maka guru perlu menyadari bahwa siswa harus merasa membutuhkan matematika. keindahan matematika perlu dirasakan oleh siswa, sehingga rasa ingin tahu dan keinginan untuk memecahkan masalah pada diri siswa dapat tumbuh. Jika rasa ingin tahu itu ada maka siswa akan termotivasi belajar matematika

    ReplyDelete
  16. Seorang guru memang dituntut untuk memiliki pengetahuan matematika yang baik, tetapi itu saja tidaklah cukup, seorang guru juga dituntut untuk memiliki kemampuan untuk menciptakan strategi dalam proses pembelajaran yang dapat membuat siswa belajar matematika. membuat siswa mendapatkan pengetahuan matematika yang baik.

    Nama : Frenti Ambaranti
    NIM : 17709251034
    Kelas : S2 Pendidikan Matematika B

    ReplyDelete
  17. Ilania Eka Andari
    17709251050
    S2 pmat c 207

    Guru yang lemah pada matematika, akan lemah pula dalam mengajarkan matematika. Namun, guru yang sangat menguasai matematika murni, belum tentu akan hebat pada saat mengajar matematika. Karena matematika murni sangatlah berbeda dengan matematika sekolah. Guru tidak boleh otoriter saat mengajar. Guru adalah seoarang fasilitator dan juga motivator untuk siswa dalam mengembangkan kemampuannya. Semua diserahkan pada siswa, guru hanya bertanggung jawab untuk memberi fasilitas dan motivasi pada siswa, agar mereka dapat belajar dengan senang hati.

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  18. Metia Novianti
    17709251021
    PPs P.Mat A

    Menguasai materi matematika dan mengajarkan materi matematika adalah dua hal berbeda namun sangat diperlukan dalam proses pembelajaran di kelas. Guru yang menguasai materi matematika namun tidak bisa menyampaikannya dengan tepat dan efektif akan kesulitan membantu siswanya memahami suatu materi. Guru yang pandai dalam mengajarkan matematika akan tetapi tidak menguasai materi matematika akan kesulitan dalam pembelajaran, apalgi jika siswa-siswa bertanya hal-hal yang belum dipahaminya, bisa-bisa siswa jadi tersesat. Jadi, guru harus menguasai keduanya dalam porsi yang seimbang agar dapat membantu siswa dalam proses pembelajaran.

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  19. Firman Indra Pamungkas
    17709251048
    S2 Pendidikan Matematika 2017 Kelas C

    Assalamualaikum Warohmatullah Wabarokatuh
    Guru matematika memang harus menguasai ilmu matematika yang akan digunakan pada pembelajaran. Mengapa? Guru matematika yang tidak menguasai materi matematika tidak akan mampu untuk menjadi fasilitator dengan maksimal. Namun hal ini tiak berarti guru matematika menggunakan matematika murni dalam pembelajaran. Matematika murni berbeda dengan matematika sekolah. Jadi, dalam pembelajaran, guru harus menggunakan pandangan matematika sekolah, yang mana salah satunya berarti bahwa memandang matematika sebagai suatu aktivitas

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  20. Kartika Pramudita
    17701251021
    PEP S2 B

    Peran guru dalam pembelajaran khususnya pembelajaran matematika menjadi topik yang tidak pernah habis untuk diperbincangkan. Yang menjadi pertanyaan adalah apa yang harus dilakukan guru untuk menggapai kesuksesan dalam pembelajaran. Jawaban dari pertanyaan tersebut tentunya beragam karena dapat dilihat dari berbagai macam sudut pandang. Berdasarkan bacaan tersebut, guru seharusnya memperluas wawasan dan pengetahuannya tentang pengajaran dan bagaimana dapat memfasilitasi siswa dengan baik, selalu melibatkan siswa dalam mengambil keputusan pembelajaran sehingga tidak mendominasi dan menjadi determinis bagi siswa.

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  21. Fitri Ni'matul Maslahah
    17709251058
    PPs PM C

    Banyak pengajar yang menyatakan bahwa mereka dapat mengajar dan berdiri di depan para sisw bukan karena beliau-beliau adalah orang yang pintar dan genius, melainkan orang yang sudah pernah melewati proses yang saat ini siswa mereka lalui. sehingga pengalaman guru tersebutlah yang menjadi senjata yang terus diasah bagaimana menyampaikan matematika yang mudah kepada siswa. Allahu a'lam

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

    Assalamualaikum wr.wb

    Pada diri manusia terdapat istilah yang disebut dengan “potensi”. Hal ini merupakan sebuah keistimewaan yang diberikan oleh Allah SWT untuk dijaga dan dikembangkan agar dapat dipergunakan sebagaimana mestinya. Potensi yang dimiliki harus disesuaikan dengan konteks ruang dan waktunya agar tidak menjadi momok (hantu). Begitu juga ketika menjadi guru matematika, kita harus mampu membuat pembelajaran matematika menjadi pembelajaran yang menyenangkan bagi siswa agar matematika itu sendiri tidak menjadi momok yang menakutkan.

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  23. Elsa Susanti
    17709251024
    S2 Pendidikan Matematika 2017 Kelas B

    Minat belajar siswa tidak hanya dipengaruhi oleh subyek dan materi pelajarannya, namun aspek penting yang ikut mempengaruhi adalah sosok guru. Tidak sedikit mereka yang memandang guru matematika adalah guru yang killer/otoriter. Inilah salah satu penyebab matematika dianggap matapelajaran yang sulit. Padahal guru yang baik mampu menfasilitasi siswanya sekalipun pada materi yang sulit. Tidak jarang ada siswa yang menyukai matematika karena menyukai pribadi gurunya. Dengan demikian, diharapkan guru yang baik dan kaya akan pengetahuan sehingga dapat meningkatkan motivasi siswa dalam belajar.

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  24. Firman Indra Pamungkas
    17709251048
    Pend. Matematika S2 Kelas C

    Dari artikel di atas, yang dapat saya simpulkan ialah bahwa perubahan paradigma pembelajaran dari tradisional menuju inovatif sangat diperlukan. Untuk dapat mewujudkan perubahan tersebut terdapat 2 hal pokok yang dapat menjadi dasar, yaitu accountability dan sustainability. Keduanya sangat berhubungan erat dan tidah bisa dipisahkan. Accountability merupakan keadaan dimana seseorang dapat dipercaya, dalam hal ini adalah dipercaya untuk membagi ilmunya pada peserta didik. Accountability menimbulkan Sustainability, yaitu suatu kepercayaan yang dapat dijaga secara terus-menerus.

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  25. Atik Rodiawati
    17709251025
    S2 Pendidikan Matematika B 2017

    Bahasa merupakan sarana komunikasi, dalam hal ini merupakan sarana komunikasi matematis yang dilakukan antara guru dan murid. Bentuk komunikasi ideal yang diharapkan tentu saja adalah membuat siswa tidak hanya mengerti tentang materi matematika namun lebih jauh dari itu, diharapkan siswa dapat menyukai matematika. Di sinilah peran guru sebagai fasilitator harus ditingkatkan kemampuan komunikasinya. Dalam mencapai tujuan agar siswa menyukai matematika, guru semestinya dapat memberi perlakuan baik namun wajar, pemberian motivasi dan melakukan pendekatan personal.

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  26. Ibnu Rafi
    14301241053
    S1 Pendidikan Matematika Kelas I 2014

    Bahasa merupakan alat penting yang dapat digunakan oleh guru untuk memaksimalkan usahanya dalam memfasilitasis siswa belajar matematika secara bermakna. Pentingnya penggunaan bahasa tercermin dari bagaimana cara guru memotivasi siswa dan mengembangkan permasalahan matematika.Seorang guru matematika sebaiknya mampu memahami penggunaan bahasa yang tepat untuk digunakan dalam membelajarkan matematika sekolah.

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