Saturday, 6 October 2018

EVIDENCE-BASED POLICY AND PRACTICE


EVIDENCE-BASED POLICY AND PRACTICE
(From The Book Philosophy of Education By Richard Pring)

to complete the task of Science Philosophy Course
Lecturer : Dr. Laila Fitriana, M.Pd.




By :
                                                                   Heri Satriawan






POSTGRADUATE OF MATHEMATICS EDUCATION
FACULTY OF TEACHER TRAINING AND EDUCATION
UNIVERSITAS SEBELAS MARET
SURAKARTA
2018





CHAPTER I
INTRODUCTION
A.    Background
Giving policy in every case is not always based on evidence because the evidence is an implication of the policy. A marely superficial acquaintance with the philosophy of science makes it clear that science grows from constant (and often successful) attempts to negate the current state of scientific knowledge. All such knowledge is, as it were, provisional-to be accepted until such time as it is refuted and replaced by more comprehensive and better corroborated scientific propositions.
An important part of refining the evidence lay in the systematic review of existing research, rejecting that which did not meet rigorous experimental criteria, ignoring that where the data and method were less than clear, reconciling where possible the different samples bases for the samples, identifying where further research was needed to fill the gaps in our scientifically based knowledge.
Such has been the success of the Cochrane Centre’s work that people in other areas of the public services have looked for the lessons which can be learnt from it. The Campbell Collaboration, based in the United States, but with regional centres in Canada an Denmark, has extended the work to other areas of social life – for example, education and criminology. Saw the approach of Cochrane and Campbell to be what was require to improve the quality of research to inform both goverment policy and professional practice. And this was seen to be necessary because of the criticismof that research, certainly in education.
The reaction to the transfer, to the field education, of the evidence-base approach of Conchrane and Campbell has varied from the hostile to the welcoming. But the essence of the critisms and of the differences betwen them is philosophical.
B.     Problem Formulation
1.        What is the meaning of evidence?
2.        How does the evidence related to the education policy?
C.    The Aim of Research
1.         To knows the meaning of evidence
2.         To knows the evidence related to the education policy.


CHAPTER 11
DISCUSSION

Evidence
A lot depends on how one interprets the word 'evidence' There are many different kinds of evidence, depending on the type of claim being made. Example: Evidence that the water boils at 100 degrees Celsius is different from the evidence to show whether or not someone is beautiful. Historical evidence is different from that in science, and even within science there are different sorts of discourse, each characterized by differences in what is deemed to constitute evidence. Furthermore, 'evidence' must not be confused with proof. As the previous example, the water boils at a temperature of 100  is one of the proof because it is already believed inthe truth, whereas the proof that someone who is beautiful is one of the shape  of "evidence" in which, we must prove it gradually because someone who is beauty or not, it is relative, we see from many aspect.
One can gradually build up the evidence for a belief but gradually proving it seems a little odd. On the basis of evidence, it may be probable that something is the case - although there may be counterevidence which is less persuasive. For example in education, the teacher sees the child cheating, but the teacher does not have sufficient evidence. Considering that the child is really cheating or not by looking back or reconsidering when he cheats, it is called a prima facie motive. Educational peactice requires judgesments about intentions as well as motivation
There are philosophical problems about the concept of evidence those are as follows:
1.    The logical unpredictability of all the consequences of a particular course of action or a particular policy.
2.    The irreconcilability of scientific discourse (and thus the social sciences within a particular tradition) with that concerned with persons.
3.    The logical separation of educational 'ends' or 'goals' from the 'means' of achieving them.
Unpredictability
The difficulty to predict what will happen so that the uncertainty underlies us to make a policy. So the policy will affect to what happens. For example: the school wants to get good accreditation, so the policy is made to increase the school's value, eg in the case of increasing minimum competency criteria. Given this necessary unpredictability of complex social situations, there is a limit to how far the accumulation of evidence can ensure certain consequences will follow from carefully considered interventions.
Explaining human behaviour
Educational policy and professional practice are ultimately about getting people (usually young people) to learn something - and something which is deemed to be of value. To educate is to develop the capacity to think, to value, to understand, to reason, to appreciate. These are states of mind, mental capacities, distinctively human qualities. One feature of such states of mind is that they constitute a different kind of'reality' from that which is the subject matter of the natural sciences.
Those intentions and motives presuppose a social context of rules where by the intended behaviours are going to be interpreted by others in a particular way. It is no good signalling a revolution if the fellow revolutionaries do not understand the gesture. To explain human actions requires a grasp of the social rules through which social intercourse is able to take place. Furthermore, such social rules will change from social group to social group-indeed, a social group is partly defined in terms of the social rules through which they engage with each other.
Means and ends
The concern for evidence-based policy and practice arises within a climate of 'improvement', 'raising standards', 'making schools more effective'. Knowledge is required of 'what works'. To do this, so the argument goes, there is a need to set targets, as specific as possible. That case is its goal.
Into school effectiveness - the characteristics of a school and its leadership which will ensure 'success'. Success is spelt out in terms of very precise targets (such as a given proportion of students attaining grades in public examinations). Similarly, effective teaching, for example: schools that want to achieve accreditation A, set targets to achieve accreditation A. eg increasing student's KKM score. It makes the teachers think that how to increase the value of KKM so that teachers can apply appropriate learning methods to increase the value of student’s KKM.
In the educational encounter, the teacher is the expert in knowning what ‘means’ will most effectively attain those ‘ends’. The goal, end or purpose shapes the way in which the teacher teaches – it is captured and ‘shown’ in the very act of teaching. Teaching is a transaction between the teacher and the learner, not the delivery of something to the learner. But the main educational purpose lies in the engangement with a valuable text.


CHAPTER III
CONCLUSION
1.             Evidence is of different kinds relative to the form of discourse through which a problem is being addressed.
2.             Educational policies, aiming to improve the quality of learning and to increase the number of people who successfully participate in education at different phases, need evidence to show that one policy rather than another will make things better.So, what we have already done, it must have the evidence.There are three conclusions that need to be drawn from this as we look to the future
·                The first is that evidence-based policy and practice need to look much more carefully at the diffent kinds of evidence which legimately enter into educational deliberations at the policy and professional practice levels.
·                Second, despite the rather electric nature of educational discourse, there are lessons to be learnt from the insistence by the advocates of evidence-based policy and practice for the more rigorous search for evidence.
·                Third, the political and often highly charged context of educational research needs to be recognized. It cannot be wished away. And that political context invades not only the policies and practices themselves, but also the different philosophical advocacies of different sorts of research.

Definisi Kurikulum


CHAPTER II
DISCUSSION

A.    Definisi Kurikulum
Kurikulum kata berasal dari bahasa Latin “currere”, yang berarti medan balapan berjalan. Kurikulum yang juga berasal dari bahasa Inggris adalah "kurikulum" yang berarti jalannya kursus. Dalam bahasa Prancis dikenal dengan "carter" yang artinya berlari. Menurut UU Sistem Pendidikan Nasional tahun 1989 pasal 1 ayat 1 menyatakan bahwa: “Kurikulum adalah seperangkat rencana dan peraturan mengenai konten dan materi pelajaran dan cara yang digunakan sebagai pedoman untuk penyelenggara kegiatan pembelajaran.
Kurikulum adalah program pendidikan yang berisi materi pembelajaran dan pengalaman belajar yang telah diprogram, direncanakan, dan dirancang secara sistematis berdasarkan norma-norma yang berlaku dan digunakan sebagai pedoman dalam proses mengajar dan belajar bagi pendidik atau guru untuk mencapai tujuan pendidikan. Kurikulum adalah sesuatu yang sangat penting untuk keberhasilan pendidikan, tanpa kurikulum yang tepat akan sangat sulit untuk mencapai tujuan dan target pendidikan yang diinginkan.
B.     Definisi Kurikulum Menurut Para Ahli
1.      Perencanaan teori rasional Profesor Hirst.
Profesor Hirst menyatakan bahwa teori kurikulum tidak kekurangan masalah, dan dibutuhkan waktu untuk melihat perencanaan kurikulum. Langkah pertama dari kurikulum yang rasional adalah harus dimulai dengan pemahaman yang jelas tentang tujuan yang akan dicapai. Sementara untuk mempertahankan tujuan yang ditentukan sebelumnya Hirst mengkritik dan kurang setuju bahwa perilaku perilaku yang diamati tidak sesuai dengan teori kurikulum yang sebenarnya.
Perbedaan pendapatku dengan Hirst adalah bahwa dia mengungkapkan tentang praktik berdasarkan logika murni, tetapi tidak mencoba memahami praktik tersebut secara kompleks. Selanjutnya, kelemahan dalam praktik ini terletak pada rasionalitas yang tidak diimbangi dengan berbagai dengan berbagai analisis tentang rindakan yang rasional atau alasan praktis. Yang dimaksud dengan rasional disini adalah memilih berbagai proses mental yang tidak sesuai dengan paradigma untuk mengambil sebuah kesimpulan. Contohnya, dalam sebuah seminar atau diskusi dengan anak, kegiatan yang mengajarkan untuk menandai hal-hal yang sesuai dengan kebiasaan-kebiasaan, (seperti berpendapat secara logis, menunjukkan apresiasi terhadap siswa atau anak, menghubungkan berbagai keterangan) daripada membuat maksud tertentu untuk mencapai maksud akhir yang diinginkan.
2.      Dua macam kurikulum Michael Young
Menurut Michael Young, masalah pendidikan adalah hasil pengalaman dari orang-orang yang berada dilingkungan  sekolah. Mengenai fakta tersebut tersirat bahwa mempelajari kurikulum secara terus-menerus bertujuan pada pengalaman praktik yang akan dihasilkan, tentunya hal tersebut  menggambarkan situasi dimana banyak teori kurikulum yang sudah berlaku, serta cara-cara yang akan digunakan untuk kepentingan siswa. Michael Young menekankan bahwa didalam kurikulum termuat rencana mengenai isi dan bahan pembelajaran yang dapat dipedomani dalam aktivitas belajar mengajar.
Selanjutnya, Michael Young mengemukakan tentang konsep kurikulum yang berbeda, yaitu Kurikulum sebagai praktik adalah kurikulum sebagai alat untuk mencapai tujuan pendidikan yang harus mampu menghantarkan anak didik menjadi manusia yang terampil, berilmu, dan bermoral, tidak hanya sebagai mata pelajaran yang harus diberikan kepada murid semata-mata, melainkan sebagai aktifitas pendidikan yang direncanakan untuk dialami, diterima, dan dilakukan.
            Pada dasarnya Michael young bersandar pada dua konsep kurikulum yaitu kurikulum adalah fakta (pandangan komoditi)dan kurikulum adalah praktek. Contoh pertama adalah sebuah konsep yang diwujudkan melalui pemahaman pembelajaran yang dimiliki oleh siswa sehingga membentuk suatu pengetahuan yang akan dikuasainya. Contoh kurikulum adalah praktek. menurut penulis praktek adalah semacam penyelidikan interdisipliner terbuka(IDE) yang dibuat oleh Charity James dalam laboratorium Goldsmith. Dalam kurikulum seperti yang dimaksud, tidak ada persoalan yang dipaksakan, semua persoalan didiskusikan antara guru dan siswa yang mencoba menguasai persoalan.
Pada pandangan ini yang melihat kurikulum sebagai sebuah inisial dalam bentuk pengetahuan/ pengalaman guru dalam memilih dan melaksanakan metode mengajar yang sesuai dengan kemampuan anak. Dengan pemilihan metode yang sesuai diharapkan siswa tidaka hanya sekedar mempelajari dan menghafal tetapi menjadi sebuah jalan dalam pemahaman pengetahuan sesuai dengan apa yang telah direncanakan dan menjadi tujuan dari kurikulum.
Penulis menyambut teori praktek kurikulum Michael Young dengan baik.Itu semua sangat mengecewakan. Suatu konsep sempit yang bervariasi.
3.      Klasifikasi dan pembingkaian pengetahuan Bernstein
Pengetahuan pendidikan umum ada tiga sistem didalamnya, yaitu kurikulum (apa yang dianggap sebagai pengetahuan yang valid), pedagogi (cara apa yang dianggap valid untuk mentransmisiskan pengetahuan) dan evaluasi (apa yang dianggap realisasi yang valid dalam pengetahuan). ‘Kode pengetahuan pendidikan’ adalah prinsip-prinsip dari bentuk pengetahuan pendidikan. Oleh karena itu proses membentuk (mengatur atau menentukan) cara dimana masyarakat untuk menerima pengetahuan kurikulum adalah ada prinsip-prinsip sosial yang membentuk kode pendidikan tertentu yang pada gilirannya membentuk pengetahuan pendidikan. Sebagaimana dijelaskan dalam bab sebelumnya, suatu masyarakat dapat menerima dengan cara mengklasifikasi, mentransmisikan, dan mengevaluasi pengetahuan ditentukan (dibentuk atau diatur) oleh prinsip-prinsip sosial dan ini bisa ditunjukkan secara teoritis, yaitu melalui konstruksi teoritis kode pendidikan.
Konstruk teoritis ditetapkan sebagai berikut, yaitu ada dua jenis kurikulum yaitu koleksi dan terintegrasi. Ini berarti bahwa kurikulum sekolah dapat dibagi menjadi satuan waktu. Kurikulum tipe koleksi adalah isi mempunyai hubungan tertutup satu sama lain (masing-masing tiap pelajaran mempunyai hubungan), kurikulum tipe “integrasi” adalah tempat konten dalam hubungan terbuka. Tutup relasi (closed relations) didefinisikan kejelasan batas antara satu unit dan lain (sudah ditentukan secara pasti). Buka relasi (open relations) ditentukan sebaliknya, yaitu tanpa ada batas-batasnya (memiliki kebebasan untuk mengembangkan kurikulum sesuai keinginan dan kemampuannya).
Bernstein memperkenalkan teori istilah yang menggambarkan variasi dalam keterbukaan atau ketertutupan hubungan antara isi yaitu klasifikasi kuat dan lemah. Klasifikasi kuat dari kode koleksi adalah di mana ada kejelasan batas erat antar perbedaan isi. Berdasarkan definisi ini, kita memiliki satu prinsip kode pendidikan, yaitu prinsip yang memberi kita 'struktur dasar dari sistem pesan: kurikulum'. Struktur dasar kurikulum disediakan oleh pembagiannya ke dalam jenis luas terkait dengan definisi ketentuan untuk istilah yang tetap tidak terdefinisi dan tidak dapat dijelaskan.
Untuk membantu menggambarkan kurikulum konten adalah perubahan konseptual lebih lanjut untuk membantu menggambarkan proses penyaluran konten ini. Kerangka adalah bentuk hubungan guru/murid di mana konten kurikulum ditransmisikan. Oleh karena itu, Bernstein menyimpulkan bahwa 'dari perspektif analisis ini, struktur dasar dari sistem pesan 'kurikulum' ditentukan oleh konten (materi), dan dasar sistem pesan 'pedagogi' ditentukan oleh kerangka'.
Sejauh ini hanya ada serangkaian definisi. Definisi ini memiliki dua efek yang penting. Pertama, mereka akhirnya mengarah kembali ke istilah Isolasi dan batasan antar unit (closed dan open relations). Kedua, mereka bergantung pada perangkat logis untuk semua umum (yaitu kurikulum sekolah dasar dan menengah dengan semua variasinya) menjadi dua kategori yang lengkap namun saling terpisah.

MATHEMATICAL EXPLANATION: WHY IT MATTERS


MATHEMATICAL EXPLANATION: WHY IT MATTERS
(From The Book Philosophy of Mathematical Pratice ByPaolo Mancosu)

to complete the task of Science Philosophy Course
Lecturer : Dr. LailaFitriana, M.Pd.



Heri Satriawan





POSTGRADUATE OF MATHEMATICS EDUCATION
FACULTY OF TEACHER TRAINING AND EDUCATION
UNIVERSITAS SEBELAS MARET
SURAKARTA
2018



CHAPTER 1
INTRODUCTION
A.    Background
The last two decades have witnessed a significant increase of attention to mathematical explanation, both in science and mathematics. This has led to novel joint work between philosophers of science and philosophers of mathematics.The philosophical analysis of mathematical explanations concerns itself with two different, although connected, areas of investigation.
The first area addresses the question of whether mathematics can play an explicit role in the natural and social sciences (mathematical explanation of scientific facts). A mathematical explanation of scientific facts based on a clear phenomenon with detailed information and minimizing the significance of error by explanation. The second offering with the problem of whether mathematical explanation occurs in mathematics itself (mathematical explanation of mathematical facts). Mathematical explanation of mathematical facts based on formal evidence and informal evidence. Formal evidence is basically the logical form of the set of premises and axioms. Whereas, informal proof is a form of proof that aims to conclude a new statement based on a previously known statement. The difference in informal proofs is more likely to be in the form of paragraphs than in logical forms. Thus, these entries survey contributions to both fields, it shows their relevance to the history of philosophy, mathematics, and science, it articulates their connections, and points to the expected philosophy by deepening our understanding of the topic. Mathematical applications occur in a natural phenomenon studied science is then described using mathematics or when the problem to be solved by science is answered by using mathematical techniques.
Based on consideration, the evidence of mathematical explanation can be generalized rationally, those that are not so the 'generalization' stipulations which fail to illuminate those areas of mathematics which have already been developed are not rationally acceptable. The process of generalization rationality in mathematics, also need a mathematical explanation. Only significant generalizations can explain rational changes in mathematics. Thus, it should be clear how the need for a mathematical explanatory theory appears in generalization. The explanatory theory requires that the explanatory evidence be generalizable. To uncover these problems, then all will be explained in this paper.
B.     Problem Formulation
Based on the background issues, the problems discussed can be formulated as follows:
1.      Why is the matter of mathematical explanation of scientific facts to mathematical explanations of mathematical facts?
2.      What is the connection of explanation and generalization in mathematics?

C.    The Aim of Research
Based on the problem formulation, the aims of the research are:
1.      To know the matterof mathematical explanation of scientific facts to mathematical explanations of mathematical facts.
2.      To know the connection of explanation and generalization in mathematics.




















CHAPTER 2
FINDINGS AND DISCUSSION

5.1 Mathematical Explanations of Scientific Facts
The mathematical explanations of natural phenomena are widely recognized in the literature. However, until now very little attention has been devoted to them. The mathematical explanations of natural phenomena articulate how the explanation of how mathematics relates to reality, for example the explanation of the application of mathematics to reality. Illustrative example: a bunch of sticks thrown into the air with many turns and falls. When the stick falls freely, the stick is more near the horizontal than near the vertical. Broadly speaking, the facts of geometry can explain the cause.
The mathematical explanation of empirical phenomena can not be used to infer the existence of a mathematical entity, since existence is presupposed in the factual description to be explained. In other words, the existence of a mathematical entity can not be observed by the human senses. Example: "there are two cows in the field" the existence of number two can not be explained in a physical statement, since clear reference to number two can be explained by using the standard of the numeric number. The mathematical explanations of the empirical facts have not been sufficiently studied, as much detailed case study is needed to understand the various uses of explanations that mathematics can play in an empirical context.
Philosophical advantage may come from at least three different directions. First, towards a better understanding of the application of mathematics to the world, that is how mathematics can help in scientific explanations. Second, the study of mathematical explanations of scientific facts will serve as a causal explanation (causation). Thirdly, the arguments and forcing nominalist discussions take a stand on how to explain the clarity of mathematics in empirical science

5.2  From Mathematical Explanations of Scientific Facts to Mathematical Explanations of Mathematical Facts
In an interesting note to her paper Leng says:
Given the form of Baker and Colyvan’s argument, one might wonder whyit is mathematical explanations of physical phenomena that get priority. For ifthere are, as we have suggested, some genuine mathematical explanations [ofmathematical facts] then these explanations must also have true explanans. Thereason that this argument can’t be used is that, in the context of an argument forrealism about mathematics, it is question begging. For we also assume here thatgenuine explanations must have a true explanandum, and when the explanandumis mathematical, its truth will also be in question. (2005, p. 174)
This comment reflects the general use to which indispensability arguments. just as standard indispensability arguments address those who are realists abouttheoretical entities in science, so here the intended audience for the argumentwould consist of those who are realists about a certain realm of mathematicalentities (say, the natural numbers) and in addition are not already committedto a foundational position (such as predicativism) which forbids entertainingthe entities being postulated by the explanationhave been put.
The indispensability argument is not endorsed for mathematics. The original form of the indispensability argument relied on a form of confirmational holism. This left the argument open to the objection, raised forcefully by Maddy, that scientific practice proceeds otherwise or to the objection that other accounts of confirmation block the conclusion (Sober, 1993). In response, advocates like Colyvan and Baker have argued that explanatory considerations lead to platonism even if we drop conformational holism. But, as pointed out, nobody really has an account of mathematical explanations of scientific phenomena.
Quine originally used the indispensability argument to argue that we should believe in sets because they do the best job in tracking all our commitments to abstract objects. For Quine the appeal to empirical science was essential. Maddy’s realism drops the connection to empirical science and tries to obtain the same conclusion just by focusing on pure mathematics. In Chapter 4 of Maddy (1990) we find a lengthy discussion of theoretical virtues, including explanatory ones, that play a role in ‘extrinsic’ justifications for axiom choice in set theory. And althoughMaddy herself gave up the attempt in favor of ‘naturalism’ (see Maddy (1997)), mathematical explanation can still play an important role in this debate.
For those who believe that her realism can be revived perhaps the detour through indispensability arguments that appeal to mathematical explanations might provide a more persuasive type of argument than the other varieties of ‘extrinsic’ justifications mentioned in 1990. Moreover, those who are persuaded by the ‘naturalist’ approach of her latest book will as a matter of fact have to welcome investigations into mathematical explanation as they are part and parcel of the kind of work the methodologist in this area ought to carry out. So both these options call for an account of mathematical explanations of mathematical facts.

5.3  Mathematical explanations of mathematical facts
The history of the philosophy of mathematics shows that a major conceptual role has been played by the opposition between proofs that convince but do not explain and proofs that in addition to providing the required conviction that the result is true also show why it is true. This philosophical opposition between types of proof also influenced mathematical practice and led many of its supporters often to criticize existing mathematical practice for its epistemological inadequacy. Steiner’s model of explanation, to be discussed below, although not relying on the Aristotelian opposition, aims at characterizing the distinction between explanatory and non-explanatory proofs.
Hence what distinguishes an explanatory proof from a non-explanatory one is that only the former involves such a characterizing property. In Steiner’s words: ‘an explanatory proof makes reference to a characterizing property of an entity or structure mentioned in the theorem, such that from the proof it is evident that the result depends on the property’. Furthermore, an explanatory proof is generalizable in the following sense. Varying the relevant feature (and hence a certain characterizing property) in such a proof gives rise to an array of corresponding theorems, which are proved—and explained—by an array of ‘deformations’ of the original proof. Thus Steiner arrives at two criteria for explanatory proofs, i.e. dependence on a characterizing property and generalizability through varying of that property (Steiner,1978a, pp. 144, 147).
According to Steiner, an explanatory proof always makes reference to a characterizing property of an entity or structure mentioned in the theorem. Furthermore, it must be evident that the result depends on the property (if we substitute the entity for another entity in the family which does not have the property, the proof fails to go through) and that by suitably `deforming' the proof while holding the `proof-idea' constant, we can get a proof of a related theorem. Though many of Steiner's concepts (family, deformation, proof-idea) are vague, we can construct examples which beyond any doubt would classify as explanatory proofs by his criterion.Steiner’s model was criticized by Resnik and Kushner (1987) who questioned the absolute distinction between explanatory and non-explanatory proofs and argued that such a distinction can only be context-dependent.

5.4  Kitcher about explanations and generalizations
According to Curnot generalizations are generated because they reveal a common basic principle in thinking derived from certain truths, connections and common origins that had not previously been seen, found in all sciences, and especially in mathematics. Such generalization is the most important generalization, and their invention is a work of genius. Sterile generalization consists of the expansion of nonessential cases populated by inventive people for important cases, leaving the rest on a visible analogy. In such cases, a further step towards abstraction and generalization does not mean improvement in the order of mathematical explanations. What distinguishes both is that the first is the second temporary explanation not. According to Cournot generalizations were able to express the sequence of explanations in accordance with the mathematical truths that are structured.
Mandelbrojt says that generalizations are informative and clear. Such generalizations can be obtained with the proper level of abstraction and should show the learned object in the "natural setting". Eventually both Cournot and Mandelbrojt regarded it as having a mathematical explanation.
Kitcher discusses the problem of generalization in his book The Nature of Mathematical Knowledge from 1984. In his book according to Kitcher One of the most easily visible patterns of mathematical change, so far not explicitly discussed is the extension of the Mathematical Language by generalizations. For example Kitcher mentions Reimann's redefinition of definite integrals, Hamilton's search for hypercomplex numbers, and generalization of citing of limited arithmetic. Kitcher's goal is to try to understand the generalization process that takes place and to see how generalization's search becomes rational. However, not all generalizations are significant. Significant generalization is explanation. They explain to us exactly how, by modifying certain rules which are constitutive of the use of some Language expressions.
Kitcher tries to distinguish between rational and non-acceptable generalizations. The generalization provisions that fail to explain the areas of mathematics that have been developed can not be accepted rationally. In other words, to explain the rationality of the generalization process in mathematics, we need mathematical explanations. Thus it must be clear how the theoretical need for mathematical explanations arises on generalizations. Only significant generalizations can explain rational change in mathematics and that is explanatory. From the above description then the mathematical explanation should be generalizable, with evidence generally applicable.


CHAPTER III
CONCLUSION
The conclution of the explanation about mathematical explanation: why it matters are
1.         The mathematical explanation of scientific facts is an explanation of how mathematics relates to reality and is informed in detail of its physical form, whether in the natural sciences or in the social sciences. Meanwhile, a mathematical explanation of mathematical facts based on explanatory proof and non-explanatory proof. What distinguishes an explanatory proof from a non-explanatory one is that only the former involves such a characterizing property. An explanatory proof always makes reference to a characterizing property of an entity or structure mentioned in the theorem.
2.         Generalization is informative and clear. Such generalizations can be obtained with the proper level of abstraction and should show the learned object in the "natural setting", which in general there is a mathematical explanation with general evidences.