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FROM DELAYED TO REFLEXIVE MODERNIZATION
Maria Nikolakaki
University of Peloponnese
HYPERLINK "mailto:manikolak@yahoo.gr" manikolak(@)yahoo.gr
INTRODUCTION
Modernization as a particular model of societal development refers to a total transformation of a traditional or premodern society into a society that features the types of technology and social organization that characterize the advanced, economically prosperous, and relatively politically stable nations of the western world (Moore, 1963, p. 89). This theory of social change implies that change occurs in evolutionary stages and that underdeveloped countries should use as a model the developed (say western) countries to evolve inexorably to ever higher levels of development and civilization. The more modern a country is, the wealthier and more powerful it is, and its citizens have greater measures of personal freedom and enjoy a higher standard of living. Behind modernization theory were both political and ideological concerns: many of the theorists were from the United States, and they were involved in governmental advisory roles and explicitly committed to the curtailment of socialism and communism in the Third World (Collins, 1991, p. 405).
This theory had an effect on and implications for education since education was regarded as the key to modernization (Kazamias, 1995). One of the basic school subjects to be transformed were mathematics as the basis of science and technology and as the cornerstone for rationality itself (Walkerdine, 1988). Modern Mathematics (MM) was originally introduced in the West in the context of national competitiveness and under the shadow of the Cold War. In fact, it was presented as the peak of modernized mathematics education (Nikolakaki, 2000).
Conversely, various theorists of culture and society describe and comment on the dramatic social transformations that are taking place. They express the concern that the societies we live in are crossing borders. Rethinking modernity has been the immanent challenge of sociological theory during the last three decades. Becks analysis of a new type of (second) modernity is at peril to de-historicize the transition from one epoch to a new one by addressing it in terms of a rupture. Levy (2004, p. 5) proposes to think about the relationship of first and second modernity in terms of a continuum of changes. This notion entails a transformative element and suggests that meaningful political-cultural premises are informed by a significant past as well as by a changing present. On this view, collective modes of identification and the claims that are perceived as legitimate may change over time. However, the respective meanings those claims carry remain linked by a long continuum of changes. We are moving from an industrial to a postindustrial society or information society, from a Fordist to a post-Fordist society, from a modern to a postmodern society, or from a classically modern society to a risk society (Bell, 1973; Beck, 1992; Beck, Bonss, & Lau, 2003; Giddens, 1990; Castells, 1996). The 1980s concept of the risk society with its unanticipated side effects is returning. . Here uncertainty takes centre stage, and with it a transformation of the reflexive, from reflection to reflex, occurs (Levy, 2004). Beck supports that, on this view, development is no longer perceived in terms of a Bruch (rupture) but as Umbruch (transformation). One objective of this essay then is to elaborate on the necessity to historicize developments of modernization, rather than to delineate categories of modernity (Beck, 2000).Beck refers to the times we live in with the notion of reflexive modernity. According to Beck et. al.(2003), there has been a plurality of the boundaries within and between societies, between society and nature, between the self and Other, between life and death. This plurality also changes the inherent nature of boundaries. Boundaries between collectivities cease to be given in our era; they rather become choices of groups or individuals, while the boundaries between traditional collectivities collapse. They do not become boundaries so much but rather a variety of attempts to draw boundaries. Border conflicts become transformed into conflicts over the drawing of boundaries.
The existence of multiple boundaries changes not only the collectivity defined by them but the nature of boundaries themselves. They become not boundaries so much as a variety of attempts to draw boundaries. (Beck, Bonss, & Lau, 2003, p. 19)
According to Beck et.al. (2003, pp. 133), when modernization reaches a certain point, it radicalizes itself. Radical social change has always been part of modernity. What is new is that modernity has begun to modernize its own foundations. This interpretation suggests that modernity has become reflexive. It has become directed at itself. Thus, first modern society emerges as a prerequisite for second modern society. Simple modernization becomes reflexive modernization to the extent that it disenchants and then dissolves its own taken for granted premises.
Reflexivity, Beck argues, has more to do with reflex than with reflective. Reflexes are intermediate, on the self in the sense of a self-organizing system. Beck (1992, pp. 176177) denies that his notion of reflexivity is of cognitive character. He distinguishes reflexivity from reflection. Whereas reflection refers to a subjects more or less conscious thinking, reflexivity refers to unintentional self-dissolution and action directed towards or against one self. Consequently, reflexivity in modernity does not mean conscious reflection on modernity, but it primarily involves a paradoxical auto-dynamic modernization process. In other words, "modernization undercuts modernity," unintended and unseen, and reflection-free, with the force of autonomic modernization. Therefore "reflexive modernization" does not automatically lead to reflection on modernity.
According to Skovsmoses thesis of the formatting power of mathematics (Skovsmose, 1994, 1999), this reflexivity implies also an expression of mathematical activity. Mathematics is the formatting power in society in the sense that modern technological society relies on abstract symbolic mathematical instruments (intellectual technology in the words of Daniel Bell). Moreover, as the risk society can be associated with the further development of mathematically based technologies, mathematics becomes a constituent of reflexive modernization. Thus, critical reflections with respect to mathematics become of particular importance (Skovsmose, 1999, p. 16). Unquestionable assumptions as mathematical definitions and axioms are no longer unquestionable. As it happens with any cognitive basis of modern society, they become a matter of choice (Beck et al., 2003).
According to Elias (1992), the past shapes the present order and the legitimacy of claims, implicitly as one of its conditions, explicitly through the picture, which living generations carry of the past of their country; it (the past) has, like the future, the character and the function of the present. As determinants of behavior the past, present and future operate together. The methodological insight from Eliass figuration approach forbids looking at the political culture of a society from a contemporary perspectiveboth in the sense of focusing merely on the present as well as projecting current sensibilities back into history. In each case, the new, global narrative has to be reconciled with the old, national narratives, and the result is always distinctive.
As Levy (2004) states, The importance of memory and the reflexive nature of how collectivities and individuals comes to terms with their past is a key factor in the process of reflexive modernization. Similarly, Beck (2000) observes, It is interesting to note that vision of this kind not only must precede historically the emergence of analytic effort in any field but also may re-enter the history of every established science each time somebody teaches us to see things in a light of which the source is not to be found in the facts, methods, and results of the preexisting state of science.
B. THE CENTRALIZATION OF THE GREEK EDUCATIONAL SYSTEM
The Greek educational system consists of a primary school of six years. In this type of school a teacher teaches all subjects, and it is followed by junior (gymnasium) and senior high school (lykeum); both of them last for three years. In high school, teachers, who have been educated in specific scientific fields, provide the instruction. The Greek centralized educational system is indeed highly centralized. It is characterized by absolute control of the content of education through the teaching of one and only textbook for each subject of education throughout the whole country. Textbooks are produced and distributed freely by the Greek state. It is common that each grade has at least 14 books per year. The Pedagogical Institute, which is the institution that is responsible for the content of education, curriculum, and textbook writing, is subordinated to the Ministry. The Institute must not produce new content depending on the needs and the rhythm that evolve "from the inside, i.e., the school systems needs, but the needs imposed by the outside," i.e., the political environment. Thus, there are periods of inactivity in its work and periods of intensive work that take place under the pressure of the Ministry. Margaret Archer (1984), describing the pathology of centralized systems of educational administration, points out that because the political center thinks long and slowly before it makes decisions and passes laws, long periods of relative stagnation of educational reforms are typical in centralized systems. Then reforms follow the explosion of complaints of teachers. Members of the governmental elite attempt to blow out the proverbial fire. Now that a sense of urgency exists, politicians hastily pass laws in order to neutralize the reactions and to restore the order. This pattern repeats itself indefinitely.
Due to the high centralization in Greece, reforms of the content of education in general and mathematics education specifically are scarce. In Greece, between 1913 and 1982 essentially the same mathematics curriculum was taught in primary school without major changes. From 1982 to 1985 new books were written based on the set concept as considered in modern mathematics. When modern mathematics (MM) was y first taught in Greece, it was claimed to be an innovation. Modern mathematics was taught in Greek primary education until 2007. Thus the teaching of MM lasted twenty-two years in Greece. It endured for forty years after it was first invented. To be precise, MM continued to be taught in Greece for twenty more years after it was withdrawn in other countries, and different teaching methods were applied around the world. This extended use is a phenomenon that is tied to the institutions and the powers that control educational policy in Greece. In 2007 a new set of people took control of the educational system in Greece. A cross-thematic curriculum was introduced, and corresponding textbooks were made available in order to modernize mathematics education in Greek primary education. As had happened before, this reform was sloganeered with the rhetoric that it was a major change to mathematics education.
To highlight how changes occur in Greek mathematics education and to understand the abnormalities of the centralized educational system, the reforms made from 1964 onwards in Greece are put into historical perspective. . Because of the inertia of the centralized educational system and because Greece is a country of the periphery of the West, educational reform episodes are typically belatedly implemented or delayed. Then spasmodic motions are made in order to fill the gap between the Greek educational system and other industrialized countries and to justify them to the public. This dynamic is symptomatic of reflexive reforms. In each case, the modernizing elements are combined with traditional ones to create a new educational strategy.
Specifically, this article explores the teaching of mathematics in Greek primary education in the context of delayed modernization and reflexive modernization. The method to be used is historical-comparative. First, to trace its ideological principles and functions, there will be an analysis of delayed modernization, i.e., the politics and the content of the modern mathematics reform and the reactions to it in the Greek educational system. . In the second part, core issues of the reflexive mathematics reform and the truth within are explored. This reform has been implemented with new textbooks since 2007. It has been introduced with much fanfare and a sloganeered discourse. . Even so, the centralization of the Greek educational system has always created asphyxiating conditions that have led to reflexivity. Reflexivity does not contribute to a greater certitude and control in social and personal life but actually subverts reason by its effects, thus causing instability in practical routines and established outlooks on the world (Giddens, 1990). The juxtaposition of these reforms reveals the particularities of mathematics education in Greece. Finally, a reflective approach to mathematics education in Greece is suggested instead.
C. REFORM EPISODES OF THE DELAYED MODERN MATHEMATICS MODERNIZATION IN GREEK PRIMARY EDUCATION
In the present section the delayed reform of modern mathematics education in Greek primary schools and the attempts that followed the reform to explain the phenomenon of the delayed modernization of mathematics in education are discussed. Four phases or periods in modern mathematics reform in Greek primary education are easily discernible:
A Phase: 19651966.
Up to 1965 the content of mathematics education was the curriculum of 1913. In the context of the educational reform in 1964 and the liberal modernization project of the educational system of the time, the Pedagogical Institute (PI) was founded (Andreou, 1999, pp. 141160). OECD organized the 3rd International Congress of Mathematics Education in Athens in 1963, and modern mathematics was taught the following year as part of the secondary education curriculum in Greece. The situation was suitable for the modernization of mathematics in primary schools. An experimental teaching of mathematics began in 1965 in the 1st grade of primary schools. This attempt ended with the dictatorship that ruled in Greece between 1967 and 1974. One of the early effects was the closure of the Pedagogical Institute and the firing of its employees. The (Council of Education( was to replace it during the junta period.
In 1969 the (Council of Education( under the junta regime revised the mathematics program, again using the structure and the content of the 1913 curriculum. Set theory in the 6th grade of primary schools constituted an innovation of the program, but its teaching was classified as optional. In 1973, there was a limited revision of the mathematics program of primary schools.
The program of this period is characterized by an extreme focus on providing information without taking into consideration the psychological attributes of students, the conditions of schools, or the teaching ability of teachers. As the junta prohibited criticism, the schoolteachers had to wait for the re-establishment of democracy in 1974 to denounce the program as "overloaded and inapplicable," that is to say that the program should be revised( (Troulis, 1992:23).
P h a s e : ( 1 9 7 4 1 9 7 6 )
I n 1 9 7 4 , a f t e r t h e e n d o f t h e d i c t a t o r s h i p , t h e E d u c a t i o n a l C o u n c i l t h a t h a d r e p l a c e d t h e P e d a g o g i c a l I n s t i t u t e h a d t o d e a l w i t h t h e i m m e d i a t e c h a n g e o f s c h o o l b o o k s a n d t o r e f o r m t h e c u r r i c u l u m t h a t w a s e s t a b l i s h e d d u r i n g t h e j u n t a p e riod. As to mathematics, the experimental teaching in the 1st grade that had begun in 1964 was to be continued. This experimental work was again interrupted, for political reasons again, as soon as the KEME (Center of Educational Studies) was constituted to replace the Educational Council in 1975.
C Phase: Modern Mathematics Is Taught in Primary Schools.
This phase consists of two parts: from 1976 until 1980, the period when the new curriculum was prepared, and from 1982 until 1985, when new curricula and books with the use of MM were written.
A Period (19761982): Curriculum Efforts
In 1976 the Educational Council of the junta period was replaced by KEME (Center of Educational Studies) to distinguish itself as a democratic institution. During this period, KEME was preparing a new curriculum without direction or clear orientation. Thus, because the experts were informed of the withdrawal of the MM reform, they were reluctant to produce the new post-junta mathematics curriculum based on modern mathematics.
B Period (19821985): The Writing of Textbooks
When PASOK (the socialist party) assumed power in 1981, a wide range of reform measures was introduced at all levels of education. The new governments educational policy followed demands for democratization and modernization as expressed in the post-dictatorial era, which until then had not found a suitable foundation. The reform was a response to different kinds of contingencies: social on the one side and economic on the other. One of the basic goals was the change of the content of education. The socialist air that the new government was bringing to the country had as result the production of new curricula and textbooks carrying this new state ideology. In fact, it was particularly evident at the level of primary education where curriculum reform was way overdue.
In 1982, once again, the foundation for curriculum development changed its name and its orientation, so KEME was transformed and renamed Pedagogical Institute (again). Its main objective was the modernization of educational content and methods of teaching. Thus, the members of the PI decided on the teaching of modern mathematics in primary schools, ignoring or overlooking the failure of MM reforms in other countries. In order to justify the teaching of MM by the newly appointed socialist party (PASOK), an argument used was that set theory promoted a concept of collectivity (Salvaras, 1983).
An important project of this era was to reduce the overloaded content of the curriculum of the lyceum, which was transferred to lower classes of the educational system. Since in the secondary school, MM was taught, it was considered appropriate to introduce it in primary schools. To give this new approach a legal foundation, a law was passed, according to which (the curriculum of compulsory education should have internal cohesion and gradual development of content.. To achieve this goal, new books for mathematics education were written by 1985.
D' Phase: Reform in the Gymnasium and an Effort for Co-ordination of Mathematics of Primary School and Secondary School
The fourth reform episode began in 1987 during an ideological change in the mathematics curriculum of the gymnasium. Problem solving together with the utilitarian and anti-modernist view in mathematics (as expressed in M. Klines popular 1990 book) were imported to Greece by a team of secondary and university teachers who wrote new textbooks for the gymnasium in the years 1986 and 1987. Sets and theoretical notions were reduced to a minimum, and geometry was mainly based on the use of ruler and protractor. Rational numbers were introduced with a main emphasis on decimal representation.
Modern mathematics in Greek secondary education dominated from 1964 until 1987. When first used, set theory and MM in the primary schools aimed at connecting primary and secondary mathematics education. In 1987, with the new books for the gymnasium, there was an incompatibility in the teaching of mathematics between primary school and the gymnasium. Thus, it was decided to revise the textbooks of the three last grades of primary school to ensure continuity of structure and content of the mathematics curriculum from primary school to secondary school. No attention was paid to the co-ordination between the three first and the three last grades of primary school since in the first three grades mathematics was taught using the set theory. Once again, primary school received the rung of a pre-secondary school.
D. THE POLITICS OF MODERN MATHEMATICS REFORM IN GREEK PRIMARY EDUCATION
The curriculum of MM implied assumptions about historical progress, scientific reasoning, the development of the child, and democratic assumptions for how the educational system would produce a just society. Internationally, the use of MM was attempted in the context of the theories of modernization that saw education as a factor of economic growth, leading to its instrumentalization in the dominant discourse of economic determinism (Mellin Olsen, 1981, pp. 351367). Modern Mathematics was originally introduced in the West in the context of national competitiveness and under the shadow of the Cold War. It was presented as the peak of modernizing mathematics education. However, as Moon (1986) has argued, modernization implies the antithesis to the old. Is it modernization of what? Is it the curriculum, the teaching methods, or both? One can be teaching new concepts in an old way or old concepts in a new way.
The educational reform of 1985 was attempted in the context of bureaucratic-centralized planning. Top-down change of content of education overlooks the particularities and the uniqueness of teaching. At the same time, the intellectual, the cultural, and the racial particularities of students that make teaching unique are not taken into consideration. Central planning in Greece is supported by the theory of technology of teaching, according to which certain experts, the technologists of teaching, analyze the general aims of education and relate them to the special aims of teaching units. To achieve this goal, the implementation of a certain method and concrete means of teaching, specifically manufactured books and special criteria of evaluation, are necessary. The technology of teaching reflects the adoption of rational management in education, which originated in western modernized countries. The technological perceptions in education "shift the interest from the essential problems of education, as is the quality of school knowledge, social discriminations in education, financial funds for the improvement of educational opportunities, and so on to secondary issues, that concern the teaching methodology, book writing, and other aspects of teaching.
While, however, the official discourse supported modernization with the goal to democratize schooling further, in practice, these intentions evaporated because the intended changes were not accompanied by necessary measures to assist teachers in their work. To improve the teachers standing, their training would have had to change, and the bureaucratic control present at every level of the educational system would have had to be eliminated.
Because of how MM was structured, it ceased to have a relation with the experience of children, numeracy, and mathematical reasoning. Mathematics education in this way was presented as an end in itself and not as means to accomplish certain wider goals, and it became a subject where students regurgitated answers on questions that are placed by others. Thus, however, the teaching of MM began and finished its role and its raison d etre in the classroom, without any implications for the students lives because it failed to provide them with the tools to deal with numbers outside the classroom.
The perception that existed in the curriculum of MM for primary school was platonic, i.e., mathematics constitutes a system of established knowledge that students are called to acquire. According to A. Boufi (1996:45), only if Mathematics is considered exclusively a prefabricated system from definitions, rules and techniques with general applicability, that requires students to acquire first the mathematical knowledge and then its application, this could have a meaning.( Because it is dubious that mathematics is regarded a prefabricated system, other, more suitable teaching approaches were needed.
The imposition of one textbook for each subject to be studied across the whole Greek territory within a discourse for equality is fundamentally a totalitarian approach to the content of education. At the same time, it projects the curriculum and its application as the expression of the textbook authors opinion without leaving room for different or even opposing views for discussion. Thus, the mathematics textbook had a negative impact on a large number of good teachers who, because of the symbolic violence that the implementation of the unique textbook implied, could not follow the developments of mathematics education at large, but were forced to teach the particular textbook. Consequently, it led to the deskilling of teachers, who could not question their role as reformers and agents of change in society and, even if they did, could not do anything about it.
THE REFLEXIVE CROSS-THEMATIC REFORM
By 1997, curriculum reform in Greece was way overdue. Teachers were teaching from the same books that were used when they were students. This status quo led to pressure for reform; at the same time studies about boredom and burnout had started to appear (Koliades, 2000). The new curriculum reform was attempted with the assistance of European funds. Since the European funds in Greece were granted originally for training and not for education, curriculum development by the Pedagogical Institute started with the lyceums, followed by the gymnasiums, and then the primary schools.
The first new curriculum effort was initiated in 1997 and lasted until 2000 when the director of the Pedagogical Institute was to be replaced. This new curriculum was to remain unused after the directors departure, and a new curriculum development effort was to be undertaken from 2001 to 2003, this time with a cross-thematic approach to it. The replacement of the director resulted in change of politics, an aspect that is connected to the centralization of the educational system. Once again, as researchers have supported, most differentiations, curriculum change cannot be interpreted as a result of conscious political choices or application of educational policy with reliable ideological goals" (Toumasis, 1989).
To upgrade teaching in primary schools, the Pedagogical Institute worked on the development of the new Cross Curricular/Thematic Framework (C.C.T.F):
Although in the new curriculum for primary education the traditional school subjects are maintained, a holistic approach to content learning is followed, whereby cross-disciplinary connections and relationships rather than delineations between academic disciplines are promoted. In order that high levels of internal cohesion of school knowledge are achieved, the content in the new curriculum is organized around disciplinary and cross-disciplinary concepts. These help the curriculum to overcome fragmentation of content.
Cross-thematic curriculum and cross-curricularity are reflexive concepts in that they attempt to cross boundaries in school knowledge. Rationalization of teaching leads to seeking re-invented concepts in reflexivity such as team working, cooperation, imagination, and critical thinking. However, these concepts come across as theoretical objectives because only few incidents of these declarations are evident.
Conversely, Ruthven (2009) suggests the understandable valorisation of mathematical aspects over others has had the unfortunate corollary of encouraging an insularity which poorly equips the field to enter the kinds of collaboration necessary to address many issues of educational policy and practice through bringing the specialized perspectives of several fields to bear with them.
The basic cross-thematic curriculum was based on the curriculum developed between 1997 and 2000. One of the aims for teaching mathematics is cited below:
The aim of teaching Mathematics, which can be placed among the general aims of school education, is to facilitate the pupils personal development and provide them with the necessary skills for their smooth social integration. Mathematics can help pupils develop structured and critical thinking abilities and improve their reasoning abilities of analysis, abstraction and generalization that will enable them to express their thoughts in a neat, clear, simple and accurate way. Mathematics also sharpens pupils abilities of observation, self-concentration and persistence, stimulate their initiative, creative imagination and freethinking and develop their sense of order, harmony and beauty. Mathematics is a necessary tool in everyday life, especially at the workplace, and has a significant contribution to the development of other scientific fields, especially Technology, Economics and Social Studies. (Cross Curricular/Thematic Framework; emphasis added)
Therefore, the teaching of mathematics in primary school has among its major aims to integrate students smoothly into society, to assist students to express their thoughts in a neat, clear, simple, and accurate way, to develop their sense of order, harmony, and beauty, and to contribute to the development of students interest in other scientific fields. There is no mention of everyday life, development of citizenship, and challenging social conditions.
The presence of this new curriculum is supported with a discourse about postmodernity: Successful living in post-modern times presupposes that one is fully literate in many areas, such as reading, science, technology and mathematics (Cross Curricular/Thematic Framework). There are notices for multiculturalism, but they are vague at the same time: All these are necessary for individuals in order to become creative and contributing members of multicultural societies in times of dramatic changes (Cross Curricular/Thematic Framework).
The new mathematics curriculum for primary education has a different form to the previous one. One of the basic differences of the new curriculum is that it is based on Content Guiding Principles (problem solving, geometry, measurement, and so on), and then it has the General Goals to be achieved (knowledge, skills, attitudes, and values). In the end it has Indicative Fundamental Concepts (change, system, space, time, similarity, difference). Despite the different form, as noted above, most of the cross-thematic mathematics curriculum includes the previous curriculum, but with an added emphasis placed on cross-thematic notes. In essence, this new curriculum attempts to combine both fragmentation of units and the spiral approach with some elements of cross-disciplinarity.
From 2003 to 2007, since there was a new curriculum but not a new book to teach it, teachers were instructed to use the new curriculum but to use the old book. However, in a book-centered, centralized educational system, teachers have no connection to the curriculum. Thus, in Greece only book authors are involved with the curriculumthe curriculum is just a medium, a guide for authors to write a textbook. Furthermore, since teachers were not trained in numeracy and how to teach it,, it is doubtful that they are able to teach it, even if they had a curriculum conscious professional attitude.
The production of the new books to confirm the new curricula lasted from 2003 until 2007 without evaluating the quality of the books. The Ministry declared that the books that proved untenable through teaching would be withdrawn. In other words, books had to be proven bad to be withdrawn rather than their high quality established to be used! The casualties of bad books are not mentioned, of course. It is striking that in order for new books to be taught in the Greek educational system, more than ten years had to pass without any experimental period.
F. CROSS-THEMATIC MATHEMATICS EDUCATION: MYTH OR REALITY?
The cornerstone of the recent reform has been cross-thematic teaching. To comprehend the changes, a research project investigated the degree of integration the textbooks promoted. Thus, the number of the monothematic and cross-curricular exercises that are given to the students through textbooks were recorded with the goal to investigate cross-thematic teaching of mathematics in primary schools. . The analysis is based initially on quantitative data to calculate the percentage of the cross-thematic approach of the school knowledge, as it is presented in the school textbooks (Matsagouras, 2003).
More specifically, the methodology outlined below served as a guide:
1. Precise recording of all exercises that were contained in the textbooks. The recording concerns all copies of students books and exercise books pertaining to mathematics for all primary school grades. The exercise is considered to be a unit of measurement for the present analysis. Where an exercise has two or more parts, then each part is considered to be a different unit.
Categorization of units o f a n a l y s i s a c c o r d i n g t o t h e f o r m :
% m o n o t h e m a t i c
% c r o s s - c u r r i c u l a r
% d i s c u s s i o n i n c l a s s .
F o r t h e c l a s s i f i c a t i o n o f u n i t s i n t h e c o r r e s p o n d i n g c a t e g o r y , c o n c r e t e i n d i c a t o r s w e r e c r e a t e d c r i t e r i a a i m e d a t t h e m a i n t e n a n c e o f t h e r e s e a r c h w i t h i n t h e frame of the initial planning.
Collection and analysis of the data.
Monothematic is considered all the work in which the knowledge that is presented by the textbook is placed in the cognitive region of a particular discipline, mathematics. It is the work that facilitates the promotion of objectives of a particular discipline within the context and management of basic significance of the particular science. Such work usually focuses on the management of mathematic action, completion of tables, solution of exercises, and so on. Either the special labeling in the book, or the absence of a connection of the exercises with other cognitive objects, is considered to be an indicator of classification. These exercises can be completed either individually or in group work.
D. Cross-curricular activities are based on the treatment of cross-curricularity concepts, exercises that require the answers to be an extension and the management of other cognitive subjects, too. The special labeling present in the book, in combination with the connection of a corresponding exercise with other cognitive objects, is considered to be an indicator of classification. This exercise, too, can be practiced individually or in a team.
E. Discussion in the order. The exercises with the signal we discuss are recorded as a separate category because usually they concern the management of text (thematic) in the context of its understanding, while at the same time they create the adequate conditions of interaction and creation of a dynamic in the class without, however, promoting the collaboration and the feeling of belonging to a team. The special labeling that exists in the book is considered to be an indicator of classification.
Monothematic
Cross-thematic
Discussion in class1st Grade48342162nd Grade 42020553rd Grade 246197374th Grade 6669555th Grade 59556856th Grade 6293757Total3039343305
Table 1. Cross-thematic teaching in mathematics books in Greek primary schools.
The results are revealing: out of 3382 exercises in mathematics books of all six grades of primary school, only 343 (about 10%) are cross-curricular and 3039 exercises are monothematic, making the reform a rhetoric without essence. However, it was advertised as such. The myth and the reality of education in Greece have a distance between them and they do not coincide (Nikolakaki, Moraiti, Dossa, 2011).
The first evidence of reflexivity in the new curriculum is the cross-thematic approach, as a crossing borders process. Reflexivity in education, also, relates to the loss of certainties of the effectiveness of a whole teaching package. In mathematics education, the loss of boundaries appears as an inconsistency between aims and objectives, curriculum, methods, textbooks, and evaluation. By directing the modernization to its content, it loses its position. These inconsistencies or even contradictions are a consequence of reflexivity rather than conscious reflections in the process of educational change.
The reflexive reform was dealt with suspicion and controversies between the teachers. The teachers were protesting in the streets that they were not adequately trained to teach the new books. Teachers felt they were left out in the making of the reform, yet they were called to implement it. As Ruthven observes (2009)): Strong collaboration with teachers is (also) important in research aiming to develop professional practice, because of the person-embodied, tool mediated, and setting embedded practitioner craft that is key to making such practice realizable (p. 226).What the Modern Mathematics reform taught us is that without appropriately trained teachers, no reform is viable. The Greek educational politicians again did not take this requirement into consideration.
G. CONCLUDING REFLECTIONS
Because of the high level of centralization of the educational system in Greece, there is an interplay dynamic between delayed and reflexive modernization. Choices in mathematics education, and more generally, the contents of education are political actions and consequently are selected or rejected depending on the political regime. The first attempt to teach modern mathematics in Greek primary schools started in 1964. However, it was only between 1982 and 1985 that it was possible to have modern mathematics introduced into primary schools. This innovation, however, occurred at a time when most western countries had already withdrawn modern mathematics and had begun to find more effective ways to teach mathematics. Although countries like Greece that are on the periphery of western countries and slow to adopt changes have the benefit of learning from the mistakes of others, it seems that those in charge of mathematics education in Greece ignored the comparative example. According to G. Flouris, the continuous party interventions deprived the Greek education from a long-term planning, that would be conformed to the increasing and complex socio-economic needs of the country, (1992, p. 139). The lack of long-term planning, of course, is not only a Greek phenomenon, but the case of Greece can be characterized as extreme, and the Greek approach is by no means effective.
In 2007 there was a cross-thematic reform of mathematics education in Greek primary schools. The cross-thematic framework rhetorically supports that mathematics is taught by crossing boundaries between academic disciplines. In reality, though, it is an empty shirt, or as a Greek Nobel Prize winning poet would say that it is a vessel without essence. This reflexive reform occurred as a reaction to the delayed modernization of mathematics education.
The cross-curricularity, as has been shown, does not alter the asphyxiating content and teaching method that has been suggested, with proposals made of even what kind of pedagogy and teaching method and approach to use. It is not enough to refer to other subjects that have some connection to the unit taught and that alone does not lead to an integrated curriculum but consists of a cross-disciplinary approach. In Shakespeares words: So quick bright things come to confusion.
Beck (2003) has suggested that reflexivity is more an attempt to draw boundaries than co-existing with boundaries and crossing them. Moreover, the past co-exists with the present in that the pasts effects continue to be felt. Reflexive knowing needs to be compared to reflective knowing, which, according to Skovsmose is knowing about the complexity of the social implications of technological problems (1994, pp. 97124). While reflexive knowing refers only to the internal language of mathematics (by experiencing it as a situated concrete representation), reflection, and reflective knowing of mathematics refer both to internal and to external meaning of the language of mathematics and to the complex relations of this language with modern society These relations are mutual and dialectic: mathematics not only structures society, but it also indirectly reflects social developments by restructuring its own theories and applications. Thus reflective knowing of mathematics extends to mathematics itself (as a field of social activity) and does not confine to external social implications of mathematics. To put it in more concrete terms, mathematical activity is not a self-contained entity or a black box outside social processes, but mathematics plays an integral part in society.
More specifically, learning in the context of uncertainty, complexity, instability, uniqueness, and value conflict, presupposes another in-built learning mechanism. It presupposes a second learning level, which involves, next to a simple comparison of goals and means a questioning of the goals themselves. The question is raised whether the goals are appropriate in a given situation. If not, the action is given a completely new direction. Furthermore, there should be a third learning level that should refer to processes of learning social responsibility. Issues of social responsibility are articulated in the theories emphasizing critical reflection, which is a core issue of critical mathematics education (Nikolakaki, 2010).
The question is whether schooling in the context of Greek society is prepared to undertake the changes necessary to foster such critical analysis. Are those who prepare the mathematics education curriculum not only to cut across disciplinary lines to produce new content but also to seek out new practices in dealing with this content? Such shifts should be oriented to a reflective approach regarding curriculum and structure, textbook content, and teaching practices. In addition, as Ruthven (2009) points out: The last lesson then is that success in developing research-informed practice is likely to depend both on dialogical development of scholarly and craft knowledge at the stage of research, and on productive interplay between reflective and effective approaches at the stage of wider dissemination.
The present paper has critiqued the reforms in teaching mathematics in primary schools in Greece. However, beyond the reforms themselves the basic problem of the Greek educational system is centralization. Centralization in the Greek state is authoritarian and hegemonic. Furthermore, centralization of the educational system leads to experts deciding and re-deciding again and again, with no real results, since the key actors of education are not taken under consideration. Mathematics education is another form of mis-education in this sense (Nikolakaki, 2011). It is as if they want to kill the childrens soul through boredom. At the same time it shows that they do not trust teachers to decide how to teach. Teachers are dealt with suspicion and mistrust. However, this approach is no way to treat education. If education is to flourish, it must be based on respect and freedomand reflection.
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Special appreciation is owed to Tasos Patronis for the elaboration of ideas in this article.
Students in the first grade have 14 textbooks.
Actually, rather than being part of the West, Greece is between the East and the West.
In the early 1960s the Greek publisher Pataki released a book that contained the curriculum of 1913.
The effect of OECD is evident in this reform. In the book for A class of secondary school with the title Experimental Teaching of Mathematics in the A Class of Secondary Schools, vol. 2, which was written with the collaboration of H. Fehr, it says: "It is an experiment, whic X Y i j ®|j|j|j\NC8 hU h} mH sH hU hX3 mH sH hU h} 6]mH sH hU hX3 5\mH sH #h y hX3 <OJ QJ ^J mH sH ,j h y hX3 <OJ QJ U^J mH sH 4j h y h} 0J OJ QJ U\^J aJ mH sH 'h y h} OJ QJ \^J aJ mH sH .h y hX3 5CJ OJ QJ \^J aJ mH sH hU h} 5\mH sH .h y h} 5CJ OJ QJ \^J aJ mH sH 0 X Y j $ *$ ]^ gdX3 $a$gdX3 $d $
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O P s t z { B C , L, , , , , , , , , , , , A- B- R- S- - - - - ĸĸĸĸĸĩĚ~v hU mH sH h} CJ ]aJ mH sH hU 6CJ ]aJ mH sH h} 6CJ ]aJ mH sH Uh} 6CJ aJ mH sH hU CJ aJ mH sH h} CJ aJ mH sH j h} CJ H*UaJ hU mH sH h} mH sH j h} 0J Uh} mH sH . h aims to the achievement of strengthened modernized mathematics education, which is judged essential for the aims, particularly the technical and economic progress of all member countries of OECD" (emphasis added).
Lyceum is the senior high school.
Gymnasium is the junior high school and follows the three years of primary school.
The books were the same, but they were presented now without the extreme set theory.
Small-scope revisions of the mathematical textbooks of primary school take place in the years 1984, 1985, and 1988.
Teachers are obliged to teach the unique textbook, willing or not.
Teachers are deskilled because they do not construct the course of their teaching, but the textbook does.
The curriculum was prepared between 1997 and 2001.
Shakespeare, W. A Midsummer Nights Dream, I, iv, p. 149.
This is the case, for example, with set theory, which had to be restructured and reformulated several times from the beginning of the 20th century (first formally by Zermel and Faenkel) until the 1970s, when it was reformulated by Nelson in order to include post-formal developments in nonstandard analysis.
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