International Journal of Human Rights Education (IJHRE)

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1. EDUCATION AND LYCEUM
   Author Mircea Orasanu
   ABSTRACT

   After finishing with the book on Differentiation of Integrals, Miguel undertook a more complex project as he collected in a single volume the techniques most commonly used at that time in harmonic analysis
   and considerations pf prof dr mircea orasanu and prof horia orasanu that have suffered repression more in romania and sinceThe book describes not only the classical techniques, but also shows in subsequent chapters how one can apply them to several topics. A chapter on differentiation of integrals along curves (a new topic not covered in his previous book) and the disproof by Charles Fefferman of the disk multiplier conje
   cture are two examples of this.
   1 INTRODUCTION
   here we remember thatIt was in the mid 1980’s when he realised that his goal of bringing Spanish mathematics to the international forefront had been a success and he decided to move forward to yet another project. As several of his Ph.D. students and his “students abroad” had started to publish papers on their own in international journals, Miguel felt that he had another important task to accomplish, that of modernising the teaching of mathematics not only at the university level but at other levels of the educational system in Spain. In collaboration with José Colera and Adela Salvador, Miguel wrote several textbooks for middle and high school students. These books represent a landmark in the way mathematics is taught today at this level. The original motivations of the each of the subjects, the approach to deducing results from a variety of examples, and the final notes of each chapter showing the many applications that mathematics provide, have been imitated by many high school textbooks to follow. and these are not recognized in romania where prof dr mircea orasanu suffered incredible repressionWhat role is mathematics given in the general Core Curriculum?
   One should know that the general curriculum for LK06, also called the Core Curriculum, has been accepted and used, almost without any changes, by shifting governments over a broad political spectre for the last 13 years (TRMERCA, 1999). It describes what may be called Bildung-like elements, as six different ‘human beings’, characterised by the following adjectives ‘spiritual’, ‘creative’, ‘working’, ‘liberally-educated’, ‘social’ and ‘environmentally aware’ (… human beings). These aspects are supposed to be brought together in the integrated human being (which some interpreters see as a seventh human being).
   However, this chapter in LK06 explicitly warns that an integration is inevitably dilemmatic in its nature. 17 concrete oppositions are described to illustrate this challenge. Furthermore, there is nothing in the 17 dual aims formulated in the chapter The integrated human being that directly makes room for the overall aims in mathematics in LK06. The 17 dilemmatic ‘aims’ are supposed to problematise and raise awareness of how aims within and between different school subjects can actually be in conflict when one attempts to integrate them.
   While one can easily find many roles and important connections for all the other school subjects in this overall and (over-?)ambitious educational enterprise, mathematics has a very weak position in this (con-)text, if any. One of the few places it actually is mentioned is under ‘the creative human being’. Here, three main traditions are mentioned (practical, theoretical and cultural): Learners meet the theoretical tradition in subjects where new knowledge is won through theoretical development, tested by logic and facts, experience, evidence and research. It is presented in the study of languages, mathematics, social and natural sciences (RMERCA, 1999:29). The text is accompanied by an illustration of a page from the writings of Pythagoras in (old) Greek.
   If this very loose (or almost non-existent?) connection between mathematics and the general curriculum reflects a general situation, mathematics education seems to have severe problems developing convincing didactic arguments for in which sense and to which degree it actually intend to contribute to ‘Bildung’ or to ‘the integrated human being’. It is not necessarily so that it does not, but the didactic responsibility of making this explicitly aware for students and other readers seems not to be adequately met.
   One can therefore even speculate whether the seemingly strong position mathematics has had, both on a rhetorical level in society and in the mind of students and parents, may actually be counter-productive to the idea of school subjects making students ‘spiritual’, ‘creative’, ‘working’, ‘liberally-educated’, ‘social’, ‘environmentally aware’ and finally ‘integrated’.
   The inspection of the mathematics section in the 2006 plan revealed that integration of communicational and disciplinary dimensions is almost no-existent. This pattern seems to be in line with a general impression of mathematics as more ‘isolated’ relative to curricular ambitions to create Bildung (Ongstad, 2006b). Hence there might be good reasons for doubting the will in the field of mathematics education to see mathematics as a means and rather as an independent discipline.
   Language and communication in the former curriculum, L97?
   If we compare the curriculum for mathematics in L97 with the one in LK06, the main pattern is similarity, especially focusing on the format and the key curricular categories. However, if we look at the introduction in L97, where the subject and the educational aims are described, many formulations reflect a deeper and a more explicit understanding of the interface between mathematics and the lifeworld than we found in the 2006 plan: Mathematics has many modes of expression and is undergoing constant development. It is a science, an art, a craft, a language and a tool (RMERCA, 1999:165). And: The syllabus seeks to create close links between school mathematics and mathematics in the outside world. Day-to-day experience, play and experiment help to build up its concepts and terminology (RMERCA, 1999:165). Further: Mathematical insight and skills are needed in order to understand and utilise new technology, and it is also a key to communication in modern societies (RMERCA, 1999:165). Finally: Familiarity with its language and symbols, and a clear grasp of its concepts, are important prerequisites for progress in mathematics (RMERCA, 1999:166).
   It is probably not fair to make a direct comparison between the two curricula, since the groups that wrote the 2006 plan were not ‘allowed’ to use more than one page on the general description of the subject. Besides, as mentioned, a (traditional) chapter on ‘approaches’ was left out (seemingly to allow for more local freedom). Nevertheless, the L97 had a conscious will to establish a constructivist view of the relationship between important aspects of language and mathematics that is more or less left behind in LK06: Pupils have already developed some mathematical concepts when they start school although they may have difficulties expressing them in words (RMERCA, 1999:166). Also: learners construct their own mathematical concepts. In that connection it is important to emphasise discussion and reflection (RMERCA, 1999:167). It should be added, though, that the language conception we find here seems mainly connected to a traditional conceptual or basically a semantic view of which aspects of language are most relevant for the teaching and learning of mathematics.
   Disciplinarity and discursivity as separated and the LAC approach
   Formally, the descriptions of language and communication under Basic skills in LK06 are strong and explicit. But in general and in practice the separation is more or less total. Readers/teachers are left with few, if any, ideas about what a reasonable, valid and didactic relevant connection should or could be. This ‘lesson’ is very important for further CoE work on LAC for several reasons. Firstly, because there might even exist an ‘alien’ attitude to the idea of seeing mathematics as language in some circles, refuting the integration as a waste of energy. This position is often accompanied by a disciplinary pride in the many victories of mathematics, this universal language of real sciences that avoids the Babel of language.
   Secondly, it will in any case be a challenge to persuade (communicatively) teachers and curriculum designers in mathematics who are positive to the importance of the connection, that it would be possible to find a sensible ‘balance’ of mathematics and language. Thirdly, it is, of course, not given which conceptual framework, which set of notions and perspectives, can be seen as adequate, given the many (relevant) disciplinary and cultural contexts for such a framework. Finally, the quite different disciplinary ‘profiles’ of teachers (Ongstad, 2006b) in different countries may influence, in different ways, to what extent an LAC approach can be introduced in primary and secondary mathematics education in Europe.
   Conclusion?
   An overall conclusion, then, can be that some agents/actors seem to want an interwoven curriculum, while others prefer to keep the dimensions/aspects separate. Some may not really want to interfere with ‘language’ other than as practical communication. If there is any significant ‘developmental line’ from 1997 to 2006 on this matter, a conceptual orientation seems lost. But a formal ‘communicative’ approach seems to have won a pyrrhic (formal) victory, and just in isolated parts of the curriculum. However, if future evaluations will focus on the basic skills rather than the detailed mathematical skills, the communication-related ambitions might be of practical relevance.

   References
   Ongstad, S. (2006a) Mathematics and Mathematics Education – Language and/or Communication? Triadic Semiotics Exemplified. Educational Studies in Mathematics, 61/1-2.
   Ongstad, S. (2006b) Teacher of mathematics or Teacher Educator? Positionings and problematisations. In
   Hudson, B. and J. Fragner (Eds.) Researching the Teaching and Learning of Mathematics II . Linz: Trauner.
   TRMERCA (1999) The Curriculum for the 10-year Compulsory School in Norway. Oslo: The Royal Ministry of Education, Research and Church Affairs (L97).
   Utdanningsdirektoratet (2007) Mathematics Subject Curriculum. (LK06). Visited 31.05.07.
   http://www.udir.no/…/Fastsatte_lareplaner_for_Kunnskapsloeftet/english/Mathematics_subject_curriculum.rtf

   Culture, language and mathematics education:
   aspects of languages in English, French and German mathematics education

   Birgit Pepin, University of Manchester, United Kingdom

   Abstract
   This paper analyses the ways in which the particular culture of the mathematics classroom and the culture at large influence language in mathematics education in three countries: England; France; and Germany. Drawing on two recent comparative studies of mathematics education in the three countries, the findings of the research demonstrate that ideas, beliefs and principles, and every system’s cultural and philosophical traditions, penetrate the educational enterprise in terms of language forms. It is argued that language in mathematics education needs to be understood in terms of the larger cultural context, and that an understanding of the wider meaning of language forms in a particular context can enhance communication between students and teachers, as well as between those involved in mathematics education across countries.
   Introduction
   Although language and communication factors have long been recognised to significantly influence mathematics teaching and learning, the question of how thought and language are related, and how this varies in different cultural systems, is currently been re-examined. Lean, for example, studied indigenous counting systems of Oceania, Polynesia and Melanesia. He found out that every distinct language has an associated unique counting system. Furthermore, he asserts counting systems are an integral part of language and that language is inextricably bound to culture (Lean, 1992 and 1995). Harris (1987) investigated measurement concepts in Aboriginal communities in Australia. She supports the view that aspects of indigenous mathematics, language and culture are inseparable. Other scholars, such as Zevenbergen (1995) and Stephen Harris (1990), also emphasise the importance of locating education theory and practice within cultural framework, and they distinguish between ‘European’ and ‘Aboriginal’ mathematics and culture. Harris (1990) claims that
   “the nature and degree of the difference between Aboriginal and European culture is so great that the only honest conclusion we can arrive at is that they are largely incompatible. … The degree of difference is so great that it is harder to find what they have in common in cultural terms than it is to see the differences.”
   (Harris 1990, p.9)
   Thus, whilst comparisons have been made between Aboriginal and European cultures, both are in themselves viewed to be homogeneous. What are the characteristics of a so-called ‘European’ culture? No distinction is made between the different European countries’ cultural and educational traditions. This paper helps to fill that gap. Differences of language in mathematics education are identified in the English, French and German context. Furthermore, whilst it is re-emphasised that culture influences mathematics teaching and learning, and in particular mathematics language and communication, it is claimed that there are differences between English, French and German cultural traditions which help to understand the ways in which mathematics is communicated in classrooms.
   Culture, language and mathematics education
   Mathematicians and mathematics educators communicate using language; they use it to teach mathematics, to share their understandings and to clarify and test understanding. Von Glasersfeld points out that communication is not as straightforward as one might assume.
   “Educators have spent and are rightly spending much time and effort on curriculum. That is, they do their best to work out what to teach and the sequence in which it should be taught. The underlying process of linguistic communication, however, the process on which their teaching relies, is usually simply taken for granted. There has been a naïve confidence in language and its efficacy. Although it does not take a good teacher very long to discover that saying things is not enough to ‘get them across’, there is little if any theoretical insight into why linguistic communication does not do all it is supposed to do.”
   (von Glasersfeld 1983, p.43)
   Bishop (1992) challenges the naïve view of the curriculum as an instrument for instruction regardless the cultural perspective.
   “The cultural perspective requires us to culturalise the curriculum at each of the levels, and demonstrates that no aspect of mathematics teaching can be culturally neutral. The cultural ‘messages’ in the educational enterprise are created and manifested by people. People create the national and local curriculum statements, people write the books and computer programs, people bring their cultural histories into the classroom, and people interpret and reconstruct the various messages.”
   (Bishop 1992, p.185)
   This confirms that people, with their beliefs, principles and practices that are underpinned by the ‘culture’ of the individual system or country, are central to the educational enterprise. Mathematical knowledge is constructed, interpreted and mediated by people, and language is part of communicating mathematical knowledge.
   In mathematics cl

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