The School of European Education, Heraklion

Back in September I wrote a post for www.philhellenes.org, a website which celebrates Greece and Greek culture, and which is run by my partner Keith Frankish. The post is about the School of European Education, Heraklion, where I work and where my children study.

At the time I wrote the post, the school was going through a difficult phase, owing to administrative problems outside the control of the school’s management. I am happy to report that this difficult phase has now passed. The school is now functioning normally, and the students have largely made up for the time lost at the beginning of term.

However, I am linking to the post here, both because I stand by all the positive things I say in it about the school, and because I want to remind everyone how precious this institution is, and how important it is that it survives and flourishes.

Here is my post on the school.

18 comments to The School of European Education, Heraklion

  • merniu

    here we have as say prof dr mircea orasanu and prof horia orasanu

  • merniu

    and also we consider as say prof dr mircea orasanu and prof horia orasanu as

    The first Hans Freudenthal Medal of the International Commission on Mathematical Instruction (ICMI) is awarded to Professor Celia Hoyles. This distinction recognises the outstanding contribution that Celia Hoyles has made to research in the domain of technology and mathematics education, both in terms of theoretical advances and through the development and piloting of national and international projects in this field, aimed at improving through technology the mathematics education of the general population, from young children to adults in the workplace.

    Celia Hoyles studied mathematics at the University of Manchester, winning the Dalton prize for the best first-class degree in Mathematics. She began her career as a secondary teacher, and then became a lecturer at the Polytechnic of North London. She entered the field of mathematics education research, earning a Masters and Doctorate, and became Professor of Mathematics Education at the Institute of Education, University of London in 1984

    Her early research in the area of technology and mathematics education, like that of many researchers, began by exploring the potential offered by Logo, and she soon became an international leader in this area. Two books published in 1986 and 1992 (edited) attested to the productivity of her research with Logo. This was followed, in 1996, by the publication of Windows on Mathematical Meanings: Learning Cultures and Computers, co-authored with Richard Noss, which inspired major theoretical advances in the field, such as the notions of webbing and situated abstraction, ideas that are well known to researchers irrespective of the specific technologies they are studying.

    From the mid nineties, her research on technology integrated the new possibilities offered by information and communication technologies as well as the new relationships children develop with technology. She has recently co-directed successively two projects funded by the European Union: the Playground project in which children from different countries designed, built and shared their own video games, and the current WebLabs project, which aims at designing and evaluating virtual laboratories where children in different countries build and explore mathematical and scientific ideas collaboratively at a distance. As an international leader in the area of technology and mathematics education, she was recently appointed by the ICMI Executive Committee as co-chair of a new ICMI Study on this theme.

    However, Celia Hoyles’ contribution to research in mathematics education is considerably broader than this focus on technology. Since the mid nineties, she has been involved in two further major areas of research. The first, a series of studies on children’s understanding of proof, has pioneered some novel methodological strategies linking quantitative and qualitative approaches that include longitudinal analyses of development. The second area has involved researching the mathematics used at work and she now co-directs a new project, Techno-Mathematical Literacies in the Workplace, which aims to develop this research by implementing and evaluating some theoretically-designed workplace training using a range of new media.

    In recent years Celia Hoyles has become increasingly involved in working alongside mathematicians and teachers in policy-making. She was elected Chair of the Joint Mathematical Council of the U.K. in October 1999 and she is a member of the Advisory Committee on Mathematics Education (ACME) that speaks for the whole of the mathematics community to the Government on policy matters related to mathematics, from primary to higher education. In 2002, she played a major role in ACME’s first report to the Government on the Continuing Professional Development of Teachers of Mathematics, and contributed to the comprehensive review of 14-19 mathematics in the UK. In recognition of her contributions, Celia has recently been awarded the Order of the British Empire for “Services to Mathematics Education”.

    Celia Hoyles belongs to that special breed of mathematics educators who, even while engaging with theoretical questions, do not lose sight of practice; and reciprocally, while engaged in advancing practice, do not forget the lessons they have learned from theory and from empirical research. Celia Hoyles’ commitment to the improvement of mathematics education, in her country and beyond, can be felt in every detail of her multi-faceted, diverse professional activity. Her enthusiasm and vision are universally admired by those who have been in direct contact with her. It is thanks to people like Celia Hoyles, with a clear sense of mission and the ability to build bridges between research and practice while contributing to both, that the community of mathematics education has acquired, over the years, a better-defined identity.

    (Document for a press release issued on April 4, 2004)

    The Fifteenth ICMI Study:
    The Professional Education and Development of
    Teachers of Mathematics

    Discussion Document

    Contents
    1. Introduction
    2. Why Conduct a Study on the Professional Education of Mathematics Teachers?
    3. Scope and Focus of the Study
    4.1. Strand I: Teacher Preparation Programs and the Early Years of Teaching
    4.2. Strand II: Professional Learning for and in Practice
    5. Design of the Study
    6. Contributions to the Study
    7. Study timeline
    8. International Programme Committee and Contacts

    1. Introduction
    This document announces a new Study to be conducted by the International Commission on Mathematical Instruction (ICMI). The focus of this Study, the fifteenth to be led by ICMI, will be the professional education and development of mathematics teachers around the world. The premise of this Study is that the education and continued development of teachers is key to students’ opportunities to learn mathematics. What teachers of mathematics know, care about, and do is a product of their experiences and socialization both prior to and after entering teaching, together with the impact of their professional education. This impact is variously significant: In some systems, the effects of professional education appear to be weak or even negligible, whereas other systems are structured to support effective ongoing professional education and instructional improvement. The curriculum of mathematics teacher preparation varies around the world, both because of different cultures and educational environments, and because assumptions about teachers’ learning vary. Countries differ also in the educational, social, economic, geographic, and political problems they face, as well as in the resources available to solve these problems. A study focused on mathematics teacher education practice and policy around the world can provide insights useful to examining and strengthening all systems.

    We recognize that all countries face challenges in preparing and maintaining a high-quality teaching force of professionals who can teach mathematics effectively, and who can help prepare young people for successful adult lives and for participation in the development and progress of society. Systems of teacher education, both initial and continuing, are built on features that are embedded in culture and the organization and nature of schooling. More cross-cultural exchange of knowledge and information about the professional development of teachers of mathematics would be beneficial. Learning about practices and programs around the world can provide important resources for research, theory, practice, and policy in teacher education, locally and globally. Study 15, The Professional Education and Development of Teachers of Mathematics, is designed to offer an opportunity to develop a cross-cultural conversation about mathematics teacher education in mathematics around the world.

    Because the professional education of teachers of mathematics involves multiple communities and forms of expertise, the Study also explicitly welcomes contributions from individuals from a variety of backgrounds. Mathematicians and school practitioners are particularly encouraged to submit proposals for contributions.

    The Study will proceed in three phases: (a) the dissemination of a Discussion Document announcing the Study and inviting contributions; (b) a Study Conference, to be held in Brazil, 15-21 May 2005; and (c) publication of the Study Volume — a Report of the Study’s achievements, products and results.

    First is this Discussion Document, defining the focus of the Study and inviting proposals for participation in a Study Conference. We welcome individual as well as group proposals; focusing on work within a single program or setting, as well as comparative inquiries across programs and settings. In order to make grounded investigations of practice in different countries possible, we invite proposals in three formats: papers, demonstrations, and interactive work-sessions. Details are provided below.

    Second, a Study Conference will be held in Brazil in May 2005, bringing together researchers and practitioners from around the world. The Conference will be deliberately designed for active inquiry into professional development of teachers of mathematics in different countries and settings. Some sessions will offer paper presentations; other sessions will engage participants in direct encounters with particular practices, materials and methods, or curricula.

    Third, a Study Report — the Study Volume — will be produced, representing and reporting selected activities and results of the Study Conference and its products. This Report will be useful to the mathematics education community, as well as for other researchers, practitioners, and policymakers concerned with the professional education of teachers.

    2. Why Conduct a Study on the Professional Education of Mathematics Teachers?
    Three main reasons underlie the decision to launch an ICMI Study focused on teacher education. One reason rests with the central role of teachers in students’ learning of mathematics, nonetheless too often overlooked or taken for granted. Concerns about students’ learning compel attention to teachers, and to what the work of teaching demands, and what teachers know and can do. A second reason is that no effort to improve students’ opportunities to learn mathematics can succeed without parallel attention to their teachers’ opportunities for learning. The professional formation of teachers is a crucial element in the effort to build an effective system of mathematics education. Third, teacher education is a vast enterprise, and although research on mathematics teacher education is relatively new, it is also rapidly expanding.

    The timing is right for this Study. The past decade has seen substantial increase in scholarship on mathematics teacher education and development. A growing number of international and national conferences focus on theoretical and practical problems of teacher education. Publication of peer-reviewed articles, book chapters, and books about the development of teachers of mathematics is on the rise. Centers for research and development in teacher education exist increasingly in many settings. A Survey Team led by Jill Adler will report on the development of research on mathematics teacher education as part of the program at the tenth International Congress on Mathematics Education (ICME-10) in July 2004 in Copenhagen. In addition, it is significant that the past decade has also included the launching of a new international journal (in 1996): the Journal of Mathematics Teacher Education (JMTE) is published by Kluwer, and edited by an international team of scholars. Seven volumes later, JMTE hosts a thriving international discourse about research and practice in teacher education.

    Mathematics teacher education is a developing field, with important contributions to make to practice, policy, theory, and research and design in other fields. Theories of mathematics teachers’ learning are still emerging, with much yet to know about the knowledge, skills, personal qualities and sensibilities that teaching mathematics entails, and about how such professional resources are acquired. The outcomes of teacher education are mathematics teachers’ practice, and the effectiveness of that practice in the contexts in which teachers work. Yet we have much to learn about how to track teachers’ knowledge into their practice, where knowledge is used to help students learn. And we have more to understand about how teacher education can be an effective intervention in the complex process of learning to teach mathematics, which is all too often most influenced by teachers’ prior experiences as learners, or by the contexts of their professional work.

    Study 15 aims to assemble from around the world important new work –– development, research, theory, and practice –– concerning the professional development of teachers of mathematics. Our goal is to examine what is known in a set of critical areas, and what significant questions and problems warrant collective attention. Toward that end, the Study aims also to contribute to the strengthening the international community of researchers and practitioners of mathematics teacher education whose collective efforts can help to address problems and develop useful theory.

    3. Scope and Focus of the Study
    This Study focuses on the initial and continuing education of teachers of mathematics. Our focus is the development of teachers at all levels, from those who teach in early schooling to those who teach at the secondary school. (In this Discussion Document, we use “primary” to refer to teachers of students of ages 5 – 11; “middle” to refer to ages 11 – 14, and “secondary” for ages 14 and older.) Teacher development is a vast topic; this Study focuses strategically on a small set of core issues relevant to understanding and strengthening teacher education around the world.

    The Study is organized in two main strands, each representing a critical cluster of challenges for teacher education and development. In one strand, Teacher Preparation and the Early Years of Teaching, we will investigate how teachers in different countries are recruited and prepared, with a particular focus on how their preparation to teach mathematics is combined with other aspects of professional or general academic education. In this strand, we will also invite contributions that offer insight into the early phase of teachers’ practice. In the second strand, Professional Learning for and in Practice, we will focus on how the gap between theory and practice is addressed in different countries and programs at all phases of teachers’ development. In this strand, we will study alternative approaches for bridging this endemic divide, and for supporting teachers’ learning in and from practice. This strand may be explored at any of the developmental stages –– preservice, early years, and continuing practice –– of teachers’ practice. In both strands, we seek additionally to learn how teachers in different countries learn the mathematics they need for their work as teachers, and how challenges of teaching in a multicultural society are addressed within the professional learning opportunities of teachers.

    Table 1 provides a graphic representation of the scope and focus of the Study. The table makes plain that for Strand 1, the focus will be on the preservice and early years of teaching only; the Study will not focus on issues of recruitment, program structure and curriculum for experienced teachers. However, Strand II, focused on professional learning in and from practice, may be studied at all phases of teachers’ development.

    Phases of teacher development
    Initial teacher education
    (preservice and early years of teaching)
    Continuing practice
    Strands
    Programs of teacher education (recruitment, structure, curriculum, first years)

    yes

    no

    Professional learning for and in practice

    yes
    yes

    Table 1: Scope and focus of the Study

    4.1 –– Strand I: Teacher Preparation Programs and the Early Years of Teaching
    This strand of the Study will examine a small set of important questions about the initial preparation and support of teachers in countries around the world, at the preservice stage, and into the early years of teaching. How those phases are structured and experienced varies across countries, as does the effectiveness of those varying structures. Questions central to the investigation of initial teacher preparation and beginning teaching will include:

    a) Structure of teacher preparation: How is the preparation of teachers organized — into what kinds of institutions, over what period of time, and with what connections with other university or collegiate study? Who teaches teachers, and what qualifies them to do so? How long is teacher preparation, and how is it distributed between formal study and field or apprenticeship experience? How is the preparation of teachers for secondary schooling distinguished from that of teachers for the primary and middle levels of schooling?

    b) Recruitment and retention: Who enters teaching, and what are the incentives or disincentives to choose teaching as a career in particular settings? What proportion of those who prepare to teach actually end up teaching, and for how long? How do teachers’ salaries and benefits relate to those of other occupations?

    c) Curriculum of teacher preparation: The Study seeks to probe a small set of key challenges of teacher preparation curriculum and investigate whether and how different systems experience, recognize, and address these issues. Two such issues are:
    • What is the nature of the diversity that is most pressing within a particular context –– for example, linguistic, cultural, socio-economic, religious, racial –– and how are teachers prepared to teach the diversity of students whom they will face in their classes?
    • How are teachers prepared to know mathematics for teaching? What are the special problems of subject matter preparation in different settings, and how are they addressed? Is interdisciplinarity in teacher education commonplace, and if so, how is managed? How do faculty in education interact with faculty in mathematics over issues of teacher education?
    In addition, we invite proposals that identify and examine other specific central challenges for the curriculum of teacher preparation.

    d) The early years of teaching: What are the conditions for beginning teachers of mathematics in particular settings? What supports exist, for what aspects of the early years of teaching, and how effective are they? What are the special problems faced by beginning teachers, and how are these experienced, mediated, or solved? What is the retention rate of beginning teachers, and what factors seem to affect whether or not beginning teachers remain in teaching? What systems of evaluation of beginning teachers are used, and what are their effects?

    e) Most pressing problems of preparing teachers: Across the initial preparation and early years, what are special problems of teaching mathematics within a particular context and how are beginning teachers prepared to deal with these problems?

    f) History and change in teacher preparation: How has mathematics teacher preparation evolved in particular countries? What was its earliest inception, and how and why did it change? What led to the current structure and features, and how does its history shape the contemporary context and structure of teacher education?

    Proposals for this Strand may offer descriptions accompanied by analyses of practices, programs, policies, and their enactment and outcomes. This is a scientific Study, and thus, we seek papers based on systematically-gathered information and analyses.

    In order to maximize the range of systems of teacher preparation about which we can learn through this Study, we seek proposals from a variety of countries. The Study’s investigation will be improved if the countries represented on the Program differ in size, population diversity (language, culture, race, socioeconomic), performance in mathematics, centralization of curricular guidance and accountability, and level of societal and economic development.

    Contributions to Strand I will be organized into a coherent section of the Study, with an overview and one or more analytic comparative commentaries to extend what can be learned from the individual cases and studies.

    4.2 –– Strand II: Professional Learning for and in Practice
    This strand of the Study adds substantive focus, in complement to the first. Whereas the first Strand examines programs and practices for beginning teachers’ learning, the focus of the second relates to teachers’ learning across the lifespan. This strand’s central focus is rooted in two related and persistent challenges of teacher education. One problem is the role of experience in learning to teach; a second is the divide between formal knowledge and practice. Both problems lead to the central question of Strand II: How can teachers learn for practice, in and from practice?

    Researchers and practitioners alike know that, although most teachers report that they learned to teach “from experience,” experience is not always a good teacher. Prospective teachers enter formal professional education with many ideas about good mathematics teaching formed from their experience as pupils. Their experience learning mathematics has often left them with powerful images of how mathematics is taught and learned, as well as who is good at mathematics and who not. These formative experiences have also shaped what they know of and about the subject. These experiences, along with many others, affect teachers’ identities, knowledge, and visions of practice, in ways which do not always help them teach mathematics to students.

    Moreover, teacher education often seems remote from the work of teaching mathematics, and professional development does not necessarily draw on or connect to teachers’ practice. Opportunities to learn from practice are not the norm in many settings. Teachers may of course sometimes learn on their own from studying their students’ work; they may at times work with colleagues to design lessons, revise curriculum materials, develop assessments, or analyze students’ progress. In some countries and settings, such opportunities are more than happy coincidence; they are deliberately planned. In some settings, teachers’ work is structured to support learning from practice. Teachers may work with artifacts of practice –– videotapes, students’ work, curriculum materials –– or they may directly observe and discuss one another’s work. We seek to learn about the forms such work can effectively take and what the challenges are in deploying them.

    Strand II of the Study asks how mathematics teachers’ learning may be better structured to support learning in and from professional practice, at the beginning of teachers’ learning, during the early years of their work, and later, as they become more experienced. Central questions include:

    a) What sorts of learning seem to emerge from the study of practice? What do teachers learn from different opportunities to work on practice –– their own, or others’? In what ways are teachers learning more about mathematics, about students’ learning of mathematics, and about the teaching of mathematics, as they work on records or experiences in practice? What seems to support the learning of content? In what ways are teachers learning about diversity, about culture, and about ways to address the important problems that derive from social and cultural differences in particular countries and settings?

    b) In what ways are practices of teaching and learning mathematics made available for study? How is practice made visible and accessible for teachers to study it alone or with others? How is “practice” captured or engaged by teachers as they work on learning in and from practice? (e.g., video, journals, lesson study, joint research, observing one another and taking notes)

    c) What kinds of collaboration are practiced in different countries? How are teachers organized in schools (e.g., in departments) and what forms of professional interaction and joint work are engaged, supported, or used?

    d) What kinds of leadership help support teachers’ learning from the practice of mathematics teaching? Are there roles that help make the study of practice more productive? Who plays such roles, and what do they do? What contribution do such people make to teachers’ learning from practice?

    e) What are crucial practices of learning from practice? What are the skills and practices, the resources and the structures that support teachers’ examination of practice? How have ideas such as “reflection,” “lesson study,” and analysis of student work been developed in different settings? What do such ideas mean in actual settings, and what do they involve in action?

    f) How does language play a role in learning from practice? What sort of language for discussing teaching and learning mathematics –– professional language –– is developed among teachers as they work on practice?

    Examining how some systems and settings organize teachers’ work or their opportunities for continued learning close to the work of teaching can offer images and resources for grounding the ongoing development of professional practice educatively in practice.

    5. Design of the Study
    The Study on the Professional Education of Teachers of Mathematics is designed to enable researchers and practitioners around the world to learn about how teachers of mathematics are initially prepared and how their early professional practice is organized in different countries. In addition, the Study takes aim at an endemic problem of professional education –– that is, how learning from experience can be supported at different points in a teacher’s career, and under different circumstances. Toward this end, the Study is designed to invite a variety of kinds of contributions for collective examination and deliberation at the Conference: research papers; program descriptions accompanied by analysis; conceptual work; demonstrations of practice; and interactive work on important common problems of teacher education and teacher learning.

    The Study Conference will be organized to be different from a conventional research meeting. Although research papers will be part of the program, substantial time will be designed for direct engagement with artifacts and materials of practice, for critique and deliberation, and for collective work on significant problems in the field. The Program Committee will design the Conference using the proposals we receive, and add, as needed, commentators, activities, and other resources so that the Conference enables participants to work together at the meeting, and to generate new insights, ideas, and questions important to the professional education of teachers of mathematics around the world. We anticipate that participants will be organized into working groups that will meet regularly across the Conference, affording the opportunity for joint discussion, work, and possible plans for future collaborative activity. Working groups’ ideas will be shared across the Conference; we will experiment with useful formats for such exchange of and repression of education and on prof dr mircea orasanu here

  • pagiu

    in romania there a hard repression against of education , of teachers and learning these facts can be proved by prof dr mircea orasanu

  • giogionu

    also here from long time there are a repression against of those men teachers or scientists or learning that publish works or articles in mathematics so a hard and fury repression as say prof dr mircea orasanu and often from 1970 and threats cruel permanently with stupids pretext

  • ceampu

    also there are more repression from the so called democracy institutions as cruel manifest of they behavior as say prof dr mircea orasanu and other and huge against of mind or thinking or profound studies as militia so called police a brutal organization and these continued and extension for universities

  • daceanu

    here we consider some solutions as say prof dr mircea orasanu and other concerning some aspects as LEARNING and TEACHING
    ABSTRACT
    We recognize that all countries face challenges in preparing and maintaining a high-quality teaching force of professionals who can teach mathematics effectively, and who can help prepare young people for successful adult lives and for participation in the development and progress of society. Systems of teacher education, both initial and continuing, are built on features that are embedded in culture and the organization and nature of schooling. More cross-cultural exchange of knowledge and information about the professional development of teachers of mathematics would be beneficial. Learning about practices and programs around the world can provide important resources for research, theory, practice, and policy in teacher education, locally and globally. Study 15, The Professional Education and Development of Teachers of Mathematics, is designed to offer an opportunity to develop a cross-cultural conversation about mathematics teacher education in mathematics around the world.

    Because the professional education of teachers of mathematics involves multiple communities and forms of expertise, the Study also explicitly welcomes contributions from individuals from a variety of backgrounds. Mathematicians and school practitioners are particularly encouraged to submit proposals for contributions.

    The Study will proceed in three phases: (a) the dissemination of a Discussion Document announcing the Study and inviting contributions; (b) a Study Conference, to be held in Brazil, 15-21 May 2005; and (c) publication of the Study Volume — a Report of the Study’s achievements, products and results.

    First is this Discussion Document, defining the focus of the Study and inviting proposals for participation in a Study Conference. We welcome individual as well as group proposals; focusing on work within a single program or setting, as well as comparative inquiries across programs and settings. In order to make grounded investigations of practice in different countries possible, we invite proposals in three formats: papers, demonstrations, and interactive work-sessions. Details are provided below.

    Second, a Study Conference will be held in Brazil in May 2005, bringing together researchers and practitioners from around the world. The Conference will be deliberately designed for active inquiry into professional development of teachers of mathematics in different countries and settings. Some sessions will offer paper presentations; other sessions will engage participants in direct encounters with particular practices, materials and methods, or curricula.

    Third, a Study Report — the Study Volume — will be produced, representing and reporting selected activities and results of the Study Conference and its products. This Report will be useful to the mathematics education community, as well as for other researchers, practitioners, and policymakers concerned with the professional education of teachers.

    2. Why Conduct a Study on the Professional Education of Mathematics Teachers?
    Three main reasons underlie the decision to launch an ICMI Study focused on teacher education. One reason rests with the central role of teachers in students’ learning of mathematics, nonetheless too often overlooked or taken for granted. Concerns about students’ learning compel attention to teachers, and to what the work of teaching demands, and what teachers know and can do. A second reason is that no effort to improve students’ opportunities to learn mathematics can succeed without parallel attention to their teachers’ opportunities for learning. The professional formation of teachers is a crucial element in the effort to build an effective system of mathematics education. Third, teacher education is a vast enterprise, and although research on mathematics teacher education is relatively new, it is also rapidly expanding.

    The timing is right for this Study. The past decade has seen substantial increase in scholarship on mathem

  • voiududu

    here we remark that in lyceum there exist more repression as in case of prof dr mircea orasanu and oppressed for true teaching and teach with ardor and loving for these activity and these in specially for summer Except for formulations such as mathematics provides opportunities for pupils to develop the key skills of: Communication, through learning to express ideas and methods precisely, unambiguously and concisely and Working with others, through group activity and discussions on mathematical ideas language and communication is not really a significant issue in the national curriculum for mathematics in England but here tgere exist a bestial oppression against of activity of teaching ). Study programmes in the National Curriculum describe ‘what pupils should be taught’. What are called ‘attainment targets’ (AT) give expected standards of performance (as an outcome of teaching and learning). In mathematics there are four: using and applying mathematics; number and algebra; shape, space and measures; and handling data (HD). All the ATs consist of eight level descriptions of increasing difficulty, plus a description for exceptional performance above Also:
    The subject aims at developing the pupil’s interest in mathematics, as well as creating opportunities for communicating in mathematical language and expressions. It should also give pupils the opportunity to discover aesthetic values in mathematical patterns, forms and relationships,

  • fosadiu

    sure we see that in romania there is not exist a real education as say prof dr mircea orasanu but exist a simulacrum and hypocrisy in all situations that is obviously

  • drintu

    also here we mention and we see as say prof dr mircea orasanu as followed with deformation and false learning of acknowledges in school romanian as obviously due to oppression

  • ciompogiu

    so here we consider that appear some as say prof dr mircea orasanu and prof horia orasanu as followed with
    CONSIDERATIONS OF ANALYTIC MECHANICAL
    ABSTRACT
    To find the average value of a function of two variables, let’s start by looking at the average value of a function of one variable. Note that, over the interval , the integral gives the total area of the region. We could also get the total area of the region by treating the region as a rectangle of length b-a and height equal to the average value of the function.

    Thus,

    Rearranging this formula, we see that

    We can perform a similar “trick” for functions of two variables. The volume under f is given by . But we could treat this volume as a solid whose cross section is shaped like R and whose height is the average value of f over the region R:

    To envision this, think of building the volume under z=f(x,y) as a solid mass of wax. Trap the wax inside a tube whose cross-section looks like R. As the wax melts, it will eventually form a solid whose height is equal to the average value of f on the region R.

    Click here to view an animation of a function made of wax melting to its average value. This will open a new window. To return to this window, simply close the new one. To view the animation repeatedly, use the “reload” feature of your browser.
    How do we calculate the area of a region? The area of a region R will be the same as the integral of a uniform density of 1 over the region. Thus,

    Volume under a surface
    Just like the integral gives the area between y=0 and y=f(x) from x=a to x=b, the integral gives the total volume of the solid which lies between z=0 and z=f(x,y) with a cross section shaped like R.

  • buivu

    here prof dr mircea orasanu and prof horia orasanu have

  • siudasu

    also as say prof dr mircea orasanu and prof horia orasanu these can be extended

  • didaciu

    then here sure that we consider some with education and suffering for thus questions as say prof dr mircea orasanu and prof horia orasanu as followed
    EDUCATION AND MATHEMATICS IN LAGRANGIAN
    ABSTRACT The Fundamental Theorem of Calculus
    Theorem 13 (Rolle’s theorem):
    Let be continuous on and differentiable on . If , then there exists at least one number c in at which .
    [Intuitions:]
    (1)

    (2)

    (3)

    If (3) (figure), in . Thus, . If
    (1) or (2), suppose takes on some positive values in . Intuitive, there is a number in , such that , where M is the maximum value of in . Then, .

    Theorem 14 (mean-value theorem):
    Let be continuous on and differentiable on . If , then there exists at least one number c in at which
    .

    [justifications of theorem 14:]

  • buidu

    yje above are indeed very true and as say prof dr mircea orasanu considerable as represio9n in teaching and learning in schools specially here for education or professors and teachers as me in all learning system that is oppression totally and as example we have
    ABSTRACT The Lagrangian formulation has been extended so far to handle constraints on that lower the dimension of the tangent space. The formulation can also be extended to allow nonconservative forces. The most common and important example in mechanical systems is friction. The details of friction models will not be covered here; see [681]. As examples, friction can arise when bodies come into contact, as in the joints of a robot manipulator, and as bodies move through a fluid, such as air or water. The nonconservative forces can be expressed as additional generalized forces, expressed in an vector of the form . Suppose that an action vector is also permitted. The modified Euler-Lagrange equation then becomes
    (13.181)

    A common extension to (13.142) is
    (13.182)

    in which generalizes to include nonconservative forces. This can be generalized even further to include Pfaffian constraints and Lagrange multipliers,

  • gingiu

    in these aspects we consider some as say prof dr mircea orasanu and prof horia orasanu since important as
    repression of learning and teaching of me and our activities ,and these appear from earlier years as more and many during

  • gcioagu

    we have new aspects of repression and oppression here against of learning and teaching as say prof dr mircea orasanu and prof horia orasanu and against of many work and papers written by prof dr mircea orasanu between 1970 and present by so called authorities and EU European publication and AMS and more ,and these can be as concern a hate more . The term “application”, on the one hand, focuses on the opposite direction mathematics  reality and, on the other hand and more generally, emphasises the objects involved — in particular those parts of the real world which are accessible to a mathematical treatment and to which corresponding mathematical models exist. In this comprehensive sense we understand the term “applications and modelling” More specifically, in the first dimension we discern three different domains, each forming some sort of a continuum. The first domain consists of the very notions of applications and modelling, i.e. what we mean by an application of mathematics, and by mathematical modelling; what the most important components of applications and modelling and more results are ignored

  • banciu

    also here we mention that the men specifies continues to same activities and as say prof dr mircea orasanu abs there an hysteria and despairing for their facts for many results our in hard conditions concerning learning and teaching ,therefor these used repression as method since these have a great fear and we observe that FOUNDATIONS OF PHYSICS AND ANALYTIC MECHANICS Along a phase transition line, the pressure and temperature are not independent of each other, since the system is univariant, that is, only one intensive parameter can be varied independently.

    When the system is in a state of equilibrium, i.e., thermal, mechanical and chemical equilibrium, the temperature of the two phases has to be identical, the pressure of the two phases has to be equal and the chemical potential also should be the same in both the phases.

    Representing in terms of Gibbs free energy, the criterion of equilibrium is:

    at constant T and P

    or,

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