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 موضوع: (The Paradigm shift in Mathematics Education Scenario for change )Dr.William Ebeid السبت يوليو 10, 2010 2:25 pm  

(The Paradigm shift in Mathematics Education Scenario for change )
Dr.William Ebeid
Study on this link http://mbadr.net/articles/view.asp?id=18
Prof. Mathematics Education
Faculty of education,
Ain Shames university.
With the advent of third millennium , the mathematics education institutions seem to be in a “tempestuous” zone. While there is a great progress in mathematics as a discipline and as a recognized effective tool for the advancement in science and technology to the extent that high technology is considered as mathematical technology (David,1984) , there is a distress and dissatisfaction with mathematics education in terms of its content, pedagogies and delivery system at all levels. In general there are poor outcomes in spite of the rich mandated objectives>
Symptoms of Dissatisfaction:
International and local levels of students attainment indicate poor results in the mathematics examination papers. Perceived problems are: serious lack of essential technical facility such as the ability to undertake numerical and algebraic calculation with fluency and accuracy , deficiency in spatial abilities and visual thinking ,decline in analytical powers when faced with problemsolving situations. International Olympiads and competitions such as the third international mathematics study (TIMSS) confirm many of these perceptions. The serious problem , as recorded by the London Mathematics Society (1995) , is not just that some students are less well prepared, but that many high attaining students are lacking in fundamental notations of subjects.
The beliefs about mathematics tend to perceive it as a tough subject to learn . Skemp (1971) mentioned that mathematics in a subject to be endured , not enjoyable and to be dropped. Cockroft (1982) reported that mathematics is known as difficult subject both to teach and to learn . Jensen and Others (1989) indicated that because of their intrinsic abstractness and generality of their issues , concepts and methods both mathematics and physics are hard subjects to study . There are no roads to their acquisition that do not involve hurdles to overcome and hardship to be endured . Frudental (in Ebeid ,1995) alluded to two devils menacing geometry : its absorbtion in a system of mathematics or strangulating it by rigid axioms.
Negative attitudes towards mathematics are reflected in many attitudinal studies. Mathematics is a generally disliked subject (Ernest,1991) . Mathematics leads away from the things of life and estrange men fro the perception of what conduces to the common “weal” (Howson,1982). Mathematics dries out the heart (Winslow,1998). Mathematics is a black forest of symbols, it requires to prove the obvious, its professors look arrogant (Ebeid,1999). Math phobic social environment has its impact on some students to have mathematics anxiety which causes aversion from learning mathematics or turn them to poor achievers.
There is an evidence of a general decline in enrolments to tertiary education in mathematics during the last decade (Jorgensen,1998). This is also noted on upper grades of the secondary stage where courses follow the elective system or profilization into different streams of study.
A Half Century of Swirl Progress:
Prior to the panic reaction to the Sputnik incident in the midfifties of the twentieth century, mathematics education enjoyed a reasonable state of stability and “linear” amendments in its content. The dominating content was the canonical syllabus as manifested in arithmetic of numbers, AlKhouarizmytype algebra concentrating around solving equations and – in some places – manipulation of determinants, Euclidean geometry moving from practical constructions to theoretical proof. Later more new topics or branches were introduced here and there. In Egypt for example: trigonometry and solid geometry were introduced as early 1874, coordinate geometry in 1908 as part of algebra then as a separate branch in 1953, history of mathematics in 1953 but was dropped in 1961, statistics in 1957 , descriptive geometry in 1961 but was dropped two years later ,differentiation and integration (as more related to algebraic functions) in 1961 (Ebeid, 1992). In UK and many other countries mathematics was seen as a training for discipline of though and for logical reasoning (Dainton,1968) . The most profound change in mathematics curricula in the twentieth century is synonymous with introduction of modern mathematics in the 1960s. The changes envisioned in that era were intended to bring mathematics through in schools into line with that of university mathematics including changes in language symbolism , treatment and topics so as to give preuniversity students a sense of what was preached as honest mathematics emphasizing the “logicaxiomatic” approach to unifying context – free mathematics systems. The enthusiasm for modern mathematics infused most of the countries even those which lacked enough resources, repetoire of experienced teachers , and cognitive readiness to early abstractness on the part of the learners. However the enthusiasm for modern mathematics had faltered by the seventies (in spite of the fact that some third world countries were just being ignited by movement , encouraged by some international and regional organizations and some commercial agents ). The reforms enshrined by the modern mathematics had refutable outcomes. A world wide concern about the inadequacies of modern mathematics was expressed in the “Back to Basic” wave of new changes . The lack of recognizing what is basic caused swirl changes in different places. Some sought a mixture of traditional and modern topics and approaches others restricted the content to traditional computations and manipulations. A reconciliation agenda for change was proposed by the National Council of Teachers of Mathematics in U.S.A. (NCTM,1980) recommended eight priorities :
(1) Problem solving be the focus of School Mathematics,
(2) Basic skills must encompass more than computational facility,
(3) Mathematics programs must take full advantage of power of calculators and computers at , all grade levels
(4) Stringent standards of both effectiveness and efficiency must be applied to the teaching of mathematics ,
(5) The Success of mathematics programs student learning must be evaluated by a wider range of measures than conventional testing
(6) More mathematics for all and greater range of options ,
(7) A high level of professionalism for teachers,(8) Public support must be raised to commensurate with the important of mathematics to individuals and society.Thus we find shift in change to encompass multidimensional aspects of improvement and involve all stakeholders. In different projects the pendulum has swung oscillating between emphasizing mathematical skills and between trails for the infusion of thinking abilities while teaching mathematical topics. Paul Ernest (1991) distinguished five interest groups in Britain showing that each has different aims and views about mathematics education as shown:(1) Radical conservatives and Bourgeois: Back to basics numeracy ,social training in obedience. (2) Meritocratic industrycentered Industrialists and Mangers: Useful mathematics to appropriate level and certification.(3) Conservative Mathematicians: Preserve rigour of proof and purity of mathematics. Transmit body of pure mathematical knowledge.(4) Professionals , Liberal educators, Welfare state supporters: Creativity ,selfrealization through mathematics.(5) Democratic Socialists and Radical Reforms concerned with justice and inequality: Critical awareness and democratic citizenship via mathematics.Ernest (1998) reports that aims(1) and (3) are conservative, with the lower elements of knowledge and skills together with external testing achieving in aims (1), and the higher elements of knowledge and skill directed for the few elite in aim (3). The aims are directed at “good” external to the students. They embody views of knowledge and skills as decontextualized . Aims (2) and (4) support the inclusion of a progressive knowledge application dimension. The two aims support the using and application relevant to the learner for using knowledge productively. Aim (5) is concerned with the development of critical citizenship and empowerment for social change and equality through mathematics. Ernest considered that making mathematics relevant to critical citizenship is neglected in most of the countries.With the increasing availability and access to calculators and computers , there have been demand to benefit from this technology in mathematics education leading to eliminate some traditional skills and inject new concepts and topics which are relevant to the need to live with complexity. Thus , mathematics educators are more riding wave of interest to create new and innovative approaches that capitalize on using technology. Some, for example, are calling to approach mathematics as an experimental science ,within visual thinking , but not as language or as liturgy (Davis et al. ,1994) . However , Ernest (1998) reports that in technology education , curriculum theories distinguish between developing technological capability and appreciation and awareness (Jeffery,1998). Capability consists of the knowledge and skills in planning and making artifacts and systems. Appreciation and awareness comprise of high level skills, knowledge and judgment necessary to evaluate the significance ,important and value of technological artifacts and systems within the social, environmental ,ecological and moral education.Kahan (1998) asserts that the educational project of our time cannot be Bourbaki type. Rather it should be inspired by the web system. Webbing mathematical knowledge would be to allow everyone ,starting from his own culture and interest to find a short track in mathematics forest.Example of Paradigmatic shifts. The above motioned trails and suggestion reflect the fact that “modern “ societies as they are contending to socioeconomic prosperity and advancementneed numerate citizens, top mathematicians , authentic scientists and creative engineers and technologist. This implies compelling and imperative necessity to make paradigmatic shift in course of mathematics education so as to tune it to the appropriate content , delivery systems and learning theories.In this context the following projects give examples of indigenous shifts , not just changes through addition and deletions. I. Chinese Perspective (ErSheng,1998).The perspective of mathematics education (PME) in china in st century calls for a shift based on changes ins: the social needs for mathematics , nature of mathematics and its applications and the understanding of how studens learn mathematics. These changes imply the following :(a) Adaptation to the needs of economy of information age and market economy. This requires useful mathematics to be learnt at mastery level so as to: interpret computer – controlled processes , acquire analytical rater than merely mathematical skills, deal with daily activities such as cost , profit ,tock ,forecast ,risk evaluation … which in turn needs the study of ratio and proportion , operational research and optimization , systematic optimization , analysis and decision theory (and complexity and chaos).(b) Inclusion of applications from the real world in such areas like environmental and ecologyical sciences , social sciences , art music ... (in addition to biology and other biosciences). This requires more statistics and probability , dynamic systems , mathematization, modeling patterns as manifested in number , data shape , arrangements ... this also need to use of appropriate packages of software to facilitate and empower students work. (c) Approach Learning mathematics through constructivism , where the students approach each new task with some prior knowledge , assimilate the new information and construct their own meaning to the extent that new knowledge be integrated to their own cognitive structure via creative activities … instead of learning (if any ) through passive absorption of information and storing it in easily retrievable fragments fragments as result of repeated practices. II. A view From South Africa : OutComes Based Education (OBE) (Volmik,1989).South Africa has adopted a National Qualification Framework and Curriculum 2005 as the focus for systematic transformation of the education and training system. Future , an outcomes based education approach was chosen as the vehicle to implement the objectives of the NQF. Eight generic outcomes have been chosen to ensure that learners would be prepared for life in global society. The eight crosscurriculum outcomes are:1. Identifying and solving problems in which responses display that responsible decisions, using critical and creative thinking , have been made.2. working effectively with others.3. organizing and managing onself and ones activities responsibility and effectively.4. Collecting, analyzing, organizing and critically evaluating information .5. Communicating effectively , using visual and / or language skills in the modes of oral and / or written persuation.6. Using science and technology effectively and / or critically , showing responsibility towards the environments and health of others.7. Demonstrating an understanding of the world as a set of related systems by recognizing that problem solving contexts do not exist in isolation .8. Contributing to the full personal development of each learner and social and economic development of the society at large. The specific outcomes for learning for learning mathematics are stated as follows: (1) Demonstrate understanding about ways of working with numbers. This outcomes is intended to develop an intuitive understanding of number concept and to extend that understanding to include the tools needed to solve problems and handle information.(2) Manipulate number patterns in different ways. This involves observing , representing and investigation patterns in social and physical phenomena.(3) Demonstrate understanding of the historical development of mathematics in various social and cultural contexts. Mathematics must be seen , not as a European product, but as a human activity to which all people of the world have contributed in significant ways.(4) Critically analyze how mathematical relationships are used in social , political and economy relation. This outcome is intended to allow learners to develop the critical capacity to participate in the decisions that effect their lives and to be aware of how issuses such as race , gender and class playout in their lives and their communities .(5) Measure with competence and confidence in variety of contexts. This outcome is intended to develop the skills of measurements with due regard to accuracy and relevant units.(6) Use data from various contexts to make informed judgments. In order to have the skills to make informed decisions within the context of a technologically advanced global system , learners must understand how information is processed.(7) Describe and represent experience with shape , space , time and motion , using all available senses. This outcome is intended to help learners to visualize and represent phenomena within the context of space and time more effectively.(8) Analyze natural forms , cultural products and processes as representations of shape, space and time. This will allow learners to make sense of aesthetic forms, relationship and processes in their communities and beyond.(9) Use mathematical language to communicate mathematical ideas, concepts , generalization and thought processes. Learners will acquire the algebraic skills to process and communicate the ideas.(10) Use various logical processes to formulate , test and justify congecturecs , and to develop their reasoning skills to construct and evaluate arguments.Volmink (1998) comments that the curriculum of the past had been contentdriven and extremely sterile. The new specific outcomes encourage educators and learners to focus on outcomes aiming at helping people to understand and act the world they live in.III. U.S.A. Standards 2000 (NCTM,1998).A draft document has been issued by the American National Council of Teachers Of Mathematics (NCTM). It is concerned with principles and standards for mathematics classrooms which are viewed as places where thinking about and doing mathematics is the central focus for the 21 st century.Guiding Principles:Mathematics instructional program should:(1) promote the learning of mathematics by all students.(2) emphasize important and meaningful mathematics through curricula that are coherent and comprehensive.(3) depend on competent and caring teachers who teach all students to understand and use mathematics.(4) Enable all students to understand and use mathematics.(5) Include assessment to monitor , enhance and evaluate the mathematics learning of all students and to inform teaching.(6) Use technology to help all students understand mathematics and prepare them to use mathematics in an increasingly technological world.Content and Processes. Ten standard followed the guiding principle which describe the knowledge base through a connected body of mathematics understanding and competencies. The last five address the processes which represent ways of acquiring and using that knowledge . All the ten standards are to be developed spirally through preK12 grades:(St.1) Number and operation:Mathematics program should foster the development of number and operation sense so that all students :(a) understand numbers , ways of representing numbers , relationships among numbers and number systems.(b) Understand the meaning of operations and how they relate to each oter.(c) Use computational tools and strategies fluently and estimate appropriately.(St.2) Patterns , Functions and Algebra : mathematics programs should include attention to patterns ,functions ,symbols and models so that all students :(a) understand all various types of patterns and functional relationships.(b) Use symbolic forms to represent and analyze mathematical situations and structures.(c) Use mathematical models and analyze change in both real and abstract contexts.(St.3) Geometry and Spatial Sense: Mathematics programs should include attention to geometry and space sense so that all students:(a) analyze characteristics and properties of two and three dimensional geometric objects.(b) Select and use different representational systems , including coordinate geometry and graph theory .(c) Recognize the usefulness of transformations and symmetry in analyzing mathematical situation.(d) Use visualization and spatial reasoning to solve problems both within and outside matematics.(Std.4) Measurement : Mathematics programs should include attention to measurement so that all students :(a) understand attributes , units and systems of measurements .(b) apply a variety of techniques , tools and formulas for determining measurements. (St.5) Data Analysis , Statistics and Probability : Mathematics programs should include attention to data analysis , statistics and probability so that all students:(a) pose questions and collect , organize and represent data to answer those question .(b) interpret data using methods of exploratory data analysis .(c) develop and evaluate inferences , predictions and arguments that are based on data. (d) Understand and apply basic notions of chance and probability .(St.6) Problem solving:Mathematics programs should focus on solving as part of understanding mathematics so that all student :(a) build new mathematical knowledge through their work with problems.(b) Develop a disposition to formulate , represent , abstract and generalize in situations within and outside mathematics.(c) Apply a wide variety of strategies to solve problems and adapt the strategies to new situations.(d) Monitor and reflect on their mathematical thinking in solving problems.(St.7) Reasoning and Proof: Mathematics programs should focus on learning to reason and construct proofs as part of understanding mathematics so that all students :(a) recognize reasoning and proof as essential and powerful parts of mathematics.(b) Make and investigate mathematical conjectures.(c) Develop and evaluate mathematical arguments and proofs.(d) Select and use various types of reasoning and methods of proof as appropriate.(St.8) Communication:Mathematics programs should use communication to foster understanding of mathematics so that all students:(a) organize and consolidate their mathematical thinking to communicate with others.(b) Express mathematical ideas coherently and clearly to peers ,teachers and others.(c) Extend their mathematical knowledge by considering the thinking and strategies of others.(d) Use the language of mathematics as precise means of mathematical expression .(St.9) Connections:Mathematics programs should emphasize to foster understanding mathematics so that all students:(a) recognize and use connections among different mathematical ideas.(b) Understand how mathematical ideas build on one another to produce a coherent whole.(c) Recognize , use and learn about mathematics in contexts put side mathematics.(St.10) Representation:Mathematics programs emphasize mathematical representations to foster understanding of mathematics so that all students:(a) create and use representations to organize , record and communicate mathematical ideas. (b) Develop a repertoire of mathematical representations that can be used purposefully , flexibly and appropriately .(c) Use representations to model and interpret physical , social and mathematical phenomena. IV. The Swedish “ ADM” project (Björk and Brolin,1998).The ADMproject is a research and development project for analysis of the consequences of the computer for mathematics education which has been imitated at the department of teachers training at the university of Uppsala in Sweden. In experimental materials , for secondary school calculus , the amount of time for skills development and procedural knowledge was reduced in favor of conceptual knowledge and enhancing problem solving learning environment. Computers and later on graphing calculators were used to perform all routine operations in analysis of graphs of functions. In longitudinal study (195792), the results indicated that the use of computing and graphing technology in calculus courses can have many positive effects when compared to traditional paper and pencil methods. In particular, students will be better problem solvers , have a deeper and richer understanding of fundamental concepts, be better able to model word problems with functions , to interpret given functions and equations and to change between different representations, more often use their own methods for solving problems. In 1996/97 the ADM project launched a TEMA (Technology in Mathematics) study Secondary school teachers assessed the changes in the new courses and called for:(a) less emphasis on exact integration and curve construction using derivatives. (b) Greater emphasis on problem solving , discussion , reporting solutions , lines of thought , understanding concepts , using and interpreting derivatives , setting up and interpreting integrals , properties of families of functions… V. An Australian Curriculum and standards Framework (CSF). (board of studies,1995).This framework is a policy about mathematics education for the eleven years of schooling in State of Victoria , Australia. Its content is adopted from Australian wide national profiles , CSF provides an outline of the mathematics curriculum. It leaves to the schools to be responsible for detailed development and delivery. It encompasses : goals , activities , content as structured into strands and sub strands , learning outcomes expected at each level, and guidelines to approaches to teaching and learning in addition to time allocation for the strands at different levels.Content : The content is structured in the following strands and sub strands:(a) Space : interpreting , drawing and making , location , shapes, transformation.(b) Number : number, counting and numeration , mental computation and estimation ,written computation , applying numbers , number patterns and relationships.(c) Measurements: choosing units , measuring , estimating . time , using relationships.(d) Chance and data : chance , posing questions and collecting data ,summarizing and presenting data , interpreting data.(e) Algebra : expressing generality , equations and inequalities , function.(f) Mathematical Tools and procedures : mathematical tools , communicating mathematics, strategies for mathematical investigation, contexts of mathematics.Access To Technology: CSF places clear emphasis upon sensible use technology in: concept development, problem solving , modeling and investigative activates. It encourages schools to ensure that calculators and computers are available for mathematics lessons. Four functions or Scientifics calculators are recommended to all students. Schools are to avail graphing calculators at levels 6 and 7. Improved access to computer resources is necessary : free stand computer with overhead projector in each class, computer labs and range of appropriate software.Competencies and Learning Outcomes: The following is summarized example of the learning outcomes expected by the end of the first level from of the five strands, such that children can:1. (Space): Draw , build and describe shapes and objects that they see and handle , note simple similarities and inferences , match congruent shapes, recognize symmetry in picture , follow and give directions of position and movement. .2. (Number) : Make , count , record and estimate small collections of objects and order and compare them , relate numbers using partwhole imaginary , deal with numbers , copy , continue and devise repeating and counting patterns , recall simple facts , count forwards and backwards to make simple mental calculations, represent number stories using materials and drawings , exchange money for goods in play situations.3. (Measurement): use everyday language to describe ,order and compare length , mass and capacity for familiar objects , compare length and capacity by repeated use of informal units , understanding the purpose of clocks and relate time to familiar recurring events , link the days of the week and months of the year with events , their lives.4. (chance and data): Recognize elements of chance in familiar situation, collect and classify objects , pose questions and represent information to make comparisons.5. (Mathematical Tools): Recognize ways in which mathematics is part of their family’s everyday life, communicate and discuss mathematical ideas in natural language , explore and test conjectures about problems that arise in their everyday experience ,detect and correct inconsistence in simple patterns , reassess nonnumerical estimates of size , use calculators to represent numbers and explore counting.A Scenario For Change In Mathematical Education (case study : Egypt).Guiding and Controlling Rules:
Following a holistic perspective away from fragmentation and piecemeal changes.
Consider the complexities regarding school buildings , classroom densities , teachers reactions and competencies , centralized curriculum development, line authority , physical facilities and the flow of increasing students enrolment in all stage.. etc.
Learn from past experience whether failed or succeeded.
Benefit from others experiences and innovative projects and the patheays to smooth implementation.
Simulate the realities using systems analysis.
Share and interact in dialectical dialogues with mathematicians , mathematics educators, teachers ,students , parents , consumers and users of mathematics.
Look for policies , rather polities, in the process of change so as to serve the society’s current and future real changes and needs.
Avoid generalization before scientific experimentation and formative evaluation.
Consider the cost and benefit expectations in the light of hard equation of financing and obligation for free education.
Be aware of consistency among different levels of decision making. Avoid passive or antireform executive through convincing dialogues.
Guiding Feature For Change:
Mathematics instruction should free itself from the classic taxonomy of Bloom and shift standard and outcomesbased philosophy.
Levels of achievement ought to be raised to the international benchmarks.
Soften centralized curriculum development by adopting “core” mathematics program which covers 6080% of the allocated time to be mandated all over the country. And leave the rest to be differentiated by the educational zones so as to contextualize and socialize it to local situations.
Delete routine skills and operations along with increasing sensible use of technologies.
incorporate new mathematics concepts at relevant levels. Examples come from data analysis , sampling techniques , probability concepts and new applications , linear programming , game theory , topological maps , operational research ,patterns , recursions and fractal geometry which reflect the aesthetics of mathematics , history of mathematical discoveries and roles of mathematicians including Egyptians and Arabs discrete mathematics wherever , quantity is to be counted.
Add integrated societal application module by the end of each grade to address the mathematics of : the farm , the building , the factories , the family budgeting … etc as related to local communities , environments and ecological situations.
The students use of technology are to be beyond just “pick and click”. It must be interactive aiming at development of concepts , discovering relations and verifying generalizations.
Create a new culture in the teachinglearning classroom environment , so as to avoid the culture of “talk and chalk” by solo “sage on the stage” . Staff development is an urgent and prerequisite necessity.
Create a center for producing relevant mathematics software in Arabic language.
Mathematical knowledge is not a sort of sports to be watched or limited nor they are mere scripts to be psychometric skills which need to be constructed through productive actions. Its main media for development ought to be very close to real life situations and problems , targeted to career preparation and self realization. In general , change out not to be merely contentdriven.
Guidelines For Pathways to Change:
Sustainable change for reform must be institutionalized , not depending on mere a top authority initiative. It must be directed to foster on classroom work.
Change has to pass through main four phases: initiation , pilot experimentation and evaluation , implementation and followup , and dissmetation with flexible continuity.
There must be a programmed time schedule until changes reach classrooms.
A strategy has to be chosen from alternatives in the light of governing rules and possible intervening factors. Literature in this context projects two influential variables (1) the degree and intensity of aimed change, (2) the extent to which the educational community is ready to accept the intended change. This of course goes along with the availability of relevant facilities. Within these variables , one of the following pathways can be taken :
(a) Successive Development : that is to implement certain parts of the change, one after the other , all over the country. Be time , complete change will be implemented at large.(b) Increasing Expansion: that is to implement the complete change in limited number of schools which can be gradually expanded.(c) Cautious Change: this is to do partial reforms , or implement some innovative projects in limited number of schools without specific plants for dissemention or generalization on large scale.(d) Pilot Experimentation : that is to experiment with some facets of reform in some schools , without having enough facilities nor sufficient support. Thus it will depend on convenient circumstance to be done now and then , here or there .In general , it may not be easy to choose the appropriate choice without being engaged openly and creatively with the context of the total view identified. This needs democratic sharing , professional preparation, human and material resources and courage perseverance and faith on part of leadership.References: (1) Birk, LarsEric and Bolin (1998) : “Which Traditional Algebra an Calculus are Still Important”? A paper presented in Fourth UCSMP Intl Conference on Mathematics Education, University of Chicago , August 1998, U.S.A.(2) Board of Studies (1995,96) “Curriculum and Standards Framework ,Mathematics; Mathematics Study Design” , Victoria, Australia. (3) Cokroft, W.H. (1982): “Mathematics Counts”, Report of Inquiry Committee in the Teaching of Mathematics , HSMO, London, U.K.(4) Daintor , F.S. (1968):”Enquiry into the flow of Candidates in Science and Technology into Higher Education” ,HSMO, London(5) David, Jr. E. (1984): “Renewing U.S. Mathematics : Critical Resource for the Future”, National Academy , Washington , D.C.,U.S.A.(6) Davis , B. Porta and Uhl,J. (1998): “Is the Mathematics we teach the Same as Mathematics we Do?” A paper distributed at the Rosilde University Conference in 1997,Denmark.(7) Ebeid , William (1999): “Current and Future Trends in Learning and Teaching Mathematics”; MOR,Dubai, United Arab Emirates.(8) Ebeid , William (1999): “Societal Mathematics as a Futuristic Trend” , International Conference on Mathematics Education into the 21 st Centaury “. (Roerson ed.) Cairo, Egypt , Nov. 1999.(9) Ebeid , William (1998): “Enrolment in Mathematics , Problems and Aspiration in Kuwait University” , in Proceedings of Conference on Justification and Enrolment Problems Involving Mathematics or Physics”, Roskilde University , Denmark.(10)Ebeid , William (1999): “Mathematics for All in Egypt :Adoption and Adaptation “, Proceeding of Fourth UCSMP Conference (1998), Chicago , U.S.A.(11)Ebeid , William (1996): “Difficulties in Learning Geometry”, 8^{th} ICMI, Seville, Spain.(12)Elaisawy and Others (1999): “ The Theoretical and Methodological Bases for “Egypt 2020” Scenarios” , (in Arabic ) , Third International Forum , Cairo, Egypt >(13)ErShing , Ding (1998) : “Mathematics Reform Facing the New Century in China “ , in Proceeding of UCSMP Conference ,op.cit.(14)Ernest , Paul (1998): “Why Teach Mathematics ?” in Justification Conference , op. cit.(15)Frudental , Hans (1971): “Geometry Between Devil and Deep the Deep Blue Sea”,in Educational Studies in Mathematics Vol.3 No. 34 , Boston , Mass. U.S.A.(16)Horwon , G. (1991): “National Curricula in Mathematics”, Leister, The Mathematics Association, U.K.(17)Jeffery , J.(1988) : “Technology Across the Curriculum”, Exter School of Education, Exter , U.K.(18)Jenssen , J. , Niss and Wedge , T. (1998) : Introduction “ in the Justification Conference , op. cit.(19)Jorgensen , Bent (1998): “Mathematics and Physics Education in Society” “ in the Justification Conference , op. cit.(20)Jorgensen , Bent (1998): “Mathematics and Physics Education in Society” “ in the Justification Conference , op. cit.(21)Kahan , J.P. (1998) : “Mathematics and Higher Education Between Utopia and Realism “, in the Justification Conference , op. cit.22)London Mathematical Society(1995) : “Tacking the Mathematical Problem”,LM S,IM and RSS, Burlington House , Pecadilly , London.(23)Moris and Arora (eds. ) (1992): “Moving into Twenty First Centaury”, Studies in Mathematics Education , Vol.8.Unesco,Paris.(24)NCTM (1980) :” An Agenda for Action : Recommendations for school Mathematics of Eighties”, NCTM, Virginia ,U.S.A.[/size](25)Skemp , R. (1971) : “The Psychology of Learning Mathematics “ Penguin , Harmondswarth , U.K.(26)Standard s2000 Group (1998): “Principles and Standards for School Mathematics Discussion Draft”, NCTM, Virginia, U.S.A.(27)Usiskin , Zalman, (1999): “Is there a worldwide Mathematics Curriculum ?”, in UCSMP Conference (August 1998) op. cit.(28)Volmink , John (1999): “School Mathematics and OutcomesBased Education – A view From South Africa “ , in the UCSMP Conference op. cit.(29)Winslow , Carl (1998): “Justifying Mathematics as a way to Communicate”, in Justification Conference , op. cit. _________________ مع خالص شكري وتحياتي د.وسام محمد إبراهيم علي مدير"وسام المنتدي التربوي" http://wessam.allgoo.us/ مدرس المناهج وتقنيات تعليم التاريخ كلية التربيةجامعة الإسكندريةجمهورية مصر العربية mobile:00201069785672 emails:dr_wessam2006@hotmail.com wessam_mohamed_4ever@yahoo.com With my sincere thanks and greetings Dr.Wessam Mohammed Ibrahim Ali Lecturer and faculty member PHD Doctor in curricula and Technology of history Education College of Education  University of Alexandria  Arab Republic of Egypt

