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 (The Paradigm shift in Mathematics Education Scenario for change )Dr.William Ebeid

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(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
    problem-solving 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,
Al-Khouarizmy-type
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
pre-university students a sense of what was preached as honest
mathematics
emphasizing the “logic-axiomatic” 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 stake-holders. 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 industry-centered 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 ,self-realization 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 socio-economic prosperity and
advancement-need 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 (Er-Sheng,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 bio-sciences). 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 : Out-Comes
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
cross-curriculum 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 content-driven
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
pre-K12
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 ADM-project 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
(1957-92),
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

part-whole 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 non-numerical 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
    piece-meal
    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 anti-reform executive through convincing dialogues.







Guiding Feature For Change:


  • Mathematics instruction should free itself from the classic taxonomy
    of Bloom
    and shift standard and outcomes-based philosophy.

  • Levels of achievement ought to be raised to the international
    benchmarks.

  • Soften centralized curriculum development by adopting “core”
    mathematics
    program which covers 60-80% 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 teaching-learning classroom environment ,
    so as to
    avoid the culture of “talk and chalk” by solo “sage on the stage” .
    Staff
    development is an urgent and pre-requisite 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
    content-driven.




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 follow-up , 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, Lars-Eric 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”,
8th 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)Er-Shing , 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. 3-4 ,
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
Outcomes-Based 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.


_________________

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