CHAPTER IDETERMINANTS OF THE CURRICULUM

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Before discussing areas of innovations of the mathematics curriculum, it is necessary to have a glimpse of the different views of curriculum in general. In this chapter, recent views of the school curriculum have been critically discussed and its relevant determinants drawn out. In the light of this discussion, contemporary issues and problems in the mathematics curriculum have been elaborated and analysed in subsequent chapters.

1.1 Towards a Definition

The meaning of 'curriculum' has evolved over time and assumed different nuances according to the various schools of thought. Below are some interpretations of the meaning of curriculum, in general.

1.1.1 Layman's view

Formalized education usually takes the shape of courses of study. Children attend school and pursue a course of studies, the guidelines of which are usually laid down. Without aforementioned guidelines, education would be haphazard, and the outcome would be questionable. In the layman's parlance, curriculum usually means this course of study - the subjects to be studied and the contents to be purveyed to students.

1.1.2 Holistic view

Over the last 2 centuries, formalized education has become indespensable to society. No more is learning at the 'Mother's knee' or apprenticeship in the father's trade deemed sufficient. The school has to hand down certain skills to all children. Over and above this, modem society has placed many of the responsibilities, previously dealt with in the family, at the school. Instead of being the narrow dispensgr of skills, the school has the responsibility of developing the minds and bodies of its pupils. It therefore has to devise the correct activities while imparting skills, so that its pupils may benefit totally from their education.

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The school therefore, does not only have to devise the best ways of imparting certain skills, but consider how the means of doing so actually affect the activities of its pupils. It also has to consider the values of certain activities of pupils on their own merit. So, from the narrow confines of the content as a definition of the curriculum has evolved a term like 'educational experiences' ( 2.75" ). Curriculum therefore, is now viewed not only as the content or the subject matter, but as the concepts to be imparted, the ways and means of doing so, and the application of the concepts to several aspects of life.

The curriculum is not only confined to the hallowed campus of the school. Children's educational experiences are not solely restricted to their educational experiences in the class-room. However, from the class-room activities the every day experiences of children are largely affected. They learn certain ways of reacting socially, of adjusting themselves to their environment, of judging their experiences logically and critically, and of forming ethical principles as well as many other qualities and types of behaviours. Many of these traits are not instilled consciously in the class-room. Nevertheless, class-room experiences do affect their transactions with the world, and theenvironment becomes their laboratory for testing the truth of much of what they learn in school. This aspect of learning, though not totally intended by the school, may also be regarded as part of curriculum, and is hence termed the 'hidden curriculum'. It may not always be possible for curriculum developers to take note of this aspect in its entirety. Nevertheless, it cannot be disregarded all together. In fact, the hidden curriculum may be utilized to the advantage of thecurriculum in its entirety by attempting to connect the content as far as possible to children's everyday experiences (76,2.73).

Curriculum, therefore, includes academic content from the discipline as well as all other educational experiences, both intended or unintended, that are related to the content. Curriculum then becomes nothing less than a guideline for initiation into a way of life. Incidental experiences of children during the course of the curriculum are almost impossible to envisage in advance and at best depends on

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the foresight of the curriculum planner. Taylor and Richards ( 275) therefore see the curriculum as the content of Education, the course of study,, the educational experiences, the subjects to be studies, thesubject matter, and the educational activities. To this may be added the role of the 'feedback systems', i.e., the diagnostic and assessment qualities of the curriculum.

Kerr (150) has defined curriculum as all the learning which is planned and guided by the school, whether it is carried on in groups or individually, inside or outside the school. This definition is all- encompassing , and takes into account the incidental or unintended experiences that may be encountered by the pupils during the course of the curriculum. However, it may pose problems for curriculum planners, should thay be on the lookout for an origin to a framework for planning. Rugg's definition of the curriculum (2.35) may be more suitable for the above purpose '... the entire program of theschool's work .... It is everything that the students and their teachers do. Thus it is two fold in nature, being made up of the activities, the things done, and the materials with which they aredone'.

1.1.3 Curriculum as a dynamic issue -the changing face of the curriculum

The very utility of curriculum studies is justified by the fact that the curriculum is never static. Changing circumstances, changes in society, technological developments and novel requirements make an impact on education and forces rethinking and readjustments of the curriculum. This has been amply demonstrated by Benjamin's satire 'The Saber - Toothed Curriculum' ( 22. ). Renewal is required forsurvival and rejuvination. This ever-changing aspect of the curriculum is demonstrated by its definition as 'the amorphous product of generations of tinkering' (272). As riders to this definition, are value laden definitions of curriculum, like that of Lawton's (161) 'the school curriculum (in a wider sense) is essentially a selection from

I

the culture of a society, and the Open University's (i4-8) "A curriculum is the offering of socially valued knowledge, skills and attitudes made

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available to students through a variety of arrangements during the time they are at school, college or university. The values indicated can only be that of the particular society at the particular time in which the curriculum is to be implemented.

1.2 Different views of the curriculum - Classical vs.Progressive curriculum. Idealists vs. Romanticists

Once the rationale for the total school curriculum is established, that for the different disciplines within it fall into place. This rationale is influenced by the views of educationists and their interpretations of the definition of the curriculum. Traditionalists hold the classical view of the curriculum i.e., it is subject centred, and is essentially a tool for incorporating pupils to established ways of thinking, and handing down to them certain skills and informationregarded as essential by society. Traditionalists, therefore, demand obedience, conformity and discipline from students. They advocate competition and the striving for excellence and rationality, as well as initiation into the established culture of society (162). Phenix (I0)0!) is an example of a leading classicist. He contends that general education, encompassing certain aspects of knowledge, is essential for the curriculum. Hirst ( I2.» ) is another advocate of the classical curriculum. His seven forms of knowledge are

1. Formal logic and Mathematics.2. The Physical Sciences.3. Our awareness and understanding of our

and other people's minds.4. Moral judgement and awareness5. Aesthetic experience.6. Religious.7. Philosophical.

These forms of knowledge, according to Hirst, may most efficiently be taught in the traditional subject based time - table of the school.

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In contrast, the Romantic view of the curriculum favours child- centric education that nurtures creativity, awareness, originality and freedom, and encourages discovery and the interpretation of experiences and discovery (162-). This view arose as a result of the upsurge of psychological and developmental studies, and the demand of the 1960's to shift focus from elite to mass education (73 ). The Romantic view point sparked a liberal educational upsurge in the West. Already, however, an economically hard-hit West is reconsidering this standpoint. Too much child-centricism as practiced in the U.S.A., for example, is being blamed for a fall is educational standards.

The Indian curriculum, for practical and cultural reasons, has veered towards the traditional curriculum. The accent is on obedience and acquiescence. Iconoclasm and innovations are infrequent. Government agencies, which are usually the propagators of change, have initiated child centricism and the play way method in the primary school. However, in practice, lack of proper implementation and technical back-up have resulted in half-hearted efforts. Lack of opportunity, in general, has forestalled risk taking associated with liberal movements, and education has precipitated to rote-learning of 'hand-me-downs' to a large extent. As a result, leadership in progressive or novel ideas is

rare.The following queries emerge in the light of the foregoing

discussion : Will rigid adherence to traditional views of the curriculum create a status quo? Will the resultant equanimity obstruct advancement? Conversely, will prescribing to a liberal view give too much licence for child centricism, hence producing a too diluted vision of mass education? Can a happy medium between the two schools of thought, viz., liberal and traditional be achieved in the Indian context? Can we in India, with our limited resources, provide an education that will be available to all, and can this education nurture future citizens who can think logically and (at least in some cases) who are creative and can think with ingenuity?

In the context of mathematics, the answers to the preceding questions and the solutions to the problems ingrained in them require intensive consideration by teachers, educators and mathematicians.

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Perhaps a move has to be made to realign the image of the discipline, implement innovative methods of teaching, overhaul the established content structure in schools, and so on. Still, there are bound to be a number of obstacles to change. Will it be possible to overcome these obstacles and work towards meaningful and beneficial change? The author hopes to arrive at some of the answers to these questions and problems at the end of her investigatory work.

1.3 Major areas of curriculum design

In designing a curriculum, the major areas to be considered are -

(a) Formation of aims and objectives(b) Selection of the content(c) Methods of instruction, including intended

educational experiences and material development(d) Evaluation(e) Dissemination and implementation

1.3.1 Formulation of aims and objectives

This area donotes the purpose of the curriculum - what the curriculum designers intend that the students will know, and what changes of behaviour they will undergo by the end of the course. Generically speaking, aims denote broad goals, while objectives denote more specific intentions.

Some curriculum planners prefer to specify objectives in great detail at the start of the curriculum project. They are assisted by the Taxonomies of objectives,. most popular of which is that of Bloom ( 2.8, 2.9 ). Other notable Taxonomies are Guildford's Structure of Intellect Model (JI2.), the Gagne-Merril Taxonomy (qz,94-, t75), Gerlach and Sullivan's Taxonomy (95), De Block's Taxonomy ( 63 ),Krathwohl's Taxonomy (64-,i5*,i55;t56) etc. ( 6S,I%4-, 3o i ).

Advantages of pre-specifying objectives :

a) They act as a navigational map for curriculum developers.b) They clarify the intentions of the planner.

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c) They assist in formulating an instructional design.d) They make known clearly the process of the curriculum during

implementation.e) They form the basis of the evaluation process.f) They give a rational base to the curriculum.g) They are particularly useful in programmed learning, mastery

learning, individualized learning, etc. (31,54-).

Disadvantages of pre-specifying objectives

a) Education is too complex to be entirely defined by means of

objectives (219).

b) Readily measurable objectives, i.e., behavioural objectives are

stressed (19).c) More subtle aspects of education will be ignored.d) Measurement of behavioural objectives may lead to rote learning.e) Restricts the teacher from interpreting the curriculum critically.f) Interferes in social power equations.g) A plethora of highly specific behavioural objectives may result in

a triviality in such objectives. Perhaps a few broad based objectives are required, but not so broad as to make them vacuus generalities ( I &%).

h) Teachers and students tend to take objectives as something given and not to be questioned, and hence autonomy is lost (14-B, 2150.

C.M. James (133), for example, is a strong objector to curriculum planning with prespecified objectives. She argues that the wholeness of humans and their learning is lost in trying to see education in discrete steps and independent units. Means and ends, according to James, cannot be separate. Instead of knowledge accumulation, she proposes that the curriculum be more concerned with development of attitudes, values and experiences. Therefore, she seems to value enquiry and dialogue over instruction and obedience, and activity over passivity. Increasingly, this outlook seems to be making sound sense to mathematics educators.

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Expressive and Instructional Objectives

In answer to the debate on whether objectives should rule curriculum design, Eisner ( 77 ) made a distinction between expressive and instructional objectives. Instructional objectives denote behavioural changes and may be predicted before-hand. Expressive objectives, however, cannot be predicted. They may be described as educational encounters and may be unique to the students' interests and circumstances.

Though expressive objectives have so far, by and large, been used in Humanities and Social Sciences curricula, they are gradually being used in mathematics curricula as well. By so doing, the accent on behavioural obejctives which often reduce the learning of the discipline to that of fragmented learning of certain skills, is offset. The in-depth understanding of concepts, interrelationships between different sub-areas of the discipline, or applicability may subsequently be stressed, and induction to the discipline that much more complete. Stressing expressive objectives over and above instructional objectives may prove to be no mean task for the mathematics curriculum designer. Yet, proceeding in this direction may yield mathematicians instead of robots.

1.3.2 Content Choice

This is dependent on ideas of what constitutes knowledge, i.e., the nature of knowledge. An attempt has been made to delineate some of these ideas :-

a) Culture and the curriculum :

The curriculum is seen by some as the common cultural heritage which must be transmitted to the next generation (161). The difficulty with this idea is that it gives rise to several interpretations of culture and to two or more cultures - upper class and folk culture. India has followed the British tradition of conveying the aristocrats' culture to the milieu. Yet, our tribals and other grassroots sources of culture are a reality, and this aspect has to be noted in the selection

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of the content of mathematics. Without this consideration, mathematics in schools will be alien to the masses.

b) The debate on whether the curriculum should transmit theprevalent culture of the society or attempt to transform thatculture :

If culture must be transformed by society, whose opinion should prevail ? For example, should education be used to nurture religious fundamentalism (as has already been observed in 1992 in some Indian states) or should education nurture integration and equality ? On theother hand, the problems faced by humanity imply a need forconsidered and cautious change of society by attempting to change certain ways of thinking of the masses. Mathematical topics, for example, may be selected so that they illumine women's interests. In this way, the gender bias in the discipline may be overcome andwomen may be encouraged to think independently about logistic problems.

Another aspect of this issue is that technological change leads to changes in norms, values, beliefs, customs, etc. Yet there is a timelag between technological change and changes in norms, customs, etc. A recognition of rapid social changes and the need for people to beequipped to cope with it and may be, even to have some control over it suggests the transformer's role for the curriculum developer. Thecurriculum needs not only to acculturise, but to prepare students for social change and to initiate changes in society. At the same time, unless the students are taught to think for themselves, intended acculturisation may become the imposition of an alien and irrelevant curriculum.

Therefore, as culture comes from a wide spectrum, the selection of the content for the curriculum has to be eclectic and thoroughlyjustified (I 6 2. ). Several topics, influenced by the onslaught oftechnology seek to enter the mathematics curriculum. Which of these topics may best equip the future citizen to cope with tomorrow's world has to be justified, not only for their utilitarian values, but also their use as vehicles for independent thinking.

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c) The nature of knowledge :

Opinions about the nature of knowledge influence the selection of content. Broadly speaking, this refers to the debate between the rationalists and the empiricists. The former believe in the supremacy of the intellect over other senses. Knowledge, to them, is acquired independently by the mind, regardless of information provided by the senses. Knowledge, to the rationalists, is therefore, permanent, derived from pure reason, and has nothing to do with human feelings or passions. Proponents of this way of thinking quote Plato, Descartes and Kant (lZo,t2.l,\Q1, 198").

Empiricists, beginning with LQcke, take the view that knowledge cannot be acquired without the senses. To acquire knowledge, reflection, introspection and interpretation of perceptions is required. Knowledge, therefore, is hypothetical, transient, and subject to modification and evolution. This view questions the value of any body of knowledge or its inclusion in the curriculum. Rather, it encourages knowledge to be equated with experience. Knowledge, according to this view, must not be imposed on a child, but the latter must be assisted to develop his/her own knowledge. This makes knowledge personal and subjective, and suggests that 'common sense knowledge' should form the basis of the content of the curriculum. Proponents of this view of knowledge among others are Dewey, Illich and Friere (67a.,<35,132.).

Till recent times, mathematics has drawn from the rationalist view, and a halo has formed about the discipline, illuminating the sanctified permanence of knowledge. Yet, these days, the empiricist view of mathematics is increasingly gaining ground, especially in school education. As a result, every child is treated as a mathematician in the making. Experimental and investigative ways of discovering mathematics are being encouraged. Moreover, the myth of absolute truth through mathematics is being exploded in the school room through introduction of topics such as Probability and Approximation.

d) High and low status disciplines :

The traditional view, stemming from the time of Plato is that the more the abstraction the higher the status of the discipline. This view

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leads to elitism in education. Opposed to this view is the utilitarian view. Mathematics has traditionally been viewed as a high status subject, primarily because of its abstract nature. Students, not able to grasp the abstraction inherent in the discipline, have fallen by the wayside. The consequent elitism is not in keeping with the demands of modem life which requires proficiency and familiarity with mathematical ideas by all and sundry. School mathematics, therefore, has to shed its elitism by relating to the average student, and this probably requires integrating Mathematics with other subjects, i.e., breaking down subject barriers. This may help in applications, and in preventing the feeling of irrelevancy associated with so much of school Mathematics. However, initiating these changes involve administrational difficulties, new relationships between teachers and students, new modes of time tabling, and perhaps novel tacties like team-teaching.

The case against high and low status disciplines has been further high-lighted by sociologists such as Berger and Luckman ( 2.3 ) v/ho have proposed that knowledge itself is socially constructed and subject to different views of reality according to the social class of the receipient of the knowledge. Whether this is true in the case of mathematics today is open to question.

Left leaning sociologists like M.F.D. Young ( 3o5") have further worked along these lines and concluded that all knowledge and even rationality itself is socially constructed, that subject barriers are artificial and arbitrary, and are impositions by those in control of education, and finally, that the content of education, i.e., the knowledge to be transmitted by schools, cannot be taken for granted, but that its choice must be made into a problem.

1.3.3 Methods of instruction, including intended educational experiences and material development

1.3.3.1 Instructional methods

While selecting the content for the curriculum, the curriculum developer needs to keep in mind the way in which the knowledge is to be imparted to the learner. This will be a result of considerations about theories of learning and of instruction as well as about specific

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circumstances for which the curriculum is designed, for example, the age of the pupils, their social milieu, etc.

Among the different schools of psychology, the theories of learning that have gained prominence in the past have been that of the Behaviourists (2>o, i63(262-,3o2) and the Gestaltists (30,14-5,163,22.1,2.93). Among the cognitivist theories of learning that have gained in credibility in recent years are Information Processing theories of learning like that of Gagne (89, <90,43,2.37), Developmental theories like that of Piaget (10,87,2P3-2.I3,229,2.5o)and Bruner (163,23"?), and Humanistic theories of learning like that of Maslow (172.) and Rogers ( ).

The theories of learning that bear special relevance to the teaching of school Mathematics in recent times are developmental theories like that of Piaget {11 MTo,2<yv)2-ta,215,2.60,26t) and developmental theories that have been influenced by Piaget, for example like that of Bruner (33-31), Dienes (68-7t) etc.

The stages of development as expounded by Piaget (83,163,237)show that children benefit from experiencing knowledge in a concrete manner before being able to make abstract concepts. Piaget believed thatchildren learn, by experimenting, questioning, seeking answers comparing experiences, sharing and cooperating with peers and hence forming individual 'cognitive frameworks' 036).

Piaget stressed on the child's experience and that too, not artificially. He also stressed the need for structure in introducing the child to the learning experience. Herein he felt was the true' expertise of the teacher. Piaget distinguished 'discovery' and'invention' in learning. Though the two terms are interlinked, it helps in being conscious of the distinct traits of these two kinds of learning. Piaget laid great stress in building up the child's autonomy of thinking, so that he/she is not limited by the teacher's 'right' answer (136).The child must be made1 a critical judge of experiences. Theteacher's role, therefore, is ; not that of an authority figure whoenforces standards from outside, but one who knows her pupils very well, and who can maintain the delicate balance between being an authority figure and encourager of active learning.

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Piaget's influence on Dienes can be seen from Dienes' 'Six stages in the learning of Mathematics' (/7o).Dienes, while favouring experience and play also lays stress on presenting structured environments to the child. The child will proceed from the concrete to the abstract by being able to generalize from a range of experiences and express this in symbolic form. Dienes'work (76,7%, ) shows how all mathe­matical concepts can be internalised by students in this manner.

Bruner was influenced both by cognitive development theories like that of Piaget and information processing theories of learning. The acquisition of knowledge, to Bruner, is not a product, but a process, and the duty of teachers is to ensure that students actually participate in the process of getting knowledge C$3).

Bruner's theory of instruction consists broadly of 4 parts :-Predisposition to learn, structure of knowledge, effective sequences, form and pacing of reinforcement (237). Each part consists of several sub-parts. Bruner's theory of instruction also states that knowledge can be represented in the 'enactive', 'iconic' and 'symbolic' forms.

The first form may lead to the second and the second to the third and final form. Thus, enactive representation may be the first representa­tion leading ultimately to symbolic (and abstract) understanding of knowledge. This seems to be the antecedent to Bruner's earlier writing ( 32,3 6 ) where he favoured learning by discovery. Learning, toBruner, is motivated by the very job of 'discovering' the treasury ofknowledge.

Bruner's theory of instruction is pertinent to the teaching ofmathematics because it attempts to answer many of the problems faced by the learner. In particular, for problem solving and enquiry based lessons, the outline of the tactics involved is almost mapped out - understand the problem, experiment with specific situations, generalise.

Gagne's theory of instruction ( 9 I ) is based on information processing theories of learning. It attempts to map out the teacher intervention process in learning.

Rogers' theory is a Humanistic theory of learning. His terminology includes 'significant learning' which is knowledge (or 'relationship') which when acquired may be used for 'growth, and change and

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personal development' (2.32.). Instead of teachers, Rogers calls for ' facilitators of learning', who will not be the familiar authoritarian figures of the classroom, but equal partners in the acquisition ofknowledge. Students must not be hemmed in by set curricula, but must learn how to leam. The 'facilitator' is the rescource person, who may set up an atmostphere for learning and share empathetically the feelings and thoughts with the learner. In other words Rogers' theory requires both learner and teacher to be emotionally involved in thelearning process.

Changes in the mathematics classroom place less stress on skills and rote memory and so relationships in the classroom are less that of the authoritarian teacher and submissive pupils and more democratic in nature. The classical view of education is slowly changing to the romantic view of education, so that less teacher intervention is preferred. In this light, and with the ongoing stress on problemsolving skills, investigative work, group and individualised learning, etc., the mathematics classroom bears special relevance to Rogers' theory of learning. The teacher is fast becoming the facilitator who needs to take account of her students individually and help them to understand and internalize Mathematical concepts instead of dinning in half-understood facts.

The preceding discussion highlights, some of the emerging trends in instructional methods for teaching mathematics. These include concrete representation of concepts, activity, experiencing, experimentation, use of dialogue and contoversy, mathematical modelling, recreational mathematics, etc. Perhaps this means a complete reorientation of theteacher-pupil relationship in the mathematics classroom.

1.3.3.2. Material development

Quite often the syllabus or the nearest equivalent to the curriculum is a loose ensemble of phrases denoting the concepts to be acquired by the student in order to prepare for a certain examination. So, the curriculum is often determined by the examination. But how does the examinee prepare ? By taking recourse to the popular books in the market that cater. to the particular examination. It is these books

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that give the student, and nearly quite as often the teacher, a guide to what the curriculum is all about. Text books, therefore, became the representation of the curiculum and quite often what text book is used will reflect on the suocess of the new curriculum. Ideally, then, these text books should be prepared by people who have an inside view of the curriculum. However, this is usually not the case. Therefore, the intended curriculum is frequently distorted through the medium of the book. Over and above this, publishers in the free-market. favour books that are advantageous to them (as opposed to being educationally beneficial) and promote these books. Governmental control of textbooks as in erstwhile East Germany (G.D.R.) means imposition of the opinions of the bureaucrats and politicians instead of the intentions of the curriculum developer. Apart from these restrictions, there is the genuine financial consideration which might restrict publication of books which could truly depict the curriculum - for e.g. colour plates or too many tables or diagrams may raise the cost of books. This last point is particularly applicable to school mathematics textbooks. Without the necessary illustrations and tables,graphs, etc. they became dull and intimidating collections of 'sums' for drill and practice.

Apart from textbooks, well thought out teachers'guides may prove useful in the actual implementation of curricula. Examples of teachers' guides are those of SMP (U.K.) (J2.G*r).

Many curricula might require material other than textbooks or teachers guides. These have to be made with an eye on finances. Nowadays, different equipment for teaching mathematics are commercially available. However, the most convenient apparatus are those that can be locally made (like little 1 cm by 1 cm by 1 cm cubes to teach volume), or better still, made by the teacher and/or the students (for e.g. dice). Whatever the apparatus, the curriculum designer has to keep in mind that these must be simple, easy to manipulate, and easily and cheaply available.

1.3.4. Evaluation of the curriculum

1.3.4.1. What curriculum evaluation means

This area of curriculum development deals with validation of the curriculum, i.e. whether it has achieved its 'stated intents' (24-8).

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Sanders (T.'iS) defines curriculum evaluation as thB 'process of studying the merit or worth of some aspect, or the whole, of a curriculum. Cooper's (53 ) definition of evaluation is broader : 'Curriculumevaluation is the collection and provision of evidence, on the basis of which decisions can be taken about the feasibility, effectiveness and educational value of curriculum. In other words, 'curriculum evaluation' may indicate evaluation of the product of the curriculum or evaluation of the programme of the curriculum itself ( 4- ). The former evaluates textbooks, syllabuses, the actual course of study, etc. Curriculum programme evaluation does not evaluate the curriculum itself, but how the curriculum works in its broader context, i.e. as part of an established instructional setting. Therefore, curriculum evaluation may focus on several aspects of the programme: student needs and rate of progress, teacher effectiveness, changes in learning attitudes, the learning environment, resource usage, etc.

Lewy ( 16 7 ) has identified six issues / stages in curriculumevaluation :

(1) the developmental stage of the programme(2) the entity to be evaluated(3) criteria(4) data type(5) mode of data summary and(6) role

1.3.4.2. Roles played by curriculum evaluation

Obviously, the role played by the curriculum evaluation programme is dependent on the state of the curriculum development programme. Lewy (16 7) has identified the different roles played by evaluation programmes according to the stage of development of the curriculum programme. At the initial stage of the curriculum programme, general aims are determined. The evaluation programme can assist at this stage by assessing the expected changes, cultural valued, social forces, present level of achievement, and feasibility of programmes. At the tryout stage of the curriculum development programme, evaluatory studies can assist by collecting, evidence by observing, judging,

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discussing with teachers and students, etc. On the basis of these evaluations, the curriculum programme may be modified in successive stages of development.

1.3.4.3. Different types of evaluation

Evaluation may take place at different stages of the curriculum development programme, and may focus on different aspects of the curriculum. Different types of evaluation are therefore necessary, and their use depends on the judgement and inclinations of the evaluator. Among the different types of evaluation that may be carried out are formative evaluation, summative evaluation, and new wave evaluation, e.g., illuminative evaluation, democratic evaluation C29 8,

Formative evaluation is carried out during the course of the curriculum development programme with a view to clarifying and adjusting the curriculum programme or the materials produced by the curriculum project. It is 'a part of making a curriculum' (296). Formative evaluation is, therefore, a necessity throughout the tenure of the curriculum development project.

Summative evaluation is carried out on the completed curriculum project, usually by potential users who want to decide which curriculum to elect, or whether a certain curriculum will suit their purpose. The method employed for this is usually the biological model, ' where experimental groups of pupils use the project's materials and teaching methods, whereas controlled groups of pupils are subjected to traditional materials and teaching methods. The two types of groups are subsequently tested and the results compared. However, this method quite often does not yield significant results. Also, the project material is often radical, and so does not have a comparable counterpart in the existing syllabuses. In any case, the merit of the curriculum, as judged by summative evaluation, is subject to the needs of the user C298).

Illuminative evaluation is a result of the increasing realisation of the necessity to place more emphasis on the context of the curriculum and the learning situation than on measurable aspects of evaluation and psychometric methods so common as in the biological model mentioned

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earlier. The task of the illuminative evaluator is to take note of the complex situation in which the curriculum project takes place, and thence to identify causes for certain situations or happenings and their effects, and to find out the different relationships and patterns between various aspects of the curriculum from different participants of the curriculum process (194). In order to undertake this complex task, the evaluator takes recourse to several strategies and procedures, which may include psychometric methods. The evaluator, therefore, strives to illuminate aspects of the curriculum rather than only measure whatever can be measured. For this he/she adopts the 'Social Anthropological Paradigm'

Though this method of evaluation has been criticized for lack of organized methodology ( I “b*r ), Parlett ( I 9 5 ) has argued against overemphasis on measurability, and even suggested ways in which the criticisms against qualitative contextual analysis of data may be circumvented. Bryman ( 4-1 ) too, has suggested ways in whichqualitative and quantitative research can together give a general picture of the subject of research.

Qualitative (or illuminative) curriculum evaluation can also be termed as 'educational criticism' (2.97). According to Willis (197) the empiricist and naturalistic forms of enquiry required for qualitative evaluation can be of case study format and can incorporate four phases of criticism.: observation, description, interpretation, and appraisal. Willis emphasises that in carrying out qualitative evaluation, a total evaluation of the curriculum programme is possible by taking account of its total context. Hence, reflection about the causes, effects, etc. about the curriculum is possible. However in carrying out this subtle form of evaluation, Willis does not disregard the role of ennumeration. According to Willis, one of the main advantages of qualitative evaluation is that it can be carried out by almost anybody who cares - teachers, students, etc.

Case-Study method of evaluation - is a non-traditional method of evaluation. It is essentially qualitative-contextual in nature, though it does not preclude quantitative measurements. Adelman, Jenkins and

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Kemmis ( I ) have listed some of the advantages of case-studies :

Case-studies are based on direct experience, and therefore credible, though difficult to organize. Their credibility base gives a natural basis to 'generalisations'.

Case-studies handle the complexities of the situation, and give a subtle picture offering alternative interpretations.

Case-studies form a bank of descriptive material that may give opportunities for alternative interpretations at a later date or to different evaluators or for different purposes.

Case-studies orginate from the 'action' and so are capable of a variety of uses as feedback, for example, for administrative purposes * improvement of teacher efficiency, information about socio-economic status of students etc.

Case-studies, though often lengthy, contain information more accessible to people and so contribute to the democratisation of thecurriculum process.

The case-study method of evaluation is therefore interpretative and subjective and suggests 'participant observation' C^l,,

Democratic evaluation is another version of a 'new-wave' evaluation technique. Like illuminative evaluation it is also based on social anthropological models. It emphasises participation and right toinformation of other parties apart from the sponsors of the curriculum development programme about the evaluation process.

Apart from the above mentioned techniques of evaluation, other techniques are often applied as per necessity.- One of. these islongitudinal evaluation, by which the curriculum is evaluated over a period of time to judge its effects on students and on education, ingeneral, and to propose changes where necessary.

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1.3.4.4. Illuminative evaluation for the mathematics curriculum

In the case of m athematics, it often seems an attractive and easy proposition to evaluate in a manner that is totally dependant on

measurements of stated objectives. Yet, the way in which certain concepts may be imbued by pupils may be of greater interest to the curriculum developer than the actual skill learning of those concepts. For this reason, and others, illuminative evaluation, preferably by means of case studies may prove to be more effective than the analysis of test scores alone. Collecting data for qualitative evaluation, however, may prove to be difficult to procure. It may involve observing classes, talking to individual students, teachers, and even parents, as well as going through pupils'written work very carefully (66,76).

1.3.5. Dissemination and implementation

When a curriculum has been developed to stated intent, it is sought to distribute or propagate it among potential users. This distribution or propagation of the new curriculum is referred to as dissemination. Dissemination is often affected in stages, starting from pilot studies and gradually increasing the range to include the anticipated clientele. However, dissemination without adequate implementation may result in faulty adaptation of the new curriculum, and hence cause unwarranted dissatisfaction. To implement, therefore, is to see that the new curriculum is adopted according to the guidelines and ethos of the curriculum development project. It is like 'after sales' service (IT.7).

1.3.5.1. Strategies of dissemination

Dissemination and implementation of a curriculum is often dependent on the managing style of the curriculum project or the educational system for which the innovation is intended. For example, the curriculum can be determined and or developed at the centre and then transmitted by channels to the periphery. This is usually the case when the curriculum is centrally controlled or the government supports the curriculum development project and mandates its use. This is

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termed as a 'power-coercive' strategy. 'Modem Maths' was introduced to France and FDR (VI. Germany) in this way. By the term 'centre'here, is meant the origin of the power of fiat. It can be the national, state or local government, for example O'2-"*)*

Another instance of development and dissemination can be seen when a curriculum is developed at the periphery, and then transmitted to other users through the centre. This is usually the case whenteacher-made materials are put into a central collection bank and managed by a curriculum development centre. Examples of this type of dissemination can be seen in the case of IMU (IT^).

A third form of dissemination may be seen in cases wherecurricula are developed outside the educational system and then spread through the system via the centre from points on the periphery.Experts in the discipline prepare materials which are then made available to teachers.

De-centralized systems like in the U.K. often develop curricula at the periphery, and these are then gradually disseminated from the periphery. This happens when the status of the teacher includes that of decision-maker, and so teachers in these systems are expected to take on the role of innovators.

Other de-centralized systems may develop curricula at the centre for possible emulation by individual schools.

Apart from power coercive strategies employed by disseminaters is the use of emotive pressure on teachers. For example, using phrases like 'If you care for the pupils you would adopt the new innovations'.

In centrally controlled educational systems, innovations are often encouraged by advertising a higher pass rate for the new syllabus in the public examinations; or, teachers teaching the innovative syllabii are given higher pay.

The 'rational-empirical' ( 12.7 ) strategy for dissemination is to demonstrate the educational benefits following the adoption of the innovations. This is possible usually in de-centralized systems of education. An example of a project following this mode of implemen­tation is I.M.U. (Mdividualiserad Matematik Undervisning, Sweden).

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The re-educative strategy ( 12.7 ) of dissemination is generally associated when in-service training is closely associated with the curriculum development project (as in case of the strategy adopted by the Association of Teachers of Mathematics in U.K.). It involves attempting to change the attitude, skills, and values of those within the educational system, so that they may be favourably disposed towards the objectives of the innovation.

1.3.5.2 Indian Context

In India, curricula in the main are devised and implemented by centralized national and State boards of education. Grassroots retros­pections about the curriculum and initiation of reforms are therefore minimal. Therefore development of curricula at the periphery is not

common.Rigid hierarchies, massive population and far-reaching areas often

make implementation of new curricula a nonreality. In-service training is hardly forthcoming, and paucity of funds limit the use of specialized apparatus. Intentions behind innovations are remote from the average school/college and teacher. Innovations, therefore, whenever accomplished (not always in the true sense) are through dicta from the powers that be. This is not to conclude that centralized systems of curriculum development are always defective'( 12.7 ). The most obvious benefit of centralized systems is uniformity of methods and standards, and the pooling of scarce resources , (usually through Government patronage) for maximum efficiency.

Implementational difficulties are sought to be alleviated by different Government agencies. The NCERT publishes material on innovative course work, while the U.G.C. is sponsoring refresher courses for practising teachers.

The Central Committee of the Government has recently recommended decentralizing the 'process of curriculum formulation and preparation of text books', supervised only at the local level, so that teachers can have the opportunity to develop curricula that are best suited to the local circumstances and needs (2.77).

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In the case of mathematics, curricula that is tailor-made to the abilities, interests and resources of children in a particular school can be the ideal. It would mean that teachers participate actively in the curriculum process and hence bear the true brunt of being professionals. At the same time, with scare resources, total abdication of control may spell mayhem.

1.4 Designs for curriculum change

The order in which the different areas of a curriculum develop­ment project may be tackled is decided by means of curriculum models. These models indicate plans for curriculum designs. The optimum model to be adopted depends on the circumstances and purposes of the curriculum, as well as the inclination of the curriculum developer. The older idea of the curriculum may be illustrated by the following model :

TeacherCurriculum------------ ->-j

MethodPupil (162.)

One of the earliest models of curriculum planning is that- ofTyler's (%!&)',-

(Arrows indicate the order in which the areas are to be developed-).

: 31 :Wheeler's model develops from the above ( 1^1 )

-----

1. Aims, goals and objectives

2. Selection oflearning experiences

/ \5. Evaluation

\ 3. Selection of content

4. Organisation and ‘ integration of learn­ing experiences andcontent

Both models preempt the need to make clear-cut objectives at the outset of the curriculum project, so that students and teachers may know what exactly is required of them, and that evaluation may take place likewise. The cyclical order of the arrows indicates scope for readjustment of objectives following evaluation.

Criticisms about the above models :-

Predominance given to formation of exact objectives leading to noting behavioural objectives which can be readily observed and measured, to the exclusion of other objectives (like affective objectives).

The validity of the initial set of objectives is taken for granted.

Continuous interaction of the different areas of the curriculum process is ignored.

' Ends1 are decided before deciding on the ' means'. It is as if the models are removed from the complexities that make up the real life school situation.

Restriction of teacher flexibility.

Make education fragmentary and encourage rote learning (i4-8,'2.4-c?/n5).

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Leaving evaluation until the final stage of the curriculum process is rather like doing military intelligence after the war is over - in other words, evaluation should take place at every stage (162.).

The criticisms encountered by the above models were sought to be countered by more recent models like that of Kerr (f 50 ).

A Mode! for Curriculum Theory

This model strives to formulate objectives, and at the same time relate the different areas of curriculum development, and take note of the external and internal forces that affect the curriculum.

The three models discussed above, viz., that of Tyler, Wheeler and Kerr give pre-eminence to the formulation of objectives. These models are basically meant for the " research —^ development --^ diffusion"design of curriculum. Yet, later developments show a tendency

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to veer away from models of the whole process of the curriculum to concentrate on its diffusion and acceptance. This is because, no matter how good an idea is, it has to be made acceptable to the clients, who are mainly the teachers. Therefore, a combination of a 'social interaction' model and a 'problem-solving' model which involves teachers and educational administrators in a more active way and tackles the problem of overcoming individual and group resistance to change is often proposed (2.75). This type of model may be tactfully utilized in bringing change in sensitive areas of the curriculum or in backward areas of the country.

Stenhouse's 'process model' of the curriculum (267), while not rejecting the objectives model altogether, stresses on specifying content and principles of procedure. Content, according to this model, is to be selected on the grounds that it represents a particular form of knowledge which is intrinsically worthwhile, or that it reflects key concepts and criteria inherent in a field of knowledge. Teaching methods, consistent with the principles, concepts and criteria of the discipline are to be devised, so that the pupil may be inducted into the knowledge of the discipline, and know it from the 'inside'. The end-product of this process cannot be specified, but can be evaluated after the event by the criteria built into it.

Stexihouse claimed that this model was not appropriate for areas of the curriculum centering on knowledge and understanding and demonstrated its usefulness in the Humanities Curriculum Project (268). Though allowing for limited use of the objectives model, this design of curriculum development proved to be close to teachers' concerns, and at the same time, not presuppose any form of linear treatment of its components as in Tyler's and Wheeler's models. However, it did come under a number of criticisms, viz., lack of planning, difficulty in assessing pupils, problems of teacher competence (because teachers needed to have a deep understanding of the subject), demanding on the teacher and therefore difficult to implement (2.75).

The 'process model' may partially be used in the field of mathematics. Arriving at relations between different mathematical topics, realizing the structure of a certain field, arriving at

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mathematical formulae by inducting from real-life situations, for

example, may require persuation of the process model. In other words, the mathematical way of thinking may be induced by the 'process model'. However, to understand mathematics, students need a certain amount of deftness in handling concepts learnt. For this the objectives model may not be abandoned altogether.

The situational model was developed by Lawton and Skilbeck.Lawton's idea is illustrated in [162. ). It acknowledges curriculum as a drivative of cultural values and circumstantial concern. Lawtonand Skilbeck's model has five major components :-

(a) Situational analysis;- external factors - social changes, ideological shifts, parental and community expectations, changing nature of subject/discipline, teacher support like colleges and universities etc.

Internal factors - pupils and their attributes, teachers and their knowledge, skills, interests, etc. Also, school ethos, political structure, materials, resources, felt problems, etc.

(b) Goal formulation;- embracing teacher and pupil actions (not necessarily behavioural).

(c) Programme building ; - selection and sequencing of content andexperiences, staff, choice of supplementary material, media, etc.

(dj Interpretation and implementation -

(e) Monitoring, assessment, feedback and reconstruction - This is a wide concept of evaluation.

Advantages of the situational model are :-

* A comprehensive model, encompassing the process model and objective model - depending on which aspects of the curriculum are being designed.

* Flexible, adaptable, open to interpretation in the light of changing circumstances.

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* Does not presuppose linear progression of components.

* Forces the curriculum developer to consider the context and link decisions to wider social considerations.

1.4.1. Some other models

Several views of the process of curricular reform are held. Some of them have been mentioned above. Yet other views have been given precisely in diagramatic form

INPUT OUTPUT(14*0

(5-2.)

(c)

(d)

(272)

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▼(f) Subject

Child ( | q )

(g) Committees

MediaCommittees

( 2.7 )

(h)

(i)

The above figures show the interactions with the different factors

influencing innovations in the curriculum and curriculum development. They require to be kept in mind whilst initiating pragmatic and

effective changes in the curriculum.

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1.4.2 The mathematical context

When considering a model for curriculum development in mathematics, the objectives model presents an easy option. Yet, with increased awareness about the importance of understanding, applicability, problem-solving and enjoyment in learning mathematics over and above skill learning and knowledge accumulation, the importance of adopting some sort of process model is often desired. Also, awareness of the differences between different populations and increased teacher professionalism is culminating in grass-roots and peripheral origins of curriculum development, which in turn, are taking up the situational model of curriculum design. Perhaps the latter may be only a realization and realistic reaction to curriculum design. All the same, clear thinking is desirable, and if this means chalking out objectives at the start of the project, so be it - so long as these objectives are not allowed to dominate the entire process creating inflexibility, or allowed to fritter down to trivialities realized through hackneyed questions and answers and subsequent rote learning.