This paper argues that
The term Technocratic Ideology has been used to describe an idealised (not an ideal) set of beliefs and values along lines similar to the following. Schools are like pieces of machinery which produce outputs with certain standards. The outputs are students who have demonstrated outcomes at specified levels of achievement. The machine is adjusted by defining these outcomes, levels and standards. Teachers have unwanted interests in education and need controlling. They are variables in the machine and must be `set' to produce the required outputs. Teachers are further controlled by ensuring that there is a close link between the achievement of the outputs, and remuneration. Educational costs are reduced and quality is maintained by ensuring schools compete for students and therefore for funding. Schools are required to decide how they are going to produce the required outputs and prove that they are in fact doing so. Curriculum development within this ethos is based on the definition of outputs, standards and methods of national testing.
This set of beliefs and values about education and schooling, and others similar to them, have been widely condemned by educationalists (see for example, Ellerton and Clements, 1994; and Ernest, 1991). But they continue to have an impact on mathematics education. In a growing number of countries this approach to education (perhaps in a slightly watered-down form) is being imposed on schools by government and quasi-government organisations. Two of the down-stream effects are worrying.
Firstly, some teachers seem to be increasingly accepting that this approach to education is the only approach. The mechanism for this change in belief and values shouldn't be described as conversion; it is more like absorption. Some teachers are being coerced into changing their language and practices; their beliefs and values are subtly changing in accordance with this. These teachers could well be unaware that their language, practice and increasingly their beliefs and values are changing to correspond to the imposed view. Secondly, classroom environments are changing. I see increasing movement towards output driven, neo-behaviourist, approaches to pedagogy, and an increasing distortion of assessment practices towards those which aim solely at assessing and recording the achievement of objectives. I call this combined effect the `magnet effect' of the imposed ideology - teachers beliefs and practices become drawn into those of the dominant ideology. It is caused by the dominance of the language associated with neo-behaviourism; the language of objectives, outputs, accountability, levels of attainment, and so on. This language, especially when ordained as orthodoxy by powerful governmental organisations, becomes common in teacher talk, gradually leading to the marginalisation of those concepts which remain unexpressed.
So, can anything be done to protect the integrity of mathematics teaching under this threat? I don't know. But if anything can be done it must involve classroom teachers. Teachers would be better able to creatively interpret the legislative restrictions placed on them if they knew more about, and could better articulate, other competing ideologies and their related approaches to classroom teaching. Teachers could learn to adapt the anti-educational components of the dominant ideology, and write policy statements for their schools which protect good teaching practice.
A framework outlining a range of ideologies and practices can be used by teachers to help them analyse their working environment. But what sort of framework will be most suitable? The choices would seem to include: A framework of educational ideologies, such as Skilbeck's (1976), a framework of ideologies of mathematics education, such as Ernest's (1991); a framework of learning theories in mathematics education, such as Resnick and Ford's (1981); or a framework of curriculum approaches to mathematics teaching. Whichever framework is chosen for initiating discussion, the resulting analysis should eventually include discussion of the key elements of the others. The question is, which of the options is the best point of entry for teachers? Which framework best uses the `currency' of teacher dialogue, concepts and practice as its basis? I believe that a framework of curriculum approaches, although more complex and difficult to describe, is the one which best links with the intuitions, culture and practice of teachers. Such a framework uses, as the point of entry, the complex of integrated concepts and perceptions which teachers routinely use in the classroom. It recognises that teachers have a unique feel for classroom activities, and for the student work and dialogue which accompany them.
So what is a curriculum category? It is a description of a particular approach to classroom mathematics teaching. It is a loose amalgam of a number of features with one or two being dominant. It is not a neat, systematic, arrangement; just as the nature of teaching and learning is not neat and systematic. However, it does need to be recognisable to teachers. They should be able to say "Yes I have seen a lesson which clearly comes from this category" or "I can imagine using the activities from this category to teach mathematics." Each category in the framework needs to be distinct from others, and each needs to have one or two features which give it a focus.
The framework I shall propose will include classroom approaches which are consistent with the Technocratic ideology; it will also include approaches which are inconsistent with this ideology. I shall identify a set of beliefs and values with each category by linking them with the ideolgical positions used by Ernest (1991) in his analysis of beliefs and values in mathematics education. Ernest identified five ideological positions and associated them with five social groups: Industrial Trainers, Technological Pragmatists, Old Humanists, Progressive Educators, and Public Educators.
The framework of curriculum approaches I am going to propose will use Keitel's classification, outlined in Howson, Keitel and Kilpatrick (1981), Howson (1983) and Bishop (1988), as its basis. My intention is not so much to classify actual curriculum projects, but to present `ideal types' which represent particular approaches to curriculum design and teaching practice. Accordingly, I will modify Keitel's classification to make it more idealised. For example, when I describe the New Maths category below, I will not be attempting to describe the features of all the classroom programmes which developed under this name, but to describe the features of the purest new maths projects. I have `purified' Keitel's other categories in a similar way.
The New Maths approach had its origins in mathematics itself and showed little concern for pedagogical matters. It arose as an extension of the emphasis, within the philosophy of mathematics at the time, on set theory and logic. Meaningfulness was viewed as subsidiary to the presentation of mathematics as a fully unified structure based on sets, operations and relations. The intention was that this would give the subject a foundational, conceptual, unity and better facilitate the achievement of the goals of mathematics education. Although the intention might have been to make mathematics accessible to all, in reality it was often directed at the more able, and others were left floundering, with some eventually moving into courses on basic numeracy for citizenship. There was little concern for individual learning styles or needs, and it lost acceptance because of this, the abstract nature of the mathematics, and the emphasis on symbols and jargon. It could be argued that the New Maths category is no longer needed in a framework. I believe it should remain because many teachers and parents, and some school texts, are still influenced one way or another by its approach to mathematics education. This approach links with Ernest's Old Humanist ideology.
This category has origins in educational psychology and adopts a mechanistic approach to teaching. It has links with Thorndike's Associationism, Skinner's Behaviourism, Bloom's Mastery Learning and Taxonomy, Keller's Personalised Instruction, and Gagne's Learning Hierarchies. There is an emphasis on outputs, the end points of learning. Mathematics is broken down, some would say trivialised, to a sequence of tasks to be mastered. It is viewed as facts to be learned and skills to be acquired. The focus tends to be on what people can do, rather than on what understandings and meanings have been achieved. This approach can be seen as anti-mathematical if one sees mathematics as rule challenging, or rule transcending, rather than rule learning, and as a process of conjecture and refutation based on the formation and analysis of mathematical models and proofs. In a secondary form behaviourism appears as mastery learning. Standards based assessment with levels of achievement is often described in precise `output' terms, and is therefore also behaviourist in form.
Behaviourist approaches were, in part, designed to transform education from a labour intensive process to a capital intensive one, and for this reason, and the emphasis on outputs, it is closely linked to the Technocratic ideology mentioned above, to Ernest's Industrial Trainer ideology, and to a lesser extent the Technological Pragmatist ideology.
The early proponents of behaviourist and mastery learning approaches had sound educational goals. In contrast to some other approaches, which had pre-deterministic overtones, they believed that students can learn almost anything given enough time and proper prerequisite learning. Therefore if teachers arrange all instructional tasks into their proper learning sequence, almost all students will eventually be able to accomplish each and every one. If each student is taught until she masters a particular learning unit, she can then be sent on to the next task or goal. Students are not compared with each other, and teachers wait until a student has mastered all the material before reporting this fact. Accordingly, students do not experience the feelings of failure associated with norm referenced test results.
Unfortunately, the benefits of these educational goals were overshadowed by the negative side-effects of the means adopted to reach them, and behaviourism has been widely condemned by mathematics educators (see for example, Freudenthal, 1978; Erlwanger, 1973; Clements and Ellerton, 1993; Ellerton and Clements, 1994). Behaviourist techniques do enable the efficient achievement of low order skills. Their use more widely inhibit the intuitive construction of ideas and the examination of misconceptions, and is therefore of dubious merit as an approach to education.
So, in view of the strength of the criticism of behaviourist approaches, why are they still so common? The reason for this cannot be completely laid at the door of recent developments in output driven curricula. Skemp (1976) listed three `apparent' advantages of using a form of teaching which emphasises the learning of rules:
However, technocratic approaches to education are at the heart of many other reasons for the ubiquity of neo-behaviourism. Freudenthal (1978) points out that mathematics `appears' to lend itself easily to being described as a sequence of behavioural objectives, and so those who wish to technologise instruction use mathematics as the exemplar of this approach. He describes this atomism of mathematics as a `caricature'. The `accountability industry', is in full swing. Teachers are required to demonstrate to outside agencies that they are teaching the curriculum, and that students are learning the objectives. Faced with these requirements it is small wonder that teachers adopt reductionist and behaviourist approaches, and use as a rule-of-thumb the adage `if you can't measure it, don't teach it.' Much of our educational thinking is still influenced by reductionist notions of knowledge. The language and concepts associated with holistic, and socially and culturally based views of knowledge remain uncommon. Many teachers are aware of the `miraculous' claims made by proponents of mastery learning approaches (for example, that 90% of students will score at the level formerly reached by only 10% (Bloom, 1968), but are unaware that these assertions have no validity whatsoever (see Freudenthal (1979), Ellerton and Clements (1994) and Neyland (forthcoming) for discussion of the validity of Bloom's claims, and the validity of other claims made in support of mastery learning). Many teachers greatly overrate the importance of sequencing and hierarchy in mathematics teaching (see Freudenthal (1978 and 1979) for a discussion of the application of mathematics to Bloom's Taxonomy, and Ernest (1991) for a critique of hierarchy in mathematics more generally.)
This approach has origins in both mathematics and cognitive development theories in psychology, and arose out of the work of genetic epistemologists who were examining the process of concept formation. The focus here is on the identification of the structures and processes intrinsic to the way mathematicians operate. It is believed that these structures and processes will promote learning in an optimal way. Learners explore and `discover' these structures via a series of embodiments and through a `spiral' programme. The structures provide the subject with an internal cohesiveness and unity. The best known proponent of this approach within mathematics education is Dienes who built on the work of Bruner. Bruner was interested in both the processes and concepts which underpinned the discipline concerned. Dienes, however, focused more on the concepts. I will use this latter feature as the distinctive approach in this category and leave the consideration of processes to other categories. The concern for the personal learning styles of the individual and attempts through the use of equipment and `games' to help students to explore and hence form mathematical structures, links this category most closely to Ernest's Progressive Educator ideology. This approach can be criticised for sometimes using unlikely and often confusing embodiments and equipment in the attempt to help students to `discover' predetermined structures. It can also be criticised for not putting enough emphasis on students forming their own structures.
This approach has origins in developmental psychology and focuses on the natural structures of personal development. It is learner-centred and aims to match learning opportunities in mathematics with natural cognitive abilities and affective characteristics. Formativists built on the work of Piaget, a genetic-epistemologist, who emphasised that learners actively construct their own knowledge, and that the nature of these constructions depends on the stage of development of the learner's natural thinking structures. There is a focus on beginnings, rather than on ends, and so this approach sharply contrasts with behaviourism. Learners are expected to construct their own mathematical structures and this requires the teacher to give high levels of individual attention. Where this cannot happen the learning can become unfocussed. The approach is not based on a didactic mathematical structure, and so there is a danger that it can degenerate into a variety of unrelated activities, particularly if the teacher lacks the mathematical knowledge to help the students make connections between their emerging ideas. This approach is aligned to the Progressive Educator ideology, too. Mathematics is viewed as a body of knowledge `out there' to be discovered through personal experience.
This approach is based on the attempt to teach mathematics in contexts integrated with other subjects and the environment. This concept of integration is stronger than mere applications or context based learning. The approach is multi-disciplinary. Subject barriers are down-played, knowledge is seen as an integrated web. The environment provides both a source of inspiration and of meaning for the learner. The learner must work with more general structures than those of mathematics. Teachers using this approach commonly rely on mathematical modelling, statistics, thematic units, and project work, in an attempt to develop mathematical ideas in context. Mathematics is seen as a strand of concepts which can be discovered by exploring problems in the environment. The emphasis is on the individual learner constructing concepts from contexts familiar to them. Accordingly, this approach is consistent with the Progressive Educator ideology. This approach has all the difficulties associated with the Formative approach.
I propose, now, to extend the above framework of five categories to include three more. In each case I need to justify the inclusion of a new category by outlining the unifying features and showing that it is sufficiently distinct from other categories. The three categories I propose to add are the
This approach became a focus for attention during the early 1980s when NCTM published the Agenda for Action (1980) which called for problem solving to become the basis for all learning of mathematics. Mathematics educators were able to use the work of Polya (1945) as a source of strategies for problem solving and investigation. Many countries have now joined the move to incorporate problem solving approaches into their curricula. The term `problem solving approach' means different things to different people and I am going to describe the narrowest of these approaches within this category. Other approaches to problem solving are able to be identified as combinations of categories in the framework.
The Problem Solving category has problem `solving strategies' as key processes within mathematics; in fact, ones that make many of the procedures taught in other approaches redundant. Typically, these strategies, such as `solve a simpler problem first', `work backwards', and `try extreme cases', are taught as a series of techniques which are then used within a range of content areas to solve problems. There are a number of books on the market which are devoted to teaching problem solving in this way. Mathematics is viewed as an area of knowledge to be explored using these generic strategies; content becomes a context within which the problem solving processes are developed. The problem solving strategies enable the students to solve traditional mathematical problems using more generic approaches.
Because Problem Solving, as defined here, appeals to those with utilitarian goals, and mastery approaches are often used, this category is consistent with the Technocratic ideology, and with Ernest's Industrial Trainer and Technological Pragmatist ideologies. A smaller group of teachers take a more learner-centred approach and develop the key concepts in ways more consistent with the Progressive Educator ideology.
The approach has weaknesses. It can be criticised for reducing mathematical thinking to just another series of skills to be learned and practised. It is not clear exactly how content knowledge is to be dealt with. This latter difficulty has led many teachers to treat problem solving as a stand-alone unit alongside other content units.
Is this approach distinct from the other approaches mentioned? It has similarities with the Structuralist approach, but I characterised that approach as being concerned mainly with content structures. This approach is based around a process structure, and the teaching approaches seldom resemble those typified by Structuralism. There are similarities with the Behaviourist category because many teachers use mastery approaches for teaching problem solving strategies. However, the Problem Solving approach tends to down-play many of the rules and procedures learned through the behaviourist methods and replace them with more generic strategies, and it has counter-hierarchical elements (for example, it emphasises that there are many ways of solving problems). The approach as defined is not similar to the Integrated Environmentalist approach, because the teacher has a well defined didactic structure for presenting the strategies.
This category is not reflected in typical classroom practice. It is placed in the framework for two reasons. Firstly, it represents the educational aims of the small group of teachers who work closely with indigenous peoples. Secondly, the beliefs and values underpinning this category are at variance with those in other categories and provide a contrast. Some minority cultural groups are asserting their cultural identity and seeking to improve the achievement of their members by establishing firm links between their culture and mathematics. One approach is to acknowledge that all cultural groups have developed their own approach to mathematics, with western mathematics being only one case (and even this has been enriched by other cultures).
This approach views mathematics as a cultural product based on certain unifying activities. It is not essential that one particular set of activities be universal for all cultures. What is important is that the activities reflect mathematical thinking, and be harmonious with the cultural context concerned. Bishop (1988) suggests six groups of activities: counting, locating, measuring, designing, playing, and explaining. A particular cultural group identifies the ways it uses these activities within its culture to develop mathematics, and uses these as the structural elements upon which mathematics can be constructed. The concepts developed would eventually include those of school mathematics. This approach views mathematics as a social product. The Cultural approach has emancipatory aims and so aligns with Ernest's Public Educator ideology.
This approach has much to recommend it. The learning is context based and grounded in the cultural practices most familiar to the students. It recognises as valid, mathematical activities which might otherwise have been disregarded by the dominant culture. The emphasis on mathematics as a cultural product is empowering, reduces the mystery many associate with mathematics, and gives traditional school mathematics a useful base upon which to build.
Is this approach distinct from other approaches? There are similarities to the Structuralist approach with the structures being the groups of activities. But the emphasis on using contexts drawn from the culture, the conception of mathematics as a social product, and the emancipatory aims give this category a different flavour. There are also similarities with the Integrated Environmentalist category, but this lacks the structure provided by the unifying activities. The approach has many difficulties associate with it. The whole curriculum has to be redesigned from scratch. The teachers need to be very confident and secure with their own mathematical knowledge in order to develop it afresh with a new structure.
This approach is also placed in the framework as a contrast to others, in particular Behaviourism and the practices leading from the Technocratic ideology. I shall argue, later, that practices associated with Social Constructivism resist the `magnet effect.'
Mathematics is seen as a social construct. Mathematics education is seen as a sense-making, activity through which students socially reconstruct the knowledge of the past for the new age. This process is similar to what has been termed enculturation (Bishop, 1988; Schoenfeld, 1992) and socialisation (Resnick, 1989), and has a base in an area of cognitive psychology which sees cognition as a social phenomenon, and another in social constructivist and quasi-empirical approaches to the development of mathematical knowledge (Ernest, 1991; Lakatos, 1976).
Enculturation, as I have defined it, is a reconstructive and not a reproductive activity. The group of students is thought of as a fledgling mathematical community which is being enculturated into the expert mathematical community. The teacher has the responsibility of aiding this process and is seen as an agent of cultural renewal. This expert community encompasses more than academic mathematicians; it includes all competent users and developers of mathematics in all its conceptualisations. Enculturation involves learning the concepts, orientations, values and processes of the expert community, and seeing none of these as beyond examination and revision. It involves learning the way ideas are explored; the processes of generating, justifying and validating knowledge; the way it uses criticism-aimed-at-consensus, falsification, proof and observation; and the way knowledge has a taken-as-shared quality. Mathematics is presented as an integrated web of knowledge, and as a problem posing, problem solving and investigative activity, through which students build on existing achievements to create new understandings and a better future. They reflect on their mathematical experiences, and take a major role in validating their mathematical ideas. They form their own structures for the discipline using their interpretations of the existing ones as part of their reference frame. They examine the value base, and learn about the way mathematical ideas have emerged in the past. They take on the perspectives of the mathematical community, but do it with a measure of understanding about the place of mathematics in society. Full use is made of advances in technology, both as a way of reducing the time spent on routine mathematical procedures, and as a tool for exploring mathematical ideas. The Social Constructivist approach to knowledge is emancipatory. It is consistent with the Public Educator ideology.
Is this approach distinct from the other approaches? The approach has similarities with the Structuralist, Integrated Environmentalist, Problem Solving and Cultural approaches, but there are sufficient differences from each of these to warrant treating the Social Constructivist approach as a valid alternative. The process of enculturation is similar to the process of forming structures in the Structuralist, Problem Solving and Cultural approaches. However the process of enculturation is more comprehensive than any of these approaches involving a combination of processes, ideas, values, and perspectives, and there is an emphasis on recreating structures, not imposing them. The Integrated Environmentalist approach also encourages learners to form their own structures, but the Social Constructivist approach is much more based around the discipline of mathematics and its dynamic interaction with the community of mathematicians than the multi-disciplinary Integrated Environmentalism. The emancipatory aims of Social Constructivism, and the conception of mathematics as a social construction, also make it distinct from most other categories.
The approach has shortcomings. It requires the teacher to be aware of the perspectives, approaches, attitudes and processes of the expert community, although this is not saying that the teacher has to be as technically proficient as a working mathematician. There can be problems when applying the notion of `fledgling community' to young children who may not have the social skills to act as a mini-community.
Firstly, Social Constructivism is inconsistent with output, technique-oriented, mechanistic, and behaviourist approaches. Social Constructivism makes full use of all the developments in computer technology. It is recognised that computers perform most of the techniques formerly taught using behaviourist approaches. Accordingly, less attention is placed on technique development and more is put into the aspects of mathematics which do not fit under the `technique' umbrella. Social Constructivism emphasises mathematics as a way of seeing the world - a way of knowing - rather than a way of doing and a series of techniques. Enculturation is an interactive, interpersonal process involving the use of humanistic and organismic, rather than mechanistic, approaches. It focuses on shaping ideas and meanings not behaviours and techniques. Social Constructivism has emancipatory aims, the Technocratic ideology has, at best reproductive, and at worst elitist, aims.
Secondly, the Behaviourist approach inhibits the achievement of Social Constructivist goals. Teachers, pressured by time constraints, often give priority to the objectives which can be tested as outputs and leave out other goals, even though many of these other goals, for example, metacognitive goals (such as learning to plan a problem solving activity) would lead to more effective learning. Within Social Constructivism different points of view, errors and misconceptions become a focus for discussion and resolution. But with behaviourism skills are taught in sequence with little attempt to search for, or explore, misconceptions. Behaviourism encourages the students to learn that mathematics is a `one way' subject - the teacher's way; that mathematics is a series of facts and skills; that answers and methods will be provided by the teacher; that mathematics is handed down by experts; that one should always have a ready method of solution for any problem; that learning mathematics involves mostly memorisation; and that doing mathematics involves a lot of practice in rule-following. These are contrary to the aims of the enculturation process.
Thirdly, the aims of Social Constructivism cannot be met in output terms. The development of values, perspectives, metacognitive approaches, and other higher order concepts, all inherent in Social Constructivism, is not a linear process with well defined steps which are easy to assess. Social Constructivism seeks to help students learn number sense, a feeling for reasoning u0nder uncertainty, a predilection to quantify, and so on. These cannot be taught as a sequence of skills. Social Constructivism aims to have students engage in the continual reconstruction of mathematical knowledge for the new age. This cannot be described in output terms.
Tables 1, 2, 3 and 4 summarise the curriculum categories and the beliefs and values which are consistent with them.
| New Maths | Behaviourist | Structuralist | Formative | |
| Focus | Structure of mathematical knowledge | Neo-behaviourist approach to learning; learning hierarchies | Structures of mathematical knowledge; theory of concept formation; discovery | Natural processes of learning and child development |
| Ernest's Mathematical Education Ideologies | Old Humanist | Industrial Trainer; Technological Pragmatist | Progressive Educator | Progressive Educator |
| Aim of Mathematics Education | Aid learning of mathematics by presenting it as a highly structured discipline | Present mathematics as a hierarchy of small components; students master each component before moving on to the next | Present structures of mathematics in spirals of increasing complexity, and as embodiments which students explore and hence discover the underlying structure | Provide rich learning experiences to enable students to learn the mathematical concepts which are developmentally appropriate for them |
| Nature of Mathematics | A structured discipline based on an axiomatic foundation | A hierarchical sequence of skills and concepts | A structured discipline | A body of knowledge `out there' |
| New Maths | Behaviourist | Structuralist | Formative | |
| Approach to Learning | No particular approach to learning | Neo-behaviourist; associationist; learning hierarchies; mastery learning | Concept formation by discovery of the underlying structures embedded in embodiments | Constructivist |
| Role of the Teacher | Transmitter of knowledge | Administrator of a learning system | Presents mathematics as a series of embodiments; assists students `discover' structures | Provides learning experiences appropriate to the student's stage of development; helps them form structures |
| Strengths | Mathematics presented as a highly structured discipline | Students can learn at their own pace; no feelings of failure caused by norm referencing | Based on a concept of mathematical knowledge and a theory of learning | Based on a theory of learning; learning experiences as the starting point for learning |
| Weaknesses | Not based on any learning theory; meaningfulness seen as secondary to purity of structure | Trivialises mathematics; emphasises rule following and memorisation | Sometimes embodiments and equipment are confusing; predetermines structures | Can degenerate into a collection of unrelated activities; teachers need sound mathematical knowledge |
| Integrated Environmentalist | Problem Solving | Cultural | Social Constructivist | |
| Focus | Integrated knowledge; environment as context for learning | Problem solving strategies | Holistic and cultural reconceptual-isation of mathematical knowledge; context based learning | Students as a fledgling mathematical community; problem posing pedagogy; recreating structures |
| Ernest's Mathematical Education Ideologies | Progressive Educator | Technological Pragmatist; Industrial Trainer; (Progressive Educator) | Public Educator | Public Educator |
| Aim of Mathematics Education | Students learn mathematical concepts in their own way by exploring problems and situations found in their environment; modelling | Mathematics presented as a series of generic problem solving strategies; these are used to develop other mathematical concepts | Mathematics presented as integrated strands; strands are harmonious with the cultural context concerned; students develop mathematical concepts within this context | Mathematics presented as a problem posing activity; students are fledgling mathematical community; introduction, and reconstruction for own generation, of concepts, orientations, values and processes of the expert community; build on past achievements to create better future |
| Nature of Mathematics | Strand of concepts which can be discovered by exploring problems in the environment | An area of knowledge which can be explored using generic problem solving strategies | Develops from key activities carried out within a cultural context | A social construct involving values, orientations, processes and concepts, none of which are beyond examination; arising from problem posing, problem solving and investigation; using criticism-aimed-at-consensus, falsification, proof and observation for validation |
| Integrated Environmentalist | Problem Solving | Cultural | Social Constructivist | |
| Approach to Learning | Students construct knowledge from familiar contexts; contexts provide motivation | Mastery learning with a loose hierarchy; (Learning strategies via meaningful contexts) | Students construct knowledge from own cultural contexts; this provides motivation and self-esteem | Learning is a constructive activity based on problem posing, problem solving, investigation, and collaborative sense making; learning is an emancipatory form of enculturation |
| Role of the Teacher | Assists student to explore problems in meaningful contexts and hence form mathematical structures | Administrator of a leaning system; (Aid students develop strategies from contexts) | Assists student develop mathematical concepts using cultural context as starting point | Agent of cultural and social renewal; link between the expert and fledgling communities |
| Strengths | Uses meaningful context as a source and motivation for learning; students construct own concepts | Uses generic problem solving strategies instead of the larger number of rules and procedures; many problems set in context; allows `many-ways' approach | Uses student's cultural concepts, practices and orientations as the starting point for learning; these provide motivation and self-esteem | Students empowered by learning process because knowledge is not treated as something `out there' to be learned and accepted, but as something to be constructed and reconstructed; students fully involved in validating knowledge; problem posing allows the students to have control over their learning |
| Weaknesses | Can degenerate into a collection of unrelated activities; teachers need sound mathematical knowledge | Likely to degenerate into rule following and memorisation; content areas can be neglected | Teachers need sound mathematical knowledge and a sound knowledge of the culture concerned | Teachers do not need to be competent users of all the techniques of the expert community, but they do need to familiar with the basic orientations, processes, and values; they need an understanding of mathematics as an activity which occurs in a number of social contexts; they need to be secure with their own mathematical knowledge to allow an open exploration of ideas |
So how might teachers use curriculum categories such as the ones proposed above? The goal is the empowerment of teachers. The method suggested is based on the idea that teachers need to start with their own practice and move from there to consider the beliefs, values, habits and coercions which constrain this practice. Planned action for change must be based on such an analysis. However, the proposed framework is only one approach to an examination of teaching. I prefer to see a preliminary analysis of practice, structured entirely by teachers, before the framework is considered. I suggest a four phase cycle.
A number of objections could be raised about the ideas outlined above.
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© 1996 The Author