The following are complementary rationales for mathematical education.

Mathematics as a valuable aspect of our human heritage
Mathematics in a variety of forms is part of our multicultural inheritance. All learners should therefore enjoy an induction into the practice of mathematics in pragmatic, scholarly and recreational contexts and should gain some experience of and a taste for personally engaging in these forms of mathematical activity. It may be argued that personal engagement in mathematics is an intrinsically rewarding human activity adding to the quality of life. This implies that mathematical learning experiences should be intrinsically worthy experiences contributing to the quality of learners' and teachers' lives, i.e. the pursuit of quality knowledge and range and extent of learning outcomes should not be the sole criterion driving the design and realisation of mathematics education practice.

Mathematics for economic/social competence
It is often argued that mathematical competence contributes to economic development and provides access to rewarding economic activity. This view is debatable and should not be allowed to dominate the mathematics curriculum. However the mathematics curriculum should provide pupils with effective opportunities to attain basic mathematical literacy, i.e. to master the basic numerical and mathematical understandings and skills required for dealing critically with quantitative and spatial information in everyday life, and which provides access to further training and development.

Mathematics as a vehicle for personal development
Certain personal values, attributes, thinking and social skills facilitate productive mathematical activity Likewise, individual and collective engagement with mathematics may provide valuable opportunities for the development of a variety of values and personal and interpersonal skills In addition to this, there are a number of mathematical processes (e.g. hypothesising, logical argumentation) which may be of use beyond the subject discipline..

These three rationales should inform the mathematics curriculum at all levels. This means that if students choose to abandon mathematics at the end of General Education Certificate level they would do so having acquired basic mathematical literacy, and having experienced some authentic mathematical activity in a range of contexts.

A variety of stakeholders in society exert demands on school mathematics. These include parents, learners, teachers, mathematics educators employers, professional mathematicians, tertiary institutions, professions cultural and political lobbies and organisations. Stakeholders influence mathematics classrooms to a greater or lesser extent. There is always a potential for conflict between the rationale for mathematical education embedded in mathematical education as envisioned by mathematics educators, and demands exerted by other stakeholders.

We envision a practice in which the influences on mathematical education are constantly and explicitly re-negotiated in order to ensure coherent and effective learning opportunities in classrooms. Specifically, we would work for the ongoing clarification of the implications of demands exerted by different stakeholders.

Malati should therefore participate in the process of negotiating and articulating demands made on pre-school, school and ABET mathematics, as well as on the mathematical training of primary and secondary school and ABET teachers and teacher educators

The quality of learning in mathematics, as manifested in the assignment of authentic meanings, the appreciation of genuine purposes, the generation of sound justified and the acquisition of substantial know-how needs radical improvement. Apart from appropriate choices of content and clear and justified articulation of outcomes, this requires addressing one of the greatest challenges of the present time, namely rebuilding and reshaping the culture of learning in mathematics classrooms. Such a culture should address the challenges of social justice issues posed by sexism, racism and multilingual and multicultural education, and would involve acceptance of the following obligations by teachers and learners.

• provide opportunities for learners to engage with meaningful problems and to reflect, plan, explore, investigate, check, express, articulate, justify listen, discuss and practice, at their own pace without impatient, disturbing or intellectually-repressive interference;
• recognise the difference between mathematical conventions, observations and rational designs and to set up learning contexts which ensure that pupils experience each mathematical knowledge element for what it really is;
• provide ample opportunities for individual and interactive reflection by pupils;
• set tasks which enable learners to make meaningful connections between new and existing knowledge;
• view learners' mistakes and misconceptions as valuable opportunities for discussion and learning;
• cultivate learners' perseverance and willingness to tackle challenging tasks;
• encourage pupils to explain and justify their arguments rationally at appropriate levels, and require this of others in an atmosphere of mutual regard;
• engage with the mathematical ideas offered by pupils through questioning, discussion, negotiation and re-articulation; promote rationality and appropriate levels of rigor in justification;
• inform pupils at appropriate times about conventional forms of expression of mathematical ideas and information and encourage them to utilise these forms, taking care that pupils perceive these interventions as referring to forms of expression and not the substance expressed;
• seriously engage with their custodial responsibilities, namely ; the needs of learners, the organisation of safe, orderly and creative learning environments and, as members of a wider mathematical community, towards mathematics as a discipline.

• tackle novel and routine tasks with vigour and perseverance
• validate their own work individually and co-operatively by learning and using appropriate validation techniques, and by rational critical argumentation
• justify their methods and arguments

Assessment should be seen as a vehicle for promoting quality of learning, both as a vehicle for providing feedback to teachers on the quality and extent of learning achieved and hence on the effectiveness of their own efforts and as an instrument for accrediting the achievement of desirable outcomes. The actual effects of present and alternative assessment practices should be analysed and taken seriously.

At the present time, classroom culture and practices are driven in large measure by formal, examination-based assessment, often focusing on prescriptively-stylised performances that suppress real intellectual engagement and promoting narrow proceduralisation in an effort to "beat the test"

An alternative understanding of assessment seeks to move away from the idea that assessment only serves to provide a basis for accreditation, (differentiating performance), towards a more interactive, "every-time-along-the-way" view in which assessment is seen as a mechanism for teachers to reflect on their own practice, to monitor the performance of learners, and as away of encouraging learners to reflect on their own progress. Such an understanding also seeks to move away from question types and marking techniques that promote the proceduralising of stylised performance, towards question types and marking techniques which identify and reward genuine mathematical performance.

1. Content should be varied and rich to prevent learners from constructing a narrow and distorted perspective of the nature and significance of mathematics.

2. Choice of content should not be dictated by ease of assessment.

3. Learning contexts should be selected with a view to enable learners to assign appropriate meanings to and experience authentic purposes for mathematical contents.

4. The choice of content should provide access to different mathematical processes e.g.:

• modelling
• designing algorithms
• making definitions
• transformation (replacement with
• organising data
• making rational explanations and proofs
• structural abstraction
• choosing axioms
• organising and systematising
• error analysis

5. The choice of content should provide access to fundamental mathematical ideas e.g.:

• variables
• functions
• equivalence
• iteration (recursion)
• finite situations
• infinitesimal processes - inferences through a limit process
• invariance provide access to fundamental

6. Powerful ideas and processes should be introduced even if only implicitly in early grades but the age, culture and environment of the child should be considered when the context is chosen to introduce content . This does not imply that learners should be restricted only to non-mathematical and localised contexts within their immediate environment.

7. The choice of content should allow access to organising notions, e.g. formal definitions and axiomatic deductive organisation in mathematics.

8. The choice of content should allow access to mathematical technology. This has certain implications:

• Access to certain topics is required to facilitate appreciation of the use of (need for) mathematical technology.
• Access to certain topics (e.g. numerical methods) is required to facilitate understanding of how mathematical technology works.
• New content, e.g. numerical analysis, substantive linear programming, becomes possible.
• The danger that the availability of mathematical technology could impede learners' willingness to try to understand mathematics rationally needs to be combated.
• Mathematical technology provides substantial opportunities for mathematical explorations/investigations. Mathematical technology provides access to specific learning activities which promote understanding.
• The availability of mathematical technology affects the relative importance of various mathematical procedures and algorithms.

9 The choice of content should allow access to mathematics which will serve as a basis for further mathematical study and to the mathematics which will serve as a basis for the mathematics used in other subjects eg. economics, physical science, computer science.

Historically there has been a range of demands on teacher education. These have changed over time and according to shifting contexts. For the purposes of this document, we will argue that teacher education should support the purposes and rationale mathematics education spelled out in the first section and should empower teachers to be critical participants in curriculum processes at both local and national levels. Teachers are the carriers of the vision for school mathematics and their education is therefore crucial for its successful achievement.

Both preservice and inservice education have a role to play in the ongoing involvement of teachers in this mission although there should be different expectations of these two interventions. Preservice education should concentrate on developing competence in classroom practice wile inservice education should be able to engage teachers critically with this practice and with issues in mathematics education as a whole. We do not envisage this process of teacher development as finite - we take seriously notions of life long learning. Nor does teacher education depend on the intervention of outside agencies - teachers might well take initiative for their own development. Preservice education should inculcate the principle of life-long learning and professional growth.

What teacher education should not do is prescribe for teachers sets of procedures for "good teaching". Where demands about practice are made, these should be justified and underlying assumptions should be made explicit and subjected to critique. The emphasis should be on teachers developing their own theories based on mathematical knowledge, pedagogical knowledge and an engagement with mathematics education discourses.

Mathematical knowledge
Teachers need to know the basis for the selection of criteria for mathematical content. Teachers themselves need to be induced into the practices of the discipline - this is important at all school levels as primary teachers too will carry this responsibility in the schools. It would be important for teachers to engage in an in-depth study of a (any, albeit limited) area of mathematics. This should be a real sense making experience and it should not entail merely reproductive learning. Such in-depth encounter with at least one area of mathematics is needed in order that teachers develop a knowledge of what such an experience entails. Teachers should also develop a deep understanding of all school mathematics curriculum strands, of where mathematical ideas develop to, how they can be used to reflect recursively on each other and how they link horizontally to one another.

Teachers need to have access to knowledge of applied mathematics within reach of the school curriculum (this Is both important in terms of applied mathematics as a human heritage and as information which would assist in local curriculum development of contextual problems)

Pedagogical knowledge
Teachers need to be given access to experiences, research information, and scholarly debate in order to develop their own personally actualised theory and to contribute these ideas to the mathematics education community. This should be a two-way communication between teachers-in-training and scholars/researchers.

For example, teachers need:

• a knowledge of learning theories
• to think through how different kinds of mathematics are experienced epistemologically and therefore made sense of by learners
• to engage critically with a range of practices, methodologies, and forms of classroom organisation
• engage critically with the principles which underlie the design of mathematical curricula and activities
• access to substantive information about children, their mathematical intuitions, recurring misconceptions
• a knowledge of perspectives on mathematical education gained from other disciplines. Teachers need to engage actively and interactively with philosophical, sociological, psychological, historical, linguistic and political perspectives on the teaching and learning of mathematics. These perspectives impact on the selection of mathematical content, the organisation of learning experiences and the use of resources in the learning context.