Case Studies in Agents and Processes of Mathematics Curriculum Development and Reform

Abstract

This chapter is based on concrete experiences of curriculum design, development, and reforms in school mathematics across a range of social and cultural settings in different countries: Lebanon, Chile, Australia, South Africa. These were presented as a panel at the conference. Each focuses on concrete experiences of curriculum reforms to illustrate how the reform processes – from design to effective implementation – developed, and how the different agents – teachers, teacher educators, education researchers, mathematicians – influenced them. They all also explained how both specific and general features of different cultural and political situations in each country deeply affected such processes featuring both the on-going evolution and the final product of the curriculum reform. The four panelists raise issues, whose general meaning is discussed in the subsequent chapters of theme E based on further examples and theoretical documents. Below, the main issues raised by the four panellists are first highlighted, then their contributions are presented.

Introduction

The four contributors to this chapter represent diverse educational contexts and their case studies emphasise the complex connections between cultural and political factors and aspirations to the learning of mathematics.

Iman Osta discussed the relevance of documentation in the development of curricular reform in Lebanon as a strongly centralised action, its role in guiding the production of textbooks as the main guide for teachers, as well as teachers’ role as major agents for the enactment and resha** of the curriculum, and finally the national examinations as a central focus for curriculum development. She also pointed out the dramatic differences in the whole curricular reform between what happened in private schools with respect to public ones.

Fidel Oteiza discussed how the curricular reform in Chile sought quality education for all, and specifically how this aim was declined for mathematics curriculum, for example adapting it to the international standards within the globalised world, considering also what kind of mathematics should be pursued in an environment where technology offers the capabilities to do that properly. He highlighted how the curriculum development in Chile happened through a wide consultation process in the country which involved many agents. However, such in principle useful and democratic process in the end risked to produce too rigid rules, difficult to apply and almost impossible to modify when convenient on the light of application results.

Peter Sullivan, basing on the new mathematics curriculum developed in Australia, pointed out how a curriculum should be seen as an agent of reform with the emphasis being on documentation that both assumes and creates a focus for teachers as active learners about curriculum and pedagogy. He discussed how the Australian commission appointed for this had to face and decide between a series of six basic dichotomies to develop such a curricular philosophy: (1) teacher-proofing or teachers as learners; (2) documenting everything possible versus including just enough information; (3) practitioner versus specialist writers; (4) mathematics as preparation for later study or mathematics as experience; (5) general versus specific descriptions of expected mathematical actions; (6) mathematics for elite or mathematics for all.

John Volmink discussed features of curriculum reforms in South Africa, pointing out the dramatic differences between the racist curriculum before 1994 and the one after that date which was a key project in the transformation of the post-apartheid South African society. He summarises the main challenges that the appointed commissions had to face into three main items: the post-apartheid challenge; the global competitiveness challenge; the challenge of develo** critical citizens. The aim was a curriculum where inclusiveness of mathematics curriculum was devised according two concurrent dimensions: the necessity of a mathematics for all and of a mathematics by all. The former means that a curriculum must design the same quality of mathematics for all; the latter that everyone is engaged in a quality mathematical experience.

The Math Curriculum Reform in Lebanon: Achievements, Problems and Challenges – Iman Osta

The educational system in Lebanon is characterised by a high level of centralisation and a national curriculum that is binding to both, public and private schools. Decision making and developments are exclusively under the jurisdiction of the Center for Educational Research and Development (CERD), overseen by the Ministry of Education (MoE). While public schools apply only the national curriculum and textbooks, private schools may apply other programs and may use different series of textbooks, local or foreign. They are, however, bound to cover the national curriculum. A major tool of governmental control is the national examinations, referred to as official exams.

The Lebanese Ministry of Education proposed, in 1994, a project for overhauling the educational sector, as stipulated by the Taïf Agreement (1989), which has put an end to the fifteen-year-long war. In October 1995, the government approved a plan for develo** the new curricula. Starting 1995, a reform process of the educational system and national school curricula began, after a stagnation that lasted more than twenty-five years, partly because of the war that hit the country. The older national curriculum initially created in 1946, just after the independence of Lebanon, was partially revised in 1968 and 1971 to include instances of the worldwide, New Math wave, such as the set theory. An extremely abstract, procedural and directive spirit has always characterized the old, long lasting math curriculum, setting up an educational culture guided by, and revolving around stereotypical national examinations (Osta, 2007). In those curricula, conceptualisation was neglected and students were seen as passive receivers of information and executors of algorithms.

Between 1995 and 1999, the reform efforts mobilised politicians, educators, teachers, textbook developers, and other constituents of the Lebanese society. The educational ladder has been organized into two main levels: Basic Education (BE) and Secondary Education (SE). The BE consists of three cycles, three years each – Elementary cycle 1 (grades 1–3), Elementary cycle 2 (grades 4–6) and Intermediate cycle (grades 7–9). Secondary education includes grades 10–12. The main curriculum document, delineating general objectives and objectives of the cycles, as well as the scope-and-sequence and contents to be taught in every grade level, was issued in 1997. The national textbooks were gradually developed and applied over three years thereafter (every year, the new curriculum and textbooks were implemented in one more grade level of each cycle), till the year 2000 that witnessed full implementation at all grade levels, and culminated into the first national exams under the new reformed curriculum.

After a long period of adoption of an old traditional curriculum, the reform of the LMC constituted a revolution. It changed the ways the nature of mathematics and its teaching are perceived by the educational community. The intention was to align the new curricula with the worldwide curricular trends at that time. The methods adopted are defined as constructivist and active, the learner being the ‘centre of the teaching/learning operation’, and the capacities of ‘reasoning and problem solving’ outweighing algorithmic procedures and memorisation of facts. Compared with the old curricula, a real revolution was announced and expected.

The major question remains: has this revolution been maintained throughout the curriculum development and implementation processes? An essential claim of this paper is that, with the marginal role of teachers, absence of internal coherence of the curriculum, and lack of suitable resources, the high-stake national exams determine, to a large extent, the orientations of the curriculum enactment and make it revert back to the deeply rooted old practices.

Reflection on the Lebanese Math Curriculum

In the rest of the text, four of the main components of the LMC will be discussed, namely: (1) the foundational documentation of the curriculum – the role of this documentation was to act as a guiding roadmap for the development of textbooks and an interface between the curriculum philosophical/pedagogical foundations and the educational community; (2) textbooks as the main guiding resource for teachers; (3) teachers as the main agent for the enactment and resha** of the curriculum; (4) the national examinations as the central focus and determinant factor of the curriculum development, implementation and reorganisation.

Foundational Documentation

The foundational curriculum documentation consists of: (1) the main curriculum document issued by an official governmental decree (CERD, 1997) delineating the aims of the curriculum, its pedagogical recommendations, general objectives (GOs), and objectives of cycles (OCs); (2) the details of content, published gradually in three volumes over three years (1997 for the first year of every cycle, 1998 for the second years, and 1999 for the third years). They include the specific objectives (SOs) and detailed information about the contents of the mathematics subject for each grade-level year.

Osta (2003) investigated the internal coherence of the LMC documentation using map** tables and text analysis of the curriculum documents above. The analysis of the main curriculum document showed a high level of coherence between the general objectives GOs and the philosophical and pedagogical foundations announced in the introduction. They both use a language focused on the development of cognitive abilities, the importance of problem solving, and the appreciation of mathematics as a practical tool related to everyday life. Below are a few examples.

Mathematics is defined in the introduction as, “a fertile field for the development of critical thinking, for the formation of the habit of scientific honesty, for objectivity, for rigor and for precision. It offers to students the necessary knowledge for the social life and efficient means to understand and explore the real world”. As for the recommended teaching methods, “[they] consist of starting from real-life situations, lived or familiar, to show that there is no divorce between mathematics and everyday life”. As described, the recommended teaching methods are clearly constructivist and focus on problem solving. “The stress is mainly on the individual construction of mathematics; it no longer consists of teaching already made mathematics but of making it by oneself. Starting with real-life situations in which the learner raises questions, lays down problems, formulates hypotheses and verifies them, the very spirit of science is implanted and rooted.”

The General Objectives (GOs) are clearly consistent with this approach; they insist on the importance of ‘the construction of arguments’ and on ‘develo** critical thinking, and emphasising mathematical reasoning’, the latter being presented as the first GO. Problem solving is presented as the second GO and described as “perhaps the most significant activity in the teaching of mathematics. On the one hand, every new mathematical knowledge must start from a real-life problem. On the other hand, students must learn to use various strategies to tackle difficulties in solving a problem”. The student must also “encode and decode messages, formulate, express information orally, in writing and/or with the help of mathematical tools”, which makes mathematical communication a third main GO. We will refer to these three objectives by “cognitive objectives”, to distinguish them from objectives purely related to the factual and procedural mathematical content.

The curriculum therefore proposes a progressive teaching approach. A constructivist approach, focused on reasoning, problem solving and communication, is reflected in the teaching method and general objectives advocated in the first curriculum declarations. It is to be noted that the three highlighted cognitive objectives are mostly in line with the American “Standards” (NCTM, 1989) which have profoundly affected modern international trends in mathematics education at that time.

However, only partial consistency is found between the cycles’ objectives COs and the GOs, with a deviation in the discourse that reflects a beginning of separation from the pedagogical foundations above. Indeed, the COs continue to reiterate the importance of the three cognitive objectives, which systematically appear as the three first objectives for every cycle, followed by content-related, factual and procedural objectives.

One example, where we can touch upon the deviation of discourse, is found in the objectives cited under ‘problem solving’ for the secondary cycle: “Find the solution of a problem following a given algorithm”. Requiring that solving the problem should follow a ‘given algorithm’ is in opposition to the very meaning of problem solving. It also defeats the purpose stated in the GOs, delineating the traits of the learner as being ‘an individual with a critical mind who questions, doubts, proposes solutions’, and who ‘must learn to use different strategies’.

The deviation from the curriculum’s foundations and GOs increases and becomes more serious at the level of the specific objectives in the SOs in the details of content volumes. The three cognitive objectives are not maintained in the SOs. Not only have they disappeared as independent objectives, but they are also very rarely reflected in the contents. The analysis of the SOs shows that they mostly represent declarative knowledge and procedural skills related to formal mathematical content, emphasizing the execution of predetermined and automated steps and overlooking conceptual understanding. Very few SOs are linked to the cognitive GOs, which are supposed to perpetuate the link to the constructivist intentions of the curriculum.

The tension is evident in the Details of content. The phrase ‘to train the student’ is frequently used. The learner is seen as a passive receiver of information and executer of algorithms, and the teaching style that is detected from the teaching tips is extremely directive. Consider for example the case of problem solving: Even though the GOs insist on problem solving as a context “from real, lived or familiar situations” for both, learning and applying concepts, we find in the details of content clear reluctance to actual situations and mistrust of learners’ abilities as problem solvers.

The details of content were later used as the main basis for the development of the subsequent documents and tools, including the student textbooks, pedagogical guides and evaluation guides.

Textbooks

School textbooks are the main interface between teachers and the curriculum foundations, as well as the main tool for their educational practices. The question raised here is: considering the fact that the Details of Content drifted away from the innovative spirit of the intended curriculum, and the fact that school textbooks are the main tools in the hands of teachers, how can teachers maintain the link between the tools available to them in their professional practice, and the GOs and OCs which ensure the true reflection of the intended curriculum’s foundations?

Knowing the fact that the textbooks for the first year of each cycle (grades 1, 4, 7 & 10) were authored just after the development of the foundational documents in 1997 and that the textbooks for the third year of each cycle (grades 3, 6, 9 & 12) were authored two years later in 1999, it may be legitimate to assume that the textbook authors have gradually deviated from the reformed curriculum’s foundations and reverted back to the old approaches.

Teachers and the Reform

A radical reform requires involving and preparing the teachers for the enactment of the intended change. It also offers opportunities for teachers’ professional development in view of modifying their beliefs about the nature of mathematics they will be teaching and the approaches to its teaching. Educators agree that teachers are main agents for any educational change.

The MoE and CERD have conducted ‘training’ workshops in the new curriculum, involving a large number of mathematics teachers, especially in the public sector. These workshops proved to be too directive. They mainly revolved around providing information on the new content, as well as the recommended pedagogical approaches.

In a study that solicited teachers’ reflections on the reformed Lebanese curriculum and their feedback about the workshops (Osta, 2006), all participants reported that they were not sufficiently prepared to apply the recommended teaching methods. They requested more practice on techniques such as group dynamics, group work management, active methods, design of didactical situations, development of students’ autonomy, use of calculator and computer for teaching/learning purposes. Teachers expressed their belief that the educational authorities which ‘impose’ such methods should provide support to teachers up to the classroom level, such as providing ‘model lessons’, activity sheets or additional exercises to respond to certain learning problems that may arise.

National Exams

In Lebanon, national (known as official) exams take place every year at two grade levels: the end of the intermediate cycle of study (grade 9), for the ‘Brevet’ certificate, which gives access to secondary school, and the end of secondary level (grade 12), for the ‘Baccalaureate’ certificate and graduation from pre-college education. They are high-stakes exams and have an imposing power. In the Lebanese culture, a major goal for schools is to raise their students’ test scores in the official exams. It is as well an indicator of school improvement. Teachers whose students pass the official exams gain in reputation and receive good offers with high salaries from private schools. This leads to the observed fact that teachers tend to teach to the test, and that school administrators shape their school policies and focus their academic activities around that goal. As a result, the official tests determine the valued mathematics that should be taught, and the ways it should be taught.

This study showed that the official exams under the old curriculum kept a stable structure and addressed a specific body of mathematical content. It was noticed that many topics in the curriculum were never addressed in the official exams. The topics frequently occurring in test items defined a ‘mini-curriculum’ that gradually replaced the original one, and was reinforced every year and in every test. This ‘mini-curriculum’ fosters memorization of answers to stereotyped test items, through drill and practice rather than conceptual understanding.

The study led to a hypothesis expecting that the extremely procedural nature that has always characterised the old math official exams, has established a deeply rooted testing “culture” focused on direct procedural skills. Consequently, the new official exams could not, over the years, reflect the real change intended by new curriculum. This ‘culture’ was nurtured by the long-lasting old curriculum, and its official exams are still influencing the new official exams. The hypothesis above was confirmed by three studies (Safa, 2013; Shatila, 2014; Sleiman, 2012) that used the framework developed by Osta (2007) to investigate the extent of alignment between the official exams over 10–12 years, with the reformed curriculum. The results of the three studies converged to confirm the hypothesis above. A mini-curriculum was identified, and low levels of alignment are found between the exams and the curriculum guidelines, especially as pertains to the cognitive general objectives. They found, however, that the alignment improved gradually over the years. Global alignment remained, however, lower than enough to reflect actual change in the testing culture.

The nature, scope and structure of the official tests send a clear message to the educational community (teachers, administrators, parents and students) over the years. This implicit “contract” among all involved parties binds, in return, the committees in charge of constructing the tests. Even if they want to include modifications or additions, they find themselves bound to the “mini-curriculum”. This closed cycle is sustained by the “doctrine of no surprise” that English and Steffy (2002, p. 46) explain as being the idea that students should not be taken by surprise by any test question.

In Summary: From Design to Implementation

In general, while teachers are in direct contact with the implementation tools, among which the textbooks and the official exams, they are at a distance from the other foundational components, including the pedagogical foundations and general objectives of the curriculum. Those are particularly absent from their direct perception and their day-to-day professional practice, if they are not actively implicated in the reform movement. Even an attentive reading by the teachers and their participation in informational workshops are not enough to guarantee a modification of their professional practices that they developed over many years according to the old curriculum.

Processes and Agents of Curriculum Design, Development and Reforms in Three Decades of School Mathematics in Chile – Fidel Oteiza

Since the beginning of the nineties, Chile has experienced continuous economic and social growth. This process has been slow but sustained. There has been a significant improvement in economic and social development indicators. Reduction of poverty and a substantial improvement in the quality of life are unmistakable signs of a positive change. The continuous clamour for a better education, “quality education for all”, has forced the above-mentioned period of repeated reform efforts. National and international tests show little progress in learning. These small gains are not compatible or sustainable when compared to the development of the country in other areas. Another driving force is the pervasive and perverse gap between the haves and the have-nots. A single and driving force is inequity as shown by learning results. Evidence shows that learning outcomes in public schools are significantly inferior to the ones in private educational institutions. This gap has shown to be the most difficult barrier to trespass in the Chilean educational system. The search for more equitable educational outcomes may be the most important driving force behind a thirty-year effort to reform the national educational system in the country.

Some Milestones

The reform of school mathematics curriculum is to be understood as embedded in a broader process: the reform of the educational system. The following are some of the milestones in the reform process along the relevant milestones dates, which are major decisions that might impact the educational system as a whole: [as a list] the creation – as a result of a multi-sector consultant committee- of the National Council of Education (CNE), (1996–1998); the extension of compulsory education up to 12 years of schooling (2003); a major reform of the framework defining the education for the country (2009); the creation of the Quality Agency (2011), responsible for the national test as applied in various school levels; a new definition for elementary, secondary and technical education and the creation – in process- of regional entities responsible of the administration of public schools which are accountable for the implementation of the national curricula, a policy that promotes decentralisation of the educational system.

In a minor scale – but significant because they are some of the major results of reform efforts – the following can be mentioned: new infrastructure for schools throughout the country; new standards for teacher selection and teacher preparation; an improvement, although still insufficient, of working conditions and professional development for teachers; the almost universal access to digital technologies; free, newly designed, textbooks for all students in public schools; the extension of school schedules; especially relevant to the subject of this analysis, a renewed and more demanding school curriculum. National tests applied to the entire system, at various school levels, are mentioned separately because, although considered to be a guaranty of quality control, have become, at the same time, the operational definition of school curriculum and the latter competes with the official national curriculum.

Tendencies in the Process of Reform of the National Mathematics Curricula

There has been a remarkable effort to bring the national curriculum closer to international standards. Simultaneously, ideas, themes or content, before reserved for the last two years of schooling or the beginning of university courses, are now included in lower levels. This tendency can be observed in the treatment of functions, previously reserved for grades 11 and 12, now initiated in grade 7 or 8. The same occurs with probability and statistics or patterns and algebra, beginning now in first grade. Geometry includes, now, coordinate geometry and vectors. Another tendency is the emphasis of skills over content. The national curriculum in Chile promotes modelling, problem solving, communication and argumentation, and multiple representation skills. Mathematical reasoning has been of major concern among policy makers of the mathematics curriculum. The new curriculum points to classroom management that encourages the formulation, analysis and verification of conjectures. Modelling skills are emphasised throughout the curriculum. The proposed intense use of digital technologies is another new emphasis.

Agents, Institutions and Driving Forces

The above-mentioned division, which is responsible for school curriculum, has specialized teams in different areas of the curriculum, particularly in mathematics.

What is the role of the mathematics team at the Ministry of Education? When involved in a reform process, the main responsibilities are the analysis of existing curriculum, the compilation and analysis of evidence about curriculum implementation, the search and analysis of the demands and proposals of specific leading actors, the search for significant results of research and, in the field and international experience in mathematical curriculum, the interpretation of general directives as generated by educational authorities within the Ministry of Education.

Moreover, there is the formulation of proposals for the new curricula, the participation in different consultations and validation processes and the incorporation, to proposed curriculum, of the results of the consultation process. Once the new curriculum has been approved, several other tasks are in order: textbook specifications; the search for and the evaluation of different resources including digital ones and digital support; the participation both in the process for the diffusion of new curricula and the implementation of several actions related with diffusion and teacher preparation. Also, there is participation in actions related to the impact of new curricular proposals in teacher preparation and national tests which include the university entrance procedures and their corresponding exams.

The Consultation Process, Its Major Contributors and the Role of the National Education Commission

Several consultations precede the presentation of the curricular proposal to the National Council. The consultation process and the action of the National Council are the mechanism that seek to balance or counterbalance the action of the technical teams of the Curriculum Unit.

Consultation has been shown to be a powerful instrument in the definition of new curricula. Who is addressed in the process of consulting on the new proposals and how consultation instances are organised, are important issues subject to analysis and improvement. Teachers, research centres, researchers, mathematics and mathematics education associations, leaders of private educational organisations and the general public have all been consulted. Consultations have been done, mostly, in the modality of focus-groups, also with small groups of experts and public web questionnaires. Face to face feedback was effective in all the consultation meetings that were organized. Public consultations on the web proved to be more effective in making the proposals be known than in generating a specific contribution. The fact that a reform has been consulted and has received more than 15,000 public reactions is a powerful factor for face validity and acceptance.

A generalised statement can be made for both faces to face and web consultations. Most of the feedback and sometimes the whole of it were about teaching methods or teacher preparation. In a smaller proportion, reactions focused on teachers’ abilities needed to put into practice what was proposed and also on the necessary conditions for implementation. A generalized reaction was: “what is proposed is too much; the amount of content exceeds what is possible in the time available to treat it”. Those responsible for the proposal often agreed with this evaluation. When authors of this comment where asked about what to remove from the proposal, the most frequent answer was “nothing” and in many opportunities, “nothing, but there are many things missing”. It is clear that the entire process of curriculum innovation and the way it has been implemented in the country lead very naturally to a growing curriculum. This is one of the questions to be addressed in the next section.

The role of the National Council of Education is now mentioned because it addresses two important needs of a reform that leads to a new formulation of the curriculum: the decision-making regulation and necessary institutional counterweight. The national curriculum in Chile is law enforced. Before a new curricular proposal becomes compulsory, a complex – also a matter of law procedure – needs to be implemented. Proposals are generated in the Unit of Curriculum previously mentioned. Once the design has been approved within the Curriculum Unit, they are subject to approval by de National Council. This process is a guaranty of quality, pertinence and proper formulation.

Two additional consequences of this process are mentioned later as open questions: one is – and this is a statement that reflects only the author ́s point of view – the exaggerated weighing that has the opinion of one or very few experts when summoned as reviewers by the Council. This delicate situation has generated distortions or imbalances in the curricula that it has acted on. It is a question to be analysed. The second issue to be considered is the excessive rigidity that the whole reform process gives to the curriculum. Once constituted by law, a change, an improvement, no matter how minor, must go throw the same procedure. The result is unnecessary rigidity that inhibits needed systematic and permanent revisions and makes almost impossible consequent corrections.

Main Social, Cultural and Technical Factors Sha** School Mathematics Curricula, New Questions and Pending Issues

The gap factor shows that there is a significant, odious and until now permanent difference between the learning outcomes of students attending public and private schools. This non-solved situation poses the question of who we are formulating the curriculum for. During decades, national curricular requirements have been growing. Results, in national tests, show that students attending public schools, close to 85% of school population, are not fulfilling those standards. How does mathematical school curriculum contribute to this gap? How might mathematical curriculum be a factor in the reduction of these differences? Topics such as function, systems of inequations or homothetic figures are increasingly lower in the curriculum. Is it advancing topics that make a curriculum be better? Does the maturity of the student matter when deciding these advances? There is tremendous and extremely valuable talent diversity. Can we justify the existence of only one curriculum and only one way to evaluate it through standardised tests?

Testing gives solid information and has impact on the gap between stated and actual curricula. From one point of view, national tests are very much valued as indicators of learning outcomes. Simultaneously, they act as an operational definition of the mathematics curriculum. Teachers, schools, local educational authorities and parents give high value to SIMCE results. In consequence, what is measured ends up being a guide for teachers when making subject matter decisions. As it is very simple to guess, higher-level learning and skills as promoted by reformed mathematics curricula, therefore, are often not covered by classroom teachers.

This is an unsolved dilemma: to test or not to test. Mathematical modelling, argumentation skills, guessing and testing of one’s own ideas or those of peers are difficult to measure and, thus, they lose importance for the teachers. What are adequate relations between national curriculum and national tests? How may skills in argumentation, modelling and enquiry be evaluated?

Globalisation has influenced national mathematics curriculum in several ways: media generates access to news, cultural issues, tendencies and frequent expert opinions on educational results; international tests have proved to be very influential. Another factor is the almost universal and instant access to any nation’s curriculum, including those of leading countries and economies.

There are strong questions we have not yet addressed in designing the national mathematics curriculum: what is it that mathematics students need know in order to do mathematics in an environment where technology offers the capabilities to do so? What are the skills a person needs to learn to take all the advantages of existing digital technologies when doing mathematics? Information and communication technologies have shaped our culture. The second derivative of this change grows. Is computational thinking a necessary knowledge for everyone? What should a mathematics teacher know about computer science?

Currently there are new social and cultural requirements: gender, the inclusion of those showing physical or learning disabilities and personal and environmental care. All of these pose new questions. How is curriculum worded to promote inclusion? How does one formulation for the curriculum take care of the diversity in talent? How is personal and environmental care included in the school mathematics curricula? How is the mathematics classroom organized and monitored, if handicapped students are to be included?

Another issue to be analysed refers to when and why reforms are initiated. These have begun in a casuistic, not predictable agenda, the opposite to a planned systematic process. Search for long term, periodically evaluated curriculum proposals has been an issue in Chile. A one-year educational committee was appointed (in 2016) to deal with this issue. Nationally recognised educational authorities were asked to generate proposals to create a “National policy of curriculum development”. The purpose of the Committee was to make reforms of school curricula less vulnerable to political or conjectural factors. These are important questions: What is an appropriate-long- term policy in school mathematics reforms? What are the conditions that make a reform needed? Is there a way to apply significant and defensible school curriculum diagnosis? How is a new reform decided?

There is a fundamental role played by researchers, and research and development centres and institutions. The period of school mathematics curriculum considered in these pages is the first in Chile where researchers – both in mathematics and in mathematics education or didactics – have had significant influences on school mathematics. In another publication, (Rojas & Oteiza, 2014) the authors refer to this as “new actors”. However, questions remain:

How does the knowledge generated by the research reach the classroom?

How do the questions that originate in the classroom reach a research centre or a graduate program?

The Aspirations of the Australian Curriculum in Prompting Reform of School Mathematics – Peter Sullivan

This contribution and the associated presentation provide an opportunity to reflect on the intention and processes for the design and writing of the Australian curriculum: mathematics (AC: M) and to reflect on subsequent developments. The argument is that curriculum reform can be an agent and process for prompting teacher professional learning but whether this happens or not depends on whether the structure of the curriculum documentation and associated support foster such knowledge creation.

In any curriculum reform process there are many dilemmas or dichotomies about which active decisions are taken. One of the meanings of dichotomy is that there are two mutually exclusive, opposed, or contradictory positions. This contribution outlines some of the dichotomies in any curriculum reform process and reflects on ways that such dichotomies were and are being resolved in the Australian curriculum.

The Process of Development

Even though there are broader definitions of curriculum, including terms such as intended, planned and enacted (see, for example, Porter, 2004), this discussion focuses on documentation associated with centrally developed curriculums and decisions on the form and substance of that documentation. Of course, the real curriculum results from the ways that such documentation is interpreted, implemented and experienced in schools and classrooms, but the main opportunity for governments to intervene meaningfully is at the level of documentation.

Prior to the creation of a single national curriculum, there were eight Australian jurisdictions that each had their own curriculums and associated supporting resources. The responsibility for such curriculums was jealously guarded. In most cases the curriculums were informed by earlier national profiles so there was substantial overlap in the substance of the content specifications in the various jurisdictions but the extent of collaboration on aligning the documentation was minimal.

The motivation for creating national curriculums in all domains was essentially political. The Australian curriculum started from four domains, mathematics being one. The first step was the development of a discussion paper that set the goals and processes of the curriculum. This was described as the shape paper (ACARA, 2009) and outlines the principles, the aims, the terms used, the focus of the respective levels of schools, various issues such as connectedness and clarity, and a discussion of pedagogy and assessment especially as they related to equity and inclusion. The paper was developed by a broadly-based writing team and sought online and face to face feedback nationally. The following discussion describes some of the dichotomies and is intended to raise some of the considerations in the documentation of curriculums generally.

Dichotomy 1: Teacher Proofing or Teachers as Learners

Curriculum reform and associated teacher learning are integrally connected to views that curriculum developers and system decision makers have of teachers. There is a clear dichotomy of perspectives apparent in the ways that the initial curriculum was designed and has been interpreted.

On one hand, if teachers are seen as unreliable and unable to interpret curriculum documents then the curriculum will be written and supported in a particular way. On the other hand, if teachers are viewed as thinking, flexible and creative agents, then the curriculum documentation and associated support can reflect those perspectives.

The shape paper and the initial curriculum design opted explicitly for the latter position. The underlying assumption is that if systems place trust in teachers, they will come to see the underlying principles of the curriculum. In this process, teachers can become better educators, while increasing the detail of the documentation can be counterproductive to the mathematical intention and also to the learning of teachers.

Another decision taken was to seek to reduce the breadth of the specified content so that the more important aspects were presented. Each time jurisdictions increase the level of detail and breath of expected content, they reduce teacher decision making and the potential for teachers to learn about the broader goals of mathematics learning.

Dichotomy 2: Documenting Everything Possible vs. Including Just Enough Information

One of the initial decisions in the creation of the AC: M was that the curriculum should be described clearly and succinctly. Indeed, the intention was that the content for any one year be presented on a notional single ‘page’, described parsimoniously and presented flexibly via a dynamic web-based environment to emphasise the need for teachers to make active decisions (ACARA, 2009). The dichotomy is that, on one hand, comprehensive documentation would provide teachers with guidelines of what to teach, while on the other hand it would have the effect of restricting teacher decision making, causing it to be harder for teachers to see the ‘big picture’.

The early consensus in the creation of the AC: M was that mathematics is much less a set of isolated micro skills to be learned independently of each other than it is sets of connected concepts and processes and that it is better for teachers to see the connections. An excessive compartmentalisation and documentation can reduce the possibilities of teachers seeing connections. The tendency in some jurisdictions in Australia, subsequent to the initial publication, has been to increase the level of detail in and complexity of curriculum descriptions which has the effect of limiting the extent to which teachers can imagine the bigger picture or even consider seeing the broader perspective as important.

A related aspect is the ways that the curriculum fosters connections between and within strands and sub-strands. A key international perspective which emphasises the importance of connections is Variation Theory (Kullberg et al., 2013). Watson and Mason (2006) outlined the importance of thoughtfully constructed sequences of learning experiences out of which the underlying concepts can be extracted. Similarly, Dibrenza and Shevell (1998) described number strings as an example of the ways that sequences of related exercises can emphasise number properties. Sinitsky and Ilany (2016) explained that considering both change and invariance illustrates not only the nature of the mathematics but also the process of constructing concepts. In other words, finding ways to support teachers in seeing and using connections between and within concepts can support teacher learning and effective teaching. To achieve this, the curriculum needs to be clear and concise.

One of the disadvantages of having the content determined by a student text is that teachers are less required to think about their own broader purposes. The same is true for curricula in which the teachers are ‘told’ which tasks to teach without having to appreciate the goals, both content and processes, associated with the tasks. One of the critical foci for teacher learning is to enhance their capacity to make their own decisions using the curriculum documents and other resources to which they have access.

A further central aspect that relates to the nature of the documentation is the expectations that teachers will collaborate with colleagues in their planning of sequences of learning. It seems that in some countries the textbook serves as the curriculum and teachers need only to turn to the next page in planning their lessons. In Australia, it is common for groups of teachers to plan sequences of lessons together. Not only does this allow teachers to learn from each other but also planning together encourages them to anticipate how students might respond, identify potential blockages and misconceptions, share the development of supporting resources, and so on.

Dichotomy 3: Practitioner vs. Specialist Writers

Another early dichotomy relates to whose voice should be heard. One of the initial considerations was whether the curriculum should be written by experts or by practitioners, with the latter option being chosen. The process for creating the curriculum and associated documents was collaborative involving extensive, indeed exhaustive, consultation. Subsequently curriculum writers, predominantly classroom teachers, were employed and an advisory committee formed. There were extensive consultations around successive drafts, piloting in schools across the nation, map** of the drafts against the various state and international curricula, and many other actions as well. The advantage of this process is that a curriculum was developed which was familiar to many teachers. The disadvantage is that the writing was informed by many and diverse contributions. In other words, there is a tension between seeking consensus and maximising coherence that is not generally acknowledged by commentators.

Dichotomy 4: Mathematics as Preparation for Later Study or Mathematics as Experience

One of the key dichotomies in determining a mathematics curriculum is related to the nature of the mathematics to be described. One perspective refers to the structure and content of many mathematics curricula that create the impression that the main goal of learning mathematics is preparation for study in a subsequent year level. An alternate perspective is that curricula should inform an experience of learning that is like being a mathematician, in which the learning about and using mathematics is the primary goal. Of course, a balanced curriculum will consider both perspectives but the intention in the AC: M was to move away from a curriculum that focused only on the former.

The AC: M took an explicit stance that the mathematics and numeracy that should be experienced by school students is much more than the emphasis on procedures and computational processes that seemed to constitute much of the teaching of mathematics in Australia at the time (Hollingsworth et al., 2003; Stacey, 2010). It is unfortunate that much of the subsequent discussion of the curriculum starts from the perspective that the primary rationale for the inclusion or emphasis on an aspect of content is that it will be used in subsequent study. This tendency is especially evident at senior levels with the pressure from interest groups being to increase the emphasis on procedures and routines and to include additional topics exacerbating the already crowded curriculum.

Dichotomy 5: General vs. Specific Descriptions of Expected Mathematical Actions

The first aspect of the AC: M that teachers gain access to is the descriptions of the concepts or content that form the focus of learning experiences. There are achievement standards available that give advice to teachers of the expected standards of performance. The key device for broadening teacher-focus to encourage them to value specific mathematical actions was described as proficiencies.

ACARA (2009) proposed that the content be arranged in three strands that can be thought of as nouns, and four proficiency strands that can be thought of as verbs. The content strands – number and algebra; measurement and geometry; statistics and probability – represent a conventional statement of the ‘nouns’ that are the focus of the curricula worldwide.

These four were adapted from the recommendations in Kilpatrick et al. (2001). The first of these, understanding (the Kilpatrick and colleagues’ term was ‘conceptual understanding’), was described as follows:

Students build a robust knowledge of adaptable and transferable mathematical concepts, they make connections between related concepts and progressively apply the familiar to develop new ideas. They develop an understanding of the relationship between the ‘why’ and the ‘how’ of mathematics. Students build understanding when they connect related ideas, when they represent concepts in different ways, when they identify commonalities and differences between aspects of content, when they describe their thinking mathematically and when they interpret mathematical information.

A second proficiency strand, fluency (the Kilpatrick and colleagues’ term was ‘procedural fluency’), was described as:

choosing appropriate procedures, carrying out procedures flexibly, accurately, efficiently and appropriately, and recalling factual knowledge and concepts readily. Students are fluent when they calculate answers efficiently, when they recognise robust ways of answering questions, when they choose appropriate methods and approximations, when they recall definitions and regularly used facts, and when they can manipulate expressions and equations to find solutions.

A third such strand, problem solving (the Kilpatrick and colleagues’ term was ‘strategic competence’), was described as:

the ability to make choices, interpret, formulate, model and investigate problem situations, and communicate solutions effectively. Students formulate and solve problems when they use mathematics to represent unfamiliar or meaningful situations, when they design investigations and plan their approaches, when they apply their existing strategies to seek solutions, and when they verify their answers are reasonable.

The fourth proficiency, reasoning (the Kilpatrick and colleagues’ term was ‘adaptive reasoning’), included:

analysing, proving, evaluating, explaining, inferring, justifying and generalising. Students are reasoning mathematically when they explain their thinking, when they deduce and justify strategies used and conclusions reached, when they adapt the known to the unknown, when they transfer learning from one context to another, when they prove that something is true or false and when they compare and contrast related ideas and explain their choices.

The proficiencies are represented as intersecting with each of the three sets of content descriptions, illustrating that the proficiencies are not only a focus of learning of all aspects of mathematics, but also can be the vehicle for that learning. There was an explicit intention to support teachers in seeing mathematics learning as incorporating all of these actions.

It is noted that while the first two proficiencies, understanding and fluency, can be prompted by explicit teacher instruction, while the latter two, problem solving and reasoning, require student-centred approaches, further communicating to teachers about the breadth of pedagogies needed and the nature of learning experiences that they can create.

Dichotomy 6: Mathematics for Elite or Mathematics for All

A further key element of the AC: M, which was intended to inform teacher learning is related to the challenge of equity.

ACARA (2018) argued that all students should experience the full range of mathematics in the compulsory years. Mathematics learning creates employment and study opportunities and all students should have access to these opportunities. This is both an equity and a national productivity issue. The curriculum makes the explicit claim that all students should have access to all of the mathematics in the compulsory years.

A fundamental educational principle is that schooling should create opportunities for every student. There are two aspects to this. One is the need to ensure that options for every student are preserved as long as possible, given the obvious critical importance of mathematics achievement in providing access to further study and employment and in develo** numerate citizens. The second aspect is the differential achievement among particular groups of students (ACARA, 2009). An explicit goal of education in Australia is the intention to build an inclusive society in which all citizens can participate.

In summary, the claim here is that the initial intentions of the AC: M were that the curriculum should be seen as an agent of reform with the emphasis being on documentation that both assumes and creates a focus for teachers being active learners about curriculum and pedagogy. This intention was also evident in the processes used to communicate to teachers that doing mathematics is as important as skill development, and that not only is it possible to structure classrooms to be inclusive of all students but also that that is an expectation.

School Mathematics Reform in South Africa: A Curriculum for All and By All? – John Volmink

The term curriculum reform, like any other concept, always has a contextual ancestry. It also has a career that needs to be recognised and understood within a particular setting. But while there is general acceptance that curriculum reform grows into its own career and takes shape within a context, we often need to be reminded that this evolution is not bound by some transcendent, universally applicable set of laws which are independent of people. The political aspirations and ideological commitments of the drivers of the reform and the social forces that shaped the reform cannot be ignored and omitted from its ancestral biography. I see the purpose of this discussion as an attempt to understand how we can influence the development of these contextual careers of mathematics curriculum reform by understanding how choices were made within the various contexts and to what extent there was a willingness to embrace the complexity and ambiguity for the greater public good.

Curriculum inertia occurs when we choose to ignore the complexity inherent in making educational choices and retreat to the false safety of the universality of mathematics. Behind this wall we see our task as creating access to fixed, universally accepted ways of knowing and learning mathematics, stripped of all the clutter of ideological and cultural expectations.

South Africa is a society in transition. We have moved away from what was a stable but cruel past to a new and dynamic present. The conventional signposts have been swept away and we have been travelling on largely unchartered waters since 1994. One way of describing the new, democratic, educational reality in South Africa is that of celebrating the chaos and turbulence of a new beginning. It has been exciting to be part of this wonderful and dynamic period of our history and for me it has been particularly rewarding to be asked by both the previous and present Ministers of Education to play a key part in educational reform in post-apartheid South Africa.

Challenges Facing Curriculum Reform in South Africa

Over the many years of apartheid two education systems coexisted – one predicated on the goals of a first-world education, the other intended to be merely reproductive. The one was seen to be sufficient to produce enough high-level skills to support the larger economy, the other to reproduce people who were just sufficiently functional to serve the low-level skills demand of the extractive-metals economy. Race was the main determinant of educational access, provision and quality. Throughout the years of apartheid, there was a continuous groundswell of resistance to “Bantu education” culminating finally in the 1976 Soweto uprising. Since that time the Mass Democratic Movement (MDM) and the politics of confrontation in education, became increasingly organised until it established the National Education Crisis Committee (NECC) in 1980.

The failure of the then government to respond to the crisis in education led the MDM to resolve to strive for People’s Education for People’s Power at its first Education Crisis Conference, in December 1985. People’s Education (PE) would lead to educational practices that would enable the oppressed to understand and resist exploitation in the workplace, school and any other institution in society. It would also encourage collective input and active participation by all in educational issues and policies, by facilitating appropriate organisational structures. These ideals found expression in the work of three commissions, one each in the fields of History, English and Mathematics. When it became clear that PE would be introduced in schools by mid-1986, the apartheid government moved in very quickly to restrict its impact. The momentum for PE, during the years after the restrictive measures, was sustained for a while in large part, by the work of the Mathematics Commission, but this momentum also finally ground to a halt for a variety of reasons.

An underlying assumption in educational policy in South Africa is that the achievement of democracy requires a (national) curriculum to realise its goals. Curriculum change in post-apartheid South Africa thus started immediately after the election in 1994. So, the genesis of new curriculum thought in South Africa finds its roots in the debate within the Mass Democratic Movement over previous decades. The first major curriculum statement of a democratic South Africa was known as Curriculum 2005 launched in 1997. It signalled a dramatic break from the past with its narrow visions and concerns for the interests of limited grou**s at the expense of others. But it was also bold and innovative in its educational vision and conception. It introduced new skills, knowledge, values and attitudes for all South Africans and stands as the most significant educational transformation framework in South African education.

At the dawn of democracy in 1994, South Africa had nineteen different educational departments separated by race, geography and ideology. While these were merged into nine provincial departments, there was also a need for a single core syllabus. It did not touch the core content since a part of its brief was not to necessitate new textbooks. So, beyond the rationalisation and consolidation of the existing syllabi, the process could at best sanitise the syllabus by removing overtly racist and other insensitive and offensive content forms from the syllabi.

After the completion of the syllabus revision process in late 1994 the national Department of Education (DoE) set in place a new vision for education through a series of policy initiatives in 1995. This included a vision for curriculum development and design. At the same time South Africa adopted a National Qualifications Framework (NQF) as the focus for systematic transformation of the education and training system. Some of the objectives of the NQF are to create an integrated national framework for learning achievements and to accelerate the redress of past unfair discrimination in education, training and employment opportunities.

Furthermore, an outcomes-based education approach was chosen as the vehicle to implement the objectives of the NQF at all levels and sectors of education and training in the country. When the Minister of Education announced the introduction of a new curriculum framework in 1995, there were plans to introduce it into all grades by 2005. In line with this timetable, the new National Curriculum Statement (NCS) became known as Curriculum 2005 (C2005). At a broader level, eight critical outcomes have been chosen to ensure that learners would be prepared for life in a global society. These generic, cross-curriculum outcomes also reflect the aims of the Constitution.

C2005 was inspired, not so much by the theories of others, nor on experiences elsewhere, but was an attempt to respond in an authentic manner to the realities facing the South African classroom. But it was also flawed in several ways. Some of these were design flaws while others were directly attributable to the rate and scope of implementation. None of these however, outweigh the significance or detract from the impact of C2005 as the curriculum policy that would forever change the landscape of education in South Africa.

The development of an NCS was seen as a key project in the transformation of South African society. The thrust of the project is towards achieving, in the words of the DoE, a prosperous truly united, democratic and internationally competitive country with literate, creative and critical citizens leading productive, self-fulfilled lives in a country free of violence, discrimination and prejudice (DoE, 1997, p. 4).

Curriculum reform since 1994 faced several challenges. These include:

• The post-apartheid challenge: to provide awareness and the conditions for greater social justice, equity and development. This is the challenge of develo** new values and attitudes.

• The global competitiveness challenge: to provide a platform for develo** knowledge, skills and competences to participate in an economy of the twenty first century.

• The challenge of develo** critical citizens: citizens in a democracy need to be able to examine the many issues facing society and where necessary to challenge the status quo and to provide reasons for proposed changes.

The view taken by the curriculum designers was that the best route to greater social justice and development is through a high-knowledge and high skills curriculum and that mathematics education can play a vital role in the realisation of this vision. The general expectation was that the NCS would result in learners who are literate, numerate and multi-skilled, but who are also confident and independent, compassionate, environmentally respectful and able to participate in society as critical and active citizens.

Review Committee on Curriculum 2005 recommended major changes to the NCS (C2005) in May 2000 and the Revised National Curriculum Statement (RNCS) was implemented immediately thereafter. The vision adopted by the Review Committee in 2000 keeps in focus the dual challenge for C2005 of addressing the legacies of apartheid on the one hand and preparing learners to participate in the global village on the other – these two are taken as indivisible. The RNCS has been further refined in 2011 through a new statement called the Curriculum and Assessment Policy Statement (CAPS) (DoBE, 2011) that specifies content and assessment criteria in a more integrated manner.

Mathematics Curriculum Pre-1994

During the apartheid period the canonical syllabus for mathematics, although compartmentalised by race, had remained roughly invariant for everyone over decades. In a sense, the content was almost immaterial and by itself, made very little difference to the way mathematics, as a school subject, was used as a means of control and social stratification. Some attempt was made to revise the mathematics syllabus every eight years or so, but this rarely made any substantive change to the core content. Even in the current South African curriculum parlance, mathematics is referred to as a ‘gateway subject’ precisely because it provides access as a gatekeeper. More than any other subject, mathematics will decide who will stay behind and who will go ahead. Although some may feel that mathematics has only been able to assume this central position in the curriculum because it is over-admired and over- privileged, very few will question the need for all learners to be ‘mathematically literate’.

In fairness it must be acknowledged that a feature of school mathematics during the late 1980s and early 1990s was a concerted effort by some mathematics educators to adopt a different approach to the teaching and learning of mathematics at school. The impetus for this change came largely from the world-wide swing towards a constructivist perspective that was implemented mainly in white primary schools in South Africa. Euphemistically called the ‘problem-centred approach’, this perspective came across in the South African context as a prescriptive methodology, a new orthodoxy, which dismissed and replaced any set of ideas mathematics teachers may have had about the teaching of the subject. Nevertheless, few will deny that where this constructivist approach was piloted, it made a significant change to the classroom culture. Pupils at these schools developed very positive attitudes to mathematics and there is strong evidence that they also developed powerful ways of learning mathematics. It would therefore be unfair to say that this “socio-constructivist” approach to mathematics did not have a beneficial effect on classroom practice. It is however the case that the classroom of majority population in South Africa, where the teacher typically has to cope with a large class and poor resources, was left virtually unreached and therefore unaffected by this approach.

During the pre-1994 period People’s Mathematics developed independently and indigenously rather than an attempt to embrace the “loudest fad from the West”. In addition to facilitating discourses around mathematics in the communities, People’s Mathematics also developed a unique emphasis and character. Cyril Julie (1991) argues that the four major distinguishing features of People’s Mathematics were:

• its ability to reveal how school mathematics can be used to reproduce social inequalities;

• its rejection of absolutism in school mathematics and its contribution towards seeing mathematics as a human activity and therefore necessarily a fallibilist one;

• its incorporation of the social history of mathematics into mathematics curricula and its belief in the primacy of applications of mathematics.

Julie acknowledges that People’s Mathematics did not have the desired effect on the development of a mathematics culture at the time. This he claims, is partly due to the preoccupation of the advocates of People’s Mathematics to design mathematical activities that had a direct bearing on the day-to-day political struggles of the people. Another reason for its lack of efficacy was the sense of scepticism and even distrust about the notion of People’s Mathematics as a poor substitute for the ‘real mathematics’. People’s Education (PE) failed to re-direct its focus away from a struggle in the streets to a struggle within the classroom. While it may be the case that it was too overtly political or even woolly at times, the People’s Mathematics Movement did provide a focus for mathematics curriculum debate and indeed for PE itself and it was encouraging to how the spirit and core ideas of PE became mainstreamed in the National Curriculum Statement.

Mathematics Curriculum Reform Post-1994

In the post-apartheid era, mathematics curriculum reform continues to be influenced by two main considerations namely, a call for mathematics for all and the need to ensure mathematics by all. The first deals with the legacy of the past and considerations of equity, while the second is response to a renewed focus on quality of provision and global economic challenge of participating in a global village.

Mathematics for All

In a country where there has been a neglect of provision for decades, the need for massification of provision remains a major challenge for the future of education in general, and of mathematics in particular. The legacies of gross discrimination of the past meant that blacks were actively discouraged from taking mathematics as a subject. Historically between 30% and 40% of secondary schools in the country simply did not offer any mathematics beyond grade nine. We now have a policy that requires that everyone must take some form of mathematics. “Mathematics for all” is fundamentally a statement of policy, and as such it is a statement of provision. Of course it is a statement about curriculum, but essentially it signals that every learner should have the opportunity to learn mathematics.

But mathematics for all does not necessarily mean the same content for all. It is a truism that what content is used must be tied to purpose. It is therefore perfectly reasonable to assume that while all learners need mathematics, not all need the same mathematics. Mathematics for all however, must mean the same quality of mathematics for all. Although this seems to be an educationally defensible position, the idea of a differentiated approach to subject offerings at school (including mathematics) was rejected in favour of a single undifferentiated approach to mathematics. This decision should be seen within its historical and political context. During the pre-democratic era and up until 2007, more than 10 years into the new democracy, mathematics, like all other subjects was offered at two levels namely Higher Grade (HG) or Standard Grade (SG). At the dawn of democracy only twenty percent of blacks were taking HG mathematics while seventy percent of whites took mathematics at the same level. A Ministerial Committee on Differentiation (DoE, 2003a) recommended that curriculum reform in South Africa move away from differentiation at subject level.

In order to comply with the new policy that all learners to take some form of mathematics, mathematics literacy was introduced as a high-school subject from grade ten level in 2006 as part of the field of mathematics. Although seen as part of the ‘field of Mathematics’, it had a very different purpose from that of mathematics. While mathematics is important as a foundation for those with an interest to pursue work and further study in fields that require mathematics (such as business, science and engineering), mathematical literacy is about hel** people to participate more fully in the choices that affect their lives. Mathematical literacy may help individuals to engage in discussion with employers over what constitutes fair wages and conditions of service, make sense of even participate in national debates on issues such as health, crime etc., particularly where quantitative arguments are used.

Generally, mathematical literacy was intended to assist learners to take charge of their own experiences as self-managing individuals and critical citizens in a democracy, crucial for nation-building and the strengthening of the new democracy. However, it was never meant to be a dead-end low-level subject that represents a kind of watered-down mathematics in the same way that SG mathematics differed from HG mathematics. In short, the difference between mathematics and mathematical literacy is a difference in kind rather than level or degree. Initially, many more learners opted for mathematical literacy, but in recent years there has been a more even split with 56% of the 617, 982 grade 12 candidates enrolled for mathematics literacy in 2018.

One of the points of departure is that the South African school curriculum is composed of ‘learning areas’ rather than subject disciplines. Integration within and across learning areas is another important building stone of the curriculum.

In the learning area of mathematics there are five learning outcomes (DoE, 2003b). They are:

• 1. Numbers, operations and relationships: The learner is able to recognise, describe and represent numbers and their relationships and can count, estimate, calculate and check with competence and confidence in solving problems.

• 2. Patterns, functions and algebra: The learner is able to recognise, describe and represent patterns and relationships, and solves problems using algebraic language skills.

• 3. Space and shape: The learner is able to describe and represent characteristics and relationships between 2-D shapes and 3-D objects in a variety of orientations and positions.

• 4. Measurement: the learner is able to use appropriate measuring units, instruments and formulae in a variety of contexts.

• 5. Data handling: The learner is able to collect, summarise, display and critically analyse data in order to draw conclusions and make predictions, as well as interpret and determine chance variation.

As in the case of the other learning areas, the mathematics learning area is based on the principles of high knowledge, high skills and integrates within mathematics and with other learning areas. It infuses concerns of human rights and inclusivity throughout the assessment standards.

There is however always a danger that there would be a lack of fit between the intended curriculum and the actual or implemented curriculum. This danger is of course very great in South Africa where the biggest challenges for implementation are the lack of resources and adequate teacher training, infrastructure and leadership capacity. Teachers implementing C2005 indicated that although they believed it to be beneficial to their learners and were eager to implement it, they were undermined in their efforts to do so in the absence of the necessary support.

Mathematics By All

While mathematics for all is a statement of provision, mathematics by all is a statement of participation and a statement of mathematical engagement. If we are concerned only with provision of opportunity and the construction of mathematics curricula, without considering who is engaged in mathematics and how they are engaged, we will be giving ourselves a false sense of comfort. There is very little point in laying a table with the best food without inviting those around the table to participate in the eating and enjoyment that goes with it. There is a recognition that if we are going to effect change in South Africa, we have to accept that both “mathematics for all” and ‘mathematics by all’ are essential ingredients of a transformation agenda. The focus in education generally has been shifting from provision and access to quality.

At the same time the educational measurement industry both locally and internationally has, with its narrow focus, taken the attention away from the things that matter and has led to a traditional approach of raising the knowledge level. South Africa performs very poorly on the TIMSS study. In the 2015 study South Africa was ranked 38th out of 39 countries at grade 9 level for mathematics and 47th out of 48 countries for grade 5 level numeracy. Also, in the Southern and Eastern Africa Consortium for Monitoring Educational Quality (SACMEQ), South Africa was placed 9th out of the 15 countries participating in Mathematics and Science – and these are countries which spend less on education and are not as wealthy as we are. South Africa has now developed its own Annual National Assessment (ANA) tests for grades 3, 6 and 9. In the ANA of 2011, grade 3 learners scored an average of 35% for literacy and 28% for numeracy while grade 6 learners averaged 28% for literacy and 30% for numeracy.

Although these performances are pertinent in assessing educational quality of mathematics in the country, we have become pre-occupied with the political pressure to ‘do better’ and to improve our relative standing in relation to other countries using the comparative construct provided by these studies. In this process our focus has been fixed on the ‘knowing of mathematics’ instead of the ‘doing of mathematics’. In our attempt to get teachers and learners to demonstrate knowledge we forget sometimes that teaching and learning are actions and that people rather than knowledge must be at the center. Mathematics by all is about changing the focus away from provision and compliance towards engagement and taking charge of our own mathematical experiences. This is not being reckless about the importance of knowledge but to see the key challenge facing mathematics teachers and learners as that to engage with the subject and to get them to believe that mathematical engagement could be part of their ‘possible selves’.

Mathematics by all means that everyone is engaged in a quality mathematical experience. Quality of mathematical teaching and learning depends on whether the teacher can select cognitively demanding tasks and plan the learning experiences by encouraging learners to go beyond the “answer” to seek elaborations and generalisations whenever appropriate to do so through these tasks. This will require learners and teachers alike to commit to extra time on task and be engaged cognitively, socially and mathematically.

Allocating sufficient time for the learners to engage in and spend time on mathematical tasks in an already overcrowded curriculum presents a significant challenge. To address this challenge policy makers are currently in engaged in develo** a new Mathematics Teaching and Learning Framework for South Africa: Teaching Mathematics for Understanding (DoBE, 2018). It is not intended to be a new curriculum but supports the implementation of the existing CAPS curriculum by introducing a model to help teachers change the way they teach. Taking its bearing from the work of Kilpatrick et al. (2001), the model of teaching mathematics has four dimensions: conceptual understanding, mathematics procedures, learners’ own strategies and reasoning while each of these takes place in a dynamic classroom culture. In addition, the topics in the existing mathematics curriculum will be re-sequenced and even where necessary, removed to make space and time for deeper mathematics engagement.

While it is recognised that one of the major problems in mathematics education in South Africa is the level of teacher knowledge, it is felt that there has been too much emphasis on “teacher blame” when trying to explain the poor level of learner proficiency in mathematics. While teachers with strong content knowledge are more likely benefit from high level interventions and they therefore are more likely to lead their learners into richer mathematical experiences, strength in content knowledge does not always transfer to pedagogical knowledge. However, we need now to go beyond this and ask what we can do within the current reality. To wait until teachers’ knowledge has all radically improved would drive us into paralysis. Transformation of the classroom practice must begin with an enabling framework. Teachers’ re-socialization into the new mathematics landscape envisaged in the new framework would have to start with unfreezing and deconstructing existing notions of working mathematically. The work of Leone Burton (1999, 2004) and Jo Boaler (1998, 2002) illustrate how important it is for teachers to themselves be immersed in mathematical experiences that will give them an insight into the practice of mathematicians.

In summary, South Africa has a new set of values: democracy, social justice and equity, equality, non-racism and non-sexism, ubuntu (human dignity), an open society, accountability (responsibility), the rule of law, respect, and reconciliation are the ten fundamental values of our Constitution. The promotion of these values is seen as important, not only for the sake of personal development, but also for the evolution of a national South African character. These values have been infused in all learning areas and school mathematics in particular is expected to respect these values. The need is to develop a mathematics curriculum that will not only recognise the global competitiveness challenge by providing a platform for develo** the knowledge, skills and competences to participate in an economy of the twenty first century, but also to show how our fundamental values can be lived out in our everyday experience while at the same time illuminating and exposing violations of these values. The mathematics curriculum reform in South Africa holds in tension the need to provide mathematics for all on the one hand, while creating opportunities to ensure that mathematics achievement is seen and experienced as part of the ‘possible self’ of every learner.

Final Conclusions

The four experiences illustrated above show some commonalities as well some crucial differences insofar the same issue is sometimes faced from unlike and even opposite standpoint because of context historical and social differences.

Among the commonalities is the theme of documentation as an agent of reform: however, also this is approached in different ways. For example, the report from Lebanon underlines this as a centralised action that can guide the production of textbooks as a main guide for teachers, who in their turn can so become agents of reform. On a different stream, the report from Australia argues that documentation can assume and create a focus for teachers as active learners about curriculum and pedagogy. Also, the rationale of documentation can be different, as pointed out in curricular dichotomies, which can determine the philosophy of curricular reforms: for example, documentation of everything is possible vs. including just enough information.

A theme which is faced in very different ways is that of inclusion, which may have opposite or at least different results with respect to the waited ones. For example, in Chile this issue was interpreted as the aim of having quality education for all, in compliance with the international standards of the globalised world, and with a wide consultation process in the country, which involved many agents; however, the result consisted in having too rigid rules, which made it difficult to apply them. Another issue concerning the possibility of having an effective inclusion was also linked to socio-economic features of the countries, which can strongly influence the way different types of agents can realize the curricular reforms.

This aspect was underlined in the report from Lebanon a discussing the differences between public and private schools as agents of reforms, and from South Africa, in the crucial difference in the realisation of curricular reforms and in the concrete actions of their agents, namely that between a mathematics for all and a mathematics by all: the former meaning that a curriculum must design the same quality of mathematics for all, while the latter that everyone is engaged in a quality mathematical experience. This last aspect highlights a subtle but important aspect in the way agents can be really effective in the implementation of curricular reforms.

In a sense, this issue is present also in another feature of the complex landscape produced by the variety of curriculum reforms agents, that is in what one of the curricula dichotomies in the Australian report formulates as the difference between mathematics for elite or mathematics for all. This conflict is present in different forms in almost all the contributions to the panel, and assumes interesting connotations in other curricular conflicts, represented in different curricular dichotomies: from the possible differences between the writers of the curriculum, practitioners vs. specialists, to the aims of the mathematics curriculum, as preparation for later study vs. mathematics as experience, to the ways, general versus specific, which describe the concepts or content that form the focus of learning experiences.

In general, what appears as a main question from the different contributions, and is explicitly pointed with reference to the curricular reform in Lebanon, is the way teachers are involved in the processes of curricular reforms. In fact, the success of a curricular reform heavily depends on the way they concretely interpret and apply it in their classrooms, as well as how their students react to it. Theme C expresses this issue as the ‘law of alignment’ (Chap. 19): the effective implementation of a curricular reform in the classrooms depends on the way its mathematical content and pedagogical assumptions, materially written in the official documents, are effectively interpreted by the ‘terminal’ chain of the curriculum agents, that is the teachers, and how the interpretation determine/change their beliefs and practices.

All this happens in concrete historical and social contexts, which produce different, possibly opposite effects, to apparently similar actions: this poses delicate and very difficult questions for researchers. In the following chapters of theme E, this point emerges as a relevant stream of discussion and analysis and, as it will pointed out in the final comments to the theme-related chapters, the issue of resilience of a curriculum plan remains as one of the main problems to be solved in the face of possible disruptions to the planning and enactment of curriculum reforms.