AMESA (the Association for Mathematics Education of South Africa) has prepared this submission to the Department of Education through its Curriculum Committee and in consultation with a wide range and large number of members.

Introduction

AMESA would like to thank the Department of Education for the opportunity to participate in this Public Consultative session. Given the time available, we have chosen to use this opportunity to make our most salient observations and comments with respect to the two subject statements: Mathematics and Mathematical Literacy. Our fuller commentary, which will critique the statements at a paragraph-by-paragraph level and which is structured according to the headings in the Minister’s invitation for comment, will be submitted by 31 January 2003.

Our remarks today are made in three parts:

• Mathematical Literacy
• Mathematics
• General and concluding remarks

We stand on the threshold of an exciting time. Our country and our education department are about to embark on a most exciting, commendable and entirely necessary project. The introduction of Mathematical Literacy as a compulsory subject for all learners in the FET Band is new, it is critical for the development of this country and it is a brave decision that few countries have taken. It is therefore important that we succeed in this our first attempt, or the opportunity may be lost for years to come.

In 2001, commenting on the SAQA unit standards for Mathematical Literacy (NQF levels 2, 3, and 4), AMESA wrote:

1. We recognise the need for and support the introduction of Mathematical Literacy as a compulsory credit for all learners at NQF levels 2, 3 and 4. That is, we see the learning of Mathematics as the right of all learners.
2. We believe that all learners at NQF levels 2, 3 and 4 have the ability to learn Mathematics to the extent that it has meaning in their lives.
3. Agreeing with SAQA’s stated aim of the FETC, namely that: the primary purpose of the FETC "is to equip learners with the knowledge, skills and values that will enable meaningful participation in and offer benefits for society as well as providing a basis for continuing learning in higher education and training, and enable learners to be productive and responsible in the workplace," we believe that relevance should be the measure against which the inclusion and exclusion of content is determined.
4. We are concerned that Mathematical Literacy should not be a "watered down" academic Mathematics but rather Mathematics with a different emphasis. If the purpose of the FETC is among other things to benefit society then the Mathematics needed by the learner is not necessarily more (in terms of knowledge) than that covered at the GET level, but rather the Mathematical thinking skills—habits of mind—to be able to apply that learning in various contexts.

We are, as an association, extremely concerned that we have not yet, in this country, conceptualised properly what Mathematical Literacy is. This lack of understanding of what constitutes Mathematical Literacy is reflected in the subject statement at both a definitional and assessment standard level. Mathematical Literacy is not:

Although Mathematical Literacy and Mathematics are quite clearly related, they are also quite different. In the conceptualisation of Mathematical Literacy as presented in the subject statement it remains too much a watered down version of Mathematics. The outcomes are essentially the same as those of Mathematics. It may be that this is done intentionally, to address matters of portability and mobility between Mathematics and Mathematical Literacy, but we argue that the two subjects are so dissimilar in philosophy and purpose that such portability and mobility should not be a consideration.

If literacy is the ability to read and write, then Mathematical Literacy should be the ability to read, write, and engage with information and situations that are numerical in nature and mathematical in structure. While the mathematically literate person may draw on mathematical algorithms or knowledge, their mathematical literacy is reflected in habits and behaviours and ways of engaging with problems and situations. Mathematical literacy is further reflected in the confidence with which a person uses the mathematical algorithms or knowledge. That is, the person does so without fear of the algorithms and knowledge or of their capacity to be able to do so. A Mathematical Literacy curriculum should therefore be much more about habits and ways of behaving than about "content." The proposed Mathematical Literacy curriculum reads too much like a Mathematics curriculum and does not reach the target set for it. In its present form we predict that it will not make the important social contribution that we hope for.

While recognising the sincere and concerted efforts of those who have the task of developing the Mathematical Literacy subject statement, we believe that the statement is poorly conceptualised.

1. The statement does not adequately communicate a clear image of what Mathematical Literacy is or a philosophy with respect to how it should be taught.

Given the nature of Mathematical Literacy we believe that the subject statement should foreground the teaching approach. That is, rather than listing mathematical skills in the form of assessment standards, the statement should suggest contexts and then demonstrate how those contexts can be exploited to develop the habits and behaviours that constitute Mathematical Literacy in learners. Outcomes could be framed as behaviours rather than topics in Mathematics, and assessment standards as demonstrations of ways of behaving in contexts rather than lists of mathematical skills. Written the way the statement is, we are concerned that the teaching of Mathematical Literacy can and will be algorithmic in nature, rather than developing attitudes and behaviours. It may be that the GET Natural Science statement gives an image of how the subject statement could be revised.

2. With respect to the assessment standards, we have several concerns.

What the authors have done well, in writing the assessment standards, is to recognise that the learner, on completion of the GET band, probably has an adequate repertoire of mathematical skills and that what is needed in Mathematical Literacy is the opportunity to learn to use these skills with increasing confidence and in increasingly sophisticated contexts. As a result we notice that the assessment standards do not build significantly on the mathematical skills of the GET Mathematics standards, and this is as it should be. However, given the design brief of the subject statement and the desire to show progression from one grade to the next, we observe frequent instances of what can be described as "arbitrary" allocations of assessment standards to grades.

In general the assessment standards overemphasise algorithmic behaviour and underemphasise relationships between assessment standards. Several of our members, in commenting on the subject statement, have noted that it is "overloaded." Whether the statement is overloaded or not, is very much a function of how it is understood and taught. If it were possible for the authors to make more explicit the links and underlying connections between the different standards, then the statement may not seem quite as overloaded or to consist so much of a collection of lists.

At a more detailed level, members have expressed concern that there may be an overemphasis on financial contexts, and an under-emphasis on other concepts useful for the functioning of the individual in a democratic society. Concern has also been expressed that many of the assessment standards are too vague. The authors have clearly struggled with the tension between over specifying to teachers what should be taught, and leaving to teachers some of the decisions with respect to scope and content selection. The result is understandably that the reader will at times feel that the assessment standards are not specific enough, and at other times too dogmatic and full of "check-lists." We believe that this may not be easily resolved.

IMPLEMENTATION

We have several grave concerns with the implementability and proposed implementation timeframe of this subject statement. We need to recognise that all FET learners (with the exception of those who do Mathematics) will be doing Mathematical Literacy. At present a little more than one half of all matriculants offer Mathematics and we predict that more than half of current Standard Grade Mathematics students will opt for Mathematical Literacy. By implication we anticipate some 400000 Mathematical Literacy students per grade. There are simply not enough well qualified or trained teachers able to respond to this need, and certainly not by January 2004.

If Mathematical Literacy and Mathematics are indeed as different as we suggest then even current Mathematics teachers are not adequately prepared to teach Mathematical Literacy. Current teachers, in the main, lack the capacity both to connect their mathematics to real contexts and struggle to see the internal connections between mathematical concepts. Given that every learner will have to pass this (rather demanding) subject in order to matriculate, we cannot afford to allow this shortage of qualified teachers to impact negatively on school leaving pass rates.

To train teachers to teach Mathematical Literacy and to implement Mathematical Literacy both require, among other things, high quality resource materials. In the absence of a well-articulated philosophy on the nature and teaching of Mathematical Literacy it is not possible to develop such materials and by implication to train teachers and implement the subject.

We believe that to proceed with the proposal of implementing Mathematical Literacy in 2004 will certainly prevent the achievement of the high hopes we have for the subject. The introduction of Mathematical Literacy by the Department and the country is a most exciting and socially important initiative. Let us not lose the opportunity to make a very positive change, by being under-prepared for it.

TECHNOLOGY

The subject statement makes frequent references to the use of technology, and in particular to the use of calculators, spreadsheets and even software such as Geometer’s Sketchpad. We believe that rather than leaving this to chance, by suggesting that technology should and could be used in certain instances, its use should be enforced. We recognise that this necessarily places on the Department the burden of ensuring that every child has access to such technology, to avoid further exaggeration of the differences that exist in a stratified society. The need to ensure equitable and easy access to technology by all learners, coupled with the importance of technology in a Mathematical Literacy curriculum, gives further support to the argument that the implementation of this curriculum should be delayed.

UNIVERSITY ENTRANCE

Some questions have been raised by members about the role that Mathematical Literacy will play in the acceptance of students into university. This is an important issue, in that teachers will need to guide students in making the choice between Mathematics and Mathematical Literacy in the next few months, should implementation in 2004 be pursued. We urge that clarity in this regard be developed as a matter of urgency. This also provides a further argument for delay. Teachers have also asked for clarity on the need or not to do Mathematics if a learner wants to do Physical Science.

CONCLUSIONS WITH RESPECT TO MATHEMATICAL LITERACY

It is our hope that the Mathematical Literacy subject statement offered for public comment will be seen as a first iteration in a debate on developing such a statement. It is our sincere hope that the Department will avoid the temptation to "fix" the statement and publish it by the end of February, but rather to allow sufficient time for a substantial revision. We would propose that, given the completely new and untested nature of this subject, as well as its critical role in social upliftment and nation building, a focussed debate on what constitutes Mathematical Literacy and how it should be taught be convened. Such a debate should inform the next iteration of the subject statement. There are few international precedents that we can look to: we need to develop this uniquely South African curriculum for South Africa

We have the capacity and the determination: please grant us, as a country and education community, the time. We acknowledge that we as an association do not have a clear and uniform understanding of the answers in this debate, but we believe that the Association could play a significant role in bringing together players and facilitating the discussion for the Department if invited to do so.

MATHEMATICS

The Mathematics subject statement reflects interesting shifts in focus, content and approach to the current Mathematics curriculum and the authors are to be commended on their courage in this regard.

Given our comments on Mathematical Literacy, and in particular our belief that Mathematics and Mathematical Literacy are quite different, even if quite clearly related, we note with some concern that the Department has opted to allow learners to not take Mathematical Literacy if they take Mathematics. Of course this represents a compromise that is quite possibly a necessary one. However we know well that the study of Mathematics at Higher and/or Standard Grade level has not guaranteed mathematical literacy. It is therefore important that opportunities for developing mathematical literacy be provided for in the Mathematics subject statement. While there are several references to Mathematical Literacy scattered through the document, commitment to Mathematical Literacy is not sufficiently evident in the assessment standards. One area in particular that we feel is too sophisticated at this stage is LO4 (Data Handling and Probability). This outcome could be significantly reduced in the extent to which it is developed (in particular with respect to the combinatorial analysis) to allow for a teaching approach that is more in line with the philosophy of Mathematical Literacy.

One member in commenting on the Mathematics Subject statement says: "It seems to me that there is a reorganisation of the existing curriculum into a new form, with data added in. So the question of overloaded content remains." It is very easy to see how the member can make this remark. Very few topics in the current syllabus have been omitted and several new ones added. However, it would appear (particularly from reading the purpose and scope paragraphs) that while the authors have retained most topics (and added others) they do not see the topics as being isolated, and in most cases are not expected to be covered to the same extent. That is, the philosophy that has guided the selection of skills and content for inclusion in the statement is one of modelling. The authors appear to have foregrounded mathematical modelling and expect it to be used as the vehicle to learning Mathematics. In this sense many topics are included as tools for mathematical modelling rather than as topics in their own right.

If our assumption about the philosophy guiding the selection of skills and content is correct then the subject statement may not be overloaded. However if we are incorrect in this regard then the subject statement is clearly overloaded.

Furthermore, assuming that modelling is indeed the underlying philosophy of this subject statement, we must also raise the concerns that a Mathematics programme must do more than provide skills for modelling, it must also teach mathematical enquiry and mathematical processes as well. That is, learners should come to understand the structure of and abstract nature of Mathematics itself. This is a serious concern with respect to the subject statement and we urge the authors to attend to this matter. The roles of proof and generalisation are two issues that come to mind in this regard. At the same time we run the risk of missing the opportunity of including more modern topics in Mathematics such as matrices, iterative processes and aspects of fractal geometry.

With those remarks as background our most significant concern with this subject statement is that the underlying philosophy with respect to the teaching and learning of Mathematics that has guided the development of the statement is simply not apparent enough. As a result, the reader is left without a clear understanding of the scope and extent to which each assessment standard needs to be addressed. To suggest that the assessment standards are minimum standards is not really an adequate response to this concern. In most cases the reader is left to interpret the reason for inclusion of assessment standards, and the extent to which they need to be applied.

We urge the authors, in revising this subject statement to be significantly more transparent about the philosophy that has guided its development. We suggest that the "learning outcome focus" paragraphs be developed significantly to help the reader understand the inclusion of and extent of assessment standards and that a way be found to make the links and interrelationships between the different assessments standards within different learning outcomes more explicit.

By making the philosophy with respect to teaching and learning more explicit, by showing the links between assessment standards and by being significantly more explicit in terms of exactly what is expected with respect to each assessment standard (reducing vagueness) we believe that the curriculum may not be as overloaded as it seems.

We want to caution against the possible response that the ambiguity we allude to will be addressed in a Learning Programme guideline document that is to be published alongside the subject statement. The subject statement should be able to stand on its own and not be as open to interpretation as this one is in its current form.

Ultimately teachers "see" a subject statement translated in two significant ways; firstly, in the learning support materials developed for the subject (in the case of Mathematics—most often a textbook) and secondly, through the assessment and in particular the external assessment of the subject. We believe that a lot of clarity could be gained with respect to the role of and extent to which topics should be covered by providing an image of the exit examination. It is true that the provision of the examination guideline document with respect to the current Higher and Standard Grade Mathematics syllabi has played a very useful role in this regard.

IMPLEMENTATION

We believe this subject statement represents an attempt at a significant departure from the current Mathematics programme. Furthermore we also believe that Mathematics in this form; not consisting of Higher and Standard Grade (a decision we are quite happy with) will have a much larger participation rate than the current Higher Grade programme. In light of these remarks, we believe that significant teacher training will be required, to ensure the effective implementation of the new Mathematics curriculum. Such teacher training will have to deal with, in particular, the shift in approach to Mathematics and its role. In addition training will have to attend to topics such as transformations and probability that are both new to the curriculum and unfamiliar to most teachers. Teacher training will also have to deal quite explicitly with the importance of learners also becoming mathematically literate while studying Mathematics.

The introduction of this subject statement will place high demands, in terms of change, for teachers of Mathematics in the FET Band. Many of these teachers are already involved in changes in the GET Band and in particular with the changes associated with the introduction of a GETC. We are concerned that, should the implementation of Mathematics in Grade 10 proceed as planned in 2004, teachers may simply be overwhelmed by all the concurrent changes. This would compromise the effective introduction of an exciting subject statement.

For these reasons, we urge the Department to re-consider the current implementation timeframe.

TECHNOLOGY

We consider technology to be an important tool of teaching and learning Mathematics in the 21^st century. We urge that the subject statement ensures the use of technology rather than leaving it to chance, by suggesting opportunities for its use. We would like to suggest that a greater range of technologies (beyond spreadsheets and dynamic geometry software) be described in the document and its use in exploration, experimentation, and testing of conjectures be explored. Finally, we would consider the provision of appropriate technology in all schools critical, both to effective implementation of this subject statement and to ensuring that we do not create further division in access to Mathematics by learners, according to the poverty index of their school.

CONCLUSION WITH RESPECT TO MATHEMATICS

This subject statement is brave in the shift that it represents and exciting in terms of its potential. For the successful interpretation of the statement and implementation of the subject, much work remains in making the teaching, learning and knowledge selection philosophy considerably more explicit.

GENERAL AND CONCLUDING REMARKS

As indicated in the opening remarks AMESA will submit, before the end of the month, a further document with comments on the various features of the subject statements and detailed remarks with respect to the individual assessment standards.

AMESA is deeply committed to the transformation of Education and Mathematics Education as well as the nation building that is implied and intended by the changes in curriculum being proposed. Our comments are intended to strengthen that process and AMESA is proud to be a social partner of the Department of Education on this journey. We hereby commit the Association to supporting the further development of these documents and to contributing to both teacher training and the development of learning support materials if invited to do so.

We stand on the threshold of exciting and important changes in Education and Mathematics Education. We sincerely hope that we will, by heeding the cautions expressed in this submission, gain all the benefits we can from the opportunity that these changes provide.

Thank you for the opportunity to participate in this process and all strength to the Department of Education as you lead us forward in this work.

Compiled by the AMESA Curriculum Committee
21 January 2003