SA NCS:Mathematics
Introducing the Subject
DEFINITION
The curriculum for Mathematics is based on the following view of the nature of the discipline.
Mathematics enables creative and logical reasoning about problems in the physical and social world and in the
context of Mathematics itself. It is a distinctly human activity practised by all cultures. Knowledge in the
mathematical sciences is constructed through the establishment of descriptive, numerical and symbolic
relationships. Mathematics is based on observing patterns; with rigorous logical thinking, this leads to theories
of abstract relations. Mathematical problem solving enables us to understand the world and make use of that
understanding in our daily lives. Mathematics is developed and contested over time through both language and
symbols by social interaction and is thus open to change.
PURPOSE
In an ever-changing society, it is essential that all learners passing through the Further Education and Training
band acquire a functioning knowledge of the Mathematics that empowers them to make sense of society. A
suitable range of mathematical process skills and knowledge enables an appreciation of the discipline itself. It
also ensures access to an extended study of the mathematical sciences and a variety of career paths.
The study of Mathematics contributes to personal development through a deeper understanding and successful
application of its knowledge and skills, while maintaining appropriate values and attitudes. Mathematics is a
discipline in its own right and pursues the establishment of knowledge without necessarily requiring
applications in real life. Competence in mathematical process skills such as investigating, generalising and
proving is more important than the acquisition of content knowledge for its own sake.
Mathematical competence provides access to rewarding activity and contributes to personal, social, scientific
and economic development. It is understandable, therefore, that a variety of stakeholders in society exert
demands on school Mathematics. These stakeholders include parents, learners, educators, Mathematics
educators, employers, professional mathematicians, tertiary institutions, and cultural and political organisations.
Individual and collective engagement with Mathematics will provide valuable opportunities for the
development of a variety of values, as well as personal and interpersonal skills.
Mathematics enables learners to:
- communicate appropriately by using descriptions in words, graphs, symbols, tables and diagrams;
- use mathematical process skills to identify, pose and solve problems creatively and critically;
- organise, interpret and manage authentic activities in substantial mathematical ways that demonstrate responsibility and sensitivity to personal and broader societal concerns;
- work collaboratively in teams and groups to enhance mathematical understanding;
- collect, analyse and organise quantitative data to evaluate and critique conclusions; and
- engage responsibly with quantitative arguments relating to local, national and global issues.
An important purpose of Mathematics in the Further Education and Training band is the establishment of
proper connections between Mathematics as a discipline and the application of Mathematics in real-world
contexts. Mathematical modelling provides learners with the means to analyse and describe their world
mathematically, and so allows learners to deepen their understanding of Mathematics while adding to their
mathematical tools for solving real-world problems. Mathematics can be used in a wide variety of physical,
social and management sciences. An appreciation of the manner in which Mathematics has developed over time
establishes its origins in culture and the needs of society.
SCOPE
Learners in the Further Education and Training band who are interested in the subject or who intend to follow a
career path requiring Mathematics will, while ensuring that they are mathematically literate, work towards
being able to:
- competently use mathematical process skills such as making conjectures, proving assertions and modelling situations;
- calculate confidently and competently with and without calculators, and use rational and irrational numbers with understanding;
- competently produce useful equivalents for algebraic expressions, and use such equivalents appropriately and with confidence;
- use Mathematics to critically investigate and monitor the financial aspects of personal and community life and political decisions;
- work with a wide range of patterns and transformations (translations, rotations, reflections) of functions and solve related problems;
- describe, represent and analyse shape and space in two and three dimensions using various approaches in geometry (synthetic, analytic transformation) and trigonometry in an interrelated or connected manner;
- collect and use data to establish basic statistical and probability models, solve related problems, and critically consider representations provided or conclusions reached;
- use and understand the principles of differential calculus to determine the rate of change of a range ofsimple, non-linear functions and to solve simple optimisation problems;
- solve problems involving sequences and series in real-life and mathematical situations;
- solve non-routine, unseen problems using mathematical principles and processes;
- investigate historical aspects of the development and use of Mathematics in various cultures; and
- use available technology (the minimum being a modern scientific calculator) in calculations and in the development of models.
Such mathematical skills and process abilities will, where possible, be embedded in contexts that relate to
HIV/AIDS, human rights, indigenous knowledge systems, and political, economic, environmental and
inclusivity issues.
EDUCATIONAL AND CAREER LINKS
Mathematics is an essential element in the curriculum of any learner who intends to pursue a career in the
physical, mathematical, computer, life, earth, space and environmental sciences or in technology. Mathematics
also has an important role in the economic, management and social sciences. It is an important tool for
creating, exploring and expressing theoretical and applied aspects of the sciences. Mathematics is also
important for the personal development of any learner.
The learning achieved in Mathematics in the General Education and Training band provides an essential base
from which to proceed into the demands of Mathematics in the Further Education and Training band. The
essentials of numeracy developed in the General Education and Training band are taken further, working in
more symbolic ways. The General Education and Training engagement with space and shape becomes more
formalised. The methods and uses of statistics and chance are dealt with in greater depth. How Mathematics
can contribute to an understanding of financial issues is taken beyond dealing with budgets. The emphasis on
contexts and integration within Mathematics and across the curriculum is maintained, while mathematical
modelling becomes more prominent.
The subject Mathematics in the Further Education and Training band will provide a platform for linkages to
Mathematics in Higher Education institutions. It will also provide for linkages to Mathematics of a
complementary nature but specific to the needs of the individual, in appropriate Further Education and Training
sites of learning. Learners proceeding to institutions of Higher Education should be mathematically literate, so
that they are able to progress effectively in whatever discipline they decide to follow.
Mathematics is being used increasingly as a tool for solving problems related to modern society. The financial
aspects of dealing with daily life are informed by mathematical considerations. Mathematical ways of thinking
are often evident in the workplace. The Learning Outcomes and Assessment Standards in Mathematics are
designed to allow all learners passing through this band to develop into citizens who are able to deal with the
Mathematics that impinges on the society they live in and on their daily lives.
If a learner does not perceive Mathematics to be necessary for the career path or study direction chosen, the
learner will be required to take Mathematical Literacy.
LEARNING OUTCOMES
Learning Outcome 1: Number and Number Relationships
When solving problems, the learner is able to recognise, describe, represent and work confidently with numbers and their relationships to estimate, calculate and check solutions.
The range of numbers encountered in the Further Education and Training band will include irrational numbers
as they occur in contextual problems. Learners will develop an understanding that not all numbers are real.
In this band learners will:
- expand the capacity to represent numbers in a variety of ways and move flexibly between representations;
- develop further the ability to estimate and judge the reasonableness of solutions and the ability to give solutions to an appropriate degree of accuracy, depending on the accuracy of measuring instruments and on the context;
- calculate confidently and competently, with and without a calculator, guarding against becoming over-dependent on the calculator;
- develop the concepts of simple and compound growth and decay;
- solve problems related to arithmetic, geometric and other sequences and series, including contextual problems related to hire-purchase, bond repayments and annuities; and
- explore real-life and purely mathematical number patterns and problems which develop the ability to generalise, justify and prove.
Learning Outcome 2: Functions and Algebra
The learner is able to investigate, analyse, describe and represent a wide range of functions and solve related problems.
A fundamental aspect of this outcome is that it provides learners with versatile and powerful tools for
understanding their world while giving them access to the strength and beauty of mathematical structure. The
language of algebra will be used as a tool to study the nature of the relationship between specific variables in a
situation. The power of algebra is that it provides learners with models to describe and analyse such situations.
It also provides them with the analytical tools to obtain additional, unknown information about the situation.
Such information is often needed as a basis for reasoning about problem situations and as a basis for decision
making.
Learners should:
- understand various types of patterns and functions;
- investigate the effect of changing parameters on the graphs of functions;
- use symbolic forms to represent and analyse mathematical situations and structures; and
- use mathematical models and analyse change in both real and abstract contexts.
The mathematical models of situations may be represented in different ways  in words, as a table of values, as
a graph, or as a computational procedure (formula or expression). The information needed is mostly acquired in
the following ways:
- finding values of the dependent variable (finding function values);
- finding values of the independent variable (solving equations);
- describing and using the behaviour of function values, periodicity;
- considering the increasing and decreasing nature of functions, rates of change, gradient, derivative, maxima and minima;
- finding a function rule (formula); and
- transforming to an equivalent expression (`manipulation' of algebraic expressions).
It is important that the Learning Programme provides for appropriate experiences of these problem types, and
that it develops the underlying concepts and techniques to enable learners to experience the power of algebra as
a tool to solve problems. The emphasis is on the objective of solving problems and not on the mastery of
isolated skills (such as factorisation) for their own sake.
Learning Outcome 3: Space, Shape and Measurement
The learner is able to describe, represent, analyse and explain properties of shapes in 2-dimensional and 3-dimensional space with justification.
The teaching and learning of space, shape and measurement in the Further Education and Training band must
build on experiences from the General Education and Training band to make more formal and extended levels
of knowledge accessible. Aspects that are important for the attainment of this Learning Outcome include
location, visualisation and transformation. Learners' previous knowledge becomes deeper, they engage with
new tools that can be used in a range of applications, and they become more proficient in processes leading to
proof.
The study of space, shape and measurement enables learners to:
- explore relationships, make and test conjectures, and solve problems involving geometric figures and geometric solids;
- investigate geometric properties of 2-dimensional and 3-dimensional figures in order to establish, justify and prove conjectures;
- link algebraic and geometric concepts through analytic geometry;
- link the use of trigonometric relationships and geometric properties to solve problems;
- use construction and measurement or dynamic geometry software, for exploration and conjecture;
- analyse natural forms, cultural products and processes as representations of shape and space;
- investigate the contested nature of geometry throughout history and develop an awareness of other geometries;
- use synthetic, transformation or other geometric methods to establish geometric properties; and
- connect space, shape and measurement to other Learning Outcomes within Mathematics and where possible to other subjects.
Learning Outcome 4: Data Handling and Probability
The learner is able to collect, organise, analyse and interpret data to establish statistical and probability models to solve related problems.
The focus of teaching and learning data handling in this band builds on what has been learned in the General
Education and Training band. Learners will master further methods of organising, displaying and analysing
data. Measures of central tendency and spread will be explored. A basic appreciation of the difference between
data that is normally distributed about a mean and data that is skewed will be developed. Learners will become
critically aware of the deliberate abuse in the way data can be represented to support a particular viewpoint.
Learners will carry out practical research projects and statistical experiments. At least one project each year
will involve the selection of a random sample of a specific population with a view to determining statistics that
predict the corresponding parameters of the population.
The basic understanding of probability and chance gained at General Education and Training level will be
deepened so that, for example, learners can compare the actual odds in winning popular games of chance with
the odds offered by gaming houses. A basic understanding of the way the probability of everyday events can be
calculated and used in prediction will be developed.
Wherever possible, contexts that are investigated will focus on human rights issues, inclusivity, current matters
involving conflicting views, and environmental and health issues.
Acronyms
2D/2-D - Two Dimensions (or two-dimensional)
3D/3-D - Three Dimensions (or three-dimensional)
AIDS - Acquired Immune Deficiency Syndrome
CASS - Continuous Assessment
DO - Developmental Outcome
FET - Further Education and Training
GET - General Education and Training
HIV - Human Immunodeficiency Virus
IKS - Indigenous Knowledge Systems
NCS - National Curriculum Statement
NQF - National Qualifications Framework
OBE - Outcomes-Based Education
SAQA - South African Qualifications Authority
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