RENTABILIDAD DE EJECUCION DEL PROYECTO

Before articulating what the South African curriculum suggests in terms of the teaching and learning of functions, it is important to understand what is meant by the word “curriculum”. The curriculum or (curricula) is what the state deems to be important for its citizens to learn and consists of four components. The first component is “Goals”, which include the

“benchmarks or expectation for teaching and learning and are designed in a form of a scope and sequence of skills to be addressed”. The second component is “Methods” which includes specific instructional methods for the teacher that appears in Teacher’s Editions of the

curriculum. Thirdly, the “Materials” consist of media and tools to be used for teaching and learning. And the fourth component is “Assessment”, which aims to measure the student’s progress (Eisner, 1994, p. 96).

However, “the South African curriculum takes as its starting point a clear political agenda and the need to transcend the curriculum of the past, which perpetuated race, class, gender and ethnic divisions. It emphasised separateness, rather than common citizenship and nationhood”. (Department of Education, 1997, p. 1). The old curriculum included the four components: goals; methods; material and assessment as described above. The South African

curriculum is based on the seven critical outcomes as described by the South Africans

Qualifications Authority (SAQA) of 1995 whose primary objective is the promotion of a high quality education and training system in South Africa that embraces the concept of lifelong learning for all.

According to these critical outcomes, learners are required to be able to:

 Identify and solve problems and make decisions using critical and creative thinking

 Work effectively with others as members of a team, group, organisation and community

 Organise and manage themselves and their activities responsibly and effectively

 Collect, analyse, organise and critically evaluate information

 Communicate effectively using visual, symbolic, and/or language skills in various modes

 Use science and technology effectively and critically, showing responsibility towards the environment and the health of others

 Demonstrate an understanding of the world as a set of related systems by recognising that the problem solving contexts do not exist in isolation.

The teaching of the concept of a function satisfies all of these critical outcomes as outlined by the SAQA of 1995. The next section outlines what the South African curriculum expects teachers to concentrate on when teaching functions. It also defines what the learners should know in this regard in the Further Education and Training (FET) phase (Grades 10 to 12).

2.3.1 FUNCTIONS FOR GRADE 11’S IN THE SOUTH AFRICAN

CURRICULUM.

The South African curriculum is divided into Learning Outcomes and related Assessment Standards. Functions are part of learning outcome 2 (LO2). Learning outcome 2 and the assessment standards pertaining to it (as prescribed by the National Curriculum Statement (NCS) of South Africa for the learning area of Mathematics in the FET phase for the topic on functions) are detailed and represented in Appendix H.

The aspects of functions that are included in the South African curriculum require learners to understand various types of functions, to find values of the dependent variable by finding

function values and finding the independent variable by solving equations. Learners are also required to be able to describe and use function values, finding a function rule/ formula and be able to transform equations of a function to equivalent expressions through the

manipulation of functional equations. In the case of quadratic functions, learners are expected to use the manipulations of factorisation and completing a square to transform a quadratic function into its different forms.

Learners are expected to be able to understand the relationships between variables in terms of numerical, graphical, verbal and symbolic representations of functions and be able to flexibly convert between these representations. They should be able to generate graphs using the point-wise and the global approaches, identify the properties of functions including the domain and range, turning points, intercepts with the axis, maxima and minima, asymptotes, average gradient, intervals on which the functions decrease/ increase and discrete or

continuous nature of functions. They should also be able to provide descriptions of situations focussing on trends and features of algebraic functions.

In summary, in Grade 10, the learners should have been introduced to the differences between a relation and a function and be in a position to define these different aspects. The learners also should have been introduced to the sketching and interpretation of the straight line function, the parabolic function, the hyperbolic function and the exponential function. They should also have looked at the vertical movement, the stretch and the intercepts of these functions, but they would not have looked at the horizontal movement of these functions. They would have concentrated on the following parameters: ; ; , where the represents the shape of the quadratic/parabolic function. For example, when the parabola is concave upwards and when the parabola is concave downwards, the movement of the straight line and the hyperbolic functions from the first quadrant to the third quadrant when and from the second quadrant to the fourth quadrant when . Their lessons should also have included the exponential function, where the movement from the first quadrant to the second quadrant for and the movement from the second quadrant to the first quadrant when as well as the vertical movement of the straight line, the quadratic, the exponential and the hyperbolic functions upwards when and downwards when .

In Grade 11, the curriculum requires that the teachers and learners look at more of the

parameters when dealing with functions and also that they look at the exponential function in more detail. The curriculum for Grade 11 emphasises the multiple representations of

algebraic functions and the shifting in both the vertical and the horizontal axes. For instance, the teaching of functions should put more emphasis on algebraic functions that include all parameters represented by the following algebraic functions: for linear

functions; ( ) and ( )( ) for quadratic functions; for hyperbolic functions and _{for exponential} functions. These parameters should be dealt with in more detail as they involve all

transformations as required by the Grade 11 curriculum.

The Grade 11 curriculum also emphasises representations of functions, for example assessment standard 11.2.1(b) which states that learners should be able to recognise relationships between variables in terms of numerical, graphical, verbal and symbolic

representations and be able to flexibly convert between these representations (tables, graphs, words and formulae), intervals on which the function increases/decreases and the discrete and continuous nature of graphs.

The curriculum also focusses on the characteristics and features of the functions, describing a situation by interpreting graphs, drawing graphs from a description of a situation with special focus on trends and pertinent features of the graphs including the application of the function concept to contextual situations. The formal definition of a function is only dealt with at length in Grade 12. The detail is described in Appendix H which is based on the curriculum across the FET phase.

Based on the Umalusi (2012) report and the results of the diagnostic test given to my learners at the beginning of the study, it is clear that these sections are the sections that the learners have problems with. In particular, learners struggle with the aspects of functions that deal with multiple representations and interpretations of algebraic functions.

In the South African curriculum, as detailed in Appendix H, functions are grouped together with algebra and equations as these are interrelated. The main focus of this study is the intertwining of the quadratic equation with the quadratic function. The correspondence between these concepts is such that in order for one to simplify an equation representing a function, one needs to be conversant with the manipulations involved in algebraic equations

in the form of factorisation of these equations. The algebraic manipulations include, more specifically, factorisation of the quadratic equations and completing the square of the quadratic equation.

In order for learners to find the specific features of a quadratic/parabolic function, the learners have to know that by factorising the quadratic function in the general form: , they will be in a position to see the features of a parabolic function represented by the factorised form of the quadratic/parabolic equation:

( )( ). The features presented by this general form of the parabolic function are the - intercepts of the parabola namely: and . By completing the square on the general quadratic/parabolic function: , the quadratic/parabolic

function becomes: ( ) , with the features of the parabolic function represented being the turning point at ( ) and the axis of symmetry represented by the equation: . The parameters and give the dimensions of the quadratic/parabolic function, namely the shape (concave upwards for and concave downwards for ) and the -intercept of the parabolic function respectively.

Similarly, the Grade 11 curriculum expects the learners to be able to determine the equation of the parabolic function when they are given a parabolic graph. The learners should be able to substitute the given values of the given variables in the appropriate form of the

quadratic/parabolic equation in order to find the equation of the parabola.

This forms the focus of this study and answers the critical question ‘which aspect of the function appears to be the most problematic?’ for learners.

It’s interesting to see what researchers have found about learners’ difficulties in working with functions in general. Literature on the aspects of functions that learners find problematic and the misconceptions that learners have on these aspects of functions and what contributes to learners developing these misconceptions are discussed in the next section.