Elementos para la evaluación en la nueva práctica

The literature on the constant function, functional notation, and multiple representations of functions, interpretations of functions and the dual nature of functions is used in the analysis of the diagnostic/pre-test. The errors and misconceptions that the learners have regarding these aspects of functions are categorised according to the content in the literature.

The literature on multiple representations of functions, interpretations of functions and the dual nature of functions is used in this study to enhance learner understanding of how to interpret the parabolic graphs in the different representations. The appropriate forms of the parabolic equations in order to find the parabolic equations of the given parabola are used to reinforce learner interpretation of the parabolic graphs. The intervention lessons are used in the application of the suggestions from the researchers on the multiple representations of functions, interpretations of functions and the dual nature of functions.

The learners are helped in acquiring the knowledge of when and how each of the equations of the parabolic function should be used to derive the equation when the graphical form is given. They should be able to use the appropriate forms of the quadratic function namely:

( ) ( ) when a graph of the parabola with the turning point and another point on the graph of the parabola is given; ( ) ( )( ) when a graph of the parabola with the -intercepts and another point on the graph of the parabola is given; when a graph of the parabola with any three points on the graph of the parabola is given. The learners should also be able to draw the graph of a parabolic function when they are given the parabolic equations in the following forms:

( ) ( ) , which puts emphasis on the functional character of the parabola and informs the learners of the turning point and the axis of symmetry and the shape due to the value of the parameter ; ( ) ( )( ) which informs the learners of the - intercepts and the shape due to the value of the parameter ; ( ) which informs the learners that they have to sketch the graph of a parabolic function and should apply the operational processes on the equation to calculate the turning point, the axis of symmetry and the - intercepts.

The key focus in this study is the interpretation of functions by the Grade 11 learners. In order to explore this focus, it is necessary to ascertain whether the learners have the structural view or not. It is interesting to note whether the problem the learners have relate to this or not. Another difficulty in the teaching and learning of the concept of a function is that it is an abstract idea which can only be given in different representations. Learners have difficulty relating to the idea and rather treat each representation as a thing in itself and are seldom able to flexibly move from the graphical representation to the equation form and from the equation form to the graphical representation of algebraic functions in particular the parabolic

function.

Issues pertinent to this study are around the different operations that can be performed to the different quadratic forms of the quadratic equations of a function and how these different forms of the quadratic functions can be linked to the graphical representation in order to generate the equations of the functions represented graphically. This study also looks at how learners interpret these different forms of the quadratic functions in order for them to generate the graphical representations of the quadratic function from these different forms.

This study does not focus so much on domain and range, on the definition of the function or the functional notation. The knowledge that the learners’ problems concerning functions could be due to their misconceptions emanating from these aspects of functions is worth

contemplating in order to answer the critical question ‘What aspects of functions do learners find problematic?’

It is evident for this study that if learners do not fully understand the difference between a relation and a function in the early years of dealing with functions, they will not be in a position to interpret the different representations and notations of a function. It is therefore important to make learners aware of the link between the definition of a function and the functions they deal with in class.

The use of the definition and the teaching of learners in the different approaches are useful in this study. This study focusses on the three algebraic forms of the quadratic function: ( ) ( ) ( ) ( ) ( )( ) . These different forms of the quadratic functions are used in the different representations of the parabolic function. These are the graphical form and the equation form. The graphical form is used where learners are expected to interpret the graphical form given particular features of the parabolic graph, including the -intercepts and another point on the graph, the -intercept and another point on the graph and the turning point and another point on the graph. Learners are also expected to interpret the graph given three points on the parabola and the choice of the appropriate form of the equation of the parabola to be used in finding the equation. In addition to these approaches, the study investigates how the structural approach can be used using the theory of variation. The theory of variation is discussed in the following chapter.