Elementos para la planeación de la nueva práctica

The misconceptions concerned with the interpretation of functions as required by the study are all related to the misconceptions described above. They are all a result of, and stem from, the misconceptions of the definition of a function, the misconceptions with the constant function and the misconceptions with the representations and notation of functions above.

The interpretations of functions are included in the expectations of the South African curriculum on functions and as the Umalusi report and the pre-test given to the learners discussed in chapter one, learners have problems with the interpretation of functions. Their problems include the translation of functions (the horizontal and vertical shifts), interpretation of local processes (regarding point-to-point attention), global interpretation processes

(detecting trends), global and general (what happens to as increases and visa-versa), continuation (interpolation/extrapolation), rate (how fast should a car travel to reach a certain distance in one hour), qualitative interpretation (looking at an entire graph to gain the general meaning of the situation) and quantitative (a collection of isolated points).

The literature of misconceptions of the interpretation of functions serves as a guide to understanding the causes of the learners’ misunderstanding of the concept of functions. In this study, it is evident in learners’ interpretation of the quadratic function when it is represented in different algebraic forms. These different algebraic forms of a quadratic function are represented in general form by using the parameters and variables, for example: ( ) ( ) and ( ) ( )( ) which highlight different features of a quadratic function. These different forms of the quadratic function form the basis of this study and are elaborated on in the methodology. The intervention lessons undergone in this study identify this as the most problematic section on functions that the learners in the sample seem to have.

This study focuses on the correct interpretation of these different forms of the quadratic functions as they are key to the learners’ ability to determine the equations of quadratic functions when they are given the quadratic functions in graphical form. It also helps the learners to interpret the quadratic functions and to represent the quadratic function in graphical form.

An example of the interpretation of an equation in the form, ( ) ( ) in numerical form, ( ) ( ) immediately informs the learner that the graph is one of a quadratic function that is concave downwards with the axis of symmetry at and a maximum value at .

As discussed in the Umalusi report and as it happened with the learners in the diagnostic/pre- test, learners cannot interpret quadratic functions given as parameters only. This is evident in a question where the learners, according to the report, were required to draw a sketch graph of the function where and . Some learners could not access the appropriate form of a prototype that could deal with a general case regarding the parameters , and of the required sketch as there were no numerals in the quadratic function.

Some misconceptions noted by Oehrtman, Carlson & Thompson (1998) in their research on A-students who have completed college algebra was that when these students attempted to find the value of ( ), about 43% of them added instead of substituting ( ) into the function. When probed for their reasoning behind this process, they provided some rule or procedure, showing that they were not thinking and interpreting “ ” as a value at which the function is being evaluated but as “ ” being added to the function ( ) This form of evaluating also appeared in the Umalusi report and could be due to the students’

misconceptions of the interpretation of the function notation.

In other studies by Thompson (1994) and Carlson (1998) it is clear discovered that students cannot distinguish between algebraically-defined functions and equations. Students interpret and view functions as two equations separated by an equal sign. In order to help students distinguish between the use of an equal sign as a means of defining a relationship between two varying quantities and a statement of equality of two expressions, the two authors developed instructional interventions that promoted students’ thinking about algebraically- defined functions. They explain that the use of an equal sign as a means of equating the output values of two functions and the act of solving an equation as a means of finding the input value(s) where the output values of these functions are equal makes it clearer

This misconception has been reiterated in the Umalusi report where the learners solved quadratic functions as if they were quadratic equations. When learners were given a function like they solved it as they would solve an equation where one side was equal to zero and divided each term by negative one without considering that this was not an equation equal to zero. When they multiplied the right hand side by a negative number they should get the answer of ( ) instead of the answer as it would be the case if this was an equation with .

Carlson, Oehrtman & Thompson (1998) assert that there are different types of interpretations that can be expected from the learners. Some of the interpretations referred to by the authors are translation (the horizontal and vertical shifts), interpretation of local processes (regarding point-to-point attention), global interpretation processes (detecting trends), global and general (what happens to as increases and visa-versa), continuation (interpolation/extrapolation), rate (how fast should a car travel to reach a certain distance in one hour), qualitative

interpretation (looking at an entire graph to gain the general meaning of the situation) and quantitative (a collection of isolated points).

In this study, the diagnostic/pre-test required learners to determine the values of for the given functions ( ) ( ), where . Chapter 6 discusses the fact that learners in this study could not interpret the global processes when they were expected to detect trends where one function was greater than the other in a specified interval.

The literature informs about the misconceptions that learners have with regards to the definition of the function and how the definition of a function should be introduced to the learners. It further explores the misconceptions of functional notation and how these

misconceptions are derived, how they could be alleviated and what misconceptions emanate from these different functional notations, the representation of functions and how these different representations should be linked together in order to enhance deeper learner

understanding of the function concept and its interpretations. Keeping the literature in mind, the following section discusses the suggested teacher’s thinking of function while teaching the concept. In doing so, the literature is elaborated on and an indication of what the teacher’s focus should be on to help learners reach a level where they can understand functions in all its representations.