2.4 LEARNER MISCONCEPTIONS
2.4.2 MISCONCEPTIONS RELATED TO THE CONSTANT FUNCTION
the function the author means that functions do not have to be described by any specific expression, follow some regularity or be described by a graph with any particular shape (Even, 1990, p. 528). The author further elaborates on the definition of a function as to include that: “there is an assignment of a single value to each number”, (p.529).
In other words, the misconceptions that learners had as far as the definition of a function is concerned is that a function that is not defined by an equation, which has both the
independent and the dependent variable, was not a function. This misconception is elaborated on in the section relating to misconceptions of learners with regards to the constant function in the following section.
The following section deals with the misconceptions relating to the interpretation of the constant function with regards to a function’s definition.
2.4.2 MISCONCEPTIONS RELATED TO THE CONSTANT
FUNCTION.
In this study, the misconception with regards to the constant function is evidenced in the results of the diagnostic/pre-test when the learners were asked to give the equations of the asymptotes of the parent graph of an exponential function and the vertically transformed version of the same exponential function. The learners gave the equations of the exponential functions as their answers to these questions requiring them to give the equations of the asymptotes. As the equations of the asymptotes are special cases of the equations of the straight line, this sheds light on the fact that the learners do not understand what an asymptote is.
In other words, the equation of the asymptote is problematic to the learners as it is a straight line which does not conform to the general form of a straight line . The important issue is that the learners could possibly point to the asymptote, draw the asymptote, but
actually saying what the equation of the asymptote is, interpreting that and writing it in equation form was something they were unable to do. In order to do this they need to know the global view of the function and also need to know the special case of the linear function or equation. From the definition of a function, it appears as if a function should be
represented by two variables and since the equations of the asymptotes consist of only one variable, the learners could not give the correct answers.
As already noted, the definition of a function has been reported by researchers to be difficult for learners to comprehend and interpret. The constant function, as it is represented, appears to differ from the general form of this algebraic function representing a linear function that the learners are familiar with in that some of the parameters are missing from the general form of a linear function.
Research by Sfard (Sfard, 1992; Markovitz et al., 1986) revealed that students generally had difficulties with the constant function as they believed that a function involved a change in the independent variable which then influences a change in the dependent variable according to the definition of a function. This was confirmed in a study by Carlson et al. (1998), where about 7% of A- students in college algebra could produce a correct answer to a question requiring them to give examples of functions. In this case, all of its output values were equal to each other as is the case with a constant function like while the remaining 93% of A- students could only give an example of as an example of such a function as they justified their answers with the fact that did not represent a function because it does not vary.
This misconception on a constant function prevailed and is reiterated by a majority of students in the research undertaken by Monk (1994). These students viewed a constant function as a non-function as it did not vary. As the definition of a function involves a dependent and independent variable, the constant function was viewed as having only one variable and therefore could not be a function. Similarly, in the study conducted by Tall & Bakar (1991), a constant function like: was generally considered to be a non-function as the notion of a function was that it should be represented by a formula with two variables. This formula should be compiled in such a way that one variable, usually the -variable, is the independent variable and the other variable, usually the - variable, is dependent variable.
These findings lead to the fact that when teaching the definition of a function, one has to be aware of the pitfalls that are associated with the definition. The way in which functions are taught, beginning with the prototypes that are used in order to make the learning of the notion of a function possible, to how the definition of a function should be introduced to the learners are of incredible importance
In the following section, the misconceptions of learners regarding the functional notation will be stated and discussed.
2.4.3 MISCONCEPTIONS WITH FUNCTIONAL NOTATION.
In this study, the misconceptions of functional notation are proven by the learners’ inability to answer a related question in the diagnostic/pre-test. This question required the learners to calculate the distance of a boy on skates after three seconds where the distance of the boy at the beginning of the event is defined by the function: ( ) . The learners who attempted this question, changed the equation and substituted for ( ) and found the value of : . This misconception of learners is in line with the findings by Carlson et al. (1998) and shows that the learners could not express distance ( ) as a function of time ( ).
This misconception is shown by Carlson et al. (1998) in a study that shows that one of the recurring misconceptions and errors among students is that when they are asked to express speed ( ) as a function of time ( ), many high-performing pre-calculus students could not represent this as ( ) This is not unexpected as this is the state of affairs in South Africa as the Umalusi report has proved functional notation is a problem for learners in general. Additionally, the results of the pre-test (which are discussed in Chapter 6) also confirm that functional notation presents a problem for learner. The Umalusi report of 2011 also refers to functional notation as one of the areas that learners found problematic and is therefore included as part of the FET curriculum in South Africa. These misconceptions are also apparent in this study as the learners could not relate the functional notation to the contextual problem presented to them. This is discussed in detail in Chapter 6.
One of the reasons given by researchers for the misconception of functional notation is that the different notations of functions make the function concept look like different concepts instead of one ‘unifying’ concept (Maclane, 1986). The author further gives many examples of functions like algebraic operations that give rise to examples of functions of numbers, geometric definitions that produce trigonometric functions, exponential functions and their inverses that result in logarithmic functions that are numeric functions. The author further concedes that in space, distance is a function with real values that function in pairs of points in Boolean algebra, intersections and unions are functions of pairs of sets and in geometry, length is a real valued function of curves.
Coady & Pegg (1993) in their research on tertiary students’ difficulties with the function notation, discovered that students responded to the question “If is increased by , find an expression for ” as if they were required to find the expressions for the following:
1. ( ) 3. ( ) ( ) 4. ( ) 5. ( ) ( ) 6.
7. .
Some of these students ignored the phrase ‘ is increased by ’, some treated the expression as either a single term or two independent terms. Others simply added or
multiplied a as they saw fit. These responses indicate that the students had misconceptions with the function notation.
The companion brief of the pitfalls identified by the Commission of Mathematical Instruction in the international seminar known as the PCMI makes comments on other common notations of functions and elaborates on what these representations draw the attention of the students to (Hazzan & Goldenberg , 1997, p. 263). These are the representations that were in the brief. Firstly, ( ) which is a notation that draws attention to the functional nature of the relationship under consideration? Secondly, which is a functional notation that supportsthe graphical representation in the Cartesian plane hirdly, ( ) which puts emphasis on the functional character of the graphical form as a notation that is meant to represent the idea of a mapping from one set to another.
Finally, ( ) { } as the notation that emphasises the idea of a set of points (Vinner, 1983, p. 301).
These different functional notations are included in the South African curriculum for the FET phase and the students are expected to be familiar with all of them. These different functional notations is one of the problem areas that the Umalusi report covers and is one facet of the notion of a function that the learners generally have difficulties with. This is further proven by the following studies and the results of the pre-test that informs this study.
From engaging with literature that defines mathematical functions, it emerged that functions have different functional notations which include, among others, the function represented by
( ) ∑ ( ) for is viewed by learners as two separate functions: ( ) and ∑ ( ), although they produce the same results on the set of natural numbers (Carlson et al., 1998; Sfard, 1992).
To put it more simply, the different functional notations have different and important uses in different situations. It is important for learners to understand how they should be applied in the different situations. As noted above, these notations place different emphasis on the various ideas that are placed on the functions. These ideas can range from those representing the functional nature of the relationship under consideration, the graphical representation in the Cartesian plane, the mapping from one set to the next, the set of points on the function, computing the value of the function at a particular point and the value of the point – coordinate(s) corresponding to a particular value of the function.
The following section deals with the misconceptions of learners with regards to the multiple representations of functions, which forms the focus of this study.