Prototypes and exemplars of functions

In the Turkish context, the set-correspondence digram is used in a prototypical way to

explain the colloquial definition as presented in Figure 1.1 in section 1.2.2. On the other

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curriculum. For instance, students study linear functions then trigonometric functions and

then move onto logarithmic functions and so on (table 1.2). The data in this study indicated

that students’ cognitive development of learning the function concept followed a similar

structure. As seen in the grid in table 8.1, the colloquial definition is mostly used for the

set-correspondence diagram and the sets of ordered pairs. In other words, in these contexts,

more students focused on the definitional properties. The successful students are the ones

who could focus on the definitional properties even for the exemplars of functions, namely

the graphs and expressions. Less successful students such as students in the second

category, could use the colloquial definition for the set-correspondence diagrams and the

sets of ordered pairs but not for the graphs and expressions.

The least successful students, such as students in the third and fourth category, could not

focus on the definitional properties for any aspects of functions. In other words, the set-

correspondence diagram and set of ordered pairs were also exemplars for them as well as

the graphs and expressions.

So, as one of the research questions tried to find out, how do these students who could

coherently use the colloquial definition both for the prototypes and exemplars achieve this?

Responses from successful students (Ali, Ahmet, Aysel and Arif) in the first category

revealed that they used particular aspects of functions in a prototypical way to check the

definitional properties for the other aspects of functions. They directly used the colloquial

definition for the prototypes of functions, the set-correspondence diagram. For the

exemplars of functions, they did not use the colloquial definition directly as summarized in

the grid in table 8.1. For instance, they used the graphs in a prototypical way to focus on

the definitional properties when dealing with the expressions. While less successful

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overcame these complications by drawing the graphs of the given expressions and then

focusing on the definitional properties on those graphs. One student, Ahmet used the set-

correspondence diagram in a prototypical way. He drew the set diagram and put a few

elements in the sets and checked whether every element in the domain was assigned to a

unique element in the range.

Lakoff (1987a) states that according to the classical view of categorization, category

members only have definitional properties and all category members have those properties.

However, as discussed in the literature review, human beings do not categorize in such a

way that all category members share the same definitional properties. In other words,

people do not treat categories as clear-cut entities. Lakoff (1987a) distinguishes between

essential and incidental (accidental) properties. Essential properties are ‘those properties

that make the thing what it is, and without which it would not be that kind of thing’ (p.

161). Other properties are incidental. They are the properties that things happen to have but

not the ones that capture the essence of the thing. The function concept, being in a well-

defined category, has essential properties called definitional properties, that is given two

non-empty sets each element in the first set is assigned to a unique element in the second.

The analysis of the data indicated that, while successful students focused on these essential

properties, less successful students focused on the incidental properties such as the visual

hints from the graphs and diagrams etc. In other words, exemplar based responses are the

ones which focused on the incidental properties of different aspects of functions. It is

claimed that the graphs and expressions being introduced in clusters carry more incidental

properties. Therefore, they caused more complications especially for the students, as

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based ways for a few questions. However their overall responses indicated that they

focused on the essential properties of different aspects of functions.

The data indicated not only that the students developed exemplars of graphs and

expressions, but also that they developed exemplars in separate clusters. Since graphs and

expressions have different incidental properties, students developed exemplars in separate

symbolic and graphical clusters. In other words, students might reject a graph as a function

yet accept it as a function in symbolic form. The results reveal that graphically

x x

f( )=−sin acted like an exemplar while the graph of f(x)=sinx−2 acted as a non-

exemplar. There is a big difference between the number of students who accept the former

and latter graphs (table 6.3, section 7.10 and section 7.1.11). On the other hand,

symbolically this is not the case. More students tend to accept the expression

2 sin )

(x = x−

f as a function compared to its graph. This concludes that a function can be a

non-exemplar as one aspect and an exemplar as another aspect.