Second category

In the second category, there are two students (Belma and Belgin) who could focus on the

definitional properties only for the set-correspondence diagrams and the sets of ordered

pairs but not for the graphs and expressions. These two students gave complicated

responses for the graphs and expressions. They focused on the properties of the graphs and

expressions which are irrelevant to the core concept of function.

8.4.2.1 The case for Belma

Overall responses from Belma indicated that she could not focus on the core concept of

function. She could focus on the definitional properties for only two aspects of functions,

the set-correspondence diagram and the set of ordered pairs. In her responses, the

complexity of the function concept reveals itself as complications in the context of graphs

and expressions.

In the questionnaire Belma gave her personal concept definition as follows:

‘Let f :A→B. If every element in A is assigned to B then this is called a function’

(Belma).

Although her personal concept definition does not focus on all properties of the definition,

Belma used the colloquial definition correctly for the set-correspondence diagram and for

the set of ordered pairs. She could focus on the elements in the domain which are not

assigned to any elements in the range and the elements in the domain which are assigned to

two elements in the range. However, she could not focus on the definitional properties for

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irrelevant to the core concept of function. She relied on the appearances of the graphs or

the specific hints from the graphs. Therefore, her evoked concept images were not

connected to the core concept of function which, in the end, revealed itself through

complicated explanations. For instance, she rejected the graph of f(x)=sinx−2 since it has

only one intercept of the axes as discussed in section 7.1.11 or she simply considered the

smiley face graph (as discussed in 7.1.7) as a function since it looked like a graph of a

parabola.

Belma’s responses to the expressions were very complicated. Her focus of attention was

not the definitional properties. Mainly speaking, she approached the expressions in two

different ways; either substituting a few values in the expressions or trying to draw the

graphs of the expressions. However, she was not successful in doing these. For instance,

for the split-domain function she focused on the numbers -1, 0, 1 as the elements of the

domain, and then substituted them in 2₋2 ₊1 x

x , the expression of the condition on the domain (section 7.1.12). She was not successful to draw the graphs of the expressions

either. The subtle differences between the two notations for the constant functions, y=5

and f(x)=5 caused a lot of complications. She drew two different graphs for them.

Although she drew the graphs for the three cases of constant function along the line y=5,

when transforming “ f(x)=5” to its graph she drew the graph as the line x=5 as shown in

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Figure 8-1. Belma’s drawings for the two constant functions.

When the constant function is given in the form of f(x)=5, she could not focus on x as a variable although she tried to assign the numbers on the x axis to the x=5 line.

8.4.2.2 The case for Belgin

Similar to Belma, Belgin was more successful with the set-correspondence diagrams and

the set of ordered pairs and could not focus on the definitional properties when dealing

with the graphs and expressions.

In the questionnaire Belgin did not write anything for the definition of a function.

However, she used the colloquial definition for the set-correspondence diagram (both in

the interview and some items for the questionnaire) and set of ordered pairs (in the

questionnaire). However, she did not give any explanations for the set of ordered pairs in

the interview. For the graphs, she either used the colloquial definition wrongly or focused

on the graphs as exemplars (see the grid in table 8.1). For instance, she rejected the graph

of f(x)=sinx−2 since the general appearance of it was different and considered the graph

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focus on the definitional properties for the expressions either. As seen in the grid in table

8.1, she either focused on expressions as exemplars or confused the domain with the range.

For instance, she considered f(x)=sinx−2 as a trigonometric function without referring to

the definitional properties. For ‘y=5’ and ‘y=5 for x≤2’ she did not find the symbol

) (x

f in the expression therefore did not consider them as functions.

In summary, Belgin’s overall responses revealed that she gave more complicated responses

to the graphs and expressions.