Non exemplar graph 1

In the interview, all students were presented with a graph as shown in Figure7–43 below:

Figure 7-43. Non-exemplar graph 1.

Students were asked whether it was a function or not. To realize that it is not a function,

Chapter 7 –Results from the interviews

112

asked to explain the reasons for their answers. The table 7–8 below summarizes all

students’ responses and explanations:

Function or not Explanation

Ali Not a function Vertical line test/colloquial definition Aysel Not a function Vertical line test/colloquial definition

Ahmet Not a function Vertical line test/use of set-correspondence diagram/colloquial definition

Belma Not a function Numbers on axes are irrational.

Belgin Function Finding corresponding values of numbers on x axis Arif Not a function/

change to function

Finding corresponding values of numbers on x axis

Cem Function Visual hints. Numbers on x axis (-3, -2, -1, 1, 2, 3) are inside the graph

Deniz Not a function General appearance of the graph Demet Not a function General appearance of the graph

Table 7-8. A summary of students’ responses to the non-exemplar graph 1.

Ali did not consider the graph as a function using the vertical line test as shown in Figure

7–44 below:

Figure 7-44. Ali’s written explanations for the non-exemplar graph 1.

He first wanted to know whether or not the part of the graph between x values of 3.5 and 4

Chapter 7 –Results from the interviews

113

‘Isn’t it passing through the same point, is it? In other words, it’s not vertical, is it? Does it have a slope?’ (Ali).

When he was told that it has a slope, he did not consider the graph as a function:

‘For instance, here, for two x values, there are different values of y. For instance,

for 3.5…it is 1, ½, –½…it’s not a function’ (Ali).

Aysel did not consider the graph as a function using the vertical line test and the colloquial

definition as shown in Figure 7–45 below:

Figure 7-45. Aysel’s written explanations for the non-exemplar graph 1.

‘This isn’t a function, because …it’s the rule of a function. In the domain, it can’t go to more than one in the range…’ (Aysel).

Ahmet did not consider the graph as a function using the vertical line test. He first asked

whether or not the graph bends onto itself. When he was told that it did bend onto itself, he

did not consider it as a function by using the vertical line test as shown in Figure 7–46

Chapter 7 –Results from the interviews

114

Figure 7-46. Ahmet’s written explanations for the non-exemplar graph 1.

When he was asked the reason why, he drew a set diagram picture as shown in Figure 7–

46 above and used the colloquial definition. He first said that two elements cannot be

assigned to one element in the range. He then changed his mind and said:

‘one element in the domain is not given to two different values in the range. When I draw verticals here, x gives two different y values, like a parabola, pardon opposite parabola. It’s not a function’ (Ahmet).

Belma did not consider the graph as a function since she considered the numbers on the

axes are irrational:

‘From R to R…the values between this and this are rational values. I mean between

2

− and −3 (referring to y-axis). From rational numbers to rational numbers. This

isn’t a function…because there isn’t an integer between −2 and −3, not 2 or 1 for

instance. There are normally irrational numbers between these numbers. That’s why (it’s not a function)’ (Belma).

Belgin considered the graph as a function since she could find corresponding values of

some values of x as shown in Figure 7– 47 below:

‘I have looked at the numbers. They have certain values, therefore it’s a function’ (Belgin).

Chapter 7 –Results from the interviews

115

Figure 7-47. Belgin’s written explanations for the non-exemplar graph 1.

Demet did not consider the graph as a function due to the general appearance of the graph:

‘It’s impossible, this can’t be a function…I can’t think of a function like this…function can be on the same plane, and can be proportional, but it starts here then goes wavy’ (Demet).

Deniz did not consider the graph as a function due to the general appearance of the graph:

‘This is not a function. First of all, the lines didn’t go straight. It goes shape by shape. To be able to intersect exactly, it shouldn’t be like this shape’ (Deniz).

He was then asked to draw how the graph should have been drawn. He then drew straight

lines close to the graph as shown in Figure 7–48 below:

Chapter 7 –Results from the interviews

116

Cem focused on some irrelevant visual hints. He considered the graph as a function since

1

− , −2, −3, 1, 2, 3, (on x axis) are inside the graph (between the x intercepts).

Arif did not at first consider the graph as a function since he could not find corresponding

values of x:

‘For x value, it’s passing through 3 and 4, nearly 3.5 (on the x axis). In y, it’s

passing through −2 and −3…I think this isn’t a function…I don’t think I can find

values by looking at this shape…when I look at the graph, I should be able to find the corresponding values for some x. For instance f(x)…I should be able to find the image of f(x), but here I can’t find’ (Arif).

He considered the graph as a function by finding the corresponding values for a few

numbers as shown in Figure 7–49 below:

Figure 7-49. Arif’s written explanations for the non-exemplar graph 1.

‘Before, I said that it’s not a function…because I couldn’t find integer values. But then I realized that it shouldn’t be integer values. Because it says that it’s from reals to reals…I didn’t take this into account. Since it’s from reals to reals, it’s a function’ (Arif).