Table of Values

A table of values presents the x and y values that satisfy a given equation. The x-values represent the input and the y-values the output. All these values need to satisfy an equation that represents a particular function. All 24 participants identified the graph as an exponential function, although their justifications were different. The responses of 8 (eight, 33 %)

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participants focused on the asymptote; 10 (42%) sketched the graph to prove that the table of values represented the exponential function; and the last 6 (six, 25%) spoke of the proportionality of the numbers and the common ratio. These participants were responding to the question below:

1.1 Name the function represented by the above table of values, state reasons for your answer 1.2 How would you identify the asymptote from the table of values?

The table below shows the ordered pairs for a certain function: 𝑥 ∈ ℝ Table 7.1: Ordered pairs of the function 𝒙 ∈ ℝ

x -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5

y -0,998 -0,996 -0,99 -0,98 -0,97 -0,94 -0,88 -0,75 -0,5 0 1 3 7 15 31 63

The participants expected the asymptotes on the table of values to denote the asymptotes by an error or undefined on the output section of the table of values. In the study, the participants were presented with an exponential function whose asymptote was horizontal and required an undefined on the input. For example when Participant LL was asked how she would identify an asymptote from the table of values, she responded as follows:

Participant LL: It maybe an error

Interviewer: Only that? If there is no error or undefined, then there is no asymptote?

Participant LL: No sir, that what’s the graphs, if there is an error in y, then the y values will be the asymptote.

The intention of the question was for participants to state how they would identify the asymptotes from a table in general. The error which Participant LL refers to, appears in a table of values only for the vertical asymptote when the undefined part of the denominator is not a zero of the numerator. Otherwise, it will be just a removable discontinuity. Participant LL’s assertion that an error always denotes an asymptote is partly correct. It is not always a certainty that in instances of discontinuity, there would be an asymptote and exhibits ritualised routines. Accordingly, the researcher then classified Participant LL’s narratives as “memorisation based on visuals”, because her assertions were based on what she had seen at face value; rather than from her investigations and explorations on the mathematical object. Participant Y, who is Participant LL’s interview partner identified the table of values above as an exponential function. When the researcher asked for her reasons, she responded that the function had only one asymptote. Further probing by the researcher resulted in the following response by Participant Y:

Interviewer: Why do you say exponential?

161 Interviewer: What is the asymptote?

Participant Y: It is zero. Interviewer: What is it?

Interviewer: And where did you get your zero?

Participant Y: From the .... From negative three (-3), when x is negative three (-3) or negative one (-1), it is negative one (-1). Because the y values, when you sketch it, it decreases, it will never touch negative one (-1). And by looking, when you look at the positive side, the graph is increasing...it not the same as this one and when it gets to negative one (-1), when it gets close to negative one (-1) it becomes -0.9998, as it decreases, it does not get to the actual number negative one (-1).

Emanating from the statements above, it is axiomatic that Participant Y was able to identify the function as representing the table of values, but could not justify herself for stating so. At first, Participant Y justified her statement on the basis that the function had only one asymptote. When she was probed further to explain how she identified that asymptote from the table of values, she mentioned another representation of the function. She then explained that the graph she sketched had an asymptote of -1 (negative one) and not the 0 (zero) she talked about. Participant Y’s use of words was nearly mathematical because she did not talk of the relationship between the x and y coordinates, but referred to a decrease on the negative side and an increase on the positive side. An exponential function either increases or decreases, but couldn’t be both increasing and decreasing on the same function. Her interpretation of the exponential function was not construed because it did not fit the description of an exponential function. The researcher then classified her routines as ritualised mathematical, because she knew that an exponential function has one asymptote despite her failure to explain how she could identify it from the table of values. She could not give a mathematically acceptable explanation of the function’s movement. Her narratives were based on what she saw. There was a side of the graph increasing, but the coordinates were not both negative. For Participant Y, it meant the graph was decreasing, and for the part of the positive coordinates, she mistook it to be an increase. She used a table of values as a means of arriving at a graph, rather than seeing it as a representation of a function in its own right.

Participant Y was not the only participant who identified the function as an exponential but could not provide mathematical reasons for stating so. For instance, Participant OO also stated that the table of values represented an exponential function. When asked to explain how he knew that the function was exponential, Participant OO spoke of the graph not touching 1 (one). While an asymptote is related to an exponential function, there was no need for Participant OO to use it in that instance. Below is the researcher’s depiction of Participant OO’s interpretation of the table of values.

162 Participant OO: It is an exponential

Interviewer: Yes exponential. However, why do you say so?

Participant OO: If we represent this data on a graph. The graph will not touch one. It will come close to it. You can see that it means the asymptote is one.

Participant OO also used the table of values as a means of drawing a graph rather than a representation of a function. His explanation shows that he did not regard a table of values as one of the representations of a function which could be interpreted such that the features of a function could be brought to the fore. From the table of values, one could derive such features as the intercepts, asymptotes, the point at which the graph was increasing or decreasing, points of discontinuity, and the name of the graph. This is the curriculum’s underlying intentions for learners to be able to convert the four representations of a function flexibly (DBE, 2011). Participant OO used the presence of an asymptote as the means by which the presence of the asymptote is shown. The presence of the asymptote does not mean that the function is exponential as there are other functions which have asymptotes as part of their features. Mentioning the presence of an asymptote is not the most convincing reason. Furthermore, Participant OO stated that the asymptote was 1 (one), yet an exponential function y is equal to one (y= 1) features in the middle of the function and cannot be the asymptote.

Participant OO’s use of words was non-mathematical. He mentioned that the graph would not touch 1 (one) instead of stating that the graph would not intersect with the asymptote or line y was equal to 1 (one) (y=1). Participant OO mentioned an asymptote as though it were a number by stating that it is 1 (one). The researcher, therefore classified his use of words as “non-mathematical”. Participant OO had challenges in interpreting the table of values, to such an extent that she had to sketch a graph in order to obtain a picture of what function it was. The researcher then classified her interpretation of the table of values as “not construed”, because he interpreted the function using another representation. Participant OO’s narratives were classified as “memorisation based on visuals” because his talk was based on what he saw rather than on generalisations he found through the exploration of the table of values. Therefore, the researcher classified his routines as ritualised.

As one of the four representations of a function, the table of values needed to be interpreted with some key features of that function being recognised. Participant AA tried to explain the function from the table of values themselves, rather than using other representations such as the graph. This is demonstrated in the following explanation by Participant AA:

Participant AA: It is exponential Interviewer: Why do you say so?

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Participant AA: You can tell that the graph is approaching the asymptote from the table by because we can see, you can see the difference between these terms it's not the way that it is eemm between the y- axis, between the y terms it is decreasing. The difference between the terms it is decreasing. At first, Ok the first term let us say the first position x value it is 5, then you can have the y value is 63, then the second term it is 4, second term of x is 4, then the y term or the y value it is 31 then the difference between them is 32. Then moving between ok the difference. Ok the fourth. OK the third term which means x on x=3, y=15, then the difference between the y values is 16, therefore the decrease shows that it is an exponential function.

While Participant AA could not clearly articulate himself, it was not difficult to understand his ideas. At first, he spoke of the function approaching the asymptote. He went on to show that the differences between the successive output values had a pattern, which was provided with the number that had exponents of 2 (two). He mentioned 32, and then the next output value was 16. He then concluded that the difference showed that the graph was exponential. When Participant AA mentioned that the “difference between the terms was decreasing”, it was due to his interpretation of the graph from the right to the left. He mentioned the differences from the largest numbers to the smallest. He also referred to the last term as the first, but he moves in the opposite direction. When reading the table of values backwards, he named some terms first when they were far from being the first. He also alluded to the y-values as decreasing, yet the graph was increasing. The researcher therefore classified his use of words as colloquial. Participant AA explained the table of values as if the function moved from right to left. He described the function as decreasing and because of that; the researcher classified her interpretation of the iconic visual mediator as being not construed. His narratives were memorisation based on what she could see. Therefore, his routines were ritualised non- mathematical.

In this section, the researcher discussed how participants interpreted the table of values that represented an exponential function. The participants attempted to explain the table of values using the asymptote. Their use of words was predominantly “colloquial”. Their interpretation of the table of values was largely “not construed”, and their routines were mostly “ritualised mathematical”. In the next few paragraphs, the researcher discusses the mathematical discourse of participants who sketched the graph as a way of identifying the function represented by the table of values.

While Participant U and Participant V claimed to have drawn the graph represented by the table of values, they did not have the evidence to show that they skethched the graph. The other reason was that Participant U and Participant V did not answer follow-up questions directly,

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and that made it difficult for their exposure to mathematical discourse. The following responses show how Participant U and Participant V responded to the question: What function was represented by the table of values given above.

Participant V: It is an exponential function