Graphical Representation

When some learners talked of functions, they are referring to the graphical representation. This was brought about by the fact that when functions were learnt, these functions culminated with the drawing or interpretation of the graph. As expected, 21 (87%) of the 24 learners who participated in this study positively identified the two graphs, namely the hyperbola and the exponential function. The three (13%) participants who did not positively identify the graphs did so because they did not respond to the questions posed to them. In this part of the analysis, the researcher specifically discusses participants’ mathematical discourse on a statement which required learners to “Explain what happens to the function as x approaches infinite”.

Functions are distinguished and named according to their behaviour, which is conspicuous through the graphical representation of the function. For instance, the characteristics of hyperbolas are such that they had two parts which are a reflection of each other and asymptotes which are perpendicular to each other. As stated in the previous paragraph, participants could identify the two (hyperbola and asymptote). However, 13 participants (54%) had challenges describing the behaviour of the graph as the graph approached the infinite from either side. In the next paragraphs, the researcher discusses to the question: “What happens to the graph as x or y approaches both positive and negative infinite”?

Participants encountered problems when describing the behaviour of the graph as it became closer to infinity. Eleven participants stated that the graph would approach the asymptotes for the hyperbola. The researcher analysed Participant GG’s and Participant NN’s responses to the question on the behaviour of the graph because they came close to explaining the behaviour of the graph at the required instances. The following statements reflect Participant GG’s and Participant NN’s response to the question: Which is the asymptote, explain what happens as function x approaches infinity.

Participant GG: In the hyperbola as x is approaching the infinity, there are going to touch the asymptote sir

Participant GG: And Participant NN: Closer

Participant GG: Yes, it will be closer; I think it is supposed to be closer. Yes. Participant GG: Ok, and then the exponential, same applies.

Participant NN: It will get closer to the asymptote. Participant GG: Yeah closer.

Participant GG had initially stated that the hyperbola graph was going to touch the asymptote, but then Participant NN interjected to mention that it was closer rather than a touch. When

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Participant GG mentioned, “They are going to touch the asymptote”, she actually meant that the asymptote and the graph would intersect. As stated by these participants already, the graph became closer to the asymptote as the x values tend to infinite. The following diagrams (Figure 7.1) from geogebra indicates that the asymptote and the graph appear to intersect.

Figure 7.1: Representation of the hyperbola and the exponential function

In Figure 7.1 above, the exponential function and its asymptote appear to be coinciding. In actual fact, they do not. This diagram may have affected Participant GG’s utterances, but she was corrected quickly and accepted the correction. This situation shows that her narratives on the behaviour of the graph was affected by what she saw.

After Participant NN corrected Participant GG, she went on to state that the same applied to the exponential function. As x approaches infinity in a hyperbola, the graph approached the asymptote. However, the graph appears exponentially large in an exponential function. Accordingly, Participant GG’s routines were classified as ritualised. While her utterances were correct for a hyperbola, the same was not true for an exponential function. This meant that Participant GG’s routines on functions appeared to be ritualised since she did not distinguish the behaviour of the hyperbola graph from that of an exponential function. What Participant GG did was common in the learning of Mathematics. Learners exhibit objectification of the mathematical object only on one aspect. When the researcher asked her to examine her utterances, Participant GG described the behaviour of the graph as x approached negative infinity. This shows that Participant GG conducted some self-correction. Initially, she said both graphs would become closer to the asymptote as x approached infinity. The dialogue below illustrates that the two learners’ mathematical discourse was ritualised. Participant GG: So now am talking about exponential. As x approaches the negative infinity, the graph will get closer to the asymptote sir.

Interviewer: Ok fine.

Participant GG: And then it approaches the positive infinity, it will go away from the. Participant GG: And Participant NN: It will move away from the x-axis.

Participant NN: Closer to the y. Participant GG: Yes

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Participant GG clarified that the behaviour of the graph x approached negative infinity rather than the positive side. In explaining how the exponential graph would behave as x approaches infinity, Participant GG and Participant NN mentioned that the graph would move away from the x-axis. It did not show how the graph behaved. Both these participants also agreed that the graph would be closer to y. Any graph is composed of the x and y-coordinates. It was difficult following her statement that the graph would be close to the y. At first, Participant GG and Participant NN seemed to have the mathematically acceptable narratives on the behaviour of the hyperbola graph, but their utterances indicated that their routines were ritualised. These two participants’ words were not mathematical as they used utterances such as: “The graph moves away from the x-axis”, or “The graph would be closer to y”. It is unlikely that the community of mathematicians agrees with the two learners’ explanations of the behaviour of the exponential function. Participant GG’s and Participant NN’s explanation of the behaviour of the hyperbola as x approaches infinity was classified as “construed”. As x approaches infinity, the graph gets closer to the asymptote. At the same time, when explaining the same for an exponential function, the two participants’ responses contradicted the earlier classification and interpretation of “construed”. This led to the researcher classifying their routines as “ritualised” because they did not objectify the mathematical object – the function. These two participants successfully described the behaviour of the hyperbola, but could not do the same with the exponential function. They were not the only participants who struggled to explain the behaviour of the exponential function.

Meanwhile, Participant FF admitted that it was difficult describing what happened to the exponential graph as x increased towards infinity. It would seem that little attention was paid to the behaviour of the graph, especially on the extremes when learning about the functions. Participant EE did not show a distinction between an exponential function and a hyperbola in her utterances. Both Participant EE and Participant FF demonstrated some degree of a growing mathematical discourse by partly explaining the behaviour of the hyperbola. Below is an excerpt of their response to questions on the behaviour of the hyperbola and an exponential function.

Interviewer: What happens to the graph as x approaches infinity? Participant FF: As x approaches infinity? Which graph is x?

Interviewer: No, x in any graph. What happens to the graph as x approaches infinity? Let us begin maybe with an exponential function. As x approaches infinity, what happens to the graph?

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Participant FF seemed not to understand the question. For her, x was supposed to be a graph. In functions, it is axiomatic that x is a coordinate, and it cannot be confused with a graph. In fact, Participant FF did not respond to the question. She only stated that the question was tough. While she was able to identify the graphs as the hyperbola and the exponential function, she could not explain the behaviour of the graph towards the extremes. As a result, the researcher classified her word use as “not mathematical”, because she thought that x was a graph. Her interpretation of the graph was “not construed “,because she could not describe the behaviour of the graphs that she identified. Additionally, her narratives were not in concord with those of the community of mathematicians. She thought x was a graph. While Participant FF could name the graphs, she did not only describe the behaviour of the graph but also showed some confusion with some components of the graph as she referred to x as a graph.

Participant EE: Ok so the exponential is such as so and then we increasing function so it could go higher. I think

Interviewer: The x -values will go higher or y values will go higher or the graph will go Participant EE: Sorry, the exponential

Participant EE’s mathematical lexicon may have hindered her from freely articulating his thoughts on the behaviour of the exponential function. He gestured with his hands as he indicated that the function would increase exponentially. Evidently, language became a barrier for Participant EE as he tried to explain himself. He stated, “…so it could go higher”, referring to the exponential function. Any function could get higher. His ambiguous description could fit in any function. When it came to the hyperbola, both these participants were more comfortable in their narratives, and found the appropriate mathematical language to express themselves.

Interviewer: Ok, its fine, it fine, what about the hyperbola.

Participant FF: Ok sir, so as x approaches the infinity in the hyperbola, it goes closer and closer to the asymptote even though it will never touches it, but it will go closer and closer. Interviewer: Ok and the. Yes as x approaches negative infinity.

Participant FF: In the hyperbola

Interviewer: Yes in all of them, hyperbola and exponential Participant FF: It does the same thing but in the negative side Interviewer: Where do you get that?

Participant EE: It is what we were taught. Participant FF: Textbook.

Participant FF described the behaviour of the hyperbola as it approaches the asymptote. She emphasised that the graph would approach the asymptote but would not come into contact with

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it. While that was true that the the exponential function and the hyperbola graph did not intersect the asymptote, it was not true for all asymptote functions. Participant FF mentioned that the same would happen as x approaches negative infinity. When the interviewer asked them to justify their narratives, they attributed their utterances to authority. Participant FF explained that she was taught that graphs behaved that way, while Participant EE attributed his narratives to the authority of the textbooks. When learners rely on the textbook or teacher for their narratives, it indicates that they have not yet explored the mathematical object. As a result, their routines are ritualised (Sfard, 2012).

7.7 Conclusion

The empiricalluy-based discussions in this chapter revealed that learners’ mathematical discourse on the hyperbola and exponential function was still developing. There were signs of growth in their mathematical discourse because they could communicate mathematically, although they could not properly name some of the objects.

While learners could identify the two algebraic representations of the functions, they could not explain the relationship between the parent function and the transformed function. None of the learners mentioned the translations in their explanations. Instead, they explained the translations in terms of asymptotes.

Learners often justified their actions by imitating their communication from their teachers, and attributed their reasons for their talk to the very teachers. The learners did not give reason for their action, except that it was how they were taught. That gave credence to the view that learners’ narratives were based on authority. In the event that the teacher gave the rule, that rule had to be followed. The discourse of learners attributing their actions to their teachers still shows a discourse of others instead of a discourse of their own.

Learners managed to show that the equation of a geometric series was the same as the equation of the exponential function. When learners are able to recognise the relationship between two separate topics in Mathematics, it shows that their mathematical discourse is still developing. Notwithstanding the latter, learners still struggled with the hyperbola expressed in words. Once more, the challenge was premised on the parent function. Most of the learners thought that the hyperbola was also an exponential function, posing the problem of saming two distinct functions.

While learners’ routines were generally ritualised, their narratives were mathematical. Most of what learners wrote or said would earn them marks in an examination. However, the challenge would prevail in the event that reasons were required for why they wrote what they have written. This is an area in which learners were found most wanting. In questions that are more than just recall of facts, learners’ mathematical discourse was found to be lacking and needed

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some improvement. There were few instances in which flexibility and corrigibility were noticed from learners’ mathematical discourse.

In the next chapter (Chapter 8), the researcher concludes the study and discusses some relevant and critical recommendations accruing from the findings of the study.

174 CHAPTER 8

SUMMARY OF MAIN CONCLUSIONS, FINDINGS, AND RECOMMENDATIONS