TIPOS NÚMERO ESTUDIANTES

In general, many learners face the challenges of distinguishng a vertical asymptote from a removable discontinuity, especially that both are calculated at the zeros of the denominator. It is impossible for learners to get an undefined on the horizontal asymptote because it is normally an output, rather than an input (for vertical asymptote). Fourteen participants (58%) in this study used the word “undefined” 72 times. To these participants, the term “undefined” signifies an asymptote. While this may be true for the rational functions that they have been exposed to, it is not necessarily true for all the functions. For further clarity on the undefined asymptote,

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the researcher deemed it necessary to explore the participants’ responses to responses to the question: Explain how you would identify key features like the asymptotes in a table of values for the above functions f= 3𝑥+1− 9 and g= 2

𝑥+3+ 1.

The researcher selected four participants and interrogated their use of the term “undefined” in relation to the asymptote. Participant Z is one of those participants who used the word “undefined” more than four times in three phrases. For Participant Z “undefined” and “asymptote” are used interchangeably, because she thinks that “undefined” means “asymptote”. Following is an extract confirming Participant Z’s interchangeable usage: Undefined means asymptote because it states the graph never touches the asymptote because if it touches it, it becomes undefined, so conclude and state undefined is the asymptote. In a tangent function, for example in y=t and x, there is an asymptote. You get the undefined when you punch it on the calculator.

According to Participant Z, “undefined” means “asymptote”. When Participant Z sees “undefined” on the table of values, she immediately knows that there must be an asymptote. “Undefined” in this case is a result of a zero on the denominator for certain values. It is not always the case that an asymptote is present wnenever there is a zero. In some instances, an asymptote will be a removable discontinuity, which happens when the denominator is a factor of the numerator. In such a case, there is ‘undefined’ at the zero of the denominator, but the asymptote will not pass through that coordinate. Undefined can only be used to determine a vertical asymptote, and not a horizontal one. It is not mathematical to then conclude the presence of an asymptote whenever there is an undefined. Participant Z further explains ‘undefined’ on the basis of a graph touching the asymptote. The function is only defined in terms of an input and output, otherwise it is undefined in all other instances. The asymptote does not necessarily show where the graph is undefined; but explains the behaviour of the graph as the function tends to infinity. Participant Z relates the question of the asymptotes to trigonometry, where there is a vertical asymptote for the tangent function. Participant Z further shows the universality of the asymptote at the zeros of the denominator. Participant Z shows that it is not only with the iconic visual mediator that an undefined will result in an asymptote, but also in symbolic visual mediator when she narrates the calculator. She explains and puts it beyond any reasonable doubt that for her, “undefined” means “asymptote”.

Based on Participant Z’s utterances, the researcher classified her use of words for this question as “colloquial”, because it is not mathematical to always associate “undefined” with “asymptote” as synonymous, which is based on what she has seen. The researcher then classified her interpretation of the visual mediator routines as “ritualised mathematical”, because her utterances clearly demonstrate both the ability to draw mathematically acceptable

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graphs and further interpret the table of values and formulae. Since Participant Z relates the table of values or ordered pairs to graphs to formulae, the researcher then classified her use of routines as “flexible” substantiations that are not approved by the community of mathematicians. She justifies her statements by relating the three representations of a function. Participant KK is one of those participants to whom the word “undefined” was synonymous with asymptote. He uses the term “undefined” eleven times in response to the question pertaining to his identification of an asymptote from a table of values. The following extract represents his response: The graph will never touch this line because when it touches this line it will be undefined. If it is undefined, it means that it does not touch this line. This line gives you a… then it becomes, our what, our asymptote. I will state asymptote means undefined because, on undefined, the graph, the graph there is not undefined, the graph does not have values. I will state it means undefined. If you present a time graph, you know a tan graph, you have something like this, a time graph, this is 0, come to this line to 180. You find out that in 45, you have this number and when you press it on your calculator, it is what. It is undefined. But this graph, it will go on the same manner like this but it will not. So that's why I'm stating, it is undefined.

Participant KK responded further thus: If you have a tan graph, tan, tan it begins it 0, and this is step 45, Yeah, this will become a graph. A tan graph actually it is 0, 45, 90, it's from 35, 180. it ends here. Then, you will find in your 90 is what is n. You draw this line and there is what, there is error undefined. and in your 180, in your 270 you can add and your graph, will have it in this manner and your graph, never touches this line. Because in this line it is what, it is error meaning it is equal, it is undefined. So this line becomes the asymptote. So I'm still stating undefined means it is the asymptote.

According to Participant KK, in the event that the graph and the asymptote intersect, the graph then becomes either invalid or undefined, based on the graph touching the asymptote line. Furthermore, Participant KK mentioned that ‘asymptote’ and “undefined” were synonymous, mainly due to the graph not having undefined values. What he actually means is that there is no corresponding y-value at the zero of the graph. Unfortunately, the language barrier induces a narrative which suggests there are many more coordinates, yet he speaks of only one x-value; which does not even have a corresponding y-value. To buttress his point, Participant KK uses another function, a tangent, to illustrate that the function is undefined at the zero of the denominator. He does not show the relationship between undefined and the asymptote. Participant KK uses what he sees to interpret mathematical actions. When performing calculations, he commits an error on the calculator. According to him, there is no corresponding y-value at the point of calculation. Participant KK realises that when proceeding from the

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ordered pairs to the graph, the disproportionate part of the x-value and the y- value is then sketched as a dotted line and named the asymptote. According to Participant KK still, whenever there is an undefined on the table of values, we automatically have an asymptote. What Participant KK does not explain is how he arrived at the horizontal asymptote.

The researcher classified Participant KK’s word use as “colloquial”, because undefined does not necessarily mean there is an asymptote. Sometimes, it might be a removable discontinuity. Participant KK did not explain how that undefined translated to an undefined. His interpretation of the iconic visual mediator is then classified as “not construed”. His routines are ritualised because he could perform all the mathematical processes without showing some understanding of the reasons for the mathematical process. He has not objectified the mathematical object. Accordingly, the researcher classified his use of routines as “flexible”, because he is able to use the tangent function to explain the hyperbola. Using the DPHEF analytic tool, the researcher further classified Participant KK’s narratives as “memorisation based on visuals” due to his memorisation ofwhat he has seen in both the table of values and the graphs.

A mathematical error occurs on a calculator when the calculator does not recognise the mathematical manipulation performed on it. For example, some calculators show an error on the screen when a square root of a negative number is written. The square root of negative nine is three I (3i), but the calculator may display an error. This means that an error on a calculator is all about the programming of thecalculator, than a mathematical error per se. At the zeros of denominators of rational functions, calculatotors normally display an error. Participant KK and Participant Y are examples of learners who used the word error for an asymptote. The following excerpt by Participant Y used the word error to explain the presence of the asymptote: For asymptotes, we will see where there is an error. For asymptote, that’s what’s the graph states, if there is an error in Y, then the X values will be the asymptote.

As explained in the above paragraph, an error does not necessarily mean the prevalence of an asymptote passing through that coordinate. Participant Y states that the identification of the table of values should be shown by an error. Undefined is usually written on the table of values not as an error. As explained earlier, undefined does not necessarily mean the graph has an asymptote. Participant Y’s response shows that she relied most on what she saw on the calculator without necessarily making explorations on the mathematical object. An error does not necessarily mean that the asymptote will pass through that particular point.

Using the DPHEF analytical tool, the researcher classified Participant Y’s word use as “colloquial”, because an asymptote is not identified by an error on the graph; but the behaviour of the graph as the function tends to infinity. The iconic visual mediator is not construed

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because a calculator is supposed to be a calculation aid. What is found on the calculator should be used without processing. When a calculator uses words such as error, participants should be able to interpret their meaning and use mathematical language. Given her responses, the researcher classified her routines as “ritualised non-mathematical”, because on a table of values, the word “error” should not be written or found. The researcher accordingly classified her use of routines as “applicability with a visual trigger”. Participant Y justifies her answers based on what she sees. On that account, her narratives are classified as “memorisation based on visuals”.

According to CAPS, learners should be able to work flexibly between four representations of the function. These representations are verbal (words), equations (formulae), numeric (ordered pairs), and diagrammatic (graph). These four representations of the function are the same phenomenon expressed in different formats. Participants should be able to identify key features of the function from whichever representation. A table of values or ordered pairs also represents a function. The choice of the input values for the vertical asymptote determines whether key features such as intercepts, turning points, or asymptotes could be identified. Three of the participants stated that the asymptote could not be identified from the table of values. These participants (Participant MM, Participant FF and Participant OO) indicated that an asymptote could not be identified from the table of values. Their statements were virtually the same, suggesting that an asymptote cannot be identified from a table of values. Since the four representations of an asymptote are an expression of the same function, there should be a way of expressing the table of values such that key features such as the intercepts and asymptotes are visible. Based on the nearly similar statements of the participants, the researcher classified their word use as “colloquial”, because it is possible to identify an asymptote from the table of values; especially when non-integral values are used for the input (Moalosi, 2014). Furthermore, the researcher classified the interpretation of the iconic visual mediator as “not construed”, due to their failure to recognise that the asymptote could be construed from the table of values. These types of routines are “ritualised mathematical”. While participants are able to draw tables of values, they are unable to interpret these tables fully.

6.3 Conclusion

In this chapter, the researcher demonstrated that the mathematical discourse of the participants was still developing and characterised mostly by colloquial and ritualised routines, as well as narratives that were mainly based on what participants could see, rather than the mathematical explorations (Mpofu & Pournara, 2018). The researcher utilised the DPHEF analytic tool and discussed participants’ representation of the term “asymptote”. Language posed a barrier for most of the participants. Such a state of affairs is inimical to their capacity to fully represent

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their knowledge, understanding and experiences. Inadvertently, this leads to learners referring to an asymptote as a point.

While most of the words used by learners were mathematical, there were instances of their confusing communication as accepted in the community of mathematicians; for example, when they named an asymptote as a number, an asymptote of an exponential function, or as a vertical line. Learners’ visual mediators were generally construed, meaning that they were able to interpret both the iconic and symbolic mediators of the two functions. It was difficult to obtain reasons for these mathematical actions performed by the participants, which are also largely premised on what they saw and/or what they have been taught. Learners deduced meaning from what they saw from symbolic and iconic visual mediators.

The lerners’/ particpants’ representation of the asymptote of the hyperbola and exponential functions in this chapter was of critical importance to the broader domain of the study’s objectives. Similarly, the next chapter (Chapter 7) provides a conclusive link to learners’ mathematical discourses and the community of mathematicians. Such a link is necessary, since the learning Mathematics is not a end in itself, neither is such learning peripheral to learners connection with their daily realities (Bradley et al., 2013; Caduri et al, 2015)

134 CHAPTER 7

FOUR REPRESENTATIONS OF A FUNCTION