An exponential function is the steepest graph (Webber, 2002). Furthermore, the exponential function is mostly useful in practical situations such as representing growth in infections, births or businesses. In sequence and series, the exponential function is a counterpart of geometric sequences. It is characterised by a fixed base and a variable exponent, hence the name “exponential function”. In this study, the research participants were required to use ordered pairs for purposes of calculating the fixed base, or to sketch the graph using the points. In this study, all participants opted for calculation of the fixed base. In this sub-question, there were more positive responses than in the previous question on the hyperbola. While only 1 (one) participant identified and gave a mathematical explanation on the hyperbola, approximately 13 participants (54%) managed to deduce that the verbal statement was an exponential function. Two participants (8%) thought it was a linear function, while another two (%) thought it was a hyperbola, and 1 (one) was of the view that it was a quadratic. Six participants (25%) did not respond to the question below.
Given this scenario: Mr Mkhize, a chicken farmer, begins his poultry business with 200 chicken, in the second year, he plans to have 400 chicken, increasing them to 800 in the third, 1600 chicken in the fourth. He plans to continue growing the number of chicken in that manner for a long time.
Based on the above scenario: Name the function represented by the above information, why do you say so?
There were two main categories of responses from the participants. One group calculated the equation of the function, and the other used the geometric sequence of the common ratio. Three (3) participants (13%) formulated the equation using the statement presented earlier of chicken that increased yearly. Participants’ utterances were similar in structure. Participant MM’s response are referred to below due to the level of detail entailed.
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Participant MM: Ok. Here. Ok in this graph, this function is an exponential function. Interviewer: Why do you say so?
Participant MM: Because of, when I calculated it using the scenario given which is above, then I have calculated it. I noticed that there is. Ok. Firstly, exponential function, says it is equation says y is equal to a times b to the power of x plus or negative p, negative q or positive p, (𝑦 = 𝑎𝑏𝑥−𝑝+ 𝑞) then here, we are given, here our, Ok. When I have calculated it, our 'a' was 20, then which we multiply 2 to the power x, then we have 200. 200 is made up of 20 times5, oh 20 times 2 to the power of 3 that is 200. Then 400 here it is 20 times 2 to the power of 4, then we have 800 which is equal to 20 times 2. It is 25 not 20. 25. It is 25
An exponent denotes algebraic representation of the exponential function (Webber, 2002. Subsequent to realising that the function was exponential, Participant MM used a general equation of an exponential function (𝑦 = 𝑎𝑏𝑥−𝑝+ 𝑞), to show that she was correct in her assertion. She managed to calculate the base of the exponent and then tried to prove her equation was correct. Participant MM did not use the explicit name for the base. She then attempted to show that her algebraic equation was correct by making substitutions into her formula. She used appropriate mathematical language, because she named the function correctly and used terms such as “power” for “an exponent”. She also showed some corrigibility by show the correctness of her equation by means of substitutions into the formula. Participant MM also showed that her mathematical discourse is developing because she was able to move from verbal expression of a function to an algebraic form of expression. Most of the participants (n=12, 50%) decided to use the geometric sequence to show that the function was exponential.
Of the 12 participants that used the geometric sequence formula to identify the verbally expressed function, the researcher discusses the responses of the 2 (two) participants who correctly stated that there was a common ratio. A geometric series was provided in the form of 𝑓(𝑥) = 𝑎𝑟𝑛−1, where a was the first term; r was the common ratio; and n the position of the term on the sequence. A number of participants concentrated on the common ratio and did not overly focus on the other component of the geometric sequence. In her explanation, Participant HH showed all the components of the geometric sequence.
Interviewer: Yes, what is the equation?
Participant HH: Y is equal to 100 multiplied by 2 raise to the exponent x (𝑦 = 100. 2𝑥) Interviewer: How did you find it?
Participant HH: I said when you times a hundred by 2, you get 200 to the power 1, which is what you get , then you get the, and then you make x=1, when you want to work for the first 200 chickens and then when you say, and I then say 100 to the power 2, then 2 to the power 2
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which gives you a 400, and then you say 100 times 2 then we put a 3, which gives you a 800, and then you say 100 times 2 to the power 4, which gives you 1600.
Participant HH calculated the equation of the function, which was an indication that she could proceed from one representation of the algebraic function to the other with ease. She thus exhibits flexibility by not only naming the function as an exponential function, but also stating how she could prove it by showing the equation of the function. She also exhibited corrigibility by showing how the equation satisfied the statement given for this function. In addition, she also demonstrated some exploratory routines by not only stating the equation, but also proving that the equation works. Participant HH also demonstrates a developing mathematical discourse, as she has objectified the mathematical object, the exponential function. Contrary to Participant HH’s mathematical discourse, Participant KK’s utterances showed some semblance of growth, but not at par with Participant HH’s.
Although Participant KK made a number of unmathematical statements, there was evidence of his familiarity with geometric sequences. While Participant KK’s counterpart thought the function was a parabola because of a “common second difference”, Participant KK maintained his stance and insisted that the function was exponential, as indicated below. Interviewer: How did you respond to the question?
Participant KK: I said it is a what, it is an exponential graph because my first term which is 200, when he began his farm and then they multiply by 2 and get what? 400. And in the next year, you multiply also by 2 and get what? You get 800. Then you multiply and so on and so on. meaning each year, each year this chicken farmer, what he is going to do is his chicken are multiply by 2, so if I can say the common difference here it is, common ratio, it is 2, it is 2, it is 2. Then I can reverse that by saying such as this the nth term or y is equal to a times plus x plus q (Tn= or y = a.𝑏𝑥 + q), and then I don't have my q then you take, then my first is 400 times two to the power x (400.2𝑥).
Participant KK clearly used the geometric sequence as a base for his contention that the verbal function was exponential. He noted that 2 (two) multiplied the next term, thereby noting that the sequence has a common ratio. Participant KK also stated the general formula for the geometric sequence. When he tried to apply his formula, he did not succeed because he had a first term as 200, and it was highly improbable that he could have 200 from 400. 2x. While Participant KK has some knowledge of the relation between the geometric sequence and the exponential function, he could not perform the algebraic manipulation. He was able to interpret the symbolic mediator expressed in the form of words because he recognised that the worded statement represented an exponential function. He also managed to recognise that the equation used to find the nth term of the geometric sequence was similar to the equation by stating the
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general formula and its counterpart 400. 2𝑥. However, he could not perform correct algebraic manipulation because 400. 2𝑥 does not represent the worded statement. On that basis, the classified his routines as ritualised. As much as he had developed his mathematical discourse, his algebraic manipulation was not fluid (Sfard, 2012).
When interpreting the symbolic visual mediator y = a.𝑏𝑥 + q, Participant KK stated that there was no q in 400(2)𝑥. That the q value was not written, does not mean the q value is absent. It only means that there is no vertical translation. If the function is not translated vertically from the parent function, then the value of q is zero and does not need to be written. Participant KK did not interpret the symbolic visual mediator mathematically (Sfard, 2012). He continued to explain himself as captured below.
Participant KK: I say my nth term is same as a times b to the power n minus 1 (𝑎𝑏𝑛−1) Participant JJ: Yeah
Participant KK: Then I will, if I can change this to be in the form of y is equal to a times b to the exponent x minus 1 (𝑎𝑏𝑥−1), I get my y is equal to 200 times 2 to the exponent x minus 1 (𝑦 = 200. 2𝑥−1). Because if I can say 200 times two divided 1, therefore I get my first term which is 200 and then saying, 200 times two divided by two minus 1, I get my 400, which is my second term, and saying 200 times two to the power three I get my second which will be 800.
Participant KK emphasised the relationship between the geometric sequence and the exponential function by stating that the nth term is the same as that of an exponential function, to which his counterpart Participant JJ agreed to. Participant KK then recovered from his earlier utterances where he had written an incorrect formula. The researcher could not attribute this change to corrigibility routines as there was no evidence of something he did to recover. He used the unsimplified geometric sequence formula. The algebraic manipulation of y = 200. 2x−1 yields y = 100. 2x. Participant KK went further to prove that his formula was correct when used to show the number of chickens in each successive year. Participant KK in this instance showed some flexibility and corrigibility routines. Flexibility routines were shown by taking the worded statement and expressing it in algebraic form. Corrigibility was also displayed by showing that his equation was correct using three of the four coordinates provided in the worded statement (Ben-Yahuda et al., 2005). Participant KK demonstrates that he was moving towards objectification of exponential function by expressing the worded function as an equation and proving how the ordered pairs were calculated. He further demonstrates exploratory routines by formulating the equation and showing how it works.
During the interviews Participant KK’s partner (Participant JJ) was of the view that the function was quadratic, due to a common second difference. This showed that for the participants, there is a strong link in learners’ mathematical discourse between sequences and
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functions. Participant JJ did not have convincing reasons for stating that the function was a parabola. Following is how Participant JJ argued her case.
Participant JJ: I said, it is the quadratic function or a parabola. Interviewer: Why? I am interested in the reason.
Participant JJ: … because it has the second common difference. Interviewer: Second common difference, ok
Participant JJ: Yes …
Participant JJ’s assertion of the second difference indicates that the function is quadratic in nature. It also shows that her narratives could be classified as substantiation because they are the same as those of the community of mathematicians. However, Participant JJ made an unproven assertion earlier. She could not prove that there was a common second difference. The researcher gave her the opportunity to prove her assertion, but she could not. This attests to Participant JJ’s challenges in arithmetic manipulation as her utterances show below. Participant JJ: Then. Press it on the calculator. Then first, he got here, we got negative 200, then it is 400, then it is 600.
Interviewer: Are you sure.
Participant JJ: Yes, I am sure sir. Interviewer: What is 1600 – 800?
Participant JJ knew that a sequence with a common second difference is quadratic. As shown above, her main challenges were located in the area of subtraction. It is not that she could not subtract, but she did not check the correctness of her answers. This has been a challenge noticed throughout the study, in terms of which participants made assertions without checking for the correctness of what they thought the functions presented. Therefore, Participant JJ is not the only participant whose mathematical discourse is at granular stage, but has room for further development (Caspi & Sfard, 2012). In the next few paragraphs, the researcher discusses the respective participants’ responses to the question on exponential growth.
Participant NN and Participant GG were of the view that the function was linear, but they did not have corroborating evidence for their assertion. After their conversations with the researcher, these two above-mentioned changed their stance to adopt the stance that the function was exponential.