The decreasing temperature

The third question of the functions test was a non-routine question which presented a practical decreasing exponential function. The question required participants to complete a table of values, as well as drawing and naming the graph. The phrase exponential function, did not appear in the question, but the equation was exponential. The participants were also supposed to state the asymptote of the graph and explain what the asymptote meant in real-life terms. Two-thirds of the participants (67%) successfully completed the table of values and plotted the corresponding graph. Just over a third of the participants (33%) named the function successfully. The number of participants that successfully named the asymptote went down to 26% (n=29) and the meaning of the asymptote in real life terms went down to 9% (n=10). When comparing the performance or success rate of the first and the third questions, a significant decline was observed with regard to participants’ performance (response) in the context of the third question. The researcher attributes this decline to the general ritualisation of learning of functions. The difference in these two questions was that the familiar mathematical language was used in the first question, whereas practical language was used for the third question’s responses.

The completion of a table of values is a routine task for participants in grade 11, especially that it is a requirement in earlier grades where learners are required complete a linear function’s table of values. In the context of this study, the task given to the participants was not a daunting one. Notwithstanding, two factors were observed as contributing to the level of difficulty experienced by learners. Firstly, the question was not written in terms of x and y variables. Secondly, the exponent was in the form of a fraction. Figure 5.23 below illustrates Participant W’s response to the completion of the table of values.

Figure 5.23: Participant W’s response to completion of the table of values

Participant W was one of the 74 participants who successfully completed the table. As stated earlier, the use of t for time and θ in the equation, posed a challenge to some participants. Participant W rounded off her output to two decimal places, which helped her to plot the

89

graph, unlike some participants who did not do so and could not even produce a smooth curve. Participant W showed some flexibility in her routines by not writing numbers as emanating from the calculator; but rounded them off in a manageable manner, especially for plotting points on the graph (Ben-Yahuda et al., 2005). That did not mean that those who did not round off their values for the output were incorrect in their responses.

Figure 5.24: Participant UU’s response to completion of the table of values

In Figure 5.24 above, Participant UU rounded off her output values to the nearest whole number. While marking their work, the researcher awarded full marks to her although she produced a smooth curve due to the high degree of approximation. In Figure 5.25 below, the participants were required to draw a continuous graph of the data on the table provided.

Figure 5.25: Participant W’s response to drawing a continuous graph

In Figure 5.25 above, Participant W, drew a smooth decreasing exponential function. As indicated in the previous paragraph, rounding off the output coordinates contributed to the type of graph that Participant W drew and indicated the location of the asymptote of the graph on the Cartesian plane. Her graph and table of values did not show the asymptote of the function.

90

In Figure 5.26 below, Participant UU’s graph showed that rounding off the output values to the whole number caused her graph not to be as smooth. The situation was compounded further as Participant UU chose to use writing paper instead of the grid that was provided; hence the faintness in the diagram.

Figure 5.26 Participant UU’s response to drawing a continuous graph

Although Participant UU thought this was a straight line, the diagram did not resemble a straight line at all. Rounding off to the nearest whole number and the choice of a scale on a writing pad affected Participant UU’s accuracy. Some of the participants also thought the input and output values should be integral values. Such thinking was based on the fact that in most cases, integral values are used for convenience rather than as a rule when sketching or plotting a graph. Therefore, in the functions discourse, learners should not just sketch or draw a graph without realising that a graph is characteristicallycontinuous. This means that its coordinates are elements of real numbers. In this regard, Participant UU’s diagram was then more of a sketch than a drawing.

Any kind of paper could be used for sketches. However, Participant UU’s situation was compounded by the fact that the graph was drawn on paper of very poor quality, which resulted in its faintness. Nonetheless, there was a modicum of accuracy with the coordinates. In most instances, beginners’ drawings involve many coordinates before the shape of the functions becomes clearly visible. Participant UU’s sketch was more of an approximation, and accuracy of the scale was not important as only key features were used to show the path of the graph. Some participants did not complete the table of values accurately. Following are a few examples of those who did not complete the table of values during the tests.

91

Figure 5.27: Participant VV’s responses to completing the table and drawing a continuous graph

From Figure 5.27 above, it is clear that Participant VV failed to complete the table of values due to the negative values of the output from an exponential equation. His main challenge was primarily located on the failure to interpret the symbolic visual mediator (Ben-Zvi & Sfard, 2007). From the literature review, it is posited that mathematical symbols were important in communicating mathematical information (Bradley et al., 2013; Flesher, 2003). Participant VV did not recognise the t variable as an exponent. He worked on his answer as though the variable was part of the base, hence the output of negative numbers. Secondly, when drawing his graph, Participant VV mixed the -ve (negative) and the +ve (positive) numbers on the same side of the axis; resulting in positive (+24), negative (-12) and so on, on the same quadrant. Instead of his graph showing a decrease as the table of values suggests, it showed a decrease. This shows that in spite of Participant VV drawing a table of values and the axes of the graph, he did not understand their meaning. Participant VV understood the plotting of points as the diagram suggests, but his problem was with the axes and their meaning. It was difficult for Participant VV to draw any graph, because he did not understand how to place coordinates in the four quadrants.

Participants were presented with the equation, the table of values and the graph of the function in question 3.3, which required them to name the function. Forty-five participants (37.5%) could name the function as exponential. The most common response was that it was a line graph. In a line graph, a straight line joins different points. This means that there was no expectation of a defined function to be produced. Participants did not see a pattern emerging from either their table of values or the graph. Of the 74 participants (61.6%) who named the function as something other than the exponential function, 54 of them (45%) stated that it was

92

a line graph. Despite that some participants such as Participant VV had not drawn a line graph (as seen in Figure 5.27 (p. 91) above), they stated that the graph was linear. Figure 5.28 below shows Participant UU’s naming of the type of graph.

Figure 5.28: Participant UU’s response to naming the type of graph

Participant UU’s graph does not look like a line graph, although she names it as such. A possible reason for misnaming the graph was that she could not find any regular function which could fit the graph. Forty five participants (40%) named the function as exponential. In Figure 5.29 below, the graph was named correctly by Participant W.

Figure 5.29: Participant W’s response to naming the type of graph

While Participant W’s response is acceptable in Figure 5.29 above, she did not give a reason for her particular response. It was very difficult to speculate as to why she decided to name the function as exponential. Only 31 (25.8%) of the participants named the asymptote correctly, while the rest of the participants (89, 74.2%) did not. The participants did not provide reasons for their choice of answers. The researcher noticed that the number of participants who responded positively decreased as the questions increased.

An asymptote is characterised by the graph’s behaviour as the x or y values approach infinity. At infinity, the graph approached a straight line. In case of the cooling curve of coffee, an asymptote approached 20℃. As indicated by the wording in question 3, the cooling curve meant that as time increases, the temperature drops to 20℃elcius. Only 9% (n=11) of the

93

participants were able to provide an explanation that was close to an acceptable response. Figure 5.30 below is a representation of Participant WW’s responses to question 3.5: What is the real-life meaning of the asymptote?

Figure 5.30: Participant WW’s response to the real-life meaning of the asymptote

Participant WW indicated that the temperature of the coffee “should not pass this point”, instead of stating that “the temperature of the coffee cannot be lower than the position of the asymptote”. The point he referred to, was the asymptote. Participant WW spoke of an asymptote as though it was a point, when it was in fact a line. His view was that the closer the temperature was to the asymptote, the lesser the possibility of the temperature decreasing beyond that point. This shows further that Participant WW only understood the asymptote partially. While he knew that the graph would not pass through the asymptote, Participant WW wrote as though there was a point known as the asymptote. In the same vein as Participant WW, Participant XX also mentioned the word “point”. Figure 5.31 below represents Participant XX’s response to question 3.5 (the real life meaning of the asymptote).

Figure 5.31: Participant XX’s response to the real life meaning of the asymptote

An extrapolation of Figure 5.31 above indicates that Participant XX took θ to represent the amount of money to be paid for the coffee, instead of the temperature. Participant XX stated that the amount of money had a limit at R20. 00. In her explanation, Participant XX stated that the price of coffee should not be more than R20.00. Participant XX drew a graph with a decreasing exponential function, but she wrote the price of coffee as not exceeding R20. 00. This contradiction indicates ritualised routines, in terms of which participants acted without thorough meaning-making of what was actually happening (Gcasamba, 2014).

The following figure (Figure 5.32, p. 94) further indicates participants’ deviation from the asymptote. In this regard, Participant GG’s response to the meaning of the asymptote in the context of cooling coffee had nothing to do with the asymptote

94

Figure 5.32: Participant GG’s response to the real-life meaning of the asymptote

Participant GG described the phenomenon of what happened to the coffee over time, which had no direct bearing or effect on the asymptote. This particular participant provided the correct asymptote for question 2.2 (writing the equation of the asymptote) and drew relevant asymptotes for question 1.3 (sketching the asymptote of the graph), as reflected in Figure 5.33) below.

Figure 5.33: Participant GG’s response to sketching the asymptotes

In Figure 5.33 above, Participant GG could sketch asymptotes and state their equation in the hyperbola and the exponential function. Questions 1.3 (sketching the graph) and question 2.2 (writing the equation of the asymptote) represented the daily mathematical realities that participants were exposed; while question 3.5 (the real-life meaning/ implications of the asymptote) represents an unseen question. Asking learners to explain the meaning of an asymptote from a contextual situation was not a regular occurrence. Failure to respond to application questions reinforced the view that participants’ mathematical discourse was mostly ritualised routines. This question had the least number of positive responses of all the questions in the test.

95 5.4.2 The intersection of two graphs

Question 4 of the test required participants to sketch both an exponential and a quadratic function on the same set of axes by means of the global method. When using the global method, sketching the graph is undertaken by calculating key features such as intercepts, turning points and asymptotes. A global method is applicable when participants have been grounded in the plotting of graphs. In addition they should knowing the kind of shape to expect from each equation. In this case, the question required the sketching of the graph by using key features. About 60% (n=72) of the participants successfully responded to the questions. The quadratic function was better responded to than the exponential function was.

While participants could sketch the two graphs (the parabola and the hyperbola), the interpretation thereof was not as impressive. Only 20 (16.7%) of the 120 participants could interpret the intersection of the two graphs as the solution to the equation

1 2 1 12 4 2 _{ }          x x

x . Most of the participants tried to solve it algebraically, but failed due to their inability to simplify the exponent x. The question required participants to mark the solution using the letters A and B. Figure 5.34 below represents Participant R’s sketching of the exponential and quadratic functions. Participant R’s work was an example of participants who succeeded in completing the task of sketching the two graphs and showing the location of the solution to the equations. Although the graphs do not look smooth, they conveyed the essential message.

Figure 5.34: Participant R’s sketch of 2 graphs and location of the equation

Figure 5.34 above indicates that Participant R had many coordinates on her graph. This reflects a state of uncertainty regarding the shape of the graph. The interpretation of the graphs

96

by the community of mathematicians was the same. Using the DPHEF analytical tool, the iconic visual mediator was considered as represented, since Participant R showed all the key features of the two graphs; viz: the turning points, the intercepts, the asymptotes, and the shape of each graph. The researcher also classified her use of routines as “applicability”, based on her correct use of the key features when producing the two graphs (Ben-Yahuda et al., 2005; Kidron, 2011).

The solution to the equation 1

2 1 12 4 2 _           x x

x is at the intersection of the two graphs. The point of intersection was the solution because at that point, the x and y values of the two graphs coincide. Participant R marked the point of intersection of the two graphs in bold letters. Her iconic visual mediator was “construed” because she could tell the point of intersection of these two graphs was the solution to the two equations. Participant R displayed flexibility routines because she demonstrated there was more than one way of solving equations (Ben-Yahuda et al., 2005). Other participants in the study did not respond to the question in the manner that Participant R did. Figure 5.35 below illustrates an almost similar response of a participant whose graph is well sketched, but with few details.

Figure 5.35: Participant W’s sketch of 2 graphs and location of the equation

In Figure 5.35 above, Participant W’s response is almost the same as that of Participant R in Figure 5.34 (p. 95). The difference between these two participants is only that Participant W’s graph was not as detailed as Participant W’s, whose graphs elaborately showed the key features of the quadratic and the exponential functions. Unlike Participant R who showed each of the key coordinates of the graph, Participant W further located her points on the axes. Participant W’s diagram is an example of good sketching, because her graph was not overly crowded or cluttered. The graph also demonstrates that she knew the shapes of the two

97

functions and did not rely on plotting coordinates. Using the DPHEF analytic tool, Participant W’s mathematical discourse on sketching of the graph was characterised as “depicted”, based on her competent sketching of the graph. To the extent that he did not rely on plotting points for her shapes, displaying confidence and knowledge, Partipant W’s routines are then characterised as “routine”.

While Participant W exhibited flexibility in sketching her graph, she did not indicate the solution to the point of intersection of the two graphs. During the interview, when the researcher asked her why she did not respond to question 4.2, she responded that she did not know how to solve the equation. She was able to draw the graphs, but was unable to explain that the interpretation of the graphs indicated ritualised routines. Participant W’s routines were “ritualised”, because she did not realise that the point of intersection guided the solution to the graphs.

Participant YY was one of the few participants who sketched the parabola well, but experienced difficulties regarding the exponential function. Figure 5.36 below is an illustration of Participant YY’s sketch of the parabola and the exponential function.

Figure 5.36: Participant YY’s sketch of 2 graphs and location of the equation

In Figure 5.36 above, Participant YY sketched the parabola showing the intercepts and the turning point, although she did not show the coordinates of the turning point. Participant YY identified and drew the asymptote of the exponential function y = -1. She also marked the intercept with the axes, which was the point of origin. Participant YY went on to draw an increasing function. She did not check the correctness of her work by finding another point on the graph that would have acted as a guide. Another point would have helped her recognise that the graph was decreasing. The action of not checking the correctness of the answer (or lack of it) indicates lack of flexibility in her routines.

In Figure 5.36, Participant YY marked one of the points of intersection B, which indicates that she knew the point of intersection of the two graphs was the solution to the algebraic

98

equation of these two graphs. However, Participant YY did not mark the other point of intersection. When the researcher asked for the reason, she stated that she forgot to write it. She further pointed to the solution on the graph, indicating that she knew how to find the solution for the equation. In Figure 5.37 below, Participant YY’s calculation of the intercepts is reflected.

Figure 5.37 Participant YY’s calculation of the intercepts

In Figure 5.37 above, there are two calculations for the intercepts of the exponential function. Participant YY’s calculation was correct, but there were some evident elements of ritualised routines. After calculating the x-intercept and getting a zero, Participant YY should have known that the coordinate (0; 0) is both the y and x intercept. The ritualised aspects of Participant YY’s work would have shown some flexibility if she had checked the correctness of her answer (Ben-Yahuda et al., 2005). In the next few paragraphs, the researcher shows examples of participants who tried to solve the equation 1

2 1 12 4 2 _           x x x

algebraically, as shown in Figure 5.38 below.

99

In Figure 5.38 above, Participant AA tried to solve the equation 1 2 1 12 4 2 _           x x x

algebraically, yet the instruction in the question had clearly stated they should mark the solution on the graph. Participant AA eventually had the exponent x as a denominator. There is no mathematical explanation to support his action. In short, Participant AA moved from an exponential function to a fraction, and then concluded with a linear equation. In this regard, Participant AA’s routines were non-mathematically ritualised since he just wrote