During the data analysis process, there was one instance which involved interpretive dilemmas, in particular deciding how the respondent made
107 sense of a particular mathematics statement. Below is the mentioned instance.
Forwhatvaluesissinxdecreasing?Explainwhyitisdecreasingfor thesevalues?
One might argue that this respondent (Respondent A) was making sense through operation because she was increasing the angle of xto make sense of this mathematics statement. However if we reflect deeply then we will notice that the essence of this sense‐making process is not the operation but is the perception. She got her conclusion through noticing the changes of opposite side of the right angled triangle which is related to perception. 5.6Relatingempiricalevidencetothetheoreticalframework.
The data has shown that most of the respondents’ evoked concept images are in the context of triangle trigonometry or circle trigonometry or a blending of both except for respondent B who has a strong link to analytic trigonometry with some aspects in circle trigonometry. This also indicates the sophisticated nature in human thinking. The evoked concept image is only part of a conceptual structure. It is sensible to say that the evoked
108 concept image is the most direct and strongest link in humans’ mind to a particular stimulus.
The initial theoretical framework is based on the theory of three world of mathematics proposed by Tall (2004). However based on the data collected from the pilot studies and special case study, it shows that when the respondents made sense of mathematics, they didn’t necessary based on the formal world of mathematics. Therefore the theoretical framework of making sense of trigonometry was refined to be making sense through
perception, operation and reason. In this case, perception means making sense through sensory input. Operation means making sense through physical actions.
Reason means making sense through conceptions, definitions and deductions. In this case, the reasoning is related to triangles and to dynamic relationships shape of the graph, symmetries, periodicity etc. In fact, most of the respondents reason based on the properties of visual images such as the sine graph. In general, reasoning involves verbalizing the relationships between different things such as verbalizing the relationships between certain perceptions and operations. For instance, respondent SC knows sin x
is decreasing for 2 3 2 ,
2 n x 2 n n
because as he varies the
angle (operation) in the unit circle and sees (perception) the length of the opposite side of the right angled triangle varies as a consequence of this action. On the other hand, respondent B who reasons some of the properties of sine by Taylor series without going into the details of it because he has a
109 supportive conception for Taylor series (reason based on conceptions). Apparently this is a S1 response. Respondent SC also reasons based on definition when he responded that the hypotenuse is defined as 1 in the unit circle therefore sin can never equal 2. This idea will be discussed in more detail in section 3.2.
The data has shown that different respondents have responded differently for certain items. For instance, respondent A was operating in triangle trigonometry when attempting to explain why sincan never equal 2 by saying “the hypotenuse is the longest side since it is a right‐angled triangle”. Alternatively, respondent B was operating in analytic trigonometry by using the Taylor expansion without going into the details of its computation. Meanwhile, respondent C was operating in circle trigonometry probably by evoking the sine graph in order to answer this item. This shows the diversity of contexts in trigonometry.
Further reflection on the data also revealed that the conceptions that the respondent possessed were related to the notion of met‐before as suggested by Tall (2005). Met‐before is used to indicate the effect of previous experience in new situation that affects our current thinking. Lakoff and Nunez (2000) proposed a similar notion as metaphor which means speaking of new or abstract ideas in terms of familiar ideas. The notions of met‐before
and metaphor are related to the supportive conceptions and problematic conceptions of this study. Supportive conceptions support generalization in new context whereas problematic conceptions impede generalization in new context. As we can notice from the data, all the respondents have
110 supportive or problematic conceptions in making sense of trigonometry. For instance, respondent A who has a strong link to triangle trigonometry, she tries to conceive sin 270in triangle trigonometry which is clearly a problematic conception because there is no way that she can construct this triangle in triangle trigonometry.
Supportive conception might contain problematic aspects in it and problematic conception might contain supportive aspect. For instance, respondent B who has a supportive conception with problematic aspects on Taylor series, he uses this series to explain why sin 270 1but his explanation is not correct and he does not go into the details of the computation. Meanwhile, respondent C who has a supportive conception with problematic aspects on the sine graph, she draws the sine graph and says sin 270 1because she can see from the sine graph. In fact, she didn’t offer an explanation of why the sine graph would look like that. Similarly respondent SC also has a supportive conception with problematic aspects. For instance he can state the formula for sin (A+B) but he couldn’t prove it. In this case, it may be hypothesize that he couldn’t prove this formula in the three distinct contexts of trigonometry. This idea of supportive and problematic conceptions is discussed in detail in section 3.4. Relevant literature on this idea is in section 2.5.
Apparently the three respondents in the pilot study didn’t exhibit the coherent links between the three distinct contexts of trigonometry. For instance, respondent A didn’t link triangle trigonometry to circle trigonometry and ended up drawing a weird figure by thinking of sin 270in
111 triangle trigonometry. Respondent B never evoked triangle trigonometry in answering the items and apparently he didn’t build coherent links across the three contexts. He has strong links to analytic trigonometry. Meanwhile, respondent C has a strong link to the sine graph however there is no evidence showing that she has linked it to the unit circle and the triangle trigonometry. Respondent SC does exhibit coherent links between triangle trigonometry and circle trigonometry. He could link the sine graph to the unit circle and justify its properties. In short, the three respondents in the pilot study know the concepts of trigonometry but they don’t grasp the relationships between them. On the other hand, respondent SC has a much better grasp of these relationships than the other three respondents.
112 Chapter6
TheStoriesofFiveStudentTeachers 6.1Introduction.
This study concerns with how student teachers make sense of trigonometry. Based on the theoretical framework as discussed in Chapter 3, humans make sense of mathematics through perception, operation and reason. There are three distinct contexts in trigonometry, namely triangle trigonometry, circle trigonometry and analytic trigonometry. The main issue would be how do the student teachers cope with the changes of meaning across different contexts as they learn more sophisticated ideas. The student teachers involved in this study were taking PGCE Secondary Mathematics at a British university when the data was collected.
This chapter presents the main data analysis of five student teachers on how they made sense of trigonometry. The responses of these five student teachers show a spectrum of responses for the collected data. Data was first collected through questionnaires, then follow‐up interviews were conducted in order to gain greater insight into how the interviewees make sense of trigonometry through four important and interrelated aspects as follows:
(a)The ways that interviewees make sense of trigonometry, (b)The contexts of trigonometry that the interviewees were
operating while making sense,
(c)The supportive or problematic conceptions involved in making sense,
113 (d)The nature of knowledge possessed by the respondents: whether
they know it or they grasp it.
In order to guide the reader through the analysis of this chapter, the items of the questionnaire are presented first followed by relevant evidence gathered from the student teachers’ responses. A summary of each student teacher is given at the end of each case and these summaries form the basis of an overall analysis. These summaries are written in a way to answer the proposed research questions in particular question 1 to 8 of chapter 4 (see page 62‐63).