PRESENTACIÓN DE LA INFORMACIÓN

Hiebert (1986) distinguished two different kinds of mathematics knowledge as conceptual knowledge and procedural knowledge. Conceptual knowledge is rich in relationships and is characterized as a connected web of knowledge as a single coherent knowledge. Procedural knowledge consists of formal language or symbol representation system and rules or algorithms to solve mathematical tasks. To grasp a particular piece of mathematical knowledge requires conceptual relationships to place that knowledge in context, but it also requires a fluent ability to perform the necessary procedures where necessary.

Skemp (1976) proposed a similar notion as instrumentalunderstandingand

relational understanding. In this case, instrumental understanding can be summarized as knowing what to do and how to do whereas relational understandingis concern with knowing why it works. Again, grasping ideas requires relational understanding, with a fluent ability to perform the operations.

57 Skemp (1979) distinguished between conceptual links (C‐links) and associative links (A‐links) possessed by the learners. C‐links have conceptual qualities which relate one idea to another whereas A‐links are associative in nature which may be formed through rote learning and memorizing. Leron and Hazzan (2006) reported dual processing theory where the immediate response (S1 response) operates at a non‐analytic or intuitive level, which is immediate, effortless and inflexible. In contrast, S2 response operates at an analytic level, which is slow, effortful and relatively flexible. Grasping ideas requires C‐links which usually are S2 responses. Kahneman (2011) spoke of a similar notion by suggesting two different ways the brain forms thoughts namely system 1 which is fast, automatic, subconscious, frequent, stereotypic whereas system 2 is slow, logical, effortful, calculating, conscious and infrequent. When a C‐link is used in a regular basis, it might be transformed into a A‐link without consciously recognizing the underlying ideas therefore it will become a S1 response. In this case, a blending of links occurs to reduce the cognitive strains in doing mathematical tasks.

3.6Summary.

The theoretical framework of this study is the consequence of three important ideas in this thesis. The idea of how human make sense of mathematics has motivated the work of reviewing extensive literature which leads to the hypothesized theory of making sense through perception, operationand reason. The idea of changing of meaning in mathematics has resulted in the framework of looking at trigonometry in three distinct

58 contexts namely triangle trigonometry, circle trigonometry and analytic trigonometry. Based on the notion of extensional blend in this study, the factor which impedes the shifting between triangle trigonometry and circle trigonometry is proposed. In this context, the proposed factor is the changing of meaning between Euclidean geometry and modern Cartesian. Finally the idea of how humans cope with the changes of meaning in mathematics has resulted the notion of problematic conception and supportive conception. Human may suppress problematic conceptions and problematic aspects of supportive conceptions in order to keep on learning mathematics at a higher level. Problematic conceptions impede the sense making in new context and thus prohibit the building of coherent knowledge structure. The idea of knowingand grasping is important in the sense that it could provide a powerful explanation for the nature of knowledge possess by a respondent. The more able learners would know an idea at one level and grasp the same idea at a higher level. They can look at it from different angles and speak about it as a coherent entity in its own right. On the other hand, the less able learners would know an idea at one level but couldn’t grasp the idea at a higher level. This is evident when they couldn’t think of it and speak of it as a coherent entity.

59 Chapter4

ResearchDesignandMethods 4.1Introduction.

In general, this study concerns how student teachers make sense of trigonometry. Based on the review of literature and the collected data, trigonometry can be categorized into three distinct contexts namely triangle trigonometry, circle trigonometry and analytic trigonometry. In these three contexts, student teachers use different combinations of perception, operation and reason to make sense of trigonometry. The transition in different contexts of trigonometry involves supportive and problematic conceptions. In the later stages of the sense‐making process, some student teachers grasp the concept while others only manage to know the essential skills to progress to learning higher‐level concepts. The focus is to explore how the student teachers cope with the changes of meanings in trigonometry after learning triangle trigonometry and circle trigonometry in school and analytic trigonometry in university.

According to the review of literature, there are a few themes that are strongly related to this study and should be taken into consideration. The central ideas relate to the nature of the knowledge structures and their difficulties in learning trigonometry together with the importance of subject matter knowledge and the level of confidence in responding to the mathematics items of the questionnaire. This chapter discusses the research design, methods of data collection and method of data analysis, together with issues related to validity and reliability.

60 4.2Researchquestions.

In this section, the overall development will be considered to draw out important aspects for study that will be denoted initalics. According to Tall & Vinner (1981), the concept image is the total cognitive structure of an individual’s mind that is associated to a concept. In this case, the questionnaire begins with an item to describe the concept of sine so as to exploretheevokedconceptimageofstudentteachers,whichislikelytoinvolve triangletrigonometry. Then relevant items in circle trigonometry are set to explore the conceptions of student teachers in this context and to seehow theywillmakesenseofcircletrigonometry in particular with reference to the changes of meaning from the triangle trigonometry to circle trigonometry. For instance, item 3 and item 4 of the questionnaire are about sense making of sin200and sin270. Item 4 explores further by asking the respondents to explain why sin270 _{has certain value,} which is likely to evoke a graphical explanation. Besides, these items also aim to explore the conceptions of student teachers on the given expressions.

The student teachers are then asked to explainthereasonforusing radians insteadofdegrees. This item is important in the sense that radians only start to arise in circle trigonometry, in particular using the unit circle and measuring the angle in terms of the length of the arc, and this idea also links triangle trigonometry and circle trigonometry in a visually meaningful way. Since the generation of the sine curve is from the unit circle, radians can be considered as the starting point of calculus in trigonometry.

61 Next, some of the properties of sine are explored. For instance the questionnaire asks the respondents to state and explain why sinx is decreasing for certain values. This item is set to explore whether the respondents have developed a coherent link between the unit circle and the sinegraph. The questionnaire also asks the respondents to explain why sin



can never equal 2. For this item, the respondent will have a freedom to respond in either one of the three trigonometry contexts. I also expect that some of the respondents might respond in more than one context.

Then the questionnaire asks about the ideas in calculus. For instance the questionnaire asks “What does dy/dx mean?”, “What would d/dx [sinx] mean? What is d/dx [sinx]? Explain why.” Allthese itemsare settoexplore theconceptionsofrespondentsincalculus.

After that the respondents are asked to describesin30, sin120andtan90.

The first may invoke triangle trigonometry, the second may involve circle trigonometryandthethirdinvolvesapossiblesingularcasewheretheangleis nolongpartofaproperEuclideantriangle.

The last item of the questionnaire asks the respondents to explain the relationshipsbetweentheconceptofsineandconceptssuchasfunction,series, complexnumbersandy=mx. This item is set to see whethertherespondents have coherent links between different aspects of mathematics that arise in triangle,circleandanalytictrigonometry.

In general, the questionnaire was designed to cover the full range of development of trigonometry encountered in school including triangle

62 trigonometry in terms of ratio and proportion, circle trigonometry in terms of radians, angles in a circle and graphical representations of trigonometric functions, with more sophisticated topics in the calculus. I was particularly interested in how students who had spent several years studying more formal analytic mathematics may respond to these questions and how this related to the development of trigonometry in school.

Follow‐up interviews were conducted with selected student teachers on a voluntary basis in order to gain further insights on their written responses. For instance during the follow‐up interviews, the interviewees were asked whether they can visualize a triangle with sin200and sin270or not. These questions are asked in order to explore the conceptions of interviewees and to expand the written responses for items in the questionnaire. Meanwhile it is impossible to list all the questions which were asked in the follow‐up interviews because different interviewees will be asked different questions, were based on the verbal responses during the follow‐up interviews. The main aim of the interviews is to gain further insights on the written responses of the interviewees.

Based on the description above, it should be evident that I am interested to research the following things:

1. What is the evoked concept image of sine?

2. How do respondents make sense of trigonometry?

3. Is there any evidence that shows student teachers working in different contexts of trigonometry in making sense of trigonometry?

63 4. What are the supportive and problematic conceptions involved in

making sense of trigonometry and how do these conceptions affect the sense making of trigonometry?

5. What are the student teachers’ conceptions on using degrees and radians in trigonometry?

6. Do the student teachers have a coherent link between the unit circle and the sine graph?

7. What are the conceptions of student teachers in calculus in trigonometry?

8. Do the student teachers grasp the knowledge of trigonometry or they just know it?

9. What is the perceived level of importance for the subject matter knowledge tested by the mathematical items?

10.What is the level of confidence in responding to the mathematical items?

11.What are the difficulties in learning trigonometry as perceived by the student teachers?

The research questions above can be regrouped into two types which are specific and general research questions. Some research questions are specific in the sense that they can be answered by using a particular item in the questionnaire. For instance, the first research question (What is the evoked concept image of sine?) can be answered by analyzing the data collected from item 1 of the questionnaire (Describe sin x in your own

64 words.). Similarly, the fifth research question (What are the student teachers’ conceptions on using degrees and radians in trigonometry?) can be answered by the data collected from item 6 (What do radians mean? Why do we need radians when we have degrees?) of the questionnaire. Likewise for the seventh research question (What are the conceptions of student teachers in calculus of trigonometry?) can be answered by the data collected from item 10 (What does dy/dx mean?) and item 11 (What would d/dx [sin x] mean? What is d/dx [sin x]? Explain why.) Research question no 9 is answered by the data collected from the part B of the questionnaire. As with research question no 10, it can be answered by analyzing the data from part C of the questionnaire. The follow‐up interviews also contribute to the generation of additional data to answer these research questions.

On the other hand, research question no. 2, 3, 4, 6 and 8 as stated above are regarded as general because these questions can only be answered through analyzing data collected from a series of items in the questionnaire. Additionally, the follow‐up interviews are also an important source of data to answer these research questions.

In document Influencia de la mensajería instantánea en las actitudes y nivel de conocimiento de las TICs en los estudiantes del 5° año, de la especialidad de relaciones públicas de la escuela de ciencias de la comunicación unsa, segundo semestre año 2015 (página 153-158)