Although trigonometry is an essential field of mathematics, research in this field is quite limited. Research done in this field shows that students may have difficulties linking different representations in trigonometry due to fragmented understanding of the subject (Kamber & Takaci, 2018; Moore et al. 2014; Moore et al., 2016; Weber, 2005; Weber, 2008).

Research found that students who were taught trigonometry using interactive computer graphics showed better results than those taught trigonometry through conventional teaching methods. Furthermore, research could relate the use of digital learning materials to high mathematics self-efficacy as well as positive attitudes towards learning mathematics (Blackett and Tall, 1991; Merrill , 2010; Naidoo & Govender 2014; Prabowo et al. 2018).

Conventional trigonometry teaching has often been linked to procedural understanding based on memorizing rules and applying algorithms, instead of conceptual understanding based on establishing relationships between different mathematical concepts. Procedural knowledge is defined as being the knowledge of procedures, algorithms and a sequence of steps executed to solve a familiar task correctly without necessarily involving reflection and deep understanding. Conceptual knowledge is the knowledge of abstract concepts and general principles that is rich in relationships and ‘a connected web of knowledge, a network in which the linking relationships are as prominent as the discrete pieces of information’. A conceptual understanding involves the ability to switch between different representations of a concept. Conceptual knowledge can even play the role of validating critic, improving the student’s ability to judge the validness of an answer and to check whether it makes any sense. However, not all knowledge needs to be classified into either conceptual or procedural knowledge, some knowledge is a mixture of both or even neither of them (Baroody et al., 2007; Carpenter, 1986; Hiebert & Lefevre, 1986; Rittle-Johnson & Alibali, 1999; Rittle-Johnson & Schneider, 2015; Rittle-Johnson, 2017).

Both procedural knowledge and conceptual knowledge are connected by dynamic interplay; they complete and improve each other and are both needed to achieve a deep learning and a deep understanding. In order to achieve this co-transition between procedural and conceptual understanding, the present study is based on a model with three pillars developed by Rittle-Johnson (2017): Comparing, Exploring before instruction, Self-explaining

The focus of this paper is to investigate a student’s understanding of the standard trigonometric limit sin(x)/x=1 as x tends to zero. The student is encouraged in a further step to explore and understand the reason behind this limit according to the model developed by Rittle-Johnson (2017). This standard limit is very useful to know and understand in future classes in mathematics, but also in, e.g., engineering, physics, chemistry and architecture. Conceptual errors made by Chemical Engineering students show that they did not grasp the concept of limit sin(x)/x=1 as x tends to zero. Thus the need to study this standard limit further.

This standard limit is first encountered by the Swedish students when they are dealing with the proof of the derivative of the function f(x)=sin(x). At this stage, the students are asked to accept this fact. Sometimes the teacher shows a numerical explanation for the trigonometric limit using a table were the value of x decreases until the student can see the pattern converging towards 1. Regardless of which, the students did not see the reason behind this limit converging towards 1.

This paper investigates if visualization and interactive technology environments can contribute in lifting the student’s understanding from a procedural understanding to a combination of conceptual and procedural understanding. The following research question is addressed:

• How does visualization support a student’s understanding of the standard trigonometric limit sin(x)/x =1 as x tends to zero?

This study is based on a controlled observation, under arranged conditions. The student and the researcher met in a quiet room, where they could talk undisturbed. The room was equipped with a computer as well as recording camera. The researcher explained the purpose of the study and the student was aware that he was being observed and that the conversation was video recorded. The observation lasted approximately 20 minutes. The student, i.e., the object of this study, is a Swedish upper-secondary high school student, 18 years old. The student was selected according to one single criterion: the student should be familiar with trigonometry and with the concept of limits. After an introductory meeting with the researcher, the classroom teacher asked the class for volunteers for this study. The student was picked after showing interest in participating in the study. All the ethical aspects were followed according to the Swedish principles of ethical research. The observation starts with the participating student being shown the detailed proof of how to get the derivative of f(x)=sin(x). The student is then told that one of the reasons for f’(x)=cos(x) is due to the standard trigonometric limit sin(x)/x=1 as x tends to zero. This step is expected to be a repetition of what has already been taught by a teacher conducting a conventional teaching. The observation is then divided into 4 phases: 1: A paper and a pen are handed to the student. The student is asked to interpret the same standard trigonometric limit. 2: The student is handed a paper illustrating the unit circle. The same question as in Phase 1 is asked. The student is expected to illustrate the reason behind the same standard trigonometric limit through graphical representations of the sine function using the unit circle. 3: The student is shown a Geogebra application with the unit circle and a draggable point P on the unit circle periphery. None of the parameters in the application is labelled. The student is invited to interpret what he sees and to associate it with the studied object. 4: The student is shown the same Geogebra application. All the parameters are now labelled, and the application is dynamic in the meaning that all the parameters change values when the student drags the point P. The entire observation is recorded, then transcribed and analyzed using a thematic analysis.

Phases 1 and 2: Inability to connect different representations mixed with frustration Those two phases witnessed clearly that the participant was unable to connect the different representations of sine. The student shows no sign of conceptual knowledge. The lack of conceptual knowledge prevents him from going further in his understanding. This phase is associated with strong feelings of frustration due to the fact that the student is unable to ‘see’ patterns and associate concepts. Phase 3: Validating critic In this case, the student is paying attention to small details and important information using visualization through the interactive technology environment Geogebra. When equipped with conceptual knowledge a student can tell when a procedure is inappropriate, or when it is violating the conceptual principles. The student’s conceptual knowledge is acting as a validating critic and tells the student that his answer does not make any sense, prompting the student to re-evaluate the choice of procedure. In this phase, even though the student is showing conceptual knowledge, he is not yet associating all the factors and the representations forms. Phase 4: Conceptual knowledge mixed with self-efficacy The student is using visualization through the interactive technology environment Geogebra. He is showing conceptual understanding when he managed to link together the two pieces of information: the arc of the circle being the same as the angle, when one is dealing with the unit radians. This is in line with Hiebert & Lefevre’s research (1986) stating that the linking process between two pieces of information stored in the memory or between an existing and one newly learned displays a distinct sign that the student has acquired a conceptual understanding of the problem (Hiebert & Lefevre, 1986, p.4). This phase is also associated with self-efficacy and satisfactory feelings, due to the sense of achievement.

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