Trigonometry, technology, and didactic objects (original) (raw)
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Riding the Mathematical Merry-Go-Round to Foster Conceptual Understanding of Angle
Teaching children mathematics, 2006
tering students’ conceptual understanding of angle that revolve around the Mathematical Merry-Go-Round game. We focus on activities for two reasons. On one hand, NCTM’s Principles and Standards for School Mathematics (2000) stresses the central role of student activity in coming to understand mathematics. This emphasis is consistent with a constructivist stance (Piaget 1971) about learning as an active process. On the other hand, typical activities used for teaching angle, in which an introduction of the definition is followed by operations on angles, such as measuring, adding, comparing, and classifying, seem to leave many students unclear about the “thing” on which they operate. For example, students are often confused about which scale on a protractor they should use when measuring an angle. Another example can be found in students’ solutions to a problem involving two pairs of scissors with blades of different lengths (see fig. 1). When asked, “Which is
Artefacts teach-math. the meaning construction of trigonometric functions
Atti della Accademia Peloritana dei Pericolanti : Classe di Scienze Fisiche, Matematiche e Naturali, 2021
Trigonometry is an area of the high school mathematics curriculum strictly related to algebraic, geometric, and graphical reasoning. In spite of its importance to both high school and advanced mathematics, research has shown that trigonometry remains a difficult topic for both students and teachers. A new approach the teaching of trigonometry – calibrated for the 21 st century - opens the question of the place and nature of trigonometry in contemporary high school mathematics. In this prospective, the authors have carried out a study aimed at explaining connections between research and teaching practice of trigonometric functions emphasizing conceptual understanding, multiple representation and connections, mathematical modelling, and mathematical problem-solving. This paper shows a teaching approach for meaning construction of trigonometric functions with the aid of technological artefacts.
2013
This thesis is concerned with how a group of student teachers make sense of trigonometry. There are three main ideas in this study. This first idea is about the theoretical framework which focusses on the growth of mathematical thinking based on human perception, operation and reason. This framework evolves from the work of Piaget, Bruner, Skemp, Dienes, Van Hiele and others. Although the study focusses on trigonometry, the theory constructed is applicable to a wide range of mathematics topics. The second idea is about three distinct contexts of trigonometry namely triangle trigonometry, circle trigonometry and analytic trigonometry. Triangle trigonometry is based on right angled triangles with positive sides and angles bigger than 0 [degrees] and less than 90 [degrees]. Circle trigonometry involves dynamic angles of any size and sign with trigonometric ratios involving signed numbers and the properties of trigonometric functions represented as graphs. Analytic trigonometry involves...
Enhancing Conceptual Understanding of Trigonometry Using Earth Geometry and the Great Circle
Australian Senior Mathematics Journal, 2011
T rigonometry is an integral part of the draft for the Senior Secondary Australian National Curriculum for Mathematics, as it is a topic in Unit 2 of both Specialist Mathematics and Mathematics Methods, and a reviewing topic in Unit 1, Topic 3: Measurement and Geometry of General Mathematics (ACARA, 2010). However, learning trigonometric ideas is difficult for students and the causes of the difficulties seem to be multifaceted and interrelated. First, trigonometric functions are perhaps students' first encounter with operations that cannot be evaluated algebraically, the kinds of operations about which they have trouble reasoning (Weber, 2005). Those introduced to the subject via triangle trigonometry, which is more comprehensible than circle trigonometry at the early stage of learning (Kendal & Stacey 1997), have (i) to relate triangular pictures to numerical relationships, (ii) to cope with trigonometric ratios, and (iii) to manipulate the symbols involved in such relationships (Blackett & Tall, 1991). To facilitate the memorisation of these ratios, students are often taught the mnemonic SOHCAHTOA with the detrimental effect that they stop trying to make sense of the work because they have a simple rule to follow (Cavanagh, 2008). Teaching triangle trigonometry before circle trigonometry also leads to students' understanding of trigonometric functions as taking right triangles, not angle measures, as their arguments (Thompson, 2008). In fact, students may never develop a coherent concept of angle measure. To a certain extent, the same can be said about teachers (Thompson, Carlson, & Silverman, 2007). As such, both teachers and students have trouble transferring to circle trigonometry (Bressoud, 2010; Thompson, 2008), which is the foundation for more advanced topics in science and mathematics, as evidenced by the fact that they have to rely on yet another mnemonic, "All Students Take Calculus," to determine the signs of trigonometric functions in different quadrants (Brown, 2005). Some of the studies cited above also suggested remedies for the problems of their concern. Blackett & Tall (1991) employed a computer program that draws the desired right triangles to facilitate students' exploration of the relationship between numerical and geometric data. Quinlan (2004) and 54
Teaching and learning trigonometry with technology
Modern school classrooms have access to a range of potential technologies, ranging from calculators to computers to the Internet. This paper explores some of the potential for such technologies to affect the curriculum and teaching of trigonometry in the secondary school. We identify some of the ways in which the teaching of trigonometry might be supported by the availability of various forms of technology. We consider circular measures, graphs of functions, trigonometric identities, equations and statistical modeling and focus on activities that are not possible without the use of technology. Modern technology provides an excellent means of exploring many of the concepts associated with trigonometry, both trigonometric and circular functions. Many of these opportunities for learning were not available before technology development and access within schools we enjoy today. This paper suggests some of the avenues for exploration.
Find the Measure of Angle X: Teacher's Technology Investigation
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This paper discusses strategies that two high school geometry teachers employed to solve a challenging geometry problem with the aid of dynamic geometry software (The Geometer's Sketchpad (GSP). The teachers were part of a group of teachers who participated in a summer professional institute funded by the National Science Foundation. The paper demonstrates the use of technology to construct, explore, and prove, incorporating the measurement tool in GSP. Although the two teachers showed similar approaches, they displayed interesting proof strategies. Introduction In recent years, there have been examples of exploration of open geometry problems in Dynamic Geometry Environments (DGEs). Research indicates “that a DGE impacts students’ approach to investigating open problems in Euclidean Geometry, contributing particularly to students’ reasoning during the conjecturing phase of open problem activities” (Baccaglini-Frank, Antonini, Leung, & Mariotti, 2017, p. 103). Dynamic geometry t...
An Examination of Pre-service Secondary Mathematics Teachers’ Conceptions of Angles
The concept of angles is one of the foundational concepts to develop of geometric knowledge, but it remains a difficult concept for students and teachers to grasp. Exiting studies claimed that students’ difficulties in learning of the concept of angles are based on learning of the multiple definitions of an angle, describing angles measuring the size of angles, and conceiving different types of angles such as 0-line angles, 1- line angles, and 2-line angles. This study was designed to gain better insight into pre-service secondary mathematics teachers’ (PSMTs) mental constructions of the concept of angles from the perspective of Action-Process-Object-Schema (APOS) learning theory. The study also explains what kind of mental constructions of angles is needed in the right triangle context. The four PSMTs were chosen from two courses at a large public university in the Midwest United States. Using Clements’ (2000) clinical interview methodology, this study utilized three explanatory interviews to gather evidence of PSMTs’ mental constructions of angles and angle measurement. All of the interview data was analyzed using the APOS framework. Consistent with the existing studies, it was found that all PSMTs had a schema for 2-line angles and angle measurement. PSMTs were also less flexible on constructions of 1-line and 0-line angles and angle measurement as it applied to these angles. Additionally, it was also found that although PSMTs do not have a full schema regarding 0-line and 1-line angles and angle measurement, their mental constructions of 1-line and 0-line angles and angle measurement were not required in right triangles, and the schema level for 2-line angles was sufficient for constructions of right triangle context.
The High School Students’ View of Trigonometry Problems to Develop Their Mathematics Reasoning
2017
Many people do not like learning Mathematics because it requires highly analytical skills to solve for only one problem. It also same in mastering the field of algebra, it’s already difficult, moreover to learn trigonometry which involves many calculations of formulas in the form of corners in the triangle. It’s really not easy because it allows the combination between algebra and geometry. Though, the universe of the world is full of models of trigonometry measurements, then by studying trigonometry means that we have applied much different knowledge. The student difficulties in understanding trigonometry material are seen when students are faced with trigonometry problems that require reasoning based from real life. Students have lack of opportunities to solve the problems with this type of reasoning is a factor that limits the opportunities and positive perceptions of students in learning mathematics. Based on research, by the development of trigonometry problems that are relevan...
Students understanding of trigonometric functions
In this article students' understanding of trigonometric functions in the context of two college trigonometry courses is investigated. The first course was taught by a professor unaffiliated with the study in a lecture-based course, while the second was taught using an experimental instruction paradigm based on Gray and Tall's (1994) notion of procept and current process-object theories of learning. Via interviews and a paper-and-pencil test, I examined students' understanding of trigonometric functions for both classes. The results indicate that the students who were taught in the lecture-based course developed a very limited understanding of these functions. Students who received the experimental instruction developed a deep understanding of trigonometric functions. * Responses to question 1a (approximate sin 340˚) were judged to be "correct" if they were between 0 and -0.5.
ZDM, 2010
We describe the process followed to design representations of mathematics teaching in a community college. The end product sought are animated videos to be used in investigating the practical rationality that community college instructors use to justify norms of the didactical contract or possible departures from those norms. We have chosen to work within the trigonometry course, in the context of an instructional situation, "finding the values of trigonometric functions," and specifically on a case of this situation that occurs as instructors and students are working on examples on the board. We describe the design of the material needed to produce the animations: (1) identifying an instructional situation, (2) identifying norms of the contract that are key in that situation, (3) selecting or creating a scenario that illustrates those norms, (4) proposing alternative scenarios that instantiate breaches of those norms, and (5) anticipating justifications or rebuttals for the breaches that could be found in instructors' reactions. We illustrate the interplay of contextual and theoretical elements as we make decisions and state hypothesis about the situation that will be prototyped.