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Historical Reflections on Teaching Trigonometry
... Bir çemberin çevresini yani çember yayının tamamını gören merkez açının derece ölçüsü 360°dir. Bu durumda bir derece, uzunluğu çemberin çevresinin 1 360 'ı olan bir çember yayı olacaktır (Abramson, 2018;Bressoud, 2010;Thompson, 2008). O yüzden 1° çemberin çevresinin 1 360 'ına eşit olan çember yayını gören merkez açının ölçüsü olarak tanımlanır (Moyer ve Ayers, 2018). ...
... Radyan kelimesi radyal açı için kısaltma olarak kullanılır (Bressoud, 2010). Radyan, merkez açının gördüğü yayın uzunluğunun çemberin yarıçap uzunluğuna oranıdır. ...
... Bir çemberin çevresini yani çember yayının tamamını gören merkez açının radyan ölçüsü diğer deyişle tam dairesel dönüş 2 'dır (Moyer ve Ayers, 2018). 1 herhangi bir çemberde yarıçap uzunluğunda yayı gören açıdır ve çember çevresinin 1 2 'lik kısmına eşit olan çember yayını gören merkez açının ölçüsü olarak tanımlanır (Moore ve LaForest, 2014). Kıyaslama yapmak gerekirse öğrenciler için 1 2 'ye göre 1 360 'ı kavramlaştırmak daha kolay görünebilir (Bressoud, 2010). Ancak derece ve radyan bu şekilde yay uzunluğuna bağlı ifade edilirlerse tam olarak aynı tür şeyler olurlar (Thompson, 2008). ...
Bu çalışmanın amacı Türkiye’deki matematik ders kitaplarında yer alan radyan tanım ve şekil temsillerinin incelenmesidir. Bu nitel araştırmanın verileri 2019-2020 eğitim-öğretim yılında 11. sınıf ortaöğretim matematik derslerinde okutulması kararlaştırılan beş ders kitabından toplanmıştır. Bu belgelerden elde edilen veriler Alyami’nin (2020) radyan açı ölçüsünün tanım ve şekilleri için nitel ve nicel açı görüşü ayrımına dayanan analiz çerçevesi kullanılarak betimleme çözümlemesine tabi tutulmuştur. Araştırmanın sonuçlarına göre ders kitaplarındaki radyan tanımları ve tanımı açıklayan radyan şekil temsilleri bir radyana odaklanmaktadır. Bu tanım ve şekillerde yay uzunluğu ile yarıçapın eşitlik dışındaki orantısal ilişkiye yeterince vurgu yapılmamıştır. Araştırmanın sonuçları doğrultusunda ders kitaplarındaki radyan tanımı ve radyan temsili şekillerde yay uzunluğu ile yarıçap uzunluğunun eşitlik dışındaki nicel ilişkisinin ön plana çıkarılması önerilebilir.
... K. Moore discuss that in literature teachers' and students' patchy understandings of angle measure lead to lack of understandings of trigonometric functions (Moore 2013). D. Bressoud reflects on the history of trigonometry and remarks that history has much to teach us about the mathematical ideas (Bressoud 2010). He concludes his paper with Henri Poincaré's quote "The task of the educator is to make the child's spirit pass again where its forefathers have gone, moving rapidly through certain stages but suppressing none of them. ...
... He concludes his paper with Henri Poincaré's quote "The task of the educator is to make the child's spirit pass again where its forefathers have gone, moving rapidly through certain stages but suppressing none of them. In this regard, the history of science must be our guide'' (Bressoud 2010). L. Laren deliberate in (van Laren 2012) how teachers of trigonometry point to obstacles in using a variety of representations. ...
Trigonometry is always a challenging subject to students. It requires more that the algebra background to understand it. In this paper, we discuss our perspective and experience in which we revisited the teaching approach after making several observations. The course is a mixed mode, delivered in a large lecture hall, and the class size is about 300 to 400 students in each section. Lab hours are required and only a single line scientific calculator is allowed. The results show a significant decrease in failing and withdrawal rates after the revamp. 1. Course Structure The Mathematics Department at the University of Central Florida (UCF) has implemented a Modified Emporium Model (MEM) for college algebra, intermediate algebra, precalculus algebra and college trigonometry. They are considered mixed mode courses. Students in these courses are required to attend one fifty-minute lecture per week and to spend at least three hours per week working in a computer lab called the Mathematics Assistance and Learning Lab (MALL). Because of the short lecture, most of students' engagement happen before class, after class, and in the MALL. The lecture hall format allows for some engagement yet it is limited. Instructors rely on technology to help with progress monitoring and weekly announcements. Tutors, graduate students and instructors staff the MALL. An online system is used for assignments like homework, quizzes and proctored tests. Students were allowed multiple attempts on the homework and quizzes prior to due date, and one attempt on the proctored tests. In both classroom and MALL, students' attendance is tracked. Their participation score in class depends on their responses to the asked questions. Their MALL attendance depends on completing at least three hours per week (except on testing week in which they are lifted), with the rule of 100% for at least three hours and 0% for anything less. In the MALL, students are expected to work only
... Weber, Knott, and Evitts (2008) used geometric approach to trigonometry and found that this can lead students to understand trigonometric operations as functions while traditional instruction does not. Bressoud (2010) presented the historical reflection on teaching geometry suggesting it to use history in teaching to show to the students that mathematics comes from real people who struggled with the same concepts. Franz and Pope (2005) suggested the use of children's stories to deepen the students' understanding of mathematics. ...
... Although students learn a variety of techniques for solving triangles, manipulating trigonometric expressions, and graphing functions, they miss many opportunities for reasoning and sense making. A number of authors have called for more meaningful connections in the teaching and learning of trigonometry (Bressoud, 2010;Thompson, 2008;Weber, 2005).Another study was designed to investigate the extent to which using technology for visualization (Arcavi, 2003) would affect students' understanding of the different trigonometric concepts. A mixed method approach (Johnson & Onwuegbuzie, 2004) consisting of a modified experimental design and a qualitative component was used with two high school precalculus classes. ...
Teacher-education institutions are established to provide quality and holistic pre-service education to prospective teachers. This study was conducted to specifically look into the Pre-service-Teacher Education (PTE) in terms of personal preparation, professional preparation and classroom management and Practice-Teaching Performances (PTP) among the fourth year Bachelor in Elementary Education (BEEd) students of Mindanao State University-Maigo School of Arts and Trades (MSU-MSAT). This study measured the level and relationship of PTE of the students and their PTP. It employed descriptive method which involved all the 30 fourth year BEEd students. The data were derived through a questionnaire; and chi-square was used for the statistical computation. Results revealed that the fourth year BEEd students had an outstanding performance in personal - professional preparations, classroom management and had good performance in practice teaching, but average in English and Filipino languages. Since the computed chi-square values were more than the critical values with one degree of freedom, the null hypotheses were rejected implying that PTE and PTP had significant relationship. In conclusion, PTE teachers have an appreciable performance in terms of personal -professional preparation, classroom management and average in practice teaching. PTE influences the performance in practice-teaching. As recommendations, pre-service education teachers should attend seminar workshop on language and personality development to be cautious in the medium of communication and on their mannerisms for practice teaching. The department should conduct PTE seminar to make them more prepared for teaching. Future researchers should conduct related study on pre-service- teacher education efficacy.
... The unit circle is central to the study of trigonometric functions, with many historical developments in and applications of trigonometry occurring in circle settings (Bressoud, 2010). Yet, researchers (e.g., Akkoc, 2008 Moore, 2013 Moore, , 2014 Thompson, 2008; Weber, 2005) have argued that students' and teachers' difficulties in trigonometry partially stem from impoverished connections between trigonometric functions and the unit circle. ...
... These ways of thinking are inaccessible to students (or teachers) when they understand trigonometry as being about triangles, and understand sine, cosine, and tangent in terms of SOH-CAH-TOA (Sine is opposite over hypotenuse, etc.), a mnemonic that is common in the United States. When students learn sine, cosine, and tangent as SOH-CAH-TOA, then sine, cosine, and tangent are not functions to students, or if students do think of sine, and so forth, as something like a function, they are functions that have triangles as their arguments (Bressoud, 2010; Thompson, 2008a). The key difference between triangle trigonometry and trigonometric functions in trigonometry is that trigonometric functions have angle measures as their arguments, and thus to understand trigonometric functions productively, students must have an appropriate conception of angle measure. ...
... The unit circle is central to the study of trigonometric functions, with many historical developments in and applications of trigonometry occurring in circle settings (Bressoud, 2010). Yet, researchers (e.g., Akkoc, 2008; Moore, 2013 Moore, , 2014 Thompson, 2008; Weber, 2005) have argued that students' and teachers' difficulties in trigonometry partially stem from impoverished connections between trigonometric functions and the unit circle. ...
We discuss a teaching experiment that explored two pre-service secondary teachers’ meanings for the unit circle. Our analyses suggest that the participants’ initial unit circle meanings predominantly consisted of calculational strategies for relating a given circle to what they called “the unit circle.” These strategies did not entail conceiving a circle’s radius as a unit of measure. In response, we implemented tasks designed to focus the participants’ attention on various measurement ideas including conceiving a circle’s radius as a unit magnitude. Against the backdrop of the participants’ actions on these tasks, we characterize shifts in the participants’ unit circle meanings and we briefly describe how these shifts influenced their ability to use the unit circle in trigonometric situations.
... This results in a dichotomy in their understanding of trigonometric functions. As Bressoud (2010) and Thompson (2008) noted, traditional approaches to trigonometry and angle measure contribute to students' difficulties with the subject. Students' first experiences with angles often involve using a prelabeled protractor to determine their measure, classify them as acute or obtuse, and relate them as supplementary or complementary. ...
A connected introduction of angle measure and the sine function entails quantitative reasoning.
... Algunos autores nos refieren que la enseñanza de la trigonometría se aborda en un inicio desde el triángulo rectángulo y eso lo vemos en los libros de texto escolares, después se comienza con el círculo unitario para las funciones trigonométricas y eso genera confusiones en los alumnos (Grabovskij y Kotel'Nikov, 1971; De Kee, Mura y Dionne, 1996; Montiel y Buendía, en prensa). En contraste con esta organización Bressoud (2010) muestra, en una breve revisión histórica, cómo la Trigonometría nace del estudio del triángulo en el círculo y son los problemas de ―estudio de sombras‖ los que detonan el uso de la razón trigonométrica. Hace énfasis en que si los estudiantes transitan por el estudio de la trigonometría desde el círculo para determinar longitudes de cuerdas y arcos con problemas en contexto astronómico, para darle un sentido a las funciones trigonométricas, sería más fácil para ellos transitar del círculo al triángulo que ...
Presentamos los antecedentes y las consideraciones teóricas que fundamentan el diseño didáctico con el que nos proponemos analizar cómo estudiantes del nivel medio superior construyen las razones trigonométricas en un contexto geométrico. Comenzamos por plantear una problemática con base en una experiencia didáctica previa, para ubicar nuestra propuesta de investigación entre los resultados y las aportaciones de algunas investigaciones relacionadas con la enseñanza-aprendizaje de la Trigonometría; de las cuales, además, se han tomado tanto actividades como elementos de organización didáctica para nuestro diseño. Esbozamos las consideraciones teóricas y didácticas que fundamentarán el avance del diseño didáctico, que a su vez servirá como instrumento para la obtención de datos de nuestra investigación.
Presentamos el avance de una investigación que busca estudiar la transición de la trigonometría en un contexto estático-geométrico (cuerdas y razones trigonométricas) a la trigonometría en un contexto dinámico-variacional (función trigonométrica), en el nivel medio superior. El avance se centra en un análisis documental de fuentes históricas relativas a la Geometría y a la Trigonometría, con la finalidad de establecer una base de conocimientos necesarios y problemáticas que contextualicen su construcción y resignificación. La finalidad de esta problematización es devolver los procesos de construcción geométrica que le dan sentido y razón de ser al aprendizaje de la Trigonometría.
Palabras clave: geometría, trigonometría, génesis histórica, análisis documental
Using expository text and examples available in 10 college textbooks we identify two conceptions of angles, trigonometric functions, and inverse trigonometric functions that rely on either a static or a dynamic definition of angle. Although the textbooks favor a conception of trigonometric functions that is based on a dynamic conception of angle, they split in their definition of inverse trigonometric functions. We argue that transparency in making explicit how these conceptions can be bridged might be useful in understanding difficulties that emerge when solving problems with inverse trigonometric functions.
This is a survey on the recent developments (since 2000) concerning research on the relations between History and Pedagogy of Mathematics (the HPM domain). Section 1 explains the rationale of the study and formulates the key issues. Section 2 gives a brief historical account of the development of the HPM domain with focus on the main activities in its context and their outcomes. Section 3 provides a sufficiently comprehensive bibliographical survey of the work done in this area since 2000. Finally, section 4 summarizes the main points of this study.
Processes of knowledge construction are investigated. A learner is constructing knowledge about the trigonometric functions and their geometric meaning on the unit circle. The analysis is based on the dynamically nested epistemic action model for abstraction in context. Different tasks are offered to the learner. In his effort to perform the different tasks, he has the opportunity to understand the process used to create unit circle representations of trigonometric expressions. The theoretical framework of abstraction in context is used to analyse the evolution of the learner's construction of knowledge in the transition from ‘triangle’ trigonometry to ‘circle’ trigonometry.
I first provide a bit of historical background on a theory of students’ development of algebraic reasoning through quantitative reasoning. The quantitative reasoning part of the theory gained some popularity, but its most important features (at least in my thinking), the parts explicitly related to algebraic reasoning, received little notice. I then point to important work that extended the quantitative reasoning part of this theory in important ways (e.g., Lobato, Reed, Ellis, Norton, Castillo-Garsow, and Moore) and discuss how the “little noticed” aspects could inform that work in useful ways. I also discuss how a focus in school mathematics on quantitative reasoning and its extension into algebra could be leveraged with regard to students’ engagement in authentic mathematical modeling. Finally, I extend the theory of quantitative reasoning, with a focus on quantitative covariation, to include reasoning with magnitudes and discuss examples of how rich, coherent understandings of magnitudes can be foundational for advanced reasoning in analytic geometry, calculus, differential equations, and analysis.
While the unit circle is a central concept of trigonometry, students' and teachers' understandings of trigonometric functions typically lack connections to the unit circle. In the present work, we discuss a teaching experiment involving two pre-service secondary teachers that sought to characterize and produce shifts in their unit circle notions. Initially, both students experienced difficulty when given a circle that did not have a stated radius of one. The students relied on memorized procedures, including "unit-cancellation," to relate the unit circle to given circles. In an attempt to foster more robust connections between novel circle contexts and the unit circle, we implemented tasks designed to foster thinking about a circle's radius as a unit of measure. We report on the students' progress during these tasks.
I discuss a teaching experiment that sought to characterize precalculus students’ angle measure understandings. The study’s findings indicate that the students initially conceived angle measures in terms of geometric objects. As the study progressed, the students formed more robust understandings of degree and radian measures by constructing an arc length image of angle measures; the students’ quantification of angle measure entailed measuring arcs and conceiving multiplicative relationships between a subtended arc, a circle’s circumference, and a circle’s radius. The students leveraged these quantitative relationships to transition between units with a fixed magnitude (e.g., an arc length’s measure in feet) and various angle measure units, while maintaining invariant meanings for angle measures in different units. These results suggest that quantifying angle measure, regardless of unit, through processes that involve measuring arc lengths can support coherent angle measure understandings.
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.
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