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Pre-service mathematics teachers’ concept images of radian
International Journal of Mathematical Education In Science & Technology
Abstract
This study investigates pre-service mathematics teachers’ concept images of radian and possible sources of such images. A multiple-case study was conducted for this study. Forty-two pre-service mathematics teachers completed a questionnaire, which aims to assess their understanding of radian. Six of them were selected for individual interviews on the basis of theoretical sampling. The data indicated that participants’ concept images of radian were dominated by their concept images of degree. As the data in this study suggested, pre-service mathematics teachers were reluctant to accept trigonometric functions with the inputs of real numbers but rather they use value in degrees. More interestingly, they have two distinct images of π : π as an angle in radian and π as an irrational number.
... Since radian angle measure is a central concept in secondary and post-secondary mathematics, exploring the impact of textbooks' treatment is important. I build on research that explored conceptions of radian using Tall and Vinner's (1981) concept definition/concept image construct (Akkoc, 2008;Fi, 2003;Topcu, Kertil, Akkoc, Yilmaz, & Onder, 2006). However, I focus on concept definitions to provide insights from a curricular perspective on the contribution of different textbook representations on prospective mathematics teachers' (PMTs') conceptions of radian angle measure. ...
... Despite its age, many researchers (e.g. Akkoc, 2008;Fi, 2003;Topcu et al., 2006) have found the Concept Image/Concept Definition framework an appropriate tool for exploring PMTs' conceptions of radian angle measure. Tall and Vinner (1981) define a concept definition as the "form of words used to specify that concept," where a formal concept definition is "accepted by the mathematical community at large" (p. ...
... Since learners may consult their concept image to formulate a concept definition (Vinner, 1991), researchers (Akkoc, 2008;Fi, 2003;Topcu et al., 2006) attended to PMTs' concept images of radian. However, considering the fundamental role mathematics definitions play for learners' experiences of mathematical concepts (Edwards & Ward, 2008;Zandieh & Rasmussen, 2010), it is essential to explore how different sources inform concept definitions. ...
Despite the use of axiomatic definitions for mathematical concepts, scholars struggle to produce a single definition of angle that does not have some limitations. Textbooks’ treatment of angle and its measure reflects this multifaceted nature of the angle concept while informing the different ways learners conceive of these concepts. In this study, I investigated the concept definitions provoked by textbook representations (e.g., diagrams) of radian angle measure. Seven preservice mathematics teachers (PMTs) participated in a think-aloud semi-structured interview and defined radian angle measure through examining six diagrams. A thematic analysis of concept definitions indicated that interacting with different representations provoked a variety of concept definitions. Some provoked concept definitions focused on defining one radian. Other definitions indicated different quantitative operations (e.g., iteration, partitioning, proportionality). To provide learners opportunities to conceptualize radian angle measure quantitatively, teachers and curriculum developers are encouraged to incorporate multiple representations of radian angle measure into instructional materials.
... However, learners typically struggle to reason about trigonometry and radian angle measure (Moore 2014; Tallman and Frank 2020; Thompson 2013). Many students understand radians instrumentally and can successfully perform radian-related tasks (Fi 2003) but struggle to connect radians to angle measure (Moore 2013) and real numbers (Akkoc 2008;Çekmez 2020). Conceptions of radian angle measure are dominated by measurement in terms of π (e.g., Akkoc 2008;Fi 2003), where π is often perceived as the unit of measure (Fi 2003). ...
... Many students understand radians instrumentally and can successfully perform radian-related tasks (Fi 2003) but struggle to connect radians to angle measure (Moore 2013) and real numbers (Akkoc 2008;Çekmez 2020). Conceptions of radian angle measure are dominated by measurement in terms of π (e.g., Akkoc 2008;Fi 2003), where π is often perceived as the unit of measure (Fi 2003). Using the Radian Lasers activity challenges these conceptions by allowing students to input any real number, thus supporting their understanding of radian angle measure. ...
... Although research indicates that students might conceptualize radian angle measure in terms of π (e.g., Akkoc 2008;Cekmez 2020;Fi 2003), they might also depend on educated guesses (see figure 3). To build on prior knowledge and to discourage random guessing, I asked students to share some values of angles measured in radians. ...
During a Desmos activity, students adjust the measures of angles in radians to reposition a laser and a mirror so the beam passes through three stationary targets. The Radian Lasers activity can be extended to simulate project-based learning.
... Some scholars have posited that the concept of functions is fundamentally significant in mathematics and that a strong understanding of functions contributes greatly to the further study of mathematics and related subjects (Even, 1998;Watson & Harel, 2013). Functions are relevant in topics such as trigonometry, and central to the study of pre-calculus, calculus, and subjects such as Physics (Akkoc, 2008), and are considered to be foundational in secondary school mathematics. However, research has shown that students struggle to understand the concept of functions (Bloch, 2003;Even, 1993;Spyrou & Zagorianakos, 2010). ...
... Furthermore, trigonometry has applications in life areas such as navigation and in topics such as differentiation and integration, and affords the transition from algebra to geometry (Abdulkadir, 2013;Fi, 2003). In spite of its purported significance, scholars contend that trigonometry has not received much attention in mathematics education research (Akkoc, 2008;Fi, 2003Fi, , 2006. Moreover, some of the most recent research articles on the topic only deal with components of trigonometry such as degrees and radian measures, and trigonometric functions (Abdulkadir, 2013;Hiebert, 2013;Moore, 2012Moore, , 2013Moore, LaForest, & Kim, 2012;Ogbonnaya & Mogari, 2014). ...
... Bromme"s (1994) argument suggests that knowledge of advanced university mathematics is not a guarantee that a teacher will have in-depth understanding of the content and pedagogy of secondary school mathematics. The assumption that a teacher who has studied high level mathematics automatically understands secondary school mathematics and can teach it effectively does not always hold (Akkoc, 2008;Ball, 1990;Even, 1990Even, , 1993Fi, 2003Fi, , 2006. ...
... One of the school mathematics topics which are receiving attention in mathematics education research is trigonometry (Abdulkadir, 2013;Akkoc, 2008;Malambo et al., 2018;Ogbonnaya & Mogari, 2014). This topic presents advantages to learners such as improving their ability to reason and represent information in different forms (Abdulkadir, 2013). ...
... Besides, trigonometry is useful to the study of calculus and navigation. Among what has been researched on under trigonometry are units of angle measures, and general trigonometric concepts (Abdulkadir, 2013;Akkoc, 2008;Malambo, 2016;Malambo et al., 2018;Ogbonnaya & Mogari, 2014). Trigonometric concepts have been researched through case study designs that allow for relating of results to other contexts without room for generalization. ...
... Trigonometric concepts have been researched through case study designs that allow for relating of results to other contexts without room for generalization. A dominant finding of those studies though is that student teachers exhibit conceptual difficulties in understanding trigonometric concepts (Abdulkadir, 2013;Akkoc, 2008;Malambo et al., 2018). ...
Mathematics teachers’ ability to translate and make connections between representations of functions requires investigation. Consequently, this qualitative case study article focuses on pre-service mathematics teachers’ nature of understanding of the tangent function; a function bearing unique characteristics compared with the sine and cosine functions. Twenty-two finalist pre-service teachers were conveniently selected and assessed concerning the ability to translate a tangent function to the graphical representation. Likewise, participants’ abilities to correctly explain this function and provide appropriate justifications for espoused perspectives were investigated. Although the teachers learned higher mathematics, their assessment was school mathematics-based. Descriptive analyses showed that only one teacher (5%) accurately completed the task. Eleven teachers (50%) did not provide graphs; suggesting a lack of knowledge required to change representation. Ten (45%) drew flawed graphs depicting a lack of understanding of discontinuity of the tangent function at certain angles and the role of a domain. Essentially, these demonstrated mere memorization of the appearance of the tangent graph. A purposive sub-sample of size six then participated in exploratory semi-structured interviews. The interviews allowed participants to elucidate their answers to the initial task. Content analysis of the transcripts corroborated the earlier finding as the interviewees could not coherently explain the tangent function, and failed to justify their reasoning. The teachers demonstrated a superficial understanding despite having studied advanced mathematics. This reinforces the view that studying advanced mathematics does not assure a relational understanding of school mathematics. Therefore, it is necessary for trainee teachers to explore school mathematics.
... Studies of teachers' conception of radian angle measure have also illustrated an influence of the qualitative view of angle. Akkoc (2008), Fi (2003), and Topcu et al. (2006) have indicated that mathematics teachers' conceptions of radian angle measure were dominated by degrees. Fi's (2003) study showcased teachers' instrumental understanding of radian angle measure. ...
... Some of Fi's participants believed that π is the unit of radian measure. Akkoc (2008) reported similar results, as the participants in her study believed that radian angle measures are always expressed in terms of π. This also indicates a qualitative view of angle transferring to angle measure through viewing π as a unit of measure rather than being a real number that represents the angle measure. ...
... Many of the previously mentioned studies utilized Tall and Vinner's (1981) Concept Definition and Concept Image construct (Akkoc, 2008;Fi, 2003;Keiser, 2004;Topcu et al., 2006). Through Vinner and Hershkowitz's (1980) curricular approach to concept image, I review in the following section literature on how textbook representations influence the development of angle conceptions. ...
Empirical studies of angle, angle measure, and radian angle measure have primarily focused on documenting students' and teachers' conceptions. Representations of angle measure have the potential to inform and build on these conceptions. To gain insight into the concept of radian angle measure from a curricular perspective, I investigated the representations of radian angle measure in eight widely used U.S. high school mathematics textbooks. The data for this study are the 20 figures and the 8 definitions of radian angle measure in the textbooks. Inductive content analysis was used to identify the frequency of observable characteristics in the representations. Textbook figures of radian angle measure most frequently focus on a one‐radian angle. Textbook figures also demonstrated radian angle measures in terms of π, in standard position, even when the Cartesian‐coordinate plane was not in the figure. The frequently used characteristics of radian representations could inadvertently reinforce a qualitative concept of angle, instead of fostering the quantitative attributes of radian angle measure. Moving forward, analyzing conceptions that are evoked by the frequently used representations of radian angle measure could build a link between evidence from this study and studies of learning.
... Although there are studies in the relevant literature which explore the concept of radian in particular, they are quite limited and are usually conducted either with students or preservice teachers. Studies concerning mathematics teachers' ways of understanding of the concept of radian, on the other hand, are very few (Akkoç, 2008;Doerr, 1996;Fi, 2003). However, the concept of radian is one of the important concepts that constitute the basis of trigonometry (Erdem and Man, 2018) and the studies on the concept are of great importance. ...
... Almost 90% of pre-service teachers used phrases such as "The expression of degree in terms of π", "The unit of length of degree", "I just know the formula of D/180 = R/π", "I do not know what radian is" when defining radian. Akkoç (2008) obtained parallel results from her study conducted on 42 pre-service teachers with the aim of revealing pre-service teachers' concept images of radian and the possible origins of these images. In the end, she concluded that pre-service teachers had more concept images of degree than concept images of radian and therefore, they had difficulty making sense of trigonometric functions of real numbers. ...
... These elements can be expressed as; the radian cannot be defined as the ratio of two lengths and accordingly cannot be predicted to correspond to a real number, the radian cannot be related to unit circle and trigonometric functions, and π cannot be interpreted related with the concept of radian. It appears that the elements mentioned here are also presented as basic points for the correct understanding of the radian concept in different studies (Akbaş, 2008;Akkoç, 2008;Akkoç ve Gül, 2010;Fi, 2003;Maor, 1998;Moore, 2012;Tuna, 2013;Topçu, Kertil, Yılmaz, Akkoç ve Önder, 2006;Thompson, Carlson & Silverman, 2007). These elements were used for the construction of subproblems of this study. ...
... La importancia de la comprensión de las relaciones entre los elementos geométricos y analíticos con las medidas angulares radica en que la fragmentación entre ellos produce dificultades en la comprensión de las funciones trigonométricas (Akkoc, 2008;Maldonado, 2005;Moore, 2013Moore, , 2009, pero a pesar de que se pueden introducir y emplear las funciones trigonométricas desde distintos contextos estas funciones poseen bases comunes como es la medida del ángulo (Moore, 2009) y se identifica como un punto neural el entendimiento de las mediciones angulares para construir y entender la noción funcional de las relaciones trigonométricas, y al no hacerse explícita al estudiante le es indistinto el tratamiento como razón o como función (Maldonado, 2005). ...
... Por ejemplo, Akkoc (2008) reporta que al poseer una idea fuerte del radián se logra establecer conexiones entre el círculo unitario y las funciones trigonométricas y tener una empobrecida comprensión de esta medida angular contribuye a que surjan dificultades en el entendimiento de dichas funciones. Siendo necesario retomar el contexto circular de forma más amplia para el desarrollo del pensamiento trigonométrico (Montiel, 2013(Montiel, , 2005Moore, 2013Moore, , 2009Torres, 2014;Weber, 2005) que involucre una reflexión en torno al trabajo geométrico (desde las relaciones entre los arcos, ángulos y cuerdas) para culminar con el trabajo analítico. ...
En diversas investigaciones se reconoce que el tratamiento del ángulo es una actividad fundamental en el desarrollo del pensamiento trigonométrico, pero que resulta complejo por su propia naturaleza y por su tratamiento indiferenciado dado en la trigonometría. Por lo cual, el objetivo de este documento es mostrar los usos y las siete funcionalidades del ángulo identificados durante la resolución de un diseño didáctico centrado en el desarrollo del pensamiento trigonométrico que se ejecutó como parte de la investigación en el 2020. Para tal fin se examinan, desde un enfoque cualitativo y desde el marco teórico de la Socioepistemología de la Matemática Educativa (TSME), las soluciones dadas por 12 educandos de secundaria de la Ciudad de México al responder al diseño didáctico elaborado por Scholz en el 2020. El análisis de la información se establece en dos fases: en la fase 1 se realiza una triangulación de las evidencias (provenientes del diario de campo, grabaciones y notas del estudiantado) y se identifican las acciones y actividades (desde la TSME); en la fase 2 se identifican los usos del ángulo según su naturaleza polifacética, posteriormente se determinan las funcionalidades de acuerdo con la intencionalidad del conjunto de usos, acciones y actividades. Entre los principales hallazgos de la investigación del 2020 se encuentran la identificación de siete funcionalidades del ángulo, “como referente para: aplicar una herramienta aritmética-algebraica, aplicar una herramienta aritmética-trigonométrica, estudiar las relaciones dadas en el modelo, operar aritméticamente, emplear una herramienta empírica-métrica, clasificar triángulos” (Sánchez, 2020, p. iv), y como herramienta de construcción. Si bien estos usos y funcionalidades del ángulo son empleados por el estudiantado de manera indirecta y, principalmente, en contextos geométricos, se reconocen como necesarios al ser utilizados como referentes para el tratamiento de otras nociones. Por último, se recomienda para el desarrollo del pensamiento trigonométrico dar especial consideración a la conceptualización y al tratamiento del ángulo, de manera que sea más explícito y visible su empleado, al ser referente necesario para el tratamiento de otras herramientas, operaciones y constructos matemáticos.
... Er stellt in Fallstudien mit Lehramtsstudierenden fest, dass bestimmte Winkelmaße fest mit konkreten geometrischen Objekten verknüpft und von den Studierenden nicht als Ergebnisse eines Messprozesses verstanden werden. Der Umgang mit dem Bogenmaß ist Untersuchungsgegenstand mehrerer Studien (Thompson et al. 2007;Akkoc 2008;Cetin 2015;Katter 2019). Akkoc (2008) stellt fest, dass Lehramtsstudierende überwiegend das Gradmaß dem Bogenmaß vorziehen. ...
... Der Umgang mit dem Bogenmaß ist Untersuchungsgegenstand mehrerer Studien (Thompson et al. 2007;Akkoc 2008;Cetin 2015;Katter 2019). Akkoc (2008) stellt fest, dass Lehramtsstudierende überwiegend das Gradmaß dem Bogenmaß vorziehen. Viele Studierende hatten generell Schwierigkeiten damit, reelle Zahlen als Definitionsbereich für trigonometrische Funktionen zuzulassen und akzeptierten nur Werte im Gradmaß. ...
... Otra investigación sobre la medida angular es la efectuada por Akkoc (2008), donde analiza las imágenes del concepto de la medición angular (en particular la del radián) que poseen los maestros de matemáticas en formación. Como parte de la información obtenida, se resalta que: a. Los profesores saben convertir grados y radianes, pero no logran dar una explicación de las relaciones de las longitudes involucradas, b. ...
... Por lo tanto, proponemos para futuras investigaciones un estudio más a profundidad de la medición angular en el contexto de la matematización del movimiento debido a que se da una articulación entre los aportes del estudio de los fenómenos celestes (de lo geométrico-trigonométrico, trabajado principalmente en grados) con los modelos mecánicos (a lo funcional, trabajado en radianes-reales), probablemente vinculado a un cambio contextual-funcional asociado en el uso de la medición angular y (Montiel, 2005 y Buendía y Montiel, 2009) y en particular de la obra Introduction in Analysin Infinitorum de Euler ya que empleó y transitó entre grados y radianes (en términos de π) sin complejidad, uso cantidades trigonométricas como relaciones funcionales trascendentes introduciendo las funciones trigonométricas al Cálculo asociando las propiedades de lo periódico y lo acotado a la función trigonométrica (Buendía y Montiel, 2009), se le atribuye el uso del círculo unitario (Mesa y Goldstein, 2016) y midió ángulos como la longitud de corte de círculo unitario (Akkoc, 2008). ...
Este estudio considera elementos de la Teoría Socioepistemológica de la Matemática Educativa para identificar y analizar la noción de ángulo desde sus usos y funcionalidades que emergen naturalmente en el tránsito de las razones a las funciones trigonométricas en un diseño no tradicional elaborado por Scholz (2018) donde se plantean contextos de significación geométrico-variacional y la articulación de diversas herramientas empíricas, geométricas, algebraicas y trigonométricas. Además, se emplean elementos del método etnográfico para el registro de las observaciones de la implementación del diseño a un grupo de 12 alumnos.
Como parte de los resultados se reportan siete funcionalidades de la noción de ángulo, las cuales son: como herramienta de construcción, referente para aplicar una herramienta aritmética-algebraica, referente para aplicar una herramienta aritmética-trigonométrica, referente para estudiar las relaciones dadas en el modelo, referente para operar aritméticamente, referente para emplear una herramienta empírica-métrica y referente para clasificar triángulos. Además, se presenta una discusión de los fenómenos didácticos identificados en esta investigación: uso del transportador como regla, significado lineal, comprensión de las medidas angulares y el paso del grado al radián-real; con la finalidad de proporcionar una perspectiva integral.
... Radyan kavramına ilişkin literatür; öğrencilerin (Akbaş, 2008;Akkoç ve Akbaş Gül, 2010;Güntekin ve Akgün, 2011;Orhun, 2004;Özaltun Çelik ve Bukova Güzel, 2016;Steckroth, 2007), öğretmen adaylarının (Akkoç, 2008;Fi, 2003;Tuna, 2013;Topçu, Kertil, Akkoç, Yılmaz ve Önder, 2006) ve öğretmenlerin (Topçu vd., 2006) radyan kavramı ile ilgili yanılgılara sahip olduklarını göstermiştir. Bu çalışmaları şu şekilde ...
... Yapılan araştırmalarda da, öğrencilerin (Akbaş, 2008;Akkoç ve Akbaş Gül, 2010;Güntekin ve Akgün, 2011;Orhun, 2004;Özaltun Çelik ve Bukova Güzel, 2016;Steckroth, 2007), öğretmen adaylarının (Akkoç, 2008;Fi, 2003;Topçu vd., 2006;Tuna, 2013) ve öğretmenlerin (Topçu vd., 2006) radyan kavramı ile ilgili yanılgılara sahip oldukları ortaya çıkmıştır. Bu bulgular detaylandırıldığında şu sonuçlara ulaşılmıştır. ...
Bu çalışmanın amacı ortaokul matematik öğretmenlerinin radyan kavramı ve özelde π sayısına ilişkin kavramsal bilgilerini incelemektir. Araştırma, Türkiye’nin dört ilindeki farklı sosyo-ekonomik çevrelerde bulunan ortaokullarda görev yapan ve farklı mesleki deneyime sahip 43 matematik öğretmeninin katılımıyla gerçekleştirilmiştir. Araştırmanın verileri, katılımcıların radyan ve π sayısı hakkındaki bilgilerini ortaya çıkarmaya yönelik hazırlanan ve beş açık uçlu sorudan oluşan bir form vasıtasıyla toplanmıştır. Verilerin analizinde içerik analizi tekniği kullanılmıştır. Araştırma sonucunda, ortaokul matematik öğretmenlerinin π’yi 22/7 kesri, çevre/çap ve 3,14 sabit sayısı olarak düşündükleri, alan ve çevre hesabı ve kolay işlem yapmak için kullandıkları görülmüştür. Bazı öğretmenler radyan ve derece arasında eşleme yapamamış ve sabit bir sayı olan π’nin iki farklı değerinin olamayacağını fark edememişlerdir. Ayrıca katılımcıların çoğunun merkez açının radyan olarak ölçüsünün gördüğü yayın uzunluğuna sadece birim çemberde eşit olduğunu bilmedikleri ve detaylı açıklama yapamadıkları ortaya çıkmıştır. Öte yandan, az sayıda katılımcı açıların sayı doğrusunda derece olarak gösterilemeyeceğini bunun yerine radyan karşılıklarının yazılacağını belirtmiştir. Bu bulgulardan hareketle, katılımcıların çoğunun radyan kavramına ilişkin eksik, yanlış ya da kavram yanılgılı bilgilere sahip olduğu söylenebilir.
... This approach does not lend itself to coherent ways of thinking about trigonometry as the study of angle measures, ratios and values on the unit circle, and relations within right triangles. Unfortunately, incoherent thinking about trigonometry is often shown by secondary school teachers, which can hinder their students' quantitative and covariational reasoning with these topics [4,5]. ...
... So that makes the slope still the same right here, since we're basing it off the slope line. 4 Yes. No? Charlotte: Well . . . ...
Including opportunities for students to experience uncertainty in solving mathematical tasks can prompt learners to resolve the uncertainty, leading to mathematical understanding. In this article, we examine how preservice secondary mathematics teachers’ thinking about a trigonometric relationship was impacted by a series of tasks that prompted uncertainty. Using dynamic geometry software, we asked preservice teachers to compare angle measures of lines on a coordinate grid to their slope values, beginning by investigating lines whose angle measures were in a near-linear relationship to their slopes. After encountering and resolving the uncertainty of the exact relationship between the values, preservice teachers connected what they learned to the tangent relationship and demonstrated new ways of thinking that entail quantitative and covariational reasoning about this trigonometric relationship. We argue that strategically using uncertainty can be an effective way of promoting preservice teachers’ reasoning about the tangent relationship.
... To estimate the value of a function such as sine or cosine for a given input, students must make connections between triangle ratios, the unit circle, and the shape of the appropriate graph (Bressoud, 2010; P. W. Thompson, 2008; P. W. Thompson, Carlson, & Silverman, 2007;Weber, 2005). Difficulties in understanding trigonometric functions are partly attributable to fragmented understanding of angle measure (Akkoc, 2008;Moore, 2012b; P. W. Thompson et al., 2007). Weber (2005) found that students enrolled in a lecture-based college trigonometry course thought that they could only approximate specific outputs of the sine function, such as sin(340º), if given a labeled triangle to use. ...
... For them, periodic functions were seemingly synonymous with trigonometric functions, and therefore "chunky" reasoning was enough to produce a graph with the appropriate shape. Supporting students' knowledge of trigonometric functions, then, seems to be not only a matter of supporting understanding of angle measures (Akkoc, 2008;Moore, 2012aMoore, , 2012bMoore, , 2014 but also a matter of introducing contexts that require different periodic representations. Providing such contrasts has the potential to problematize questions of covariation for students. ...
In support of efforts to foreground functions as central objects of study in algebra, this study provides evidence of how secondary students use trigonometric functions in contextual tasks. I examined secondary students' work on a problem involving modeling the periodic motion of a Ferris wheel through the use of a visual programming environment. Using the cKji model of conceptions, I characterized 5 conceptions of sine and cosine functions in students' work that differed in the ways that students interpreted the given problem, their use of semiotic resources, and the control structures that motivated their work. This study illustrates the range of prior knowledge and resources that students may draw on in their use of trigonometric functions as well as how the goals of students' work inform their reasoning about trigonometric functions. © 2018 National Council of Teachers of Mathematics. All rights reserved.
... 57). While a number of empirical studies demonstrate that Thompson's claim applies to a spectrum of mathematics courses and topics (e.g., Ma 1999;Stigler and Hiebert 1999), several researchers have noted that pre-and in-service teachers' personal understandings of angle measure, as well as their instruction, tend to be particularly lacking in coherence and conceptual meaning (Akkoc 2008;Moore et al. 2016;Tallman 2015;Thompson 2008;Thompson et al. 2007). Others have observed that trigonometry is a notoriously difficult subject for students and that underdeveloped meanings for angle measure often underlie these difficulties (Moore 2012(Moore , 2014Weber 2005). ...
... Several researchers have documented the problematic nature of trigonometry teaching and learning, both in the USA and internationally (Akkoc 2008;Moore et al. 2016;Tallman 2015;Thompson 2008;Thompson et al. 2007, Weber 2005. Scholars have also observed that quantitative reasoning plays a critical role in supporting students' learning of angle measure and trigonometric functions (Hertel and Cullen 2011;Moore 2012Moore , 2014Tallman 2015;Thompson 2008). ...
This paper reports findings from a study that establishes empirical support for Harel’s (Zentralblatt für Didaktik der Math 40:893–907 2008b) inclusion of mathematical ways of thinking as a component of teachers’ professional knowledge base. Specifically, we examined the role of quantitative reasoning (Smith and Thompson, in: Kaput, Carraher, Blanton (eds) Algebra in the early grades, Erlbaum, New York 2007; Thompson, in: A theoretical model of quantity-based reasoning in arithmetic and algebra, Center for Research in Mathematics & Science Education: San Diego State University 1990; Thompson, in: Hatfield et al (eds) New perspectives and directions for collaborative research in mathematics education, University of Wyoming, Laramie 2011) on the quality and coherence of an experienced secondary teacher’s instruction of angle measure. We analyzed 37 videos of the teacher’s instruction to characterize the extent to which he attended to supporting students in reasoning about angle measure quantitatively, and to examine the consequences of this attention on the quality and coherence of the meanings the teacher’s instruction supported. Our analysis revealed that the inconsistent meanings the teacher conveyed were occasioned by his lack of awareness of the conceptual affordances of students’ quantitative reasoning on their ability to construct coherent, meaningful understandings of angle measure. Our findings therefore support Harel’s notion that teachers’ mathematical ways of thinking constitute an essential component of their specialized content knowledge.
... This research can be divided into two parts. The rst part has investigated the di culties in their learning (Akkoc, 2008 Weber, 2005Weber, , 2008. In what follows, we will focus solely on the second part. ...
This research aims to investigate how to support the development of high school students’ covariational reasoning of sine and cosine functions. To achieve this goal, we conducted classroom design research where we experimented with a hypothetical learning trajectory aimed at fostering students’ continuous smooth covariational reasoning of sine and cosine functions. The Freddy’s actual learning trajectory confirms our general learning hypotheses, indicating that the visualization and manipulation of the simulation of a Ferris wheel’s movement in GeoGebra applets incorporating measuring instruments foster the development of covariational reasoning of sine and cosine functions. The results of this research are consistent with previous studies that have found that the use of interactive activities in dynamic digital environments, simulating a Ferris wheel, promotes the development of students’ covariational reasoning regarding sinusoidal functions at a smooth continuous level. Integrating covariational reasoning with hypothetical learning trajectories offers promising implications for educational practices which are discussed at the end.
... Trigonometry is a challenging topic for learners at school (Akkoc, 2008). Gür (2009) confirms that "trigonometry is an area of mathematics that students believe to be particularly difficult and abstract compared with other subjects of mathematics" (p. ...
This paper discusses how researchers explored the mathematical meanings held by Grade 10 teachers on a selected trigonometric topic of ratios and functions. The study began by addressing the concepts of teacher knowledge and teacher knowing, considered to have a bearing in unpacking the notion of mathematical meaning. Proper conceptualisation of these concepts are conceivably obligatory in mapping out a study of teachers’ mathematical meanings. This is a case study approach embracing elements of exploratory research design. Twelve Grade 10 mathematics teachers from four secondary schools in the Tshwane South District in the Gauteng Province of South Africa were selected for the study using convenience sampling. Participants were given a Trigonometry task to reflect on their conceptualisation and understanding of the mathematical content in the task, which the researchers considered as constituting an initial phase to access the experts’ mathematical meanings. A purposive sampling of two teachers set the stage for the subsequent semi-structured interviews, largely probing teachers on their task responses. The study found that participants presented incoherent and unclear mathematical meanings for teaching the sine and cosine of an angle.
... Building on these descriptions, we interpret ways of thinking as the thought patterns a learner demonstrates when reasoning about a particular concept given a specific situation that evokes such reasoning. For example, researchers demonstrated that PMTs think of radian angle measure as angles expressed in terms of π (Akkoc, 2008;Fi, 2003). Additionally, Moore et al. (2016) reported that PMTs' thinking about radian angle measure incorporates a unit circle diagram ( Figure 2) angle measure. ...
Integrated science, technology, engineering, and mathematics (iSTEM) education allow learners to utilize multiple disciplinary perspectives. However, the discipline of mathematics remains underrepresented in iSTEM curriculum. To explore the nature of mathematical thinking with an iSTEM curricular approach that emphasizes mathematics, we investigated the thinking of a preservice mathematics teacher, Alex (pseudonym), who engaged in a task-based digital activity involving radian angle measure in the context of light reflection. Findings suggest that Alex's ways of thinking comprise mathematical terminology, concepts, and processes, including mathematical ways of thinking about light reflection. The findings in this report suggest that emphasizing mathematics in this iSTEM context provided an opportunity for new ways of thinking about radian angle measure, and about how angle measure relates to light reflection.
... The role of the teacher, is essential if they are to support children to develop a more mature abstract angle concepts from their work in Logo by linking those meanings to other perspectives and angle contexts (Mitchelmore and White, 2000). Some research has shown that teacher's personal understanding of angle measure tend to be lacking in coherence and conceptual meaning (Akkoc, 2008;Thompson, Carlson and Silverman, 2007). Although notably the most recent research is related to secondary teachers' knowledge in more advanced topics such as radians and trigonometry (see Tallman and Frank (2020) ...
This thesis is based on research to explore the role of primary school teachers’ mathematical and pedagogical knowledge in their engagement with computer-based microworlds that formed part of ScratchMaths (SM). SM is a two-year mathematics and computing curriculum designed for pupils aged nine to eleven years old. The aims of the research were to trace the evolution of teachers’ mathematical knowledge, as they taught SM microworlds designed for exploration and reasoning about place value, variable and angle through computer programming. The study adopted a multiple-case study approach with augmenting teacher episodes situated in the English primary school setting. The thirteen Year 6 teachers of the study were selected from national participants of the second year of the two-year SM intervention. Data collection involved video-recorded classroom observations, audio-recorded post-lesson semi-structured teacher interviews, and ‘think aloud’ while engaging with computer-based tasks. The conceptual framework for the thesis incorporated the Mathematical Pedagogical Technology Knowledge (MPTK) framework and the Instrumental Orchestration model. The findings reveal the knowledge required to teach at the intersection of programming and mathematics, and crucially, how the ideas mediate and are mediated by engagement with the SM curriculum. The findings also illustrate how teaching mathematics through computer programming requires the teacher to bridge between the computing and the mathematics domains and how some teachers managed to do this while creating new connections within and between the knowledge domains. The study contributes to the literature of teachers’ mathematical knowledge of place value, variable, and angle as well as teachers’ ability to (re-) express mathematics through computer programming. The thesis makes an original contribution to the literature with the specification of a theoretical model for analysing teachers’ knowledge for teaching mathematics through programming in the primary setting.
... Otras investigaciones en el nivel superior se ubican en programas de formación inicial docente en matemáticas, que esperan cierto dominio de conceptos como razón y función trigonométrica por parte del estudiante, pues lo asumen como 'conocimiento del contenido a enseñar'. Entre las dificultades que se reportan están algunas asociadas con la medición angular (Akkoc, 2008;Fi, 2003;Topçu et al., 2006), ya sea por la resistencia a considerar valores distintos a los grados como variable de las funciones o dificultades para definir al radián como relación multiplicativa entre el arco y el radio de un círculo; mismas que se explican por el uso predominante que hay de la medida angular en grados en el contexto del triángulo rectángulo y en la enseñanza introductoria de la trigonometría. ...
Se reporta una investigación de diseño fundamentada en la Teoría Socioepistemológica (TS), con la que se responde a la pregunta ¿cuál es el proceso de resignificación de la razón trigonométrica en estudiantes de primer año de Ingeniería en un escenario de construcción social de conocimiento? Con el análisis transversal de una anidación de prácticas y el reconocimiento de los usos del conocimiento matemático, se identificó el proceso de resignificación en tres categorías. Estas caracterizan cuando los estudiantes (1) actúan matemáticamente en forma distinta, (2) responden con herramientas que van más allá de los procesos memorísticos y (3) evocan alguna noción relacionada. Aunado a ello, se reconoce que los modelos (a escala o bosquejo) de las tareas de trigonometría permiten un ir y venir entre el objeto simulado, el análisis del modelo y el cálculo numérico; así también, que para distinguir la relación trigonométrica de una relación proporcional es necesario un estudio covariacional de la relación ángulo-cuerda de objetos en diferentes ubicaciones.
... Furthermore, it was revealed in some studies that preservice teachers have insufficient content knowledge regarding circle geometry and have difficulties in explaining the concepts related to a circle (e.g., Akkoc, 2008;Bekdemir, 2012;Erdem & Man, 2018;Gok Colak & Tugluk, 2017;Sudihartinih, 2018;Ünlü, 2021). In this context, Bekdemir (2012) determines that the conceptual knowledge of preservice teachers regarding the concept of a circle is low, and that they predominantly only possess operational knowledge. ...
The purpose of this study was to examine the effects of using GeoGebra software in a computer-supported collaborative learning (CSCL) environment on seventh grade students’ geometry achievement, retention of learning, and attitudes toward geometry. The study was designed using a quasi-experimental research method with pretest, post-test and delayed post-test. This study was carried out with 62 seventh grade students in a city in western Turkey. CSCL activities using GeoGebra software were implemented in the experimental group, while instruction in the control group continued with textbook-based direct instruction. The Geometry Achievement Test (GAT) and Geometry Attitude Scale (GAS) were applied to groups as pretest and post-test. A retention test was applied to both groups eight weeks after the post-test. Data were analyzed through SPSS 17.0 statistical software by using a t-test and ANCOVA test. It was indicated in this study that CSCL using GeoGebra software significantly increased seventh grade students’ geometry achievement and retention of learning in comparison to textbook-based direct instruction. It was also determined that the CSCL environment with GeoGebra software significantly increased students’ attitudes toward geometry.
... Largely, the findings discussed in this article corroborate findings of previous studies to the extent that mathematics student teachers' understanding of school mathematics concepts is superficial (Akkoc, 2008;Malambo, 2020;Malambo et al., 2018Malambo et al., , 2019Marbán & Sintema, 2020). The findings are also consistent with the theory of didactical situations (Brousseau, 1997). ...
The likelihood of misunderstanding and misrepresenting trigonometric ideas motivated an investigation of implicit misconceptions in prospective mathematics teachers' reasoning about particular trigonometric concepts. To access the participants' implicit misconceptions, a qualitative approach and case study design in particular were employed. Three prospective teachers chosen purposefully composed the sample. The student teachers majored in mathematics and were in the final year of training. Four diagnostic questions based on trigonometry were administered followed by semi-structured interviews. Qualitative analyses of calculations and interview transcripts revealed implicit misconceptions in participants' reasoning. The prospective teachers reasoned that trigonometric equations can be resolved in the same way as conventional algebraic equations. Likewise, they demonstrated an erroneous notion that inverse trigonometric functions are evaluated just like indices. Besides, the participants incorrectly considered elements of domains of trigonometric functions to be synonymous with such functions' extreme values. Overall, prospective teachers' reasoning demonstrated didactical obstacles. It is therefore proposed that mathematics teacher education should include opportunities for prospective teachers to reason about mathematics concepts in a manner that prevents didactical obstacles. Furthermore, mathematics educators should engage in instructional practices which facilitate prospective teachers' acquisition of in-depth understanding of mathematics conceptual relationships and differences.
... Fakat bu konuda Türkiye de birkaç çalışmanın dışında çalışma bulunmamaktadır. Akkoç (2008) tarafından öğretmenlerin, öğretmen adaylarının ve öğrencilerin trigonometrik fonksiyonların tanımlanmasında önemli olan radyan kavramı ile ilgili öğrenme güçlüklerine sahip oldukları ortaya konmuştur. Literatüre katkı sağlamak ve öğrencilerdeki güçlüklerini zamanında belirlenip gidermek (Duval, 2002) için trigonometri ve trigonometrik fonksiyonlar konusunda çalışmalar yapılması önemlidir (Kutluca ve Baki 2009;Kültür, Kaplan ve Kaplan, 2008). ...
ZET Bu çalışma; ilköğretim matematik öğretmenliği programı öğrencilerinin koordinat düzleminde birim çemberi kullanarak tanjant ve kotanjant fonksiyonları-nın grafiklerinin çiziminde sayı doğrusu kullanımlarını araştıran betimsel bir çalış-madır. Çalışmada nicel veri toplama yöntemlerinin yanında gözlem ve görüşme gibi nitel veri toplama yöntemleri de kullanılabilen tarama modeli kullanılmıştır. Çalış-ma, Türkiye'nin Doğu Anadolu Bölgesinin nüfusça orta ölçekli bir ilinde yapılmış-tır. Çalışma grubunu ilköğretim matematik öğretmenliği programına kayıtlı 56 bi-rinci sınıf öğrencisi oluşturmaktadır. Çalışmaya katılan öğrencilere on bir adet beşli Likert tipi ve altı adet açık uçlu soru sorulmuş ve veriler analiz edilerek elde edilen bulgular yorumlanmıştır. Sonuçlar, öğrencilerin koordinat düzleminde tanjant ve kotanjant fonksiyonlarının grafiklerinin çiziminde sayı doğrusunu doğru bir şekilde kullanamadıklarını ve birbirine bağlı olarak tanımlanan kavramları birbirinden ba-ğımsız gibi kullandıklarını göstermektedir. Buna göre öğrencilerin kavramlar ara-sındaki ilişkileri kurabilecekleri etkinliklerin yaptırılması önerilmiştir. ABSTRACT This is a descriptive study analyzing use of numerical axis in the graphic drawing of tangent and cotangent functions on coordinate plane by means of unit circle by primary education mathematics teaching department students. In addition to quantitative data gathering methods, scan model which has the capacity to apply qualitative data gathering methods like survey and interview have also been employed. The study has been conducted in a moderately populated Eastern Anatolian Region city in Turkey. Research group consists of 56 freshmen students registered to primary education mathematics teaching department. Participant students in the research have been directed eleven 5-Likert type and six open-ended * Yrd. Doç. Dr., Eğitim Fak. İlköğretim Bölümü, Erzincan Üniv.
... From a review of the literature, it can be seen that teachers and preservice teachers experience certain difficulties in defining basic geometric concepts, exemplifying the definitions and recognizing them, and they may even have certain misconceptions (Cunningham & Roberts, 2010;Fujita & Jones, 2007;Gutiérrez & Jamie, 1999;Leikin & Zazkis, 2010;Shaughnessy & Burger, 1985;Vighi, 2003;Ward, 2004). The literature appears to be mostly concerned with the concepts of triangle, circle, and cylinder (Tsamir et al., 2015), tangent (Tsamir et al., 2015), radian (Akkoc, 2008), area (Tossavainen et al., 2017), angle, circle, geometric location, and metric (Gülkılık, 2008), quadrilaterals (Aktaş & Aktaş, 2012;Alaylı & Türnüklü, 2014;Erşen & Karakuş, 2013;Okazaki & Fujita, 2007;Türnüklü & Berkun, 2013), cylinder and cone (Avgören, 2011;Ertekin et al., 2014;Karakuş, 2018;Koçak et al., 2017), prism and pyramid (Avgören, 2011;Ünlü & Horzum, 2018), sphere (Gökbulut, 2010;Gökkurt, 2014;Yılmaz, 2015), and also circular region (Aydoğdu-İskenderoğlu & Akşan-Kiliçaslan, 2017;Bekdemir, 2012). Gökbulut (2010) stated that among geometric objects, preservice elementary teachers faced the greatest difficulty in defining the concept of sphere. ...
The aim of this study was to analyse preservice elementary mathematics teachers’ concept definitions of circle, circular region, and sphere concepts in geometrical terms. The research was conducted during the 2016–2017 academic year with 56 preservice mathematics teachers enrolled within a teacher education programme at a state university in Turkey. A test consisting of three open-ended questions regarding circle, circular region, and sphere was used as the study's data collection tool. The preservice teachers’ test answers were investigated according to correctness and generalization criteria. Answers regarding each concept were compared with those of the two other concepts. Following the data analysis, semi-structured interviews were conducted with nine of the participant preservice teachers. The findings of the research revealed that the preservice teachers mostly experienced difficulty in defining the sphere. In addition, it was determined that most preservice teachers provided either insufficiently detailed or incorrect answers for the concepts of circle, circular region, and sphere. When the examples of the concepts were examined, it was determined that only a few of the preservice teachers gave appropriate examples of the related concepts.
... Yet, relative to other quantities like length, area, and volume, very little scholarly literature addresses how students and teachers understand angle measure (Smith & Barrett, 2017). From the scant extant literature, it is clear that developing productive conceptions of angle measure is non-trivial for students and teachers alike (Akkoc, 2008;Lehrer, Jenkins, & Osana, 1998;Smith & Barrett, 2017). In the U.S., individuals' challenges in quantifying angularity may be partially attributed to instructional approaches that (a) overemphasize the use of conventional protractors to measure angles and (b) fail to address how the design of these conventional tools renders them appropriate for measuring angles (Moore, 2012). ...
... Trigonometry is a unit in which algebraic techniques, geometrical realities and trigonometric relations come together. However, it was found that students had difficulty in understanding some basic concepts of trigonometry and trigonometry is not interested for them (Akkoc, 2008). ...
The Organization for Economic Co-operation and Development (OECD) (2016) released the results of the Programme for International Student Assessment (PISA) 2015, highlighting significant differences in mathematics performance between students in Singapore and Indonesia. There is a strong relationship between the textbooks used and students' mathematics performance. If textbooks differ, students will have different learning opportunities, which in turn influence their achievement. The purpose of this study was to analyze the trigonometry content and cognitive demand levels in Singaporean and Indonesian mathematics textbooks. The data were primarily qualitative. Horizontal and vertical analyses were used in this study.
The results showed that Singaporean textbooks emphasize all the concepts of trigonometry in right triangles and further trigonometry (sine/cosine rules), while Indonesian textbooks provide more discussion on angles and their concepts, trigonometry in right triangles, and trigonometric graph functions (which are more complex than sine/cosine rules). Additionally, Singaporean textbooks contain more mathematical questions requiring higher cognitive demand levels, while Indonesian textbooks feature more questions requiring lower cognitive demand levels. The differences in textbook content and cognitive demand levels likely influence students' mathematics performance in the two countries. It is hoped that these results will inform curriculum designers and textbook authors in Indonesia, Singapore, and other countries as they review and update their mathematics curricula and textbooks.
... Numerous studies have found that students leave trigonometry classrooms with poor understanding of the subject. Studies have depicted struggles among secondary students (Kendal & Stacey 1998), undergraduate students (Moore 2013;Weber 2005) pre-service teachers (Akkoç 2008;Tuna 2013), and even in-service teachers (Topçu, Kertil, Yilmaz & Öndar 2006). Kendal and Stacey examined secondary students who had been taught trigonometry through right triangle representations and found that they had difficulty solving problems that involved non-acute angles. ...
Trigonometry is integral to mathematics education. The field of trigonometry plays a crucial role in the study of mathematics and its applications. Despite the importance of the subject, students struggle to understand trigonometric constructs such as angle measure. It has also been noted how students struggle to understand transformations of functions generally. Our review of the literature found few studies specifically on students’ understanding of transformations of trigonometric functions, but evidence exists showing students have difficulties with the concept. Here, a MATLAB program called 'TrigReps' is discussed. 'TrigReps' accepts four inputs for the algebraic representation '(a)sin(bx' + 'c) + d', and provides three additional representations as outputs. Students are presented with a graphical representation, an auditory representation, and a dynamic representation of a radius rotating around a unit circle. 'TrigReps' has potential to be a useful tool for teaching transformations of trigonometric functions. In particular, it may be able to help students justify why combinations of horizontal transformations are counterintuitive. 'TrigReps' is analytically sound in its design: it is interactive, dynamic, and displays Multiple External Representations (MERs) simultaneously. Initial data support its usefulness in a trigonometry classroom, but more research must be conducted to draw firm conclusions.
... Researchers have also shown that when students are limited to only a static definition of an angle, other difficulties emerge, such as not being able to measure angles, a challenge that persist even into secondary level (Akkoc, 2008;Browning & Garza-Kling, 2009;Devichi & Munier, 2013;Keiser, 2004;Mitchelmore, 1998;Moore, 2009Moore, , 2013Moore & LaForest, 2014;Smith, 2017;Topcu et al., 2006;Wilson, 1990). As Smith (2017) noted, "angle measurement relates the static geometric figure to rotational motion," where "an angle's measure is its amount of rotational sweep -the amount one ray has been rotated to coincide with the other" (p. ...
A strong foundation in students’ understanding of the multifaceted nature of the angle concept is of paramount significance in understanding trigonometry and other advanced mathematics courses involving angles. Research has shown that sixth-grade students struggle understanding the multifaceted nature of the angle concept (Keiser, 2004). Building on existing work on students’ understanding of angle and angle measure and instructional supports, this study asks: How do sixth-grade students conceptualize angle and angle measure before, during, and after learning through a geometry unit of instruction set in a miniature golf context? What instructional supports contribute to sixth-grade students’ conceptualization of angle and angle measure in such a context? I conducted a retrospective analysis of existing data generated using design-based research methodology and guided by Realistic Mathematics Education (RME) theory. Using Cobb and Yackel’s (1996) Emergent Perspective as an interpretive framework, I analyzed transcripts of video and audio recordings from nine days of lessons in a collaborative teaching experiment (CTE), focusing on two pairs of students in sixth-grade mathematics classes. I also analyzed transcripts of pre-interviews before instruction, midway interviews during instruction, and post-interviews after instruction with each student in the two pairs. To answer research question one, I developed codes from data guided by the existing literature. For research question two, I used Anghileri’s (2006) levels of supports framework. Overall, the findings revealed that sixth-grade students conceptualized an angle as a static geometric figure defined by two rays meeting at a common point, and conceptualized angle measure through their body turns. In addition, Anghileri’s three levels of supports, such as the use of structured tasks, teacher’s use of probing questions, generation of conceptual discourse were evident in contributing to students’ conceptualization of angle and angle measure during the miniature golf geometry unit of instruction. The findings of this study have implications for the school mathematics curriculum, and how to teach and to prepare teachers to teach angle and angle measure. This study emphasizes the need to redefine the angle concept in the curriculum documents, the need to increase activities involving body turns and the use of Anghileri’s (2006) levels of supports in the teaching and learning of angle and angle measure in a real-world context. Further research is needed to identify instructional supports, in particular activities that can support students’ conceptualization of slopes and turns as angles in a real-world context.
... Researchers have also shown that when students are limited to only a static definition of an angle, other difficulties emerge, such as not being able to measure angles, a challenge that persist even into secondary level (Akkoc, 2008;Browning & Garza-Kling, 2009;Devichi & Munier, 2013;Keiser, 2004;Mitchelmore, 1998;Moore, 2009Moore, , 2013Moore & LaForest, 2014;Smith, 2017;Topcu et al., 2006;Wilson, 1990). As Smith (2017) noted, "angle measurement relates the static geometric figure to rotational motion," where "an angle's measure is its amount of rotational sweep -the amount one ray has been rotated to coincide with the other" (p. ...
A strong foundation in students’ understanding of the multifaceted nature of the angle concept is of paramount significance in understanding trigonometry and other advanced mathematics courses involving angles. Research has shown that sixth-grade students struggle understanding the multifaceted nature of the angle concept (Keiser, 2004). This study asks: How do sixth-grade students conceptualize angle and angle measure before, during and after learning through a geometry unit of instruction set in a miniature golf context? What instructional supports contribute to sixth-grade students’ conceptualization of angle and angle measure in such a context? I conducted a retrospective analysis of existing data generated using design-based research methodology and guided by Realistic Mathematics Education (RME) theory. Using Cobb and Yackel’s (1996) Emergent Perspective as an interpretive framework, I analyzed transcripts of video and audio recordings from nine days of lessons in a collaborative teaching experiment (CTE), focusing on two pairs of students in sixth-grade mathematics classes. I also analyzed transcripts of pre-interviews before instruction, midway interviews during instruction, and post-interviews after instruction with each student in the two pairs. To answer research question one, I developed codes from data guided by the existing literature. For research question two, I used Anghileri’s (2006) levels of supports framework. Overall, the findings revealed that sixth-grade students conceptualized an angle as a static geometric figure defined by two rays meeting at a common point, and conceptualized angle measure through their body turns. In addition, Anghileri’s three levels of supports, such as the use of structured tasks, teacher’s use of probing questions, generation of conceptual discourse were evident in contributing to students’ conceptualization of angle and angle measure during the miniature golf geometry unit of instruction. The findings of this study have implications for the school mathematics curriculum, and how to teach and to prepare teachers to teach angle and angle measure. This study emphasizes the need to redefine the angle concept in the curriculum documents, the need to increase activities involving body turns and the use of Anghileri’s (2006) levels of supports in the teaching and learning of angle and angle measure in a real-world context. Further research is needed to identify instructional supports, in particular activities that can support students’ conceptualization of slopes and turns as angles in a real-world context.
... Para números como -7, o 0.357 algunos no sabían interpretar el argumento. Mientras, Akkoc (2008) reporta que profesores en formación tuvieron dificultades con el concepto de radián, además de evidenciar que la imagen del concepto de grado predomina sobre la de radián concluyendo que estas dificultades podría causar problemas para comprender las funciones trigonoméricas. ...
The title of this paper causes a priori bewilderment, because the reader might consider that the concept of angle and its measurement is understood by students starting a bachelor’s degree; however, although some difficulties reported in investigations has been overcome, others still persist. In mathematics, the concept definition is relevant, however, in practice, students usually resort to the evoked concept image, which often makes it difficult for them to perform a specific task. A questionnaire was designed based on the notion of angle and its measurement. These questions were answered by 22 Mexican under-graduate students. To identify students’ images and definitions, the Tall and Vinner’s model was applied. We found a wide variety in the students’ personal definitions of angle, which are not memorization of definitions given in their courses or textbooks. In addition, they do not have a single image concept of angle and its measurement, because they evoked different images according to the problem to be solved.
... Relacionado con la medida angular, en su investigación Akkoc (2008) menciona que los docentes saben convertir entre radián y grados, pero no pueden definir la relación de dos longitudes. Es decir, se tiene una compresión poco profunda de lo que significa la medida radián. ...
Se exponen los principales resultados de la revisión bibliográfica, identificando consideraciones preliminares para el estudio de la angularidad en el desarrollo del pensamiento Trigonométrico. Principalmente se encuentra el tratamiento de la trigonometría en la institucionalización del mismo conocimiento, relacionado a ello se identifican dificultades y fenómenos en el aprendizaje de las nociones de ángulo, trigonometría y el ángulo-trigonometría. De modo que permita responder, grosso modo, por qué surge el tema, cuál es la importancia de investigar al respecto y cuáles consideraciones son importantes al iniciar el estudio del desarrollo del pensamiento Trigonométrico. Permitiendo extraer elementos propicios para analizar el uso de la noción de ángulo en el desarrollo de este tipo particular de pensamiento matemático, para dar cuenta de su transversalidad, en etapas futuras de la investigación.
... To begin with, item a ( Figure 6) is intended to recognise the behaviour of the movement of the points on each circumference. For this question, it is probable that the idea of angular velocity will generate difficulties, which was evidenced in the piloting and the different staging because the conception of radian itself causes difficulties, as reported in different investigations (Akkoc, 2008;Moore, 2009). Item b (Figure 6) is intended to verify the understanding of the behaviour of the system using a static system model (for a given time). ...
ABSTRACT The aim of this article is to show a didactic proposal that favors the signification of the Trigonometric Fourier Series through a learning situation, based on the Socio-epistemological Theory of Mathematics Education, on research where this knowledge has been problematized. The Fourier Trigonometric Series is a complex topic for learning at the higher level, where the process is usually mechanized without fully understanding its operation and characteristics. We want to verify that with activities that support the relationship between algebra and geometry, making use of GeoGebra-dynamic geometry software-as a control variable, the series and its convergence can be signified using a physical-geometric context. RESUMEN El objetivo de este artículo es mostrar una propuesta didáctica que propicie la significación de la Serie Trigonométrica de Fourier a través de una situación de aprendizaje, cuyo fundamento se basa en la Teoría Socioepistemológica de la Matemática Educativa, en investigaciones donde se ha problematizado este saber. La Serie Trigonométrica de Fourier es un tema complejo para su aprendizaje en el nivel superior, donde por lo general se mecaniza el proceso sin comprender del todo su funcionamiento y características. Se quiere comprobar que con actividades que apoyen la relación entre lo algebraico y lo geométrico, haciendo uso de GeoGebra-software de geometría dinámica-como variable de control, se puede significar a la serie y su convergencia mediante un contexto físico-geométrico. | HOW TO CITE? Farfán, R. M. & Romero, F. (2017). Learning situation for the trigonometric Fourier series from a Socio-epistemological stand point. Acta Scientiae, 21(2), 28-48. DOI: https://doi.org/10.17648/acta.scientiae.v21iss2id5019
... Matematik eğitim teorilerinden kavram imajı ve kavram tanımı teorisi son yıllarda birçok araştırmada ele alınmakta ve önemsenmektedir (Akkoç, 2008;Nordlander ve Nordlander, 2012;Erşen ve Karakuş, 2013). Bu teori Tall ve Vinner tarafından 1981 yılında yayınlanan makale çalışmasıyla teorik çerçevesi şekillenmiştir. ...
Amaç: Bu çalışmada kavram imajı teorisi temel alınarak ilköğretim matematik öğretmenliği öğrencilerinin “rasyonel sayı ve kesir” kavramlarına ilişkin kavram imajlarının belirlenmesi amaçlanmıştır. Yöntem: Bu çalışma nitel bir araştırma yöntemine uygun olarak tasarlanmıştır. Araştırmanın çalışma grubu, 2014-2015 eğitim öğretim yılında Doğu Anadolu’da bir üniversitenin İlköğretim Matematik Öğretmenliği Programında öğrenim görmekte olan ve çalışmaya gönüllü olarak katılan 110 öğrenciden oluşmaktadır. Verilerin toplanması amacıyla araştırmacı tarafından geliştirilen “Rasyonel Sayı ve Kesir Kavram İmajı Anketi” kullanılmıştır. Elde edilen verilerin analizinde araştırma desenine uygun olarak içerik analizi kullanılmıştır. Bulgular: Öğrencilere uygulanan anket neticesinde, kesir kavramı konusunda çoğunlukla parça-bütün imajına, rasyonel sayı kavramı için çoğunlukla oran imajına sahip oldukları görülmüştür. Öğrencilerin rasyonel sayı ile kesir kavramı arasındaki fark imajı da kesir kavramının negatif olamayacağı ama rasyonel sayı kavramının negatif olabileceği şeklindedir. Sonuçlar ve Öneriler: Çalışma sonunda öğrencilerin rasyonel sayı ve kesir kavram imajlarının yeterince net olmadığı sonucuna ulaşılmıştır. Öneri olarak daha az öğrenciyle görüşme yaparak daha ayrıntılı olarak incelenebilir. Sonuçlar farklı bölüm ya da farklı fakülte öğrencileriyle karşılaştırılabilir.
... These difficulties that students showed could be related to a lack of angles and trigonometric knowledge. A growing body of research highlights that students have difficulties with angle measurements and trigonometric functions of angles [26][27][28]. It can be concluded that the difficulties with polar coordinates for students could be attributed to students' difficulties in dealing with prerequisite concepts; e.g. the unit circle, trigonometric functions of real numbers, trigonometric graphs, angle measurements, and trigonometric functions of angles. ...
The aim of this study is to analyse undergraduate students’ understanding of polar coordinates based on two theories, Action, Process, Object and Schema (APOS) and Onto-Semiotic Approach (OSA). These two theories complement each other and each of them separately has been used in many research to explore students’ performance of mathematical concepts. However, the combined use of APOS and OSA in educational research in mathematics is scarce. Seven tasks of polar coordinates were designed to characterize the schema development of thirty-four undergraduate students in terms of intra, inter and trans. Results showed that most of the students had difficulties in doing the practical tasks and developing the mental constructions needed to solve polar coordinates’ tasks, particularly those mental constructions that have to be made to sketch polar curves and to plot the point (r,θ) in the plane where r and θ were negative. Also, a lack of trigonometric knowledge caused the students to be unsuccessful in solving polar coordinate tasks. The complementary use of APOS and OSA helped us to have a better insight regarding students’ understanding of the concept of polar coordinates.
... In fact, their training usually includes high level mathematics. However, there are researchers who, over the last thirty years, have become convinced that the notion that a teacher who has studied advanced mathematics automatically understands secondary school mathematics is a faulty one (Akkoc, 2008;Ball, 1990;Even, 1990Even, , 1993Fi, 2003Fi, , 2006Wilburne & Long, 2010;Wilson, 1994;Wood, 1993). While the study of advanced mathematics with future engineers and mathematicians can create a foundation for understanding school mathematics, this is not a guarantee that the students will know and understand the school mathematics they have to teach (Bryan, 1999;Cooney, 2003;Hiebert, 2013). ...
This study investigated student mathematics teachers' ability to recognise and explain their understanding of school level functions. We challenged the assumption that studying advanced mathematics automatically develops an understanding of school mathematics that is sufficient to explain concepts and justify reasoning. This case study tested this assumption by exploring the depth of pre-service mathematics student teachers' understanding of school function concepts at the University of Zambia. The test items required calculation, as well as justification of the answers, and an explanation of the concepts. Of the 22 participants, all final year mathematics education students, 18 student teachers scored below the 50% pass mark. The average mark was 8 out of a possible 28 (27%). The majority of the participants found it difficult to explain and justify their reasoning. This study resulted in the development of a new school mathematics module for prospective mathematics teachers at the University of Zambia.
... Many researchers have studied students' and teachers' conceptions of advanced mathematical topics, such as functions, derivatives, limits and continuity, as well as some basic geometrical concepts (e.g., Akkoç, 2008;Gutiѐrrez & Jaime, 1999;Przenioslo, 2004;Tall & Vinner, 1981;Vinner & Dreyfus, 1989;Vinner & Hershkowitz, 1980;Wawro, Sweeney & Rabin, 2011;Yanık, 2014). For instance, Vinner and Hershkowitz's (1980) study, where they tested students' images in grades 7, 8 and 9 regarding some simple geometrical concepts (including obtuse angle, straight angle, right triangle and altitude of a triangle), showed that students' concept images contain "only obtuse angles with horizontal rays, only horizontal straight angles, a right triangle with a vertical side and a horizontal side" (p.181-182). ...
In recent years, researchers have become more interested in quadratic equations. This study explores the conceptions high school students have concerning quadratic equations with one unknown, while using concept definition and images as theoretical framework. The data was gathered through semi-structured interviews with 14 eleventh grade high school students. Analysis of the data revealed that none of the students could provide an exact and correct definition of a quadratic equation and their attempted definitions were not consistent with the formal (standard) definition of quadratic equation. Moreover, the findings showed that students' concept image of quadratic equation was quite limited and dominated by ideas concerning factoring. Students' conceptions of quadratic equations also showed that participating students lacked three types of prerequisite knowledge: degree of a polynomial, variable and equals sign. In order to enrich students' concept definition and images, both procedurally and conceptually, lessons from this study need to be used in teaching. The implications of the findings are discussed.
... Although trigonometry is a fundamental topic and has a very large area of application, some studies have revealed that it is a hard topic for students (Akkoc, 2008;Fi, 2003;Steckroth, 2007;Topcu, Kertil, Akkoc, Yilmaz, & Onder 2006). Durmus (2004) stated that the difficulty of trigonometry stems from its abstract nature, including algebraic equations and formulas such as addition and sum-to-product formulas. ...
The purpose of the present study is to investigate the relationship between teacher efficacy to student trigonometry self-efficacy and student trigonometry achievement. The study included 16 high school teachers and their tenth grade students (n=571). Teacher efficacy was studied in terms of general teaching efficacy, mathematics teaching efficacy, and trigonometry teaching efficacy. No significant relationship was found between general teaching efficacy or mathematics teaching efficacy and student-related variables. The t-test results show that students of teachers who had high trigonometry teaching efficacy got higher scores on the trigonometry self-efficacy scale than students of teachers with low trigonometry teaching efficacy. Between these two groups of students’ achievement test scores, on the other hand, no significant difference was found. The results underline the importance of teachers’ trigonometry teaching efficacy for students’ trigonometry self-efficacy, as well as the importance of measuring self-efficacy in a task-specific way.
... This result is in agreement with the result of Erdogan and Dur (2014)'s research. Therefore, it is important to detect these difficulties in early phases and contribute to enriching mathematical concepts by reviewing the school mathematics during teacher training programs in line with the developing pedagogical concept knowledge (Akkoc, 2008). Considering school mathematics, other types of representations of geometrical objects should be used in addition to prototype figures while instructing quadrilaterals to ensure variety. ...
This study attempts to reveal pre-service teachers’ conceptions, definitions, and understanding of quadrilaterals and their internal relationships in terms of personal and formal figural concepts via case of the parallelograms. To collect data, an open-ended question was addressed to 27 pre-service mathematics teachers, and clinical interviews were conducted with them. The factors influential on pre-service teachers’ definitions of parallelograms and conceptions regarding internal relationships between quadrilaterals were analyzed. The strongest result involved definitions based on prototype figures and partially seeing internal relationships between quadrilaterals via these definitions. As a different result from what is reported in the literature, it was found that the fact that rectangle remains as a special case of parallelogram in pre-service teachers’ figural concepts leads them not to adopt the hierarchical relationship. The findings suggested that learners were likely to recognize quadrilaterals by a special case of them and prototypical figures, even though they knew the formal definition in general. This led learners to have difficulty in understanding the inclusion relations of quadrilaterals.
... 57). While a number of empirical studies demonstrate that Thompson's accusation applies to a spectrum of mathematics courses and topics (e.g., Ma, 1999;Stigler & Hiebert, 1999), several researchers have noted that pre-and in-service teachers' personal understandings of trigonometry, as well as their instruction, tend to be particularly lacking in coherence and conceptual meaning (Akkoc, 2008;Moore et al., in press;Tallman, 2015;Thompson, 2008;Thompson, Carlson, & Silverman, 2007). Others have observed that trigonometry is a notoriously difficult subject for students (Moore, 2012(Moore, , 2014Weber, 2005). ...
This paper reports findings from a study that explored the effect of a secondary mathematics teacher’s level of attention to quantitative reasoning on the quality and coherence of his instruction of angle measure. I analyzed 37 videos of an experienced teacher’s instruction to characterize the extent to which he attended to supporting students in reasoning quantitatively, and to examine the consequences of this attention (or lack thereof) on the quality and coherence of the meanings the teacher’s instruction supported. My analysis revealed that the incoherencies in the teacher’s instruction were occasioned by his inattention to quantitative reasoning. This study therefore demonstrates that when teachers do not possess a disposition to attend to quantities and their relationships, the circumstances are ripe for instruction that emphasizes inconsistent, incoherent, and sometimes incompatible, mathematical meanings.
... Trigonometry is a unit in which algebraic techniques, geometrical realities and trigonometric relations come together. However, it was found that students had difficulty in understanding some basic concepts of trigonometry and trigonometry is not interested for them (Akkoc, 2008). ...
Organization for Economic Co-operation and Development (OECD) (2016) released the results of Programme for International Student Assessment (PISA) 2015 and reported that the students' performance in mathematics of Singapore and Indonesia had significant differences. There is a strong relationship between textbooks used and mathematics performance of the students. If textbooks differ, students will get a different opportunity to learn and the opportunity to learn influences students' achievement (performance). The purpose of this study was to analyze the trigonometry contents and cognitive demand levels in Singaporean and Indonesian mathematics textbooks. The data were primarily qualitative. Horizontal and vertical analyses were used in this study. The result showed that Singaporean textbooks put more emphasis on all the concepts of trigonometry on right-triangle and further trigonometry (sine/cosine rules) while Indonesian textbook provided more discussions on angle and its concepts, trigonometry on right-triangle, and graph function of trigonometry (which is a lot more difficult than sine/cosine rules). In addition, Singaporean textbook provided more mathematical questions requiring higher cognitive demand levels while, Indonesian textbook provided more questions requiring lower levels. The differences of textbooks contents and required cognitive demand level probably influenced students' mathematics performance in the two countries. It is hoped that the results will inform curriculum designers and/or textbooks' author(s) in Indonesia, Singapore, and other countries as they review and update the mathematics curriculum and/or mathematics textbooks.
... Tall ve Vinner'ın (1981) kavram tanımı ve kavram imajı modelini kullanarak yaptıkları çalışmanın ardından, bu modeli baz alarak günümüze kadar hem yurt dışında hem ülkemizde pek çok araştırmanın yapıldığı görülmektedir (Vinner, 1983;Wilson, 1990;Furinghetti & Paola, 1991;Vinner, 1991;De Villiers, 1994;Nakahara, 1995;Hasegawa, 1997;Monaghan, 2000;Toluk vd., 2002;Olkun & Aydoğdu, 2003;Ward, 2004;Aktaş, 2005;Erez & Yerushalmy, 2006;Akuysal, 2007;Fujita & Jones, 2007;Okazaki & Fujita, 2007;Pickreign, 2007;Akkoç, 2008;Fujita, 2008;Usiskin, Griffin, Witonsky & Willmore, 2008;Okazaki, 2009;Ergün, 2010;Aktaş & Aktaş, 2011;Aktaş & Aktaş, 2012;Fujita, 2012;Nordlander & Nordlander, 2012;Türnüklü, Akkaş & Gündoğdu-Alaylı, 2012). Bu alanda yapılan çalışmalar incelendiğinde de literatürün önemli bir bölümünü dörtgenler konusuna yönelik yapılan araştırmalar oluşturmaktadır (Wilson, 1990;De Villiers, 1994;Nakahara, 1995;Monaghan, 2000;Toluk vd., 2002;Olkun & Aydoğdu, 2003;Aktaş, 2005;Erez & Yerushalmy, 2006;Akuysal, 2007;Fujita & Jones, 2007;Okazaki & Fujita, 2007;Pickreign, 2007;Fujita, 2008;Okazaki, 2009;Ergün, 2010;Aktaş & Aktaş, 2011;Aktaş & Aktaş, 2012;Fujita, 2012;Türnüklü vd., 2012). ...
The aim of this study was to determine preservice elementary teachers' images for some special quadrilaterals. The case study was conducted with 6 preservice elementary teachers. The data were collected by clinical interviews to evaluate preservice teachers’ images for quadrilaterals. Hence, initially, the questionnaire consisting of two parts was given to preservice teachers. In the first part, preservice teachers were asked to draw 3 different squares, rectangles, trapezoids and parallegroms; they were asked to identify these quadrilaterals in the second part. Clinical interviews were conducted by preservice teachers about the questionnaire; thus, preservice teachers’ concept images and own definitions for some special quadrilaterals were defined. The data were analyzed by descriptive analysis. It was revealed that preservice teachers drew quadrilaterals incorrectly not to show notation, know the properties, and classify the relationship between them. Furthermore, it was identified that they had wrong images for especially trapezoid about their individual definitions.Key Words: Quadrilaterals, concept image, concept definition, preservice teachers
With the shift of learning modality from face-to-face to online distance learning, teaching Mathematics has been linked to new challenges and constraints. This paper investigates how radian measure can be taught via distance learning delivery modality. An Online Lesson Study was conducted, and the research lesson was implemented for first-year college students taking Bachelor of Secondary Education major in Mathematics at a state university in the Philippines. The recorded dialogues during the post-lesson discussion were transcribed and analyzed by the researchers. Through these processes, three themes emerged: (1) logistical issues in online teaching, (2) alignment of purpose and approach in online teaching, and (3) theoretical underpinnings for online teaching. Hence, online lesson study provides a venue for teachers to pursue professional development amidst the pandemic.
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.
This paper reports on concept images of 38 secondary school mathematics prospective teachers, regarding the evenness of numbers. Written assignments, individual interviews, and lesson transcripts uncover salient, erroneous concept images of even numbers as numbers that are two times “something” (i.e., 2i is an even number), or to reject the evenness of zero. The notion of concept image serves in the analysis of the findings, and the findings serve in offering two refinement notions: mis-in concept images that mistakenly grant non-examples the status of examples (e.g., 2i is an even number), and mis-out concept images that mistakenly regard examples as non-examples (e.g., zero is not an even number). We discuss possible benefits in distinguishing between these two refinement notions in mathematics education.
The aim of this study is to investigate the concept images of 8th grade students for algebraic expressions, equations and identities. In this direction, the students’ concept images, their explanations and the consistency of their answers to the application questions were investigated. The participants of the research, which is designed as a case study from qualitative research methods, consist of 36 8th grade students in Sakarya in 2020-2021 academic year. In order to determine the concept images of the students as data collection tools, the Concept Information Form (CSF) consisting of 4 open-ended questions and the Application Form (UF) consisting of 2 sample questions were used. The obtained data were analyzed according to the content analysis method. In this context, the explanations of the students about the concepts were coded, then the themes were created and the findings were defined and interpreted. According to this; The majority of the students participating in the research had difficulty in using verbal expressions while defining the concepts, and expressed their explanations about the concepts on mathematical operations. In addition, it was determined that the students made explanations about the concept images in the applications in the examples.
Understanding trigonometry relational system is a school mathemat-ics demanding topic. The angle, the unit circle and the trigonometric functions are its foundational notions. Trigonometric contents mean-ing and their understanding involve these three concepts and their re-lationships. This research aims to deepen in the pre-service teachers’ understanding about the angle, the unit circle and the trigonometric function when converting notions between two trigonometric repre-sentation systems based on the unit circle and the trigonometric func-tions. The results indicate that pre-service mathematics teachers’ pre-sent a lack of connections between the goniometric and the analytical representation system.
Given the vast experiences pre-service mathematics teachers (PSMTs) have with functions, one may assume that they have existing personal definitions and concept images. The purpose of this paper is to build an understanding of the variety of images that one might associate with a definition of function that includes ‘for each input there is one output’ and how these images are coordinated with the definition to determine whether an object is a function or not. Two cases are presented to describe PSMTs who were working from the same personal definition of function, yet used that definition differently within a task designed around the function concept. The task itself was novel in that it is presented in a non-algebraic context, namely a vending machine. As such, PSMTs were required to coordinate their personal definition and images within the novel context. Attending to PSMTs’ concept images revealed that though they were working from the same personal definition, images they drew upon to determine whether or not an object is a function were varied.
The aim of this study is to examine prospective mathematics teachers’ generalizations of trigonometric functions from the unit circle to the Cartesian coordinate system. The researcher developed a test that aimed to reveal students’ generalizations, as well as the possible differences between them. The test was administered to 30 students who were near completion of their university degree program. The findings showed that the students were unable to establish the link between the unit circle and the Cartesian coordinate representation system; and therefore, they were not able to interpret the outputs of trigonometric functions with input of a real number that is not a multiple of π. The researcher also found that the students had developed certain misconceptions regarding the properties of trigonometric functions. To improve the teaching of trigonometric functions an instructional sequence and an alternative definition for trigonometric functions is proposed.
In this paper, research on some problematic aspects high school students have in learning trigonometry is presented. It is based on making sense of mathematics through perception, operation and reason in the case of trigonometry. We analyzed students' understanding of trigonometric concepts in the frame of triangle and circle trigonometry contexts, as well as the transition between these two contexts. In the conclusion, we present some new problematic aspects we noticed.
The research was carried out with two groups of high school students, one of them at the beginning of their trigonometry learning (17 years old) and the other at the end of their high school education (19 years old). The students were given a questionnaire similar to that of Chin and Tall, and we analyzed the students' response. In our research, we noticed that students have difficulties with properties of periodicity and the fact that trigonometric functions are not one-to-one. In addition, there is poor understanding of radian measure and a lack of its connection to the unit circle.
This study investigates pre-service and in-service mathematics teachers' subject knowledge of radian. Subject knowledge is investigated under the theoretical frameworks of concept images and cognitive units. Qualitative and quantitative research methods were designed for this study. Thirty seven pre-service and fourteen in-service mathematics teachers' completed a questionnaire which aims to assess their understanding of radian. Three pre-service and one in-service teachers were selected for individual interviews on the basis of theoretical sampling. The data indicated that participants' concept images of radian were dominated by concept images of degree.
This empirical study tests the hypothesis that the versatile learning of trigonometry using interactive computer graphics would lead to a greater improvement in the performance of girls over boys. The experiment was carried out with 15 year old pupils in two schools with matched entry standards, each subdivided by ability into four corresponding mixed gender groups. In every case, experimental boys improved more than control boys and experimental girls improved more than control girls. However, whilst the control girls performance deteriorated compared with the control boys, the experimental girls performance improved in comparison with the experimental boys, eventually becoming superior in all but the least able group.
Mathematical proof seems attractive to some, yet impenetrable to others. In this paper a theory is suggested involving "cognitive units" which can be the conscious focus of attention at a given time and connections in the individual's cognitive structure that allow deductive proof to be formulated. Whilst elementary mathematics often involves sequential algorithms where each step cues the next, proof also requires a selection and synthesis of alternative paths to make a deduction. The theory is illustrated by considering the standard proof of the irrationality of 2 and its generalisation to the irrationality of 3.Cognitive units and connections
The concept image consists of all the cognitive structure in the individual's mind that is associated with a given concept. This may not be globally coherent and may have aspects which are quite different from the formal concept definition. The development of limits and continuity, as taught in secondary school and university, are considered. Various investigations are reported which demonstrate individual concept images differing from the formal theory and containing factors which cause cognitive conflict.
In this case study, we have investigated the construction of understanding of the motion of an object down an inclined plane which takes place through the process of model building. This study was conducted in an integrated algebra, trigonometry, and physics class at an alternative public school. The components of the modeling process explored in the study are the action of building representations and relationships from physical phenomena, the use of a simulation environment to explore conjectures, and the iterative process of developing and validating a solution through the use of a multirepresentational analytic tool. Four major results related to student model building emerged from this study. First, students pursued problems with far more diversity in approaches than the problem itself might have initially suggested. Second, this analysis challenges conventional notions of closure and completeness. Third, the integration of the simulation environment provided access to an expert's model that could be used as the students built their own model of the phenomena being investigated. The fourth theme is that of progressive complexity in the student model as a structure that was built over an extended period of time. The implications of these results for both instruction and curriculum are discussed.
This paper describes the process of inducting theory using case studies-from specifying the research questions to reaching closure. Some features of the process, such as problem definition and construct validation, are similar to hypothesis-testing research. Others, such as within-case analysis and replication logic, are unique to the inductive, case-oriented process. Overall, the process described here is highly iterative and tightly linked to data. This research approach is especially appropriate in new topic areas. The resultant theory is often novel, testable, and empirically valid. Finally, framebreaking insights, the tests of good theory (e.g., parsimony, logical coherence), and convincing grounding in the evidence are the key criteria for evaluating this type of research.
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