Experiencing Geometry: Euclidean and Non-Euclidean with History
... No matter how an angle is defined, it is "likely that no formal definition can capture all the aspects of our experience of what an angle is" (Henderson & Taimina, 2005, p. 38). Despite the variability in definitions of angles, there are at least three perspectives from which an angle can be defined: angle as a geometric shape, angle as a region of space, and angle as movement (angle as an amount of rotation) (see Browning et al. 2008;Casas-García, & Luengo-González 2013;Henderson & Taimina 2005). ...
... Angles as geometric shapes or regions of space do not contain movement, and are static. On the other hand, the dynamic conceptualization of angles, namely angle as movement, considers the amount of turn by rotating a ray clockwise or counterclockwise about a fixed point to the position of the other (Browning et al. 2008;Casas-García, & Luengo-González 2013;Henderson & Taimina 2005). Influenced by how the quadrants are located on the Cartesian coordinate system, there is a common belief that the rotation is made counterclockwise unless the direction is specified. ...
... Influenced by how the quadrants are located on the Cartesian coordinate system, there is a common belief that the rotation is made counterclockwise unless the direction is specified. Angles with direction (directed angles) contain arrows that show the direction (clockwise or counterclockwise) and the amount of rotation (Henderson & Taimina, 2005). ...
This case study investigates prospective middle school mathematics teachers’ use of the Angle tool in GeoGebra, focusing on reflex angles in a geometric construction task. They constructed a parallelogram with individual heights for any two consecutive sides. We analyzed their screencasts to identify the ways in which they used reflex angles in GeoGebra. The results indicated that most participants concealed reflex angles in their geometric constructions. They most often showcased the ways in which reflex angles were eliminated utilizing software operations (e.g., keeping angles between 0° and 180°). Implications for future research are provided.
... The idea for a lesson on symmetry and the art of Escher came from a university geometry textbook entitled Experiencing Geometry: Euclidean and Non-Euclidean with History (Henderson & Taimina, 2005), where the authors of this text outline the seven different types of symmetry of line. The authors described symmetry using a definition of isometry, stating that, "an isometry is a transformation that preserves distance and angle measures" (Henderson & Taimina,p. ...
... Initially, the students were limiting symmetry to only reflections. But as I offered more pictures and the students discussed the examples as a group, they were able to informally agree on a definition for symmetry that was similar, albeit simplified, to the definition of isometry offered by Henderson and Taimina (2005). In particular, the students agreed that they needed to look at the length and distances between the lines and the angles of the pictures. ...
... With this agreement on what to look for when searching for symmetry, I then demonstrated for the classes the seven different types of symmetry of a line on the overhead (Henderson & Taimina, 2005), using simple geometric shapes like triangles. ...
... The existence of various definitions of the concepts in the literature is shown as one of the most important reasons for students' difficulties (Butuner & Filiz, 2017;Henderson & Taimina, 2005;Keiser, 2004). Definitions as the building blocks of mathematical thoughts undertake a fundamental task in forming a concept and distinguishing it from other concepts (Cakiroglu, 2015). ...
... She stated that the definitions could generally include one of the three. Another classification is defined angle with the different perspective as "a geometric shape", "a changing and dynamic structure", and "a measurable attribute" (Henderson & Taimina, 2005). In these perspectives, the dynamic concept of angle includes action in the form of a rotation, rotation point, or direction between two lines. ...
... In these perspectives, the dynamic concept of angle includes action in the form of a rotation, rotation point, or direction between two lines. The angle as a measure is explained as the arc length, or the ratio between the areas of the circle segments and angle as a geometric shape is explained by two lines intersecting in space (Henderson & Taimina, 2005). ...
This study aims to determine the misconceptions and difficulties of sixth-grade students on the subject of angles. The study participants are 25 sixth grade students from a public school in a city in western Turkey during the 2017-2018 academic year. This qualitative study used 17 open-ended questions designed by the researchers for data collection to examine the students' misconceptions and difficulties. Data were examined by implementing content analysis. It has been analysed that students cannot define the angle due to difficulties and misconceptions in determining the corners and edges of a symbol. Besides, they also find it difficult to compare the measures of the angles, adjacent angles, complementary and supplementary angles.
... Indeed, the problems that students encounter regarding the concept of angle also lead to other problems in future topics (trigonometric functions, etc.) (Moore, 2013). Henderson and Taimina (2005) list the following conceptions of angle: angle as a geometric shape, union of two rays with a common end point (static); angle as movement; angle as rotation (dynamic); angle as measure; and, amount of turning (also dynamic). Older students may be able to conceptualize angles in turns, ray pairs, or regions, but may not be able to relate them. ...
... In coding the angle as a static or dynamic concept, the opinions in the literature are taken as basis (Clausen-May, 2005;Henderson & Taimina, 2005;Mitchelmore & White, 2000;Wilson & Adams, 1992). The coding criteria for the angle as a dynamic and static concept is given in Table 4. ...
... angle as movement; angle as rotation; angle as measure; and, amount of turning (Henderson and Taimina, 2005) angle as a geometric shape, union of two rays with a common end point (Henderson and Taimina, 2005) How is the angle represented? ...
The present study compared Turkish and Singaporean textbooks with respect to their instructional contents on a difficult topic for most students: the concept of angle. The study used the 3 rd and 4 th grade mathematics textbooks taught in Turkish and Singaporean schools. The analysis showed that Turkish textbooks defined the angle as a static concept, and Singaporean textbooks defined it as both a static and dynamic concept. The definitions of the concept of angle included in the textbooks reflect on the representation of the angle, instructional tools and problems. Turkish students learn angle from textbooks only as a static concept, so they may have difficulties and misconceptions about the subject and related concepts. The findings showed that the contents of Singaporean textbooks offer students more opportunities than Turkish textbooks in learning about the angle as a static and dynamic concept.
... Also, they cannot fully comprehend concepts [7,8]. Finally, students may only learn one of the dual meanings of a concept such as the angle concept [10][11][12][13][14][15]. ...
... In other words, the amount of turn needed to carry one of the angle's arm over the other one without moving out the plane [19]. Since the static and dynamic of definitions represent two diverse situations and have unique limitations [10], the union of these definitions cannot be determined. Due to this issue, students may harbour some misconceptions on the angle concept. ...
The aim of this research was to investigate high achievers’ erroneous answers and misconceptions on the angle concept. The participants consisted of 233 grade 6 students drawn from eight classes in two well-established elementary schools of Trabzon, Turkey. All the participants were considered to be current achievers in mathematics, graded 4 or 5 out of 5, and selected via a purposive sampling method. Data were collected through six questions reflecting the learning competencies set out in the grade 6 curriculum in Turkey and the findings of previous studies that aimed to identify students’ misconceptions of the angle concept. This questionnaire was then applied over a 40-minute period in each class. The findings were analysed by two researchers whose inter-rater agreement was computed as 0.97, or almost perfect. Thereafter, coding discrepancies were resolved, and consensus was established. We found that although the participants in this study were high achievers, they still held several misconceptions on the angle concept such as recognizing a straight angle or a right angle in different orientations. We also show how some of these misconceptions could have arisen due to the definitions or representations used in the textbook, and offer suggestions concerning their content in the future.
... 45). They also have more in common with the approach to geometry taken by Henderson and Taimina (2005) in their book Experiencing Geometry, where the first task is to explore why folding a piece of paper produces a straight line, using concepts of symmetry. In these points of inquiry, we are asked to step out of the bounds of fixed two-dimensional space to explore multidimensional dynamic spaces. ...
... none of them being three! Henderson and Taimina (2005) list the following different conceptions of angle: angle as a geometric shape, union of two rays with a common end point (static); angle as movement; angle as rotation (dynamic); angle as measure; and, amount of turning (also dynamic). Much of the research conducted on the development of the concept of angles has focused on children of age 9 and higher. ...
This paper outlines the new opportunities that that will be changing the landscape of geometry education at the primary school level. These include: the research on spatial reasoning and its connection to school mathematics in general and school geometry in particular; the function of drawing in the construction of geometric meaning; the role of digital technologies; the importance of transformational geometry in the curriculum (including symmetry as well as the isometries); and, the possibility of extending primary school geometry from its typical emphasis on vocabulary (naming and sorting shapes by properties) to working on the composing/decomposing, classifying, comparing and mentally manipulating both two- and three-dimensional figures. We discuss these opportunities in the context of historical developments in the nature and relevance of school geometry. The aim is to motivate and connect the set of papers in this special issue.
... 178). Researchers (Henderson & Taimina, 2005;Matos, 1990aMatos, , 1990bMitchelmore & White, 1998, 2000Rotaeche & Montiel, 2008) found additional classifications of angle in school mathematics: a dynamic rotation or turn involving movement, a fixed geometric figure or configuration, and a measure of size. Each of these views serves a purpose, with advantages and disadvantages depending on context. ...
This article investigates evoked concept images (CIs) of seven secondary preservice mathematics teachers (PMTs) as they interact with different types of representations of radian angle measure. A thematic analysis indicates that representations emphasizing quantitative aspects of angle measure evoke conceptions that are attuned to quantitative elements of radian angle measure. Specifically, quantitative representations evoked CIs relating the length of the arc subtending the angle to the radius. Non-quantitative representations evoked CIs of radian in terms of π or in relation to degrees, even when the representation did not contain angles in these measurements. To foster a quantitative understanding of radian angle measure, teachers and curriculum developers are encouraged to consider how visual representations might support students in conceptualizing radian angle measure quantitatively.
... Angle is another one of those concepts being at the centre of curricula in primary and early secondary education and yet causing a lot of confusion among learners. According to Henderson and Taimina (2005), angle can be defined from at least three different perspectives: (a) angle as a directional relationship between two geometric shapes, i.e. formed between two geometrical objects which can be either segments or 2D geometrical figures; (b) angle as a dynamic notion, indicating a change of one direction both as a turn and as the result of a turn; and (c) angle as a measure represented by a number. In typical school education, angle is approached as a static geometric figure (Freudenthal, 1983), disconnected from real-world contexts. ...
Classification is a complex process that involves scientific concepts and higher-order mental processes such as abstraction, generalization, and pattern recognition. Even though it is an important competence for understanding the world, dealing with data and information, and solving complex problems, the education system embeds just its simplest operations and only in very early schooling. This study examines six middle school students’ activities as they play, modify, and redesign two Tetris-like classification games on the mathematical concepts of number sets and angle in an online authoring system called Sor.B.E.T (Sorting Based on Educational Technology). The qualitative data analysis of students’ dialogues aimed to bring in the foreground the classification processes students applied and the way these processes were entangled with the development of meanings and ideas on the mathematical concepts embedded in the games. According to the results, the play and modding of the two classification games enabled the development of higher-order classification processes such as objects’ properties comparison, properties discrimination, and classes’ encapsulation. They also supported meaning-making processes and triggered discussions about abstract mathematical notions, such as the concept of angle in various typical mathematical or physical contexts and the concept of number sets, the boundaries of each one, and the relationships among them through exploration and learner-generated exemplification.
... Esférica de Menelao, con tres libros, donde el libro I se entiende como una analogía del libro I de los Elementos de Euclides en la superficie de la esfera, se centra en el triángulo esférico y sus propiedades (Henderson y Taimina, 2004). ...
¿CÓMO CITAR ESTE ESCRITO?: Cruz-Amaya, M. y Montiel-Espinosa, G. (2023). Un acercamiento histórico-epistemológico a la geometría esférica en sus inicios. Anais do IX Congresso Iberoamericano de Educação Matemática, 1566 - 1574. São Paulo, Brasil. | RESUMEN: Desde la literatura se cuestiona la poca actualización y pertinencia de la geometría escolar y buscando atender dicha problemática, en algunos proyectos educativos, se han incorporado las geometrías no euclidianas en el currículo. Las causas y consecuencias de esta incorporación se convirtieron en fenómenos de interés, de investigaciones que exponen la necesidad de profundizar en la naturaleza misma de esas geometrías. Así, proponemos los avances de un estudio histórico-epistemológico desde la socioepistemología, buscando identificar y caracterizar las prácticas que anteceden y acompañan la emergencia de la geometría esférica en su génesis, cuyos resultados sirvan de posicionamiento epistemológico inicial para el diseño didáctico.
... Early activities in the course explore the range of what Henderson and Taimina (2005) call the historical "strands" of geometry. Problems and inspirations arise from navigation, visual art, dance, architecture, and mechanical engineering. ...
Introduction
This article illustrates a pedagogical approach to integrating models and modeling in Geometry with mathematics teacher-learners (MTLs). It analyzes the work of MTLs in a course titled “Computers, Teaching, and Mathematical Visualization” (or “MathViz”), which is designed to engage MTLs in making mathematics together. They use a range of both physical and virtual models of 2-manifolds to formulate and investigate geometric conjectures of their own.
Objectives
The article articulates the theoretical basis and design rationale of MathViz; it analyzes illustrative examples of the discourse produced in collaborative investigations; and it describes the impact of this approach in the students’ own voices.
Methods
MathViz has been iteratively refined and researched over the past 6 years. This study focuses on one iteration, aiming to capture the phenomenological experience of the MTLs as they structured and pursued their own mathematical investigations. Video data from two class sessions of the Fall 2021 iteration of the course are analyzed to illustrate the discourse of collaborating students and the nature of their shared inquiry. Excerpts from this class’s Learning Journals are then analyzed to capture themes across students’ experience of the course and their perspectives on its impact.
Results
Analysis of students’ discourse (while investigating cones) shows how they used models and gesture to make sense of geometric phenomena; forged connections with investigations they had conducted throughout the course on different surfaces; and articulated and proved mathematical conjectures of their own. Analysis of students’ Learning Journals illustrates how experiences in MathViz contributed to their conceptualization of making mathematics together, using a variety of models and technologies, and developing a set of practices that that they could introduce with their future students.
Discussion
An argument is made that this approach to collective mathematical investigation is not only viable and valuable for MTLs, but is also relevant to philosophical reflections about the nature of mathematical knowledge-creation.
... Plusieurs auteurs ont abordé la question des perspectives de l'angle en tant que statique ou dynamique (Close, 1982 ;Freudenthal 1973Freudenthal , 1983Krainer, 1993 ;Magina 1994 ;Douek, 1998 ;Mitchelmore, & White, 1998Henderson & Taimina, 2005 ;Fyhn, 2010 ;Kontorovich & Zazkis, 2016a, 2016b. Freudenthal (1973Freudenthal ( , 1983 propose une approche plurielle du concept d'angle. ...
In the scientific literature, the angle is a concept that is difficult to define and teach in mathematics. As part of our research, we explored the two meanings of angle : figure and magnitude. However, the figure of the angle does not allow spontaneous access to its magnitude. We then asked ourselves about a way that would allow students at the end of primary school to apprehend the angle as a magnitude. We hypothesized that the grid could be this means based on Piaget and Inhelder’s (1947) theory of orthogonality and Duval’s (1988) closing law. We proposed to CM2 schoolchildren, before formal instruction, to compare angles using an angle grid (QA), an artifact that we created for our study. It is a half A4 sheet of a square mesh grid, movable and transparent. For our analysis, we used Duval’s (1988) cognitive approach to geometric figures, Vergnaud’s (1990) conceptual field theory, and Rabardel’s (1995) instrumental approach. The results of the students’ written productions show that the number of students who have apprehended the magnitude in terms of the ratio of measurements of two lengths or in terms of measuring the length between the two sides of the angle increased from 3/29 students to 11/28 students six months later. These results show, on the one hand, that the artifact named QA was, within the framework of our experiments, a means for some students to apprehend the angle as a magnitude. On the other hand, that the QA instrumentation has shown to be long- lasting in some students, which seems to support Rabardel’s statement to this effect. In addition to the analysis of all the students’ written productions, two case studies provided a detailed analysis of the apprehension of the angle as a magnitude with respect to the properties of the QA. The results of our study allow us to consider a didactic engineering on a larger number of students as a perspective for our research.
Au regard de la littérature scientifique, l’angle est un concept difficile à définir et à enseigner en mathématiques. Dans le cadre de notre recherche, nous avons exploré les deux acceptions de l’angle, soit la figure et la grandeur. Or, la figure de l’angle ne permet pas d’accéder spontanément à sa grandeur. Nous nous sommes alors interrogés sur un moyen qui permettrait à des élèves de la fin du primaire d’appréhender l’angle en tant que grandeur. Nous avons fait l’hypothèse que le quadrillage pouvait être ce moyen, en prenant appui sur la théorie de l’orthogonalité de Piaget et Inhelder (1947) et la loi de clôture de Duval (1988). Nous avons proposé à des élèves de CM2, avant enseignement formel, de comparer des angles à l’aide d’un quadrilleur d’angle (QA), un artéfact que nous avons créé pour notre étude. Il s’agit d’une demi-feuille A4 d’un quadrillage à maille carrée, mobile et transparent. Pour notre analyse, nous avons pris appui sur l’approche cognitive des figures géométriques de Duval (1988), la théorie des champs conceptuels de Vergnaud (1990), ainsi que l’approche instrumentale de Rabardel (1995). Les résultats des productions écrites des élèves font apparaitre que le nombre d’élèves ayant appréhendé la grandeur en termes de rapport de mesures de deux longueurs ou en termes de mesure de la longueur entre les deux côtés de l’angle passent de 3/29 élèves à 11/28 élèves six mois plus tard. Ces résultats montrent, d’une part, que l’artéfact quadrilleur d’angle (QA) a été, dans le cadre de nos expérimentations, un moyen pour certains élèves d’appréhender l’angle en tant que grandeur. D’autre part, que l’instrumentation du QA s’est révélée pérenne chez certains élèves, ce qui semble appuyer l’affirmation de Rabardel (1995) à cet effet. Outre l’analyse de l’ensemble des productions écrites des élèves, deux études de cas ont permis une analyse fine de l’appréhension de l’angle en tant que grandeur au regard des propriétés du QA. Les résultats de notre étude permettent d’envisager une ingénierie didactique sur un plus grand nombre d’élèves comme perspective à notre recherche.
... The courses required for PTs have included conventional offerings at the undergraduate level (e.g., linear algebra), as well as two types of courses for teachers described by Murray andStar (2013, p. 1298) as "secondary mathematics from an advanced standpoint and … tertiary mathematics with connections." In terms of geometry courses for teachers Moise's (1974) text written during the New Math movement and the more recent text by Clark (2012) illustrate courses in which the Euclidean geometry material for secondary schools is taught from an advanced standpoint, while Greenberg (1993) or Henderson and Taimina (2019) illustrate courses that offer advanced geometry with connections. In her examination of what types of courses for secondary teachers are valued by mathematicians, Lai (2019) noted that some of these courses also include development of pedagogical content knowledge (PCK; Shulman, 1986) though this is not frequent. ...
This paper contributes to understanding the work of teaching the university geometry courses that are taken by prospective secondary teachers. We ask what are the tensions that instructors need to manage as they plan and teach these courses. And we use these tensions to argue that mathematics instruction in geometry courses for secondary teachers includes complexities that go beyond those of other undergraduate mathematics courses–an argument that possibly applies to other mathematics courses for teachers. Building on the notion that the work of teaching involves managing tensions, and relying on interviews of 32 instructors, we characterize 5 tensions (content, experiences, students, instructor, and institutions) that instructors of geometry for teachers manage in their work. We interpret these tensions as emerging from a dialectic between two normative understandings of instruction in these courses, using the instructional triangle to represent these.
... For this curve is a straight line, otherwise is a circle, which is called Appollonian circle. Subsequently Appollonius proved that a circle (with center and radius belongs to the family only if (1.2) where and are on the same ray [1]. The nineteenth-century has great importance in view of the history of geometry. ...
The purpose of this paper, first, is to give a definition of the focal surfaces of the inverse of a given regular surface in E3. Second, some new characteristic properties of the focal surfaces are to express depending on the algebraic invariants of the inverse surface of a given regular surface. In the last part of the study, we gave examples supporting our claims and plotted their graphics with the help of Maple software.
... The fundamental idea of the Comparative Geometry project is to teach and learn two (or later more) different systems of geometry simultaneously. Students compare and contrast concepts and theorems in two or more different worlds of geometry, as in the book of David W. Henderson and Daina Taimina for university students [3], or my book for upper elementary and secondary school [4], or the book of Anna Rybak and István Lénárt for the interested layman and the professional teacher as well [5]. ...
I have been working on an educational project called Comparative Geometry for decades. The project is based on teaching and learning plane geometry and spherical geometry simultaneously, mainly through direct experimentation with hands-on tools, and intensive use of discussion between classmates. This work has gravely been affected by the changes that occurred two months ago because of the pandemic, still causing emergencies in many areas of education. In this article, I describe how I tried to adapt the work of a university course to an emergency; the methods by which I enabled direct experimentation and personal communication that were not possible in the given situation; my efforts to reduce the drawbacks of the situation and take advantage of the potential benefits.
... The authors propose some representations of the angle (Henderson D.W. & Taimina D., 2005), referred to static and dynamic ones to be introduced into primary school, such as: rotation angle (turning), sector angle (the quantity shared by the set of all superimposed angular sectors), and a pair of half lines that extend from a common point (openness or inclination). They are fully aware, however, that the term "angle" must be considered in relation to the various definitions that have been proposed over time. ...
... Many publications deal with the topic, including spherical geometry, both for its own sake and as a bridge to other types of geometry. We mention here Experiencing Geometry, Euclidean and Non-Euclidean, with History by Henderson and Taimina (2001), and Heavenly Mathematics: The Forgotten Art of Spherical Trigonometry by van Brummelen (2012). It was a pleasant surprise to note that a South African schoolbook for Grade 8 also deals with spherical geometry (Johnson, Davidson, Jaffer and Galant, 2000). ...
... Many publications deal with the topic, including spherical geometry, both for its own sake and as a bridge to other types of geometry. We mention here Experiencing Geometry, Euclidean and Non-Euclidean, with History by Henderson and Taimina (2001), and Heavenly Mathematics: The Forgotten Art of Spherical Trigonometry by van Brummelen (2012). It was a pleasant surprise to note that a South African schoolbook for Grade 8 also deals with spherical geometry (Johnson, Davidson, Jaffer and Galant, 2000). ...
... Many publications deal with the topic, including spherical geometry, both for its own sake and as a bridge to other types of geometry. We mention here Experiencing Geometry, Euclidean and Non-Euclidean, with History by Henderson and Taimina (2001), and Heavenly Mathematics: The Forgotten Art of Spherical Trigonometry by van Brummelen (2012). It was a pleasant surprise to note that a South African schoolbook for Grade 8 also deals with spherical geometry (Johnson, Davidson, Jaffer and Galant, 2000). ...
This chapter first seeks to provide evidence of ongoing social exclusion at
school level in relation to quality mathematics attainment in South Africa. It
does that by drawing on secondary sources that point to exclusion along class
and race lines.
Second, this chapter approaches the sociopolitical aspects of mathematics
education in South Africa from a mathematical perspective. It achieves that
by setting up as the problem the question of how we define the object of
mathematics. It approaches that question by offering a personal, historical
perspective on the current definition of mathematics. Within that
perspective, school mathematics is defined as a dynamic product of human
reasoning. This chapter explores how that reasoning has been shaped by
South Africa’s past and present as well as the interpenetration of the local
with the national and the global.
... Angles are very complex concepts that can be defined in many different ways. [Henderson and Taimina (2005) have written that depending on your definition of an angle, a triangle might have 3, 6, 9, or 12 angles!] A dynamic approach to angle involves defining angle as the amount of turn, an approach to early angle learning that does not depend on the use of degrees and that does not confuse the size of the angle with the length of its arms (Kaur, 2017). ...
This commentary explores ways in which cognitive psychologists and mathematics educators could jointly contribute to better understanding the temporal dimension of spatial reasoning, where temporality refers primarily to the movement of mathematical objects and relations.
... 37. For example Van Maanen [1992], Bartolini Bussi [2000], Isoda [2003], Sangaré [2003], Henderson and Taimina [2005]. ...
In "La Géométrie," Descartes proposed a “balance” between geometric constructions and symbolic manipulation with the introduction of suitable ideal machines. In particular, Cartesian tools were polynomial algebra (analysis) and a class of diagrammatic constructions (synthesis). This setting provided a classification of curves, according to which only the algebraic ones were considered “purely geometrical.” This limit was overcome with a general method by Newton and Leibniz introducing the infinity in the analytical part, whereas the synthetic perspective gradually lost importance with respect to the analytical one—geometry became a mean of visualization, no longer of construction.Descartes’s foundational approach (analysis without infinitary objects and synthesis with diagrammatic constructions) has, however, been extended beyond algebraic limits, albeit in two different periods. In the late 17th century, the synthetic aspect was extended by “tractional motion” (construction of transcendental curves with idealized machines). In the first half of the 20th century, the analytical part was extended by “differential algebra,” now a branch of computer algebra.This thesis seeks to prove that it is possible to obtain a new balance between these synthetic and analytical extensions of Cartesian tools for a class of transcendental problems. In other words, there is a possibility of a new convergence of machines, algebra, and geometry that gives scope for a foundation of (a part of) infinitesimal calculus without the conceptual need of infinity. The peculiarity of this work lies in the attention to the constructive role of geometry as idealization of machines for foundational purposes. This approach, after the “de-geometrization” of mathematics, is far removed from the mainstream discussions of mathematics, especially regarding foundations. However, though forgotten these days, the problem of defining appropriate canons of construction was very important in the early modern era, and had a lot of influence on the definition of mathematical objects and methods. According to the definition of Bos [2001], these are “exactness problems” for geometry.Such problems about exactness involve philosophical and psychological interpretations, which is why they are usually considered external to mathematics. However, even though lacking any final answer, I propose in conclusion a very primitive algorithmic approach to such problems, which I hope to explore further in future research.From a cognitive perspective, this approach to calculus does not require infinity and, thanks to idealized machines, can be set with suitable “grounding metaphors” (according to the terminology of Lakoff and Núñez [2000]). Thisconcreteness can have useful fallouts for math education, thanks to the use of both physical and digital artifacts (this part will be treated only marginally).
... Activities were purposefully designed to ensure that a clear maths concept was explored in such a way that whole body movement was integral to the task. This drew on Henderson and Taimina's work [35] which sets out three ways of defining angles: an angle as a geometric figure (a pair of rays with a common endpoint), an angle as a dynamic figure (a turn or rotation), and an angle as a measure. Activities were mapped to the curriculum, and tailored to the curriculum requirements specific to the age group of each set of participants. ...
An increasing body of work provides evidence of the importance of bodily experience for cognition and the learning of mathematics. Sensor-based technologies have potential for guiding sensori-motor engagement with challenging mathematical ideas in new ways. Yet, designing environments that promote an appropriate sensori-motoric interaction that effectively supports salient foundations of mathematical concepts is challenging and requires understanding of opportunities and challenges that bodily interaction offers. This study aimed to better understand how young children can, and do, use their bodies to explore geometrical concepts of angle and shape, and what contribution the different sensori-motor experiences make to the comprehension of mathematical ideas. Twenty-nine students aged 6–10 years participated in an exploratory study, with paired and group activities designed to elicit intuitive bodily enactment of angles and shape. Our analysis, focusing on moment-by-moment bodily interactions, attended to gesture, action, facial expression, body posture and talk, illustrated the ‘realms of possibilities’ of bodily interaction, and highlighted challenges around ‘felt’ experience and egocentric vs. allocentric perception of the body during collaborative bodily enactment. These findings inform digital designs for sensory interaction to foreground salient geometric features and effectively support relevant forms of enactment to enhance the learning experience, supporting challenging aspects of interaction and exploiting the opportunities of the body.
... İlgili alanyazında, açının tanımını, statik (durağan) ve dinamik olmak üzere ikiye sınıflandırmışlardır. Statik tanıma göre, başlangıç noktaları aynı iki ışının birleşimi şeklinde tanımlanmakta (Mitchelmore ve White, 1998) ve dinamik tanıma göre, iki doğru arasındaki dönme miktarının ölçüsü (dönme tanımı) şeklinde tanımlanmıştır (Clements ve Burns, 2000). Sonuç olarak, öğrenciler açı kavramının ikili anlamından birini öğrenebilirler (Henderson ve Taimina, 2005;Keiser, 2014). ...
Bu çalışmada, ilköğretim matematik öğretmeni adaylarının bilgisayar destekli matematik öğretimi (BDMÖ) dersi kapsamında temel Logo komutlarını kullanarak geometrik şekilleri çizerken açı ve dönme kavramlarını içeren problemlere yönelik yaptıkları hataları ve olası kavram yanılgılarını tespit etmeyi amaçlanmıştır. Bu amaçla, 2016-2017 eğitim-öğretim yılı güz döneminde 37 ilköğretim matematik öğretmeni adayına 10 saatlik Logo programlama dili eğitimi verilmiştir. Daha sonra adaylardan Logo komutlarını kullanarak açı ve dönme kavramıyla ilgili performans gösterebilecekleri 8 sorudan oluşan açık uçlu bir sınav yapılmıştır. Bu çalışmada 8 soru içinden seçilen 5 soru üzerinde durulmaktadır. Çalışmanın verileri, nitel veri toplama yöntemlerinden doküman analizi ve klinik mülakat ile elde edilmiştir. Elde edilen bulgular neticesinde, öğretmen adayları kâğıt-kalem ortamında problemleri çözerken Logo programlama komutlarından daha çok dönme açısı ve geometrik şekillerin özellikleriyle ilgili hatalar yaptıkları tespit edilmiştir. Bu hatalar şu üç noktada yoğunlaşmaktadır: Dönme açısı, açı-kenar bağıntısı ve eksik kodlama. Logo ’da performans gösterebilecekleri problemleri çözerken bazı öğretmen adaylarının programlama becerilerinin sınırlı; diğerlerinin kabul edilebilir seviyede olduğu belirlenmiştir. Araştırma sonucunda elde edilen bulgular dâhilinde Logo programlama dili belirli geometrik kavramların öğrenilmesinde kullanılırken sınırlı geri bildirim veren yapılandırmacı yaklaşımla ile hayata geçirilmelidir.
... Many publications deal with the topic, including spherical geometry, both for its own sake and as a bridge to other types of geometry. We mention here Experiencing Geometry, Euclidean and Non-Euclidean, with History by Henderson and Taimina (2001), and Heavenly Mathematics: The Forgotten Art of Spherical Trigonometry by van Brummelen (2012). It was a pleasant surprise to note that a South African schoolbook for Grade 8 also deals with spherical geometry (Johnson, Davidson, Jaffer and Galant, 2000). ...
This book offers an overview of the current position of teaching and learning Mathematics in South Africa. It poses the question whether there is (or should be) a unifying logic informing the way mathematics is taught and learnt. Chapters written by a number of eminent local and international mathematics educators and researchers contribute ideas towards creating deeper understandings of mathematics, developing learners with productive mathematical identities, and ways of nurturing abstract reasoning.
... Aside from this, during the past decades, many researchers investigated the concept of angle, which is the basic study area of geometry, with different viewpoints, and made some definitions. Henderson and Taimina (2005) defined the angle concept in three different viewpoints as (i) a geometric shape, (ii) a changing and dynamic structure, (iii) a measurable attribute. The modern definition of the angle concept on the other hand, is (1) the measurement of the revolving of a beam from one point to another in a position change (dynamic dimension) (2) a geometrical shape formed by two beams whose starting points are common (static dimension) and (3) the gap between two beams (static dimension) (Keiser, 2004). ...
... Serious games is a very successful and rapidly expanding area of learning research, which is driven by the fact that, according to data reported by the Pew Internet & American Life Project [5], 97% of teens in the U.S. play video games. Furthermore, there is a well documented theoretical basis that indicates that there is a significant increase in the learning outcomes when the students build associations between mathematical concepts and other real world equivalent metaphors [10,23,24,30,34,36]. The link between action and cognition related to mathematical concepts has been well studied and this project is founded on the theoretical basis of the reported findings [21,35,37,32,33,30]. ...
... The other necessary component of the definition for a square is right angle. As a multifaceted concept, angle may be defined as: a geometric shape, the union of two rays with a common end point, a movement, a rotation, a measure, and the amount of turns (Henderson & Taimina, 2005). Since angle derived from the Greek word gonia, to mean 'corners or knees', the word corner is acceptable for younger children whereas a formal and precise term is preferred for secondary years. ...
This study surveyed and analysed four secondary school students’ writing about a square. Sfard’s discursive approach to understanding mathematical discourse was used to analyse the responses collected from 214 Australian secondary school students. The results showed that geometric knowledge was developed experientially and not developmentally. This in turn helps refining the development of a geometric learning progression, with the accompaniment of a set of validated assessment tools and learning tasks that seeks to deepen teachers’ understanding of geometric reasoning and support student learning.
... angle as a measurement (e.g., Henderson & Taimina, 2005). Accordingly, research in mathematics education has focused on developing the pedagogy for bridging these perspectives on the concept of angle (e.g., Mitchelmore, 1998;Smith, King, & Hoyte, 2014). ...
This study is concerned with tensions between the two different perspectives on the concept of angle: angle as a static shape and angle as a dynamic turn. The goal of the study is to explore how teachers cope with these tensions. We analyze scripts of 16 in-service secondary mathematics teachers, which feature a dialogue between a teacher and students around the following statement: “The sum of the exterior angles of a polygon is 360°.” The findings show that while addressing a variety of intellectual needs of their student characters, in many cases, the teachers compromise the mathematical rigour of the concept of angle.
... For example, to foster an intuitive understanding of geodesics on non-Euclidean surfaces to support more mathematically rigorous notions of geodesics, the professor introduced a variety of metaphors. One metaphor was the idea of imagining the experience of a bug as it walks along a given surface (Henderson and Taimina, 2005). As the bug walks in a straight path, from its perspective, the bug is walking along a geodesic of that surface. ...
The purpose of this study was to investigate the discourse elicited by a multi-representational view of non-Euclidean surfaces, the artifacts used to model these surfaces, and the metaphorical discourse used to construct mathematical understanding. A multi-tiered teaching experiment was conducted in a 15-week undergraduate course in non-Euclidean geometry. The results suggest that the careful use of metaphor helped provide an intuitive base for a more conceptual understanding of geometric concepts. Introduction Given the communicative nature of mathematical learning, semiotics as a theoretical perspective has become a viable framework for research in mathematics education. While the term semiotics in its root form refers to the study of signs in communication, the breadth of the body of mathematics education research spans a variety of issues related to semiotic theory. Some semiotic issues investigated are the role of representations (Doerfler, 2000; Presmeg, 1992, 2002; Sfard, 2000b), discourse (Presmeg, 1997, 1998; Sfard, 2000a, 2001), and cultural artifacts (Hoyos, 2002) in the semiotic mediation of the construction of mathematical knowledge. Representations can be thought of as the form, perceptual or cognitive, which the mathematical concept takes. Furthermore, mathematical representations may take the form of a literal symbol (Sfard, 2000) or exist in the mind of the learner (Doerfler, 2000: Presmeg, 1992). Cultural artifacts refer to physical objects that are external to the cognizing being, and mediate the internal construction of psychological constructs (Mariotti, 2000). Cultural artifacts may be technologically advanced or complex (e.g. -dynamic-geometry software, computer algebra systems, or applets), or technologically simple (e.g. – pencil, paper, Lenart Spheres, or everyday objects). In an instructional setting, the teacher constructs an activity utilizing the artifact in order to promote the construction of a mathematical concept. At the same time, the student uses the artifact to accomplish the given activity. The artifact serves as a semiotic mediator to elicit meaningful mathematical discourse on a representational level; in other words, artifacts are tools that help us talk and write about mathematical objects. Furthermore, the relationship between representations, discourse and cultural artifacts is reflexive in which there is an active interaction between the three strands in the construction of one's mathematical understanding. Within this three-fold framework or representations, discourse, and cultural artifacts is intuitive process that the learner utilizes to make sense of the mathematical investigation and discourse. While the term intuition has many different connotations, Fischbein, Tirosch, and Melamed (1981) characterize intuition as a direct acceptance without the necessary support of an explicit detailed justification. Furthermore, there is an immediacy to this form of knowledge. In this study, the lessons and instruction were designed to enable an intuitive thought process to develop a deep conceptual understanding of non-Euclidean geometry. To help guide this intuitive process, the literary notion of metaphor (Presmeg, 1992, 1997, 1998) served as an important construct to help foster the students' mathematical intuition, and to better understand the discursive activity in the meaning making process in this study. The use of _____________________________ Alatorre, S., Cortina, J.L., Sáiz, M., and Méndez, A.(Eds) (2006). Proceedings of the 28 th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. Mérida, México: Universidad Pedagógica Nacional.
... Detailed course notes (including all of the projects) for both of these courses are available from the Journal of Inquiry-Based Learning in Mathematics [3,4]. These notes loosely follow the first several chapters of the textbook Experiencing Geometry by Henderson & Taimiņa [1], and cover a mixture of Euclidean and non-Euclidean geometry. ...
This paper describes a mentoring program designed to prepare novice instructors to teach a college geometry class using inquiry-based methods. The mentoring program was used in a medium-sized public university with approximately 12,000 undergraduate students and 1,500 graduate students. The authors worked together to implement a mentoring program for the first time. One author was an associate professor and experienced using inquiry-based learning. The other author was a graduate student in mathematics education. During the course of the year the graduate student first observed and then taught a college level inquiry-based geometry course for pre-service teachers. This article describes the details of this mentoring program and our reflections on how the program went.
Argument
The Fourth Postulate of Euclid’s Elements states that all right angles are equal. This principle has always been considered problematic in the deductive economy of the treatise, and even the ancient interpreters were confused about its mathematical role and its epistemological status. The present essay reconsiders the ancient testimonies on the Fourth Postulate, showing that there is no certain evidence for its authenticity, nor for its spuriousness. The paper also considers modern mathematical interpretations of this postulate, pointing out various anachronisms. It further discusses the validity of the ancient proof by superposition of the Fourth Postulate. Finally, the article proposes an interpretation of the history of the concept of angle in Greek geometry between Euclid and Apollonius, and puts forward a conjecture on the interpolation of the Fourth Postulate in the Hellenistic age. The essay contributes to a general reassessment of the axiomatic foundations of ancient mathematics.
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.
Embodied cognition has been a useful means to support learning in science education. In this article, we describe a model of embodied cognition through analogical mapping that is supported through the participatory simulation and mathematical description of a link and pivot system. We look at the propensity for this model of embodied cognition to support student mechanistic reasoning. While these link and pivot systems provide students with immediate perceptual accessibility to their components, they do not provide perceptual accessibility to the way that forces are transmitted through them. In an afterschool STEM program, students engaged in a participatory simulation (the Rope Walk) of the link and pivot system that allowed them to simulate the parts and forces present. Then, students progressively re‐described these physical experiences as a mathematical system; they then described the simulation and mathematical system as the physical link and pivot system. In this study, this model of embodied cognition was validated through a microgenetic analysis of student talk and gesture during the first two instructional sessions as well as during pre‐ and post‐assessments. These data show that students were supported to construct analogical mappings between three analog conceptual domains: the participatory simulation (the Rope Walk), the physical system (link and pivot system), and the mathematical system (the mathematics of circles). In addition, students made gains on measures of mechanistic reasoning and explanation.
This paper is a self-independent continuation of my article on Comparative geometry between the plane and the sphere that was presented at the previous edition of ICon-MaSTEd Conference 2020. Below I discuss the possibility of adding a third geometry to the plane and the sphere, namely, the hyperbolic geometry on the hemisphere. I describe my own path to the subject, then the content of the syllabus which contains basic concepts of hyperbolic geometry for future preschool, elementary school, and secondary school teachers. Finally, I give reasons to introduce the subject into primary and secondary schools, not just for the “talented” but also for the “average” students.
Ignoring the remedial incapacity ensuring future suffering
Retreat to an eternal stronghold from present responsibility
Holy fatherhood and paternal responsibility
Necessary incompleteness
Cognitive mystery of holes, lacunae and incompleteness
Nature of metaphysical and theological holes
Cognitive and experiential black holes
Holiness and unholiness -- an unholy complementarity?
Holiness framed by a triangulated configuration of holes
Cultivating "holiness" and "unholiness" in all their forms?
Vital hole dynamic: embracing error, otherness and neglect
Missing linking process to enable the global resolutique
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.
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.
This paper examines the effect of the use of dynamic geometry environments on young children’s (ages 5–6, kindergarten/grade 1) thinking about angle. It provides a detailed description of introductory sessions of a geometry unit about angle, during which children worked in a whole classroom setting in which they could interact directly with Sketchpad through an interactive whiteboard. Using Sfard’s [2008, Thinking as communicating: Human development, the growth of discourses and mathematizing. Cambridge: Cambridge University Press] communicational approach, an attempt is made to show how children developed a reified discourse on angles and were able to develop an understanding of angle as ‘turn’, that is, of angle as describing an amount of turn. This discourse focused on the behaviour of a pre-constructed sketch which instantiated both dynamic and static representations of angle. Extending prior research on children’s difficulties in unifying static and dynamic conceptions of angles, this study provides one way of establishing a relationship between angle-as-turn and angle-as-shape conceptions. Gestures and motion played an important role in children’s developing conceptions of angles. It presents implications of considering young children’s embodied forms for communications along with their verbal communication for understanding their mathematical thinking.
The aim of this study is to clarify the typological structure and existence of regularity in contemporary architecture that is made of a square plan. The results can be summarized in the following statements: Explication of linkage structure between the compositional elements.
Discovery of latent structures that define types.
Symmetry and Asymmetry of the compositional elements
Unity and plurality of the compositional elements.
Explication of the character of the morphological structure in consideration of size.
International standardized tests are showing an underperformance of students in geometry and spatial ability relative to other content domains. In this research, we examined the relationship between a teacher’s judgment of third‐ and fourth‐grade students’ geometry and spatial ability (i.e., grades), three spatial ability tasks (Water‐Level‐Task, the Rod‐and‐Frame Test, and the Mental‐Rotations‐Test), composite spatial ability score, and standardized test scores of geometry and spatial ability. Results showed that this teacher’s judgment of a student’s geometry and spatial ability was more ambitious than evidence from the other measures. While two of the spatial ability tasks and the composite spatial ability score showed improvement over time, the teacher’s judgment of students’ geometry, and spatial ability did not show improvement over time. Implications for learning will be discussed.
Bu çalışmanın amacı ortaokul matematik öğretmenlerinin öğrencilerde açı kavramı ile ilgili var olan kavram yanılgıları ile ilgili farkındalık durumlarını belirlemektir. Çalışmanın örneklemini, çalışmaya katılmaya gönüllü olan 16 matematik öğretmeni oluşturmaktadır. Veri toplama aracı olarak “demografik bilgiler” ve “kavram yanılgılarının farkındalıkları” olmak üzere iki ana kısımdan oluşan bir form kullanılmıştır. Veriler üzerinde içerik analizi yapılmış, analiz sonuçları karşılaştırıldığında ise iki araştırmacının kodlamaları arasındaki uyum indeksi 0.88 olarak bulunmuştur. Çalışmaya katılan öğretmenlerin tamamının, derslerde sadece açının statik tanımı üzerinde durdukları ve öğrencilerin açılar konusunda yaşadıkları kavram yanılgılarını tespit etmede zorluklar yaşadıkları tespit edilmiştir. Çalışmaya katılan öğretmenler, kavram yanılgılarını gidermek için kavramları yeniden anlatma, açının statik tanımına ek olarak açı konusunda ki bazı yaygın kavram yanılgılarına vurguda bulunma ve somut materyal kullanma gibi öğretim yöntemlerine başvurabileceklerini belirtmişlerdir. Çalışmanın sonucuna dayalı olarak, öğretmenlere, öğrencilerde oluşabilecek kavram yanılgılarını gidermek için nasıl öğretim faaliyetleri tasarlayabileceklerine yönelik hizmet içi eğitim kursları düzenlenebilir.
The astronomical instrument known as the astrolabe has been called star taker, mirror of the sun, and recently, the first personal computer. First developed by the ancient Greeks, it was refined and improved by scientists working in medieval Islamic lands. Studying it reveals the many ways in which art, science and mathematics were employed in its making and its use. It is a product of all three disciplines: it works through stereographic projection, making trigonometric calculations possible. By rotating the movable parts that indicate the relative positions of heavenly bodies, the user can determine exact times and distances on earth. It is also an example of the fine art of metalworking. The essay explores the astrolabe's functioning and significance with a particular focus on a fourteenth-century astrolabe in the Aga Khan Museum collection in Toronto. Its potential for use in museum education is examined, with suggestions for in-gallery interpretive strategies.
The purpose of the present study was to determine whether there are visual-spatial gender differences in two-year-olds, to investigate the environmental and cognitive factors that contribute to two-year-olds’ visual-spatial skills, and to explore whether these factors differ for boys and girls. Children (N = 63; Mage = 28.17 months) were assessed on their visual-spatial skills and on measures related to visual-spatial skills: intelligence, quantitative reasoning, working memory, and home spatial activity engagement. Children’s mothers were assessed on mental rotation ability. Results found no difference between boys’ and girls’ visual-spatial skills at age two. Quantitative reasoning contributed the most to girls’ visual-spatial skills. No variables were predictive for the boys, though boys with higher spatial activity frequency had higher visual-spatial skills. The differential predictors have implications for the development and fostering of visual-spatial skills, particularly for girls, who may be at a disadvantage in this area when they are older.
Elementary students' difficulties with angles in geometry are well documented, but we know little about how they conceptualize angles while solving problems and how their thinking changes over time. In this study, we examined 26 third and fourth grade students completing a body-based angle task supported by the Kinect for Windows. We used fine-grained, multimodal data detailing students' actions and language to identify three common patterns of interactions during the task: the explore, dynamic, and static clusters. We found that students with higher learning gains spent significantly more time in the dynamic cluster than students with low learning gains. Implications for mathematics teaching and research using body-based tasks are discussed.
Tactile mathematics, defined as recognizing deep mathematical concepts through engagement with physical objects, can be used to help students discover mathematics for themselves. This paper discusses the design of tactile learning activities, the insertion of such activities into existing courses, and special considerations for courses to be taught almost entirely with tactile activities. We explain a specific example activity for a group theory course. A collection of mathematics faculty members experienced in tactile learning contribute their thoughts on the implementation of largely tactile mathematics courses. We end with the role of tactile mathematics in the author’s career.
The concept of angle emerges in numerous forms as the learning of mathematics and its applications advances through the high
school and tertiary curriculum. Many difficulties and misconceptions in the usage of this multifaceted concept might be avoided
or at least minimized should the lecturers in different areas of pure and applied mathematics be aware of the way their students
have learned the concept in their previous studies. The article presents an analysis of the literature on the mathematical
and didactical origins of the concept of angle. The purpose of the analysis is to identify the principal characteristics of
the concept required in various contexts of pure-mathematical and applied courses, and to trace the way these appearances
might have been previously presented. Attentiveness to students’ possible lack of mastery of the notion will hopefully help
to prevent or at least minimize difficulties related to it.
ResearchGate has not been able to resolve any references for this publication.