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Recognising the relevance of learning Geometry, and in particular 3D Geometry, this study aims to discuss the contributions that digital technology and paper and pencil approaches can bring to students' learning. We seek, therefore, to identify the differences between the two approaches, and specifically: What factors are relevant in one and the other approach? What does one approach facilitate over the other? A quantitative and a qualitative and interpretive methodology was adopted, and based on a didactic intervention, the students' resolutions of the proposed tasks were analysed. The results obtained show that the experience and prior knowledge of the students with each of the solids involved seems to be decisive in the approach with paper and pencil. However, technology emerges as an enhancing resource when prior knowledge is more fragile. The study also shows differences between the representations supported by the two resources, suggesting the mobilisation of different knowledge by the students in relation to each of the resources.

This article focuses on the possible relationship between predictions, developed during the resolution of geometrical tasks within a Paper-and-Pencil Environment (PPE), and subsequent explorations within a Dynamic Geometry Environment (DGE). Building on Fischbein's Theory of Figural Concepts, and to gain insight into the transition of predictions from the PPE to the DGE, I introduce an analytical scheme that includes the identification of the solvers' figural concepts in focus and geometric predictions; dragging modalities that solvers make use of in the DGE; reactive and proactive dimensions of the interaction with dynamic objects in the DGE. I present a fine-grained qualitative analysis of data collected during task-based interviews as high school students reason about a geometrical task--first on paper-and-pencil and then in a DGE. The analyses of three representative examples show that the interaction with DGE objects can lead students to rethink the geometrical configuration and renegotiate predictions, depending on the circumstances that offer the opportunities to experiment with continuities or discontinuities during the transition of prediction from one to the other environment.

This study aims to examine the roles of tactile materials and origami models in the perception of concept definitions and representations in geometry applications of individuals with visual impairments. This case study draws on data from eight individuals with visual impairments selected according to the purposive sampling method in a mathematics village. Video recordings of focus group interviews conducted with tactile materials and origami models were analyzed by content analysis. Tactile materials made it possible for them to perceive concept definitions and visual representations, while the construction of origami models made every component designable, such as the relationship between concepts, visual prototypes and sub-concepts. Individuals with visual impairments have been quite successful in designing different representations and strategies for geometric concepts with perfect folding. Thus, it has been determined that origami is an accessible material for individuals with visual impairments to do geometry with creases and models.

This study aims to investigate the similarities and differences between prospective mathematics teachers' creative thinking skills in paper-pencil test and on a GeoGebra-supported environment in terms of problem-posing. This case study used purposive sampling method for determining the participants. Findings revealed that the activities carried out in the GeoGebra-supported environment were insufficient to produce creative problems, and GeoGebra's main utility to prospective teachers was in identifying their mistakes related to mathematical concepts and discrepancies among numerical values of the problems posed. The reasons for the low achievement in posing problem were discussed: These were: (1) lack of problem-posing experience; (2) the structure of problem-posing activity; and (3) prospective teachers' mathematical content knowledge.

The main purpose of this study is to investigate mathematics teachers' mathematical thinking process while they are constructing the length of [cube root of 2] by paper folding. To carry out this aim, two teachers--who are PhD. students--were interviewed one by one. During the construction, it was possible to observe the consolidation process of Pythagorean and Thales Theorem. All interviews were videotaped and analyzed through descriptive methods, according to abstraction in context, characteristics of consolidation and mathematical habits of mind. It was found that both two teachers constructed the knowledge of Pythagorean and Thales Theorems before the study and also these knowledges were consolidated. In addition, it was determined that these two approaches (abstraction in context and mathematical habits of mind) were close and corroborated each other. Moreover, consolidation process corroborated mathematical habits of mind.

This study examined pre-service teachers' views and experiences of building geometric constructions on paper and with the interactive whiteboard. The study group consisted of 26 pre-service elementary teachers in a state university in the Black Sea region of Turkey who took the Geometry Teaching course. The data were obtained from an opinion form consisting of open-ended questions and field notes. Findings revealed that almost all of the teachers had no experience with geometric constructions in their previous education and the majority of pre-service teachers' opinions about geometric construction activities are positive. They experienced more problems when building geometric constructions on the interactive whiteboard so their opinions about building geometric constructions on paper are more positive than building on the interactive whiteboard. Moreover, it was determined that pre-service teachers' views imply that building geometric constructions on the interactive whiteboard makes sense in the affective learning domain, whereas building on paper contributes more to the cognitive learning domain. [This study was presented as a verbal presentation at the 1st International Congress on Social Sciences Humanities and Education held in Istanbul, Turkey on 22-32 December, 2017.]

This study examined pre-service teachers' views and experiences of building geometric constructions on paper and with the interactive whiteboard. The study group consisted of 26 pre-service elementary teachers in a state university in the Black Sea region of Turkey who took the Geometry Teaching course. The data were obtained from an opinion form consisting of open-ended questions and field notes. Findings revealed that almost all of the teachers had no experience with geometric constructions in their previous education and the majority of pre-service teachers' opinions about geometric construction activities are positive. They experienced more problems when building geometric constructions on the interactive whiteboard so their opinions about building geometric constructions on paper are more positive than building on the interactive whiteboard. Moreover, it was determined that pre-service teachers' views imply that building geometric constructions on the interactive whiteboard makes sense in the affective learning domain, whereas building on paper contributes more to the cognitive learning domain.

The aim of this study was to determine pre-service elementary school teachers' capabilities of deciding the viewpoint and perspective in geometrical drawing. The study examined geometrical drawings the participants did on isometric paper. This is a case study, a qualitative study method, and the study data were analyzed using written documents. The study sample included 34 senior students in the Education Faculty's Elementary School Teaching Program at a Turkish public university. They were selected from approximately 205 pre-service teachers. The students participated in the study on a voluntary basis. The study results revealed that the pre-service teachers generally had weak skills in geometrical drawing. The participants had better skills in building structures which they saw from the air using unit blocks, and forming a structure of which they had the direct views (left and right; front and back) using unit blocks. However, the participants had weaker skills in drawing the direct views of the structures they had built, recognizing the connections between the left-right views and front-back views of the structures and drawing the structures they built on isometric paper in perspective.

The purpose of this article is to describe a couple of challenging mathematical problems that involve paper folding. These problem solving tasks can be used to foster geometric and algebraic thinking among students. The context of paper folding makes some of the abstract mathematical ideas involved relatively concrete. When implemented appropriately these activities have the potential to address many of the mathematical proficiencies, as delineated by Australian Curriculum and Assessment Reporting Authority (ACARA, 2014). [This paper was first published in the "Australian Senior Mathematics Journal," in 2014 (EJ1093386).]

This paper focuses on pre-service mathematics teachers' geometrical construction processes while using a duo of artefacts, namely the paper-and-pencil environment (PPE) and a dynamic geometry system (DGS). The participants of this case study were two pre-service mathematics teachers, who had limited DGS experiences in solving geometry tasks. The data were collected through task-based interviews in which the participants were asked to complete angle bisector construction first with PPE and then in DGS, using together with the PPE to support the development of their construction strategies in DGS. Data analysis was carried out to examine (dis)continuities that the pre-service teachers faced while connecting the duos and the feedback received during such process. Findings indicated that the pre-service teachers had difficulties in comprehending ( 1) the concept of robustness of the construction and (2) the dependency relationships between the objects in DGS, which created discontinuities for them to differentiate between a static representation on paper and dynamic representation in DGS. In this, the use of duos and the feedback different artefacts generated -- i.e., direct manipulation feedback, evaluation feedback and tool operation feedback from the DGS, and strategy feedback scaffolded by the interviewer --, promoted their conception of angle bisector construction.

Although we move and live in three-dimensional space, many prefer "two-dimensional" thinking in mathematics lessons. As a result, the transition between physical objects and their graphical representations can be considered one of the most common difficulties in learning and teaching solid geometry. This paper describes results of our quantitative-qualitative research focused on solving a non-standard spatial visualisation problem. Demanding mathematical operations are not required in its solution, but creating the right mental images of 3D objects and the ability of their mental manipulation are key to the correct solution. Students' paper-and-pencil solutions were the main source of research data and the principal aim of the study was to examine the relation between the number of correct external representations (iconic or non-iconic) and the numerical scores in student solutions. In this article, we focus on the qualitative findings and supplement them with quantitative results.

A popular physics legend holds that scissors can cut paper with a speed faster than light. Here this counter-intuitive myth is investigated theoretically using four simple examples of scissors. For simplicity, all cases will involve a static lower scissors blade that remains horizontal just under the paper. In the first case, the upper blade will be considered perfectly rigid as it rotates around and through the paper, while in the second case, a rigid upper blade will drop down to cut the paper like a guillotine. In the third case, the paper is cut with a laser rotating with a constant angular speed that is pointed initially perpendicular to the paper at the closest point, while in the fourth case, the uniformly rotating laser is pointed initially parallel to the paper. Although details can be surprising and occasionally complex, all cases allow sections of the paper to be cut faster than light without violating special relativity. Therefore, the popular legend is confirmed, in theory, to be true.

The purpose of this note is to discuss how paper folding can be used to find the exact trigonometric ratios of the following four angles: 22.5°, 67.5°, 27°, and 63°.

This article discusses how paper folding can be used in the classroom to introduce the standard results of school geometry, such as the transversal and parallel lines results, along with results concerning angles in convex polygons and centres of triangles, for example. Angle bisectors, midpoints, perpendiculars are all straightforward "constructions" for the paper folder. If translucent paper is used it renders easy tasks such as duplicating an angle or a segment to any position desired. Paper folding encourages students toward conjecture and invites the teacher to reflect upon the meaning of proof. Results are made plausible by simple observations associated with the folding of paper. A significant benefit of the paper folding technique is its accessibility to students and the affective benefits this confers. (Contains 1 table.)

This paper presents an example of how paper folding can be used in a geometry class to support conceptual understanding. Specifically, it explains an activity that constructs a rhombus and explores its attributes by using paper folding. The steps of constructing a rhombus are described and some discussion questions are given to consolidate understanding.

Various studies have indicated that incorporating origami activities into a geometry lesson can promote elementary and middle school students' mathematical reasoning, enhance their ability to use precise mathematical terminologies, and increase their communication skills. The authors wondered if origami activities have a place in high school geometry. They designed a series of one-straight-cut activities aimed at deepening high school students' essential understanding of geometry; developing their communication skills by producing, interpreting, and explaining diagrams; and thinking about questions embedded into PowerPoint® slides and shown in activity sheets. Links to the one-straight-cut activity sheets, PowerPoint slides and videos are accessible to classroom teachers through the links included in the article. The authors begin by presenting the introductory tasks they used before illustrating one-straight-cut heart activities to preservice teachers. Then they share how these activities planned and implemented by co-authors Dr. Ko and preservice teacher Holden supported their preservice teachers to work collaboratively, think critically, and communicate effectively, along with representative dialogues and reflections from preservice teachers Caroline, (co-author) Devon W. Maxwell, Elisa, (co-author) Runge, and Kevin (first names only indicate pseudonyms). Although the implementation of their activities was conducted with preservice teachers, similar results would be true in the high school setting.

This paper discusses the relevance of using physical situations to introduce the concept of angles at elementary school. We compare the effectiveness of two geometry teaching sequences. In the first one (physical sequence), the pupils learned the angle concept by experimenting on the playground (i.e. mesospace) and then modelling the situation. In the second one (paper-and-pencil sequence), the pupils worked solely in the space of a sheet of paper (i.e. microspace). In both sequences, pupils compared areas of space delineated by an angle between two directions. Pupils in two Grade 3 classes were exposed to one of the two teaching sequences. The unfolding of these sequences was videotaped and analyzed, and the pupils were tested individually, before and after teaching, to measure each sequence's effectiveness. Results showed that both sequences are effective to grasp the angle concept: Most pupils overcame the common erroneous conception of comparing angles' sides' lengths instead of angle openness. The comparison of areas of space delineated by an angle between two infinite directions, which is the two sequences' common core, seems to be the key factor underlying angle conceptualization. This paper ends with a discussion of these results' teaching implications and the merits of each sequence. [ABSTRACT FROM AUTHOR]

Explains how conic sections can be approximated by paper-folding activities and proves why they work. (Author/NB)

This study aims to investigate plane geometry problem-solving strategies of prospective mathematics teachers using dynamic geometry software (DGS) and paper-and-pencil (PPB) environments after receiving an instruction with GeoGebra (GGB). Four plane geometry problems were used in a multiple case study design to understand the solution strategies developed by 2 prospective teachers. The results revealed that although the participants mostly used algebraic solutions in the PPB environment, they preferred geometric solutions in the GGB environment even though algebraic solutions were still possible (the software did not preclude them). Furthermore, different proofing strategies were developed in each environment. This suggests that changing the environment may prompt students to seek for additional solutions, which, in turn, results in a deeper understanding of the problem. As such, using both environments simultaneously in solving the same problems appears to bring about important benefits.

The purpose of this article is to describe a couple of challenging mathematical problems that involve paper folding. These problem-solving tasks can be used to foster geometric and algebraic thinking among students. The context of paper folding makes some of the abstract mathematical ideas involved relatively concrete. When implemented appropriately these activities have the potential to address many of the mathematical proficiencies, as delineated by Australian Curriculum and Assessment Reporting Authority (ACARA 2014).

This contribution focuses on paper models of architectural surfaces, in particular on some roofing systems describable by developable ones. Drawing on an interdisciplinary approach, between Architecture and Mathematics, potentialities and criticalities of these models in explicitly conveying Geometry are investigated, in relation to educational and communicative tasks, both when they are used in a direct, tangible way, and when the use is mediated by images generated by them (thus indirect); we discuss on the possibility for models and images to communicate their explicit and implicit Geometries. The main issue discussed is that a material or analytical description unequivocally allows to grasp all the peculiarities of geometrical shapes, while other representations are subject to critical selection of data and are therefore affected by subjective interpretations; similarly, the translation of the physical model into images is the result of choices which emphasize certain object peculiarities over others and is thus less objective. Hence the importance of the physical model (as well as its digital counterparts) which, even if not used directly, can be complementary to a content that, alone, would be partial and/or misleading.

Introduction: The present paper shows the importance of a joint use of pencil and paper activities and of technology so that students may develop a complete understanding of ratio and proportion. A previous experience with strategy use when solving ratio and proportion problems provided background. Prompted by a recognition of the cognitive components in students, we decided to offer support through the observation and manipulation of representations made using dynamic geometry software. Method: Application of three activities to be solved by a group of 29 sixth-grade pupils public, primary education in Mexico City. The activities were solved with paper and pencil and using dynamic geometry. Results: Students built figures proportional to others they were given, using comparison by superimposing one figure over another, by dragging them, and by enlarging or reducing a figure by click-and-drag on one of its vertices. They also established numerically equivalent ratios by obtaining measurements for the figures given and making the comparison as a quotient. They also were able to recover data from the problems given and represent them using drawings, then they made use of these drawings to help them solve the assigned activity. Discussion and Conclusion: Most of the students could establish figures that were proportional to the ones given by using registers of representation, such as using drawings with the data that they were able to extract from the story problems, using a table which they filled in with both external and internal ratios, as well as equivalence relationships between the ratios and the operations used to get the numeric values. They also showed development in their qualitative proportional thinking as a support to quantitative proportional thinking. (Contains 11 figures, 1 table and 1 footnote.)

Instructions are given for constructing two models, one to illustrate the general principles of symmetry in T=1, T=3, and T=4 viruses, and the other to illustrate the disposition of protein subunits in the T=3 plant viruses and the picornaviruses. (Author/CW)

This article contains investigative activities to assist students in constructing formulas out of an understanding of the area of geometric shapes. Included with this article are a "Finding Areas on Square Dot Paper Activity Sheet" and a "Finding Areas on Triangle Dot Paper Activity Sheet." (Contains 1 table and 11 figures.)

This article describes two alternative coordinate systems and their use in graphing conic sections. This alternative graph paper helps students explore the idea of eccentricity using the definitions of the conic sections.