Geometric Reasoning Research Articles

Disrupting a learning environment for promotion of geometry teaching

ABSTRACTCreating a classroom learning environment that is suitably designed for promotion of learners’ performance in geometry, a branch of mathematics that addresses spatial sense and geometric reasoning, is a daunting task. This article focuses on how grade 8 teachers’ action learning changed the learning environment for the promotion of geometry teaching. This was an exploratory study in which a sample of 13 grade 8 mathematics teachers from the rural schools of a district in the Eastern Cape Province in South Africa participated in a project aimed at creating mathematical learning environments through action learning while supporting and promoting the teaching of geometry. The study was qualitative and data was collected using questionnaires, classroom observations and semi-structured interviews with the teachers on a participatory action research conducted in two cycles. Results indicated that 92% of teachers changed their classroom environments by modifying instructional strategies, learner -interactions, and engagements, but could not change how they managed the classrooms due to some factors related to power dynamics andthe education policy. It is recommended that teachers be workshopped on modifying their classroom learning environments while they undergo learning in action on the promotion of teaching and learning geometry in their school defining contexts.

Extracting the turning volume and features for a mill/turn part with multiple extreme faces

This paper presents a geometric reasoning approach for extracting the maximum turnable state (MTS) and its turning features for a mill/turn part which has multiple extreme faces. Here, an extreme face refers to a vertical planar face whose adjacent faces are all located on the same side of the plane. It is more difficult to construct the MTS for such a part because the positions of its cross-sectional planes do not change monotonically on the principal axis, and the concave regions formed by its profile need special treatments in its feature recognition. The proposed method addresses the issue of multiple extreme faces with a graph decomposition strategy for the attributed adjacency graph of CAD models, and handles the feature recognition problem by matching the feature shape descriptors with the descriptors of local element combinations out of the delta volume mesh. The method not only produces all the turning features for a part but also generates the feature precedence graph as well from reasoning on their adjacency relations. At the end of the paper, an example is presented to demonstrate the method’s effectiveness.

Physical conditions for Jupiter-like dynamo models

The Juno mission will measure Jupiter’s magnetic field with unprecedented precision and provide a wealth of additional data that will allow us to constrain the planet’s interior structure and dynamics. Here we analyse 66 different numerical simulations in order to explore the sensitivity of the dynamo-generated magnetic field to the planets interior properties. Jupiter field models based on pre-Juno data and up-to-date interior models based on ab initio simulations serve as benchmarks. Our results suggest that Jupiter-like magnetic fields can be found for a number of different models. These complement the steep density gradients in the outer part of the simulated shell with an electrical conductivity profile that mimics the low conductivity in the molecular hydrogen layer and thus renders the dynamo action in this region largely unimportant. We find that whether we assume an ideal gas or use the more realistic interior model based on ab initio simulations makes no difference. However, two other factors are important. A low Rayleigh number leads to a too strong axial dipole contribution while the axial dipole dominance is lost altogether when the convective driving is too strong. The required intermediate range that yields Jupiter-like magnetic fields depends on the other system properties. The second important factor is the convective magnetic Reynolds number radial profile Rmc(r), basically a product of the non-axisymmetric flow velocity and electrical conductivity. We find that the depth where Rmc exceeds about 50 is a good proxy for the top of the dynamo region. When the dynamo region sits too deep, the axial dipole is once more too dominant due to geometric reasons. Extrapolating our results to Jupiter and the result suggests that the Jovian dynamo extends to 95% of the planetary radius.The zonal flow system in our simulations is dominated by an equatorial jet which remains largely confined to the molecular layer. Where the jet reaches down to higher electrical conductivities, however, it gives rise to a secondary αΩ dynamo that modifies the dipole-dominated field produced deeper in the planet. This secondary dynamo can lead to strong magnetic field patches at lower latitudes that seem compatible with the pre-Juno field models.

ANALYSIS OF THE STUDENTS’ GEOMETRIC REASONING ABILITY

This research was a descriptive research with quantitative approach that aimed to analyze or describe the geometric reasoning ability of SMA Negeri 1 Wundulako students in solving mathematical problems related to the building of the three dimensions. Indicators used to describe students' geometric reasoning abilities include visualization, analysis, abstraction, and formal deduction. Data collection techniques consisted of interviews to deepen information about students 'geometric reasoning abilities and written tests to get an idea of students' cognitive abilities. Subjects in this study, namely students class XI.IA SMA Negeri 1 Wundulako as many as 40 students who have studied three dimensional materials. Data analysis in this research used descriptive analysis. Based on the data analysis obtained average value of 49.86, median 53.33, mode 66.67. The standard deviation was obtained at 22.39, it showed that there was a considerable difference of value between the students who mean in high ability category with the average of 91,11, medium ability category equal to 66,27, and low ability equal to 31,82 , With minimum value is 3.33 and maximum 93,33. In general, based on descriptive analysis, the geometric reasoning ability of SMA Negeri 1 Wundulako students was still low.

UTILIZING THE UNCERTAINTY OF POLYHEDRA FOR THE RECONSTRUCTION OF BUILDINGS

Abstract. The reconstruction of urban areas suffers from the dilemma of modeling urban structures in a generic or specific way, thus being too unspecific or too restrictive. One approach is to model and to instantiate buildings as arbitrarily shaped polyhedra and to recognize comprised man-made structures in a subsequent stage by geometric reasoning. To do so, we assume the existence of boundary representations for buildings with vertical walls and horizontal ground floors. To stay generic and to avoid the use of templates for pre-defined building primitives, no further assumptions for the buildings’ outlines and the planar roof areas are made. Typically, roof areas are derived interactively or in an automatic process based on given point clouds or digital surface models. Due to the mensuration process and the assumption of planar boundaries, these planar faces are uncertain. Thus, a stochastic geometric reasoning process with statistical testing is appropriate to detected man-made structures followed by an adjustment to enforce the deduced geometric constraints. Unfortunately, city models usually do not feature information about the uncertainty of geometric entities. We present an approach to specify the uncertainty of the planes corresponding to the planar patches, i.e., polygons bounding a building, analytically. This paves the way to conduct the reasoning process with just a few assumptions. We explicate and demonstrate the approach with real data.

Retrieving challenging vessel connections in retinal images by line co-occurrence statistics

Natural images contain often curvilinear structures, which might be disconnected, or partly occluded. Recovering the missing connection of disconnected structures is an open issue and needs appropriate geometric reasoning. We propose to find line co-occurrence statistics from the centerlines of blood vessels in retinal images and show its remarkable similarity to a well-known probabilistic model for the connectivity pattern in the primary visual cortex. Furthermore, the probabilistic model is trained from the data via statistics and used for automated grouping of interrupted vessels in a spectral clustering based approach. Several challenging image patches are investigated around junction points, where successful results indicate the perfect match of the trained model to the profiles of blood vessels in retinal images. Also, comparisons among several statistical models obtained from different datasets reveal their high similarity, i.e., they are independent of the dataset. On top of that the best approximation of the statistical model with the symmetrized extension of the probabilistic model on the projective line bundle is found with a least square error smaller than 2%. Apparently, the direction process on the projective line bundle is a good continuation model for vessels in retinal images.

Waves and rays in plano-concave laser cavities: I. Geometric modes in the paraxial approximation

Eigenmodes of laser cavities are studied theoretically and experimentally in two companion papers, with the aim of making connections between undulatory and geometric properties of light. In this first paper, we focus on macroscopic open-cavity lasers with localized gain. The model is based on the wave equation in the paraxial approximation; experiments are conducted with a simple diode-pumped Nd:YAG laser with a variable cavity length. After recalling fundamentals of laser beam optics, we consider plano-concave cavities with on-axis or off-axis pumping, with emphasis put on degenerate cavity lengths, where modes of different order resonate at the same frequency, and combine to form surprising transverse beam profiles. Degeneracy leads to the oscillation of so-called geometric modes whose properties can be understood, to a certain extent, also within a ray optics picture. We first provide a heuristic description of these modes, based on geometric reasoning, and then show more rigorously how to derive them analytically by building wave superpositions, within the framework of paraxial wave optics. The numerical methods, based on the Fox–Li approach, are described in detail. The experimental setup, including the imaging system, is also detailed and relatively simple to reproduce. The aim is to facilitate implementation of both the numerics and of the experiments, and to show that one can have access not only to the common higher-order modes but also to more exotic patterns.

Developing Students’ Geometric Reasoning about the Derivative of Complex Valued Functions

In this article, we describe the results of a case study examining the development of two undergraduate students’ geometric reasoning about the derivative of a complex-valued function with the aid of The Geometer’s Sketchpad (GSP). Initially, our participants saw it in terms of the slope of the tangent line. Without the aid of GSP, they could describe the rotation and dilation aspect of the derivative for linear complex-valued functions, but were unable to generalize this perception to non-linear ones. Participants’ use of GSP assisted with exploring function behavior, generalizing how for non-linear complex-valued functions the derivative describes the rotation and dilation of an image with respect to its pre-image, and recognizing that the derivative is a local property.

Using expectancy-value theory to explore aspects of motivation and engagement in inquiry-based learning in primary mathematics

Inquiry-based learning (IBL) is a pedagogical approach in which students address complex, ill-structured problems set in authentic contexts. While IBL is gaining ground in Australia as an instructional practice, there has been little research that considers implications for student motivation and engagement. Expectancy-value theory (Eccles and Wigfield 2002) provides a framework through which children’s beliefs about their mathematical competency and their expectation of success are able to be examined and interpreted, alongside students’ perceptions of task value. In this paper, Eccles and Wigfield’s expectancy-value model has been adopted as a lens to examine a complete unit of mathematical inquiry as undertaken with a class of 9–10-year-old students. Data were sourced from a unit (∼10 lessons) based on geometry and geometrical reasoning. The units were videotaped in full, transcribed, and along with field notes and student work samples, subjected to theoretical coding using the dimensions of Eccles and Wigfield’s model. The findings provide insight into aspects of IBL that may impact student motivation and engagement. The study is limited to a single unit; however, the results provide a depth of insight into IBL in practice while identifying features of IBL that may be instrumental in bringing about increased motivation and engagement of students in mathematics. Identifying potentially motivating aspects of IBL enable these to be integrated and more closely studied in IBL practises.

Why Can’t Geometers Cut Themselves on the Acutely Angled Objects of Their Proofs? Aristotle on Shape as an Impure Power

For Aristotle, the shape of a physical body is perceptible per se ( da ii.6, 418a8–9). As I read his position, shape is thus a causal power, as a physical body can affect our sense organs simply in virtue of possessing it. But this invites a challenge. If shape is an intrinsically powerful property, and indeed an intrinsically perceptible one, then why are the objects of geometrical reasoning, as such, inert and imperceptible? I here address Aristotle’s answer to that problem, focusing on the version of it that he presents in De caelo iii.8. I argue that if we grant that Aristotle conceived of the shape of a sensible body as some kind of causal power, then the satisfactory resolution of that challenge pushes us to interpret him as having conceived of it as being, more specifically, an impure power—that is, as a property that is not only intrinsically powerful but also, in some way, intrinsically non-powerful as well. This is a notable result not only insofar as it illuminates Aristotle’s conception of shape but also insofar as it contributes to our knowledge of Aristotle’s theory of dunameis and his ontology more broadly.

A contact relation analysis approach to assembly sequence planning for assembly models

ABSTRACTA reasonable assembly planning can obviously reduce manufacturing time and costs. From the perspective of geometric reasoning in reverse, a contact relation analysis approach to ASP for assembly models is proposed in this paper. The main characteristic of the presented method has an ability to choose non-orthogonal assembling directions automatically. Firstly, contact relations between parts of an assembly model are automatically extracted from its CAD model. According to contact relations, contact vectors are generated. Secondly, the disassembly of a part in an assembly model is checked. Based on extracted contact vectors, instantaneous movable directions for parts are identified by solving inequality constraints. And Gaussian sphere is introduced to express a set of movable directions. Subsequently, currently removable parts are judged by checking the existence of feasible disassembly paths along instantaneous movable directions. Thirdly, different layers of removable parts in an assembly are ob...

Managing the Uncertainty in the van Hiele Levels of Geometric Reasoning

Student group total possibilistic uncertainty and probabilistic uncertainty are calculated for evaluating the student acquisition of the van Hiele levels of geometric reasoning and an example is presented illustrating our results.

An Application of the Generalized Rectangular Fuzzy Assessment Model to the van Hiele Level Theory of Geometric Reasoning

The Generalized Rectangular Fuzzy Assessment Model (GRFAM) is applied for the evaluation of student group acquisition of the van Hiele levels of geometric reasoning and an example is presented illustrating our results. The GRFAM is a variation of the Center of Gravity defuzzification technique created in an effort to treat better the ambiguous assessment cases being at the boundaries between any two successive assessment grades.

Vesicle formation by proton transfer driven short-tailed fatty acids of C4–C8 chain length in water

Ultrashort single-chain fatty acids self-assemble to form vesicles under certain proton-driven conditions. The protonation provides a larger charge area around the hydrophilic carbonyl headgroups, and proton shift as the key driving parameter was studied. The ultrashort fatty acids (C4-C8) formed stable unilamellar vesicles predominantly through out the whole range of tested pH levels (6.5-9.5). A proton-driven self-assembly process and effects on the phase transition were characterized by dynamic light scattering, transmission electron microscopy and cryo-transmission electron microscopy. In particular, we studied in greater detail the molecular packing characteristics of FA vesicles for geometric reasons and the protonation effect changes the molecular surface charge and further carboxylic acid headgroup motion. This study enhances the understanding of the physicochemical specificity of these membrane vesicles, and may facilitate the alteration of membrane function caused by FAs.

Van Hileova teorija geometrijskog mišljenja u nastavi matematike

Although the twentieth century was marked research in algebra, the emergence of the theory of geometric thinking encourages interest in geometric content. With increasing interest in geometric idea is important to research about the nature of geometrical reasoning and how it develops. This paper summarizes the theoretical framework for the study of geometrical reasoning van Hiele model of thinking in geometry. First, we give van Hiele theory of geometrical reasoning, levels of understanding of spatial relationships and geometric phase of learning content. Second, we present and analyze the interpretation of van Hiele theories. Then, we give a brief overview of research based on van Hiele theory. The paper uses the methods of theoretical analysis. Based on the researched literature we conclude that still need a lot of research on deep process of development and learning of geometrical reasoning.

Writing Code to Assess Geometric Reasoning

Engage students and promote thinking in a student-centered environment that is rich with technology.

Breaking of parallelograms in presence of torsion: An equivalent alternative approach to detect gravitational waves

The equations for gravitational plane waves produced by a typical binary system as a solution of linear approximation of Einstein equations are derived. The dynamics of the corresponding gravitational field is analyzed in a four-dimensional space-time manifold, endowed with a metric and taking into account torsion. In this context, the geometrical reason of the existence of torsion due to the presence of gravitational waves (GW) is highlighted and the geodesic deviation is obtained taking into account both curvature and torsion. In a laser interferometer gravitational detector, the delay time between the arrivals of the two laser beams traveling back and forth along the two arms in presence of gravitational waves is interpreted from this point of view. This delay is calculated for the NS–NS binary pulsar (1913 + 16) in two specific orientations with respect to the experimental device, corresponding to different polarizations of gravitational waves. In the specific case of this example, it is shown that the results obtained in the context of the standard general relativity (GR) and in the framework of teleparallel gravity are equivalent.

A Geometric Dynamic Temporal Reasoning Method with Tags for Cognitive Systems

Temporal reasoning is one of the cognitive capabilities humans involve in communicating with others and everything appears related because of temporal reference. Therefore, in this paper a geometric dynamic temporal reasoning algorithm is proposed to solve the temporal reasoning problem, especially in autonomous planning and scheduling. This method is based on the representation of actions in a two dimensional coordination system. The main advantage of this method over others is that it uses tags to mark new constraints added into the constraint network, which leads the algorithm to deal with pending constraints rather than all constraints. This characteristic makes the algorithm suitable for temporal reasoning, where variables and constraints are always added dynamically. This algorithm can be used not only in intelligent planning, but also computational intelligence, real-time systems, and etc. The results show the efficiency of our algorithm from four cases simulating the planning and scheduling process.

Prompting Teacher Geometric Reasoning through Coaching in a Dynamic Geometry Software Context

This study investigated the ways in which four middle grades teachers developed mathematical knowledge for teaching (MKT) geometry as they implemented dynamic geometry software in their classrooms with the assistance of a coach. Teachers developed various components of MKT by observing coaches teach, by dynamic discourse with students, which is discourse with respect to dynamic geometry software images, and by discussions with coaches. The dynamic geometry software environment proved productive as coaches guided teachers’ growth in explanations, examples, and definitions. The environment also helped teachers discover unnoticed abilities among their low achievers. Moreover, teachers developed confidence to teach with more expert uses of dynamic geometry software.

Limits of RNA 2'-OH Mimicry by Fluorine: Crystal Structure of Bacillus halodurans RNase H Bound to a 2'-FRNA:DNA Hybrid.

RNase H1 cleaves the RNA strand of RNA:DNA hybrids. Replacement of RNA 2'-hydroxyls by fluorine (FRNA) is commonly used to stabilize aptamers and siRNAs. However, FRNA:DNA hybrids fail to elicit RNase H activity. The underlying reasons are unclear, as 2'-OH groups are not directly involved in cleavage. We determined the crystal structure of Bacillus halodurans RNase H bound to a FRNA:DNA hybrid. The structure points to dynamic (slippage of the FRNA:DNA hybrid relative to the enzyme), geometric (different curvatures of FRNA:DNA and RNA:DNA hybrids), and electronic reasons (Mg(2+) absent from the active site of the FRNA:DNA complex) for the loss of RNaseH activity.