No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
Complexity of Geometry Problems as a Function of Field-Dependency and Asymmetry of a Diagram
Abstract
Solving geometry problems is challenging for students of all ages. The complexity of geometry problems is multidimensional and is linked to visualization, auxiliary constructions required for solutions, computational and proof skills, and deep and robust knowledge of geometry concepts and their properties (definitions, axioms, and theorems). In this paper, we hypothesized that the field-(in)dependency and (a)symmetry of the geometric diagrams given in geometry problems affect the complexity of the problems. We designed 168 short verification problems: 21 sets of 4 geometric diagrams with a systematic variation of the examined conditions accompanied by a correct and an incorrect property. The hypotheses were examined using the accuracy and response times of 45 undergraduate and graduate Taiwanese students who were asked to solve these problems in a computerized setting. Based on the results of this test, we demonstrated that both asymmetry and field-dependency of diagrams increase problems’ complexity, as indicated by a decrease in the accuracy and an increase in reaction time of responses.KeywordsField-(in)dependency(A)symmetryGeometry ProblemsCognitive complexity
ResearchGate has not been able to resolve any citations for this publication.
The so-called semantic interference effect is a delay in selecting an appropriate target word in a context where semantic neighbours are strongly activated. Semantic interference effect has been described to vary from one individual to another. These differences in the susceptibility to semantic interference may be due to either differences in the ability to engage in lexical-specific selection mechanisms or to differences in the ability to engage more general, top-down inhibition mechanisms which suppress unwanted responses based on task-demands. However, semantic interference may also be modulated by an individual’s disposition to separate relevant perceptual signals from noise, such as a field-independent (FI) or a field-dependent (FD) cognitive style. We investigated the relationship between semantic interference in picture naming and in an STM probe task and both the ability to inhibit responses top-down (measured through a Stroop task) and a FI/FD cognitive style measured through the embedded figures test (EFT). We found a significant relationship between semantic interference in picture naming and cognitive style—with semantic interference increasing as a function of the degree of field dependence—but no associations with the semantic probe and the Stroop task. Our results suggest that semantic interference can be modulated by cognitive style, but not by differences in the ability to engage top-down control mechanisms, at least as measured by the Stroop task.
Symmetry is a basic geometry property that affects people's aesthetic experience in common ways across cultures and historical periods, but the origins of the universal preference for symmetrical patterns is not clear. We assessed four-year-old children's and adults' reported aesthetic preferences between symmetrical and asymmetrical visual patterns, as well as their spontaneous attentional preferences between the patterns. We found a striking dissociation between these two measures in the children: Children looked longer at the symmetrical patterns, relative to otherwise similar but asymmetrical patterns, but they showed no explicit preference for those patterns. These findings suggest that the human's aesthetic preferences have high postnatal plasticity, calling into question theories that symmetry is a "core feature" mediating people's aesthetic experience throughout life. The findings also call into question the assumption, common to many studies of human infants, that attentional choices reflect subjective preferences or values.
In this paper, we investigate children’s learning of reflectional symmetry in a dynamic geometry environment. Through a classroom-based intervention involving two 1-h lessons, we analyse the changes in the children’s thinking about reflectional symmetry: first, they developed dynamic and embodied ways of thinking about symmetry after working with a pre-constructed sketch called the “symmetry machine”. Secondly, they moved from distinguishing symmetrical and asymmetrical figures statically to generalising about properties of symmetry. This was evident in the way children expressed symmetric movement through words, gestures and diagrams during the computer-based lessons as well as in the follow-up paper-and-pencil tasks. We highlight the specific roles of the teacher and of the digital technology in supporting the process of semiotic mediation through which the children learned symmetry.
Computer-based assessment can provide new insights into behavioral processes of task completion that cannot be uncovered by paper-based instruments. Time presents a major characteristic of the task completion process. Psychologically, time on task has 2 different interpretations, suggesting opposing associations with task outcome: Spending more time may be positively related to the outcome as the task is completed more carefully. However, the relation may be negative if working more fluently, and thus faster, reflects higher skill level. Using a dual processing theory framework, the present study argues that the validity of each assumption is dependent on the relative degree of controlled versus routine cognitive processing required by a task, as well as a person's acquired skill. A total of 1,020 persons ages 16 to 65 years participated in the German field test of the Programme for the International Assessment of Adult Competencies. Test takers completed computer-based reading and problem solving tasks. As revealed by linear mixed models, in problem solving, which required controlled processing, the time on task effect was positive and increased with task difficulty. In reading tasks, which required more routine processing, the time on task effect was negative and the more negative, the easier a task was. In problem solving, the positive time on task effect decreased with increasing skill level. In reading, the negative time on task effect increased with increasing skill level. These heterogeneous effects suggest that time on task has no uniform interpretation but is a function of task difficulty and individual skill.
School geometry is the study of those spatial objects, relationships, and transformations that have been formalized (or mathematized) and the axiomatic mathematical systems that have been constructed to represent them. Spatial reasoning, on the other hand, consists of the set of cognitive processes by which mental representations for spatial objects, relationships, and transformations are constructed and manipulated. Clearly, geometry and spatial reasoning are strongly interrelated, and most mathematics educators seem to include spatial reasoning as part of the geometry curriculum. Usiskin (Z. Usiskin, 1987), for instance, has described four dimensions of geometry: (a) visualization, drawing, and construction of figures; (b) study of the spatial aspects of the physical world; (c) use as a vehicle for representing nonvisual mathematical concepts and relationships; and (d) representation as a formal mathematical system. The first three of these dimensions require the use of spatial reasoning.
Human adults from diverse cultures share intuitions about the points, lines, and figures of Euclidean geometry. Do children develop these intuitions by drawing on phylogenetically ancient and developmentally precocious geometric representations that guide their navigation and their analysis of object shape? In what way might these early-arising representations support later-developing Euclidean intuitions? To approach these questions, we investigated the relations among young children's use of geometry in tasks assessing: navigation; visual form analysis; and the interpretation of symbolic, purely geometric maps. Children's navigation depended on the distance and directional relations of the surface layout and predicted their use of a symbolic map with targets designated by surface distances. In contrast, children's analysis of visual forms depended on the size-invariant shape relations of objects and predicted their use of the same map but with targets designated by corner angles. Even though the two map tasks used identical instructions and map displays, children's performance on these tasks showed no evidence of integrated representations of distance and angle. Instead, young children flexibly recruited geometric representations of either navigable layouts or objects to interpret the same spatial symbols. These findings reveal a link between the early-arising geometric representations that humans share with diverse animals and the flexible geometric intuitions that give rise to human knowledge at its highest reaches. Although young children do not appear to integrate core geometric representations, children's use of the abstract geometry in spatial symbols such as maps may provide the earliest clues to the later construction of Euclidean geometry.
In this article we examine students' perspectives on the customary, public work of proving in American high school geometry classes. We analyze transcripts from 29 interviews in which 16 students commented on various problems and the likelihood that their teachers would use those problems to engage students in proving. We use their responses to map the boundaries between activities that (from the students' perspective) constitute normal (vs. marginal) occasions for them to engage in proving. We propose a model of how the public work of proving is shared by teacher and students. This division of labor both creates conditions for students to take responsibility for doing proofs and places boundaries on what sorts of tasks can engage students in proving. Furthermore we show how the activity of proving is a site in which complementarity as well as contradiction can be observed between what makes sense for students to do for particular mathematical tasks and what they think they are supposed to do in instructional situations.
A study with 114 young adults investigated the correlations of intelligence factors and working-memory capacity with reaction time (RT) tasks. Within two sets of four-choice RT tasks, stimulus–response compatibility was varied over three levels: compatible, incompatible, and arbitrary mappings. Two satisfactory measurement models for the RTs could be established: A general factor model without constraints on the loadings and a nested model with two correlated factors, distinguishing compatible from arbitrary mappings, with constraints on the loadings. Structural models additionally including factors for working memory and intelligence showed that the nested model with correlated factors is superior in fit. Working-memory capacity and fluid intelligence were correlated strongly with the nested factor for the RT tasks with arbitrary mappings, and less with the general RT factor. The results support the hypothesis that working memory is needed to maintain arbitrary bindings between stimulus representations and response representations, and this could explain the correlation of working-memory capacity with speed in choice RT tasks.
The construct of field independence (FI) remains one of the most widely cited notions in research on cognitive style and on learning and instruction more generally. However, a great deal of confusion continues to exist around the definition of FI, its measurement, and the interpretation of research results, all of which have served to limit our understanding of and practice in education.
This study reviews research evidence on FI and highlights key issues to frame a more informed agenda for future research. ARGUMENTS: Caution needs to be exercised over the interpretation of the evidence around FI and field dependence (FD). In tests measuring FI only, it is inappropriate to use the term FD. FI is clearly correlated with measures of spatial ability; however, whether FI is just a measure of perceptual and more specifically spatial ability is a matter of debate. Furthermore, whether FI is just a cognitive ability or a cognitive style is not the central issue, as both can be developed. FI has a significant relationship with aspects of working memory and with other variables. It is especially important in the management and interpretation of complex cognitive tasks.
Field independence has an important role to play in the navigation of the complex and information-rich learning environments of the 21st century. It is therefore important to move beyond the present narrow focus on FI as a style or trait by acknowledging, embracing, and exploring the complexity of the interaction between individual and contextual variables.
The report discusses the use of reaction time measures in modern experimental psychology. Methodological and theoretical issues are raised concerning the logic of experimentation in which reaction time is the major dependent variable and the limitations of interpretation of reaction time in the presence of variable error rates. The relationship between the speed and the accuracy of performance and theoretical models underlying this relation are also discussed.
Two questions are dealt with: (1) Can those strategies or behaviors which enable experts to solve problems well be characterized, and (2) Can students be trained to use such strategies? A problem-solving course for college students is described and the model on which the course is based is outlined in an attempt to answer these questions. The five-step model provides a guide to the problem-solving process. The stages are analysis, design, exploration, implementation, and verification. Pretest to posttest comparisons indicate that the students became more fluent at generating plausible approaches to problems. (MP)
We present a theory of human artistic experience and the neural mechanisms that mediate it. Any theory of art (or, indeed, any aspect of human nature) has to ideally have three components. (a) The logic of art: whether there are universal rules or principles; (b) The evolutionary rationale: why did these rules evolve and why do they have the form that they do; (c) What is the brain circuitry involved? Our paper begins with a quest for artistic universals and proposes a list of Eight laws of artistic experience -- a set of heuristics that artists either consciously or unconsciously deploy to optimally titillate the visual areas of the brain. One of these principles is a psychological phenomenon called the peak shift effect: If a rat is rewarded for discriminating a rectangle from a square, it will respond even more vigorously to a rectangle that is longer and skinnier that the prototype. We suggest that this principle explains not only caricatures, but many other aspects of art. Example: An evocative sketch of a female nude may be one which selectively accentuates those feminine form-attributes that allow one to discriminate it from a male figure; a Boucher, a Van Gogh, or a Monet may be a caricature in colour space rather than form space. Even abstract art may employ supernormal stimuli to excite form areas in the brain more strongly than natural stimuli. Second, we suggest that grouping is a very basic principle. The different extrastriate visual areas may have evolved specifically to extract correlations in different domains (e.g. form, depth, colour), and discovering and linking multiple features (grouping) into unitary clusters -- objects -- is facilitated and reinforced by direct connections from these areas to limbic structures. In general, when object-like entities are partially discerned at any stage in the visual hierarchy, messages are sent back to earlier stages to alert them to certain locations or features in order to look for additional evidence for the object (and these processes may be facilitated by direct limbic activation). Finally, given constraints on allocation of attentional resources, art is most appealing if it produces heightened activity in a single dimension (e.g. through the peak shift principle or through grouping) rather than redundant activation of multiple modules. This idea may help explain the effectiveness of outline drawings and sketches, the savant syndrome in autists, and the sudden emergence of artistic talent in fronto-temporal dementia. In addition to these three basic principles we propose five others, constituting a total of eight laws of aesthetic experience(analogous to the Buddha's eightfold path to wisdom).
The translation principle allows students to solve problems in different branches of mathematics and thus to develop connectedness in their mathematical knowledge.
In our society, the recognition of talent depends largely on idealized and entrenched perceptions of academic achievement and job performance. Thinking Styles bucks this trend by emphasizing the method of our thought rather than its content. Psychologist Robert Sternberg argues that ability often goes unappreciated and uncultivated not because of lack of talent, but because of conflicting styles of thinking and learning. Using a variety of examples that range from scientific studies to personal anecdotes, Sternberg presents a theory of thinking styles that aims to explain why aptitude tests, school grades, and classroom performance often fail to identify real ability. He believes that criteria for intelligence in both school and the workplace are unfortunately based on the ability to conform rather than learn. He takes the theory a step further by stating that 'achievement' can be a result of the compatibility of personal and institutional thinking styles, and 'failure' is too often the result of a conflict of thinking styles, rather than a lack of intelligence or aptitude. Sternberg bases his theory on hard scientific data, yet presents a work that remains highly accessible.
Interference of irrelevant salient variables may cause difficulties for students. This study focused on eye tracking during the comparison of perimeters task, in which area is the interfering irrelevant salient variable. There were three trial types: congruent (larger area—larger perimeter), incongruent inverse (larger area—smaller perimeter), and incongruent equal (larger area—equal perimeter). Behavioral findings corroborated previous studies: congruent trials yielded higher accuracy and a shorter reaction time than did incongruent trials. Surprisingly, the area saliency could not be revealed in fixation location and duration measurements in incongruent inverse trials nor in the heat maps for incongruent inverse or incongruent equal trials, suggesting that such processing does not require overt attention; measures of attention shift from one geometrical shape to another were higher for incongruent equal trials, as was pupil dilation, suggesting that greater effort is associated with solving incongruent equal trials.
Young children are exposed to symmetrical figures frequently before they are taught the concept of symmetry, which is a valuable experience for the development of geometry; however, limited research has explored how this concept develops. This study investigated the developmental sequence of "general symmetry" concept and "specific symmetry" concepts (i.e., bilateral, rotational, and translational symmetry) with 106 4-6-year-old children using a symmetry deviant detection task. The test examined children's conception of general symmetry against asymmetry, specific symmetry against asymmetry, and discrimination of specific symmetries. The results suggested that the concept of symmetry develops as a differentiation process. The concept of general symmetry was acquired first, followed by specific symmetries which were acquired in sequential order.
Students experience difficulties in comparison tasks that may stem from interference of the tasks’ salient irrelevant variables. Here, we focus on the comparison of perimeters task, in which the area is the irrelevant salient variable. Studies have shown that in congruent trials (when there is no interference), accuracy is higher and reaction time is shorter than in incongruent trials (when the area variable interferes). Brain-imaging and behavioral studies suggested that interventions of either activating inhibitory control mechanisms or increasing the level of salience of the relevant perimeter variable could improve students’ success. In this review, we discuss several studies that empirically explored these possibilities and their findings show that both types of interventions improved students’ performance. Theoretical considerations and practical educational implications are discussed.
This paper focuses on reasons students’ give while introducing auxiliary lines in geometry proofs. Three cases of pairs of high school students participating in proving activities are presented. The overarching theme of the tasks proposed to the students is the comparison of areas of triangles and/or parallelograms. The students gave a wide spectrum of reasons when introducing an auxiliary line. Two main groups of reasons are discerned: The students introduced auxiliary lines recalling some known results or definitions and modified the given diagrams accordingly, as a part of a learned procedure. The students introduced auxiliary lines anticipating to receive more information from a modified situation. Students may combine reasons of recalling and anticipating nature when introducing an auxiliary line.
https://www.sciencedirect.com/science/article/pii/S073231231830021X
This survey on the theme of Geometry Education (including new technologies) focuses chiefly on the time span since 2008. Based on our review of the research literature published during this time span (in refereed journal articles, conference proceedings and edited books), we have jointly identified seven major threads of contributions that span from the early years of learning (pre-school and primary school) through to post-compulsory education and to the issue of mathematics teacher education for geometry. These threads are as follows: developments and trends in the use of theories; advances in the understanding of visuo spatial reasoning; the use and role of diagrams and gestures; advances in the understanding of the role of digital technologies; advances in the understanding of the teaching and learning of definitions; advances in the understanding of the teaching and learning of the proving process; and, moving beyond traditional Euclidean approaches. Within each theme, we identify relevant research and also offer commentary on future directions.
It seems obvious to contrast physical representation (a drawing on paper or on a screen) of an object with mental images of this same object. However, there is an important contrast which attracts less attention, between a drawing of a physical object (a house, for example, outlined by a square with a triangle at the top and rectangles for the door…) and a drawing of a geometrical figure (square, circle, triangle…).
The relationship between the cognitive style variable of field dependence/independence and instructional treatments was investigated using high or low guidance in a unit on networks. The 97 prospective elementary teachers were tested on cognitive style (using the Group Embedded Figures Test) and on mathematical achievement (a measure of crystallized ability), and randomly assigned to treatments. Following instruction, students were tested for immediate achievement and then retested 5 weeks later. There were no interactions with field dependence/independence, but there was a significant ( $p<.05$ ) interaction with crystallized ability on the retention test.
The interaction of two aptitude variables, field independence and general reasoning, with treatments that differed in sequence of instruction was investigated using either an inductive or deductive approach to the learning of properties of equivalence relations. The 66 students came from an advanced class in mathematics for elementary school teachers. Students were randomly assigned to treatment groups where they worked independently on a packet of programmed materials. Students were then given an achievement test, followed two days later by a transfer test. Four weeks later, the same tests were readministered as measures of retention. There was a significant interaction ($p) with field independence on the transfer test and with general reasoning on both the immediate achievement test and the transfer retention test. All significant interactions were in the direction predicted by the theory.
Abilities are always abilities for a definite kind of activity ; they exist only in a person's specific activity. Therefore they can show up only on the basis of an analysis of a specific activity. Accordingly, a<span style=background-color: #ffff00;> mathematical ability exists only in a mathematical activity and should be manifested in it. (S. 66)
The purpose of this study was to explore the role of spatial orientation skill in the solution of mathematics problems. Fifty-seven tenth-grade students who scored high or low on a spatial orientation test were asked to solve mathematics problems in individual interviews. A group of specific behaviors was identified in geometric settings, which appeared to be manifestations of spatial orientation skill. Spatial orientation skill also appeared to be involved in understanding the problem and linking new problems to previous work in nongeometric settings.
The concept of symmetry is one of the great universal principles used to comprehend the enormous amounts of data encountered in both the worlds of natural phenomenon and of abstract knowledge. With the advent of computers, methodology has evolved to process and generate huge amounts of information. This information is often inconsistent and ambiguous and is similar to that encountered by human perception. This article develops some commonalities between applications of symmetry and applications of computer methodology to visual perception (robotic vision), to explore the impact of developing technology on general understandings about human knowledge. These commonalities suggest that advances in robotic vision will enlarge the study of symmetry, reveal astonishing new types of symmetry, and produce unexpected applications of philosophical interrelationships between abstract and perceptual knowledge.
Symmetry is an aspect of mathematics that is strongly linked to art and design. We chose to explore this connection in the context of a liberal arts mathematics course. Here we present a brief description of this course, including an outline of the curriculum and specific features of the course. We subsequently present the results of a study we conducted to examine students' understanding of symmetry in this context. The findings suggest that when working with repetitious geometric art designs students' approach to symmetry often goes beyond the one-to-one correspondence between specific pieces of art and symmetry concepts, to the development of a correspondence between types of art designs and abstract types of group symmetry concepts. Further, familiarity with symmetry concepts allows students to use abstraction as a tool in their artistic creations. Finally, students reported improved attitudes towards mathematics and its relevance to their lives in general - a bonus finding for a liberal arts course.
Symmetry is an important mathematical concept which plays an extremely important role as a problem-solving technique. Nevertheless, symmetry is rarely used in secondary school in solving mathematical problems. Several investigations demonstrate that secondary school mathematics teachers are not aware enough of the importance of this elegant problem-solving tool. In this paper we present examples of problems from several branches of mathematics that can be solved using different types of symmetry. Teachers' attitudes and beliefs regarding the use of symmetry in the solutions of these problems are discussed.
It has been proposed that field-independent and field-dependent learners differ more in the processes they use than in the effectiveness of their learning or retention. The present article discusses alternative explanations which emphasize developmental differences and differences in efficiency of performance between field-independent and field-dependent learners. Review of the concept learning literature and some new data suggested that the greater effectiveness of field-independent learners was related to memory efficiency and the ability to conduct combinatorial analysis. Research concerning short-term memory and free recall was also examined. High information load, greater interference potential, and less subjective organization were suggested as factors which contribute to the less efficient memory of field-dependent learners.
In this paper we present a theorem which extends the results of an inequality originally due to Franz Rellich [4]. The theorem by Rellich es-tablishes an inequality widely used in the spectral theory of partial differential operators. Our theorem allows for a broader range of application by extend-ing the class of functions to which the theorem is applicable. Many authors call upon inequalities similar to the one established in our theorem in dealing with problems concerning essential self-adjointness of Schrödinger operators and other problems arising in oscillation theory of elliptic operators. In the first part of the paper we present Rellich's inequality and discuss some problems dealing with symmetric operators on Hubert spaces where Rellich's inequality is a useful tool. We shall also discuss some important extensions of Rellich's work which were established by other mathematicians. One such extension was proved by W. Allegretto [1] in dealing with elliptic equations of order 2/¡. Another extension was established by U. W. Schmincke [5] in considering essential self-adjointness criteria of Schrödinger operators. Schmincke's extension is of par-ticular interest to us due to his elegant proof. We follow Schmincke's method of proof. We then state and prove our generalization of Rellich's inequality along with a useful corollary. The paper concludes with a few brief comments on our result and other work which could be done with Rellich's inequality. Rellich's Inequality Define the operator T by Tu = AAu -c\x\~ u on L (R") with domain 3¡(T) = C^°(R"\{0}). Then T is symmetric on 3¡(T). As Rellich mentions in [4] his inequality can be viewed as a means of finding the largest positive real number c such that the eigenvalues for the Friedrich's extension of T are nonnegative. Also, this could be viewed as find-ing the least point in the spectrum of AAu, u E C^°(R"\{0}), in the weighted Received by the editors August 5, 1987 and, in revised form, August 18, 1988.
The literature on sources of individual differences in field dependence-independence is reviewed, and findings on ontogenetic development and cross-cultural differences are incorporated into the theory of psychological differentiation. During the growth years, individuals develop toward greater field independence. Hormonal and X-linked genetic factors may influence development of specific cognitive components of the field-dependence-independence dimension. It is clear from a wide variety of data that the development of field independence is enhanced by appropriate training programs, child-rearing practices which encourage separation from parental authority, and "loose" social structures which permit individual autonomy. Among subsistence-level societies, members of nomadic hunting groups tend to be more field independent than members of sedentary farming groups. Viewed from an ecocultural perspective, this finding suggests a progression from relative field independence toward relative field dependence as cultural forms changed from early hunting to later agricultural economies. That developmental changes may proceed in opposite directions in the course of ontogeny and during cultural history is compatible with the bipolar conception of cognitive styles; the cognitive restructuring skills of field-independent individuals and the social-interpersonal competencies of field-dependent individuals have adaptive value in different life circumstances. (Author/MV)
Discusses students' use of visual considerations in doing geometrical proofs. Studies ninth-grade students (n=17) in an academically-selective high school in Jerusalem. Concludes that only three out of 16 students who turned in their papers chose symmetry considerations when working on a proof. Contains 16 references. (ASK)
Conducted 3 experiments to assess the development of symmetry perception in children between the ages of 4 and 6 yrs. Exp I employed a learning task in which 72 Ss were asked at different times to discriminate vertically symmetrical, obliquely symmetrical, and horizontally symmetrical holistic patterns from asymmetrical ones. Results reveal a developmental progression: 4-yr-olds discriminated only vertical; 5-yr-olds, vertical and horizontal; and 6-yr-olds, vertical, horizontal, and oblique. Exp II retested the 18 6-yr-olds with fragmented patterns of the different symmetries; these Ss regressed to the performance level of 4-yr-olds and only discriminated vertical. Exp III, conducted with 18 Ss, used a memory-production task with new vertical, oblique, horizontal, and asymmetrical patterns constructed to 4, 5, or 6 elements. Measures of the goodness and accuracy of Ss' reproductions were consistent with data from the discrimination-learning experiments in terms of age, stimulus orientation, and stimulus complexity. These studies support the view that vertical symmetry is special perceptually and developmentally and that, after vertical, horizontal predominates, followed by oblique. The role of symmetry in early perceptual development and the value of child–adult perceptual comparisons are discussed. (55 ref) (PsycINFO Database Record (c) 2012 APA, all rights reserved)