Visual‐Spatial Representation,Mathematical Problem Solving,and Students of Varying Abilities

The purpose of this study was to investigate students' use of visual imagery while solving mathematical problems. Students with learning disabilities (LD), average achievers, and gifted students in sixth grade (N= 66) participated in this study. Students were assessed on measures of mathematical problem solving and visual‐spatial representation. Visual‐spatial representations were coded as either primarily schematic representations that encode the spatial relations described in the problem or primarily pictorial representations that encode persons, places, or things described in the problem. Results indicated that gifted students used significantly more visual‐spatial representations than the other two groups. Students with LD used significantly more pictorial representations than their peers. Successful mathematical problem solving was positively correlated with use of schematic representations; conversely, it was negatively correlated with use of pictorial representations.     

A cross-sectional study of mathematics achievement, estimation skills, and academic self-perception in students of varying ability

This study investigated students' mathematics achievement, estimation ability, use of estimation strategies, and academic self-perception. Students with learning disabilities (LD), average achievers, and intellectually gifted students (N = 135) in fourth, sixth, and eighth grade participated in the study. They were assessed to determine their mathematics achievement, ability to estimate discrete quantities, knowledge and use of estimation strategies, and perception of academic competence. The results indicated that the students with LD performed significantly lower than their peers on the math achievement measures, as expected, but viewed themselves to be as academically competent as the average achievers did. Students with LD and average achievers scored significantly lower than gifted students on all estimation measures, but they differed significantly from one another only on the estimation strategy use measure. Interestingly, even gifted students did not seem to have a well-developed understanding of estimation and, like the other students, did poorly on the first estimation measure. The accuracy of their estimates seemed to improve, however, when students were asked open-ended questions about the strategies they used to arrive at their estimates. Although students with LD did not differ from average achievers in their estimation accuracy, they used significantly fewer effective estimation strategies. Implications for instruction are discussed.     

The Effect of Visual‐Chunking‐Representation Accommodation on Geometry Testing for Students with Math Disabilities

Students who struggle with learning mathematics often have difficulties with geometry problem solving, which requires strong visual imagery skills. These difficulties have been correlated with deficiencies in visual working memory. Cognitive psychology has shown that chunking of visual items accommodates students’ working memory deficits. This study investigated the effects of visual‐chunking representation as a testing accommodation for improving students’ geometry problem‐solving performance. Participants were four third‐graders with difficulties in mathematics. An adapted reversal design was employed to examine the students’ performance changes during standard testing conditions and accommodated testing conditions. During the accommodated condition, students were presented with visual‐chunking images. Results suggested that the visual‐chunking representation accommodation improved students’ performance on geometry problem‐solving tasks, and an interview confirmed students’ preference for the visual‐chunking representation approach.     

Performance on Middle School Geometry Problems With Geometry Clues Matched to Three Different Cognitive Styles

ABSTRACT— This study investigated the relationship between 3 ability‐based cognitive styles (verbal deductive, spatial imagery, and object imagery) and performance on geometry problems that provided different types of clues. The purpose was to determine whether students with a specific cognitive style outperformed other students, when the geometry problems provided clues compatible with their cognitive style. Students were identified as having a particular cognitive style when they scored equal to or above the median on the measure assessing this ability. A geometry test was developed in which each problem could be solved on the basis of verbal reasoning clues (matching verbal deductive cognitive style), mental rotation clues (matching spatial imagery cognitive style), or shape memory clues (matching object imagery cognitive style). Straightforward cognitive style–clue‐compatibility relationships were not supported. Instead, for the geometry problems with either mental rotation or shape memory clues, students with a combination of both verbal and spatial cognitive styles tended to do the best. For the problems with verbal reasoning clues, students with either a verbal or a spatial cognitive style did well, with each cognitive style contributing separately to success. Thus, both spatial imagery and verbal deductive cognitive styles were important for solving geometry problems, whereas object imagery was not. For girls, a spatial imagery cognitive style was advantageous for geometry problem solving, regardless of type of clues provided.     

Spatial ability,visual imagery,and mathematical performance

116 Foundation Year Engineering Students, at the University of Technology, Lae, Papua New Guinea, were given a battery of mathematical and spatial tests; in addition, their preferred modes of processing mathematical information were determined by means of an instrument recently developed in Australia by Suwarsono.Correlational analysis revealed that students who preferred to process mathematical information by verbal-logical means tended to outperform more visual students on mathematical tests. Multiple regression and factor analyses pointed to the existence of a distinct cognitive trait associated with the processing of mathematical information. Also, spatial ability and knowledge of spatial conventions had less influence on mathematical performance than could have been expected from recent relevant literature.     

The problem‐solving methods of mathematically gifted and older average‐attaining students

This paper examines some of the ways gifted students are said to be different from non‐gifted students by comparing the responses of 475, 9‐year‐old “gifted” students with those of 230 average‐attaining 13‐year‐old students on the same mathematical problem‐solving questions. The questions were specifically written for mathematically gifted 9 year olds as part of the World Class Tests project. The performance and approaches used by students in the two samples were found to be very similar, as was the frequency of different responses to the questions, suggesting that many of the mathematically “gifted” are not qualitatively different in their problem‐solving approaches from students of average ability, but are merely precocious.     

Visuospatial anatomy comprehension: The role of spatial visualization ability and problem‐solving strategies

The present study explored the problem‐solving strategies of high‐ and low‐spatial visualization ability learners on a novel spatial anatomy task to determine whether differences in strategies contribute to differences in task performance. The results of this study provide further insights into the processing commonalities and differences among learners beyond the classification of spatial visualization ability alone, and help elucidate what, if anything, high‐ and low‐spatial visualization ability learners do differently while solving spatial anatomy task problems. Forty‐two students completed a standardized measure of spatial visualization ability, a novel spatial anatomy task, and a questionnaire involving personal self‐analysis of the processes and strategies used while performing the spatial anatomy task. Strategy reports revealed that there were different ways students approached answering the spatial anatomy task problems. However, chi‐square test analyses established that differences in problem‐solving strategies did not contribute to differences in task performance. Therefore, underlying spatial visualization ability is the main source of variation in spatial anatomy task performance, irrespective of strategy. In addition to scoring higher and spending less time on the anatomy task, participants with high spatial visualization ability were also more accurate when solving the task problems. Anat Sci Educ 7: 280–288. © 2013 American Association of Anatomists.     

PROBING THE RELATIONSHIP BETWEEN PROCESS OF SPATIAL PROBLEMS SOLVING AND SCIENCE LEARNING: AN EYE TRACKING APPROACH

There were two purposes in the study. One was to explore the cognitive activities during spatial problem solving and the other to probe the relationship between spatial ability and science concept learning. Twenty university students participated in the study. The Purdue Visualization of Rotations Test (PVRT) was used to assess the spatial ability, whose items were divided into different types of problems with respect to the rotation angles and levels of plane invisibility. The eye tracking technology and the interview technique were employed to analyze subjects’ the online cognitive processes and problem solving strategies. Students’ concept gains were examined by content analysis after reading a science report. The result shows that, first, the interview analysis shows that students of different PVRT performances employed different problem solving strategies. Second, rotation angles as well as levels of plane invisibility inserted significant effects on the online processes and performances of the spatial problem solving. Third, the accuracy performance of PVRT was correlated with eye movement patterns. At last, it was found that concept performance was not correlated with PVRT performance but associated with spatial memory and problem solving strategies.     

Self‐Regulation and Mathematics Instruction

Abstract The purpose of this article is to provide an overview of research‐based interventions that incorporate self‐regulation strategies to improve mathematics performance of students with learning disabilities (LD). Self‐regulation is a metacognitive function essential to academic success. Students with LD are notoriously poor at self‐regulation and must be taught explicitly to monitor and control their cognitive activities as they engage in academic tasks such as mathematical problem solving. This article describes intervention studies that use self‐regulation strategies to improve mathematics performance of students with LD at the elementary, middle, and secondary school levels. Several techniques to facilitate effective implementation of self‐regulation instruction in the classroom are presented.     

The role of visualization in learning from computer‐based images

Among the sciences, the practice of geology is especially visual. To assess the role of spatial ability in learning geology, we designed an experiment using: (1) web‐based versions of spatial visualization tests, (2) a geospatial test, and (3) multimedia instructional modules built around QuickTime Virtual Reality movies. Students in control and experimental sections were administered measures of spatial orientation and visualization, as well as a content‐based geospatial examination. All subjects improved significantly in their scores on spatial visualization and the geospatial examination. There was no change in their scores on spatial orientation. A three‐way analysis of variance, with the geospatial examination as the dependent variable, revealed significant main effects favoring the experimental group and a significant interaction between treatment and gender. These results demonstrate that spatial ability can be improved through instruction, that learning of geological content will improve as a result, and that differences in performance between the genders can be eliminated.     

Visual spatial representation in mathematical problem solving by deaf and hearing students

This research examined the use of visual-spatial representation by deaf and hearing students while solving mathematical problems. The connection between spatial skills and success in mathematics performance has long been established in the literature. This study examined the distinction between visual-spatial "schematic" representations that encode the spatial relations described in a problem versus visual-spatial "pictorial" representations that encode only the visual appearance of the objects described in a problem. A total of 305 hearing (n = 156) and deaf (n = 149) participants from middle school, high school, and college participated in this study. At all educational levels, the hearing students performed significantly better in solving the mathematical problems compared to their deaf peers. Although the deaf baccalaureate students exhibited the highest performance of all the deaf participants, they only performed as well as the hearing middle school students who were the lowest scoring hearing group. Deaf students remained flat in their performance on the mathematical problem-solving task from middle school through the college associate degree level. The analysis of the students' problem representations showed that the hearing participants utilized visual-spatial schematic representation to a greater extent than did the deaf participants. However, the use of visual-spatial schematic representations was a stronger positive predictor of mathematical problem-solving performance for the deaf students. When deaf students' problem representation focused simply on the visual-spatial pictorial or iconic aspects of the mathematical problems, there was a negative predictive relationship with their problem-solving performance. On two measures of visual-spatial abilities, the hearing students in high school and college performed significantly better than their deaf peers.     

Visual processing during mathematical problem solving

This study investigated, in the context of mathematical problem solving by secondary school students, the nature of the visual schemata which Johnson (1987) hypothesises mediate between logical propositional structures and rich specific visual images. Four groups of grade 10 students were studied, representing all combinations of high and low operational ability in mathematics (equivalent to Johnson's logical propositional structures) and high and low vividness of visual imagery (corresponding to Johnson's rich images). The results suggested first, that success at problem solving was related to logical operational ability, but not to vividness of visual imagery. Second, a variety of visually based strategies were used during problem solving which differed in their level of generality and abstraction, and use of these strategies appeared related to either logical operational ability or vividness of visual imagery, depending on their level of abstraction. The results supported Presmeg's (1992b) continuum of abstraction of image schemata.Throughout the paper, the first High or Low denotes logical operational ability, and the second, vividness of visual imagery.     

Gender differences in spatial ability: Relationship to spatial experience among Chinese gifted students in Hong Kong

Spatial ability based on measures of mental rotation, and spatial experience based on self‐reported participation in visual‐arts as well as spatial‐orientation activities were assessed in a sample of 337 Chinese, gifted students. Consistent with past findings for the general population, there were gender differences in spatial ability favoring boys. However, other results provided little support for the idea that gender differences in spatial experience might foster gender differences in spatial ability; specifically, results showed modest gender differences in visual‐arts experience favoring girls, and variation in visual orientation experience favoring secondary‐school boys. Nonetheless, the role of spatial experience had a more marked effect on girls, suggesting that encouraging female students to gain spatial experience might help bridge the gap in spatial ability between the genders.     

The role of working memory in spatial text processing: What benefit of imagery strategy and visuospatial abilities?

This study investigated the construction of a spatial model in relation to working memory (WM) and visuospatial abilities. Participants were trained to use either imagery or verbal strategies to process route spatial texts. Results obtained on a free recall task, a verification test and a graphic representation task showed the beneficial effect of using a strategy based on mental images. When imagery strategies were used, a concurrent articulatory task produced interference effects on recall performance, and a spatial tapping task also impaired performance as compared to the control condition. These interference effects suggest that both visuospatial and verbal WM were involved in construction of the spatial model. When repetition strategies were used, however, only the articulatory task produced interference effects, highlighting the role of the verbal WM. To elucidate how the involvement of the visuospatial component may differ in relation to visuospatial abilities, participants with good or poor ability in generation of visual images and spatial manipulation of objects were compared. The benefit of the imagery strategy was found in both groups, but whereas low-visuospatial imagery participants were sensitive to spatial interference, their high ability counterparts were not. These results question the role of imagery processes in the construction of spatial models and their relation to the visuospatial WM.     

Perception,imagery, visualization and engineering graphics

We have conducted action research involving an instructional intervention over a 20-year period. This has demonstrated that spatial ability influences academic performance in engineering, and can be increased through instruction focused on using perception and mental imagery in three-dimensional representation. Prior to our intervention, the first-year engineering graphics course at our university had a failure rate of 36% for all engineering students and failure rates of 80% for African students studying at our university. Similar high failure rates were reported in engineering drawing and design courses at other Southern African universities, and similar association between low scores on tests of spatial ability and academic performance, suggesting that the problem was one encountered by many engineering students, not just by students at our university. Over the initial 2 years of the intervention, pass rates for the first-year engineering graphics course increased from 64 to 76%. With further changes in teaching, and the training of senior students as tutors to support the lecturing and practical activities provided in the course, the pass rates have risen to 88% annually, over a period in which the composition of the first-year student has become increasingly diverse, with greater numbers of students entering the university from disadvantaged educational backgrounds. The instructional model we have used is based on Piagetian principles, and confirms Piaget's theories with respect to the trainability of spatial ability in adulthood. Our findings suggest the importance of early identification of students with difficulty, as well as the potential value of an intervention aimed at training the processes involved in visualization through three-dimensional modelling and representation of objects. While spatial ability appears to be trainable through the methods we have developed, our research also indicates that level of spatial ability at time of intake to university is an important influence on academic performance, suggesting the value of instruction in visualization and three-dimensional representation at school level. Social factors are also important influences on academic performance, suggesting the value of tutorial-based interventions aimed at improving spatial ability in those university and technikon courses for which visualization and three-dimensional representation are a requirement.     

Virtual connections,personal resources,loneliness, and academic self-efficacy among college students with and without LD

The goals of the study were first to compare the social and academic well-being (loneliness and academic self-efficacy (ASE) among college students with and without learning disabilities (LD), as well as three personal strengths (hope, optimism and sense of coherence (SOC). The second goal was to identify the predicting factors to their loneliness and ASE. The sample consisted of 178 female students from the special education programme in a teachers’ college, divided into two subgroups: 59 students with LD and 119 students without LD. The following questionnaires were used: the loneliness scale; ASE; Internet and smartphone activities; hope, optimism and SOC. As expected, students with LD reported higher levels of loneliness and lower levels of ASE. Multiple hierarchical regression analyses showed that the use of the Internet for avoidance coping was a significant predictor for the loneliness and ASE. Results of the mediation analysis further demonstrated that LD was positively associated with online avoidance coping. However, hopeful thinking had a significant mediation effect between LD and online avoidance coping. Students with LD who have hopeful thinking were less engaged in avoidance coping than their peers, focusing attention on the importance of hope as a mediation factor.     

Motivational Preconditions for Girls Gifted and Highly Gifted in Physics

During their socialisation process, many girls gifted in physics acquire a reality construction inconsistent with their objectively measurable competencies. In comparison to boys they rate their action and problem solving competencies unrealistically low, which results, for example, in extremely low participation rates in scientific and technical studies and professions. For this reason differences in motivation and self‐related cognitions become the focus of interest in explaining achievement differences. The present study was carried out prior to initial physics instruction. Students in the 7th grade of the German Gymnasium (243 girls and 282 boys) were divided according to their KFT 4‐13+ results into “average”, “gifted” or “highly gifted groups”. Prior to commencement of physics instruction, boys in general, as well as gifted male and female students, already possessed more knowledge of physics and more favourable motivation for the subject than girls or male and female students of average ability. In addition, domain specific measures and self‐related cognitions were evaluated in accordance with Dweck's model of achievement motivation.     

Using Number Lines to Solve Math Word Problems: A Strategy for Students with Learning Disabilities

Students with learning disabilities (LD) consistently struggle with word problem solving in mathematics classes. This difficulty has made curricular, state, and national tests particularly stressful, as word problem solving has become a predominant feature of such student performance assessments. Research suggests that students with LD perform poorly on word problem‐solving items due primarily to deficits in problem representation. Therefore, it is imperative that teachers provide these students with supplemental problem‐solving instruction that specifically targets the development of representational strategies. This article describes how one representational strategy, using number lines, can be used to model word problems as part of a comprehensive problem‐solving intervention to improve the conceptual understanding of math word problems and, subsequently, the problem‐solving performance of students with LD.     

Spatial ability and its role in organic chemistry: A study of four organic courses

The relationship between spatial ability and performance in organic chemistry was studied in four organic chemistry courses designed for students with a variety of majors including agriculture, biology, health sciences, pre-med, pre-vet, pharmacy, medicinal chemistry, chemistry, and chemical engineering. Students with high spatial scores did significantly better on questions which required problem solving skills, such as completing a reaction or outlining a multi-step synthesis, and questions which required students to mentally manipulate two-dimensional representations of a molecule. Spatial ability was not significant, however, for questions which could be answered by rote memory or by the application of simple algorithms. Students who drew preliminary figures or extra figures when answering questions were more likely to get the correct answer. High spatial ability students were more likely to draw preliminary figures, even for questions that did not explicitly require these drawings. When questions required preliminary or extra figures, low spatial ability students were more likely to draw figures that were incorrect. Low spatial ability students were also more likely to draw structures that were lopsided, ill-proportioned, and nonsymmetric. The results of this study are interpreted in terms of a model which argues that high spatial ability students are better at the early stages of problem solving described as “understanding” the problem. A model is also discussed which explains why students who draw preliminary or extra figures for questions are more likely to get correct answers.     

How spatial abilities enhance,and are enhanced by,dental education

In two studies with a total of 324 participants, dentistry students were assessed on psychometric measures of spatial ability, reasoning ability, and on new measures of the ability to infer the appearance of a cross-section of a three-dimensional (3-D) object. We examined how these abilities and skills predict success in dental education programs, and whether dental education enhances an individual's spatial competence. The cross-section tests were correlated with spatial ability measures, even after controlling for reasoning ability, suggesting that they rely specifically on the ability to store and transform spatial representations. Sex differences in these measures indicated a male advantage, as is often found on measures of spatial ability. Spatial ability was somewhat predictive of performance in restorative dentistry practical laboratory classes, but not of learning anatomy in general. Comparisons of the performance of students early and late in their dental education indicated that dentistry students develop spatial mental models of the 3-D structure of teeth, which improves their ability to mentally maintain and manipulate representations of these specific structures, but there is no evidence that dental education improves spatial transformation abilities more generally.