Individual differences in mental animation during mechanical reasoning

In three experiments we tested the effects of spatial visualization ability on performance of a motion-verification task, in which subjects were shown a diagram of a mechanical system and were asked to verify a sentence stating the motion of one of the system components. We propose that this task involves component processes of (1) sentence comprehension, (2) diagram comprehension, (3) text-diagram integration, and (4) mental animation. Subjects with law spatial ability made more errors than did subjects with high spatial ability on this task, and they made more errors on items in which more system components had to be animated to solve the problem. In contrast, the high-spatial subjects were relatively accurate on all trials. These results indicate that spatial visualization is correlated with accuracy on the motion-verification task and suggest that this correlation is primarily due to the mental animation component of the task. Reaction time and eye-fixation data revealed no differences in how the high- and low-spatial subjects decomposed the task. The data of the two groups of subjects were equally consistent with a piecemeal model of mental animation, in which components are animated one by one in order of the causal chain of events in the system.     

Individual differences in mental rotation: what does gesture tell us?

Gestures are common when people convey spatial information, for example, when they give directions or describe motion in space. Here, we examine the gestures speakers produce when they explain how they solved mental rotation problems (Shepard and Meltzer in Science 171:701–703, 1971). We asked whether speakers gesture differently while describing their problems as a function of their spatial abilities. We found that low-spatial individuals (as assessed by a standard paper-and-pencil measure) gestured more to explain their solutions than high-spatial individuals. While this finding may seem surprising, finer-grained analyses showed that low-spatial participants used gestures more often than high-spatial participants to convey “static only” information but less often than high-spatial participants to convey dynamic information. Furthermore, the groups differed in the types of gestures used to convey static information: high-spatial individuals were more likely than low-spatial individuals to use gestures that captured the internal structure of the block forms. Our gesture findings thus suggest that encoding block structure may be as important as rotating the blocks in mental spatial transformation.     

小学生视觉-空间表征类型和数学问题解决的研究

本研究考察并比较了四至六年级儿童的视觉－空间表征策略、数学问题解决和空间视觉化能力。结果表明：五、六年级儿童的解题正确率、使用图式表征策略的程度显著高于四年级儿童;使用图像表征策略的程度各年级无显著差异。将数学问题分成三个难度等级,发现年级差异主要表现在难度等级1的题目上。另外,六年级儿童的空间视觉化能力显著高于四年级儿童。     

Gender differences in advanced mathematical problem solving

Strategy flexibility in mathematical problem solving was investigated. In Studies 1 and 2, high school juniors and seniors solved Scholastic Assessment Test-Mathematics (SAT-M) problems classified as conventional or unconventional. Algorithmic solution strategies were students' default choice for both types of problems across conditions that manipulated item format and solution time. Use of intuitive strategies on unconventional problems was evident only for high-ability students. Male students were more likely than female students to successfully match strategies to problem characteristics. In Study 3, a revised taxonomy of problems based on cognitive solution demands was predictive of gender differences on Graduate Record Examination-Quantitative (GRE-Q) items. Men outperformed women overall, but the difference was greater on items requiring spatial skills, shortcuts, or multiple solution paths than on problems requiring verbal skills or mastery of classroom-based content. Results suggest that strategy flexibility is a source of gender differences in mathematical ability assessed by SAT-M and GRE-Q problem solving.     

Productive Failure

This study demonstrates an existence proof for productive failure: engaging students in solving complex, ill-structured problems without the provision of support structures can be a productive exercise in failure. In a computer-supported collaborative learning setting, eleventh-grade science students were randomly assigned to one of two conditions to solve problems in Newtonian kinematics. In one condition, students solved ill-structured problems in groups followed by well-structured problems individually. In the other condition, students solved well-structured problems in small groups followed by well-structured problems individually. Finally, all students solved ill-structured problems individually. Groups who solved ill-structured problems expectedly struggled with defining and analyzing the problems, resulting in poor quality of solutions. However, despite failing in their collaborative efforts, these students outperformed their counterparts in the well-structured condition on individual near- and far-transfer measures subsequently, suggesting a latent productivity in what initially seemed to be failure.     

The nature of gestures' beneficial role in spatial problem solving

Co-thought gestures are hand movements produced in silent, noncommunicative, problem-solving situations. In the study, we investigated whether and how such gestures enhance performance in spatial visualization tasks such as a mental rotation task and a paper folding task. We found that participants gestured more often when they had difficulties solving mental rotation problems (Experiment 1). The gesture-encouraged group solved more mental rotation problems correctly than did the gesture-allowed and gesture-prohibited groups (Experiment 2). Gestures produced by the gesture-encouraged group enhanced performance in the very trials in which they were produced (Experiments 2 & 3). Furthermore, gesture frequency decreased as the participants in the gesture-encouraged group solved more problems (Experiments 2 & 3). In addition, the advantage of the gesture-encouraged group persisted into subsequent spatial visualization problems in which gesturing was prohibited: another mental rotation block (Experiment 2) and a newly introduced paper folding task (Experiment 3). The results indicate that when people have difficulty in solving spatial visualization problems, they spontaneously produce gestures to help them, and gestures can indeed improve performance. As they solve more problems, the spatial computation supported by gestures becomes internalized, and the gesture frequency decreases. The benefit of gestures persists even in subsequent spatial visualization problems in which gesture is prohibited. Moreover, the beneficial effect of gesturing can be generalized to a different spatial visualization task when two tasks require similar spatial transformation processes. We concluded that gestures enhance performance on spatial visualization tasks by improving the internal computation of spatial transformations. (PsycINFO Database Record (c) 2010 APA, all rights reserved).     

Contribution of sex-differentiated experiences to spatial and mechanical reasoning abilities

For 137 women and 115 men first-year college students tested spatial visualization and mechanical reasoning were most strongly correlated with four everyday spatial abilities--understanding mathematics/science and graphs/charts, drafting and drawing things, and arranging objects. Despite greater practice on only 2 of 10 activities, men uniformly judged they had significantly better spatial ability compared to their same-gender peers than did the women.     

Conceptual Knowledge,Procedural Knowledge,and Metacognition in Routine and Nonroutine Problem Solving

When, how, and why students use conceptual knowledge during math problem solving is not well understood. We propose that when solving routine problems, students are more likely to recruit conceptual knowledge if their procedural knowledge is weak than if it is strong, and that in this context, metacognitive processes, specifically feelings of doubt, mediate interactions between procedural and conceptual knowledge. To test these hypotheses, in two studies (Ns = 64 and 138), university students solved fraction and decimal arithmetic problems while thinking aloud; verbal protocols and written work were coded for overt uses of conceptual knowledge and displays of doubt. Consistent with the hypotheses, use of conceptual knowledge during calculation was not significantly positively associated with accuracy, but was positively associated with displays of doubt, which were negatively associated with accuracy. In Study 1, participants also explained solutions to rational arithmetic problems; using conceptual knowledge in this context was positively correlated with calculation accuracy, but only among participants who did not use conceptual knowledge during calculation, suggesting that the correlation did not reflect “online” effects of using conceptual knowledge. In Study 2, participants also completed a nonroutine problem-solving task; displays of doubt on this task were positively associated with accuracy, suggesting that metacognitive processes play different roles when solving routine and nonroutine problems. We discuss implications of the results regarding interactions between procedural knowledge, conceptual knowledge, and metacognitive processes in math problem solving.     

When do spatial abilities support student comprehension of STEM visualizations?

Spatial visualization abilities are positively related to performance on science, technology, engineering, and math tasks, but this relationship is influenced by task demands and learner strategies. In two studies, we illustrate these interactions by demonstrating situations in which greater spatial ability leads to problematic performance. In Study 1, chemistry students observed and explained sets of simultaneously presented displays depicting chemical phenomena at macroscopic and particulate levels of representation. Prior to viewing, the students were asked to make predictions at the macroscopic level. Eye movement analyses revealed that greater spatial ability was associated with greater focus on the prediction-relevant macroscopic level. Unfortunately, that restricted focus was also associated with lower-quality explanations of the phenomena. In Study 2, we presented the same displays but manipulated whether participants were asked to make predictions prior to viewing. Spatial ability was again associated with restricted focus, but only for students who completed the prediction task. Eliminating the prediction task encouraged attempts to integrate the displays that related positively to performance, especially for participants with high spatial ability. Spatial abilities can be recruited in effective or ineffective ways depending on alignments between the demands of a task and the approaches individuals adopt for completing that task.     

Cognitive processes in the solution of three-term series problems.

This study investigated the nature of strategies used in solving the three-term series problem. Three presentation modes (auditory, visual/sequential, and visual/simultaneous) were crossed with two question positions (before-premises and after-premises), for a total of six methods of problem presentation. Both high-spatial/imagal and low-spatial/imagal problems were employed, the assumption being that better performance on high-spatial/imagal problems reflected the use of a spatial/imagal strategy, while equal performance on both types of problems indicated the use of an alternative, perhaps verbal, strategy. It was hypothesized that different presentations would lead to differences in memory demands, input/processing interference, and mathemagenic behaviors, and thus to different problem-solving strategies. Response data and subjective reports confirmed this prediction. Results were discussed in terms of the Clark-Huttenlocher controversy (H. H. Clark, Linguistic processes in deductive reasoning, in Psychological Review, 1969, 76, 387--404; J. Huttenlocher, Constructing spatial images: A strategy in reasoning, Psychological Review, 1968, 75, 550--560).     

Gesture Supports Spatial Thinking in STEM

The present article describes two studies that examine the impact of teaching students to use gesture to support spatial thinking in the Science, Technology, Engineering, and Mathematics (STEM) discipline of chemistry. In Study 1 we compared the effectiveness of instruction that involved either watching gesture, reproducing gesture, or reading text. The results indicate that students in the reproducing gesture condition produced significantly more gestures while problem solving than students in the other two groups and significantly outperformed the other groups on study measures. In Study 2 we compared the effectiveness of gesture to an instructional approach that involved manually handling concrete models without gesture. Students performed equally well in both conditions; however, students taught with concrete models performed significantly worse if concrete models were not available during assessment. These studies show that gesture is an effective strategy for supporting spatial thinking in STEM disciplines and that this benefit may result from physically simulating spatial transformations.     

数学学习不良儿童视觉-空间表征与数学问题解决

采用临床访谈的方法,考察了30名数学学习不良(MD)儿童和31名一般儿童的数学问题解决、视觉-空间表征策略和空间视觉化能力。结果发现：图式表征能促进数学问题的解决,图像表征则起妨碍作用;空间视觉化能力与解题正确率及图式表征策略有显著正相关,与图像表征策略有显著负相关。MD儿童的解题正确率以及使用图式表征策略的程度显著低于一般儿童,使用图像表征策略的程度则显著高于一般儿童。在解题正确率和图式表征策略这两个变量上,MD儿童和一般儿童的年级发展趋势相同,都随年级的升高而提高。但在图像表征策略的使用上,一般儿童有随年级的升高而下降的趋势,MD儿童却没有下降的趋势。两类儿童的空间视觉化能力都随年级的升高而提高。     

Distribution of Spatial Visualization and Mathematical Problem Solving Scores: A Test of Stafford's X-linked Hypotheses

This study investigated distribution of spatial visualization scores (Space Relations test of the Differential Aptitude Test) and mathematical problem solving scores (Mental Arithmetic Problems) obtained by 161 male and 152 female, 9th grade, white students for fit to the distributions predicted by the X-linked hypotheses of recessive inheritance of these skills. Data did not support the X-linked hypotheses. No significant sex-related differences were found between mean scores of tests of spatial visualization or mathematical problem solving.     

Creativity,consciousness, and “becoming wise fast enough”

Thirteen 11th-grade students in enriched mathematics and 9 in a regular course solved nine word problems that varied in the ease with which they could be solved by verbal-logical or visual strategies. A pervasive use of verbal-logical solution methods was noted for both groups. Significant differences were found in the final solution strategies. In the course of solving each problem, students tried different strategies. Enriched students alternated almost exclusively between verbal-logical and visual solution methods, whereas regular students alternated equally between verbal-logical and visual solution methods or between verbal-logical strategies and trial and error.     

Who Benefits from Diagrams and Illustrations in Math Problems? Ability and Attitudes Matter

How do diagrams and illustrations affect mathematical problem solving? Past research suggests that diagrams should promote correct performance. However, illustrations may provide a supportive context for problem solving, or they may distract students with seductive details. Moreover, effects may not be uniform across student subgroups. This study assessed the effects of diagrams and illustrations on undergraduates' trigonometry problem solving. We used a 2 (Diagram Presence) × 2 (Illustration Presence) within‐subjects design, and our analysis considered students' mathematics ability and attitudes towards mathematics. Participants solved problems more accurately when they included diagrams. This effect was stronger for students who had more positive mathematics attitudes, especially when there was an illustration present. Illustrations were beneficial for students with high mathematics ability but detrimental for students with lower ability. Considering individual differences in ability and attitude is essential for understanding the effects of different types of visual representations on problem solving. Copyright © 2017 John Wiley & Sons, Ltd.     

Measurement Skills in Low-Income Elementary School Students: Exploring the Nature of Gender Differences

In this research, we examined overall performance and gender differences in measurement skills in elementary-school students from low-income families. In Study 1, accuracy and error patterns were analyzed in a large sample of fourth-graders; in Study 2 error patterns and strategy usage were examined with a smaller sample of fourth-graders. Study 1 showed no main effect of gender on students’ performance. Instead, as predicted, the direction of gender difference varied as a function of problem type: boys outperformed girls on spatial/conceptual measurement, whereas girls outperformed boys on formula-based measurement, as well as on a test of computation skills. Study 2 revealed both similarities and differences in the way boys and girls approached measurement problems. Girls appeared to have specific difficulty with spatial/conceptual problems where objects and measurement units were not pictorially presented. When recording their solutions, girls generally wrote down calculations while boys made drawings. Overall, the students performed poorly in measurement; strategy analysis allowed for examination of common weaknesses, indicating possible ways of improving performance of underserved groups.     

The Effects of Presenting Multidigit Mathematics Problems in a Realistic Context on Sixth Graders' Problem Solving

Mathematics education and assessments increasingly involve arithmetic problems presented in context: a realistic situation that requires mathematical modeling. This study assessed the effects of such typical school mathematics contexts on two aspects of problem solving: performance and strategy use. Multidigit arithmetic problems presented in two conditions—with and without a realistic context—were solved by 685 sixth graders from The Netherlands. Regarding performance, the same (latent) ability dimension was involved in solving both types of problems, and the presence of a context increased the difficulty level of the division problems but not of other operations. Regarding strategy use, strategy choice and strategy accuracy were not affected by the presence of a problem context. In sum, the presence of a typical context in multidigit arithmetic problems had no marked effects on students' problem-solving behavior, which held for different subgroups of students with respect to language ability and gender.     

Number-word sequence skill and arithmetic performance

In two studies, the role of the number‐word sequence skill for arithmetic performance was investigated. In the first, children between 4 and 8 years of age were asked to count forward and backward on the number‐word sequence and to solve arithmetic problems followed by post‐solution interviews about solution procedures. The results demonstrated that the number‐word sequence skill predicted both number of problems solved and strategy to solve the problems. In Study 2 it was found that solving doubles (e.g., 2 + 2 = ?) problems served as a link between the number‐word sequence skill and the number of arithmetic problems solved. The findings suggest that counting on the number‐word sequence may be an early solution procedure and that, with increasing counting skill, the child may detect regularities in the number‐word sequence that can be used to form new and more accurate strategies for solving arithmetic problems.     

Verbalization of discovered principles for solving match-stick problems

Abstract.— Problem solving performance of subjects who were taught part of a principle was studied under three experimental conditions, differing with respect to the amount of hints given as to the nature of the missing part of the principle, and compared with performance of suhjects who were taught the full principle. After practice, those who were taught the principle partially were asked to state the missing part of the principle, or any other general method found for solving the problems. Subjects who were taught the full principle solved most problems, while performance under experimental conditions varied with the preciseness with which the stated principle specified relevant parts of the problems to be solved. The hints failed to influence performance with respect to both problem solving and ability to state principles.     

Spatial visualization and sex differences in quantitative ability

The hypothesis that sex differences in spatial visualization ability might account for sex differences in mathematical ability was supported for a group of 183 male and 81 female college students. With spatial visualization statistically controlled, no significant sex differences in Quantitative Scholastic Aptitude Test (QSAT) scores was found; including sex as a predictor variable increased the variance explained by less than 1%. Although the slope of the regression of mathematics on spatial visualization did not differ as a function of sex, males were somewhat more predictable than females. As the OSAT of both males and females high on spatial visualization was more predictable than the QSAT of those scoring less well, it appears that the sex difference in predictability is due to males having higher spatial ability than females.