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.     

Improving students’ proportional thinking using schema-based instruction

This study investigated the effectiveness of an instructional program (schema-based instruction, SBI) designed to teach 7th graders how to comprehend and solve proportion problems involving ratios/rates, scale drawings, and percents. The SBI program emphasized the underlying mathematical structure of problems via schematic diagrams, focused on a 4-step procedure to support and monitor problem solving, and addressed the flexible use of alternative solution strategies based on the problem situation. Blocking by teacher at three middle schools, the authors randomly assigned the 21 classrooms to one of two conditions: SBI and control. Classroom teachers provided the instruction. Results of multilevel modeling used to test for treatment effects after accounting for pretests and other characteristics (gender, ethnicity) revealed the direct effects of SBI on mathematical problem solving at posttest. However, the improved problem solving skills were not maintained a month later when SBI was no longer in effect nor did the skills transfer to solving problems in new domain-level content.     

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.     

EFFECT OF THE PRESENCE OF EXTERNAL REPRESENTATIONS ON ACCURACY AND REACTION TIME IN SOLVING MATHEMATICAL DOUBLE-CHOICE PROBLEMS BY STUDENTS OF DIFFERENT LEVELS OF INSTRUCTION

This study explores the effects of the presence of external representations of a mathematical object (ERs) on problem solving performance associated with short double-choice problems. The problems were borrowed from secondary school algebra and geometry, and the ERs were either formulas, graphs of functions, or drawings of geometric figures. Performance was evaluated according to the reaction time (RT) required for solving the problem and the accuracy of the answer. Thirty high school students studying at high and regular levels of instruction in mathematics (HL and RL) were asked to solve half of the problems with ERs and half of the problems without ERs. Each task was solved by half of the students with ERs and by half of the students without ERs. We found main effects of the representation mode with particular effect on the RT and the main effects of the level of mathematical instruction and mathematical subject with particular influence on the accuracy of students’ responses. We explain our findings using the cognitive load theory and hypothesize that these findings are associated with the different cognitive processes related to geometry and algebra.     

The effects of using drawings in developing young children’s mathematical word problem solving: A design experiment with third-grade Hungarian students

The present study aims to investigate the effects of a design experiment developed for third-grade students in the field of mathematics word problems. The main focus of the program was developing students?? knowledge about word problem solving strategies with an emphasis on the role of visual representations in mathematical modeling. The experiment involved five experimental and six control classes (N?=?106 and 138, respectively) of third-grade students. The experiment comprised 20 lessons with 73 word problems, providing a systematic overview of the basic word problem types. Teachers of the experimental classes received a booklet containing lesson plans and overhead transparencies with different types of visual representations attached to the word problems. Students themselves were invited to make drawings for each task, and group work and teacher-led discussion shaped their beliefs about the role of visual representations in word problem solving. The effect sizes of the experiment were calculated from the results of two tests: an arithmetic skill and a word problem test, and the unbiased estimates for Cohen??s d proved to be 0.20 and 0.62. There were significant changes also in experimental group students?? beliefs about mathematics. The experiment pointed to the possibility, feasibility, and importance of learning about visual representations in mathematical word problem solving as early as in grade?3 (around age 9?C10).     

Make a drawing. Effects of strategic knowledge,drawing accuracy,and type of drawing on students’ mathematical modelling performance

Drawing strategies are widely used as a powerful tool for promoting students’ learning and problem solving. In this article, we report the results of an inferential mediation analysis that was applied to investigate the roles that strategic knowledge about drawing and the accuracy of different types of drawings play in mathematical modelling performance. Sixty-one students were asked to create a drawing of the situation described in a task (situational drawing) and a drawing of the mathematical model described in the task (mathematical drawing) before solving modelling problems. A path analysis showed that strategic knowledge about drawing was positively related to students’ modelling performance. This relation was mediated by the type and accuracy of the drawings that were generated. The accuracy of situational drawing was related only indirectly to performance. The accuracy of mathematical drawings, however, was strongly related to students’ performance. We complemented the quantitative approach with a qualitative in-depth analysis of students’ drawings in order to explain the relations found in our study. Implications for teaching practices and future research are discussed.     

Do realistic contexts and graphical representations always have a beneficial impact on students’ performance? Negative evidence from a study on modelling non-linear geometry problems

The aim of the present study was to investigate the influence of authentic contexts and of self-made graphical representations on students’ well-documented tendency to improperly apply the linear model to represent and solve non-proportional word problems about area and volume. A paper-and-pencil test on this kind of geometrical problems was collectively administered in different experimental groups of 13–14- and 15–16-year old students. Problem authenticity was experimentally enhanced for half of the students by prefacing the test by an assembly of well-chosen video fragments telling the story of Gulliver’s visit to the world of the Lilliputians and by linking all test items directly to these video fragments. The impact of self-made graphical representations was examined by asking half of the students to draw a reduced copy of the geometrical figure described in the problem before actually solving it. None of the experimental manipulations yielded the expected results. To the contrary, both factors even yielded a negative effect on students’ performance. Several explanations for these unexpected results are discussed.     

Themes and Interplay of Beliefs in Mathematical Reasoning

Upper secondary students’ task solving reasoning was analysed with a focus on arguments for strategy choices and conclusions. Passages in their arguments for reasoning that indicated the students’ beliefs were identified and, by using a thematic analysis, categorized. The results stress three themes of beliefs used as arguments for central decisions: safety, expectations and motivation. Arguments such as ‘I don’t trust my own reasoning’, ‘mathematical tasks should be solved in a specific way’ and ‘using this specific algorithm is the only way for me to solve this problem’ exemplify these three themes. These themes of beliefs seem to interplay with each other, for instance in students’ strategy choices when solving mathematical tasks.     

Schema-based instruction: Effects of experienced and novice teacher implementers on seventh grade students' proportional problem solving

This study examined the effects of a research-based intervention, schema-based instruction (SBI), implemented by experienced- (taught SBI in previous study; Jitendra et al., 2015) and novice-teacher implementers (taught SBI for the first time with professional development) on the mathematics outcomes of seventh-grade students. SBI is a multicomponent intervention that emphasizes the mathematical structure of problems through the use of schematic diagrams and incorporates problem solving and metacognitive strategy instruction. Results indicated that both experienced- and novice-teacher implementers delivered SBI with similar levels of fidelity; there was no SBI experience effect on the immediate and 10-week retention tests of proportional problem-solving, on a general measure of problem solving, or on the end of the year state mathematics achievement test. These results provide evidence that the effectiveness of SBI generalizes over time to different cohorts of teachers and that the impact of SBI on student mathematics outcomes is maintained over time without additional PD.     

Signs,symbols and schemas: understanding meaning in a child’s drawings

This paper examines the schematic underpinnings in the drawings of a four-year-old girl, Thea. The paper reviews literature on graphic representations, signs and meaning-making before discussing schematic form in children’s drawings, the theoretical background for the study. The paper discusses ethical issues and methodological approaches to the study where data include drawings made at home and school, Thea’s recorded talk about drawings, and video recordings of her drawing sessions over a four-month period. These were coded manually and using NVivo to identify schemas. The paper discusses examples of Thea’s exploration of enclosure and trajectory schematic form, which are represented by rich content derived from her experiences and imagination. The paper concludes that Thea’s drawings included many schematic signifiers with clear evidence of complex thinking around enclosures and of vertical and horizontal trajectories. The paper evidences the importance of listening to children’s talk as they draw in order to understand more fully, the meanings they are making. Through signs, symbols and personal narratives, Thea used drawing as a meaningful semiotic space where her persistent schematic concerns were manifest.     

What is the unknown? The ability to identify the semantic role of the unknown from word problems longitudinally predicts mathematical problem solving performance

The current study aimed to investigate children’s difficulties in word problem solving through assessing their ability to mathematize, or to identify the semantic role of the unknown from word problems. Fifth graders (n = 213) were given an advanced word problem reasoning task in which they had to match word problems with schematic diagrams that depict different processes (multiplication versus division) and the unknown being in different semantic roles (e.g., unit size, number of units, or total in an equal group problem). They were also tested on their mathematical problem solving as well as some potential confounding variables (i.e., intelligence and working memory) and mediators. The ability to identify the semantic role of the unknown was shown to be longitudinally predictive of children’s mathematical problem solving performance even after controlling for the effects of covariates and autoregressor. Such a relation was partially mediated by children’s ability to convert word problems into the correct number sentences/equations. The findings not only highlight the importance of unknown identification in mathematical problem solving process, but also provide a practical tool to assess such an ability.     

When procedures discourage insight: epistemological consequences of prompting novice physics students to construct force diagrams

ABSTRACT

One aim of school science instruction is to help students become adaptive problem solvers. Though successful at structuring novice problem solving, step-by-step problem-solving frameworks may also constrain students’ thinking. This study utilises a paradigm established by Heckler [(2010). Some consequences of prompting novice physics students to construct force diagrams. International Journal of Science Education, 32(14), 1829–1851] to test how cuing the first step in a standard framework affects undergraduate students’ approaches and evaluation of solutions in physics problem solving. Specifically, prompting the construction of a standard diagram before problem solving increases the use of standard procedures, decreasing the use of a conceptual shortcut. Providing a diagram prompt also lowers students’ ratings of informal approaches to similar problems. These results suggest that reminding students to follow typical problem-solving frameworks limits their views of what counts as good problem solving.     

规范楼梯结构平面图的图示方法探讨

在教学和工程实践中存在楼梯结构图图示混乱无章、不规范的情况,文章从《建筑结构制图标准》的作用、要求的角度,提出楼梯结构图图示方法应列入《建筑结构制图标准》的观点,并阐述它在该标准中的图示方法、内容。     

Multiple Representation in Learning About Evaporation

There has been extensive research on children’s understanding of evaporation, but representational issues entailed in this understanding have not been investigated in depth. This study explored three students’ engagement with science concepts relating to evaporation through various representational modes, such as diagrams, verbal accounts, gestures, and captioned drawings. This engagement entailed students (a) clarifying their thinking through exploring representational resources; (b) developing understanding of what these representations signify; and (c) learning how to construct representational aspects of scientific explanation. The study involved a sequence of classroom lessons on evaporation and structured interviews with nine children, and found that a focus on representational challenges provided fresh insights into the conceptual task involved in learning science. The findings suggest that teacher‐mediated negotiation of representational issues as students construct different modal accounts can support enriched learning by enabling both (a) richer conceptual understanding by students; and (b) enhanced teacher insights into students’ thinking.     

Common features of expertise in working life: implications for higher education

ABSTRACT

We need to know what it means to be an expert in working life today. Universities are often accused of neglecting the basic idea that higher education should be relevant to working life, and research on the subject of expertise in today’s workplace is lacking. Thirteen experts from different fields were interviewed and the obtained data were analysed using grounded theory as an analytical approach. The research questions were: (1) How do experts define expertise? (2) What kind of problems persist in their work? and (3) How are ‘routine’ and ‘adaptive’ expertise performed? The results revealed that expertise is more a social and collaborative phenomenon than an individual property. Experts develop new solutions and seek constant learning in their work. They excel at spontaneous problem solving. These findings suggest that, to prepare students to become experts, deliberate learning and practice should be provided within a framework of collaborative problem solving.     

Teaching students with LD to use diagrams to solve mathematical word problems

This study examined the effectiveness of instruction focused on teaching students with learning disabilities (LD) to solve 1- and 2-step word problems of varying types. Three students with LD in Grade 8 participated in the study. During the treatment, students received instruction in diagram generation and a strategy that incorporates diagrams as a part of the procedure to solve word problems. The results indicated that all students improved in the number of diagrams they used and in their ability to generate diagrams. Their word problem solving performance increased. Moreover, the students generated and used diagrams to solve other types of problems. Overall, the students were very satisfied with the instruction and would continue to use the diagrams and the strategy to solve word problems in other classroom settings.     

Gendered Career Choice: is sex‐stereotyping the cause or the consequence?

The interaction between school pupils’ schematic representations of ‘social’ and ‘technical’ roles and the impact that the ‘sex‐typing’ of careers has on occupational choice were investigated using stimulus vignettes. Questionnaire data pertaining to occupational choice were collected from first‐year university students enrolled on courses designated as either ‘social’ or ‘technical’. The prototypical in‐group positions for the two occupational areas were calculated and used as the basis of vignettes depicting either a male or female school pupil experiencing difficulty in deciding whether to pursue a technical or socially oriented career. The vignettes were presented to 107 school pupils aged 16‐18 who were asked which career area they thought the target pupil was likely to choose. Results indicate that subjects were able to identify correctly the prototypical characteristics utilised in the vignettes and that these were of greater importance in their expectation of course choice than the ‘sex’ of the target pupil.     

Varying effects of subgoal labeled expository text in programming,chemistry, and statistics

Originally intended as a replication study, this study discusses differences in problem solving performance among different domains caused by the same instructional intervention. The learning sciences acknowledges similarities in the learners’ cognitive architecture that allow interventions to apply across domains, but it also argues that each domain has characteristics that might affect how interventions impact learning. The present study uses an instructional design technique that had previously improved learners’ problem solving performance in programming: subgoal labeled expository text and subgoal labeled worked examples. It intended to replicate this effect for solving problems in statistics and chemistry. However, each of the experiments in the three domains had a different pattern of results for problem solving performance. While the subgoal labeled worked example consistently improved performance, the subgoal labeled expository text, which interacted with subgoal labeled worked examples in programming, had an additive effect with subgoal labeled worked examples in chemistry and no effect in statistics. Differences in patterns of results are believed to be due to complexity of the content to be learned, especially in terms of mapping problem solving procedures to solving problems, and the familiarity of tools used to solve problems in the domain. Subgoal labeled expository text was effective only when students learned more complex content and used unfamiliar problem solving tools.     

Text editing in chemistry instruction

In two experiments, differential performance onchemistry problems was obtained for twotraining strategies: text editing andconventional problem solving. Text editingrequires students to scan the text of problemstatements and specify whether it providessufficient, missing or irrelevant informationfor solution. It was hypothesized that textediting, which emphasizes gaining familiaritywith schematic knowledge, would lead to higherachievement than conventional problem solving.Experiment one indicated that text editing wassuperior to conventional problem solving inlearning to solve molarity and dilutionproblems. In particular, students who weretrained in text editing skipped someintermediate steps while solving molarityproblems. In contrast, using stoichiometryproblems, experiment two showed that students whowere trained in text editing performed worsethan students given conventional problems tosolve. An error analysis suggested that becauseof its failure to direct students' attention tothe coherent problem structure in the firstinstance, text editing has no advantage overconventional problem solving in the domain ofstoichiometry problems. It was concluded thatthe suitability of a text editing trainingstrategy depends on the learning materials.     

Holding the severed finger: Korean students’ understanding of historical significance

This study investigated students’ ideas about historical significance in Korea. Using qualitative, task-based interviews, I interviewed 28 secondary students who had taken Korean and world history courses. In response to various historical contexts, these Korean students relied primarily on a schematic narrative template consisting of ‘tragedy,’ ‘struggle,’ and ‘freedom and equality’ to judge historical significance. This schematic narrative template reflected the national identity which Korea established in the postcolonial context after independence in 1945. By applying this template to world history, moreover, students showed strong sympathy towards the oppressed, and this reflected their resistance to colonial and class hegemony. However, imposing this template resulted in significant omissions and distortions that might hinder the development of critical historical thinking. These findings suggest that a variety of cultural tools are necessary for Korean students to develop more complete and nuanced historical understanding.