Prairie Dog Optimization is a population-based optimization method that uses the behavior of prairie dogs to find the optimal solution. This paper proposes a novel optimization method, called the ...Opposition-based Laplacian Distribution with Prairie Dog Optimization (OPLD-PDO), for solving industrial engineering design problems. The OPLD-PDO method combines the concepts of opposition-based Laplacian distribution and Prairie Dog Optimization to find near-optimal solutions. This causes faster convergence to the optimal solution and reduces the chances of getting stuck in a local minimum. The OPLD-PDO method was tested on several benchmark problems to validate its performance. The results were compared with other methods, and the OPLD-PDO method was superior regarding solution quality. The results of this study demonstrate the potential of the OPLD-PDO method as a useful tool for solving industrial engineering design problems and photovoltaic (PV) solar problems.
Students' difficulties with word problems have been the subject of research for decades. Many studies identified students' ability to construct a situation model that reflects the word problem's ...situation structure correctly as a major factor. To overcome such difficulties, prior works suggested to provide learners with strategies, which comprise to restructure the situation model by integrating different perspectives on the presented situation. Corresponding trainings have not been investigated systematically yet.
We report on an experimental feasibility study investigating a training targeting the proposed strategies. Students from ten grade 2 classrooms (N = 115) in Germany participated in the study. The ten-day training focused on generating and comparing different perspectives on given situations but did not include any word problem solving.
Students participating in the training showed significantly higher progress in their ability to restructure situation models and their word problem solving skills from pre-to follow-up test than students from the control-group. The effect of the training was not influenced by students’ language skills.
The results indicate that it is feasible to foster word problem solving skills by solely training how to restructure the initial situation model generated from a word problem.
Since the experimental group received additional support in contrast to the control group, it is impossible to draw conclusions about the importance of the training for regular mathematics lessons, beyond the fact that the training is effective in principle.
The approach should be compared to other approaches to foster word problem solving.
•Investigates a training for flexibility in dealing with additive situations.•Training flexibility is feasible without solving word problems.•The training has effects on word problem solving skills.•Similar effects for learners with high and low language skills.•Development of flexibility explains development in word problem solving.
Let G be a profinite group. The coprime commutators γj⁎ and δj⁎ are defined as follows. Every element of G is both a γ1⁎-value and a δ0⁎-value. For j≥2, let X be the set of all elements of G that are ...powers of γj−1⁎-values. An element a is a γj⁎-value if there exist x∈X and g∈G such that a=x,g and (|x|,|g|)=1. For j≥1, let Y be the set of all elements of G that are powers of δj−1⁎-values. The element a is a δj⁎-value if there exist x,y∈Y such that a=x,y and (|x|,|y|)=1.
In this paper we establish the following results.
A profinite group G is finite-by-pronilpotent if and only if there is k such that the set of γk⁎-values in G has cardinality less than 2ℵ0 (Theorem 1.1).
A profinite group G is finite-by-(prosoluble of Fitting height at most k) if and only if there is k such that the set of δk⁎-values in G has cardinality less than 2ℵ0 (Theorem 1.2).
This study proposes a theoretical view for bridging mathematical modeling and word problem-solving activities. We introduce and elaborate on two theoretical ideas of the fictionality of word problems ...and the creation of possible fictional worlds. A world described by a word problem exists only fictionally (or potentially). A fictional world includes any imaginable world, any model for the real world, and any mathematical model. We developed a semi-open problem based on these theoretical ideas and observed Japanese eighth-grade students’ activity when solving it in an experimental lesson. Consequently, we identified a theoretically overlooked type of validation: considering the cultural relevance of solutions. The most important implication we draw from our observation is that the current definition of validation as a comparison between two stages in modeling should be extended to consider the integration of a target into a base possible fictional world.
•Bridging authentic mathematical modeling and artificial word problem-solving.•The fictionality of word problems is explored as a theoretical construct.•Activities can prompt students’ problem-solving processes and validation.•Validation should consider the cultural relevance of solutions.•Validation should integrate a target into a base possible fictional world.
Prior work on teachers’ mathematical knowledge has contributed to our understanding of the important role of teachers’ knowledge in teaching and learning. However, one aspect of teachers’ ...mathematical knowledge has received little attention:
strategic competence for word problems.
Adapting from one of the most comprehensive characterizations of mathematics learning (NRC,
2001
), we argue that teachers’ mathematical knowledge also includes strategic competence, which consists of
devising a valid solution strategy
,
mathematizing the problem
(i.e., choosing particular strategies and presentations to translate the word problem into mathematical expressions), and
arriving at a correct answer
(executing a solution) for a word problem. By examining the responses of 350 fourth- and fifth-grade teachers in the USA to four multistep fraction word problems, we were able to explore manifestations of teachers’ strategic competence for word problems. Findings indicate that teachers’ strategic competence was closely related to whether they devised a valid strategy. Further, how teachers dealt with known and unknown quantities in their mathematization of word problems was an important indicator of their strategic competence. Teachers with strong strategic competence used algebraic notations or pictorial representations and dealt with unknown quantities more frequently in their solution methods than did teachers with weak strategic competence. The results of this study provide evidence for the critical nature of strategic competence as another dimension needed to understand and describe teachers’ mathematical knowledge.
Some elementary students may exhibit challenging externalizing or internalizing behaviors in addition to difficulty with mathematics. In this study, we explored the behavioral patterns of 441 ...third-grade students with and without mathematics difficulty (MD). Compared with students without MD, students with MD demonstrated higher rates of externalizing and internalizing behaviors. We then randomly assigned 162 third-grade students with MD to receive a 10-week word-problem intervention or to be in a business-as-usual comparison group. Within the word-problem intervention, students with MD who exhibited higher occurrences of externalizing behaviors performed significantly lower on a word-problem measure than students without as many occurrences of externalizing behaviors. Response to the word-problem intervention did not differ based on internalizing behavior patterns.
•The production of visual-schematic representations determines word problem solving performance.•Relational processing makes it possible to represent the correct relations in a visual-schematic ...representation.•Both visual–spatial and linguistic–semantic processes foster word problem solving.•Both spatial ability and reading comprehension are related to word problem solving.
Two component skills are thought to be necessary for successful word problem solving: (1) the production of visual-schematic representations and (2) the derivation of the correct relations between the solution-relevant elements from the text base. The first component skill is grounded in the visual–spatial domain, and presumed to be influenced by spatial ability, whereas the latter is seated in the linguistic–semantic domain, and presumed to be influenced by reading comprehension. These component skills as well as their underlying basic abilities are examined in 128 sixth grade students through path analysis. The results of the path analysis showed that both component skills and their underlying basic abilities explained 49% of students’ word problem solving performance. Furthermore, spatial ability and reading comprehension both had a direct and an indirect relation (via the component skills) with word problem solving performance. These results contribute to the development of instruction methods that help students using these components while solving word problems.
Inconsistent operations: A weapon of math disruption Jarosz, Andrew F.; Jaeger, Allison J.
Applied cognitive psychology,
January/February 2019, 2019-00-00, 2019-01-00, 20190101, Letnik:
33, Številka:
1
Journal Article
Recenzirano
Summary
Word problems embed a math equation within a short narrative. Due to their structure, both numerical and linguistic factors can contribute to problem difficulty. The present studies explored ...the role of irrelevant information in word problems, to determine whether its negative impact is due to numerical (foregrounding hypothesis) or linguistic (inconsistent‐operations hypothesis) interference. Across three experiments, participants solved multiplication and division word problems containing irrelevant numerical information, which was either associated or disassociated with the protagonist. Results demonstrated increased solution errors on division problems when irrelevant numbers were disassociated with the protagonist. When memory for numerical information was emphasized, disassociation was specifically impacted low‐working memory individuals. The effect of disassociation on division performance persisted even when irrelevant numbers, but not words, were removed from problems. These results suggest that, even in the presence of numerically interfering information, it is the language of word problems that often drive their difficulty.
The success or failure of education systems in promoting student problem-solving skills depends on attitudinal, political, and pedagogical variables. Among these variables, the design of mathematics ...textbooks is thought to partially explain why students from high-achieving countries show better problem-solving ability in international assessments. In the current study, we delved into this question and compared the frequency and characteristics of arithmetic word problems (AWPs) contained in primary school math textbooks in two countries with different levels of performance in international assessments—Singapore and Spain. In our analyses, we focused on (1) the quantity of arithmetic word problems, (2) the variety of problems in terms of their additive or multiplicative structures and semantic-mathematical substructures, and (3) the quantity and nature of illustrations that were presented together with arithmetic word problems. Although a larger proportion of AWP activities was found in Singaporean textbooks, the results showed a similar variety of AWPs in both Singaporean and Spanish math textbooks. Furthermore, in both countries, math textbooks emphasized the structures classified as (additive) combine 1 and (multiplication) simple rate in AWPs. Notably, the Singaporean textbook contained a larger percentage of illustrations that reflected the semantic-mathematical structures of the problems and helped students learn how to solve AWPs (e.g., bar models). The findings are discussed in light of theories that posit that textbooks constitute a fundamental part of the teaching–learning process in the classroom.
Word problems are among the most difficult kinds of problems that mathematics learners encounter. Perhaps as a result, they have been the object of a tremendous amount research over the past ...50 years. This opening article gives an overview of the research literature on word problem solving, by pointing to a number of major topics, questions, and debates that have dominated the field. After a short introduction, we begin with research that has conceived word problems primarily as problems of comprehension, and we describe the various ways in which this complex comprehension process has been conceived theoretically as well as the empirical evidence supporting different theoretical models. Next we review research that has focused on strategies for actually solving the word problem. Strengths and weaknesses of informal and formal solution strategies—at various levels of learners’ mathematical development (i.e., arithmetic, algebra)—are discussed. Fourth, we address research that thinks of word problems as exercises in complex problem solving, requiring the use of cognitive strategies (heuristics) as well as metacognitive (or self-regulatory) strategies. The fifth section concerns the role of graphical representations in word problem solving. The complex and sometimes surprising results of research on representations—both self-made and externally provided ones—are summarized and discussed. As in many other domains of mathematics learning, word problem solving performance has been shown to be significantly associated with a number of general cognitive resources such as working memory capacity and inhibitory skills. Research focusing on the role of these general cognitive resources is reviewed afterwards. The seventh section discusses research that analyzes the complex relationship between (traditional) word problems and (genuine) mathematical modeling tasks. Generally, this research points to the gap between the artificial word problems learners encounter in their mathematics lessons, on the one hand, and the authentic mathematical modeling situations with which they are confronted in real life, on the other hand. Finally, we review research on the impact of three important elements of the teaching/learning environment on the development of learners’ word problem solving competence: textbooks, software, and teachers. It is shown how each of these three environmental elements may support or hinder the development of learners’ word problem solving competence. With this general overview of international research on the various perspectives on this complex and fascinating kind of mathematical problem, we set the scene for the empirical contributions on word problems that appear in this special issue.
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