Mathematical knowledge involving whole number multiplication and division is integral to understanding multiplicative structures such as ratios, slope, rate of change, and proportions, which are ...important in subsequent mathematical learning. In this study, we investigated the effectiveness of a research-based intervention, schema-based instruction (SBI), to teach multiplicative whole number word-problem solving to three-fifth-grade students with mathematics disabilities. SBI emphasizes the underlying mathematical problem structure, uses visual-schematic diagrams to show how quantities in a problem are related, focuses on problem solving procedures grounded in reasoning, and highlights metacognitive strategy knowledge to monitor and reflect on the problem-solving process. We used a multiple probe across participants design to evaluate the functional relation between the SBI intervention and word-problem solving performance. Immediately following treatment as well as 1–3 weeks later, results indicated acquisition and retention of word problem-solving skills by all three students. In addition, the percentage of student use of representations (i.e., drawing a diagram, writing a number sentence) increased from the baseline to the intervention phases. These findings and their relation to theory, research, and practice are discussed.
Problem-solving ability is a primary skill students should acquire in learning mathematics and is often connected to Adversity Quotient (AQ). This qualitative descriptive study describes the ability ...to solve PLSV problems regarding AQ (climbers, campers, and quitters) in 7th-grade students of SMP Negeri 21 Surabaya. Data were collected through questionnaires, tests, and interviews using instruments such as the AQ questionnaire sheet, Word Problem Solving Test (TPWP), questionnaires, and interview guidelines. The problem-solving ability in this study refers to the four stages of Polya's problem-solving ability including understanding the problem, planning, implementing, and evaluating. The results of this study showed that (1) the climbers subject was able to accomplish all of Polya's stages, (2) the campers subject, was able to accomplish three stages of Polya's problem solving, which were able to understand the problem, plan the problem solving, and carry out the problem-solving. Whereas, (3) the quitters subject, was unable to complete Polya's problem-solving stages as it showed that the subject was poorly understanding the problem, planning to solve the problem, implementing the solution plan, and incapable of looking back at problems that had been solved.
Studying worked examples impose relatively low cognitive load because learners' attention is directed to learn the schema, which is embedded in the worked examples. That schema encompasses both ...conceptual knowledge and procedural knowledge. It is well-documented that worked examples are effective in facilitating the acquisition of problem-solving skills. However, the use of worked examples to develop problem-solving expertise is less known. Typically, experts demonstrate an efficient way to solve problems that is quicker, faster, and having fewer solution steps. We reviewed five studies to validate the benefit of worked examples to develop problem-solving expertise for word problems. Overall, a diagram portrays the problem structure, coupled with either study worked examples or complete multiple example-problem pairs, facilitates the formation of an equation to solve words problems efficiently. Hence, an in-depth understanding of conceptual knowledge (i.e., problem structure) might contribute to superior performance of procedural knowledge manifested in the reduced solution steps.
Background
The ability to translate concrete manipulatives into mathematical formulas can aid in the solving of mathematical word problems among students, and metacognitive prompts play a significant ...role in enhancing this process.
Aims
Based on the concept of semantic congruence, we explored the effects of metacognitive prompts and numerical ordinality on information searching and cognitive processing, throughout the process of solving mathematical word problems among primary school students in China.
Sample
Participants included 73 primary school students (38 boys and 35 girls) with normal or corrected visual acuity.
Methods
This study was based on a 2 (prompt information: no‐prompt, metacognitive‐prompt) × 2 (number attribute: cardinal number, ordinal number) mixed experimental design. We analysed multiple eye‐movement indices, such as fixation duration, saccadic amplitude, and pupil size, since they pertained to the areas of interest.
Results
When solving both types of problems, pupil sizes were significantly smaller under the metacognitive‐prompt condition compared with the no‐prompt condition, and shorter dwell time for specific sentences, conditional on metacognitive prompts, indicated the optimization of the presented algorithm. Additionally, the levels of fixation durations and saccadic amplitudes were significantly higher when solving ordinal number word problems compared with solving ordinal number problems, indicating that primary school students were less efficient in reading and faced increased levels of difficulty when solving ordinal number problems.
Conclusions
The results indicate that for Chinese upper‐grade primary school students, cognitive load was lower in the metacognitive prompting condition and when solving cardinal problems, and higher when solving ordinal problems.
For autistic students receiving special education services, little is known about their relative strengths, weaknesses, and enjoyment across different math content areas; their overall math interest ...and persistence are also not well-studied. Using the 2017 eighth-grade National Assessment of Education Progress data, this study finds, relative to general education peers with the same math proficiency level, autistic students scored higher and exhibited faster speed in solving visuospatial problems (e.g. identifying figures), but scored lower on math word problems with complex language or social context. Autistic students reported a higher level of enjoyment in solving math problems related to finding areas of shapes or figures but a lower level of persistence than their non-autistic, general education peers. Our work points out the need to help autistic students overcome their weaknesses in word problems and develop their mathematical persistence.
To investigate student difficulties in solving word problems in algebra, we carried out a teaching experiment involving 51 Indonesian students (12/13 year-old) who used a digital mathematics ...environment. The findings were backed up by an interview study, in which eighteen students (13/14 year-old) were involved. The perspective of mathematization, i.e., the activity to transform a problem into a symbolic mathematical problem, and to reorganize the mathematical system, was used to identify student difficulties on the topic of linear equations in one variable. The results show that formulating a mathematical model--evidenced by errors in formulating equations, schemas or diagrams--is the main difficulty. This highlights the importance of mathematization as a crucial process in the learning and teaching of algebra.
This study examined effects of gamified comparison on sixth graders' performance of algebra word problem solving and attitude toward algebra learning. Seventy-two sixth graders were invited to ...participate in a four-week experimental instruction and assigned to three groups: gamified comparison, comparison, and control. The results showed (1) a significant effect on solving similar problems: the gamified comparison group performed significantly better than the comparison group and the control group respectively, and the comparison group performed significantly better than the control group; (2) a significant effect on solving transfer problems: the gamified comparison group gained significantly higher scores than the comparison group and the control group respectively, and the comparison group gained significantly higher scores than the control group; (3) a significant effect on students' learning attitude: while no significant differences found on students' confidence, the gamified comparison group made significantly more positive responses than the comparison group and the control group in terms of enjoyment, motivation, and perceived value. This study proposed a feasible combination of game rule and comparison strategy, as well as exploring implications for teachers' teaching design and students' gamified learning activities.
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.
Diagnosing students’ mistakes in solving word problems using Newman’s Error Analysis (NEA) is time-consuming. To address this limitation, NEA was incorporated in the Cognitive Diagnostic Assessment ...(CDA) with Ordered Multiple Choice (OMC) word problem testlets developed in this study for assessing students’ attribute mastery level in solving word problems involving ‘Time’. The assessment instrument was developed based on the principled assessment design which comprised four building blocks: (i) construct map, (ii) item design, (iii) outcome space, and (iv) measurement model. The construct map was adapted from NEA for guiding the development of the assessment instrument. The assessment instrument was administered to a sample of 90 Year Four students in three elementary schools. The results indicated that the instrument showed appropriate difficulty level and discrimination power, satisfactory model-data fit, and high reliability. Thus, the CDA with OMC word problem testlets developed could be a valuable instrument for pinpointing students’ mistakes and their mastery level of word-problem-solving attributes involving ‘Time’.
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.
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