Problem-solving Phase Research Articles

The professional practice of designing tasks: how do pre-service early childhood teachers promote mathematical processes in early algebra?

AbstractSpanish educational curriculum adopts a mathematical process-based approach, which encompasses problem solving, reasoning and proof, communication, connections and representation. A fundamental role in the integration of these processes in mathematics teaching is played by teachers’ professional practice of designing tasks. According to this, our aim is to analyze the ways in which pre-service early childhood teachers understand the mathematical processes in the professional practice of designing early algebra tasks and to identify how they intend to promote these processes through the tasks. Content analysis techniques were used to examine the designed tasks. To illustrate the data analysis and results, six tasks are presented. As a result, pre-service early childhood teachers associate problem solving with challenges and questions. They understand problems as unfamiliar situations but ignore the relationships between students and tasks. Moreover, they do not encourage exploration of phases of problem solving and tend to use strategies more suitable for routine tasks. Communication is identified in all the tasks designed, encouraging interaction and discussion. However, only one task explicitly promotes mathematical language. For reasoning and proof, pre-service teachers begin to use questions to elicit explanations and justifications, but do not encourage verification strategies and various modes of reasoning. The process of connections is only present in one task, reflecting the fragmented nature of mathematics teaching. We conclude that the professional practice of designing mathematical tasks is a powerful in teacher education. However, training programs should place greater emphasis on the meaningful use of mathematical processes.

How to Optimize Self-Assessment Accuracy in Cognitive Skill Acquisition When Learning from Worked Examples

The present study was designed to understand and optimize self-assessment accuracy in cognitive skill acquisition through example-based learning. We focused on the initial problem-solving phase, which follows after studying worked examples. At the end of this phase, it is important that learners are aware whether they have already understood the solution procedure. In Experiment 1, we tested whether self-assessment accuracy depended on whether learners were prompted to infer their self-assessments from explanation-based cues (ability to explain the problems’ solutions) or from performance-based cues (problem-solving performance) and on whether learners were informed about the to-be-monitored cue before or only after the problem-solving phase. We found that performance-based cues resulted in better self-assessment accuracy and that informing learners about the to-be-monitored cue before problem-solving enhanced self-assessment accuracy. In Experiment 2, we again tested whether self-assessment accuracy depended on whether learners were prompted to infer their self-assessments from explanation- or performance-based cues. We furthermore varied whether learners received instruction on criteria for interpreting the cues and whether learners were prompted to self-explain during problem-solving. When learners received no further instructional support, like in Experiment 1, performance-based cues yielded better self-assessment accuracy. Only when learners who were prompted to infer their self-assessments from explanation-based cues received both cue criteria instruction and prompts to engage in self-explaining during problem-solving did they show similar self-assessment accuracy as learners who utilized performance-based cues. Overall, we conclude that it is more efficient to prompt learners to monitor performance-based rather than explanation-based cues in the initial problem-solving phase.

Fostering Learning from Errors—Computer-Based Adaptivity at the Transition Between Problem Solving and Explicit Instruction

When learners acquire new content by working on a problem-solving task prior to explicit instruction, their attempts to solve the problem usually represent only partial steps on the way to the target concept. Both, theoretical assumptions on conceptual change as well as empirical findings on effective instructional formats with incorrect solutions, suggest that it is beneficial to address incorrect student solutions in a (subsequent) instruction phase by comparing incorrect and correct solutions. There is initial evidence that learning is most successful when learner compare the correct solution to an incorrect solution that reflects the learners’ conceptual understanding from the problem-solving phase. In the present study, we investigated in a highly controlled experimental design the relevance of this fit between the learners’ individual solution type from the problem-solving phase and the incorrect solution type in the instruction phase for learning success. In a computer-based learning environment, sixth graders worked on a problem-solving task to compare fractions. In the subsequent instruction phase, students in three conditions were given 1) an adaptive comparison, 2) a contra-adaptive comparison, 3) only the correct solution. Overall, there were no differences across conditions regarding the learning success. Further exploratory analyses revealed that only learners with an intermediate prior knowledge benefited from the adaptivity. This finding can be interpreted as indicator that our short intervention only induces conceptual change when basic knowledge is already available.

Student visual attention during group instruction phases in collaborative geometry problem solving

AbstractTeacher gestures support mathematics learning and promote student collaboration. Aligned with speech, gestures can help students to notice the important visual information of geometry tasks. However, students’ visual attention to the teacher’s gestural cues during collaborative problem solving remains a largely unexplored field in mathematics education research. This mixed-method case study investigated relations between students’ visual attention, teacher gestures, and students’ collaborative problem-solving process on a geometry task. The data were collected with video cameras and mobile gaze trackers on four students simultaneously in two Finnish 9th-grade mathematics lessons with the same teacher. The findings show that the students attended to their own papers most of the time during the teacher's gestures, but differences in student attention between the gesture types emerged. The qualitative analysis showed that the teacher’s tracing, pointing, and representational gestures helped in directing student attention to targets relevant to the situational learning process. We conclude that teacher gestures can both convey mathematical contents and direct student attention, which intermediates mathematical thinking in problem solving.

Types of formative feedback provided by technology teachers during practical assessment tasks

Technology learners require formative feedback from their teachers to aid them in structuring and solving design problems. Without such feedback, learners often do not explore sub-problems and are inclined to reproduce known and existing solutions. Due to a paucity of literature on formative feedback in technology classrooms, this study investigated the formative feedback types that teachers provide to learners during the problem-structuring and problem-solving phases. A qualitative case study design was employed to investigate such feedback from five technology teachers who supported their learners in solving ill-structured design problems. The findings indicate that teachers’ formative feedback consists predominantly of low-level questions, while generative design questions and low-level comments were the least observed. A deeper understanding of how formative feedback unfolds in a technology classroom may help teachers guide learners through problem structuring and solving. Further research is needed to determine the influence of formative feedback on learners’ design outcomes.

Understanding the Solution-Making Ability of Senior High School Students in Solving Photosynthesis Problems

This study further explains the errors made by senior high school students when trying to solve photosynthesis problems. The method used in this research is descriptive qualitative, involving a sample of four senior high school students. The study results will be discussed with Polya's problem-solving procedure, which includes understanding, planning, execution, and checking. Errors at the comprehension stage are in understanding the relationship between photons and photosynthetic light reactions and in drawing genealogies. Errors in the planning stage, in making wrong assumptions or inaccuracies in interpreting those derived from the understanding stage. In addition, this study found errors in identifying chemical reactions in photosynthesis and errors in mathematical reasoning competence. The conclusion of this study shows the failure in making photosynthesis problem-solving solutions due to the interaction between the problem-solving phase and the student's general competence.

Innovating Household Food Waste Management: A User-Centric Approach with AHP-TRIZ Integration.

Food waste management remains a paramount issue in the field of social innovation. While government-led public recycling measures are important, the untapped role of residents in food waste management at the household level also demands attention. This study aims to propose the design of a smart system that leverages sensors, mobile terminals, and cloud data services to facilitate food waste reduction. Unlike conventional solutions that rely on mechanical and biological technologies, the proposed system adopts a user-centric approach. By integrating the analytical hierarchy process and the theory of inventive problem solving, this study delves into users' actual needs and explores intelligent solutions that are alternatives to traditional approaches to address conflicts in the problem solving phase. The study identifies five main criteria for user demands and highlights user-preferred subcriteria. It determines two physical conflicts and two technical conflicts and explores corresponding information and communications technology (ICT)-related solutions. The tangible outcomes encompass a semi-automated recycling product, a mobile application, and a data centre, which are all designed to help residents navigate the challenges regarding food waste resource utilisation. This study provides an approach that considers users' genuine demands, empowering them to actively engage in and become practitioners of household food waste reduction. The findings serve as valuable references for similar smart home management systems, providing insights to guide future developments.

Conflicting Problem Representations and Creativity: Effects on Problem Construction

ABSTRACT Contradictions and competing demands are common in everyday problems. To address these problems, individuals need to both recognize and integrate these contradictions during problem-solving to find creative solutions. We propose that recognition and integration take place in the problem construction phase of the creative problem-solving process. During problem construction, individuals work to develop their own understanding of the problem at hand. To do so, they sort through and combine problem representations activated through cues in the problem. We argue that the integration of conflicting and paradoxical elements in their understanding of the problem during problem construction is related to the originality and quality of proposed solutions beyond their initial ability to identify these elements. In a laboratory study, participants solved two different problems. We found that individuals identified more conflicting elements in the problem construction phase, but retained fewer of them in the problem-solving phase. Larger conflict between elements retained for problem solving resulted in more original and higher quality solutions to the problems.

ANALISIS KEMAMPUAN PEMECAHAN MASALAH SISWA DALAM MENYELESAIKAN SOAL MATEMATIKA TIPE HOTS PADA MATERI POKOK PERBANDINGAN

This study aims to find out the description of the ability of students in solving mathematical questions type Higher Order Thinking Skill (HOTS) comparative material in class VII. This research uses descriptive research using a qualitative approach. The subjects selected in this study were 3 students, the selection technique is based on each 1 student from a high-capability student group, 1 students from a medium-capable group and 1 student of a low-capacity group. Students with high cognitive abilities at C4 (analysing), C5 (evaluating), and C6 (creating) level are able to solve HOTS issues, as well as meet all phases of problem solving. But at the cognitive level C5 (evaluating) is not able to complete the entire phase of problem solving (3) Students with low cognitiveness at the level C4 (analysing), C 5 (assessing), and C 6 (creating) are unable to solve the HOTS issue and are only able to meet the stage of solving the problem of information compilation alone.

Promoting future teachers’ evidence-informed reasoning scripts: Effects of different forms of instruction after problem-solving

Pre-service teachers face difficulties when dealing with problem situations in the classroom if their evidence-informed reasoning script (EIRS) is not adequately developed. An EIRS might be promoted by demonstrating how to implement evidence-informed reasoning after a problem-solving activity on an authentic case. However, it is unclear what form of instruction is appropriate to promote pre-service teachers in the development of an EIRS. The present 2×3-factorial experimental intervention study investigated how different forms of instruction on functional procedures (example-free vs. example-based) and on dysfunctional procedures (without vs. example-free vs. example-based) affect the development of an EIRS. N = 384 pre-service teachers worked on a written case vignette of a problem situation in a problem-solving phase, in which the crucial steps of the EIRS were prompted externally. In the subsequent instruction phase, the participants compared their own solution with an example-free or example-based instruction on functional procedures, which was either supplemented by an example-free or example-based instruction on typical dysfunctional procedures or not at all. The participants’ learning success (declarative EIRS; near and far transfer problem-solving performance) and error awareness were assessed. The results revealed that the example-based instruction on functional procedures led to a higher learning success than the example-free instruction. Both forms of instruction on dysfunctional procedures improved learning success compared to learning without one. During learning, error awareness was higher for learners who worked with an example-free instruction on dysfunctional procedures. In order to promote the development of an EIRS in pre-service teachers, it is promising to provide instruction after problem-solving that presents a functional example of evidence-informed reasoning for the given problem and that also points out typical dysfunctional approaches to solving the problem. The results highlight the importance of selecting appropriate scaffolds in case-based learning approaches that aim to develop cognitive schemata. The mechanisms that explain when and why instructions on dysfunctional procedures work need to be further explored.

Development and differences in mathematical problem-solving skills: A cross-sectional study of differences in demographic backgrounds

Problem-solving skills are the most applicable cognitive tool in mathematics, and improving the problem-solving skills of students is a primary aim of education. However, teachers need to know the best period of development and the differences among students to determine the best teaching and learning methods. This study aims to investigate the development and differences in mathematical problem-solving skills of students based on their grades, gender, and school locations. A scenario-based mathematical essay test was administered to 1067 students in grades 7–9 from schools in east Java, Indonesia, and their scores were converted into a logit scale for statistical analysis. The results of a one-way analysis of variance and an independent sample t-test showed that the students had an average level of mathematical problem-solving skills. The number of students who failed increased with the problem-solving phase. The students showed development of problem-solving skills from grade 7 to grade 8 but not in grade 9. A similar pattern of development was observed in the subsample of urban students, both male and female. The demographic background had a significant effect, as students from urban schools outperformed students from rural schools, and female students outperformed male students. The development of problem-solving skills in each phase as well as the effects of the demographic background of the participants were thoroughly examined. Further studies are needed with participants of more varied backgrounds.

Comparing metacognitive regulation and socially shared metacognitive regulation in face-to-face and online learning settings in ill-structured problem solving

For collaborative groups to succeed, problem solvers need to agree on what the problem is and how to solve it. In ill-structured problems, the problem solvers switch back and forth between redefining the problem and generating solutions. This lack of parameters and structure implies that solving ill-structured problems is a complicated process, which can present problems in several different phases of problem solving. Efficient metacognitive regulation (MR) and socially shared metacognitive regulation (SSMR) skills benefit ill-structured problem solving. Online environments often lack the necessary social dimension to foster MR and SSMR. In the current article we report on a natural experiment caused by COVID-19 which forced a classroom-based workshop into an online version, thus contrasting face-to-face and audio-synchronous online learning setting in an Educational Sciences course of the Open University of the Netherlands. The student groups were presented with an ill-structured problem during which MR and SSMR processes were analyzed. We found that groups from the online setting demonstrated more MR processes than the face-to-face groups whereas the face-to-face groups engaged in more SSMR than the online groups.

Use of Technological Resources for the Development of Computational Thinking Following the Steps of Solving Problems in Engineering Students Recently Entering College

In this work, the authors propose the use of technological resources to develop computational thinking following the steps or phases of problem-solving for first-year students. During the development of the activities using technological resources (Arduino board, sensors, electronic devices, and mBlock) the students carried out activities, such as algorithm development and programming of the Arduino board and sensors from a friendly and playful interface such as the mBlock, as well as the debugging of programs until obtaining the expected results. These activities had an impact on the cognitive processes, practices, and technological perspectives of the students. Causality has been shown to exist between computational thinking skills and problem-solving phases in an environment of engineering students entering college. For the analysis of the relationship between computational thinking skills and problem-solving, Pearson’s statistical correlation test was used through SPSS software.

Assessment of domain-specific prior knowledge: A development and validation of mathematical problem-solving test

Science, technology, engineering, and mathematics (STEM) is a complex problem-solving that depends on the deep structures of domain-specific prior knowledge (DSPK) in mathematics. However, there is a lack of mathematics DSPK tests measuring several mathematics topics in every problem-solving phase in conceptual and procedural knowledge. This study aims to develop a mathematical problem-solving test as a mathematics DSPK test and investigate the content and construct validity. The product of test development is a 30-multiple-choice-item test in six mathematics topics. Every topic underlined all problem-solving phases in conceptual and procedural knowledge in a science or individual context. There were six experts performed the content validity sheets which analyzed using the content validity index (CVI) and intraclass correlation coefficient (ICC). The construct validity was examined using the Rasch model from 175 data of 7th grader students in Indonesia (Mage=12.66, SD=55). The result of content validity revealed overall items were valid (CVI≥83) and reliable (α=863; rxx=513). The construct of all items indicated fit (90≤weigheted MNSQ≤1.16) and were reliable (α=74) with various levels of difficulties and six low discrimination items. The recommendation for improvement is emphasized in language aspects. The absence of knowledge of facts could be an improvement for further study.

Do students learn more from failing alone or in groups? Insights into the effects of collaborative versus individual problem solving in productive failure

Productive Failure (PF) is an instructional design that implements a problem-solving phase which aims at preparing students for learning from a subsequent instruction. PF has been shown to facilitate students’ conceptual knowledge acquisition in the mathematical domain. Collaboration has been described as a vital design component of PF, but studies that have investigated the role of collaboration in PF empirically so far, were not able to confirm the necessity of collaboration in PF. However, these studies have diverged significantly from prior traditional PF studies and design criteria. Therefore, the role of collaboration in PF remains unclear. In an experimental study that is based on the traditional design of PF, we compared a collaborative and an individual problem-solving setting. It was hypothesized that collaboration facilitates the beneficial preparatory mechanisms of the PF problem-solving phase: prior knowledge activation, awareness of knowledge gaps, and recognition of deep features. In a mediation analysis, the effects of collaborative and individual problem solving on conceptual knowledge acquisition as mediated through the preparatory mechanisms were tested. In contrast to the hypotheses, no mediations or differences between conditions were found. Thus, collaboration does not hold a major preparatory function in itself for the design of PF.

Understanding Student Navigation Patterns in Game-Based Learning

Research on learning analytics (LA) has focused mostly at the university level. LA research in the K–12 setting is needed. This study aimed to understand 4,115 middle school students’ learning paths based on their behavioural patterns and the relationship with performance levels when they used a digital learning game as their science curriculum. The findings showed significant positive relationships between various tool uses and performance measures and varied tool use patterns at different problem-solving phases by high- and low-performing students. The results indicated that students who used tools appropriately and wisely, given the phase they were at, were more likely to succeed. The findings offered an insightful glimpse of learners’ navigation patterns in relation to their performance and provided much-needed empirical evidence to support using analytics for game-based learning in K–12 education. The findings also revealed that log data cannot explain all learners’ actions. Implications for both research and practice are discussed.

Realizing the problem-solving phases of Pólya in classroom practice

When teaching mathematical problem-solving is mentioned, the name of Pólya György inevitably comes to mind. Many problem-solving lessons are planned using Pólya's steps and helping questions, and teachers often rely on his heuristics even if their application happens unconsciously. In this article, we would like to examine how the two phases, Making a plan and Looking back, can be realized in a secondary school mathematics lesson. A case study was designed to observe and analyse a lesson delivered using cooperative work.
 Subject Classification: 97B10, 97C70, 97D40, 97D50

The Use of Communication Strategies by Second Language Learners of Chinese in a Virtual Reality Learning Environment

Virtual reality (VR) for Language learning has been shown effective in improving students’ communication skills. However very few studies reveal the communication strategies students used in communicating in a VR environment, especially in learning Chinese as a foreign or second language (CFL/CSL). In this pilot study, a prototype Virtual Experiential Language Learning Environment (VELLE) was used to provide immersive learning experiences in developing CFL students’ communication strategies. Productive Failure was employed as an instructional approach to support the learning design. The participants were six CFL students learning in an Australian university. The main data sources included audio recordings of the students’ communication in VR and interview data. The findings showed that participants’ use of various types of communication strategies was influenced by the VELLE and Productive Failure design. In particular, the use of L1 in the problem-solving phase enriched the participants’ learning opportunities and all participants used a lesser number of resource deficit-related strategies after the instruction. Second, Students reported enhanced engagement in language use in the VR context.

Analysis of Students' Mathematical Problem-Solving Ability on Trigonometry

This investigation aims to examine problem-solving ability (PSA) in Polya's problem-solving phases in the material of sine and cosine number and difference. This study used the descriptive qualitative research method. This study looked at 33 students in Class XI MIPA 5 from SMA Negeri 2 Bantul. Data was collected using written tests and interviews. The written test in this study takes the form of a two-point description question. The study's findings were used to calculate the mathematics PSA of students in class XI MIPA 5 at SMA Negeri 2. In mathematics, the average percentage of students who meet the low requirement is 59.72 percent.
 Furthermore, children with high expectations can thoroughly, meticulously, and systematically solve difficulties. Understanding the problem remains a struggle for pupils who reach intermediate standards and have yet to complete the problem's solution. Meanwhile, students who scored low on the criteria had trouble understanding the problem and were unable to solve it using the prepared plans, as well as students who were unable to double-check the response and complete the solution to the problem that had been solved.

Understanding high school students’ errors in solving genetic problems

ABSTRACT This study aimed to provide a deeper understanding of high school students' errors while solving genetics problems. Four students were asked to solve two genetics problems and participate in a modified think-aloud interview. Data from the interview, video clips of the problem-solving process, and test sheets were used to analyse participants’ problem-solving processes and identify errors during each of the following problem-solving phases: understanding, planning, execution, and checking. The error of Misinterpreting evidence was observed in the understanding phase for Problem 1 (represented verbally), whereas the error of Making incorrect assumptions was observed for Problem 2 (students had to draw a pedigree). Errors of Incorrect conclusion occurred in the planning phase, characterised by problem-solving strategies based on incorrect assumptions or misunderstanding the problem due to errors in the understanding phase. Further, Using incomplete representation and Mathematical errors were observed in the execution phase due to representational competence and the application of mathematical principles, respectively. The checking phase was characterised by Not monitoring the problem-solving process and Disregarding evidence, wherein participants failed to check the logical consistency of genetic principles with the information embedded in the problem. The results indicated that problem-solving errors were related to the interaction between the students’ domain-general competencies and domain-specific knowledge.