Mathematical Tasks Research Articles

An Analysis of Students’ Mathematical Reasoning in Solving Probability Problems Judging from Learning Styles: The Converger

[Background] Several challenges are encountered by students when attempting to solve probability problems, one of which relates to reasoning about the given problem scenario. This phenomenon may arise due to variations in students’ learning styles. [Objective] Thus, the current study aimed to examine students’ reasoning abilities when dealing with probability problems, focusing on those with converging learning styles. [Method] This study employed a qualitative method, involving two students with a converging learning style who were invited to solve reasoning tasks. Three instruments were utilized: learning style questionnaire, mathematical reasoning tasks, and semi-structured interviews. The research data was analyzed through data reduction and display. [Results] The study’s findings revealed that, in solving probability problems, convergers (students with a converging learning style) primarily relied on imitative reasoning. These students demonstrated four patterns. First, they accurately and comprehensively articulated known information from a question and correctly identified the question being asked. Second, they possessed a conceptual understanding of probability but provided inaccurate explanations regarding arithmetic sequences. Although they could devise problem-solving strategies, their articulation of these strategies lacked precision. Third, they utilized the concept of probability and employed problem-solving strategies to solve the given problem and explain the steps involved in reaching the solution. Four, they derived answers from problem-solving strategies and drew accurate conclusions, often exhibiting a tendency to trust their answers and subsequently verify them. [Conclusions] Students with converging learning styles predominantly used imitative reasoning when solving probability problems.

Caracterización de tareas sobre inecuaciones en libros de textos escolares en educación primaria

[Objective] This study characterized mathematical inequation tasks in Chilean primary school textbooks for 4th and 5th graders (9 and 10 year olds). [Methodology] A qualitative descriptive study was conducted considering the following six categories: algebraic practices involved, syntactic structure of the expressions, meaning of the concept of inequation, resolution strategies, level of cognitive demand, and context. The sample is comprised of 206 tasks from math textbooks. [Results] The characterised tasks have few practices leading to algebraic thinking, and it is the alphanumeric symbolic representation the one that predominantly stands out within these practices. The structure of the expressions shows little variety for generalising and reasoning, the latter having repercussions on the resolution strategies, which are not progressively addressed in order to improve their comprehension. The proposed tasks present different meanings of inequation. Regarding the cognitive demand, most of the tasks consist of connected and unconnected procedures. Finally, the most frequent contexts are mathematical, while the ones with the least presence are personal, social, and occupational contexts. [Conclusions] It is concluded that improving task design is necessary. The proposed categories are a tool to visualise which elements are taken into consideration and to improve inequation tasks.

E-Module for Problem-Based Learning Integrating Batak Toba Culture (PBL BTC) to Improve Problem-Solving Ability

This research aims to analyze: 1) the validity, practicality and effectiveness of the PBL e-module based on Batak Toba Culture (PBL BTC) which was developed to increase students' problem-solving abilities at SMP Negeri 1 Simanindo; 2) increasing students' problem-solving abilities through the developed e-module. This research is using the ADDIE. From the tested of class IX 2 was obtained: 1) the PBL BTC e-module have the criteria of being valid, effective, and practical; 2) Increased problem-solving abilities using the PBL BTC e-module as seen from the N-gain value is 0.506, it is in the "medium" category. This research recommendations might be made the E-Module with mathematical problem-solving tasks on the topic of congruency should be continued for future effectiveness evaluations.

Development of an Evaluation Tool for Mathematical Communication Based on RPG Educational Games

This research aims to develop an innovative and interesting evaluation tool in assessing and improving students' mathematical communication through Role-Playing Game (RPG) based educational games. RPG educational games were chosen because they can provide an in-depth context for students' learning experiences, while facilitating interaction and collaboration. This evaluation tool includes a variety of mathematical tasks that students must complete through interaction in a carefully designed virtual world. The research method used is the research and development method, with the research model being the ADDIE model which consists of five steps, namely: (1) analysis, (2) design, (3) development (development), (4) implementation, and (5) evaluation. This research was tested for practicality on 15 respondents consisting of 8 students and 7 students, and validated by one supervisor. The results of this research were obtained as follows: (1) the practicality test assessment for the game operation indicator was 82.22% (very good), the game presentation indicator was 83.96% (very good), for the game benefit indicator it was 90.00 % (very good), and for the overall indicator obtained 85.83% (very good). In the validity test, the percentage obtained for the game presentation indicator was 93.75% (very valid), the percentage for the game benefit indicator was 84.88% (very valid) (2) he cognitive ability assessment obtained was 90.53% (very good) for the value student average, the student's achievement of completeness in each indicator has met very well, seen from the average of each question, the first question obtained a score of 98%, the second question was 86.3% and the third question was 87.3% of maximum score of 100% for each indicator. This research contributes to the mathematics education literature by combining aspects of technology, educational games, and mathematical communication.

МАТЕМАТИКАЛЫК МАСЕЛЕЛЕРДИ АР ТҮРДҮҮ ЖОЛДОР МЕНЕН ЧЫГАРУУ ОКУУЧУЛАРДЫН ЧЫГАРМАЧЫЛЫК ИШМЕРДҮҮЛҮГҮН ӨСТҮРҮҮНҮН КАРАЖАТЫ КАТАРЫНДА

The article discusses solution of mathematical tasks in different ways and teaching students to solve them applying different methods, which helps to increase the level of cognitive interest of students. The goals of solving tasks in different ways are: ✓ to expand students' knowledge of various ways of solving math tasks; ✓ to teach different methods of mathematical calculations; ✓ to provide with the advanced capacity to solve problems in different ways. The goals, role and significance of solving mathcalculations in different ways are also reflected and accompanied by examples. Problem solving using a variety of methods plays a key role in successfully achieving goals and personal and professional development. Several roles of this method can be listed.

Integrating vision and somatosensation does not improve the accuracy and response time when estimating area and perimeter of rectangles in primary school

BackgroundProblem-solving and learning in mathematics involves sensory perception and processing. Multisensory integration may contribute by enhancing sensory estimates. This study aims to assess if combining visual and somatosensory information improves elementary students' perimeter and area estimates. Methods87 4th graders compared rectangles with respect to area or perimeter either solely using visual observation or additionally with somatosensory information. Three experiments targeted different task aspects. Statistical analyses tested success rates and response times. ResultsContrary to expectations, adding somatosensory information did not boost success rates for area and perimeter comparison. Response time even increased with adding somatosensory information. Children's difficulty in accurately tracing figures negatively impacted the success rate of area comparisons. DiscussionResults suggest visual observation alone suffices for accurately estimating and comparing area and perimeter of rectangles in 4th graders. ImplicationsCareful deliberation on the inclusion of somatosensory information in mathematical tasks concerning perimeter and area estimations of rectangles is recommended.

Self-regulated learning while solving mathematical problems among mathematically gifted and talented students’

ABSTRACT Self-regulated learning (SRL) represents a necessary fundamental 21st century skill for children and adolescents and is essential for the development of gifted students. This study explored mathematically gifted (MG) junior high school students’ SRL capabilities while solving mathematical problems compared to typical achievers (TAs). A sample of 71 ninth-grade students from three junior high schools were divided into two study groups (MG and TA). Participants completed Motivated Strategies for Learning Questionnaires (MSLQ) and a Metacognitive Strategy Usage Questionnaire (MSUQ). In addition, students underwent Think-Aloud Interviews while solving mathematical problems (quadratic function) at two levels of difficulty. The results only revealed significant differences between MG and TA students on the organization subscale of the MSLQ. No differences were found between the study groups in the metacognitive strategies used in the questionnaire. MG students outperformed TA students on the Mathematical Problem-Solving Task and generally used more SRL strategies during the qualitative Think-Aloud Interviews. Moreover, the differences in the between-group effect sizes were higher on the high-difficulty level of the problem-solving task than on the low-difficulty level of the task. The findings constitute a significant milestone in the study of SRL capabilities of MG in junior high school.

Remote assessment of the association between early executive function and mathematics skills

AbstractExecutive function (EF) skills are consistently associated with global mathematics assessments. However, less is known about which specific mathematics skills invoke EF in early childhood. We adapted batteries of EF, numerical, and patterning tasks to be conducted via synchronous video conferencing with typically developing 4‐year‐old children (N = 115, 56.6% female, 85.2% White, non‐Hispanic, 7.8% Multiracial, 0.9% Hispanic, 0.9% Black, 1.7% American Indian/Alaska Native, 2.6% Asian and 0.9% missing) from primarily upper middle class households (maternal education range: some college to graduate or professional degree, mode = graduate or professional degree) in the United States. We found feasibility of remote adaptation varied greatly by task. Just over half of enrolled children (57.02%) completed all tasks and there were higher rates of missingness compared to an in‐person sample with a similar age range collected in schools. Nonetheless, working memory predicted various mathematics skills, including patterning, verbal counting, and number comparison, controlling for age and maternal education. Relations between cognitive flexibility and patterning varied across model specifications. Overall, these findings provide insight into the costs and benefits of remotely administered direct assessments of EF and mathematics in early childhood and add to the growing body of literature suggesting working memory plays an important role in several foundational early mathematics skills. Suggestions are offered for remote assessment with preschoolers.Highlights Executive function and mathematics tasks were adapted for remote use and associations among skill sets were examined. Remotely administered data collection sessions had higher rates of missingness than comparable in‐person studies. Findings replicate and extend prior work on the associations between working memory and patterning, verbal counting, and number comparison in a remote setting.

Comparison of Vector Tasks in Indonesian and Singaporean Mathematics Textbooks

Vector is one of the most important subjects in mathematics, but many students face difficulties in learning vector. To find out the cause, this study is conducted to find the differences and similarities in learning opportunities presented in the vector tasks from Indonesian and Singaporean mathematics textbooks. Therefore, vector tasks from the two books are compared with the aspects of representations, contextuality, and response types. Firstly, this study finds out that the chosen Indonesian textbook provides fewer learning opportunities than the chosen Singaporean textbook in terms of vector tasks. Secondly, this study also finds out that the Indonesian textbook tends to present tasks in visual representation, while the Singaporean textbook tends to present tasks in pure mathematics form. Thirdly, this study also finds out that Indonesian textbook usually uses applicative tasks that correlate with real-life situations, while the Singaporean textbook does the otherwise. Lastly, this study also finds out that both textbooks mostly use close-ended tasks.

Designing learning trajectory to support preservice mathematics teachers' skills in creating and implementing realistic mathematics tasks

In mathematics teaching and learning, mathematics tasks embedded in realistic contexts are crucial for developing mathematical concepts, procedures, and the application of mathematical knowledge. Despite this, mathematics teachers often encounter challenges in designing and implementing such realistic mathematics tasks. Therefore, this study aims to construct a learning trajectory to enhance preservice mathematics teachers' abilities to create and implement Realistic Mathematics Tasks (RMTs). Employing a design research methodology, the study comprises three phases: preliminary design, teaching experiment, and retrospective analysis. The data presented in this article are from the first cycle, encompassing all three phases. The research involved four fourth-year preservice mathematics teachers from Sanata Dharma University in Yogyakarta, Indonesia, participating in a preliminary experiment conducted over three months outside their regular coursework. Data were collected through pre-course written tests, observations, interviews, journal reflections, and analyses of the preservice teachers' work. The findings indicate that the learning trajectory effectively raised awareness among preservice teachers about key characteristics of realistic mathematics tasks. This heightened awareness was evident in their reflections and work. Additionally, observations of their teaching practices revealed that preservice teachers valued student reasoning in solving RMTs. However, they faced difficulties in classroom management and guiding students to construct new knowledge actively through their experiences in solving RMTs.

Primary education students’ beliefs about the nature of mathematics, self-efficacy, and mathematics educators; a descriptive study from Indonesia

ABSTRACT The significant role of mathematical belief studies has been recognised several decades later. These beliefs have been found to be significantly associated with performance and problem-solving. However, empirical studies of these beliefs in Indonesia are scarce. This study investigates students’ beliefs on the nature of mathematics, self-efficacy, and their perceptions of mathematics educators. The findings of this study revealed that Indonesian students strongly believe anyone could attain proficiency in mathematics. At the same time, students had less confidence that they could understand difficult mathematical tasks and less confidence in showing that they were good at mathematics. Notably, boys’ students exhibit stronger beliefs in their understanding of the course, including the most challenging topics in mathematics, compared to their girls’ counterparts. The implications of this study significantly contribute to the existing literature and inform teaching practices.

A professional development course inviting changes in preservice mathematics teachers’ integration of technology into teaching: the lens of instrumental orchestration

Information and Communication Technologies (ICTs) can provide rich learning opportunities in classroom teaching, which requires teachers’ effective instrumental orchestrations. However, there is a lack of research regarding the development of teachers’ instrumental orchestrations. In this study, we designed and implemented a professional development course, aiming to explore whether the course could change preservice mathematics teachers’ instrumental orchestrations. To this end, an analytical framework of instrumental orchestrations within a GeoGebra environment was established. Data were collected from Chinese preservice mathematics teachers, including the videotapes of their simulated lessons, GeoGebra files, written documents, self-reflections, online speaking, etc. In the results, three participants’ changes in instrumental orchestrations were presented as main cases. It was found that the three preservice mathematics teachers’ types of instrumental orchestrations did not change, but the frequencies of those increased significantly. The changes in didactical configurations were mainly reflected in the layout of GeoGebra, and the changes in exploitation modes were mainly manifested in the techniques teachers use and the management of mathematics tasks. In addition, it was revealed that theoretical support (instrumental approach), technical support (knowledge about using GeoGebra), and community support (“Same Content Different Designs” activity and communicating and discussing with others) were useful course elements inviting changes in instrumental orchestrations. In this regard, this study contributes to providing a promising professional development course that can be used in teacher training so as to improve teachers’ instrumental orchestrations.

Fostering engineering freshmen’s shifts of attention by using Matlab LiveScript for solving mathematical tasks

Fostering engineering freshmen’s shifts of attention by using Matlab LiveScript for solving mathematical tasks

CLASSROOM SOCIAL ENVIRONMENT AND STUDENT ENGAGEMENT IN MATHEMATICS: A CORRELATIONAL STUDY

This study investigated the association between classroom social environment and student engagement in mathematics. It utilized the descriptive correlational and causal-comparative research design and applied the following statistical tools: mean, Pearson-r, independent T-test, and Analysis of Variance (ANOVA). This investigation used an adapted, content-validated, and reliability-tested survey questionnaire. A proportional stratified random sampling technique was used to obtain a sample of 282 respondents from Manat National High School – Junior High School. The result of the study revealed that students often show high engagement with mathematics in emotional, social, and cognitive domains, with cognitive engagement peaking. The classroom social environment is favorable, with teachers providing support and mutual respect, even when performance goals are not as important. Moreover, the study revealed a significant correlation between engagement dimensions and classroom social environment, with a minor association between social environment quality and emotional and social engagement and a moderate correlation with cognitive engagement. The study found no significant differences in engagement based on sex and age, but observed substantial variation in engagement across different grade levels, suggesting that grade level significantly influences students’ mathematics engagement. Based on the findings, it was highly recommended that teachers use techniques like acknowledging student accomplishments, offering constructive criticism, and creating a friendly learning environment to promote healthy mathematical learning. They should also encourage group projects, peer tutoring, and engaging mathematical tasks to enhance critical thinking and problem-solving skills. A respectful classroom culture and tailored engagement strategies were also suggested. School administrators should also conduct professional development workshops for teachers.<p> </p><p><strong> Article visualizations:</strong></p><p><img src="/-counters-/soc/0779/a.php" alt="Hit counter" /></p>

Computationally Hard Problems for Logic Programs under Answer Set Semantics

Showing that a problem is hard for a model of computation is one of the most challenging tasks in theoretical computer science, logic and mathematics. For example, it remains beyond reach to find an explicit problem that cannot be computed by polynomial size propositional formulas (PF). As a model of computation, logic programs (LP) under answer set semantics are as expressive as PF, and also \(\mathtt{NP}\) -complete for satisfiability checking. In this paper, we show that the PAR problem is hard for LP, i.e., deciding whether a binary string contains an odd number of \(1\) ’s requires exponential size logic programs. The proof idea is first to transform logic programs into equivalent boolean circuits, and then apply a probabilistic method known as random restriction to obtain an exponential lower bound. Based on the main result, we generalize a sufficient condition for identifying hard problems for LP, and give a separation map for a logic program family from a computational point of view, whose members are all equally expressive and share the same reasoning complexity.

Preservice Secondary Teachers’ Mathematical Knowledge for Teaching: Focus on Reflective Practice

ABSTRACT Mathematics education researchers have highlighted that preservice secondary mathematics teachers (PSTs) need opportunities to learn how their content knowledge is contextualized during instruction. Mathematical Knowledge for Teaching (MKT) subdomains provide a way to understand mathematics discipline-specific knowledge that is unique to the profession of teaching. Using a single case study design, we investigated how 11 secondary PSTs described MKT subdomains as they engaged in approximations of practice in the form of lesson planning, peer teaching, and associated guided reflections. The findings demonstrated that the PSTs referenced the MKT subdomains related to planned teacher actions (e.g., posing scripted questions) in their initial reflections. As they progressed through the approximations of practice and associated reflections, they began reflecting on MKT subdomains related to contingency actions and decisions (e.g., modifying mathematical tasks based on students’ thinking). Findings also demonstrated that, after experiences reflecting, the PSTs described a broader set of MKT subdomains (e.g., curricular trajectory) in addition to their responses to the specific prompts. We argue that providing PSTs with opportunities to reflect on their practices through the lens of MKT is an effective way to develop their understanding of how content knowledge is contextualized during mathematics teaching.

Towards an advanced mathematical language task for the early primary school years: preliminary results on reliability and validity

Towards an advanced mathematical language task for the early primary school years: preliminary results on reliability and validity

Mathematical Work of a Future Teacher in Teaching the Box Plot Diagram

Objective: The study aims to characterize the teaching proposals of the future Mathematics teacher in the context of their practical training in the final year of their university studies. Theoretical Framework: To achieve this, we consider the theory of Mathematical Workspaces, which allows for the analysis of both the mathematical activity that an individual engages in while solving a mathematical task, and the activity that is promoted during teaching. Method: A qualitative methodology is adopted through the design of an instrumental case study. The case pertains to a future teacher conducting a class on constructing a box plot. This class was observed and transcribed for analysis in light of the proposed mathematical work. Results and Discussion: The mathematical work exhibited by the future teacher includes a strong semiotic component and the use of non-material artifacts for quartile calculations. Students' prior knowledge is utilized in this context, with procedural aspects taking precedence over statistical thinking. Implications of the Research: The study raises concerns about the statistical education of mathematics teachers and its impact on future teaching proposals. Originality/Value: This research contributes to the study of statistics and initial teacher training in their influence on the future practices of Mathematics teachers. It provides a characterization of the mathematical work promoted by a future teacher and offers insights into concerns regarding the development of statistical thinking.

What is the room for guessing in metacognition? Findings in mathematics problem solving based on gender differences

Introduction. Understanding the interaction of metacognitive strategies with guessing in problem-solving is a focus point in educational psychology, especially in mathematics education, especially with gender differences. These provide different nuances in understanding students' cognitive dynamics in guessing and metacognitive strategies. These interactions must be explained in depth to understand the unique phenomenon of second-guessing that sometimes arises in metacognitive activities based on students' gender. This research aims to reveal the process of metacognition and students' guessing thinking in solving mathematical problems. This research seeks to contribute to an empirical of metacognitive processes and their practical implications for educators and curriculum designers by uncovering these activities. Study participants and methods. The participants in this study were (30) 12th-grade high school students, consisting of (16) female students and (14) male students who had studied mathematics with statistical material. The primary method in this research is a case study with a qualitative descriptive approach. The indepth interview technique using the MAI protocol is based on the results of mathematical problem-solving to explore information about students' natural metacognitive processes according to what they think when working on problems. Semi-structured interviews used the Indonesian language to eliminate the influence of differences in regional language proficiency levels as much as possible. Results. The results of interviews on mathematical problem-solving tasks show a 'guessing' strategy in students' metacognitive activities. However, there is a striking difference between female and male students' use of the 'guessing' strategy when solving mathematical problems. Female students use the 'guessing' method more often than male students; even the 'guessing' method used by male students cannot solve the problem, thus changing the guessing strategy to estimation. This means that most of the students' answers used the guessing method. These findings underscore the dynamic nature of the cognitive processes used by students, revealing diverse interactions between metacognitive strategies and guessing during problem-solving efforts. Conclusion. This research shows that rather than standing in isolation, guessing has a place within metacognitive processes, with metacognitive regulation guiding and shaping the deliberate application of guessing in mathematical problem-solving contexts. This can be a factor for reconsidering the perceived dichotomy between precision-driven methodologies and intuitive guessing.

Examination of Grade 7 Ratio Concept Tasks Designed by ChatGPT Based on Cognitive Demand Levels

The process that enables students to achieve objectives while performing mathematical tasks is defined as an activity. The task creation process is a lengthy and challenging journey involving many complex stages. Artificial intelligence technologies can facilitate this process. For instance, ChatGPT can assist teachers on various issues and help them create tasks. This study aimed to evaluate the tasks prepared for the Grade 7 ratio concept using ChatGPT, focusing on cognitive demand levels. It employed a case study design to examine the cognitive demand levels of tasks created with ChatGPT 3.5 and ChatGPT 4, identifying any potential deficiencies. The content analysis was performed to analyze the data. It was concluded that the created tasks could be used with necessary improvements, allowing teachers to enrich their course content while saving time and energy.