Mathematical Tasks Research Articles

The Mathematics-Language Proficiency: The Learners’ Perspecctive

Mathematics is considered a difficult school subject by majority of learners. For many learners, mathematics is a series of hurdles and challenges-a task made with continued failure and seeming irrelevance in spite of the value that mathematics plays in society. The effect of this has been unwilling class participation, disinterestedness, haphazard solving of mathematical tasks, plus low achievements and failure to communicate mathematics. The latter effect is important in application of mathematics in occupations such as building, construction, engineering and accounting. This state of affairs propagated the topic of the paper: “The Mathematics-Language Proficiency: The Learners’ Perspective”. The objective was to find out the nature of the relationship between proficiency in mathematics vocabulary and conceptual understanding of mathematics. The study drew on Vygotskian Socio-Cultural Theory (SCT) and used a multiple-case study design. The sample size of the study was 1353 participants comprising of 1339 form three learners and 14 mathematics teachers drawn from Sub-County Schools (SCS), County Schools (CS) and Extra-County Schools (ECS) in Bungoma South Sub- County. Data were collected by questionnaires, classroom observations and interviews. Analysis was done using Pearson correlation with two tailed tests for all the tests with a level of significance of 0.01. The study found out that there exists a strong positive relationship between mathematics vocabulary and conceptual understanding across all cases, that is, 0.798, 0.778 and 0.709 in SCS, CS and ECS respectively. Further learners were unable to interpret the meaning of some mathematics solutions after solving the question correctly. The study concluded that proficiency in mathematics vocabulary is necessary but not sufficient for conceptual understanding of mathematics. The study recommends learners to be supported in communicating mathematics ideas both verbally and in writing during mathematics lessons to broaden their conceptual understanding of mathematics.

Brain Imaging in Mathematics Education: An Analysis of KCI Studies

This study reviews the application of EEG, fMRI, and fNIRS in mathematics education research in South Korea, analyzing 16 significant papers from the last 25 years. The results demonstrate a significant relationship between brainwave patterns and various aspects of mathematical learning. EEG studies highlighted the role of different brainwaves in math anxiety, computation, and problem-solving. fMRI research revealed the importance of whole brain activation in spatial and numerical processing. Studies using fNIRS emphasized the prefrontal cortex's involvement in mathematical tasks, particularly in spatial and computational tasks. Conclusions highlight the need for math learning programs that enhance brain connectivity and consider problem difficulty and learner preferences. The study advocates for more research on the prefrontal cortex, particularly in relation to math anxiety, and emphasizes the importance of brain imaging techniques in understanding cognitive processes in math education, suggesting an integrated approach in future research.

МОДЕЛІ ВИБОРУ РОЗТАШУВАННЯ ЛОГІСТИЧНОГО ЦЕНТРУ

Abstract. Allocating a certain territory for transportation, logistics and distribution of goods involves spatial planning and infrastructure rationalization. The decisions regarding the geographical location of the logistics center itself and its structural elements are extremely important, because their impact on the indicators of its activity is long-term. And if mis-take is made during the construction of production or logistics facilities, and this or that facility is placed in an unfortu-nate place, then to correct the situation (moving to a new, better place), a large amount of material, financial, labor and other resources will be needed, which will lead to to the loss of customers, time, freezing of capital and, accordingly, a decrease in the competitiveness of enterprises, industries, countries, etc. The work summarizes the criteria for choosing the location of the logistics center and reviews practices for making decisions about its optimal location, substantiating proposals for choosing the best method. Making a decision about choosing the optimal location of a logistics center is a very difficult economic and mathemat-ical task, because decision-makers have to rationalize the combination of quantitative and qualitative factors, the mass of factors and the trade-offs between them on the basis of socio-economic, political and geographical criteria and the condi-tion of the motor transport infrastructure. When choosing the optimal location of the logistics center, it is proposed to apply the method of analytical hierarchy (AI), which includes procedures for synthesizing many statements, obtaining the priority of criteria and finding alternative solutions. Keywords: multi-criteria models, hierarchical structure, logistics center, routing, motor transport infrastructure.

Using Mathematical Models in Secondary Grammar Schools

On February 11, 2021, I gave a motivational lecture at the Tamás Esze High School in Mátészalka, and I completed a series of test tasks with the students. In this article, I looked for an answer to how students who are about to graduate approach to solving math problems. The main objective was to investigate whether they prefer to solve purely mathematical or application-oriented tasks, and in which one they achieve better results. In this paper, I analyze the results obtained.

Predictive Data Modeling: Student's Obstacle in Mathematical Literacy Tasks Focusing on Ratio and Proportion using The K-Nearest Neighbor Algorithm

Predictive Data Modeling: Student's Obstacle in Mathematical Literacy Tasks Focusing on Ratio and Proportion using The K-Nearest Neighbor Algorithm

Stochastic linear regularization methods: random discrepancy principle and applications

The a posteriori stopping rule plays a significant role in the design of efficient stochastic algorithms for various tasks in computational mathematics, such as inverse problems, optimization, and machine learning. Through the lens of classical regularization theory, this paper describes a novel analysis of Morozov’s discrepancy principle for the stochastic generalized Landweber iteration and its continuous analog of generalized stochastic asymptotical regularization. Unlike existing results relating to convergence in probability, we prove the strong convergence of the regularization error using tools from stochastic analysis, namely the theory of martingales. Numerical experiments are conducted to verify the convergence of the discrepancy principle and demonstrate two new capabilities of stochastic generalized Landweber iteration, which should also be valid for other stochastic/statistical approaches: improved accuracy by selecting the optimal path and the identification of multi-solutions by clustering samples of obtained approximate solutions.

Mathematical Tasks and training for teaching Mathematics in High School

No segundo semestre de 2022, na disciplina de Geometria para o Ensino II, no Curso de Licenciatura em Matemática da Universidade de Brasília, cinco licenciandos colaboraram com o objetivo de desenvolver uma Tarefa Matemática (TM) relacionada à deformação de ângulos e áreas em projeções cartográficas. Após o estudo teórico sobre Tarefas Matemáticas, o grupo aprimorou a TM por meio de apresentações e debates coletivos na disciplina. Esse processo também envolveu análise conceitual, simulação do desenvolvimento da TM em sala de aula e planejamento para antecipar os possíveis desafios dos alunos. A TM foi desenvolvida em uma turma de 3º ano do Ensino Médio Técnico Integrado ao Meio Ambiente no Instituto Federal de Brasília, Campus Estrutural. Os resultados revelaram que os alunos do ensino médio enfrentaram dificuldades em associar as observações do experimento aos conceitos de distorção, tangência e projeção. Ao longo do processo de construção, validação, desenvolvimento e avaliação da TM, os licenciandos reconheceram a importância do cuidado no planejamento e preparação de ações de ensino de matemática, especialmente quando se busca uma maior interação dos alunos com os conceitos e sua aplicação na realidade. Palavras-chave: Tarefa Matemática; Projeções cartográficas; Distorções; Investigação; Geometria.

Visual analysis of commognitive conflict in collaborative problem solving in classrooms.

In today's knowledge-intensive and digital society, collaborative problem-solving (CPS) is considered a critical skill for students to develop. Moreover, international education research has embraced a new paradigm of communication-focused inquiry, and the commognitive theory helps enhance the understanding of CPS work. This paper aims to enhance the CPS skills by identifying, diagnosing, and visualizing commognitive conflicts during the CPS process, thereby fostering a learning-oriented innovative approach and even giving the script of technology-assisted feedback practices. Specifically, we utilized open-ended mathematical tasks and multi-camera video recordings to analyze the commognitive conflicts in CPS among 32 pairs, comprising 64 Year 7 students. After selecting the high-quality, medium-quality, and low-quality student pairs based on the SOLO theory, further investigations were made in the discourse diagnosis and visual analysis for the knowledge dimensions of commognitive conflict. Finally, it was discovered that there is a need to encourage students to focus on and resolve commognitive conflicts while providing timely feedback. Visual studies of commognitive conflict can empower AI-assisted teaching, and the intelligent diagnosis and visual analysis of CPS provide innovative solutions for teaching feedback.

How pedagogical content knowledge sharpens prospective teachers’ focus when judging mathematical tasks: an eye-tracking study

Teachers’ ability to accurately judge difficulties of mathematical tasks is an essential aspect of their diagnostic competencies. Although research has suggested that pedagogical content knowledge (PCK) is positively correlated with the accuracy of diagnostic judgments, experimental studies have not been conducted to investigate how PCK affects perception and interpretation of relevant task characteristics. In an intervention study with a control group, 49 prospective mathematics teachers judged the difficulty of 20 tasks involving functions and graphs while an eye tracker tracked their eye movements. Some of the tasks included characteristics well known to be difficult for students. Participants’ domain-specific PCK of typical student errors was manipulated through a three-hour intervention, during which they learned about the most common student errors in function and graph problems. We found that the process of perception (relative fixation duration on the relevant area in the tasks) was related to judgment accuracy. Pre-post comparisons revealed an effect of the intervention not only on participants’ domain-specific PCK of typical student errors but also on their perception and interpretation processes. This result suggests that domain-specific PCK of typical student errors allowed participants to focus more efficiently on relevant task characteristics when judging mathematical task difficulties. Our study contributes to our understanding of how professional knowledge makes teachers’ judgment processes of mathematical tasks more efficient.

Teachers pose and design context-based mathematics tasks: what can be learned from product evolution?

Teachers pose and design context-based mathematics tasks: what can be learned from product evolution?

Ways of Developing Creative Thinking and Reasoning of Students in Mathematics Learning

Reasoning skill is one of the main components of mathematical competency (NCTM, 2014). Mathematics students use mainly two types of reasoning skills namely imitative reasoning and creative reasoning in solving mathematical tasks. The textbook tasks in our context are of routine types which are solvable using imitative reasoning. Most mathematics teachers use prescribed textbooks in lesson planning and teaching (Thomson & Fleming, 2004) which ultimately promotes imitative reasoning among the students as compared to creative reasoning. Moreover, many mathematics teachers still follow the traditional chalk-and-talk banking pedagogy which does not support the enhancement of creative thinking and reasoning skills. This literature-based research study has identified some of the ways to develop the creative thinking and reasoning skills of mathematics students through the intense study of the available research articles, thesis, and books in the field of mathematical thinking and creativity. This study has found the positive impact of productive struggle and adversity quotient on students’ creative thinking and reasoning skills. Likewise, problem-posing and problem-solving skills have been found to be essential skills for enhancing creativity. This study has also found modern student-centric teaching approaches like the ADI (Argument-Driven Inquiry) approach and the RME (Realistic Mathematics Education) approach beneficial in improving reasoning skills. This study also found CMR (Creative Mathematically Founded Reasoning) tasks to be helpful in increasing the creative reasoning skills of mathematics students.

The Contribution of Technology to Understanding Shared Phenomena in Math and Music

In this paper, we present a computerized tool developed specifically for a study that examined the development of creative thinking among third and fourth graders, following multidisciplinary instruction that combined mathematics and music. The computerized tool included closed and open questions. The closed questions allowed the students to perform separated mathematical and musical tasks by typing a specific melody, rhythm, pattern, or mathematical form; The open questions required the students to respond freely and creatively using the computer keyboard or a virtual piano keyboard. The programming was written in Max/MSP environment using BACH package that enables generating, modifying, and playing musical scores either with a mouse or keyboard. The software was built in a modular way by using math and music units, which makes it possible in the future to expand the number of tasks as needed. The article reviews the necessity of developing a technological tool for the research, the deliberations that accompanied the development of the technological tool, the technology's contribution to monitoring creative thinking process and drawing conclusions for further research.

GPS: Composing and Decomposing Shapes Across the Grades

Growing Problem Solvers provides four original, related, classroom-ready mathematical tasks, one for each grade band. Together, these tasks illustrate the trajectory of learners' growth as problem solvers across their years of school mathematics.

Forefronting Equitable Mathematical Tasks that Empower Student Teachers in the 4ir Era

Forefronting Equitable Mathematical Tasks that Empower Student Teachers in the 4ir Era

초등학교 수학 교과서에 나타난 수학적 추론 양상 탐색

The purpose of this study was to explore how mathematical reasoning, one of the most important topics in school mathematics, is implemented in elementary school mathematics textbook. To this end, this study analyzed the type of reasoning, level of cognitive demands, and structural characteristics of mathematical reasoning tasks provided in the elementary school mathematics textbooks. As a result of this study, it was revealed that the mathematics textbooks had the strength of providing sufficient tasks dealing with pattern recognition type, but were deficient in providing tasks dealing with multiple types of mathematical reasoning, such as making a conjecture, investigating a conjecture, developing a justification, finding a counterexample, and evaluating an argument. Regarding the level of cognitive demands of mathematical reasoning tasks, the mathematics textbooks provided sufficient tasks of procedures with connections, but insufficiently provided tasks of doing mathematics requiring complex thinking. In relation with the structural characteristics of mathematical reasoning, the mathematical reasoning tasks in the mathematics textbooks embodied only the aspects of pattern recognition-making a conjecture and pattern recognition-developing a justification, but were insufficient in embodying the structural flow of mathematical reasoning. In addition, this study suggested that elementary school mathematics textbooks need to provide a wealth of tasks dealing with multiple types of mathematical reasoning, tasks of doing mathematics, and tasks reflecting the structural characteristics of mathematical reasoning.

초등학교 수학 교과의 원격수업에서 나타나는 교사의 수학적 주목하기

This study aims to explore the impact of the characteristics of remote learning on the overall mathematics classroom. Adapting Lobato et al.'s (2013) notion of “noticing as a phenomenon distributed across individual cognition, social interactions, material resources, and normed practices (p. 809),” the present study reports on similarities and differences in mathematical noticing of two elementary teachers who conducted remote learning on fractions division and how what each teacher noticed was shaped by other elements of their classroom. Although the two teachers taught the same lesson with similar tasks, there were distinct differences between the two teachers' classrooms. Similar to previous studies reporting on students' mathematical noticing, the differences in teachers' centers of focus emerged as a result of focusing interactions, features of mathematical tasks, and the nature of mathematical activities during remote learning. We concluded the study by suggesting the importance of focusing on teachers’ mathematical noticing, implications for future research, and effective professional development design for remote learning.

Tareas matemáticas escolares diseñadas por docentes de educación secundaria: un análisis desde la visión funcional de las matemáticas

Background: After mathematics curricular reform in 2012 in Costa Rica, which accentuate the functional vision of mathematics, it is worth inquiring if teachers designed tasks respond to established curricular guidelines and what mathematical and didactic characteristics that these tasks exhibit. Objectives: The aim of the study was to describe and analyze the mathematical and didactic elements, linked to the curricular proposal for Secondary Education in Costa Rica, used by in-service mathematics teachers for designing or selecting school tasks oriented to the development of mathematics skills on the subject of functions. Design: It corresponds to a descriptive-qualitative research, based on an intrinsic study of cases. Setting and Participants: Participants were five mathematics teachers from different educational institutions in Costa Rica during the year 2022 were studied, all teachers participated in a training program to promote teacher reflection competence regarding the design and analysis of school mathematics. Data collection and analysis: Data was collected from school mathematical tasks designed by the participants. The analysis of the information was carried out by defining categories and units of analysis based on the curricular provisions established for the teaching and learning of mathematics in Costa Rica, and the components of didactic analysis, as a methodology for the design of school mathematics tasks. Results: The analyzed tasks exhibit strengths in the curricular component, as well as in the didactic component. However, they lack elements that allow understanding how and when to put the designed tasks into practice, among other aspects. Conclusions: The training of in-service teachers takes a significant enhancement to strengthen knowledge and capacities oriented to the design, selection, and analysis of school mathematics tasks, associated with the functional vision of mathematics that articulates the curricular plan regarding student learning in Secondary Education in Costa Rica.

Real and illusionary difficulties in conceptual learning in mathematics: comparison between constructivist and inferentialist perspectives

AbstractDue to the learning paradox, students cannot have real difficulty in understanding a mathematical concept that they have not yet understood. There is a gap between real difficulties, directly experienced by students, and illusionary ones, only observed by researchers. This paper aims to offer a critical reflection on our understanding of the term difficulty in mathematics education research. We start this paper by arguing that a constructivist perspective, which has often been adopted in researches on mathematical task design, can deal with difficulties in solving a mathematical problem, but it cannot theoretically deal with those in understanding a mathematical concept. Therefore, we need the alternative philosophy of Robert Brandom’s inferentialism to capture students’ real difficulties in conceptual learning. From an inferentialist perspective, we introduce the idea of illusionary and real difficulties. The former is defined as what students cannot do, but they are not conscious of what they should do, while the latter is defined as what students cannot do despite their consciousness of what they should do. Through an eighth grade classroom episode, we argue that it is important in mathematics education research to focus not only on illusionary difficulties but also on the transition from illusionary to real difficulties. Researchers are encouraged to design a learning environment in which students become conscious of what they cannot do and to observe their mathematics learning in such an environment.

The Task and Goal Alignment Rubric: A tool for assessing the alignment of mathematical tasks and goals

AbstractMathematics education researchers suggest that it is important for teachers to select mathematics tasks that align with their goals for students; however, this can be challenging as there are many facets of alignment to consider. In this article, I present a rubric that details the alignment between tasks and learning goals. This rubric was developed by adapting the Achieve protocol and making modifications based on responses of a Delphi model panel that applied the rubric. I then discuss the use of this rubric with preservice teachers in a course about teaching mathematics with technology and the positive impact it had on their understanding and ability to align tasks and learning goals. I conclude with possible future uses of this rubric for researchers, teachers, and teacher educators.

Exploring Candidates' Initial Images of Similar Figures

This article investigates preservice candidates' knowledge of similar figures while engaged in teacher-related mathematical tasks (TRMT) with the primary goal of developing a lesson plan on similar figures. This investigation was part of a larger study investigating candidate knowledge of what and how to teach proportional reasoning concepts while engaged in TRMT. The data collection and TRMT cycle included individual discussions and reflections, group planning and discussions, and group presentations. Individual candidates also wrote their teaching philosophies and math autobiographies, which were reviewed. While candidates appeared to grasp many of the aspects of the concept of similar figures and similarity, there was a lack of precision and strength in their representations. The knowledge base of the candidates in both mathematics and teaching varied. As the study progressed, candidates relied heavily on their subject matter knowledge to justify their images and ideas.