Mathematical Tasks Research Articles

Unpacking associations among children’s spatial skills, mathematics, and arithmetic strategies: decomposition matters

Several studies revealed links between mental rotation and mathematical tasks, but the intervening processes in this connection remain rather unexplored. Here, we aimed to investigate whether children’s mental rotation skills relate to their accuracy in solving arithmetic problems via their usage of decomposition strategies, thus probing one potential intervening process. To this end, we examined a sample of 6- to 8-year-olds (N = 183) with a chronometric mental rotation task, and asked children to solve several arithmetic problems while assessing their solution strategies. After each arithmetic problem, children were asked about their strategy to solve the respective arithmetic problem and these were classified as either counting, decomposition, or retrieval strategies. Analyses were controlled for age, sex, fluid and verbal reasoning. Results indicated that children’s response times and accuracy in the mental rotation task were best explained by linear functions of rotation angle, suggesting the usage of dynamic mental transformation strategies. A multiple mediation model revealed that children with higher mental rotation skills were more inclined to use higher-level mental strategies such as decomposition which in turn increased their accuracy of solving arithmetic problems. None of the other arithmetic strategies revealed significant indirect effects. These findings suggest that children with higher mental rotation skills may profit from visualizing and flexibly transforming numerical magnitudes, increasing the frequency of decomposition strategies. Overall, decomposition may play a unique role in the connection between children’s mental rotation and arithmetic skills, which is an essential information for planning future training and experimental studies.

It is a "small world": Relations between performance on five spatial tasks and five mathematical tasks in undergraduate students.

One of the most robust relations in cognition is that between spatial and mathematical reasoning. One important question is whether this relation is domain general or if specific relations exist between performance on different types of spatial tasks and performance on different types of mathematical tasks. In this study, we explore unique relations between performance on five spatial tasks and five mathematical tasks. An exploratory factor analysis conducted on Data Set 1 (N = 391) yielded a two-factor model, one spatial factor and one mathematical factor with significant cross-domain factor loadings. The general two-factor model structure was replicated in a confirmatory factor analysis conducted in a separate data set (N = 364) but the strength of the factor loadings differed by task. Multidimensional scaling and network-based analyses conducted on the combined data sets reveal one spatial cluster, with a central node and one more tightly interconnected mathematical cluster. Both clusters were interconnected via the math task assessing geometry and spatial sense. The unique links identified with the network-based analysis are representative of a "small-world network." These results have theoretical implications for our understanding of the spatial-mathematical relation and practical implications for our understanding of the limitations of transfer between spatial training paradigms and mathematical tasks. (PsycInfo Database Record (c) 2024 APA, all rights reserved).

Педагогические приёмы устранения трудностей при решении текстовых задач четвероклассниками

The article examines the current problem of difficulties in solving word problems by fourth grade students, and describes the main types of difficulties. The importance of educational modeling, planning and control activities within the framework of the requirements of the Federal educational program for primary general education in mathematics is described. Two pedagogical techniques are presented: “Model the situation” and “Plan and control”. Examples of mathematical tasks are given, the use of which will allow fourth-graders to successfully overcome difficulties in solving word problems.

Contextual mathematical tasks as a means of developing students� functional literacy in school mathematics course

Relevance. The relevance of the research is conditioned upon the existence of a contradiction between the predominantly abstract-theoretical nature of the conventional school course of mathematics and the objective need to develop students� skills in the practical application of mathematical knowledge and tools to solve specific applied problems in real-life situations. Purpose. The purpose of the research is to identify and comprehensively justify the role and place of contextual mathematical tasks in the development of students� functional literacy, understood as the ability to apply mathematical knowledge in life situations. Methodology. The study employed methods of historical and comparative analysis, curriculum and educational policy analysis, and the generalization of best practices. Results. The work analyses the historical evolution of advanced national systems of mathematics education from the perspective of integrating the practice-oriented contextual approach into them. It is demonstrated that in many countries there has been a gradual transition to more applied mathematics education through the inclusion of realistic contextual tasks. The high effectiveness of school students solving mathematical problems contextualised in everyday scenarios for developing competence in applying abstract mathematical knowledge and skills in real-life situations has been proved. A set of factors stimulating the introduction of the contextualised approach in mathematics teaching, such as the results of international assessments, has been explored in detail. Conclusions. Key pedagogical, methodological and organisational conditions for the successful implementation of this approach in the school system, including the development of teaching materials, teacher training and assessment tools, are identified. The results expand the understanding of the possibilities of contextual tasks in the development of students� functional literacy, which is significant for the improvement of mathematics education. Keywords: mathematics education; practice-oriented learning; application of knowledge; pedagogical practices; curriculum

Maze-solving in a plasma system based on functional analogies to reinforcement-learning model.

Maze-solving is a classical mathematical task, and is recently analogously achieved using various eccentric media and devices, such as living tissues, chemotaxis, and memristors. Plasma generated in a labyrinth of narrow channels can also play a role as a route finder to the exit. In this study, we experimentally observe the function of maze-route findings in a plasma system based on a mixed discharge scheme of direct-current (DC) volume mode and alternative-current (AC) surface dielectric-barrier discharge, and computationally generalize this function in a reinforcement-learning model. In our plasma system, we install two electrodes at the entry and the exit in a square lattice configuration of narrow channels whose cross section is 1×1 mm2 with the total length around ten centimeters. Visible emissions in low-pressure Ar gas are observed after plasma ignition, and the plasma starting from a given entry location reaches the exit as the discharge voltage increases, whose route converging level is quantified by Shannon entropy. A similar short-path route is reproduced in a reinforcement-learning model in which electric potentials through the discharge voltage is replaced by rewards with positive and negative sign or polarity. The model is not rigorous numerical representation of plasma simulation, but it shares common points with the experiments along with a rough sketch of underlying processes (charges in experiments and rewards in modelling). This finding indicates that a plasma-channel network works in an analog computing function similar to a reinforcement-learning algorithm slightly modified in this study.

Students' Perspectives on Selecta Capita of Mathematics Online Course on Their Ability to Develop Mathematics Task

The main problem in implementing online lectures is how to adjust what students expect with the achievement of learning quality. This study examines students’ perspectives on effective lectures on the ability to develop mathematics task. Quantitative descriptive research has been carried out on students of the mathematics education study program Universitas Negeri Semarang who took the Selecta Capita of Mathematics course. The subjects in this study were 101 first-year students in the mathematics education study program at Semarang State University for the 2021/2022 academic year. The data were obtained through questionnaires and analyzed by the descriptive statistics. The data that had been obtained and then tabulated in a table, the average value is searched, and presented in the appropriate graph. The results of the study indicate that the availability of learning modules and videos as learning resources is the most important thing that must be prepared during lectures. Furthermore, several other things such as synchronous lectures, discussions using the WhatsApp Group, the use of the learning management system, group assignments, virtual face-to-face meetings, and project assignments need to be prepared during lectures. These results are needed to be applied on team-based project learning models to develop students’ abilities in developing math tasks.
 Keywords: students’ perspectives, selecta capita, mathematics online course, mathematics task

The role of mathematical vocabulary in the development of mathematical skills for Spanish-speaking students

Does mathematical vocabulary predict the change in students’ performance on mathematical tasks from one academic year to the next? Chilean Spanish-speaking students (N = 87) completed measures of mathematical vocabulary, mathematical skills (i.e., arithmetic fluency, calculation, and applied problems), receptive vocabulary, and working memory in Grade 2 (T1, Mage = 7:11 years:months, SD = 0:5, 46% girls). One year later (T2) they completed the same mathematical measures. Concurrent relations were found between mathematical vocabulary and the three mathematical skills at both time points. Together, general and mathematical vocabulary at T1 explained significant unique variance in the change in applied problems and calculation from T1 to T2. For calculation however, only mathematical vocabulary predicted significant unique variance in the change from T1 to T2. Change in arithmetic fluency was only predicted by working memory. These results address the roles of general and mathematical vocabulary in students’ mathematical development in elementary school.

GPS: The Arithmetic and Harmonic Triangles

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.

초등학생의 수학 과제 수행 특성에 대한 인지적 요구 수준별 분석

Mathematical tasks are a key factor that affects students' learning process and outcomes in mathematics classes. Each mathematical task requires a specific cognitive demand level to be solved. This study presented mathematical tasks requiring different cognitive demand levels to elementary school students and analyzed the characteristics that emerged during task performance, allowing students to make modifications in their own ways. The analysis showed that the students recognized the cognitive demand levels of mathematical tasks that were appropriately modified for elementary students and solved and modified them. However, the study found that passive modifications, which changed the components of the task, were more common than active modifications. Active modifications would involve adjustments related to the mathematical concepts, competencies required, or the cognitive demand level. The analysis of performance characteristics allowed for the identification of elementary school students' perceptions of high cognitive demand level tasks and various task types. Based on these findings, pedagogical implications were derived.

Assessing Concepts, Procedures, and Cognitive Demand of ChatGPT-generated Mathematical Tasks

In November 2022, ChatGPT, an Artificial Intelligence (AI) large language model (LLM) capable of generating human-like responses, was launched. ChatGPT has a variety of promising applications in education, such as using it as thought-partner in generating curricular resources. However, scholars also recognize that the use of ChatGPT raises concerns, such as outputs that are inaccurate, nonsensical, or vague. We, two mathematics teacher educators, engaged in a collaborative self-study using qualitative descriptive approaches to investigate the procedures, concepts, and cognitive demand of ChatGPT-generated mathematical tasks focused on fraction multiplication using the area model approach. We found that the ChatGPT-generated tasks were mostly procedural and not cognitively demanding. Moreover, despite ten variations of input prompts, ChatGPT did not produce any tasks that used the area model approach for fraction multiplication. Rather, it generated tasks focused on procedural approaches. Alarmingly, some tasks were conceptually and/or procedurally inaccurate and vague. We suggest that educators cannot fully rely on ChatGPT to generate cognitively demanding fraction multiplication tasks using the area model. We offer recommendations for educators’ strategic use of ChatGPT to generate cognitively demanding mathematical tasks.

Formation and Development of Mathematical Literacy in the Context of PISA-2021 Program Concep

This article presents the results of pedagogical experiments, which revealed educational problems that teachers face in the process of teaching mathematical literacy to schoolchildren: a low proportion of practice-oriented tasks in the total number of educational tasks in school mathematics intended for 15-year-old schoolchildren, a passive attitude of teachers to the formation of mathematical literacy among schoolchildren. The analysis of the experimental data made it possible to conclude that the knowledge and skills of teachers on the issues of reducing the standard task of the school mathematics course are insufficient to problem situations in the context of the PISA assessment tasks, compiling task systems for these tasks, containing an algorithm for solving the assigned problem, description of a visual object that allows the visualization of the task. To solve these problems, the paper proposes to apply a modular learning technology in the process of teaching schoolchildren mathematical literacy, an algorithmic method for reducing a standard task to assessment-learning tasks in the context of the PISA-2021 program concept. In the work, examining the task to the problem situation as an object of mental activity of students, the main components of the "Mathematical Reasoning" of the student are highlighted. Namely: recognizes the area of the task; singles out the main components of the tasks; to bring data under known concepts; establishes interdisciplinary links; applies induction, interdisciplinary knowledge in the preparation of a mathematical model; solves a mathematical problem; interprets, evaluates the solution in the context of the phenomenon under consideration. The analysis of algorithms for solving tasks made it possible to identify the following key structural components of mathematical literacy: understanding the context of tasks; the ability to establish correspondences between various objects, apply induction and knowledge, build mathematical models, find solutions to mathematical models of tasks, interpret and evaluate the solutions obtained; have mathematical reasoning skills. The final experimental study confirmed the effectiveness of the developed methodology for the formation and development of mathematical literacy at 15 year old schoolchildren.

How the relationship between individual and social factors informs the narrowing of gender gaps in elementary mathematical achievements

Few scholarly studies have examined gender gaps vis-à-vis various types of mathematical problems by controlling for individual factors (e.g., general intelligence and self-regulated learning [SRL]) and social factors (e.g., the school’s socioeconomic status [SES]) among elementary school students, as addressed in this study. Achievements on three types of mathematical tasks (operations with numbers, geometry, and word problems) and general intelligence scores were obtained from elementary school students. Information regarding students’ SRL was obtained from a rating scale designed for the teachers. Results showed that boys’ achievements were significantly higher in word problem-solving and geometry tests but not in operation with numbers tests. The results concerning word problems suggest that the effect of the school’s SES level on the gender gap decreased in accordance with the increase in the school’s SES level. The significant interaction between gender and SRL indicated that the effect of students’ SRL level on the gender difference in the students’ performance on the geometry test increased in accordance with the increase in the SRL level. General intelligence contributed to individual differences in word problems and geometry, but not on operations with numbers. The analyses showed that SRL contributed to mathematical performance in all tasks. Directions for future research and implications for narrowing these gender gaps are discussed.

Urban-Rural Differences in Early Arithmetic Performance are Accounted for by Phonological Processing

ABSTRACT Urban and rural children have different levels of performance in arithmetic processing. This study investigated whether such a residence difference can be explained by phonological processing. A total of 1,501 Chinese primary school students from urban and rural areas were recruited to complete nine cognitive tasks: two in arithmetic performance (subtraction and multiplication), one in phonological processing (word rhyming), two in quantity processing (numerosity and number comparison), one in visuo-spatial processing (mental rotation), as well as a mathematical reasoning task (number-series completion) as a contrast with arithmetic processing and two general cognitive tasks (choice reaction time and Raven’s progressive matrices). The results showed that Chinese urban children had superior performance in arithmetic, which was eliminated after controlling for phonological ability, but not after controlling for quantity or visuo-spatial performance or general cognitive factors. Urban children also showed better performance in mathematical reasoning than did rural children, but residence difference in mathematical reasoning was not eliminated by any of the cognitive factors. The above result suggests that phonological processing explains the relationship between residence and arithmetic performance.

The Effect of Level-Marked Tasks on Students’ Mathematics Self-Efficacy and Persistence—Investigating a Tiered Activity

In school mathematics, tiering tasks according to students’ readiness levels is a common approach to adapting content to address students’ varying abilities. To distinguish between different tiers, the tasks are commonly marked by their level of difficulty, as in so called level-marked tasks – a strategy used in differentiated instruction to help students gain mastery and improve performance. However, less is known about whether and to what extent the marking of the tasks works as intended. Regarded as a significant predictor of students’ performance in mathematics, the level marking of mathematics tasks and their effects on students’ self-efficacy and persistence were investigated. The online survey part of an experimental design was used to collect data from 436 lower secondary students in Norway. Independent sample t-tests and Mann–Whitney U tests revealed that, when students across groups were given the same mathematics tasks with and without level markings, tasks marked as difficult reduced both students’ mathematics self-efficacy and their persistence, even for students with high baseline self-efficacy. Although more research is needed to fully understand how the level markings of mathematics tasks affect students’ self-efficacy and persistence, the reported results have implications for both mathematics teachers and textbook authors.

Utilizing a third order difference scheme in numerical method of lines for solving the heat equation

Obtaining the numerical solution for partial differential equations (PDEs) is a fundamental task in computational mathematics and scientific computing. In this work, we propose a third order numerical method of lines (MOL) based on difference schemes for the efficient and accurate approximation of heat equation. The method combines the flexibility of MOL with the superior accuracy of higher order difference schemes. The proposed approach begins by discretizing the spatial domain, and then applies higher order difference schemes to approximate the spatial derivatives in the PDE. The MOL framework is utilized to transform the resultant set of ordinary differential equations to a system of algebraic equations, facilitating their solution through an appropriate numerical method of choice. We have utilized the Adams method for this purpose. We analyze the stability of the proposed method and establish conditions under which it yields accurate and reliable solutions. We present the theoretical foundations of the method and discuss its implementation. We also present numerical results that demonstrate the procedure’s precision and effectiveness. The results show that the method is able to achieve high accuracy with relatively few grid points, and the results are compared with those obtained by some existing numerical schemes accessible in the literature.

Students’ Verification and Elaboration in Outdoor Mathematics: The Role of Digital Feedback in MathCityMap

In this article, the role of digital feedback that was provided in an outdoor mathematics education setting is taken into consideration. Using the app MathCityMap (2020) in the context of a mathematics trail, the influence of positive and/or negative feedback is examined in relation to how it influences the processes of verification and elaboration. In this context, special emphasis is placed on the students’ verification and elaboration and their relation to reasoning. In this qualitative study, 19 secondary students were filmed while solving mathematics tasks outdoors without digital support, as well as in indoor settings to enable a comparison. The results show that negative feedback in particular leads to a verification of the result. Still, an elaboration and explanation of why a result was incorrect was not often explicitly formulated by the students. Therefore, the potential of feedback is mainly seen in giving students a clear idea about the correctness of the result and searching for an alternative strategy to solve the task when in an outdoor setting.

Learning analytics in mathematics education: the case of feedback use in a digital classification task on reflective symmetry

Learning Analytics is concerned with the use of data collected in educational settings to support learning processes. We take a Learning Analytics approach to study the use of immediate feedback in digital classification tasks in mathematics. Feedback serves as an opportunity for learning, however its mere existence does not guarantee its use and effectiveness, as what matters is how learners interact with it. Therefore, our research questions are focused on that interaction. The data consisted of 266 object movements for classifying polygons, and 524 shape movements for classifying traffic signs, under the topic of symmetry. Participants included 29 elementary school students (9–12 years old) from Israel and Germany. Analyzing students’ success, feedback use, and the associations between them, we demonstrate how not acting upon feedback is negatively associated with success, and how this undesired behavior slightly reduces along the learning process.

GPS: Supporting Use of AI in the Math Classroom

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.

Appraisement of mathematics tasks—a point of view of Norwegian eighth-grade students

In the large body of literature that is available about the role of motivation in mathematics education, the focus is mostly on students’ attitudes and feelings toward mathematics in general. In the current small-scale explorative study, we zoomed in on what students think about the tasks they can come across in mathematics education. To investigate students’ appraisement of mathematics tasks, that is, their judgment of how much they like mathematics tasks, we used a mixed method approach with an online questionnaire including closed and open questions. We asked 67 Norwegian eight graders to give an appraisal score for 24 tasks and to express in their own words their reasons for liking/disliking them. In addition, the students also had to indicate for each task whether they think they can solve the tasks. The analysis of the data indicated that the students may prefer bare number problems over context problems. Similarly, students inclined to like puzzle-like tasks more than estimation or straightforward tasks. The reasons for liking/disliking a task most often refer to the difficulty or easiness of the task. The data about the perceived solvability revealed that the more the students believed that they can solve the task, the more they liked the task. Given the different appraisals that the students granted to the different task types and the comments they came with, there is much reason to afford students a stronger voice in mathematics education research, particularly when it is about the tasks used in mathematics education.

EEG alpha and theta time-frequency structure during a written mathematical task.

Since the first electroencephalogram (EEG) was obtained, there have been many possibilities to use it as a tool to access brain cognitive dynamics. Mathematical (Math) problem solving is one of the most important cortical processes, but it is still far from being well understood. EEG is an inexpensive and simple indirect measure of brain operation, but only recently has low-cost equipment (mobile EEG) allowed sophisticated analyses in non-clinical settings. The main purpose of this work is to study EEG activation during a Math task in a realistic environment, using mobile EEG. A matching pursuit (MP)-based signal analysis technique was employed, since MP properties render it a priori suitable to study induced EEG activity over long time sequences, when it is not tightly locked to a given stimulus. The study sample comprised sixty healthy volunteers. Unlike the majority of previous studies, subjects were studied in a sitting position with their eyes open. They completed a written Math task outside the EEG lab, wearing a mobile EEG device (EPOC+). Theta [4 Hz-7.5 Hz], alpha (7.5 Hz-13 Hz] and 0.5 Hz micro-bands in the [0.5 Hz-20 Hz] range were studied with a low-density stochastic MP dictionary. Over 1-min windows, ongoing EEG alpha and theta activity was decomposed into numerous MP atoms with median duration around 3s, similar to the duration of induced, time-locked activity obtained with event-related (des)synchronization (ERS/ERD) studies. Relative to Rest, there was lower right-side and posterior MP alpha atom/min during Math, whereas MP theta atom/min was significantly higher on anteriorly located electrodes, especially on the left side. MP alpha findings were particularly significant on a narrow range around 10 Hz-10.5 Hz, consistent with FFT alpha peak findings from ERS/ERD studies. With a streamlined protocol, these results replicate previous findings of EEG alpha and theta activation obtained during Math tasks with different signal analysis techniques and in different time frames. The efficient application to real-world, noisy EEG data with a low-resolution stochastic MP dictionary shows that this technique is very encouraging. These results provide support for studies of mathematical cognition with mobile EEG and matching pursuit.