Mathematical Problem-solving Processes Research Articles

Analisis Proses Pemecahan Masalah Matematika Berdasarkan Kerangka Kerja Mason pada Siswa SMP dengan Kategori Kemampuan Berpikir Kritis Matematis yang Berbeda

Problem-solving ability is an ability that affects the success of learning mathematics. One that affects problem-solving ability is critical thinking ability. Teachers must know the problem-solving process that students have to improve the problem-solving-oriented learning process. The purpose of this qualitative descriptive study was to analyze the mathematical problem-solving process based on the Mason framework for junior high school students with different categories of mathematical critical thinking abilities. The subjects of this study were 4 grade VII E students of SMP Negeri 5 Tuban with high and low mathematical critical thinking skills categories. The instruments used are written test questions for comparison materials with FRISCO indicators, written test questions for comparative materials in the form of HOTS, and interview guidelines. The results of the study will be analyzed descriptively and qualitatively based on the Mason framework. From the results of the study, it can be concluded that: (1) subjects with the category of high critical thinking skills can solve HOTS questions with cognitive levels C4, C5, and C6 correctly and fulfill all Mason's framework, namely the entry, attack, and review stages; (2) subjects with low mathematical critical thinking skills were unable to solve all HOTS questions with cognitive levels of C4, C5, and C6 and only met Mason's framework, namely the entry stage with complete aspects.

Self-control and self-monitoring behaviour of gifted learners in the mathematical problem-solving process: A case study

In the study reported on here we used a qualitative case study design to examine the self-control and self-monitoring behaviour of gifted learners in problem-solving processes. We selected 3 gifted secondary learners using the purposeful sampling method. For the study, each learner completed 10 individual problem-solving sessions. A think-aloud protocol as well as observations and interviews were used in each problem-solving session. The gifted learners displayed various and intertwined self-control and self-monitoring behaviour to read, understand, and solve the problems, and to find and verify the answer. They also displayed this behaviour much more frequently in problems that required using visual drawings and/or had long texts. The gifted learners left or adapted self-control behaviour when the behaviour did not work for solving mathematical problems. They made decisions regarding self-control behaviour by means of the self-monitoring process. The participants presented insistent, quick, flexible, and fluent actions for both self-control and self-monitoring processes. Based on our findings, we propose a portrait of gifted learners’ self-regulative behaviour in the mathematical problem-solving process.

Exploring the Characteristics of Digital Pedagogy Model for Developing Computational Thinking in Mathematical Problem Solving

Challenges in the 21st century are increasingly complex, technology is developing rapidly and competition is getting tougher. Therefore we need quality human resources that can keep up with and anticipate the times. The use of technology involves computational thinking (CT) skills which are closely related to the problem-solving process. The stages in computational thinking are part of mathematical thinking, meaning that learning mathematics can support students' CT skills. Through the development of digital pedagogical models in CT integrated mathematics learning, it can improve problem-solving skills. This research uses  design based implementation research with 4 phases including; preliminary research, prototyping, results, and design principle. The participants were 28 grade 8 junior high school students who took part in two rounds of experiment in direct CT activities and digital CT activities. In this paper, we present an iterative mathematical problem-solving process in the digital pedagogy model. The computational task, environment, tool and practices were iteratively improved over two rounds to incorporate CT effectively in mathematics. The results from CT environment demonstrated that direct CT activities are more effective than digital CT activities in mathematical problem-solving.  Based on empirical research, we summarize the characteristic of the digital pedagogy model from computational tasks, computational environment and tools, and computational practices in mathematical problem solving.

Evaluating ChatGPT-Generated Linear Algebra Formative Assessments

This research explored Large Language Models potential uses on formative assessment for mathematical problem-solving process. The study provides a conceptual analysis of feedback and how the use of these models is related in the context of formative assessment for Linear Algebra problems. Particularly, the performance of a popular model known as ChatGPT in mathematical problems fails on reasoning, proofs, model construction, among others. Formative assessment is a process used by teachers and students during instruction that provides feedback to adjust ongoing teaching and learning to improve student’s achievement of intended instructional outcomes. The study analyzed and evaluated feedback provided to engineering students in their solutions, from both, instructors and ChatGPT, against fine-grained criteria of a formative feedback model that includes affective aspects. Considering preliminary outputs, and to improve performance of feedback from both agents’ instructors and ChatGPT, we developed a framework for formative assessment in mathematical problemsolving using a Large Language Model (LLM). We designed a framework to generate prompts, supported by common Linear Algebra mistakes within the context of concept development and problem-solving strategies. In this framework, the instructor acts as an agent to verify tasks in a math problem assigned to students, establishing a virtuous cycle of learning of queries supported by ChatGPT. Results revealed potentialities and challenges on how to improve feedback on graduate-level math problems, by which both educators and students adapt teaching and learning strategies.

Conceptualizing Flexibility in Programming-Based Mathematical Problem-Solving

Computational thinking (CT) has received much attention in mathematics education in recent years, and researchers have begun to experiment with the integration of CT into mathematics education to promote students’ CT and mathematical thinking (MT) development. However, there is a lack of empirical evidence and new theoretical perspectives on the mechanisms of interaction between CT and MT. To address this research gap, this study analyses the participants’ thinking processes in solving programming-based mathematical problems from a flexibility perspective, focusing on the interplay between computational and mathematical thinking, that is, how CT and MT work together to influence and determine the problem-solver’s choice of solution strategy. Using data collected from a large design-based study, we summarise two types of flexibility and six subtypes of flexibility demonstrated by participants in the programming-based mathematical problem-solving process using thematic analysis. These different types of flexibility provide researchers and mathematics educators with new theoretical perspectives to examine the interplay of CT and MT. Findings will also contribute toward student learning characteristics in programming-based mathematical problem-solving to sketch the big picture of how CT and MT emerge in complementary or mismatching ways.

Quantifying structural distortion manipulation for desired perovskite phase: Part I. Paradigm demonstration in tungsten oxides

Slight distortions can cause dramatic changes in the properties of crystalline perovskite materials and their derivatives. Due to the numerous types of distortions and unclarified distortion-structure relations, a quantitative distortion manipulation for the desired crystalline phase of perovskite materials suitable for various application remains challenging. Here, by establishing parameter sets to systematically describe the types, magnitudes and positional relations involved in distortions, we are able to interpret the structural regulations and manipulation strategies in 7 reported crystal systems. Through the construction of distortion-phase-property functional curves, we further propose a paradigm to quantify the structural distortion manipulation for desired perovskite phases. Using the example of perovskite-like tungsten oxides, we successfully quantify their volume shrinkage and symmetry increase during lithiation. This work verifies that the complicated research and development of perovskite materials can be simplified into a mathematical problem solving process, which will inspire researchers with different backgrounds to participate, especially mathematicians and computer scientists.

What is the unknown? The ability to identify the semantic role of the unknown from word problems longitudinally predicts mathematical problem solving performance

The current study aimed to investigate children’s difficulties in word problem solving through assessing their ability to mathematize, or to identify the semantic role of the unknown from word problems. Fifth graders (n = 213) were given an advanced word problem reasoning task in which they had to match word problems with schematic diagrams that depict different processes (multiplication versus division) and the unknown being in different semantic roles (e.g., unit size, number of units, or total in an equal group problem). They were also tested on their mathematical problem solving as well as some potential confounding variables (i.e., intelligence and working memory) and mediators. The ability to identify the semantic role of the unknown was shown to be longitudinally predictive of children’s mathematical problem solving performance even after controlling for the effects of covariates and autoregressor. Such a relation was partially mediated by children’s ability to convert word problems into the correct number sentences/equations. The findings not only highlight the importance of unknown identification in mathematical problem solving process, but also provide a practical tool to assess such an ability.

Using Machine Learning to Detect SMART Model Cognitive Operations in Mathematical Problem-Solving Process

Using Machine Learning to Detect SMART Model Cognitive Operations in Mathematical Problem-Solving Process

Exploring students’ mathematical computational thinking ability in solving pythagorean theorem problems

This study explores students' mathematical computational thinking ability in solving the Pythagorean Theorem problem. This research method used a qualitative approach with a phenomenological design. The subjects involved in this study were 12 junior high school students. Six students in grade 7 had not studied the Pythagorean Theorem, and six students in grade 8 were studying the Pythagorean Theorem. This study's results indicate several problems with students' mathematical computational thinking skills in mathematics learning. The first problem is seen from the aspect of abstraction. Students are given problems with the help of digital-based teaching aids. Then the researcher provides procedures containing questions so students can digest the information and follow their intuition to find a solution strategy. Still, students have not decided what information should be stored or ignored. The second problem is seen from the aspect of decomposition. Students have not been able to decompose complex problems into simpler and more manageable ones. Student responses are also still not according to the researchers' predictions. However, with the scaffolding technique, researchers can direct students' intuition or thought processes to focus more on the problem being asked. The third problem is seen from the aspect of generalization. Students have not been able to generalize the problem and have not been able to conclude from the steps that have been taken. These three problems indicate that students cannot recognize and identify patterns well, thereby reducing the efficiency of the mathematical problem-solving process.

Habermas’ construct of rationality in the analysis of the mathematical problem-solving process

Habermas’ construct of rationality in the analysis of the mathematical problem-solving process

Using sequence mining to study students’ calculator use, problem solving, and mathematics achievement in the National Assessment of Educational Progress (NAEP)

Using appropriate tools strategically to aid in problem solving is a crucial skill identified in K-12 mathematics curriculum standards. As more assessments transition from paper-and-pencil to digital formats, a variety of interactive tools have been made available to test takers in digital testing platforms. Using onscreen calculators as an example, this study illustrates how process data obtained from student interactions with a digitally-based large-scale assessment can be leveraged to explore how and how well test takers use interactive tools and unveil their mathematical problem-solving processes and strategies. Specifically, sequence mining techniques using the longest common subsequence were applied on process data collected from a nationally representative sample who took the National Assessment of Educational Progress (NAEP) mathematics assessment to examine patterns of eighth-grade students’ calculator-use behaviors and the content of calculator input across a series of items. Sequences of keystrokes executed on the onscreen calculator by test takers were compared to reference sequences identified by content experts as proficient and efficient use to infer how well and how consistently the calculator was used. Results indicated that calculator-use behaviors and content differed by item characteristics. Students were more likely to use calculators on calculation-demanding items that involve intensive and complex computations than on items that involve simple or no computation. Using the calculator on more calculation-demanding items and using it in a manner that is more efficient and more similar to reference sequences on these items were related to higher mathematical proficiency. Findings have implications for assessment design and can be used in educational practices to provide educators with actionable process-related information on tool use and problem solving.

Making Thoughts Visible through Formative Feedback in a Mathematical Problem-Solving Process

This study aims to elicit the role of formative feedback in the development of students in a mathematical problem-solving process. For this purpose, the study's primary process is to investigate the development of elementary school students (aged 10 to 11) through feedback given during a problem-solving process. While visually engaged in the sub-processes expressing a problem situation and describing their thinking structures in writing, three different dimensions are addressed: communicating visually what they understood from the problem; expressing their thoughts about solution; and creating explanations regarding their solution process. The six-week embedded mixed method study reveals that students' explanations of their thinking processes developed towards the expectations. They were able to depict the problem and the relationships involved in the problem more clearly in their drawings to understand the problem. They made fewer mistakes in mathematical operations.

Matematik Öğretmen Adaylarının Problem Çözme Sürecinde Sergiledikleri Olumsuz Zihinsel Davranışlar

The aim of this study is to reveal various reasons that prevent the solution of pre-service mathematics teachers in their mathematical problem-solving process. The case study was used in the research in accordance with the qualitative research approach. Participants of the study consisted of 24 pre-service teachers who were selected by purposeful sampling method and studying mathematics education department of education faculty in a state university. Participant observation process and interview form were used to collect the data. According to the results of the study, it was determined that the pre-service mathematics teachers used monotonous approaches for the solution in the problem-solving process, they had insufficient knowledge, had anxiety and prejudices, and also displayed some mental errors.

Recall and mathematical problem solving for students with disabilities

This article reports on the ability of feeble-minded or mentally retarded students in mathematical recall and problem-solving processes. In particular, this study investigated whether recall ability can affect feeble-minded students' ability to work on mathematical problems. This research was conducted using a mixed-method approach with a sequential explanatory research design. The sample of this study consisted of 34 feeble-minded students. The findings show that the ability of feeble-minded students to solve mathematical problems and their recall abilities were categorized as fair Another finding is that the ability to recall had a 62.1% effect on the ability to work on mathematical problems. The contribution of this study is that feeble-minded students' ability to work on mathematical problems is greatly influenced by their ability in the mathematical recall process.

A Study on the Awareness of the Teachers Working in Special Education Schools towards Mathematical Problem-Solving Process

This study aims to examine the awareness of teachers working in special education schools towards mathematical problem-solving process. The study was carried out in two stages. In the first stage of the study, a scale was developed for the awareness of problem-solving process. The draft scale form was administered to 215 teachers working in special education schools. In order to determine the structural integrity of the scale, exploratory factor analysis (EFA) was conducted for the data obtained. The findings of the EFA indicated that the scale, which contains twenty-nine items, has a three-factor structure. It was calculated that the three factors explained 52.40% of the total variance. The Cronbach's alpha value of the scale is .93. The three-factor structure revealed by EFA was tested with confirmatory factor analysis (CFA). As a result of CFA, it was calculated as SRMR=.06, RMSEA=.06, CFI=.90, IFI=.90. In the second stage of the study, the teachers’ awareness of mathematical problem-solving process was examined in terms of several variables. For this purpose, the scale was administered to 181 teachers working in special education schools. As a result of the analysis of the obtained data, there is no statistically significant in teachers’ awareness of mathematical problem-solving process in terms of their gender and professional seniority; however, there is a statistical significance in terms of the variables such as graduation, the school and the group taught. The results were interpreted based on the previous studies in the literature.

Assessment Tests in the Mathematics Teaching Guides in Spain. Analysis of the Content Blocks and the Treatment of Arithmetic Word Problems

The teaching guides that complement textbooks have key importance in the assessment of competence in problem solving, because these materials contain the assessment tools that teachers frequently use to quantify the achievements of their students. In this paper, we set two aims: to analyze which curriculum contents are given priority in the assessment tests of the teaching guides; and to check to what extent these tests assess the steps of the mathematical problem solving process. For this, an analysis of the initial and final assessment tests of six Spanish publishers was conducted. The results show that the distribution of mathematical tasks by type of content does not fully conform to the theoretical framework proposed by TIMSS. In addition, only one of the six publishers considered the problem-solving process as evaluable.

Young children’s mathematical problem solving and thinking in a playworld

In the literature in recent years, a number of developmental studies have demonstrated the importance of children entering the school environment with a solid foundation of mathematical content knowledge and argued that problem solving, as an important mathematical process, should be acknowledged in early childhood mathematical education. However, there is less research on how children process mathematics information through problem solving in play-based early childhood education settings. This paper draws upon a cultural-historical concept of play, motives and pedagogical process of a playworld approach to investigate how Mathematical Playworld creates the motivating conditions for young children to achieve a meaningful learning experience about repeating patterns? We argue that Mathematical Playworld, as a new pedagogical approach within the worlds of imaginary situations, should be promoted, as it builds the motivating conditions that support meaningful learning of mathematical concepts in the double sense created in children. This study also contributes to understanding young children’s mathematical problem-solving processes in the collective imaginary situation, considering how learning processes become personally meaningful for children and capturing teachers’ role in play for supporting children’s mathematisation.

Cognitive Enhancement Through Mathematical Problem-Solving.

Nowadays, modern lifestyle along with the ever-growing technological and scientific advancement, especially in developed countries, has led to a significant increase of the middle age and elderly population. As a consequence, a substantial rise in neurodegenerative diseases has been reported, such as mild cognitive impairment, Alzheimer's disease, and other types of dementia. The aim of this paper is to study the impact of mathematical problem-solving on cognitive enhancement of adults. Moreover, the possible applicability of mathematical problem-solving as part of a treatment in cases of adults suffering from different levels of cognitive decline due to neurodegenerative diseases has been taken into consideration. For the needs of this study, a qualitative research was conducted in a sample of 16 participants, in order to confirm the effect of the mathematical problem-solving process on their cognition and its reinforcement, as stated by the participants. The results of this research, as interpreted, indicate a positive effect of mathematical problem-solving on adult cognition and on cognitive processes in general. Nonetheless, there are certain limitations to this research, deriving from its nature, while others derive from the measures enforced during the SARS-CoV-2 outbreak. Therefore, this chapter suggests that a more thorough quantitative research should be conducted in order to specifically measure the magnitude, as well as the time span of the impact of mathematical problem-solving on adult cognitive enhancement. Moreover, it is proposed that future studies should concentrate on the usefulness of mathematical problem-solving as a component in both invasive and noninvasive cognitive treatments used to cure or alleviate the aforementioned cognitive impairments.

Mathematical Problem-Solving Processes of Students with Special Needs: A Cognitive Strategy Instruction Model Solve It

A Cognitive Strategy Instruction Model namely 'Solve It!' which involves cognitive and metacognitive elements, was developed by Montague (1992) and it is one of the process based teaching strategies. Solve It! is a strategy which aims to teach seven cognitive strategy steps (i.e., read, paraphrase, visualize, hypothesize, predict, calculate, and check) and in each of the cognitive strategy step it contains three metacognitive steps (ask, say, and check). This strategy was used to teach solving word problems to students with special needs. This study aims to evaluate the studies using Solve It!. The reviewed studies are located by conducting searching in the electronic databases, journal indexes and references of studies. A total of 48 studies were found. The researchers reviewed them in terms of inclusion and exclusion criteria and retained 12 of them for descriptive analysis. Analyses showed that Solve It! was found to be effective in teaching mathematical problem solving skills to students with special needs. Findings are discussed, suggestions for researchers and practitioners are made.

A practice for using Geogebra of pre-service mathematics teachers' mathematical thinking process

We aim to examine the pre-service mathematics teachers' mathematical problem-solving processes by using dynamic geometry software and to determine their evaluations based on experiences in this process. The design is document analysis, one of the qualitative research approaches. In the fall semester of the 2019 - 2020 academic year, a three-problem task was carried out in a classroom environment where everyone could use geogebra individually. A total of 65 pre-service mathematics teachers enrolled in the course of educational technology. This task includes questions that they would use, their knowledge of basic geometric concepts to construct geometrical relations and evaluations related to this process. Besides the activity papers of the prospective teachers, geogebra files were also examined. The result is pre-service mathematics teachers who are thought to have a certain level of mathematical background are found to have incorrect/incomplete information even in the most basic geometric concepts and difficulties with regard to generalisation. Keywords: Dynamic geometry, geogebra, instructional technologies, mathematical thinking, teacher education.