Mathematical Thinking Research Articles

Investigating preservice teachers’ feedback to students’ mathematical solutions

ABSTRACT Teachers require support in developing the pedagogical skill of interpreting students’ mathematical thinking and providing effective feedback that promotes learning. However, there is a lack of research assessing the quality of feedback based on teacher noticing in mathematics classes. This study analysed 20 preservice teachers’ (PSTs) feedback approaches through two perspectives: mathematical alignment and the responsiveness of feedback to students’ mathematical approaches. The findings indicated alignment of the focused mathematical concepts/processes in noticing and feedback. PSTs paid more attention to mathematical accuracy than the strength or potential of students’ strategies. Furthermore, there was a relationship between the complexity of student solutions and PSTs’ depth of noticing and the cognitive demands of feedback.

Cultivating Students’ Agency by Infusing Number Talks with Notice and Wonder

ABSTRACT Instructional routines, like Number Talks and Notice and Wonder, engage students in doing and then talking about their mathematical thinking. Number Talk routines are short mathematical discussions where students mentally solve and share the strategies they used to make sense of the problem. In comparison, the Notice and Wonder routine asks students to reflect on a high-interest visual prompt with multiple entry points, share their noticings and wonderings, and then the teacher steers the discussion toward a mathematical question to explore further. But what happens when the Notice and Wonder routine is used in conjunction with a Number Talk? In this descriptive case study, we analyze whether and how this variation of a Number Talk provides a class of third-grade students opportunities to exercise agency. Findings from this study yielded two major themes: 1) agency is built when authority is shared during a mathematical discussion, and 2) Notice and Wonder is an instructional routine that supports sharing authority. We then discuss why this matters and next steps for research and practice.

Reification, Curry-Howard Correspondence, and Didactical Consequences

The roles of languages, processes, and objects in mathematical thinking have led to many theories, yet no consistent big picture has evolved from this. This paper puts forth the hypothesis that the Curry-Howard correspondence from computer science and the theories it is built on provide a unification framework. This correspondence asserts that (formal) proofs and programs (in functional programming languages) do not only have some similarities, but can, at least if formalized in an appropriate way, be mapped to each other by an isomorphism such that proofs are programs and vice versa. Moreover, objects can be realized as function evaluation strategies, and this provides a model of the reification process. The paper explores all this and discusses the didactical relevance; especially, the reification theories are revisited. Computer-based realizations of concepts are used as a tool to show the consistency of ideas and the practicability of concepts.

Strategic Factors that Influence Critical Thinking Ability in Mathematics Learning in Elementary Schools

<p>Mathematical critical thinking skills are a person's ability to understand the thought process in order to achieve rational thinking to do something or believe in what should be done. Many factors influence students' critical thinking abilities, including physical condition, motivation, intellectual development, anxiety and habits. This research aims to determine and analyze the factors that influence critical thinking abilities in mathematics learning. This type of research is quantitative research with a survey method. The population in this study were all fifth grade elementary school students in Kartasura District, taking 2 schools as research samples. The number of samples in this research was 200 students selected using purposive sampling technique. Data collection techniques in this research used instruments consisting of: (1) physical condition questionnaire, (2) motivation questionnaire, (3) anxiety questionnaire, (4) habits questionnaire, (5) intellectual development questionnaire and (6) thinking ability questionnaire critical. The data analysis used was Structural Equation Modeling (SEM) using the Smart AMOS 23 computer program. The results of this study showed that 1) physical condition and intellectual development had a significant effect on anxiety, 2) physical condition, motivation and intellectual development had a significant effect on habits, 3) habits and anxiety have a significant effect on critical thinking. Meanwhile, motivation has no significant effect on anxiety. Based on the research results, it can be seen that physical condition, motivation, intellectual development, anxiety and habits influence students' critical thinking abilities.</p>

Analysis Of Junior High School Students' Mathematical Creative Thinking Ability Reviewed By Gender

This research is motivated by the importance of creative mathematical thinking skills for students studying mathematics. This study aims to analyze students' mathematical creative thinking skills in terms of gender. This study used a qualitative descriptive method. The population in this study was 20 grade IX students of SMP Negeri 2 Cimahi who were used as a sample, consisting of 10 male students and 10 female students who were randomly selected. The instrument used in this study is to provide four mathematical creative thinking questions consisting of indicators of fluency, flexibility, originality, and elaboration. The data in this study were analyzed using the value of the percentage of students who had done mathematical creative thinking problems that had been given and then compared based on gender. The results of the research that has been conducted revealed that overall, the ability of female students to think creatively mathematically is valued higher than the ability of male students to think creatively mathematically. This can be seen in terms of the proportion of overall value or certain aspects. Regarding fluency, male students score higher than female students, while female students score higher on flexibility, originality, and elaboration. It can be concluded that gender can affect students' ability to creative mathematical thinking skills.

Model Pembelajaran Learning Cycle 5E Terhadap Keterampilan Critical Thinking Matematis Materi Fungsi Ditinjau dari Kemampuan Awal Siswa

This research aims to improve students' critical thinking or critical thinking skills which are divided into three levels, namely students who have high level, medium level, and low-level critical thinking through the application of the Learning Cycle 5E learning model in class X. This research is quasi-experimental research namely the development of true experimental design with Nonequievalent control group design research design through Posttest-Only Control Design carried out at Sukake 1 Public High School. The sample selected used Cluster random sampling, namely class X MIA 1 which was used as the experimental class. Data collection was carried out by trial at SMA Negeri 2 Menyuke followed by research school at SMA Negeri 1 Menyuke using the Learning Cycle 5E learning model and post-test final test to determine students' mathematical critical thinking skills at high, medium, and low levels, interviews to find out student responses regarding the results of the answers. The t-test results for both normal and homogeneous distributions obtained a significance value (2-tailed) = 0.031 < 0.05 with the decision H_1 being accepted. It was proven that there was a significant difference between the learning outcomes in the control class and the learning outcomes of students in the experimental class. Descriptively, students show improvement. Students also experienced quite good improvement. So, it can be concluded that the Learning Cycle 5E learning model provides an increase in critical thinking skills

The Effective Application of Metacognitive Strategies in High School Mathematics Teaching

Metacognitive strategies, as an educational method that focuses on the metacognitive knowledge and skills of learners' thinking and learning process, have attracted widespread attention in the field of high school mathematics teaching in recent years. This article aims to analyze the effective application of metacognitive strategies in high school mathematics teaching, and deeply explore their role in improving students' mathematical learning achievements and developing mathematical thinking abilities. Firstly, this article provides an overview of the current situation and problems in high school mathematics teaching, including the difficulty of teachers' teaching concepts keeping up with the times, the lack of novelty in teaching methods, and the solidification of students' thinking. Then, the article pointed out the importance of metacognitive strategies for mathematical learning and introduced the basic overview and classification of metacognitive strategies. Next, this article explores in detail how metacognitive strategies can be applied to high school mathematics teaching, including carefully preparing teaching based on metacognition and finding the best teaching methods, guiding students to reflect through metacognitive monitoring, and improving teaching evaluation systems based on metacognition. The effective application of these strategies helps students better understand mathematical concepts, improve problem-solving abilities, and enhance mathematical confidence.

Fostering critical thinking and self-efficacy in mathematics students

This research addresses the low critical mathematical thinking abilities of students when facing mathematical problem-solving challenges. The study aims to evaluate the impact of implementing various teaching approaches, including Contextual Teaching and Learning (CTL), Problem-Based Learning (PBL), and Direct Instruction (DI) models, on students' critical mathematical thinking abilities. Participants included 276 eighth-grade students from SMP La Tansa Mashiro 1, 2, and 3 in Lebak Banten during the academic year 2022/2023, divided into three classes. Samples were selected using simple random sampling, resulting in 94 students for CTL, 92 students for PBL, and 90 students for DI instructional models. Data collection involved self-efficacy questionnaires and critical mathematical thinking ability tests, analyzed using a two-way ANOVA. The findings suggest that implementing CTL and PBL models positively influences students' critical mathematical thinking abilities. Additionally, students' self-efficacy levels contribute significantly to their critical mathematical thinking abilities. Furthermore, a significant interaction was observed between the implementation of CTL, PBL, and DI models and students' efficacy levels, collectively influencing students' critical mathematical thinking abilities. These results emphasize the importance of employing active approaches, such as contextual, problem-based, and direct instructional methods, while also enhancing students' self-confidence to improve their critical mathematical thinking abilities. It is hoped that implementing these recommendations will enhance the effectiveness of mathematics learning and foster the development of students' critical thinking skills.

Improving quantitative literacy through a general education acoustics course

Most Colleges and Universities have some form of quantitative reasoning requirement for graduation. A narrow view of this requirement is that students need to do some mathematics in the course to demonstrate some facility at calculating quantities. Additionally, students often discuss their discomfort with mathematics when discussing their anxiety about registering in a non-majors physics course. In this presentation, I will discuss strategies used in a musical acoustics course for non-science majors to address proportionalities, graphical representations, and approximations to help students get past equations to think about the underlying physical relationships. Often, I find that students actually are much better at using mathematics than their self-assessments. By giving students different contexts for mathematical thinking, the quantitative elements make more sense to them and calculations become less intimidating. The examples I will discuss focus on using classroom-based experiments to identify relationships between dependent and independent variables on topics such as musical scales, simple oscillators, and fourier analysis.

O pensamento computacional, como suporte ao pensamento matemático, na Instituição Educacional Santo Domingo Sávio de Chinchiná (Caldas – Colômbia)

The present research evaluated the improvement of mathematical thinking, from a computational skills intervention in eighth grade adolescent students of the Santo Domingo Savio Educational Institution of Chinchiná (Caldas-Colombia), of socio-economic stratum 1, 2 and 3, with ages between 12 and 17 years old. The work was prepared in the years 2020 and 2021, it was conceptually based on Computational Thinking proposed by Jeannette Marie Wing, for the process it was considered as the independent study variable and the Theory of Cognitive Development of Jean William Fritz Piaget was referenced, specifically on mathematical thinking, which in the study is the dependent variable. The research developed is of a quasi-experimental type, with an intentional nonprobabilistic sampling, we worked with experimental groups, one group for each year, to whom the “intervention program” was applied for four months; Before and after the intervention program, pretest and posttest tests were applied, using international tests called the “Binet-Simón and Wechsler mathematical intelligence test,” for the purposes of comparing evidence. Finally, through hypothesis tests, using the normal and Student’s t distributions, it is concluded that computational thinking (independent variable) has a statistically significant effect on the improvement of mathematical thinking (dependent variable).

An Exploration of Interdisciplinary Thinking Patterns in Secondary English Grammar Teaching-- Tenses

Interdisciplinary thinking is a way of thinking that utilizes the intersection and fusion between internal and external knowledge of disciplines to solve problems through cross-border integration of knowledge. The penetration of interdisciplinary thinking in secondary school grammar teaching is of great significance, which is conducive to change the conception of teacher’s focusing on the teaching and research of their own disciplines merely, enhances students' interest in grammar learning, and promotes the development of students' multiple intelligence. In this paper, the author takes the tenses in secondary school English grammar as an example to explore the penetration of interdisciplinary thinking in grammar teaching. Specifically, the author takes the combination of mathematical and physical thinking in mathematics as an starting point, combining the "coordinate axes" with the "union sets" in the set to discuss the use of single mathematical model and the multiple mathematical models during the teaching of English tenses.

Characteristics of Students Engaged in Spontaneous Problem-Posing

Spontaneous problem posing is a basic concept of spontaneous mathematical thinking and science learning. Students generate problems without systematic encouragement and pose problems based on the student's desire to develop their skills. As a result, they can serve as important markers of constructive mathematical and science engagement, particularly affective engagement, for problem solvers and their classroom communities. We used a qualitative approach to analyze student characteristics, especially in the affective domain, when presenting random problems. We used an observational approach and experience sampling in each class to observe students' engagement in spontaneous problem posing both individually and in groups. The findings revealed that each student showed different characteristics when presenting problems suddenly (spontaneous problem posing). The submission of the first subject problem was categorized as problem-as-exercise, satisfying the characteristics of spontaneous originality, where constructive emotional experiences impressed more on the teacher, while negative emotional experiences impressed more on oneself (self), classmates, and mathematical activities. The submission of the second subject problem is classified as a problematic problem, fulfilling the characteristics of spontaneous originality. Negative emotional experiences are more visible in me (myself), while constructive emotional experiences are more visible in teachers, classmates, and math activities

A model of the process for forming the readiness of future mathematics teachers to teach schoolchildren in the context of the PISA-2021 concept

Relevance. The relevance of the research topic lies in the fact that educational tasks in geometry contribute to the formation and development of mental abilities, spatial imagination, which are necessary in logistics, architecture and construction, mechanical engineering, cartography and other fields of applied science. Purpose. The purpose of the study is to study the stage of training of future mathematics teachers in the direction of implementing the PISA-2021 concept. Methodology. Among the theoretical research methods, analysis and synthesis, the method of comparison and generalization were used. At the stage of the empirical part, an experimental pedagogical study was conducted on the basis of three universities of Kazakhstan. Results. The article examines the issue of designing the learning process in the context of forming the readiness of future mathematics teachers to carry out pedagogical activities. A system of content components of the future teacher's general readiness to teach schoolchildren was determined. The signs and stages of formation of the readiness of future mathematics teachers to teach schoolchildren in the context of the PISA-2021 concept have been identified. A model of the process of forming the professional readiness of future mathematics teachers to teach schoolchildren in the context of the formation and development of their mathematical thinking, spatial ideas, and divergent thinking has been built skills of the 21st century. Conclusions. The study confirms the effectiveness of a new model for preparing future math teachers, showing high student proficiency in mathematical and pedagogical skills, while noting potential for broader application and challenges in implementation due to current school curriculum limitations. Keywords: readiness components; stages of the learning process; thinking abilities; geometric problems; section of polyhedra

Analisis Kemampuan Berpikir Kreatif Matematis Siswa Kelas VII SMP Pada Materi Persamaan Dan Pertidaksamaan Linear Satu Variabel Dengan Menggunakan Hands on Activity

Analisis Kemampuan Berpikir Kreatif Matematis Siswa Kelas VII SMP Pada Materi Persamaan Dan Pertidaksamaan Linear Satu Variabel Dengan Menggunakan Hands on Activity

Advantages and prospects of constructive learning of school mathematics epistemes in a pedagogical higher educational institution

Relevance. In the context of constant changes in the educational sphere, the introduction of constructive teaching to the epistemes of mathematics is an urgent task, the study of this topic helps to adapt teaching methods to modern requirements and effectively introduce innovations into pedagogical practice. Purpose. The purpose of the study was to analyse the effectiveness of constructive teaching of mathematics epistemes in a pedagogical higher educational institution in order to identify and evaluate the results obtained. Methodology. The following methods of scientific cognition were used: literary analysis, synthesis, deduction, and induction. Results. The study identified specific epistemic competencies needed by future mathematics teachers, such as the ability to analyse mathematical concepts, identify connections between them, and develop systematic mathematical thinking. Specific elements of successful educational programmes were also highlighted, such as precisely tuned courses, systematic practical exercises, and effective feedback that contribute to the development of professional competence of future mathematics teachers. The methods contributing to the effective mastery of educational material by students and to the development of pedagogical competence in the field of mathematics education were considered. Tools and criteria were also proposed to assess the effectiveness of the implementation of constructive teaching of epistemes of mathematics, contributing to the development of systematic mathematical thinking and deep awareness of students. Conclusions. Constructive learning based on an epistemic approach is aimed at developing students� holistic understanding of mathematics, supporting flexibility, creative problem solving and interest in the subject. The study proposed strategies and methods aimed at developing the professional competence of future mathematics teachers, including special educational courses, practical exercises, and feedback from experienced teachers. Keywords: educational practice; professional competence; applied sciences; methodology; teaching; learning; abilities

Developing creative thinking abilities in mathematics education: A systematic literature review

Mathematical creative thinking ability is the ability to think logically and divergently to produce ideas or various kinds of solutions to given mathematical problems. This systematic review of literature aims to look at research trends related to creative thinking abilities in mathematics education, indicators, research methodology, and learning models used in developing mathematical creative thinking abilities. A Systematic Literature Review (SLR) was carried out using the PRISMA method. This paper reviews 30 research articles from 2019 to 2023 from which such data were obtained from Scopus. The findings show that research themes such as tests, abilities, and creative thinking have been widely carried out. Indicators of creative thinking abilities that are often used are fluency, flexibility, and originality. The learning models or approaches that are widely used in developing creative thinking skills in mathematics education are contextual learning models such as problem-based learning and STEM/STEAM. These findings are useful for researchers and teachers. Keywords: Creative thinking; learning model; mathematics education; PRISMA; systematic literature.

Analysis of Students' Mathematical Creative Thinking Ability in Solving Open-Ended Questions Based on Their Self-Concept

To find out students' mathematical creative thinking abilities, this can be done by giving students open-ended problems. The aim of this research is to describe students' mathematical creative thinking abilities in solving open-ended problems when viewed from their self-concept. This type of research is qualitative descriptive research. The subject selection technique used in this research was purposive sampling. The subjects of this research consisted of one student with a high self-concept category, one student with a medium self-concept category, and one student with a low self-concept. Data collection methods in this research include questionnaire methods and tests. And the data analysis techniques used include: analysis of mathematical self-concept questionnaires, analysis of open-ended mathematics questions. The research results show that students with a high self-concept category can fulfill all aspects of creative thinking abilities, including fluency, flexibility and originality, so that their mathematical creative thinking abilities are also high. In contrast to students with moderate self-concept, it was found that their creative thinking abilities were low and students with low self-concept actually had moderate mathematical creative thinking abilities. So it can be concluded that based on this research, it shows that students' self-concept does not always describe students' mathematical creative thinking abilities in solving open-ended problems.

Risks in Identifying Gifted Students in Mathematics: Case Studies

Abstract The article discusses the risks involved in identifying gifted students in mathematics in the school environment. Although the theory of gifted students is widespread and available today, school practice shows many cases of gifted students who have not been identified in the school environment. The reasons for this are various. The article will present the theory related to this topic. The research was conducted as two case studies using qualitative research methods. Semi-structured interviews with parents and pupils were used, and the pupils were observed for a long time by the researcher. In the first case, it was a mathematically gifted pupil with dyslexia, and in the second case, it was a mathematically gifted pupil with an extremely high IQ. The result of the study was to find out in which moments the school did not provide support to the pupil in the two cases mentioned and what kind of help the pupil would appreciate at school. Both pupils were involved in a 1-year intervention in which the intention was to cultivate their mathematical thinking and expressions with the help of appropriately chosen tasks. The stated findings are important for practice and discussion about the training of future mathematics teachers, for which consistent professional training focusing, among other things, also on the education of gifted students is essential. Both case studies are part of long-term research dealing with the identification of gifted students.

Students’ Creative Thinking Ability in Solving Mathematical Problems

This research aims to determine the description of students' creative thinking abilities in solving mathematical problems in learning Trigonometry. The research design is descriptive qualitative. The research subjects were Class X high school students. Data collection instruments included Creative Thinking Ability TESTs and interviews. The data analysis technique used is time triangulation. This research explores students' creative thinking abilities in solving mathematical problems. The results show that the ability to think creatively Students' mathematics is described based on a review of the answers to the mathematical creative thinking ability test which consists of 4 aspects, namely fluency, flexibility, novelty and detail. Based on the average results of students' answers in solving questions regarding mathematical creative thinking abilities, there were 5 questions, which had different averages, although none of their creative thinking abilities reached the very high category. This shows that students are less able to solve problems with mathematical creative thinking skills. The ability to think creatively at the high strata is at high criteria with an average of 28.6. The highest score achieved by students was 32 and the lowest score was 23.

Preservice teachers' mathematical knowledge for teaching: Focus on lesson planning and reflection

AbstractMathematics teacher educators have suggested preservice mathematics teachers' (PMTs') practices provide evidence of their mathematical knowledge for teaching (MKT). PMTs engage in approximations of practice to rehearse teaching. To explore connections between MKT and PMTs’ practices, we operationalized six MKT subdomains in terms of approximations of practice. We used that operationalization to investigate, which subdomains were evidenced in PMTs’ lesson plans and their lesson plan reflections. Eleven secondary PMTs participated in our study. Using a qualitative content analysis method, we found Knowledge of Content and Teaching most evidenced, suggesting that the PMTs demonstrated MKT when they described mathematical tasks and associated instructional activities. Also, the PMTs demonstrated Knowledge of Content and Students when they described how their students would engage with proposed mathematical tasks. However, the PMTs struggled to plan for activities by building on students’ potential mathematical thinking and reasoning. Only limited instances of Horizon Content Knowledge suggest that the PMTs had limited opportunities to demonstrate how their advanced mathematical knowledge connected with their mathematics teaching in the approximations of practice provided. We propose alternative forms of approximations of practice to optimize PMTs’ opportunities to demonstrate and develop MKT.