Mathematical Thinking Research Articles

What are explanatory proofs in mathematics and how can they contribute to teaching and learning?

This paper will examine several simple examples (drawn from the mathematics literature) where there are multiple proofs of the same theorem, but only some of these proofs are widely regarded by mathematicians as explanatory. These examples will motivate an account of explanatory proofs in mathematics. Along the way, the paper will discuss why deus ex machina proofs are not explanatory, what a mathematical coincidence is, and how a theorem's proper setting reflects the naturalness of various mathematical kinds. The paper will also investigate how context influences which features of a theorem are salient and consequently which proofs are explanatory. The paper will discuss several ways in which explanatory proofs can contribute to teaching and learning, including how shifts in context (and hence in a proof’s explanatory power) can be exploited in a classroom setting, leading students to dig more deeply into why some theorem holds. More generally, the paper will examine how “Why?” questions operate in mathematical thinking, teaching, and learning.

A decade of PISA: student perceived instructional quality and mathematics achievement across European countries

This study examines the relationship between instructional quality and mathematics achievement, taking into account socioeconomic status. Data from ten European countries participating in PISA 2012 and 2022 were analysed using the framework of the Dynamic Modelling of Educational Effectiveness. In PISA 2012, instructional quality was defined by student-oriented instruction, classroom management, teacher support, disciplinary climate, and cognitive activation. The finding indicated that teacher support, disciplinary climate, and cognitive activation positively predict student achievement, but dimensions of student-oriented instruction and classroom management were negatively related in several countries. Instructional quality was captured by four dimensions in PISA 2022, namely, teacher support, disciplinary climate, cognitive activation in mathematics of encouraging mathematical thinking and of fostering reasoning. Teacher support and disciplinary climate had a positive effect on achievement, but two aspects of cognitive activation had opposite effects on achievement. SES also had significant indirect effects on achievement through various instructional quality dimensions. Across two PISA cycles, teacher support, disciplinary climate, and cognitive activation consistently benefited low-SES students, while classroom management and student-oriented instruction tended to harm their performance. These results underscore the role of instructional quality in promoting equity for disadvantaged students. Trend data from these two cycles of PISA, give an indication of specific instructional strategies that do consistently contribute to student achievement. Implications and findings are discussed in relation to the policy and practice context in different education systems.

Mastery of the Concept of Percentage and Its Representations in Finnish Comprehensive School Grades 7–9

This research was conducted in 2019 in collaboration with Japanese colleagues, with the research tasks translated from the original Japanese. In Finland, a total of 1112 students from grades 7–9 in primary school participated in the study. In our article, we examine Finnish students’ mastery of the concept of percentages and how they express their mathematical thinking about percentages in a multimodal way. A multimodal expression of mathematical thinking can typically take the form of natural language, written mathematics, pictorial representations, mathematical symbolic language, or combinations of these, which can be used to manage information, such as through tables. Our focus is on Finnish primary-school students’ mastery of the concept of percentages, and we then narrow our analysis to tasks where students have used a multimodal approach. We analyze the representations of the concept of percentages and the related thought processes. The concept of percentages was best mastered in grade 9 and weakest in grade 7. Students mostly used a combination of pictorial language, natural language, and mathematical symbolic language to describe their solution processes. However, describing their thinking in different ways did not necessarily highlight possible errors in their reasoning.

An examination of the cognitive demand levels posed by problems in the middle grades mathematics textbooks

This research investigated the cognitive demand levels of mathematics tasks within the middle school mathematics textbooks endorsed by the Ministry of National Education for the academic year of the 2023-2024. To achieve this, a qualitative research method was utilized, and document analysis was performed. In this direction, the textbooks were analyzed using descriptive analysis. The tasks within the textbooks were examined through the lens of Smith and Stein’s (1998) theoretical framework. The findings indicate that the majority of mathematics tasks in middle school mathematics textbooks exhibit a low level of cognitive demand. It was found that over 80% of the tasks fell into the categories of memorization and procedures without connections levels. It has been revealed that tasks at the levels of procedures with connections and doing mathematics, which are high cognitive demand levels, are uncommon. This shows that textbooks do not sufficiently support the potential of students to develop mathematical thinking and problem-solving skills. Based on these results, various suggestions have been made for textbooks. The most important of these suggestions include ensuring that students are cognitively exposed to higher level tasks and increasing the number of tasks that will improve their mathematical thinking skills. Thus, it will be possible for students to have a more effective learning process by deepening their mathematical understanding.

Management pedagogy:

It's a study with a qualitative approach through action research. With the aim of teaching addition using the Mariela-Lehren-addieren method to students in 3rd and 4th grade. The Method of teaching addition is from the teaching of the basic skills of mathematical thinking, design and evaluation of the teaching strategy, the concept of number positional value, sign, concept of grouping, regrouping of numbers, the algorithm of addition, the practice of it with numerical exercises and reasoned problems. It is concluded that 18 students out of 21, achieved the learning of addition up to 4 digits with the Mariela-Lehren-addieren method through action research. It's inferred that the problems that arise in the learning of the addition, can be rooted by lack of systematic approach in previous educational cycles of the student, absences, learning problems, teaching models, motivation and school synergy.

Entangled Mathematics as a Tool of Reasoning in the Mid-Twentieth-Century UK Electricity Industry

In the mid-twentieth century, the identity of those who oversaw the UK electricity grid tentatively and slowly began to shift from those who joined the electricity industry directly from secondary school to a university-educated elite with a higher level of technical education. At the same time, electricity infrastructure became increasingly centralised, leading to the creation of a national grid in 1938, meaning that control of electricity became concentrated in the hands of an ever-smaller group and increasing the stakes in debates over strategy. By using the case studies of power engineer Paul Schiller and mathematician Maurice Davies, this paper traces how the boundaries of what mathematics could or could not be used for were renegotiated between factions within the UK electricity industry. Schiller unsuccessfully attempted to use mathematics, including the authority of academic mathematicians, to reinforce controversial arguments about the strategic direction of the industry in the 1940s. Such arguments failed to resonate with a class of, as Schiller put it, “practical men” that still held considerable influence. Davies on the other hand was employed by the industry in the 1950s to improve efficiency, and was able to implement lasting measures in this effort by using the weather to forecast electricity demand, and successfully defended the usefulness of his work in the 1970s against colleagues who felt that forecasting could be done better with mathematical models without weather inputs. The contrast between these two stories shows how the influence of mathematical thinking was contingent upon existing power dynamics, and reinforce how the history of mathematics should be socially embedded.

Impact of Differentiated Learning Management on Enhancing Students' Creative Thinking in Mathematics

In the era of globalization, innovation in education is essential, particularly in fostering students' creative thinking abilities. Mathematics, as a core subject, requires teaching strategies that stimulate creativity. This study explores the influence of differentiated learning management in mathematics on students' creative thinking at MIN 1 Sidoarjo. A quasi-experimental design with a quantitative approach was used. The research population included all students at MIN 1 Sidoarjo, with the sample drawn from class VI students. Data were collected using pre-tests, post-tests, and questionnaires, and analyzed using t-tests to assess the impact of the differentiated learning approach on creative thinking. The findings reveal that differentiated learning management significantly enhances students' creative thinking in mathematics. The experimental group, which received differentiated instruction, showed greater improvements in creative thinking compared to the control group, which followed a traditional teaching approach. The results highlight the positive influence of differentiated learning strategies in promoting creativity within mathematics education. The experimental group's superior performance underscores the need for more innovative teaching practices to meet the demands of modern education. Differentiated learning management in mathematics has a significant positive effect on students' creative thinking. These findings offer important insights for educators seeking to enhance creativity through tailored instructional methods.

EKSPLORASI HUBUNGAN ANTARA LITERASI NUMERASI DAN KEMAMPUAN BERPIKIR KREATIF MATEMATIS DI MADRASAH TSANAWIYAH

Numeracy literacy and mathematical creative thinking are essential skills for students to tackle learning challenges and solve problems effectively. However, there are gaps in their understanding of these skills. This study aims to examine the relationship between numeracy literacy and mathematical creative thinking among students. A simple regression and correlation analysis was used as the research method. The sample consisted of 33 students randomly selected from a population of 113. Data were gathered through tests on numeracy literacy and mathematical creative thinking skills after a trial phase. Findings reveal a significant relationship between numeracy literacy and mathematical creative thinking, with a correlation coefficient of 0.857. The R-value of 0.857 indicates that 73.50% of the variation in mathematical creativity skills is explained by numeracy literacy. The study identifies four patterns: students with high literacy and creativity excel in analysis, while those with mixed levels show varying strengths and weaknesses in data interpretation and innovation.

ANALISIS KEMAMPUAN BERPIKIR KREATIF DITINJAU DARI ADVERSITY QUOTIENT PADA PEMBELAJARAN CREATIVE PROBLEM SOLVING PENDEKATAN OPEN-ENDED

This research aims to determine the quality of creative mathematical thinking abilities in terms of AQ in CPS learning with an open-ended approach assisted by G. Classroom. The type of research is quantitative research. The research population was all class VII. The research sample used a cluster random sampling technique. The conclusions of the results of the analysis were: (1) the mathematical creative thinking ability of participants who received the CPS learning model with an open-ended approach assisted by Google Classroom achieved BTA (2) Students with the ability to think creatively mathematically in terms of AQ in CPS learning with an open-ended approach assisted with Google classroom achieves learning completeness of more than 75%, (3) the mathematical creative thinking ability in the CPS learning model with the open-ended approach assisted by G. Classroom is better than the PBL learning model, (4) the proportion of mathematical creative thinking ability completeness in the CPS learning model approach open-ended with the help of Google Classroom is more than the proportion of students in the PBL learning model, (5) there is a difference in the average increase in mathematical creative thinking abilities of students in the CPS learning model with the open-ended approach assisted with G. Classroom and students in the PBL learning model, (6) There is a positive influence on the adversity quotient on the ability to think creatively mathematically in mathematics learning

Possibilities of using chatbots in the teaching of grammar in teacher training

Education and the process of teaching and learning is undergoing a major transformation today. 21st century skills increasingly emphasize the importance of collaboration, self-development and lifelong learning. In addition, new information and communication technologies are emerging and the potential for the use of artificial intelligence in education is growing. In teacher education, the teaching of semantics and grammar (vocabulary, morphology, syntax) in the Hungarian Language II course is combined with the development of algorithmic thinking in mathematics. This provides space for integration between the subjects and the theory of multiple intelligences. By rethinking traditional teaching, we gave the students the task of playing the role of a chatbot. This paper summarizes the description and results of this experimental research, as well as the complex support for students based on multiple intelligences.

Analysis of Mathematical Critical Thinking Ability based on Students' Ability Level

Analyzing and testing mathematical critical thinking skills based on the ability level of class VIII students of junior high school is the aim of this research. By understanding the level of students' abilities, educators can determine students' mathematical critical thinking abilities and develop appropriate teaching strategies to improve students' abilities. This study uses a qualitative descriptive research method. This research was conducted at one of the junior high schools in Cipatat, West Bandung Regency with heterogeneous abilities of 8 students. This research uses data collection techniques by providing mathematical critical thinking ability questions on SPLDV material which consists of 5 essay test questions based on the following indicators: (1) Checking the completeness of problem solving steps. (2) Look for alternative solutions to problems. (3) Identify spldv equations with certain characteristics. (4) Choose the best way to solve the problem from alternative ways of solving it. (5) Identifying the adequacy of data on a problem. In this study, the results of the analysis have shown that students at a high level of ability can check the completeness of problem solving steps, students with a medium level of ability have little difficulty in identifying spldv equations with certain characteristics, and students with a low level of ability have difficulty finding alternatives problem solution.

Analysis of Mathematical Creative Thinking Ability in View of Students' Confidence in Number Pattern Material

In the 21st century, learning mathematics requires students to master 4 skills, one of which is creative mathematical thinking. This ability will encourage students to have different points of view in solving a problem. The low ability to think creatively in mathematics is shown by the large number of students who are unable to express creative ideas when studying mathematics. One of the things that underlies creativity is self-confidence because there is a relationship between the two. The aim of this research is to describe mathematical creative thinking abilities in terms of self-confidence in number pattern material. This research is descriptive qualitative research. The subjects in this research were 6 students with high, medium and low self-confidence categories in each class VIII A and VIII B. The data collection techniques used were questionnaires, tests and interviews and analyzed using triangulation of sources and techniques. Apart from that, there are 3 stages of data analysis techniques, namely data reduction, data presentation, and drawing conclusions. The research results show that students with high self-confidence have a level of creative thinking ability (TKBK) 4 (very creative) by mastering the indicators of fluency, flexibility and novelty. Then students with moderate self-confidence have a creative thinking ability level (TKBK) 3 (creative) by mastering fluency and flexibility indicators. Finally, students with low self-confidence have a creative thinking ability level (TKBK) 3 (creative) by mastering indicators of fluency and flexibility.

Investigating the use of ChatGPT to solve a GeoGebra based mathematics+computational thinking task in a geometry topic

ChatGPT is a chatbot with potential educational benefits, particularly in enhancing computational thinking (CT) proficiencies such as programming, debugging, and algorithmic thinking for students. Despite its promise, there is limited research on how ChatGPT can specifically support the integration of CT into mathematics education using tools like GeoGebra. The researchers implemented plugged-computational thinking in mathematics (Math+CT) lessons by means of the utilization of GeoGebra, an application that requires students to input commands in order to generate mathematical objects. The present investigation employed an educational design research (EDR) methodology in which the researchers incorporate ChatGPT into our Math+CT lessons to assist students in accomplishing the task. We purposely selected the participants who are mainly postgraduate students and collected data from the participants’ conversation with ChatGPT and recorded their screens while interacting with ChatGPT and our Math+CT task. We analyzed the data through descriptive qualitative method on the participants’ prompts, the final codes and the number of iterations. The researchers examined how ChatGPT could be utilized to assist the participants in writing GeoGebra commands in terms of its benefits and limitations. ChatGPT assisted most participants in completing the task successfully, with only a basic need for proficiency in GeoGebra commands, mathematics, and critical thinking. However, it revealed that participants did not yet utilize an affective prompt to ChatGPT. Furthermore, ChatGPT has the potential to be utilized for differentiated instruction due to the fact that its responses to individual users vary significantly based on the input prompts. Limited understanding of basic GeoGebra commands, and mathematical concepts could hinder the participants from training ChatGPT or prevent them from arguing with ChatGPT. This study enhances the existing literature by illustrating that ChatGPT can facilitate critical CT aspects, including programming and debugging, in a mathematics education context. This suggests that AI tools such as ChatGPT can contribute to the development of independent problem-solving skills, provide tailored support based on the needs of individual students, and enhance personalized learning experiences. Additional research involving students in school is required in order to gain a deeper understanding of the integration of ChatGPT into Math+CT lessons.

Impacts of a Summer Transition Program on Kindergarten Readiness for Children from Disadvantaged Backgrounds

ABSTRACT Kindergarten transition programs support successful school transitions for families and may reduce disparities in academic readiness upon school entry. Yet, access and participation in high-quality summer transition programs are lower among children from disadvantaged backgrounds. This study examined the impact of a summer kindergarten transition program on children’s readiness in language and literacy, mathematical thinking, and social foundations and the strength of their home-school relationships. Assessment scores of 416 kindergartners within 30 classrooms in 28 elementary schools in one southeastern state were analyzed. Children who participated demonstrated significantly more readiness in their language and literacy, mathematical thinking, and social foundation skills compared with children who were eligible but did not participate in the program. Further, children who participated in the program had significantly stronger home-school relationships compared to children who were eligible but did not participate and those who were not eligible to participate in the program. The findings provide preliminary evidence that summer transition programs are an equitable strategy to reduce achievement gaps upon kindergarten entry. School administrators should consider implementing summer transition programs to support academic readiness and develop strong home-school relationships in communities with a high population of children who are at risk for later school difficulties.

Competencia docente del profesorado de educación primaria en el área de medidas en un contexto de reforma curricular

[Objective] The aim is to explore how in-service teachers professionally view curricular materials and students' mathematical thinking in the area of measurement. [Methodology] The participants were 18 in-service primary school teachers, who completed a 12-item questionnaire with three focuses: (i) interpret students' mathematical thinking, (ii) propose activities to support students' understanding, and (iii) analyze activities related to curriculum references regarding magnitudes and their measurements. The analysis was conducted in two stages. In the first stage, the characteristics of the responses by in-service teachers to each of the items were identified, and in the second stage those characteristics were grouped according to the three focuses in order to identify any peculiarities on how teachers interpreted students' mathematical thinking and proposed and analyzed activities using the curriculum. [Results] Results indicate that teachers have difficulties in identifying errors because they do not consider the real context of the problem. Furthermore, although they were able to propose different activities, they had difficulties when asked to place the activity into an everyday context. The curricular difference between mathematical process and ability proved to be difficult for teachers in the area of measurement. [Conclusions] Teachers’ limitations show the need to develop continuing education proposals for in-service primary teachers that foster the development of teaching competences.

유아의 놀이 상호작용이 언어 및 문해력과 수리적 사고에 미치는 영향 -사회적 유능감의 매개효과-

The purpose of this study is to examine the mediating effect of social competence in the structural relationship between child's play interactions and social competence, language and literacy, and mathematical thinking. For this purpose, data from the 7th year of the Korean Child Panel(2014) provided by the Korea Institute for Childcare Policy(KICE) were used, and a total of 667 young children and their classroom teachers were analyzed. Descriptive statistics, correlation analysis, confirmatory factor analysis, exploratory factor analysis, structural model analysis, and path analysis were conducted using SPSS 23.0 and AMOS 22.0. The results of this study are as follows. First, children's play interactions had an effect on language and literacy, but not on mathematical thinking. Second, children's play interactions had a statistically positive effect on social competence. Third, children's social competence had a statistically positive effect on language, literacy, and mathematical thinking. Fourth, social competence was found to partially mediate the relationship between children's play interactions and language and literacy skills, and social competence was found to fully mediate the relationship between children's play interactions and mathematical thinking.

Analysis of students' mathematical communication skills in terms of learning styles

The purpose of this research is to obtain a description of the mathematical communication skills in terms of the learning styles of Grade XI TKR 3 students at SMK Negeri 1 Tanara for the 2023-2024 academic year. This research is qualitative in nature. Six subjects were selected, representing each learning style, from Grade XI TKR 3 students at SMK Negeri 1 Tanara, Tanara District, Serang Regency. The data collection techniques used were observation, interviews, documentation, and tests based on the indicators of mathematical communication skills, namely: (1) the ability to express mathematical symbols through everyday events, (2) the ability to determine the solution set by relating real objects to mathematical ideas, (3) the ability to explain mathematical problems related to elimination, (4) the ability to analyze mathematical problems related to substitution, and (5) the ability to evaluate mathematical thinking and strategies used in problems according to the observed object. The results showed that (1) subjects with a visual learning style have good mathematical concept comprehension on indicators 1, 2, 3, 4, and 5; (2) subjects with an auditory learning style have good mathematical communication on indicators 1, 2, and 5, but performed poorly on indicators 3 and 4; and (3) subjects with a kinesthetic learning style have good mathematical communication skills on indicators 1 and 2, but performed poorly on indicators 3, 4, and 5.

Who Puts the ‘Active’ into ‘Active Learning’?

AbstractLearning is here considered to have taken place when someone has developed the habit, propensity, and disposition to attend productively to things not previously noticed, and in ways not previously experienced, to do with some specific and particular content. ‘Active learning’ sounds like a tautology, but was introduced as a contrast to the more passive activity of students sitting in lectures listening and transcribing mathematics written on a board or screen onto their own paper. The stance taken here is that effective and efficient learning involves active engagement in activity, but includes enculturation through being in the presence of a relative expert1 who themselves is manifesting mathematical thinking, not simply passing on the records of the results of previous mathematical thought. Such ‘passivity’ does not necessarily require intention. Following Bennett2 actions are here taken to involve three agents or impulses: initiating, responding, and reconciling or mediating. All three agents are thus active, but in different ways. Interactions intended to contribute to learning are considered to be actions, and so involve three agents: learner, teacher (in some manifestation), and mathematical content, all within a culture or ethos. Since there are six different ways in which the triple of agents can be assigned to the triple of impulses, six different modes are possible. Analysing these modes sheds light on different ways in which learning could be said to be ‘active’. Activity takes place within a mode of interaction. Again following Bennett, effective activity is here taken to require appropriate relationships among the gap between current state and intended goal, the resources available, and the tasks set.1Vygotsky (1978) pointed out that ‘higher psychological processes’ are first encountered in others.2Bennett (1993); see also Shantock Systematics Group (1975)

Mathematical Computational Thinking Skills In The Context of Critical Thinking Based On Gender

Computational thinking is related to critical thinking skills. The purpose of this research is to find out how students' ability in mathematical computational thinking in the context of critical thinking and the differences in the abilities of women and men. This research is a qualitative research. Data collection techniques used written tests, interviews, and documentation. The subjects of this study were selected through purposive sampling technique and totaled 34 students. After that they did a written test with 7 questions on the material of the system of linear equations of two variables and obtained 6 students with the category of 2 high ability students, 2 medium ability students, and 2 low ability students. The data were analyzed by using Triangulation technique. The results showed that female students are superior in solving problems according to the indicators of computational thinking because they pay more attention to accuracy in the work. So it must often work on problems to train students' abilities so that all students can do it better and become balanced for each student's ability. So it is hoped that students will get used to thinking in a stepped, systematic and logical manner.

Analysing the Mathematical Critical and Creative Thinking Abilities in Problem Solving Introverted Students

This study aims to evaluate the critical and creative thinking abilities in the context of mathematics when solving problems related to arithmetic sequences and series among introverted students in the 11th grade at Xaverius 1 High School in Jambi City. This issue arises due to differences in thinking skills and cognitive processes among students based on personality types, where it was found that introverted students tend to have low mathematical critical and creative thinking abilities. The research method employed is a descriptive qualitative approach, collecting data through personality questionnaires, tests, and interviews. Research subjects were selected from introverted students based on personality questionnaire results and answer sheet analysis, with three students representing the various response groups. The results of this study indicate that introverted students demonstrate good mathematical critical thinking skills in understanding problems, planning, and implementing plans. However, they tend to overlook rechecking their solutions. The creative mathematical thinking abilities of introverted students mainly involve fluency and flexibility in understanding problems, planning, and implementing plans, although they lack in terms of originality and elaboration in problem-solving