Mathematical Ideas Research Articles

Developing ‘deep mathematical thinking’ in geometry with 3- and 4-year-olds: a collaborative study between early years teachers and university-based mathematicians

ABSTRACT Mathematics in early years settings is often restricted to learning to count and identifying simple shapes. This is partly due to the narrow scope of many early years curricula and insufficient teacher training for exploring deeper mathematical concepts. We note that geometry is an area particularly neglected. In an innovative year-long project, a group of university-based mathematicians and early years teachers collaborated on a child-led exploration of ‘patterns in nature.’ The early years teachers ran the project within the setting, meeting regularly with the mathematicians to discuss potential areas of interest, and to highlight the children’s mathematical thinking. We found that, with the appropriate environment and guidance, the children naturally displayed deep levels of geometrical thinking and found enjoyment and satisfaction in the exploration of mathematical ideas. We define what we mean by the term ‘deep geometrical thinking’ and demonstrate this by looking at three excerpts through the lens of the van Hiele levels of geometric thought, finding that the children are capable of exhibiting thought at level 3 (abstraction), more advanced than previously thought of children of this age. Using a second taxonomy we also assess the range of skills across which they are demonstrating such geometric thought.

Mathematics learning in the early years through nature play

ABSTRACT In Australia, there have been a growing number of nature-based early childhood education initiatives [Alme, H., and M. A. Reime. 2021. “Nature Kindergartens: A Space for Children’s Participation.” Journal of Outdoor and Environmental Education 24: 113–131]. Research into early years’ nature education internationally remains under-theorised particularly in STEM teaching and learning. The nature or bush kindergarten programme (often referred to as ‘bush kinders’ in the Australian context), is one such programme where nature play and education occur. This paper considers how children’s mathematical thinking develops through their time spent in bush kinder settings. Using an ethnographic lens, this longitudinal study is the first of its type in Australia with a focus on STEM in bush kinders. In this paper, eight ‘powerful mathematical ideas’ are used to analyse how educators can approach teaching mathematics with only what nature provides. The paper reports one main finding, that bush kinders provide opportunities for children through educator support to build mathematical understandings through nature play. Drawing on fieldwork data and the analysis of research literature, this paper contributes to the conceptualisation of mathematics in early childhood nature play and bush kinder programmes.

Manipulative Materials in Teaching Mathematics among Junior High School Teachers: A Literature Review

This review article visualizes the use of manipulative materials in teaching mathematics among junior high school teachers. The review correlated to the study in the following areas: The concept of manipulative materials, types of manipulative materials teachers use in teaching mathematics, How teachers obtain manipulative materials for teaching mathematics, teaching methods teachers use for teaching mathematics with the use of manipulative materials, teachers’ perceived benefits of using manipulative materials in teaching mathematics, and challenges of using manipulative materials in teaching mathematics among teachers. In order to develop every student’s mathematical proficiency, leaders and teachers must systematically integrate the use of concrete and virtual manipulative materials into classroom instruction at all grades. Manipulative materials not only allow students to construct their own cognitive models for abstract mathematical ideas and processes, but they also provide a common language with which to communicate these models to the teacher and other students.

Learning Mathematics in an After-school Mathematics Club

Learning mathematics is as much about developing relationships with mathematics as the development of mathematics concepts and skills. Mathematics clubs usually support learners to develop relationships with mathematics through engaging with challenging mathematics problems, exploring mathematical ideas, and communicating and sharing their ideas. This paper reports on research on such a mathematics club. Data sources included learners’ mathematics results, short written responses after club sessions and interviews with club learners. Data were analysed quantitatively and qualitatively. The study shows that club learners’ school results improved relative to non-club learners, that learners’ experiences of mathematics in club and class were very different from each other and that the club supported different relationships with mathematics than learners experienced in class. The study suggests that mathematics clubs can support learners’ positive relationships with mathematics.

A Gender-Related Analysis of a Robots' Math Class

Using robot programming activities for learning in the classroom is one way to drive interest and engagement in the STEM field among students, especially girls. And this is a field that is particularly characterized by an underrepresentation of women. Accordingly, many countries are increasingly integrating activities related to computer science concepts into their education systems. The EU also sets the goal of considering the connections between STEM disciplines in schools and having students gain experience with robots as well. The use of robots for teaching purposes creates opportunities for motivating and meaningful mathematics lessons that are linked to the fundamental concepts of computer science. Mathematics teaching in such a context offers possibilities for an experimental and problem-oriented approach to the content and a deep insight into mathematical concepts. Research in this area shows that the use of robots can promote understanding of mathematical concepts, change attitudes and motivation, and develop metacognitive and problem-solving skills. However, as for gender differences in this context, little is known to date. Addressing this gap, for this work, we investigated learners' performance, mathematical and computational ideas and experiences, problem-solving strategies, and help used in an ER (Educational Robotics) activity. In addition, the learners’ mathematical competence and computational thinking skills as well as possible correlations of these measures with the learners’ performance on an ER activity were examined. For these purposes, an ER activity on the topic of plane geometric figures was designed, which was carried out in a 6th grade (11-12 years) class (n=24) of an Austrian middle school in the city of Salzburg using the TI-Innovator Rover. The comparison of six female and six male student groups, each consisting of two students, made it possible to address the above research questions. For this purpose, a mixed-methods approach was chosen. Qualitative data, consisting of the audio recordings of the student groups' conversations during the ER activity, the constructions made on the posters, the student notes, and the saved programs, form the basis for thematic analysis. The quantitative data include the number of tasks solved during the ER activity by the student groups, the mathematics grade of the last school year by the students, and the results of a test on the students' computational thinking skills with the related self-assessments. Appropriate quantitative methods for analysis include the Wilcoxon rank-sum test (Mann-Whitney test), the Welch Two Sample t-test, and Kendall's tau and Pearson's correlation coefficient to test for differences and correlations. The main results indicate that groups with female students perform better while showing high engagement in the activity, exhibit a more systematic approach to problem-solving and at the same time use less intensive help from the teachers than their male counterparts in this class. The paper concludes by giving future directions for research and the limits of the present work.

Philosophical concepts of creative insubordination in ethnomodelling research

The application of modelling methods makes sense to researchers and educators when these professionals examine the mathematical patterns developed by members of distinct cultures. Currently, an important issue in mathematics education is its tendency towards valuing a local orientation in its research paradigm. Thus, a search for innovative methodologies such as ethnomodeling is necessary to record the historical forms of mathematical ideas, procedures, and practices developed in diverse cultural contexts. Yet, ethnomodeling is not an attempt to replace globalized school/academic mathematics, however, it is necessary to recognize the existence of local mathematical knowledge in the school curriculum. This context enabled the emergence of creative insubordination of ethnomodeling as it evoked a disturbance that caused a revision of the rules and regulations in the mathematical modelling process. This process triggered a debate about the nature of mathematics in relation to culture by proposing a dialogue between local and global approaches in dialogical ways. Thus, the main objective of this article is to discuss the philosophy of ethnomodeling as a creative insubordination of mathematics education and also as a process of glocalization.

Analysis of mathematical communication ability reviewing from learning style of students class VIII in the think-pair-share learning

This study aims to determine and describe the mathematical communication skills of class VIII students in the cooperative learning model on the think-pair-share type reviewed based on learning styles, namely visual, auditory, and kinesthetic. The type of research used is descriptive-qualitative research with six students of class VIII D SMPN 7 kota Serang selected by purposive sampling. Test, questionnaire, and interview techniques were used for collecting the data. The findings of this research showed that : (1) the subject with visual learning style capable to connecting real objects in the form of mathematical ideas, capable in presenting mathematical ideas in the form of mathematical symbols, capable of explaining ideas, everyday situations and mathematical relationships in writing with pictures, quite capable to understand and evaluate mathematical ideas in solving problem, and capable of communicating the conclusions of the answers that are in accordance with the problems given. (2) The subject with auditory learning style quite capable for the first indicator and capable for other indicators of communication skills . (3) The subject with kinesthetic learning style capable to connecting real objects in the form of mathematical ideas, and quite capable for other indicators of communication skills.

Analysis of Mathematical Communication Skills to Students Based on Problem-Solving Construction

This research aims to systematically analyze students' mathematical communication skills based on the given problem-solving construction from the given math problems. The students involved in this study are class IX SMPN 1 Mangarabombang. The type of research applied is qualitative research with a descriptive approach. The research instruments are Mathematical Ability Test, Mathematical Communication Skills Guidelines, and Interview Guidelines. In this study, triangulation techniques were used to check the validity of the data that had been obtained. Data analysis techniques in this study use model with the following stages: 1. Data Reduction, 2. Data Presentation, and 3. Making Conclusion. The results show that mathematical communication skills can be demonstrated by mathematical modeling, representation of ideas, development of the ability to use mathematical object devices, and connecting algebraic expression to mathematical ideas. To realize this, the researcher understood and mastered mathematical concepts to support solving the given problems. Understanding, in this case, is characterized by indicators not only knowing mathematical concepts but knowing at the time when they are used and how to apply them. Then mastery is to place the mathematical concept in the context of the problem.

Research on the Design of Community Museums Based on the Fuzzy Comprehensive Evaluation Method

The new museum movement of the twentieth century has driven the development of the form and function of museums around the world. Museums began to be actively open to the public, and some new concepts of museums, such as eco-museums and community museums, emerged. The aim of these museums is to build a cultural bridge between people and the city and to promote the harmonious development of society, economy, and culture. The visitors, as the service targets of the museum, will directly influence the popularity of the museum among the masses by their evaluation of the museum experience; however, at present, there is a clear gap between the design of many museums and the feedback of subsequent visitors’ experiences. Only by understanding visitors’ feelings and preferences can subsequent museum design be improved; this paper will focus on demonstrating the application of the mathematical idea of the fuzzy comprehensive evaluation method to community museums; it establishes a community museum quality evaluation system based on human-centered design principles from the perspective of urban community museums and constructs a fuzzy comprehensive evaluation model of a community museum experience. Finally, the design of a community museum in Nottingham, UK, is used as an example to make a comprehensive evaluation of its quality. According to the analysis, the fuzzy comprehensive evaluation method has practical value in scientifically evaluating the quality of urban community museums through data on the visitor experience.

Early childhood educators’ beliefs about mathematics education for children under three years of age

ABSTRACT Early childhood educators play a central role in fostering children’s mathematics learning; however, few studies have focused on educators’ beliefs about very young children’s capabilities in mathematics. Understanding these beliefs is critical for gaining insight into the mathematics education provided in birth to three settings. In light of this dearth of research, this article presents data from a study of Australian early childhood educators’ beliefs about mathematics education for children under three years of age. 506 educators responded to an online survey which generated both quantitative and qualitative data regarding educators’ beliefs. The findings indicate that this large sample of Australian educators seem to be positive and perceptive about mathematics. The educators demonstrate a good level of recognition of very early mathematical development, and the majority of respondents indicate a belief that children begin exploring a range of mathematical ideas before two years of age. The findings contribute to understandings of early childhood educators’ beliefs regarding mathematics education for children under three years of age and provide positive indications about the quality of mathematics education opportunities for very young children.

Characterizing whole class discussions about data and statistics with conversation profile analysis

Whole class discussions (WCDs) are an important pedagogical tool for mathematics classes but are challenging to characterize across large numbers of observations because of their dynamic and complex nature. In this paper, we report on an exploratory method to characterize WCDs in mathematics classes across large numbers of observations that we refer to as Conversation Profile Analysis (CPA). CPA uses Latent Class Modeling (LCM) with live observation data to generate profiles of WCDs in middle-grade mathematics classes. We report on our exploratory use of CPA to analyze observation data from 259 WCDs about data and statistics in middle school classes making use of an innovative approach to instruction called Data Modeling. We identified 4 profiles of WCDs and found that these profiles varied in likelihood across time and were associated with different ways students talked about key mathematical ideas. We also discuss broader implications of the CPA approach to studying WCDs in math classes.

Technology-Immune/Technology-Enabled Problem Solving as Agency of Design-Based Mathematics Education

The paper is written as a reflection on the author’s online teaching of a graduate elementary mathematics content and methods course through the asynchronous modality. The need for developing skills in posing problems that integrate non-technological and technological problem-solving strategies is emphasized. Such problems are considered agency of design-based mathematics education. The importance of revealing hidden mathematical ideas stemming from teacher candidates’ willingness to explore self-posed problems within the forum components of the course is discussed. Using these hidden ideas, the paper demonstrates connectivity among different mathematical concepts including historical connections. Teacher candidates’ reflections on the course have been included and reviewed.

Comparing Mathematics Early Years Education in Spain, Portugal and Slovenia

This work aims to examine how the learning of mathematics in early childhood is developed in different policies, particularly within the processes of formal education and care in early years institutions. A comparative analysis of early mathematics education policies across countries must consider cultural differences, teaching practice, structural differences and institutional framework conditions, as well as the initial training and professional knowledge of teachers and educators. Extracted from the official country regulations, the following pages include some of the main characteristics of the national systems of early childhood education and care (ECEC) in Portugal, Slovenia and Spain, as well as a comparison of the ECEC guidelines concerning mathematics education between these three countries. There is an international consensus on an approach to early mathematics education inspired by realistic mathematics education (RME), i.e., on the importance of working mathematically in context, as well as on the idea of doing so through play, developing the language to communicate mathematical ideas. However, we found that these three aspects are reflected very differently in the official regulations of the three countries: while in Spain the development is very detailed and emphasizes the holistic approach and the role of mathematics in exploring the environment, the Portuguese curriculum emphasizes the role of mathematics as a form of language. The Slovenian curriculum, at last, focuses on the concepts and procedures associated with each mathematical sense. Furthermore, there are structural features concerning the regulation and type of ECEC system that have an influence in the implementation of the curriculum.

On the Epistemological Relevance of Social Power and Justice in Mathematics

In this paper we argue that questions about which mathematical ideas mathematicians are exposed to and choose to pay attention to are epistemologically relevant and entangled with power dynamics and social justice concerns. There is a considerable body of literature that discusses the dissemination and uptake of ideas as social justice issues. We argue that these insights are also relevant for the epistemology of mathematics. We make this visible by a journalistic exploration of relevant cases and embed our insights into the larger question how mathematical ideas are taken up in mathematical practices. We argue that epistemologies of mathematics ought to account for questions of exposure to and choice of attention to mathematical ideas, and remark on the political relevance of such epistemologies.

Attending to others’ mathematical ideas: a semiotic alternative to logocentrism in bilingual classrooms

Educational research tends to overemphasize spoken language as the main resource when attending to others’ ideas. In this paper, we problematize hegemonic assumptions of such a logocentric approach and its narrow scope. As an alternative, we propose a semiotic ideology, which focuses on the interplay among multiple semiotic resources, both linguistic and nonverbal. We argue that a semiotic ideology broadens what attending to others’ mathematical ideas in whole class settings entails. Recognizing bilingual children’s heightened sensitivity to nonverbal semiotic resources, we center bilingual classrooms as privileged sites to learn about a semiotic ideology. We draw on data from a third-grade Spanish-immersion classroom to illustrate considerations of temporality, materiality, and participant roles. We call for a continued search for intentional and creative ways to attend to the ideas of those with whom we do research.

DESKRIPSI KEMAMPUAN KOMUNIKASI MATEMATIS SISWA DALAM PENYELESAIAN PERMASALAHAN PROGRAM LINIER

This study aims to describe the mathematical communication skills of high school students in class XI on linear program material. The research method used is descriptive research. The subjects in this study consisted of 5 students of class XI one of the public high schools in Karawang Regency which were determined using purposive sampling technique. The research data collection technique was obtained from a test instrument in the form of 3 validated description questions which contained 3 indicators of mathematical communication ability. The results of this study indicate that the mathematical communication ability of class XI high school students in Karawang Regency in understanding linear programming material is still very low. This can be seen from the average percentage of the three questions of mathematical communication skills is 48.3℅. In indicator 1, students have not been able to express mathematical ideas through writing, demonstrate and visually describe them. For indicator questions 2 students are able to understand, interpret, and evaluate mathematical ideas both in writing and in other visual forms, but there are still some errors. Meanwhile, on indicator questions 3 students have not been able to use terms, notations, mathematical symbols and their structures to present ideas, describe relationships with various situation models.

LEARNING FOR TRANSFORMATION: CURRICULAR INTERVENTIONS IN MATHEMATICS AND ENGLISH LANGUAGE FOR ENSURING VALUES THROUGH EDUCATION

Transformative learning is just aligning our self with co existential system. While we are struggling in this alignment we feel disharmony which finally resulted in discomfort and a sense of disrespect to self and others. Education is provided to sustain these alignments with the Co existential system. Life is not only to be alive but to live too. Education is to develop a sense of understanding about self and others. Education is to understand the relationship of one with self, family, society, state and then with nations. Understanding of the orders in systems of co existential reality is sole aim of all educational systems. Education must provide answers of questions such as why and how aspects of education. Education must prepare child to have skills of co existential living. Prepared to live happily and harmoniously with all around that exist is the golden objective of providing education. All schools curriculums and subjects must be visualized in true sense of co existential reality. This research article visualized the scope of curricular intervention in mathematics and English for ensuring the goal of quality education which could be able to produce human beings and not just machines. Mathematics itself is of the nature of patterns seeking, rational thinking, well organisation and independent thinking which capable children to take their own decisions for living happily. The concepts of number sense, understanding the concept of zero, abstractions of algebra and inceptions of geometrical shapes and ideas had infinite scope of providing life skills education through mathematics ideas. Whereas English being a common communicative language with four basic skills i.e. listening, speaking, reading and writing can be made more sensible, meaningful and supportive in imparting values to develop responsible and understanding human beings. The skills of decisions making with co existential understanding, living in harmony with physical and living entities can be developed with mathematical ideas and with supportive help of language (English) very naturally and effectively. The ultimate aim of living a worthy life can be achieved through these curricular interventions across all subjects of school and universities education. May the transformative power of education leads to a world of understanding and happiness!

MATHEMATICAL REPRESENTATION ABILITY OF MATHEMATICS EDUCATION STUDY PROGRAM STUDENTS

This research is descriptive because it aims to describe students' mathematical representation ability. Mathematical representation is related to ways of expressing mathematical ideas. Mathematical representation is divided into verbal, visual/pictorial, and symbolic representations. The subjects of this study were 40 new Mathematics Education study program students. Data were collected using test and interview techniques.
 Furthermore, the data were analyzed using descriptive and qualitative statistics. The results showed that only 10% of the students had the mathematical representation ability in the good category, 12.5% in the moderate category, 10% in the poor category, and 67,5% in the very poor category. In detail, for verbal skills, there is 2.5% in the very good category, 7.5% in the good category, 17.5% in the moderate category, 15% in the poor category, and 55% in the very poor category. For the ability of visual/pictorial representation, there is 2.5% in the very good category, 7.5% in the good category, 12.5% in the moderate category, 7.5% in the poor category, and 70% in the very poor category. Furthermore, for the symbolic representation ability, 10% is in a good category, 10% in the moderate category, 7.5% in the poor category, and 72.5% in the very poor category. Based on the classroom observation and interview, the low mathematical representation ability is caused by two main things, namely (1) relatively low prerequisite knowledge and (2) mathematics learning, which is dominated mainly by teachers; learning does not provide opportunities for students to develop mathematical representation skills

Students’ Worksheets Based on Problem Based Learning In Composition and Inverse Functions to Enhance Conceptual Understanding

This research aims to design student worksheets based on problem-based learning in composition and inverse functions to facilitate conceptual understanding is valid. This research uses a PPE model by Richey and Klein; there are three parts: planning, production, and evaluation. The planning stage is a product planning process that will be developed for a specific purpose. The production stage is the process of making a product according to a previously made design. The evaluation stage is the process of testing and assessing the product that has been developed. The data obtained from the expert validator is then processed using a Likert scale and calculated using Aiken's formula to conclude the validity of the product developed. The Student Worksheets based on Problem Based Learning to aid students' understanding of mathematical ideas on composition functions and inverse functions were valid, have a 0.84 average validity value, and were classified as high. The PBL model familiarizes students with understanding concepts through problem-solving processes. On the other hand, learning mathematics using PBL effectively improves students' understanding and abilities because students apply mathematical concepts in everyday life.

High school students’ mathematical representation ability: Evaluation of disposition based on mastery learning assessment model (MLAM)

Mathematical representation is the ability of students to explore and use mathematical ideas explicitly. This study aimed to learn about the effectiveness of the Mastery Learning Assessment Model (MLAM) in mathematics learning at the Assyfa Learning Center Foundation Pasuruan, which then described students' abilities in solving mathematical representation problems evaluated from mathematical dispositions. MLAM is based on repeating a similar assessment through an overhaul of mastery learning from previous researchers. This study used a descriptive qualitative method. In the study presented here, 20 subjects were pooled and surveyed using a continuous assessment tool based on high, low, and moderate math ability. The results showed that almost all students could use representational images to solve mathematical problems, create mathematical models, and solve problems related to mathematical expressions. On the other hand, most students could not complete the two indicators of mathematical representation, namely making sketches and mathematical models based on the given situation and data. This result is indicated by the 75% of students who completed this lesson. Based on the Teaching Evaluation (TER) results, most students are also satisfied with this approach. It is expected that the MLAM assessment can improve students' mathematical representation skills.