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In this paper, we offer a comparative review of research on understanding and promoting students' mathematical thinking. The sources for the review are papers that were published in Educational Studies in Mathematics (ESM) during two windows of time: 1994–1998 and 2014–2018. Selection of these two time periods enables us to comment on the "state of the art" in research as well as identify changes over the past 25 years. The review is guided by an analysis of conceptualizations of "mathematical thinking" proposed in the research literature, selected curriculum documents, and international assessment programs such as the OECD's Programme for International Student Assessment (PISA). The review not only documents salient features of research studies, such as the country of origin of the authors, educational level of the participants, research aims, theoretical perspectives, and methodological approaches, but also identifies the contribution to knowledge made by this body of work as well as future research directions and opportunities. [ABSTRACT FROM AUTHOR]

Tablet computers (tablets) are positioned to be powerful, innovative, effective, and motivating research and assessment tools. We addressed two questions critical for evaluating the appropriateness of using tablets to study number-line estimation, a skill associated with math achievement and argued to be central to numerical cognition. First, is performance with paper and pencil comparable with performance on a tablet? Second, is comparability affected by students’ preference for one method of presentation? Thirty-two students in Grade 6 estimated targets on a number line; half estimated with paper and pencil and half with a tablet. For both presentation methods, students’ performance was comparable. Students liked both presentation conditions equally but, when asked to choose, most students preferred the tablet. Preference did not influence comparability of results across presentation methods. Finally, students’ reasons for their preferences were explored, along with implications for using tablet applications in research and educational assessment. [ABSTRACT FROM AUTHOR]

Computational thinking, defined as a way of thinking that can be applied to various fields that require problem-solving skills, has become prevalent in education. Students, i.e., future specialists, have to be prepared for complex thinking competence, necessary for solving business and societal problems, for which a combination of mathematical thinking and computational thinking is essential. The preliminary premise is that there is a correlation between ability in specific mathematical and computational fields. Therefore, this paper aims to highlight the significance of investigating the relations between those fields from linguistic point of view. In order to better understand the relationship between abilities in specific mathematical and computational fields, this paper presents an analysis of a new approach, namely, developing hypotheses for exploring the relationship between metalanguages of different fields of Mathematics and Computer Science. Additionally, the paper describes the first stage of a study on a doctoral level in an attempt to suggest possible statistical analyses suitable for testing hypotheses based on meta-analysis of the current literature. [ABSTRACT FROM AUTHOR]

Assessment is a crucial factor in the implementation of curriculum reform. Little is known, however, on how curriculum changes can be reflected adequately in assessment, particularly if the reform concerns process skills. This issue was investigated for the case of assessing mathematical thinking in a mathematics curriculum reform for 15–18-year-old students in the Netherlands. From 2011 until 2017, these reform curricula were field tested in pilot schools, while other schools used the regular curricula. The research question is how this reform was reflected in national examination papers and in student performance on corresponding assignments. To address this question, we developed a theory-based model for mathematical thinking, analyzed pilot and regular examination papers, and carried out a quantitative and qualitative analysis of students' work on assignments that invite mathematical thinking. The results were that the pilot examination papers did address mathematical thinking to a greater degree than the regular papers, but that there was a decrease over time. Pilot school students outperformed their peers in regular schools on assignments that invite mathematical thinking by 4–5% on average and showed more diversity in problem-solving strategies. To explain the decreasing presence of mathematical thinking in examination papers, we conjecture that conservative forces within the assessment construction process may push back change. [ABSTRACT FROM AUTHOR]

International curriculum and policy directions have called to embed critical thinking across discipline areas including mathematics; however, conceptually, this is under-theorised and under-researched in the field of mathematics education. This paper presents the conceptualisation of critical mathematical thinking (CMT) and the application of a literature informed conceptual framework; in particular, it examines what CMT capabilities young students exhibit as they enter formal schooling. We present the findings from one-on-one task-based interviews, undertaken with 16 young students (aged 5–6) as a means to investigate their CMT capabilities and refine the CMT framework. The interview data were analysed using the new critical mathematical thinking conceptual framework. The data confirms the definition and understanding of CMT in young students, indicating a need for curriculum refinement, improved teaching practices, and further research in this area. [ABSTRACT FROM AUTHOR]

Recently, computational thinking (CT) has attracted much research attention, especially within primary and secondary education settings. However, incorporating in mathematics or other disciplines is not a straightforward process and introduces many challenges concerning the way disciplines are organised and taught in school. The aim of this paper is to identify what characterises CT in mathematics education and which CT aspects can be addressed within mathematics education. First, we present a systematic literature review that identifies characteristics of CT that have been explored in mathematics education research. Second, we present the results of a Delphi study conducted to capture the collective opinion of 25 mathematics and computer science experts regarding the opportunities for addressing CT in mathematics education. The results of the Delphi study, which corroborate the findings of the literature review, highlight three important aspects of CT to be addressed in mathematics education: problem solving, cognitive processes, and transposition. [ABSTRACT FROM AUTHOR]

The paper discusses a certain type of competitions based on distance interaction of a participant with simulation models of concepts from discrete mathematics and computer science. One of them is the "Construct, Test, Explore" (CTE) competition, developed by the authors, the other is the Olympiad in Discrete Mathematics and Theoretical Informatics (DM&TI). The tasks presented in this paper are generally devoted to the concept of a graph isomorphism. Most of the tasks are verified automatically. [ABSTRACT FROM AUTHOR]

A critical part of supporting the development of students' algorithmic thinking is understanding the challenges that emerge when students engage with algorithmatizing tasks--tasks that require the creation of an algorithm. Knowledge of these challenges can serve as a basis upon which educators can build effective strategies for enhancing students' algorithmic thinking skills. This paper presents three illustrative cases of emergent challenges evident as students grapple with the process of creating an algorithm. The first challenge highlights discrepancies between the method with which students solve a problem and the algorithm they create, and claim would, when implemented, solve the same problem. The second challenge pertains to the persistence of students' normatively incorrect algorithms, despite going through multiple iterations of testing and revising. Finally, the third challenge concerns issues around the use of test problems for supporting students in their creation of generalized algorithms. These three challenges are discussed using student data (illustrative cases) from three different mathematical algorithmatizing tasks. Suggestions are put forth for addressing some of these challenges, with a particular emphasis on practical pedagogical suggestions for cultivating students' mathematical thinking in the context of algorithmatizing tasks. [ABSTRACT FROM AUTHOR]

As we move further into the digital age, the acquisition of digital literacy (DL) and computational thinking (CT) skills is emerging internationally as an essential goal for students in contemporary school curricula. As the world becomes more uncertain and volatile due to impacts of artificial intelligence (AI), international unrest, climate change, global economic instability and unforeseen catastrophes such as the Coronavirus (Covid-19) pandemic, it is necessary to review, revise and refine school education curricula and policies. The issue of what is essential for students to learn, and how they learn it, is of growing importance to international organisations such as the Organisation for Economic Co-operation and Development (OECD) and the United Nations Educational, Scientific and Cultural Organisation (UNESCO) and is emerging as a significant driver for educational reform across the globe. The growing prominence of CT and DL skills across many industry sectors has prompted recent changes to international assessment frameworks such as the Programme for International Student Assessment (PISA) and the Trends in International Mathematics and Science Study (TIMSS). This paper will briefly discuss specific examples of alternative approaches to addressing CT in national curricula for the compulsory years of schooling and explain how CT has been adopted as a driver for mathematics curriculum change in Australia. [ABSTRACT FROM AUTHOR]

Research in mathematics education is a discursive process: It entails the analysis and production of texts, whether in the analysis of what learners say, the use of transcripts, or the publication of research reports. Much research in mathematics education is concerned with various aspects of mathematical thinking, including mathematical knowing, understanding and learning. In this paper, using ideas from discursive psychology, I examine the discursive construction of mathematical thinking in the research process. I focus, in particular, on the role of researchers’ descriptions. Specifically, I examine discursive features of two well-known research papers on mathematical thinking. These features include the use of contrast structures, categorisation and the construction of facts. Based on this analysis, I argue that researchers’ descriptions of learners’ or researchers’ behaviour and interaction make possible subsequent accounts of mathematical thinking. [ABSTRACT FROM AUTHOR]

Many researchers and philosophers recognize the importance of critical thinking and reasoning skills in the life of all students. However, they failed to address the question of how such skills may be incorporated into mathematics lessons or activities. While such skills are often included in mathematics curriculum, they are not always seen in the lesson. This paper presents the integrations of critical thinking and reasoning skills into a sample lesson through block factor game for teaching factors of a number to young learners to play and list factors of a given number. Scholars have agreed that making mathematics practical using concrete materials in the form of game makes it interesting and fun for learners. Two participants in primary 2 and primary 6 were involved in this study through qualitative research design of case study. Data collection was done through audio-andvideotaped. The students' responses to questions and tasks indicated conceptual understanding and acquisition of critical thinking and reasoning skills. Critical thinking and reasoning skills are incorporated into lesson through critical questions, and games or well constructed tasks in appropriate context. Also, the participants were able to list factors of a given number correctly and also determine the divisibility of a number by another number using block factor game. [ABSTRACT FROM AUTHOR]

This article presents two different classroom situations in which two pairs of students, one in grade 2 and the other in grade 6, attempted mathematically challenging problems through GeoGebra applets. Their reasoning, conjecture making, and argumentation is analyzed using Pea's (1985, 1987) theory of technology as amplifier and reorganizer and the notion of internal and external representations. GeoGebra played the role of amplifier and reorganizer in enabling students' explorations and led them to make conjectures. Furthermore, it was observed that students' static internal representations were enhanced to more dynamic external representations and this played a significant role in developing students' thinking. The three aspects of fidelity have been discussed with regard to the use of GeoGebra based applets and how such applets need to be critically designed and assessed in order to support students' learning. [ABSTRACT FROM AUTHOR]

Copyright of Hellenic Journal of Research in Education / Ereuna Stīn Ekpaideusī is the property of Laboratory of Research in Pedagogy & Educational Practices and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

This paper presents an experiment that attempts to mobilise an arithmetic-algebraic way of thinking in order to articulate between arithmetic thinking and the early algebraic thinking, which is considered a prelude to algebraic thinking. In the process of building this latter way of thinking, researchers analysed pupils' spontaneous production using a triangular numbers activity. Based on a specific collaborative learning methodology, this study explores the possibility of constructing an Arithmetic-Algebraic Work Space around the process of constructing signs as framed by both activity theory and a technological approach, showing the spontaneous representations produced by seventh grade pupils and their evolution in a socio-cultural environment. [ABSTRACT FROM AUTHOR]

Copyright of Larq Journal for Philosophy, Linguistics & Social Sciences is the property of Republic of Iraq Ministry of Higher Education & Scientific Research (MOHESR) and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

We researched how the transition to remote learning during the COVID-19 pandemic affected the development of mathematical thinking and reading comprehension skills of high school students based on the analysis of the results of a standardized test for higher education admission. The total number of participants was N = 17,942, out of which N1 = 10,611 were pre-pandemic applicants and N2 = 7,331 were post-pandemic applicants. The pre-pandemic group took into consideration the exam results from September 2016 to June 2020. The post-pandemic group took into consideration the exam results from September 2020 up to May 2023. We conclude that emergency remote teaching decreased the average mathematical competence of incoming university students by 12.8%. Similarly, emergency remote teaching decreased the average reading achievement of students entering university by 8%. Therefore, the impact of emergency remote teaching was greater on mathematics skills than on reading achievements. [ABSTRACT FROM AUTHOR]

This paper establishes semantic conceptions for the terms mathematical thinking and mathematical culture, which are present in the Theory Mathematics in the Context of Sciences. We used the method called comparative constant. This is fundamental to avoid misconceptions or carry out contradictory analyses in regard to the precepts of the referential adopted. The acknowledgement of the elements that characterize these conceptions can enable the design and implementation of adequate programs of Mathematics to develop mathematical thinking, as well as mathematical culture in students of Higher Education. [ABSTRACT FROM AUTHOR]

The nature of the area of mathematical competitions as a design science is considered, historical roots of mathematical problem-solving competitions are traced, the complementary aspects of mathematics as theory building and as problem solving are touched upon in relation to the practice of competitions. Two historical figures, Euler and Erdős, emerge, and the appropriateness of many of Euler's mathematical ventures are seen as role models for competition mathematics as first put into practice in mathematical competitions towards the end of the nineteenth century. Distinctions and definitions are made, a venture into identifying competition syllabi and the principal types of reasoning employed in solving competition problems is explored, and a description of the many different types of competitions is considered. Interaction between the field of mathematics itself and problem-solving competitions is briefly explored, as are the possibilities that open when competitions and their access to huge amounts of data, both national and international, are taken into account in research belonging to mathematics education. Finally, the range of topics addressed in this special issue of ZDM is covered, along with some possible conclusions relating to the components of the overview. [ABSTRACT FROM AUTHOR]

This paper deals with the nature of teaching mathematical thinking and presents a case study of a single Japanese lesson where the characteristics of mathematical thinking and the teaching thereof are identified in relation to multiplication. The raison d'être for this teaching is questioned and investigated by looking at how multiplication is described in the curriculum and representative textbook material. It is seen how Japanese teachers are institutionally conditioned to incorporate mathematical thinking in the context of multiplication, something which may appear in contrast to other countries. The lesson is analysed using the notion of praxeologies and didactic co-determination conceptualised in the Anthropological Theory of the Didactic. [ABSTRACT FROM AUTHOR]

This paper reports on a research study conducted with a group of practising primary school teachers (n = 24) in North East Scotland during 2011–2012. The teachers were all participants in a newly developed Masters course that had been designed with the aim of promoting the development of mathematical thinking in the primary classroom as part of project supported by the Scottish Government. The paper presents the background for this initiative within the context of the Scottish Curriculum for Excellence reform. Particular attention is given to the epistemological positioning of the researchers as this influenced both the curriculum design process and also the theoretical framing of the research study which are both described. The project was set up within a design research framework, which aimed to promote classroom-based action research on the part of participants through the course and also research by the university researchers into the process of curriculum development. The research questions focused on the teachers’ confidence, competence, attitudes and beliefs in relation to mathematics and their expectations and experiences of the impact on pupil learning arising from this course. Empirical data were drawn from pre- and post-course surveys, interviews and observations of the discussion forums in the online environment. Findings from this study highlight the way the course had a transformational and emancipatory impact on these teachers. They also highlight ways in which the ‘framing’ of particular aspects of the curriculum had an oppressive impact on learners in the ways that suppressed creativity and limited the exercise of learner autonomy. Furthermore, they highlight the ways in which a number of these teachers had experienced mathematics as a school subject in very negative ways, involving high levels of ‘symbolic violence’ and of being ‘labelled’. [ABSTRACT FROM PUBLISHER]

In this paper we report on our aims to design a mathematical unit in line with Trends in International Mathematics and Science Study, TIMSS, see Appendix-1, and to assess the impact of this proposed design on the thinking of the second intermediate students in Najaf. The unit was designed according to: i) the two steps of the original educational design and ii) the skills included in the global project. The mathematical content was: relative numbers, real numbers, and algebra. The sample consisted of 60 students from the secondary schools in Najaf, divided into 2 groups: the experimental group of 30 and the control group of 30 students. The selection was done in a simple random way. The research tools were pre-test and post-test, both of which were built to measure the forked thinking according to 4 strategies: i) correlation analysis, ii) analysis of views, iii) complementarity, and iv) network analysis of relations. We used Kuder and Richardson Formula 20, see Appendix-2, T- test for two independent samples and T- test for statistical samples interconnecting means. The results showed: i) there were statistically significant differences between the pre and post test of the experimental group, ii) the existence of statistically significant differences between the post-test of the two research groups, iii) both in favor of the design of the proposed unit. The results are followed by a set of recommendations. [ABSTRACT FROM AUTHOR]

A rigorous and axiomatic-deductive approach is emphasized in teaching mathematics at university-level. Therefore, the secondary-tertiary transition includes a major change in mathematical thinking. One viewpoint to examine such elements of mathematical thinking is David Tall's framework of the three worlds of mathematics. Tall's framework describes the aspects and the development of mathematical thinking from early childhood to university-level mathematics. In this theoretical article, we further elaborate Tall's framework. First, we present a division between the subjective-social and objective sides of mathematics. Then, we combine Tall's distinction to ours and present a framework of six dimensions of mathematics. We demonstrate this framework by discussion on the definition of continuity and by presenting a visual construction of a nowhere differentiable function and analyzing the way in which this construction is communicated visually. In this connection, we discuss the importance to distinguish the subjective-social from the objective side of mathematics. We argue that the framework presented in this paper can be useful in developing mathematics teaching at all levels and can be applied in educational research to analyze mathematical communication in authentic situations. [ABSTRACT FROM AUTHOR]

This paper reports on a research project which aims to improve prospective mathematics teachers’ relational understanding and pedagogical beliefs for teaching in second-level Irish classrooms. Prospective mathematics teachers complete their teacher education training with varying pedagogical beliefs, and often little relational understanding of the mathematics they are required to teach at second level. This paper describes a course designed by the authors to challenge such beliefs and encourage students to confront and possibly transform their ideas about teaching, while simultaneously improving their subject knowledge and relational understanding. Both content and pedagogical considerations for teaching second-level mathematics are integrated at all times. The course was originally optional and was piloted and implemented in a third-level Irish university. Apart from offering an insight into the design considerations when creating a course of this type, this paper also addresses some of the challenges faced when evaluating such a course. Overall participant feedback on the course is positive and both qualitative and quantitative results are provided to support this and also highlight the efficacy of the programme. [ABSTRACT FROM PUBLISHER]

Copyright of Annals of Philosophy / Roczniki Filozoficzne is the property of John Paul II Catholic University of Lublin, Faculty of Philosophy and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

There is a long standing history of mathematical games and puzzles being used to support mathematical thinking and concepts. The Maths Arcade is a recent initiative designed to provide an environment for students to play games in order to communicate with fellow students whilst developing a range of skills. The purpose of this study was to determine how these skills are developed and how they link to different games, we have found the evidence for this through reviewing a variety of sources focussing on game-play. We found that the playing of board games is fundamentally linked to the development of mathematical thinking and skills. These are developed in many ways including trial and improvement and discussion. [ABSTRACT FROM AUTHOR]

In this paper we document and discuss how the use of digital technologies in problem solving activities can help students to develop mathematical competences; particularly, we analyze the characteristics of reasoning that students develop as a result of using Cabri Geometry software in problem solving. We argue that the dynamical nature of representations constructed with Cabri, and the availability of measure tools integrated to it are important elements that enhance students' ability to think mathematically and foster the implementation of several heuristic strategies in problem solving processes. [ABSTRACT FROM AUTHOR]

This paper describes the implementation of extended analysis tasks (EATs) in a required mathematics course for prospective high school mathematics teachers, and investigates the mathematical discovery promoted through engagement with these tasks. Extended analysis tasks are designed to move students beyond the problem context to the underlying mathematical structure of the problem. The results of this study revealed that with careful implementation, these tasks were effective for generating powerful mathematical thinking in preservice mathematics teachers, improving their problem solving, and generally enhancing the habits-of-mind exhibited by successful mathematicians (Carlson & Bloom, ). [ABSTRACT FROM AUTHOR]

This paper examines the process through which students learn to make functional use of computer algebra systems (CAS), and the interaction between that process and the wider mathematical development of students. The result of ‘instrumentalising‘ a device to become a mathematical tool and correspondingly ‘instrumenting’ mathematical activity through use of that tool is not only to extend students' mathematical technique but to shape their sense of the mathematical entities involved. These ideas have been developed within a French programme of research – as reported by Artigue in this issue of the journal – which has explored the integration of CAS – typically in the form of symbolic calculators – into the everyday practice of mathematics classrooms. The French research –influenced by socio-psychological theorisation of the development of conceptual systems- seeks to take account of the cultural and cognitive facets of these issues, noting how mathematical norms – or their absence – shape the mental schemes which students form as they appropriate CAS as tools. Instrumenting graphic and symbolic reasoning through using CAS influences the range and form of the tasks and techniques experienced by students, and so the resources available for more explicit codification and theorisation of such reasoning. This illuminates an influential North American study– conducted by Heid – which French researchers have seen as taking a contrasting view of the part played by technical activity in developing conceptual understanding. Reconsidered from this perspective, it appears that while teaching approaches which ‘resequence skills and concepts’ indeed defer – and diminish –attention to routinised skills, the tasks introduced in their place depend on another –albeit less strongly codified – system of techniques, supporting more extensive and active theorisation. The French research high lights important challenges which arise in instrumenting classroom mathematical activity and correspondingly instrumentalising CAS. In particular, it reveals fundamental constraints on human-machine interaction which may limit the capacity of the present generation of CAS to scaffold the mathematical thinking and learning of students. [ABSTRACT FROM AUTHOR]

The paper discusses a certain type of competitions based on distance interaction of a participant with simulation models of concepts from discrete mathematics and computer science. One of them is the “Construct, Test, Explore” (CTE) competition, developed by the authors, the other is the Olympiad in Discrete Mathematics and Theoretical Informatics (DM&TI). The tasks presented in this paper are generally devoted to the concept of a graph isomorphism. Most of the tasks are verified automatically.

Mathematical thinking (MT) has been one of the most important goals for mathematics education as it can support sustainable mathematics learning. Its role in school mathematics has recently been explicitly identified as one of "Four Basics" in the latest national curriculum standard for compulsory education, which is seen as one of the prominent features of Chinese mathematics education. This paper reviewed definitions, descriptions, and explanations from a historical perspective and aimed to provide a comprehensive and contemporary conceptualization for MT in a Chinese context, which can be applied as a comparison to MT in English works. To achieve this, document analysis was applied in this study. Major official documents, papers, and books were reviewed to see the process of MT conceptualization given by the policy makers and researchers. Results indicated that MT places more emphasis on the process of mathematical methods application in problem solving, such as the method of combination of symbolic and graphic mathematics. Mathematical thought is also recommended by Chinese researchers to help students think like mathematicians. Another major characteristic is that the classification of major types of MT is usually focused on that which can make the concept more understandable. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Engaging critical thinking and mathematical thinking in solving engineering problems is a way of approaching the engineering criteria of Engineering Accreditation Council, Board of Engineers Malaysia. Thus, it is timely and crucial to inculcate the critical thinking and mathematical thinking into the current engineering education. Unfortunately, information about these two modes of thinking in real-world engineering practice is found lacking in the literature. Therefore, this paper focuses on explaining an analytic process in identifying pertinent elements of critical thinking and mathematical thinking used in real-world civil engineering practice. The analytic process, namely open coding is a part of coding process in modified grounded theory analysis. Data consist of semi-structured interviews with eight practicing civil engineers from two different consultancy firms. A total of fifty three pertinent elements emerged during the analytic process. The selection of these pertinent elements was based on the predominant pattern and frequency of the informants and open codes. The pertinent elements were eventually integrated to develop a substantive theory. The substantive theory provides useful information for the engineering education. [ABSTRACT FROM AUTHOR]

This paper presents a case study of a first grade student to illustrate the diversity of her understandings related to variables and variable notation. While prior research has documented secondary school students' difficulties with variables and variable notation, we identify many productive understandings in this much younger student, leading us to question the prevailing argument that students might have difficulties with variables due mostly to their own limitations. We draw our data from a teaching experiment that explored functional relationships. Individual interviews were carried out with a subset of the students in the experiment prior to, as well as mid-way through and at the end of the experiment. This paper focuses on a set of three interviews with one of the first grade students. We illustrate the shifts that occurred in the student's understandings about variables and variable notation across as well as within each of the three interviews. [ABSTRACT FROM AUTHOR]

Copyright of Iskolakultúra is the property of University of Szeged, Faculty of Arts & Social Sciences and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Programming in undergraduate mathematics is an opportunity to develop various mathematical skills. This paper outlines some topics covered in a second year, optional module 'Programming with Mathematical Applications' that develop mathematical thinking and involve mathematical activities, showing that practical programming can be taught to mathematicians as a mathematical skill. [ABSTRACT FROM AUTHOR]

Copyright of Turkish Journal of Education is the property of Mehmet Tekerek and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

This paper examines the perceptions and understandings of ten grades 1 and 2 Singapore mathematics teachers as they learned to use clinical interviews (Ginsburg, Human Development 52:109-128, ) to understand students' mathematical thinking. This study challenged teachers' pedagogical assumptions about what it means to teach for student understanding. Clinical task-based interviews opened a window into students' knowledge, problem-solving and reasoning, and helped teachers reflect on their teaching and assessment of student learning. Teachers also learnt about what it means to establish a culture of thoughtful questioning in the classroom and developed an emerging awareness that this requires a readiness to hear students' ideas and connect informal or invented strategies with classroom mathematics. [ABSTRACT FROM AUTHOR]

Linear algebra with its rich theoretical nature is a first step towards advanced mathematical thinking for many undergraduate students. In this paper, we consider the teaching approach of an experienced mathematician as he attempts to engage his students with the key ideas embedded in a second-year course in linear algebra. We describe his approach in both lectures and tutorials, and how he employed visualization and an emphasis on language to encourage conceptual thinking. We use Tall's framework of three worlds of mathematical thinking to reflect on the effect of these activities in students’ learning. An analysis of students’ attitudes to the course and their test and examination results help to answer questions about the value of such an approach, suggesting ways forward in teaching linear algebra. [ABSTRACT FROM PUBLISHER]

This paper describes the prisoner dilemma and applied it three times during three examples in the contemporary economics on frequent or usual situations. After a short introduction, the first part of the article is an attempt to detailed the prisoner dilemma, which still has an important influence in the contemporary theory of economics as an interesting problem of applied mathematics that indicates the absence of a solution, and thus the game theory offer an image of complexity and simplicity in mathematics thinking. The dilemma appears more realistic in economic phenomena. Three examples constitute the second section, constructed around the economic way of thinking of three important problems. A final remark closes the paper with the idea of a necessary and excellent approach to other scientific worlds through mathematics, games' theory, theory of complexity and other multi-disciplinarity options. [ABSTRACT FROM AUTHOR]

Individual mathematics teachers may value different aspects of teaching and learning mathematics, but at the same time their value systems are under the influence of socially shared values. This paper describes such values in Japanese mathematics education from a historical and normative perspective. After the introduction of Western mathematics into the modern school system in the Meiji period (1868-1912), the people of Japan struggled to adapt and absorb it onto the foundation of Japanese tradition. In the subsequent development of Japanese mathematics education, the integration of both practical and theoretical aspects have been issues, alongside changes in educational focus and in society at large, which are symbolically represented by the enrolment rates at all of elementary, secondary, and tertiary education levels. Mathematics education in Japan has also been subject to international influences, such as the reform movement and the modernization movement, at critical junctures in its development. Key concepts such as mathematical ideas, mathematical thinking, and mathematical activities are traces of such historical efforts by the Japanese mathematics education community, and represent their socially shared values. [ABSTRACT FROM AUTHOR]

Design is the fundamental soul to all branches of engineering. It is a prime context for understanding how civil engineers use critical thinking and mathematical thinking in engineering problem-solving. However, information about the interrelation between these two types of thinking in real-world engineering practice is found lacking in the literature. This paper presents the first-hand experience of developing a substantive theory which relates both critical thinking and mathematical thinking used by practicing engineers in the civil engineering design process. The qualitative research using modified grounded theory method was employed in this study. Data were generated from semi-structured interviews with practicing engineers from two engineering consultancy firms. Six essential processes of justifying decisions reasonably in engineering design process were identified, namely complying requirements, forming conjectures, drawing reasonable conclusions, defending claims with good reasons, giving alternative ways and selecting and pursuing the right approach. Findings of this study may advise prospective civil engineers of the applicability and indispensability of critical thinking and mathematical thinking in making and justifying decisions during the engineering design process. The study also contributes useful information to engineering education on fulfilling the expectations of engineering program outcomes set by the Engineering Accreditation Council. [ABSTRACT FROM AUTHOR]

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In an effort to innovate teaching and learning process to prepare for a new generation for the demands of the new era, many educators have discovered the value of metacognition. Students will need both mathematical skills and problem-solving skills, therefore, teachers should focus on both math content and thinking processes in students' math learning. This paper presents the importance of metacognition in solving mathematical problems. A project was conducted for students at the age of thirteen, and the findings suggest that students used four problem-solving steps highlighted by George Polya. However, students feel better when they adjust their thinking processes or use metacognitive skills in the process of solving math problems. [ABSTRACT FROM AUTHOR]

Problem Statement: The acquisition of certain mathematical skills is part of many curricula around the world. These skills involve problem-solving, effective use of mathematics in daily life, thinking logically and systematically, taking risks and making decisions. The US National Council of Teachers of Mathematics ([NCTM], 1991) relates these skills to mathematical power. Knowledge construction and abstraction might have an impact on mathematical power. In this paper, the primary concern is with the construction of new knowledge structures rather than with consolidation or abstraction for that matter. Recognizing, Building-with, and Constructing (RBC hereafter) theory of abstraction provides a particularly useful framework in achieving a detailed examination of the new mathematical constructions through epistemic actions. Therefore, this paper, with reference to the RBC epistemic actions, examines the knowledge construction process of students with different mathematical power. Purpose of Study: The purpose of the study is to explain the similarities and differences between the knowledge construction processes of 6th grade students with different mathematical power. Methods: A case study was used as the main research approach. The mathematical power scale was applied to 282 students. The case study was conducted with four students who were chosen purposefully among them. The RBC theory of abstraction was used as an analytical tool. The patterns noticed in the case study were determined and interpreted. Findings and Results: According to findings, it was seen that the students with low mathematical power, whose knowledge constructing processes were examined, could recognize structures while they could not building with and constructing. When observing that the students with low mathematical power have low communication, connection and reasoning skills, in general, it can be said that for building-with and constructing, it is important to have these three skills. The students who reflected themselves and used the feedback to continue had high mathematical power. They also were able to construct mathematical knowledge more quickly. Conclusions and Recommendations: The RBC theory of abstraction can be useful in explaining the relationship between the components of mathematical power and other mental activities. For this purpose, the usage of the RBC theory of abstraction in the mathematical knowledge construction process in the classroom can be proposed as a new study. [ABSTRACT FROM AUTHOR]