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It is well-known to historians of mathematics in East Asia that the tianyuan shu 天元術 was one of the distinct features of mathematics in the late Joseon period. Joseon Confucian scholars' studies on the tianyuan shu have yet to be analyzed. From the late seventeenth to the early eighteenth century, there were two extant mathematical works by Korean Confucian scholars concerning the tianyuan shu: Sanhak wonbon 算學原本 by Bak Yul (朴繘, 1621-1668) and Sanhak bonwon 算學本源 by Hwang Yun-seok (黃胤錫, 1729-1791). By analyzing these two works, this paper examines the characteristics of their views and interpretations of the tianyuan shu. The Confucian authors extended the usage of the tianyuan shu into other mathematical fields to which it had never been applied before. They also preferred to use mathematical tables that showed the relationships between the numbers popping up during the performance of the method or the visual diagrams that explained the algorithms. [ABSTRACT FROM AUTHOR]

Early modern Europe went through two distinct types of revolution ᅳ scientific and military. Military historians have employed the term, Military Revolution, to account for several significant changes in the seventeenth century, such as advent of new weapons, enlargement of military forces, and extension of battle length. Under these circumstances, it is reasonable to assume a certain degree of correlation between the Scientific Revolution and the Military Revolution, both of which took place in the same place at the same time. This paper features Maurits van Nassau and Simon Stevin to show the interactions between two revolutions. They were patron and client in private, and general and officer in public. Maurits is considered to have brought about the Military Revolution, while Stevin is valued highly as a mathematician that invented decimal fraction system and laid the foundation for the new mathematics through innovating traditional number concept. Maurits is well-known for the 'scientific war,' particularly the scientific siege warfare. I suggest that his scientific wars were carried out through systematic deployment of armed forces, detection of the weakest point based on mathematical analysis, calculation of safe attacking routes, and intensive attack on target points. Stevin's publications on the art of war will show how much they influenced on Maurits' strategies and tactics. This shows that the clear correlation between science and war, characterized by goal-driven research for warfare tactics, elevated social status of relevant scientists, and strong recognition of victor nations as the centre of science, already started slowly in the seventeenth century. [ABSTRACT FROM AUTHOR]

This paper aims to present how video production is impacting the classroom, mathematics students and teachers in Brazil, as well as mathematics knowledge production developed with this media. Digital video Festivals with mathematical content are being implemented in Brazil, locally and nationally. One of them is Mathematics Education Digital Video Festival, organized by the Research Group in Informatics, Other Media and Mathematics Education, GPIMEM. In this century, video is earning space as a pedagogical approach in either face-to-face or distance education, even before the COVID-19 pandemic. Currently, considering social distance, the use of videos became an imposed reality to students and teachers. Students and teachers with different level of experience or age started to realize possible positive effects regarding the reorganization of the classroom by the presence of mathematical videos. However, other aspects provided negative effects: for example, inequalities in homes and mobile access impacted equal opportunity to all. This paper will summarize the use of mathematical video in the last century and discuss what is happening this century. We will revise some research developed, mainly in Brazil, that shows how the classroom does not fit in a parallelepiped model. Internet, videos and software combined are "things" that have agency and are co-participating in learning and teaching mathematics. The organization of festivals of videos have become important not only in Brazil but elsewhere in many formats either nationally or internationally, in a way to create challenges, exhibitions, cultural mix that aim to show mathematics applications to the general public. In particular, this article reports on parts of a qualitative research that investigated participants of the first edition of the referred national festival. As a result, we look for the comprehension of following the discussions of qualitative transformations regarding the use of videos in classrooms and mathematics knowledge production, as this new setting suggests that digital technologies arrived in XXI century to collaborate with learning. This article will be supported by a theoretical perspective on technology based on the notion of humans-withmedia. In such a perspective knowledge is seen as being constructed by humans and different media and different artifacts. This article shows how such a perspective may help us to cope with the classroom of the XXI century, including the classroom that during the pandemic include students' and teachers' homes.

In order to understand the research trends in mathematical reasoning since 2000, this study analyzed 262 papers published in seven KCI journals and 381 papers published in five SSCI journals via topic modeling. The overall research topics were compared and contrasted between domestic journals and international journals. A more detailed analysis was conducted by considering different publication periods. The results showed that the main domestic research topics included, in order, geometry proof, mathematical justification, problem solving, pattern generalization, proportional reasoning, and statistical reasoning, whereas the main international research topics included, in order, proof and argument, teacher education, geometric reasoning, pattern generalization, problem solving, and statistical reasoning. The results of this study also showed that gifted students represented the most popular research target of domestic studies, while the process of mathematical reasoning was the main focus of international studies. This paper closes with implications on research targets including teachers, attention to the mathematical reasoning process, diversity of research topics, and new research topics that may guide future research on mathematical reasoning.

In light of a growing emphasis on the relationship between fractional knowledge and algebraic thinking, the purpose of this study was to compare algebraic thinking about reverse fraction problems between fifth graders who had not been taught fraction division and sixth graders who had been taught fraction division. For this purpose, we conducted cognitive interviews with 30 students (15 fifth graders and 15 sixth graders) who used different solution methods in a paper-and-pencil test dealing with reverse fraction problems. We analyzed their algebraic thinking in terms of problem structure reasoning, generalization-representation, and justification. The results showed that fifth graders could generalize their solution methods but sixth graders who learned only the algorithm of fraction division had difficulties with generalization and justification. However, sixth graders who reasoned about the problem structure demonstrated proficiency in algebraic thinking. Although some students used the same solution methods in the paper-and-pencil test, they had different capacities in generalizing, representing, and justifying them during the cognitive interviews. The results of this study provide implications for teaching fraction operations in a way that develops algebraic thinking by attending to the mathematical structure and relation.

This paper looks at the evolution of ideas on the role of proof in mathematics education from 1980 to 2020, examining in particular the contributions of both theoretical and empirical research to the teaching of mathematical proof. In so doing, it describes some of the major epistemological themes that emerged in the last forty years, primarily in the philosophy of mathematics and in mathematical education, and informed both the mathematics curriculum and research in mathematics education. The paper also discusses selected research studies that shed light on the opportunities and limitations students face when they engage in proof. Finally, it describes briefly a number of developments in proof technology and the potential of automated theorem provers for enhancing the teaching of proof.

Students often have difficulty understanding the proofs that they read. Some mathematics educators have suggested using generic proofs to improve students' proof comprehension. However, few empirical studies about students' perceptions of generic proofs or whether students understand generic proofs better than traditional proofs have been conducted. In this paper, we describe a qualitative interview study and a quantitative experimental study exploring these issues. Our first finding is that generic proofs are popular with students, who believe they have the potential to improve proof comprehension. Our second finding is that in a randomized controlled experiment, students did not learn more from reading a generic proof than a traditional proof. The significance of these findings is discussed at the end of the paper.

The concept of proof is central to every undergraduate mathematics curriculum, but it is also a difficult concept for university students to understand and for university instructors to teach. Prior studies have contributed useful research knowledge about the nature and sources of undergraduate students' difficulties with proof, but they have paid less attention to the development of effective instructional practices to enhance student learning of proof. As a step towards this direction, this paper makes a contribution to the limited body of case studies of promising or potentially effective teaching practices in undergraduate mathematics by reporting a case study of an instructor's teaching, locally characterized as "effective," in an undergraduate analysis course at a leading Indian university. The instructor did not deviate from the so called "Definition-Theorem-Proof" (DTP) format that is followed in most proof-oriented university mathematics courses, but her teaching presented a set of features that, collectively, form a rather innovative teaching practice at the undergraduate level. Specifically, our case study shows that, even in the context of a rather crowded university classroom, proof can be demystified for students through structured interaction between the instructor and the students, that is, an interactive, conversational style of proof instruction invoking the participation of students. This is based on a solid foundation in symbolic logic at the very outset and on a significant time investment in definitions being explained in depth using informal language, visual aids, and real-life analogies. In addition to contributing to existing images of potentially effective teaching practices in undergraduate mathematics, this paper draws attention to the Indian educational context that has had little representation in international forums of mathematics education research thus far.

Mathematics is at the root of coding and AI convergence education, emphasized in recent school education. This paper proposes an education content and education method that can combine elementary and secondary coding education that is being introduced recently with school mathematics in a developmental way. Specifically, in this paper, through the mathematising of the code in coding education, the contents of education to develop from arithmetical coding to algebraic coding are designed, as well as the minimum coding mapping strategy that induces development into algebraic coding. Additionally, this study addresses and analyzes the process in which learners develop their algebraic abilities with educational content and guidance strategies that converge coding education in elementary and secondary schools into math education, especially algebra education. This convergence education is achieved through a Papert's constructionism learning theory-based artifact design, and the minimum code strategy is designed to enable learners to form a knowledge structure of algebra and gain experience in exercising computational thinking.

Clarity and consistency of terms are more important in mathematics than in any other discipline. Mathematics terminology can be an essential clue to understanding a mathematical concept. Moreover, it can evoke images related to the concept both during the learning process and after learning. However, the evoked image may fail to match the original mathematical concept accurately if a student has a misconception about a specific mathematical term. This paper specified the ambiguity and inconsistency of the expression “the function represented by a parameter” used in textbooks. Further, it investigated the teachers’ perceptions caused by the phrase. Teacher responses in a survey showed that teachers perceived the words in various ways, and some teachers demonstrated mathematical misconceptions. Additionally, the teachers recognized the need to revise the phrase, “the function represented by a parameter.” Our study implicates that unclear and inconsistent uses of mathematical terms in textbooks and teacher’s guide books might contribute to teacher misunderstandings, which can directly or indirectly affect students.

In this paper we report on a case study involving four Chinese students; the aim of the study was to develop a conceptual framework for investigating student participation in a collaborative task in mathematics. Building on previous research on student participation, we defined student participation in a collaborative task in mathematics as the process of taking part in student interactions and task completions. A video recording and transcript of students involved with collaborative task in mathematics were analyzed. Four categories (initiation, response, evaluation, and non-interactive) were created for analyzing the students' interactions. The four students showed different approaches to communicating with other group members. The content of the group's discussion was examined in terms of the negotiation process. By looking at the content of the group's discussion, the process how the group work on the task could be revealed. We identified that students showed involvement in the group discussion by proposing and revisiting topics. It was found that through forming an interactive pair, students might be able to take up the leading role in working on the task and controlling the conversation of the group.

Characterised by increased automation and digitalisation of work processes, the Fourth Industrial Revolution (4IR) has displaced and redesigned many existing jobs, and will create new occupations that are currently non-existent. To prepare a future workforce that is adaptive amid a volatile employment landscape, schools should provide the necessary learning experiences to help students today develop transferrable competencies, which encompass deep conceptual understanding of domain-specific knowledge and 21st century competencies in the cognitive, intrapersonal, and interpersonal domains. In this paper, we study this possibility in the context of mathematics learning and propose a constructivist learning design (CLD) that affords students to engage in deeper learning processes. In the proposed CLD, students first work collaboratively to solve a complex problem targeting a math concept that they have yet to learn, before being engaged in instruction that builds upon their solutions in the teaching of the concept, and practices that reinforce these ideas. Testing CLD in mathematics learning at secondary level via a quasi-experimental design, we found out that (1) CLD facilitates deeper learning as it encouraged students to apply their cognitive, intrapersonal, and interpersonal competencies, and (2) CLD students (n=23) outperformed their Direct Instruction counterparts (n=18) on mathematical conceptual understanding and transfer. Overall, this study suggests that the CLD has the potential to cultivate competencies that allow students to transfer in novel situations, rendering it as a possible learning environment to better prepare students for the 4IR.

This article studies some central developments in the propagation and teaching of mathematical logic in 1930s China. Focusing on the emergence of a twofold disciplinary approach to mathematical logic, namely as a discipline studied and disseminated by Chinese philosophers on the one hand and mathematicians on the other, this paper explores one of the key turning points in the development of the academic notion of mathematical logic in China. Apart from casting some light on the teaching of mathematical logic in the framework of both philosophical as well as mathematical spheres of inquiry, this article also provides some preliminary insights into the circumstances surrounding the first systematic introduction of mathematical logic into the modern standardized system of education, which gradually took shape over the late-1920s and early 1930s in China.

Measurement involves actual measuring or estimating to generate a quantitative understanding of a phenomenon. Estimation in particular refers to the process of obtaining measurements without a measuring tool. This paper aims to investigate didactical implications for estimation by analyzing the estimation activities of textbooks. To this end, we focused on estimation activities in elementary school mathematics textbooks from the seventh to the 2015 revised national curriculum, analyzing them based on activity quantity, estimation type, and estimation strategy. As regards length, the estimation activities were presented appropriately in terms of quantity, type, and strategy. However, other attributes were insufficient in quantity while some activities showed biases and limitations with few estimation strategies to use in a variety of contexts. Based on these results, we propose the inclusion of achievement standards for time, area, and volume and argue that the textbook should address various types of and strategies for estimation activities.

This paper is a study on the improvement of classroom instructions for enhancing algebra learning in slow learning middle school students. Based on the study of Watson & De Geest(2012), the algebraic properties presented in textbooks have been detailed in the present study. Also, the aim was to verify the effect of each level when the detailed instructional data was put into a general class where students with different achievement levels were mixed. The effect of the detailed instructional treatment was quantitatively analyzed using the SPSS statistical tool, and a qualitative analysis was attempted based on the specific examples of answers written by students. In addition, post-test 1 and post-test 2 were performed to analyze the retention effect. As a result of the analysis, no difference in all the levels of the comparison group and the experimental group in post-test 1 was observed; however, in post-test 2, the algebraic achievement of the lower group of the test group was statistically significant, thereby confirming the retention effect. Moreover, there was no statistically significant difference in algebraic achievements above the middle group. As a result of the average comparison, it was confirmed that the refined instructional treatments had a positive effect on learners above the middle group. These findings provide pedagogical implications for a lesson plan that can be helpful to students of all levels with poor math skills in the mixed mathematics classroom.

This study reports on a classroom-based intervention that lasted eight lessons with the particular focus of exploring whether and how specific activities, strategies, and questions may promote an enquiring atmosphere in a mathematics classroom, i.e., a classroom environment in which students feel safe to investigate new mathematical ideas in order to produce, share, and test their conjectures in search for a proof. Although many researchers agree on the central role that conjectures and proofs should play in all students' mathematical experiences, many students find it difficult to engage with conjecturing and proving. In this paper we argue that the classroom atmosphere has a crucial influence in helping students to overcome these difficulties and we focus our investigation on an English secondary mathematics classroom taught by the first author. Students' responses to questionnaires, prompts, tasks, and interview questions were analysed in order to evaluate the impact of the intervention. The analysis suggests that specific activities, strategies, and questions can be used by teachers to facilitate an enquiring classroom atmosphere. However, for these to be effective, the teacher needs to explicitly teach students how to listen, question, and ponder in order to give students the tools to formulate conjectures and proofs.

Over years of collaborating with elementary-school teachers to research students' thinking about the "big ideas" of K-6 mathematics, particular attention was given to generalizations about the operations-addition, subtraction, multiplication, and division-and arguments that explain why these generalizations are true. Through this work, we created a model of five phases that separate different points of focus in the complex process of formulating and proving such generalizations: 1) noticing patterns, 2) articulating conjectures, 3) representing with specific examples, 4) creating representation-based arguments, and 5) comparing and contrasting operations. In this paper, we illustrate the phases with classroom examples as students investigate a set of generalizations. We then present assessment results from classrooms of project teachers who engaged their students in this content.

Along with the significance of distributive property playing a fundamental role in providing explanations for multi-digit multiplication of whole numbers, the array model has been widely used to teach distributive property effectively. Given this, the purpose of this study was to investigate how third graders could understand multiplication in (two digits) × (two digits) and how their understanding could be expanded and applied to algebraic ideas after being taught reconstructed units of multiplication using the array model. The results of this study showed that most students represented the principle of computation between two-digit number and two-digit number using the array model as well as reasoned which picture represented the product of two numbers. Also, the participating students successfully applied the principle to the computation between three-digit number and two-digit number, and some students represented the product of two numbers by using variables (e.g., □, △). This paper concludes with some implications on how to teach multiplication of whole numbers in elementary mathematics.