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The aim of this pilot study is to evaluate the relevance of the use of history in mathematics education. This paper presents an experiment carried out in France with sixth-grade students (n=108) in which an ancient number system is used, an approach that is commonly suggested in French sixth-grade textbooks but has previously been unassessed. Based on the data of a pretest and a post-test surrounding an activity on an ancient Chinese numeration system, a statistical analysis using Rasch modeling shows a specific added value of the history of mathematics for students with low abilities in mathematics. For these students, a significant increase in observed abilities of +0.67 logit in mean is measured with a large effect size (Cliff delta +0.52). This effect is then weighted by considering the regression to the mean (RTM) effect, leading to a value around +0.14 logit in mean and a negligible effect size (Cliff delta +0.10). So, this pilot study shows the important effect of RTM, which suggests a very strong rebalancing of students' results. In the last part of the paper, we discuss how RTM can nonetheless be positively interpreted in this specific context where students' disorientation is one of the purposes of history in mathematics education.

Understanding the concepts related to real function is essential in learning mathematics. To determine how students understand these concepts, it is necessary to have an appropriate measurement tool. In this paper, we have created a web application using 32 items from conceptual understanding of real functions (CURF) item bank. We conducted a psychometric analysis using Rasch model on 207 first-year students. The analysis showed that CURF is a dependable and valid instrument for measuring students' CURF. The test is uni-dimensional; all items are consistent with the construct and have excellent item fit statistics. The results indicate that the items are independent of each other and unbiased towards the gender and high school background of the students.

Purpose: This paper defines and assesses student learning patterns under the influence of problem-based learning (PBL) and their classification into a reasonable minimum number of classes. Study utilizes PBL implemented in an undergraduate Statistics and Operations Research course for techno-management students at a private university in India. Design/methodology/approach: Study employs an in situ experiment using a conceptual model based on learning theory. The participant's end-of-semester GPA is Performance Indicator. Integrating PBL with classroom teaching is unique instructional approach to this study. An unsupervised and supervised data mining approach to analyse PBL impact establishes research conclusions. Findings: The administration of PBL results in improved learning patterns (above-average) for students with medium attendance. PBL, Gender, Math background, Board and discipline are contributing factors to students' performance in the decision tree. PBL benefits a student of any gender with lower attendance. Research limitations/implications: This study is limited to course students from one institute and does not consider external factors. Practical implications: Researchers can apply learning patterns obtained in this paper highlighting PBL impact to study effect of every innovative pedagogical study. Classification of students based on learning behaviours can help facilitators plan remedial actions. Originality/value: 1. Clustering is used to extract student learning patterns considering dynamics of student performances over time. Then decision tree is utilized to elicit a simple process of classifying students. 2. Data mining approach overcomes limitations of statistical techniques to provide knowledge impact in presence of demographic characteristics and student attendance.

Formal teacher leader programs that develop, position, and reward teachers to work with peers to improve instruction are a growing reform effort in the United States, yet there are few published studies of their efficacy. In this paper, we examine the impacts of one district's teacher leadership program on students' annual state test performance. The program placed full-time instructional coaches and partly released English language arts (ELA) and mathematics content specialists in each of 11 district schools. To assess the program's impact, we examined five years of student-level state test data; two years before the adoption of the intervention and three years afterwards. Using an interrupted time series design, we examined trends in performance before and after the adoption of the intervention. Overall, there were no significant effects in ELA, and a small negative effect in mathematics. By contrast, in the stable sub-sample of students who were in the district for the five years examined in the study, there was a large significant positive effect in mathematics and large but non-significant positive effect in ELA. We conclude with a discussion the implications of these findings for research and policy.

The main purpose of this study is to investigate mathematics teachers' mathematical thinking process while they are constructing the length of [cube root of 2] by paper folding. To carry out this aim, two teachers--who are PhD. students--were interviewed one by one. During the construction, it was possible to observe the consolidation process of Pythagorean and Thales Theorem. All interviews were videotaped and analyzed through descriptive methods, according to abstraction in context, characteristics of consolidation and mathematical habits of mind. It was found that both two teachers constructed the knowledge of Pythagorean and Thales Theorems before the study and also these knowledges were consolidated. In addition, it was determined that these two approaches (abstraction in context and mathematical habits of mind) were close and corroborated each other. Moreover, consolidation process corroborated mathematical habits of mind.

Problem solving in chemical kinetics poses substantial challenges for university students since it often involves significant use of mathematics as a tool and language, with challenging translations and transitions between chemical phenomena and mathematical representations. In this paper, we present key findings from a study investigating chemistry students solving tasks centred around the steady-state approximation. Building upon the mathematical modelling cycle (MMC), qualitative analysis of the data collected using a think-aloud protocol led to the development of the extended MMC. This empirically derived extended MMC offers a more detailed account of the processes involved in mathematical modelling of chemical phenomena, highlighting aspects such as the occurrence of deliberation and evaluation throughout the modelling cycle, as well as the varying characteristics, points of activation and roles of extra-mathematical resources during problem solving. We further introduce and use problem-solving trajectories as a tool for visualising and analysing the complex and diverse approaches used by students in their attempts at reaching a solution. Overall, the extended MMC provides a finer-grained model of the cognitive and metacognitive activities that students engage in, offering further insights for research and practice.

This paper provides evidence on the effect of local prescription opioid use on academic achievement of 3rd-8th graders between 2009 and 2018. Using county fixed effects models, I find that when counties have higher levels of prescription opioid use, students score lower on standardized assessments two to three years later, with variation by student subgroups and magnitudes comparable to effective interventions. I find the largest magnitudes in counties with higher poverty rates and states with below-median state education spending. As test score effects predict adult outcomes, these findings point to economic and public health challenges when affected children become adults.

In this paper, I analyze the impact of culture and metaphor on cryptology education. I will compare and contrast the historically grounded metaphors of cryptology-is-warfare and encryption-is-security to a set of counter-metaphors: cryptology-is-privacy and encryption-is-communication. Using this explicit understanding of conceptual metaphor, I present design recommendations and metaphorizing activities for educators building their undergraduate cryptology course.

This paper describes how the notion of the strongly didactic contract can serve to characterize the teaching adopted to implement a task in probability. It is particularly focused on the reality of mathematical work performed by students and teachers. For this research, classroom sessions were developed in an in-service teacher training course designed (and adapted) according to the Japanese Lesson Study model. Through the combined use of the Theory of Didactical Situations (TDS) and the Theory of Mathematical Working Spaces (ThMWS), a coding of the sessions observed was developed. Based on this coding, different patterns emerged which gave each session a specific rhythm and identity from which it was possible to recognize and characterize different strongly didactic contracts. The study highlights the difference between the potential contracts intended by the teachers and those observed in practice. The tools, and especially the coding, developed for the study could be used for future research on instructional situations or in-service teacher training.

The present paper aims at investigating the relationship between students' attitudes toward online homework use and mathematics anxiety among secondary school students. In this study, a model has been proposed by integrating the technology acceptance model (TAM) with mathematics anxiety theory. Three hundred and forty-five secondary school students participated in the instrument of the study that included 20-item based on a 5-point Likert scale. The subsequent analysis using Partial Least Squares Structural Equation Modeling (PLS-SEM) supported the negative relationship between attitude toward an online homework (OHW) tool and mathematics anxiety. The results revealed that perceived usefulness and perceived ease of use are predictors of attitude toward the use of OHW. This study maximized the capability of acceptance of the OHW tool in mathematics learning. The findings provided by the study will encourage mathematics educators to implement mathematics OHW tools in the learning process.

This paper reports on students' conceptions of minima points. Written assignments and individual interviews uncovered salient, concept images, as well as erroneous "mis-out examples" that mistakenly regard examples as non-examples and "mis-in examples" that mistakenly grant non-examples the status of examples. We used Tall and Vinner's theoretical framework to analyze the students' errors that were rooted in mathematical and in real-life contexts.

In this paper we discuss the types of knowledge used by six middle school students as they engaged with a debugging task designed to integrate ideas from computer science, mathematics and science. Our findings show that the computational thinking practice of debugging is a rich source of opportunities to integrate these different disciplines. The analysis illustrates how the types of knowledge the students did and did not use at each step of the debugging process were related to their ability to succeed at each step. Our work contributes to theory and practice by uncovering implications for studying debugging through two refined frameworks and for designing debugging tasks to support transdisciplinary learning.

Mathematics anxiety can be defined as negative feelings towards mathematics and mathematical operations in general. Math anxiety is seen in many students, and even in parents. In fact, parents' own anxiety regarding mathematics may lead to their children experiencing angst over the subject as well. This can be a problem considering parents are considered one of the basic components of education and have important responsibilities to guide the education process of their children. Mathematical literacy is one of the research topics that have come into prominence in recent years due to PISA. Consequently, the purpose of this study is to determine the relationship between the parents' mathematical literacy self-efficacy and mathematics anxieties. Additionally, through examining the participation of parents in mathematic teaching during the COVID-19 pandemic, it is hoped to better understand their views and methodologies. All participants in this study are parents themselves. The study is designed with an explanatory sequential design among mixed method research designs. The grade in which the students are educated does not make any difference in the self-efficacy and anxiety perceptions of the parents regarding their mathematics literacy. Also, the parents' mathematical literacy self-efficacy and anxiety levels were high. [Paper presented at International Congress of Educational Research (13th, Hatay, Turkey, Sep 3-6, 2020).]

Implementing an equity agenda in the classroom is both necessary and challenging for classes containing pre-service teachers. For this intervention, we chose to begin our History of Mathematics course for preservice middle school teachers with a cultural simulation training exercise: Bafa Bafa. After participating in the exercise, pre-service teachers were asked to write a reflection paper and were later interviewed about their experiences in the course. Although participants found Bafa Bafa an uncomfortable experience, it was not an unsafe one, and the preservice teachers agreed that this exercise helped them better understand, articulate, and notice experiences with microaggressions.

Computational Thinking (CT) has emerged in recent years as a thematic trend in education in many countries and several initiatives have been developed for its inclusion in school curricula. There are many pedagogical strategies to promote the development of elementary school students' CT skills and knowledge. Unplugged learning tasks, block-based programming projects, and educational robotics are 3 of the most used strategies. This paper aimed to analyze the effect of Scratch-based activities, developed during one scholar year, on the computational thinking skills developed and concepts achieved by 4th-grade students. The study involved 189 students from two school clusters organized into an experimental group and a control group. To assess students' computational knowledge, the Beginners Computational Thinking Test developed by Several Zapata-Cáceres "et al." (2020) was used. The results indicate statistically significant differences between the groups, in which students in the experimental group (who performed activities with scratch) scored higher on the test than students in the control group (who did not use Scratch).

Problem-solving and enquiry-based learning are integral in the Mathematics and Science curriculum in South Africa through online-based assessment. Online assessment has rapidly gained recognition because of technology. In this paper, the authors explored the views of online baseline assessment among Senior Phase First-year Mathematics Student Teachers. The researchers adopted a qualitative research method. Data was collected from the participants using an online Google form developed into a questionnaire. The participants in this study were first-year students enrolled for the Bachelor of Education in Mathematics at a university in a rural province of South Africa. The first-year student teachers were exposed to the senior phase baseline assessments through the licensed online Computer Aided Mathematics Instruction (CAMI) tool. One hundred and sixteen (116) senior-phase student teachers completed the online questionnaire. This study used convenience sampling since it was the most appropriate method to conveniently invite the participant. The findings revealed that first-year student teachers were enthusiastic and motivated to write the baseline assessment on computers for the first time. The result also showed that several first-year student teachers experienced difficulties solving mathematics problems using computers. This study recommends more intensive research on student teachers' views on online assessment regarding gender and time.

What do ethical relations look like in the context of the many injustices that pervade mathematics education? In this paper, I argue, first, that violence is the relation that characterizes much of contemporary mathematics education and, second, that understanding ethical relations requires considering mathematics as an equal actor in creating possible relations rather than simply treating it as a context for human relations. I examine how literature in care theory, emancipatory pedagogies, and mathematics education have framed ethical relationality and suggest that the feminist new materialist conceptualization of response-ability offers several contributions for rethinking agency, justice, and praxis for mathematics teachers concerned with addressing mathematical violence.

Educational communities have expanded 3D printing in education, emphasizing the educational value of "creation" to develop the competencies required in the future. This study was conducted for the purpose of identifying the effect size by synthesizing prior studies on the effect of 3D printing in education. To achieve the goal, 26 research papers on 3D printing in education conducted in South Korea from 2014 to 2021 were selected and meta-analysis was performed. The results of meta-analysis are as follows. First, the overall effect size of 3D printing in education was 0.759, which was found to be a medium effect size. Second, the effect sizes for each categorical variable are as follows: both the affective domain and the cognitive domain showed medium effect sizes for the learning effects. Middle school showed a large effect size, and elementary school and high school had a medium effect size. Gifted students had a large effect size, and general students had a medium effect size. In terms of the concentrated subjects, mathematics showed a large effect size and science and technology showed a medium effect size. Single subject showed a large effect size and convergence education had a medium effect size. Both offline classes and blended classes showed increased effect sizes. Both the team activity and the individual activity showed a median effect size. The out-of-school program showed a large effect size, and the school program showed a medium effect size. 3D pens showed a large effect size, and 3D printers showed medium effect size. In terms of the modeling type, unstructured modeling showed a large effect size, as did semi-structured modeling, while semi-structured modeling showed a medium effect size. This study has implications for instructional design that can increase the effect of 3D printing in education.

Collaboration between Mathematicians and Mathematics Educators is crucial in advancing knowledge on the teaching and learning of Mathematics, particularly in advanced Mathematics pedagogy. However, there is a need for the type of collaborations where Mathematicians and Mathematics Educators can find common ground, and the synergy of their expertise results in new, hybrid meanings and understandings that can benefit practice. This paper aims to help researchers from these communities come together by presenting a novel methodology for collaborative inquiry and qualitative data analysis--"dialogical inquiry"--based on Mikhail Bakhtin's theory of dialogism. In this framework, "truth" is conceived as never finalised but always coevolving through dialogue that involves participants with different "voices" and "languages". We describe the three principles that underpin "dialogical inquiry": motivation, power balance and a process for solving disagreements. These three principles interact together to create a space where critically productive dialogue allows for meanings to coevolve and new, hybrid understandings to emerge. We illustrate our operationalisation of these principles (i.e., the methods of "dialogical inquiry") in two areas: understanding solutions to linear ordinary differential equations and making meaning of the Bakhtinian concept of superaddressee. We reflect on our use of this methodology in Mathematics Education research and invite the readers to create their own dialogic spaces of collaboration.

Understanding children's attitudes towards mathematics provides insights into their lived mathematical experience and engagement. Despite the considerable amount of research into students' attitudes toward mathematics, limited research has been conducted into "young children's" attitudes toward mathematics (YCATM). Within this limited research, investigating YCATM has certain challenges. From a methodological perspective, limitations exist regarding the type of research techniques that can be employed to study the nuances of the issue. The foci of this paper are to present and evaluate a methodological approach that used children's drawings (N = 106) and interview responses as the primary sources of data. Findings indicate that the strategy of "Draw yourself doing mathematics", when used with other research methods, generated rich attitudinal data in the form of personal stories about YCATM.

This study focuses on a case study that highlights the mathematical discourse developed by two pairs of students when dealing with a specific transition task, i.e., an activity leading to the construction of a graph of a function based on the exploration of another representation of the same function. Such a task was designed to work on the "transition beyond" that involves moving from the graph of a function in a dynamic geometry environment to the Cartesian graph of the same function in the paper-and-pencil environment. In this case study, I analyze in fine-grained detail the discourse developed by two pairs of high-school students (ages 15-16) and describe how they translate the dynamism of the proposed representation into the paper-based context. The analysis aims at investigating the potentialities of transition tasks for supporting the building of bridges between multiple representations of the same function. The analysis also showcased the important role dragging routines played for making the transition.

This study aims to examine adaptability for educational games in terms of adaptation elements, components used in creating user profiles, and decision algorithms used for adaptation. For this purpose, articles and full-text papers in Web of Science, Google Scholar, and Eric databases between 2000-2021 were searched using the keywords "educational games", "serious games", "game-based learning", "adapt*", "player modeling", "user modeling". After applying the inclusion and exclusion procedures of studies accessed in the search, 26 studies were included in the study. The studies were analyzed in line with the themes determined for the components used in the adaptation of educational games. According to the results, adaptive educational game design was made for a wide variety of fields such as programming teaching, physics, mathematics, computational thinking, and logic. As for adaptive factors; It was determined that adaptations were made for the game, educational content, interface, and non-player character (NPC) behaviors. It is understood that pre-game adaptation and in-game adaptation methods are used as adaptation types. Finally, it is seen that Bayesian networks, artificial neural networks, fuzzy logic, deep learning, item response theory, and decision tree methods are preferred as decision systems in the adaptation process. The findings of this literature review can facilitate the design process by providing a roadmap for researchers interested in adaptive educational game design.

This paper reports on a transdisciplinary approach to science with a Year 4/5 class incorporating citizen science through the Birds in Backyards project. This transdisciplinary approach created opportunities for student engagement through science, mathematics, design and technology, humanities and social sciences (HASS), arts and English, while also creating meaningful connections to nature and the local environment.

This paper outlines the development of a comparative research framework in subject didactics and applies this in the process of analysing the transformations from academic disciplines across different school subjects. The theoretical framework builds on the concepts of 'powerful knowledge' and 'transformation' and 'epistemic quality' within which transformation processes from the classroom to the societal level are considered as 'trajectories of powerful knowledge and epistemic quality'. The framework is used to analyse the findings from recent empirical studies across school subjects that have been reported on in publications arising from the Knowledge and Quality across School Subjects and Teacher Education (KOSS) network. The paper then focuses on analysing the transformations from disciplines across school subjects, given that the first boundary in defining powerful knowledge concerns knowledge that is specialized in both how it is produced and transmitted. To analyse this boundary, the findings from the empirical studies are grouped into broad subject categories. These are then compared with the corresponding disciplines by using the widely cited Biglan classification scheme of academic disciplines in higher education. Finally, we consider the implications for curriculum planning and teacher education policy and reflect on the concept of subject-specific educational content knowledge (SSECK).

This special issue introduces recent research on mathematics in interdisciplinary STEM education. STEM education is widely promoted by governments around the world as a way of boosting students' interest and achievement in science, technology, engineering, and mathematics and preparing STEM-qualified workers for twenty-first century careers. However, the role of mathematics in STEM education often appears to be marginal, and we do not understand well enough how mathematics contributes to STEM-based problem-solving or how STEM education experiences enhance students' learning of mathematics. In this survey paper, we present a narrative review of empirical and conceptual research literature, published between 2017 and 2022. These literature sources are organised by a framework comprising five thematic clusters: (1) interdisciplinary curriculum models and approaches; (2) student outcomes and experiences; (3) teacher preparation and professional development; (4) classroom implementation and task design; and (5) policy, structures, and leadership. We use the framework to provide an overview of the papers in this issue and to propose directions for future research. These include: investigating methods and rationales for connecting the constituent STEM disciplines so as to preserve the disciplinary integrity of mathematics; clarifying what is meant by student "success" in interdisciplinary STEM programs, projects, and other educational approaches; moving beyond classroom practices that position mathematics as just a tool for solving problems in other disciplines; understanding what makes a STEM task mathematically rich; and asking how STEM education research can productively shape STEM education policy.

Background: The presence of African women in mathematics has been nearly invisible. The underrepresentation of African women in this field is a result of their historical socio-political marginalisation. The mathematics discipline is politicised, racialised, and gendered to systematically oppress African women. The mathematics fields continue to be a masculine and white male dominated field, which reinforces and preserves masculine culture which is hostile and unwelcoming to women. African women mathematicians are further oppressed because of their racial and gendered identities in fields that are ideologically founded on proving the racial, gendered, social, cultural, and intellectual inferiority of Africans. Aim: The article aims to exemplify the lived experiences of African women doctoral students in the mathematics disciplines in South African universities. The article critically interrogates the factors that influence the participation, progression, and retention of African female doctoral students in the mathematics disciplines. Setting: The article comes from a larger study which investigated the reasons why African doctorate students do not become academics after they receive their doctorates in science, technology, engineering and mathematics (STEM) disciplines in South African universities. This paper focuses on the experiences of 10 African female doctoral students at five universities in South Africa. The universities were selected because they ranked in the top five in South Africa. Two of the institutions are historically Black universities and the rest are historically white institutions, with one historically Afrikaner-speaking university--between 2019--2021. Methods: This article employed a qualitative research methodology, where semi-structures interviews were conducted with 10 African female doctoral participants in mathematics disciplines in five South African universities and is underpinned by the theory of intersectionality. Results: The findings reveal how interlocking systems of oppression continue to influence the recruitment, retention, and progression of women in the mathematics discipline, thereby providing insight into the mechanisms that need to be altered and/or put in place to actively recruit African female doctoral students and retain them in academic positions. Conclusion: The article concludes that despite the mathematics field proclaiming neutrality and objectivity nevertheless, African women still experience racism, sexism, and classism. The experiences of African women in mathematics are vital to understanding the reasons why there is a high attrition rate of African women in the mathematics discipline in academia and why they do not become academics when they could transform this discipline.

When performing regression analysis, one way to examine the normality of data is to screen outliers. Outliers, on the other hand, do not always have an effect on regression results. In reality, cases with a large amount of residuals that affect regression analysis results are referred to as influential cases. It is important to detect them in the dataset because they can lead to erroneous conclusions. The influence of influential cases has already gotten a lot of attention in the regression literature, while it has gotten a lot less attention in factor analysis. The aim of this paper is to show how influential cases affect factor analysis results when they are detected using the Forward Search algorithm. The data was collected from 686 university students ranging in age from 17 to 30. The data was gathered using the Self-Regulation Scale (SRS). The results revealed that the removal of influential cases had an effect on the observed correlation matrice for the SRS items, the factorability results, the number of dimensions extracted, CFA fit indices, and the amount of factor loadings and associated errors. Later, in light of related literature, these results were discussed and the researchers were recommended to consider the effect of influential when applying factor analysis.

Technology education varies across countries depending on the goals they set to achieve. Therefore, comparative research on different technology education approaches can provide a holistic perspective and contribute to the literature. This paper compared the technology-focused courses offered by primary schools in Turkey and Scotland. A qualitative research design was adopted. Data were collected using document review and analyzed using descriptive analysis. The results pointed to differences in technology policies, manifesting themselves in the curricula of the courses offered by the schools. However, the courses also had something in common in terms of structure, goal, content, and approach to learning and teaching. We discussed the differences and similarities based on literature. In order to reveal different dimensions of technology education, comparative education studies that address different countries can be suggested.

Topology optimization (TO) has been a useful engineering tool over the last decades. The benefits of this optimization method are several, such as the material and cost savings, the design inspiration, and the robustness of the final products. In addition, there are educational benefits. TO is a combination of mathematics, design, statics, and the finite element method (FEM); thus, it can provide an integrative multi-disciplinary knowledge foundation to undergraduate students in engineering. This paper is focused on the educational contributions from TO and identifies effective teaching methods, tools, and exercises that can be used for teaching. The result of this research is the development of an educational framework about TO based on the CDIO (Conceive, Design, Implement, and Operate) Syllabus for CAD engineering studies at universities. TO could be easily adapted for CAD designers in every academic year as an individual course or a module of related engineering courses. Lecturers interested in the introduction of TO to their courses, as well as engineers and students interested in TO in general, could use the findings of this paper.

E-learning is renowned as one of the highly effective modalities of learning. Social learning, in turn, is considered to be of major importance as it promotes collaboration between learners. For properly managing learning resources, recommender systems have been implemented in e-learning to enhance learners' experience. Whilst recommender systems are of widespread concern in online learning, it is still unclear to educators how recommender systems can improve the learning process and have a positive impact on learning. This paper seeks to provide an overview of the recommender systems proposed in e-learning between 2007 and the first part of 2021. Out of 100 initially identified publications for the period between 2007 and the first part of 2021, 51 articles were included for final synthesis, according to specific criteria. The descriptive results show that most of the disciplines involved in educational recommender systems papers have approached e-learning in a general way without putting as much emphasis on social learning, and that recommender systems based on explicit feedbacks and ratings were the most frequently used in empirical studies. The synthesis of results presents several recommender systems types in e-learning: (1) content-based recommender systems; (2) collaborative-filtering recommender systems; (3) hybrid recommender systems; and (4) recommender systems based on supervised and unsupervised algorithms. The conclusions reflect on the almost lack of critical reflection on the importance of addressing recommender systems in social learning and social educational networks in particular, especially as social learning has particular requirements, the weak databases size used in some research work, the importance of acknowledging the strengths and weaknesses of each type of recommender system in an educational context and the need for further exploration of implicit feedbacks more than explicit learners' feedbacks for more accurate recommendations.

In this article, the author presents a paper folding activity that can be used for teaching fractions. This activity can be used to describe areas of folded polygons in terms of a standard unit of measure. A paper folding fractions worksheet and its corresponding solutions are also presented in this article. (Contains 2 figures.)

Several researchers have noted how children's whole number schemes can interfere with their efforts to learn fractions. An Australian study found that children who were successful with the solution of rational number tasks exhibited greater whole number knowledge and more flexible solution strategies. Behr and Post (1988) indicated that children needed to be competent in the four operations of whole numbers, along with an understanding of measurement, for them to understand rational numbers. This paper describes a "hands on" approach developed by researchers that focuses on the use of paper folding, fraction walls and number lines to develop an understanding of fractions using a measurement model. (Contains 8 figures.)

With the rapid application of blended learning around the world, a large amount of literature has been accumulated. The analysis of the main research topics and development trends based on a large amount of literature is of great significance. To address this issue, this paper collected abstracts from 3772 eligible papers published between 2003 and 2021 from the Web of Science core collection. Through LDA topic modeling, abstract text content was analyzed, then 7 well-defined research topics were obtained. According to the topic development trends analysis results, the emphasis of topic research shifted from the initial courses about health, medicine, nursing, chemistry and mathematics to learning key elements such as learning outcomes, teacher factors, and presences. Among 7 research topics, the popularity of presences increased significantly, while formative assessment was a rare topic requiring careful intervention. The other five topics had no significant increase or decrease trends, but still accounted for a considerable proportion. Through word cloud analysis technology, the keyword characteristics of each stage and research focus changes of research were obtained. This study provides useful insights and implications for blended learning related research.

Mathematical competitions feature in most developed countries as a part of the secondary school experience, but to a lesser extent in developing countries. In this paper we investigate how widespread these competitions are in Africa, both historically as well as currently, focussing both on national Olympiads and on the participation of African countries in competitions beyond their borders, such as the continent-wide Pan African Mathematical Olympiad and the global International Mathematical Olympiad. Since we could not find a consolidated source of information on African competitions, we include many details and links, hoping that this paper will serve as a valuable starting point for future investigators. Next, we present some survey findings on the attitudes towards and experiences of mathematical competitions, both from the perspective of teachers in secondary schools as well as from secondary school students, and whether these experiences have served to improve or degrade their perceptions of mathematics in general. Among other results, we find that teachers view their students' participation in mathematics competitions as being important, with the biggest barrier to participation being that there are not enough competitions at appropriate levels available, and that students enjoy mathematics competitions and are more likely to pursue a STEM career because of them. Finally, we present some suggestions on how this research may be taken further, and on how the state of mathematical competition culture in Africa may be improved.

The power and value of mathematics cannot be underestimated in today's society. Thus, it is pertinent that education systems worldwide help students to develop an appreciation of the subject. However, research has shown that students often lose interest in science, technology, engineering and mathematics subjects, particularly mathematics, as they cannot see the relevance of it in their lives. This, in turn, leads to the development of negative attitudes towards mathematics, which have been shown to have a knock-on effect on students' academic performance, their participation and engagement with and motivation for the subject. The purpose of this paper is to describe and evaluate the impact of an innovative programme, known as Career Mathways, designed to improve secondary school students' attitudes towards mathematics by highlighting the applicability and usefulness of the subject in a multitude of careers. Career Mathways was piloted in five Irish secondary schools in the 2018/2019 academic year. It involved professionals (n = 13) from a variety of industries sharing insights with students (n = 156) and teachers (n = 6) about how they use mathematics in their day-to-day working lives. Based on these insights, a suite of resources for use in the classroom was developed and implemented with students across the five schools. This paper reports on the impact that this pilot study had on students' attitudes towards mathematics. The results from pre- and post-student surveys and teacher interviews show that Career Mathways had a moderate but positive influence on students' appreciation of the usefulness of mathematics and this impact was more profound among lower-achieving students.

Compared with the application of Digital Twin (DT) in the industrial field, the application of DT in the field of education is still in its infancy. In this paper, a Digital Twin Campus (DTC) for teaching and learning is proposed. It is argued that DTC possesses two characteristics. First, DTC has a wide variety of employment orientations for students or teachers. Second, teaching-learning resources in DTC system is huge but divisible. These characteristics result in the optional difficulty for teaching-learning objects, i.e., curriculum resources, book information resources, and electronic-article resources. To solve this problem, this paper proposes a Double-layer Collaborative Filtering Algorithm Framework (DCFAF) to recommend teaching-learning objects for digital twin teachers or students in DTC. The recommended objects will be further optimized by simulation and prediction in the virtual space of DTC. DCFAF is designed based on the principle that similar teachers (or students) may prefer similar teaching-learning resources, and for the first time, it is given double-layer property by applying the divisible characteristics of teaching-learning resources. The double-layer property can effectively solve the problem of data sparsity in collaborative filtering algorithms. Finally, the superiority of DCFAF is verified on benchmark data sets including MovieLens100k and MovieLens1M which possess the above two characteristics of DTC. DTC constructed in this way can be expected to carry out simulation, prediction, optimization and feedback continuously with the help of DCFAF, so as to realize deep integration of the teaching-learning process between real campus and virtual campus to some extent.

Numerous preservice primary school teachers begin their training with a negative attitude towards mathematics: this phenomenon can have important consequences for their training path and for their future teaching. How teachers' training programs could affect preservice teachers' attitudes is one of the main issues in research on future teachers in the field of mathematics education. However, longitudinal studies developed during specific teacher development programs are still rare in the research field of teacher development. In this frame, this paper reports on the results of a longitudinal 2-year study carried out within the initial training of 35 preservice primary school teachers. According to the three-dimensional model of attitude towards mathematics, we developed the analysis of future teachers' emotional disposition towards mathematics, vision of mathematics, and perceived competence in mathematics, at the beginning of the training path and after 2 years. We found two main results on two different levels. First, a positive change in future teachers' attitude towards mathematics, generalized over all the three dimensions, occurred during the training period, and this result opens the issue of the relationship between formative choices and change in attitude. Second, our data confirm the strong relationship between the three dimensions of attitudes: every change in one dimension affects the other two dimensions.

Multicultural or integrative art education helps students understand themselves and the diverse changing visual world around them. This article highlights the the tradition of Chinese joss paper and shows how it can be added to the repertoire of a multicultural art curriculum or an integrative art curriculum that enables the teacher to infuse social studies, math, and science by offering probable connections embedded in this artistic practice. Exploring the context and content of Chinese joss paper through both artistic and spiritual lenses, the authors first examine this tradition and its origins and discuss its visual characteristics and the associated meanings of both traditional and modern joss paper. Next, they suggest several classroom activities for high school students to give them a better understanding and appreciation of this cultural practice.

Conventional approaches to assessment involve teachers and examiners judging the quality of learners work by reference to lists of criteria or other 'outcome' statements. This paper explores a quite different method of assessment using 'Adaptive Comparative Judgement' (ACJ) that was developed within a research project at Goldsmiths University of London between 2004 and 2010. The method was developed into a tool that enabled judges to distinguish better/worse performances not by allocating numbers through mark schemes, but rather by direct, holistic, judgement. The tool was successfully deployed through a series of national and international research and development exercises. But game-changing innovations are never flaw-less first time out (Golley, Jet: Frank Whittle and the Invention of the Jet Engine, Datum Publishing, Liphook Hampshire, 2009; Dyson, Against the odds: an autobiography, Texere Publishing, Knutsford Cheshire, 2001) and a series of careful investigations resulted in a problem being identified within the workings of ACJ (Bramley, Investigating the reliability of Adaptive Comparative Judgment, Cambridge Assessment Research Report, UK, Cambridge, 2015). The issue was with the 'adaptive' component of the algorithm that, under certain conditions, appeared to exaggerate the reliability statistic. The problem was 'worked' by the software company running ACJ and a solution found. This paper reports the whole sequence of events--from the original innovation, through deployment, the emergent problem, and the resulting solution that was published at an international conference (Rangel Smith and Lynch in: PATT36 International Conference. Research & Practice in Technology Education: Perspectives on Human Capacity and Development, 2018) and subsequently deployed within a modified ACJ algorithm.

Reflection on the theme of the paper makes one to remember Sir Snow's 1959 lecture on The Two Cultures in which Snow had advanced the thesis that the sciences and the humanities had become split into two cultures and had argued that this division had become a major handicap to solving the world's problems. Mathematics education itself has always been negatively be influenced by this rift. For example, in educational context it is frequently claimed that "mathematics is a language since it provides both a conveyance for and a substantiation of our thoughts. It is that aspect of mathematics that explains the key role it plays in modern science" (EFFROS 1998, p.132). But mathematics is no mere language, as everyone will find out if he tries to order a new door to his apartment with words alone and without measurements. The paper will argue historically and in semiotic terms proposing a methodology based on the conception of a complementarity of meaning and reference of knowledge representations.

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.

Quantitative reasoning and interdisciplinary skills are central to real-world environmental problem solving. Enhancing those skills for students in environmental programs will help them succeed as future environmental professionals. This paper describes an approach that uses an applied geophysical imaging course to enhance quantitative reasoning and interdisciplinary learning in an environmental geography program. To adapt the course to geography students, applied learning is emphasized through the high-impact educational practices (HEPs) of undergraduate research and service learning. Throughout the course, students learn the theories of, and utilize electrical resistivity (ER), ground penetrating radar (GPR), and electromagnetic (EM) induction methods to answer real-life environmental questions in the local community. Course evaluations indicate that the course produced positive learning outcomes consistent with the course objectives. Similarly, students appreciate the unique opportunity to learn and utilize these technologies that are not commonly found within geography programs. The teaching strategies described in this paper can benefit other faculty contemplating curricular integration for interdisciplinary learning and quantitative reasoning outcomes.

In this paper the challenge of promoting computational thinking for all by contextualized computing education is addressed. The two phases learning session we designed was implemented in the AlgoRythmics environment which includes ten algorithmic dance choreographies (and attached interactive computer animations). In addition to previous studies that had focused only on supporting students in assimilating the strategy of the studied algorithms, this study examined whether they are able to build on this knowledge by extracting from visualizations some algorithm efficiency related concepts too. To this end, the learning environment was complemented/supplemented with targeted questioning (with and without teacher guidance). Participants (first year undergraduate students) were grouped based on the number of years they had learned programming in high school (0, 1/2 or 4 years). We performed two analyses: (1) group-0 vs. group-1/2 and group-4 (N=181; self-paced setting: algorithm visualization complemented with teacher prepared questions); (2) group-0 (one-group pretest-posttest design, N=46; supplementary Socratic questioning). Research results revealed that the AlgoRythmics environment, if complemented/supplemented with tutorial questionasking, could be an effective instrument in introducing students with no prior knowledge in computing (group-0), even with deeper Computer Science concepts such as algorithm efficiency.

The research on which we report in this paper was framed within the socioepistemological theory and dealt with the fundamental role of variational practices in the understanding of COVID-19 pandemic graphs. Given the proliferation of mathematical graphs related to the pandemic in the media, we proposed to analyse the variational practices in use in the interpretation of graphs of exponential behaviour that a Chilean newscast of high rating used to inform the population about the pandemic in Chile. For this purpose, a thematic analysis intertwined with documentary analysis techniques was carried out, in order to describe the types of graphs shown in the newscast, and a variational reference system was studied to analyse the use of variational practices. To obtain our results, the types of graphs most frequently shown in the newscast at the beginning of the pandemic were identified and the use of the variational practices of "comparison," "seriation" and "prediction" in such graphs was analysed. In conclusion, the need is discussed for the conceptual evolution of the exponential function to be accompanied by a pragmatic evolution in school that favours the use and development of variational practices in the student body.

Background: The coronavirus pandemic has caused a shift in how many teaching, learning and research activities are conducted internationally. Lockdowns compelled all education sectors, including higher education (HE), to adopt a variety of online learning practices at short notice. As these changes in practice have implications beyond the pandemic, more needs to be understood about settings across the globe where the move to online may pose particular challenges. This study brings a focus to how lecturers in a college of education in Nigeria responded to the professional demands of lockdowns and developed strategies for managing educational activities that have relevance during and beyond the pandemic itself. Purpose: The aim of the study reported in this paper was to examine lecturers' perceptions of the management of teaching, learning and research activities during a lockdown, drawing out longer term implications. Methods: Data were collected through in-depth, one-to-one interviews with six lecturers from a college of education in Nigeria. Interview data were analysed qualitatively. Findings: Detailed analysis indicated that lecturers' strategies included encouraging students to engage in self-directed learning, as they uploaded learning materials on messaging groups and the college's e-learning platform. However, lecturers felt that the quality of teaching and assessment was compromised by factors including: irregular power supply, poor internet connection, high data costs and some students' limited digital skills. A particular concern was students' lack of interaction on the online platforms, which lecturers perceived to result in a more passive teaching and learning process. Lecturers also experienced difficulties undertaking research activities. A range of strategies and implications for the ongoing management of educational activities was discussed. Conclusions: The study offers insights into how some of the challenges encountered could be addressed to support sustainable online and blended teaching and learning in the longer term, in settings internationally where there are barriers to accessing online teaching and learning. Crucial aspects include strengthening educational technology training for lecturers and students, enhancing e-learning platforms for teaching practical courses, and enabling lecturers to develop beneficial and valuable online resources.

Teachers' and students' values influenced the teaching and learning process in the mathematics classroom. Few studies were conducted to compare teachers' and students' value in mathematics learning. This paper explores Malaysian mathematics teachers' and students' values in mathematics learning. Fifty-three mathematics teachers and 354 secondary school students participated in the online survey. This study reports ten semantic differential items and an open-ended question related to mathematics learning. An independent t-test was conducted on the semantic differential items. Thematic coding analysis was performed to code the open-ended questions based on the framework of three intersecting values: mathematical values (the discipline), mathematics educational values (the pedagogy), and general educational values (the ethical and moral principles). The findings showed significant mean differences in mathematics educational values between teachers and students. No significant difference was found in mathematical values between teachers and students. Moreover, the qualitative data confirmed the similarities in mathematical values and differences in mathematics educational values between teachers and students. The current study also discusses possible reasons for the differences in teachers' and students' values. The findings imply that teachers' and students' values are crucial in enhancing teaching and learning in the mathematics classroom.

Purpose: The case study is inspired in Interpretative Phenomenological Analysis (IPA) to make sense of a Relational Model within Artistic Creation (RMAC) developed by the author a long time in their artistic and research practices. Design/methodology/approach: An Interpretative Phenomenological Analysis (IPA) is considered within a case study where the author is simultaneously researcher and participant, using old unsolicited diaries for more than a decade as raw material. Findings: RMAC was previously presented as a tool for creating (maker) as well as a tool for research (researcher) in specific solo performance art projects, being the first time that it is approached through an Interpretative Phenomenological Analysis, bringing new insights into artistic practice and research within their intersections. Originality/value: This paper brings them the use of IPA in a case study with a researcher-participant to make sense of how RMAC, a model already presented formally and developed for many years, appeared and changed the author's professional and personal life. Also, it is given a new study using diaries as material within IPA.

We report on a series of task-based interviews in which nine mathematicians were asked to evaluate a series of six mathematical arguments, purportedly produced either by fellow mathematicians or undergraduate students. In this paper, we attend to the role of context in mathematicians' responses, leading to four themes in expectations when evaluating the proofs that they read. First, mathematicians' evaluations of identical arguments were sensitive to researchers' manipulation of authorship, with most accepting arguments purportedly produced by a colleague while taking a more critical view of that same argument if produced by an undergraduate student. Our thematic analysis of interview responses led to three context-based factors influencing mathematicians' responses when evaluating student-produced texts: "course goals, instructors' expectations, and assessment type." In the final section, we consider implications for researchers focused on understanding common practice amongst mathematicians as well as the pedagogic consequences of our findings for practice in the classroom.

The advantages and disadvantages of ability grouping for student achievement are strongly contested, with studies presenting different results. However, much of this research has focused on class-level or subject-level ability grouping. Relatively less research has focused on school-level ability grouping. More importantly, the role of teacher support has often been neglected in the ability grouping literature. The aim of this study was to shed light on the under-investigated area of school-level ability grouping, with teacher support examined as a crucial theoretical mechanism. We examined whether teacher support plays a mediating and/or moderating role in terms of how school-level ability grouping is related to student achievement in English and mathematics. The participants were 554 Hong Kong secondary students from the high- (Band 1), medium- (Band 2), and low- (Band 3) ability groups. The results indicate that students from high-ability groups enjoyed greater levels of teacher support, which partly explains their higher levels of achievement. This finding supports a mediation mechanism. Moreover, the results also provide support for a moderation mechanism. When students in low-ability groups received high levels of teacher support, they were able to achieve as much as students in the high-ability groups, in English but not for math.

It is generally accepted that teachers' beliefs impact their instructional choices, but characterizations of that relationship are limited in college settings. Furthermore, examinations of instructor beliefs, instruction, and student beliefs together in one setting are rarely described. Based on interviews with two Abstract Algebra instructors, classroom video from three units of instruction, and survey and interview data from students in the classes, this paper examines instructors' stated beliefs, ways these beliefs manifested in their teaching, and students' beliefs across the course. Both instructors made curricular choices clearly aligned with their stated views of the nature of mathematics, learning, and teaching. Day-to-day instructional choices reflected these stated beliefs as well, but the difficulty of material and tensions with other beliefs like the importance of interactivity manifested. Characterizations of the interactivity of classes and placement of the mathematical authority in class are provided through descriptive and quantitative measures. These characterizations of instruction provide nuanced portrayals of classroom norms and changes in those norms throughout the semester. Furthermore, subtle shifts in student beliefs about teaching and learning are noticeable, suggesting students' beliefs about teaching and learning mathematics can be influenced even by modest changes in instructional practice.