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Educational research is repeatedly confronted with the question of its relevance. Current interpretations of relevance narrowly focus on outcomes and impact of research. In this essay, we propose an alternative, ontological conceptualization of relevance, arguing that more is at stake than outcomes and impact. We characterize the ontology of education and learning in terms of people’s meaningful movements in an always changing world and propose that relevance of educational research resides in what we call “ontological synchronization”—continuous attunement to what is happening and matters at hand, and what future is being generated, including what values and judgments researchers themselves perpetuate in society. Such synchronization, we conclude, hinges on a disciplinary and ethical commitment to principles of actuality and generativity. We discuss what such conceptualization of relevance implies for educational research.

This paper reports a systematic literature review of the balance model, an often-used aid to teach linear equations. The purpose of the review was to reportwhysuch a model is used,whattypes of models are used, andwhenthey are used. In total, 34 peer-reviewed journal articles were analyzed, resulting in a comprehensive overview of described rationales for using the balance model, its appearances, situations in which it was used, and the gained learning outcomes. Some trends appeared about how rationales, appearances, situations, and learning outcomes are related. However, a clear pattern could not be identified. Our study shows that this seemingly simple model actually is a rather complex didactic tool of which in-depth knowledge is lacking. Further systematic research is needed for making informed instructional decisions on when and how balance models can be used effectively for teaching linear equation solving.

To celebrate that Hans Freudenthal (1905–1990) founded Educational Studies in Mathematics (ESM) about 50 years ago (in 1968), and that our journal has reached its 100th volume, we asked former editors (Table 1) to select highly significant articles previously published in ESM and to explain their choices. This, admittedly, was a cruel request, and many former editors were in despair which papers to propose given the many excellent publications in the years of their editorship. Yet, on the basis of their own personal criteria, they all responded. Below you see their choices. Their accompanying texts give a flavor of what they valued as editors and would like to see more of in ESM. Because Freudenthal of course could not make his own choice, we chose for him, namely the very first paper in the journal, his introductory address titled Why to teach mathematics so as to be useful. Taken together, they give a sense of the impressive intellectual achievements of mathematics education researchers over the past 50 years.

In this paper, we combine dialogic and embodied theories of learning to create a unified analytic lens. Embodied cognition is a theoretical approach operating under the premise that thinking and communication are multimodal activities. Under this premise, dialogue between learners needs to be conceptualized using a multimodal lens. We identify multimodal voices as speech and movement bundles situated within a learning context and describe a phenomenon that we call Multimodal Dialogue – multimodal interaction between different multimodal voices. To demonstrate this phenomenon, we analyze a learning sequence by two third-grade students who participated in a mathematics lesson aimed to foster embodied learning of proportion. Our analysis zooms in on the phenomenon of a multimodal voice as a speech-and-movement bundle situated within a learning context. We further show how multimodal dialogic gaps – differences between multimodal voices within and between modalities – drive communication and eventual changes in voices.

Digital technology is indispensable for doing and learning statistics. When technology is used in mathematics education, the learning of concepts and the development of techniques for using a digital tool are known to intertwine. So far, this intertwinement of techniques and conceptual understanding, known as instrumental genesis, has received little attention in research on technology-supported statistics education. This study focuses on instrumental genesis for statistical modeling, investigating students’ modeling processes in a digital environment called TinkerPlots. In particular, we analyzed how emerging techniques and conceptual understanding intertwined in the instrumentation schemes that 28 students (aged 14–15) develop. We identified six common instrumentation schemes and observed a two-directional intertwining of emerging techniques and conceptual understanding. Techniques for using TinkerPlots helped students to reveal context-independent patterns that fostered a conceptual shift from a modelofto a modelfor. Vice versa, students’ conceptual understanding led to the exploration of more sophisticated digital techniques. We recommend researchers, educators, designers, and teachers involved in statistics education using digital technology to attentively consider this two-directional intertwined relationship.

In many countries, assessment and curriculum reforms came into being in recent decades. In Iran, an important educational assessment reform took place called Descriptive Assessment (DA). In this reform, the focus of student assessment was moved from a more summative approach of providing grades and deciding about promotion to the next grade to a more formative approach of providing descriptive feedback aimed at improving student learning. In this study, we evaluated how seven fourth-grade mathematics teachers used the principles of DA. Data were collected by a questionnaire on assessment practices and beliefs, lesson observations, and interviews. Although the teachers varied in how they assess their students, in general their assessment practice is by and large in line with the DA guidelines. Nevertheless, in some respects we found differences. When assessing their students, the teachers essentially do not check the students’ strategies and when preparing the report cards, they still use final exams because they do not sufficiently trust the assessments methods suggested by DA. The guideline to use assessment results for adapting instruction is also not genuinely put into action. The article is concluded by discussing dilemmas the teachers may encounter when implementing DA.

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.

This article is based on the panel on inquiry based mathematics education and the development of learning trajectories held at the VARGA 100 Conference. After an introduction presenting the theme and organization of the panel, this article focuses on the diversity of conceptualizations of inquiry based education existing today in mathematics education and their influence on the vision and development of learning trajectories. More precisely, it considers the conceptualizations respectively associated with Realistic Mathematics Education, Genetic Constructivism, Tamás Varga’s educational approach and the Anthropological Theory of the Didactic, presented by the panellists, and also shows the efforts undertaken in European projects to reach consensusal visions. Subject Classification: 97C30Q, 97D10, 97D20, 97D30, 97D40, 97D50

In this paper, we present an escape box as a means to introduce the escape room concept into classrooms. Recreational escape rooms have inspired teachers all over the world to adapt the popular entertainment activity for education. Escape rooms are problem-based and time-constrained, requiring active and collaborative participants, a setting that teachers want to achieve in their classroom to promote learning. This paper explores the adaptation of the escape room concept into educational escape game boxes. These technology-enhanced escape boxes have become hybrid learning spaces, merging individual and collaborative learning, as well as physical and digital spaces. The design of the box with assignments on each side puts users face to face with each other and requires them to collaborate in the physical world, instead of being individually absorbed in a digital world. The developed box is a unique concept in the field of escape rooms; the content is adaptable. This paper describes the process leading to the design criteria, the design process, test results and evaluation, and provides recommendations for designing educational escape rooms.

This paper reports an overview of contemporary research on early childhood mathematics teaching and learning presented at recent mathematics education research conferences and papers included in the special issue (2020–4) of ZDM Mathematics Education. The research covers the broad spectrum of educational research focusing on different content and methods in teaching and learning mathematics among the youngest children in the educational systems. Particular focus in this paper is directed to what lessons can be drawn from teaching interventions in early childhood, what facilitates children’s mathematical learning and development, and what mathematical key concepts can be observed in children. Together, these themes offer a coherent view of the complexity of researching mathematical teaching and learning in early childhood, but the research also brings this field forward by adding new knowledge that extends our understanding of aspects of mathematics education and research in this area, in the dynamic context of early childhood. This knowledge is important for future research and for the development of educational practices.

In electronic learning environments, information about a student's performance can be provided to the student in the form of an inspectable student model. While relatively easy to implement, little is known about whether students use the feedback provided by such models and whether they benefit from it. In this study, the use of inspectable student models in an introductory university statistics course by 599 first-year social science students was monitored. Research questions focused on whether students sought feedback from the student models, which decisions for subsequent study steps they made, and how this feedback seeking and decision making related to results on their statistics exams. Results showed a large variety among students in feedback-seeking and decision-making behavior. Lower student model scores seemed to encourage students to practice more on the same topic and higher scores seemed to evoke the decision to move to a different topic. Viewing frequency and amount of variety in decision making were positively related to exam results, even when controlling for total time students worked. These findings imply that inspectable student models can be a valuable addition to electronic learning environments and suggest that more intensive use of inspectable student models may contribute to learning.

In this study we investigated the structure of quantitative competence of kindergartners by testing a hypothesized four-factor model of quantitative competence consisting of the components counting, subitizing, additive reasoning and multiplicative reasoning. Data were collected from kindergartners in the Netherlands (n = 334) and in Cyprus (n = 304). A confirmatory factor analysis showed that the four-factor structure fitted the empirical data from the Netherlands. For the Cyprus data a one-factor structure was found to have a more adequate fit. Regarding the effect of country on performance, a comparison at item level showed that the kindergartners in the Netherlands outperformed those in Cyprus in the majority of quantitative competence items. Analyses of variance revealed for each country a significant effect of kindergarten year on performance, with children in K2 (second kindergarten year) outperforming those in K1 (first kindergarten year). A statistical implicative analysis at item level revealed that in both countries the relevant implicative chain, showing what successful solving of an item implies for correct solving of another item, reflects by and large the sequential steps mostly followed in teaching kindergartners early number. This sequence starts with counting and subitizing, then continues with additive reasoning and finally multiplicative reasoning. These implicative chains also clearly show that the development of early quantitative competence is not linear. There are many parallel processes and cross-connections between the components of quantitative competence.

Thanks to digital technology, methods for finding and analysing research literature have become dramatically more powerful over the last decades. Also, new bibliometric techniques have been developed and applied to the results of such literature search queries. The application of these bibliometric tools to mathematics education research, however, is rare. In this paper, we explore the value of these techniques for mathematics education research through triangulating bibliometrics and expert findings. To do so, we address the case of instrumental orchestration, and want to know how this notion developed over time and was used in research practices. The results show that bibliometric clustering techniques provided a sense-making sketch of the ‘landscape’ of instrumental orchestration research. Triangulating the bibliometric findings with expert interpretations seemed an appropriate method to set up compact ‘identity cards’. In the case of instrumental orchestration, we identified five main clusters in research literature, characterized by the following labels: Managing teaching complexity, Designing living resources, Teaching with technology, Adult learners, and Interacting with computers. The paper ends with some reflections on the potential of bibliometrics in our field and on future research on instrumental orchestration.

The balance model is often used for teaching linear equation solving. Little research has investigated the influence of various representations of this model on students&rsquo, learning outcomes. In this quasi-experimental study, we examined the effects of two learning environments with balance models on primary school students&rsquo, reasoning related to solving linear equations. The sample comprised 212 fifth-graders. Students&rsquo, algebraic reasoning was measured four times over the school year, students received lessons in between two of these measurements. Students in Intervention Condition 1 were taught linear equation solving in a learning environment with only pictorial representations of the balance model, while students in Intervention Condition 2 were taught in a learning environment with both physical and pictorial representations of the balance model, which allowed students to manipulate the model. Multi-group latent variable growth curve modelling revealed a significant improvement in algebraic reasoning after students&rsquo, participation in either of the two intervention conditions, but no significant differences were found between intervention conditions. The findings suggest that the representation of the balance model did not differentially affect students&rsquo, reasoning. However, analyzing students&rsquo, reasoning qualitatively revealed that students who worked with the physical balance model more often used representations of the model or advanced algebraic strategies, suggesting that different representations of the balance model might play a different role in individual learning processes.

In a three-phase study, with a total of 40 third-grade teachers and their 830 students, teachers were supported to use classroom assessment techniques (CATs) to reveal their students’ knowledge of number operations. In phase I, four teachers and 66 third-grade students participated in five monthly workshops in which CATs were co-designed and their use was discussed. In phase II, the first phase was replicated with four workshops with six different teachers and 148 third-grade students. In these two exploratory phases, we evaluated student achievement on a standardized national mathematics test in a pre-/posttest design and compared changes herein to changes in the national norm sample. In phase III, a control condition was added to the design to experimentally investigate the effect on student achievement with 30 teachers and 616 third-grade students. Teachers were randomly assigned to participate in 0, 1, 2, or 3 1-hour workshops. In all three phases, we found a significant increase in students’ mathematics achievement scores on the standardized mathematics test. In phase III, the increase was significantly larger in the classes of teachers participating in three workshops than in classes with less workshops. Additionally, results from the analysis of classroom observations, feedback forms, and interviews indicate that teachers could easily integrate the CATs into their practice and could gather valuable information on their students. The results from the different phases of this study combined indicate that supporting teachers in their development and use of classroom assessment in mathematics may contribute to the improvement of students’ mathematics achievement.

Teachers are increasingly using video in their lessons, with various aims (e.g., to raise students’ levels of conceptual knowledge or interest). Videos that can be used for educational purposes are numerous, ranging from instruction videos to fiction films. Such videos have different characteristics, for example regarding the amount and structure of information, and the audio-visual presentation. However, guidelines on which video characteristics can help to achieve specific teacher aims are lacking. As a first step towards composing such guidelines, we added a film theory perspective to educational research on videos. The study included seven science teachers, thirteen videos, and 233 students (aged 13–18 years). We used teacher interviews, video analyses, student questionnaires, and a cross-case analysis connecting all the data. Data analysis followed a grounded theory approach, including open and axial coding to structure the data, and the constant comparative method to interconnect them. The results showed that videos that posed questions were associated with an increase in students’ interest, and that highly informative videos with authoritative speakers were associated with an increase in students self-reported conceptual knowledge gains. Moreover, teachers often did not have explicit aims for using a particular video, and they selected and used videos in their lessons intuitively. Stimulating teachers to use videos in a more aim oriented way may make video usage more effective. From these findings we developed a framework to assist teachers in selecting or making videos that match their aims, and a model of possible connections between teacher aims and film types as a first step towards guidelines for teachers using educational videos.

Discovery learning continues to be a topic of heated debate. It has been called a zombie, and this special issue raises the question whether it may be a phoenix arising from the ashes to which the topic was burnt. However, in this commentary I propose it is more like an elephant—a huge topic approached by many people who address different aspects. What is needed in the discussion about discovery learning and related approaches, I argue, is sublation: the kind of lifting up from the one-dimensional discussion between two extremes (minimal guidance vs. direct instruction) that puts an end to the everlasting tug of war by integrating justified concerns from both opposite positions. I evaluate how the different contributions to the special issue help to sublate the discussion about discovery learning. In particular, the case study presented by Trninic illustrates how strong guidance and repetition may be needed for the discovery of something that cannot be told. I further suggest scaffolding, inferentialism, and design research as potential theoretical and methodological ways forward.

Embodied learning environments have a substantial share in teaching interventions and research for enhancing learning in science, technology, engineering, and mathematics (STEM) education. In these learning environments, students’ bodily experiences are an essential part of the learning activities and hence, of the learning. In this systematic review, we focused on embodied learning environments supporting students’ understanding of graphing change in the context of modeling motion. Our goal was to deepen the theoretical understanding of what aspects of these embodied learning environments are important for teaching and learning. We specified four embodied configurations by juxtaposing embodied learning environments on the degree of bodily involvement (own and others/objects’ motion) and immediacy (immediate and non-immediate) resulting in four classes of embodied learning environments. Our review included 44 articles (comprising 62 learning environments) and uncovered eight mediating factors, as described by the authors of the reviewed articles: real-world context, multimodality, linking motion to graph, multiple representations, semiotics, student control, attention capturing, and cognitive conflict. Different combinations of mediating factors were identified in each class of embodied learning environments. Additionally, we found that learning environments making use of students’ own motion immediately linked to its representation were most effective in terms of learning outcomes. Implications of this review for future research and the design of embodied learning environments are discussed. Paid Open Access

Contains fulltext : 201324.pdf (Publisher’s version ) (Closed access) The importance of promoting mathematical creativity in education is increasingly acknowledged. Several strategies have been recommended to foster mathematical creativity such as creating an open atmosphere in the classroom, offering open lessons and possibly enriching mathematics education with ideas from other disciplines and experiences from out-of-school contexts. However, it is not yet clear in what way recommended pedagogical strategies promote pupils' mathematical creativity. Therefore, the purpose of this case study was to gain an in-depth understanding of promoting mathematical creativity in educational practice. To this end, interactions between a teacher and her 22 fourth-grade pupils in three different types of mathematics lessons were investigated. An 'open' in-school mathematics lesson, an 'open' out-of-school mathematicslesson and a regular ('closed') mathematics lesson were video recorded and interactions were transcribed verbatim. Subsequently, dialogic episodes were identified in the video transcripts and were coded on mathematical creative expressions of pupils and strategies used by the teacher. Furthermore, after each lesson the teacher was interviewed regarding her experiences with the given lesson. Interviews were audiotaped, transcribed verbatim and analysed by using constant comparison analyses. Findings indicate that mathematical creativity was only promoted in the two open mathematics lessons. More specifically, mathematical creative expressions were related to longer whole class dialogues in which the teacher created an open atmosphere; she created opportunities for pupils to express their ideas and took these ideas seriously. Although in some episodes of the regular mathematics lesson this open atmosphere was also created, no mathematical creativity occurred. 12 p.

Recent developments in cognitive and educational science highlight the role of the body in learning. Novel digital technologies increasingly facilitate bodily interaction. Aiming for understanding of the body’s role in learning mathematics with technology, we reconsider the instrumental approach from a radical embodied cognitive science perspective. We highlight the complexity of any action regulation, which is performed by a complex dynamic functional system of the body and brain in perception-action loops driven by multilevel intentionality. Unlike mental schemes, functional systems are decentralized and can be extended by artifacts. We introduce the notion of a body-artifact functional system, pointing to the fact that artifacts are included in the perception-action loops of instrumented actions. The theoretical statements of this radical embodied reconsideration of the instrumental approach are illustrated by an empirical example, in which embodied activities led a student to the development of instrumented actions with a unit circle as an instrument to construct a sine graph. Supplementing videography of the student’s embodied actions and gestures with eye-tracking data, we show how new functional systems can be formed. Educational means to facilitate the development of body-artifact functional systems are discussed.

Due to increased use of technology, the workplace practices of engineers have changed. So-called techno-mathematical literacies (TmL) are necessary for engineers of the 21st century. Because it is still unknown which TmL engineers actually use in their professional practices, the purpose of this study was to identify these TmL. Fourteen semi-structured interviews were conducted with engineers with a background in different educational tracks in higher professional education (e.g. civil, chemical, biotechnical and mechanical engineering). As a result of the data analysis, 7 commonly used TmL are identified: data literacy, technical software skills, technical communication skills, sense of error, sense of number, technical creativity and technical drawing skills. Engineers also noted a discrepancy between their education and workplace needs; they characterized mathematics in their education as an island with limited relevance. These findings lead to recommendations for the future of science, technology, engineering and mathematics (STEM) in higher technical professional education that can help students learn STEM for the future.

Reasoning about graphical representations representing dynamic data (e.g., distance changing over time), including interpreting, creating, changing, combining, and comparing graphs, can be considered a domain-specific operationalization of the general twenty-first century skills of creative, critical thinking and solving problems. This paper addresses the issue of how these 21st century skills of interpreting and creating graphs can be supported in a six-lesson teaching sequence about graphing motion. In this teaching sequence, we focused on the potential of an embodied learning environment to facilitate the development of primary school students’ reasoning about motion graphs by having primary school students (9–11 years) ‘walk’ graphs in front of a motion sensor to generate distance-time graphs. We asked: How does students’ reasoning about graphing motion develop over a six-lesson teaching sequence within an embodied learning environment? Based on the collected data, we examined changes in students’ level of reasoning on graph interpretation and graph construction tasks using a repeated measurement design. Additionally, we present two teaching episodes showing instances of how perceptual-motor experiences during the lessons aided students’ reasoning about graphical representations of motion. Results show that students went from iconic understanding towards understanding in which they reasoned based on one or two variables when interpreting and constructing graphical representations of motion events. At these higher levels of reasoning these students showed understanding of modelling motion in line with the intended 21st century skills of generating, refining, and evaluating graphs. Paid Open Access

Histograms are widely used and appear easy to understand. Research never the less indicates that students, teachers and researchers often misinterpret these graphical representations. Hence, the research question addressed in this paper is: What are the conceptual difficulties that become manifest in the common misinterpretations people have when constructing or interpreting histograms? To identify these conceptual difficulties, we conducted a narrative systematic literature review and identified 86 publications reporting or containing misinterpretations. The misinterpretations were clustered and—through abduction—connected to difficulties with statistical concepts. The analysis revealed that most of these conceptual difficulties relate to two big ideas in statistics: data (e.g., number of variables and measurement level) and distribution (shape, centre and variability or spread).These big ideas are depicted differently in histograms compared to, for example, case-value plots. Our overview can help teachers and researchers to address common misinterpretations more generally instead of remediating them each individually. ispartof: Educational Research Review vol:28 pages:1-23 status: published

Vygotsky conceptualized the teaching/learning process as inherently collaborative. We extend prior evaluations of this claim by enlisting eye-tacking instruments to monitor the perceptual activity of four teacher–student dyads, as the student solves a challenging manipulation problem designed to ground the scientific notion of parabolas in their new sensorimotor routines. Analyzing each dyad’s gaze paths led us to model the teaching/learning process as the emergence and dynamic transformation of intersubjective coupling between the student and tutor perception–action systems. While the student’s sensory-motor coordination gradually gravitates toward an effective routine, the tutor’s perception is iteratively launched from the student’s current action, until the tutor detects an optimal moment for verbal intervention. In this micro-zone of proximal development, the student’s motor action comes to align with the tutor’s cultural-perspective strategy. Our elaboration of the cultural–historical approach to teaching/learning draws on research on joint attention and joint action from the cognitive sciences as well as the embodied-design approach from the educational sciences and demonstrates a compatibility of Vygotsky’s heritage and complex dynamic systems theory. Finally, we discuss the educational value of the observed student–tutor intersubjective coupling phenomena, thus grounding the contribution of this multidisciplinary study within educational concerns.

Given the growing interest in the scaffolding process, it is worthwhile to address competing accounts about the origin of this term. The concept was empirically introduced by Wood, Bruner, and Ross in 1976 and has often been associated with the “zone of proximal development” in the writing of L.S. Vygotsky. We trace the origins of it in instances of the term being used by Nikolai Bernstein and Alexander Luria, as well as in Vygotsky’s notebooks. Our historical search helps to highlight the theoretical connection between this metaphor and the teaching/learning versus development opposition, and its relation to motor control development.

In this study, we investigated the development of algebraic reasoning in 65 fifth-grade students who never had algebra instruction before. In the six-lesson teaching sequence on solving linear equations, a hanging mobile, a physical balance model, played the central role. We expected students’ perceptual-motor experiences with this hanging mobile to be beneficial for the development of their reasoning related to linear equation solving. To investigate how students’ reasoning developed, we analysed videos of classroom interactions, students’ written work during the lessons and students’ responses to lesson-specific assessment tasks. Our results reveal that students showed progress in their level of algebraic reasoning and in their writing down of strategies. While working with the hanging mobile, students applied algebraic strategies such as restructuring, isolation and substitution. They later used these algebraic strategies for solving linear equations in new contexts. This suggests that the experiences students gained in the embodied learning environment provided a basis for algebraic reasoning, which appeared to support them when solving systems of symbolically presented linear equations. Paid Open Access

Introduction:Braille readers encounter difficulties when reading mathematical expressions. In this exploratory study, we created a setting to investigate these difficulties.Method:Using a motion-capturing system, we analyzed the tactile strategies of three braille readers while they read mathematical expressions. To compare tactile with visual reading strategies, we also analyzed the oculomotor performance in five print readers.Results:The analysis showed that the two experienced braille readers needed about 3.5 times as much time as print readers to read and solve four items involving mathematical expressions. The braille readers used personal reading strategies for all items with little use of the structure of the expression. In contrast, the reading strategies of print readers showed item-dependent and structure-related characteristics.Discussion:The braille readers had difficulties, within the constraints of tactile reading, to align their reading strategies with the solution procedures required by the mathematical structure of the items.Implications for practitioners:Teachers need to become aware of the kind of problems that braille readers confront when they try to comprehend and solve mathematical problems.

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.

In this study, we aimed to gain further knowledge about Brazilian primary schoolteachers' assessment practices and beliefs in mathematics education. For this, we carried out an analysisof curriculum documents and reviewed recent research literature on assessment in Brazil, permitting usto answer the following research question: How are the mathematics assessment practices and teachers' beliefs, asrevealed in recent research, and how are these related to the conceptualization of assessment in Brazilian mathematicscurriculum documents? We found that teachers' assessment practices, as described in the reviewed researchbetween 2010 and 2017, did, in general, not reflect the conceptualization of assessment in thecurriculum documents. Furthermore, despite that the beliefs of most teachers on assessment reflected itto be a process to improve teaching and learning, their mathematics assessment practices did not favorassessment for learning. Still, they were mostly used as an instrument to determine students'classification. We recommend a public policy in which teachers are offered opportunities forprofessional development, focusing on formative assessment in mathematics education. En este estudio se busca obtener un mayor conocimiento sobre prácticas y creencias enrelación a la evaluación en educación matemática de docentes de primaria. Realizamos un análisis de losdocumentos curriculares y revisamos la literatura reciente sobre investigación en evaluación en Brasil,respondiendo a la siguiente pregunta de investigación: ¿Cuáles son las prácticas de evaluaciónmatemática y las creencias de los docentes, reveladas en investigaciones recientes, y cómo se relacionancon la conceptualización? evaluación en documentos curriculares brasileños? Encontramos que lasprácticas de evaluación de los docentes, según encuestas revisadas entre 2010 y 2017, en general noreflejaban la conceptualización de la evaluación presentada en los documentos curriculares. Aunque lamayoría de los profesores conjeturan que la evaluación es un proceso para mejorar la enseñanza y elaprendizaje, sus prácticas de evaluación en matemáticas no favorecieron la evaluación para elaprendizaje, sino que se utilizaron principalmente como un instrumento para determinar lascalificaciones de los estudiantes. Recomendamos una política pública de formación docente en la que seofrezcan oportunidades de desarrollo profesional con un enfoque en la evaluación formativa eneducación matemática. Neste estudo, procuramos obter mais conhecimento sobre as práticas e as crenças emrelação à avaliação em educação matemática de professores dos anos iniciais. Realizamos uma análisedos documentos curriculares e revisamos a literatura recente sobre pesquisas em avaliação no Brasil, respondendo a seguinte questão de pesquisa: Como são as práticas de avaliação matemática e as crençasdos professores, reveladas em pesquisas recentes, e como essas estão relacionadas à conceituação deavaliação em documentos curriculares brasileiros? Verificamos que as práticas de avaliação dosprofessores, conforme pesquisas revisadas entre 2010 e 2017, em geral não refletiam a conceitualizaçãoda avaliação apresentada nos documentos curriculares. Apesar da maioria dos professores conjeturaremque a avaliação seja um processo para melhorar o ensino e a aprendizagem, suas práticas de avaliaçãoem matemática não favoreciam a avaliação para aprendizagem, mas foram usadas principalmente comoum instrumento para determinar a classificação dos alunos. Recomendamos uma política pública deformação de professores na qual sejam oferecidas oportunidades de desenvolvimento profissional comfoco na avaliação formativa em educação matemática.&nbsp

Since the late 1960s, a reform in mathematics education, which is currently known under the name Realistic Mathematics Education (RME), has been taking place in the Netherlands. Characteristic for this approach to mathematics education is that mathematics is not seen as ready-made knowledge but as an activity of the learner. Although much has been written about the big ideas and intentions of RME, and multiple RME-oriented textbooks have been published, up to now the development of this approach to mathematics education has not been thoroughly investigated. In the research reported in this article, we traced how RME has evolved over the years. The focus in our study was on early addition and subtraction in primary school. For this, we studied RME core curriculum documents and analyzed RME-oriented textbooks that have been published between the onset of RME and the present. We found that the big ideas and teaching principles of RME were clearly reflected in the learning facilitators for learning early addition and subtraction and were steadily present in curriculum documents over the years, although some were made concrete in further detail. Furthermore, we found all RME learning facilitators also to be present in all RME-oriented textbooks, though in some cases in other ways than originally intended. Our research shows the complexity of a curriculum reform process and its implementation in textbooks.

Statistics is a challenging subject for many university students. In addition to dedicated methods of didactics of statistics, adaptive educational technologies can also offer a promising approach to target this challenge. Inspectable student models provide students with information about their mastery of the domain, thus triggering reflection and supporting the planning of subsequent study steps. In this article, we investigate the question of whether insights from didactics of statistics can be combined with inspectable student models and examine if the two can reinforce each other. Five inspectable student models were implemented within five didactically grounded online statistics modules, which were offered to 160 Social Sciences students as part of their first-year university statistics course. The student models were evaluated using several methods. Learning curve analysis and predictive validity analysis examined the quality of the student models from the technical point of view, while a questionnaire and a task analysis provided a didactical perspective. The results suggest that students appreciated the overall design, but the learning curve analysis revealed several weaknesses in the implemented domain structure. The task analysis revealed four underlying problems that help to explain these weaknesses. Addressing these problems improved both the predictive validity of the adjusted student models and the quality of the instructional modules themselves. These results provide insight into how inspectable student models and didactics of statistics can augment each other in the design of rich instructional modules for statistics.

The importance of language in mathematics learning has been widely acknowledged. However, little is known about how to make this insight productive in the design and enactment of language-oriented mathematics education. In a design-based research project, we explored how language-oriented mathematics education can be designed and enacted. We drew on genre pedagogy to promote student proficiency in the language required for interpreting line graphs. In the intervention, the teacher used scaffolding strategies to focus students’ attention on the structure and linguistic features of the language involved in this particular domain. The research question addressed in this paper is how student proficiency in this language may be promoted. The study comprised nine lessons involving 22 students in grades 5 and 6 (aged 10–12); of these students, 19 had a migrant background. In light of the research aim, we first describe the rationale behind our design. Next, we illustrate how the design was enacted by means of a case study focusing on one student in the classroom practice of developing proficiency in the language required for interpreting line graphs. On the basis of pre- and posttest scores, we conclude that overall their proficiency has increased. Together, the results indicate that and how genre pedagogy may be used to help students become more proficient in the language required in a mathematical domain.

Introducing the slope of a curve as the limit of the slope of secant lines is a well-known challenge in mathematics education. As an alternative, three other approaches can be recognized, based on linear approximation, based on multiplicities, or based on transition points. In this study we investigated which of these approaches fits students most by analyzing students’ inventions during a lesson scenario revolving around a design problem. The problem is set in a context that is meaningful to students and invites them to invent methods to construct a tangent line to a curve: an implementation of the guided reinvention principle from Realistic Mathematics Education (RME). The teaching scenario is based on the phased lesson structure of the Theory of Didactical Situations (TDS). The scenario was tested with 44 groups of three students in six grade 9 or 10 classrooms. We classified the strategies used by students and, using the emergent models-principle from RME, investigated to which of the four approaches the student strategies connect best. The results show that the groups produced a variety of strategies in each classroom and these strategies contributed to a meaningful institutionalization of the notion of slope of a curve.

Although learning scientific language is crucial for learning science, many primary school teachers lack the knowledge and skills to support this. The present case study reports on a primary school teacher who learned to use a repertoire of scaffolding strategies for stimulating pupils’ scientific language development in inquiry-based science lessons (14 pupils; grade 4). Teacher support included an instructional sequence, participation in interviews and writing reflective reports. The aim of this study is to identify how the teacher used the scaffolding strategies in a classroom with native speakers and which challenges she experienced during the process. Analysis of lesson transcripts showed that the teacher applied all scaffolding strategies suggested to her. Analysis of interview transcripts gave insight into five categories of challenges the teacher experienced while using scaffolding strategies, including her expectations regarding pupils’ scientific language level and dealing with differentiation in the classroom. The findings show that a teacher can learn to apply multiple scaffolding strategies for stimulating scientific language development. Patterns in the use of scaffolding strategies arose related to the aim of the strategy, the situation (i.e., phase of the empirical cycle and teaching approach) and the required pedagogical content knowledge (and skill) of the teacher.

This study investigated whether providing opportunity-to-learn can improve Indonesian students’ performance in solving context-based mathematics tasks. On the basis of an inventory of Indonesian students’ difficulties with these tasks and an analysis of textbooks and classroom practices, an intervention program for mathematics teachers was developed. This program contained tasks with relevant and essential contexts with missing or superfluous information, but without explicitly given mathematical procedures. The program also comprised guidelines for a consultative teaching approach with metacognitive prompts and questions for discussion to promote reflection in class. A field experiment with a pretest-posttest control-group design was carried out in six junior high schools in Indonesia involving 299 eight-graders. Students in the experimental group made significantly more progress on solving context-based mathematics tasks than students in the control group. Furthermore, an analysis of students’ errors revealed that experimental students made significantly fewer task comprehension errors than control students. These results show that providing opportunity-to-learn, that is offering context-based tasks to students, which require mathematical modeling, and having teachers knowing the characteristics of such tasks and using a consultative teaching approach, can improve students’ ability in solving context-based tasks.

This article reports on a post hoc study using a randomised controlled trial with 31,842 students in the Netherlands and an instrument consisting of 21 paired problems. The trial showed a variability in the differences of students’ results in solving contextual mathematical problems with either a descriptive or a depictive representation of the problem situation. In this study the relation between this variability and two task characteristics is investigated: (1) complexity of the task representation; and (2) the content domain of the task. We found indications that differences in performance on descriptive and depictive representations of the problem situation are related to the content domain of the problems. One of the tentative conclusions is that for depicted problems in the domain of measurement and geometry the inferential step from representation of the problem situation to the mathematical problem to be solved is smaller than for word problems.

To gain insight into how teachers' judgment accuracy can be improved, we investigated effects of cue-type availability. While thinking aloud, 21 teachers judged their fourth grade students' (n = 176) decimal magnitude understanding. Sensitivity (correctly judging what students did understand) did not improve from availability of both answer cues (students' answers to prior practice problems) and student cues (knowledge of students triggered by knowing their names), and was lower when only answer cues were available, compared to only student cues. Specificity (correctly judging what students did not understand) was higher when only answer cues were available, compared to only student cues or both student and answer cues.

Research on solving mathematical word problems suggests that students may perform better on problems with a close to real-life representation of the problem situation than on word problems. In this study we pursued real-life representation by a mainly depictive representation of the problem situation, mostly by photographs. The prediction that students perform better on problems with a depictive representation of the problem situation than on comparable word problems was tested in a randomised controlled trial with 31,842 students, aged 10–20 years, from primary and secondary education. The conclusion was that students scored significantly higher on problems with a depictive representation of the problem situation, but with a very small effect size of Cohen's d = 0.09. The results of this research are likely to be relevant for evaluations of mathematics education where word problems are used to evaluate the mathematical capacity of students.

The purpose of the design-based research reported here is to show – as a proof of principle – how the idea of scaffolding can be used to support primary teachers in a professional development programme (PDP) to design and enact language-oriented science lessons. The PDP consisted of six sessions of 2.5 h each in which twelve primary school teachers took part over a period of six months. It centralised the language support that pupils need to reason during science lessons. In line with the idea of scaffolding, the structure of the PDP targeted teachers' gradual independence in designing lessons. The first research question is how scaffolding was enacted during the PDP. The analysis of video recordings, field notes, researcher and teacher logs, and teacher design assignments focused on the enactment of three scaffolding characteristics: diagnosis, responsiveness and handover to independence. The second research question concerns what teachers learned from the participation in the PDP that followed a scaffolding approach. The data analysis illustrates that these teachers had learned much in terms of designing and enacting language-oriented science lessons. In terms of diagnosis and responsiveness, our PDP approach was successful, but we problematise the ideal of scaffolding approaches focused on handover to independence.

An instantly graphable formula (IGF) is a formula that a person can instantly visualize using a graph. These IGFs are personal and serve as building blocks for graphing formulas by hand. The questions addressed in this paper are what experts’ repertoires of IGFs are and what experts attend to while recognizing these formulas. Three tasks were designed and administered to five experts. The data analysis, which was based on Barsalou and Schwarz and Hershkowitz, showed that experts’ repertoires of IGFs could be described using function families that reflect the basic functions in secondary school curricula and revealed that experts’ recognition could be described in terms of prototype, attribute, and part-whole reasoning. We give suggestions for teaching graphing formulas to students.

Each volume in the series presents state-of-the art research on a particular topic in mathematics education and reflects the international debate as broadly as possible, while also incorporating insights into lesser-known areas of the discussion. Each volume is based on the discussions and presentations during the ICME-13 conference and includes the best papers from one of the ICME-13 Topical Study Groups, Discussion Groups or presentations from the thematic afternoon.

This study reports on a way to address twenty-first-century skills in mathematics education by organizing one-day mathematics challenges in the Netherlands. During such a day, students work in teams in school on an open-ended problem which aims to elicit skills like problem-solving, modeling, collaboration, and communication. The framework and the methodological approach of the maker movement are used to describe and analyze the design of these learning spaces for students and the practices they become engaged in. In this study, two design teams are interviewed and two assignments, including student work, are analyzed. The results show that the maker perspective bears similarities with the problem-solving perspective, but also enriches the problem-solving perspective by emphasizing the importance of tinkering, making something, and working as a community of practice. Emerging task characteristics that afford students’ making processes are the use of a context that is meaningful for students, the low-floor-high-ceiling character of the open problem, and the request for a product. The extent to which the requested product is more context-related or more mathematical depends on the intentions of the task and the interest of the target group. Maker characteristics of the design teams elicit the importance of brainstorms with professionals, time for tinkering with the problem situation, and time for exploring possible student strategies, before the final assignment is developed.

Mathematical and scientific knowledge are integral to preparing our population to be actively engaged and responsible citizens. Science and mathematics education, however, has mainly focused on concepts and skills detached from societal implications. In this paper we present an interdisciplinary international design research study in which we developed teaching and professional development materials connecting mathematics and science education to citizenship education. We outline the design research process, its theoretical basis as well as the design products. The study shows that it is indeed possible to develop such approaches supporting active citizenship and thereby the development of 21st century skills in mathematics education, thus strengthening the role of mathematics education in the STEM field.

Inferentialism, as developed by the philosopher Robert Brandom (1994, 2000), is a theory of meaning. The theory has wide-ranging implications in various fields but this special issue concentrates on the use and content of concepts. The key idea, relevant to mathematics education research, is that the meaning of concepts is understood in terms of their role in reasoning practices. In line with the anti-representationalist literature in mathematics education (e.g., Cobb et al. 1992), Brandom explains the meaning of representations in terms of reasoning practices rather than the possibility of reasoning or making inferences on the basis of representations. This view does by no means diminish the significance of representations (signs, diagrams, graphs, symbols…). Rather, understanding how representations come to be is enriched by appreciating that they gain their meaning in human activities in which, as a matter of course, people exercise reason that relies on particular inferences.

The purpose of this article is to draw the attention of mathematics education researchers to a relatively new semantic theory called inferentialism, as developed by the philosopher Robert Brandom. Inferentialism is a semantic theory which explains concept formation in terms of the inferences individuals make in the context of an intersubjective practice of acknowledging, attributing, and challenging one another’s commitments. The article argues that inferentialism can help to overcome certain problems that have plagued the various forms of constructivism, and socioconstructivism in particular. Despite the range of socioconstructivist positions on offer, there is reason to think that versions of these problems will continue to haunt socioconstructivism. The problems are that socioconstructivists (i) have not come to a satisfactory resolution of the social-individual dichotomy, (ii) are still threatened by relativism, and (iii) have been vague in their characterization of what construction is. We first present these problems; then we introduce inferentialism, and finally we show how inferentialism can help to overcome the problems. We argue that inferentialism (i) contains a powerful conception of norms that can overcome the social-individual dichotomy, (ii) draws attention to the reality that constrains our inferences, and (iii) develops a clearer conception of learning in terms of the mastering of webs of reasons. Inferentialism therefore represents a powerful alternative theoretical framework to socioconstructivism.

To understand how statistical and other types of reasoning are coordinated with actions to reduce uncertainty, we conducted a case study in vocational education that involved statistical hypothesis testing. We analyzed an intern’s research project in a hospital laboratory in which reducing uncertainties was crucial to make a valid statistical inference. In his project, the intern, Sam, investigated whether patients’ blood could be sent through pneumatic post without influencing the measurement of particular blood components. We asked, in the process of making a statistical inference, how are reasons and actions coordinated to reduce uncertainty? For the analysis, we used the semantic theory of inferentialism, specifically, the concept of webs of reasons and actions—complexes of interconnected reasons for facts and actions; these reasons include premises and conclusions, inferential relations, implications, motives for action, and utility of tools for specific purposes in a particular context. Analysis of interviews with Sam, his supervisor and teacher as well as video data of Sam in the classroom showed that many of Sam’s actions aimed to reduce variability, rule out errors, and thus reduce uncertainties so as to arrive at a valid inference. Interestingly, the decisive factor was not the outcome of a t test but of the reference change value, a clinical chemical measure of analytic and biological variability. With insights from this case study, we expect that students can be better supported in connecting statistics with context and in dealing with uncertainty.

Proportional reasoning is important and yet difficult for many students, who often use additive strategies, where multiplicative strategies are better suited. In our research we explore the potential of an interactive touchscreen tablet application to promote proportional reasoning by creating conditions that steer students toward multiplicative strategies. The design of this application (Mathematical Imagery Trainer) was inspired by arguments from embodied-cognition theory that mathematical understanding is grounded in sensorimotor schemes. This study draws on a corpus of previously treated data of 9-11 year-old students, who participated individually in semi-structured clinical interviews, in which they solved a manipulation task that required moving two vertical bars at a constant ratio of heights (1:2). Qualitative analyses revealed the frequent emergence of visual attention to the screen location halfway along the bar that was twice as high as the short bar. The hypothesis arose that students used so-called "attentional anchors" (AAs)-psychological constructions of new perceptual structures in the environment that people invent spontaneously as their heuristic means of guiding effective manual actions for managing an otherwise overwhelming task, in this case keeping vertical bars at the same proportion while moving them. We assumed that students' AAs on the mathematically relevant points were crucial in progressing from additive to multiplicative strategies. Here we seek farther to promote this line of research by reanalyzing data from 38 students (aged 9-11). We ask: (1) What quantitative evidence is there for the emergence of AAs?; and (2) How does the transition from additive to multiplicative reasoning take place when solving embodied proportions tasks in interaction with the touchscreen tablet app? We found that: (a) AAs appeared for all students; (b) the AA-types were few across the students; (c) the AAs were mathematically relevant (top of the bars and halfway along the tall bar); (d) interacting with the tablet was crucial for the AAs' emergence; and (e) the vast majority of students progressed from additive to multiplicative strategies (as corroborated with oral utterances). We conclude that touchscreen applications have the potential to create interaction conditions for coordinating action and perception into mathematical cognition.

In this study, we explored the insights that Chinese primary mathematics teachers gained into their students&rsquo, mathematical understanding from using classroom assessment techniques (CATs). CATs are short teacher-initiated targeted assessment activities proximate to the textbook, which teachers can use in their daily practice to make informed instructional decisions. Twenty-five third-grade teachers participated in a two-week program of implementing eight CATs focusing on the multiplication of two-digit numbers, and filled in feedback forms after using the CATs. When their responses described specific information about their students, emphasized the novelty of the gained information, or referred to a fitting instructional adaptation, and these reactions went together with references to the mathematics content of the CATs, the teachers&rsquo, responses were considered as evidence of gained insights into their students&rsquo, mathematics understanding. This was the case for three-quarters of the teachers, but the number of gained insights differed. Five teachers gained insights from five or more CATs, while 14 teachers did so only from three or fewer CATs, and six teachers showed no clear evidence of new insights at all. Despite the differences in levels of gained insights, all the teachers paid more attention to descriptions of students&rsquo, performance than to possible instructional adaptations.

Hypothesis testing involves a complex stepwise procedure that is challenging for many students in introductory university statistics courses. In this paper we assess how feedback from an Intelligent Tutoring System can address the logic of hypothesis testing and whether such feedback contributes to first-year social sciences students’ proficiency in carrying out hypothesis tests. Feedback design combined elements of the model-tracing and constraint-based modeling paradigms, to address both the individual steps as well as the relations between steps. To evaluate the feedback, students in an experimental group (N = 163) received the designed intelligent feedback in six hypothesis-testing construction tasks, while students in a control group (N = 151) only received stepwise verification feedback in these tasks. Results showed that students receiving intelligent feedback spent more time on the tasks, solved more tasks and made fewer errors than students receiving only verification feedback. These positive results did not transfer to follow-up tasks, which might be a consequence of the isolated nature of these tasks. We conclude that the designed feedback may support students in learning to solve hypothesis-testing construction tasks independently and that it facilitates the creation of more hypothesis-testing construction tasks.

This study investigated Chinese primary school mathematics teachers’ views on assessment in an effort to determine their assessment profiles. A large-scale questionnaire survey with 1101 teachers from 12 Chinese provinces and regions was carried out. The teachers reported to use assessment on a daily or weekly basis for different purposes. They recognized the importance of assessing different types of skills and knowledge and considered assessment useful for improving teaching and learning. To determine teachers’ assessment profiles, we used several latent variable modeling techniques. With exploratory factor analyses, we identified eight factors in the teachers’ responses: general instructional decision-making assessment purposes [1], specific instructional decision-making assessment purposes [2], assessment methods [3], diversity of assessment problem format [4], importance of assessing skills and knowledge [5], importance of assessing extra-curricular skills [6], Perceived usefulness [7], and acceptance of assessment [8]. When these factors were used to interpret the results of a latent class analysis, three distinct assessment profiles could be distinguished. One fifth of the teachers were in the Enthusiastic assessors profile. These teachers not only reported to use assessment frequently [3, 4] and purposefully [1, 2], but also highly endorsed its importance [5, 6] and usefulness [7, 8]. Around half of the teachers were in the Mainstream assessors profile; these teachers scored close to the mean on all factors. The remaining teachers held the relatively negative views on assessment and were therefore in the Unenthusiastic assessors profile. This profile characterization sheds light on Chinese primary school mathematics teachers’ assessment culture.