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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.

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

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 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

In this study, we aimed to gain further knowledge about Brazilian primary school teachers' assessment practices and beliefs in mathematics education. For this, we carried out an analysis of curriculum documents and reviewed recent research literature on assessment in Brazil, permitting us to answer the following research question: How are the mathematics assessment practices and teachers' beliefs, as revealed in recent research, and how are these related to the conceptualization of assessment in Brazilian mathematics curriculum documents? We found that teachers' assessment practices, as described in the reviewed research between 2010 and 2017, did, in general, not reflect the conceptualization of assessment in the curriculum documents. Furthermore, despite that the beliefs of most teachers on assessment reflected it to be a process to improve teaching and learning, their mathematics assessment practices did not favor assessment 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 for professional development, focusing on formative assessment in mathematics education.

Bodily experiences are associated with powerful forms of understanding, yet not much research has investigated to what extent bodily experiences benefit the development of graphical reasoning. We examined the effectiveness of providing embodied support in a teaching sequence of six lessons on motion graphs, including both graph interpretation and graph construction activities, on fifth-grade students' reasoning about graphically represented motion. Divided over nine classes 218 students took part in our study. Students in three classes received lessons on graphing motion with direct embodied support, three classes received lessons on graphing motion with indirect embodied support, and three classes served as a baseline condition and received lessons on a different mathematics topic. Development of students' graphical reasoning was measured on four measurement occasions. All students were given these same tasks four times with two months intervals. The teaching sequence on graphing motion took place either after the first, second, or third measurement. We used a cohort-sequential design to assess the intervention effect, the condition effect and the fading effect. Results showed that students improved their graphical reasoning at post-intervention-measurements when compared to their performance before the intervention. Moreover, students in the teaching sequence with direct embodied support showed a slightly larger gain in their graphical reasoning than students in the teaching sequence with indirect embodied support. These results suggest that embodied support as a learning facilitator can improve reasoning about graphing motion in primary school classrooms.

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.

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

Formatief toetsen is hot in het onderwijs. Binnen het basis- en voortgezet onderwijs zijn al legio initiatieven om het gebruik ervan onder de aandacht te brengen. In het mbo wordt hier nog minder gebruik van gemaakt. Toch biedt ook hier formatief evalueren kansen voor docenten en studenten. In het praktijkonderzoek Grip op leerpotentieel, waaraan docenten Nederlands en rekenen van verschillende roc’s deelnemen, worden de mogelijkheden van formatief evalueren in het mbo verkend.

Many students have difficulties giving meaning to algebra. In this study, we investigated how graphing formulas by hand can help grade 11 students to give meaning to algebraic formulas, defined as the ability to “read through” an algebraic formula: i.e., to recognize its structure and to link it to graphical features. During five 90-minute lessons, 21 students worked on graphing tasks focusing on recognition and heuristic search. To assess the effects of this intervention, a graphing task, and a cardsorting task were administered to the students. In addition, six students were asked to think aloud during the graphing task. The results of the card-sorting task showed that 14 students used categories similar to the ones experts use, but had trouble in consistently categorizing all formulas. The thinkingaloud protocols showed that the students improved their recognition of basic functions and graph features, and their qualitative reasoning, which allowed them to give meaning to algebraic formulas

In this paper, we describe four studies that have been done in the project Improving Classroom Assessment in China (ICA-C). The project was a sequel to the ICA project in the Netherlands. To shed light on the current situation of Chinese primary school mathematics teachers’ perception and practice of classroom assessment, we first did a review study based on teacher-written papers addressing classroom assessment. Then a large-scale questionnaire survey was conducted by means of which we could identify different assessment profiles of teachers. With the gained knowledge, two other studies were set up to explore the possibility for improving Chinese primary school mathematics teachers’ assessment activities, with a particular focus on using classroom assessment techniques in the domain of division and multiplication.

Interventions aimed at fostering students’ growth mindsets have been carried out in 21 classrooms (with a focus on Grade 7 and Grade 10) in the Netherlands. The intervention consisted of three main elements: introduction to neuroplasticity, the importance of learning from errors in learning processes, and growth-mindset feedback. Before and after the intervention, the students finished a questionnaire, and during the intervention, students and their teachers were interviewed. Results indicate that the students’ mindsets tended to be more towards a growth mindset after the intervention, in particular for the Grade 10 students. Students were very positive about the intervention, especially the neuroplasticity and the attention for and learning from errors. Teachers and students valued the changes in attitudes and interactions with a new language for teaching and learning mathematics.

The scarce research on teachers indicate that they often misinterpret histograms. We conjecture that the confusion of histograms with case-value plots is the source of many misinterpretations. Therefore, the question for this study is: what are the most common strategies of secondary school teachers when interpreting histograms and case-value plots? To answer this question, we use a method that allows for a more in-depth analysis of twelve teachers’ interpretations of graphical representations than was ever possible before: eye-tracking combined with retrospective verbal reports. Preliminary results show that several teachers apply a case-value plot interpretation strategy on a histogram. Furthermore, some participants use an area interpretation strategy or histogram interpretation strategy applied to a case-value plot. In addition, gaze data suggest that teachers use strategies that did not reach awareness and therefore will not be reported during thinking aloud protocols.

Rationales for interdisciplinary STEM courses are often based on the fact that the problems we face in today’s world call for perspectives and knowledge from many different areas. In many cases this includes mathematics because it is used in many research fields and because it is part of everyday life. At the same time interdisciplinary literature suggests that mathematics gains the least from integration. In this paper we use a successful interdisciplinary STEM course in the Netherlands to illustrate how students and teachers think about the value of mathematics. To analyse teacher and student statements concerning the value of mathematics a model is introduced for a disciplinary mathematics perspective for interdisciplinary STEM courses and the opportunities this model ca provide are discussed.

Drawing statistical inferences (SI) is essential in a society where data are of increasing importance. Understanding the relation between data and chance, necessary to understand statistical inference, is however challenging for students. Technological innovations – such as the Sampler in TinkerPlots (TP) – enable students to investigate this relationship by modeling a population and simulating repeated samples. Along this line, the research reported here presents the results of a pilot with fourteen 9th-grade students, inexperienced with sampling, in a Learning Lab. The pilot focuses on how students use TP as a digital environment for exploring data and chance – i.e. what strengths and constraints they encounter – and how they subsequently use this information for SI. The results suggest that the participating students encountered difficulties in modeling the population, however, they were able to simulate, explore and reason inferentially through repeated sampling in TP.

In this paper, we describe the impact of a mathematics game programming project on the intrinsic motivation of eighth grade students (n = 8). We investigate which aspects of the project contributed to student motivation based on a taxonomy that distinguishes individual (e.g., challenge) and interpersonal motivation aspects (e.g., recognition). We also employ these aspects to compare the programming project to regular mathematics education. The findings reveal that students appreciated the project because it was challenging and they had freedom of choice. All group members had a positive attitude towards learning. In regular mathematics classes, they experienced a lack of challenge while the freedom of choice was minimal.

There is increasing attention for the embodied and extended nature of mathematical cognition but the bodies of literature on embodied and on extended cognition have developed mostly separately. We propose a new step in the tradition of networking theories: design research from two theoretical perspectives to promote integration. Embodied design and instrumental genesis inspired us to elicit embodied instrumentation: learning via techno-physical interaction with digital artifacts. A case study illustrates a design and subsequent problem solving by a student (aged 16), who uses her body and the designed artifact to solve trigonometric equations. We reflect on the benefits of not only analyzing but also designing from different theories to network these theories.

Educational research is often of limited use to designers, as it typically offers descriptions of or explanations for how education was or currently is. In this paper we argue that generalizability is not enough; what deserves more attention is generativity. Designers and educators need actionable knowledge on how education could be. To substantiate the criterion of generativity, we here develop the construct of a design genre—a theory-driven class of designs targeting a particular form of learning. As an example we discuss the evolution of an action-based genre of embodied design through iterative theory-design clarifications. Initially instantiated in the form of activities for promoting the understanding of proportional reasoning, the genre has been elaborated through the process of applying it to additional notions such as parabolas and trigonometry. We argue that this genre was generative for design, theory development, and methodological innovation.

The aim of this study was to investigate the effect of a six-lesson teaching intervention on fifth-grade students’ linear equation solving abilities. A hanging mobile, a balance model consisting of a horizontal beam with on each side a number of bags hanging on a chain, played a central role in this intervention. In total, 213 fifth-graders participated in one of the two intervention conditions or in the control condition. The intervention conditions differed with respect to the type of hanging mobile; either a physical hanging mobile that students could manipulate or a static version on paper. Preliminary analyses of the scores on the pre- and post-test seemed to show an improvement in students’ linear equation solving abilities for students in both intervention conditions. Students, who worked in an embodied learning environment with a physical hanging mobile, seemed to show more improvement than students who worked with a paper-based mobile.

Making connections between variables and proficiently constructing graphical representations is key to higher-order thinking activities within mathematics and science education. In our research, we make use of a learning environment informed by embodied cognition theory to promote students’ graphical understanding. In the teaching sequence offered to fifth graders, students created distance-time graphs describing their own movements in front of a motion sensor. In a pretest-intervention-posttest design, we investigated whether task-related bodily movements enhanced students’ understanding of graphs of motion, as reflected in their competence in interpreting and constructing graphs. Preliminary results point to important links between students’ motion experiences and their ability to reason about the relationship between distance and time as represented in graphs.