Search

This article brings critical and postcolonial theories together with extended ethnographic research in a predominantly Latinx high school in California's "South Bay" to theorize the co-production and co-naturalization of mathematical and racial essentialization. Analysis of vignettes and interview excerpts illuminates both student uptake and resistance to homogenizing narratives of mathematics and racial personhood. Student voices from Sierra High School both evidenced the existence, stakes, and personal consequences of narrowly bounded, taken for granted, conceptions of mathematics and racial personhood and challenged these homogenizing categories. This paper contributes theory illuminating the co-construction of mathematical and racial essentialization along with examples of local critique and resistance from youth and their adult allies at one high school in the USA. Implications suggest that decolonial work in mathematics education must jointly address the narrow and essentializing frames for race and racialization and for mathematics itself as co-producing and co-naturalizing each other. This study contributes insight into mechanisms that perpetuate and also resist or disrupt these processes. [ABSTRACT FROM AUTHOR]

The article discusses the evolving field of mathematics education research and how it is being influenced by changes in society and academia. Topics include the evolving scope of mathematics education research, expanding research boundaries, and the importance of addressing contemporary issues in the field to reflect the evolving identity of mathematics education research.

This paper is a commentary on the classroom interventions on the teaching and learning of proof reported in the seven empirical papers in this special issue. The seven papers show potential to enhance student learning in an area of mathematics that is not only notoriously difficult for students to learn and for teachers to teach, but also critically important to knowing and doing mathematics. Although the seven papers, and the intervention studies they report, vary in many ways-student population, content domain, goals and duration of the intervention, and theoretical perspectives, to name a few-they all provide valuable insight into ways in which classroom experiences might be designed to positively influence students' learning to prove. In our commentary, we highlight the contributions and promise of the interventions in terms of whether and how they present capacity to change the classroom culture, the curriculum, or instruction. In doing so, we distinguish between works that aim to enhance students' preparedness for, and competence in, proof and proving and works that explicitly foster appreciation for the need and importance of proof and proving. Finally, we also discuss briefly the interventions along three dimensions: how amenable to scaling up, how practicable for curricular integration, and how capable of producing long-lasting effects these interventions are. [ABSTRACT FROM AUTHOR]

In this paper, we present a way of describing variation in young children's learning of elementary arithmetic within the number range 1–10. Our aim is to reveal what is to be learnt and how it might be learnt by means of discerning particular aspects of numbers. The Variation theory of learning informs the analysis of 2184 observations of 4- to 7-year-olds solving arithmetic tasks, placing the focus on what constitutes the ways of experiencing numbers that were observed among these children. The aspects found to be necessary to discern in order to develop powerful arithmetic skills were as follows: modes of number representations, ordinality, cardinality, and part-whole relation (the latter has four subcategories: differentiating parts and whole, decomposing numbers, commutativity, and inverse relationship between addition and subtraction). In the paper, we discuss particularly how the discernment of the aspects opens up for more powerful ways of perceiving numbers. Our way of describing arithmetic skills, in terms of discerned aspects of numbers, makes it possible to explain why children cannot use certain strategies and how they learn to solve tasks they could not previously solve, which has significant implications for the teaching of elementary arithmetic. [ABSTRACT FROM AUTHOR]

Social institutions function not only by reproducing specific practices but also by reproducing discourses endowing such practices with meaning. The latter in turn is related to the development of the identities or subjectivities of those who live and thrive within such institutions. Meaning and subjectivity are therefore significant sociological categories involved in the functioning of complex social phenomena such as that of mathematical instruction. The present paper provides a discursive analysis centered on these categories of the influential OECD's PISA mathematics frameworks. As we shall see, meaning as articulated by the OECD primarily stresses the utilitarian value of mathematics to individuals and to society at large. Furthermore, molding students' subjectivities towards endorsing such articulation of meaning is emphasized as an educational objective, either explicitly or implicitly, as connected to the OECD's definition of mathematical literacy. Therefore, the OECD's discourses do not only serve to reproduce the type of mathematical instruction implied in the organization's services concerning education, but also concomitantly provide a potentially most effective educational technology through which the demand of these very services may be reproduced. [ABSTRACT FROM AUTHOR]

Part of the international reflection on the use of history in mathematics teaching consists in a quest of frameworks and models suitable for empirical studies. Following this demand, this paper explores the way Balacheff’s cKȼ model, a model taken from the didactics of mathematics, can be used in the analysis of learning at student level. In the first part of this paper, Balacheff’s cKȼ model (conceptions, knowledge, ȼoncepts) is shortly presented, and in the second part, the relationship between the epistemological background of the model and the use of the history of mathematics is explored in order to show a possible suitableness. The third part addresses an example of a school activity (about ancient Indian geometry) in which the model is applied and the historical issues clarified. Questioning the role of problems both in the cKȼ model and in the use of history, the last part shows how a study at the students’ conception level enlightens the way in which historical elements can interact with contemporary mathematical learning. [ABSTRACT FROM AUTHOR]

In contrast to existing research that has typically addressed the process from example generation to proof construction, this study aims at enhancing empirical examination after proof construction leading to revision of statements and proofs in secondary school geometry. The term 'empirical examination' refers to the use of examples or diagrams to investigate whether a statement is true or a proof is valid. Although empirical examination after proof construction is significant in school mathematics in terms of cultivating students' critical thinking and achieving authentic mathematical practice, how this activity can be fostered remains unclear. This paper shows the strength of a particular kind of mathematical task, proof problems with diagrams, and teachers' roles in implementing the tasks, by analysing two classroom-based interventions with students in the eighth and ninth grades. In the interventions, the tasks and the teachers' actions successfully prompted the students to discover a case rejecting a proof and a case refuting a statement, modify the proof, properly restrict the domain of the statement by disclosing its hidden condition, and invent a more general statement that was true even for the refutation of the original statement. [ABSTRACT FROM AUTHOR]

In this paper, we analyse a large, opportunistic dataset of responses (N = 219,826) to online, diagnostic multiple-choice mathematics questions, provided by 6–16-year-old UK school mathematics students (N = 7302). For each response, students were invited to indicate on a 5-point Likert-type scale how confident they were that their response was correct. Using demographic data available from the online platform, we examine the relationships between confidence and facility (the proportion of questions correct), as well as gender, age and socioeconomic disadvantage. We found a positive correlation between student confidence and mean facility, higher confidence for boys than for girls and lower confidence for students classified as socioeconomically disadvantaged, even after accounting for facility. We found that confidence was lower for older students, and this was particularly marked across the primary to secondary school transition. An important feature of the online platform used is that, when students answer a question incorrectly, they are presented with an analogous question about 3 weeks later. We exploited this feature to obtain the first evidence in an authentic school mathematics context for the hypercorrection effect (Butterfield & Metcalfe J EXP PSYCHOL 27:1491–1494, 2001), which is the observation that errors made with higher confidence are more likely to be corrected. These findings have implications for classroom practices that have the potential to support more effective and efficient learning of mathematics. [ABSTRACT FROM AUTHOR]

This paper draws on data from the development of annual national mathematics assessment materials for 7-year-old pupils in Wales for use during the period 2000–2002. The materials were developed in both English and Welsh and were designed to be matched. The paper reports on item analyses which sought items that exhibited differential performance in relation to whether the materials were English medium or Welsh medium. The items that exhibited consistent differential item functioning in relation to language during pre-testing are reviewed in order to discuss the linguistic factors that could affect such behaviour. [ABSTRACT FROM AUTHOR]

This paper focuses on the kinds of notations young children make for fractional numbers. The extant literature in the area of fractional numbers acknowledges children's difficulties in conceptualizing fractional numbers. Some of the research suggests possibly delaying an introduction to conventional notations for algorithms and fractions until children have developed a better understanding of fractional numbers (e.g., Hunting, 1987; Sáenz-Ludlow, 1994, 1995). Other research (e.g., Empson, 1999), however, acknowledges the interaction between conceptual understandings and the representations for those understandings. Following the latter line of thinking, this paper argues that children's notational competencies and conceptual understandings are intertwined, addressing the following research questions: (a) What kinds of notations do five and six-year-old children make for fractional numbers?; and (b) What can be learned about young children's learning of fractional numbers by analyzing the notations they make? This paper presents data from interviews carried out with twenty-four children in Kindergarten and first grade (five and six-year-old children), exploring the nuances of their notations and the meanings that they attach to fractions. [ABSTRACT FROM AUTHOR]

In this paper, motivated by the desire to understand which forms of parent-initiated activity are productively implicated in young children's mathematics learning, we present a methodological critique of recent research. Many such studies, based on assumptions that parent-initiated activities can be categorised as formal or informal, direct or indirect, or advanced or basic, exploit surveys to elicit how frequently parents engage their children in various predetermined activities. While such survey data have the potential to yield important insights, the analytical procedures typically employed prevent them. Studies involving factor analyses yield uninterpretable factors, which are then used to create summative variables based on the scores of individual activities. Other studies, drawing on untested preconceptions, simply create summative variables. In all cases, these summative variables are based on such a wide range of qualitatively different activities that labels like formal or informal become arbitrary and the potential of individual activities to support learning gets lost beneath colleagues' desires for statistical significance. In closing, we ask colleagues, albeit somewhat rhetorically, what is the purpose of such research? Is it to identify those activities that support learning or to offer statistically robust factors, which, due to the diversity of activities embedded within them, offer few useful insights? [ABSTRACT FROM AUTHOR]

This paper argues that rationality and belief are mutually formative dimensions of school mathematics, where each term is more politically embedded than often depicted in the field of mathematics education research. School mathematics then presents not so much rational mathematical thought distorted by irrational beliefs but rather a particular mode of activity referenced to the performance of certain substitute skills and procedures that have come to represent mathematics in the school context as a result of social management. The paper considers alternative modes of apprehending mathematical objects. Firstly, two accounts of how a young child might learn to point at mathematical entities are presented, where alternative interpretations of this act of pointing are linked to conceptions of enculturation. This comparison then underpins a discussion of how mathematics is produced as entities to be acquired according to certain ideological schema. The resulting cartographic definition of mathematics steers the production then selection of learners according to arbitrary curriculum or assessment criteria. Secondly, some trainee teachers report on shared experience in a spatial awareness exercise concerned with exploring alternative apprehensions of geometric objects. This provides an account of my own teaching and explains why I find teaching mathematics so exciting if it can be linked to the generation of multiple perspectives. The paper's central argument is that rational mathematical thought necessarily rests on beliefs set within a play of ideological framings that within school often partition people in terms of their proxy interface with mathematics. The challenge is to loosen this administrative grip to allow both students and teachers to release their own powers to generate diversity in their mathematical insights rather than conformity. [ABSTRACT FROM AUTHOR]

Mathematics discussions are important for helping students to develop conceptual understanding and to learn disciplinary norms and practices. In recent years, there has been increased attention to teaching prospective teachers to lead discussions with students. This paper examines the possibilities of designing a formative assessment that gathers information about prospective elementary teachers' skills with leading problem-based mathematics discussions and makes sense of such information. A decomposition of the practice of leading discussions was developed and used to design the assessment. Nine first-year teachers who graduated from a range of different teacher education programs participated in the study. The findings reveal that our formative assessment works to gather information about teachers' capabilities with leading discussions and that the associated tools support making sense of the information gathered. This suggests that such tools could be useful to support the formative assessment of the developing capabilities of prospective teachers. [ABSTRACT FROM AUTHOR]

The work that mathematics teachers do is frequently mathematical in nature and different from other professions. Understanding and describing common ways that teachers draw upon their content knowledge in the practice of teaching is important. Building on the descriptions by McCrory et al. (Journal for Research in Mathematics Education 43(5) 584-615, ) of two primary ways teachers use their content knowledge-decompressing and trimming-this paper differentiates between micro- and macro-levels of mathematical complexity as a way to extend and further conceptualize the moves that teachers make at a more nuanced grain size. This paper explains and portrays their counterparts, micro-level trimming and macro-level decompressing, which we discuss as concealing and foreshadowing complexity, respectively. Support for their use in teachers' work is provided through specific examples. Implications for the mathematical preparation and professional development of teachers are discussed. [ABSTRACT FROM AUTHOR]

Research on learner errors in mathematics education is beginning to focus on how teachers can learn to identify and engage with the reasoning behind these errors. Research on professional learning communities is beginning to show that they present powerful opportunities for on-going teacher collaboration and learning. In this paper, I bring the two areas of research together. Drawing on data from one professional learning community in the Data Informed Practice Improvement Project, I show how teachers in this community came to understand key concepts about learner errors and shifted their ways of talking about learner errors. I identify three important shifts that the teachers made in their learning about learner errors: from identifying to interpreting errors; from interpreting to engaging errors; and from focusing on learner errors to focusing on their own knowledge. I argue that these three shifts suggest a deepening of teachers' thinking in relation to learner errors. [ABSTRACT FROM AUTHOR]

A proof is a connected sequence of assertions that includes a set of accepted statements, forms of reasoning and modes of representing arguments. Assuming reasoning to be central to proving and aiming to develop knowledge about how teacher actions may promote students' mathematical reasoning, we conduct design research where whole-class mathematical discussions triggered by exploratory tasks play a key role. We take mathematical reasoning as making justified inferences and we consider generalizing and justifying central reasoning processes. Regarding teacher actions, we consider inviting, informing/suggesting, supporting/guiding and challenging actions can be identified in whole-class discussions. This paper presents design principles for an intervention geared to tackle such reasoning processes and focuses on a whole-class discussion on a grade 7 lesson about linear equations and functions. Data analysis concerns teacher actions in relation to design principles and to the sought mathematical reasoning processes. The conclusions highlight teacher actions that lead students to generalize and justify. Generalizations may arise from a central challenging action or from several guiding actions. Regarding justifications, a main challenging action seems to be essential, while follow-up guiding actions may promote a further development of this reasoning process. Thus, this paper provides a set of design principles and a characterization of teacher actions which enhance students' mathematical reasoning processes such as generalization and justification. [ABSTRACT FROM AUTHOR]

The concept of proof has attracted considerable research attention over the pastdecades in part due to its indisputable importance to the discipline of mathematics and tostudents' learning of mathematics. Yet, the teaching and learning of proof is an instructionallyarduous territory, with proof being recognized as a hard-to-teach and hard-to-learn concept atall levels of education. Prior research has examined, documented, and cast light on theprocesses underpinning different problems of classroom practice in the area of proof, buthas paid less emphasis on acting upon such problems to generate possible solutions throughresearch-based interventions in mathematics classrooms. In this Editorial, we first situate thecontributions in this Special Issue in a brief chronological account of scholarly work on research-based interventions in the area of proof, and we conclude with a proposal for somehigh-leverage directions for future research. [ABSTRACT FROM AUTHOR]

In this theoretical paper, we present a framework for conceptualizing proof in terms of mathematical values, as well as the norms that uphold those values. In particular, proofs adhere to the values of establishing a priori truth, employing decontextualized reasoning, increasing mathematical understanding, and maintaining consistent standards for acceptable reasoning across domains. We further argue that students' acceptance of these values may be integral to their apprenticeship into proving practice; students who do not perceive or accept these values will likely have difficulty adhering to the norms that uphold them and hence will find proof confusing and problematic. We discuss the implications of mathematical values and norms with respect to proof for investigating mathematical practice, conducting research in mathematics education, and teaching proof in mathematics classrooms. [ABSTRACT FROM AUTHOR]

As part of a larger study of student understanding of concepts in linear algebra, we interviewed 10 university linear algebra students as to their conceptions of functions from high school algebra and linear transformation from their study of linear algebra. An overarching goal of this study was to examine how linear algebra students see linear transformation as related to high school function. Analysis of these data led to a characterization of student responses into three categories of mathematical structures used to discuss function: properties, computations, and a series of five interrelated clusters of metaphorical expressions. In this paper, we use this analytic framing for exploring the question: to what extent does each of the students in this study have a unified concept image of function across two mathematical contexts, high school algebra and their study of linear algebra? We found that students who expressed a unified notion of function used metaphorical language to bridge any gaps between the notion of function from high school and the notion of linear transformation from linear algebra. We conjecture that the framing of computations, properties, and metaphorical clusters could be extended to discussions of functions in contexts with other mathematical domains. Future research that further examines the extent to which undergraduate students develop a unified concept image of function could then lead to various design research efforts aimed at explicitly fostering such a unified understanding. [ABSTRACT FROM AUTHOR]

In this paper, we analyze and compare two didactical designs for introducing primary school pupils to proportional reasoning in the context of plane polygons. One of them is well-documented in the literature; the other one is based on our own data and is accordingly presented and discussed in more detail in this paper. The two designs come from different cultural and intellectual environments: lesson study in Japan (implicitly based on the “open approach method”) and “didactical engineering” in France (based on the theory of didactical situations). The general aim of our paper is to compare these two environments and their approaches to didactical design, basing our discussion on the concrete designs mentioned above. Clear differences among them are presented, while we also identify links which hold potential for integrating research and practice. [ABSTRACT FROM AUTHOR]

This paper describes elementary students' awareness and representation of the aggregate properties and variability of data sets when engaged in predictive reasoning. In a design study, 46 third-graders interpreted a table of historical temperature data to predict and represent future monthly maximum temperatures. The task enabled students to interpret numbers in context and apply their understanding of inherent natural variation to create a generalised data set. Student predictions, representations, and written and verbal descriptions were analysed using two frameworks—Awareness of Mathematical Pattern and Structure (AMPS), and Data Lenses. While 54% of students used the variability of the given data table to predict temperatures that were within the historical range for each month, only 20% described the table by focusing on aggregate properties. Student representations varied from highly structured line and bar graphs to idiosyncratic drawings on weather-related themes. In total, 83% of student representations were either idiosyncratic or direct copies of the data table. These findings suggest a progression in students' predictive reasoning, with an awareness of range and seasonal patterns emerging before a multifaceted aggregate view. [ABSTRACT FROM AUTHOR]

Proportional reasoning, the ability to use ratios in situations involving comparison of quantities, is essential for mathematical competence, especially in the middle school years, and is an important determinant of success beyond school. Research shows students find proportional reasoning and its foundational concepts difficult. Proportional reasoning does not always develop naturally, however some research suggests that with targeted teaching, its development can be promoted. This paper reports on a large Australian study involving over 130 teachers and their students. A major goal of the study was to investigate the efficacy of ongoing teacher professional development for promoting middle years students' proportional reasoning. A series of professional development workshops was designed to enhance the teachers' understanding of proportional reasoning and to extend their repertoire of teaching strategies to promote their students' proportional reasoning skills. The workshop design was informed by research literature on proportional reasoning teaching and learning as well as the results of a diagnostic instrument administered to over 2500 middle years students prior to the professional development. Between workshops, the teachers implemented a variety of targeted teaching activities. This paper reports on pre- and post- instrument student data collected at the beginning and end of the first year of the project (i.e., after completion of half of the workshops). The findings suggest that targeted professional development and explicit teaching can make a difference to students' proportional reasoning. [ABSTRACT FROM AUTHOR]

Mathematics-related affect comprises an individual's attitudes, beliefs, emotions and motivations towards mathematics. These affective constructs have been widely studied in mathematics education and cognitive psychology and have been shown to be related to cognitive outcomes such as performance on a range of mathematical tasks. However, it is not yet clear how these constructs develop, or how they relate to cognitive factors in young children who are in the early stages of learning mathematics. As such, the current special issue focuses on mathematics-related affect in primary school children aged 4 to 10 years. It brings together five recent empirical studies and two discussant articles looking at the development of attitudes, beliefs, emotions and motivations towards mathematics, and the relations between affective and cognitive factors in these young age groups. In this introductory paper, we provide some brief historical context, followed by a rationale for the special issue, and an overview of its structure and scope. [ABSTRACT FROM AUTHOR]

At the start of mathematics education children are often presented with addition and subtraction problems in the form of pictures. They are asked to solve the problems by filling in corresponding number sentences. One type of problem concerns the representation of an increase or a decrease in a depicted amount. A decrease is, however, more difficult to represent in paper pictorially than an increase. Within the Cognitive Load Theory framework, we expected that commonly used decrease problems would be harder and slower to solve than increase problems because of higher intrinsic cognitive load. We also hypothesised that combining the pictorial information with auditory information would reduce this load and, as a result, would lead to improved performance. We further expected that these effects would be most prominent in children who score below average on a general mathematics test. We conducted an experiment with sixty children attending the first grade of primary school, who were divided into a higher and lower mathematics-achieving group. As expected, children performed worse on the decrease problems compared to the increase problems. Also, the combination of pictorial and auditory information reduced the accuracy lag of the decrease problems compared to the increase problems. With response time this effect occurred only in the group of lower mathematics achieving children. These results are in line with the cognitive load theory framework, but we also offer an alternative explanation regarding attention. [ABSTRACT FROM AUTHOR]

In this paper, we present and discuss a framework for considering the quality of primary teachers' mediating of primary mathematics within instruction. The 'mediating primary mathematics' framework is located in a sociocultural view of instruction as mediational, with mathematical goals focused on structure and generality. It focuses on tasks and example spaces, artifacts, inscriptions, and talk as the key mediators of instruction. Across these mediating strands, we note trajectories from error and a lack of coherence, via coherence localized in particular examples or example spaces, towards building a more generalized coherence beyond the specific example space being worked with. Considering primary mathematics teaching in this way foregrounds the nature of the mathematics that is made available to learn, and we explore the affordances of attending to both coherence and structure/generality. Differences in ways of using the framework when either considering the quality of instruction or working to develop the quality of instruction are taken up in our discussion. [ABSTRACT FROM AUTHOR]

Decades of research has documented the consequences of allocating school students into a hierarchy of classes with narrow ranges of mathematics attainment, a process known as streaming, tracking, setting, or "ability" grouping. The purported benefits of streaming are inconsistent and disputed, but the harms are clear, in particular, (1) the limiting curriculum often available in low streams and (2) the loss of self-confidence that results from being positioned in a low stream. Building on this foundation, we discuss streaming in mathematics as tied to systemic racism in Aotearoa New Zealand, where the harmful effects of streaming fall most heavily on Māori and Pasifika students. Previous analyses of race and streaming have focused primarily on the racial composition of streamed classes, bias in stream allocation, and racialised teacher expectations in streamed settings. In contrast, we focus on the emotional consequences of streaming, arguing that streaming produces racialised emotions of shame and entitlement as unintended but predictable consequences. We illustrate the racialised production of entitlement and shame through collaborative storying, interweaving our own biographies with a re-analysis of student interviews from two prior studies. [ABSTRACT FROM AUTHOR]

As in many countries, for decades in Aotearoa (New Zealand), we have heard the story of the Pacific and Māori achievement gap in mathematics. This has become a widely accepted part of beliefs constructed across multiple communities about students and schools and who can do and learn mathematics successfully. A common response by policy makers and educators alike is to fix the problem of those deemed academically bleak by putting in place a range of interventions. In this article, we challenge the positioning of Pacific students as a problem to be fixed and instead focus on how we can address the practices inherent in historical forms of institutionalised racism related to colonisation. We use an Indigenous research model—Tivaevae—to develop an exemplary case study of the teachers and students from one low socio-economic urban school as they were involved in conscientisation and the reconstitution of educational practices to privilege indigenous knowledge systems. The findings highlight one model of how teachers and students can change institutionalised Western world practices in the mathematics classroom. We argue that the shift to honouring indigenous knowledge systems and a strength-based approach provided opportunities for Pacific students to learn mathematics in ways that supported them to build strong mathematical dispositions, and rather than being assimilated, retain their cultural identity. [ABSTRACT FROM AUTHOR]

Building on the research of Vygotsky regarding the role of social interaction and the zone of proximal development (ZPD) in learning and development, this paper explored the relation between students' oral thought processes and written thought processes. It is argued that the practice of writing provides a context for a new learning zone: the `zone of proximal practice' (ZPP). In this new zone, students independently organize their thinking about mathematical concepts and ideas. An interpretative case study of seven middle grade students is presented to support this contention. The case study describes the strategies and procedures students employed while solving mathematical problems and documents students' oral and written thought processes through interview protocols and writing samples. The position that students' mathematical understanding is further developed through writing as a communicative tool, while taking advantage of mediated social practices, is discussed to make clear the rationale for introducing a new learning zone. [ABSTRACT FROM AUTHOR]

This paper deals with the competence of Cultural (epistemological, historical and anthropological) Analysis of the Content (CAC), which is important for teachers' awareness and autonomy when dealing with educational choices in a changing cultural and institutional context. We report on an 18-hour intervention in a teacher education course at the undergraduate level attended by 12 participants during the fall of 2013. The goal of the intervention was to enhance participants' CAC in the case of Elementary Arithmetic Theorems by using the rationality construct derived from Habermas' work. Qualitative analyses of participants' discussions showed how the construct was gradually appropriated by them and contributed to their CAC. Trends emerging from quantitative comparisons with previous implementations of the same course provided some evidence regarding the effects of the intervention on the quality of participants' CAC performances at the end of the course. The method chosen for participants' assessment and their cultural background will be discussed as conditions for the success of the intervention. [ABSTRACT FROM AUTHOR]

Studies on identity in general and mathematical identity in particular have gained much interest over the last decades. However, although measurements have been proven to be potent tools in many scientific fields, a lack of consensus on ontological, epistemological, and methodological issues has complicated measurements of mathematical identities. Specifically, most studies conceptualise mathematical identity as something multidimensional and situated, which obviously complicates measurement, since these aspects violate basic requirements of measurement. However, most concepts that are measured in scientific work are both multidimensional and situated, even in physics. In effect, these concepts are being conceptualised as sufficiently uni-dimensional and invariant for measures to be meaningful. We assert that if the same judgements were to be made regarding mathematical identity, that is, whether identity can be measured with one instrument alone, whether one needs multiple instruments, or whether measurement is meaningless, it would be necessary to know how much of the multidimensionality can be captured by one measure and how situated mathematical identity is. Accordingly, this paper proposes a theoretical perspective on mathematical identity that is consistent with basic requirements of measurement. Moreover, characteristics of students' mathematical identities are presented and the problem of 'situatedness' is discussed. [ABSTRACT FROM AUTHOR]

The study reported in this paper concerns the tensions and conflicts that teachers experience while they enact a new set of reform-oriented curricular materials into their classrooms. Our focus is οn the interactions developed in two groups of teachers in two schools for a period of a school year. We use Activity Theory to study emerging contradictions and we elaborate on the construct of dialectical opposition to understand the nature of these contradictions and their potential for teacher learning. We provide evidence that discussions about contradictions and their dialectical character in the two groups support teachers to engage differently in mathematics teaching and learning and carry potentials for shifts in the practices that evolve in their classrooms. Our study addresses empirically in the context of mathematics teaching the philosophical claim about the role of contradictions as a driving force for any dynamic system. [ABSTRACT FROM AUTHOR]

The hypothesis that mathematically superfluous brackets can be useful when teaching the rules for the order of operations is challenged. The idea of the hypothesis is that with brackets it is possible to emphasize the order priority of one operation over another. An experiment was conducted where expressions with mixed operations were studied, focusing specifically on expressions of the type a ± (b × c) with brackets emphasizing the multiplication compared to expressions of the type a ± b × c without such brackets. Data were collected from pen and paper tests, before and after brief (about 7 min) instructions, of 169 Swedish students in year 6 and 7 (aged 12 to 13). The data do not seem to support the use of brackets to detach the middle number ( b) from the first operation ( ±) in a ± b × c type of expressions. [ABSTRACT FROM AUTHOR]

The part-whole multiplicative relationship, as a topic that gives rise to the concept of fraction, is fundamental in education at the primary school level, and must therefore be included in training courses for prospective primary school teachers (PSTs). In this paper, we introduce a first study of a larger project, which aims to understand the usefulness of a semantic triangle in studying prospective teachers' knowledge of various mathematical topics. In particular, we present the results of a study focused in the starting level of the fraction concept, based on the multiplicative part-whole relationship. We carried out this study with PSTs by means of a questionnaire. We analyzed the collected responses using a framework of three components that form a semantic triangle, in terms of their conceptual structure, system of representations, and contexts and modes of use of the part-whole relationship. The results show different typologies of meaning expressed by the participants in terms of the semantic triangle. Each typology emphasizes some aspects of the meaning of the part-whole relationship such as the equality of parts, the model of area as representation and the context of division. [ABSTRACT FROM AUTHOR]

It is now well known that fractions are difficult concepts to learn as well as to teach. Teachers usually use circular pies, rectangular shapes and number lines on the paper as teaching tools for fraction instruction. This article contributes to the field by investigating how the widely used three external graphical representations (i.e., circle, rectangle, number line) relate to students' fractional knowledge and vice versa. For understanding this situation, a test using three representations with the same fractional knowledge framed within Fractional Scheme Theory was developed. Six-hundred and fifty-six 4th and 5th grade US students took the test. A statistical analysis of six fractional Problem Types, each with three external graphical representations (a total of 18 problems) was conducted. The findings indicate that students showed similar performance in circle and rectangle items that required using part-whole fractional reasoning, but students' performance was significantly lower on the items with number line graphical representation across the Problem Types. In addition, regardless of the representation, their performance was lower on items requiring more advanced fractional thinking compared to part-whole reasoning. Possible reasons are discussed and suggestions for teaching fractions with graphical representations are presented. [ABSTRACT FROM AUTHOR]

This paper describes the importance of developing students' reasoning about samples and sampling variability as a foundation for statistical thinking. Research on expert-novice thinking as well as statistical thinking is reviewed and compared. A case is made that statistical thinking is a type of expert thinking, and as such, research comparing novice and expert thinking can inform the research on developing statistical thinking in students. It is also posited that developing students' informal inferential reasoning, akin to novice thinking, can help build the foundations of experts' statistical thinking. [ABSTRACT FROM AUTHOR]

In this paper, we explored how engagement developed over time during a proportional reasoning unit for a group of US bilingual Latino/a students, with particular attention to aspects of social and cultural activity that supported students' engagement. Our findings suggest that student mathematical engagement developed primarily as a relational process characterized by students' social relations across time, their understandings about their relationships with mathematics, and the important relations emerging across proportional reasoning ideas. [ABSTRACT FROM AUTHOR]

Mathematics educators have long emphasised the importance of attitudes, beliefs, and feelings towards mathematics, as crucial in motivating (or not) its learning and use, and as influenced in turn by its social images. This paper is about images of mathematics. Our search for advertisements containing such images in UK daily newspapers, during 2006-2008, found that 4.7 % of editions included a 'mathematical' advert, compared with 1.7 % in pilot work for 1994-2003. The incidence varied across type of newspaper, being correlated with class and gender profiles of the readership. Three quarters of advertisements were classified as containing only very simple mathematics. 'Semiotic discursive' analysis of selected advertisements suggests that they draw on mathematics not to inform, but to connote qualities like precision, certainty, and authority. We discuss the discourse on mathematics in advertising as 'quasi-pedagogic' discourse, and argue that its oversimplified forms, being empty of mathematical content, become powerful means for regulating and 'pedagogising' today's global consumers. [ABSTRACT FROM AUTHOR]

This paper explored (1) the developmental trend of student level of reasoning across grade level in pattern generalization; (2) the mediatory role of task variables in the developmental trend of student level of reasoning within and across tasks; and (3) developmental trend of student level of reasoning associated with strategy use across grade level. A test designed to measure student level of reasoning was given to a sample of 1232 students from grades 4 to 11 from 5 schools in Lebanon. The Structure of the Learned Outcomes or Responses (SOLO) was used as a theoretical model. Results show that student level of reasoning exhibited an increasing trend across clusters of grade levels and that there were several SOLO levels in each cluster of grade levels. Type of task (immediate, near, far) and function type (linear, non-linear) seem to mediate the development of level of reasoning across grade level but the complexity of the task (simple, more complex) did not. Students used several strategies in each cluster of grade levels and the developmental trends of student level of reasoning associated with strategy use were not uniform and varied across clusters of grade levels, thus supporting a neo-Piagetian interpretation of the results. [ABSTRACT FROM AUTHOR]

In 2008, the research center 'Individual Development and Adaptive Education' was constituted by the Goethe University, the German Institute for International Educational Research, and the Sigmund Freud Institute, all located in Frankfurt am Main, Germany (). The research of the center focuses on the development of children in preschool, kindergarten, and primary school age by including and combining cognitive, neuro-cognitive and socio-emotional factors. The Special Issue 'Early Years' reports about the research activities concerning mathematics learning in the early years. [ABSTRACT FROM AUTHOR]

This paper deals with one aspect of the endeavor to generate a theory of the development of mathematical thinking of children in the early years ages 3 to 10. By comparing two scenes, one from preschool and one from a first grade mathematics class, the relationship between diagrammatic and narrative argumentations among children and teachers is reconstructed and related to possible developmental trajectories of mathematical thinking. Theoretically, I attempt to implement these developmental paths in a concept of an 'Interactional Niche in the Development of Mathematical Thinking.' [ABSTRACT FROM AUTHOR]

This paper is methodologically based, addressing the study of mathematics teaching by linking micro- and macro-perspectives. Considering teaching as activity, it uses Activity Theory and, in particular, the Expanded Mediational Triangle (EMT) to consider the role of the broader social frame in which classroom teaching is situated. Theoretical and methodological approaches are illustrated through episodes from a study of the mathematics teaching and learning in a Year-10 class in a UK secondary school where students were considered as “lower achievers” in their year group. We show how a number of questions about mathematics teaching and learning emerging from microanalysis were investigated by the use of the EMT. This framework provided a way to address complexity in the activity of teaching and its development based on recognition of central social factors in mathematics teaching–learning. [ABSTRACT FROM AUTHOR]

This paper examines the change in young children’s understanding of ‘equal’, ‘more’, ‘less’, and ‘between’, words commonly used in equivalent and non-equivalent situations, over a 3-year period. Seventy-six children participated in the longitudinal study. Each year they were asked to share their understanding of these four words. Past research has indicated that many children have limited understanding of ‘equal’ as quantitative sameness. The results of this research suggested that many children also have limited understanding of ‘more’ and ‘less’ and that these understandings did not significantly change over the 3-year period. [ABSTRACT FROM AUTHOR]

This paper is based on a long-term didactic engineering about division problems (only in a numerical setting) at primary school. Situations and students’ work are analyzed by means of a double theoretical framework: the theory of situations and the theory of conceptual fields (Vergnaud 1991). The analysis focuses mainly on classroom interactions and on the didactic memory from both the teacher perspective and the learner perspective: in particular, it not only investigates how didactic memory is managed by the teacher, but also how students recall past events or reread those events in a-didactic situations. [ABSTRACT FROM AUTHOR]

This paper examines a theoretical perspective on the ways in which children progress in learning mathematics. It suggests that there is a difficulty in associating teaching discourses with the mathematics they locate. This can result in an incommensurability between alternative perspectives being offered. The paper resists attempts to privilege any particular account but rather demands an analysis of these discourses and their presuppositions. In developing these themes the paper invokes Ricoeur‘s analysis of time and narrative as an analytical approach to treating notions such as transition, development and progression in mathematical learning. His notion of semantic innovation is introduced. This embraces both the introduction of a new metaphor into a sentence or the creation of a new narrative which reorganises events into a new ’plot‘. The notion is utilised in arguing that the shift in the student‘s mathematical development from arithmetic to first order linear equations with unknowns reconfigures the contextual parameters governing the understanding of these mathematical forms. It is also utilised in showing how alternative approaches to accounting for such transitions suit different and perhaps conflicting outcomes. For example, demonstrating awareness of generality or performing well in a diagnostic test featuring the solution of linear equations. [ABSTRACT FROM AUTHOR]

In this article, we demonstrate how I poems, part of the Listening Guide method, can help researchers to better understand the multiple voices that might comprise a student’s mathematical identity. We provide an analysis of one student’s interview as an example of this method. We argue that the use of the Listening Guide method illuminates the complexity of students’ mathematical identities, especially their fluid, negotiated, and sometimes contested character. In the example shared here, we show how others’ discourses become part of a student’s developing mathematical identity, as well as highlight how the addition of drawings enhances the Listening Guide method. [ABSTRACT FROM AUTHOR]

This study situates children's early notions of average within an inquiry classroom to investigate the rich inferential reasoning that young children drew on to make sense of the questions: Is there a typical height for a student in year 3? If so, what is it? Based on their deliberations over several lessons, students' ideas about average and typicality evolved as meaning reasonable, contrary to atypical, most common (value or interval), middle, normative, and representative of the population. The case study reported here documents a new direction for the development of children's conceptions of average in a classroom designed to elicit their informal inferential reasoning about data. [ABSTRACT FROM AUTHOR]

This study looks at a mixed ability group of 21 Year 5 primary students (aged 9-10 years old) who had previously never had formal instruction using letters to stand for unknowns or variables in a mathematics context; nor had they been introduced to formal algebraic notation. Three lessons were taught using the computer software Grid Algebra where they began working with formal notation and were solving linear equations with some degree of success by the end of the lessons. The teaching was such that nothing was explained or justified by the teacher explicitly. The students appeared either not to meet, or to overcome quickly, some of the difficulties identified within previous research studies. They demonstrated remarkable confidence working with complicated linear algebraic expressions written in formal notation. A key feature of the software activities was that formal notation continually needed to be used and interpreted, and the software provided neutral feedback which enabled the students to educate their interpretation of the notation. [ABSTRACT FROM AUTHOR]

This study examines the collective mathematical reasoning when students and teachers in grades 3, 4, and 5 explore fractions derived from length comparisons, in a task inspired by the El´konin and Davydov curriculum. The analysis showed that the mathematical reasoning was mainly anchored in mathematical properties related to fractional or algebraic thinking. Further analysis showed that these arguments were characterised by interplay between fractional and algebraic thinking except in the conclusion stage. In the conclusion and the evaluative arguments, these two types of thinking appeared to be intertwined. Another result is the discovery of a new type of argument, identifying arguments, which deals with the first step in task solving. Here, the different types of arguments, including the identifying arguments, were not initiated only by the teachers but also by the students. This in a multilingual classroom with a large proportion of students newly arrived. Compared to earlier research, this study offers a more detailed analysis of algebraic and fractional thinking including possible patterns within the collective mathematical reasoning. An implication of this is that algebraic and fractional thinking appear to be more intertwined than previous suggested. [ABSTRACT FROM AUTHOR]

The article introduces a mathematics education measure in response to the COVID-19 epidemic in China and explores students' degree of approval and perception of digital equity towards the response. After the outbreak of the COVID-19, the Chinese New Century Primary School Mathematics Textbook (NCPM) committee had developed a series of micro classes (abbreviated as NCPM micro classes), and more than 25 million teachers and students in China watched the NCPM micro classes during the 3 months social isolation. Then, students' degree of approval towards the NCPM micro classes and perception of digital equity were examined after social isolation. A total of 132,740 pieces of data were collected from Chinese primary school students. Quantitative analysis of student's degree of approval towards different parts of NCPM micro classes indicated that the introduction, interaction, summary and consolidation, curriculum characteristics, and goal achievement parts of the NCPM micro classes have received high approval from students, and students with higher former achievement perceived a higher degree of approval towards the NCPM micro classes. Furthermore, we found that gender, socioeconomic status, school location, and learning location had no significant impact on students' degree of approval, indicating a digital equity exists. This study helps researchers or educators understand the mathematics education response to the COVID-19 outbreak in China and extends our understanding of primary students' degree of approval and perception of digital equity with these online classes. [ABSTRACT FROM AUTHOR]

In the context of early algebra research and as part of a classroom teaching experiment (CTE), we investigated fourth grade (9- to 10-year-old) students' justifications of how they performed tasks involving the functional relationship y = 2x. We related their written justifications (part of the task) to the task characteristics, which included various semiotic systems (verbal, numerical and alphanumeric, among others) and the demand of different type of justifications. The role of classroom discussion in helping express the functional relationship orally in more sophisticated terms was also investigated. The findings showed that students' written justifications changed with the semiotic system involved in the task. Oral discussion helped students generalize in more sophisticated terms than in their written justifications, in which they omitted information or used less precise language. [ABSTRACT FROM AUTHOR]