Abstract: Abstract Even in primary school, mathematics achievement depends upon the efficiency of cognitive, metacognitive and self-regulatory processes. Thus, for pupils to carry out a computation, such as a written calculation, metacognitive mechanisms play a crucial role, since children must employ self-regulation to assess the precision of their own thinking and performance. This assessment, in turn, can be helpful in the regulation of their own learning. In this regard, a body of literature suggests that the application of psychoeducational interventions that promote the development of mathematics-related metacognitive (e.g., control) processes, based on the analysis of the students’ errors, can successfully influence mathematics performance. The main objective of the current study was to investigate the impact of a metacognitive and cognitive training program developed to enhance various arithmetic skills (e.g., syntax, mental and written calculation), self-regulatory and control functions in primary and secondary school students exhibiting atypical mathematical development. Sixty-eight Italian children, 36 male and 32 female (mean age at pretest = 9.3 years, SD = 1.02 years), meeting the criteria for the diagnosis of dyscalculia or specific difficulties in mathematics, took part in the study. Of these, 34 children (i.e., experimental group) underwent the cognitive and self-regulatory intervention enhancing mathematics skills training for 16 weekly sessions. The remaining students were assigned to the control group. For a pre-test and post-test, a battery of standardized mathematical tests assessing different mathematics skills, such as written and mental operations, digit transcription and number ordering skills, was administered and provided a series of measures of calculation time and accuracy (i.e., number of errors). In the post-test, the experimental group exhibited better accuracy in written calculation and in digit transcription. Overall, the current outcomes demonstrate that psychoeducational interventions enriching metacognitive and mathematical achievements through error analysis may be an effective way to promote both the development of self-regulatory and control skills and mathematical achievement in children with atypical mathematical development. PubDate: 2019-03-21

Abstract: Abstract Although there is no generally accepted list of criteria for the acceptance of proofs in mathematical practice, judging the acceptability of purported proofs is an essential aspect of handling proofs in daily mathematical work. For this reason, school students, university students, and mathematicians need to hold certain acceptance criteria for proofs, which are closely tied to their epistemology, notion of proof, and concept of evidence. However, acceptance criteria have so far received little academic attention and results have often been related to mathematicians’ acceptance criteria in a research context. Still, as the role of proof changes during mathematical education and differs from research, it can be assumed that acceptance criteria differ substantially between educational levels and different communities. Thus, we analyzed and compared acceptance criteria by school students, university students, and mathematicians in the context of teaching. The results obtained reveal substantial cross-sectional differences in the frequency and choice of acceptance criteria, as well as major differences from those acceptance criteria reported by prior studies in research contexts. Structure-oriented criteria, regarding the logical structure of a proof or individual inferences, are most often referred to by all groups. In comparison, meaning-oriented criteria, such as understanding, play a minor role. Finally, social criteria were not mentioned in the context of teaching. From an educational perspective, the results obtained underline university students’ need for support in implementing acceptance criteria and suggest that a more explicit discussion of the functions of proof, their evidential value, and the criteria for their acceptance may be beneficial. PubDate: 2019-03-18

Abstract: Abstract Assessment data from the United States and international reports of student achievement indicate that upper elementary students are failing to meet basic levels of proficiency in fractions and writing, and that this is particularly prevalent with students with or at-risk for learning disabilities in mathematics. Proficiency with fractions has been identified as foundational for learning higher-level mathematics but remains one of the most difficult skills for students to learn. In addition, students’ difficulty with fractions is exacerbated because of increased chances of comorbidity with language learning problems, particularly difficulties constructing arguments and communicating using writing. We describe FACT + R2C2, a language-based, metacognitive instructional intervention that was designed using the Self-Regulated Strategy Development model (SRSD) for teaching foundational concepts of fractions. The results from two studies in which the intervention was administered to upper elementary students who exhibit mathematics difficulties indicated selected increases in students’ computational accuracy, quality of mathematical reasoning, number of rhetorical elements, and total words. With evidence of improved performance in these areas, FACT + R2C2 holds promise for helping these students become proficient self-regulated learners. PubDate: 2019-03-18

Abstract: Abstract Whole Number Arithmetic (WNA) appears as the very first topic in school mathematics and establishes the foundation for later mathematical content. Without solid mastery of WNA, students may experience difficulties in learning fractions, ratio and proportion, and algebra. The challenge of students’ learning and mastery of fractions, decimals, ratio and proportion, and algebra is well documented. Most of this research has focused on either fractions, decimals, ratio and proportion, algebra, or WNA. There is a lack of research that addresses the connection between these relevant topics. Within WNA, most research focuses on counting, computation, or solving word problems. There is a lack of research that investigates connections within WNA. This special issue is intended to bridge this research gap by explicitly highlighting the conceptual knowledge of counting, calculations, and quantity relationships, as well as the structure of word problems within and beyond WNA. PubDate: 2019-03-14

Abstract: Abstract In this paper we argue that the choice of research methods reflects the theoretical framework even before these methods have been put to use in case studies. We understand the term ‘case study’ broadly in this paper and argue that neither thinking of them as cherry-picked cases to support preconceived ideas about mathematical practices nor thinking of them as inductive leaps from (too) few cases to general features is suitable. By realising the deep entanglement of our case studies with our theoretical framework we propose to view case studies as an invitation for critical reflection upon one’s own assumptions. We discuss an example taken from the philosophy of mathematical practices. The upshot is threefold: (1) we provide an argument that case study based research strategies can be successful; (2) we delineate how an awareness of the methodological difficulties of case study based research strategies can positively influence the way case studies are conducted; (3) we suggest that case studies are not dispassionate examinations that deliver cold facts. PubDate: 2019-03-07

Abstract: Abstract Drawing on students’ language repertoires for conjecture, we here focus on communication acts to consider students’ mathematical identities in action. Our attention to authority and positioning in mathematics classrooms led us to interrogate the intertwined nature of discourses and how they may make identities available. We use an interaction among a group of grade ten students in a problem-solving situation to illustrate and develop our conceptualization of identities in contexts of intertwining discourses. We consider how the participants position themselves in relation to each other, to mathematics, and to other available discourses, and we ask how their communication acts might serve their purposes within the discourses. We discuss the implications of intertwined discourses for students’ developing identities when communicating in a mathematical context. We argue that one cannot understand students’ mathematical identities without understanding that their communication acts are also part of their repertoires for a range of discourses. This motivates intentional work to support the development of identities accessing available discourses in mathematics learning. PubDate: 2019-03-04

Abstract: Abstract The aim of this paper is on the one hand to discuss from an APOS (Action–Process–Object–Schema) theory perspective the mental constructions involved in the learning of linear algebra, through examples concerning the linear transformation concept and related notions. On the other hand, methodological issues related to the design of research instruments and implementation of didactic interviews are discussed, supported by empirical data. Detailed analysis of transcripts from an interview with a student focuses on strategies used in interviewing as well as the mental stages involved in the construction of some linear algebra concepts. Due to the strategies employed, during the interview it is possible to witness the transition between different conceptions. A discussion of the relationships and interactions between different mental structures and mechanisms that play a role in the development of knowledge is provided, including theoretical considerations on the matter. Recommendations about pedagogical strategies are included. PubDate: 2019-03-01

Abstract: Abstract Scholars who take a situative perspective on mathematics identity have focused on different kinds of resources and mechanisms that are available for identity development, particularly for students from minoritized backgrounds. Although these approaches are generally not contradictory, they are not always consistent. Differences in what is foregrounded and backgrounded in these studies mean that at times it can be difficult to develop a comprehensive picture of identity as it develops in the mathematics classroom. Our goal in this paper is to connect these literatures with an eye towards offering a design heuristic for mathematics educators interested in accounting for students’ identity development in mathematics teaching and learning. To do so, we present a model of mathematical identities as they are realized in the classroom, articulate the elements involved in the development of such an identity, and then use the model to analyze approaches to intervene in classroom systems by focusing on different elements as starting points. PubDate: 2019-02-21

Abstract: Abstract In the study presented, as we report in this paper, we describe our theoretical and practical consideration to engage first-year pre-service teachers in proving activities in the context of a transition-to-proof course. We investigated how students argued to verify a claim of elementary number theory on entering university and compared the results to their performance in the final examination of the course. Subsequently, we elaborate on the following results: On entering university, students do not seem to be capable of using algebraic variables as a heuristic to engage in reasoning. However, after learning about different kinds of proofs and the symbolic language of mathematics, students give evidence of starting to value mathematical language and of enhancing their proof competencies. PubDate: 2019-02-11

Abstract: Abstract We present exploratory analyses of three cases in which groups of four (quartets) worked with video recordings of choreographed performances from the opening ceremony of the 2016 Rio Olympic Games. We asked quartets to view the recordings, explain what performers were doing by reenacting what they noticed in the video, and create their own performances using props akin to those in the recording. In response, participants explored symmetries and transformations of quadrilaterals and triangles. We contribute to research on distributed, embodied mathematical learning at four levels. First, we argue for design research that engages in creative re-use by foraging in public media for performances with mathematical potential, then designing activities that invite learners to dissect and reenact these performances to explore that potential. Second, we analyze quartets’ work as a form of ensemble learning that hybridizes dance and mathematics. Third, we describe interactions that produced intercorporeality in the material work of quartets. Finally, we argue for reenactment as a supplement to methods of Interaction Analysis, using our own analysis as an illustration of this novel approach. PubDate: 2019-02-09

Abstract: Abstract There is relatively little research specifically about student understanding of basis. Our ongoing work addresses student understanding of basis from an anti-deficit perspective, which focuses on the resources that students have to make sense of basis using everyday ideas. Using data from a group of women of color in the United States, we previously developed an analytical framework to describe student understanding about basis, including codes related to characteristics of basis vectors and roles of basis vectors in the vector space. In this paper, we utilize the methods of the previous study to further enrich our findings about student understanding of basis. By analyzing interview data from students in Germany, we found that this group of students most often used ideas that we describe by the roles generating, structuring, and traveling, and the characteristics different and essential. Some of the themes that emerged from the data illustrate common pairings of these ideas, students’ flexibility in interpreting multiple roles within one everyday example, and the ways that the roles and characteristics motivate students to create additional examples. We also discuss two ways that differences between the German and English languages were pointed out by students in the interviews. PubDate: 2019-02-08

Abstract: Abstract Although many scholars in the field of mathematics education are aware that identity discourses are highly political, research in the field usually lacks a framework theoretically and methodologically to address the political dimension of identity research. Based on Laclau and Mouffe’s discourse theory and the case of a female secondary school student at a German public school, the present paper analyses identity as a socio-political process of identity work articulated around the discourse of ‘refusing school mathematics’ in our contemporary times. Her refusal of mathematics is constituted around issues related to a series of noted classroom practices such as collective work, being together and having fun, relevance of mathematics in society and life, respect of one’s own dignity instead of becoming humiliated, and bodily activity instead of seated work. We illustrate how discourse theory allows us to see the identity work of refusing mathematics as a contingent process in a discursive field of socio-political struggle. In this process the subject moves beyond an essentialist ‘refusal’ of mathematics learning towards articulating her refusal of a particular mathematics education socio-materiality that needs to become subverted and reworked into more affirmative terms. PubDate: 2019-02-06

Abstract: Abstract The academic field of Disability Studies (DS) offers theoretical tools to understand how social practices intersect with embodiment, long a critical issue in DS because disability is a category of human difference that is always already embodied. I review two theories that seek to resolve this dichotomy between the body and social worlds: complex embodiment (Siebers, Disability theory, University of Michigan Press, Ann Arbor, 2008) and the political/relational model (Kafer, Feminist, Queer, Crip, Indiana University Press, Bloomington, 2013). I use these theories to analyze ethnographic data and narratives of a Latina named Desi around disability and mathematics. Desi’s narratives explored experiences relating to Attention Deficit Hyperactivity Disorder, Learning Disabilities, and mathematics anxiety. Desi’s narratives described disabilities as socio-political constructs, involving relations of power and exclusion, as well as acknowledging the physiological, embodied experience of some differences in relation to mathematics. Through this analysis, I argue for the inclusion of emotion in embodiment, and the use of narrative analysis paired with ethnography as a tool to understand embodied experience. PubDate: 2019-02-02

Abstract: In our study, 32 German and Swiss 8th/9th-grade classes of lower-secondary school worked with their teacher on the same proving problem. The sample belongs to the Swiss-German study “Quality of Instruction, Learning Behavior and Mathematical Understanding”. Our data analyses relate to the teachers’ approaches to generating a specific form of evidence with their classes when dealing with a particular elementary number-theory problem. We address the question of how the different strategies can be characterized as manifestations of a certain approach to proving and try to clarify in which way the observed approach can be interpreted as adaptive teaching behavior. For this purpose, we searched for possible correlations between three main strategies or types of generating a specific form of evidence (experimental, operative, formal-deductive approach) on the one hand and (a) the teachers’ beliefs and personal characteristics and (b) the students’ prior knowledge of algebra and mathematics in general on the other hand. As our analyses show, three main approaches to proving occurred but not in equal proportions: there is a predominance of the approach that entails the highest extent of formalization and abstraction. Nevertheless, an operative way of proving is widespread too. On the whole, the findings indicate that one particular approach to proving can be interpreted as a personal preference of a specific group of teachers and, at the same time, with respect to the students’ mathematical skills as a manifestation of adaptive teaching behavior. PubDate: 2019-02-01

Abstract: Abstract In this article we outline the role evidence and argument plays in the construction of a framing theory for Proof Based Teaching of basic operations on natural numbers and integers, which uses tiles to physically represent numbers. We adopt Mariotti’s characterization of a mathematical theorem as a triple of statement, proof and theory, and elaborate a theory in which the statement “The product of two negative integers is a positive integer” can be proved. This theory is described in terms of a ‘toolbox’ of accepted statements, and acceptable forms of argumentation and expression. We discuss what counts as mathematical evidence in this theory and how that evidence is used in mathematical arguments that support the theory. PubDate: 2019-01-28

Abstract: Abstract The aim of this research is to analyse students’ sense-making regarding matrix representation of geometric transformations in a dynamic geometry environment (DGE) within the perspective of semiotic mediation. In particular, the focus is on students’ reasoning on the transition from the notion of function to transformation and to matrix representation of geometric transformations in \({{\mathbb{R}}^2}\) . Along these lines, the theory of semiotic mediation is referred to as a theoretical framework in both the design of a teaching and learning environment and the emergence of mathematical thinking. Epistemological analysis was employed to elaborate the semiotic potential of the DGE and, thereafter, two specific tasks were considered. Task-based interviews were conducted with a pair of undergraduate linear algebra students, with the students working in front of a computer installed with a specific DGE: GeoGebra. The data sources are video-recorded interviews, screen recorder software, field notes and the students’ production analysed through a semiotic lens. According to the results, the dragging tool evokes a sense of understanding of covariation and independent/dependent variables. In addition, the simultaneous use of the dragging tool and grid function evokes a sense of the geometric transformation and the notion of matrix representation of geometric transformations, while the ApplyMatrix construction command plays a key role in linking the notions of function, transformation and matrix transformation. PubDate: 2019-01-28

Abstract: Abstract Mathematical connections are widely considered an important aspect of learning linear algebra, particularly at the introductory level. One effective strategy for teaching mathematical connections in introductory linear algebra is through inquiry-based learning (IBL). The demands of IBL instruction can make it difficult to implement such strategies in courses in which the instructor faces various constraints. The findings presented here are the product of an action research study in which IBL instructional materials were designed for a large-enrolled introductory linear algebra course with limited class time. This resulted in IBL being presented in a limited capacity alongside traditional lecture in what will be described as adapted inquiry. Specifically, these IBL materials were designed as vehicles through which students could form mathematical connections. This study was conducted with the goal of determining what mathematical connections students appear to be able to exhibit within the context of an adapted inquiry approach to IBL instruction. PubDate: 2019-01-24

Abstract: Abstract This study investigates the teaching and learning of single-digit whole number multiplication in China. Analysis of data from documents, classroom teaching, and semi-structured interviews revealed three salient characteristics of emphasizing oral calculation, calculation speed, and understanding across standards, textbook and classroom practices. It also showed how mathematics teachers enact these features in their teaching practice to help students develop their computational skills. The study particularly elaborates the role played by the nine-times table, or Chengfa Kou Jue Table (CKJ Table) in teaching practices, as well as how teachers treat memorization of CKJ and how understanding of operations contributes to their better understanding the relationship between the two in the teaching process. PubDate: 2019-01-23

Abstract: Abstract Previous studies have reported the influence of professional development (PD) on participating teachers’ identities. However, what goes on in PDs, how and why they shape particular identities require further investigation. This study contributes in this direction by drawing on the notions of practice-linked identities and identity resources to examine how two teachers’ mathematics teaching identities developed following their interactions with the resources offered in a particular PD. We argue that their developing mathematics teaching identities appeared to be linked to their backgrounds and initial motivations for joining the PD, which in turn influenced their selective interaction with resources. Implications for research and PD are discussed. PubDate: 2019-01-21

Abstract: Abstract In this paper, we offer a theoretical account of the emotion–cognition dialectic (i.e. the unit of feeling and cognition in thought) in identity formation (or identification), focusing on early childhood and mathematics. We consider how contradictory (emotional-cognitive) experiences which arise in different forms of mathematical activity (a mathematical play activity versus formal classroom mathematics) produce and are produced by contradictory acts of mathematical identification. We illustrate this perspective using the case of Daniel (aged 6 years) who expressed the emotion of ‘astonishment’ at seeing himself on video engaged in a play activity. We highlight how this experience of emotion–cognition enables Daniel to articulate a juxtaposition between play that is ‘fun’ and his classroom mathematics that makes him ‘tired’, which we associate with contrasting mathematical identifications. This recognises that social/cultural symbolic forms of emotion–cognition and identification are present in dialogue with others/adults even at this young age. We therefore, call for a research agenda that explores how such identifications might become sedimented into mathematical identities as the child develops. PubDate: 2019-01-18