‘When I am being rushed it slows down my brain’: constructing self-understandings as a mathematics learner

Understanding learning disabilities (LDs) as constructed through multiple cultural practices including discourse, this paper focuses on a Latino middle school student with a LD named Elijah. This study documents both the discourses and practices used to position Elijah as a mathematics learner, as well as his use of similar discourses as he constructs a complex set of self-understandings as a mathematics learner. Elijah is positioned by discourses that prioritise speed as an indicator of mathematical ability, as well as discourses that construct students with LD as having both intelligence and differences such as processing speed. An analysis of interview and observational data suggests that Elijah constructed a unique set of self-understandings as a mathematics learner. Like his sixth-grade special education teacher, Elijah seems to differentiate between knowledge and the performance of knowledge in school. He created a unique identity as both ‘fast’ and ‘slow’ in mathematics, rejecting the binary found in many mathematics classrooms. These findings suggest that multiple discourses circulate in schools about ability and disability in mathematics.     

Designing a Bridging Discourse: Re-Mediation of a Mathematical Learning Disability

Students with disabilities present a unique instructional design challenge. These students often have qualitatively different ways of processing information, meaning that standard instructional approaches may not be effective. In this study I present a case study of a student with a mathematical learning disability for whom standard instruction on fractions had been ineffective. With regard to theory, I draw on Lev Vygotsky’s framing of disability and then use Anna Sfard’s conceptualization of mathematics as a discourse to design a fraction re-mediation that provided a bridge from the student’s discourse to the canonical mathematics discourse. This bridging discourse was used in 5 videotaped re-mediation sessions with the case study student. A fine-grained analysis of the re-mediation sessions traced the ways in which the student’s discourse shifted over time, which enabled her to solve problems she had previously been unable to solve. This study provides a proof of concept for reconceptualizing remediation and illustrates the potential utility of a bridging discourse to help students who have a history of failure gain access to the canonical mathematics discourse and content.     

The co-construction of learning difficulties in mathematics—teacher–student interactions and their role in the development of a disabled mathematical identity

Leaning on a communicational framework for studying social, affective, and cognitive aspects of learning, the present study offers a new look at the construction of an identity of failure in mathematics as it occurs through teaching–learning interactions. Using the case of Dana, an extremely low-achieving student in 7th grade mathematics, I attempt to unearth the mechanisms of interaction between Dana and myself, her teacher, that instead of advancing Dana, perpetuated her failure. Through examining the interactional routines followed by Dana and me, I show how Dana’s deviations from normative routines resulted in my identification of Dana as “clueless” in mathematics. This identification, shared both by Dana and by me, was accompanied by adherence to ritual rule following that did not enable Dana’s advancement in mathematical discourse. This case points to the need to re-examine permanent difficulties in mathematics in light of the reciprocal nature of such difficulties’ (re)construction in teaching-and-learning interactions.     

Mathematics and its other: (dis)locating the feminine

In this article, the author examines how school mathematics maps itself onto the body, delineating the contours of a gendered learner. The author draws on Judith Butler to discuss the process by which school mathematics contributes to the stabilising/legitimating of a cultural arbitrary like gender, and how mathematics inscribes otherness onto the feminine. A student narrative entitled ‘Im/perfect Other’ is offered as a form of disruptive writing that might provisionally trouble or unfix the binary between the feminine and mathematics. The focus of the narrative is on the passionate attachment to mastery and submission in/through school mathematics, and the possibility of an emergent autonomy erupting through and against the process of subjectification.     

On the Relationship of Mathematics to the Real World

In this article, I discuss the relationship of mathematics to the physical world, and to other spheres of human knowledge. In particular, I argue that mathematics is created by human beings, and the number π cannot be said to have existed 100,000 years ago, using the conventional meaning of the word ‘exist’.     

The construction of word meaning in a multicultural classroom. Mediational tools in peer collaboration during mathematics lessons

In this article, we examine the construction of word meaning by students during collaborative activities in a multicultural classroom at a Dutch primary school. The analysis is based on recordings of student talk in small groups of four or five students during mathematics lessons. Difficulties with specific terms and expressions frequently arose during group work. Students could easily ask each other about the meaning of difficult words as a part of the collaborative activities they were accustomed to. In the groups with both Dutch and minority children, the minority students addressed their Dutch classmates as language experts. Conversations about language difficulties also occurred in the groups with only minority students. The conversations about word meaning revealed four patterns: (1) ignoring a question about the meaning of a word, (2) showing the meaning using gestures, (3) explaining, or (4) discussing word meaning. In none of the cases were the language problems solved by referring to the everyday meaning of the word. Instead, the conversations focussed directly on the specialised meaning which the word had in the context of the mathematics lesson. This means that the children used the mathematical discourse as a mediational tool for constructing a mathematical meaning of the words.     

Educational forms of initiation in mathematical culture

A review of literature shows that during the history of mathematics education at school the answer of what counts as ‘real mathematics’ varies. An argument will be given here that defines as ‘real mathematics’ any activity of participating in a mathematical practice. The acknowledgement of the discursive nature of school practices requires an in-depth analysis of the notion of classroom discourse. For a further analysis of this problem Bakhtin’s notion of speech genre is used. The genre particularly functions as a means for the interlocutors for evaluating utterances as a legitimate part of an ongoing mathematical discourse. The notion of speech genre brings a cultural historical dimension in the discourse that is supposed to be acted out by the teacher who demonstrates the tools, rules, and norms that are passed on by a mathematical community. This has several consequences for the role of the teacher. His or her mathematical attitude acts out tendencies emerging from the history of the mathematical community (like systemacy, non-contradiction etc.) that subsequently can be imitated and appropriated by pupils in a discourse. Mathematical attitude is the link between the cultural historical dimension of mathematical practices and individual mathematical thinking.     

In Pursuit of a Connected Way of Knowing: The Case of One Mathematics Teacher

In this paper, we offer illustrations of a mathematics teacher’s difficulties with content knowledge when trying to find connections between school mathematics and science; we do so by describing the development of this teacher’s thinking and learning in her pursuit of connections between the concepts of slope of a line and density of matter. The paper is based on a sub-study that is part of a larger Colombian project, PROMESA (Creating Science and Mathematics Connected Learning Experiences that Open Opportunities for the Promotion of Algebraic Reasoning), which incorporated a Professional Learning Programme (PLP) seeking to integrate school science and mathematics teachers into working teams, in order to create science and mathematics connected learning experiences that considered the promotion of algebraic reasoning. The ‘challenging questions’ that emerged for this teacher, during the workshops of the induction stage of the PLP, became the driving force for her continued engagement in learning mathematics content in a connected way, as opposed to the compartmentalised content-item thinking that she had experienced as a school student. We provide illustrations of first steps in the development of a teacher’s mathematical understanding, which can support growth of mathematical knowledge for teaching.     

Building the wall brick by brick: one prospective teacher’s experiences with mathematics anxiety

Mathematics education researchers have investigated mathematics anxiety in prospective elementary teachers. While many of these studies have focused on the bodily sensations and emotions of mathematics anxiety, particularly those felt in assessment situations, opportunities remain to investigate how prospective elementary teachers interpret their experiences with mathematics anxiety and connect them over time to compose personal histories of mathematics anxiety. Currently, over 90 % of elementary teachers in US schools are women, and women have been shown to suffer more from mathematics anxiety than do men. In this article, I analyze how one woman prospective elementary teacher described, explained, and related her experiences of mathematics anxiety across her personal narratives of learning mathematics as a K-12 student and of learning to teach mathematics as a college student in a teacher preparation program. My research demonstrates that experiences of mathematics anxiety may persist beyond assessment situations to influence women prospective elementary teachers’ larger mathematical histories. I also show that women prospective elementary teachers may interpret mathematics anxiety as specific fears (e.g., loss of opportunities for social participation) and may develop particular coping strategies related to those fears. Finally, I point out that while a coping strategy may be used consistently across K-12 mathematics learning and undergraduate teacher preparation, and may even offer a woman prospective elementary teacher some relief from mathematics anxiety, it may also limit her mathematics learning and professional development. To conclude, I present implications of my research for mathematics teacher educators.     

Analysing the correlation between secondary mathematics curriculum change and trends in beginning undergraduates’ performance of basic mathematical skills in Ireland

The phenomenon in which students enter university under-prepared for the mathematical demands of their undergraduate courses, regularly referred to as the ‘Maths Problem’, has been widely reported in Ireland, UK, Australia, and the US. This issue has been of particular concern in Ireland recently, with beginning undergraduates’ performance of basic mathematical skills showing signs of significant decline in recent years. New mathematics curricula, commonly referred to as ‘Project Maths’, were gradually introduced into the Irish secondary school education system from 2010 onwards. These new curricula aim to place greater emphasis on student understanding of mathematical concepts, use of contexts, and applications of mathematics. This study analyses, through mathematics diagnostic test data gathered between 2008 and 2014, the change in the basic mathematical skills of beginning undergraduates over the time period in which the new mathematics curricula were introduced to secondary education in Ireland. It was found that students’ basic mathematical skills have declined over the time period during which the new mathematics curricula were introduced. Significant declines in beginning undergraduates’ performance of basic mathematical skills were observed during the period 2008–2014, particularly among students who have achieved Higher Level C grades and Higher Level D grades.     

How inconvenient assumptions affect preservice teachers’ uptake of new interactional patterns in mathematics: analysis and aspiration through a bifocal lens

In this paper, I highlight the inadequacies of contemporary theoretical and philosophical orthodoxies to fully address pedagogic change. The required change is in mathematics education, and it has to do with enabling preservice teachers, upon graduation, to rework extant power relations in implementing new interactional patterns that centre the mathematics and the learner in dynamic, productive interaction. I interpret data from published research and my own teaching using psychological, overlaid with poststructuralist, constructs of the relationship between knowledge and action. In data interpretation I read through the words for a psychological interpretation (meaning), and I look at the words for poststructuralist indications of subjectification and identity formation (related to how well students recognise themselves as full participants in the combined discourses of mathematics and education). I have my cake and eat it; contradictory notions of the learner and learning to teach in innovative ways are held together to demonstrate (a) how it can happen that a teacher educator’s aspirations can be held ransom to constituted assumptions that inconveniently work against change, and (b) how the recognition that humanist assumptions, mathematical proficiency and agency are discursively constituted (Davies, 1990) can suggest avenues for change in teacher education.     

Reading as Gender Inquiry in English Language Learning

Using Bourdieu’s forms of symbolic capital as the analytical framework, this article reports how a Taiwanese female university student, Huei, played out her linguistic, social and cultural capital in reading and responding to female-centred texts in a course entitled ‘Gender and Reading’. Journal entries were used as key sources for data analysis, while oral interviews and classroom observations were applied to verify the data analysis and interpretations. The results indicate that Huei exercised different forms of symbolic capital to revalue herself, leading her to changed perspectives and new understandings about her implicit beliefs as an English learner, as well as a female reader. Some pedagogical implications for classroom practices are also addressed.     

Teachers’ assessment practices in mathematics: classrooms in the context of reform

This article responds to calls that have been made for research into how teachers incorporate new assessment ideas into their practice. We draw on a large‐scale study that examined the implementation of reform in mathematics in grades 7–10 in Ontario, Canada. We present teacher questionnaire data, and focus on data gathered from case studies for details of what new assessment practices look like in classrooms. We show teachers using a variety of forms of assessment and using assessment to improve student learning. Observed assessment practices went beyond tests to include the use of journals, observation, questioning, self‐assessment and unique forms of ‘quizzes’. These practices allowed teachers to examine students’ mathematical thinking and provided feedback to students and teachers to improve student learning. We also observed the important role of collaboration, coherence, and the teachers’ beliefs in supporting new assessment practices.     

A new direction?

Digital literacy is now defined as a key area of competence in the new national curriculum for schools in Norway. For policy makers the terms ‘information society’ and ‘knowledge society’ has been used to argue for implementing new technologies in education, and for improving learning. These views have been highly problematic, partly because they do not take into consideration how new technologies are used by young people, or how schools work as social practices. This article will focus on how we conceptualize a student perspective in schools related to the use of digital technologies. Combining an increased focus on digital literacy in school curricula with an increased focus on student participation challenges our conception of the school-aged learner. In discussing these issues I will draw on results from a number of school-based ICT projects that I have been involved in since 1998.     

Complicating ‘student behaviour’: exploring the discursive constitution of ‘learner subjectivities’

When educators consider ‘student behaviour’, they usually think about ‘problem behaviour’ such as disruption or defiance. This limited and limiting view of ‘student behaviour’ not only fails to acknowledge children as educational actors in a wider sense, but also narrowly positions educators as either in control or out of control of their classroom. Mainstream educational psychology’s responses to ‘challenging behaviour’ point educators to numerous ways to prevent its occurrence, through, for example, changing their disciplining approaches and techniques. However, much of the advice directed at improving student behaviour fails to interrogate the core notion of ‘student behaviour’ itself, as well as the conceptual baggage that it carries. The focus is squarely on eliminating ‘problem behaviour’ and often resorts to a pathologisation of students. Meanwhile, when considering ‘student behaviour’ through a Foucauldian post-structuralist optic, behaviour emerges as something highly complex – as spatialised, embodied action within/against governing discourses. In this opening up, it becomes both possible and critical to defamiliarise oneself with the categorisation of ‘challenging behaviour’ and to interrogate the discourses and subject positionings at play. In this paper, we pursue this task by asking: what happens with the notion of ‘behaviour’ if we change focus from ‘fixing problems’ to looking at the discursive constitution of ‘learner subjectivities’? What does it become possible to see, think, feel and do? In this exploration, we theorise ‘behaviour’ as learning and illustrate the constitution of ‘learner subjectivities’. Drawing on two case scenarios, we explore how children accomplish themselves as learners and how this accomplishment links the production of subjectivity and embodied action, and illustrate how ‘student/child behaviour’ appears significantly different to what mainstream educational psychology would have us see.     

Storying Joanne, 1 an undergraduate mathematician

In this article two different accounts are juxtaposed. In one, we use a variety of texts to narrate the story of Joanne, a woman undergraduate student of mathematics. Like many of our mature students Joanne came to the university with a ‘non‐traditional’ academic background. We describe how Joanne developed as a learner of mathematics and connect this to our ways of working in the undergraduate mathematics classroom. We believe that our pedagogy is unusual outside (some) school classrooms and suggest it allows our students to develop positive ‘disciplinary relationships’. In the other, we grapple with the issues raised by telling other people’s stories especially when we are also characters within it. Our intention has been that, in interweaving these two threads, each helps us understand more about the other.     

Social acceptance of students with Down syndrome and students without disability

We investigated the influence of teacher feedback on the social acceptance of peers with intellectual disabilities and peers without disabilities. A computer task was administered to 601 students in grades 3 and 4. Twenty-six per cent of the students attend an inclusive school; the others are in regular schools without students with special educational needs. Participants are introduced to ‘new’ virtual classmates, one student with Down syndrome (DS), and one control student with no obvious disability. Additionally, teacher feedback and feedback about fun playing with the new classmates is given. Social acceptance is evaluated by asking if one would like to sit next to him/her. Both feedbacks showed a strong effect. The child with DS was less socially accepted than the child without disability. No difference regarding the social acceptance of the students with DS was found between students from inclusive and regular classes. Students from regular classes rate the social acceptance of the student without disabilities significantly higher than students from inclusive classrooms.     

Meta-Representation in an Algebra I Classroom

We describe how 1 Algebra I teacher and her 8th-grade students used meta-representational knowledge when generating and evaluating equations to solve word problems. Analyzing data from a sequence of 4 lessons, we found that the teacher and her students used criteria for evaluating equations, in addition to other types of knowledge (e.g., different interpretations of the equal sign) previously reported in the literature. Moreover, the teacher and her students had trouble understanding one another's proposed algebraic models of problem situations due to differences in the criteria that each applied, and this impeded learning. These findings (a) extend an accumulating body of evidence for the role meta-representation plays in mathematics and science learning and (b) add a new dimension to researchers' growing understanding of what teachers must know in order to teach algebra and other complex mathematics and science topics effectively.     

Why School English Needs a ‘Good Enough’ Grammatics (and Not More Grammar)

At the dawn of a national curriculum for English in Australia, grammar has appeared without any serious interrogation of the terms of its re-entry and against ambiguous evidence about its value for teaching writing. What kinds of knowledge about language do teachers need in rhetorically productive teaching? This article investigates the potential of Halliday’s notion of grammatics for understanding students’ writing as acts of meaning in context. Drawing on systemic-functional linguistics, I show how teachers can assess writing achievement using ‘big picture’ tools like genre, register and ‘small picture’ tools like Expansion. I apply these tools to two student texts that call for attention to creative uses of language and to excursions and to difficulties with logic and coherence. The paper concludes that a ‘good enough’ grammatics will enable teachers to recognize playful developments in students’ texts and also to foster their control of literate discourse.