Basic Irony: Examining the Foundations of School Mathematics With Preservice Teachers

This article reports on an inquiry into what tends to be taken for granted with regard to the teaching and learning of mathematics. The inquiry, undertaken in the context of a course on methods for mathematics teaching, was developed around an examination of the mathematical notions that infuse conventional theories of cognition and that permeate the structures and practices of school mathematics. In particular, concepts drawn from or aligned with Euclidean geometry were examined. Specifically, alternatives drawn from fractal geometry were explored. The importance of interrogating the often-transparent figurative underpinnings of our thinking about thinking is highlighted.     

An emerging community in online mathematics teacher professional development: an interactional perspective

Online collaborative and content-focused professional development (PD) is becoming an increasingly important setting for supporting mathematics teachers’ professional learning. The purpose of this study was to better understand the process by which a community emerges in such a PD setting by examining how the cohesiveness of 21 mathematics teachers’ social network evolves and associated shifts in the quality of mathematics teachers’ mathematical discourse. We employed social network analysis (SNA) to examine the evolving cohesiveness of mathematics teachers’ social network and coding procedures to examine teachers’ mathematical discourse. A key finding was the documentation of an emergent divide between participation in the core and periphery during initial weeks of the PD and then a reduced divide and emergence of a social network that resembles a community. We argue that the instructor’s pattern of participation that included distributing their interactions across the subgroups while sending a common message regarding expectations for mathematical discourse in the PD may have contributed to the community formation process. We propose the Interaction Assessment Model, which outlines an approach for PD facilitators to use SNA as a feedback mechanism to differentiate facilitation of online collaborative and content-focused PD and build online communities.

     

Let them eat Shakespeare: prescribed authors and the National Curriculum

In this article, we examine the debate that surrounds prescribed reading lists in the English National Curriculum. In particular, we attempt to locate the role which ideas about heritage and social and moral values have played in constructing this debate. We begin by examining the English National Curriculum's origin in the 1980s as a conservative exercise in stemming cultural crisis, and the discourse about literature's role in the curriculum which this helped construct. We then examine how this discourse has influenced, and continues to influence, the educational policy of prescribing a list of authors and consider the assumptions that are embedded in this policy. Finally, we reflect upon how the material conditions of the classroom provide a site of resistance, or difficulty, for the officially sanctioned discourse concerning literature's role in the curriculum.     

Additional workload or a part of the job? Icelandic teachers' discourse on inclusive education

The aim of this article is to examine the discourse of Icelandic compulsory school teachers on inclusive education. From 1974 and onwards, the education policy in Iceland has been towards inclusion, and Iceland is considered to be an example of a highly inclusive education system with few segregated resources for students with special educational needs. In particular, the article focuses on what characterises and legitimises teachers' discourse on inclusive education, the contradictions in the discourse and how teachers have involved themselves in the process. We use the approach of historical discourse analysis to analyse the discourse as it appears in interviews with teachers and media articles on education as well as in key documents issued by the Parliament. The article provides an insight into the complexities of this topic and draws attention to underlying issues relevant to inclusive education.     

Heuristic literacy development and its relation to mathematical achievements of middle school students

The relationships between heuristic literacy development and mathematical achievements of middle school students were explored during a 5-month classroom experiment in two 8th grade classes (N = 37). By heuristic literacy we refer to an individual’s capacity to use heuristic vocabulary in problem-solving discourse and to approach scholastic mathematical problems by using a variety of heuristics. During the experiment the heuristic constituent of curriculum-determined topics in algebra and geometry was gradually revealed and promoted by means of incorporating heuristic vocabulary in classroom discourse and seizing opportunities to use the same heuristics in different mathematical contexts. Students’ heuristic literacy development was indicated by means of individual thinking-aloud interviews and their mathematical achievements – by means of the Scholastic Aptitude Test. We found that heuristic literacy development and changes in mathematical achievements are correlated yet distributed unequally among the students. In particular, the same students, who progressed with respect to SAT scores, progressed also with respect to their heuristic literacy. Those students, who were weaker with respect to SAT scores at the beginning of the intervention, demonstrated more significant progress regarding both measures.     

Animation as an aid for the teaching of mathematical concepts

Mathematics can be perceived as being a difficult subject to learn due to the conceptual leaps required to understand particular mathematical topics. In some areas of mathematics, part of the difficulty may be associated with applying sufficient imagination to visualize a particular mathematical concept, and applying sufficient visio–spatial abilities to apply the concept to practical use. In this paper we examine the potential use of animation as an aid for the teaching of mathematical concepts at undergraduate level. An experiment was conducted with a group of UK undergraduate computing students to ascertain if and how animation can potentially be used to support the teaching of mathematical concepts in areas such as rotational symmetry and matrices. In general, the results of the experiment appeared to indicate that animated learning materials can prove more useful than static representations for such topics.     

Algebraic methods in plane geometry

In this article we examine the role of mappings in elementary geometry. After making some comments about the Erlangen programme initiated by Felix Klein in 1872, wherein he proposed a way of studying geometries based on the underlying transformation groups, we see how theorems like Von Aubel’s theorem and Napoleon’s theorem can be proved in an elegant manner using similarity mappings, and how some construction problems may be solved using isometries. At the end we present a recent proof by Alain Connes of “Morley’s Miracle”, based on affine transformations. Shailesh Shirali heads a Community Mathematics Center at Rishi Valley School (KFI). He has a deep interest in teaching and writing about mathematics at the high school/post school levels, with particular emphasis on problem solving and the historical aspects of the subject.     

Analysis of students’ ideas and conceptual artifacts in knowledge-building discourse

This study proposes a new methodological approach to evaluate students’ knowledge-building discourse. First, we discuss the knowledge–creation metaphor of learning. In the metaphor, theories mention that learners should consider their collective knowledge objects or artifacts that materialize as a result of their collaborative discourse. Second, we argue the necessity of developing new analytics to evaluate student learning. We describe how students’ ideas and their conceptual artifacts can be examined in discourse analysis. Third, we demonstrate the application of our analytics to real discourse data. We conducted a new type of social network analysis of discourse to examine how students continuously improve their ideas. Further, we conducted another network analysis of discourse, called the Epistemic Network Analysis, by coding students’ epistemic actions as conceptual artifacts to create and examine their ideas.     

SITUATIONS D'INITIATION AU RAISONNEMENT DÉDUCTIF

We examine classroom situations which allow the learning of rules of mathematical reasoning. Drawing on an epistemological analysis and G. Brousseau's theory of situations, we define a priori theoretical characteristics which such situations must possess. We then use this analysis, and hypotheses about the cognitive functioning of pupils of this age, to establish learning situations for the rule of the excluded middle, and for the use of the figure in geometry. We report the results of experimentation with two of thses situations and show that, through the study of a mathematical problem, debate about these rules can be generated amongst students aged twelve to thirteen. We explain why, in our opinion, for epistemological reasons linked to the two particular rules concerned, it is necessary for the teacher to conclude the debate, and we examine our cognitive hypotheses against the actual results of experimentation.     

Gender equality in German universities: vernacularising the battle for the best brains

We examine how global pressures for competitiveness and gender equality have merged into a discourse of ‘inclusive excellence’ in the twenty-first century and shaped three recent German higher education programmes. After placing these programmes in the larger discourse about gender inequalities, we focus on how they adapt current global concerns about both being ‘the best’ and increasing ‘gender equality’ in locally specific ways, a process called vernacularisation. German equality advocates used ‘meeting international standards’ as leverage, drew on self-governance norms among universities, used formal gender plans as mechanisms to direct change, and set up competition to legitimate intervention. This specific incremental policy path for increasing women's status in German universities also mobilised the national funding agency and local gender equality officers as key actors, and placed particular emphasis on family friendliness as the expression of organisational commitment to gender equality.     

EFFECT OF THE PRESENCE OF EXTERNAL REPRESENTATIONS ON ACCURACY AND REACTION TIME IN SOLVING MATHEMATICAL DOUBLE-CHOICE PROBLEMS BY STUDENTS OF DIFFERENT LEVELS OF INSTRUCTION

This study explores the effects of the presence of external representations of a mathematical object (ERs) on problem solving performance associated with short double-choice problems. The problems were borrowed from secondary school algebra and geometry, and the ERs were either formulas, graphs of functions, or drawings of geometric figures. Performance was evaluated according to the reaction time (RT) required for solving the problem and the accuracy of the answer. Thirty high school students studying at high and regular levels of instruction in mathematics (HL and RL) were asked to solve half of the problems with ERs and half of the problems without ERs. Each task was solved by half of the students with ERs and by half of the students without ERs. We found main effects of the representation mode with particular effect on the RT and the main effects of the level of mathematical instruction and mathematical subject with particular influence on the accuracy of students’ responses. We explain our findings using the cognitive load theory and hypothesize that these findings are associated with the different cognitive processes related to geometry and algebra.     

Building awareness of dispositions: enhancing moral sensibilities in teaching

The purpose of this paper is to explain why and how dispositions can operate as a mechanism for enhancing teacher candidates’ moral sensibilities. Dispositions conjoin the knowledge and skills of teaching with the commitments one has to achieve intended purposes. Dispositions build candidates’ awareness of their own perceptions (and misperceptions) and how they can best connect their intentions with their practice, given their perception of the specific teaching situation. Teacher education programs foster candidates’ moral sensibilities when they help candidates connect their intention with their perception and with their practice. After exploring a conceptual rationale, three teacher candidates’ case study analyses are presented, illustrating a window into candidates’ inclinations toward particular behaviors and their awareness, or lack of awareness, into how they perceive teaching situations and connect their intention and practice.     

Changing policy discourses: Constructing literacy inequalities

This paper explores the ways in which policy discourses have constructed rationales for addressing adult literacy over the last 50 years. In particular, we examine how policy positions the literacy learner as citizen within discourses of rights and equity. Taking the case of the UK, we compare two key documents produced at different historical moments. Our discourse analysis reveals a long standing discourse of individual deficit within a functional model of literacy. This is now overlaid by a discourse of social exclusion that views adult learners as entrepreneurial global citizens who must compete within a market economy.     

Mathematics engagement in an Australian lower secondary school

The importance of actively engaging in mathematics discourse in order to learn mathematics is well recognized. In this paper, I use Basil Bernstein’s concepts of pedagogic discourse to document and analyse academic learning time of students in Years 8 and 9 at a suburban lower secondary school: in particular, for what proportion of class time students reported being academically engaged, their explanations for this engagement and how they felt about the discourse. It was found that many students had disengaged from mathematical endeavour as a result of the failure of the instructional discourse either to engage students or to serve the purpose of developing discipline-specific content knowledge. The reasons for this relate to the overemphasis on mundane mathematics resulting in some students lacking the cognitive tools to engage with the concepts and having neither the intrinsic nor instrumental motivation to persist with secondary school esoteric mathematics. The implications for mathematics curriculum development are discussed.     

Using an Undergraduate Geometry Course to Challenge Pre-service Teachers' Notions of Discourse

This investigation uses classroom discoursein an undergraduate mathematics course tochallenge pre-service secondary mathematicsteachers' notions about mathematical discourse,what it might resemble in the classroom, andhow its various forms can be cultivated byclassroom teachers. The research setting was arequired geometry course taught by the author.Eleven pre-service mathematics teachers intheir junior or senior year of an undergraduate program participated. Results indicated participants made a transition toward an image of discourse as an active process by which students use the collective knowledge of their peers to build mathematical understanding and developed in their ability to participate in such discourse.This awareness, along with participants'analyses of their own habits of discourse asclassroom teachers, prompted shifts in theirprojected image of the role of discourse intheir future practices of teaching. Resultssuggested further that the undergraduatemathematics classroom (as opposed to themethods classroom) offers a powerful and uniqueforum in which pre-service secondary teacherscan practice, articulate, and collectivelyreflect on reform-minded ways of teaching.     

Empowering Dialogues in Humanistic Education

Abstract

In this article I propose a conception of empowering educational dialogue within the framework of humanistic education. It is based on the notions of Humanistic Education and Empowerment, and draws on a large and diverse repertoire of dialogues—from the classical Socratic, Confucian and Talmudic dialogues, to the modern ones associated with the works of Nietzsche, Buber, Korczak, Rogers, Gadamer, Habermas, Freire, Noddings and Levinas. These forms of dialogue—differing in their treatment of and emphasis on the cognitive, affective, moral and existentialist elements—have become more dominant in recent educational discourse and practice—an intellectual phenomenon that calls for a more analytic and reflective elaboration of the essential elements that constitute educational dialogues. Hence it is the purpose of this article to elucidate the distinguishing marks of true dialogues, to set them within the normative discourse of humanistic education and empowerment, and to offer a normative and stipulative conception of empowering educational dialogue that can be utilized in the various intellectual and practical spheres of humanistic education—a paradigm,working definition, and outline for contemporary teachers in their quest to develop their students’ sensibilities and sensitivities, and empower their ability to live complete, autonomous, authentic, moral and dignified human lives.     

Conjecturing via reconceived classical analogy

Analogical reasoning is believed to be an efficient means of problem solving and construction of knowledge during the search for and the analysis of new mathematical objects. However, there is growing concern that despite everyday usage, learners are unable to transfer analogical reasoning to learning situations. This study aims at facilitating analogy use for conjecturing in discourse-rich mathematics classrooms. We reconceptualized one of the traditional perspectives on analogical reasoning, called classical analogy, as a more dynamic one by providing learners with the opportunity to choose a target object and its property. While shifting attention to particular aspects of mathematical activity, we observed and analyzed how students became aware of hidden relational similarities and utilized them while weakening others to make conjectures. The detailed analysis of the constructs and processes of several similarity-making and conjecturing activities supports the significance of reconceived classical analogy use in mathematics classrooms.     

Practice and conceptions: communicating mathematics in the workplace

The study examined the experience of communication in the workplace for mathematics graduates with a view to enriching university curriculum. I broaden the work of Burton and Morgan (2000), who investigated the discourse practices of academic mathematicians to examine the discourse used by new mathematics graduates in industry and their perceptions of how they acquired these skills. I describe the different levels of perception of discourse needs and of how they gained the necessary skills. At the lowest level, they learnt through trying out different approaches. At the next level, they were assisted by colleagues or outside situations. At the highest level, a small group viewed communication and interpersonal skills as a scientific process and stood back and used their “mathematical” observation skills to model their behaviour. These graduates did not appear to have systematically studied communication as part of their degree and they were unaware of the power of language choices in the workplace. Those who were working as mathematicians had to come to grips with explaining mathematical concepts to a wide range of people with varying mathematical skills but who generally were considerably less skilled in mathematics. The study revealed that these graduates were seriously underprepared in many aspects for joining the workforce. Many found it difficult to adapt to dealing with colleagues and managers, and developing communication skills was often a matter of trial and error.     

To know and to teach: Mathematical pedagogy from a historical context

Historical texts tell us much about how mathematical concepts and techniques were conceived and evolved. But they can also tell us even more. Usually, their contents embody a pedagogy, specific organizational forms and methods used for teaching mathematics. This article isolates and examines several particular pedagogical techniques: the use of an instructional discourse; a logical sequencing of mathematical problems and exercises and employment of visual aids, evident in historical works. It concludes that much of present day mathematical pedagogy evolved from distant historical antecedents.