Using tasks to explore teacher knowledge in situation-specific contexts

This article was published in the journal, Journal of Mathematics Teacher Education [© Springer] and the original publication is available at www.springerlink.comResearch often reports an overt discrepancy between theoretically/out-of context expressed teacher beliefs about mathematics and pedagogy and actual practice. In order to explore teacher knowledge in situation-specific contexts we have engaged mathematics teachers with classroom scenarios (Tasks) which: are hypothetical but grounded on learning and teaching issues that previous research and experience have highlighted as seminal; are likely to occur in actual practice; have purpose and utility; and, can be used both in (pre- and in-service) teacher education and research through generating access to teachers’ views and intended practices. The Tasks have the following structure: reflecting upon the learning objectives within a mathematical problem (and solving it); examining a flawed (fictional) student solution; and, describing, in writing, feedback to the student. Here we draw on the written responses to one Task (which involved reflecting on solutions of x+x−1=0 of 53 Greek in-service mathematics teachers in order to demonstrate the range of teacher knowledge (mathematical, didactical and pedagogical) that engagement with these tasks allows us to explore

Developing mathematical thinking in the primary classroom: liberating students and teachers as learners of mathematics

This paper reports on a research study conducted with a group of practising primary school teachers (n = 24) in North East Scotland during 2011–2012. The teachers were all participants in a newly developed Masters course that had been designed with the aim of promoting the development of mathematical thinking in the primary classroom as part of project supported by the Scottish Government. The paper presents the background for this initiative within the context of the Scottish Curriculum for Excellence reform. Particular attention is given to the epistemological positioning of the researchers as this influenced both the curriculum design process and also the theoretical framing of the research study which are both described. The project was set up within a design research framework, which aimed to promote classroom-based action research on the part of participants through the course and also research by the university researchers into the process of curriculum development. The research questions focused on the teachers’ confidence, competence, attitudes and beliefs in relation to mathematics and their expectations and experiences of the impact on pupil learning arising from this course. Empirical data were drawn from pre- and post-course surveys, interviews and observations of the discussion forums in the online environment. Findings from this study highlight the way the course had a transformational and emancipatory impact on these teachers. They also highlight ways in which the ‘framing’ of particular aspects of the curriculum had an oppressive impact on learners in the ways that suppressed creativity and limited the exercise of learner autonomy. Furthermore, they highlight the ways in which a number of these teachers had experienced mathematics as a school subject in very negative ways, involving high levels of ‘symbolic violence’ and of being ‘labelled’

Representing addition and subtraction : learning the formal conventions

The study was designed to test the effects of a structured intervention in teaching children to represent addition and subtraction. In a post-test only control group design, 90 five-year-olds experienced the intervention entitled Bi-directional Translation whilst 90 control subjects experienced typical teaching. Post-intervention testing showed some significant differences between the two groups both in terms of being able to effect the addition and subtraction operations and in being able to determine which operation was appropriate. The results suggest that, contrary to historical practices, children's exploration of real world situations should precede practice in arithmetical symbol manipulation

Confidence trick: the interpretation of confidence intervals

The frequent misinterpretation of the nature of confidence intervals by students has been well documented. This article examines the problem as an aspect of the learning of mathematical definitions and considers the tension between parroting mathematically rigorous, but essentially uninternalized, statements on the one hand and expressing imperfect but developing understandings on the other. A small-scale study among schoolteachers sought comments on four definitions expressing differing understandings of confidence intervals, and these are examined and discussed. The article concludes that some student wordings could be regarded as less inaccurate than they might seem at first sight and presents a case for accepting a wider range of more intuitive understandings as a work in progress

A diagrammatic view of the equals sign: arithmetical equivalence as a means, not an end

It is recommended in the mathematics education literature that pupils be presented with equality statements that can be assessed for numerical balance by attending to notational structure rather than computation. I describe an alternative, diagrammatic approach in which pupils do not assess statements but instead use them to make substitutions of notation. I report on two trials of a computer-based task conducted with pairs of pupils and highlight two findings. First, the pupils found it useful to articulate the distinct substitutive effects of commutative (‘swap’, ‘switch’) and partitional (‘split’, ‘separate’) statements when working on the task. Secondly, the pupils did not notice that some of the statements presented were in fact false, which suggests their substituting activities were independent of numerical equivalence conceptions. This demonstrates that making substitutions offers task designers a mathematical utility for equality statements that is distinct from, but complementary to, assessing numerical balance

Troubling "understanding mathematics-in-depth": Its role in the identity work of student-teachers in England

Copyright @ The Author(s) 2013. This article is published with open access at Springerlink.comThis article has been made available through the Brunel Open Access Publishing Fund.In this paper, we focus on an initiative in England devised to prepare non-mathematics graduates to train as secondary mathematics teachers through a 6-month Mathematics Enhancement Course (MEC) to boost their subject knowledge. The course documentation focuses on the need to develop “understanding mathematics in-depth” in students in order for them to become successful mathematics teachers. We take a poststructural approach, so we are not interested in asking what such an understanding is, about the value of this approach or about the effectiveness of the MECs in developing this understanding in their participants. Instead we explore what positions this discourse of “understanding mathematics in-depth” makes available to MEC students. We do this by looking in detail at the “identity work” of two students, analysing how they use and are used by this discourse to position themselves as future mathematics teachers. In doing so, we show how even benign-looking social practices such as “understanding mathematics in-depth” are implicated in practices of inclusion and exclusion. We show this through detailed readings of interviews with two participants, one of whom fits with the dominant discourses in the MEC and the other who, despite passing the MEC, experiences tensions between her national identity work and MEC discourses. We argue that it is vital to explore “identity work” within teacher education contexts to ensure that becoming a successful mathematics teacher is equally available to all.King’s College Londo

Negotiating different disciplinary discourses: biology students’ ritualized and exploratory participation in mathematical modeling activities

Non-mathematics specialists’ competence and confidence in mathematics in their disciplines have been highlighted as in need of improvement. We report from a collaborative, developmental research project which explores the conjecture that greater integration of mathematics and biology in biology study programs, for example through engaging students with Mathematical Modeling (MM) activities, is one way to achieve this improvement. We examine the evolution of 12 first-semester biology students’ mathematical discourse as they engage with such activities in four sessions which ran concurrently with their mandatory mathematics course and were taught by a mathematician with extensive experience with MM. The sessions involved brief introductions to different aspects of MM, followed by small-group work on tasks set in biological contexts. Our analyses use the theory of commognition to investigate the tensions between ritualized and exploratory participation in the students’ MM activity. We focus particularly on a quintessential routine in MM, assumption building: we trace attempts which start from ritualized engagement in the shape of “guesswork” and evolve into more productively exploratory formulations. We also identify signs of persistent commognitive conflict in the students’ activity, both intra-mathematical (concerning what is meant by a “math task”) and extra-mathematical (concerning what constitutes a plausible solution to the tasks in a biological sense). Our analyses show evidence of the fluid interplay between ritualized and exploratory engagement in the students’ discursive activity and contribute towards what we see as a much needed distancing from operationalization of the commognitive constructs of ritual and exploration as an unhelpfully dichotomous binary