97 research outputs found
Studentsâ Evolving Meaning About Tangent Line with the Mediation of a Dynamic Geometry Environment and an Instructional Example Space
In this paper I report a lengthy episode from a teaching experiment in which fifteen Year 12 Greek students negotiated their
definitions of tangent line to a function graph. The experiment was designed for the purpose of introducing students to the
notion of derivative and to the general case of tangent to a function graph. Its design was based on previous research results on
studentsâ perspectives on tangency, especially in their transition from Geometry to Analysis. In this experiment an instructional
example space of functions was used in an electronic environment utilising Dynamic Geometry software with Function
Grapher tools. Following the Vygotskian approach according to which studentsâ knowledge develops in specific social and
cultural contexts, studentsâ construction of the meaning of tangent line was observed in the classroom throughout the
experiment. The analysis of the classroom data collected during the experiment focused on the evolution of studentsâ personal
meanings about tangent line of function graph in relation to: the electronic environment; the pre-prepared as well as
spontaneous examples; studentsâ engagement in classroom discussion; and, the role of researcher as a teacher. The analysis
indicated that the evolution of studentsâ meanings towards a more sophisticated understanding of tangency was not linear. Also
it was interrelated with the evolution of the meaning they had about the inscriptions in the electronic environment; the
instructional example space; the classroom discussion; and, the role of the teacher
Emergences and affordances as opportunities to develop teachersâ mathematical content knowledge
Teachersâ mathematical content knowledge has been under scrutiny for some time. This development is in the wake of learnersâ unsatisfactory performance in national examinations and international achievement tests. A widely held belief is that one, if not the most important, of the efforts to improve and enhance the performance and achievement in mathematics of learners is addressing teachersâ mathematical content and pedagogical content knowledge through continuous professional development initiatives. The focus of this article is on the former. It describes how emergent and affording opportunities are brought to the fore from classroom observations and interactions in workshops and institutes with practising teachers. It concludes that this in situ dealing with mathematical content knowledge holds much promise for buy-in by teachers because it addresses an immediate need related to their practice
Development of intuitive rules: Evaluating the application of the dual-system framework to understanding children's intuitive reasoning
This is an author-created version of this article. The original source of publication is Psychon Bull Rev. 2006 Dec;13(6):935-53
The final publication is available at www.springerlink.com
Published version: http://dx.doi.org/10.3758/BF0321390
Justifications-on-demand as a device to promote shifts of attention associated with relational thinking in elementary arithmetic
Student responses to arithmetical questions that can be solved by using arithmetical structure can serve to reveal the extent and nature of relational, as opposed to computational thinking. Here, student responses to probes which require them to justify-on-demand are analysed using a conceptual framework which highlights distinctions between different forms of attention. We analyse a number of actions observed in students in terms of forms of attention and shifts between them: in the short-term (in the moment), medium-term (over several tasks), and long-term (over a year). The main factors conditioning studentsÂŽ attention and its movement are identified and some didactical consequences are proposed
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 âunderstanding mathematics in-depthâ 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 âunderstanding mathematics in-depthâ makes available to MEC students. We do this by looking in detail at the âidentity workâ 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 âunderstanding mathematics in-depthâ 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 âidentity workâ within teacher education contexts to ensure that becoming a successful mathematics teacher is equally available to all.Kingâs College Londo
Doctoral studentsâ use of examples in evaluating and proving conjectures
The final publication is available at: www.springerlink.comThis paper discusses variation in reasoning strategies among expert mathematicians,
with a particular focus on the degree to which they use examples to reason about
general conjectures. We first discuss literature on the use of examples in understanding and
reasoning about abstract mathematics, relating this to a conceptualisation of syntactic and
semantic reasoning strategies relative to a representation system of proof. We then use this
conceptualisation as a basis for contrasting the behaviour of two successful mathematics
research students whilst they evaluated and proved number theory conjectures. We observe
that the students exhibited strikingly different degrees of example use, and argue that
previously observed individual differences in reasoning strategies may exist at the expert
level. We conclude by discussing implications for pedagogy and for future research
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