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Students’ Evolving Meaning About Tangent Line with the Mediation of a Dynamic Geometry Environment and an Instructional Example Space

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Abstract

In this paper I report a lengthy episode from a teaching experiment in which 15 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 students’ 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 students’ knowledge develops in specific social and cultural contexts, students’ 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 students’ personal meanings about tangent line of function graph in relation to: the electronic environment; the pre-prepared as well as spontaneous examples; students’ engagement in classroom discussion; and, the role of researcher as a teacher. The analysis indicated that the evolution of students’ 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.

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Notes

  1. We call inflection point of a function graph, a point in which the concavity of the curve changes and there is a tangent line at this point; edge point a point in which the function is continuous and the derivative from the left and the right exist without being equal; and, cusp point a point in which the function is continuous and the limit of the rate of change from the left and the right is infinite.

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Acknowledgments

I would like to thank the students who participated in this study and their teacher, the anonymous reviewers for their useful feedback, Theodossios Zachariades and Despoina Potari for their support throughout the study and Lulu Healy, Victor Giraldo and the Research in Mathematics Education Group at the University of East Anglia, particularly Elena Nardi, for their helpful comments on drafts of this paper.

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  1. Mathematics Education Centre, Loughborough University, Loughborough, LE11 3TU, UK

    Irene Biza

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Correspondence to Irene Biza.

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Biza, I. Students’ Evolving Meaning About Tangent Line with the Mediation of a Dynamic Geometry Environment and an Instructional Example Space. Tech Know Learn 16, 125–151 (2011). https://doi.org/10.1007/s10758-011-9180-3

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