Abstracts of Research Papers:

• Representational Versatility in Learning Statistics
• Classifying Students’ Graphics Calculator Strategies
• Enhancing Conceptual Insight: Plane Curves in a Computerised Learning Environment

Representational Versatility in Learning Statistics
Alan T. Graham¹ and Michael O. J. Thomas^2
1Centre for Mathematics Education, The Open University, Milton Keynes, United Kingdom a.t.graham@open.ac.uk
²Mathematics Education Unit, Department of Mathematics, The University of Auckland, Auckland, New Zealand,
moj.thomas@aukland.ac.nz

Statistical data can be represented in a number of qualitatively different ways, the choice depending on the following three conditions: the concepts to be investigated; the nature of the data; and the purpose for which they were collected. This paper begins by setting out frameworks that describe the nature of statistical thinking in schools, and suggests how they can be used explicitly by teachers and students to support work on statistical investigations. It goes on to describe some of the ways in which students at school level can interact with different statistical representations using technology, and illustrates the interactions using common examples of statistical diagrams, graphs and tables. The descriptions of these interactions are based on a theory of representational versatility that espouses the importance of engaging with statistical representations in a conceptual as well as a procedural manner. Some implications for the teaching and learning of statistics are also considered.

Classifying Students’ Graphics Calculator Strategies
John Berry¹, Ted Graham¹ and Andy Smith^2
1Centre for Teaching Mathematics, The School of Mathematics and Statistics The University of Plymouth, Drake Circus Plymouth, PL4 8AA
jberry@plymouth.ac.uk , egraham@plymouth.ac.uk
²Andy Smith, Stockport College of Further Education, South Stockport, SK1 3UQ. Andy.smith@stockport.ac.uk

When students are working with hand held technology, such as graphics calculators, we usually only see the outcomes of their activities in the form of a contribution to a written solution of a mathematical problem. It is more difficult to capture their process of thinking or actions as they use the technology to solve a problem. In this paper we report on a case study that followed the progress of twelve first year university students as they solved twelve mathematical problems associated with the investigation of functions and their graphs. We used software, which we refer to as the key recorder software. This software runs in the background of the graphics calculator, discreetly capturing the students’ keystrokes as they make use of the calculator. The aim of the research study described in this paper was to provide insights into the problem solving strategies of these students. Through a detailed analysis of their graphics calculator keystrokes and associated written solutions, we have proposed a classification of different strategies that identifies the effectiveness of the solution strategies and the efficiency of the use of technology. This classification has important implications for the teaching of functions and their graphs with such technology.

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Enhancing Conceptual Insight: Plane Curves in a Computerised Learning Environment
Thierry Dana-Picard
Department of Applied Mathematics, Jerusalem College of Technology, Havaad Haleumi Str. 21, POB 16031, Jerusalem 91160 - Israel
email: dana@jct.ac.il

For questions in calculus, tables containing either intermediate or final results provide the student with a semi-graphical guide both to enhance their intuition and to help complete tasks. The joint usage of (hand-made) tabular presentations and of a CAS, in particular its graphical features, is particularly valuable in step-by-step working: such a learning process provides a more profound understanding of the mathematics involved and helps to develop better skills for solving problems. The examples considered include plane curves, either graphs of functions or parametric curves and space curves in implicit, parametric or polar form.