Learning Number and Algebra in Computer-Based Environments: Selected Papers from ICTMT-7

Guest editors:
Keith Jones, University of Southampton, UK
Federica Olivero, University of Bristol, UK

• Exploring Links across Representations of Numbers with Young Children
• Designing to See and Share Structure in Number Sequences
• Experiments with Aplusix in Four Countries
• T-algebra: Adding Input Stage to Rule-Based Interface for Expression Manipulation
• New Tools for Mathematical Learning: Dynamic Number Lines

Exploring Links across Representations of Numbers with Young Children
Tony Harries and Jennifer Suggate
School of Education, University of Durham, Leazes Rd., Durham DH1 1TA
a.v.harries@durham.ac.uk jennifer.suggate@dur.ac.uk

Numbers can be represented in a variety of ways – through pictures, diagrams, symbols. Each representation highlights different features of the number and the number system. This study aims to explore pupil understanding of number both within and across representations. A computer environment (suite of programmes) was created within which representations could be generated and manipulated. This study focuses on one of the programmes within which activities were developed for pupils in years 1, 2 and 3 of English primary schools (ages 5 to 8 years). The results were analysed across year groups and across attainment levels. The study found that not all representations are equally well understood. Reading figures accurately, often comes before an understanding of place value. Over the first three years of schooling there is improvement in understanding all representations although the number line and the beads seem to cause some difficulties. An ability to count in tens and ones is associated with greater understanding of many representations.

Designing to See and Share Structure in Number Sequences
Yishay Mor, Richard Noss, Celia Hoyles, Ken Kahn and Gordon Simpson
London Knowledge Lab, Institute of Education, University of London, UK
yishaym@gmail.com r.noss@ioe.ac.uk c.hoyles@ioe.ac.uk k.khan@ioe.ac.uk g.simpson@ioe.ac.uk

This paper reports on a design experiment in the domain of number sequences conducted in the course of the WebLabs project. We iteratively designed and tested a set of activities and tools in which 10-14 year old students used the ToonTalk programming environment to construct models of sequences and series, and then shared their models and their observations about them utilising a web-based collaboration system. We report on the evolution of a design pattern (or programming method) called ‘streams’ which enables students to engage in the process of summing and ‘holding the series in their hand’, and consequently make sophisticated arguments regarding the mathematical structures of the sequences without requiring the use of algebra. While the focus of this paper is mainly on the design of activities, and in particular their epistemological foundations, some illustrative examples from the work of one group of students indicate the potential of the activities and tools for expressing and reflecting on deep mathematical ideas.

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Experiments with Aplusix in Four Countries
Jean Francois Nicaud¹, Marilena Bitta², Hamid Chaachoua¹, Parimala Inamdar³ and Laura Maffei^4
1Université Joseph Fourier and IUFM, Grenoble, France Nicaud@imag.fr Chaachoua@imag.fr
²Universidad Federal de Mato Grosso do Sul, Campo Grande, Brazil, marilena@nin.ufms.br
³Center for Research in Cognitive Systems, NIIT, Mumbai, India, ParimalaI@NIIT.com
⁴University of Siena, Italy, laura.maffei@libero.it

The Aplusix system has been designed for helping students to learn algebra. Its capacity to tell the students whether their calculations are correct or not, to provide families of exercises of a chosen level, and to give scores after tests allows this system to be used in the regular functioning of the class. Its capacity to record the students’ actions and to replay the students’ work is a valuable feature for the researcher in mathematics education. In this paper, we describe four experiments conducted in four different countries with different goals: remediation piloted by researchers in Italy; remediation integrated in the regular functioning of classes in Brazil, collaborative learning in India, and learning and use during the entire school year in France.

T-algebra: Adding Input Stage to Rule-Based Interface for Expression Manipulation
Marina Issakova, Dmitri Lepp and Rein Prank
University of Tartu, Institute of Computer Science, Estonia
marina.issakova@ut.ee dmitri@ut.ee rein.prank@ut.ee

T-algebra is a project for creating an interactive learning environment for expression manipulation tasks of elementary algebra. Our main didactical principle has been that all the necessary decisions and calculations at each solution step should be made by the student, and the program should be able to understand the mistakes. This paper describes the design of our Action-Object-Input dialogue and different input modes as an instrument to communicate three natural attributes of the steps: choice of conversion rule, operands and result.

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New Tools for Mathematical Learning: Dynamic Number Lines
Alison Clark-Wilson
The Mathematics Centre, University of Chichester, UK
a.clarkwilson@chi.ac.uk

Teachers and mathematics educators strive to develop tools for learning which support pupils to visualise mathematical structures and emphasise mathematical relationships. This paper describes the development of a new ICT tool for mathematical learning which it is planned be made freely available to UK schools through the National Grid for Learning portals. The number line tool evolved from a similar microworld developed with dynamic geometry software. Some of the design issues are described which, by their inherent mathematical nature turned into rich opportunities for exploration.

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