# Tensions in the Design of Mathematical Technological Environments: Tools and Tasks for the Teaching of Linear Functions

## Abstract

The design of tasks for the exploration of mathematical concepts involving technology can take several starting points. In many cases the ‘tool’ is predefined as an existing mathematics application with an embedded set of design principles that shape the mathematical tasks that are possible. In other cases, the tool and tasks are designed through a more dynamic process whereby designers and educators engage in a discourse that influences the resulting tasks. The chapter will begin with a brief description of a longitudinal study, and its theoretical framework that resulted in a rubric to inform the design of tasks that privilege the exploration of mathematical variants and invariants (Clark-Wilson and Timotheus in *ICMI
study 22 task design in mathematics education*, UK: Oxford, 2013; Clark-Wilson in
How does a multi-representational mathematical ICT tool mediate teachers’
mathematical and pedagogical knowledge concerning variance and invariance?
2010). This rubric is then used as a construct for the post-priori analysis of two tasks that introduced the concept of linear functions and that use different technologies. Conclusions will be drawn that highlight subtle tensions that relate to the mathematical knowledge at stake and to the design principles of the underlying technology and task.

## Keywords

Mathematics Digital technology Task design Linear functions## Notes

### Acknowledgments

This chapter draws on research from two studies. The *TI*-*Nspire* evaluation study was funded by Texas Instruments and has been subsequently reported in Clark-Wilson (2008). *Cornerstone Maths* was generously funded by the Li Ka Shing Foundation as a multi-year collaborative project between London Knowledge Lab, UCL Institute of Education and SRI International, USA, and directed by Celia Hoyles, Richard Noss, Jeremy Roschelle and Phil Vahey.

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