# Developing students’ functional thinking in algebra through different visualisations of a growing pattern’s structure

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## Abstract

Spatial visualisation of geometric patterns and their generalisation have become a recognised pathway to developing students’ functional thinking and understanding of variables in algebra. This design-based research project investigated upper primary students’ development of explicit generalisation of functional relationships and their representation descriptively, graphically and symbolically. Ten teachers and their classes were involved in a sequence of tasks involving growing patterns and geometric structures over 1 year. This article focuses on two aspects of the study: visualising the structure of a geometric pattern in different ways and using this to generalise the functional relationship between two quantifiable aspects (variables). It was found that in an initial assessment task (*n* = 222), students’ initial visualisations could be categorised according to different types and some of these were more likely to lead either to recursive or explicit generalisation. In a later task, a small number of students demonstrated the ability to find more than one way to visualise the same geometric structure and thus represent their explicit generalisations as different but equivalent symbolic equations (using pronumerals). Implications for the teaching of functional thinking in middle-school algebra are discussed.

### Keywords

Algebra Functional thinking Pattern generalisation Visualisation Middle years of schooling## Notes

### Acknowledgments

The authors would like to acknowledge with appreciation the teacher participants and School Mathematics Leaders from the *Contemporary Teaching and Learning of Mathematics* project who contributed to the study on which this article is based. The CTLM project was conducted by the Mathematics Teaching and Learning Research Centre, Australian Catholic University, and funded by the Catholic Education Office, Melbourne, for 5 years (2008–2012).

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