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.
Increasing numbers of young people experience disruption to their schooling owing to chronic illness. Absence from the day-to-day life of their school for prolonged or accumulative periods of time can erode their sense of belonging and create anxiety about falling behind academically. Maintaining positive connections to school can meet their desire for normalcy and realisable educational goals. Part of a project called Link ‘n’ Learn, funded by an Australian Research Council Linkage grant (2008–2010), this in-depth qualitative case study of 22 participants — senior secondary students and their mathematics teachers — investigated academic continuity: students’ access to and utilisation of opportunities to learn effectively so that academic progress is made despite disruption to full-time schooling. The students experienced diverse types of chronic illness, medical interventions, and patterns of absence from school. They all sought to continue their school studies. Their teachers highlighted surprise, concern and discomfort related to students studying during serious illness, and school workload issues. Ambiguities about educational responsibility for students during absence were widespread. Teachers demonstrated hesitance to initiate contact with students, but students nevertheless expressed their desire for teachers to remain involved with them. Implications for the educational support of young people with chronic illness are presented.
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