Abstract
In this study, we identify ways in which a sample of 18 graduates with mathematics-related first degrees found the nth term for quadratic sequences from the first values of a sequence of data, presented on a computer screen in various formats: tabular, scattered data pairs and sequential. Participants’ approaches to identifying the nth term were recorded with eye-tracking technology. Our aims are to identify their strategies and to explore whether and how format influences these strategies. Qualitative analysis of eye-tracking data offers several strategies: Sequence of Differences, Building a Relationship, Known Formula, Linear Recursive and Initial Conjecture. Sequence of Differences was the most common strategy, but Building a Relationship was more likely to lead to the right formula. Building from Square and Factor Search were the most successful methods of Building a Relationship. Findings about the influence of format on the range of strategies were inconclusive but analysis indicated sporadic evidence of possible influences.
Highlights
In some educational contexts, finding a general rule for a data set, often presented as pairs of independent and dependent values or as sequential data, is a typical task in school algebra
The closest work we have found to our enquiries is from Yeşildere and Akkoç (2010) who gave 147 pre-service teachers two sequences of diagrams generated by quadratic growth, and identified two generalization approaches in participants’ self-reports
We investigate strategies in finding the nth term for simple quadratic sequences adopted by those who are already confident with numbers and elementary algebra
Summary
In some educational contexts, finding a general rule for a data set, often presented as pairs of independent and dependent values or as sequential data, is a typical task in school algebra (see Rivera, 2013). Crisp, Inglis, Watson and Mason (2012) analysed the ways in which a small sample of participants looked at data pairs presented in a vertical tabular form when identifying linear and quadratic rules from the first few terms of a sequence. All participants had degrees in mathematics-related subjects, having secure relevant numerical and algebraic competence in arithmetical operations and algebraic expressions They had all been successful in a mathematics curriculum which included generalization of sequence data at elementary and advanced levels, but we did not know if they had used taught or ad hoc methods for non-linear data. The technology requires participants to carry out the task ‘on screen’, without access to pencil and paper, as this would result in us not being able to follow the eye movements Aware of this limitation, we designed sequences (linear and monic two-term quadratics) which we felt were accessible to be completed mentally for these participants, given their academic background.
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