Pattern generalisation in secondary school mathematics : students’ strategies, justifications and beliefs and the influence of task features

Number pattern generalisation is often regarded a difficult topic for students to learn. To explore this perception, the present study undertakes an empirical investigation with the main aim of providing a comprehensive description of how 14-year-old secondary school students in Singapore generalise figural patterns and justify their generalisations when varying the formats of pattern display and the types of function. Comprising two interrelated parts, the study first examines 515 students’ strategies and justifications and probes systematically the influence of the formats of pattern display and the types of function on their generalisations through a specially developed paper-and-pencil test. The other part, through a specially designed questionnaire, looks at their beliefs about which strategy would best help them to derive the rule for predicting any term of a figural pattern as well as their ability to construct the rule using their choice of strategy. The first part uses an independent-measures research design to examine whether different formats of pattern display have any effect on students’ rule construction and a repeatedmeasures research design to determine whether their rule construction is influenced by the different types of function. In the second part, a survey study is employed with all students asked to identify their choice of best-help generalising strategy. This is then followed by interviews with 16 of the 515 students to probe whether they are able to derive a correct functional rule using their chosen strategy. This study complements many previous studies mainly undertaken in the west in that its findings indicate that the more academic students are competent in developing a functional rule for linear patterns but falters when working with quadratic patterns. There is a widespread failure of the less academic students in both linear and quadratic patterns, confirming the oft-regarded view that expressing generality is elusive. Successful students perceive the patterns in several ways and generate wide-ranging functional rules, predominantly symbolic, to describe them. They employ a variety of generalising strategies, especially the figural type, and some of which are new in the literature. Both the test and the survey confirm that the figural strategy involving the breaking up of the whole configurations into non-overlapping parts is their clear favourite. For rule justification, verifying it using the numerical cues and drawing diagrams to explain its development are their favourite approaches. Task features such as the format of pattern display and the type of functions do contribute to student difficulties in generalisation. Based on these findings, some useful teaching strategies for teachers and teacher educators are then suggested to help them improve their teaching of pattern generalisation. The findings also point the direction for future research studies on pattern generalisation by suggesting some recommendations for researchers