Young children’s approaches to additive missing number equations: A longitudinal study

Abstract

This doctoral research explores young children’s solving approaches for additive missing number equations, during their first and second years of formal school instruction (Year 1 aged five to six years, and Year 2 aged six to seven years, in England). A context for age-related expectations is provided by the inclusion of additive missing number equations in the Statutory Guidance for Mathematics: Addition and Subtraction in the National Curriculum for England, for Year 1. Central to this study is the qualitative longitudinal research design and the holistic approach taken. This research project explores how the same participant group of ten children approached additive missing number equations through qualitative, individual, task-based interviews. Data were collected at three collection points: when the children were at the end of Year 1, halfway through Year 2 and again at the end of Year 2. An analytical framework was developed to analyse the data, which came from audio-visual recordings and children’s written work and jottings. A holistic view of ‘solving approaches’ was operationalised to include observable solving behaviours, including verbal utterances and any use of mathematical models and manipulatives during each task solution. Areas of mathematics drawn on by the participants were quantitative relations, knowledge of part-whole structure and relations, addition and subtraction principles and interpretations of the equals sign and the notion of equivalence. The thread that runs through this study is mathematical structure. How participants read aloud the equation revealed that reading aloud the equation syntax incorrectly was not necessarily associated with an incorrect solution. When the surface structure (syntax) of canonical and non-canonical equations was verbally reordered, conserving the underlying part-whole relations underpinned successful solving approaches. From the finegrained analysis of the observed solving behaviours, the study found that the use of mathematical models and manipulatives provided evidence of structuring part-whole relations, and addition and subtraction counting strategies. From the holistic approach, a key finding was that children generally drew on combinations of mathematical knowledge during a solving approach, further, that convincing evidence of drawing on knowledge of part-whole relations and structure underpinned correct solutions, even when evidence of drawing on other aspects of mathematical knowledge was less strong. These findings remained true over time and reflected individual differences. Changes in participants’ solving approaches over time were analysed individually. Unsuccessful solutions for equation tasks, particularly for those with the structure ‘missing whole and using the symbol ‘-‘ for subtraction, showed evidence of the recall of associated incorrect number triplets, also backwards working in tasks with a non-canonical syntax. Early algebraic reasoning has roots in aspects of mathematics encountered in preformal learning experiences. The outcomes of this study bring to the fore the strength of evidence of early algebraic reasoning in solving missing number equations. Convincing knowledge of part-whole structure and relations was important, and evidence of drawing on combinations of mathematical knowledge was observed in successful solving approaches