The importance of additive reasoning in children's mathematical achievement: a longitudinal study
Keywords: Mathematical reasoning
Mathematics learning
Mathematical development
Children

The aim of this thesis is to examine the relative importance of working memory, counting ability, and additive reasoning in children’s mathematics learning. One hundred and fifteen 6-year-old Chinese children in Hong Kong participated in two waves of assessments. At the first time point (T1 – first grade), they were assessed using non-verbal intelligence, working memory (central executive, phonological loop, and visuospatial sketchpad), counting ability (procedural counting and conceptual knowledge of counting), additive reasoning (knowledge of the commutativity and complement principles), and mathematical achievement (calculation and story problem solving). Approximately 10 months later (T2 – second grade), children's mathematical achievement in calculation and story problem solving were evaluated once again. The extent to which various cognitive factors longitudinally predicted children's mathematical achievement was evaluated in this study. Several key findings were identified through two sets of analyses – multiple regression models and latent profile analysis.
The multiple regression analyses showed that counting ability accounted for a significant amount of variance in T1 and T2 calculation beyond the effects of age, IQ, and working memory, in which conceptual knowledge of counting, but not procedural counting, was a unique predictor. However, counting ability did not contribute significantly to story problem solving at both time points. When additive reasoning was also included in the regression model, counting ability made a unique contribution to T1 calculation only, but not T2 calculation.
By contrast, additive reasoning and working memory appeared to be more stable and stronger predictors of children’s performance in calculation and story problem solving at both time points than counting ability. Additive reasoning explained a substantial and significant amount of variance in calculation and story problem solving at both time points after the effects of age, IQ, working memory, and counting ability were controlled for – Both knowledge of the commutativity and complement principles were unique predictors. Similarly, working memory also accounted for a significant amount of variance in calculation and story problem solving at both time points beyond the influence of age, IQ, counting ability, and additive reasoning. Among the three components of working memory, only the central executive was a unique predictor for all measures of mathematical achievement.
Autoregressive analyses provided strong evidence for the longitudinal predictive powers of additive reasoning and working memory. The analyses showed that both additive reasoning and working memory remained significant predictors of T2 mathematical achievement (calculation and story problem solving) even after the effects of children's previous performance were taken into account (i.e. T1 mathematical achievement). Overall, additive reasoning accounted for the greatest amount of variance in mathematical achievement both concurrently and longitudinally among all the other factors. This finding underscores the importance of additive reasoning in the teaching and learning of mathematics in young children.
Because additive reasoning (as indicated by the knowledge of the commutativity and complement principles) is a critical variable in this thesis and relatively scarce research has examined this construct, particular concern was paid to the measurement of additive reasoning. It was measured in two ways in the present study: with the support of concrete materials (the concrete condition) and without the support of concrete materials (the abstract condition). Latent profile analysis showed that all children who performed well in the abstract conditions also did well in the concrete conditions, whereas it did not reveal a group of children who performed well in the abstract conitions, but not in the concrete conditions as well. Another interesting finding was that all children who obtained high scores on tasks that assessed their knowledge of the complement principle also obtained high scores in tasks that assessed their understanding of the commutativity principle. The overall pattern of profiles provides initial evidence suggesting that additive reasoning may develop from thinking in the context of specific quantities to thinking about more abstract symbols, and children acquire the knowledge of the commutativity principle in abstract tasks before they start to acquire the knowledge of the complement principle.
This finding demonstrated that patterns of individual differences are present in the development of different aspects of additive reasoning. If teachers possess some knowledge about the particular strengths and weaknesses of each child, it would be easier for them to devise teaching strategies that are tailored to the needs of different children, which may relate to the developmental order of the commutativity and complement principles, and the role of concrete materials in this development. Thus, this study contributes to the literature by showing that assessing additive reasoning in different ways and identifying profiles with classification analyses may be useful for educators to understand more about the developmental stage where each child is placed. It appears that a more fine-grained assessment of additive reasoning can be achieved by incorporating both concrete materials and relatively abstract symbols in the assessment.