How to Model the Development of Strategy Use: The Case of Multiplication Learning and Working Memory

Variability in strategy selection is an important characteristic of learning new skills, such as mathematical skills: strategies gradually come and go during this development. Siegler (1996) described this phenomenon metaphorically as overlapping waves. Although this hypothesis of overlapping waves has received increasing interest in developmental research, it has thus far not been directly tested. Testing the overlapping waves hypothesis requires a microgenetic study: frequent, dense observations, and a modeling technique that allows modeling of growth based on categorical responses (i.e., strategies). The graded response model (Samejima, 1969), in combination with latent growth modeling, allows exactly this. A further advantage of such an approach is that it enables a quantification of the strategy ability, as all strategies that a child uses to solve various problems are used to derive an ability estimate on one single latent scale. This score can then be used to analyze and quantify the relationship between strategy use and other variables. To test the validity of this approach, we applied this model to the strategies that children applied when learning single-digit multiplication. Strategy use was modelled and then related to accuracy, but also to working memory. We expected that children with poor working memory are limited in their possibilities to progress to more mature strategies. This would explain the often-found relationship between working memory and mathematical abilities (for a review, see Raghubar, Barnes, & Hecht, 2010). Once a week, for eight weeks, 98 children (7-8 years) solved a series of multiplication problems, and their multiplication strategy use and accuracy was assessed. Strategy was coded as wrong, counting (verbally, on fingers, or drawing and counting), repeated addition (e.g., 4 x 6 = 6 + 6 + 6 + 6), derived facts (e.g., 9 x 7 = 10 x 7 – 7), or retrieval. A graded response model was fitted with Marginal Maximum Likelihood estimation (two example curves are shown in Figure 1). This model showed a good fit, confirming that strategies come and go like waves during development. It also shows that individual differences (and improvement over time) can be mapped onto an underlying continuous latent ability. This is in line with the overlapping waves hypothesis. The curves in Figure 1 show that strategy use is also problem-specific: e.g., 3 x 7 is solved earlier with retrieval while for 9 x 3 repeated addition dominates longer. Strategy use and improvement was also significantly related to accuracy: children that used more advanced strategies were also more accurate, and children that improved more in strategy also improved more in accuracy. Finally, working memory predicted both strategy selection and accuracy: children with high working memory scores used more advanced strategies and made fewer mistakes, even when correcting for their strategy use. The results show that the Graded Response Model in combination with growth modeling is a promising technique to model the development of strategy use, that can potentially also be used in many other domains.