Two Plus Three Is Five: Discovering Efficient Addition Strategies without Metacognition.

Two Plus Three Is Five: Discovering Efficient Addition Strategies without Metacognition Steven S. Hansen (sshansen@stanford.edu) Cameron McKenzie (crlmck@stanford.edu) James L. McClelland (mcclelland@stanford.edu) Department of Psychology Stanford University, Stanford, CA 94305 USA Abstract several early models did have this character (Neches, 1987). However, the paucity of solution paths that involved faulty strategies appear to rule out the „take a random step‟ style exploration used by most reinforcement learning models (Crawley, Shrager, & Siegler, 1997). Trial and error accounts were thus rejected, and replaced by a theory that posited a metacognitive mechanism with explicit, domain- specific content knowledge to filter out flawed strategy proposals. This mechanism allows the discovery of new strategies while producing only reasonable errors. However, it remains unclear how children could acquire the complex knowledge required to judge the appropriateness of their own strategy proposals. The acquisition of the metacognitive filter is neither explained nor explicitly modeled. The approach taken in this paper is to circumvent this difficulty by proposing that a metacognitive filter may not be necessary in the first place. We accomplish this by modifying a standard trial and error, reinforcement-learning- based paradigm to be biased towards previously learnt actions. We note that children learning the addition task have already learnt to count out numbers of objects, count on their fingers, and perform addition using a finger- counting strategy (Siegler & Jenkins, 1989). As we shall demonstrate, instantiating a model with biases towards these actions obviates the need for a metacognitive filter. We also expand the notion of retrieval – a „strategy‟ that circumvents the need to engage in a structured sequence of behaviors by simply recalling the correct final answer to a problem – by suggesting that retrieval might also occur for appropriate subparts of a larger problem. Our model makes several novel predictions about the discovery process and questions the notion that selection and discovery processes necessarily take place at the level of complete strategies. When learning addition, children appear to perform a remarkable feat: as they practice counting out sums on their fingers, they discover more efficient strategies while avoiding conceptually flawed procedures. Existing models that seek to explain how children discover good strategies while avoiding bad ones postulate metacognitive filters that reject faulty strategies. However, this leaves unexplained how the domain- specific knowledge required to evaluate a strategy might be acquired prior to addition being mastered. We introduce a biased exploration model, which demonstrates that new addition strategies can be discovered without invoking metacognitive filtering. This model provides a fit to data comparable to previous models, with the considerable advantage of avoiding an appeal to knowledge whose source is not itself explained. Specifically, we fit the pattern of changes in strategy use over time as children learn addition, as well as the overall error rate and error types reported empirically. The model suggests that the critical element allowing strategy discovery may be prior learning, rather than metacognitive strategy evaluation. We close by offering several empirical predictions and propose that what others have called strategies might often be decomposable into elements that can be assembled on the fly as problem solving unfolds in real time. Keywords: M athematical Cognition; Strategy Discovery; Reinforcement Learning; M etacognition. Introduction Single-digit addition is one of the first hurdles children master on their way to mathematical competence. Given the importance of mathematics to educational attainment, it is unsurprising that the process by which children learn addition has received considerable attention (eg, Siegler & Jenkins, 1989). A remarkable observation from these studies is that, once they are equipped with the ability to count out sums on their fingers, children spontaneously (without instruction) exhibit faster strategies. Despite this willingness to innovate, children rarely arrive at a strategy that, when executed correctly, leads to the wrong answer. This poses a real problem for trial and error theories of learning. As they acquire new, faster strategies, how do children know which strategies to avoid? Several attempts have been made to model the evolution of children‟s approaches to solving simple addition problems. The apparent absence of explicit instruction in the use of particular observed s trategies would normally make reinforcement learning a candidate mechanism, and indeed Background The data our model attempts to account for comes from a study examining 4 and 5 year olds‟ discoveries of new finger addition strategies (Siegler & Jenkins, 1989). Children are assumed to come to the task knowing a correct but inefficient strategy, and are observed in a series of sessions spread over approximately three months as they solve simple addition problems . Over this time, children gradually acquire strategies that lead to faster completion of the task.