Modeling Acquisition of a Torque Rule on the Balance-scale Task

Modeling Acquisition of a Torque Rule on the Balance-scale Task Frederic Dandurand (frederic.dandurand@univ-provence.fr) Laboratoire de psychologie cognitive, CNRS & Universite de Provence, UMR 6146, Case D, Bâtiment 9, 3 Place Victor Hugo, 13331 Marseille Cedex 3 France Thomas R. Shultz (thomas.shultz@mcgill.ca) Department of Psychology and School of Computer Science, McGill University, 1205 Penfield Avenue, Montreal, QC H3A 1B1 Canada Abstract We present a new model of development of children’s performance on the balance-scale task, one of the most common benchmarks for computational modeling of development. Knowledge-based cascade-correlation (KBCC) networks progress through all four stages seen in children, ending with a genuine torque rule that can solve problems only solvable by comparing torques. A key element in the model is injection of a neurally-implemented torque rule into the recruitment pool of KBCC networks, mimicking the explicit teaching of torque in secondary-school science classrooms. Keywords: Cognitive development; balance scale; neural networks; knowledge-based learning; KBCC; SDCC. Introduction The ongoing competition between symbolic and neural- network models of cognition often focuses on development of children’s performance on balance-scale problems, one of the most modeled tasks in developmental psychology. The symbolic view is that knowledge is represented in rules containing propositions referring to things in the world, that processing occurs as rules are selected and fired thus generating new propositions, and that knowledge is acquired by learning such rules. In neural-network accounts, active knowledge is represented in rapidly changing unit activations and long-term knowledge by excitatory and inhibitory connections between units, processing involves activations being passed from one layer of units to another, and knowledge acquisition results from adjustment of connection weights and occasional recruitment of new units into the network. The symbolic approach is sometimes referred to as rule use, and the neural-network approach as rule following (Shultz & Takane, 2007). At first glance this may seem to be a rather subtle distinction, but there are important differences between the two viewpoints that have consistently guided research over the last few decades. The rule-use approach assumes that people literally have and use rules to guide their reasoning and behavior, often affording the perfect generalization that symbolic rules sometimes allow. Rule-use is consistent with the idea that human cognition is often quite regular. In contrast, the rule-following approach assumes that such regularities may be approximated by neural networks that adapt to regularities in the environment. This affords graded generalizations whose regularity approximates the extent to which the environment is consistently regular, with the advantage that both regularities and exceptions, which are quite common in the complex phenomena that humans encounter, can be handled within the same neural network. In rule-use approaches, exceptions are instead typically memorized by a separate system from the rules themselves. Such differences are highlighted as researchers build precise computational models of psychological theories (Shultz, 2003). Computational models with artificial neural networks are quite different from those that represent and use symbolic rules. One of the most frequently modeled domains in developmental psychology focuses on the balance-scale task, used by Siegler (1976) and several other developmental researchers. The balance-scale is considered to be representative of the many tasks requiring integration of information across two separate quantitative dimensions. Results have been consistently replicated and include an interesting stage progression. Here we report an extended computational model of balance-scale development that addresses a recent criticism affecting most of the computational models – namely ensuring that the final stage consists of a genuine, multiplicative torque rule and not a simpler rule based on addition (Quinlan, van der Maas, Jansen, Booij, & Rendell, 2007). We first describe the basic balance-scale task and its stages before presenting our new computational model. The Balance-scale Task In this task, a participant is presented with a rigid beam balanced on a fulcrum (Siegler, 1976). There are several pegs positioned on the beam at regular distances to the left and right of the fulcrum. An experimenter places some identical weights on a peg on the left side and some other identical weights on a peg on the right side of the scale. The participant is asked which side of the scale will drop, or whether the scale will remain balanced, when the beam is released from its supports, often consisting of a block placed under each end of the beam. Archimedes’ (c. 287-212 BC) principle of the lever describes a rule that yields a correct answer to balance-scale problems: multiply the weight and distance from the fulcrum on each side and predict the side with the larger product (torque) to drop. A neural-network simulation using the cascade- correlation (CC) algorithm (Shultz, Mareschal, & Schmidt, 1994) captured the four stages seen in children (Siegler,