Computational Modelling in Mathematical Cognition
4.6 Plan of Work
The review of the ample experimental data available on mathematical cognition in chapter 2, and the related computational modelling efforts in chapter 3, followed by an overview of the relevant methodological aspects in chapter 4, set the scene for the presentation of the original research conducted in this thesis. The PhD research will investigate, with the aid of the tools provided by developmental cognitive robotics, two classes of embodied phenomena connected with learning to count.
The first class of the phenomena refers to a relatively short period of the de-velopment of human numerical knowledge, and focuses on the contribution of the counting gestures to learning to count. As reviewed in chapter 2, section 2.2.2, the effect of the counting gestures on counting accuracy in children is profound, and
various lines of experimental evidence suggest that such gestures are an important embodied cue, which aids, in some way, the acquisition of the conceptual under-standing of counting. Inspired by this, the aim of the first modelling experiment, presented in chapters 5 and 6, will be to replicate this phenomenon in the iCub humanoid robot. I am going to propose an artificial neural network model that will make it possible for the robot to learn to count with and without gestures, and I will assess the obtained counting accuracy according to a methodology based on that used in the studies with children. By simulating learning to count in the robot I will try to answer the question if the proprioceptive information connected with the counting gestures represented in the form of values of joint angles that change over time can be helpful to an artificial learning agent (research question 2). Further-more, inspired by some of the hypotheses about the mechanism of the contribution of the counting gestures put forward by the psychologists (cf. section 2.2.2) I will also conduct simulations aimed at investigating the significance of the simultaneous correspondence of the counting gestures to the number words being recited in the temporal domain, and to the items being enumerated in the spatial domain (research question 3). Although the model has been designed primarily with the above two research questions in mind, it turns out the approach adopted makes it possible to address also the research question 1.
The second group of experiments (chapters 7 and 8) assumes a broader per-spective in terms of the period of the development that is being looked at. More specifically, using an analogous methodology of simulating embodied phenomena with the aid of an artificial body provided by the iCub humanoid robot platform, I will investigate if it is possible to replicate the acquisition of spatial-numerical associations in an artificial learning agent as the result of systematic spatial bi-ases present in the development environment (research question 4). The evidence I quoted in chapter 2, section, 2.2.3, suggests that the degree to which humans intern-ally link the representations of numbers and space is striking. Although quite a lot is known about spatial-numerical associations in general, the nature of the
mech-anisms behind their ontogeny is still an open research question (cf. section 2.3.3).
In my simulation experiments I am going to extend the previous efforts in the com-putational modelling of the interactions between numbers and space (reviewed in section 3.4), by setting up a developmental process inspired by the progress of the acquisition of numerical knowledge by children. This process will involve the con-struction of spatial representations, as well as a simulation of the aforementioned spatial biases in learning to count, with the use of the iCub robot. In the subsequent behavioural simulations I will assess if the simulated developmental sequence leads to the manifestation of the association of numbers and space in the robot. This will be achieved with the classical tasks used in the same context for humans, such as timed number comparison, parity judgement and visual target detection (see section 2.2.3).
The motivations behind the focus on embodied numerical cognition are twofold.
From the point of view of developmental robotics, the investigation of mathemat-ical cognition is attractive because numermathemat-ical knowledge in general, and the concept of number (the understanding of which is believed to be acquired in the course of learning to count) in particular, can be put forward as prime examples of abstract concepts. The question how abstract concepts can be represented and acquired by an artificial learning agent is still open, yet it is of fundamental importance for the artificial intelligence research (Barsalou, 1999). The replication of the acquisition of numerical knowledge in robots, in a way that is plausible from the point of view of cognitive science, can therefore be expected to provide useful pieces of information that will contribute toward solving this puzzle. In turn, from the point of view of cognitive science, numerical thinking is so widespread in our lives and so crucial for many of its aspects that hopefully it is not necessary to convince the reader about the importance of the investigation of the mechanisms which enable humans to first of all come up with the mathematical ideas, and then to reason, understand, and create with the use of them (Campbell, 2005). As discussed in chapter 3, compu-tational modelling is a useful tool often employed in the research on cognition, and
section 4.2 of this chapter introduced cognitive developmental robotics as a method-ology that elegantly supports computational modelling of the embodied phenomena.
Considering the ample evidence for the embodied roots of mathematical thinking quoted in chapter 2, it is somewhat surprising that, to the best of my knowledge, at the time of writing no efforts to employ cognitive robotics as an aid in the study of mathematical cognition have been made. Hopefully, the efforts in this direction undertaken in the present thesis will pave the way for the future research, which will further our understanding of the nature of our mathematical thinking.