Cognitive arithmetic studies the mental representation of numbers and arithmetic facts (counting, addition, subtraction, multiplication, division) and the processes that create, access and manipulate them. Why does a task which seems so straightforward and is indeed trivial for computer architectures (including the symbolic level of ACT-R) take years of formal schooling for humans to master? This suggests that human cognition at the subsymbolic level embodies some assumptions about the changing, approximate and adaptive nature of its environment which are at odds with the precision and immutability of formal mathematical theories.
This chapter presents a number of simulations of basic results of Cognitive Arithmetic. The most common is the ubiquitous problem-size effect, which states that larger problems are harder than smaller ones, both in terms of latency and percentage of errors. We present a straightforward symbolic model, which together with the common assumption that smaller problems are more common than larger ones, allows ACT-R's subsymbolic learning and partial matching mechanisms to reproduce a wide range of data including the problem-size effect in adults, the evolution of the problem-size effect over time, the pattern of errors in addition retrieval in four-year-olds and the pattern of errors in multiplication computation by repeated addition in fourth-graders. Those simulations assume certain distributions of knowledge strengths over time and directly use ACT-R's equations instead of Monte Carlo simulations to efficiently generate predictions.
While those simulations produced both tractable analyses and excellent fits, they use separate parameter values, require additional assumptions about the state of knowledge over time, and fail to provide a full understanding of the complex interactions between each arithmetic skill over time. To address those issues, this chapter also presents a lifetime simulation which traces in a single learning model the evolution of knowledge and performance through the hundreds of thousands of problems of the entire development cycle from childhood to adulthood. This suggests a view of cognitive systems which are not only determined by the statistics of the environment but also by the dynamics of the system itself.