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Cognitive Arithmetic

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Christian Lebiere
John R. Anderson

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Abstract

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.

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Models

Static Simulations
Lifetime Simulation