Mental multiplication is an abstract and advanced cognitive skill that requires applying systematic algorithms to structured numerical data, but it remains underexplored in neural models. In this paper, we propose a psychologically plausible model of how anatomical circuits in the human brain may perform mental multiplication of small natural numbers and implement it in biologically plausible spiking neurons. Our model uses similar strategies to those used by humans, specifically repeated addition and rule use; matches various human performance levels, including that of both children and adults; and qualitatively replicates multiple human performance patterns, such as problem-size and outlier effects. Our novel model sheds further light on how structured cognitive processes, like the rule-based algorithms underlying mathematical reasoning, may be implemented in the substrate of spiking neurons in the human brain.