Mental multiplication is an abstract and advanced cognitive skill requiring the application of systematic algorithms to structured numerical data. Although mental addition has previously been modeled in a biologically plausible substrate (Choo, 2018; Eliasmith et al., 2012), mental multiplication has not yet been explored. We present a spiking neuron implementation of a psychologically plausible model capturing how the mental multiplication of small natural numbers may be performed in the human brain. Our model uses many of the same strategies as humans, replicates multiple performance trends observed in experiments on humans, exhibits a level of performance tunable by the amount of resources at its disposal, and, ultimately, is capable of near-perfect performance (similar to adult humans). Our model sheds further light on how structured cognitive processes may be realized in a relatively unstructured substrate, and how mathematical reasoning and rule-based algorithms may be implemented in the human brain.