Thermodynamic quantities such as the average energy, heat capacity, and entropy are calculated using a Monte Carlo method based on the Metropolis algorithm. This method is illustrated with reference to the harmonic oscillator but is particularly useful when the partition function cannot be evaluated; an example using a one-dimensional spin system is examined. The calculations are performed entirely using a computer algebra package, and the calculations are straightforward enough for undergraduates to investigate for themselves in the laboratory. (Contains 4 figures.)