We apply collaborative filtering (CF) to dichotomously scored student response data (right, wrong, or no interaction), finding optimal parameters for each student and item based on cross-validated prediction accuracy. The approach is naturally suited to comparing different models, both unidimensional and multidimensional in ability, including a widely used subset of Item Response Theory (IRT) models which obtain as specific instances of the CF: the one-parameter logistic (Rasch) model, Birnbaum's 2PL model, and Reckase's multidimensional generalization M2PL. We find that IRT models perform well relative to generalized alternatives, and thus this method offers a fast and stable alternate approach to IRT parameter estimation. Using both real and simulated data we examine cases where one- or two-dimensional IRT models prevail and are not improved by increasing the number of features. Model selection is based on prediction accuracy of the CF, though it is shown to be consistent with factor analysis. In multidimensional cases the item parameterizations can be used in conjunction with cluster analysis to identify groups of items which measure different ability dimensions. (Contains 7 figures.) [This paper was also supported by the Massachusetts Institute of Technology. For the complete proceedings, "Proceedings of the International Conference on Educational Data Mining (EDM) (5th, Chania, Greece, June 19-21, 2012)," see ED537074.]