Item response model applications to National Assessment of Educational Progress (NAEP) data specifically aimed at the uses of item response models in mathematics item banking are discussed. Approaches for addressing goodness of fit were organized into three categories: Checks on model assumptions, expected features, and additional model predictions. Within the categories, several new methods were also advanced and several older methods which were not in common use for determining item response model-data fit were described. Many of these methods were then used to determine the fit of the one- and three-parameter models to six NAEP mathematics booklets (three booklets for 9-year-olds and three booklets for 13-year-olds) in the 1977-78 assessment. There were some inconsistent findings but it did appear that the three-parameter model provided an excellent fit to the data sets whereas the one-parameter model did not. When a bank of content valid and technically sound test items is available, and goodness of fit studies reveal a high match between the chosen item response model and the test data, item response models may be useful to NAEP in test development, detection of biased items, score reporting, equating test forms and levels, item banking, and other applications as well. Primary type of information provided by the report: Results (Secondary Analysis). (Author/PN)