"1.95 for the fitted parameters and 1.63 for the clustered parameters. A paired t-test showed a significant difference: t(181) = 3.2, p < 0.01. This may be due to the fact that clustering tends to move parameters away from extreme values, bringing them closer to delivered parameters, which generally avoid extremes. Another advantage of clustering is to avoid overfitting with smaller amounts of data. To test this, we developed 23 new clusters, using 1561 skills and 1312 students. We then found the best-fitting cluster for each of the 275 skills that were not used in developing the clusters, using varying numbers of students. We also found best-fitting parameters for these 275 skills on the subsets of students and tested the fit with another set of 200 students. As Figure 4 shows, when there are a small number of students contributing to the data, the clusters provide a substantially better fit to the data than the best-fit estimates. This provides evidence both that clusters developed with one set of skills will generalize to another set and that, with small amounts of student data, clusters can help prevent overfitting. Figure 4: Comparison of clustered vs. best-fit estimates with differing numbers of students. 5 Conclusion. Previous work has shown that modeling student learning and performance parameters based on prior-year student data results in improved system efficiency. This paper explored the issue of how sensitive such effectiveness is to the particular sets of parameters used. Our results have shown that tutor performance is relatively insensitive to the particular parameter sets that are used. We were able to show that, using only 23 sets of parameters, we could produce virtually the same system behavior as we would see if we had used parameters found through exploring the full parameter space. This result does not argue against fitting these parameters based on data; rather it suggests that a quick estimate of such parameters can be sufficient to produce near-optimal behavior. It is worth pointing out that the parameters we are setting act as population parameters, which would likely benefit from adjustment for individual differences [1]. Indeed, these results may suggest that a more profitable route to accurate student modeling is to focus on individual differences, rather than population characteristics. We see clustering as complementary to both the Dirichet priors approach [2] and the use of contextual guess and slip [1]. The fact that we can model student behavior with a very small set of parameters helps us to extend the knowledge tracing model beyond simply a mathematical model of student behavior; we now have a better chance to interpret individual parameters within the set. For any knowledge component, we could calculate the goodness of fit to the data for each of the 23 parameter clusters. If we only see a good fit to one cluster, and that cluster has a high plearn parameter, then we can reasonably conclude that that the knowledge component is easily learned. Such a conclusion would be computationally expensive to reach in the full parameter space since, since we would need to explore a large part of the space before we could conclude that there is an almost-as-good fit to the data to be found with a low-plearn parameter set. Clustering parameters thus provides us a way to quickly examine knowledge components and determine which ones are problematic. Knowledge components with low plearn might suggest areas where we should refine our instruction. Ones with high pguess or high pslip might indicate areas where we need to reconsider the user interface. Ones with high pinitial might indicate areas where instruction is unneeded. We are optimistic that our work in reducing the parameter space for knowledge tracing will provide us with new ways to more quickly and confidently use knowledge tracing parameters to interpret student behavior."