Abstract: Mechanisms for automatically grading a large number of solutions provided by learners in response to an open response mathematical question. Each solution is mapped to a corresponding feature vector based on the mathematical expressions occurring in the solution. The feature vectors are clustered using a conventional clustering method, or alternatively, using a presently-disclosed Bayesian nonparametric clustering method. A representative solution is selected from each solution cluster. An instructor supplies a grade for each of the representative solutions. Grades for the remaining solutions are automatically generated based on their cluster membership and the instructor supplied grades. The Bayesian method may also automatically identify the location of an error in a given solution. The error location may be supplied to the learner as feedback. The error location may also be used to extract information from correct solutions. The extracted information may be supplied to a learner as a solution hint.

Abstract: A mechanism for discerning user preferences for categories of provided content. A computer receives response data including a set of preference values that have been assigned to content items by content users. Output data is computed based on the response data using a latent factor model. The output data includes at least: an association matrix that defines K concepts associated with the content items, wherein K is smaller than the number of the content items, wherein, for each of the K concepts, the association matrix defines the concept by specifying strengths of association between the concept and the content items; and a concept-preference matrix including, for each content user and each of the K concepts, an extent to which the content user prefers the concept. The computer may display a visual representation of the association strengths in the association matrix and/or the extents in the concept-preference matrix.

Abstract: Mechanisms for automatically grading a large number of solutions provided by learners in response to an open response mathematical question. Each solution is mapped to a corresponding feature vector based on the mathematical expressions occurring in the solution. The feature vectors are clustered using a conventional clustering method, or alternatively, using a presently-disclosed Bayesian nonparametric clustering method. A representative solution is selected from each solution cluster. An instructor supplies a grade for each of the representative solutions. Grades for the remaining solutions are automatically generated based on their cluster membership and the instructor supplied grades. The Bayesian method may also automatically identify the location of an error in a given solution. The error location may be supplied to the learner as feedback. The error location may also be used to extract information from correct solutions.

Abstract: A mechanism for discerning user preferences for categories of provided content. A computer receives response data including a set of preference values that have been assigned to content items by content users. Output data is computed based on the response data using a latent factor model. The output data includes at least: an association matrix that defines K concepts associated with the content items, wherein K is smaller than the number of the content items, wherein, for each of the K concepts, the association matrix defines the concept by specifying strengths of association between the concept and the content items; and a concept-preference matrix including, for each content user and each of the K concepts, an extent to which the content user prefers the concept. The computer may display a visual representation of the association strengths in the association matrix and/or the extents in the concept-preference matrix.

Abstract: A mechanism for facilitating personalized learning. A computer receives graded response data including grades that have been assigned to answers provided by learners in response to a set of questions. Output data is computed based on the graded response data using a latent factor model. The output data includes at least: an association matrix that defines a set of K concepts implicit in the set of questions, wherein K is smaller than the number of questions, wherein, for each of the K concepts, the association matrix defines the concept by specifying strengths of association between the concept and the questions; and a learner knowledge matrix including, for each learner and each of the K concepts, an extent of the learner's knowledge of the concept. The computer may display a visual representation of the association strengths in the association matrix and/or the extents in the learner knowledge matrix.