Abstract: A method for processing formulae includes encoding a formula by: training, with a server, a model by using a machine learning algorithm with a data set that includes a plurality of formulae; transforming, with a processor, a first formula into a tree format using the trained model; converting, with the processor, the tree format of the first formula into a plurality of lists; and encoding, with the processor, the plurality of lists into a fixed dimension vector by leveraging a stacked attention module; and generating one or more formula candidates by: obtaining, with the processor, input information; and generating, with the processor, one or more second formula candidates based on input information by using the stacked attention module with a tree beam search algorithm.

Abstract: A method for processing formulae includes encoding a formula by: training, with a server, a model by using a machine learning algorithm with a data set that includes a plurality of formulae; transforming, with a processor, a first formula into a tree format using the trained model; converting, with the processor, the tree format of the first formula into a plurality of lists; and encoding, with the processor, the plurality of lists into a fixed dimension vector by leveraging a stacked attention module; and generating one or more formula candidates by: obtaining, with the processor, input information; and generating, with the processor, one or more second formula candidates based on input information by using the stacked attention module with a tree beam search algorithm.

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: 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 is disclosed for tracing variation of concept knowledge of learners over time and evaluating content organization of learning resources used by the learners. Computational iterations are performed until a termination condition is achieved. Each of the computational iterations includes a message passing process and a parameter estimation process. The message passing process includes computing a sequence of probability distributions representing time evolution of concept knowledge of the learners for a set of concepts based on (a) learner response data acquired over time, (b) state transition parameters modeling transitions in concept knowledge resulting from interaction with the learning resources, (c) question-related parameters characterizing difficulty of the questions and strengths of association between the questions and the concepts.