A Propagation-Based Method of Estimating Students’ Concept Understanding

Abstract

In this paper, we introduce a method to estimate the degree of students’ understanding of concepts and relationships while they learn from digital text materials online. To achieve our goal, we first define a semantic network that represents the knowledge in a material. Second, we define students’ behavior as the sequence of relationships they read in the material, and we create a probabilistic model for relationship understanding. We also create inference rules to include new relationships in the network. Third, we simulate the propagation of the new concept understanding through the network by using a method based on Biased PageRank, extending it with a method to represent prior knowledge and weighting the contribution of every concept according to the uniqueness of its relationships. Finally, we describe an experiment to compare our method against a method without propagation and a method in which propagation is inversely proportional to the distance between concepts. Our method shows significant improvement compared to the others, providing evidence that propagation of concept understanding through the entire network exists.

Appendix

In order to prove the convergence of our method, we remind that in case of having an iterative method with transition matrix M, convergence is proven if $\lim_{y \to \infty } {\mathbf{M}}^{y} {\mathbf{u}} = {\mathbf{u}}$. Equivalently, to grant we will achieve a steady state distribution, we need to ensure that the biggest dominant eigenvalue of M is 1. In our case, we need to verify $\alpha {\mathbf{Du}} + \left( {1 - \alpha } \right){\mathbf{k}} = {\mathbf{u}}$. If we rewrite this equation as ${\mathbf{u}} = {\mathbf{Mu}}$, and we let S be the matrix that verifies s _ii  = pk(c _i ) and s _ij  = 0 for i ≠ j, we have that ${\mathbf{M}} = \alpha {\mathbf{D}} + \left( {1 - \alpha } \right){\mathbf{S}}$. Since it is difficult to prove that the dominant eigenvalue of M is 1, we will prove that the dominant eigenvalue for M ^T is 1 instead. Then, we can use the Perron-Frobenius theorem, which states that the dominant eigenvalue is the same for M and M ^T. We remember that in the concepts where the prior knowledge had been set, we had established that rund(r _ii ,t) = 1 and rund(r _ij ,t) = 0 for all j ≠ i. By using this, we can see that the dominant eigenvector u for the matrix M ^T is precisely k, the vector representing prior knowledge and the eigenvalue for that vector is 1.

As in the case of the original PageRank [27], convergence of is only granted if the transition matrix is (1) irreducible and (2) aperiodic. We know that matrices verifying that one diagonal element is non-zero are aperiodic [34]. This is our case because we had set the pk(c _i ) for at least one concept c _i and we also stated that in such case rund(r _ii ,t) = w _ii  = 1 for all t. The problem is that the transition matrix is not irreducible because we allowed rund(r _ij ,t) = 0 for all j in a concept whose prior knowledge has not been set. However, by using the principle “each relationship has its inverse” in Sect. 3.1, we have that if rund(r _ij ,t) = 0, then rund(r _ji ,t) = 0 too, so the concept c _i is completely isolated from the rest of the graph. In that case, we can just set u(c _i ,t) = 0 for all t, and we can remove the concept from the graph, having that the remaining graph is strongly connected and therefore its matrix is irreducible.