The Rasch model characterizes the probability that a learner answers a question correctly using two scalars: the question’s difficulty, and the learner’s ability.

The paper constructs the embedding of the question $q_t$ from concept ct at time step $t$ as $x_t = c_{c_t} + μ_{q_t} · d_{c_t}$, where $c_{c_t} ∈ R^D$ is the embedding of the concept this question covers, and $d_{c_t} ∈ R^D$ is a vector that summarizes the variation in questions covering this concept, and $μ_{q_t} ∈ R$ is a scalar difficulty parameter that controls how far this question deviates from the concept it covers.

The question-response pairs $(q_t , r_t )$ from concept $c_t$ are extended similarly using the scalar difficulty parameter for each pair: $y_t = e(c_t ,r_t ) + μ_{q_t} · f(c_t ,r_t )$, where $e(c_t ,r_t ) ∈ R^D$ and $f(c_t ,r_t ) ∈ R^D$ are concept-response embedding and variation vectors.