A mathematical model of peer-instruction including stochastic uncertainty

A mathematical model of peer-instruction including stochastic uncertainty is presented. By using the master equation describing stochastic transition among different states, a stochastically modified version of Nitta’s peer-instruction model is obtained. It is shown that moment equations with a simple closure reproduce the expectation and the variance obtained by using direct numerical simulations of the resultant model. Such a mathematical model will provide insights to the real data beyond the standard statistical analysis.


Introduction
Peer-instruction [1,2] is a well-recognized method for active learning, in which students' learning gain (peer instruction efficiency; PIE [3,4]) is evaluated by using pre/post tests (questions before/after discussions).Nitta [3,4] proposed a minimal mathematical model of peer-instruction which learning strongly depends on prior knowledge of students.Peer-instruction induces learning processes including stochastic transition between students choosing correct answers and those choosing incorrect answers.In Nitta [3,4], such a transition process is modeled by using the master equation for the (normalized) number of students choosing correct answers.Under the insight that transition to the correct answer disappears when no student chooses correct answers in pre-tests, the transition matrix for the peerinstruction is set to depend only on the prior knowledge, resulting in a simple deterministic model [3,4].
Such a stochastic transition process also appears in different fields such as chemical reaction [5,6], a population system [7], and voter dynamics [8].In this study, we start from two state diffusion model [7] which includes the normalized number of students choosing correct answers (  ) and incorrect answers (  = 1 −   ) as state variables.If each incorrect answer should be divided, we should apply multi-state model [8].It is important that the nondeterministic behavior in these models comes from the finite size of the system, which effects are not included in Nitta's model.In this sense, the deterministic model in Nitta [3,4] corresponds to the case that the sufficient number of students is included in each peerinstruction, while the number of students in each classroom is often small.The purpose of this study is to give a theoretical base of stochastic uncertainty of PIE through the stochastically modified version of Nitta's model.We start from the master equation of the stochastic transition among different states and a stochastic differential equation for PIE is presented.
By expanding the master equation to Δ, we obtain the Fokker-Planck equation and the stochastic differential equation (SDE) [7,8].When the outgoing process (transition from   to   through interactions among students) [3,4] is negligible (  = 0), we obtain where  is the standard Wiener process,   = , and ϵ  = ϵ  = ϵ.Terms including ϵ in the drift term have similar roles to terms related to memory and tutoring in the deterministic model [9].In the present model, terms including ϵ indicate the random transition (mutation), which occurs even if there is no interaction among students.Such a random transition statistically mimics other factors affecting student's choice of answers except for interactions among students.In this study, we assume the random transition with the equal probability (ϵ  = ϵ  ).Actually, we can also consider the case that there is no change of the expectation of   without interactions among students ( = 0).In this case, the ratio between ϵ  and ϵ  should be given by the pre-test score (  ( = 0)), otherwise ϵ  = ϵ  = 0.
Note that the deterministic model in Nitta [3,4] is derived from the master equation of   itself and we can obtain Nitta's deterministic model from eq.( 3) in the limit  → ∞ and ϵ = 0.Although we can directly obtain a simple stochastically modified version of Nitta's model by taking  =  = 1/(Δ), in this study we calculate   ( = ) by using the SDE (3) with  ≪  = 1.We also set ϵ = 0.1 not to have the bimodal distribution [6][7][8] when   =   =  and ϵ  = ϵ  = ϵ.In this study, we set   = 0 as done in the deterministic model in Nitta [3].The case with   ≠ 0 is shortly commented at the last paragraph of this section.
By using the Ito � formula [10,11], we find   2 = 2    + (Δ) 2 (  (1 −   ) + ϵ), (4) Since the drift term of the SDE is nonlinear, some closures are necessary to close the evolution equation for moments.In the present study, a simple closure (the limit of the small second and third cumulants) [12] is used.By using the closure, at the first and the second orders, we find equations for the expectation ([  ] = μ) and the variation ( where [ ] indicates the ensemble average.Filled-circles in Fig. 1 show the (a)expectation and (b)variance of   at  =  obtained by the numerical integration of (3) with a simplified second order Taylor scheme [13].The number of paths is 4 × 10 4 .The horizontal axes indicate the initial condition   ( = 0) (  before discussion).Solid lines indicate the solutions of moment equations (5)(6), which is solved by using the fourth order Runge-Kutta method.Figure 1(b) shows that the variance has the maximum value at   ( = 0) = 0.1.Figure 2 shows solutions with same parameters as those in Fig. 1 except for .If ϵ = 0, the dependence of the expectation on the pre-test score is more stressed.In contrast, the expectation becomes independent of the pre-test score with increasing ϵ .The present choice of ϵ (= 0.1) is the case that effects of interactions among students are dominant.