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# Modeling student engagement using optimal control theory

• * Corresponding author: Debra Lewis

Dedicated to Professor Anthony Bloch on the occasion of his 65th birthday.

• Student engagement in learning a prescribed body of knowledge can be modeled using optimal control theory, with a scalar state variable representing mastery, or self-perceived mastery, of the material and control representing the instantaneous cognitive effort devoted to the learning task. The relevant costs include emotional and external penalties for incomplete mastery, reduced availability of cognitive resources for other activities, and psychological stresses related to engagement with the learning task. Application of Pontryagin's maximum principle to some simple models of engagement yields solutions of the synthesis problem mimicking familiar behaviors including avoidance, procrastination, and increasing commitment in response to increasing mastery.

Mathematics Subject Classification: Primary: 49N90, 91E40, 93B50.

 Citation: • • Figure 1.  Left: graphs of $u_c(\mu, \cdot )$ for sample values of $\mu$; the lowest curve is the graph of $u_c(1, \cdot )$. Right: contour plot of $u_{\rm opt}$, with contour values $\frac j {10}, j = 0, \ldots, 10$. Black: u = 0; white: u = 1

Figure 2.  Left: $K_{\frac \pi {10}}(r)$ for $r$ near $\tan \frac \pi {10}$, right: $\upsilon_\circ(r)$ associated to $\xi_\circ(u) = \sqrt{1 - u^2}$

Figure 3.  Graphs of $X_{h_0}^\pm$ for $\mathcal{S} = [0, 1]$, $\gamma(m) = (1 - m)^2$, $\mu(m) = {1 \over 2} \left ( 1 - \frac m 2 \right )^2$, constant functions $\psi_{\rm pe} = {1 \over 2}$ and $\phi = \frac \pi {12}$, and representative values of $h_0$ for which $\breve C_{h_0}$ has a strict global minimum ${c_{h_0}}$. Solid curves: $X_{h_0}^+$; dashed curves: $X_{h_0}^-$

Figure 4.  Representative vertically aligned graphs of $K_\phi(r)$ and $\breve C_{h_0}(m)$ for constant $\phi$ and three values of $h_0$. Solid: projections of the trajectories with initial data $(m_0, r_0)$; dotted: points on the graphs outside those projections. Upper row: $r_0 < r_\phi$ and $m_0 > m_c$; initially, $m$ decreases and $r$ increases. Lower row: $r_0 > r_\phi$ and $m_0 < m_c$; initially, $m$ increases and $r$ decreases. Left: $m'$ changes sign as $r$ passes through $r_\phi$; center: $m$ and $r$ asymptotically approach the equilibrium $(m_c, r_\phi)$; right: $r'$ changes sign as $m$ passes through $m_c$

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