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doi: 10.3934/jgm.2021032
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Modeling student engagement using optimal control theory

Mathematics Department, University of California Santa Cruz, Santa Cruz, CA 95064, USA

* Corresponding author: Debra Lewis

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

Received  June 2021 Revised  November 2021 Early access January 2022

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.

Citation: Debra Lewis. Modeling student engagement using optimal control theory. Journal of Geometric Mechanics, doi: 10.3934/jgm.2021032
References:
[1]

M. C. AndersonR. A. Bjork and E. L. Bjork, Remembering can cause forgetting: Retrieval dynamics in long-term memory, Journal of Experimental Psychology. Learning, Memory, and Cognition, 20 (1994), 1063-1087.  doi: 10.1037/0278-7393.20.5.1063.  Google Scholar

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[11]

T. L. GriffithsF. Lieder and N. D. Goodman, Rational use of cognitive resources: Levels of analysis between the computational and the algorithmic, Topics in Cognitive Science, 11 (2015), 217-229.  doi: 10.1111/tops.12142.  Google Scholar

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K. HallE. FawcettK. Hourihan and J. Fawcett, Emotional memories are (usually) harder to forget: A meta-analysis of the item-method directed forgetting literature, Psychonomic Bulletin & Review, 28 (2021), 1313-1326.  doi: 10.3758/s13423-021-01914-z.  Google Scholar

[13]

D. Kirk, Optimal Control Theory: An Introduction, Dover Publications, Inc., 2004. Google Scholar

[14]

S. KoesslerH. EnglerC. Riether and J. Kissler, No retrieval-induced forgetting under stress, Psychological Science, 20 (2009), 1356-1363.  doi: 10.1111/j.1467-9280.2009.02450.x.  Google Scholar

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D. Lewis, A soothing invisible hand: Moderation potentials in optimal control, in Geometry, Mechanics, and Dynamics: The Legacy of Jerry Marsden, Springer, 2015,257–284. doi: 10.1007/978-1-4939-2441-7_12.  Google Scholar

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I. Lyons and S. Beilock, When math hurts: Math anxiety predicts pain network activation in anticipation of doing math, PLOS ONE, 7. doi: 10.1371/journal.pone.0048076.  Google Scholar

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E. MaloneyM. Schaeffer and S. Beilock, Mathematics anxiety and stereotype threat: Shared mechanisms, negative consequences, and promising interventions, Research in Mathematics Education, 15 (2013), 115-128.  doi: 10.1080/14794802.2013.797744.  Google Scholar

[19]

A. Mattarella-MickeJ. MateoM. KozakK. Foster and S. Beilock, Choke or thrive? The relation between salivary cortisol and math performance depends on individual differences in working memory and math anxiety, Emotion, 11 (2011), 1000-1005.  doi: 10.1037/a0023224.  Google Scholar

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R. Montgomery, Optimal control of deformable bodies and its relation to gauge theory, Math. Sci. Res. Inst. Publ., 22 (1991), 403-438.  doi: 10.1007/978-1-4613-9725-0_15.  Google Scholar

[22]

J. Murre and J. Dros, Replication and analysis of Ebbinghaus' forgetting curve, PLOS ONE, 10 (2015).  doi: 10.1371/journal.pone.0120644.  Google Scholar

[23]

L. Pontryagin, V. Boltyanskii, R. Gamkrelidze and E. Mishchenko, The Mathematical Theory of Optimal Processes, Interscience Publishers, 1962.  Google Scholar

[24]

R. RydellA. McConnell and S. Beilock, Multiple social identities and stereotype threat: Imbalance, accessibility, and working memory, Journal of Personality and Social Psychology, 96 (2009), 949-966.  doi: 10.1037/a0014846.  Google Scholar

[25]

A. ShenhavS. MusslickF. LiederW. KoolT. GriffithsJ. Cohen and M. Botvinick, Toward a rational and mechanistic account of mental effort, Annual Review of Neuroscience, 40 (2017), 99-124.  doi: 10.1146/annurev-neuro-072116-031526.  Google Scholar

[26]

F. Sirois and T. Pychyl, Procrastination and the priority of short-term mood regulation: Consequences for future self, Social and Personality Psychology Compass, 7 (2013), 115-127.  doi: 10.1111/spc3.12011.  Google Scholar

[27]

N. Slamecka, On comparing rates of forgetting: Comment on Loftus (1985), Journal of Experimental Psychology. Learning, Memory, and Cognition, 11 (1985), 812-816.  doi: 10.1037/0278-7393.11.1-4.812.  Google Scholar

[28]

E. Sontag, Integrability of certain distributions associated with actions on manifolds and applications to control problems, in Nonlinear Controlability and Optimal Control, Marcel Dekker, Inc., 133 (1990), 81–131.  Google Scholar

show all references

References:
[1]

M. C. AndersonR. A. Bjork and E. L. Bjork, Remembering can cause forgetting: Retrieval dynamics in long-term memory, Journal of Experimental Psychology. Learning, Memory, and Cognition, 20 (1994), 1063-1087.  doi: 10.1037/0278-7393.20.5.1063.  Google Scholar

[2]

L. Averell and A. Heathcote, The form of the forgetting curve and the fate of memories, Journal of Mathematical Psychology, 55 (2011), 25-35.  doi: 10.1016/j.jmp.2010.08.009.  Google Scholar

[3]

J. Baillieul and J. Willems, Mathematical Control Theory, Springer, 1999. doi: 10.1007/978-1-4612-1416-8.  Google Scholar

[4] A. Bandura, Self-Efficacy in Changing Societies, Cambridge University Press, 1995.   Google Scholar
[5]

A. M. Bloch, Nonholonomic Mechanics and Control, Springer, 2003. doi: 10.1007/b97376.  Google Scholar

[6]

H. Chang and S. L. Beilock, The math anxiety-math performance link and its relation to individual and environmental factors: a review of current behavioral and psychophysiological research, Current Opinion in Behavioral Sciences, 10 (2016), 33-38.  doi: 10.1016/j.cobeha.2016.04.011.  Google Scholar

[7]

K. ChoeJ. JeniferC. RozekM. Berman and S. Beilock, Calculated avoidance: Math anxiety predicts math avoidance in effort-based decision-making, Science Advances, 5 (2019).  doi: 10.1126/sciadv.aay1062.  Google Scholar

[8]

R. Dorfman, An economic interpretation of optimal control theory, The American Economic Review, 59 (1969), 817-831.   Google Scholar

[9]

H. Ebbinghaus, Über das Gedächtnis, Dunker, 1885. Google Scholar

[10]

E. O. Finkenbinder, The curve of forgetting, The American Journal of Psychology, 24 (1913), 8-32.  doi: 10.2307/1413271.  Google Scholar

[11]

T. L. GriffithsF. Lieder and N. D. Goodman, Rational use of cognitive resources: Levels of analysis between the computational and the algorithmic, Topics in Cognitive Science, 11 (2015), 217-229.  doi: 10.1111/tops.12142.  Google Scholar

[12]

K. HallE. FawcettK. Hourihan and J. Fawcett, Emotional memories are (usually) harder to forget: A meta-analysis of the item-method directed forgetting literature, Psychonomic Bulletin & Review, 28 (2021), 1313-1326.  doi: 10.3758/s13423-021-01914-z.  Google Scholar

[13]

D. Kirk, Optimal Control Theory: An Introduction, Dover Publications, Inc., 2004. Google Scholar

[14]

S. KoesslerH. EnglerC. Riether and J. Kissler, No retrieval-induced forgetting under stress, Psychological Science, 20 (2009), 1356-1363.  doi: 10.1111/j.1467-9280.2009.02450.x.  Google Scholar

[15]

D. Lewis, A soothing invisible hand: Moderation potentials in optimal control, in Geometry, Mechanics, and Dynamics: The Legacy of Jerry Marsden, Springer, 2015,257–284. doi: 10.1007/978-1-4939-2441-7_12.  Google Scholar

[16]

G. R. Loftus, Evaluating forgetting curves, Journal of Experimental Psychology, 11 (1985), 397-406.  doi: 10.1037/0278-7393.11.2.397.  Google Scholar

[17]

I. Lyons and S. Beilock, When math hurts: Math anxiety predicts pain network activation in anticipation of doing math, PLOS ONE, 7. doi: 10.1371/journal.pone.0048076.  Google Scholar

[18]

E. MaloneyM. Schaeffer and S. Beilock, Mathematics anxiety and stereotype threat: Shared mechanisms, negative consequences, and promising interventions, Research in Mathematics Education, 15 (2013), 115-128.  doi: 10.1080/14794802.2013.797744.  Google Scholar

[19]

A. Mattarella-MickeJ. MateoM. KozakK. Foster and S. Beilock, Choke or thrive? The relation between salivary cortisol and math performance depends on individual differences in working memory and math anxiety, Emotion, 11 (2011), 1000-1005.  doi: 10.1037/a0023224.  Google Scholar

[20]

H. Melville, Bartleby, the scrivener: A story of Wall Street, Putnam's Magazine, November and December (1853), 446–457 and 609–615. doi: 10.1093/oseo/instance.00209193.  Google Scholar

[21]

R. Montgomery, Optimal control of deformable bodies and its relation to gauge theory, Math. Sci. Res. Inst. Publ., 22 (1991), 403-438.  doi: 10.1007/978-1-4613-9725-0_15.  Google Scholar

[22]

J. Murre and J. Dros, Replication and analysis of Ebbinghaus' forgetting curve, PLOS ONE, 10 (2015).  doi: 10.1371/journal.pone.0120644.  Google Scholar

[23]

L. Pontryagin, V. Boltyanskii, R. Gamkrelidze and E. Mishchenko, The Mathematical Theory of Optimal Processes, Interscience Publishers, 1962.  Google Scholar

[24]

R. RydellA. McConnell and S. Beilock, Multiple social identities and stereotype threat: Imbalance, accessibility, and working memory, Journal of Personality and Social Psychology, 96 (2009), 949-966.  doi: 10.1037/a0014846.  Google Scholar

[25]

A. ShenhavS. MusslickF. LiederW. KoolT. GriffithsJ. Cohen and M. Botvinick, Toward a rational and mechanistic account of mental effort, Annual Review of Neuroscience, 40 (2017), 99-124.  doi: 10.1146/annurev-neuro-072116-031526.  Google Scholar

[26]

F. Sirois and T. Pychyl, Procrastination and the priority of short-term mood regulation: Consequences for future self, Social and Personality Psychology Compass, 7 (2013), 115-127.  doi: 10.1111/spc3.12011.  Google Scholar

[27]

N. Slamecka, On comparing rates of forgetting: Comment on Loftus (1985), Journal of Experimental Psychology. Learning, Memory, and Cognition, 11 (1985), 812-816.  doi: 10.1037/0278-7393.11.1-4.812.  Google Scholar

[28]

E. Sontag, Integrability of certain distributions associated with actions on manifolds and applications to control problems, in Nonlinear Controlability and Optimal Control, Marcel Dekker, Inc., 133 (1990), 81–131.  Google Scholar

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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