Education System Intervention Modeling Framework
3.2 Model Design
Changes in the agents ’ attributes and composite system state are modeled as a Discrete-Time Markov Chain ( DTMC ). It is assumed that change can be modeled as taking place in discrete time steps and that changes in the agents ’ attributes depend only on the current attribute level and current relationships , not on past system states . The discrete time step for this intervention is generally defined as one1 month . However , for the first three3 months of the school year starting in year three3 , the time step is reduced to one1 week to accommodate increased frequency of interactions among the agents during active implementation .
The following equation represents the general structure of the changes taking place in the model :
p change
( t )= w internal
∙p internal
( t ) ∙f s
+ w external ∙pexternal ( t ) ( 1 ) where p change is the overall probability of change at time t . The overall probability of change includes both an internal and external component . The internal component includes aspects of change affected directly by the intervention , and its weight is a function of the scale of the intervention . The structure of this equation is based on the assumption that every school environment has factors that lie beyond the scope of the intervention . However , the weighting allows a modeling decision to be made ; if the intervention is school-wide or district-wide , w _ internal will be greater than for an intervention involving only a single class or teacher . The effects of the weighting can be tested using sensitivity analysis . The probabilities are vector quantities representing the three possible state changes ,[ p improve
, p stay , p worsen
], which sum to 1 . In addition , the weights ,
w internal and w external , are non-negative and sum to 1 . Finally , the internal part of the equation also includes a multiplicative factor f s which captures an ‘“ S ”’ curve pattern in learning . This curve represents cumulative adoption of change in a complex adaptive system , meaning that it is harder to enact state change when an agent is already in a very high or very low state ( Rogers , Medina , Rivera , & Wiley , 2005 ).
The internal probability of change
p internal
( t ) is further divided into two parts : transient and steady state . It is assumed that a school is in steady state prior to an intervention . As implementation of the intervention begins , the school enters a temporary transient phase where change is more likely to occur , and then again enters a steady state after a certain number of time steps . The following equation represents the structure of the internal component of the change equation :
p internal
( t )= e -kt . p transient +( 1-e -kt ). p steady
( 2 )
where k is the transient parameter affecting how long the system stays in the transient phase , and p transient and p steady are the transient and steady steady-state sub-components , respectively , of the internal component of the change probability , p internal
( t ). In the transient state , the absolute values of the agents ’ attributes affect change in other agents , whereas in steady state , change in an agent only occurs when there is a change in other agents ’ attributes for those agents with whom there are relationships . This modeling choice dampens the effects of positive and negative feedback loops over time . Note that :
lim p i ( t )= p steady
( 3 )
( t→∞ )
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