When Time Makes a Difference: Addressing Ergodicity and Complexity in Education

Abstract

The detection of complexity in behavioral outcomes often requires an estimation of their variability over a prolonged time spectrum to assess processes of stability and transformation. Conventional scholarship typically relies on snapshots to analyze those outcomes, assuming that group means and their associated standard deviations, computed across individuals, are sufficient to characterize the educational outcomes that inform policy, and that time does not matter in this context. In its statistically abstract form, the assumption that you can rely on snapshots is referred to as the ergodicity assumption. This paper argues that ergodicity cannot be taken for granted in educational data. The first section discusses artificially generated time series trajectories to illustrate ergodicity (white noise) and three types of non-ergodicity: short-term correlations between observations, long-term correlations (pink noise) and infinite correlations (Brownian motion). A second section presents daily attendance data observed in two urban high schools over a seven year period to show that these data are non-ergodic and suggest complexity. These findings offer a counter-example to the efficacy of using time-independent measures (‘snapshots’) to measure educational outcomes.

Exploring the Effects of Creating Small High Schools on Daily Attendance: A Statistical Case Study

“…The research reported here is part of a program of ongoing study of the long range patterns of daily attendance in New York City high schools, and the identification of signs of complex adaptive behavior (fractality, meta-stability) in those patterns. Previous findings were reported by Koopmans (2015a;2015b;2016;2017a;2017b;2018).…”

Perspectives on Complexity, Its Definition and Applications in the Field

“…CDS argues that both need to be considered in conjunction in order to understand systemic stability and change. The methodological predisposition that follows from these insights is that many repeated observations of the behavior of a system are needed to understand its causal underpinnings (Koopmans, 2015).…”

“…Based on examples from various disciplines, Nicolis and Prigogine seek to find the commonalities between systems, and while their emphasis is on chemistry, physics and biology, their ability to demonstrate of the applicability of transformative models that are specific in the abstract across different fields of knowledge is what makes their search of particular interest. The extension of such change models to education can be readily appreciated by the fact many have already been investigated empirically, such as hysteresis (Stamovlasis, 2006;Stamovlasis & Tsaparlis, 2012), non-periodic attractors (Koopmans, 2015), emergence (Hussain et al 2014) and self-organization and fractality (Fromberg, 2010(Fromberg, , 2016. A discussion of complexity in terms of its transformative capabilities can also be found in Davis and Sumara (2006), who distinguish the construct defined in this way from complicity, i.e., systemic novelty due to repeated interactions, and simplexity, the emergence of simple linear features in complex systems (Stewart, 2007).…”

“…One of the broader implications of the complexity angle in education is that we need to investigate the details of the dynamical processes that result in transformation, the determinants of change and the baseline settings of the systems of interest. A wide range of statistical approaches is available to deal with these questions, such as catastrophe modeling (Stamovlasis, 2016a), fractional differencing (Koopmans, 2015(Koopmans, , 2016b, state space grids (Pennings & Mainhard, 2016) and orbital decomposition (Garner & Russell, 2016;Stamovlasis, 2016b).…”

Socio-Emotional Concern Dynamics in a Model of Real-Time Dyadic Interaction: Parent-Child Play in Autism

“…According to large swathes of empirical and theoretical work, this assumption is invalid for most measurable variables of psychological processes occurring within and between individuals (see, e.g. Molenaar, 2004; Kelderman and Molenaar, 2007; Molenaar and Campbell, 2009; Hamaker, 2012; Koopmans, 2015; Beltz et al, 2016; Hamaker and Wichers, 2017; Fisher et al, 2018). In fact, group statistics rarely represent individual cases and processes (a misconception also referred to as the ecological fallacy).…”