The power law as an emergent property

Recent work has shown that the power function, a ubiquitous characteristic of learning, memory, and sensation, can emerge from the arithmetic averaging of exponential curves. In the present study, the forgetting process was simulated via computer to determine whether power curves can result from the averaging of other types of component curves. Each of several simulations contained 100 memory traces that were made to decay at different rates. The resulting component curves were then arithmetically averaged to produce an aggregate curve for each simulation. The simulations varied with respect to the forms of the component curves: exponential, range-limited linear, range-limited logarithmic, or power. The goodness of the aggregate curve's fit to a power function relative to other functions increased as the amount of intercomponent slope variability increased, irrespective of component-curve type. Thus, the power law's ubiquity may reflect the pervasiveness of slope variability across component functions. Moreover, power-curve emergence may constitute a methodological artifact, an explanatory construct, or both, depending on the locus of the effect.     

Artifactual power curves in forgetting

Recent studies of the mathematical relationship between time and forgetting suggest that it is a power function rather than an exponential function, a finding that has important theoretical consequences. Through computational analysis and reanalyses of published data, we demonstrate that arithmetic averaging of exponential curves can produce an artifactual power curve, particularly when there are large and systematic differences among the slopes of the component curves. A series of simulations showed that the amount of power artifact is small when the slopes of the component curves are normally or rectangularly distributed and when the performance measure is noise free. However, the simulations also showed that the artifact can be quite large, depending on the shape of the noise distribution and restrictions in the performance range. We conclude that claims concerning the form of memory functions should consider whether the data are likely to contain artifact caused by averaging or by the presence of range-restricted noise.     

The powers of noise-fitting: reply to Barth and Paladino

Barth and Paladino (2011) argue that changes in numerical representations are better modeled by a power function whose exponent gradually rises to 1 than as a shift from a logarithmic to a linear representation of numerical magnitude. However, the fit of the power function to number line estimation data may simply stem from fitting noise generated by averaging over changing proportions of logarithmic and linear estimation patterns. To evaluate this possibility, we used conventional model fitting techniques with individual as well as group average data; simulations that varied the proportion of data generated by different functions; comparisons of alternative models' prediction of new data; and microgenetic analyses of rates of change in experiments on children's learning. Both new data and individual participants' data were predicted less accurately by power functions than by logarithmic and linear functions. In microgenetic studies, changes in the best fitting power function's exponent occurred abruptly, a finding inconsistent with Barth and Paladino's interpretation that development of numerical representations reflects a gradual shift in the shape of the power function. Overall, the data support the view that change in this area entails transitions from logarithmic to linear representations of numerical magnitude.     

Genuine power curves in forgetting: A quantitative analysis of individual subject forgetting functions

Wixted and Ebbesen (1991) showed that forgetting functions produced by a variety of procedures are often well described by the power function, at^?b, where a and b are free parameters. However, all of their analyses were based on data arithmetically averaged over subjects. R. B. Anderson and Tweney (1997) argue that the power law of forgetting may be an artifact of arithmetically averaging individual subject forgetting functions that are truly exponential in form and that geometric averaging would avoid this potential problem. We agree that researchers should always be cognizant of the possibility of averaging artifacts, but we also show that our conclusions about the form of forgetting remain unchanged (and goodness-of-fit statistics are scarcely affected by) whether arithmetic or geometric averaging is used. In addition, an analysis of individual subject forgetting functions shows that they, too, are described much better by a power function than by an exponential.     

Event-learning and robust policy heuristics

In this paper we introduce a novel reinforcement learning algorithm called event-learning. The algorithm uses events, ordered pairs of two consecutive states. We define event-value function and we derive learning rules. Combining our method with a well-known robust control method, the SDS algorithm, we introduce Robust Policy Heuristics (RPH). It is shown that RPH, a fast-adapting non-Markovian policy, is particularly useful for coarse models of the environment and could be useful for some partially observed systems. RPH may be of help in alleviating the ‘curse of dimensionality’ problem. Event-learning and RPH can be used to separate time scales of learning of value functions and adaptation. We argue that the definition of modules is straightforward for event-learning and event-learning makes planning feasible in the RL framework. Computer simulations of a rotational inverted pendulum with coarse discretization are shown to demonstrate the principle.     

Decomposing serial learning: What is missing from the learning curve?

Our current understanding of serial learning relies on the form of the learning curve and on changes in the serial position curve over repeated study-test trials (Ward, 1937). The averaging of data that produces these functions obscures the detailed history of memory for individual items over the course of study-test trials. Extending Tulving’s (1964) analysis of free recall learning, we present a new analysis of serial learning that tracks the acquisition and forgetting of item and order information at the level of individual items. Applying this analysis to two large data sets on serial list learning allows us to discern among hypotheses that are indistinguishable solely on the basis of the learning curve.     

Forgetting curves: implications for connectionist models

Forgetting in long-term memory, as measured in a recall or a recognition test, is faster for items encoded more recently than for items encoded earlier. Data on forgetting curves fit a power function well. In contrast, many connectionist models predict either exponential decay or completely flat forgetting curves. This paper suggests a connectionist model to account for power-function forgetting curves by using bounded weights and by generating the learning rates from a monotonically decreasing function. The bounded weights introduce exponential forgetting in each weight and a power-function forgetting results when weights with different learning rates are averaged. It is argued that these assumptions are biologically reasonable. Therefore power-function forgetting curves are a property that may be expected from biological networks. The model has an analytic solution, which is a good approximation of a power function displaced one lag in time. This function fits better than any of the 105 suggested two-parameter forgetting-curve functions when tested on the most precise recognition memory data set collected by. Unlike the power-function normally used, the suggested function is defined at lag zero. Several functions for generating learning rates with a finite integral yield power-function forgetting curves; however, the type of function influences the rate of forgetting. It is shown that power-function forgetting curves cannot be accounted for by variability in performance between subjects because it requires a distribution of performance that is not found in empirical data. An extension of the model accounts for intersecting forgetting curves found in massed and spaced repetitions. The model can also be extended to account for a faster forgetting rate in item recognition (IR) compared to associative recognition in short but not long retention intervals.     

Systems for the Production of Plagiarists? The Implications Arising from the Use of Plagiarism Detection Systems in UK Universities for Asian Learners

This paper argues that the inappropriate framing and implementation of plagiarism detection systems in UK universities can unwittingly construct international students as ‘plagiarists’. It argues that these systems are often implemented with inappropriate assumptions about plagiarism and the way in which new members of a community of practice develop the skills to become full members of that community. Drawing on the literature and some primary data it shows how expectations, norms and practices become translated and negotiated in such a way that legitimate attempts to conform with the expectations of the community of practice often become identified as plagiarism and illegitimate attempts at cheating often become obscured from view. It argues that this inappropriate framing and implementation of plagiarism detection systems may make academic integrity more illusive rather than less. It argues that in its current framing – as systems for ‘detection and discipline’ – plagiarism detection systems may become a new micro-politics of power with devastating consequences for those excluded.     

Patterns of cognitive ageing

Neuroanatomical evidence suggests that normal ageing affects some brain areas, and the “local” functions they support, earlier and more severely than others. Changes appear to be especially marked in the hippocampus, temporal association and prefrontal cortex. Evidence from classical neuropsychological studies suggests that these brain areas are associated with memory and “executive” functions, respectively. We may, therefore, expect that tests purported to measure these functions may be disproportionately affected in old age and that there may be evidence for some separation of these functions even within neurologically normal populations. What we also know, however, is that measures reflecting general fluid ability also decline with increasing age, so any hypothesis relating to specific “local” deficits must acknowledge and account for any “globar” changes in performance. Volunteers (n=162) aged between 60–80 years who had completed the Cattell and Cattell Culture Fair Intelligence Test (CCF) completed the Cambridge Automated Neuropsychological Test Battery (CANTAB). The CANTAB has been administered to several patient populations and tests from the battery have been shown to be sensitive to damage in both temporal and prefrontal areas (Owen et al., 1996). Results from the test battery showed that both the Paired Associate Learning and Spatial Recognition tests were the most sensitive to normal ageing even when CCF is accounted for. In contrast, this performance on the “executive” tests, shown to be sensitive to frontal lobe damage, was not related to age, and CCF scores predicted performance on these tests. These results are discussed in relation to theories of cognitive ageing and patterns of change and in relation to several important methodological and theoretical considerations for the study of executive function. Received: 31 March 1999 / Accepted: 23 July 1999     

Spheres of Justice within Schools: Reflections and Evidence on the Distribution of Educational Goods

This article argues that there are distinct spheres of justice within education and examines a range of justice norms and distribution rules that characterize the daily life of schools and classrooms. Moving from the macro to micro level, we identify the following five areas: the right to education, the allocation of (or selection into) learning places, teaching–learning practices, teachers’ treatment of students, and student evaluations of grade distribution. We discuss the literature on the beliefs by students and teachers about the just distribution of educational goods in these five domains, and on the practices used in the actual allocation of these goods. In line with normative ‘spheres of justice’ arguments in social theory, we conclude that the ideals of social justice within schools vary strongly according to the particular resource to be distributed. Moreover, these ideals often do not correspond with the practices that actually guide resource distribution in education, which may go some way toward explaining explicit or latent conflicts in this sphere. Justice is a human construction, and it is doubtful that it can be made in only one way – Michael Walzer (1983, p. 5)     

Culture, Communication, and Competence: A Commentary on Variables Affecting Social and Academic Behavior

The editors of this special issue have recruited six papers focused on the ways that language and communication interact with culture to influence student behavior. Two themes that emerge from these papers are the fundamental role of communication in learning and living, and the impact of culture on the functions of communication. The present commentary is offered in admiration for the work of the authors, and with the goal of emphasizing common messages that may guide both future research and practice.     

The power law repealed: The case for an exponential law of practice

The power function is treated as the law relating response time to practice trials. However, the evidence for a power law is flawed, because it is based on averaged data. We report a survey that assessed the form of the practice function for individual learners and learning conditions in paradigms that have shaped theories of skill acquisition. We fit power and exponential functions to 40 sets of data representing 7,910 learning series from 475 subjects in 24 experiments. The exponential function fit better than the power function in all the unaveraged data sets. Averaging produced a bias in favor of the power function. A new practice function based on the exponential, the APEX function, fit better than a power function with an extra, preexperimental practice parameter. Clearly, the best candidate for the law of practice is the exponential or APEX function, not the generally accepted power function. The theoretical implications are discussed.     

Realization,explanation and the mind-body relation

The Morris water maze has been put forward in the philosophy of neuroscience as an example of an experimental arrangement that may be used to delineate the cognitive faculty of spatial memory (e.g., Craver and Darden, Theory and method in the neurosciences, University of Pittsburgh Press, Pittsburgh, 2001; Craver, Explaining the brain: Mechanisms and the mosaic unity of neuroscience, Oxford University Press, Oxford, 2007). However, in the experimental and review literature on the water maze throughout the history of its use, we encounter numerous responses to the question of “what” phenomenon it circumscribes ranging from cognitive functions (e.g., “spatial learning”, “spatial navigation”), to representational changes (e.g., “cognitive map formation”) to terms that appear to refer exclusively to observable changes in behavior (e.g., “water maze performance”). To date philosophical analyses of the water maze have not been directed at sorting out what phenomenon the device delineates nor the sources of the different answers to the question of what. I undertake both of these tasks in this paper. I begin with an analysis of Morris’s first published research study using the water maze and demonstrate that he emerged from it with an experimental learning paradigm that at best circumscribed a discrete set of observable changes in behavior. However, it delineated neither a discrete set of representational changes nor a discrete cognitive function. I cite this in combination with a reductionist-oriented research agenda in cellular and molecular neurobiology dating back to the 1980s as two sources of the lack of consistency across the history of the experimental and review literature as to what is under study in the water maze.     

Meta-analytic evidence for a superordinate cognitive control network subserving diverse executive functions

Classic cognitive theory conceptualizes executive functions as involving multiple specific domains, including initiation, inhibition, working memory, flexibility, planning, and vigilance. Lesion and neuroimaging experiments over the past two decades have suggested that both common and unique processes contribute to executive functions during higher cognition. It has been suggested that a superordinate fronto–cingulo–parietal network supporting cognitive control may also underlie a range of distinct executive functions. To test this hypothesis in the largest sample to date, we used quantitative meta-analytic methods to analyze 193 functional neuroimaging studies of 2,832 healthy individuals, ages 18–60, in which performance on executive function measures was contrasted with an active control condition. A common pattern of activation was observed in the prefrontal, dorsal anterior cingulate, and parietal cortices across executive function domains, supporting the idea that executive functions are supported by a superordinate cognitive control network. However, domain-specific analyses showed some variation in the recruitment of anterior prefrontal cortex, anterior and midcingulate regions, and unique subcortical regions such as the basal ganglia and cerebellum. These results are consistent with the existence of a superordinate cognitive control network in the brain, involving dorsolateral prefrontal, anterior cingulate, and parietal cortices, that supports a broad range of executive functions.     

Learning versus behavioral expression of the learned: The effects of a secondary tone-counting task on implicit learning in the serial reaction task

The replicated finding that implicit learning in the serial reaction task (SRT) is impaired when both the learning and the assessment of learning occur in the presence of a secondary tone-counting task has been interpreted as implying that the mechanism(s) underlying implicit sequence learning require(s) attention in order to operate. However, in almost all studies, learning and the assessment of learning have been confounded. It is therefore unclear whether tone counting affects learning per se, the behavioral expression of the learned, or both. The goal of the present research was to disentangle the effects of tone counting on learning and the expression of the learned. In Exps. 1a and 1b, participants performed the Nissen and Bullemer SRT under different single-task (ST) and dual-task (DT) practice schedules. In Exps. 2a and 2b, participants received different amounts of ST and DT practice. In all experiments, degree of implicit learning was then assessed under both ST and DT conditions. Results are consistent with the argument that primarily the expression of what has been learned and, to some extent, implicit learning itself, are affected by tone counting. These findings are easily understood in terms of specific interference mechanisms but are problematic for models that contain the assumption of an attentional learning mechanism. Received: 16 October 1996 / Accepted: 5 February 1997     

Power Function Forgetting Curves as an Emergent Property of Biologically Plausible Neural Network Models

Empirical forgetting curve data have been shown to follow a power function. In contrast, many connectionist models predict either an exponential decay or flat forgetting curves. This paper simulates power function forgetting curves in a Hopfield network modified to incorporate the more biologically realistic assumptions of bounded weights and a distribution of learning rates. The modified model produces power function forgetting curves. The bounded weights introduce exponential decay for individual weights, and a power function forgetting curve when summing exponential decays with different learning rates. Because these assumptions are biologically reasonable, power function forgetting curves may be an emergent property of biological networks. The results fit empirical data and indicate that forgetting curves restrict possible implementation of models of memory.     

Research utilization: The state of the art

As a field of study, “research utilization” is at a turning point. Despite an accumulation of replicable findings, robust constructs, even a “soft technology” for bridging the gap between theory and practice, we are still largely in the situation of the distance between social problems of, let us say, conflict or inequality and the ability of social science to provide credible, reliable and usable solutions. At the same time, the initial paradigms, suffering from hyperrationalism, have given way to more transactional ones, and have been shaken by the tenets of postmodernism. Shaken, but not undone, as “middle-level” constructs emerge, ones that appear to link the research community with a variety of professional communities in more meaningful and durable ways. Michael Huberman has been Visiting Professor of Education at Harvard University since 1991, where he teaches research methodologies and conducts research on knowledge dissemination. He is also Senior Research Associate at the Network, Inc.     

An empirical test of five prominent explanations for the black–white academic performance gap

The viability of five prominent explanations for the black–white performance gap (“academic engagement,” “cultural capital,” “social capital,” “school quality” and “biased treatment”) is examined using NELS data and a LISREL model that adjusts for clustering of students within schools. Empirical models have typically assessed these factors individually—a practice that probably fosters overestimation of their explanatory power. School quality and biased treatment emerge as the primary explanations for black–white high school test performance differentials. Access to better-quality schools and receipt of more stimulating interpersonal “signals” from gatekeepers ensue from racial (and socioeconomic) privilege. Enhanced test performance in turn ensues from these resources. In essence, the explanations for the racial gap that place more emphasis on what black and white students “bring to” high school (i.e., specific levels of engagement, cultural and social capital), seem less consequential to performance differentials than “what happens to” them when they get there (i.e., quality of education provided, and race-contingent treatment received).     

Talking about race in a scientific context

There are at least two approaches that assist students in understanding complexity and differing interpretations about human diversity and race. Because differing perspectives emerge from data perceived at different levels, different scales provide a tool for understanding relationships among perspectives and understanding the differential importance of specific factors. Constructivist listening, which assists students in examining their own experiences, feelings and understanding, provides a tool for digesting complex new material and learning emotional literacy. It can be applied to dialogue about race and to classroom situations. These approaches can help students master the conceptual and interpersonal skills needed for successful scientific practice. This article was adapted from a presentation made at the “Communicating Science Conference” held at Hamilton College, Clinton, NY, October 1998.