recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation. Keywords: computational modeling, cognitive modeling, instructional theory, machine learning, learning science, second language learning, mathematics learning, science learning, robust learning, learning theory, knowledge componentsExecutive Summary The volume of research on learning and instruction is enormous. Yet progress in improving educational outcomes has been slow at best. Many learning science results have not been translated into general practice and it appears that most that have been fielded have not yielded significant results in randomized control trials. Addressing the chasm between learning science and educational practice will require massive efforts from many constituencies, but one of these efforts is to develop a theoretical framework that permits a more systematic accumulation of the relevant research base. A key piece in such a theoretical framework is the development of levels of analyses that are fine enough to be supported by cognitive science and cognitive neuroscience, but also at levels appropriate to guide the design of effective educational practices. An ideal scientific solution would be a small set of universal instructional principles that can be applied to produce efficient

In the field of cognitive science, the primary means of judging a model’s viability is made on the basis of goodness-of-fit between model and human empirical data. Recent developments in model comparison reveal, however, that other criteria should be considered in evaluating the quality of a model. These criteria include model complexity, generalizability, predictive capability, and of course descriptive adequacy. The current investigation seeks to formally compare three variants of a mathematical model for performance prediction. The results raise the issue of how to go about selecting a model when formal comparison methods reveal equivalent values. A possibility briefly proposed at the end of the paper is that cognitive/neural plausibility is an appropriate tiebreaker among otherwise equivalent functional forms.

a series of technology research roadmaps, or plans, developed over a three year period by the Federation of American Scientists and the Learning Federation, a partnership among industry, academia, and private foundations to stimulate research and development in learning science and technology. The full series of research roadmaps is available at www.FAS.org/learningfederation. We thank Dr. Jan Cannon-Bowers for her major contribution in writing this roadmap. And, we thank Dr. Marianne Bakia, who left FAS just prior to completion of the roadmap, for her contributions to the Learning Federation Project and development of this roadmap. We gratefully acknowledge the funding support of the 2003 Congressional appropriation to the Federation of American Scientists for the Digital Opportunity Investment Trust (DO IT). A major part of that funding supported the Learning Federation's Learning Sciences and Technology Research and Development Roadmap, which appears in the DO IT Report to Congress. We also gratefully acknowledge the additional funding support of the organizations that sponsored this work and helped make possible the Roadmap:

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Intelligent tutoring systems seek to optimize instruction and training by adapting and individualizing the learning experience on the basis of a student model (Shute, 1995). This model represents the system’s estimate of the student’s current knowledge or skill level, established from a performance history. Knowledge tracing (Aleven & Koedinger, 2002; Anderson, Conrad, & Corbett, 1989) is a dynamic, Bayesian approach to updating the estimates of probability of skill mastery in the student model. A fundamental shortcoming of this approach is that it does not include a representation of memory decay during periods of non-practice. As a result, traditional student modeling approaches are unable to make predictions regarding knowledge and skill changes under various future training schedules or to prescribe how much training will be required to achieve specific levels of readiness at a specific future time. In this paper, we propose a new knowledge tracing equation, computationally inspired by the learning and forgetting equations in the ACT-R cognitive architecture (Anderson et al., 2004), which uses performance history to baseline student model parameters and then extrapolates knowledge state transformation to predict future performance. We explore practical issues concerning predictive models of future trainee performance and the prescription of frequency and timing of optimal learning with training systems. For instance, we investigate how much data from the training history are necessary to achieve reasonable predictive validity, and we describe the

The ACT-R computational modeling architecture has demonstrated the ability to model both recency and frequency effects in memory with much success (e.g. Anderon & Lebiere, 1998); and through the incorporation of new decay parameters at each data point, has also been shown to capture the spacing effect (Pavlik & Anderson, 2003, 2005). Stemming from the aforementioned literature, the current research sought to build an equation capable of handling the prediction of performance at later, distributed points in time, thereby breaking from the tradition of post-fitting data. As such, we integrated a single activation-based decay rate into the ACT-R General Performance Equation (Anderson & Schunn, 2000), and scaled predictions by amount of training history improvement. We tested this algorithm by extrapolating learner knowledge states from initial points in data, and predicting performance at later points in time, across different intervals of time. Implications are discussed.