The modeling of a knowledge domain in a learning environ-ment is a critical step with far-reaching implications, notonly for the system design but also for the granularity withwhich student progress can be tracked. In the [System X]learning environment, games are designed to measure one di-mension of a broad knowledge domain (e.g. clock-reading).However, dimensionality is introduced in the games throughthe practice of specific knowledge components (KC; e.g.,reading digital clocks with half hours). The assignmentof specific items to these KCs is expert-based rather thandata-driven. To investigate the validity of these mappingsof items to KCs, this study employed a hierarchical clus-tering method on person-specific performance at the itemlevel. The results revealed a clear pattern; items associatedwith the same learning goal tend to cluster together. Thissupports the notion that learning goals capture meaningfuland distinct dimensions within the game. Finally, we illus-trate the progress of some players in the game to emphasizethe importance of detailed tracking of multiple abilities foreffective instruction and timely intervention.

In adaptive digital learning environments, it is essential to track learning trajectories. The Elo rating system, known for its computational simplicity, is frequently employed for this purpose. Current Elo-based systems cannot handle rapid changes in ability or are unable to balance accuracy and speed when updating player and item ratings. Changes in Elo ratings depend on the sensitivity parameter K. Using fixed K values necessitates a trade-off: larger values facilitate the tracking of evolving ability levels but introduce greater rating volatility. Smaller values yield more stable estimates, but are slower to reflect actual ability levels. Existing modifications of the Elo system, which diminish K as the number of responses increases, are inadequate in scenarios characterized by considerable ability fluctuation, a common occurrence in digital learning environments. To address this challenge, we introduce a novel approach for dynamically adjusting K values in response to observed trends in rating changes. This method increases K during noticeable upward or downward shifts in ratings and reduces it otherwise. We present a computationally efficient implementation of this idea and validate its superiority over existing K adjustment strategies through simulation studies. Additionally, we describe the implementation of this adaptive K model in a widely-used digital learning platform, Math Garden, which leverages both accuracy and response time in its assessments. By successfully integrating speed and precision, this innovative implementation enhances the effectiveness of digital adaptive learning environments.

The popularity of learning environments is increasing rapidly. In order to make learning environments more efficient, researchers have been matching the item difficulty to the learner's proficiency, as is done in computerized adaptive testing (CAT) by means of the item response theory (IRT). Even though some researchers have already implemented ideas of CAT and IRT for adaptive item selection in learning environments, some differences between testing and learning environments have been overlooked. In this study we focus on those differences that may require an adaptation of these existing CAT and IRT methods.