[Paper Review] Let Me Try Again: Examining Replay Behavior by Tracing Students' Latent Problem-Solving Pathways

Shan Zhang, Siddhartha Pradhan+3 more|arXiv (Cornell University)|Jan 3, 2026

Educational Games and Gamification0 citations

TL;DR

The paper uses Markov Chains and Hidden Markov Models to analyze how seventh graders’ problem-solving pathways and replay behaviors in FH2T relate to proximal and distal algebra learning outcomes.

ABSTRACT

Prior research has shown that students' problem-solving pathways in game-based learning environments reflect their conceptual understanding, procedural knowledge, and flexibility. Replay behaviors, in particular, may indicate productive struggle or broader exploration, which in turn foster deeper learning. However, little is known about how these pathways unfold sequentially across problems or how the timing of replays and other problem-solving strategies relates to proximal and distal learning outcomes. This study addresses these gaps using Markov Chains and Hidden Markov Models (HMMs) on log data from 777 seventh graders playing the game-based learning platform of From Here to There!. Results show that within problem sequences, students often persisted in states or engaged in immediate replay after successful completions, while across problems, strong self-transitions indicated stable strategic pathways. Four latent states emerged from HMMs: Incomplete-dominant, Optimal-ending, Replay, and Mixed. Regression analyses revealed that engagement in replay-dominant and optimal-ending states predicted higher conceptual knowledge, flexibility, and performance compared with the Incomplete-dominant state. Immediate replay consistently supported learning outcomes, whereas delayed replay was weakly or negatively associated in relation to Non-Replay. These findings suggest that replay in digital learning is not uniformly beneficial but depends on timing, with immediate replay supporting flexibility and more productive exploration.

Motivation & Objective

• Investigate how students transition between problem-solving strategies within and across problems in FH2T.
• Identify latent problem-solving states that emerge from complete problem sequences.
• Examine how latent states predict proximal and distal learning outcomes.
• Assess how replay timing (immediate vs. delayed) relates to learning outcomes.

Proposed method

• Model within-problem and across-problem transitions with first-order Markov chains using 12 labeled attempt states.
• Apply Hidden Markov Models with 5-fold cross-validation and BIC to identify latent states from sequences of 12 attempt classes.
• Decode hidden states and analyze emission distributions and transition matrices to interpret dynamic pathways.
• Compute state percentages and run-lengths to relate latent states to proximal post-test constructs and distal state tests.
• Use regression analyses (OLS) to relate state engagement to learning outcomes, controlling for pre-scores and engagement variables.
• Decompose replay into Immediate, Delayed, and Non-Replay and compare their associations with outcomes.

Figure 1 . Example Problem 14 from FH2T with sample solution steps (Start State: $47+33+b+52+68$ Goal State: $99+b+101$

Experimental results

Research questions

• RQ1 RQ 1: How do students transition among different problem-solving strategies within and across problems?
• RQ2 RQ 2: What latent problem-solving states emerge when modeling complete problem sequences and what transitions occur between these states?
• RQ3 RQ 3: How do latent problem-solving states predict proximal and distal learning outcomes?
• RQ4 RQ 4: How are students’ replay behaviors (immediate vs. delayed) associated with proximal and distal learning outcomes?

Key findings

• Replay-dominant and optimal-ending latent states predict higher conceptual knowledge, flexibility, and state-test performance relative to the incomplete-dominant state.
• Across-problem analysis shows that states with replay-dominant and optimal-ending patterns have positive associations with multiple outcomes after FDR correction.
• Immediate replay is consistently associated with better learning outcomes, whereas delayed replay shows weaker or negative associations in relation to Non-Replay.
• Within-problem transitions show persistence in states and replay sequences, while across-problem transitions show strong persistence of strategies across problems.
• HMMs identified four latent states: Mixed, Incomplete-dominant, Optimal-ending, and Replay-dominant, with distinct transition patterns and run-lengths.

(a) Within-problem Transition Probabilities