Understanding students’ cognitive patterns in problem-solving is crucial for personalized education, yet traditional methods struggle to effectively capture and analyze these patterns. This paper presents CogChain, a novel method that synergistically combines digital pen technology with Multi-modal Large Language Models (MLLMs) to automatically construct students’ logic chains during examinations. We collected a comprehensive dataset of 87,679 handwriting trajectories in mathematics, physics, and chemistry from 25 real-world high school students. Based on the constructed logic chains of students, we conduct an in-depth analysis across three dimensions: solution, time, and course, revealing a set of findings about their problem-solving behaviors. (1) Solution Dimension: We identify four distinct solution trajectory patterns, showing that moderate-complexity approaches achieve the highest accuracy. (2) Time Dimension: We uncover three types of time-allocation patterns, showing that students who allocate more time to structured reasoning achieve higher accuracy, whereas those who prioritize writing speed tend to perform worse. (3) Course Dimension: Different subjects require distinct problem-solving and time management strategies, with mathematics benefiting from step-by-step derivation, physics from visual reasoning, and chemistry from quick solutions. These insights provide valuable guidance for developing personalized teaching strategies.