Abstract: Teachers should follow the cognitive development rules of students, stimulate their active thinking, optimize their thought processes, and develop their problem-solving wisdom. Starting from a final problem in the college entrance examination, this study focuses on exploring the interchange process between intuition and logic in students’ problem-solving difficulties. The bidirectional construction of intuition and logic is the fundamental path to the spiral upgrade of students’ thinking quality. Mathematics education should be dedicated to exploring the logic embedded in intuition and constructing new intuition based on logic. That is, teachers should guide students to systematize, scientize, and logicize valuable intuitions on one hand, and cultivate higher-dimensional intuition based on logic on the other. In mathematics instruction, every interchange between intuition and logic facilitated by teachers promotes the development of students’ thinking.