In this study, the teaching experiment methodology is used to observe firsthand a gifted student's mathematical learning and reasoning. A series of teaching experiments was conducted with 1 gifted and 1 average 7th-grade student to investigate how the gifted student's mathematical concepts and operation constructions differed from those of the average student. The teaching experiment approach provides opportunities for gifted and average students to be challenged by immersion in various advanced mathematical topics. The data analysis provides evidence that the gifted student was more adept at applying mathematical ideas to unfamiliar problems. As a result of being able to see mathematical patterns and to think abstractly, the gifted student was able to use analytical, deductive, and inductive reasoning to solve problems in more flexible and creative ways than the average student.