The purpose of the present study was to investigate the developmental progress on the mathematical understandings of speed and acceleration in children by administering problems related to them. The subjects were at the elementary level: 341, 4 to 6 graders and 364, 1 to 3 secondaryschool-students. Three kinds of problems on both speed and acceleration were administered: i.e., Which one is faster between two objects? Which one can go farther? And, which one takes more time? The results showed that in general the children could answer more correctly to questions related to distance. That is, in the questions on speed, they could understand better than any others the relations between speed and distance, and also in the questions on acceleration, the relations between acceleration and distance. Developmental progress in the correct responses by grades was also found. Fourth and fifth grade children's understanding on the function of speed or acceleration was found to be worse than in any other grades. Children above sixth grade could answer more correctly in every kind of the given problems.