What should constitute a preschool mathematics program for children ages 3 and a half to 5 years? What mathematical ideas could youngsters explore? This paper discusses six broad mathematical concepts for investigation by preschoolers: number, space and location, shape, patterns, measurement, and permutations. For each big idea, activities are presented that illustrate aspects not traditionally offered to preschoolers. These activities are designed to enlarge children's understanding of mathematics and stimulate them to make conjectures, formulate their own problems, make decisions and predictions, discover ways to keep records of their experiments, and talk about their observations. There is no specific age designated for each activity. Following an exploration of these mathematical concepts and activities, the paper discusses the nature and importance of representation and communication in mathematics learning, and the role of the teacher. The paper concludes by noting that in order to facilitate young children's mathematical development, the preschool mathematics curriculum should be changed from a collection of unrelated activities to a cohesive development of the important ideas of number, space and location, shape, patterns, measurement, and permutations. These explorations should be designed to relate to children's daily experiences and current interests. (Contains 13 references.) (EV)