This study is one of a series which attempts to arrive at generalizations about the learning of mathematics and the use of its terminology in the context of mathematical structure by young children. The first half of the document describes an experimental training program designed to integrate mathematical concepts of metric space, arc length, and transformations with Piagetan notions of conservation, transitivity, and symmetry. Subjects were 20 four-year-old and 34 five-year-old children who participated in three units of small group instruction. They were pretested and posttested individually. Seventy-four tables and nine diagrams are interspersed within the text to statistically support points as they are made. Results, conclusions, discussion, and implications are given and indicate that generally there are slight significant relationships between the psychological and mathematical constructs after training. The second half of the document contains the bibliography and three appendixes. Appendix I lists student characteristics in tabular form; Appendix II gives the details of the three instructional units; and Appendix III lists directions and materials for making the four measuring instruments used in the pre- and posttests. (WY)