Investigated were concepts of conservation (number and length), cardination, one-to-one correspondence, and counting ability, and the role of these concepts in learning first-grade mathematics. A Piagetian-type number concept test was developed and administered at the end of the school year to a sample of 38 entering first-grade students selected randomly from the students of two schools in a suburban area of Delaware. A standardized achievement test was given at the conclusion of the first grade. The findings suggested the following conclusions: (1) there appears to be a high degree of relationship between scores on the number concept test given at the beginning of school, and scores on the achievement test given at the end of the year for first-grade students, (2) conservation of number seems to be more highly related to achievement than counting or cardination, (3) students may be expected to make significant growth in number concepts during the first grade, (4) counting by ones does not seem to be highly related to achievement for first-grade students, (5) first-grade students vary greatly in their understanding of Piagetian-type number concepts at the beginning and at the end of the school year, and (6) entering first-grade students do not have an understanding of conservation of length. (RP)