It was the purpose of this study to determine how well children in grades 1 through 6 understand the concept of place value and to compare what they know as expressed orally in an interview with what is evidenced by a written test, based on the interview questions. The present study was based upon the hypothesis that a study of children's knowledge of place values through the oral interview would reveal information that had not been secured by means of written tests. Ten children from each grade, 1 through 6, were selected for the interview on two criteria: an intelligence quotient in the range of 95-105, and no failure of any grade. The questions in the interview were designed to test the concepts of place value. For the purpose of comparison a written test, based on the interview questions, was constructed. This written test was given to one entire class at each grade level. The classes that were given the written test were selected on two bases: a single grade per room, and the room which contained a majority of those interviewed. The results of the interview, written test, and the comparisons were computed and tabulated. The results of the interview and the written test seemed to indicate that the pupils did not understand the concepts of place value as well as they should have. The pupils who were given the written test scored higher on more items than the pupils who were interviewed. There was more opportunity for pupils to guess during the written test than during the interview. Several specific conclusions based on the evidence obtained by this study were made. 1. Pupils of all grade levels have a tendency to count by one's rather than by groups. 2. The ability to read and write numbers increases as grade level increases. 3. Saying numbers is easier than writing numbers. 4. Pupils possess an adequate understanding of the positional value of numbers through the thousands' place. Knowledge of the positional value of the digits in a number greater than the thousands' place was inadequate. 5. Changing the positional sequence when writing numbers from their meanings results in confusion on the part of the pupils. 6. The reasons for the operations involved in addition and multiplication are not as fully understood as the operations themselves. 7. Pupils can learn the algorithms of arithmetic without understanding the principles involved. In general those areas of place value which are emphasized throughout the grades show adequate understanding, while those areas with little emphasis show inadequate understanding. Pupils, in many cases, know the "how" of arithmetic but not the "why." When confronted with new situations in arithmetic, few pupils attempt to solve the problems with reasoning. The use of the interview reveals how well children understand the "why" of arithmetic as well as the "how." The interview method can provide information about the thinking processes of the individual child that paper-and-pencil tests cannot provide.