In this study I examined 19 preservice secondary mathematics teachers' solution processes to word problems for which the subtraction or addition of the two given numbers yields 1 more or 1 less than the correct solution. Among the aspects of their solution processes that were examined are: the modeling strategies, the type of errors, and the interpretations of the solutions produced by the procedure. It was found that about 87% of the solution processes to such problems contained formal strategies while about 13% contained counting strategies. It was also found that about 61% of the responses contained errors of which 91% were - 1 errors. That is, errors due to the interpretation that the answer provided by the addition or subtraction of the two given numbers is the solution to the problem. It is argued that some incorrect interpretations were due, at least in part, to a lack of understanding of the connection between the enumeration process needed to obtain the solution to a problem and the answer provided by the addition or subtraction of the two given numbers. (Author)