This study arises out of research in a school where the majority of students, aged between seven and nine, had experienced substantial difficulty in problem-solving in mathematics. The study was designed to discover why such a large body of students had so many difficulties in solving what, to their peers elsewhere, might be seen as simple arithmetic problems well within their capabilities. The study was designed to discover the extent of the students' difficulties and the reasons for them. 27 students were studied over a period of many months. Each was interviewed face-to-face. All interviews were videotaped and the recordings were studied and analysed and relevant data extracted. The students were presented with combine problems in the form of guessing games similar to those used by Neuman (1987) and as categorised by Riley et al. (1983) as a 'CB2” problem. The purpose of the game was to assess whether the students had an understanding of part-whole and to observe how they went about trying to solve the problem. The game gave an insight into the strategies the students adopted and the difficulties they encountered in such matters as number conception, decomposition and part-whole relationship. The findings disclosed several problematic issues regarding both procedural and conceptual knowledge, the details of which, it was felt, could help the teachers identify symptoms of poor performance and understand the reasons for them so that they could then design a suitable remedial programme.