We administered specially-designed, free-response mathematics tasks to primary students (N = 583, ages five to 12 years old). Our focus was on whether (i) the children's responses could be reliably assessed, and (ii) the responses could provide insights into children's mathematical thinking. We used a contemporary comparative judgement technique, interviews with four teachers, and analysed a sample of six responses to make inferences about the students' mathematical thinking. We found that the sampled responses' scores, interviewees' comments and qualitative features of the sampled responses led to consistent insights on the children's mathematical thinking. We argue that free-response tasks should supplement traditional assessments in primary mathematics.