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Historically speaking, students were judged long before they were marked. The tradition of marking, or scoring, pieces of work students offer for assessment is little more than two centuries old, and was introduced mainly to cope with specific problems arising from the growth in the numbers graduating from universities as the industrial revolution progressed. This paper describes the principles behind the method of Comparative Judgement, and in particular Adaptive Comparative Judgement, a technique borrowed from psychophysics which is able to generate extremely reliable results for educational assessment, and which is based on the kind of holistic evaluation that we assume was the basis for judgement in pre-marking days, and that the users of assessment results expect our assessment schemes to capture.

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There is an increasing demand from employers and universities for school leavers to be able to apply their mathematical knowledge to problem solving in varied and unfamiliar contexts. These aspects are however neglected in most examinations of mathematics and, consequentially, in classroom teaching. One barrier to the inclusion of mathematical problem solving in assessment is that the skills involved are difficult to define and assess objectively.We present two studies that test a method called comparative judgement (CJ) that might be well suited to assessing mathematical problem solving. CJ is an alternative to traditional scoring that is based on collective expert judgements of students' work rather than item-byitem scoring schemes. In Study 1 we used CJ to assess traditional mathematics tests and found it performed validly and reliably. In Study 2 we used CJ to assess mathematical problem-solving tasks and again found it performed validly and reliably. We discuss the implications of the results for further research and the implications of CJ for the design of mathematical problem-solving tasks.

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