Assessing mathematical ‘knowledge-how’ (‘ability-knowledge’): a case for the written exam

Authentic assessment aims to evaluate the ‘meaningful, real-world performance’ on an assessment rather than recall of isolated facts. In mathematics education, written exams are sometimes dismissed as inauthentic, perceived as instruments of rote evaluation. This paper seeks to challenge that automatic assumption by arguing that well-designed mathematics exams can, in fact, constitute authentic assessment. In particular, we wish to make that claim for evaluation where the target of assessment is ‘knowledge-how’ or ‘ability-knowledge’, ideas from epistemology and virtue-theory. This allows us to construct an argument that many forms of mathematical competence, such as proof construction, problem modelling, and conceptual reasoning, are inherently knowledge-how based. These cannot meaningfully be reduced to propositional recall. Through various examples, we argue that written exams can authentically assess these forms of mathematical knowledge by requiring students to demonstrate flexible, situated mathematical skills. This is not intended as a ‘case for examinations’ rather a deeper evaluation of what we aim to achieve in mathematics education and how best to achieve that. But we would suggest that far from being inauthentic, the written examination can be the most suitable method for assessing the intellectual practices which we need to develop in order to master advanced mathematics.