Some quantitative disciplines have historically prioritised knowledge transmission in lectures and large-groups teaching. With a large body of settled knowledge, active learning can seem both harder to achieve and less useful. Students may place a lower premium on attendance, as recordings can be viewed later; as one LSE student explained, “the fact that you can really slow down and listen at your own pace makes a really big difference” (Arora et al. 2026).
However, active learning does enable learning in these disciplines, and can be supported by the approaches outlined above, including increasing cognitive engagement, peer interaction, and testing understanding.
In support of students’ cognitive engagement, pedagogic theorists of quantitative subjects have outlined different forms of conceptual work students can undertake. Smith and Stein (1998) identify a continuum from ‘memorisation’ to ‘doing mathematics’.
- Memorisation, where students memorise or reproduce datum (e.g. rules, formulae, definitions, etc).
- Procedures without connections, where students carry out specific procedures or algorithms, but without either having to make direct connections between the mathematical task and the underlying theory or explain the procedure and how they are applying it. E.g. "Use the method of linear regression to..."; "Use the substitution u=… to find the integral …"
- Procedures with connections, where students are given (either implicitly or explicitly) broad general procedures which requires some cognitive effort before students can start work. E.g. "Model this situation as a differential equation and solve using appropriate methodology."
- Doing mathematics: tasks that require complex or non-algorithmic thinking. These might also require ‘metacognition’; self-monitoring or self-regulation. Examples might include complex projects or problem-based learning, with initially incomplete information about solution methods.
Their article gives examples of each (at a level below HE, but which is illustrative).
Similarly, Smith (1996) defines three groups of assessment tasks, based on the skills and approaches required:
Group A – Routine procedures
Recall of factual knowledge / fact systems, comprehension, routine use of procedures
Group B – Using existing mathematical knowledge in new ways
Information transfer, application in new situations
Group C – Application of conceptual knowledge to construct mathematical arguments
Justifying and interpreting, implications, conjectures and comparisons, evaluation
Smith suggests increasing activities in groups B and C (in assessment tasks) to improve student understanding and retention.
Either taxonomy can be useful when considering the framing of lectures, and the thinking required of students in large-group teaching. Could the nature of explanations and tasks be changed, to increase cognitive demand at key points, without overwhelming students? The articles suggest ways to identify the existing cognitive work of an activity and ways to increase it; there is also a simple early stage example in this LSE resource.