As machine learning becomes a standard part of computing and engineering curricula, teaching its core concepts effectively has become an important educational challenge. Regularization is one such concept: L1 and L2 penalties are widely used to prevent overfitting, with L1 producing sparse solutions and L2 shrinking weights more smoothly. It is commonly taught through two representational formats: an algebraic one, presenting the loss function and its penalty terms, and a geometric one, depicting constraint regions and their intersection with the loss contours. However, instructors choose between them without clear evidence on which better supports learning. This study asks whether teaching regularization algebraically or geometrically leads to different student performance, and in which kinds of understanding.

Two interactive notebooks, matched on learning objectives, length, and reading difficulty, were developed to teach the concept in each format and are released openly; they were compared in a between-subjects experiment with students who had completed an introductory machine learning course. Learning was measured with a post-test spanning conceptual understanding, problem-solving, and knowledge transfer, alongside a thematic analysis of students’ written explanations.

Both formats supported practical reasoning about regularization, but not identically: the algebraic group performed better overall, with its clearest advantage in conceptual understanding, no reliable difference on problem-solving, and an inconclusive result on transfer. The two groups largely shared the same core understanding but expressed it through different vocabularies: a penalty on the loss function versus a shrinking constraint region in weight space. Representational choice therefore appears to shape how students explain regularization more than whether they grasp it, suggesting that combining the two formats may best support learning.
... Read more