This qualitative case study in a rural school in Umgungundlovu District in KwaZulu-Natal,
South Africa, explored Grade 12 learners’ mental constructions of mathematical knowledge
during engagement with optimisation problems. Ten Grade 12 learners who do pure Mathemat-
ics participated, and data were collected through structured activity sheets and semi-structured
interviews. Structured activity sheets with three tasks were given to learners; these tasks were
done in groups, and the group leaders were interviewed. It was found that learners tended to do
well with routine-type questions, implying that they were functioning at an action level. From
the interviews it appeared that learners might have the correct answer, but lacked conceptual
understanding. Exploring learners’ mental constructions via their responses to activity sheets
and interviews enabled common errors and misconceptions to be identified. Themes that
emerged were that learners: 1) lacked the understanding of notation dy/dx, 2) had not constructed
the derivative and minima/maxima schema, 3) had some difficulty in modelling problems, 4)
preferred rules and formulas, and 5) applied algebraic notions incorrectly. Inferences are
drawn for curriculum developers and teachers. This study also formulated itemised genetic
decompositions for particular tasks, which contribute to APOS theory.

doi: 10.15700/saje.v33n2a679