Overcoming students' cognitive difficulties in mastering the concepts of linear algebra and mathematical analysis through a system of problem-oriented tasks

Abstract

Research on undergraduate mathematics has documented in detail the conceptual difficulties first-year students meet when they first work with axiomatically defined objects, and a system of problem-oriented tasks is routinely proposed as the remedy. What nobody has measured is whether the assessment materials actually in force make the cognitive demand without which no such shift can occur. This study answers that question in numbers. The corpus consists of 786 assessment items taken verbatim from 22 published course syllabi in linear algebra and mathematical analysis, issued by twenty Russian universities between 2020 and 2026. Each item is treated as one unit and assigned to one of four levels of cognitive demand according to the grammatical form of the action it requires. Coding rules were fixed before counting, applied mechanically, and followed by a full manual review that changed the assignment of five units. Recall accounts for 691 items, or 87,9% of the corpus; execution of a familiar procedure for 58 items, or 7,4%; reasoning about the object itself for 37 items, or 4,7%; transfer of a method into a task belonging to another section never occurs. Seventeen of the 22 syllabi contain no third-level item at all, and 31 of the 37 that exist come from a single syllabus. Mathematical analysis yields no third-level item across all 416 units. A further 148 formulations name a procedure through a verbal noun, turning it into a topic to be recited. The gap between the mechanism of difficulty described in the literature and the design of assessment now in force turns out to be nearly complete: tasks capable of moving a student from performing a transformation to making statements about it are absent from the syllabi examined, and the few exceptions are isolated rather than systemic.