Error patterns in first-year students' answers to mathematical assignments as observed 2010-2018

An analysis of 1600+ one-page feedback sheets written by the author (in response to hand-in answers from comparable groups of first-year engineering students to assignments in a 20 ECTS points mathematics course) seeks to identify and classify error patterns. Three major patterns appear prominent: 1) The transfer problem: failing to apply, or wrongly applying, methods erstwhile learned in other areas; 2) The contingency problem: wrongly attaching a property of an example or mnemonic to a general problem; and 3) The failure to correctly use or extend logical conditions. Along with these are others that may be considered variations, but are probably best treated on their own, such as the general issues of failure to separate related, but different, concepts; of failure to make use of specific arguments over general ones (or vice versa); the blindness to the lucky/unlucky choice; and of failure to distinguish between a mathematical entity and its name. These, too, may deserve to be called 'patterns'. Finally, there is the universal 'Thinking fast and slow' (Kahneman) phenomenon: If a task resembles a known one, a previously successful method is attempted, whatever the outcome. The pedagogical/didactical response to an error pattern should be a manifestly layered instruction, distinguishing operational guidance from invitation to reflect upon a deeper meaning, that must needs be expressed in terms somewhat distant from the subject directly at hand. This is akin to Argyris' 'single-loop, double-loop' learning theory, which may be supplemented by the simple, but useful, five-step learning model fact→relation→operation→insight→innovation to make it explicit to students that they may have wrong or unfortunate patterns to unlearn and useful ones to internalize.