Revealing the nature of mathematical procedural knowledge by analysing students’ deficiencies and errors.

In this paper, error analyses are used to reveal the nature of mathematical procedural knowledge. In particular, the aim is to empirically confirm the structure of procedural knowledge described in theory and to further differentiate the corresponding terms. In this context, the coding of the processing of procedural tasks by students at the end of their high school years (n = 455) shows that knowledge of the procedure and arithmetic/algebraic skills can be objectively reconstructed as components of procedural knowledge. The analyses show that procedural knowledge and corresponding difficulties become visible in many ways when working on relevant tasks and that anecdotal experiences about a lack of mathematical procedural knowledge among students must fall short. The didactic implication is that the support of procedural knowledge acquisition needs to be both individualised and task specific. The results also reveal interesting correlations between deficiencies and errors in the processing of procedural tasks and specific item characteristics. For example, deficiencies in knowledge of the procedure tend to occur more frequently in tasks at higher grade levels and in analysis, while specific errors in arithmetic/algebraic skills are more likely to be found in algebra and tasks that require more procedural steps.