How do the high-achieving pupils say they come to understand a mathematical concept that is new to them? How can these reports be informed by various psychological learning theories of mathematics? These are the research questions of this study. The focus is on the metacognitive awareness of ten high-achieving high school pupils of mathematics from Denmark and England. Through qualitative un-structured interviews in smaller groups, I investigate how the pupils talk about their learning of a mathematical concept that is new to them. I focus on the cognitive learning process.

To answer the research questions I develop a model for analysis to get an understanding of what the pupils tell. I call this model the “CULTIS model for analysis” where ‘CULTIS’ stands for “Consciousness, Unconsciousness, Language, Tacit, Individual, and Social”. I interpret that these are six themes in which the various learning theories express themselves. I organise the six themes in three “pairs” of what seems to be opposite themes and I use the model as a systematic way to handle the many theories and it is a way to sort the pupils’ statements into areas. Subsequently I discuss the pupils’ explanations in each theme and relate them to the other themes. The choice of having qualitative interviews with open-end questions is mainly owing to a necessity of avoiding a self-fulfilling process if both the method of interview and the method of analysis are strongly influenced by theories.

Building on work by others, the study assumes that the pupils are able to talk about their learning. The study confirms this assumption, as it is very clear that the pupils are able to speak of their learning process in a way that makes sense to them and to the pupils they are interviewed with. They explain their learning in their own words but most of the time it is quite easy to identify some theoretical notions which reflects what the pupils say. A result of the study is that it seems that the pupils each have their own way of learning. However, there are also similarities, particularly if we look at each of the three pairs of themes. Seemingly contradictory theories are furthermore often seen in function within one single pupil. Sometimes it also depends on the branch of mathematics they learn. I do therefore also discuss the concept of complementarity between theories as well as the theoretical possibility of a synthesis. Towards the end of the thesis I also discuss the effect these results may have for teacher education and education policy.