The novice mathematician's encounter with mathematical abstraction is presented here in a network of themes — briefly outlined in the Interlude — that emerged from the data analysis exemplified in Chapters 6-9. The complexity of the observed phenomena does not allow to present these themes in a linear manner because of the multiple links and interpretational overlappings between them. Below I give a brief guide through the presentation. The text in italic in this guide corresponds to the titles of the themes as presented in more detail in the next pages.

Learning in this study is seen as taking place in the encounter of the novice mathematician with a new way of thinking mathematically, mathematical abstraction. The novice's entrance into this new world occurs as an encounter with mathematical formalism in terms of reasoning and in terms of the definition of new concepts. A learning process consists of the novice's attempts to make sense of this new world; or in other words to construct meaningful images. This process is characterised by a range of tensions between the old and the new world and there is evidence in this study of a number of these tensions. Concept-image construction takes place in part due to the exposure to the concept definition and it assumes a certain amount of prerequisite knowledge. In the absence or weakness of this prerequisite knowledge concept-image construction is obstructed and this results in the problematic interaction of the learner with the concept definition.

Concept-image construction is described here as an attempt to construct meaningful metaphors and to explore the raison-d'-être of the new concepts and the new reasoning. This image construction through acquisition of — visual or other — metaphors and through existential meaning bestowing processes occurs as a battle between the old and the new ways of the two worlds. It is thus characterised by the tension between the informal-intuitive-and-verbal and formal-abstract-and-symbolic modes of thinking. This tension has been explored here as a tension between verbal and formal/symbolic language and as a tension between informal and formal modes of reasoning. The novices' difficulties in formalising have been identified to be difficulties with the mechanics of formal mathematical reasoning as well as difficulties of applying the mechanics of formal mathematical reasoning in a well-integrated and contextualised manner. Specifically in this study the embeddedness of the novice's reasoning has been linked with their ambiguous perception of what knowledge can be assumed (the tension between Proof by First Principles and Proof by Theorem Quoting is seen as a grand example of this ambiguity), and with the fragility of their knowledge as a result of this ambiguity.

In the above description my perspective on the novice mathematician's cognition is dominated by a polarisation between the old (school mathematics: informal, concrete and intuitive) and the new (university mathematics: formal, abstract and deductive), the uninitiated and the initiated, the novice and the expert. In Part II this juxtaposition of the differences between novice and expert approaches is seen within the context of describing the encounter with mathematical abstraction as an enculturation process.

In the following, the themes presented above in italics are elaborated on the grounds of the evidence given in Chapters 6-9.