Part II The Novice Mathematician's Encounter With Mathematical Abstraction as an Enculturation Process

In Part I, the novice mathematicians' encounter with mathematical abstraction was described in terms of the tensions and the difficulties of their induction to the advanced mathematical culture; so the focus was on the novice mathematician as an individual learner-in-action within the context of tutorials. A complementary, and not necessarily distinct, way of describing the novices' encounter with mathematical abstraction is to juxtapose their practices with the practices of mathematical expertise – as represented in the tutorials by the tutors. This juxtaposition can be useful because it is possible that we learn something about the novice's cognition by contrasting it (contrast as accentuation) to what the novice is expected to aim at (mathematical expertise); also because through this contrast it may be possible to explore types of interaction between the novice and the expert that are congenial to the novice's cognitive needs.

As an example I refer here to the instance from Chapter 9 where a novice's viciously circular struggle for the construction of a meaning for cosets was described as a succession of mutual misunderstandings between the student and the tutor who insisted on repeating identically the argument for the proof of the question on which the discussion was based. Eventually the circular dialectics – largely due to the student's insistence — spiralled down to an exploration of gradually more basic knowledge relating to the question and in particular to *cosets*. This spiral journey illustrated graphically the need for an emphasis on constructive learning processes — that is processes that cautiously build on solid previous knowledge or allow revisiting and reconstructing previous knowledge with facility. So, in a sense, by describing this interaction in terms of what eventually became an enculturation process, some access was gained to the optimisation of this process.

In terms of the contrast between expert and novice approaches, the brief description of their differences given in Chapter 1, Part III was confirmed: so, in Chapter 6, in dealing with the basic inequalities of Foundational Analysis, the novices, unlike their tutors, seemed to be lacking in the kind of mathematical experience that empowers hindsight, reinforces a more fruitful use of intuition and secures the embeddedness of mathematical knowledge; in Chapter 7 their finitism when dealing with sequences and series was juxtaposed with the tutors' contextualised, concise, sophisticated and, possibly, generalisable approaches to testing convergence and divergence. In the context of Continuity and Differentiability this contrast was evident in the cases where the tutors justified their preferences for Proofs Based on First Principles on historical and epistemological grounds. Actually the novices' approach could not be described as a preference because no selecting seemed to be involved in their question-solving. So in this sense the richness of their repertory is what distinguished experts from novices. Demonstrating then the potential of a rich repertory became a necessary aim of the enculturation process.

Through the tensions and conflicts of this enculturation process the novices seem to learn about certain conditions of the didactical and epistemological contract of formal mathematical activity. So, for example in Chapter 7, when it is revealed that the students have been unconsciously assuming the validity of as yet unproved theorems about limits, the novices seem to be learning about an essential aspect of formal mathematical behaviour: that they need to exercise control over their mathematical reasoning in order to avoid subconscious and unjustified decisions.

Tension in the interaction between novice and expert is generated when the difference in their facility to formalise is exposed: while the expert and the novices may agree about the method of approaching a problem, they seem to differ in terms of the implementation of the approach. As a result their interaction evolves into an initiation process during which the students, with variable ease, become familiar with the new notational tools of mathematical formalism– examples of this were given in Chapter 8 in the context of Linear Algebra. Hence their learning becomes a specific struggle for accommodating into this new tool — whose appearance maybe intimidating — the vivid intuitive ideas they have about the solution of the problem.

Another necessary aspect of the enculturation process seems to be reconciliation between intuitive mathematical practices as a way to gain mathematical insight and formal mathematical language as a way to refine and establish these insights rigorously. The novices seem to be deeply perplexed — see Part I — about the status of rigour the various approaches carry: this was extensively exemplified in Chapter 7, in the context of finding and proving limits. The current state of affairs seems to be one of a misunderstanding: the novices are advised to leave behind their school-mathematical way of thinking and start anew by trying to build mathematics on the solid foundations of mathematical formalism. The novices interpret this suggestion in an exaggerated literal manner and turn suspicious about intuitive mathematical practices. As a result they are cognitively torn between what they instinctively know as a powerful way into mathematical insight (intuition) and their desire to be accepted in the culture of mathematical formalism. So, for example, within this schizoid discourse, they perceive the Algebra of Limits as not-formal-enough-hence-avoidable or they refrain from guessing a limit by looking at a graph. The expert's enculturating role then is elevated from the strictly mathematically-topical to the meta-topical, to demystifying not only particular proofs and solutions but also the rules of the game. I stress that crossing through the mathematical contexts explored in this study is the strongly emotional/affective dimension of this demystification and its hard distinction from the purely cognitive one.

The experts then seem to carry the responsibility for convincing the novices about the necessity and the efficiency of various ways of mathematical persuasion. As explained in Part III, Socratic closed questioning, in which unproved theorems were used tacitly, seems to be less convincing than other more openly interactive approaches. Significantly Refutation by Counterexample seemed to bear strong potential of persuasion. However its persuasiveness seems to vary, for both tutors and students, depending on rather personal, and not epistemological, features of the counterexamples used by the expert. In general the expert's invitation into the new and abstract forms of advanced mathematical thinking seemed to be received by the novices with various degrees of readiness. The examples presented in Chapters 6-9, and especially within the more abstract contexts of Linear Algebra and Group Theory, accentuated the very subjective character of the enculturation process.

The perspective in this Part has been to look at cognition as an enculturation process and also to consider potential optimal features of this process. In a sense, this Part can be seen as bridging the perspectives between Part I (cognition from a learning point of view) and Part III (cognition from a teaching point of view). Below I recapitulate briefly some didactical observations made in this study with regard to the teaching of advanced mathematics in the context of tutorials.