PART III A Synthesis of the Findings in the Area of Calculus. Indications for the Cross-Topical Synthesis in Chapter 10
In this chapter the students' first experiences of fundamental calculus concepts were explored in a series of Episodes from both terms of observation. As noted in the Interlude, the concept that emerged as paradigmatically problematic, within the area of Calculus was the notion of limit as encountered in a variety of contexts (limit of a sequence, continuity, convergence).
In particular, in their introduction to the formal definition of limit [
1, 2], the students are in difficulty with the d -e mathematical formalism as well as with assigning meaning to the formal definition. In particular, their difficulty seems to be with comprehending the mechanism of the proposition that is contained in the formal definition and with how this mechanism provides a tool for proving limits. The students also demonstrate [1] a tendency to believe that all functions are continuous and this possibly implies a view of limit as a redundant concept. Moreover visualisation proves [2] to be controversial: the novices debate the use of pictures in guessing limits and see pictures as specific representations and this specificity seems to impede the students' shift towards generality. At the same time they resist formal notation and logic as vehicles for mathematical arguments: so they cannot evoke the formal definition of limit or construct a logical statement of an argument for a proof by contradiction. Their linguistic preoccupations include wondering whether the term not-continuous is equivalent to discontinuous [2]; or whether the derivative of a function is always continuous [5].The students' use of ordinary language has been variably successful in their attempts to describe limiting processes in various contexts: the definition of limit [
1] or in specific situations [5]. So even in cases where their use of language successfully conveys their general grasp of an idea, this success is not fully integrated in the process of presenting a consummate formal argument. Another illustration of this is evident in the students' proficient algorithmic behaviour with regard to differentiation whose mechanics they seem to understand but do not actually employ fully [5] in order to complete a formal proof. Also they do not demonstrate a conceptual understanding of what a derivative is and, in particular, how the algebra of derivatives is derived from and connects to the definition of derivative as the limit of a ratio. More evidence of this behaviour was given in the context of Taylor Series [7] and within the new context of convergence of a series.The students seem to have been sensitised to the requirements of university mathematics for rigour: this engenders a hesitation towards school mathematical practices which deters them from referring to and employing previous mathematical knowledge. The novices possibly need to clarify the distinction between rigorous and intuitive arguments, legitimate and illegitimate use of knowledge that is thought of as previously established. On the other hand there are occasions [
6] where the novices are still at the stage of unconsciously using knowledge that is taken as so obvious as to be not in need of establishing formally (like the propositions of the Intermediate Value Theorem and the Inverse Function Theorem). This tacit use of theorems is evidence of a perpetuation of A-level attitudes in considerable distance from the expected mathematical formalism. Similar regression to familiar-from-school modes of action was demonstrated in the novices' inclination [8] to treat infinite sums as finite quantities, possibly influenced by deeply embedded epistemological beliefs about ¥ as well as a biased exposure to infinite sums that can be harmlessly evaluated with finite techniques. Again the novices' finitism was juxtaposed with the expert's contextualised, concise, sophisticated and, possibly, generalisable approach.The students' most prominent difficulty in the context of limits seems to be finding limits either via the definition or via the algebra of limits [
1, 2, 3]. Using inequalities in order to manipulate quantities, graphing functions, guessing limits and using the algebra of limits are mathematical practices that the students question as to their rigour and, hence, as to their acceptability. So due to their growing mistrust towards the practices of school mathematics, they begin to avoid intuitive practices, such as guessing limits. So, for instance, when [2] it is not overtly justified to them why the use of the definition of limit can alternate with the use of the algebra of limits, they seem severely confused with the tension between what can be called Proof-By-First-Principles and Proof-by-Theorem-Quoting. This tension has been generally observed in these tutorials in various contexts. Similarly the novices are seriously perplexed with [3, 4] distinguishing between the practices that they are supposed to espouse in different mathematical domains: in applied mathematics, retrospective use of unproved results is allowed, as opposed to foundational courses, where it is not. This difficulty echoes the difficulty reported in Chapter 6 to distinguish between what knowledge they are allowed to assume and what has to be established by formal proof.The tension between First Principles and Theorem Quoting was also exemplified in the context of juxtaposing novice and expert practices: the expert [
3, 7] not only distinguishes between different approaches to a proof but also expresses a preference for a proof (usually via first principles) on historicoepistemological grounds. The students' occasional preference for more sophisticated tools, such as identifying the maximum of a variable quantity via differentiation, not via basic algebraic inequalities [9], is possibly due to their — unrigorous — familiarity with these tools from school.The novices tend to generalise unrigorously or react to what they perceive as tedious rigour: so they generalised [
3] the finite case of lim(a+b+c)=lima+limb+limc into limå=ålim which is actually a double limit, thus disregarding unconsciously the conditions for the exchange of limits; they also reacted with suspicion and hesitation towards considering the limited validity of a theorem via a counterexample: characteristically for novices, one of them thought it was not useful to refute a theorem by a counterexample when it holds in the particular case in which he used it; another tried to negotiate his right to use elliptic arguments in writing, as long as he can explain what he did orally [3].The tensions and the conflicts reported above can be seen as part of an enculturation process during which the novices learn about certain conditions of the didactical and epistemological contract in formal mathematical activity. So when, for instance [
3], in the interaction it is revealed that they have been unconsciously assuming the validity of exchanging double limits, using thus a theorem which is as yet unproved in the course, the novices seem to be learning that they need to exercise control over their mathematical reasoning in order to avoid subconscious and unjustified decisions and that this skill to control is a characteristic of formal mathematical behaviour.The students' perplexity with the status of rigour of the various approaches to finding and proving limits [
1, 2, 3, 5] — even when they appear alerted to the dangers in not clarifying the foundations of their proofs [3] — makes it necessary to consider it as a task of the first year course• to establish the legitimacy of school mathematical practices as a way to gain mathematical insight, and
• to introduce formal mathematical language as a way to refine and establish these insights rigorously.
This is in contrast with certain current practices according to which 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: this is a suggestion by the tutors and lecturers which ought to be understood as a manner of speaking and not as a literal translation into practice. The distinction however is not clearly made by the novices who at times perceive these suggestions as contradictory and are thus led to inefficient practices such as avoiding the algebra of limits or avoiding to intuitively find a limit by looking at a graph.
A substantial part of the novices' action seemed to concentrate on their effort to assign meaning and significance to the new concepts [
4]. This is usually a process which takes place under the severe influence of both emotional and cognitive/contextual factors [1 and 4 ]. More detailed evidence of the meaning bestowing practices in which the novices are constantly engaged follow in Chapters 8 and 9.