# Analysis Teaching and Learning

**DOI:**https://doi.org/10.1007/978-3-319-77487-9_100029-1

## Keywords

Analysis Post-Calculus Limits Completeness Transition## Introduction

In the history, applications, and current practice of the mathematical sciences, Analysis is a domain of the most central importance, even if it has been contended (Steen 2003, p. 193) that it may be overemphasized in undergraduate programs. With roots back to the “Calculus” of real variables pioneered by Newton and Leibniz, Analysis can be defined loosely as the mathematical theory of change, based on the real number system. In the Mathematical Subject Classification (MSC), roughly one quarter of the first level categories (namely the subject numbers 26–49) can be ascribed to this huge domain, which includes both more classical areas like ordinary differential equations and real functions and also more abstract topics such as operator theory and harmonic analysis. In many university mathematics programs, the latter topics are more likely to be titles of advanced undergraduate or even graduate courses, while the basic techniques related to the study of real functions are covered by undergraduate courses in “Calculus.”

Mathematically, there is no strict separation between Calculus and Analysis, but in contemporary university teaching, they are often quite distinct. Roughly speaking, Calculus is concerned with calculation problems related to the study of real functions of one and several variables given in closed form (see “Calculus Teaching and Learning”). In the MSC category 97I (Mathematics Education, Analysis), we see that most subcategories pertain to more or less to Calculus, including the teaching of elementary methods to solve differential equations (see “Differential Equations Teaching and Learning”).

At university, Calculus is taught to a very broad range of students, from business over engineering to the natural sciences. On the other hand, Analysis is based on modern, axiomatic theories and involves notions such as completeness, compactness, and normed spaces. Analysis courses at university are therefore more theoretical and cater mostly to students of mathematics and closely connected sciences. The focus in this entry is on research on post-Calculus aspects of Analysis teaching and learning, taking into account the important transition issues. Note that some aspects of Analysis are or were also taught at the upper secondary level in many countries (see, for instance, Artigue 1996); thus, this section is not strictly limited to research on University Mathematics Education.

## Foundations: Real Numbers

The real number system is at the basis of Analysis, and many fundamental challenges for students can be traced back to their conception of the real numbers, which appear more or less informally in secondary school, often in two ways: firstly, as “all numbers found on the number line,” which the students used in primary school to visualize the position of rational numbers as well as some irrational numbers like π and \( \sqrt{2} \), and secondly, as “all infinite decimals.” These approaches suffice for the basics of analytic geometry and Cartesian graphs, while avoiding most subtle features of ℝ. They also allow or create several misconceptions about the real numbers, which are easily surviving well into university (e.g., Voskoglou and Kosyvas 2012). At a deeper level, this relates to the philosophical quandaries surrounding infinity (e.g., Dubinsky et al. 2005) and the sheer difficulty of any rigorous construction of ℝ. For Analysis, the most basic property of the real number system is its completeness, and several studies (e.g., Bergé 2008) have investigated students’ grasp of this and equivalent basic properties of ℝ. As an alternative, more or less formal versions of the so-called hyperreal numbers have been experimented with some success to teach (nonstandard) Analysis, especially at the elementary level (e.g., Katz and Tall 2012). This approach continues to be a controversial issue in the foundations of Analysis. It may appear in optional graduate courses on set theory and logic, but it remains practically unused in the basic teaching of Analysis.

## Foundations: Limits and Derivatives

*algebraic*. But at the same time, the existence of a limit, and hence for a derivative at a point, ultimately relies on

*topological*properties of the real numbers, as discussed above. This means that notions such as differentiability and continuity become somewhat circular without at least some work with a more formal definition of limits. A considerable gap between students’ informal “images” and the formal definition has been revealed in several seminal studies (Robert 1982; Tall and Vinner 1981) with repercussions also on students’ knowledge of derivatives (Artigue 1991). One reason is that the standard εδ-definitions of limits rely on subtly quantified propositions; avoiding them has been an important motivation for didactical experiments with nonstandard Analysis. A very large number of creative designs for supporting students’ acquisition of the standard theory have been proposed and experimented (e.g., Roh 2010). Such designs often involve carefully designed visualizations and numerical calculation supported by technology, as in Maschietto’s (2008) design experiment on “local straightness” of smooth curves. Indeed, Alcock and Simpson (2004) demonstrated that visual images can be an important factor in successful students’ reasoning about convergence, although this varies considerably among individual students. González-Martín et al. (2011) found that the use of visualizations remains scarce and unsystematic in current post-Calculus teaching of sequences and series at the university level. But even a strong (formal and intuitive) grasp of how limits are defined does not eliminate the necessity of formal work with the topology of real numbers, both in order to study the properties of the limit notion itself (Bergé 2008) and to approach the general notions of integral, as well as more advanced subjects in Analysis (see below).

## The Calculus-Analysis Transition at University

According to Steen (2003, p. 206), introductory Calculus courses at university aim to achieve “a phase transition in students’ mathematical passage from algebra to analysis.” This is an important special case of transition problems affecting mathematics students at the beginning of their university career (see “University Mathematics Education” and “Secondary-Tertiary Transition in Mathematics Education” for the general issue). Bergé (2008) performed a praxeological analysis of the teaching related to completeness in four consecutive courses at a major university in Argentina and identifies a progressive change of focus from tasks that require students to compute (e.g., the supremum of a given subset of ℝ) to tasks which ask to prove (e.g., the equivalence of given statements involving supremum). Also based on praxeological analysis, Winsløw (2007) proposed a general model for the transition, illustrated by examples from introductory functional analysis; it was used by Gravesen et al. (2017) to design and experiment tasks with the aim of facilitating the transition Calculus-Analysis transition for students. Students’ experiences with understanding and writing formal proof are a vast research topic (see “Mathematical Proof, Argumentation, and Reasoning”). The critical role of proof in the Calculus-Analysis transition has been studied in several papers. Students’ performance on validating a given proof in introductory Analysis has been investigated by Alcock and Weber (2005), who found that a good deal of students’ difficulties can be traced to the more or less subtle logical structure involved. A related conclusion is reached by Durand-Guerrier and Arsac (2005), who studied the ways in which university teachers assess a flawed proof of a given proposition in metric space theory.

## Developments and Perspectives

The amount of mathematics education research devoted to specific content areas reflects at least to some extent their societal importance, including the volumes of student populations concerned. The latter are small when we come to the teaching and learning of more advanced subjects such as measure theory, functional analysis, partial differential equations, and so on. Such subjects often appear in research as simple contexts for the study of a more general problem, such as mathematics students’ learning of proof. In fact, trawling the literature, we have found very few works specifically focusing on any such area; two typical examples are the theses of Danenhower (2000) and Bridoux (2011), which explore, respectively, the teaching of Complex Analysis, and of point set topology in the context of Real Analysis. One can say that the teaching and learning of most areas of Analysis remain virgin territory in mathematics education research, the main exception being those parts of Real Analysis which can be considered as “the theory behind Calculus.” On the other hand, for certain elementary notions of Real Analysis, like the definitions of limits and derivatives, a large number of studies were conducted, while leaving many questions open. One of the most promising aspects of research done so far is that the theoretical and methodological developments in mathematics education at large tend to enable research that goes beyond the teaching and learning of isolated concepts. The general tendency for more mathematics PhDs to take part in research on university mathematics education is also visible in the case of Analysis and should lead to new opportunities for innovative and relevant research on the teaching of more advanced parts of Analysis.

## Cross-References

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