# Encyclopedia of Mathematics Education

Living Edition
| Editors: Steve Lerman

# Analysis Teaching and Learning

Living reference work entry
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

There is probably no subject in the teaching and learning of Analysis which has been more extensively studied than the difficulties surrounding the definition and properties of limits of real functions and sequences (see Cornu 1991, for an early review). As pointed out by Barbé et al. (2005), the notion typically appears as a preliminary to differential calculus, where it is needed to define the derivative of a function using the formula
$${f}^{\prime }(x)=\underset{h\to 0}{\lim}\frac{f\left(x+h\right)-f(x)}{h}$$
and to justify the basic calculation rules for derivatives. These rules, as well as the corresponding rules for limits, are at face value 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.

## References

1. Alcock L, Simpson A (2004) Convergence of sequences and series: interactions between visual reasoning and the learner’s beliefs about their own role. Educ Stud Math 57:1–32
2. Alcock L, Weber K (2005) Proof validation in real analysis: Inferring and checking warrants. J Math Behavior 24:125–134
3. Artigue M (1991) Analysis. In: Tall D (ed) Advanced mathematical thinking. Kluwer, Dordrecht, pp 167–198Google Scholar
4. Artigue M (1996) Réformes et contre-réformes dans l’enseignement de l’analyse au lycée 1902–1994. In: Belhoste B, Gispert H, Hulin N (eds) Un siècle de réformes des mathématiques et de la physique en France et à l’étranger. Éd. Vuibert, Paris, pp 195–216Google Scholar
5. Barbé J, Bosch M, Espinoza L et al (2005) Didactic restrictions on the teacher’s practice: the case of limits of functions in Spanish high schools. Educ Stud Math 59:235–268
6. Bergé A (2008) The completeness property of the set of real numbers in the transition from calculus to analysis. Educ Stud Math 67:217–235
7. Bridoux S (2011) Enseignement des premières notions de topologie à l’université – Une étude de cas. Doctoral thesis, University of Paris Diderot. https://halshs.archives-ouvertes.fr/tel-00660249/document. Accessed 5 Feb 2018
8. Cornu B (1991) Limits. In: Tall D (ed) Advanced mathematical thinking. Kluwer, Dordrecht, pp 153–166Google Scholar
9. Danenhower P (2000) The teaching and learning of complex analysis at two Canadian universities. Doctoral thesis, Simon Fraser University. https://www.collectionscanada.gc.ca/obj/s4/f2/dsk1/tape3/PQDD_0008/NQ61636.pdf. Accessed 5 Feb 2018
10. Dubinsky E, Weller K, Mc Michael et al (2005) Some historical issues and paradoxes regarding the concept of infinity: an APOS-based analysis: part 2. Educ Stud Math 60:253–266
11. Durand-Guerrier V, Arsac G (2005) An epistemological and didactic study of a specific calculus reasoning rule. Educ Stud Math 60:149–172
12. González-Martín A, Nardi E, Biza I (2011) Conceptually driven and visually rich tasks in texts and teaching practice: the case of infinite series. Int J Math Educ Sci Technol 42:565–589
13. Gravesen K, Grønbæk N, Winsløw C (2017) Task design for students’ work with basic theory in analysis: the cases of multidimensional differentiability and curve integrals. Int J Res Undergrad Math Educ 3:9–33
14. Katz M, Tall D (2012) Tension between intuitive infinitesimals and formal mathematical analysis. In: Sriraman B (ed) Crossroads in the history of mathematics and mathematics education. Information Age Publishing, Charlotte, pp 71–89Google Scholar
15. Maschietto M (2008) Graphic calculators and micro-straightness: analysis of a didactic engineering. Int J Comput Math Learn 13:207–230
16. Robert A (1982) L’acquisition de la notion de convergence des suites numériques dans l’enseignement supérieur. Rech Didact Math 3:307–341Google Scholar
17. Roh K (2010) An empirical study of students’ understanding of a logical structure in the definition of limit via the ε-strip activity. Educ Stud Math 73:263–279
18. Steen L (2003) Analysis 2000: challenges and opportunities. In: Coray D et al (eds) One hundred years of L’Enseignement Mathématique. L’Enseignement Mathématique, Genève, pp 193–210Google Scholar
19. Tall D, Vinner S (1981) Concept image and concept definition in mathematics with particular reference to limits and continuity. Educ Stud Math 12:151–169
20. Voskoglou MG, Kosyvas GD (2012) Analyzing students’ difficulties in understanding real numbers. J Res Math Educ 1(3):301–336Google Scholar
21. Winsløw C (2007) Les problèmes de transition dans l’enseignement de l’analyse et la complémentarité des approches diverses de la didactique. Ann Didact Sci Cogn 12:189–204Google Scholar

## Copyright information

© Springer Nature Switzerland AG 2018

## Authors and Affiliations

1. 1.Department of Science Education, Didactics of MathematicsUniversity of CopenhagenCopenhagenDenmark

## Section editors and affiliations

• Michéle Artigue
• 1
1. 1.Laboratoire de Didactique André Revuz (EA4434)Université Paris-DiderotParisFrance