Task design in Calculus and Analysis

Fabrice Vandebrouck, Erik Hanke and Rafael Martfnez-Planell

This chapter about task design in calculus and analysis is divided into three parts. In the first one, we consider different perspectives on task design in the INDRUM papers about calculus and analysis topics. In the second one, we consider the assessment of task design and students’ task performance. In the last part, we enlarge our scope by considering papers suggesting the need for task design in calculus and analysis as well as task design for the purpose of research.

Several perspectives on task design

The aim of the first part is to show, through questions about the conceptualisation of the notion of limit, that task design can be looked at from different perspectives, from constructivist views based only on the task itself to more technological and socio-culturally oriented views which focus on the role of the teacher and tools used.

From constructivist to more socio-constructivist and cultural perspectives

In France, task design was introduced a long time ago in association with didactical engineering (Artigue, 2014; Gonzalez-Martin, Bloch, Durand-Guerrier, & Maschietto, 2014). In this first sense, task design is associated with the Theory of Didactical Situations (TDS; Brousseau, 1997), mostly in a constructivist vision of learning, and this approach is still present in some research presented at the INDRUM conference. For instance, Ghedamsi (2016) has built two situations aiming at the introduction of the notion of limit via numerical approximation of real numbers by sequences of rational numbers. In this context, task design means the building of a “milieu” such that the students can reach the notion of limit through the designed task and the interactions (actions, retroactions) with the milieu by themselves. The situation evolves through action, formulation, validation and institutionalisation phases associated with the evolution of the milieu itself. The task design supposes strong epistemological and historical analyses of the notion at stake and strong a priori analyses of the designed tasks in order to analyse the findings in the a posteriori analysis and predict students’ possible activities. In her study, Ghedamsi justifies her choices for the design with a historical perspective on understanding of real numbers with attention to approximations. She suggests two situations according to the values of some didactical variables that are crucial for the learning objectives: the real numbers which are approximated (rational or irrational), the nature of the sequence which approximates (explicit or recursive, growing or not, ...), and the nature of the approximation and error. The two situations are the approximation of the square root of 2 by Euclid’s method (antiphérèse) and the solution of cos(x) = x by Newton’s algorithm. Ghedamsi’s conclusion highlights especially the success of the situations with respect to students’ understanding of real numbers as mathematical objects together with a meta-mathematical reflection about their existence and the process of their approximation. However, the situation of the fixed point of the cos function allows to go deeper into the conceptualisation of real numbers because it gives access to real numbers which cannot be expressed in terms of elementary functions or symbols (like the square root of 2) and which oblige students to use formal procedures in order to deal with these numbers.

In the epistemological work done in order to build new tasks which aim at introducing the concepts of convergence and real numbers, Rogalski (2016) recalls several dimensions which can structure students’ conceptualisations of the notion of limit and which can be some basis for task design: first of all, the need for approximations of real numbers as Ghedamsi (2016) argued, secondly the need for formalisation in order to enter in some proof paths, and thirdly the need for understanding about mathematics practices at a meta-level.

The focus of Bridoux (2016) is to respond to the second and third need. First, she argues that the Theory of Didactical Situations is not sufficient in itself to introduce notions such as limit and convergence because of their FUG aspects (formalisation, unification, generalisation) in the sense of Robert (Robert & Hache, 2013): The definition of limit formalises intuitive aspects of convergence already encountered by students, it unifies different pre-existing students’ ideas of convergence, and it permits a generalisation which tries to overcome problems about convergence and limits that students can face. This means that the notion of limit is such a complex notion that a constructivist approach of task design is not sufficient for the students to understand it deeply. This implies the need of a meta-discourse of the teacher in order to justify the need for the formalisation and the need of the limit as a formalised concept to enter in the domain of analysis at the university level, as Rogalski (2016) argues. The designed task in Bridoux’s paper comes from old research papers of Robert in the eighties (Robert, 1982; 1983). Her paper is about the classification of ten sequences, drawing graphical representations, discussions about students’ classifications and then answering two questions which need the formalisation of limit: 1) Is a positive sequence which has 0 as limit decreasing after a certain rank? 2) If a sequence has a strictly positive limit, are all the terms strictly positive after a certain rank? There is an emphasis on the role of the teacher at each step of the engineering. Indeed, the situation alternates between adidactical phases in the same sense as in the TDS and interactive phases when the teacher gives procedural and constructive assistance to students using different kinds of didactical processes such as the interplay between different settings (“jeux de cadres” in the sense of Douady, 1986), a shift between several points of view on the tasks or the shift between several registers of representations which Duval (1993) highlighted as a cognitive benefit. Bridoux shows that this engineering is still robust with some adaptations she explains more closely.

The focus of Lecorre (2016) and Ghedamsi and Lecorre (2018) is also about the formalisation of the notion of limit. However, the papers deal with functions whereas the two previous ones from Ghedamsi (2016) and Bridoux (2016) were in the context of sequences. Cognitive analyses are made to supplement the epistemological ones. For instance, Lecorre mentions the cognitive shift which has to be done by students between dynamic conceptions of limits to static conceptions (Robert, 1982; Williams, 1991). The dynamic conceptions are often associated to a cinematic point of view in the graphical register where moving the independent variable .v leads to moving offix) towards the limit, possibly reaching it. The static point of view is associated to the approximation process where the chosen degree of approximation pulls the independent variable of the function and leads to the search of a real number A (in the case of limits of functions at infinity), or a so-called 1 - or 8 in some papers - in the case of continuity. This cognitive characteristic remains strong in the domain of analysis and for the introduction of formal definitions such as continuity as we will see below. In Lecorre’s engineering, there is a focus on the need of formalisation, which is a common point with Bridoux’s paper. As a first question, the students are asked to discuss the conjecture “If the limit of/is strictly less than the limit of £ at plus infinity then for every real x,fix) is strictly less than fix)” (Lecorre, 2016, p. 85). Then Lecorre introduces a “monster function” to the students with which they should rethink their conjectures and feel the need to establish a definition of the limit of a function at plus infinity. The role of the monster function is also to enrich discussions between students and to destabilise their dynamic conceptions. Like Bridoux, Lecorre highlights the role and the discourse of the teacher. The task itself is not sufficient for the students to reach the notion of limit in its formal definition by themselves. Lecorre uses the concept of “scientific debate” to ensure the link between students’ activities with the task and the institutional process.

Task design in calculus and analysis is renewed today in line with Chevallard’s paradigm of questioning the world (Chevallard, 2015). The scope is supposed to be larger. The starting point is the global difficulty of students to relate the content of analysis courses at the university level (formal theory of limits and continuity, function spaces etc.) to the calculus they have encountered in secondary schools. As Winslow (2016) notes, the abstract theory seems to constitute a universe in itself. The new paradigm of questioning the world is based on Klein’s ideas about Plan A and Plan В (Klein, 2016). Plan A is a compartmentalised approach to mathematics whereas Plan B, “by contrast, involves a more holistic approach which emphasises and exploits connections between different sectors” (Winslow, 2016, p. 164). In the paradigm of questioning the world, as Klein recommends, Plan В is a better strategy for presenting mathematics to students. Winslow investigates the specific notions of angles, sine and cosine. Several plans A appear in secondary school with three distinct contexts: triangle trigonometry, analytic geometry in the circle, and functions through tables and periodic graphs. Winslow develops ideas for some mathematical organisations of type В in university analysis highlighting the importance of curve length as a link between the contexts; this link has never been questioned in the sense of Chevallard (2015). However, as Winslow concludes, the task design based on such a Plan В organisation can remain too ambitious (cf. also Kondratieva & Winslow, 2018).

Task design with technologies

We now turn our attention to papers which deal with technological artefacts that can be used throughout a course to support student learning in calculus and analysis.

The task design by Sghaier and Vandebrouck (2018) builds on ideas from the previous subsection and creates tasks by integrating technological devices. They develop a digital tool aiming at introducing the concept of local continuity of a function together with its formal definition. Concept image and concept definition (Tall & Vinner, 1981) together with the notion of point of view (Bkouche, 1996) are addressed with dynamic and static conceptions as introduced by Robert (1982) and used by Lecorre (2016). The tool and the associated tasks are designed in order for students to make links between two aspects of continuity: the dynamic aspect related to graphical and cinematic approaches of continuity, and the static aspect associated to more formal approaches through the lens of approximation. The knowledge to be learned about continuity is referred to conceptualisation in the sense of Vergnaud (2009). Conceptualisation is equally the cognitive product and the developmental process which occur within students’ actions over a class of situations characteristic of the concept involved, continuity in this paper. The tasks by Sghaier and Vandebrouck are analysed with didactical tools which have been developed in the French community dealing with Vergnaud’s cognitive references: technical tasks which are direct applications of the concepts involved versus tasks with cognitive adaptations of knowledge such as shifts between several registers of representations, uses of several points of view on the tasks etc. There are adidactical aspects in the designed situation in which the task is inserted, in spirit of the TDS and the constructivist approach. However, as in several previous papers, the role of the teacher remains crucial. The students are only supposed to develop some mathematical cognitive activities while linking the two points of view of continuity which are highlighted in the tool. In this paper, the discourse of the teacher during the interactive phases is theorised in term of “proximities”: teacher’s discourse elements which are supposed to be developed within the students’ zone of proximal development according to the mathematical activities that have been developed in the tool. In this respect, the task design is supposed to take into account explicitly some Vygotskian approaches occurring in the conceptualisation process (Vygotsky, 1986).

In another paper, Hogstad, Isabwe, and Vos (2016) use the software Sim2Bil which combines a simulation of two cars, velocity graphs, and an input for velocity functions. Their focus is on how engineering students use visualisations in their mathematical communication. This perspective is more socio-cultural. This means that the tool is no longer mostly a medium for students to construct new knowledge with well-designed tasks but it is rather a medium through which students can communicate knowledge in the social context, together with language, gesture, and mathematical representations. The students are supposed to observe the movement of two cars and some representations about distances and velocities in the tool according to parameters they can choose. The task is to find the velocities of the two cars (fj and th respectively) so that th is half of Pj when the cars reach the finish line simultaneously at four seconds. The students are then asked if they can prove that their answer is correct. This initiates collaboration between the students, giving them room to follow their own approaches. As the authors were interested in visualisation, they observe three different media for visualisations: first Sim2Bil-based visualisations (the cars running in the simulation window and the area under the graphs of the functions in the graph window), second some paper-based visualisations (sketches and detailed drawings), and third gestures, for instance mimicking the kinematic movement of cars or indicating a parabolic graph. They conclude that although the task could have been posed without Sim2Bil (by symbolical calculations) the tool is central in the task by offering visualisations. Such a tool offers rich opportunities to visualise mathematics and to connect different mathematical representations and applications.

In his paper, Hoppenbrock (2016) analyses the role of clicker tasks in triggering productive discussion in small groups of students (peer instruction) and in supporting the construction of new mathematical knowledge in the group. The study uses the notion of epistemological triangles (Steinbring, 2005) to analyse the discussion of one of six audio-recorded groups of three students in a large lecture hall of an analysis course. The clicker task that is analysed is a multiplechoice question in which students are presented with three choices for the formalised statement “/’has a local maximum at x()”. Two of the choices involve changing the order of quantifiers. This question is expected to be challenging for students since, as discussed by Dubinsky and Yiparaki (2000), students have difficulty distinguishing between AE (for all there exist) and EA (there exist for all) statements. The focus of Hoppenbrock’s paper is finding characteristics of clicker tasks that will promote significant discussion resulting in new knowledge construction. Based on Dekker and Elshout-Mohr (1998), Hoppenbrock proposes that tasks should be meaningful, complex, require different student abilities, and aim at level raising. The example he develops supports this proposal. The author emphasises the need to further investigate the interplay between the clicker tasks and the skills, motivation, and knowledge of the students in the group as well as the group dynamics. Finally, the implementation of similar clicker-based tasks for the definition of convergence of sequences or continuity is suggested as a possible pedagogical tool.

The article by Gaspar Martins (2018) documents the effect of using a weekly online quiz system during an entire semester for two courses: calculus of one variable and multivariable calculus. The data include responses from surveys answered by a large percentage of students in both courses. Survey results suggest that most students regularly used the quiz system, quizzes fomented studying regularly, helped students discover what material they did not know enough about, and helped students rate their own learning level and leant new material. Further, a large percentage of survey respondents reported that they found quizzes useful and fair, and that they perceived that quizzes helped them attain a higher grade. Indeed, comparative data from five academic years shows that semesters in which the quiz system was used resulted in higher passing rates on both courses. Gaspar Martins suggests the potential of task design for online quiz systems to help students keep up to date in their studies and improve passing rates.

Tasks on topics for other disciplines and preservice teacher training

Now we deal with the use of tasks about understanding of concepts from calculus and analysis (including complex analysis) of students who do not major in pure mathematics and preservice teachers in a lecture on mathematics in their study programme. These studies have different focus. The research project by Feudel (2016, 2018) deals with the interpretation of the derivative in the context of economics, Gonzalez-Martin and Hernandes-Gomes (2018) work on applications of multivariable integration in the context of engineering, and Hanke and Schafer (2018) establish a link between mathematics and mathematics education in a course on complex analysis. What is common in all these pieces of research is that the studies address mathematics in different, more or less applied contexts and tasks are used to cross boundaries between different professional practices (Akkerman & Bakker, 2011).

Feudel (2018) investigates students’ understanding of the economic interpretation of the derivative of a cost function. The literature suggested that economics students, but also students of other disciplines that apply calculus concepts, have difficulties in interpreting the derivative C '(x) of a cost function C (x measures the unit of production) in terms of recognising C'(x) as a rate of change instead of absolute change and in terms of the corresponding units. Feudel (2018) describes that in the economic context, C'(x) is often interpreted as the additional cost of the next unit since C(x + /») - C(x) ~ hC'fx) for h ~ 0, and /1 = 1 could be considered small in economics (Feudel, 2016). However, this does not resolve the problem between rate of change and absolute change. In interviews, Feudel (2018) asks economics students to determine the marginal cost and the unit for a cost function C at a given point and to determine the relationship between C'(x) and the additional cost when x is increased by one unit. Moreover, he asks explicitly about the connections the students see between the mathematical concept of derivative and the economic context. Feudel (2018) points out that the interviewer could react to the answers of the students and provoke cognitive conflicts (Tall & Vinner, 1981), and claims that it might be helpful to thematise the relationship between the mathematical notion of derivative and the economic notion of marginal cost by explicitly referring them to each other, not only introducing the students to each of the notions separately. The study suggests the potential benefit of tasks designed to help economics students link the mathematical notions and their applications or interpretations more closely.

González-Martín and Hernandes-Gomes (2018) provide a textbook analysis on how the first moment of area is presented in textbooks for engineering students. The approach is similar to Feudel’s studies (2016; 2018) since González-Martín and Hernandes-Gomes also investigate how a notion of mathematics is related to the use of this notion in an applied field. Using the Anthropological Theory of the Didactic (ATD; Chevallard, 1999; Bosch & Gascón, 2014), the authors analyse tasks in calculus and mechanics to identify praxeologies around the determination of the moment of area ||д ydA. They also refer to Castela (2016) who claims that praxeologies used in mathematics may undergo a transposition when the mathematical notions are regarded from an applied science as an effect of boundary crossing. González-Martín and Hemandes-Gom.es come to the conclusion that techniques to work with integrals of the given type, which are part of mathematics courses on this topic, are not necessarily required to be used in the applied context the students face in their courses on mechanics of materials, e.g., in their textbooks. From the first analysis of the mechanics textbook, the authors recognise that the first moment of area is defined as an integral like JJA ydA, but in the actual computation of moments of areas the evaluation of integrals is most of the time not necessary. Rather, techniques from geometry or previously derived formulae are used to evaluate the moments of area, and thus implicitly the corresponding integrals. To sum up, in this study tasks are not designed but current task design is used to identify mathematical praxeologies in calculus and engineering about the first moment of area.

The design research project “Spotlight-Y” (Hanke & Schafer, 2018) belongs to the discussion of the “double discontinuity” in mathematics teacher training: the fragmentation between the courses on mathematics and mathematics education many preservice teachers experience within their study programs (Hefendehl-Hebeker, 2013; Winslow & Gronbaek, 2014), mathematics teacher knowledge (Ball. Thames, & Phelps, 2008), and the “higher standpoint” on school mathematics that Klein (2016) called for. The authors describe the format of a lecture in complex analysis in which the last third of the lecture is split into two branches, one for pure mathematics students and one for preservice teachers. Tasks appear at different points. Firstly, the preservice teachers get the task to create and implement learning arrangements for gifted pupils from secondary schools about a phenomenon of the lecture on complex analysis, and thus, secondly, the preservice teachers develop tasks and learning materials themselves. The intention is that the preservice teachers experience the relevance of the mathematics they learn at university for their future career as teachers. But the tasks have another function. They create an institutional opportunity for the students to link the mathematics of their study programme with their courses on mathematics education, in this case a seminar on task design, which aims to help the students to prepare tasks. Thus, one can say that the initial task acts as an initial boundary object and the tasks and materials the preservice teachers develop constitute an emerging boundary object between university mathematics and mathematics education (Hanke & Bikner-Ahsbahs, 2019). Hanke’s and Schafer’s empirical findings about two learning arrangements, one on differentiation as linearisation and one on Taylor series expansion, which were developed by their students and on the planning of which these students reflected, show that it is a challenging task for the preservice teachers to work on the preparation of a mathematically demanding phenomenon for pupils, and linkages between these topics of university mathematics and mathematics education did not happen automatically.

The results of the first and third research projects described in this section show that many students need assistance when they need to bring together mathematical knowledge with knowledge from profession-specific domains. It is not enough to teach the different perspectives next to each other.

Assessment of task design and students' task performance

The assessment of solutions to calculus tasks is often associated to the task design itself and has also been addressed in several INDRUM papers. Each of them draws on elements of well-established theories but provides adjustments and refinements needed to assess students’ solutions.

The change of methods used in schools to methods required in university mathematics, one part of the transition from school to university, is investigated by Bloch and Gibel (2016). Based on TDS, they draw on a model to analyse adidac-tical situations and use it to analyse reasoning processes in 14 students’ solutions to tasks about parametric curves and ordinary differential equations in an exam. The authors “classify the objects, signs and reasoning processes they [students] have to cope with during the resolution of calculus problems” (p. 44). The model includes the consideration of functions of reasoning, levels of argumentation and signs, and the distinction between three levels of didactic milieus: heuristic/exploration level, formulation/validation level, and institutionalisation level. Bloch and Gibel arrive at the conclusion that their students remain at low level because they often get stuck in calculations or do not adapt their previous knowledge. Some students try to work at higher level but do not succeed because, again, they either fail in calculations or do not manage to interpret their calculations. The authors also recognise that a holistic view on formulas does not seem very present; instead, parts of formulas are interpreted separately. Finally, the authors hypothesise that the phenomenon that students remain at low levels of reasoning is not directly linked to teachers’ didactical choices, rather it is a lack of association between syntactic and semantic methods.

In his case study, Hanke (2018) investigates students’ written proofs of a theorem in a topology course in terms of a refinement of “object layers” (Oerter, 1982; Schafer, 2010). The task was to show that a function R —> R is continuous if and only if its graph is path-connected. Relying on the RBC-model of the Abstraction in Context methodology (Dreyfus & Kidron, 2014), Hanke reconstructs the traces of the epistemic actions of recognising and building-with which are performed on the mathematical objects introduced in the course of the written proof attempts by the students, and how implications in the course of proof are achieved. The notions of “singular”, “contextual” and “formal” layer of objects (Oerter, 1982) reflect the levels of concreteness of objects used. Schafer (2010) applied these layers in mathematics education in a combination with RBC, and Hanke introduces the specifically mathematical “propositional layer” which includes theorems (e.g., “If F: X —> Y is any function between a path-connected topological space X and any topological space Y, then the graph of F is path-connected if F is continuous”, p. 48). The propositional layer is useful because the application of a mathematical proposition is very common in proofs and relies on the fact that the requirements in the proposition have to be recognised in advance, thus the important properties of the objects appearing in the course of proof have to be abstracted, in order to justify a building-with action on the propositional layer (cf. the similar approach in terms of “structuralist praxeologies” by Hausberger, 2018). Hanke shows that recognising and building-with actions on different layers were often successful, but also that propositional building-with actions can be carried out well by the students even if the prerequisites in a theorem have falsely been recognised to be valid.

Whereas Bloch and Gibel (2016), and Hanke (2018) propose new theoretical constructs that may be used to supplement established theories (TDS and AiC respectively) in the assessment of student task performance, other papers use existing theoretical constructs to further expound on the assessment of tasks, this time from the institutional point of view afforded by the ATD. Brandes and Hardy (2018) use the ATD to assess the tasks and activities of students in a multivariable calculus course. Student success in the course they consider depends largely on their performance in a final exam. This is reflected in the knowledge to be taught (KT, mathematical knowledge listed in the sections and exercises in the course outline) and in the knowledge to be learned (KL, the subset of KT that students need to learn in order to provide solutions in final exams), which can be used to describe what is effectively done in the course. Essentially, Brandes and Hardy argue that the technological-theoretical component of the KT praxeology disappears in the KL praxeology which will now be restricted to a practical block which is a deformation of the practical block in the KT praxeology. This is in the sense that in the KL praxeology students are limited to tasks that promote reasoning based on identification of similarities (IS), as described by Lithner (2004). IS reasoning merely requires students to recognise a type of task and to apply a similar known technique, circumventing the need for technological-theoretical arguments in problem solving. Thus, the course loses its original purpose of bridging the gap between pre-university mathematics and more advanced courses as described by the transition types of Winslow, Barquero, Vleeschouwer, and Hardy (2014), where students would first transition from praxeologies restricted mainly to practical blocks to praxeologies that begin to incorporate a theoretical block, and then transition to praxeologies where past theoretical blocks turn into practical blocks that include theorem proving. Citing these transition types, Broley and Hardy (2018) argue in a similar paper for the need to investigate students’ transition from introductory calculus and multivariable calculus courses that may be modelled with praxeologies with a missing theoretical component, to the introductory analysis courses where students are expected to work with or on the theoretical component. They discuss the plan to construct a praxeological model of the knowledge to be learned in introductory mathematical analysis courses. Both papers use the ATD to discuss the types of tasks that students actually face in the multivariable calculus and analysis courses and implicitly uncover the courses’ need of a deep re-struc-turing that will perhaps require the design of Research and Study Paths as advocated in the ATD, or similar such restructuring as described in other theories.

Research with or suggesting task design

In this part we consider papers where the results suggest the need to design or renew the design of tasks for the teaching of calculus and analysis. In contrast to the first part of this chapter, task design has still to reach classroom use and is mainly put forth by the reported research. Some of these papers are based on a cognitive theory and support the development of tasks to help students perform mental constructions in which they have shown difficulty, or of tasks to take advantage of unexpected constructions that students seem able to do. In some papers, tasks are components of the research itself and reveal specific difficulties that students encounter with calculus and analysis concepts. In contrast to many of the articles considered in the first part of this chapter, these tasks are not part of the design of Didactical Situations (TDS) or of Research and Study Paths (ATD), but rather directly address student behaviour when solving problems involving a specific mathematical notion. Of course, results of cognitive studies may be used in conjunction with other task design perspectives as shown, for example, in Salgado and Trigueros (2015) where modelling activities are interweaved with APOS-based tasks (Action-Process-Object-Schema; Amon et al., 2014). Other papers in this section base the need of task design on epistemological studies, the analysis of institutional documents, or questionnaires. In these cases, the research results suggest that tasks be designed as part of didactical situations or couched in a modelling context.

Task design as one research component

In addition to derivatives, continuity and limits for real-valued functions, addressed in the majority of papers about education in calculus and analysis, some more topologically oriented topics have also been dealt with by the INDRUM participants. Durand-Guerrier and Vivier (2016) investigated students’ usage of density and completeness of the real numbers, Branchetti and Durand-Guerrier (2018) looked at concept images for the ordering of subsets of real numbers, and Hamza and O’Shea (2016) studied students’ concept images and definitions of open sets in metric spaces. These papers share that tasks were designed for their research purpose instead of a teaching purpose. Chorlay’s (2018) questionnaire, on the other hand, was originally intended to be part of a larger project about the formalisation of limit from the perspective of didactical engineering.

Durand-Guerrier and Vivier (2016) describe the first stage of on-going research into students’ learning of the notion of density and completeness of subsets of the real numbers, and how this impacts students’ understanding of the main ideas of analysis. They analysed some questions in a questionnaire to 35 students of a first course on analysis which covered the notions of continuity of functions and sets (all proper cuts of which are Dedekind cuts), (order) density and discreteness taking into account the limits of visualisation in analysis. An important set in their tasks is the set D of decimal numbers which is dense. The students are asked to give examples of intervals which contain exactly one (or two, respectively) natural numbers, and the same for decimal numbers. Also, they should answer whether f(x) = 2 necessarily has a solution in N, D, Q or R when the function f was given by its graph. Or, in a task about approximation, it was asked whether one can deduce from a table with function values, a discrete graph of the table values, and an additional requirement for the function (such as piecewise monotonicity or continuity, respectively) whether the function has one fixed point, none or several. Durand-Guerrier and Vivier claim that an adequate conceptualisation of density is important for the introduction of the real numbers as the completion of the rational numbers, and the difference between being dense in itself and completeness is fundamental for the beginnings of analysis and the intermediate value theorem. Their tasks show that close to one third of the students did not observe the density of D (or its usefulness in the application to the tasks), possibly due to the misunderstanding of D as one of the sets {k/10n:k€N} for some natural number n, or not recognising that there are subsets of the real numbers which are neither discrete nor continuous.

Branchetti and Durand-Guerrier (2018) took a closer look at the interaction between university students’ and teachers’ concept images of ordered, order dense and denumerable sets. The authors argue that concept images of one of the notions of ordering, order density and discreteness can cause cognitive conflict when dealing with another, and they aim to identify the interplay between those concept images more closely. The tasks were (paraphrased): Why can Q be ordered even if Q is order dense (i.e. consecutive rational numbers do not exist), and why is there a bijection between N which is discrete and Q which is dense? The tasks Branchetti and Durand-Guerrier prepared are guided by historical and epistemological analyses of the concepts, and were given to master students and teachers. These questions were also embedded into a didactical scenario, i.e. the master students should indicate how they would explain the answers to the questions to pupils and the teachers how they would address these questions in class. As a result, teachers often relied on definitions, whereas some of the students had difficulties in relating the concepts to each other. The authors argue that the latter is due to their concept images and the focus on the definitions of the concepts will not be sufficient to overcome the compartmentalisation (Tall & Vinner, 1981) in students’ concept images. Further, Branchetti and Durand-Guerrier claim that “this lack of awareness of the links between the concepts of density-in-itself and order of Q might prevent them [French master students] from designing appropriate learning situations, once they pass the selection procedure exam and become teachers” (p. 22).

Hamza and O’Shea (2016) start off the research on students’ concept images and concept definitions of open sets in metric spaces. This research belongs to the transition from real analysis to advanced courses. The theoretical background of their research is formed by the idea of concept image and concept definition (Tall & Vinner, 1981) and their research is grounded by the claim that concept images are formed in early contact with the notions at hand, referencing “met-befores” by McGowen and Tall (2010) and “tacit models” by Fischbein (1989). Ten students from a course on metric spaces were interviewed. The design of the tasks in the interviews incorporates different activities (defining, explaining, solving exercises). The participants were asked to define open sets and how to explain the notion to a friend (this is similar to the didactical situation Branchetti and Durand-Guerrier (2018) offer to their participants). Further, among the interview tasks, the students had to argue, whether B = {in — 1, in, ni + 1} (for some integer ni) is an open ball in Z with the standard metric, describe a certain metric on the set of all real sequences in their own words, and argue whether {(B to be open because they got mixed up with open balls in the set of real numbers and properties like connectedness, and some students had difficulties to find a visualisation for the sets in question and relied on the definitions of metric and open ball. However, such tasks as those in Hamza’s and O’Shea’s study provide opportunities to make students aware of their concept images and enable them to express these.

The questionnaire by Chorlay (2018) asks to identify true or false statements about sequences of real numbers diverging to infinity, e.g., “If lim un = +oo then VA6l<3nAGN such that u„a>A” (p. 28). The main objective is to get insight into students’ recognition and understanding of quantifier usage in propositions given through infinite limits. The students were also asked to state the converses of the propositions and to justify why they considered which of these statements to be wrong. It turned out that several students did not validate the implication given in the tasks but rather tried to adapt the right-hand side of the implication to fit the definition of infinite limit. This was done often by grade 12 students and less so by university students who had been exposed to logic before. An interpretation might be that students who adapt the original task to a comparison with the definition of infinite limits in fact know the definition and use it as a “template against which other quantified formulae ought to be contrasted” (p. 28). In summary, Chorlay’s research shows that tasks can not only be misinterpreted by the participants but that these misinterpretations may be caused by “the level of familiarity with predicate calculus” (p. 32), suggesting that other curricular issues not related to analysis are relevant for students’ engagement in these tasks.

Research suggesting task design

In two APOS-based papers, a detailed conjecture (genetic decomposition) of mental constructions students may use to understand particular topics in the multivariable calculus is posed and tested with student interviews.

In one of these papers, Trigueros, Martinez-Planell, and McGee (2018) discuss how students’ knowledge about the total differential seemed to be unrelated to their knowledge about the tangent plane. An initial genetic decomposition (a priori conjecture of possible mental constructions students may do to understand a mathematical notion) was proposed. Study results show that students have difficulty with some of the conjectured constructions and that they also make some unexpected constructions. In particular, most students did not relate vertical change on a tangent plane at a point (a,b) with the total differential at that point, df(d,&). Students did not relate processes on the function, like partial derivatives, vertical change, and directional derivatives, with the corresponding processes on the tangent plane. Further, students did not think of the total differential at a point as a function on the independent variables dx and dy. Overall, students showed much difficulty converting between different registers of representation (Duval, 2006). These results underscored the need to refine the genetic decomposition. The refined decomposition may be seen as suggesting corresponding task design to help students do the conjectured constructions. In a follow-up to their 2016 paper, Trigueros, Martinez-Planell, and McGee (2018) take advantage of the dialogue between APOS and the ATD (Bosch, Gascon, & Trigueros, 2017) to examine students’ individual and institutional constructions of the total differential, thus exemplifying the possible interplay between these two very different theories.

In another APOS-based paper, Trigueros and Martinez-Planell (2018) explore students’ understanding of the relation between integrals of functions of two variables defined on rectangles and corresponding Riemann sums. The authors only consider a partial genetic decomposition of students’ possible constructions of the integral of a function of two variables. Their findings show that students that had finished a traditional multivariable calculus course failed to interpret geometrically even the most basic ideas relating Riemann sums to double integrals. The students showed difficulty visualising the relationship between the graph of the function and its rectangular domain, both when the function was given graphically and symbolically. They also showed difficulty interpreting geometrically a term f(a,b)AxAy of a Riemann sum as the volume of a rectangular prism, and thus, as may be expected, could not distinguish in simple cases when such a term would be an underestimate, an overestimate, or produce the exact value of the double integral over the corresponding rectangle. Also, most students were not able to geometrically interpret a four-term Riemann sum, being limited to saying as a memorised phrase lacking geometric meaning that it was an approximation of the double integral. This paper underscores the need of task design to help students visualise the geometric meaning of ideas relating Riemann sums and double integrals.

Vandebrouck and Leidwanger (2016) examine students’ computations of limits in a paper based on the notions of iconic and non-iconic visualisation (Duval, 1999), and deconstructions with a pointwise, local, or global perspective (Deconstruction With Perspectives, DWP; Montoya Delgadillo, Paez Murillo, Vandebrouck, & Vivier, 2018). In Vandebrouck’s and Leidwanger’s paper, students’ work suggests that algebraic difficulties constrain their limit computations leading them to error and to the inability to treat some limits using results on indeterminate forms. In some of these cases, students were able to successfully do some DWP to compute limits. For example, some students would avoid the use of results pertaining to indeterminate forms and algebraic manipulations by arguing that since exp(x) grows much faster than ln(x), the limit at infinity of exp(—x)ln(x) is 0, or that since x grows faster than x, the infinite limit of (x —l)/(x+3) is +oo. Students also showed misconceptions when trying to apply DWP. For example, some students would erroneously argue that the limit at infinity of exp(x)(l - yjx) is +oo since the exponential dominates, not taking into account that the exponential occurs in a product where the other factor will be negative. Another type of difficulty shown by students in their limit computations is the tendency to mix together the local and pointwise perspectives. So for example, a student might erroneously argue that the limit at 0 of sin(x)/x is 0 since, for the numerator, sin(0) is 0 (pointwise perspective) while the denominator x only tends to 0 (local perspective). Vandebrouck and Leidwanger observe that students’ work on the limit problems suggest that tasks could be designed to take advantage of the knowledge of DWP that students bring from secondary school, so that they can use DWP with understanding and avoid mistakes, and so that they will be able to compute limits using DWP as an expert would.

Dealing with the same idea of perspectives while also considering the role of representation registers (Duval, 2006), Paez Murillo, Montoya Delgadillo, and Vivier (2016) discuss students’ spontaneous conceptions of the notion of tangent line. They analysed the written responses given by first year university students in Chile, Mexico, and France to one question: For you, what is a tangent line? Based on previous research, the authors propose five types of conceptions of tangent line: unique point of intersection (UPI), perpendicular to a ray (PR), slope (S), process of derivation (PD), and horizontal tangent (HT). These are presented as two blocks, one coming from the domain of geometry (UPI, PR), and directly related to the tangent line to a circle at a given point; and the other coming from the domain of analysis and directly related to derivation (S, PD, HT). The authors then describe how the punctual, local, and global perspectives may appear and interrelate in each conception type while also considering registers of representation. They suggest to interpret the geometric and the analytic conception blocks as part of Kuzniak’s (2011) Mathematical Working Space (Espace de Travail Mathématique, ETM), relating them to two of the paradigms of standard analysis (geometrical-arithmetic and calculating paradigms) as described by Kuzniak,

Montoya Delgadillo, Vandebrouck., and Vivier (2016) and Kuzniak, Tanguay, and Elia (2016). They argue that since ETMs take into account representations, visualisation, theoretical knowledge, and their evolving interrelationship, they may allow to better explain the relation between tangent line conceptions, perspectives, and representations. They finish with what may be taken as a call to design tasks that take into account students’ spontaneous conceptions of tangent lines to allow students to refine their own ideas and to avoid limiting students’ work to the calculating paradigm.

Kouki, Belhaj Amor, and Hachaichi (2016) study the teaching and learning of Taylor series expansions (développements limités). They start by presenting a historical-epistemological analysis based on Douady’s (1986) interplay between different settings (jeux de cadre) and the tool-object dialectic. In their didactic analysis, Kouki et al. also take into account the semantic/syntactic dimensions (Kouki & Ghedamsi, 2012) as well as the semiotic dimension (Duval, 1993). Kouki et al. argue that in the historical course of the development of the notion of an infinite series expansion several phases can be distinguished. They go on to show that different techniques were developed by mathematicians in transiting across these phases which required the articulation of the syntactic and semantic dimensions while also considering the geometric, algebraic, numeric, graphical, and analytic registers. The epistemological analysis by Kouki et al. also shows how the tool-object dialectic is reflected in the evolution of the Taylor series concept. Then they go on to analyse institutional documents, like programmes, manuals, and course handouts. In contrast, here they find that explorations into the semantic dimension of Taylor series are lacking and that there are also limitations in the mobilisation of the graphical and numerical registers. This suggests the development of tasks to help better articulate the semantic and syntactic dimensions, particularly concerning numerical and graphical work. Task design would use modelling and a construction of the Taylor series notion guided by the tool-object dialectic.

Bergé (2016) studies the evolution of students’ understanding of the role of the supremum as a tool in developing basic properties of analysis. The study is part of a larger project that uses the ATD as framework. Like Kouki et al. (2016), it makes reference to ideas of Douady (1986), particularly the tool-object dialectic, and starts with an epistemological analysis of the evolution of the notion of interest, in this case the completeness of the real numbers. Following that, the response to a questionnaire given to students who had been introduced to completeness through the notion of supremum in a four-course sequence of calculus and analysis were considered. The study concentrates on the analysis of one of the questions in which students were asked about the utility of the notion of supremum. Results show that most students in the second course of the sequence (multivariable calculus) either give erroneous answers, seem to be unaware of the utility of the supremum, or (the larger percentage) have a restricted understanding of supremum only as an upper bound. Students in the third course of the sequence have an increased awareness of the role of the supremum in the proof of some of the basic theorems of analysis and by the fourth course most of them are able to give specific examples of its use. These students had re-encountered the notion of supremum in the third and fourth courses as a tool to define or determine specific numbers. This suggests the need to find a situation that will integrate the use of completeness as a tool in the mathematical organisation used to introduce the real numbers, so that students will experience the reason of being of the supremum and completeness early in the sequence of calculus and analysis courses.

Kidron (2018) investigates 10th, 11th, and 12th year students’ understanding of infinite non-repeating decimals. She used a questionnaire in which a total of 188 students were to comment on an imagined discussion between two students: one who argued that it was possible to form infinite non-repeating decimal numbers (by writing digits at random after the decimal point) and the other one who argued that any infinite decimal must be formed by dividing two whole numbers. Among other results, she finds that although around 80% of the students claimed that they had studied irrational numbers in class, 55% of the students believed that any decimal (finite or infinite) is the result of the division of two whole numbers, and only 19% of students believed that an infinite non-repeating decimal is not the result of the division of two whole numbers. Kidron proposes that students’ reluctance to deal with irrational numbers is due to their intuition that the existence of numbers is beyond their control, their request to define irrational numbers by means of rational numbers alone, and the need to know the process that leads to the infinite decimal expansion. This explanation is consistent with and suggests the need of task design at university level dealing with the construction of the real numbers, such as that presented in the first part of this chapter.

Conclusion

The 2016 and 2018 INDRUM meetings show some patterns in the reported task design for analysis and calculus, and suggest possible future trends in task design. A large proportion of the research in the analysis and calculus working groups considered the problem of formalisation of basic notions, most notably the construction of the real numbers, supremum, convergence of sequences, and continuity. Given that many of these papers reported initial steps in this research direction, or implementation attempts that are not yet completed, one may expect that these topics will continue to receive as much attention in the near future.

Many of the papers attending to the basic aspects of analysis were based on the Theory of Didactical Situations, although some were based on the Anthropological Theory of the Didactic, or on Douady’s (1986) tool-object dialectic. Many of the TDS and ATD papers appeared mostly in the form of epistemological analyses that precede and suggest the design of situations, or in the evaluation of implementations of such situations. Some papers were based on cognitive theories, mainly APOS and the notion of DWP (deconstructions with perspectives, Montoya Delgadillo et al., 2018), and their results suggest the need to design tasks to improve student understanding of some specific ideas. Other theoretical frameworks like Abstraction in Context (AiC) and the Mathematical Working Space model (Kuzniak,

2011) were also present. Semiotic challenges appeared in many of the reported research and this led to the frequent incorporation of Duval’s ideas (1999; 2006) within other theoretical frameworks.

In what has been a growing tendency for some time (see for example Sfard and Cobb, 2014), research considering the role of socio-cultural aspects in the design and assessment of tasks was also widely represented, and this may be expected to continue as the role of teachers, student group work, and open class discussions are further explored.

The incorporation of different types of technology in the design and implementation of student-centred tasks may be predicted to be another direction of future research growth. While the number of papers that directly dealt with the use of technology was not large, it is only natural to expect an eventual shift in that direction as the everyday use of technological artefacts by teachers and students continues to increase and the potential and advantages of using technology for the teaching and learning of mathematics is further explored, documented, and discussed within the community.

In terms of the mathematical content of the research, much attention was given to foundational and basic aspects of analysis, and relatively less attention was given to a variety of other topics in the differential and integral calculus, multivariable calculus, differential equations, complex analysis, and topology.

Research reported in this chapter considered task designs for student use or suggested the need or improvement of such designs, as when incorporating technology, for example. We also considered research on theoretical issues dealing with task assessment and, in what may be yet another growing tendency, research on calculus and analysis for students of other disciplines, including future teachers.

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