Calculus Teaching and Learning
KeywordsCalculus key concepts Intuitive representations Formal definitions Intuition of infinity Notion of limit Cognitive difficulties Theoretical dimensions Epistemological dimension Research in teaching and learning calculus Role of technology Visualization Coordination between semiotic registers Role of historical perspective Sociocultural approach Institutional approach Teaching practices Role of the teacher Transition between secondary school and university
Definition: What Teaching and Learning Calculus Is About
The differential and integral calculus is considered as one of the greatest inventions in mathematics. Calculus is taught in secondary school and in university. Learning calculus includes the analysis of problems of changes and motion. Previous related concepts, like the concept of a variable and the concept of function, are necessary for the understanding of calculus concepts. However, the learning of calculus includes new notions like the notion of limit and limiting processes, which intrinsically contain changing quantities. The differential and integral calculus is based upon the fundamental concept of limit. The mathematical concept of limit is a particularly difficult notion, typical of the kind of thought required in advanced mathematics.
There have been efforts in many parts of the world to reform the teaching of calculus. In France, for example, the syllabus changed in the 1960s and 1970s, due to the influence of the Bourbaki group. The limit concept, on a rigorous basis, has penetrated even into the secondary school curriculum: in 1972, the classical definition of the derivative as the limit of a quotient of differences was introduced. Another change occurred in the French calculus curriculum in 1982, this time influenced by the findings of mathematics education research, and the curriculum focused on more intuitive approaches. As a result, the formalization of the limit has been omitted at the secondary school level. This is the situation in most countries today: at the high school level, there is an effort to develop an initial approach to calculus’ concepts without relying on formal definitions and proofs. An intuitive and pragmatic approach to calculus at the senior level in high school (age 16–18) precedes the formal approach introduced at university.
On the university level, calculus is among the more challenging topics faced by new undergraduates. In the United States, the calculus reform movement took place during the late 1980s. The recommendation was that calculus courses should address fewer topics but in more depth, and students should learn through active engagement with the material. The standard course syllabus was revised, and new projects arose which incorporated technology into instruction. More recently, Bressoud et al. (2016) analyzed calculus curricula in France, Germany, the United States, Uruguay, Singapore, South Korea, and Hong Kong. They note the constant revision of the calculus curriculum and the way calculus is taught in secondary school and university in the different countries. They relate to the following questions: When does the teaching of calculus start in secondary school? Is it separated into different parts: a compulsory mathematics part for all students and an extended part for students who intend to pursue further studies, which require more mathematics? They also relate to the assessment process and to the following question: Is there an evaluation of theoretical aspects of the course on the exam and not only an evaluation of routine practical procedures? The integration of graphic technology was investigated as well. The authors differentiate between the form of work at the secondary school level, in which the activities are often devoted almost exclusively to calculation based on algebraic expressions, and the required form of work at university level, which includes more formal thinking.
The book by Bressoud et al. (2015) presents a report of selected findings from the Mathematical Association of America’s (MAA’s) study of Characteristics of Successful Programs in College Calculus. The report combines both large-scale survey data and in-depth case study analysis. The report concerns college and university students and highlights the very challenging environment students encounter, as they make the transition to postsecondary education, in their learning of calculus.
In most countries, the transition toward more formal approach that takes place at university is accompanied by conceptual difficulties.
Early Research in Learning Calculus: The Cognitive Difficulties
The cognitive difficulties that accompany the learning of central notions like functions, limit, tangent, derivative, and integral at the different stages of mathematics education are well reported in the research literature on learning calculus. These concepts are key concepts that appear and reappear in different contexts in calculus. The students meet some of these central topics in high school, and then the same topics appear again, with a different degree of depth, at university. We might attribute the high school students’ cognitive difficulties to the fact that the notions are presented to them in an informal way. In other words, we might expect that the difficulties will disappear when the students learn the formal definition of the concepts. However, undergraduate mathematics education research suggests otherwise. The cognitive difficulties that accompany the key concepts in calculus are well described in Sierpinska (1985), Davis and Vinner (1986), Cornu (1991), Williams (1991), and Tall (1992), as well as in the book Advanced Mathematical Thinking edited by Tall (1991). The main source of difficulty resides in the fact that many students’ intuitive ideas are in conflict with the formal definition of the calculus concepts, such as the notion of limit.
In these early studies of learning calculus, the theoretical dimensions are essentially cognitive and epistemological. The cognitive difficulties that accompany the learning of the key concepts in calculus, such as the limit concept, are inherent to the epistemological nature of the mathematics domain. In the following, we consider some facets of the dynamic interaction between the formal and intuitive representations, as they were discussed in these early studies. We encounter the first expression of the dynamic interaction between intuition and formal reasoning in the terms concept definition and concept image. For example, the intuitive thinking, the visual intuitions, and the verbal descriptions of the limit concept that precede its formal definition are necessary for understanding the concept. However, research on learning calculus demonstrates that there exists a gap between the mathematical definition of the limit concept and the way one perceives it. In this case, we may say that there is a gap between the concept definition and the concept image (Tall and Vinner 1981; Vinner 1983). Vinner also found that students’ intuitive ideas of the tangent to a curve are in conflict with the formal definition. This observation might explain students’ conceptual difficulties in visualizing a tangent as the limiting case of a secant.
Conceptual problems in learning calculus are also related to infinite processes. Research demonstrates that some of the cognitive difficulties that accompany the understanding of the concept of limit might be a consequence of the learners’ intuition of infinity. Fischbein et al. (1979) observed that the natural concept of infinity is the concept of potential infinity, for example, the non-limited possibility to increase an interval or to divide it. The actual infinity, for example, the infinity of the number of points in a segment and the infinity of real numbers as existing, as given, is, according to Fischbein, more difficult to grasp and leads to contradictions. For example, “If one looks at 1/3, it is easy to accept the equality 1/3 = 0.33… The number 0.333…represents a potential (or dynamic) infinity. On the other hand, students questioned whether 0.333… is equal to 1/3 or tends to 1/3 usually answer that 0.333…tends to 1/3.”
Among the theoretical constructs that accompany the early strands in research on learning calculus, we mention the process-object duality. The lenses offered by this framework highlight the students’ dynamic process view in relation to concepts such as limit and infinite sums and help researchers to understand the cognitive difficulties that accompany the learning of the limit concept. Gray and Tall (1994) introduced the notion of procept, referring to the manner in which learners cope with symbols representing both mathematical processes and mathematical concepts. Function, derivative, integral, and the fundamental limit notion are all examples of procepts. The limit concept is a procept because the same notation represents both the process of tending to the limit and also the value of the limit.
Research and Alternative Approaches to Teaching and Learning Calculus
Different directions of research were investigated in the last decades. The use of technology offered a new resource in the effort to overcome some of the conceptual difficulties: the power of technology is particularly important in facilitating students’ work with epistemological double strands like discrete/continuous and finite/infinite. Visualization and especially dynamic graphics were also used. Some researchers based their research on the historical development of the calculus. Other researchers used additional theoretical lenses that include the sociocultural approach, the institutional approach, or the semiotic approach. In the following sections, we relate to these different directions of research.
The Role of Technology
A key aspect of nearly all the reform projects has been the use of graphics calculators, or computers with graphical software, to help students develop a better intuitive understanding of calculus. Since learning calculus includes the analysis of changing quantities, technology has a crucial role in enabling dynamic graphical representations and animations. Technology was first incorporated as a support for visualization and coordination between semiotic registers. The possibility of computer magnification of graphs allows the limiting process to be implicit in the computer magnification, rather than explicit in the limit concept. In his plenary paper, Dreyfus (1991) analyzed the powerful role of visual reasoning in learning several mathematical concepts and processes. With introduction of the new technologies, there was a rapid succession of new ideas for use in teaching calculus. Calculus uses numerical calculations, symbolic manipulations, and graphical representations, and the introduction of technology in calculus allows these different registers. Research on the role of technology in teaching and learning calculus is described, for example, in Artigue (2006), Robert and Speer (2001), and Ferrera et al. in the 2006 handbook of research on the psychology of mathematics education (pp. 256–266). In the study by Ferrera et al., research that relates to using CAS toward the conceptualization of limit is described. For example, Kidron and Zehavi use symbolic computation and dynamic graphics to enhance students’ ability to pass from visual interpretation of the limit concept to formal reasoning. In this research, a balance between the conception of an infinite sum as a process and as an object was supported by the software. The research by Kidron, as reported in the study by Ferrera et al. (2006), describes situations in which the combination of dynamic graphics, algorithms, and historical perspective enabled students to improve their understanding of concepts such as limit, convergence, and the quality of approximation. Most studies offer an analysis of teaching experiments that promote the conceptual understanding of key notions such as limits, derivatives, and integral. For example, in a research project by Artigue (2006), the calculator was used toward conceptualization of the notion of derivative. One of the aims of the project was to enable 11th grade students to enter the interplay between local and global points of view on functional objects.
Thompson (1994) investigated the concept of rate of change and infinitesimal change, which are central to understanding the fundamental theorem of calculus. Thompson’s study suggests that students’ difficulties with the theorem stem from impoverished concepts of rate of change. In the last two decades, Thompson published several studies which demonstrate that a reconstruction of the ideas of calculus is made possible by using computing technology. The concept of accumulation is central to the idea of integration and therefore is at the core of understanding many ideas and applications in calculus. Thompson et al. (2013) describe a course that approaches introductory calculus with the aim that students build a reflexive relationship between concepts of accumulation and rate of change, symbolize that relationship, and then extend it. In a first phase, students develop accumulation functions from rate of change functions. In the first phase, students “restore” the integral to the fundamental theorem of calculus. In the second phase, students develop rate of change functions from accumulation functions. The main idea is that accumulation and rate of change are never treated separately: the fundamental theorem of calculus is present all the time. Rate is an important, but difficult, mathematical concept. Despite more than 20 years of research, especially with calculus students, difficulties are still reported with this concept.
Tall (2010) reflects on the ongoing development of the teaching and learning calculus since his first thinking about the calculus 35 years ago. Tall’s research described how the computer can be used to show dynamic visual graphics and to provide remarkably powerful numeric and symbolic computation. As a consequence of the cognitive difficulties that accompany the conceptual understanding of the key notions in calculus, Tall’s quest is for a “sensible approach” to the calculus which builds on the evidence of our human senses and uses these insights as a meaningful basis for later development from calculus to analysis and even to a logical approach in using infinitesimals. Reflecting on the many years in which reform of calculus teaching has been considered around the world and the different approaches and reform projects using technology, Tall points out that what has occurred is largely a retention of traditional calculus ideas, now supported by dynamic graphics for illustration and symbolic manipulation for computation.
The research on the role of technology in teaching and learning calculus is still developing, and, as pointed by Bressoud et al. (2016), the role of technology is generally the main theme discussed in the topic study group of learning and teaching calculus in the last three International Congresses on Mathematical Education (ICME).
The Role of Historical Perspective and Other Approaches
The idea of using a historical perspective in approaching calculus was also demonstrated in other studies, not necessarily in a technological environment. Taking into account the long way in which the calculus concepts were developed and then defined, appropriate historically inspired teaching sequences were elaborated.
Recent approaches in learning and teaching calculus refer to the social dimension, such as the approach to teaching calculus called “scientific debate,” which is based on a specific form of discussion among students regarding the validity of theorems. The increasing influence of sociocultural and anthropological approaches toward learning processes is well expressed in research on learning and teaching calculus. Even the construct concept image and concept definition, which was born in an era where the theories of learning were essentially cognitive, was revisited (Bingolbali and Monaghan 2008) and used in interpreting data in a sociocultural study. This was done in a study which investigated students’ conceptual development of the derivative, with particular reference to rate of change and tangent aspects.
In more recent studies, the role of different theoretical approaches in research on learning calculus was analyzed. Kidron (2008) describes a research process on the conceptualization of the notion of limit by means of the discrete continuous interplay. This paper reflects many years of research on the conceptualization of the notion of limit, and the focus on the complementary role of different theories reflects the evolution of this research.
The Role of the Teacher
In the previous section, different educational environments were described. Educational environments depend on several factors, including teaching practices. As mentioned by Artigue (2001), reconstructions have been proved to play a crucial role in calculus, especially at the secondary/tertiary transition. Some of these reconstructions deal with mathematical objects already familiar to students before the teaching of calculus at university. In some cases, reconstructions result from the fact that only some facets of a mathematical concept can be introduced at the first contact with it. The reconstruction cannot result from a mere presentation of the theory and formal definitions. Research shows that teaching practices underestimate the conceptual difficulties associated with this reconstruction and that teaching cannot leave the responsibility for most of the corresponding reorganization to students.
Research shows that alternative strategies can be developed fruitfully, especially with the help of technology; however, successful integration of technology at a large-scale level is still a major problem (Artigue 2010). Technology cannot be considered only as a kind of educational assistant; it was demonstrated how it deeply shapes what we learn and the way we learn it.
Artigue points out the importance of the teacher’s dimension. Kendal and Stacey (2001) describe teachers’ practices in technology-based mathematics lessons. The integration of technology into mathematics teachers’ classroom practices is a complex undertaking (Monaghan 2004; Lagrange 2013). Monaghan wrote and co-wrote a number of papers in which teachers’ activities in using technology in their calculus classrooms were analyzed, but there were still difficulties that the teachers had experienced in their practices that were difficult to explain in a satisfactory manner. Investigating the reasons for the discrepancy between the potentialities of technology in learning calculus and the actual uses in the classroom, Lagrange (2013) searched for theoretical frameworks that could help to focus on the teacher using technology; the research on the role of the teacher strengthened the idea of a difficult integration, in contrast with research which centered on epistemological and cognitive aspects. An activity theory framework seems helpful to provide insight on how teachers’ activity and professional knowledge evolve during the use of technology in teaching calculus.
The Transition Between Secondary and Tertiary Education
A detailed analysis of the transition from secondary calculus to university analysis is offered by Thomas et al. (2014). A number of researchers have studied the problems of the learning of calculus in the transition between secondary school and university. Some of these studies focus on the specific topics of real numbers, functions, limits, continuity, and sequences and series. They were carried out in several different countries (Brazil, Canada, Denmark, France, Israel, Tunisia) and use different frameworks. Some have shown that calculus conflicts that emerged from experiments with 1st-year students could have their roots in a limited understanding of the concept of function, as well as suggesting the need for a more intensive exploration of the dynamic nature of the differential calculus. Results of the survey suggest that there is some room for improvement in secondary school preparation for university study of calculus.
The transition to advanced calculus as taught at the university level has been extensively investigated within the Francophone community, with the research developed displaying a diversity of approaches and themes, but a shared vision of the importance to be attached to epistemological and mathematical analyses.
Analyzing the transition between the secondary school and the university, French researchers reflect on approaches to teaching and learning calculus in which the consideration of sociocultural and institutional practices plays an essential role. These approaches offer complementary insights into the understanding of teaching and learning calculus. The theoretical influence of the theory of didactic situations, which led to a long-term Francophone tradition of didactical engineering research, has been designed in the last decade to support the transition from secondary school calculus to university analysis.
New Directions of Research
New directions of research in teaching and learning calculus were investigated in the last decades. We observe the need for additional theoretical lenses, as well as a need to link different theoretical frameworks in the research on learning and teaching calculus. In particular, we observe the need to add additional theoretical dimensions, such as the social and cultural dimensions, to the epistemological analyses that were done in the early research. It is important to note that the “new” theoretical dimensions do not replace the cognitive and epistemological theoretical approaches that dominated the early research. These early theoretical constructs are necessary and coexist with additional theoretical lenses offered by different theories. In some cases, we notice the evolution of research in the course of many years, with the same researchers facing the challenging questions concerning the cognitive difficulties in learning calculus. The questions are still challenging, and the researchers use different theoretical frameworks in their research. For example, González-Martín et al. (2014) use the theory of didactic situations to analyze research cases from the study of calculus. The authors discuss the roles of the students and the teacher and the use of epistemological analyses. In one of these research cases, González-Martín et al. (2014, p. 125) analyze an activity that “fosters an epistemological change in students’ conception, allowing them to consider real numbers as conceptual objects in relation to other objects- i.e., limits- within a mathematical theory.” In the last decade, we also note discursive approaches into research, including studies using the commognitive framework for the analysis of teachers’ and students’ discursive practices in calculus courses. The commognitive framework, with its hybrid term “commognition,” emphasizes the interrelatedness of “cognition” and “communication.” Nardi et al. (2014) used the commognitive approach in three studies which explore fundamental discursive shifts often occurring in the early stages of studying calculus. They illustrate, for example, the variation of discursive patterns in practices that can be perceived initially as quite similar – as in the case of introductory calculus lectures during which they observed the construction of the object of function. More examples of theoretical frameworks used in research in teaching and learning calculus are described in Bressoud et al. (2016).
The theoretical dimension is essential for research on calculus teaching and learning, but we should not neglect practice. As pointed out by Robert and Speer (2001), there are some efforts being made toward a convergence of theory-driven and practice-driven researches. More recent studies describe research on how to consider meaningfully theoretical and pragmatic issues. Biza et al. (2016) describe the increasing interest in teaching practices at university level. The authors explore the influence of teachers’ perspectives, background, and research practices on their teaching, as well as the role of resources and mathematics professional development in teaching. In the study by Bressoud et al. (2015) of characteristics of successful programs in calculus, we read how some universities coordinate calculus instruction and foster a community of practice around the teaching of calculus.
As mentioned earlier, reconstructions have been proved to play a crucial role in calculus, essentially these reconstructions that deal with mathematical objects already familiar to students before the teaching of calculus. Further research should underline the important role of teaching practices in successful reorganization of previous related concepts toward the learning of calculus.
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