Course Identification

The goal of this course is for participants to become acquainted with research on learning calculus for developing a high school calculus curriculum that focuses on the understanding of the key concepts of calculus.

Characteristics of calculus curricula in different countries will be presented.

Course participants will read and discuss research studies about students’ conceptualization of calculus notions. 40 years of research studies will be discussed. The cognitive difficulties that accompany the learning of the key concepts in calculus (real numbers, functions, limit, tangent, derivative and integral) will be analyzed with the help of the different theoretical constructs that underlie the research on learning calculus. Dynamic interaction between formal and intuitive representations will be discussed. It includes conceptual problems in learning calculus that are related to infinite processes, learners’ intuitions of infinity and the interplay between the learner’s intuitive and logical thinking as well as construction of knowledge that results from and enables progress of this interplay. As a part of this interplay, reasoning in calculus by means of analogy  will be analyzed.

Course participants will read and discuss research studies on alternative approaches to teaching and learning calculus. These approaches include the use of the historical development of the calculus and the use of technology in the effort to overcome some of the conceptual difficulties. The power of technology to facilitate students’ work with epistemological double strands like discrete/ continuous and finite/ infinite will be analyzed as well as the use of visualization and dynamic graphics in the learning of convergence processes.

The discussion of research studies will offer opportunities for participants to reflect on their own conceptualization of the key concepts of calculus. They will be ready to design worksheets for a curriculum that offers high school students an opportunity to better conceptualize the central notions of calculus.

Upon successful completion of the course students will be able tol:

1. Sharpen their own conceptualization of the central notions of calculus
2. Gain insight into high school students’ cognitive difficulties that accompany the learning of calculus and be aware of approaches that may help overcome some of the cognitive difficulties.
3. Design relevant material towards developing a high school calculus curriculum focused on conceptual understanding.

Artigue, M. (1996). Teaching and learning elementary analysis. In C. Alsina et al, 8^th international congress on mathematics education, selected lectures, pp.15-30. Sevilla: SAEM Thales.

Artigue, M. (2010). The Future of Teaching and Learning Mathematics with Digital Technologies. In C. Hoyles, J.B. Lagrange (Eds.), Mathematics Education and Technology – Rethinking the Terrain. The 17th ICMI Study. Springer, pp. 463-476.

Bressoud, D., Ghedamsi, I., Martinez-Luaces, V., Torner, G. (2016). Teaching and learning of calculus. In: Teaching and learning of calculus, ICME- 13 Topical Surveys, Springer, Cham, pp 1–37

Fischbein, E., Tirosh, D., & Hess, P. (1979). The intuition of infinity. Educational Studies in Mathematics, 10, 3–40.

Kidron, I. (2003). Polynomial approximation of functions: Historical perspective and new tools. The International Journal of Computers for Mathematical Learning,8, 299–331.

Kidron, I. (2011). Tacit models, treasured intuitions and the discrete-continuous  interplay.  Educational Studies in Mathematics 78(1), 109-126. Springer.

Kidron, I. (2019). Calculus Teaching and Learning . In Lerman S. (ed.) Encyclopedia of Mathematics Education, 69-75. Springer, Cham.

Kidron, I. & Tall, D. (2015). The roles of visualization and symbolism in the potential and actual infinity of the limit process. Educational Studies in Mathematics 88(2),183-199. Springer.  

Tall, D. (1992). The transition to advanced mathematical thinking: Functions, limits, infinity and proof. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp.495-511). New York: Macmillan.

Tall, D., & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limit and continuity. Educational Studies in Mathematics, 12, 151–169.

Vinner, S. & Dreyfus, T. (1989). Images and definitions for the concept of function. Journal for Research in Mathematics Education, 356-366.

Readings will also be assigned from the books:

Fischbein, E. (1987). Intuition in science and mathematics: an educational approach.  Dordrecht: Reidel

Tall, D. (Ed.). (1991). Advanced mathematical thinking. Dordrecht, The Netherlands: Kluwer.