Introduction Mean Value Theorem Taylor Series L'Hopital's Rule

Our aim here is to show how the framework of the three worlds of mathematics which consists of the embodied world, the symbolic world, and the formal world, can be used as an effective tool in better understanding of learning Calculus using three selected topics of Calculus, i.e., Mean Value Theorem, Taylor Series and L'Hopital's Rule.
  1. The embodied world consists of graphs, diagrams, and their properties.
  2. The symbolic world contains operations, formulas, and calculations.
  3. The formal world consists of axioms, formal definitions, and formal proofs.


These questions are of the following types:
  1. Multiple choice tests
  2. Match the items
  3. Visualize the graph of the given function using the animation and the sliding bar

Learning Outcomes

  1. To learn to move from one world to another and between representations within one world, which is important for having a rich understanding of Calculus and for finding key insights.
  2. To provide line-by-line details of learning Calculus using a person-and-computer learning mode.
  3. To determine the truth of a theorem based on the three worlds and comparing constructions across the worlds.


Tall, D. (2004). Thinking through three worlds of mathematics. In M. J. Hoines & A. B. Fuglestad (Eds.), 28th Conference of the international group for the psychology of mathematics education. vol. 4 (pp. 281-288). Bergen.
J. C.-F. WONG. Calculus for Enginner, Lecture Note, 2012.