The Fundamental Theorem of the Calculus

1. The Fundamental Theorem of the Calculus Mike Thomas The University of Auckland

2. Overview • Why is the fundamental theorem of calculus (FTC) important?? • Brief theory • Using Geogebra to build understanding of the FTC • Classroom materials?

3. Integration • This is often introduced by using an antiderivative to calculate areas. • The question is how do we persuade students that these two processes, antidifferentiation and finding area under graphs by Riemann sums are related in this way?

4. The Fundamental Theorem of Calculus I • If f is continuous on [a, b], then the function F defined by • is continuous on [a, b] and differentiable on (a, b), and

5. The Fundamental Theorem of Calculus II • If f is a function continuous on [a, b], and F is an antiderivative of f then • This is what is usually used.

6. The Three Worlds of Mathematics The symbolic world is where the symbolic representations of concepts are acted upon, or manipulated, where it is possible to “switch effortlessly from processes to do mathematics, to concepts to think about.” (Tall, 2004a, p. 30). • Embodied • Symbolic • Formal The embodied is where we make use of physical attributes of concepts, combined with our sensual experiences to build mental conceptions. The formal world is where properties of objects are formalized as axioms, and learning comprises the building and proving of theorems by logical deduction from the axioms.

7. Tall’s 3 worlds of thinking

8. Tall’s 3 worlds of thinking

9. A conclusion • A curriculum that focuses on symbolism and not on related embodiments may limit the vision of the learner who may learn to perform a procedure, even conceive of it as an overall process, but fail to be able to imagine or ‘encapsulate’ the process as an ‘object’. Tall, 2008, pp. 11, 12

10. An aim—versatile thinking • representational versatility—the ability to work seamlessly within and between representations, and to engage in procedural and conceptual interactions with them; • process/object versatility—the ability to switch at will in any given representational system between a perception of a mathematical entity as a process or an object; • visuo/analytic versatility—the ability to exploit the power of visual schemas by linking them to relevant logico/analytic schemas.

11. What does the graph of the derivative look like?

12. Method easy

13. But what does the antiderivative look like? How would you approach this? Versatile thinking is required. But what if the graph is more difficult?…

14. Application or building-with

17. Geogebra • Zero • One • Two • Three • Four • Five

18. f(x) D(x) D(q) D(p) A(x) f(q) f(p) A(q) A(p) p q Task 3 Diagram

19. The Fundamental Theorem of Calculus I • If f is continuous on [a, b], then the function F defined by • is continuous on [a, b] and differentiable on (a, b), and

20. The Fundamental Theorem of Calculus II • If f is a function continuous on [a, b], and F is an antiderivative of f then • This is what is usually used.

21. Use of the FTC • We define • Why? • So that by the FTC. • Can we show this using Geogebra?

23. Working with a graph What does the antiderivative look like?

24. Model Development Sequence: Antiderivative Model Eliciting Activity Exploration Activity Exploration Activity Adaptation Activity y x Design a method that Amit and Becky can use to solve problems like the one they found in the textbook. Your method needs to work not only for the textbook problem, but also for similar problems of its kind, like the ones shown on the next page. Write a letter to Amit and Becky in which you (1) describe your method, (2) explain why it works, and (3) show how to use your method to solve the textbook problem and the problems below.

25. Model Development Sequence: Antiderivative Model Eliciting Activity Exploration Activity Exploration Activity Adaptation Activity Problem Statement To celebrate their 40th wedding anniversary, Helen and Brendan O’Neill are planning a tramp with their children and grandchildren. The local park provided a Gradient Graph for a nearby 5 kilometre tramp, but the O’Neills want to make sure it is suitable for them. Helen wants to know if there is a summit where they can have lunch and enjoy the view, while Brendan wants to know where the tramping gets difficult. Gradient graph of tramping track

26. Bruner’s modes of representation of knowledge • These are internal • Enactive–personal physical actions • Iconic–concrete images, diagrams and pictures • Symbolic–accepted or arbitrary use of signs

27. Development of thinking

32. Point of inflection, m

35. Their solution Link to algebra

36. APOS and the 3 worlds