1. Intern: Mrs. Linda Anderson Supervisor: Dr. Douglas Lapp MTH 261 PROBLEM-BASED ALGEBRA AND CALCULUS FOR SECONDARY TEACHERS

2. WHY DO I NEED TO KNOW THIS?!? Source: http://itsgettinghotinhere.files.wordpress.com/2009/06/science_questioning2.gif

3. WHEN DID YOU FIRST START LEARNING TO DRIVE A CAR? http://superguilho.blogspot.com/2011/07/electric-porsche-911-tries-to-show.html http://www.youtube.com/watch?v=mYGusCFBt40&feature=player_detailpage

4. YOU NEED TO KNOW THIS SO YOU CAN… Source: http://forrestermaths.files.wordpress.com/2011/12/maths-image.jpg …understand BIG concepts so that you may better explain them …connect your graduate studies to what you could be teaching …continue to learn

5. The Concept of Inverse, Dr. Douglas Lapp (2008), Unit 1: Connecting Mathematical Concepts: Secondary to Undergraduate, pg 1.

6. CONCEPT MAP

7. , Dr. Douglas Lapp (2008), Unit 1: Connecting Mathematical Concepts: Secondary to Undergraduate, pg 2.

8. CONCEPT MAP – HOW TO GET STARTED Start with your BIG topic: Inverse function What are the concepts, items, or descriptive words that you can associate with this idea? Use a top down approach, working from general to specific or use a free association approach by brainstorming nodes and then develop links and relationships. Source: http://www.graphic.org/concept.html

9. WHY is 0 considered the identity element for addition and 1 considered the identity element for multiplication?

10. Properties of Real Numbers FORM AND FUNCTION ACTIVITY

13. WHAT DOES IT MEAN FOR A SET TO BE CLOSED?

14. GROUP Closed Associative Identity Inverse May or may not be commutative

15. SUBGROUP Closed Associative Identity Inverse

16. THE TEACHING AND LEARNING OF ALGEBRA

17. We believe… Mathematical Content Knowledge and Pedagogical Content Knowledge need to develop simultaneously This means... Integrate Mathematical Learning with the Theories and Practice for Teaching

18. THEORETICAL PERSPECTIVE Pirie and Kieren Model Framework to Study the Growth of Understanding Has layers like an “onion”

20. All of the learners relevant prior knowledge

21. Pirie,S & Kieren, T. (1994) Growth in mathematical understanding: How can we characterise it and how can we represent it ? Educational Studies in Mathematics, 26 (2-3), 165-190.

22. Learners engage in specific activities aimed at helping them to develop particular ideas/images for a concept

23. No longer tied to actual activities. The learner has interiorized the action.

24. Notice properties about these images

25. Begin to make “for all…” statements

26. Start developing theories

27. Full understanding

28. FOLDING BACK Source: http://www.bing.com/images/search?q=knowledge&view=detail&id=6B7BA302D9D88A7F2E264A21E571D3B2E3D5DD05

29. FOLDING BACK Occurs when a new situation requires the learner to revisit earlier images and understanding to inform their new thinking The learner reflects on their current and previous understanding The previously held image may need to be modified or broadened (creates deeper understanding)

31. QUADRATIC EQUATIONS

32. What sort of Primitive Knowledge would you expect is needed to learn about quadratic equations?

33. What sort of activities could you do to help students develop and image for quadratic equations?

34. What would be an example that would show you a learner is at the image having level for understanding about quadratic equations?

35. What sort of properties would you expect learners to notice when exploring quadratic equations?

36. Where might folding back occur when learning about quadratic equations? Give an example.

37. THEORETICAL PERSPECTIVE Pirie and Kieren Model Framework to Study the Growth of Understanding Dynamic Leveled, but not linear Recursive

38. Reaction to Vignette Revisited 4. Folding Back

39. Reaction to Vignette Revisited How would you have dealt with Peter’s comment? “I don’t get it – Why do we have to get x?”

40. INVERSES AND THE COMMON CORE STATE STANDARDS F-BF.4. Find inverse functions. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) =2 x 3 or f(x) = (x+1)/(x–1) for x ≠ 1. (+) Verify by composition that one function is the inverse of another. (+) Read values of an inverse function from a graph or a table, given that the function has an inverse. (+) Produce an invertible function from a non-invertible function by restricting the domain.

41. Suppose you are given the following algorithm: Start with a number, add 5 to it. Divide the result by 3 Subtract 4 from that quantity Double your result. The final result is 10. Working backward, find the original number.

47. All of the learners relevant prior knowledge

48. Learners engage in specific activities aimed at helping them to develop particular ideas/images for a concept

49. No longer tied to actual activities. The learner has interiorized the action.

50. Notice properties about these images

51. Begin to make “for all…” statements

52. Start developing theories

53. Full understanding

60. SIDE-BY-SIDE COMPARISON

61. MAPPING

62. 4. MAPPING NOTATION

63. 6/7. CYCLE NOTATION As you saw in the lab, cycles can be represented in several ways meansor Mapping Notation Array Notation Cycle Notation

64. CYCLE NOTATION What if S = {1, 2, 3, 4} How would you write: In cycle notation?

65. CYCLE NOTATION What do you think means? We usually write cycles as disjoint cycles. If they are not disjoint, one action follows the other.

66. MAPPING BETWEEN

69. HOMOMORPHISM In #9, you explored compositions under the mapping from T to M More specifically, This is called operation preserving, more formally….

70. In #9a you were asked, “What two properties of functions must be true to say ‘every element in one world is mapped to exactly one element on the other world and no element in any world goes unmapped’?”

71. ONE-TO-ONE A function mapping T to M is called one-to-one (1-1) if whenever φ (a) = φ (b), a = b

72. NOT ONE-TO-ONE A function mapping T to M is called one-to-one (1-1) if whenever φ (a) = φ (b), a = b

73. ONTO A function mapping T to M is called onto if for all m in M, there exists a t in T such that φ ( t ) = m [All of M is used]

74. NOT ONTO A function mapping T to M is called onto if for all m in M, there exists a t in T such that φ ( t ) = m [All of M is used]

76. A HOMOMORPHISM That is one-to-one (injective) and onto (surjective) is called an ISOMORPHISM