The process of learning algebra should ideally teach students good logic skills, the ability to compare and contrast circumstances, and to recognize patterns and make predictions. In a world with free CAS at our fingertips, the focus on these underlying skills is even more important than it used to be. Learn how to focus on thinking skills and incorporate more active learning in algebra classes, without losing ground on topic coverage.

1. Maria H. Andersen Muskegon Community College AMATYC 2008, Washington DC

3. The problems of algebra

4. The creation of a classroom resourceMaria H. Andersen Muskegon Community College AMATYC 2008, Washington DC

5. I have been thinking about how to teach math for 30 years.

6. There are many problems we face in Algebra.

7. Let’s hear them.

8. Problem #1

9. Student conceptions’ of math are fragmented, especially at the lower levels.

10. Because of this, most students only pursue surface-level learning.

11. Being a naturally curious person … I started reading and experimenting.

12. To really understand this problem, we have to get to know the brain a little better.

13. We’re going to take a little side course in Cognitive Theory.

14. Schema(pl. schemata): a mental representation of what all instances of something have in common

15. Possible student schemata for factoring GCF _ ( ) ( )( )=0 x=_ or x=_ Trinomial ( )( ) _ + _ + _ + _ _ ( ) + _( )

16. Schemata categorize your experiences.

17. Schemata help you remember and comprehend what you are experiencing.

18. Schemata are important in developing the ability to problem solve. GCF _ ( ) ( )( )=0 x =_ or x =_ _ + _ + _ + _ _ ( ) + _( ) Trinomial ( )( )

19. Let’s look at an example …

20. what happens?

21. the problem here is that the students ran the wrong “script”

23. Three terms?

25. what just happened?

26. the students “ran” the trinomial factoring script

31. So, why did the students struggle so much with this problem?

32. Not enough factoring schemata. GCF _ ( ) ( )( )=0 x=_ or x=_ _ + _ + _ + _ _ ( ) + _( ) Trinomial ( )( ) ax2 + _ + c _ + _ + _ + _

33. Successful problem solvers have a large variety of flexible schemata.

34. Many math students don’t develop this, they simply build one schema and apply it liberally.

35. For example, there’s the “distribution” schema.

36. 5 (x + 2)

37. 5 (x + 2) f (x + 2) Uh oh!

38. 5 (x + 2) f (x + 2) log(x + 2) Uh oh!

39. 5 (x + 2) f (x + 2) log(x + 2) | x + 2 | Uh oh!

40. (xy)2

41. (xy)2 (x + y)2 Uh oh!

42. (xy)2 (x + y)2 x + y Uh oh!

43. We need to realize that we recall schemata as images, not text.

44. See if you can think of some schemata for learning about lines.

46. Students need to practice recognizing and categorizing what they learn into schemata.

47. We need to bring student schemata up against situations that refine them.

48. We need to bring student schemata up against situations that refine them. abstraction gist-extraction interpretation

49. Encountering objects in new ways revises and refines schemata.

50. Remember this? How can we change the approach in a way that helps students revise and refine their schemata?

51. Why does this approach help to revise schemata?

52. Initial script: If the problems are different, then the answers are different.

53. Learners need to have multiple encounters with objects or events in different ways

54. Learners need to have multiple encounters with objects or events in different ways

55. that’s not repetition per se

56. Repetition

57. Refining

58. Abstraction

59. the gist-extraction process revises your actual experience in order to store the memory

60. Let’s say I teach my students to multiply polynomials in the following way.* *I don’t do it this way now, but let’s just use our imaginations.

64. After the student leaves class, what do they actually remember?

65. gist of today’s class: If you have to multiply polynomials, use FOIL (first, outer, inner, last).

66. Even worse, an interpretation has been made: All multiplication involves FOIL.

67. interpretation: filling in things that were not said or seen

69. One common interpretation:

70. Knowing about the gist-extraction and interpretation that will happen when the students walk out the door … we should change tactics.

75. Now … after the student leaves class, what do they actually remember?

76. What images or visual cues do you think the students store with this schema?

77. Thinking along these lines, I started writing activities.

82. At some point, I made the mistake of opening my mouth at a textbook focus group.

83. Then, I wrote a LOT of activities.

94. 11 pounds, 1 ounce

95. The aftermath

97. Problem #2

98. Math instructors are oriented towards Information-transfer Teacher-focused learning (ITTF)

99. Students are often frustrated … or bored.

100. I began recording all my lectures.

101. And then I realized that we have to change the experiences in our math classrooms.

102. So I did.

104. concept-centered student-focused (CCSF)

107. students teaching students

108. Problem #3

109. Many algebra instructors walk into their first classroom with zero expertise and almost no preparation about how to teach algebra.

110. I had 48 hours notice before I began teaching my first Beginning Algebra class.

111. At 2-year colleges, part-time faculty teach 44% of the courses and make up approximately 65% of the faculty. Source: 2005 CBMS Survey

112. Ideally, every new math instructor would get a mentor.

113. Instructor tips.

114. Instructor tips.

115. Instructor tips.

118. Metacognitive Skills

120. 5 Bring it on!!! 4 I know I will pass, grade? Not sure. 3 Well, we’ll see how it goes. 2 I think I should have attended more class. 1 Not a clue.

123. Problem #4

124. CAS Computer Algebra Systems

125. CAS is now available on any computer with Internet … for free.

126. Have you seen this?

137. Did I mention this is free?

138. and on every computer that has Internet access?

140. Click here to see the graph.

143. So what do we do with this?

147. You might write statements something like this … “When the expressions are like this: ______ , the factored forms seem to be similar in that __________ . Then provide problems and answers as support for your conjectures.

148. Do we change the content of algebra because of CAS? I don’t know, but it’s probably not going to be pretty.

150. Do we change the way we teach in the classroom? We’d better.

151. Maria H. Andersen Muskegon Community College AMATYC 2008, Washington DC

152. Thank you to my illustrator, Mat Moore, who is the best algebra illustrator ever!

153. Questions? Sample Activities and more info at: www.cengage.com/community/mariaandersen