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# Algebra Is Weightlifting For The Brain

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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 …

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.

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### Transcript

• 1. Maria H. Andersen
Muskegon Community College
AMATYC 2008, Washington DC
• 2. This is a talk with three storylines …
• My story
• 3. The problems of algebra
• 4. The creation of a classroom resource
Maria H. Andersen
Muskegon Community College
AMATYC 2008, Washington DC
• 5. I have been thinking
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 …
• 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”
• 22. Faulty Student script:
Problem says factor
• Squared term in front?
• 23. Three terms?
• 24. Run trinomial factoring script.
• my hint
• 25. what just happened?
• 26. the students “ran” the trinomial factoring script
• 27.
• 28.
• 29. Student script:
Problem says factor
• Four terms?
• 30. Run factor by grouping 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.
• 45.
• 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.
• 61.
• 62.
• 63.
• 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,
All multiplication involves FOIL.
• 67. interpretation: filling in things that were not said or seen
• 68.
• 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.
• 71.
• 72.
• 73.
• 74.
• 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.
• 78.
• 79.
• 80.
• 81.
• 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.
• 84.
• 85.
• 86.
• 87.
• 88.
• 89.
• 90.
• 91.
• 92.
• 93.
• 94. 11 pounds, 1 ounce
• 95. The aftermath
• 96.
• 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.
• 103.
• 104. concept-centered
student-focused
(CCSF)
• 105.
• 106.
• 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.
• 116.
• 117.
• 118. Metacognitive Skills
• 119.
• 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.
• 121.
• 122.
• 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?
• 127.
• 128.
• 129.
• 130.
• 131.
• 132.
• 133.
• 134.
• 135.
• 136.
• 137. Did I mention this is free?
• 138. and on every computer
that has Internet access?
• 139.
• 141.
• 142.
• 143. So what do we do with this?
• 144.
• 145.
• 146.
• 147. You might write statements something like this …
“When the expressions are like this: ______ , the factored forms seem to be similar in that __________ .