1. Behaviorism and Constructivism
What Does it Look Like in a Classroom?
Master 2-1: Behaviorism and Constructivism

2. Behaviorism Behavior can be shaped through reinforcement.
Practice promotes desired behavior.
Edward L. Thorndike, B.F. Skinner, Robert Gagne
Master 2-1: Behaviorism and Constructivism

3. Constructivism I
Learners actively create or invent (construct) their own knowledge.
 Students create (construct) new mathematical knowledge by reflecting on their physical and mental actions.
Master 2-1: Behaviorism and Constructivism

4. Constructivism
Learning reflects a social process in which children engage in dialogue and discussion with themselves as well as others as they develop intellectually.
William Brownell, Jean Piaget, Jerome Bruner, Zoltan Dienes
Master 2-1: Behaviorism and Constructivism

5. Similarity in Learning Frameworks
Active involvement, communication
Several characteristics and stages of thinking; Children progress through stages as they mature
Learning proceeds from concrete to abstract
Symbols follow conceptualization and meaning
Master 2-2: How Children Learn

6. Procedural Knowledge vs. Conceptual Knowledge
Procedural Knowledge-based on a series of actions, rules,and algorithms
Conceptual Knowledge-based on meaningful connections among existing and new concepts
Master 2-3: Procedural and Conceptual Knowledge
Adding is increasing
divide…then multiply

7. Which is it?
To divide 23 candies among four friends, Steve knows each must receive an equal amount and there may be some left.
To take 23 divided by 4, Steve knows to take 5 x 4 and subtract the result from 23.
Master 2-3: Procedural and Conceptual Knowledge

8. I have painted 6 balls, only one more to go.
Which is it?
Sara knows when counting, 7 follows 6.
Sara knows 7 represents 7 objects.
I have painted 6 balls, only one more to go.
Master 2-3: Procedural and Conceptual Knowledge

9. Which is it?
Joe knows that to find 25% of a price he can cut the price in half, then half again to find one-fourth.
 Joe knows that to find 25% of a price he can multiply the price by .25.
Master 2-3: Procedural and Conceptual Knowledge
…to find 25% I must…

10. Which is it?
Nancy knows that to find the area of a rectangle, she must find out how much space it covers.
Nancy knows that to find the area of a rectangle, she must multiply the length times the width.
Master 2-3: Procedural and Conceptual Knowledge
I must multiply…

11. Principles for Learning Mathematics I
Actively involve students.
Learning is developmental.
Build on previous learning.
Communication contributes significantly to understanding.
Good questions facilitate learning
Master 2-4: Eleven Learning Principles

12. Principles for Learning Mathematics II
Manipulatives aid learning.
Metacognition affects learning.
Teachers' attitudes are vital.
Experiences influence anxiety.
Gender aptitudes are equal.
Retention can improve.
Master 2-4: Eleven Learning Principles

13. The Staircase Problem I
Examine these staircases:
Describe in words a relationship (formula) involving the sum of the first 4 counting numbers suggested.
Master 2-5: The Staircase Problem

14. The Staircase Problem II
Examine these staircases:
Describe in words a relationship (formula) suggested. How many counting numbers are involved? What is their sum?
Master 2-5: The Staircase Problem

15. The Staircase Problem III
Examine these staircases:
The sum of the first n counting numbers is: n = _________
Master 2-5: The Staircase Problem