1. GUIDED EXPLORATION Concrete methods that make learning more concrete Discovery methods that make learning more active Inductive methods that make the learning task more familiar

2. BRUNER’S THEORY OF COGNITIVE DEVELOPMENT: MODES OF REPRESENTATION Enactive Representation - using actions to represent information, dominant mode of representation in the sensori- motor period (0 to 2 yrs), “thought is action” Iconic Representation - using visualization to represent information, dominant mode of representation during the preschool years, Piaget’s conservation tasks are good examples Symbolic Representation - using language or other symbols to represent information, middle childhood and beyond

3. CLASSIC STUDY: BRUNER’S SCREENING STUDY Participants: 5 to 8 year old children who clearly failed the conservation of liquid substance task. Step 1: Screen is placed in front of the beakers before the liquid is poured.

4. CLASSIC STUDY: BRUNER’S SCREENING STUDY Step 2: Blue liquid is poured from one of the tall beakers to the wide beaker. Step 3: Child is asked whether the liquid is still the same amount. Results: When children do not have the misleading visual information, they say “It’s the same, you only poured it”

5. DIFFERENT MODES OF REPRESENTATION 2 + 3 = ? 4 + 7 = ? 4 - 2 = ? 5 - 3 = ? SYMBOLIC MODE 9 + 7 = ? 12 + 24 32 - 17 5 x 6 = ? 2 + ? = 7 5 x ? = 45 ICONIC MODE 3 x 5 = ?

6. TEACHING POSITIONAL NOTATION Array of Beads - Concrete Representation Color Coded Labels - Intermediate Step 1 0 02 07 Superimposed Labels 1 2 7

7. ADDITION WITH CARRYING Color Coded Squares - Intermediate Step 10010 100 + Short cut Algorithm 1 2 4 6 + 1 2 7 3 7 3 10 111 111 111 111 1 111 100

8. USING CONCRETE MATERIALS TO TEACH THE UNDERLYING STRUCTURE OF QUADRATIC EQUATIONS Example: (2x + 1) (x + 2) XX1 X 1 1 X2X2 X2X2 X X X 2X 2 + 5X + 2 X X 1 1

9. IMPLICATIONS OF CONCRETE METHODS Research findings are mixed on the usefulness of concrete materials Students must also reflect on the underlying principle, perhaps through discussions with peers Once a learner grasps a concept, then introduce drill & practice to ensure increased efficiency Computer simulation programs are a good way to help students build “situation models” (e.g., ant problem)