How we think about things… Types of Reasoning

We See the Same Thing Differently

What Do We See?

Shapes?

Symmetry?

Repetition?

Lines?

Curves?

Stripes?

Squares & Cubes?

Rectangles & Prisms?

Stars?

Ovoids?

Spheres?

Parallels?

Spirals?

Cones?

Fractals?

Colors?

Any Others??? That You Know?

How Do Our Minds Process This?

Inductive Inductive Deductive Deductive Proportional Proportional Spatial Spatial 4 Types of Reasoning

Inductive Reasoning Mathematically Defined As: Making decisions by generalizing specific examples or patterns …n

Deductive Reasoning Mathematically defined as: Making decisions based on true statements by employing the rules of logic.

Inductive & Deductive Specific Observations General Idea Broad Generalization Specific Confirmation

Proportional Reasoning Part to Part Part to Whole Whole to Part

Spatial Reasoning 7 Types Eye-Motor Coordination Figure-Ground Perception Perceptual Constancy Position-In-Space Perception Perception of Spatial Relationships Visual Discrimination Visual Memory

Eye-Motor Coordination “Ability to coordinate vision with movement of the body” (p. 14). Students weak here so busy with working their pencil, they miss the lesson. Activities requiring EMC: – Staying within the lines – Tracing – Connecting dots

Figure-Ground Perception “Visual act of identifying a specific component in a situation and involves shifts in perception of figures against complex backgrounds…” (p. 15). Bouncing a ball and only “seeing” the ball Activities requiring FGP: – Identify figure among overlapping ones – Complete a figure – Assemble a figure from its parts

Perceptual Constancy “Involves the recognition of certain geometric figures presented in a variety of sizes, shadings, textures, and positions in space and discrimination from similar figures” (p. 15). Piaget – conservation of shape – Age 4 – 5 not a square – Age 6 – 7 same piece, but no longer a square – Age 8 – 9 both are squares Activities requiring PC: – Identifying similar figures – Ordering objects by size – Identifying congruent figures

Position-In-Space Perception “The ability to relate an object in space to oneself” (p. 17). Children see themselves as the center of all things – all things are above, below, behind, or in front of them Problems with PISP mean problems reading and writing as well as in math Activities requiring PISP: – Drawing congruent figures – Identifying flip, slide, and turn images b d p q

Perception of Spatial Relationships “The ability to see two or more objects in relation to oneself or in relation to each other” (p. 17). Closely related to PISP Requires strong sense of body orientation Activities requiring POSR: – Judging distances of things around you – Playing ball – Riding a bicycle

Visual Discrimination “The ability to identify similarities and differences between or among objects” (p. 18). Independent of position Activities requiring VisD: – Sorting and classifying objects – Sorting and classifying geometric shapes – Working with attribute blocks

Visual Memory “The ability to recall accurately objects no longer in view and relate their characteristics to other objects either in view or not in view” (p. 18). Extreme case: Photographic memory Most people can remember 5 – 7 objects for short periods of time Activities requiring VM: – Briefly seeing a picture and then listing all of the things seen – Copying a figure seen briefly on a Geoboard to dot paper

“Thinking about Thinking” Promoting Metacognition in Mathematics Student Recognition of Purpose in Lessons Center for Critical Thinking Sonoma, CA

The End

References Center for Critical Thinking. Sonoma, CA. Del Grande, J. (February, 1990). Spatial sense. Arithmetic Teacher, Dr. Leslie S. Jones, Biology Department, Valdosta State University