How we think about things…How we think about things…

Types of ReasoningTypes of Reasoning

We See the Same Thing DifferentlyWe See the Same Thing Differently

We See the Same Thing DifferentlyWe See the Same Thing Differently

What Do We See?What Do We See?

Shapes?Shapes?

Symmetry?Symmetry?

Repetition?Repetition?

Lines?Lines?

Curves?Curves?

Stripes?Stripes?

Squares & Cubes?Squares & Cubes?

Rectangles & Prisms?Rectangles & Prisms?

Stars?Stars?

Ovoids?Ovoids?

Spheres?Spheres?

Parallels?Parallels?

Spirals?Spirals?

Cones?Cones?

Fractals?Fractals?

Colors?Colors?

Any Others???Any Others???

That You Know?That You Know?

How Do Our MindsHow Do Our Minds

Process This? Process This?

•InductiveInductive•DeductiveDeductive•Proportional Proportional •SpatialSpatial

4 Types of Reasoning4 Types of Reasoning

Inductive ReasoningInductive Reasoning

Mathematically Defined As:Mathematically Defined As:Making decisions by generalizing specific examples or patternsMaking decisions by generalizing specific examples or patterns..

88

1313

1818

2323……

nn

Deductive ReasoningDeductive Reasoning

Mathematically defined as:Mathematically defined as:Making decisions based on true statements Making decisions based on true statements

by employing the rules of logicby employing the rules of logic..

Inductive & DeductiveInductive & DeductiveSpecific ObservationsSpecific Observations General Idea General Idea

Broad GeneralizationBroad Generalization Specific Confirmation Specific Confirmation

Proportional ReasoningProportional Reasoning

Part to Part Part to Part Part to Whole Part to Whole Whole to Part Whole to Part

Spatial ReasoningSpatial Reasoning7 Types7 Types

Eye-Motor CoordinationEye-Motor CoordinationFigure-Ground PerceptionFigure-Ground Perception

Perceptual ConstancyPerceptual ConstancyPosition-In-Space PerceptionPosition-In-Space Perception

Perception of Spatial RelationshipsPerception of Spatial RelationshipsVisual DiscriminationVisual Discrimination

Visual MemoryVisual Memory

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

–Staying within the linesStaying within the lines–TracingTracing–Connecting dots Connecting dots

Figure-Ground PerceptionFigure-Ground Perception

•““Visual act of identifying a specific Visual act of identifying a specific component in a situation and involves component in a situation and involves shifts in perception of figures against shifts in perception of figures against complex backgrounds…” (p. 15).complex backgrounds…” (p. 15).•Bouncing a ball and only “seeing” the Bouncing a ball and only “seeing” the ballball•Activities requiring FGP:Activities requiring FGP:

–Identify figure among overlapping onesIdentify figure among overlapping ones–Complete a figureComplete a figure–Assemble a figure from its partsAssemble a figure from its parts

Perceptual ConstancyPerceptual Constancy•““Involves the recognition of certain geometric Involves the recognition of certain geometric figures presented in a variety of sizes, shadings, figures presented in a variety of sizes, shadings, textures, and positions in space and textures, and positions in space and discrimination from similar figures” (p. 15).discrimination from similar figures” (p. 15).•Piaget – conservation of shapePiaget – conservation of shape

–Age 4 – 5 not a squareAge 4 – 5 not a square–Age 6 – 7 same piece, but no longer a squareAge 6 – 7 same piece, but no longer a square–Age 8 – 9 both are squaresAge 8 – 9 both are squares

•Activities requiring PC:Activities requiring PC:–Identifying similar figuresIdentifying similar figures–Ordering objects by sizeOrdering objects by size–Identifying congruent figuresIdentifying congruent figures

Position-In-Space PerceptionPosition-In-Space Perception

•““The ability to relate an object in space The ability to relate an object in space to oneself” (p. 17).to oneself” (p. 17).•Children see themselves as the center Children see themselves as the center of all things – all things are above, of all things – all things are above, below, behind, or in front of thembelow, behind, or in front of them•Problems with PISP mean problems Problems with PISP mean problems reading and writing as well as in mathreading and writing as well as in math•Activities requiring PISP:Activities requiring PISP:

–Drawing congruent figuresDrawing congruent figures–Identifying flip, slide, and turn imagesIdentifying flip, slide, and turn images

bd

pq

Perception of Spatial RelationshipsPerception of Spatial Relationships

•““The ability to see two or more objects The ability to see two or more objects in relation to oneself or in relation to in relation to oneself or in relation to each other” (p. 17).each other” (p. 17).•Closely related to PISPClosely related to PISP•Requires strong sense of body Requires strong sense of body orientationorientation•Activities requiring POSR:Activities requiring POSR:

–Judging distances of things around youJudging distances of things around you–Playing ballPlaying ball–Riding a bicycleRiding a bicycle

Visual DiscriminationVisual Discrimination•““The ability to identify similarities and The ability to identify similarities and differences between or among differences between or among objects” (p. 18).objects” (p. 18).•Independent of positionIndependent of position•Activities requiring VisD:Activities requiring VisD:

–Sorting and classifying objectsSorting and classifying objects–Sorting and classifying geometric shapesSorting and classifying geometric shapes–Working with attribute blocksWorking with attribute blocks

Visual MemoryVisual Memory•““The ability to recall accurately objects no longer The ability to recall accurately objects no longer in view and relate their characteristics to other in view and relate their characteristics to other objects either in view or not in view” (p. 18).objects either in view or not in view” (p. 18).•Extreme case: Photographic memoryExtreme case: Photographic memory•Most people can remember 5 – 7 objects for short Most people can remember 5 – 7 objects for short periods of timeperiods of time•Activities requiring VM:Activities requiring VM:

–Briefly seeing a picture and then listing all of the things seenBriefly seeing a picture and then listing all of the things seen–Copying a figure seen briefly on a Geoboard to dot paperCopying a figure seen briefly on a Geoboard to dot paper

““Thinking about Thinking”Thinking about Thinking”

Promoting Metacognition in Mathematics Promoting Metacognition in Mathematics Student Recognition of Purpose in LessonsStudent Recognition of Purpose in Lessons

Center for Critical ThinkingSonoma, CA

The The EndEnd

ReferencesReferences• Center for Critical Thinking. Sonoma, CA.• Del Grande, J. (February, 1990). Spatial

sense. Arithmetic Teacher, 14-20.• Dr. Leslie S. Jones, Biology Department,

Valdosta State University