Progression of Counting and Quantity

@MathletePearcewww.tapintoteenminds.com

tapintoteenminds.com/sudbury

GETREADY

Shared By: Jo Boaler

How Did You

The Dots?SEE

Shared By: Jo Boaler

Shared By: Jo Boaler

Shared By: Jo Boaler

Shared By: Jo Boaler

Shared By: Jo Boaler

Shared By: Jo Boaler

Shared By: Jo Boaler

Shared By: Jo Boaler

Shared By: Jo Boaler

Shared By: Jo Boaler

Shared By: Jo Boaler

Shared By: Jo Boaler

Shared By: Jo Boaler

Shared By: Jo Boaler

Shared By: Jo Boaler

SPATIAL REASONING?What is

“Spatial thinking, or reasoning, involves the location and movement of objects and

ourselves, either mentally or physically, in space. It is not a

single ability or process but actually refers to a considerable number of

concepts, tools and processes.”

(National Research Council, 2006)

“The relation between spatial ability and mathematics is so

well established that it no longer makes sense to ask whether

they are related…”

“…moreover, spatial thinking was a better predictor of

mathematics success than either verbal or mathematical skills.”

Activities to Develop Geometric and Spatial Thinking

visualizing diagramming

designing(Davis, Okamoto & Whiteley, 2015) 

orientinglocating

perspective takingslidingrotating

reflecting

modelingexploring symmetry

composingdecomposing

scalingmap-making

CountingPrinciples

Stable-Order Order Irrelevance

Conservation

One-to-One Correspondence

Abstraction

Movement is MagnitudeSubitizing

Unitizing

Cardinality

CountingPrinciples

Stable-OrderThe list of words used to count must be in a repeatable order.

This “stable list” must be at least as long as the number of items to be counted.

CountingPrinciples

Stable-OrderThe list of words used to count must be in a repeatable order.

This “stable list” must be at least as long as the number of items to be counted.

12

3

45

6

7 8 9 10

Stable-OrderThe list of words used to count must be in a repeatable order.

This “stable list” must be at least as long as the number of items to be counted.

1 2 3 4 5 6

Order IrrelevanceThe order in which items are counted is irrelevant.

1 2 3 4 5 6

1 2 3 4 5 6

ConservationUnderstanding that the count for a set group of objects stays the same no matter whether they are spread out or close together.

7 8 9 101 23 4

5 6

ConservationUnderstanding that the count for a set group of objects stays the same no matter whether they are spread out or close together.

7 8 9 101

23 4

5

6

… the quantity of five large things is the same count as a quantity of five small things or a mixed group of five small and large things.

Abstraction…we can count any collection of objects, whether tangible or not.

1 23 4

51 2 3 4 5

One-to-One Correspondence

12

34

5

Understanding that each object in a group can be counted once and only once. It is useful in the early stages for children to actually tag or touch each item being counted and to move it out of the way as it is counted.

One-to-One Correspondence

123

4

5

Understanding that each object in a group can be counted once and only once. It is useful in the early stages for children to actually tag or touch each item being counted and to move it out of the way as it is counted.

Understanding that the last count of a group of objects represents how many are in the group. A child who recounts when asked how many candies are in the set that they just counted, has not understood the cardinality principle.

Cardinality

1 2 3 4 5 6

The ability to 'see' a small amount of objects and know how many there are without counting.

Subitizing

“5”

The ability to 'see' a small amount of objects and know how many there are without counting.

Subitizing

“5”

PerceptualThe ability to 'see' groups of small amounts of objects as parts of a whole; often beyond 5.

Conceptual

“7”

Understanding that as you move up the counting sequence (or forwards), the quantity increases by one and as you move down (or backwards), the quantity decreases by one or whatever quantity you are going up/down by.

Movement is Magnitude

1 2 3 4

@MathletePearceTapintoTeenMinds.com

UnitizingUnderstanding that every quantity we measure is relative to another pre-measured group we call a unit.

4twos

For example, counting equal groups of 2.

@MathletePearceTapintoTeenMinds.com

Unitizing

threes

For example, counting equal groups of 3.

Understanding that every quantity we measure is relative to another pre-measured group we call a unit.

4

@MathletePearceTapintoTeenMinds.com

Unitizing

tens ones

2 01 1For example, our base ten place value system.

Understanding that every quantity we measure is relative to another pre-measured group we call a unit.

@MathletePearcewww.tapintoteenminds.com

0 4 91 2 3 5 6 7 8

Counting, Quantity and Number LinesStable Order and Movement is Magnitude

More Fun With Dots

More Fun With Dots

More Fun With Dots

How Many?

More Fun With Dots

More Fun With Dots

More Fun With Dots

More Fun With Dots

How Many?

More Fun With Dots