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Progression of Counting and Quantity
@MathletePearcewww.tapintoteenminds.com
tapintoteenminds.com/sudbury
“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
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
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