Pedagogy, Curriculum Links, Points for Teaching, and Resources
-Sophie Werner
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PEDAGOGY
To begin, I would like to present a useful pedagogical approach which works to a framework detailing the development of subitizing skills in children.
'The Learning Trajectory Approach'.
The ‘Learning Trajectory Approach’, put forward by Clements & Sarama (2014), uses a developmental framework guide to diagnose a child’s level of mathematical understanding and provide guidance for teaching based on this diagnosis. It follows a developmental progress guide for the learning of mathematical concepts which can be used to direct teaching and the planning of appropriate and helpful activities. The ‘Learning Trajectory for Recognition of Number and Subitizing’ is laid out below, giving the approximate age of children in relation to developmental level in subitizing ability.
Learning Trajectory Approach for Recognition of Number and Subitizing:
AGE: DEVELOPMENTAL STAGE in Subitizing:
0-1 Pre-Explicit Number - Dishabituates (responds to old stimuli as if it were new) to number, but does not yet have explicit number knowledge
1-2 Small Collection Namer - Names groups of 1 to 2, sometimes 3
3 Maker of Small Collections - Non-verbally can make a small collection (4 or less) with the same number as another reference collection
4 Perceptual Subitizer to 4 - Instantly recognises collections up to 4 and can verbally name the number
5 Perceptual Subitizer to 5 - Instantly recognises and can name collections up to 5
Conceptual Subitizer to 5 - Verbally labels all arrangements to 5, by adding smaller subitized groups within the group
Conceptual Subitizer to 10 - Verbally labels briefly shown arrangements up to 10, using smaller subitized groups
6 Conceptual Subitizer to 20 - Verbally labels arrangements up to 20, using smaller subitized groups
7 Conceptual Subitizer with Place Value and Skip Counting - Verbally labels arrangements, using smaller subitized groups, skip counting, and place value
8 Conceptual Subitizer with Place Value and Multiplication - Verbally labels arrangements, using smaller subitized groups, place value, and multiplication.
TEACHING RECOMMENDATIONS from The Learning Trajectory Approach
-Clements & Sarama, 2014
The two forms of subitizing, as previously been addressed, are perceptual and conceptual subitizing.
Perceptual subitizing is the initial form of the skill, whereby one recognises a number (of items/objects) without using other mathematical processes. Though it is a separate concept, from perceptual subitising grows conceptual subitizing; this is the ability to recognise number and number patterns as units of units, for example a die face as being composed of four dots, and also as ‘one four' (Clements, 1999).
Teaching Recommendations for PERCEPTUAL SUBITIZING
The Learning Trajectory indicates that from approximately age 0-4, children are perceptual subitizers; therefore it is imperative that we provide rich learning experiences for students in the early years. Clements & Samara (2014) suggest the following types of instructional activities for teaching (at home and at school) at this early level.
AGE ACTIVITIES
0-1 Provide a rich, sensory manipulative environment with regular use of mathematical words and actions - exposes children from a very early age to mathematical processes
1-2 Naming of numbers within small groups (1 to 2 objects), e.g. “There are two bears!”
Encourage children’s naming of the number of small sets - discuss their thinking
3 Get children to collect a given number of objects, for example enough biscuits for three children - relate maths to real life situations
Make a small group with manipulatives then hide it. Get child to make the same number group, then compare them - talk about similarities/differences
4 Get child to verbally name groups up to 4, in increasingly harder arrangements as child’s confidence grows
Teaching Recommendations for CONCEPTUAL SUBITIZING
From around the age of five, children begin to move from perceptual to conceptual subitizing. This involves a shift in abstract thinking, as children not only have to perceptually subitise, by mentally creating individual units to be counted in their own right from pieces of experience, they also have to go a step further mentally in the recognition of number and number patterns as units of units, as individual ‘ones’ which make up a whole number and as this whole number itself (Clements, 1999). The following activities are suggested for conceptual subitizers.
AGE ACTIVITIES
5 Use different arrangements of sets which encourage subitising into smaller groups
Use set cards/resources in conjunction with matching numeral cards - helps develop addition concepts
6 Use 5- and 10-frames in lessons - helps build addition concepts/’friends of 10’ (addition combinations to 10)
7 Continue games and manipulatives with pictorial sets accompanied by numerical cards - reinforce visual/numerical connections
8 Move to activities with multiple groups within sets. This encourages multiple instances of subitizing, and develops addition and estimation strategies
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CURRICULUM LINKS
The area of subitizing occurs early on in childhood. In the AusVELS, this area appears in the Foundation Level, in the Number & Algebra Strand of the Australian Curriculum in Mathematics.
The following is the progression point for subitizing:
Content description: Subitise small collections of objects
Elaborations:
-Using subitising as the basis for ordering and comparing collections of numbers
Code: ACMNA003
(Victorian Curriculum & Assessment Authority, 2014).
This is the only reference in the Early Years curriculum of subitizing; as stated in (Clements & Sarama, 2014), children’s ability to subitize (at least perceptually) should be largely developed in children of six years, therefore there should not be a need for learning standards past this level.
Links to other mathematical concepts
Arithmetic:
Subitizing at a conceptual level is a valuable opportunity for children to progress into addition and subtraction. When children conceptually subitize they are perceiving smaller groups within the larger set. This begins to develop concepts of addition and subtraction, as children can see the parts of the whole (Fuson, 1992).
Estimation:
Conceptual subitizing also helps develop estimation strategies. If children are presented with activities that show too large a set to be accurately subitized, they are forced to subitize the smaller groups within the whole set. These can be added together to form the basis of estimation (Clements & Sarama, 2014).
Multiplication:
Once children understand the basic premises of addition and how they can use it to ‘count’ objects in a set, they will get used to the idea of multiplication, as a form of repeated addition. This also can lend itself to estimation strategies (Clements & Sarama, 2014).
Counting and Cardinality:
“Perceptual subitizing is thought to be the basis for counting and cardinality” - understanding that the last number named is the number of objects that are in a group (Charlesworth & Lind, 2012, p.119). Subitizing is instinctive to individuals, and proves an efficient and accurate method of assessing numbers in small groups (Trick, 2013). It is a fundamental base concept of all mathematics.
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POINTS FOR TEACHING
Factors in the progress of developing subitizing skills.
As children progress in their learning of subitizing, there are factors which affect the difficulty of the task at hand and therefore childrens’ learning. It is important to take these into consideration in the planning of learning activities, as it is unhelpful to plan activities that are unachievable for students - we must plan to their level.
-The number of objects to be subitized needs to start at a small amount, and increase with the developmental level of the child (Clements, 1999).
Start with a small number of objects for young children (1-2 per set), and increase the number as they get older (3-10 per set).
-The arrangement of objects matters. Objects arranged in a straight line are easiest to subitize, followed by rectangular arrangements and ‘dice’ or ‘domino’ (standard) arrangements, and the most difficult to subitize are random or scrambled arrangements (Clements & Sarama, 2014).
Begin teaching young children with pictorial/physical representations of small sets with objects in a line. Expose children to many different types of representations of sets. Once children are confident, move on to harder spatial arrangements (as above).
-Keep lessons interesting! Present many and varied activities to children. Relate activities to childrens’ lives to engage them. Use manipulatives for hands-on activities, utilise sound and music to teach, play games to make maths fun.
-Keep subitizing simple. Using cards and resources that are too busy will confuse children and hinder their learning. Pictorial representations should be as simple as possible, use similar objects of similar colours, designs, sizes, and there shouldn’t be too much illustration around the page (Clements & Sarama, 2014).
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RESOURCES
Here is a directory of resources and some ideas of subitizing activities for use in the classroom! There’s a variety, as teachers should try to include a wide range of activities in their teaching of subitizing, as exposure to this variety is the best way to teach young children this fundamental concept.
Cards
-Activities with Tens Frames:
http://www.k-5mathteachingresources.com/ten-frames.html
-Clown Counting Game, which uses both pictorial and numerical representations:
http://www.teachthis.com.au/products/view-resource/link/Clown-Counting-Game-Subitising/id/226
-Simple, effective cards for use in games, activities, or displays:
http://www.pinterest.com/pin/183521753539271169/
Games
-Fun subitizing boardgame:
http://earlyyearsfun.blogspot.com.au/2013/01/subitising-game.html
-Activity using counters to subitize groups on picture mats - great for younger children, and teaches number patterns also:
http://www.pinterest.com/pin/128282289359049468/
-Subitizing Memory game:
http://www.pinterest.com/pin/386183736770920142/
-Board game, good for quick thinking and teamwork:
http://www.pinterest.com/pin/481251910152325556/
Manipulatives
-Play ‘Knuckles’ - For perceptual subitizing, use 5 ‘knuckles’ (or any objects), toss them into the air and attempt to catch them on the back of hand. For conceptual subitizing, use 5+ objects; large sets can be subitized into smaller groups depending on how they land.
-Game for young children to work on subitizing with materials:
http://www.pinterest.com/pin/128282289356850985/
Online
-A fun and challenging game with a flash card concept:
http://www.fuelthebrain.com/Interactives/app.php?ID=29
-App for iPhone/iPad - Flash Card concept, yet you can ‘design your own’:
https://itunes.apple.com/au/app/subitising-flash-cards/id515943889?mt=8
-Suggestions for ‘Best Subitizing Apps’:
http://www.pinterest.com/pin/553239135449431847/
Craft
-Involve older students in the teaching of subitizing to younger children - give the older children ownership, and this teacher was pleasantly surprised with the creativity of her students:
http://authenticinquirymaths.blogspot.com.au/2013/03/subitising-with-kindergarten.html
-Really nice craft activity for younger children, brings learning of subitizing into the home environment:
http://www.learnwithplayathome.com/2013/02/sorting-subitising-with-egg-cartons.html
Whole class
-GREAT game for spare time in class, can be done sitting on floor, involves participation and quick thinking for all class members:
http://www.pinterest.com/pin/23784704255840199/
-Using an overhead projector, have one child on one side and another child standing on the other side. Take hole-punched cards of different amounts and spatial arrangements. One child places the card on the overhead, the other child takes the card away as quickly as they can. The class is to answer with the number they subitized in the set (Clements, 1999).
This is the only reference in the Early Years curriculum of subitizing; as stated in (Clements & Sarama, 2014), children’s ability to subitize (at least perceptually) should be largely developed in children of six years, therefore there should not be a need for learning standards past this level.
Links to other mathematical concepts
Arithmetic:
Subitizing at a conceptual level is a valuable opportunity for children to progress into addition and subtraction. When children conceptually subitize they are perceiving smaller groups within the larger set. This begins to develop concepts of addition and subtraction, as children can see the parts of the whole (Fuson, 1992).
Estimation:
Conceptual subitizing also helps develop estimation strategies. If children are presented with activities that show too large a set to be accurately subitized, they are forced to subitize the smaller groups within the whole set. These can be added together to form the basis of estimation (Clements & Sarama, 2014).
Multiplication:
Once children understand the basic premises of addition and how they can use it to ‘count’ objects in a set, they will get used to the idea of multiplication, as a form of repeated addition. This also can lend itself to estimation strategies (Clements & Sarama, 2014).
Counting and Cardinality:
“Perceptual subitizing is thought to be the basis for counting and cardinality” - understanding that the last number named is the number of objects that are in a group (Charlesworth & Lind, 2012, p.119). Subitizing is instinctive to individuals, and proves an efficient and accurate method of assessing numbers in small groups (Trick, 2013). It is a fundamental base concept of all mathematics.
---------------------------------------------
POINTS FOR TEACHING
Factors in the progress of developing subitizing skills.
As children progress in their learning of subitizing, there are factors which affect the difficulty of the task at hand and therefore childrens’ learning. It is important to take these into consideration in the planning of learning activities, as it is unhelpful to plan activities that are unachievable for students - we must plan to their level.
-The number of objects to be subitized needs to start at a small amount, and increase with the developmental level of the child (Clements, 1999).
Start with a small number of objects for young children (1-2 per set), and increase the number as they get older (3-10 per set).
-The arrangement of objects matters. Objects arranged in a straight line are easiest to subitize, followed by rectangular arrangements and ‘dice’ or ‘domino’ (standard) arrangements, and the most difficult to subitize are random or scrambled arrangements (Clements & Sarama, 2014).
Begin teaching young children with pictorial/physical representations of small sets with objects in a line. Expose children to many different types of representations of sets. Once children are confident, move on to harder spatial arrangements (as above).
-Keep lessons interesting! Present many and varied activities to children. Relate activities to childrens’ lives to engage them. Use manipulatives for hands-on activities, utilise sound and music to teach, play games to make maths fun.
-Keep subitizing simple. Using cards and resources that are too busy will confuse children and hinder their learning. Pictorial representations should be as simple as possible, use similar objects of similar colours, designs, sizes, and there shouldn’t be too much illustration around the page (Clements & Sarama, 2014).
---------------------------------------------
RESOURCES
Here is a directory of resources and some ideas of subitizing activities for use in the classroom! There’s a variety, as teachers should try to include a wide range of activities in their teaching of subitizing, as exposure to this variety is the best way to teach young children this fundamental concept.
Cards
-Activities with Tens Frames:
http://www.k-5mathteachingresources.com/ten-frames.html
-Clown Counting Game, which uses both pictorial and numerical representations:
http://www.teachthis.com.au/products/view-resource/link/Clown-Counting-Game-Subitising/id/226
-Simple, effective cards for use in games, activities, or displays:
http://www.pinterest.com/pin/183521753539271169/
Games
-Fun subitizing boardgame:
http://earlyyearsfun.blogspot.com.au/2013/01/subitising-game.html
-Activity using counters to subitize groups on picture mats - great for younger children, and teaches number patterns also:
http://www.pinterest.com/pin/128282289359049468/
-Subitizing Memory game:
http://www.pinterest.com/pin/386183736770920142/
-Board game, good for quick thinking and teamwork:
http://www.pinterest.com/pin/481251910152325556/
Manipulatives
-Play ‘Knuckles’ - For perceptual subitizing, use 5 ‘knuckles’ (or any objects), toss them into the air and attempt to catch them on the back of hand. For conceptual subitizing, use 5+ objects; large sets can be subitized into smaller groups depending on how they land.
-Game for young children to work on subitizing with materials:
http://www.pinterest.com/pin/128282289356850985/
Online
-A fun and challenging game with a flash card concept:
http://www.fuelthebrain.com/Interactives/app.php?ID=29
-App for iPhone/iPad - Flash Card concept, yet you can ‘design your own’:
https://itunes.apple.com/au/app/subitising-flash-cards/id515943889?mt=8
-Suggestions for ‘Best Subitizing Apps’:
http://www.pinterest.com/pin/553239135449431847/
Craft
-Involve older students in the teaching of subitizing to younger children - give the older children ownership, and this teacher was pleasantly surprised with the creativity of her students:
http://authenticinquirymaths.blogspot.com.au/2013/03/subitising-with-kindergarten.html
-Really nice craft activity for younger children, brings learning of subitizing into the home environment:
http://www.learnwithplayathome.com/2013/02/sorting-subitising-with-egg-cartons.html
Whole class
-GREAT game for spare time in class, can be done sitting on floor, involves participation and quick thinking for all class members:
http://www.pinterest.com/pin/23784704255840199/
-Using an overhead projector, have one child on one side and another child standing on the other side. Take hole-punched cards of different amounts and spatial arrangements. One child places the card on the overhead, the other child takes the card away as quickly as they can. The class is to answer with the number they subitized in the set (Clements, 1999).
