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Abstraction Principle – Everything and Anything Can be Counted

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The abstraction principle is the last of Gelman and Gallistel’s Five Counting Principles. The one-to-one correspondence, stable-order, cardinal, and order-irrelevance principles have been addressed in previous posts.

It is vital that children learn the other principles first, because as they get older, the abstract principle will be easier to understand.

The abstraction principle states that the preceding principles can be applied to any collection of objects,
whether tangible or not.

For young children learning to count is easier if the objects are tangible and, where possible, moveable, in order to help them to distinguish the ‘already counted’ from the ‘yet to be counted’ group. To understand this principle, children need to know that they can count non-physical things such as sounds, imaginary objects or even the counting words – as happens when ‘counting on’. 

 

Have you ever had a child who looks at two sets that have the same number of objects but automatically says the set with the larger objects has more? Or a child who sorts the set first so that different sized or coloured objects are together?

Child need to understand that regardless of what is in the set or how it looks, we are only interested in the number within the sets.

 

For example: The number within each set is the same.

Set A                                                                                          Set B

                               

 

Sources: Origo One

Difficulties

  • Sometimes when presented with different objects, children state that larger items have more value, for example, they are worth more than ‘1’.
  • It is more difficult to count non-tangible objects, for example, sounds, actions or words people say (this may impact upon literacy – the numbers of words in a sentence or the syllables in a word).
  • Matching sets of different objects that have the same quantity.

Practice

Provide children with opportunities to count a range of sets, real or imagined, similar or different.

  • Use different arrangements within the sets.
  • Use different colours and encourage them to not just focus on their favourite colour.
  • Ensure sets have different sized objects.
  • Balls Roll or throw them to a partner. Count the number of times a child bounces a ball.
  • Musical instruments Use a xylophone, drum or shake and count the number of times it’s hit or shaken.
  • Counter Drop Have children count the number of counters/pebbles that are dropped one by one into a can.

 

Remember, plan lots of opportunities for students to count whether formally or informally.

Ensure they:

  • touch and move objects, where possible, to keep track of what they have and haven’t counted
  • apply one number name to each object or action
  • say the number words in the correct order
  • say the last number word after they have finished counting then ‘you’ ask ‘how many?’
  • have a variety of things to count – differing sizes and colours in the same set, objects in different arrangements, and tangible and intangible things to count
  • begin counting from different objects within the set

 

Until next time,

Carole

Order-Irrelevance Principle – The Order of the Count Doesn’t Matter

The order-irrelevance principle refers to the understanding that the order in which objects are counted is not important. It doesn’t matter whether the counting procedure is carried out from left to right, from right to left or from somewhere else, so long as every item in the collection is counted once and only once. 

For example:
In a collection of three buttons there may be a red, green and blue one. I may start counting with a blue button the first time but the next time I may begin with the red button. Whichever button I begin, with I will always arrive at the same number.
 
This seems like a simple concept to understand but children often have to recount the set if you ask them to begin counting from a different object.

Difficulties

  • When counting, some children may assume that the number they tag an object with stays with that object and is not ‘temporary’
     For example:
     If we go back to the previous button example. Children may start with counting the blue button and         tag or label it as ‘one’. Some children will insist that it is ‘one’ and that’s where they must start                   counting from.
 
   

Source: Origin One

Recent research by LeFevre and others, 2008, indicates that from Foundation through to about Year 2 the application of conventional counting rules (left to right and top to bottom) was related to higher numeration skill. At this age, the order-irrelevance principle appeared to impact negatively on children’s working memory and therefore, their ability to count proficiently.

However, from Year 3 onward, children’s numeration knowledge was unrelated to whether they had acquired order-irrelevance. Although the order-irrelevance principle is an important understanding for children to have, some children do not fully develop this understanding until they are around eleven years of age. 

The research suggested that only when counting becomes automatic might there be the opportunity for children to accept that it is not necessary to count a set in a strict order.

Thus, for children who are still developing their counting skills, the principle of order irrelevance
might be logical, but it is not necessarily practical.

Practice

Have children count objects in their every day environment in a variety of different ways. Ensuring that they touch the object as they do so.
Constant exposure to counting helps develop this skill as well as making a game out of it by “mixing up” objects in a set to see if the numbers change.
 
  •  Count left to right and right to left with the same row of objects
  •  Count the same set but starting with a different “one” each time, for example, the middle object
  •  For a greater challenge, ask the children to count all the objects, making the middle object ‘5’
  •  Re-arrange the objects so they are no longer in a row and ask the child to count from a particular object
  • Ensure that sets for counting contain, not only sets with the same objects but also sets that have miscellaneous objects without any commonality.

 

Until next time,

Carole

Cardinality – Giving Meaning to Numbers

Cardinality is the ability to understand that the last number which was counted when counting a set of objects is a direct representation of the total in that group.

Children will first learn to count by matching number words with objects   (1-to-1 correspondence) before they understand that the last number stated in a count indicates the amount of the set.

A child who understands this concept will count a set once and not need to count it again. They will automatically remember and know how many are represented.

Source: Origo One

Students who are still developing this skill need constant repetition of counting and explicit teaching through modelling so they understand they do not need to count over and over again when it will result in the same number. Students who have difficulty with their working memory may have difficulty with this concept.

So What Can You Do To Help!!

Research by Paliwal and Baroody (2017) stresses the importance of:

  •  Labelling the total number of items then counting them (Label-first).  For example, on a page with 3 elephants, saying, “Look there are 3 elephants. Let’s count them.” And counted them as, “one, two,three.” or
  • Counting the items, then emphasising and repeat the last word (Count-first). For example, on a page with 3 elephants, say, “One, two, three, t-h-r-e-e. There are three elephants.”
  • Researchers indicate that the latter is the preferred method of modelling, suggesting that the first did make a difference compared to Counting Only, where the total number of items was not emphasised.

We can help children develop the understanding of cardinality by involving them in activities where they answer questions about ‘how many’. They need not only to be able to say the counting names in the correct order, but also to count a group of, for example, seven objects and say that there are seven.

 

This video from the Connecticut Office of Early Childhood provides examples of ways to develop cardinality in the classroom.

 

Activities

Counting Collections Activities should have some basis in reality, giving a purpose to counting. For example, create a need to count by involving children in food preparation. They will need to know how many people, plates or apples in order to complete the task.

How Many? Provide opportunities for students to count using a variety of objects such as buttons, counters, shells, coins, and dot cards. Objects can be put into jars, counted then draw and recorded. 

Order Disorder Place objects to be counted in different arrangements. Firstly, perhaps, a straight line then, the same objects, in a circle then a random arrangement. Always asking children “How many?” If they need to recount the objects, they do not understand the concept of cardinality.

Show Me Provide children with a bag, box, or bucket of objects and ask them to count out a certain number of objects. For example, say, “Show me 5 buttons.” Once the child has counted out the required number of objects, again ask, How many?”

Bugged Out Children roll a number cube and put that many bugs into the jar. If they roll a fly-swatter, they have to remove a bug. If they roll the bug spray, they have to remove ALL of their bugs. The first person to get 10 bugs in their jar wins!! Printable

Count and Graph Worksheets here

Nature Scramble Engage children in activities in the school ground, beach or local park. Ask them to collect different numbers of object, for example, shells, rocks or leaves. Always referring to “How many?”

Rocket to 10 Printable Provides opportunities to talk to children about number and their thinking. Ask children, “How many cubes did you put in the rocket?” and “How many more do you need to fill the tower?”

Spot the goof from Parenting Science

Want to make your own sock puppet for Spot the goof?? The following videos may help. And neither require sewing!!

 

 

How Many Snails? a counting book by Paul Giganti Jr from the National Centre for Excellence in the Teaching of Mathematics.

 

Once a child has a sense of cardinality, then we can involve them in matching activities where a number word is matched to a quantity and the numeral that belongs to it.

 

Matching Activities (ensuring that they are still using concrete manipulatives)

Match It Provide children with opportunities to match numerals with the number of items in the set they have counted.

Count It Provide children with a numeral card and ask them to read the number. Children then count out that many items to represent the number.

Mouse Match and Thread Printable I recommend that you only use one colour of beads, otherwise children will make coloured patterns instead of thinking about the counting!! 

Activities from Proud to be Primary

 

 

 

Ladybug Match Printable

 

Activities from In My World

 

 

Until next time,

Carole

One-to-One Correspondence – A Counting Fundamental

 

Children come to school with varied experiences related to counting. Even if young children can recite the number sequence it cannot be assumed that they can apply this knowledge to counting small sets of objects. Knowing the one-to-one correspondence principle is essential for organised, meaningful counting. This leads to an eventual ability to perform higher-level calculations (McCarthy, 2009).

Source: Origo One

Difficulties

One-to-one correspondence is often difficult for young children to comprehend. In Maths recognising the number “ten,” and being able to count out “ten” items are two separate skills. Linking objects with numbers enables a child to count with understanding (McCarthy, 2009).

Common errors when counting a set of items can be:

  • Skipping an item
  • Assigning more than one number word to a single item
  • Pointing to two or more items while saying one number word (Clements and Sarama, 2014)

Ways to Develop One-to-One Correspondence

Role Play

  • Setting the table – For each plate on the table the child needs to place one fork, one knife and one spoon
  • Fruit ice-cubes – Chop pieces of fruit, for example, pineapple or strawberries. Place one piece of fruit into each space in an ice-cube tray. Add water or fruit juice. Freeze.
  • Teddy Bear’s Picnic – Set out between 3 to 5 small chairs. Place one teddy bear on each chair.

Activities

  • Rhythm Movements – Children count the number of claps an adult makes. This can be the number of beats on a drum, taps on a triangle. Children count aloud and aim for rhythm.
  • Follow Me – Children make the number of movements given by an adult, for example, clap three times, hop three times, skip five times, nod six times. Children count aloud as the actions are done. 
  • Bean Drop – An adult drops dried beans into a container. As the beans are dropped the children need to count them.
  • Jumping on the Lily Pads (From Young Mathematicians)

 

 

 

 

 

  • Count Me – Place a group of objects (e.g.: shells, leaves, counters, teddies, boats, cars) on the table. Ask the student to count how many objects there are. Watch carefully and see if you can determine how the student decides how many objects there are (DET, accessed 11/6/2018)
  • Activities From The Measured Mom
  • Auditory Activities from https://mykidsturn.org/

Source: By the Numbers (ESSDACK)

  • Hands-on and Auditory Activities from www.raepica.com

Game Boards

PreKinders is an amazing website with free resources, not just for Maths, but also fine motor skills, literacy, science and more.

  • The following board game is easy to prepare, simple for children to understand and doesn’t require many resources.
  • Roll and Collect is from Kate in Kinder  and sourced through Teachers Pay Teachers but as a free download. There are a few different versions. The game uses a 6-sided die. This might not give the children enough turns so might need to be adapted, showing the numbers 1, 2 and 3 on two sides each.

Roll And Collect Math Center

Online Games

Ladybird Spots from Topmarks.co.uk

Gingerbread Man Game from Topmarks.co.uk

Underwater Counting from Topmarks.co.uk

Counting Fish from abcya.com

My aim is to provide free resources, where possible, that support the academic research. If you find other resources or relevant information please contact me and I’ll include it on the blog.

Thank you to all the educators who supply free resources to support teachers and their children.

Until next time,

Carole

Number Sense – What is it? Why teach it?’

Number Sense is ….

“… good intuition about numbers and their relationships.
It develops gradually as a result of exploring numbers, visually them in a variety of contexts, and relating them in ways that are not limited by traditional algorithms.” (Howden, 1989)

“Research indicates that early number Sense predicts school success more than other measures of cognition, such as verbal, spatial, or memory skills or reading ability.” (Watts, Greg, Duncan, Siegler, & Davis-Kean, 2014)

Jo Boaler, a professor at Stanford University explains it in this way:

https://www.youcubed.org/resources/what-is-number-sense/

Van de Walle and others (2018) suggest Four Early Numeracy Concepts and Four Number Sense Relationships necessary for the development of Number Sense, something we will continue to explore.

Early Number Concepts

  1. Verbal Counting – to say the numerals in the correct order
  2. Object Counting – 1 to 1 correspondence
  3. Cardinality – the last object counted in a set tells how many
  4. Subitising – the ability to see how many without having to count

Number Sense Relationships

  1. Spatial relationships – having a visual to go with a number
  2. One or Two More & Less – instantly knowing the amount that is one or two more & less
  3. Benchmarks of 5 and 10 – knowing how a number relates to 5 and 10
  4. Part-Part-Whole – understanding how a whole can be broken into parts

Counting

Counting is fundamental to later maths development. Early counting predicts later mathematical success (Clements and Sarama, 2014) and even later reading fluency (Koponen, Salmi, Eklund and Aro, 2013).

Many children begin school with the ability to differentiate between sets of certain ratios (in particular, 2:1 and 3:2) enabling them to tell the difference between sets just by looking at them. However, they are unable to tell the difference between sets that are close in number (for example, 10 and 9). It should be noted, that this does not apply to all children, in particular, those that may suffer from Dyscalculia. A term referring to a wide range of life-long learning disabilities involving maths. It includes all types of maths problems ranging from an inability to understand the meaning of numbers, to an inability to apply mathematical principles to solve problems.

Those who study children’s mathematical development explain that counting involves five principles (Gelman and Gallistel, 1978):

  1. One-to-one correspondence
  2. Stable number word order
  3. Cardinality
  4. Order irrelevance
  5. Abstraction

Sound complicated? It is! Something we adults take for granted as “simple” is actually quite complex developmentally. Although we will explore each principle in future posts the following by the National Center on Intensive Intervention is a brief introduction.

 Verbal Counting – Stable Number Word Order

As children engage with nursery rhymes, songs and role-play opportunities they often develop the order of number words. However, just because children are able to rote recite the number words it does not mean that they understand one-to-one correspondence nor the knowledge of differences in the magnitude or size of numbers (for example, knowing that 7 comes after 6 doesn’t mean that the child knows that 7 is more than 6).

Counting rhymes, books and videos are ways of supporting children in developing Verbal Counting. An enabled adult who is able to support and intervene when a child experiences difficulty is significant at this stage.

I usually add the counting rhymes to the classroom library and make activities (available in the counting rhymes section) so that students can retell the rhymes. This assists with developing the rhythm of counting.

 What strategies/ideas do you use for Verbal Counting? Please feel free to comment.

 

Until next time,

Carole