Technician
Observing/Noticing
Source: Video from Videatives
Titled: Math- 'Combining Sets'
Length: 4:08
For the purpose of this blog, 'Child 1' in the observation will be referred to as Tom, and 'Child 2' as Jack.
Tom and Jack, aged 4 years old are counting long wooden blocks on a table in an early learning centre. Initially Tom is standing at the table by himself with five blocks in front of him. He lines them up by making the blocks directly touch each other, and counts each block after placing it down in the line. When he has counted five blocks he says "I've got five". The next boy, Jack, comes along holding a number of blocks in his hands and places them down on the table saying "let's see how much I have". He then starts to count them and recognises that he has four blocks. Tom says "I have five" as he was comparing his amount with Tom's, and Jack says "wow, thats a lot!!". Tom says "you can have 5 too, I'll go get you one", he goes off screen and comes back with another block. Tom says, "here, now you have five like me. Tom then takes over by pushing both piles together into one line where all the blocks are side by side. He then starts counting from right to left and points to each block as he pronounces the number, "1, 2, 3, 4, 5, 6, 7, 8, 9, 11...look we have 11". Jack then decides to count them for himself and also says "1, 2, 3, 4, 5, 6, 7, 8, 9, 11!!". Tom then puts both hands on the pile and splits them in half by sliding half to the left and half to the right. Tom then says "here we can both have 4" and counts his half and gets 5 blocks. Tom decides to take one away from his pile to make it 4 and Jack observes what Tom has done and takes one away on his pile as well. They both walk off camera and put them away. Tom then comes back and says "now i have 1, 2, 3, 4" and Jack continues by saying "and I have 1, 2, 3, 4". Tom then says "hey, maybe we can match them like this" while he pushes both piles together again. He then counts from right to left and gets a total of 8 blocks. Similar to before, Jack decides to count for himself and also gets 8 blocks and shouts "8" in an excited tone. Tom then takes a block away from either end of the line and puts them away (leaving 6 blocks). Tom suggests "lets cut them into pieces" and divides the line in half (3 blocks each), and says "so I have 3", and Jack follows on by saying "and I have 1, 2, 3". Tom takes the lead again and says "now lets match them". Tom joins them together and Jack steps in and says "now we have..", before Tom interrupts and says "no I'm going to count." By pointing to the numbers and counting out loud, Tom counts 8 blocks in front of him, and then realises his mistake and says actually we have "1, 2, 3, 4, 5, 6". Jack then counts for himself again and also gets 6. Tom then takes another block away from each end (2 blocks) and says "don't grab them, I have to grab them". When he returns, he splits them in half again leaving 2 blocks in each pile, and counts them. Jack then counts his pile and also finds 2. Tom suggests to match them up, "now we have 1, 2, 3, 4" and Jack gets excited and laughs extremely loudly. Again, following the same pattern, Tom takes two blocks away from each end of the line but then comes to a halt and says "oh oh we have to take these two", and chooses two from the left end of the line. They leave the two remaining blocks in one pile and Tom decides to count them. Finally, Tom takes one more block away leaving 1 remaining. Jack decides to pick up the last block but Tom shouts "no no, I told you not to touch it". Then Tom screams with excitement "1, now we have 1".
Recognise
During this observation it is clear that Tom and Jack are showing signs of a 'technician', as they are breaking things down and cracking a code through numerate thinking and problem solving. The children are using numbers to count the blocks they have in their possession, which shows development of number sense through counting and rote learning. Within this particular learning experience, Tom and Jack are learning about addition and subtraction by adding more blocks to the table and by also taking them away throughout the activity. Knaus (2013) states that children in everyday life become familiar with terms such as adding on and taking away, as well as sharing and multiplying. Tom was also using division throughout the task when apportioning the piles into two even amounts. For example, when he had 10 blocks, he split the row into two piles of 5, knowing that 2 lots of 5 = 10. Both children were also using one-one correspondence when they were counting, as they tagged each wooden block with their finger before pronouncing the number out loud.
Learning outcome 5, suggests that children use language to communicate thinking about quantities, to describe attributes of objects and collections, and to explain mathematical ideas (EYLF, 2009). These two children were using team work skills in order to count how many blocks they had, and often shared the blocks evenly between them. From a social aspect, pragmatics came into play in this learning experience, as the children were working together by communicating instructions and clarifying answers during their counting of the blocks. They were also taking turns throughout this experience and Jack often respected Tom's choices, which is a significant part of working as a 'technician'.
Initially, Tom decided to add blocks to the original amount that he had on the table as Jack bought over one less, which made the total an un even amount. Therefore while thinking numerally, Tom decided to collect another block to make it an even amount on the table. This shows that he could recognise that there was an uneven amount, and therefore was not able to divide the blocks into two equal piles. As the learning experience progressed, Tom (who was showing a lot more initiation throughout the activity) decided to slowly take blocks away leaving less blocks behind each time. It was interesting to see that each time he took blocks away, he would always take two with him, which shows that he could recognise that in order to divide the piles into two, they have to be even numbers. This pattern continued throughout the play with blocks, and eventually the children were left with only one block.
Tom and Jack are also beginning to understand the correspondence between voicing numbers and how many they see in front of them. This works alongside developing hand-eye coordination where the children were seen counting from where their hand was pointing. Knaus (2013) describes the 'Cardinal principle' which revolves around the final number said when a child is counting objects and being able to recognise that the last number is how many they have. Tom and Jack repeatably used this principle when counting each time they had either added or taken away a block, and were able to correspond the number of blocks they saw in front of them to the sound of the numeral in which they speak. Knaus (2013) acknowledges that subitising and counting experiences help children connect to the concept of quantity (how many are in a group). This can also get children to start to gain awareness of estimation, e.g., when they see a large amount balls, they can recognise that the result will be a higher or larger number.
Towards the start of this observation when there were 10 blocks lined up on the table, it was interesting to see Tom count the blocks up to 9 and jump straight to 11. When Jack decided to count as well, he also skipped number 10 and counted 11 blocks. This initiates their inability to perform and understand the correct number sequence, and may need some help from an educator to assist their numerical counting, especially from the number 9 onwards. Knaus (2013) says that the pattern of the numbers becomes quite difficult after number 9, which is exactly what was evident with Tom and Jack. It also showed to me that Jack (who counted second) was unable to recognise this mistake, and may have been feeding off, or copying what Tom had previously counted.
Learning outcome 5, suggests that children use language to communicate thinking about quantities, to describe attributes of objects and collections, and to explain mathematical ideas (EYLF, 2009). These two children were using team work skills in order to count how many blocks they had, and often shared the blocks evenly between them. From a social aspect, pragmatics came into play in this learning experience, as the children were working together by communicating instructions and clarifying answers during their counting of the blocks. They were also taking turns throughout this experience and Jack often respected Tom's choices, which is a significant part of working as a 'technician'.
Initially, Tom decided to add blocks to the original amount that he had on the table as Jack bought over one less, which made the total an un even amount. Therefore while thinking numerally, Tom decided to collect another block to make it an even amount on the table. This shows that he could recognise that there was an uneven amount, and therefore was not able to divide the blocks into two equal piles. As the learning experience progressed, Tom (who was showing a lot more initiation throughout the activity) decided to slowly take blocks away leaving less blocks behind each time. It was interesting to see that each time he took blocks away, he would always take two with him, which shows that he could recognise that in order to divide the piles into two, they have to be even numbers. This pattern continued throughout the play with blocks, and eventually the children were left with only one block.
Tom and Jack are also beginning to understand the correspondence between voicing numbers and how many they see in front of them. This works alongside developing hand-eye coordination where the children were seen counting from where their hand was pointing. Knaus (2013) describes the 'Cardinal principle' which revolves around the final number said when a child is counting objects and being able to recognise that the last number is how many they have. Tom and Jack repeatably used this principle when counting each time they had either added or taken away a block, and were able to correspond the number of blocks they saw in front of them to the sound of the numeral in which they speak. Knaus (2013) acknowledges that subitising and counting experiences help children connect to the concept of quantity (how many are in a group). This can also get children to start to gain awareness of estimation, e.g., when they see a large amount balls, they can recognise that the result will be a higher or larger number.
Towards the start of this observation when there were 10 blocks lined up on the table, it was interesting to see Tom count the blocks up to 9 and jump straight to 11. When Jack decided to count as well, he also skipped number 10 and counted 11 blocks. This initiates their inability to perform and understand the correct number sequence, and may need some help from an educator to assist their numerical counting, especially from the number 9 onwards. Knaus (2013) says that the pattern of the numbers becomes quite difficult after number 9, which is exactly what was evident with Tom and Jack. It also showed to me that Jack (who counted second) was unable to recognise this mistake, and may have been feeding off, or copying what Tom had previously counted.
Respond
In response to this numerate learning experience for developing numerical counting skills, it will be a good idea to try a similar activity to this, but with using different counting objects to see if they are sorted and interpreted in the same manner. Knaus (2013) states as part of the 'Abstraction principle', that sometimes children have difficulty counting things that are not identical objects. In this case, with Tom and Jack, they were always using the same shape and coloured object which were easy to compare and line up alongside each other. It will be interesting to sit down with these children using a mixture of miscellaneous objects such as forks, cups, bowls etc, or just simply different shaped or coloured blocks, and see if this effects their rote counting and the way in which these children separate, group and divide the blocks. From this experience children will widen their view and gain the concept that any object can be counted no matter what type, shape or size it is.
I believe that it will also be wise to conduct the exact same initial observation again with Tom and Jack, but this time have an educator appropriately intervene, and reverse the roles by asking Jack to count the blocks first. In the observation, Tom took the lead most of the time and had the first opportunity to count. Jack was not given an opportunity to show his rote learning skills and it seemed as though Jack was following on from what Tom had done most of the time, and may not quite understand the full concept of counting in sequence. This will help to detect if this is an issue in Jack's rote counting, and therefore other implications can then be put in place to assist his learning if needed.
There are many opportunities to further children's learning and development for numerical counting. In taking a completely different approach to the original counting experience of Tom and Jack, is an activity which focuses on subitising, by using counters to represent the number quantity. This activity located in Marianne Knaus' 'Maths is All around you textbook' lists numbers 1-5 on a page and requires the child to select counters and place how many they think the number represents (see image 1 below). This activity will allow the participants to gain awareness of the correspondence between the numeral they see and the quantity of each number. This will assist Tom and Jack's previous learning, as they begin to recognise how many blocks should be grouped to add up to a particular number. This exercise could also be modified by providing buckets with numbers labeled on the front of them, and getting the children to place pegs or balls inside, to correspond with the number that is written on the bucket. To challenge learning even more, creating boxes with numbers going up to 10 or higher will test the child's numerate counting, and (with assistance from an educator) they will have the opportunity to learn about larger or higher numbers in the number sequence.
Any of these activities can be done at home with parents or caregivers using various household materials that are appropriate for these exercises, to further assist with numerical counting in the early years.
Image 1
References
I believe that it will also be wise to conduct the exact same initial observation again with Tom and Jack, but this time have an educator appropriately intervene, and reverse the roles by asking Jack to count the blocks first. In the observation, Tom took the lead most of the time and had the first opportunity to count. Jack was not given an opportunity to show his rote learning skills and it seemed as though Jack was following on from what Tom had done most of the time, and may not quite understand the full concept of counting in sequence. This will help to detect if this is an issue in Jack's rote counting, and therefore other implications can then be put in place to assist his learning if needed.
There are many opportunities to further children's learning and development for numerical counting. In taking a completely different approach to the original counting experience of Tom and Jack, is an activity which focuses on subitising, by using counters to represent the number quantity. This activity located in Marianne Knaus' 'Maths is All around you textbook' lists numbers 1-5 on a page and requires the child to select counters and place how many they think the number represents (see image 1 below). This activity will allow the participants to gain awareness of the correspondence between the numeral they see and the quantity of each number. This will assist Tom and Jack's previous learning, as they begin to recognise how many blocks should be grouped to add up to a particular number. This exercise could also be modified by providing buckets with numbers labeled on the front of them, and getting the children to place pegs or balls inside, to correspond with the number that is written on the bucket. To challenge learning even more, creating boxes with numbers going up to 10 or higher will test the child's numerate counting, and (with assistance from an educator) they will have the opportunity to learn about larger or higher numbers in the number sequence.
Any of these activities can be done at home with parents or caregivers using various household materials that are appropriate for these exercises, to further assist with numerical counting in the early years.
Image 1
Department of Education, Employment and Workplace Relations for the Council of Australian Governments (2009). Belonging, Being and Becoming, The Early Years Learning Framework
Knaus, M. (2013). Maths is all around you (1st ed.). Albert Park, Victoria: Teaching Solutions
Videatives,. Math- Combining sets. Retrieved from http://streaming.videatives.com/assets/1180
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