Developing Subject Knowledge

Ethical Statement

The names of the participants mentioned in this assignment have been changed in accordance with The British Education Research Association (Bera, 2011) guidelines pertaining to confidentially and anonymity. In order to maintain confidentiality, the guidelines mentioned in the Data Protection Act 1998 are followed in writing the assignment. Moreover, to maintain ethics, the informed consent of the participants are taken which means that they have understood and agreed to participate to inform about their activity without any duress so that the information may be used in writing the assignment. Further, each participant is offered the authority to withdraw their consent from using their work in executing the assignment at any time.

Activity

Mandy (class teacher) worked with a group of 5-year-old children teaching them to count to five. She introduced the numbers by singing and using a hand puppet (each finger on the puppet was a duck). While singing the five little ducks song she encouraged the children to use their hands as well adding and taking away fingers.

To recall groupings within five for example 1 and 4 make 5, 3 and 2 make 5 To recognise subtraction, for example, take away 1 and you have 4 left The key role of Mandy is to encourage the children is to get them involved in the activity for understanding and counting of 5 in an easy manner by using their five fingers and five duck adding and subtracting method. This is because it is been informed that making the children to involve their five fingers to count sets an environment for ordinal and cardinal aspect to come together (Haylock and Manning, 2014, p. 67). The singing with five plastic ducks is used as additional resource.

Children’s learning

All four children were able to use their fingers while counting up and back down again from 1-5, 5-1. All children were able to subtract one duck during each verse until there were no ducks left. All children could sing the familiar number song by memory. They could count up to 5 objects reliably using 1:1 correspondence. Joshua said ‘we can take 2 away instead of just 1, or 3 or 4, or 5'. Mandy asked Joshua ‘if I take 2 ducks away from 5 how many are left?' Joshua answered correctly. Mandy explained Joshua has grasped the understanding of subtracting numbers 1-5 and he was able to use relevant vocabulary to explain his meaning.

Key mathematical ideas and vocabulary

1. Creativity in maths (singing while learning maths). By introducing a song children were engaged and had great behaviour as argued by Clark. (week 9, 4.2). This is because singing while doing maths leads the children to have better cognitive skill leading them to more easily understand and count numbers to do maths.

2. The development of understanding the number delivers opportunity for the children to develop connection between language and objects of numbers (Haylock and Manning, 2014, p. 67)

3. Key vocabulary; take away, how many left?, how many are not? How many do not? (Haylock and Manning, 2014, p. 92)

4. The comment made by Joshua informs that he has developed effective knowledge about conversation regarding numbers which is evident as he has shown proper understanding of 5 by counting with fingers and subtracting ducks (Haylock and Manning, 2014, p. 32).

By pointing to each object in turn ranking and numbering them, one, two, three, four, five the child is using ordinal aspect and by counting one, two, three, four and five with fingers the child is showing cardinal aspect (Haylock and Manning, 2014, p. 67).

Role of resources/mental imagery

The use of the hand puppet and involvement of children using their fingers and singing was a great way to get the children’s attention. This is because it was fun activity for the children to easily learn counting through mental imagery. Moreover, the activity shows good behaviour by all children and they enjoyed taking part both in acting and singing. The use of singing props was very helpful as it allows a very tactile-kinesthetic way to act out and physically move objects while doing it. The introduction of plastic ducks added to help the children learn addition and subtraction through mental imagery.

The visual number cards 1-5 could have been useful to ensure the children’s understanding of numbers for example number 4 card being shown when four ducks are left. The number is the concept represented by the numeral and therefore consists of a whole network of connections (Haylock and Manning, 2014, p.65).

The strategy of signing a song while doing a numeracy activity was a great response from the children as they all showed better cognitive learning response and efficient understating of numbers. I was interested in Haylock and Manning (2014, 0. 91) comment to different operations of subtraction structures and how each of these has their own characteristic language patterns. However, Joshua had used appropriate vocabulary. I am looking for opportunities to develop the children’s knowledge and understanding further by asking them to match the numeral card to the correct set of ducks. I think children are ready to move to execute addition and subtraction.

6. Mental calculation strategies

Activity

Miss Black (primary 6 teachers) worked with a group of nine-year-old children where her task was to teach addition and subtracting strategies. She wrote examples 7+(13+18)= on the whiteboard followed by a vertical subtraction format 201 -20 by allowing the children to use any number if needed. The learning objectives are:

To develop an understanding of place value To mentally figuring out answers without paper, pencils or calculators. To explain how they came up with the given answer. To develop an understanding regarding the way to mentally compute maths To learn subtraction and addition by using informal mental strategies with three digit numbers.

The key role of Miss Black is to develop encouragement of the children to count forward and backwards with efficiency by using informal mental strategies for working out answers. According to the National Curriculum in England, she informed that the children required being able to solve addition and subtraction of multi-step mathematical problems by deciding which operation and methods are to be used and the reason behind their use to reach the solution (DFE, 2013, p. 39) week 8 2.1.

Children’s learning

All children knew to start with the brackets first in the addition sum but doing the subtraction sum their answers varied as for instance, some students forgot to add or subtract the carry over left resulting them to reach wrong solutions. During the lesson, Jack said ‘if you write the sum again and put the brackets around the 7+13 then we would get the same answer for example (7+13)+18=. However, during subtraction, the Jack forgot to carry over left numbers resulting him to reach wrong answers. This is because while subtracting 13-7 he reached 16 as he forgot to subtract 1 carried when executing the sum.

Key mathematical ideas and vocabulary

1. The use of vocabulary such as counting backwards and forwards in ones, tens and hundreds is an essential prerequisite (Haylock and Manning, 2014, p.106).

2. The communicative and associate law of addition together give us the freedom to add a story of numbers together in any order we like (Haylock and Manning, 2014, p. 104).

3. The development of strong images of a number square/line is seen to provide support in the process of counting back as well as on (Haylock and Manning, 2014, p. 107)

4. The main language to be used in indicating the reductions made in subtraction involve reduce by and start by, go down and count back, carry one and others (Haylock and Manning, 2014, p.89).

5. The key language to be developed in the augmentation structure of addition includes start at and count on, increase by and go up by.

Jack's comment about the brackets demonstrates a grasp of meaningful-learning mindset (Haylock and Manning, 2014, p.31).

Role of resources/mental imagery

e of the array itself is powerful as the children can visibly see the sums wrote on the board. The number squares were introduced to help work out answers using mental imagery. In case paper and pencils are given to the children for working out answers it would have led them to develop greater confidence in answering out the solution loud. Furthermore, rhetorical questions could have been asked for example ‘why do you think it is important to learn maths mentally?’.

Reflections and next steps for children’s learning

The children’s responses are seemed to show that they understand the way to demonstrating communitive and associate law in executing addition. However, the children’s wrong responses informed that the vertical layout of subtraction was found by them to be difficult to execute it mentally. Thus, I think that to improve subtraction the children are to be provided a with few jolting along the way (Haylock and Manning, 2014,p. 104). The opportunities for the children to develop better understanding could be to introduce coins or base ten blocks as Haylocks (p.123) explains in his learning and teaching point. I was interested in Haylock (p.126) learning to teach point that informs in case the method for decomposition is being learnt in the process without even understanding it then the children are going to initiate exchanging one in one column for ten in next even when it is not been required. I think the children after excelling subtraction along with addition they would be able to move on to execute multiplication and division.

Jenny (classroom assistant) worked with a group of seven-year-old children. They were making play dough where they had to follow the recipe by measuring accurately using flour, water, cream of that tar, food colouring, salt and oil by using appropriate measuring utensils. The learning objectives: To teach children life skills To introduce concepts related to science and maths To reinforce important lessons about staying safe while using sharp tools, boiling water and others To understand measurements and volume, for example, grams, litres, etc

To follow instructions To learn the way basic cooking ingredients can be used to make play dough while bolstering maths learning Jenny’s role was to assure all the children got involved even though they worked in pairs she encouraged each child to measure out at least two measurements each. She explains the cooking utensils and how they can measure, for example, scales, clear jug, spoons etc. She is seen to give them a problem to solve saying ‘the recipe said 2 cups of flour and we know half a cup is equal to 170 grams how many grams do we need to measure in our clear jug?’. Jenny also encouraged the children to look at the ingredients to find the mass, volume etc to help them become a better estimator (Haylock and Manning, 2014, p. 340).

Children’s learning

The children were able to develop knowledge and skills regarding measurements through purposeful and practical activities. In this process, the children were informed about the ideas regarding accuracy and approximation which led them to develop proper selection and use of appropriate measuring devices. They also learned imperial units and metric units in this process. Moreover, they learned concepts regarding numbers and meaning of zero.

The use of correct language while explaining the meaning of measurements like mass and weight are to be used. The conflict experienced between language in everyday purpose and scientifically correct usage of language cannot be resolved just by using the word weight and mass interchangeably. This conflict between everyday language usage and the scientifically correct usage is not resolved simply by using the two words, mass and weight, interchangeably. Thus, understanding of metric units, for example, measuring the mass s1 unit, kilogram, abbreviation kg (week 9) is also required.

The use of appropriate tools stated in the recipe and making sure they measure in the correct unit asked, for example, clear jug for kilograms. This is required to let the children usefully memorize a few key approximate conversions between grams and kilograms (Haylock and Manning, 2014, p 341). The effective understanding to follow the recipes instruction was the way in which materials could be corrected mixed by the children in developing the play dough. The microwave heating was helpful for the children to understand the way materials are transformed with new properties.

The children responses show that most of them were able to correctly measure and use measuring appliances in the activity to create great results. The children felt happy to play with the dough and were surprised to know their ability to do and make things. Jessica said ‘I have played with play dough before but not play dough I made myself’. She was delighted with how it turned out due to the correct measurements of ingredients. I was interested in Haylock and Manning (2014, p338/p342) as it restricts the range of metric units used for practical work in primary school to litre, millimetre, kilogram and do not avoid use of imperial units that are still being implement in everyday purpose. I think the children are ready to move in making mathematical implication of the knowledge developed to solve measurement problems.

Nicole (class teacher) was working with a group of eight-year-old children where her duty was to teach them shapes, names the shape, counting sides, corners and sorting them into sorting hoops. Learning objectives: -To make sense of 2D shapes and developing geometric concepts -To identify an attribute common to certain shapes such as how many sides are there in the set, etc -To describe and visualise the features of 2D shapes Nicole’s role was to give the children the opportunity to explore the properties of various 2d shapes and to encourage them to use a range of relevant vocabulary and assess their understanding of 2D shapes. In this process, questioning and prompting the children as suggested in Week 9 can be used. The explorations of 2D shapes are important because classification mathematics enables children to develop and understand new concepts (Haylock and Manning, 2014, p.29).

All children were more familiar with basic shapes such as square, circle triangle etc but when it came to hexagon, pentagon, octagon etc they were not sure. They were able to count sides of all the shapes correctly and group them by the same as or same amount of sides. When teacher asked the children to count the number of sides on an octagon, Chelsea said ‘I know an octagon has eight sides just like an octopus has eight legs’. Thus, this informs that after the learning the children were able to develop ideas about complex shapes too.

The use of various attributes of shapes to be taught informally includes helping the children to look for exemplars and non-exemplars and discussing the relationship between shapes in terms of symmetry and differences (Haylock and Manning, 2014, p. 385). The comment by Chelsea about Octopus show the extent of encouragement she has in remembering the shapes. In discussing shapes, the word ‘side’ is to be restricted to mean the straight edges of the polygon (Haylock and Manning, 2014,p. 385).

The sorting hoops were a good way for the children to visibly see the shapes in their groups as they were different coloured shapes. However, the different colours could get the children be confused in grouping the shapes as per colour instead of actual shapes. Moreover, questions like ‘it has four equal sides what am I?’ could be asked to check the memorisation power of the children regarding the shapes. Further, more questions could be arranged to make the children learn about more shapes to memorise. For example, a hexagon has six sides remember the ‘x’ in hexagon and ‘x’ in six or a heptagon looks like a 50 pence piece and show the children a 50 pence piece beside the shape.

The children’s responses from the activity inform that they have effectively understood and develop knowledge about basic shapes (Haylock and Manning, 2014, p. 32). This is evident as Chelsea was able to connect that octagon has 8 sides just like an octopus has 8 legs. I think the children are ready to move to identify the shapes in real life situations.

In this assignment, I will be reflecting back on my learning during block 2 and identifying and discussing the mathematics concepts on properties of shapes or transformations and symmetry and procedures of mental calculation strategies. I will be explaining how my understanding has developed and how the online activities in block 1 and 2, online discussions with fellow students on e-groups and discussions with teachers have contributed to this understanding and to my strengths and weaknesses. During my activity properties of shapes or transformations and symmetry, I read Shapes, Space and Symmetry by Alan Holden (2012) to develop knowledge regarding the way children can be helped to learn about shapes and symmetry. The book informed me regarding the way visual models can be used to simply inform children about the shapes and symmetry of different objects. The book led me to learn how pictures can also be used to educate children about shapes. The online discussion with the tutors led me to identify that information regarding shapes and symmetry can be educated to the children in a better way by using 2D and 3D shapes models and structures. This is because presence of visual models helps the children more effectively document and memorise the symmetry and number sides of the shape without getting lost due to confusion and complexity of the shape while explained verbally (Brunner et al. 2017).

The use of 2D visual models to inform about shape and symmetry is also effective for the children is evident from the activity of Nicole where she used 2D models to help children easily learn complex shapes and symmetry such as hexagon, octagon, decagon and others. Moreover, it was helpful to let the children remember about the sides of the shape as evident from one of the students comment when he was able to relate eight sides of the octagon is like the eight legs of the octopus. This informs that using examples of things that have similar number of features like the sides of the shapes and symmetry of the objects being taught to the children it would be effective to help them memorise the properties of transformations and symmetry. My further analysis of the activity informs that while learning through 2D model regarding shapes and symmetry by the children it may make them get confused with different colour of models they are using for grouping objects according to shapes. Thus, I have developed the perception that same colour 2D shapes models are to be used while initially educating children about shapes and symmetry to avoid them get confused in grouping objects according to shapes. My discussion with teachers and children in the school informs that using 3D model to educate children regarding properties of shapes and symmetry could be more effective in nature. This is because observation of 3D models helps children to resolve the difficulty of visualising 2D models of objects to understand shapes and symmetry in real life (Christiansen et al. 2015). Thus, this nature of models is able to render photorealistic memorisation of the properties of shapes and symmetry of objects to the children. Moreover, the 3D models offer better insight to the children regarding surface patterns and have virtual image of object’s shape and symmetry rendering them better knowledge about shapes and symmetry. In the activity related to mental calculation strategies, I developed the knowledge that many children were unable to accept the process with success. On analysis, I understood that this may have happened because not all the students have equal mental efficiency to calculate mentally and provide right answers to the mathematical problems. This is because cognitive intelligence differs from one person to another (Malone and Bernstein, 2015). Thus, I think that providing paper and pencil to all the students while doing mathematics through mental strategies is effective way to resolve the hindrances. This is because the students who have lower mental efficiency to solve the problems entirely in informal mental thinking or wish to be sure of the answers regarding problems to be provided by using mental imagery would have ability to be sure of their solutions.

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My discussion with the online fellow students and tutor informed me that ABACAS can be taught to the children for helping them to solve maths through mental imagery strategy. This is because in ABACUS students are taught regarding the way to contiguate, design and visualise primers so that with speed and proper memory along with accurate they are able to solve problems in maths correctly (Brooks et al. 2018). According to Barner et al. (2018), doing maths through mental imagery strategy require huge focus and greater memorisation power regarding calculations. The ABACUS is seen to improve the cognitive intelligence of the children which boost them to use their brain more and develop better focus to informally calculate solutions for mathematical problems without hindrance (Barner et al. 2016). Thus, I feel that I require developing training in ABACUS so that I am able to help students to learn maths through mental imagery in an efficient manner. My discussion with other teachers and adults in the school informs that students can be educated maths through mental imagery strategies if they are allowed to learn ways to count fingers and used them in keeping the carried and leftover numbers while doing subtraction or addition. This is because finger counting helps the brain to process as well as represent numerical information for a long time assisting the children to memorise and calculate the solution of mathematical problems easily while doing addition and subtraction. Thus, by completing the maths audit and online activities it is seen that it has helped immensely to develop key knowledge regarding the way to teach children about properties of shapes and symmetry along with mental imagery use to calculate. The interaction with the teachers and asking questions has helped me to identify the further knowledge I required to excel as a maths teacher in the two mentioned aspects. The activity has helped learn lot of aspects that can be applied by me in real life to help children develop easy strategies to calculate numbers and figure out shapes and symmetry.

References

• Barner, D., Alvarez, G., Sullivan, J., Brooks, N., Srinivasan, M. and Frank, M.C., 2016. Learning mathematics in a visuospatial format: A randomized, controlled trial of mental abacus instruction. Child development, 87(4), pp.1146-1158.
• Barner, D., Athanasopoulou, A., Chu, J., Lewis, M., Marchand, E., Schneider, R. and Frank, M., 2018. A one-year classroom-randomized trial of mental abacus instruction for first-and second-grade students. Journal of Numerical Cognition, 3(3), pp.540-558.
• Brooks, N.B., Barner, D., Frank, M. and Goldin‐Meadow, S., 2018. The role of gesture in supporting mental representations: The case of mental abacus arithmetic. Cognitive science, 42(2), pp.554-575.
• Brunner, J., Baburin, I.A., Sturm, S., Kvashnina, K., Rossberg, A., Pietsch, T., Andreev, S., Sturm, E. and Cölfen, H., 2017. Self‐Assembled Magnetite Mesocrystalline Films: Toward Structural Evolution from 2D to 3D Superlattices. Advanced Materials Interfaces, 4(1), p.1600431.
• Christiansen, A.N., Bærentzen, J.A., Nobel-Jørgensen, M., Aage, N. and Sigmund, O., 2015. Combined shape and topology optimization of 3D structures. Computers & Graphics, 46, pp.25-35.
• Haylock,D. and Manning, R. (2014) Mathematics Explained for Primary Teachers, 5thedn London, Sage.
• Holden, A., 2012. Shapes, space, and symmetry. Dover Publications.
• Malone, T.W. and Bernstein, M.S. eds., 2015. Handbook of collective intelligence. MIT Press.