Mathematics Task 3: Developing knowledge and understanding of numbers and calculation in young children
The early learning goal for numbers (DfE, 2014: 1) is that:
Children count reliably with numbers from one to 20, place them in order and say which number is one more or one less than a given number. Using quantities and objects, they add and subtract two single-digit numbers and count forwards or backwards to find the answer. They solve problems, including doubling, halving and sharing.
This ‘numbers’ goal focuses on skills, concepts and operations (Gifford, 2014). Yet, when children start school, they are at different stages in terms of their familiarity with numbers so their experiences at primary school are aimed at developing their knowledge and understanding of numbers. Drawing on examples from the literature as well as personal observation, this discussion focuses on the strategies used to develop early skills in numbers and calculation.
Gifford (2005) has summarised how children learn mathematics through cognitive, emotional, social and physical processes. These develop a number of strategies such as reciting the number sequence (including skip counting in twos, fives and tens), keeping track of the objects counted, using pattern recognition and number games. The common aim of these activities is to develop children’s curiosity and familiarity with numbers (Gifford, 2014).
While observing interactive learning in the Reception phase, I witnessed some of these activities. Two children at the Numeracy table stated a number and then took the relevant number of bears to place on their own sheet. After watching the children doing this a few times, I interacted with them by asking how many bears were left when I took one away or added one. By relating to the concrete numbers of bears, they were able to give the relevant answer. Then one pupil represented the number of bears she had by a form of marking. Although the marks she used were not the standard symbols, her actions gave her the opportunity to contribute to her understanding of numbers (Carruthers and Worthington, 2004). This example highlights the social and physical learning processes identified by Gifford (2005) and shows the importance of having a concrete and meaningful context to develop children’s approach to numbers in the early years. This has been identified extensively in the literature (see, for example, Anghileri, 2006; Hughes 1986; QCA, 1999).
Following on from my observations, I noted that personal jottings support children’s understanding of calculation. In a primary school following the programme ‘Primary Advantage’, I observed the use of a maths book with alternating squared and blank pages. This was to encourage pupils to do some personal jotting on the blank page to demonstrate their own thinking. That this was successful was confirmed by the leadership team who identified better results in mental maths and problem solving from those obtained before implementing the programme.
Teachers have a key role to play in supporting and enhancing children’s development of calculation strategies. The Teaching mental calculation strategies guidance (QCA, 1999) suggests that teachers can raise pupils’ awareness of a wide range of strategies, model them, and guide children in using efficient methods.
From his study of the early learning of numbers, Anghileri (2006) argues that a wide range of language types should be used. This builds on the idea developed by Hughes (1986) who claims that children have to develop links between the abstract mathematical language (e.g. ‘make’ and ‘take away’) and their own concrete language. The teacher’s role is therefore to link the abstractions of formalised mathematics to the concrete of pupils’ lives and experiences. Furthermore, Anghileri (2006) explains that teachers need to expose and discuss common misconceptions in the classroom, as errors occur when children follow routes without an adequate understanding of the processes involved.
Children’s experience of early numbers and calculation can be supported by the use of concrete material, models and images when the aim of their use is to help children’s understanding. Heads and fingers, the number line, and the 100 squares apparatus are some resources used in the classroom to support pupils’ learning (Hughes, 1986). For instance, teachers can offer pupils concrete examples showing that the addition order is irrelevant and that addition and subtraction are inverses of each other.
Despite the strategies and support identified above, children can still fail to secure an understanding of the number system and early calculation strategies; the factors leading to this situation cannot be ignored. These factors can be classified into two categories: teaching strategies and children’s understanding.
A report about teaching calculation in primary schools (HMI, 2002: 3-4) explains that ‘teachers rely too much on worksheets and commercial schemes, particularly in Key Stage 1. [This] limits opportunities for pupils to develop and use their own methods of recording’. Moreover, although Anghileri (2006) argues that children should use a wide range of strategies, including their own, teachers sometimes fail to recognise children’s own reasoning due to a lack of subject knowledge and/or practice. This could lead to some children feeling lost or insecure with their own method.
Other factors leading to children’s difficulties in understanding early numbers and calculation are directly linked to children’s attitude and understanding. For instance, I observed that because pupils do not take the time to estimate before calculating, they do not have an idea of what would constitute a sensible answer. Difficulties in understanding the language of mathematics (Hughes, 1986) can also be a real barrier to understanding numbers and calculation strategies. This is particularly relevant when the language remains unfamiliar and lacks concrete referents. Some children can also struggle to identify derived facts (Carpenter and Moser, 1983). For example, thinking that 9 is 10−1 to calculate 5+9: children will see the transformation of the problem as a new process and therefore a new difficulty.
Difficulties in number development can arise for children who have English as an additional language. Indeed, Gifford (2014: 223) gives the example that ‘“fourteen”’ sounds similar to “forty” and suggests writing the “four” digit first, as 41 (Thompson, 2008). Therefore the teen numbers may be particularly challenging for children with hearing difficulties, dyslexia or English as an additional language’. In line with the conclusions of this argument, and drawing on my observation of a Year 2 lesson, I noticed that some children had difficulties blending the hundreds with the tens and units. For example, they wrote one hundred twenty-five as ‘100 25’. Their understanding of place value was affected by the sound. Although the teacher insisted on using the hundreds/tens/units table, some children could not relate what they heard to what they should have written.
This discussion has shown that when learning about numbers and calculation, young children use processes that inform strategies and practices in the classroom. Teachers support and enhance children’s learning by modelling, analysing errors and guiding pupils effectively. However, some factors, including teaching strategies and children’s specific needs, can lead to children failing to secure their understanding of the number system and early calculation strategies.
It remains vital for teachers to help children developing active thinking and an understanding of mathematics by providing a safe risk-taking environment where children can make mistakes and have opportunities to discuss their work.
Anghileri, J. (2006) Addition and Subtraction. In Anghileri, Teaching number sense, (pp 49-70). London: Continuum
Carpenter, T. P., & Moser, J. M. (1983). The acquisition of addition and subtraction concepts. In R. Lesh & M. Landau (Eds.), The acquisition of mathematical concepts and processes (pp. 7-14). New York: Academic Press.
Carruthers, E. & Worthington, M. (2004), ‘Young children exploring early calculation’, Mathematics Teaching, 187, pp. 30 – 34, June.
DfE (2014) Early Learning Goal 11: Numbers. Online. Available at https://www.gov.uk/government/uploads/system/uploads/attachment_data/file/360535/ELG11___Numbers.pdf (accessed 3 January 2015)
Gifford, S. (2005) Number. In Gifford, Teaching Mathematics 3-5 (pp77-103). Maidenhead: Open University Press
Gifford, S. (2014) ‘A good foundation for number learning for five-year-olds? An evaluation of the English Early Learning ‘Numbers’ Goal in the light of research’, Research, Mathematics Education, 16:3, pp. 219-233
HMI (2002) Teaching of calculation in primary schools, HMI 461, London, pp 3-4
Hughes, M. (1986) What’s so hard about two and two in. In Hughes, Children and number: difficulties in learning mathematics (pp 37-52), Oxford: Basil Blakewell
QCA (1999), Teaching Mental Calculation Strategies: guidance for teachers at key stages 1 and 2. Online. Available at http://www.nationalstemcentre.org.uk/elibrary/resource/4552/teaching-mental-calculation-strategies-guidance-for-teachers-at-key-stages-1-and-2 (accessed 12 December 2014)
Thompson, I. (2008). From counting to deriving number facts. In I. Thompson (Ed.), Teaching and learning early number (pp. 97–109). Maidenhead: Open University Press/McGraw Hill.)