Assignment - Place value and the role of visual representations PDF

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Assignment - Place value and the role of visual representations...

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Place value and the role of visual representations

Our number system is underpinned by ‘place value’ (Hansen, 2017). This term refers to how big a number is (Clausen-May, 2013), how it is represented in written form (Hansen, 2017) and how it allows us to ‘express numbers more concisely’ (Haylock and Cockburn, 2017:76). It is a system that allows larger numbers to be efficiently explored, (Taylor and Harris, 2013) revolving around the Hindu-Arabic method, which became popular in the 15th Century. It is the most effective method for representing numbers, although hard for children to conceptualise (Hansen, 2017). For children to understand place value and its uses within mathematics, it is crucial they recognise certain key principles (Hansen, 2017). These conventions consist of understanding digits (there are 10 – 0, 1, 2, 3, 4, 5, 6, 7, 8, 9), recognising that the position of a digit determines its value, knowing that 0 is a placeholder, and understanding grouping and exchange (once there are ten objects in a column, they can be exchanged for one object in the column to left, etc.). Both Hansen (2017:17) and Haylock and Cockburn (2017) emphasize that there are only 10 digits in the system, recognising that without knowledge of these key principles, children will struggle to progress in mathematics. This means they are unable to conceptualise place value (Hansen, 2017), and hence be incapable of handling numbers and calculations with confidence (Haylock and Cockburn, 2017). Studies show that an early years understanding of place value is a good predictor of arithmetic performance in later primary years (Moeller et al., 2011). According to Hansen (2017:17), there are various structures which children must have knowledge of if they are to ‘progress from counting to representing numbers to written calculations’. Children must be given varied opportunities to practice place value as it is not a concept that can be taught quickly (Smith, 2014), and it cannot be learnt ‘all at once’ (Thompson, 2003). With a developing understanding of place value, children can use reasoning to justify mathematical decisions when problem solving (The NRICH Primary Team, 2014), which is why it is so crucial for them to begin to comprehend the concept. Understanding the role of place value supports children in developing their number sense and fluency; being aware that zero is a placeholder allows children to be able to differentiate between numbers such as 2, 20, 200 and begin to confidently use these numbers (Smith, 2014: 439). As the National Curriculum states (DfE, updated 2021) children in Key Stage 1 should ‘develop confidence and mental fluency with numbers, counting and place value’. Ross’ (2002) work 12 years previous to Smith (2014), highlight an understanding of the importance of place value, denoting that children need to understand the concept in order to achieve good number sense, estimation and mental mathematic skills, whilst also understanding ‘multi-digit operations’.

2 Ross (2002:419) recognises that for a child to understand place value, they must be able to make ‘connections among and sense of a highly complex system for symbolizing quantities’. The mathematical properties identified include: 

Additive property



Positional property



Base-ten property



Multiplicative property

It is the role of the teacher, given the complexity of these properties, to provide an experience that involves representations, interpretation and discussion to children in order for them to fully comprehend place value (Taylor and Harris, 2013:137). The following section of this assignment will discuss Ross’ (2002) additive and base-ten properties of place value, and how they can be represented. The additive property of place value is defined as a method that uses visual representations so that children can identify numbers in their ones, tens or hundreds columns, etc. (Ross, 2002:419). It allows children to differentiate between digits, enabling them to break down larger numbers into more manageable sizes (Taylor and Harris, 2013). The method refers to the way in which numbers can be written in expanded notation, based on their place value (Rhys et al., 2012), and how a number can be split into individual components (Taylor and Harris, 2013). For example, the number 739 is worth 700 + 30 + 9. For children to be able to partition numbers into hundreds, tens and ones, they need to understand that the value of any digit is determined by its position within a number (Taylor and Harris, 2013). This is referred to as Ross’ positional property (2002). Without an understanding of the positional property, children will struggle to break down a number into its additive form as they do not have the appropriate knowledge of how the position of a digit affects its value (Taylor and Harris, 2013). Ross’ (2002) base-ten property refers to the movement of a number within the tens columns. Each digit in a number can have a value ranging from 0 to 9 depending on its position (NCTM, 2015). The rule for the property consists that each number position is 10 times the value to the right of it, and 10 times less to the left, showing how the decimal place moves either left or right depending on if the number is being divided or multiplied by 10. It allows children to understand that 0 is a placeholder, hence proving essential for progression in mathematics (Ross, 2002). A key principle of base-ten is that it enables large numbers to be represented that otherwise physically could not be. For example, the number 1251

3 cannot be easily demonstrated with a model, but when split into thousands, hundreds, tens and units, it can be partitioned into 1 thousand, 2 hundreds, 5 tens and 1 unit (Clausen-May, 2013). Children gain knowledge of numbers through watching and copying teachers and peers (Anghileri, 2006). Therefore, visual representations are important as they can help children comprehend place value. Ross’ (2002) additive property can be represented with resources such as arrow cards (Taylor and Harris, 2013) as they support children with partitioning numbers into their correct place value (Anghileri, 2006). Figure 1 shows the additive value of 297 using arrow cards, illustrating how the number can be partitioned. By placing the 90 and the 7 on top of the 200, the cards show how the symbol 297 is made up of 2 hundreds, 9 tens and 7 units. We then construct the number sentence as two hundred and ninety-seven when we say it aloud (Coles and Sinclair, 2017). The cards provide a picture-like representation for children, allowing them to differentiate between the values of different numbers (Haylock and Cockburn, 2017), using their knowledge of positional property. However, a common misconception when learning place value revolves around children believing that numbers with more digits are larger (MaLT, 2005). As arrow cards break down a number into its appropriate place value, children can misunderstand this and assume a number is bigger than it is. If teachers can generate appropriate mathematical conversations with children, activities involving arrow cards can promote connections between symbols, language and visual representations (Haylock and Cockburn, 2017), allowing children understand how the number 297 is equivalent to 200 + 90 + 7.

Figure 1: Arrow cards showing the additive value of 297, broken into 2 hundreds, 9 tens and 7 units.

Diene blocks

are an

appropriate

method to

represent the base-ten system as they can aid children's understanding of place value (NCTM, 2000). The

4 blocks are used to represent numbers by grouping them into place value columns; each flat block is worth 100, a rod 10 and each unit is 1. Figure 2 illustrates how blocks can split a number into its hundreds, tens and units – the number 125 is broken into 1 hundred, 2 tens and 5 ones. Again, Ross’ (2002) positional property is essential for children to comprehend so they can partition large numbers into their correct place value. The blocks can allow children to understand that zero acts as a placeholder. When placing the dienes in a chart (figure 2), children can see that zero maintains the values of other digits (Lawton and Hansen, 2011). For example, in a case study from Taylor and Harris (2013) a child read 206 as 26, and 60 as 6. With the use of the blocks, children can explicitly see that zero is a placeholder, maintaining the values of other digits. There are limitations to using diene blocks, however. When applying skills used with the blocks to more concrete materials like money, children can struggle to understand. According to Haylock and Cockburn (2017), using money as a model gives children a different experience as they must recognise that £1 is equal to 1 flat block, or 10p is equivalent to 1 rod or 10 individual units. Children must learn to connect materials with symbols, demonstrating their understanding by selecting appropriate blocks to match a 2 or 3 digit number (Haylock and Cockburn, 2017).

Figure 2: This figure shows diene blocks representing a lot of 1 hundred, 2 lots of tens and 5 units.

5 Because place value is so vital to developing a mathematical identity, it is incredibly important that as teachers we allow children to establish an understanding of the concept so they can connect ideas, materials and symbols together, aiding their progression in mathematics (Haylock and Cockburn, 2017). Ball, Thames and Phelps (2008) recommend understanding the common misconceptions around place value so that errors can be ‘unraveled’ to help children learn. I recognise that this has many implications for me, as I will need a solid understanding of Ross’ (2002) properties, and how to implement them with representations in the classroom. I know that place value is complex to teach; I will need to dedicate time to developing techniques and strategies to appropriately teach the topic. 1496 words Reference list Anghileri, J. (2006) Teaching Number Sense. 2nd Edn. Continuum. Ball, D., Thames, M. and Phelps, G. (2008) Content Knowledge for Teaching: What Makes It Special? Sage Journal. Clausen-May, T. (2013) Teaching Mathematics Visually and Actively. 2nd Edn. London: Sage. 27-34 Coles, A., & Sinclair, N. (2017) Re-thinking place value: from metaphor to metonym. For the Learning of Mathematics. [Available at: http://flmjournal.org/index.php? do=details&lang=en&vol=37&num=1&pages=3- 8&ArtID=1130, accessed 13th February 2021] Department for Education (updated 2021) National curriculum in England: mathematics programmes of study [Available at: National curriculum in England: mathematics programmes of study - GOV.UK (www.gov.uk), accessed 6th February 2021] Hansen, A. (2017) Number and place value. In: Children’s Errors in Mathematics. 4th Exeter: Learning Matters. Sage. Haylock, D. and Cockburn, A. (2017) Understanding Place Value. In: Understanding Mathematics for Young Children: A Guide for Teachers of Children 3-7. 5th London: Sage. Lawton, F. and Hansen, A. (2011) Children's Errors in Mathematics. Learning Matters. Sage. MaLT (2005) Test 5 PK10, Julian Williams (Mathematics Assessment for Learning and Teaching)

6 NCTM Interactive Institute (2015) Base Ten and Place Value [Available at: NCTM Illuminations, accessed 11th February 2021] NCTM (2000) Principles and Standards for School Mathematics [Available at: NCTM-Standards-2000GradesPreK-2nd.pdf (citruscollege.edu), accessed 11th February 2021] Reys, R. E., Falle, J., and Bennett, S. (2012). Helping children learn mathematics. Milton, Australia: John Wiley & Sons Australia. Ross, S (2002) Place Value: Problem Solving and Written Assessment. In: Teaching Children Mathematics. National Council of Teachers of Mathematics. [Available at: Place Value: Problem Solving and Written Assessment (oclc.org), accessed 8th February 2021] Smith, S (2014) Early Number, Counting and Place Value. In: Primary Mathematics for Trainee Teachers. London: Sage. Taylor, H. and Harris, A. (2013) Early number counting and place value. In: Learning and Teaching Mathematics. Sage (part 1) Taylor, H. and Harris, A. (2013) Early number counting and place value. In: Learning and Teaching Mathematics. London: Sage. (part 2) pg 137-144 The NRICH Primary Team, 2014 [Available at: Problem Solving (maths.org), accessed 8th February 2021] Thompson I. (2003) Place value: the English disease? In: Thompson I. (ed.) Enhancing Primary Mathematics Teaching. Maidenhead: Open University Press. (Figure 1 and 2 are my own images)...