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**Developing Mathematical Thinking In Number : Place Value**

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Aim of presentation To encourage staff reflection on approaches to teaching number. To stimulate professional dialogue. To use as a CPD activity for staff individually or collegiately. The purpose of this material is for staff to use this individually or with colleagues as part of CPD. The slides give information and also questions to prompt discussion and thinking. The PowerPoint notes give further commentary.

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**Experiences and Outcomes**

I have explored numbers, understanding that they represent quantities, and I can use them to count, create sequences and describe order. MNU 0-02a I have investigated how whole numbers are constructed, can understand the importance of zero within the system and can use my knowledge to explain the link between a digit, its place and its value. MNU 1-02a I have extended the range of whole numbers I can work with and having explored how decimal fractions are constructed, can explain the link between a digit, its place and its value. MNU 2-02a This presentation is concerned with Place Value. The experiences and outcomes which directly relate to these are shown on this slide.

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Progression The next few slides indicate some key concepts which are necessary for pupil understanding. Curriculum for excellence is looking to develop depth of understanding. Key developmental steps and considerations are highlighted in the next few slides. Throughout learning pupils will revisit these concepts. Understanding of place value is fundamental to progressing learning around number. By establishing secure understanding of place value pupils will have greater confidence when tackling addition and subtraction, multiplication and division, working with large numbers and decimal fractions.

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Progression Using practical materials and sets of objects when beginning counting and using numbers. Emphasise the link between the “number words” and the objects. One-to-one correspondence It is essential in all aspects of number that pupils have understanding of the number processes being undertaken. At the early stages this involves physically touching and counting items. Each object when being counted is given a name when this is touched of the item . These items would be familiar play objects before progressing to linking plastic cubes and unit blocks. It should be emphasised to pupils they are undertaking the process of counting. The classroom / playroom environment are vital in having formal and informal opportunities for this to take place. 1 2 3 4 5 Pictorial: What would you like to count? How many cars did you count? Create a number rich maths environment.

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Subitising This is an important early developmental step. It is the ability to recognising a small number of objects without counting. Research has shown that children who have difficulty with subitising are more likely to show the numbers difficulties associated with dyscalculia. Opportunity to practice these skills from an early age are vital. What opportunities currently are there for developing this understanding within your practice?

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**Counting challenges The English language can be confusing for children learning number names**

Gaelic word Literal Meaning Japanese word English Word 10 deich Ju Ten 11 aon dheug 1 + 10 Ju-ichi Ten-one Eleven 12 dà dheug 2 + 10 Ju-ni Ten-two Twelve 13 trì deug 3 + 10 Ju-san Ten-three Thirteen 14 ceithir deug 4 + 10 Ju-shi Ten-four Fourteen 15 cóig deug 5 + 10 Ju-go Ten-five Fifteen 16 sia deug 6 + 10 Ju-roku Ten-six Sixteen 17 seachd deug 7 + 10 Ju-shichi Ten-seven Seventeen 18 ochd deug 8 + 10 Ju-hachi Ten-eight Eighteen 19 naoi deug 9 + 10 Ju-kyu Ten-nine Nineteen 20 fichead Ni-ju Two-ten Twenty Is there a consistent pattern in English vocabulary ? Do the number names link to number names 1-10? For young children learning numbers and the associated number concepts the English language does not lend itself to making sense of the situation. The words eleven, twelve, thirteen, fifteen have little relation to the numbers they represent. This is not the same in all languages eg consider Gaelic where each word contains the meaning “+10” and in Japanese the language pattern of the literal words are very clear. In Japanese the ‘ten’ element comes first whereas the word ‘teen’ (indicating between 10 and 20) comes second in English. This is contrary to how the number is written ie in 14 – the one digit comes first, teen is said second. This is not the same issue for the numbers from 20, thirty – the first ‘word’ that is said is also the first digit we write. This may appear relatively minor and most children cope with this however when learning number in the first instance this can sometimes cause difficulty. We cannot change the number system but it is worth being aware of the potential difficulties which might be encountered.

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**Language : Counting up and down ?**

Consider counting up in 10s from 6 What direction are the numbers going when counting up? Again with language we need to be careful. If the term “counting up” is used with this 100 square the direction that pupils will go in, when following this pattern is down. Care needs to be taken eg consider “counting on” or ...

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**Language : Counting up and down ?**

Consider counting up in 10s from 6 What direction are the numbers going now when counting up? Vocabulary – counting on, counting back Using a number square which starts in the bottom left with the smallest number. Is this better for pupils experiencing at the first level?

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Starting from 0 ? 90 91 92 93 94 95 96 97 98 99 80 81 82 83 84 85 86 87 88 89 70 71 72 73 74 75 76 77 78 79 60 61 62 63 64 65 66 67 68 69 50 51 52 53 54 55 56 57 58 59 40 41 42 43 44 45 46 47 48 49 30 31 32 33 34 35 36 37 38 39 20 21 22 23 24 25 26 27 28 29 10 11 12 13 14 15 16 17 18 19 1 2 3 4 5 6 7 8 9 91 92 93 94 95 96 97 98 99 100 81 82 83 84 85 86 87 88 89 90 71 72 73 74 75 76 77 78 79 80 61 62 63 64 65 66 67 68 69 70 51 52 53 54 55 56 57 58 59 60 41 42 43 44 45 46 47 48 49 50 31 32 33 34 35 36 37 38 39 40 21 22 23 24 25 26 27 28 29 30 11 12 13 14 15 16 17 18 19 20 1 2 3 4 5 6 7 8 9 10 Depending on whether a 100 square starts at 0 or 1 will result in the 100 square going to 99 or 100. Some children as part of early number encounter difficulty with ‘crossing decades’. The square on the left maybe emphasises the crossing more by having to ‘travel’ to this point. Which 100 square emphasises the Early Level – key concept of “counting on and back from a given number including ‘crossing the decades’ ” ? e.g. 18,19, 20, 21, 22

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**A very brief history of zero**

It took the human race about years to come up with the notion of 0. This being the Indians around 500 AD. Zero was first thought of in the context of writing numbers down eg to distinguish between 4, 40 , 400, Zero was introduced as a place holder. Zero is particularly useful in calculations – allowing the alignment of digits under column headings. eg think about trying calculations without converting Roman numerals to our number system

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**Begin to understand place value and use it to compare numbers, partition numbers and in calculating.**

3 7 3 7 We can use partitioning in calculations. Some pupils may think this, others may require to jot this thinking down (30 +2) + (40 + 6) 70 + 8 The phrase “Begin to understand place value ...” comes from which gives support in identifying some key concepts in number at different stages. The use of practical material is vital in developing this understanding – ‘longs’ representing tens and ‘cubes’ representing units. Place value arrow cards which allow ‘sliding’ on top of each other are a useful support to show the how the numbers are constructed. This visualising is vital – it develops into a mental strategy which many of us may employ when adding or subtracting ie separating out the tens and units. 37 is 3 tens and 7 units

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**Understanding and using decimal notation and place value in decimal fractions.**

1.27 Our decimal money system lends itself for pupils to understand and work with place value and emphasising the column value of digits. Eg ten 1p coins being equivalent to a 10p coin One hundred 1p coins the same as £1 and ten 10p coins A hundred square can also provide the visual representation of whole items, tenths and hundredths. Developing understanding is vital rather than just ‘recall’ of facts. Secure future learning is built on strong foundations. £1.27

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**Next steps What information will you share with colleagues?**

What might you or your staff do differently in the classroom? What impact will this have on your practice? As a result of looking at this presentation what are the next steps for you and your staff ? What else can you do as to improve learning and teaching about number

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