Mathematics

**The simple answer is - lots of different ones**.
We do not use standard algorithms until the children have a firm understanding of number and are ready for the formal writing of maths. Introducing the standard algorithms too early may result in a child being able to do a sum but not understand what they are doing and why. As far as possible we use mental methods and use jottings to keep numbers (or keep them in the head). One of the most important aspects of mathematics is getting your child to **say how they did their calculation and why they chose that way to work it out.**
 * __Remember__** there are many ways of working the same problem out and there is no one right way.

**Phases children go through in Mathematics**
A lot of research into how we learn about maths and the misconceptions that children get, through their own misunderstanding or unclear teaching, has been done in Australia. As a result they introduced a recommended way of teaching Mathematics (First Steps in Maths) based on the phases the children go through and the next stage they need to develop to move on. Children need to be ready to move on and be able to understand previous concepts to be able to do so. They found that trying to do formal algorithms at a young age will result in children being able to do a process without understanding what it is they are doing. This will be a short term gain but a long term hindrance to their development.
 * The technical bit**

Most children will go through six phases during their time in school (up to the end of the comprehensive school or sooner as everyone develops at different speeds) During the **Emergent Phase** (usually aged between 3 and 5 years) your child will reason about small amounts of physical materials, learning to distinguish small collections by size and recognise increases and decreases in them. They will also recognise and repeat number words and number symbols. Children at this stage recognise words and numbers can be used to symbolise numerosity.

They then move to the **Matching Phase** (usually aged between 5 and 6 or more years). They use numbers to describe actual quantities of physical materials. They learn to do what is expected when you ask questions like how many are ......? can you give me four.........

They then move on to the **Quantifying Phase** (usually aged between 6 and 9 years). Now they can reason about numerical quantities and come to believe that if nothing is added to or removed from a collection or quantity then the total amount must remain the same, even if its arrangement or appearance is altered. They can interpret small numbers as compositions of other numbers.

Next comes the **Partitioning Phase** (usually aged between 9 and 11 years). Children now see that numbers can be used to count groups and that they can use one group as a representative of other groups. They trust too, that appropriate partitioning of quantities must produce equal portions. If you are still awake and want to know more - between 11 and 13 they will be in the **Factoring Phase** then they move on to the **Operating Phase**.

For more information on this you can visit @http://www.ecurl.com.au/uk/index.asp

=**So how can you help**= Ask questions and get your child to tell you how they arrive at an answer. Use visual aids with younger children or older children who are having difficulty seeing relationships between numbers. Work on non standard partitioning as well as standard partitioning (what is 37 - 10 + 10 + 10 + 7 or 20 + 17 or 30 + 7 or 30 + 5 + 2 and so on) When your child starts working on word problems - **act it out first** e.g. John has 20p and goes to the shop to buy a bar of chocolate for 12p How much change will he have?
 * The older the child the more difficult it becomes to help - If you do have older children then don't** introduce misconceptions when solving mental problems by saying things like .... if you multiply, the number will be bigger or if you divide the answer will be smaller (If you multiply a fraction by a fraction the answer is a smaller fraction......dividing a fraction by a fraction gives a larger fraction). If you have older children be careful when dealing with decimals - ask yourself which is the larger 0.6 or 0.123 If you think it is 0.123 then avoid helping as you, like many others, have not understood positional values.


 * Let's start at the beginning** - children need lots of experience counting objects so they make the mental link between the number of objects and the number name. They need to be able to recognise the number of objects without having to physically count each one (to 10) before moving on.

1 Count (by rote) to 5. . . then 10 and 20, and beyond (as a group) 2 Count forwards from a given number within 10 (‘What comes next?’) 3 Count backwards from 10 4 Find 1 more than / less than numbers up to 6 5 Find 1 more than / less than numbers within 10 6 Count backwards from a given number within 10 (‘What comes before?’) 7 Recall some of the number bonds, discovered through pattern-making, involving up to 10 objects (eg 6: ‘3 and 3’ or ‘2 and 2 and 2’) 8 Recall bonds to 6 9 Find 2 more than / less than numbers within 10 10 Count back in the context of subtraction from 6 11 Count back in the context of subtraction from 10 12 Find the difference between two numbers within 10 by ‘counting on’ (eg 10p – 7p: ‘. . . 8, 9, 10’) 13 Interpret + and – signs orally in the context of a number story 14 Know and use the ‘doubles’ up to ‘5 and 5’ 15 Recall bonds to 10
 * Number Bonds to 10**

1 Count forwards and backwards up to / from 20 2 Find 1 more than / less than numbers within 11 to 20 3 Count forwards and backwards starting from a given number within 20 4 Recall some of the number bonds, discovered through pattern-making, involving 11 to 20 objects (eg 12: ‘6 and 6’ or ‘4 and 4 and 4’) 5 Find 2 more than / less than numbers within 11 to 20 6 Know and use the ‘doubles’ up to 10 + 10 7 Add onto 10 (eg know that 10 + 4 = 14, 10 + 7 = 17 etc) 8 Know and use the ‘near doubles’ within 11 to 20 (eg knowing that 7 + 7 = 14, work out that 7 + 8 must be 14 + 1 = 15) 9 Find the difference between two numbers within 20 by ‘counting on’ (eg 16p and 20p: ‘17, 18, 19, 20’) 10 Find the difference between two numbers within 20, using knowledge of bonds up to 10 (eg work out the difference between 12 and 20 by knowing that 2 + 8 = 10, therefore the difference is 8) 11 Add within 20 without bridging 10, using bonds within 10 (eg 13 + 4 = [3 + 4] + 10 = 17) 12 Subtract within 20 without bridging 10, using bonds within 10 (eg 18 – 6 = 10 + [8 – 6] = 12) 13 Recognise tens in a string of numbers (eg 7 + 6 + 3, seen as 10 + 6 = 16) 14 Add onto 9 as ‘10 less 1’ (eg 9 + 6, seen as [10 + 6] – 1 = 15) 15 Find the difference between two numbers within 20, by rounding up to 10 (eg 20 – 8, seen as 2 + 10 = 12) 16 Add, by making up to 20 (eg 8 + 6, seen as 8 + 2 + 4) 17 Subtract, by reducing to 10 (eg 13 – 5, seen as 13 – 3 – 2) 18 Recall number bonds to 20 19 Complementary addition to 50 by rounding up / down to the nearest ‘ten’ first (eg 50p – 27p, seen as 3p + 20p = 23p) 20 Complementary addition to 100 (starting within 50 to 100), rounding up down to the nearest ‘ten’ first (eg £1.00 – 78p, seen as 2p + 20p = 22p) 21 Complementary addition to 100, starting below 50 (eg £100 – £36, seen as £14 + £50 = £64)
 * Number Bonds 11 to 20+**

**Addition/subtraction**
Being able to picture numbers and their relations is another important aspect of number work - i.e. what is eight (8) it is 7and 1, 6 and 2, 2 less than 10 etc How would you add 34 + 45 do you add the 4 and 5 first keep 9 in your head then add the 30 and 40 together and add on the 9 you either kept in your head or jotted down? Is there another method you used, did you start with the tens? Did you think of the 34 as 35 (add 1 on) then add the 35 and 45 to get 80 then take the 1 away? What about adding 39 and 40. How would you do that - would you use the same method? As you can see for that simple sum there are many ways of doing it and by having a flexible approach allows you to choose the best method for the numbers you are dealing with. By recognising the bonds that make 10 - you can use them to add stings of numbers 1 + 8 + 2 + 9 Did you start with the 1 and then add 8 then 2 then 9 or did you pair the 8 and 2 and 9 and 1? If you used the first method you probably don't have a trust in numbers that they will always give the same answer - if you saw the patterns you probably understand the relationships between numbers. How would you tackle 9 - 5 ? How would you do 60 - 45 ? How would you do 60 - 21 ? Would you use the same method for all those if you were working them out in your head or would you use a different method for each one?

**Tables or Skip Counting**
There is a place for learning tables but the most important aspect is using them and understand what the tables represent. We tend to do a lot of work on using tables - some children/adults can memorise them without difficulty - but can they use them? We tend to work on the traditional reciting, do skip counting i.e. 2 times table 2 4 6 8 etc count down in and play a game with the table - Wobble (which is a non alcoholic game of buzz for those who are old enough to remember it). This encourages concentration on the numbers that don't form part of the table as well as those numbers that do.

1 Understand many-to-one correspondence 2 Sort using a base-board to experience sharing out and making groups 3 Count on in 2s within 20 4 Count on in 10s within 100 5 Count on in 5s within 100 6 Link repeated addition and multiplication (eg using hoops / number-lines) 7 Internalise multiplication tables up to 5 x 5 8 Rote knowledge of the 2, 5 and 10 times multiplication tables 9 Recognise numbers divisible by 2, 5 and/or 10 10 Use understanding of the fact that if, for example 4 x 5 = 20, then 5 x 4 also equals 20 11 Internalise division tables within 5 x 5 (applying understanding of the relationship between multiplication and division) 12 Know and use the ‘squares’ (ie 1x1, 2x2, 3x3, 4x4, 5x5, 6x6, 7x7, 8x8, 9x9, 10x10) 13 Know the 3x and 4x multiplication tables and associated division facts 14 Know and use the ‘near squares’ (eg knowing that 8 x 8 = 64, work out that 9 x 8 must be 64 +8 =72) 15 Use understanding of the commutative property of multiplication (eg 8x4 = 4x8 = 32) 16 Know associated division facts (ie know, for example, that since 4 x 5 = 20, then 20 ÷ 5 = 4 and 20 ÷ 4 = 55) 17 Recognise numbers divisible by 3 (ie know that if the sum of the digits is divisible by 3, then the original number is also divisible by 3 eg 51: 5 + 1 = 6 and since 6 is divisible by 3, 51 must also be divisible by 3) 18 Know the 6x and 7x multiplication tables and associated division facts 19 Recognise numbers divisible by 6 (ie know that if a number is divisible by 2 AND 3, then it is also divisible by 6 eg 54, 342, 1044) 20 Know how to multiply by 9 by first multiplying by 10, then subtracting one ‘lot’ (eg 9 x 6, seen as [10 x 6] – 6) 21 Know how to multiply by 8 either multiplying by 10, then subtracting two ‘lots’ (eg 8 x 6 seen as [10 x 6] – 12), or by doubling the 4x table 22 Know the 8x and 9x multiplication table and associated division facts 23 Use strategies for multiplying beyond the 10 ‘times table’ 24 Know that with multiplication, you can multiply and divide number pairs by the same factor without altering the answer (eg 24 x 6 = 12 x 12 = 144, 24 x 25 = 12 x 50 = 6 x 100 = 600)
 * Multiplication and Division**

**More on Multiplication/division**
Being able to double numbers and halve them adds lots of flexibility into the way you can work out a sum and working from the previous work done on halving and doubling the more able Year 5 / Year 6 pupils will eventually work out something like 65 x 54 in the following way,.

A lot of practical work needs to be done first but eventually you could. Think about 65 × 54 × 54 is the same as × 50 + × 4. Therefore 65 × 54 is as easy as (a), (b), (c) so using jottings you may think of (a) 50 as half of 100 so 65 x 100 is 6500 half of 6000 is 3000 and half of 500 is 250 so 50 x 65 is 3250 (b) 65 x 4 is the same as 65x 2 x 2 or double 65 which is 130 and double that 260 Add the 260 and 3250 you get 3510

If you think about division how would you share 6 sweets between 2 people - firstly you would get 6 sweets and 2 people to physically share them out. Would you give 1 each then another 1 each - give 2 each then 1 each or 3 each? The same method can be used when dealing with large numbers Think about 340 ÷ 15

15| 340 | Think about maths fact you know at the tip of your fingers i.e. X10 ,,, __- 150__ | × 10 ,,,,, 190 | You may see that double 150 is 300 so double 10 to give ,,, __- 150__ | × 10 you 20 rather than take away two lots of 150. Method works the same ,,,,,,, 40 | ,,,,, __- 30__ | × 2  ,,,,,,, 10 ,,,,,,,,, Tot. 22 remainder 10

If you need more information feel free to contact our Maths Coordinator Mrs Tozer or Mr Rees