addition - wyche.worcs.sch.uk currriculum/pedagogy/… · web viewthe truth is that children need...

Addition

Contents

Mathematical Understanding: An Introduction

2

Section 1 - Understanding Place Value: A school Based Progression

4

Section 2 - Understanding Addition: A school Based Progression

7

Mental Calculation Strategies: Progression into Key Stage 2

10

Section 3 - Understanding Subtraction: A school Based Progression

12


16

Section 4 - Understanding Multiplication: A school Based Progression

18

Key Concepts

18

Prior Learning

18

A school Based Progression

19


21

Section 5 - Understanding Division: A school Based Progression

24

Key Concepts

24

A school Based Progression

25


26

Section 6 – Assessment: Use of the Spreadsheet – Notes to teachers

28

Appendix 1: Mathematical Vocabulary

31

Appendix 2: The Instant Recall of Times Tables

32

Appendix 3: Examples of Misunderstanding

34

Mathematical Understanding

An Introduction

It is a strange truth but true nevertheless that children can undertake quite complex calculations well into Key stage 2 and yet have little understanding of the mathematical processes involved or the numerical principles underpinning them. Whilst in the short term this might appear satisfactory the reality is that without the foundations in place children become vulnerable when new concepts are overlaid on to insecure understanding.

Two examples of this will suffice; (see appendix for other examples)

At the end of a series of lessons a group of more able Year 6 children were happily calculating 28% of 324 with great accuracy, however there progress was somewhat derailed when they were asked to explain what a percentage was. Two instantly responded that they did not know and the closest we came were that two girls knew it had something to do with 100.

Similarly the Year 6 child this year who could calculate TU x TU came a little unstuck when confronted with the calculation 36x38. He calculated 36x30 which he found equalled 1080, but then realising that he still needed to multiply the 8 proceeded to calculate 1080x8. Apart from the fact that he was not able to readily see that his answer was woefully inaccurate in terms of an estimate, it materialised upon further discussion that he had no idea what the sum 36x38 meant. He could not “see” that it could be interpreted as 38 people all having 36 pence and was totally at a loss as to what numerical operations he would need to undertake to gain the correct answer.

This does not bode well for his individual mathematical development and it is imperative that the school’s Maths curriculum delivers a true understanding of mathematical principles alongside a simple ability to calculate and number crunch.

In many ways therefore this Mathematical understanding document is more important than its calculation counterpart. It seeks to break down the development of calculation into its constituent parts and outlines clear areas of understanding that children need to take on board before they can truly make well grounded progress in Mathematics. It is not enough for us to teach that “If you multiply by 10 then you add a nought on”, because it is mathematically weak and in year 5 the children will find that adding a zero to 2.3x10 does not deliver the right answer. The key is that we engage children in the true, pure principles of Mathematics, which in this case is that the zero acts as a placeholder. This dovetails into the school’s work on Learning Objectives and the attempts to tease out at planning level what the “true” learning is for each lesson, rather than satisfying ourselves with the rather bland and meaningless learning objective of “Divide a two digit number of 100 by 10”. Whilst at first glance this lesson would appear to be devoted to division, the true learning will probably hinge around place value and the relationship of numbers, this purer learning objective should be expressed in the teacher’s daily planning.

The document outlines the stages of mathematical cognitive development children need to attain before moving on and this document should be read and used alongside the associated assessment spreadsheets that the school has developed. This shows an individual children’s progression in the four rules and will allow the teacher to develop clear personalised learning based on areas of relative weakness, thereby plugging the gaps in their understanding which can easily be unwittingly built upon.

The document fulfils two main purposes:

1. Primarily it provides a systematic programme of mathematical understanding that children can progress through in the early primary years. In the first instance this will be focused heavily on the early years of the National Curriculum in Key Stage 1 and then into Key Stage 2 where understanding needs to be consolidated for certain children. Whilst there is no a direct bridging point between this document and its “Calculation” counterpart teachers should be wary of developing calculation strategies too early before secure understanding is in place.

2. However the document fulfils another purpose for children who have progressed on to the calculation document. In the multiplication and division strand the progression provides a framework for the development of a range of mental strategies that the children should be fluent in alongside the more formal written methods they are being introduced to through the Calculation document. Whilst the school acknowledges the need to focus on one core calculation strategy throughout the school (the number line), there was also the recognition that we did not want to lose the wealth and richness of calculation strategies that the original Numeracy framework introduced. Therefore this document seeks to track children’s progress in these without, hopefully, detracting from the central core strategy of the number line.

The key to all this is the school’s contention that in its purest and highest form Mathematics ceases to be based around the ability to undertake basic number calculations but instead becomes increasingly focused around patterns, relationships and the development of theorems to explain these. Therefore if children are to make good mathematical progress in the latter years of their education they need to leave their primary schools with a secure foundation in Mathematical understanding not simply an ability to calculate using large numbers.

Section 1

Understanding Place Value: A School Based Progression

Stage

Understanding to Acquire

Errors and Misconceptions

Assessment Task

1

Numbers 0-10

Order consecutive cardinal numbers up to 10 orally

Children misplace the order of the numbers

Orally state the numbers

2

Order consecutive cardinal numbers up to 10 practically

Children struggle to order the numbers

Practically demonstrate the concept of consecutive numbers

3

Order consecutive cardinal numbers up to 10 written

The inability to decode the symbols of the numerical language

Write the numbers under 10 consecutively

4

Add 1 to a given number

The inability to know 4+1 may indicate poor understanding of consecutive numbers

Be able to explain why 5+1 and 8+1 are “easy sums” to know

5

Order cardinal numbers

Ability to run off the numbers 1-10 in order may mask a true understanding

Order the following numbers 7, 3, 6, 1 and 4.

6

Numbers 10-20

Order numbers 10 – 20

The children often struggle with the teens numbers where the fourteen implies the 4 comes first e.g. 41 not 14

Orally and written the children should demonstrate their ability to order the numbers from 0-20

7

Understanding of the place value of tens and units

12 = 1 ten and 2 units not vice versa

Understand 12 is one ten and 2 units, explaining the role of the tens and units column

8

Understand labels of teens numbers

Thirteen is 13 not 31 despite what the language suggests

Write and correctly label the numbers 13,14,15,16,17,18 and 19

9

Understand relationship of numbers in columns

A 9 in the units column is not a greater number than a 2 in the tens column

Explain how and why the 9 in the units column is smaller that the 2 in the tens column

10

Numbers 20-100

Appreciation of base 10 i.e. an understanding that when unit numbers reach a ten a new ten is created

Questions like “What is the next number after 19?” Explain why?

11

Order number 20-100

Ensure the children have a good grasp of the pattern in the tens numbers i.e. 1,2,3,4 tens

12

Know the number 10 more than a given number

The children should appreciate that to add one ten to 27 will create a number 1 ten bigger i.e. 37

27 + 10 =37

13

Ability to see 20 as 2 tens

Ability to see the 4 in 43 as 4 tens not just as an isolated number

14

Ability to see 63 as 6 tens and 3 units

Ability to see the 6 in 63 as 6 tens and the 3 as 3 units, and also to understand the relationship between them

15

A secure understanding that place determines value

Understanding that to take a 9 from the units and put it in the tens markedly changes its value

Place a 9 on a tens and units grid and allow children to interpret its value

16

Understand the role of zero as a place holder

That the zero in 60 is not superfluous but “places” the 6 in the tens column. Without it the 6 reverts to 6 units

Ask the children the value of 6; then ask the value of 60. What is the difference and how do they know?

17

Numbers to 1000

Ordering numbers up to 1000

The children should be able to order the numbers up to 1000

18

Ordering numbers using the highest denominator.

When the hundreds are identical then use the tens column to ascertain the highest number

Take the numbers 245, 236 and 222 and use the tens numbers to order them.

19

Calculate to the nearest 10 with numbers under 100

Ability to round numbers

The children should be able to round 34 down to 30, round up 36 to 40 and know 35 is halfway inbetween.

Section 2

Understanding Addition: A school Based Progression

Stage

Sum



Assessment Task

1

4+3=

Lack of understanding as to what happens in the process of addition

The concept of two numbers forming a new total

Place 4 cubes and 3 cubes on the table. What will happen when I add them?

2

8+0=80

8+0=0

Lack of understanding of the role of zero. Some see the zero as a number which should make a larger total, where others see zero as the dominant number and therefore the answer is zero

The value of zero in a variety of contexts (not to be confused with zero as a place holder)

8+0=

Can they explain the answer?

3

3+1=5

Cannot identify number before or after a given number

The order of the numbers needs to be established.

What is 3+1? Can the children explain why it is 4 i.e. because 4 is the next number

4

4+2=

Can only begin counting at one; Cannot hold the first number as an initial total

The initial number does not change it is the addition process that changes the total

4+2= How did you work the sum out?

5

10 can be split into 6+4

Conservation of number

Understanding that with 10 cubes you can create different addition sums without changing the total

Give the children 10 cubes and see whether they can make groups with them whilst understanding the total remains the same

6

Understanding of Number Bonds to 10

Number bonds to 10

Understand that there are a range of sums that can be created and that these are in a pattern

Put the number 10 in the middle of a piece of paper. Can they write down the number bonds to 10

7

Number Bonds

To build on number bonds of other numbers

There is an inverse pattern in the bonds i.e. 9+1 and 1+9

Teacher writes 9+1 and seeks to elucidate from the child whether they can “see” that this allows them easy access to 1+9

8

Number Bonds


Whatever the number used the family of number bonds will be one more

“If I choose 8 how many number bonds will there be?”

9

Rapid Recall of number bonds

Children unable to recall number bonds under 10 with any rapidity

The children can recall instantly all the number bonds for the family of numbers under 10

Test questions and instant recall either written or verbal

10

+2=5

Unable to translate comprehension of “usual addition” into new context. Inability to use concept of inverse operations

Introduction to algebraic maths such that concepts such as if a+b=c then

c-b=a and c-a=b

Give the children the sum +2=5 and ask if they can give the answer

11

2+8

Doesn’t use the largest number to count on from

Lack of appreciation of specific number value and the concept of “adding on”

Give the children 2+8 and see how they calculate the answer. They should also explain why they use the bigger number first

12

3+4+7

To counteract the feeling that addition is only ever two numbers

To introduce the concept as a precursor to repeated addition that the children need for multiplication

Teacher writes sum and asks whether the sum is a “real one”

13

8+10=19

Lack of tens to units correlation

Establishing the introduction of mathematical pattern

8+10=18

Why does it equal 18?

14

8+7=22

Has no feel for the answer being between 10-20

A concept of the relative value of specific numbers

Give the children 8+7 and ask them to point on a number line where the answer might lie. They must not calculate the answer

15

4+6=7+3

Inability to understand the role of the = sign within a sum. i.e. it is not simply the place where the answer goes

Understand the meaning of = not necessarily being the end of the sum

Write the sum and ask “Does it make sense?”

Can the children write their own sums?

16

7+3+5 = 10+5

Inability to build/see relationships between numbers

Ability to see number bonds in multi-step calculations

Ask the children to calculate 7+3+5 and determine whether they used number bonds in calculation.

Write it out as a two step sum what numbers would write?

17

+3 is the opposite of -3

Inability to see subtraction as the inverse

Specific links between the 4 rules should be taught at every opportunity

Write 3+2=5 then 5-2=4. Ask the children why this is wrong.

18

21+2; 31+2; 41+2

Inability to see consistent pattern in the unit column

Teacher writes down the sums 21+2; 31+2; 41+2 and asks the children what 51+2 would be

19

8+20; 8+30

Lack of extended tens and units correlation, leading to counting on

Further work on mathematical patterning

Teacher writes 8+20=28, 8+30=38, and then asks for the answer to 8+40. The children should explain why they know the answer is 48.

20

46+5, 46+15, 46+25

Unable to recognise patterns to help deduce addition facts with numbers other than “tens numbers”

Patterns relative to addition in tens needs to be securely established

Write the sums 46+5, 46+15, 46+25 with the answers. Can the children deduce the total of 46+35

Associated vocabulary: add, more, and, make, sum, total, altogether, score, double, one more, plus, near double, addition, equals,=

Mental Calculation Strategies: Progression from KS1 and into Key Stage 2

The following is a digest of the mental calculation strategies introduced by the National Numeracy Strategy. Whilst the major calculation focus in KS2 is the development of the number line, these strategies are crucial in developing children’s mental capability in calculation. The first question a child should ask before embarking on any calculation is; “Can I do this in my head?” These strategies are central to developing a clear sense of number and a fluency in calculation that will enable children to tackle an increasing number of mathematical problems mentally. To this end they supplement rather than supersede the school’s core work on the number line. (See “School Calculation” document)

Whilst these may be taught discretely there is a single sentence tucked unobtrusively at the bottom of pages 46-49 of the NNS document that simply states “Use and apply these skills in a variety of contexts, in mathematics and other subjects”. This is a vital aspect of the child’s mathematical development. The strategies must be taught through a range of contexts so that children imbibe them holistically into their learning and use them correctly in contexts beyond the constraints of the formal maths lesson.

NNS

Exemplar

Year 2

Year 3

Year 4

Year 5

Year 6

Doubles

e.g. 8+9 = 8+8 +1

U

TU

HTU

TH HTU

Odd/Even

Relationship: 2 odds=even

U

TU

Add 9/99/999

e.g. 22+9 = 22+10-1

HTU

TH HTU

Add 11/101

e.g. 22+11=22+10+1

HTU

TH HTU

Commutative Law

p34

e.g. 95+86=86+95

TU+TU

TU+TU

HTU+TU

HTU+HTU

Associative Law

p34

e.g. (25+17)+18 = 42+18

TU+TU

HTU+TU

HTU+HTU

Derived facts-place value

p38

e.g. 7+9=16 so 70+90=160

TU+TU

HTU+TU

Decimals

Counting on in steps

p40

e.g. 643+50 adding in tens

TU+TU

HTU+TU

Partition

p40

e.g. 24+58=20+70 and 4+8

TU+TU

TU+TU

HTU+TU

Identify near doubles

p40

e.g. 36+35=35+35+1

TU+TU

HTU+HTU

Decimals

HTU

Add to a multiple of 10

p40

e.g. 74+58 = 74+60-2

TU+TU

TU+TU

HTU+TU

Decimals

Addition/Subtraction

p42

e.g. 56+14=70 so 70-56+14

TU+TU

TU+TU

TH,HTU

Add several numbers

p42

e.g. 70+71+75+77=70x4+13

U

TU

TU

Use Known Facts and place value to add or subtract a pair of numbers mentally (see NNS p44-47)

Add multiples of 10

p44

e.g. 570+250

TU

TU

HTU

TH,HTU

Add multiples of 100

p44

e.g. 500+700+400

HTU

HTU+HTU

H+H+H

Add to multiple of 100

p44

e.g. 58+x=100

TU

TU

HTU

Add to multiple of 1000

p44

e.g. 3200 + x = 4000

TH,HTU

TH,HTU

Decimals

Add single digit to HTU

p46

e.g. 6745+8

TU

TH,HTU

Add pair of 2 digits

p46

e.g. 0.05+0.3

TU

TU

Decimals

Decimals

2 places

Section 3

Understanding Subtraction: A School Based Progression

Stage

Sum



Assessment Task

1

4-3=

Subtraction is the concept of “taking away”

The concept of the fact that subtraction involves some of the original total being removed.

What is subtraction? Get the children to use cubes to illustrate their understanding

2

5-10=5

Confusion of the place of the numbers within the context of the sum

In principle subtraction involves the smaller number being subtracted from a larger total

Roll 2 die and make a subtraction sum with them. get the children to explain the rationale behind the order of the numbers they chose.

3

5-1=3

Cannot identify number before a given number

The order of the numbers needs to be established.

5-1=4

Why is the answer 4?

How did you know?

4

8-0=0

Lack of understanding of the role of zero.

The value of zero in a variety of contexts (not to be confused with zero as a place holder)

8-0=

Can they explain the answer?

5

10-4=6

When you have taken 4 away 6 remain.

The principle of subtraction using 10

Ask the question 10-4=6

Get the child to illustrate with cubes

6

10-4=6

If you start with 10 and 6 remain then 4 must have been taken

Counting on principle to establish the difference

Place 10 cubes on the table. Take 4 away without the children knowing. With them being able to see the 6 remaining cubes can they calculate how many are in your hand?

7

10-6=4 to put 4 back =10

Conservation of number in relation to addition and subtraction be the inverse

Understanding that with 10 cubes you can create different subtraction sums all of which total 10 in the number remaining and the number removed

Give the children 10 cubes and allow them to illustrate the principle with the cubes

8

10-9; 10-8


There is an pattern in the bonds to 10 i.e. 10-1; 10-2; 10-3 etc.

Allow the children to design their own subtraction patterns. If this fails…

Teacher writes 10-1; 10-2; 10-3 with answers. Can you write the next sum in the sequence

9

Number Bonds

Inability to see patterns in calculation process

Whatever the number used the family of number bonds will be one more

If I have the number 8 how many subtraction sums can I make in that family?

10

10-7=3

Cannot see that 7+3 may help them solve this equation

Ability to use “known fact” i.e. 7+3=10 to calculate (at speed) 10-7=3

Give the children 7+3=10. Then ask them the answer to 10-7. How do they know the answer is 3

11

“What is the difference between 7 and 10?”

Inability to understand the Mathematical vocabulary of “Difference” in the question “What is the difference between 7 and 10?”

To understand the use of the word “Difference in a mathematical context”

“What is the difference between 7 and 10?”

12

10-=5

Unable to translate comprehension of “usual subtraction” into new context. Inability to use concept of inverse operations

Introduction to algebraic maths such that concepts such as if c-b=a then

a+b=c and b+a=c

Give the children the sum 10-=5 and ask if they can give the answer

13

10-2-3=

Unable to see that a subtraction sum can contain more than two numbers

Introducing subtraction with more than two numbers

Does it make sense? Can it be done? Explain

14

19-10=8

Lack of tens to units correlation

Establishing the introduction of mathematical pattern

Write the sum 19-10=8 and ask children to explain why it cannot be correct.

15

15-7=12

Has no feel for the answer being between less than 10

A concept of the relative value of specific numbers

Give the children the sum 15-7 and on a number line get them to guess where the answer might lie.

16

6-4=7-5

Inability to understand the role of the = sign within a sum. i.e. it is not simply the place where the answer goes

Understand the meaning of = not necessarily being the end of the sum

Write the sum and ask “Does it make sense?”

Can the children write their own sums?

17

+3 is the opposite of -3

Inability to see subtraction as the inverse

Specific links between the 4 rules should be taught at every opportunity

Write 10-7=3 then 3+8=10. Ask the children why this is wrong.

18

19-2; 29-2; 39-2 etc.

Inability to see consistent pattern in the unit column

Develop patterning skills using tens

Write 19-2; 29-2; 39-2

What will 49-2 be?

19

59-20; 49-20;39-20 etc.

Lack of extended tens and units correlation, leading to counting on

Further work on mathematical patterning

Write 59-20; 49-20;39-20

What will 29-20 be?

20

46-5, 46-15, 46-25

Unable to recognise patterns to help deduce subtraction facts with numbers other than “tens numbers”

Patterns relative to subtraction in tens needs to be securely established

Write the sums 46-5, 46-15, 46-25 with the answers.

Can the children calculate 46-35?

21

83-79=

The child takes the 83 and starts to subtract 7 tens

The understanding that when numbers are close together the best strategy is to find the difference by counting on.

Model 83-79 by taking away 7 tens and then 9 units.

Ask the children; Is there a quicker way?

Mental Calculation Strategies: Subtraction

Progression into Key Stage 2



NNS

Exemplar

Year 2

Year 3

Year 4

Year 5

Year 6

Halves

e.g. 16-8=8 as 8 is half of 16

TU

HTU

Odd/Even

Relationship: 2 odds=even

TU

HTU

TH HTU

Subtract 9/99/999

e.g. 22-9 = 22-10+1

TU/HTU

TH HTU

Subtract 11/101

e.g. 22-11=22-10-1

TU/HTU

TH HTU

Subtraction is non-commutative

p36

Understand 7-5 is not 5-7

TU

TU-TU

Derived facts-place value

p38

160-90=70 so 1600-900=700

TU-TU

HTU-TU

Decimals

Counting on in steps

p40

e.g. 387-50 count back in tens

TU-TU

HTU-TU

Partition

p40

e.g. 98-43=98-40-3

TU-TU

TU-TU

HTU-TU

Subtract 9 to multiples of 10

p40

e.g. 84-19=84-20+1

TU-TU

TU-TU

HTU-TU

Decimals

Addition/Subtraction

p42

e.g. 83-25=58 so 83-58=25

TU-TU

TU-TU

TH,HTU

Use Known Facts and place value to add or subtract a pair of numbers mentally (see NNS p44-47)

Subtract multiples of 10

p44

e.g. 130-50

TU

TU

HTU

TH,HTU

Subtract multiples of 100

p44

e.g. 1200-500

TU-TU

HTU-HTU

Subtract a multiple of 10 or 100

p44

e.g. 582-30 or 1263-400

TU-TU

HTU

HTU

Subtract single digit to HTU

p46

e.g. 900-7

HTU

TH,HTU

Find difference when numbers lie either side of a multiple of 1000

p46

e.g. 7003 -6988 Counting on

HTU

TH,HTU

TH,HTU

Subtract pair of 2 digits crossing tens boundary

p46

e.g. 0.5-0.19

TU

Decimals

Decimals 2 places

Section 4

Understanding Multiplication: A school Based Progression

Key Concepts

These frameworks are markedly different to those for addition and subtraction. There is a lot of incremental progress that can be made in addition and subtraction, but in multiplication the stages of progression appear to hinge around a few key areas. The school sees two key areas that hinder all mathematical development in relation to multiplication and both of them act as a gateway into a new arena of understanding for children. Without these it would seem that children can make little progress and will keep needing to grapple with the basics of calculating in multiplication.

The two key areas are as follows:

1. The children need to “see” the connection between multiplication and repeated addition. They need to “understand” that 3x4 is 3 groups of 4 and when these are totalled it comes to 12. They should of course be able to place this understanding into a real life context e.g. All 3 friends had £4 each so they could afford the toy that cost £12.

2. The children need to “understand” the numerical rules that govern the multiplying and dividing of numbers by powers of 10 and the associated use of the “distributive law”. This provides children with the option of using partitioning to solve calculations involving larger numbers.

Therefore, unlike addition and subtraction which can involve incremental steps which build on a child’s previous knowledge much of our teaching of multiplication and division will be spent scaffolding a range of activities and learning experiences to address these two key issues. This makes the assessment of progression easier in the sense that there is less to assess, but in another way harder because the teacher needs to be assured that the child has a full and complete 360 degree understanding of the concept before moving on.

Prior Learning

The key area of prior learning required for the understanding of multiplication is that the children are comfortable and reasonably fluid with a variety of addition calculations. There is a danger that we introduce multiplication too soon attempting to run it in tandem with addition. The truth is that children need a secure grasp of addition, especially with regard to the concept of repeated addition, before tackling the greater complexity of multiplication. I wonder in my own mind if we have a tendency to introduce the latter too early and we should therefore not be frightened to delay this until we are sure the children have made good progress in their understanding of addition. For instance the concept of 6+6+6+6 is probably harder than 44+6

Section 4

Understanding Multiplication: A school Based Progression

Sum


Stage 1

2+2+2+2 (with pictures)

The child cannot relate the repeated addition to the concept of multiplication, nor can they hold the numbers in their head.

The following should be used to lead children through the “Repeated Addition as Multiplication” barrier

1. Pictures

Pictures provide a valuable visual illustration of “sets of” or “groups of”

2. Practical

Wherever possible there can be no better modelling of the concept of multiplication than to use “real life” contexts. This allows children to see “real” multiplication in “real” meaningful contexts – a principle supported by the Renewed framework

3. Array to illustrate groups

The use of arrays are probably the most powerful visual representation of multiplication. At this juncture they should be limited to horizontal use only. To see that 3x4 can also be 4x3 will probably only seek to confuse rather than aid understanding.

4. Real life stories

The children should be able to verbalise their calculations in the form of meaningful stories. They should fully comprehend that 4x2 is about 4 people having 2 legs each and that this delivers a total of 8

5. Use of the number line (with numbers and then as an empty number line) to illustrate 2+2+2+2

This may be used as a visual strategy (not as a calculation strategy) but as a visual tool to demonstrate 4 jumps of 2. It will also serve as an introduction to the number line that the children will meet later. However the number line should not replace the array which is a more powerful visual representation of multiplication.

The following should be used to consolidate children’s understanding of the “Multiplication process”

Stage 2

2+2+2+2 (no picture) is 4 groups of 2

The child should be able to relate the principle of repeated addition and relate them clearly to the multiplying concept of “groups of”

Stage 3

2 x 4 =

At this point the child may be introduced to the formal recording of their calculation in the standard horizontal format and the use of the x sign to represent “multiply by”

Stage 4

Learning of tables

The relative “rote” learning of tables should run alongside the child’s work on understanding the true principles of multiplication. The learning of tables should start at this point in the child’s development.

Stage 5

2x4 and 4x2

The use of the array should be used to demonstrate the commutative law e.g. that 2 rows of 4 can also be represented as 4 columns of 2

Stage 6

2x4 is not 4x2

The children should appreciate that whilst 2x4 yields the same total as 4x2 they should understand the difference underlying the two calculations

Stage 7

6 x ▲ = 12

Children should be aware that symbols can stand for numbers and should thus be able to complete calculations similar to 6 x ▲ = 12

Stage 8

Doubling relates to halving

Stage 9

Relation of the inverse

The children should develop the work on doubling and halving to understand that multiplication is the inverse of division

Stage 10

Recognising Patterns

The children should increasingly be able to readily recognise patterns of multiples notably those in the 2,10 and 5 times tables

Progression:

There should be a strategic progression through the times tables. This is related both to the ease with which particular table are learnt; e.g. the 2x table and also their intrinsic usefulness relating to further developmental stages in Mathematical understanding; e.g. the 10x table. The progression is as follows: 2,10,5,3,4,9,6,8,7




Mental Calculation Strategies: Multiplication


The following are Strategies that will be used in conjunction with the “Calculation Document” in the sense that the children will have developed in their understanding to a point where they are ready to calculate using the school’s number line method

These strategies are not therefore linear in terms of their progression but should provide a useful checklist for staff looking to provide children with a breadth of mental calculation strategies that will greatly enrich their understanding of multiplication

Multiplying by 1 leaves number unchanged (Year 3)

The children need to appreciate that whilst most of their conceptual framework of multiplication is that the number gets bigger, here the number remains the same

Multiplying by 0 = 0 (Year 3)

The children need to appreciate that whilst most of their conceptual framework of multiplication is that the number gets bigger, here the number becomes zero

Doubling (Year 3)

1 x 25 = 25; 2 x 25 = 50; 4 x 25 = 100; 8 x 25 = 200 etc.

Multiplication Relationships (Year 3)

Given numbers such as 2,5,10 the child can write four multiplication/division sums

Multiply by x1 x10 x100 (Year 3)

The child can multiply by 1, 10 and 100 spotting the relationships and pattern in the answers

7 x 5 = 35 so… 35 ÷ 7 = 5 (Year 3)

The children should have an increasing understanding of the relationship between multiplication and division

The children should understand and use the principles (but not the names) of the commutative, associative and distributive laws as they apply to multiplication.

Commutative (Year 4)

8 x 15 = 15 x 8

Associative (Year 4)

6 x 15 = 6 x (5 x 3) = (6 x 5) x 3 = 30 x 3 = 90

Distributive (Year 4)

18 x 5 = (10 + 8) x 5 = (10 x 5) + (8 x 5) = 50 + 40 = 90

Know all square numbers to 10 x 10

These should be known through rapid recall

Multiply by 5 (Year 4)

To multiply by 5 then multiply by 10 and halve the answer to get to 5


Double and double again


To multiply by 20 then multiply by 10 and double the answer

8 Times Table Facts (Year 4)

Work out the 8 times table facts by doubling the 4 times table

Quarters and Eighths

To find a quarter of a number then halve the number and halve again, halve again to find an eighth

Multiply by 9 or 11

To multiply by 9 or 11, multiply by 10 and add or subtract the number


To multiply by 50 then multiply by 100 and halve the answer to get to 50

Multiplying by 5 (Year 5)

To calculate 16 x 5 the children can halve the 16 and double the 5 i.e. 8 x 10

Halve an even number (Year 5)

16 x 51 This can be shortened to 8 x 51 =408 and 408 x 2 = 816

Use of brackets (Year 5)

The children should know that brackets determine the order of the operations and that their contents are worked out first

Use Factors (Year 5)

15 x 6 This can become 15 x 3 = 45 and then 45 x 2 = 90

Multiply by 19 or 21 (Year 5)

To multiply by 9 or 11, multiply by 10 and add or subtract the number


Multiply the number by 10, halve it and add the two resultant totals together (or multiply by 30 and halve the total)


To multiply by 25 times the number by 100 and divide by 4


Work out 24 times table facts by doubling the 6 times table and doubling again

Multiply by 49 or 51 (Year 6)

To multiply by 49 or 51, multiply by 100, halve the total and add or subtract the number

Multiplication and fractions (Year 6)

Recognise that if 5 x 60 = 300 then 1/5 of 300 = 60 and 1/6 of 300 = 50

Decimal Fractions (Year 6)

Multiply a decimal fraction by a single digit e.g. 0.7 x 8

Section 5

Understanding Division: A School Based Progression

Key Concepts

As stated previously (multiplication) the frameworks for both division and multiplication are markedly different to those for addition and subtraction. However division is the Achilles heel for all children, even into KS2 and beyond and this is not without good reason. Whilst multiplication can be taught readily as “repeated addition”, this is not possible with division as there are two forms of division; grouping and sharing. So whilst “repeated subtraction” can be taught cogently in terms of grouping this is not the case for sharing. There is a ready recognition that children often access the first principles of division though the “sharing” concept as this is often easier to demonstrate concretely. This is important but in terms of calculation strategies the number line operates using the “grouping” principle. Therefore as children move out of Key Stage 1 they will be taught discretely the difference between “Sharing” and “Grouping” and how these are calculated practically. They will then be introduced to the principle of grouping which they can readily transfer as a calculation strategy through their use of the number line. The clarity of understanding they have with the two distinct forms of calculation will assist greatly in their ability to have a full and complete understanding of the concept of division.

The school is firmly of the opinion that if children are to gain full cognitive understanding of the division principles, the concepts need to be taught firmly in the context of “real life” problems. It is only in this arena that the children can fully grapple with and gain a full understanding of the principles lying behind the numerical operations. The universal weakness children have in this area may simply be down to the fact that whereas it is possible to cognitively understand the concepts of addition and subtraction abstractly, the complexity of division requires a greater cognitive appreciation of what occurs within the calculation. If this is the case then the subject is best explored practically allowing children to develop their “own understanding” of the principles involved.

As with its multiplying counterpart the children need to “understand” the numerical rules that govern the multiplying and dividing of numbers by powers of 10 and the associated use of the “distributive law” before they can calculate effectively on the number line.

Section 5

Understanding Division: A School Based Progression

Sum


Stage 1

8-2-2-2-2=0 (with pictures)

The child cannot relate the repeated subtraction to the concept of division, nor can they hold the numbers in their head. They also need to appreciate that reaching zero determines that they have run out of objects to “Share” (or divide)

The following should be used to lead children through the “Repeated Subtraction as Division” barrier

1. Pictures

Pictures provide a valuable visual illustration of “sets of” or “groups of”

2. Practical

Wherever possible there can be no better modelling of the concept of multiplication than to use “real life” contexts. This allows children to see “real” multiplication in “real” meaningful contexts – a principle supported by the Renewed framework

3. Array to illustrate groups

The use of the array is a powerful visual representation of division especially for grouping. To see that 12÷ 4=3 is a series of 3’s that move towards zero is key to a child’s understanding of the grouping strategy

4. Real life stories

The children should be able to verbalise their calculations in the form of meaningful stories. They should fully comprehend that 12÷ 4=3 is about 12 sweets being shared between 4 people and that therefore each get 3 sweets each.

5. Use of the number line (with numbers and then as an empty number line) to illustrate

Unlike multiplication where the number line can be introduced as a visual tool (rather than an effective calculation strategy in the early concept of understanding) its use for division should be a little more judicious until the children have a deep understanding of the difference between grouping and sharing.

Grouping and Sharing

The children should have a clear understanding of the difference between grouping and sharing and be able to use them, calculating accurately, in their correct context.




Mental Calculation Strategies: Division


The following are Strategies that will be used in conjunction with the “Calculation Document” in the sense that the children will have developed in their understanding to a point where they are ready to calculate using the school’s number line method

These strategies are not therefore linear in terms of their progression but should provide a useful checklist for staff looking to provide children with a breadth of mental calculation strategies that will greatly enrich their understanding of multiplication

Dividing by 1 leaves number unchanged (Year 3)

The children need to appreciate that whilst most of their conceptual framework of multiplication is that the number gets bigger, here the number remains the same

Dividing by 0 = 0 (Year 3)

The children need to appreciate that whilst most of their conceptual framework of multiplication is that the number gets bigger, here the number becomes zero

Halving (Year 3)

Know and use halving as the inverse of doubling

Know doubles of multiples and the corresponding halves e.g. 36 ÷ 2, half of 130, 900 ÷ 2

Use known facts (Year 3)

Use doubling or halving, starting from known facts

(e.g. 8 x 4 is double 4 x 4).

Know halves of multiples 10 to 100 (Year 3)

The children know for example; Half of 70


Use doubling or halving, starting from known facts.

For example: double/halve two-digit numbers by doubling/halving the tens first;

7 x 5 = 35 so… 35 ÷ 7 = 5 (Year 3)

The children should have an increasing understanding of the relationship between multiplication and division

Fractions (Year 4)

Begin to relate fractions to division e.g. ½ , ¼ etc.

Quarters (Year 4)

Find quarters by halving halves.


Use known number facts and place value to multiply and divide integers, including by 10 and then 100 (whole-number answers).

Sixths (Year 5)

Find sixths by halving thirds.

Section 6

Assessment: Use of the Spreadsheet (Notes to teachers)

The assessment of the skills outlined in this document are held centrally on two spreadsheets; Foundational Understanding and Further Understanding. The former will relate more to those children at KS1 whilst the latter (in general) will be for children in the lower years of KS2. As ever the use of the document should be child centric rather than age related so in any year group there will be a wide range of ability and this should be reflected in the spreadsheets.

Foundational Understanding

This aspect of the document relates to the understanding of basic principles and it is appropriate to use the progression outlined in the stages as a skeleton scheme of work. With this in mind it is appropriate to use the assessment tasks that are outlined throughout the document as a summative assessment to determine a child’s understanding of the principle taught. Whilst this may come in the form of an assessment at the end of the lesson, it might be more cogent to make the assessment at a later date to ensure that the concept has been securely retained. Indeed the procedure we have adopted in the past where we have taken children through large numbers of the stages in one session may well be the way forward. The only caveat to this might be the warning that “You don’t make a pig fatter by weighing it all the time”. To this end we should limit the element of testing as far as is possible, and the use of teaching assistants to speed the process would appear to be a good way forward. The results should be recorded and the gaps that occur provide a programme of personalised learning for each child.

Further Understanding

a) The Number Line and the Iceberg Principle

These concepts run alongside the teaching of the number line as the school’s chosen formal written method. The child’s ability to imbibe these further mental strategies is crucial as they are the underpinning of all mathematical understanding. To leave children with a single strategy such as the number line, no matter how effective it is, narrows the mathematical curriculum and will by definition remain relatively numerically impoverished. Whilst we would all be pleased for the child who could calculate 247+99 on a number line, the fact that they cannot see that this sum could (and indeed should) be calculated mentally by adding 100 and subtracting 1, should be a source of great concern to us. Where children have a deep understanding of the relationships between numbers and a fluidity of understanding in how these operate then they will increasingly be able to revert to a mental calculation in the first instance when presented with a given problem. Not only so but the number line is a creative strategy, where children have relative ownership of the strategies they are able to use for any given calculation. Therefore the more strategies children have access to, and the deeper their understanding the more effective and efficient their calculation processes will be. As the school’s own calculation document states: The number line acts as a good bridging point between the “Mental” and the “Formal” methods of calculation. It is interesting to note that although in Year 2 it is introduced as an introduction to “Pen and Paper methods” in the KS3 strategy it comes under the heading of “Mental Methods”. It is, in a very real sense, a mental method that is enhanced through the visual use of the line. It is incumbent upon us therefore to use it as a method that is supportive of the mental approach and a precursor to the formal written methods. (Calculation Document p2) Therefore these additional strategies must not be seen as a “bolt on” or an “optional extra” to the number line; they should be the core of every child’s mathematical understanding and are the foundational constructs that children should draw on to solve all mathematical problems. In this regard our teaching of the Maths curriculum should be viewed as an iceberg in the sense that whilst the most visible aspect may well be the number line the majority of our teaching focuses on the understanding below the surface. The goal therefore is to provide all children with a breadth of mental strategies that they can draw on and use to solve a range of calculations effectively and efficiently.

b) The Principles of Assessment

In light of the above it would be inappropriate to assess these skills in the form of a summative assessment. The key to understanding is not just to understand the principle behind a given strategy but to use it appropriately. So, using our example above of 247+99, whilst adding 100 and subtracting 1 is a very effective strategy for this calculation the child who seeks to add 247+53 by adding 100 and subtracting 47 has not only missed the principle behind the original strategy but has missed another in the fact that 53+47 is a number bond of 100 and would be much more effective strategy in this instance. So whilst there will always be a place for short term assessments at the end of each lesson to assess the child’s immediate understanding the assessments for the spreadsheet need to be different in nature. These assessments should be similar to those applied in the Foundation stage, namely that they should be child initiated and set in an appropriate context. As the school’s Foundation Assessment guidance states; this (assessment) process should be underpinned by the observation of tasks within the curriculum as the “using of supplementary tests with Reception children is unreliable”. (Jan Dubiel: Foundation Stage Profile, DFES). The same principle would hold in this context, whilst most children could be taught a strategy and probably successfully apply this to a series of set calculations given in towards the end of that same lesson, it is a completely different level of understanding to apply that principle to a calculation in an unrelated context at a later time. The latter should form the basis for our assessment of true mathematical understanding.

c) Setting Expectations

It would be my wish that 80% (to use QCA’s numerical measure of “the majority of children”) of the children should have mastered all the strategies found in the document by the end of Year 4. I am rapidly coming to the conclusion that in the upper years of KS2 the children should all have a good understanding of all areas of Mathematics and therefore the vast majority of their work in numeracy should be based on lessons that are rooted almost constantly in “Using and Applying” principles. There is very little in the Numeracy strategy (bar concepts like ration and proportion, and the principles of fundamental algebra that can be drawn down form the KS3 curriculum) that is new to children. Much of the programmes of study simply build on the children’s earlier understanding as well as encouraging the teacher to apply these concepts to ever challenging learning contexts.

d) Filling in the spreadsheet

The “Further Understanding” document is therefore completed slightly differently from its “Foundational Understanding” counterpart. The assessments are filled in using a colour coded system (found on each page of the spreadsheet) which will record which National Curriculum year the teacher believed the child fully understood and consistently used a given strategy. This will develop into a clear audit of continuity and progression for each child as they move through KS2. Whilst there is a colour code for Year 5 and 6 it is assumed that their use will be limited to those children who are either on the SEN register for Maths or have related issues with numeracy.

Appendix 1: Mathematical Vocabulary

The use of correct Mathematical vocabulary is a key to children ability to communicate their findings to others. Mathematics is a language and whilst much of its notation is abstract the communication of thoughts and ideas are best served through the use of correct terminology. It is noticeable that children at the end of Key Stage 2 who have a rich grasp of mathematical vocabulary are more able to succinctly and accurately communicate their work and conclusions to others.

ADDITION AND SUBTRACTION

add, addition, more, plus, increase

sum, total, altogether

score

double, near double

how many more to make…?

subtract, subtraction, take (away), minus, decrease

leave, how many are left/left over?

difference between

half, halve

how many more/fewer is… than…?

how much more/less is…?

equals, sign, is the same as

tens boundary, hundreds boundary

units boundary, tenths boundary

inverse

MULTIPLICATION AND DIVISION

lots of, groups of

times, multiply, multiplication, multiplied by

multiple of, product

once, twice, three times… ten times…

times as (big, long, wide… and so on)

repeated addition

array, row, column

double, halve

share, share equally

one each, two each, three each…

group in pairs, threes… tens

equal groups of

divide, division, divided by, divided into

remainder

factor, quotient, divisible by

inverse

Appendix 2: Instant Recall of Times Tables and Number Bonds

Known Facts are the tools of Creative Mathematics

The use of “Known Facts” (as the Numeracy Strategy calls them) is a crucial mathematical building block for children. To be aware of the number bonds to 10 and the times tables up to 10x10 allows children to make good progress in all areas of Mathematics. Whilst the child may have learnt them by rote and may lack a full appreciation of their understanding; these “known facts” are the gateway to learning effectively in the area of creative Mathematics. When undertaking higher ordered mathematical tasks children should be solely focused on “choosing the appropriate operation and method of calculation” (NNS p82-3), where instant recall is lacking they often struggle to fulfil the task because they are engaged in two parallel tasks; namely selecting the correct calculation method as well as developing strategies to solve that same calculation. In the sense that the former is a higher order skill the children who lack secure knowledge of “known facts” struggle to fulfil such tasks.

Learning without Understanding

Whilst it seems a little incongruous to learn these facts without a full understanding of their mathematical meaning, the rote aspect of the learning must be seen as a process not as an end goal. The key is to put these mathematical tools into the hands of children so they can explore the underlying understanding as they use them in a range of contexts.

Known and Derived Facts

The rote learning should be restricted the number bonds to 10 (addition only) and the times table up to 10x10 (multiplication only). The goal of seeking “instant recall” in facts beyond these has the potential to stunt true mathematical understanding. Research shows that “those who can make links between known and deduced facts number progress, because the known and the derived support each other.” (M. Askew “They’re counting on us” June 2008) In this sense…“these two aspects of mental mathematics – known facts and derived facts – are complimentary. Higher attaining pupils are able to use known facts to figure out other number facts.” (Gray EM 1991 – from M. Askew 2008). The reality is that if children develop an over-reliance on rote learning they tend to resort to a default mechanism for calculation and this is often “counting on”. As Askew points out; “This is one of the reasons that the national Numeracy strategy initially advocated delaying the introduction of standard algorithms. The danger is that children can appear to be confident in working with large numbers but they are actually only dealing with single digits and using counting on procedures”.

On the basis of this understanding the school would seek to adopt the following:

1. All rote learning should be limited to the number bonds to 10 and times table up to 10x10. Children should not “know” what 12x12 is they should be encouraged to use cogent strategies to derive the answer from known facts. Eventually these “derived” facts may well become “known” facts but this needs to come out of a correct philosophical understanding that seeks to take children deeper in their understanding of how numbers relate to one another in a range of calculations. The truth is that as a “child’s range of known number facts expands, the range of strategies available for deriving new facts expands as well” (M. Askew June 2008) However this richness of mathematical understanding does not occur if the process is bypassed through rote learning.

2. The “instant recall” does exactly what it says on the tin. When gauging whether children have “instant recall” teachers should work on the basis that a child should be able to answer times tables questions at a rate of one every 2 seconds in a test situation. Any speed less than this would tend to imply that the children are “calculating” the answers and they should be pushed towards an instant recall of these facts.

The school should be very careful in to bridge the gap between known and derived facts for children, gradually weaning them away from “counting on” around the end of KS1 and into the early years of KS2. Children should be actively encouraged to see the connection between known facts and derived facts and not treat each calculation as if coming to a blank canvas. The truth is many children see their times tables as the end goal not as the tools that enable them to solve more complex calculations. So children should be shown that to know 3x3=9 also provides a solution to 30x3, and 30x30. It is this network of mathematical connections that will truly foster a rich mathematical understanding in children.

3. Appendix 3: Examples of Misunderstanding

1. Two of the school’s most able mathematicians were working in a more able group of 4 looking at the concept of Algebra. They had calculated 33 – 8t = 15 with ease and were recalibrating the equation 6x – 27 = 0. They got to x = 27 ÷ 6 + 0 and debated whether they needed the zero. They concluded that they did not, they were then asked whether they would if the operation was “multiply by 0”. They suggested that 4.5 x 0 was 45. They checked it on a calculator, finding the answer was zero they assumed the calculator was broken and got a new one. Strangely the same answer appeared. It took some time before the concept hit home that 4 x 0 is “4 lots of nothing”

2. A year 4 above average child was calculating 236 x 12 – a concept comfortably into Year 5. Upon finding she needed to calculate 236 x 2 she told me the answer was 4 … 6 … and then 12 which made 4612. I pointed out to her that the 12 was in the wrong column and asked her what the 1 stood for “Units” she informed me, and the 2; “that is units as well” she said. So they are both units I suggested “Oh yes she said”

� EMBED Word.Document.8 \s ��

� EMBED MSWordArt.2 \s ��

PAGE

34

_1018457285.doc

_1018457302.bin

addition - wyche.worcs.sch.uk currriculum/pedagogy/… · web viewthe truth is that children need...

Documents