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Misconceptions with the Key Objectives

Overview 2

Aim of group 2

Misconceptions Circular 2

How to use this circular 3

Introduction 3

Making Mathematical Errors 3

The Child and Mathematical Errors 4

The Task and Mathematical Errors 5

The Teacher and Mathematical Errors 5

Mark Carpmail – Greswold Primary School 7

Why do children have difficulty with ADDITION? 7

Why do children have difficulty with SUBTRACTION? 10

Why do children have difficulty with PLACE VALUE? 12

Why do children have difficulty with ORDERING? 14

Why do children have difficulties with PROBLEM SOLVING? 16

Why do children have difficulty with AREA AND PERIMETER? 19

Louise Burnett – Shirley Heath Junior School 21

The Nature of Errors 21

Why do children have difficulty with RATIO AND PROPORTION? 22

Why do children have difficulty when using a PROTRACTOR? 25

Why do children have difficulty with MULTIPLICATION AND DIVISION? 27

Kate Chapman – Langley Primary School 30

Common Misconceptions in Mathematics: Research theory 30

Why do children have difficulty with FRACTIONS, DECIMALS AND PERCENTAGES? 32

Why do children have difficulty with DIVISION? 35

Kate Chapman - Bibliography 37

Tables Identifying Misconceptions with the Key Objectives 38

Area of Mathematics: Addition 39

Area of Mathematics: Subtraction 44

Area of Mathematics: Ordering Numbers 50

Area of Mathematics: Area and perimeter 54

Area of Mathematics: Problem Solving 57

Area of Mathematics: Ratio and Proportion 60

Area of Mathematics: Shape 62

Area of Mathematics: Multiplication 66

Area of Mathematics: Division 71

Area of Mathematics: Fractions, Decimals and Percentages 75

Area of Mathematics: Position, Direction, Movement and Angle 86

Appendices 94

Appendix 1 – Decimal Place Value Chart 94

Appendix 2 – Fraction, decimal and percentage cards 95

Appendix 3 – Protractor OHT 96

Appendix 4 – A Monster Right Angle Measurer 97

Overview

Aim of group

The aim of this working group was to produce research material and guidance for teachers to support the planning for misconceptions.

Misconceptions Circular

Contributors to this circular include

• Mark Carpmail – Greswold Primary School

• Louise Burnett – Shirley Heath Junior School

• Kate Chapman – Langley Primary School

• Donna Crowder – Primary Maths Consultant, SIAS

This circular contains

• Information on why children make errors in the different areas of mathematics.

• Misconception tables outlining possible misconceptions for the key objectives in most strands of the National Numeracy Framework from Reception to Year 6, key questions for each objective to discover difficulties and ideas for next step activities to overcome problems. In some cases, where there are no key objectives for some year groups, general objectives have been included to show progression within the table. However, misconceptions or next step activities may not have been identified for these objectives.

How to use this circular

For each area of mathematics there are 2 sections

• General information

• Misconception table

The general information sections are intended to give background information on errors children make and why these errors are made. This section is valuable in gaining a more in-depth knowledge of errors made in the various strands of mathematics.

The misconceptions tables can be used alongside assessments made against the key objectives. The key questions within the table enable further probing in order to discover areas of difficulty. Any assessments made should inform future planning and the next steps activities provide starting points for teaching ideas to tackle any misconceptions. They also act as reminders of errors or misconceptions that the children may encounter with these key objectives so that the teacher can plan to tackle them before they occur. Of course, the tables can also be used in a similar way when working with groups during the main part of any mathematics lesson focused on the key objectives.

Introduction

Making Mathematical Errors

Errors in mathematics may arise for a variety of reasons. They may be due to the pace of work, the slip of a pen, slight lapse of attention, lack of knowledge or a misunderstanding.

Some of these errors could be predicted prior to a lesson and tackled at the planning stage to diffuse or un-pick possible misconceptions. In order to do this, the teacher needs to have the knowledge of what the misconception might be, why these errors may have occurred and how to unravel the difficulties for the child to continue learning.

Cockburn in Teaching Mathematics with Insight (1999) suggests the following model to explain some of the commonest sources of mathematical errors.

DIAGRAM

The Child and Mathematical Errors

Experience – Children bring to school different experience. Mathematical errors may occur when teachers make assumptions about what children already know.

Expertise – When children are asked to complete tasks, there is a certain understanding of the basic ‘rules’ of the task. Cockburn (1999) takes an example from Dickson, Brown and Gibson (1984, p331). Percy was shown a picture of 12 children and 24 lollies and asked to give each child the same number of lollies. Percy’s response was to give each child a lolly and then keep 12 himself. Misconceptions may occur when a child lacks ability to understand what is required from the task.

Mathematical knowledge and understanding – When children make errors it may be due a lack of understanding of which strategies/ procedures to apply and how those strategies work.

Imagination and Creativity – Mathematical errors may occur when a child’s imagination or creativity, when deciding upon an approach using past experience, may contribute to a mathematically incorrect answer.

Mood – The mood with which a task is tackled may affect a child’s performance. If the child is not in the ‘right mood for working’ or rushed through work, careless errors may be made.

Attitude and confidence – The child’s self esteem and attitude towards their ability in mathematics and their teacher may impact on their performance. For example, a child may be able in mathematics but afraid of their teacher and therefore not have the confidence to work to their full potential in that area.

The Task and Mathematical Errors

Mathematical complexity – If a task is too difficult, errors may occur

Presentational Complexity – If a task is not presented in an appropriate way, a child may become confused with what is required from them.

Translational Complexity - This requires the child to read and interpret problems and understand what mathematics is required as well as understanding the language used.

“ ‘When it says here, ‘Which angel is the right angel?’ does it mean that the wings should go this way, or that way?’ “

(Dickson et al, 1984 IN Cockburn 1999)

If the task is not interpreted correctly, errors can be made.

The Teacher and Mathematical Errors

Attitude and Confidence – As with the child, if a teacher lacks confidence or dislikes mathematics the amount of errors made within the teaching may increase.

Mood – With the pressures of teaching today, teachers may feel under pressure or rushed for time and not perform to the best of their ability.

Imagination and Creativity – Where a teacher is creative, they may teach concepts in a broader manner, looking for applications and alternative approaches thus reducing the probability of error in learning.

Knowledge – Too much teacher knowledge could result in a teacher not understanding the difficulties children have whereas too little knowledge could result in concepts being taught in a limited way.

Expertise – Expertise not only in subject matter but also in communicating with children and producing effective learning environments. Without this expertise, some pupils’ mathematics may suffer.

Experience – Knowledge can be gained from making mistakes. Teachers may learn about children’s misconceptions by coming across them within their teaching.

Mark Carpmail – Greswold Primary School

Why do children have difficulty with ADDITION?

Focus: Identifying common misconceptions for the key objectives in mathematics relating to addition and identifying ways to address these misconceptions through the teaching of appropriate pre-requisite skills.

A brief history

Addition was initially carried out as a count and a counting frame or abacus was used. The Egyptians used the symbol of a pair of legs walking from right to left, for addition. In the 15th century mathematicians began to use the symbol ‘p’ to represent plus. The modern+ came into use in Germany towards the end of the 15th century.

What is addition?

Addition is regarded as a basic calculation skill which has a value for recording and communicating. Addition can be carried out by counting, but children are encouraged to memorise basic facts. Addition involving the same number leads to multiplication.

Learning to add

In order to understand the common misconceptions that occur with column addition it is important to consider the key developments of a child’s addition abilities.

1) Counting on – The first introduction to addition is usually through counting on to find one more.

2) Memorising facts – These include number bonds to ten. It is very important that children have a sound knowledge of such facts.

3) Facts involving zero – Adding zero, that is a set with nothing in it, is difficult for young children. Children need practice with examples where zero is involved.

4) The commutative property of addition - If children accept that order is not important it greatly reduces the number of facts they need to memorise.

5) Facts with a sum equal to or less than 10 or 20 - It is very beneficial to children to only learn a few facts at a time.

6) Adding tens and units – The children add units and then add tens. The process of exchanging ten units for one ten is the crucial operation here.

7) Adding mentally in an efficient way. Starting with the largest number or grouping numbers to make multiples of ten are examples of this.

8) Adding more than two numbers - Children should realise that they are only able to add two numbers at a time. So 5+8+6 is calculated as 5+8=13 then 13+6=19.

9) Adding regardless of magnitude – If the children have understood the process of exchanging they will be able to carry out this operation when dealing with hundreds and thousands.

10) Word problems – identifying when to use their addition skills and using them efficiently.

When pupils in year 6 are having difficulties with column addition, it is probably the case that they do not have sufficient understanding of one of the areas outlined above. Such ‘gaps’ in knowledge need to be filled in order that the pupil can have a proper understanding of such addition.

Other possible reasons for misconceptions in column addition

Reliance on rules

According to Koshy (2002), a large number of misconceptions originate from reliance on rules which either have been not understood, forgotten or only partly remembered. For example:

H t u h t u h t u h t u

760 729 534 546

+240 +111 +383 +364

990 839 897 899

The pupil in this example can add two digits accurately, but is ’mixed up’ with the carrying aspect of addition. He has learnt a rule on which the place value system is based: that the largest number you could have in a column is 9. The pupil has followed the rule correctly, but does not know what to do with the rest of the number, as he has not understood the underlying principle behind the rule.

Careless mistakes

Children make mistakes. These might include mistakes with simple mental addition or mistakes when setting out work. Such mistakes cause great problems when carrying out column addition, as the place value of digits is sometimes wrong. e.g.

h t u

160

+8__

960

Teaching children to estimate answers is another good way of helping them to limit the number of careless mistakes.

Place value

The calculation above was incorrect because of a careless mistake with the ‘placing’ of a digit. However, many mistakes with column addition are caused by a fundamental weakness in a child’s understanding of place value. When teaching how to add vertically, it is also useful to reinforce the principles of place value used in the operation. Children will then be more likely to relate the word ‘carrying’ to what is actually happening rather than learn it as a rule that helps to produce correct answers.

Why do children have difficulty with SUBTRACTION?

Focus: Identifying common misconceptions for the key objectives in mathematics relating to subtraction and identifying ways to address these misconceptions through the teaching of appropriate pre-requisite skills.

What is subtraction?

Subtraction can be described in three ways:

• Taking away – where a larger set is shown and a subset is removed leaving the answer for example 5 take away 2 leaves 3

• The difference between – Where both sets are shown and the answer is shown by the unmatched members of the larger set, for example, the difference between 5 and 3 is 2.

• Counting on – Where the smaller set is shown and members are added to make it up to the larger set, fro example, 3 and 2 makes 5.

It is important to remember that subtraction is the opposite of addition. Unlike addition though, subtraction is not commutative, the order of the numbers really matters. Neither is subtraction associative as the order of the operations matters too.

Learning to subtract – step by step

1) The process of taking away involving 1 to 5 – e.g. take away 1,2 etc

2) ‘Take away’ involving 0 to 10.

3) Subtraction in the range of numbers 0 to 20 – Using a range of vocabulary to phrase questions such as fifteen take away eight.

4) Difference – The formal approach known as equal additions is not a widely used method but it involves finding a number difference.

5) Subtraction of tens and units – This is where common misconceptions occur because of the decomposition method.

6) Subtraction by counting on – This method is more formally know as complementary addition. The procedure is to add on mentally in steps to the next ten, the next hundred etc.

7) Word problems - identifying when to use their subtraction skills and using them efficiently.

8) Decomposition with larger numbers

By considering the development of subtraction and consulting a school’s ‘agreed routes through’ we should be able to see where common misconceptions are likely to occur. It seems that to teach in a way that avoids pupils creating any misconceptions is not possible, and that we have to accept that pupils will make some generalisations that are not correct and many of these misconceptions will remain hidden unless the teacher makes specific efforts to uncover them.’ A style of teaching that constantly exposes and discusses misconceptions is needed, thus limiting the extent of misconceptions. This may be possible, as much research over the last twenty years has shown that the vast majority of pupil misconceptions are quite widely shared.’ Askew and Wiliam (2001)

Why do children have difficulty with PLACE VALUE?

Focus: Identifying common misconceptions for the key objectives in mathematics relating to place value and identifying ways to address these misconceptions through the teaching of appropriate pre-requisite skills.

Identifying misconceptions

Place value

Many of the mistakes children make with written algorithms are due to their misconceptions relating to the place value of numbers. ‘When considering this aspect it is worth pointing out that children tend to make more mistakes with subtraction than any other operation.’ Koshy, Ernest, Casey (2000).

Before children decompose they must have a sound knowledge of place value. To help them with this the teacher must talk about ‘exchanging a ten for ten units’ etc.

Crutch figures

Some children carry out an exchange of a ten for ten units when this is not required and some forget they have carried out an exchange. Most children are required to show an exchange with crutch figures. The children should be shown how these might be recorded neatly and clearly.

Zeros

When faced with these within formal vertical calculations, many children find them confusing. They may require a greater understanding of the meaning of zero i.e. no units, or tens, or hundreds.

Appropriateness

When should formal, written methods be used? In an experiment twenty year 6 pupils were asked to solve the following:

6000

- 9

A majority of the pupils attempted to solve this by decomposition! When questioned, it was discovered that because the calculation was written in a formal way they thought they had to answer it in a similar fashion. Pupils need to draw on all their knowledge in order to overcome difficulties and misconceptions.

Language

With younger pupils language can get in the way of what we are asking them to do. Children need to be taught to understand a range of vocabulary for subtraction e.g. take away, subtract, find the difference etc.

The larger and smaller numbers

Children should realise that in most subtractions (unless negative numbers are involved) the smaller number is subtracted from the larger.

Approximating

Sensible approximation of an answer, by a pupil, will help them to resolve problems caused by misconceptions as discovered by OFSTED.

‘Whilst teachers recognise the importance of estimating before calculating and

teach this to pupils, pupils rarely use it in practice. As a result, they do not

always have a clear idea of what constitutes a sensible answer. Teaching of calculation in primary schools - HMI (2002)

Why do children have difficulty with ORDERING?

Focus: Identifying common misconceptions for the key objectives in mathematics relating to ordering numbers and identifying ways to address these misconceptions through the teaching of appropriate pre-requisite skills.

When considering the difficulties pupils face with ordering numbers to three decimal places it is important to consider that there are two different aspects to this. Firstly, the children might have difficulties with place value and secondly, the problem might be exacerbated by a weakness in understanding of decimal numbers.

Place value

A good understanding of place value of whole numbers and its extension to decimal numbers is vital because it is the basis of both our mental and written calculations. Casey (2000)

There is a large amount of evidence to show that children at all key stages have difficulties in understanding many aspects of place value. Children who have such problems will encounter many difficulties when attempting to order a set of numbers.

Key steps in development for the learner

1) Ten – The concept of 1 having another, different value from one unit is an important principle for children to grasp.

2) Tens and units – Understanding that tens and units can exist side by side.

3) Number names – Knowing that instead of naming a number ‘one ten and three units’, we give it a name, thirteen.

4) Hundreds, tens and units – The children should be aware that when they have ten tens they have no single digit to express this in the tens column so they should consider linking the tens together to make one hundred.

5) Numbers in different forms – The number 36 is usually thought of as 3 tens and 6 units, but it can also be thought of as 2 tens and 16 units.

6) Thousands, hundreds and tens of thousands – At this stage the children should realise the repetitive nature of column values.

Decimals

Decimal fractions are sometimes a difficult concept for KS2 children to understand. They may have encountered them, without knowing it, when working on and solving money problems. Working with decimals is an extension of place value work. Most pupils have an understanding that each column to the left of another is 10 times greater. This needs to be extended so that they are aware that each column to the right is 10 times smaller. Hence

when multiplying and dividing by 10 or 100 they are able to do so accurately due to their understanding of place value. This is helpful when teaching the following Key Objective in Year 6:

‘Multiply and divide decimals mentally by 10 or 100, and integers by 1000, and explain the effect’.

Common difficulties

Concept – Some children think the point is the decimal fraction (Duncan 27). They require more experience of explaining the value of each of the digits for numbers when there is a decimal notation.

Zeros – These may cause confusion and the children will require practice in giving values to each of the digits.

The value of decimal numbers – The magnitude of numbers can be confusing, for example, when we ask ‘Put these numbers in order, smallest first: 1, 1.01, 1.11, 0.01, 0.1’ many children are uncertain of how to do this. It may be necessary to find a method of comparison. For example some children think of each of these as a number of hundredths, that is, 100,101,111,1.

Ordering decimals

An example: Order these numbers, smallest first: 21.2, 1.112, 3.1, 11.4, 0.2112

Child’s answer: 3.1, 11.2, 21.2, 1.112, 0.2112.

This child has relied on a common generalisation that, ‘the larger the number of digits, the larger the size of the number.’

There are many other misconceptions about ordering numbers and it is important that careful, targeted teaching is done to remedy such difficulties.

Why do children have difficulties with PROBLEM SOLVING?

Focus: Identifying common misconceptions for the key objectives in mathematics relating to problem solving and identifying ways to address these misconceptions through the teaching of appropriate pre-requisite skills.

According to Ernest (2000), ‘Solving problems is one of the most important activities in mathematics’. When solving problems children will need to know general strategies. General strategies are methods or procedures that guide the choice of which skills or knowledge to use at each stage in problem solving.

Problems in maths can be familiar or unfamiliar. When a problem is familiar the pupil has done something like it before and should remember how to go about solving it. When a problem has a new twist to it, the pupil cannot recall how to go about it. This is when general strategies are useful, for they suggest possible approaches that may lead to a solution.

Such general strategies might include:

• Representing the problem by drawing a diagram;

• Trying to solve a simpler approach, in the hope that it will identify a method;

• Making a table of results;

• Searching for a pattern amongst the data;

• Thinking up a different approach and trying it out;

• Checking or testing results.

There has been a great deal of debate about how to improve pupils’ problem solving skills, with some writers advocating a routine for solving problems.

One successful example of this is the ‘7 steps to solving problems’. E.g.

1) Read the question. Underline key words that help you to solve the problem.

2) Decide which operation to use.

3) Write down the calculation you are going to do.

4) Work out the approximate answer.

5) Decide if you will use a calculator.

6) Do the calculation and interpret the answer.

7) Does the answer make sense?

Others find this sort of approach too mechanical, and suggest that we cannot teach thinking skills in a vacuum since each problem has its own context and content. Advocates of this argument believe that we should be encouraging children to ‘think outside of the box’ rather than teaching them to rely on a set of prescribed rules.

John Mason and Leone Burton (1988) suggest that there are two intertwining factors in any process of mathematical thinking:

1) The process of the mathematical enquiry – specialising, generalising, conjecturing, convincing.

2) The emotional state – getting started, getting involved, getting stuck, mulling, keeping going, being sceptical, contemplating.

As far as the mathematical enquiry is concerned, the questions ‘what do I know?’ ‘What do I want?’ ‘What can I introduce?’ may help the learner during the ‘entry phase’. In the ‘attack phase’ the processes of conjecturing, collecting data, discovering patterns, then justifying and convincing may all be involved. During this process the learner should be constantly reviewing progress and reflecting on the progress.

Organisation

The children may attempt problem solving on their own but teachers usually find that young children benefit from working in pairs and older children from working in threes or fours. You can group children for a range of reasons and it is important that the group learn to work well together if they do not already do so.

If the children are working as a group, they require a place where they have space to work and the ability to talk together without being disturbed or disturbing others (easier said than done!).

What does the teacher hope to achieve?

You may wish to encourage the children to:

• Accept a challenge which at first might seem difficult to them

• Have the confidence to make reasoned ‘guesses’ and try to find a solution.

• Realise that you can fail to find a solution without being a failure.

• Interact efficiently with classmates without teacher intervention.

What does the teacher do?

Initial teaching – You need to ‘teach’ children how to go about solving problems.

Setting tasks for children – These need to be realistic and appropriate – here are a few examples:

Ages 5 – 7

• Make a parcel which will balance this one.

• In how many different ways can you show 16p using coins?

Ages 7-9

• Make a timetable for tomorrow for a new pupil coming to your school.

• Order a meal from a Chinese takeaway menu. How much change will you get from £20 etc.

Ages 9+

• Complete the number pattern 2,4,_,_,_, in three different ways.

• Write down a price list for a shop and write out various problems for your classmates.

• Decide what is the largest number you can write.

Difficulties

Some children find it difficult to think of ideas. A brain-storming session might help, for example, produce an item like a sheet of paper and ask the children to think of as many things as possible that it could be used for.

How does problem solving benefit children?

The greatest benefit is that children learn to apply the maths they learn in school to real life situations. Thus realising the importance and relevance of a subject that unfortunately is often seen to be ‘boring’ by many pupils.

In addition children will learn to :

• Interpret instructions more effectively

• Modify their behaviour to achieve the best group solution

• Evaluate what their own group, and other groups, do constructively

• Gain confidence in solving problems.

Most children get tremendous satisfaction from solving a problem with a solution that they know is acceptable without having to ask.

Why do children have difficulty with AREA AND PERIMETER?

Focus: Identifying common misconceptions for the key objectives in mathematics relating to area and perimeter and identifying ways to address these misconceptions through the teaching of appropriate pre-requisite skills.

Definitions of area and perimeter

One of the definitions of area given in the Oxford dictionary is ‘superficial extent’. Perhaps in a more child friendly language we would say it was the ‘amount of surface’. It is actually quite a difficult concept to define, but one which children meet quite early.

Perimeter is the distance around an area or shape.

Confusion can arise between perimeter and area. However, if the children have had enough practical experience to find that length is a one-dimensional attribute and area a two-dimensional one, differences should be obvious. They should also be aware that each is expressed in different standard units.

Units

The standard SI units are square metres or square centimetres and are written as m² or cm². These refer to squares of side 1m or 1cm respectively. It should be pointed out that because there are 100cm in 1m there are 100 x 100 = 10,000 cm² in 1 m². In school the square metre is really too big to be of much use, in fact square cm are much easier to handle.

In the measurement of large areas the SI unit is a hectare, a square of side 100m and area of 10,000 m².

In the imperial system the equivalent unit is an acre.

Understanding area

• The concept of surface – Young children in nursery are involved in activities such as painting. Experiences like these, where they are covering surfaces, provide opportunities to establish a concept of area.

• Direct comparison – Making comparisons of the surface of objects by placing one on top of the other is a useful experience which can lead to phrases like, ’has a greater surface’.

• Conservation of Area – The conservation of area means that if a 2D shape is cut up and rearranged, its area is unchanged.

However, it has been shown that not all children grasp this. In an investigation reported in Dickinson, Brown, Gibson (1984), it was found that one third of 11 year olds did not conserve area consistently.

• Non-standard units – If a question such as ‘How much greater is the surface of the exercise book than the storybook?’ is asked, non -standard units could be used to find an answer.

• Standard units – As discussed previously

• Formulae – Ideally children should discover a formula for themselves. It should be regarded as a short method of finding an answer. In area work the children may realise as they count squares to find the area of a rectangle that it would be quicker to find the number of squares in one row and multiply this by the number of rows. In the same way, the children might find a formula for calculating the area of a right-angled triangle as this shape can be seen by them as half a rectangle.

• Wherever possible, examples should be related to areas of places, playing fields etc.

Louise Burnett – Shirley Heath Junior School

The Nature of Errors

As teachers of mathematics, we are all acutely aware of the need to analyse written work so that error patterns in children’s work does not go unnoticed. Research shows that there are two distinct types of errors:

□ Errors as a result of carelessness or wrong recall of number facts. This may not affect all questions, and may appear quite ‘random’.

□ Errors caused by applying a defective algorithm. These are of greater consequence, and are those which the teacher needs to address. A child who has misunderstood an algorithm or invented a defective algorithm, tends to show it in all his or her working. Such methods may go undetected especially when they yield correct answers to some questions.

The nature of errors which occur in children’s mathematics work will vary greatly and will depend on different factors; the age of the child, their individual interpretation of concepts taught, past experiences, and the specific area of mathematics itself. It is however true to say that some errors are more common than others, that many children will misunderstand particular algorithms, and we will see similar patterns emerging in their working. It is these more ‘predictable’ errors, in specific areas of the mathematics curriculum, on which we will concentrate.

Why do children have difficulty with RATIO AND PROPORTION?

Focus: identify common misconceptions for the key objectives in mathematics relating to ratio and proportion and identifying ways to address these misconceptions through the teaching of appropriate pre-requisite skills.

Objectives specifically referring to ideas of ratio and proportion do not emerge within the National Numeracy Framework until Year 4. It is often an area of the curriculum which brings about many conceptual problems, children and teachers finding difficulty in ‘imagining’ the relationships between numbers given. Thompson (1984) suggests that teachers’ beliefs about maths and how to teach maths are influenced by their own experiences with mathematics and schooling. This is confirmed by findings from studies carried out by Tirosh (2000) who claims “many pre-service as well as in-service mathematics teachers essentially struggle with the same problems as children do”

In my own experience, it seems that even the teachers who are secure in their own knowledge of the content of the curriculum relating to ratio and proportion, have difficulties in translating this to the pupils. Oldham (1999) states that “going back to the basics of a concept in order to explain to pupils is difficult.” Teachers need a great deal of support in deciding how best to present the concept to children, requiring considerable pedagogical knowledge as well as content based.

One key misconception which emerges in a great deal of research carried out in this area, (among others Owens 1993, Harel 1994, Ryan & Williams 2000) is related to difficulties in the recognition of proportional problems. Owens (1993) provides two examples used in his research, outlined below;

“Andrew and Sam are running equally fast around a track. Andrew started first. When he had run 9 laps, Sam had run 3. When Sam had run 15, how many had Andrew completed?”

32 out of 33 teachers tested solved this problem using knowledge of ratio…

9 = x. therefore x = 45

3 15

The second problem involved the conversion of currencies;

If 3 US Dollars is equal to 2 British Pounds, how many British Pounds are equal to 21 US Dollars?

At first inspection, both problems appear similar, there are three known values and one unknown. In the running example however, the relationship between the two runners can be expressed with addition or subtraction,

i.e. Andrew – 6 = Sam

In the example involving currency, the relationship between the two currencies can be expressed with multiplication or division,

i.e. 2 = x.

3. 21

It is this difference in the operations used to solve the problem, which indicates a proportional problem. The relationship between given values can always be represented by multiplication or division, thus shown as equivalent fractions.

Ryan & Williams (2000) carried out extensive research into the strategies behind common misconceptions in mathematics, and refer to the “common additive misconception”. This refers to situations when children add values given, instead of finding the ratio between relevant pairs of numbers. The example provided is that of two rectangles, one an enlargement of the other.

4 6

10

?

Most children tested observed that 2 had been added to the width of the rectangle, and so performed the same operation for the length, i.e. 12. Ryan & Williams claim that we need to encourage the children to see a conflict between their method and common sense by posing problems in which the result of their method is obviously wrong. E.g. a rectangle 1 x 4 will be nothing like the same shape as a rectangle 100 x 103, which would appear almost square.

Plotting the related fractions is another simple way of demonstrating how values are proportional. Using the earlier currency example, plot dollars (x axis) against pounds (y axis) and the resulting line will be straight, and will cross through the origin.

Streefland (1985) heavily promotes the use of a ‘ratio table’, and claims that it can go some way to alleviating many of the related problems. The table can provide a structure for the pupils, acting as a visual pattern. This is useful as many find the concept of ratio and proportion to be extremely abstract. The multiplication required for these problems, which can prove to be an immediate hurdle for many children, becomes addition. These additive strategies can provide an intermediate step towards multiplicative ones, and further teaching will aim to reduce the number of steps (columns in the table).

The Rational Number Project (RNP) found that the ‘unit rate’ approach to these problems was that most frequently chosen by pupils, and the one which led to the greatest number of correct answers. An example of this;

3 apples cost 60p, so we calculate the cost of one apple (20p). This unit rate is now a constant factor that relates apples to their cost, and can be used for any number of apples in subsequent questions.

It is vital that children see a wide variety of problems relating to this area of mathematics, and not just ‘missing number problems’ which research show to be the most common.

Two other task types suggested by Owens are numerical comparison problems…

□ Requires the comparison of two rates, e.g. with orange juice strength

(shaded glasses indicate orange concentrate, non-shaded indicate water)

2 1 3 1

3 3 4 4

Use comparison of fraction knowledge to determine which orange juice will be ‘stronger’

A similar example could be shown with the mixing of paint.

… and those eliminating the need for calculation, but encourage discussion and the understanding of factors relating to each other

□ If Carol ran fewer laps than she did yesterday, in more time, is she running faster / slower / same speed / not enough information to decide

Why do children have difficulty when using a PROTRACTOR?

Focus: identify common misconceptions for the key objectives in mathematics relating to using a protractor and identifying ways to address these misconceptions through the teaching of appropriate pre-requisite skills.

The use of a protractor is not specifically mentioned until Year 5 within the National Numeracy Framework, but there are many pre-requisite skills with which children will need to be secure before reaching this stage.

Williams and Shuard (1994) state that,

“Two different types of experience of angles need to come together and to combine if children are to have a thorough understanding of the concept of angle…The first set of experiences is static. An angle is the shape of a corner. It may be sharp, or blunt, or right angled. Much more fruitful than the static conception of an angle is the dynamic conception of the measure of an angle. If a book is gradually opened, its pages make a growing angle with each other.”

It is the latter of these which appears first within the National Numeracy Framework, indeed even at Year 1, children are talking about things that turn and making whole and half turns with objects and themselves. It is within the Year 2 list of objectives that we first see the mention of ‘static’ angles, recognising right angles in squares and rectangles. By Year 4, children are being introduced to the idea of measuring angles in degrees, moving on to using a protractor with increasing accuracy in the later years of Key Stage 2.

When comparing angles (from Year 3), Askew & William (1995) claim that the most common error occurs when children state that;

“..two identical angles are unequal because the length of the arms is different in each, as a result thinking that an angle is the distance between the ends of the lines”

This may have emerged from being continually faced with examples where the ‘arms’ of an angle are nearly always equal, and thus children have internalised that as one constant ‘rule’ of angle comparison.

One suggestion for assisting children who experience difficulties with the use of a protractor was the result of research carried out at the University of Exeter (2002);

“There can be few teachers who have not seen pupils having difficulties with the traditional dual-scale 180( protractor. Which figure should you use? It was several experiences of this which eventually prompted the question ‘what if there were no numbers on the protractor and the distance had to be counted off?’”

From this, a photocopiable protractor, (see Appendix) without numbers, has been used in further research projects carried out by the University of Exeter, with the claims of much more positive results.

Why do children have difficulty with MULTIPLICATION AND DIVISION?

Focus: identify common misconceptions for the key objectives in mathematics relating to Multiplication and Division and identifying ways to address these misconceptions through the teaching of appropriate pre-requisite skills.

Multiplication and division are clearly fundamental skills that need to be mastered if a child is to be considered a ‘proficient mathematician’. As children progress throughout Key Stages 1 and 2, increasing emphasis is placed upon the use of effective and appropriate strategies to achieve appropriate answers to multiplication and division problems.

Along with the introduction of the National Numeracy Strategy, came a greater push to encourage children to explain their methods of working. I am sure all teachers of mathematics will agree that this has been an extremely positive move forward in gaining an insight into the pupils’ understanding of various concepts taught, particularly with the four main operations. Despite this change, there is still a great deal of concern within all stages of education about this ‘need’ for a final, correct answer. Cockburn (1999) states that, for some children at least,

“… the aim when doing mathematics is to get the right answer and thus please the teachers. The route to achieving the answer is not important to them.” It is this attitude amongst young pupils, which as educators, we are all trying our utmost to discourage.

The recent advice to many schools has been to introduce a ‘Route through Calculations’, where the procedural steps to be taught for each of the four operations are laid out, with examples, for all staff in a school to work from. It is important that children are introduced to the next stage when it is appropriate for them, if at all. Thompson (1999) states that; “some written calculations, particularly ‘long multiplication’ and ‘long division’, have caused difficulties for generations of children who have tried to master complex rules to solve apparently irrelevant questions.”

Key steps in development for the learner within multiplication will start with patterns of repeated addition, although research has made us aware of conceptual hurdles linked with this. A significant number of children have difficulty with the ‘dual counting system’ (Owens, 1993) required, and producing concrete or pictorial representations of groupings, e.g. 5 boxes each with 5 cakes. Whilst this is a useful link for the children in the early stages, it is limiting later, if it is the only concept of multiplication which a child has. One example might be 0.5 x 0.5. Are we asking them to add 0.5 to itself one half times?

Children will move on to learning multiplication tables, by building sets of objects, looking at patterns, grouping the facts for learning and playing related games. It is at this stage, when children are secure with the concept, and have a sound understanding of multiplication facts, that written methods are slowly introduced. The choice of written methods taught and the order in which this is done will be specific to a particular school, although the supplement of examples within the National Numeracy Framework does suggest a ‘route’. One important note, which may be more relevant to parents rather than pupils, is the change in vocabulary. ‘Long multiplication’ and ‘Long division’ no longer indicate the need for a lengthy calculation, working in several lines down the page, indeed it simply refers to the number of digits in the question.

Key steps for the learner within division will start with informal practical activities, where they are required to carry out equal sharing tasks, distributing items among a group. Initially they will not think in terms of the number but will just follow through a process of ‘one for you’ until all the items are used up. Later tasks will be related to numbers, for example, ‘If you share the 12 sweets equally among 4 people, how many will they each get?’ Owens (1993) refers to this as the ‘partitive model’, finding the size of each of a given number of equal sets.

Before children attempt formal division calculations, they need to understand the process of repeated subtraction, or the ‘measurement model’ (Owens, 1993). This model is vitally important as later written methods will be based upon it. Once children are involved with decimal numbers, the measurement model (or repeated subtraction) becomes crucial. If a child is faced with;

9 ( 1.5

then it is difficult to ‘share’ 9 equally between one and a half people! It is far easier to find how many lots of ‘one and a half’ there are in 9.

Latham and Truelove (1990) state that “the language used in posing questions involving repeated subtraction must be chosen very carefully. A boy was once asked..

“How many times can you subtract 4 from 20?” He was busy for some time filling a whole page of his book:

2 0 2 0 2 0

- 4 - 4 - 4

1 6 1 6 1 6 … … …

Finally he returned with the answer, “I’ve subtracted 4 from 20 over 60 times, and it keeps coming to 16!”

The final step is to introduce a formal written algorithm for division, and as with multiplication, this should be in structured stages, set out in a school’s mathematics policy.

The introduction of decimals with multiplication or division provides another hurdle for many pupils. Ryan & Williams (2000) discuss the errors common when working with multiplying decimals. One example used in their research carried out at Manchester University;

1.2 x 0.3

The majority of the Year 6 children tested treated the numbers as whole numbers, (ignoring the decimal point), performed the calculation, then simply put them back in. This may come from the over-generalisation of the rule for addition of decimals. A solution to this difficulty might be to challenge the children’s idea, by providing an example where this cannot work;

0.5 x 0.5

If carried out in the same way, then an answer of 2.5 is reached. Through discussion with the child, they should be able to reason that ‘half of a half’, cannot be ‘two and a half’!

Owens (1993) discusses the difficulties faced when dividing by a decimal or fractional number,

“Put yourself in the place of a child who has divided thousands of times and on every occasion found the answer to be much smaller than the number divided – he has made ‘much smaller’ as the one uniform associate of ‘divide’. He now is told that 16 ( 0.25 gives a result far greater than 16…” It is important for teachers to emphasise the practical context of the question, i.e. how many quarter apples can we cut from 16 apples?

Another related notion that many children hold is that 2 ( 4 is equal to 4 ( 2. Again this many be due to the over-generalisation of the commutative properties of + and x. The notion that the smallest number always divides into the largest, and confusion of how to read division sentences all contribute to this. Ideas of this nature need to be challenged, in a practical situation, so that the children can see the contrast between the two confusing statements. One could use string, sharing 2 metres between 4 people, and sharing 4 metres of string between 2 people. Why have they not got the same length each?

Kate Chapman – Langley Primary School

Common Misconceptions in Mathematics: Research theory

The identification of misconceptions in pupils’ work is a vital part of the process of moving towards a focus on learning rather than teaching. Teachers need to predict the misconceptions which are likely to occur with particular pieces of work. They should plan questions and approaches which would expose such misconceptions if they occured. “Why?”, “How?” and “What would happen if....?” questions enable the teacher to probe their pupils’ understanding of a topic.

Due to the nature of maths it may relatively easy to predict where and how misconceptions may occur and hence they may be addressed during the ‘whole class input’.

“Common mistakes, often predictable, can be dealt with as part of the main teaching activity, prior to the activities. It is a feature of teaching maths, as opposed to teaching language that a large proportion of mistakes are predictable - a teacher will be able to say in advance what mistakes children are likely to make in any given situation.”

(www.hamilton-trust.org.uk)

If unpredicted misconceptions occur during the children’s activity work the plenary may be adapted to deal with these problems.

“The plenary provides an ideal opportunity to deal with common misconceptions or problems experienced by the children during the activities. Although many of these were predicted and were therefore incorporated into the lesson itself, nonetheless, the plenary gives a chance to review these areas of difficulty and to further rehearse correct procedures. It also enables the teacher to look diagnostically at the children’s understanding of the concepts covered and to deal with any unforeseen problems experienced.”

(www.hamilton-trust.org.uk)

Indeed it is the plenary and its role in identifying misconceptions and errors that was highlighted by Ofsted in its interim evaluation of the national numeracy strategy.

“Teachers did not use the time to correct misconceptions or errors that occurred.”

As teachers therefore we need to ensure we make time to address common misconceptions (through relevant questioning) as they occur and even prior to their occurrence, due to our knowledge of our children when planning. This paper includes ways to identify and address the most common misconceptions within children’s learning of concepts involving the use of fractions and percentages, simple division and position and movement.

Why do children have difficulty with FRACTIONS, DECIMALS AND PERCENTAGES?

Focus: Identifying common misconceptions for the key objectives in mathematics relating to fractions, decimals and percentages and identifying ways to address these misconceptions through the teaching of appropriate pre-requisite skills

Fractions

Fractions have often been considered as one of the least popular areas of maths. Many children consider the concept of fractions as ‘difficult’ and too often children have had difficulty understanding why they are carrying out a particular procedure to solve a calculation involving fractions.

“It has been said that ‘fractions’ have been responsible for putting more people off mathematics than any other single topic. In fact the very word fraction has been known to make strong men wince!”

(Nuffield Maths 3 Teachers’ Handbook: Longman 1991)

This is probably due to confusion caused by introducing calculations involving fraction too early, when certain children still require more experience with the visual and practical aspect of creating simple fractions of shapes in order to gain a more secure understanding of what a fraction actually is.

Children need to have a firm understanding of what the denominator represents and the numerator represents through the use of visual (and kinaesthetic) resources.

“The headlong rush into computation with fractions, using such mumbo-jumbo as ‘add the tops but not the bottoms’ or ‘turn it upside down and multiply’, has often been attempted before the idea of a fraction or fractional notation has been fully understood.”

(Nuffield Maths 3 Teachers’ Handbook: Longman 1991)

Rather than simply learning a rule to remember in order to solve a certain calculation, children must also surely know why they are carrying out this rule. For example, when adding fractions of a like denominator the aforementioned rule of ‘add the tops but not the bottoms’ will apply. However if a child does not understand why this can be done they may then apply this to addition of fractions with dissimilar denominators. They remember the rule rather than the reason for using the rule. Understanding this rule concerning common denominators will also then help them to understand the next step of altering fractions so that they have a common denominator (-thus enabling them to add the fractions together or even compare the sizes and order the fractions). Similarly, using the rule ’whatever you do the bottom you must do to the top (and vice versa)’ or reducing a fraction to its simplest form would have no meaning whatsoever of the child did not understand the concept of equivalent fractions - knowing that fractions can be the same size but be split into a different number of equal parts. It is essential that children have this pre-requisite knowledge of fractions in order to use and apply their knowledge within a range of different contexts.

Decimals

When working with decimal, children appear to have an automatic tendency to see and read the digits after the decimal point as a number, e.g. 2.47 is read as ‘two point forty-seven’ rather than ‘two point four seven’ Consequently this results in numerous mistakes concerning the size of decimal numbers. Children’s comparing and ordering skills become confused. Children need to read decimal numbers correctly. Reading 25.25, for example, as ‘twenty-five point two five’ gives a vocal indication of where the whole numbers end and the fractional parts begin in the same way that the decimal point separates the ‘whole’ from the ‘parts’ visually. This is particularly important in examples like 3.09 which should be read as ‘three point zero nine’ and not as ‘three point nine’ which of course would mean 3.9.

(The exceptions to this rule occur when the figures after the decimal point are given a specific name as in ‘two and forty seven hundredths’ or when they are read as subsets of measurement or currency such as ‘ two metres and forty-seven centimetres’ (shown in decimal from as 2.47m) or ‘two pounds and forty seven pence’ (shown in decimal form as £2.47).

It is often a useful exercise to ‘read’ a decimal number carefully and ‘take it to pieces’ so that the children are reminded of the importance of position of digits:

|Ten thousands |Thousands |Hundreds |Tens |Units |tenths |hundredths |thousandths |

| | |3 |2 |6 |8 |7 |5 |

Reading off decimal numbers from a number line helps children to make links with interpretation of previous work on graduated scales and prepares for the exercises on ordering decimals, which some children find difficult. Children should be encouraged to use the following strategies to order a range of decimal numbers. First, line up the decimal points to compare the place value of the digits. Then order the numbers by looking at the whole numbers. Where whole numbers are the same, order by the tenths digits. Where wholes and tenths are the same, order by hundredths digits.

Percentages

In their analysis of the 1998 Key Stage 2 tests, QCA highlighted that questions relating to fractions, decimals, ration and proportion posed difficulties. Questions involving percentages caused particular difficulties. Just over one in ten children reaching Level 4 were able to find 24% of 525 on the calculator paper. The most common error was to calculate 24% as one twenty-fourth, indicating there was some knowledge of a connection between percentages and fractions but showing a definite lack of understanding as what a percentage actually is and subsequently how to find a percentage of a quantity in a specific context.

The initial misconception, similar to fractions, is a lack of understanding when following a calculation strategy. Even though a child may know they need to divide by 100 then multiply by the number before the % sign when finding a percentage of an amount but do not really understand why. Again this may lead to errors when applying their strategy to a different context. Using practical, hands-on materials may address this weakness.

Children often are ‘thrown’ when they come across percentages greater than 100. They find these values particularly hard to accept. Using a real-life example here could address the issue, starting with percentatges they feel comfortable working with. For example, an explanation based on a shop selling an item which has been bought for £30. A 50% profit would mean it is sold for £30 plus £15, that is £45. A 100% profit would mean it is sold for £30 plus £30, that is £60. A 150% profit would mean it is sold for £30 plus £45, that is £75.

It is important to help children to appreciate that fractions, decimals and percentages are equivalent ways of writing the same quantity. Children need to be given good experience of these ideas in varied practical contexts and the links between these areas need to be made explicit. This will help to develop children’s overall understanding of Fractions, Decimals, Percentages, Ratio and Proportion.

Why do children have difficulty with DIVISION?

Focus: Identifying common misconceptions for the key objectives in mathematics relating to division and identifying ways to address these misconceptions through the teaching of appropriate pre-requisite skills

There are two aspects of division – sharing and repeated subtraction (or ‘grouping’). These should be dealt with separately as, although the same symbol ÷ , the moving and rearrangement of objects, counters etc. in the early learning of division, is quite different in each case.

It is imperative that from Year 3 onwards children move on from sharing to making groups. It is this aspect that links division facts to corresponding multiplication facts. Children need a solid understanding of multiplication as making ‘groups of’ or ‘lots of’ (repeated addition) in order to then carry out the inverse process of making groups or lots from a given total (repeated subtraction). Children should be encouraged to visually see the link between the two operations. Lots of practical work using objects to create these groups will help the initial understanding of this aspect of division. As will ‘hopping back on number line’ and then counting the number of hops as the groups (Nuffield Maths 3 p.75). Teaching this alongside hopping forward on a number line to show multiplication as repeated addition/groups, will help children to make the links between multiplication and division. Children should also be taught to use certain phrases when reading and solving division problems – e.g. ‘How many groups of ____ in ____?’

Without this understanding, children will have difficulty applying division in a context and may also struggle later in their mathematical learning when moving on to ‘chunking’ – dividing larger numbers by using known multiplication facts to make groups.

‘A word of warning: The language used in posing questions involving repeated subtraction (as an initioal part of the process for teaching division as ‘grouping’) must be chosen very carefully. A boy was once asked, ‘How many times can you subtract 4 from 20?’ He was busy for some time filling a whole page of his book:

20 20 20

-4 -4 -4

16 16 16

Finally he returned with the answer, ‘I’ve subtracted 4 from 20 over 60 times and it keeps coming to 16!’ ‘ (Nuffield Maths p.76)

Although the two aspects of division present different types of practical problem, in both cases division is the inverse of multiplication. Once children understand all of this, they will readily make connections in identifying the four corresponding number sentences relating to an array of counters/objects.

It can certainly be seen that the solution of division problems depends so much on multiplication facts that ‘knowing your tables’ becomes even more important. In fact the emphasis should be on knowing tables ‘inside out’ so that rather than just ‘5 times 4 is what?’ related questions should be asked. For example:

5 multiplied by what number equals 20?

How many groups of 4 are there in 20?

What number of 5’s equals 20?

Kate Chapman - Bibliography

Department for Education and Employment (1999) Guide for Your Professional Development: Book 3 – Raising Standards in Mathematics in Key Stage 2 DfEE Publications

Department for Education and Employment (1999) The National Numeracy Strategy: Framework for Teaching Mathematics DfEE Publications

Department for Education and Skills (2001) The National Numeracy Strategy: Using Assess and Review Lessons DfES Publications

Latham, P. & Truelove P. (1991) Nuffield Maths 3 Teachers’ Handbook Longman

National Numeracy Project (1999) Numeracy Lessons Beam Education

Qualifications and Curriculum Authority (1999) The National Numeracy Strategy: Teaching Written Calculations QCA Publications

Teacher Training Agency Assessing Your Needs in Mathematics: Diagnostic Tasks Teacher Training Agency

Wyvill, R. (1991) Nuffield Maths 6 Teachers’ Handbook Longman

www.bbc.co.uk/education (23/03/2000) School Maths Strategy ‘Going Well’ BBC News

www.hamilton-trust.org.uk

Tables Identifying Misconceptions with the Key Objectives

Identifying Misconceptions

Area of Mathematics: Addition

|Year |Objective |Misconception |Key Questions |Teaching Activity |

|6 |Carry out column addition of numbers |Unless a pupil has a good understanding of |What is each digit worth? |Estimation – Pupils must learn to estimate |

| |involving decimal places. |place value they will continue to make | |– This way they will know when they have |

| | |mistakes with column addition. Such errors |Which is the tenths column? Which is the |made an error. |

| | |are often dismissed as careless mistakes, |hundredths etc? | |

| | |when the pupil in fact has a fundamental | |Add numbers to one and then two decimal |

| | |weakness in their understanding. When adding| |places to begin with. Use the example of |

| | |with decimals such weaknesses are | |money to teach the concept e.g. |

| | |highlighted because of the ‘decimal point’. | | |

| | | | |£3.12 + |

| | | | |£4.15 |

| | | | | |

| | | | |Then extend so that a ‘carry’ is required. |

| | | | | |

| | | | | |

| | | | |Give the children some completed questions |

| | | | |to mark. All questions need to be written |

| | | | |horizontally as well as in column form. |

| | | |Which are correct/ incorrect? How do you |Include incorrect answers. |

| | | |know? | |

| | | | | |

| | | | | |

| |Carry out column addition of positive |As numbers get larger, pupils miscalculate |Why is it important to place the digits in |Estimation |

|5 |integers less than 10,000. |because of a lack of understanding of the |the correct columns? | |

| | |place value of numbers. e.g. | | |

| | |1163 + | | |

| | |12123 |What tips would you give someone to help | |

| | |23753 |them with column addition? | |

| | | | | |

| | |Some pupils will not realise that they will | | |

| | |have to add a ‘carried’ number. | | |

| |Use known number facts and place value to |Pupils sometimes begin adding with the left |What strategies would you use to work out |These pupils could carry out more examples |

|4 |add mentally, including any pair of |hand column first. |the answers to these calculations? Could you|using Base Ten pieces and then linking each |

| |two-digit whole numbers. | |use a different method? |practical step to a recorded step. |

| | | | | |

| |Carry out column addition of two integers | |Which column do you begin with? |Estimate the answer and check that their |

| |less than 1000, and column addition of more |Not understanding the concept of a ‘carry’ | |answer is similar to the estimation. |

| |than two such integers. |when a number totals more than ten, hundred |Does your answer make sense? | |

| | |etc. | |They should realise that it means adding |

| | |e.g. |What tips would you give someone to help |‘nothing’. When they have an answer of zero,|

| | | |them with column addition? |they often need to be reminded to record it.|

| | |+ | | |

| | |101 | | |

| | |1910 | | |

| | | | | |

| | | | | |

| | | |Zero is a placeholder. What does this mean? | |

| | | | | |

| | | |Why is it important to include zeros in our | |

| | |Pupils find it difficult to add when a zero |answers? | |

| | |is involved. | | |

| | |They might not record a zero in an answer, | | |

| | |leading to the following situation: | | |

| | |+ | | |

| | |406 | | |

| | |59 | | |

| |Know by heart all addition facts for each |Pupils are sometimes confused that addition |How many different ways can you make 10 by |Challenge – How many ways can you make 20 by|

|3 |number to 20. |is associative i.e. 3+1=4 and 1+3=4. If they|adding two numbers together? |adding 3 numbers together? Demonstrate the |

| | |were to understand this concept they would |Did you find a quick way to work out the |associative rule in order to make their |

| | |find it much easier to recall the addition |answer? |working more efficient. |

| | |facts. | | |

| | | | | |

| | |If teachers use the phrase ‘near multiple of| |Demonstrate what is happening on a number |

| | |ten’ the children are often confused and |What do we mean when we say ‘a near multiple|line. |

| |Add mentally a ‘near multiple of 10’ to a |believe that they should be multiplying a |of 10?’ | |

| |two-digit number. |number. | |Use simpler terms to describe the operation |

| | |If they understand the term correctly then |Why is this a good method? Show me on a 100 |e.g. ‘add ten and take one away’. |

| | |they might still struggle with compensating,|square. | |

| | |not knowing whether to add or subtract. E.g.| | |

| | |46+19 = | | |

| | |46+20 -1 often confused as | | |

| | |46-20 + 1 | | |

| | | | | |

| | | | | |

| |Know by heart all addition facts for each | |Look at this number sentence _+_=7. What | |

|2 |number to at least 10. | |could the two missing numbers be? | |

| | | | | |

| | | | | |

| | | |I want to find the total of 2,14 and 8. tell| |

| |Use knowledge that addition can be done in | |me some different ways that we could add | |

| |any order to do mental calculations more |Pupils believe that they have to add in the |them. |Chain sums – Children to close their eyes. |

| |efficiently. |order that the question was asked in. | |Give them 3 or 4 numbers to add together. |

| | | |Would it be better to begin adding with the |Who is the fastest? What is the best way to |

| | | |largest number first, or two numbers that |add them? Why? |

| | | |can make 10? | |

| |Understand the operation of addition and use|__ + 3 = 10 |What is the missing number>? |Using blank number lines to enable pupils to|

|1 |the related vocabulary. |When pupils are faced with problems such as |What do you need to add to 3 to gat to 10? |visualise the sentence. |

| | |the above they see two numbers and add them| | |

| | |(e.g. 3+10=13) instead of reading it as a | | |

| | |sentence. | | |

| | | | | |

| |Use mental strategies to solve simple |Pupils ‘count on’ to find the difference |Which number should you start with? | |

| |problems using addition, explaining methods |between their starting number and ten. | |Practise counting. |

| |and reasoning orally. | | | |

| | |Sometimes they are unsure of number order | |Encourage children to tap their heads when |

| | |and therefore make mistakes. | |saying their starting number and them count |

| | |Sometimes they count their starting number | |on their fingers for the following numbers. |

| | |e.g. when finding the number pair 6 + - = 10| |e.g. |

| | |they begin counting with the six and say | |‘6 (tap head), 7,8,9,10 (count on fingers). |

| |Know by heart all number pairs with a total |‘6,7,8,9,10’ and therefore believe the | |They will then get the correct answer, 4. |

| |of 10. |missing number to be 5. | | |

| | | | | |

| |Begin to relate addition to combining two |Not being able to ‘hold’ the number that |Which number are you going to start with? |Using objects and visual aids whilst |

|R |groups of objects. |they started with, when adding the second |The smallest or the largest? |counting on and/or combining groups. |

| | |group. | | |

| | | | | |

| | |Not knowing the number order when counting |Questions relating to number order such as: |Children to place numbers (or groups of |

| | |on. |Is 7 larger or smaller than 5? What comes |objects) in an order and encourage them to |

| | | |after 4? |say which is the smallest/largest etc. |

| | | | | |

| | | | |Using a playhouse, start with 3 people in |

| | | | |one room and four in another – ‘which room |

| | | | |has more people in it? How do you know? |

| |In practical activities and discussion, | | |Move some people from one room to another – |

| |begin to use the vocabulary involved in | | |What has happened in this room? |

| |adding. | | | |

| | | | | |

| | | | | |

| | | | | |

| | | | | |

| | | | | |

| | | | | |

Identifying Misconceptions

Area of Mathematics: Subtraction

|Year |Objective |Misconception |Key Questions |Teaching Activity |

|6 |Carry out column subtraction of numbers |Subtractions involving zeros cannot be done.|Can a subtraction such as 203 - | |

| |involving decimals. | |132 | |

| | | |be done? How can we take 30 from 0? | |

| | | | | |

| | | |If you think that this can’t be done using a| |

| | |That calculations such as the following |written method can it be done mentally? If | |

| | |cannot be done: |it can be done mentally then surely it can |Revise decomposition. If necessary, |

| | | |be done using a written method. |reinforce the method using base ten |

| | |34 – | |materials on an OHP or by using a power |

| | |27 | |point presentation (such presentations can |

| | | | |be found using a general search on the |

| | |Pupils who cannot do these have not got a | |internet). |

| | |sufficient understanding of exchanging. | | |

| | | | |Estimation – Pupils must learn to estimate |

| | | | |– This way they will know when they have |

| | | |What is each digit worth? |made an error. |

| | |Unless a pupil has a good understanding of | | |

| | |place value they will continue to make |Which is the tenths column? Which is the |Subtract numbers to one and then two |

| | |mistakes with column subtraction. Such |hundredths etc? |decimal places to begin with. Use the |

| | |errors are often dismissed as careless | |example of money to teach the concept e.g. |

| | |mistakes, when the pupil in fact has a | |£6.32 - |

| | |fundamental weakness in their understanding.| |£4.11 |

| | |When subtracting with decimals such | | |

| | |weaknesses are highlighted because of the | | |

| | |‘decimal point’. | |Then extend so that decomposition is |

| | | |Which are correct/ incorrect? How do you |required. |

| | | |know? | |

| | | | |Give the children some completed questions |

| | | | |to mark. All questions need to be written |

| | | | |horizontally as well as in column form. |

| | | | |Include incorrect answers. |

| |Calculate mentally a difference such as 8006|Children will have been taught to use a |Why is it possible to solve this calculation|Work with number lines and ‘counting up’ to |

|5 |– 2993. |number line and should be able to visualise |mentally? How did you do it? If you counted |find a difference. |

| | |this mentally. Some pupils May fail to |backwards would it be possible to count up | |

| | |recognise the steps they need to take and |as well? | |

| | |fail to add up ‘the steps’ at the end. | | |

| |Carry out column subtraction of positive | | | |

| |integers less than 10,000. |Misconceptions occur when decomposing from a| | |

| | |‘high’ number. | | |

| | |e.g. 9000 - |How can we ensure that we remember to answer| |

| | |3654 |the question? |Give pupils a range of subtraction questions|

| | |Some pupils will attempt subtraction | |and ask them whether they would be better |

| | |calculations using the formal written | |answered mentally or by a written method. |

| | |method, failing to recognise that it would |Should you answer this mentally or using a | |

| | |be more efficient to calculate the answer |formal written method? Why? | |

| | |mentally. | | |

| | | | | |

| | |Misconceptions occur when pupils (and | | |

| | |teachers) use inaccurate language. | | |

| | |e.g. 2367 - | | |

| | |1265 |A range of questions to do with place value.|Always refer to the digits accurately i.e. |

| | |When talking about 2000 – 1000 they may |What is the 3 digit worth? Is it more than |‘take two hundred from three hundred’. |

| | |refer to it as 2 – 1. |the 2 in this number? | |

| | | | | |

| | | |What tips would you give someone to help | |

| | | |them with column subtraction? | |

| |Use known number facts and place value to | |What strategies would you use to work out | |

|4 |subtract mentally, including any pair of two| |the answers to these calculations? Could you| |

| |digit whole numbers. | |use a different method? | |

| | | | | |

| |Carry out column subtraction of two integers| | | |

| |less than 1000. |Pupils sometimes begin subtracting with the | | |

| | |left hand column first. | | |

| | | | | |

| | | | | |

| | |In tens and units and other formal vertical |Which column do we begin with? | |

| | |subtraction calculations, children sometimes| | |

| | |take the smaller unit number from the | |Practise using base ten materials and talk |

| | |larger, regardless of whether it is part of |Why isn’t it sensible to take the larger |through the calculation. |

| | |the larger or smaller number. |number from the smaller (5-7)? | |

| | |e.g. 945 - |n.b. It is important that you don’t say that|Teach composition, being careful to use the |

| | |237 |it is impossible to take 5 from 7, as this |correct vocabulary. |

| | |712 |is not true. |Demonstrate what is happening when we |

| | | | |decompose, on an OHP with base ten |

| | | |What do we need to do instead? |materials. Show the tens and hundreds |

| | | | |‘moving’. |

| | | |Where will we get our extra 10, 100 etc. | |

| | | |from? |Teach composition using the expanded layout |

| | | | |to begin with. This will help pupils who do |

| | | |What tips would you give someone to help |not have a secure knowledge of place value. |

| | | |them with column subtraction? |e.g. 64-28= |

| | | | |50 |

| | | | |60 14 - |

| | | | |8 |

| | | | |(this is often taught at the end of Yr 3) |

| |Know by heart all subtractions facts for |If teachers use the phrase ‘near multiple of|What do we mean when we say ‘a near multiple|Demonstrate the method on a number line. |

|3 |each number to 20. |ten’ the children are often confused and |of 10?’ |Use simpler terms to describe the operation |

| | |believe that they should be multiplying a | |e.g. ‘take ten away and add one’. |

| |Subtract mentally a ‘near multiple of 10’ to|number. |Is this a good way of subtracting 17 or 18? | |

| |or from a two-digit number. |If they understand the term correctly then |If not why not? |Encourage pupils to map out their |

| | |they might still struggle with compensating,| |calculations on their own number lines. This|

| | |not knowing whether to add or subtract. E.g.|Why is this a good method? Show me on a 100 |will help them to visualise what is |

| | |46-19 = |square. |happening and enable them to work more |

| | |46-20 +1 often confused as | |efficiently mentally. |

| | |46-20 – 1 | | |

| |Understand that subtraction is the inverse |Pupils not understanding the commutative law|Can we change the calculation around and |Pupils need to know why they won’t get the |

|2 |of addition; state the subtraction |and believing that it is possible to change |still get the same answer? |same answer if they change the calculation |

| |corresponding to a given addition and vice |any addition or subtraction question around.| |around. Demonstrate practically using |

| |versa. | |I thought of a number. I subtracted 19 and |materials such as counting blocks. |

| | |e.g. 9+3=12 |the answer was 30. What was my number? How | |

| | |9-12=3 |do you know? | |

| | | | | |

| | | | | |

| |Know by heart all subtraction facts for each| | | |

| |number to at least 10. | | | |

| |Understand the operation of subtraction (as |Pupils might not understand the concept of |Can we find the difference between two |Consult your schools ‘routes through’ and |

|1 |‘take away’ or ‘difference’) and use the |‘finding a difference’. This is largely due |numbers by counting? |concentrate on either counting forwards or |

| |related vocabulary. |to the fact that they can count on or back | |backwards. Pupils often find it easier to |

| | |and are unsure which method to choose. |Using a number line show me two numbers that|‘count up’ from a given number because many |

| | | |have a difference of 2. How might you write |have consolidated their addition skills. |

| | | |that? | |

| | | | | |

| | | | | |

| | |Not realising that numbers can be counted in| | |

| |Use mental strategies to solve simple |order forwards and backwards. |Which number comes before / after 17? Does | |

| |problems using subtraction, explaining | |16 always come before 17? | |

| |methods and reasoning orally. | | | |

| |Begin to relate subtraction to ‘taking |That subtraction can only be described as |Explain to me what happens when we ‘take |Teach other phrases that mean the same as |

|R |away’. |‘taking away’ |away’. |‘taking away’. e.g. How many less? |

| | | | |Be careful that these introductions are made|

| | | |Make up a ‘take away’ question and show me |carefully as to much vocabulary at once will|

| | | |how to do it. |confuse pupils. |

| | | | | |

| | | | |Using a play house, start with 3 people in |

| | | | |one room and four in another – ‘which room |

| | | | |has more people in it? How do you know? |

| | | | |Move some people from one room to another – |

| |In practical activities and discussion, | | |What has happened in this room? |

| |begin to use the vocabulary involved in | | | |

| |subtracting. | | | |

Identifying Misconceptions

Area of Mathematics: Ordering Numbers

|Year |Objective |Misconception |Key Questions |Teaching Activity |

| |Order a mixed set of numbers with up to |Numbers with more digits are larger. |What did you look for first? |Remind pupils to always order numbers |

|6 |three decimal places. |e.g. 23.456 is larger than 123.5. |Which part of each number did you look at |systematically, beginning with the left hand|

| | | |to help you? |column. |

| | | |What do you do when numbers have the same | |

| | | |digit in the same place? | |

| | | |Can you explain this to me using a number | |

| | | |line? | |

| |Order a given set of positive and negative |Children find it difficult to understand |When ordering 201 and 210 why is it |When the introduction is made, it should be |

|5 |integers. |zero because it represents, for them, |important to include the zeros? |through practical situations. This is likely|

| | |something that does not exist. Numbers which| |to be the use of negative numbers indicating|

| | |represent quantities less than zero also |What does this number read (103)? What does |direction, either ‘down or back’. The |

| | |represent the non-existent for many children|the 0 tell us? |understanding of ‘less than zero’ as a |

| | |and so are likely to pose problems for many | |negative value can come later. |

| | |of them. |Tell me two temperatures between 0°C and | |

| | | |-10°C. Which is the warmer? How can you |Order numbers on a ‘washing line’. |

| | |When ordering, the concept of 0 being |tell? |Demonstrate the concept using a thermometer |

| | |greater than –1 is difficult for children |How can something be less than zero? |and the example of a swimming pool, where |

| | |to understand. |Can you think of any real life situations |the depths are shown in negative numbers. |

| | | |where this happens? | |

| |Use symbols correctly, including less than |That > means ‘less than’ and < means |Look at this number sentence _+_=20. What |Teach children that the largest number |

|4 |(< ), greater than (>), equals (=). |‘greater than’. |could the missing numbers be? |should always be placed next to the |

| | | |What is different about the number sentence |‘largest’ end of the symbol. Some children |

| | | |_+_ = < 20? How would you choose numbers to |find it helpful to think of the symbols as |

| | | |make it correct? |crocodile mouths. |

| |Read, write and order whole numbers to at |Confusion about the place value of numbers. |When ordering numbers, where should we |Use place value ‘arrow’ cards to demonstrate|

|3 |least 1000; know what each digit represents.|Difficulties are especially apparent when |begin? The left hand column or the right |to children how to partition numbers. |

| | |ordering numbers such as 212 and 221. |hand column? |Teach how to order numbers systematically |

| | | |When ordering a set of numbers what do you |beginning with the left hand column. |

| | | |look for first? | |

| | | | | |

| | |Failure to understand that the position of |What does this number say (36)? What is the | |

| | |the numeral gives it the value. |digit 3 worth? Is it worth more than the 6 | |

| | | |in this number? If I change the digits | |

| | | |around it reads 63. Is this number smaller | |

| | | |or larger than 36? | |

| |Count, read, write and order whole numbers |Reversal of digits is a common |Show number cards 17 and 71. Which number |A range of activities designed to enable |

|2 |to at least 100; know what each digit |misconception. |says 17? How do you know? What does the |children to write all numbers to 100 |

| |represents (including 0 as a place holder). |i.e. 03 for 30 or 31 for 13 etc. |other one say? How are they the |correctly. |

| | |This creates problems when ordering numbers.|same/different? | |

| |Read, write and order numbers from 0 to at |Counting back and finding a number that is |What number is one less than _? |Practice counting backwards around the room.|

|1 |least 20; understand and use the vocabulary |‘one less than’. | |Challenge the pupils to do it faster each |

| |of comparing and ordering these numbers. | | |time |

| | | | | |

| | |Failure to understand that numbers can be |How could we write that number in words? |Use place value cards to highlight the |

| | |expressed in different ways |Can you write the number 16? Listen |differences between tens and units. |

| | | |carefully – 60 – Is this the same number? | |

| | |Pupils sometimes think they should add two | | |

| | |numbers e.g. 12 = 1+2 = 3 therefore 4 is |Does the digit ‘1’ in ‘12’ mean ten or one? | |

| | |larger than 12. |Show cards with 12 and 21 on them. Which is | |

| | | |12? How do you know? | |

| | |When counting objects, being confused by the| | |

| | |mismatch between the number being said and | | |

| | |the fact that only one object is being |If I take some objects away like this… will | |

| | |pointed to. |I have the same number left? | |

| | | | | |

| | |Being confused by ‘teen’ numbers which are |What does this number say (17)? How could we| |

| | |not read in the order that they are written,|write that number in words? | |

| | |unlike the other numbers, for example 18 is |Show cards with 15 and 51 on them. Which is | |

| | |read ‘eighteen’ not one-ty eight. |15? How do you know? | |

| | | | | |

| | | | | |

| |Use language such as more or less, greater |Confusion caused by vocabulary causes a |What would you rather have: £1 or £2? Why? |Produce a vocabulary worksheet so that |

|R |or smaller, heavier or lighter, to compare |great number of difficulties. Particular | |pupils can group words that mean the same |

| |two numbers or quantities. |problems have been highlighted by the | |thing. |

| | |confusion between ‘more’ and ‘less’. | |Put key words on the wall. |

| | | | | |

| | | | |In role play, give two children some sweets |

| | | | |– Ask ‘is that fair? Why?’ |

| | |Linking words with practical activities: It | |Practical activities involving scales and |

| | |is common for reception pupils to think that| |balances. Pupils to label balances with the |

| | |the heavier object on a balance is the one | |words ‘heavier’ and ‘lighter’. |

| | |that is ‘higher’ than | | |

| | |the other. | |Pack two shopping bags with equal quantities|

| | | | |but not equal weights. What is going to |

| | | | |happen when we pick these up? Why? |

| | | | | |

Identifying Misconceptions

Area of Mathematics: Area and perimeter

|Year |Objective |Misconception |Key Questions |Teaching Activity |

| |Calculate the perimeter and area of simple |Confusion between ‘area’ and ‘perimeter’. |How do we measure the perimeter / area? | |

|6 |compound shapes that can be split into |This might cause pupils to add lengths | | |

| |rectangles. |rather than multiply them when attempting to| | |

| | |calculate area. | | |

| | | | | |

| | |The belief that when finding the perimeter | | |

| | |of a compound shape, which has been split |What do you need to measure? | |

| | |already, they should add the internal |Why don’t you need to measure ‘internal’ | |

| | |lengths. |lengths? | |

| | | | | |

| | | |Why is it a good idea to split this shape | |

| | | |into rectangles to find the area? | |

| | | | | |

| | | |How do you go about calculating the | |

| | | |dimensions of the rectangles / the compound | |

| | | |shape? | |

| | | | | |

| | | |How do you work out the length of sides that| |

| | | |aren’t labelled? | |

| | | | | |

| | | | | |

| | | | | |

| | | | | |

| | |Some textbooks label the lengths of certain | | |

| | |sides and require the pupil to calculate the| | |

| | |others. Pupils might mistakenly believe that| | |

| | |they should only add labelled lengths. | | |

| | | | | |

| | | | |Use examples on squared paper to begin with |

| | | | |and demonstrate to pupils that they might |

| | | | |have to use their addition and subtraction |

| | | | |skills to find the ‘difference’. |

| |Understand area measured in square cm (cm²);|Confusion between ‘area’ and ‘perimeter’. |How do we measure the perimeter / area? |Practical activities such as calculating the|

|5 |understand and use the formula in words |This might cause pupils to add lengths | |perimeter of the playground or the area of a|

| |‘length x breadth’ for the area of a |rather than multiply them when attempting to|If the area of a rectangle is 32cm² what are|desk. |

| |rectangle. |calculate area. |the lengths of the sides? Are there other | |

| | | |possible answers? |Pupils could design a treasure island on |

| | | | |squared paper and ask others to calculate |

| | | |Can you tell me a rule for working out the |the perimeter and area (this could fit in |

| | | |area of a rectangle? Will it work for all |with ratio work). |

| | | |rectangles? | |

| |Measure and calculate the perimeter and area| | | |

|4 |of rectangles and other simple shapes, using| | | |

| |counting methods and standard units. | | | |

| |Read and begin to write the vocabulary | | | |

|3 |related to length. | | | |

| |Measure and compare using standard units | | | |

| |(km, m, cm,) | | | |

| |Estimate, measure and compare lengths, | |Show a 2m piece of ribbon. How long do you | |

|2 |masses and capacities, using standard units;| |think it is? How can you find out? Can you | |

| |suggest suitable units and equipment for | |find something that is longer? How can you | |

| |such measurements. | |check? | |

| | | | | |

| |Suggest suitable standard or uniform | |Which of these are suitable to use for | |

|1 |non-standard units and measuring equipment | |measuring? Why? | |

| |to estimate, then measure a length, mass or | | | |

| |capacity. | |Before you measure what are the important | |

| | | |things to remember about measuring? | |

| |Talk about, recognise and recreate patterns.| | | |

|R | | | | |

Identifying Misconceptions

Area of Mathematics: Problem Solving

|Year |Objective |Misconception |Key Questions |Teaching Activity |

| |Identify and use the appropriate operations |An over-reliance on inadequate mental |What words do you look for in the wording of|Revise vocabulary. i.e. Ensure pupils |

|6 |(including combinations of operations) to |skills. |questions? What words mean you need to |understand what they are meant to do when |

| |solve word problems involving numbers and | |add/subtract/ multiply/divide? |the problem uses phrases such as ‘how much |

| |quantities, and explain methods and |Incorrect identification of the operation(s)| |more?’ and ‘What will be left?’ |

| |reasoning. |to be used. | | |

| | | | |Pupils underline / highlight the important |

| | | | |information. |

| | | | | |

| | | | |Ask pupils to make up two different word |

| | | | |problems for each of these calculations. Ask|

| | | | |them to use a variety of words. |

| | | | |(17+5) x 6 |

| | | | |12.5 ÷ 5 – 0.25 |

| | | | | |

| | | | | |

| | |Inadequate extraction of important | | |

| | |information. Pupils miss important the | | |

| | |important information required to carry out | | |

| | |the calculation(s). | | |

| |Use all four operations to solve simple word|Incorrect identification of the operation(s)|How do you know whether you need to | |

|5 |problems involving numbers and quantities, |to be used. |add/subtract | |

| |including time, explaining methods and |In a two-step operation, some pupils will be|Multiply or divide? | |

| |reasoning. |unsure of which operation to do first. | | |

| | | |Which operation will you need to do first? | |

| | |Misconceptions arise because of confusion |Why? | |

| | |caused by different units of measurement. | | |

| | | | | |

| | | |What are the important things to remember | |

| | | |when solving word problems? | |

| |Choose and use appropriate number operations|As problems become more complex pupils omit |What steps do we need to take to ensure that|Teach the ‘seven steps to problem solving’ |

|4 |and ways of calculating (mental, mental with|important steps. |we get an accurate answer? |(as outlined in the introduction). |

| |jottings, pencil and paper) to solve | | | |

| |problems. | |How did you know that you need to | |

| | | |add/subtract/ | |

| | | |Multiply/divide? | |

| |Choose and use appropriate operations |Not understanding which operation(s) is |What are the important things to remember |Make certain that pupils know, understand |

|3 |(including multiplication and division) to |required to solve the problem. This is often|when solving word problems? |and can recall the language, so that they |

| |solve word problems, explaining methods and |due to a mis-understanding of vocabulary. | |can explain methods and reasoning. |

| |reasoning. |e.g. Find total, more then, +, difference, |How did you know that you need to | |

| | |less than, -, etc. |add/subtract/ | |

| | | |Multiply/divide? | |

| |Choose and use appropriate operations and | |Give children a range of calculations. Which|Give children a means of jotting steps. |

|2 |efficient calculation strategies to solve | |of these can you easily work out in your | |

| |problems, explaining how the problem was | |head? Which might you need to use jottings | |

| |solved. | |for? | |

| |Use mental strategies to solve simple |Not knowing what the problem is asking them |What do you need to find out? | |

|1 |problems using counting, addition, |to do. | | |

| |subtraction, doubling and halving, | |How do you know that you need to | |

| |explaining methods and reasoning orally. | |add/subtract/double etc? | |

| | | |What clues are there? | |

| | | | | |

| | | |What did you do in your head first? How did | |

| | | |you work it out? | |

| |Use developing mathematical ideas and |That most problems cannot be solved. |What is the problem? What shall we do? |With six children say ‘there aren’t enough |

|R |methods to solve practical problems. | | |chairs around the table for all of us. What |

| | | | |shall we do?’ |

| | | | |Some children might suggest getting extra |

| | | | |chairs or sitting on the carpet. They will |

| | | | |realise that problems can be solved in a |

| | | | |variety of ways. |

Identifying Misconceptions

Area of Mathematics: Ratio and Proportion

|Year |Objective |Misconception |Key Questions |Teaching Activity |

| |Solve simple problems involving ratio|Pupils will often confuse the terms |“There are 20 boys and 5 girls in class 6. |Provide children with a range of practical activities, |

|6 |and proportion |‘ratio’ and ‘proportion’, and need a |Give me a sentence using the word ‘ratio’ (or|examples where they can see the effects of ratio (one |

| | |clear understanding of when each is |‘proportion’). Ask for alternatives. |quantity compared to another) and proportion (one part |

| | |appropriate. | |compared to the whole). |

| | |Fraction and Percentage linked work |Look at this Carroll Diagram… | |

| | |may be introduced too early, before | | |

| | |the pupils have a clear understanding| |Mix paint; children see the ratio of 2:5, where 2 tsp |

| | |of the meaning of the terms. |BOYS |of red paint are mixed with 5tsp of yellow. See the |

| | | |GIRLS |effect on the colour if the ratios are swapped round. |

| | | | | |

| | | |Glasses | |

| | | |1 |Use OHP counters to show the ratio of ‘two green frogs’|

| | | |3 |to ‘five yellow fish’, so 2:5, 2 green for every 5 |

| | | | |yellow. Use mental/oral starter … children use cubes |

| | | |No Glasses |and hold a tower where the ratio of red cubes to yellow|

| | | |2 |is 1:3 (for every red cube they use, they need 3 yellow|

| | | |4 |ones) |

| | | | | |

| | | | | |

| | | |Give me a question that has the answer… | |

| | | |3:7 |Using paint example, the proportion of red paint is 2 |

| | | |40% |out of 7 (comparing to the total), so two sevenths. |

| | | |2:1 |Proportion examples can be found using the children… |

| | | |2/3 |stand 5 children at the front of the class. |

| | | | | |

| | | | |What proportion are wearing glasses? |

| | | | |What proportion are boys? |

| | | | |What is the ratio of blonde hair to dark hair? |

| | | | | |

| | | | |Introduce fractions, and decimals once the children are|

| | | | |secure. |

| | | | | |

| | |

|R - 5 | |

| |Refer to table for fractions |

| | |

Identifying Misconceptions

Area of Mathematics: Shape

|Year |Objective |Misconception |Key Questions |Teaching Activity |

| | |. | | |

|6 | | | | |

| |Recognise parallel and perpendicular |Pupils confuse the mathematical |How would you check if two lines are |Ensure children are shown examples where parallel and |

|5 |lines, and properties of rectangles |vocabulary, words such as parallel |parallel/perpendicular? |perpendicular lines are of differing lengths and |

| | |and perpendicular. Often think that |Tell me some facts about rectangles OR Give |thicknesses, to ensure pupils look for the correct |

| | |parallel lines also need to be the |me some instructions to draw a rectangle. |properties of the lines. |

| | |same length – often presented with |What is the same about a square and a |Encourage children to look for examples in the |

| | |examples that are. |rectangle? What might be different? |environment, many pupils gaining success with drawn |

| | | |Is it possible for a right angle to have only|examples find this more difficult. Rather than just |

| | | |three right angles? Why? |present pupils with pairs of lines, for them to decide |

| | | | |if they are parallel or otherwise, ask them to draw a |

| | | | |line parallel/perpendicular to one already drawn. |

| | | | |Provide with ‘nearly’ examples, so they have to use a |

| | | | |checking method – obvious examples will not be as |

| | | | |valuable to them. |

| | | | | |

| | | | | |

| | | | | |

| |Classify polygons, using criteria |Pupils often think that all polygons |Can you think of a polygon with one line of |Avoid finding lines of symmetry on only regular |

|4 |such as number of right angles, |have the same number of lines of |symmetry? How could you check? |polygons… allow children time to investigate as many |

| |whether or not they are regular, |symmetry as they do number of |What polygons can you name with at least one |different shapes as they can, draw own on squared |

| |symmetry properties. |sides/angles. |right angle? |paper, with ‘x’ lines of symmetry. Use a feely box to |

| | |Confusion with number of angles, when|Can you find a polygon here with more than |encourage correct use of vocab, to describe to others |

| | |shape is concave. |one line of symmetry? |in the class. Link with Carroll/Venn diagram work, to |

| | | | |ensure repetition of vocab. |

| | | | |Encourage children to choose a hexagon from a mixed |

| | | | |selection of shapes. |

| | | | |Use ‘sorting cards’ such as… ‘regular polygon’, |

| | | | |‘irregular polygon’, ‘no lines of symmetry’, ‘at least |

| | | | |one line of symmetry’, ‘no right angles’, ‘at least one|

| | | | |right angle’ etc… children choose a card, and try to |

| | | | |find a matching shape. Explain why that particular |

| | | | |shape was chosen. |

| |Identify lines of symmetry in simple |When continually faced with regular |Use these tiles to make a symmetrical shape. |Ensure pupils receive a wide variety of experiences of |

|3 |shapes and recognise shapes with no |shapes, pupils will draw the |Can you take one tile away so that your shape|symmetry, and not simply drawing mirror lines on |

| |lines of symmetry |conclusion that number of lines of |is still symmetrical? Add another tile so |pictures. Use geoboards with an elastic band as the |

| | |symmetry is equal to the number of |that it is no longer symmetrical. |mirror line. Pupils add counters to match a picture, |

| | |sides – and nearly always assume 4 |If this is half a symmetrical shape, tell me |and then add others to make it symmetrical. Can they |

| | |lines of symmetry for rectangle! |how you would complete it to make it |more the band to another position, keeping the |

| | |Find recognition easier with patterns|symmetrical. How do you use the line of |symmetry? Play in pairs, one add a counter, other |

| | |rather than shapes, tackle in that |symmetry to complete the shape? |‘match’ it on other side. |

| | |order! | |Encourage children to check shapes for lines of |

| | | | |symmetry, by drawing round them, cutting them out, and |

| | | | |folding – be aware of careless cutting affecting the |

| | | | |results! |

| |Use the mathematical names for common|Children internalise vocabulary |Reveal shapes from behind a ‘wall’ – what |Play ‘Reveal the shape’ – slowly reveal a card shape |

|2 |2-D and 3-D shapes; sort shapes and |extremely quickly and enjoy learning |shapes could it be? What could it not be? |from an A4 envelope, questioning children on the |

| |describe some of the features. |new names – don’t underestimate them |When feeling shapes in a box/bag, how do you |properties revealed so far. Use a feely box, |

| | |by simplifying more precise names |know it is a cone/circle etc – why can’t it |particularly one with a front window where the |

| | |such as ‘equilateral triangle’. |be a cube etc |remaining children can see the shape that the child is |

| | |Problems occur with the change of | |describing. |

| | |vocab between 2D and 3D… sides become| | |

| | |faces! | | |

| |Use everyday language to describe |See Year 2 |See Year 2 |See Year 2 |

|1 |features of familiar 3-D and 2-D | |How do you know this shape is a square – what|Play games such as ‘Guess my Shape’ – good for building|

| |shapes | |is special about it? |accurate vocabulary. |

| | | |Show two familiar 2D shapes What is the same | |

| | | |about these two shapes? What is different? | |

| | | |Repeat with a cube and a cylinder etc. | |

| |Talk about, recognise and recreate |Children usually link patterns to one|What is special about the way I have ordered |Try not to just use colour as a way of making patterns,|

|R |simple patterns |criteria, e.g. colour, or shape. Use|these counters? Can you make a different |use different shaped objects. Children could make |

| | |examples where need to consider both.|pattern using the same counters? Try to make|sounds in a pattern. The teacher could record these, |

| | | |a pattern where the third counter is blue. |and then when played back, the class could continue the|

| | | |What is wrong with my pattern here? Can you |pattern. |

| | | |put it right? |P.E. lessons can be used for patterns in movement. |

| | | | | |

| | | |Ask pupils to describe a shape | |

| | | |In a feely bag – what might it be? Why? |Allow a wide range of experiences describing shapes, |

| |Use language such as circle or bigger| |Describe it to a friend – can they guess? |the colour, size, whether they are curved, flat, bumpy,|

| |to describe the shape and size of | |Put these towers in order – how did you |whether they roll, or slide…Pupils could make models |

| |solids and flat shapes | |decide to do it? |from blocks and try to describe it. |

Identifying Misconceptions

Area of Mathematics: Multiplication

|Year |Objective |Misconception |Key Questions |Teaching Activity |

| |Multiply decimals mentally by 10 or |Misunderstand the concept of making a|“The calculator display shows 0.1. Tell me |Label chairs TH, H, T, U and choose children to sit, |

|6 |100, and explain the effect |number 10/100/1000 times bigger, |what will happen when I multiply by 100. |holding a digit card. When multiplying by 10/100/1000,|

| | |prefer to learn ‘add a zero’. Causes|What will the display show?” “What number is|the children move required no. spaces along chairs. |

| | |difficulties when working with |10 times as big as 0.01? How do you know?” |Additional children will be needed holding ‘zero’ cards|

| | |decimal numbers and fractions. |“How would you explain to someone how to |as spare chairs become available from the units. |

| | | |multiply by 10?” |Question pupils as to their value, what value do they |

| | |Ignore decimal point, perform |“The answer is 15.2. Make up some questions |have now they have moved seats? How many times larger |

| | |calculation, then ‘count how many |using multiplication with decimal numbers |are you? |

| | |digits after the point’. Effective |that could give this answer.” | |

| |Carry out short multiplication of |shortcut, but difficulty when | |Encourage the children to approximate first, e.g. |

| |numbers involving decimals |applying to mental work – encourage | |4.92x3.1 is approx 5x3, so answer should be approx 15. |

| | |‘why does it work?’ | |Start with mental strategies first…25 x 0.4 is 10 times|

| | | | |smaller than 25 x 4, i.e. 10 times smaller than 100, = |

| | |Children introduced to formal written|Give the children three or four long |10. |

| | |strategy too early, when ‘stuck’ |multiplications with mistakes. Ask them to | |

| | |reach for a calculator because have |identify the mistakes and talk through what |Use the ‘grid method’ (See supplement of examples in |

| |Carry out long multiplication of a |no strategy of their own. |is wrong and how they should be corrected. |NNS) as it is based upon partitioning, with which the |

| |three-digit by a two-digit number |Place value errors when performing |Give a multiplication question (147x32) |pupils will be extremely familiar. It is worth showing|

| | |written calculations can cause |calculated by both the grid method and long |the pupils practically, with cubes, that multiplying |

| | |problems for even able pupils. |multiplication. Ask qs like “what 2 no.s |the parts is the same as multiplying by the whole |

| | | |multiplied together give 4410? Or 294?” |number in one step |

| | |Children are taught to multiply | | |

| | |single digits and count the number of| | |

| | |zeros. 20 x 50 = 100 is a common | | |

| | |mistake as children don’t know what | |Use 20 x 5 as a key fact and then extend to 20 x 50 |

| | |to do with the ‘extra’ zero | |which is 10x bigger. Say the number sentences one after|

| | | | |the other; |

| | | | |Twenty times five is one hundred |

| | | | |Twenty times fifty is one thousand |

| | | | |Write the connected number sentences one above the |

| | | | |other, |

| | | | |20 x 5 = 100 |

| | | | |20 x 50 = 1000 |

| |Multiply any positive integer up to |Pupils do not understand that x10 and|“Why do 6x100 and 60x10 give the same |See Year 6 activity involving children moving places |

|5 |10000 by 10 or 100 and understand the|then x 10 again, is the same as x100.|answer?” |along a set of labelled chairs. |

| |effect |Prefer to learn ‘add a zero’ and so |“I have 37 on my calculator display. What | |

| | |limited understanding. |single multiplication should I key in to | |

| | | |change it to 3700? Why does it work?” | |

| | | | | |

| |Know by heart all multiplication |Children not understand the meaning |“If someone had forgotten their 8 times | |

| |facts up to 10x10 |of ‘lots of’ or ‘groups of’. |table, what tips could you give them to work |Help the children see the links between the tables. |

| | |Children see it as a test of their |it out?” |Use a multiplication grid and complete the easier |

| | |memory, not linking tables facts. |“What other links between times tables are |questions. Encourage them to learn the square numbers |

| | | |useful?” |(4x4, 5x5 etc…) They will be shocked to see that all of|

| | | | |the tables can be reduced to just a few facts to learn!|

| | | | |Play games requiring tables knowledge, use software |

| | |Children are introduced to formal | |such as ‘Developing Number’, which encourages the use |

| | |written methods before fully | |of strategies. |

| |Carry out short multiplication of a |understand the concept, becomes a |“Roughly what answer do you expect to get? | |

| |three-digit by a single-digit integer|test of their memory to remember the |How did you reach that estimate?” |When introducing multiplication with larger numbers, |

| | |‘rule’, and have no strategies to |“Do you expect your answer to be less than or|revert back to an earlier, more secure written method, |

| |Carry out long multiplication of a |rely upon when they are ‘stuck’. |greater than your estimate? Why?” |to increase confidence. Only move to a more formal |

| |two-digit by a two-digit integer |Problems with place value can cause |Give the children some worked examples that |method when secure. Try partitioning the numbers and |

| | |difficulties with written work. |are incorrect. “Is this correct? How do you |dealing with them in parts – the grid method supports |

| | | |know? How could we put it right?” |this, and many children may never advance their written|

| | | |Give qs. such as 37x14 calculated by both the|method beyond this. Encourage mental/jotting |

| | | |grid and the long mult. method. Ask qs. like|approximation before starting written work. |

| | | |“what two nos multiplied together have the | |

| | | |answer 370?” | |

| | |Children need to understand the |“How would it be different if I worked out | |

| | |connection between 6 x 3 and 60 x 3, |14x37?” | |

| | |understanding that the answer is 10x | |Use a counting stick to count in multiples of a number |

| | |bigger because the number being | |and then the corresponding multiple of 10, e.g. 3, 6, |

| | |multiplied is 10x bigger. | |9, 12….30, 60, 90, 120…. |

| | | | |Use multiplication grids with multiples of 10on, e.g. |

| | | | |instead of 3 x 4, 30 x 4 etc. |

| |Know by heart facts for the 2, 3, 4, |See Year 5 |See Year 5 |See Year 5 |

|4 |5, and 10 multiplication tables | |“The product is 40, what two numbers could |Play ‘Round the World’; One child stands behind the |

| | | |have been multiplied together?” |chair of another, the only 2 that can answer a question|

| | | | |given. Should the standing child answer first, then |

| | | | |move to next chair – aiming to get around the whole |

| | | | |class (the world) Should the seated child answer |

| | | | |first, then swap places and continue. |

| |Know by heart facts for the 2, 5 and |See Year 4/5 |See Year 4/5 |See Year 4/5 |

|3 |10 multiplication tables | |Show some missing number statements such as (|Children may need to go back to multiplication as an |

| | | |x 5 = 35 and 10 x ( = 90… “what’s the missing|array, or repeated addition, to gain security with the |

| | | |number – how do you know?” Show ( x ( = 30 |notion of multiplication. |

| | | |and ask “what could the missing numbers be?” | |

| |Understand the operation of |Children inaccurate when displaying |“How many dots are there? How can you work it|Useful to arrange objects on a grid base, so that |

|2 |multiplication as repeated addition |arrays of cubes/objects and so |out without counting them all?” What number |pupils put one object in each square, avoids confusion |

| |or as describing an array |pattern is not clear. Link not clear|sentence can you write to record your method |over rows and columns. These could be numbered at the |

| | |between the array and the seemingly |using x…+…?” “Here are 12 (or 20, 24 etc) |sides. Lots of practical examples required, turn grid |

| | |abstract number given as the answer. |counters. Can you arrange them in equal |round, so understand multiplication can be done in any |

| | | |rows? Record in a number sentence.” |order. |

| | | | | |

| | | | | |

| | |See Year 3/4/5 |See Year 3/4/5 | |

| |Know by heart facts for the 2 and 10 |Confusion occurs because spoken | |See year 3/4/5 |

| |multiplication tables |numbers e.g. sixty, seventy, eighty | |Continue the counting even if pupils get the first few |

| | |etc. follow a regular pattern which | |numbers confused as the later numbers are easier. |

| | |link to the single digit numbers. | |Encourage children to chant as a class/group, point to |

| | |Ten, twenty, thirty, do not relate | |the numbers on the 100 square. |

| | |directly to their corresponding | | |

| | |single digit number | | |

Identifying Misconceptions

Area of Mathematics: Division

|Year |Objective |Misconception |Key Questions |Teaching Activity |

| |Derive quickly division facts |Lack of understanding that division |Start with a number with at least 6 factors, |Lots of activities requiring constant repetition of |

|6 |corresponding to multiplication |is grouping as well as sharing. |e.g. 56. “How many different X and ÷ facts |tables, play ‘Round the World’; One child stands behind|

| |tables up to 10 x 10 |Lack of tables knowledge. |can you make using what you know about 56?” |the chair of another, the only 2 that can answer a |

| | | |“What if you started with 5.6?” |question given. Should the standing child answer |

| | | | |first, then move to next chair – aiming to get around |

| |Carry out short division of numbers |Ignore decimal point when |“The answer is 12.6. What questions could |the whole class (the world) Should the seated child |

| |involving decimals |calculating, then simply ‘slot back |you ask using division with decimal numbers?”|answer first, then swap places and continue. |

| | |in’. Comes from over generalisation | | |

| | |of adding decimals. | | |

| | |(inc. above) | | |

| | | | | |

| | | | | |

| |Divide decimals mentally by 10 or | | | |

| |100, and integers by 1000, and |Misunderstand the concept of making a|“Why do 5 ÷ 10 and 50 ÷ 100 give the same |See Year 5. |

| |explain the effect |no. 10/100/1000 times smaller, prefer|answer?” |When operating with decimal numbers, and whole numbers |

| | |to learn ‘knock off a zero’. When |“I divide a no. by 10, and then again by 10. |where units digit is not zero, choose another child to |

| | |the no. ends in a different digit, |The answer is 0.3. What no. did I start |sit and hold the ‘decimal point’ card. They will NEVER|

| | |simply knock that off. Ignore |with? How do you know?” |move! Additional chairs will be required for the |

| | |decimal point, or ‘move it’ - often |“The calculator display shows 0.1. Tell me |tenths, hundredths columns. |

| | |taught by parents! |what will happen when I multiply by 100? | |

| | | |What will the display show?” | |

| | | |“How would you explain to someone how to | |

| | | |multiply a decimal by 10?” | |

| |Divide any positive integer up to |Pupils do not understand that ÷ 10 |“Why do 30 ÷ 10 and 300 ÷ 100 give the same |Label chairs TH, H, T, U and choose children to sit, |

|5 |10000 by 10 or 100 and understand the|and then ÷ 10 again, is the same as ÷|answer?” |holding a digit card. When dividing by 10/100/1000, |

| |effect |100. |“I have 3700 in my calculator display. What |the children move required no. spaces along chairs. |

| | | |single division should I key in to change it |Child as zero units ‘drops off’ end. NOTE ; only used |

| | | |to 37? Explain why this works.” |when units digit is zero. Question pupils as to their |

| | | | |value, what value do they have now they have moved |

| |Carry out short division of a three | |“Roughly what answer do you expect to get? |seats? How many times smaller are you? |

| |digit by a single digit integer | |How did you come to that estimate?” | |

| | |See Year 6 |“Do you expect your answer to be less than or|Ensure that the pupils relate the division to |

| | |Pupils are introduced to written |greater than your estimate – why?” |multiplication; 27(3 … ‘how many chunks of 3 are there |

| | |method before fully understanding the| |in 27?’ Count up in 3s. Less able children use a |

| | |concept of grouping or ‘chunking’. | |tables square for multiplication facts, so not to slow |

| | |Need more concrete examples. | |down understanding of the division process. |

| | | | | |

| | |When dealing with remainders, pupils | | |

| | |have little understanding of how to | | |

| | |represent as a fraction or a decimal.| | |

| | | | | |

| | | | | |

| |Derive quickly division facts |Lack of understanding that division |How many division facts can you make using |Lots of practise repeating multiplication tables, play |

|4 |corresponding to the 2, 3, 4, 5, 10 |is grouping as well as sharing. |what you know about 24 (or 20, 30…). How did|games such as ‘round the world’ (see Year 6). |

| |multiplication tables |Lack of tables knowledge. |you work out the division facts? |Show the pupils, physically, that ‘groups of’ |

| | |Not understand the concept of | |(multiplication) and division are the same, using |

| | |‘inverse’. | |cubes. |

| | | | |Use multiplication grid for ‘facts’, so not slowing |

| |Find remainders after division |Lack of understanding of WHAT the | |down division process. |

| | |remainder actually represents, need |“Do all divisions have remainders?” “Make up| |

| | |to see calculation in concrete way, |some division questions that have a remainder| |

| | |i.e. cubes left over after division. |of 1" ”How did you do it?” “Make up some |Practical examples where children need to put objects |

| | |Ignoring the context of the question |division questions that have no remainder. |in boxes (egg boxes good!), alongside written form of |

| | |– should it be rounded up or down? |How did you do this? Why do they not have a |the division question. Lots of practical examples of |

| | | |remainder?” |remainders – e.g. coaches for swimming etc. |

| |Understand division and recognise |Children not see link between the two|“What is the answer to 20 ÷ 5? Can you make |Show children physically that 2(4 cannot be the same as|

|3 |that division is the inverse of |operations, need more experience of |up a problem, that means you need to work out|4(2… use a 2m length of wool, cut into 4 equal pieces. |

| |multiplication |chunking, as well as sharing. |20 ÷ 5 to solve it?” |Use a 4m length of wool, cut into 2 pieces – are they |

| | |Need to be taught alongside each |“Can you tell me some no.s that will ÷ |the same? |

| | |other – not separately. |exactly by 2? 5? 10? How do you know?” | |

| | |Children think that 2 ÷ 4 is the same|Give the children a set of no.s that are |Lots of examples using cubes to show that |

| | |as 4 ÷ 2. |related by x and ÷ facts along with the |multiplication and division are the inverse of each |

| | | |multiplication, division and equals signs. |other. |

| | | |Ask them to form some x and ÷ statements. | |

| | | |Ask them to match the ones that are linked in|When teaching the chunking written method, model with |

| | | |some way and to explain why. |cubes every step of the written process. |

| | | |“If I multiply a no. by 2, and then ÷ the | |

| | | |answer by 2, what happens?” | |

| | | |“Is 2 ÷ 4 the same as 4 ÷ 2?” Why not? | |

| |Know and use halving as the inverse |Children confuse the words ‘halving’ |“I’m thinking of a number, I’ve halved it and|Practise using the vocabulary with the pupils – link |

|2 |of doubling |and ‘doubling’. |the answer is 15. What number was I thinking|‘doubling’ to a ‘double decker bus’, they will remember|

| | |Lack of understanding that two |of? Explain how you know?” |which means twice as many! |

| | |operations are linked, often taught |“I’m thinking of a no. I’ve doubled it and |Show practical examples of halving an apple, and |

| | |separately. |the answer is 18. What no. was I thinking |doubling the number of apples etc. |

| | | |of. How do you know?” | |

|1 | | | | |

|R | | | | |

Identifying Misconceptions

Area of Mathematics: Fractions, Decimals and Percentages

|Year |Objective |Misconception |Key Questions |Teaching Activity |

| | | | | |

|6 |Fractions | | | |

| |Reduce a fraction to its simplest |Lack of understanding that fractions |What clues did you look for to cancel these |Demonstrate using visual and kinaesthetic resources how|

| |form by cancelling common factors. |can be ‘equivalent’ (i.e. same size |fractions to their simplest form? |and why fractions can be equivalent sizes but be split |

| | |but split into different number of |How do you know when you have the simplest |into different numbers of parts. |

| | |equal parts). Therefore children |form of a fraction? |E.g.s:- |

| | |struggle with concept of reducing to |Give me a fraction that is equivalent to 2/3,|Fraction Walls |

| | |a simpler equivalent. |but has a denominator of 18. How did you do |Fraction Rods |

| | |Some children may remember the ‘more |it? |Numberlines (0 to 1 but split into different fractions)|

| | |abstract’ rule ‘ whatever you do to | |Card strips and paper clip indicating the size of the |

| | |the bottom, do to the top’ (and vice | |fraction. |

| | |versa) but due to lack of | |NNS: ICT CD-Rom Fractions |

| | |understanding why this works, cannot | |NOF Kit – Equivalent Fractions |

| | |apply in a context. | | |

| | | | |Show how two fractions can be of equal size but one is |

| | | | |split into less parts – as a result, compare the |

| | | | |denominators and the numerators. Finally, develop the |

| | | | |rule how fractions can be simplified using same two |

| | | | |fractions. Make link to division to reduce the number |

| | |Children only see a fraction as a | |of parts. Then apply to other example, checking with |

| | |part of a whole ‘one’ (a strip or a | |visual/kinaesthetic aids. |

| | |circle)– Do not understand can be | |Demonstrate using total of objects – a ‘whole’ group of|

| | |applied to a group of objects, a | |objects (Smarties are great for modelling, as |

| | |number or a measurement greater than | |successful participants can eat their answer!). |

| | |1. |2/5 of a total is 32. What other fractions of|Finding a fraction means finding a part of that ‘whole’|

| | |Some children may remember the divide|the total can you calculate? |group (revisit concept of equal parts). Physically |

| |Use a fraction as an operator to find|by the denominator and times by the |Using a set of fraction cards (e.g. 3/5, 7/8,|show the moving of the objects to split into groups |

| |fractions of numbers or quantities |numerator’ but do not understand why|5/8, ¾, 7/10 etc.) and a set of two-digit |(the denominator). Find one ‘part’, then look at the |

| |(e.g.5/8 of 32, 7/10 of 40, 8/100 of |and hence cannot apply this to a |number cards, ask how the fractions and |numerator and determine how many of those parts are |

| |400 centimetres). |specific context. |numbers might be paired to form a question |‘needed’. |

| | | |with a whole-number answer. What clues did | |

| | | |you use? |Whilst modelling, make links to division when splitting|

| | | | |the group into equal parts and multiplication when |

| | |Lack of understanding of the place | |finding a number of the parts. |

| | |value of decimal fractions. | | |

| | |Children don’t understand that digits| | |

| | |after the decimal point represent | | |

| | |parts of a whole. | |To improve place value understanding:- |

| | |(They either see them as some | |Use ‘human’ place value chart - place value columns on|

| | |abstract idea or even read them as | |chairs or hanging from ceiling. Give certain children |

| | |whole numbers). | |digits and ask to show a number in the correct columns |

| | |Often connected to a lack of |What did you look for first? |– including decimal fractions. Ask children to say |

| | |understanding concerning fractions |Which part of each number did you look at to |what they are worth. |

| |Decimals |(and an inability to relate fractions|help you? |Ensure children use language ‘point two five’ and not |

| |Order a mixed set of numbers with up |to decimals). |Which numbers did you think were the hardest |‘point twenty five’. |

| |to three decimal places |Children may view 6.25 as greater |to put in order? Why? | |

| | |than 6.3 as they see the digits 2 and|What do you do when numbers have the same |To improve ordering numbers |

| | |5 as twenty-five. |digit in the same place? |Model by placing pair/set of numbers on place value |

| | | |Can you explain this to me using a number |chart. (Appendix 1) Emphasise need to follow steps:- |

| | | |line? |Look at whole numbers first – allows you to order some |

| | | |Give me a number somewhere between 3.12 and |numbers. |

| | |Lack of understanding as to what a |3.17. Which of the two numbers is it closer |If have same whole numbers need to look at tenths |

| | |fraction actually is:- |to? How do you know? |column. |

| | |Children view a percentage as a | |If same digit in tenths column need to look at |

| | |number rather than part of an amount.| |hundredths column, and so on…. |

| | | | | |

| | |Children haven’t made the link with | | |

| | |fractions and so struggle to find 50%| |Encourage children to remember the french words ‘per |

| | |(1/2), 25%(1/4), 75%(3/4), 40% (4/10)| |cent’ meaning ‘out of a hundred. Show as a percentage |

| | |etc. | |and a fraction, saying ‘out of a hundred’ as drawing |

| | | | |the fraction line. |

| | |Children realise a link with | |Those who really struggle can be taught visually using |

| | |fractions but use the value of the | |an empty ‘hundred square’ shaded to show the |

| | |percentage as the denominator and | |appropriate percentage – ‘so many out of a hundred’. |

| | |subsequently that divide by value. | |Making sure they also see this percentage as a fraction|

| | |E.g. They think 24% is equal to 1/24 | |and so begin to make links with the possibility of |

| | |and so to find 24% of 300 they would | |finding a percentage of an amount not just an amount |

| | |simply divide by 24. |What percentages can you easily work out in |out of a hundred. |

| | | |your head? Talk me through a couple of | |

| | | |examples. |Show children why 50% is the same as ½. Use knowledge |

| | | |When calculating percentages of quantities, |of equivalent fractions to reduce 50/100 to its |

| | | |what percentage do you usually start from? |simplest form. Repeat with other simple |

| | | |How do you use this percentage to work out |percentages/fractions. Repeat by showing fractions |

| |Percentages | |others? |like 3/5, 2/5 and 4/5 as a percentage, by finding |

| |Understand percentage as the number | |Are there any percentages that you cannot |equivalent fraction in tenths first, then converting to|

| |of parts in every 100, and find | |work out? |a percentage (out of a hundred). |

| |simple percentages of small | |Using a 1-100 grid, 50% of the numbers on | |

| |whole-number quantities. | |this square are even. How would you check? | |

| | | |Give me a question with an answer 20% (or |Draw 3 numberlines – underneath the each other– same |

| | | |other percentages). |length – 0 to 1 (100%). One is the ‘fraction line’, |

| | | |To calculate 10% of a quantity, you divide it|one the ‘decimal line’ , one the ‘percentage line’ – |

| | | |by 10. So to find 20%, you must divide by |All representing the same amount. Mark on a fraction |

| | | |20. What is wrong with this statement? |such as ¾. Show equivalent decimal 0.75on relevant |

| | | | |line. Show equivalent percentage on relevant line. |

| | | | | |

| | | | |Demonstrate how to find simple percentages of small |

| | | | |quantities by relating to fractions of quantities. |

| | | | |(See Y6 Teaching Activities for finding fractions of |

| | | | |quantities). |

| | | | | |

| | | | |Give children plenty of opportunity to experience |

| | | | |percentages in real-life contexts – tends to give more |

| | | | |meaning to their calculations and hence aid their |

| | | | |understanding. Use household packages or containers, |

| | | | |showing ‘10% off’ or ‘15% extra free’, to calculate |

| | | | |the reduction in price or the amount extra and the |

| | | | |subsequent ‘new’ total amount. |

| | | | |(See ‘Numeracy Lessons’ booklet in the yellow training |

| | | | |pack (Professional Development 1 & 2) and ‘Springboard|

| | | | |6’ folder for Year 6 lesson ideas.) |

| | | | | |

|5 |Fractions and decimals | | | |

| |Relate fractions to division and to |Children think of fractions as part |Tell me some division questions that have the|Use a group of objects and demonstrate dividing the set|

| |their decimal representations. |of a shape and are unable to relate |answer 2, 5, 10, 15, etc. How did you go |into equal parts – fractions. Make links with division.|

| | |to finding fractions of a number. |about working this out? |Show numerical representation next to demonstration to |

| | | |Tell me some fractions of numbers that are |help links. |

| | |Children’s understanding of division |equal to 2, 5, 10, 15, etc. How did you go |e.g.1/3 of 15. Can be solved using 15 Smarties, |

| | |as sharing – making equal parts |about working this out? How do these relate |dividing the set into 3 equal parts and finding how |

| | |(rather than grouping) may be |to the division questions? |many in one part. Show the calculation ‘15 ÷ 3 = 5’ |

| | |lacking. | |and 1/3 of 15=5 |

| | | | | |

| | | | |Explain and demonstrate how the fraction line can also |

| | | | |be thought of as a dividing line. |

| | | | |e.g. ¼ can be one whole split into 4 equal parts which |

| | | | |is the same as 1÷4. Extend to show improper fractions.|

| | | | |E.g. 12/3 is another way of writing 12÷3. |

| | | | | |

| | | | | |

| | | | |When working on place value and using language of so |

| | |Children regard fractions and | |many tenths/hundredths in a decimal fraction make links|

| | |decimals as two abstract ideas. | |with recording as a fraction, and vice versa…. |

| | |Unable to make links between the two.|Tell me two fractions that are the same as |When working on fractions record suitable fractions as |

| | | |0.2. Are there any other decimals that have |decimals. |

| | | |fractions that are both fifths and tenths? |Ask children to enter the fraction into the calculator |

| | | |How many hundredths are the same as 0.2? |– thinking of the fraction line as the dividing line |

| | | |You have been using your calculator to find |(see above) to change their fraction into a decimal |

| | | |an answer. The answer on the display reads |Stress both the decimal fraction and the fraction mean |

| | | |3.6. What might this mean? |the same amount – the same part of the whole. |

| | | | |Make links with more than one fraction being equivalent|

| | | | |to a decimal fraction as we have fractions that are |

| | | | |equivalent. (See Year 6 ‘equivalent fractions’ |

| | | | |activities above.) |

| | | | | |

| | | | |Draw 2 numberlines – one underneath the other– same |

| | | | |length – 0 to 1. One is the ‘fraction line’ the other |

| | | | |the ‘decimal line. Mark on a fraction such as ¾. Show |

| | | | |equivalent decimal 0.75on line below. Continue with |

| | | | |other values (including tenths and their equivalents). |

| | | | |Clearly shows fractions and decimals that represent the|

| | | | |same amount. |

| | | | | |

| | | | | |

| | | | |Create an ‘equivalent fractions and decimals washing |

| | | | |line’. (See Appendix 2 for fraction/decimal (and |

| | | | |percentage ) cards.) |

| | | | |Create an ‘equivalent fractions and decimals’ star |

| | | | |chart (mind map). |

| | | | | |

| | | | | |

| | | | | |

| | | | |To improve place value understanding:- |

| | | | |Use ‘human’ place value chart - place value columns on|

| | | | |chairs or hanging from ceiling. Give certain children |

| | | | |digits and ask to show a number in the correct columns |

| | | | |– including decimal fractions. Ask children to say |

| | | | |what they are worth. |

| | | | |Ensure children use language ‘point two five’ and not |

| | | | |‘point twenty five’. |

| | |Children view the decimal fraction as|What can you tell me about the digit 7 in | |

| | |a whole number. (See Year 6) |each of these numbers: 3.7, 7.3, 0.37, 7.07?|Stress digits before the decimal point are whole |

| | | |What if I put a £ sign in front of each of |numbers, whereas those after it reprepresent parts of a|

| | |Children consider hundredths to be |them? |whole (not yet 1 unit) – link language and value to |

| | |greater than tenths. |What if they are all lengths given in metres?|fractions by showing fraction equivalent to tenths and |

| | | |Enter 5.3 onto your calculator display. How |hundredths. |

| |Decimals | |can you change this to 5.9 in one step | |

| |Use decimal notation for tenths and | |(operation)? |Demonstrate why hundredths are less than tenths – use |

| |hundredths. | | |idea of pizza/cake/apple being split into tenths |

| | | | |compared to being split into hundredths where each |

| | | | |piece would be smaller – children visualise cutting the|

| | | | |object. |

| | | | | |

| | | | | |

| | | | | |

| | | | | |

| | | | | |

| | | | | |

| | | | | |

| | | | | |

| | | | | |

| | | | | |

| |Percentages | | | |

| |Begin to understand percentage as as | | | |

| |the number of parts in every 100, and| | | |

| |find simple percentages of small | | | |

| |whole-number quantities (e.g. 25% of | | | |

| |8) | | | |

| | | | | |

| |Fractions and Decimals | | | |

| |Express one half, one quarter, three | | | |

| |quarters and tenths and hundredths as| | | |

| |percentages | | | |

| | | | | |

|4 |Fractions | | | |

| |Recognise simple fractions that are |Unsure of the meaning of the value of|Tell me some fractions that are equivalent to|Chant rhymes to remember the significance of the |

| |several parts of a whole, and mixed |the denominator and the numerator, |½. How do you know? Are there others? |numerator and the denominator such as:- |

| |numbers; recognise the equivalence of|particularly where the numerator is |Repeat for fractions like ¼ and ¾, 1/3 and |‘How many parts have been made this time? Write this |

| |simple fractions. |greater than 1. |2/3. |below the dividing line. |

| | | |Tell me some fractions that are greater than |How many parts do we want this time? |

| | | |½. How do you know? What about fractions |Write this above the dividing line!’ |

| | | |that are greater than 1? | |

| | | | | |

| | | | | |

| |Decimals | | |Demonstrate using visual and kinaesthetic resources how|

| |Understand decimal notation and place| | |and why fractions can be equivalent sizes but be split |

| |value for tenths and hundreds and use| | |into different numbers of parts. |

| |in context. For example: | | |E.g.s:- |

| |Order amounts of money; | | |Fraction Walls |

| |Convert money from pounds to pence; | | |Fraction Rods |

| |Convert length from cm to m; | | |Number lines (0 to 1 but split into different |

| |Round a sum of money to the nearest | | |fractions) |

| |pound. | | |Card strips and paper clip indicating the size of the |

| | | | |fraction. |

| |Fractions and Decimals | | |NNS: ICT CD-Rom Fractions |

| | | | |NOF Kit - Equivalent Fractions |

| |Recognise the equivalence between the| | |Show how two fractions can be of equal size but can be |

| |decimal and fraction forms of one | | |split into a different number of parts. Compare the |

| |half and one quarter, and tenths such| | |denominators and the numerators Encourage children to |

| |as 0.3. | | |make links between the numerical value of the |

| | | | |numerators and the denominators and to remember |

| | | | |equivalents between haves, quarters and eighths; tenths|

| | | | |and fifths; thirds and sixths. |

| | | | | |

| | | | |Create an equivalent fractions star chart encouraging |

| | | | |children to spot the ‘patterns’ between equivalent |

| | | | |fractions and to add to the branches. |

| | | | | |

| | | | | |

|3 |Fractions |Children struggle with the idea of a|Which would you rather have 1/3 of £30 or ¼ |Reinforce children’s understanding of what a fraction |

| |Recognise unit fractions such as ½, |fraction being an equal part of |of £60? Why? |actually is and how we find fractions. |

| |1/3, ¼, 1/5, 1/10, and use them to |something. |What numbers/shapes are easy to find a | |

| |find fractions of shapes and numbers.| |third/quarter/fifth/tenth of? Why? |Demonstrate how we can find a quarter of a KitKat by |

| | |Children struggle moving on from | |splitting into 4 equal parts and showing each part is |

| | |finding a fraction of a shape to the | |equal and 1/4 |

| | |more abstract concept of finding a | | |

| | |fraction of a number. | |Model how to find a unitary fraction of a shape |

| | | | |kinaesthetically by folding/ cutting/labelling the |

| | | | |fraction. |

| | | | |Show how the number of equal parts made is shown in the|

| | | | |denominator. |

| | | | | |

| | | | |Demonstrate finding a fraction of number using a set of|

| | | | |objects (Multilink/ counters/Smarties!!) representing |

| | | | |that number. Split into equal number of parts as given|

| | | | |by the denominator. Find one of those parts. |

| | | | |If children struggle with the more abstract calculation|

| | | | |of finding a fraction of a number encourage them to |

| | | | |draw as objects/dots/counters, then split into equal |

| | | | |parts. |

| | | | | |

| | | | | |

|2 |Fractions | | | |

| |Begin to recognise and find one half | | | |

| |and one quarter of shapes and small | | | |

| |numbers of objects. | | | |

| |Begin to recognise that two halves or| | | |

| |four quarters make one whole and that| | | |

| |two quarters and one half are | | | |

| |equivalent. | | | |

| | | | | |

|1 | | | | |

| | | | | |

|R | | | | |

Identifying Misconceptions

Area of Mathematics: Position, Direction, Movement and Angle

|Year |Objective |Misconception |Key Questions |Teaching Activity |

| | | | | |

|6 |Position and Direction |Putting y co-ordinate before |A square has vertices at (0,0), (3,0) and |Chant phrases to aid memory e.g:- |

| |Read and plot co-ordinates in all four |the x, resulting in incorrectly|(3,3). What is the co-ordinate of the fourth|x comes before y in the alphabet, |

| |quadrants. |placed position – due to:- |vertex? |along the corridor and up the |

| | |lack of knowledge of order or |A square has vertices at (3,0), (0,3) and |stairs, |

| | |lack of knowledge concerning |(-3,0). What is the co-ordinate of the |x is across (- a ‘cross’!!). |

| | |names of axes. |fourth vertex? | |

| | | |A square has vertices at (0,0), (2,0). Give | |

| | | |two possible answers for the positions of the| |

| | | |other two vertices. | |

| | |When using 4 quadrants, |A square has vertices at (-1, 1) and (-2, | |

| | |misplaced positions due to lack|-3). Give two possible answers for the | |

| | |of understanding of order of |positions of the other two vertices. |Practise pointing and chanting negative and positive |

| | |negative numbers on a scale. | |numbers on a scale, using a ‘counting stick’ (forwards |

| | |Uncertainty where to place the | |and backwards). Hold stick both horizontally and |

| | |protractor – not sure what | |vertically to link to both the x and the y axes |

| | |they’re measuring. Comes from | |Refer to the ‘symmetrical’ quality of the numbers with |

| | |lack of understanding of what | |0 as the middle value. |

| | |an angle actually is. | | |

| | | | | |

| | | | |Reinforce understanding of what an angle actually is – |

| | | |What important tips would you give to a |an amount of turn and that this can be represented by |

| | | |person about |two ajoining lines – 1 showing the starting position, |

| | | |using a protractor? |the other showing the point after the turn. |

| | | |How do you know which scale to use on the |Children stretch out two arms and turn, leaving one arm|

| | | |protractor? |in original position and move other arm a certain |

| |Angle | |What type of angle is this? How do you know?|amount (90(, 180( etc). Children can see the two |

| |Use a protractor to measure acute and | | |‘lines’. |

| |obtuse angles to the nearest degree. | |How can you use what you know about acute and|(Further reinforcement using geostrips connected at one|

| | | |obtuse angles to check your measurement? |end using a brass tack so able to move one of the |

| | | |Is your measurement sensible? Why? |‘arms’) |

| | | | | |

| | | | |Link to measuring using a protractor – original arm is |

| | | | |‘base’ line. Identify point where child’s body was and|

| | |Uncertainty of which scale to | |finishing position is the second line. |

| | |use to measure the size of the | | |

| | |angle. | |Model positioning of protractor using OHT (use scale |

| | | | |without numbers at this point), making clear how to |

| | | | |position protractor on the angle. |

| | | | |Link movement of body and second arm to reading the |

| | | | |scale. No turn at the start means 0(. Find |

| | | | |appropriate scale starting at 0( on the base line |

| | | | |(-introduce marked protractor – OHT and individual). |

| | | | |See Appendix 3 for OHTs. |

| | | | | |

| | | | |Encourage children to make checks. Need to look at |

| | | | |angle first and identify whether acute or obtuse. If |

| | | | |measuring an acute angle, their measurement must be |

| | | | |less then 90(. If it isn’t – ‘Did you use the correct |

| | | | |scale?’ |

| | | | | |

| | | | |Devise chants to aid memory e.g ‘An acute angle is a |

| | | | |cute angle’ (i.e. less than 90()– but remember to point|

| | | | |out spelling of acute as children then tend to spell |

| | | | |‘acute’ simply as ‘cute’! |

| | |Unable to recall property of an| | |

| | |acute angle and an obtuse | |Good teaching and independent activities in NNS |

| | |angle. | |Software – ‘What’s My Angle?’ |

| | | | | |

| | | | | |

| | | | | |

| | | | | |

| | | | | |

|5 |Position and Direction | | | |

| |Recognise positions and directions: | | | |

| |Read and plot co-ordinates in the first | | | |

| |quadrant. | | | |

| | | | | |

| |Angle | | | |

| |Understand and use angle measure in | | | |

| |degrees. | | | |

| |Identify, estimate and order acute and | | | |

| |obtuse angles. | | | |

| |Use a protractor to measure and draw acute| | | |

| |angles to the nearest 5(. | | | |

| |Calculate angles in a straight line. | | | |

| | | | | |

|4 |Position and Direction | | | |

| |Recognise positions and directions: for | | | |

| |example, describe and find the position of| | | |

| |a point on a grid of squares where the | | | |

| |lines are numbered. | | | |

| |Recognise simple examples of horizontal | | | |

| |and vertical lines. | | | |

| |Use the eight compass directions. | | | |

| | | | | |

| |Movement and Angle | | | |

| |Make and measure clockwise and | | | |

| |anti-clockwise turns. | | | |

| |Begin to know that angles are measured in | | | |

| |degrees and that: | | | |

| |One whole turn is 360( or 4 right angles; | | | |

| |A quarter turn is 90( or 1 right angle; | | | |

| |Half a right angle is 45(. | | | |

| |Start to order a set of angles less than | | | |

| |180( | | | |

| | | | | |

|3 |Position and Direction | | | |

| |Read and begin to write the vocabulary | | | |

| |related to position, direction and | | | |

| |movement; for example, describe and find | | | |

| |the position of a square on a grid of | | | |

| |squares with the rows and columns | | | |

| |labelled. | | | |

| |Recognise and use the four compass | | | |

| |directions N, S, E, W. | | | |

| | | | | |

| |Movement and Angle | | | |

| |Make and describe right-angled turns, | | | |

| |including turns between the four compass | | | |

| |points. | | | |

| | | | | |

| | | | | |

| |Identify right angles. | | | |

| | | | | |

| | | | | |

| | | | | |

| | | | | |

| | | | | |

| | | |What do you look for when deciding if a |Children create a ‘Monster Right Angle Measurer’ (See |

| | | |corner (vertex) of a shape is a right angle? |Appendix 4) and use to find vertices of shapes/objects |

| | | | |that have right angles. |

| | | | | |

| | | | |NNS Software ‘Sorting 2-D Shapes’ |

| | | | | |

|2 |Postion, Direction and Movement | | | |

| |Use mathematical vocabulary to describe | | | |

| |position, direction and movement. |Lack of understanding of |How could I/ the robot get from… to…? |Model how to use appropriate vocabulary through V, A & |

| | |vocabulary. | |K activities (see NNS Supplement of Examples p.86 & |

| | | | |p.87 for vocabulary) e.g.:- |

| | | | | |

| | |Confusion between left and | |Pretend to be a robot and ask the children to give |

| | |right. | |instructions using mathematical vocabulary to move ‘the|

| | | | |robot’ through an obstacle course. |

| | | | | |

| | | | |Using a large grid, tell the children to place objects |

| | | | |according to instructions. Say, for example: Place |

| | | | |the tree to the left of the house. |

| | | | | |

| | | | |Ask the children to give instructions so a partner can |

| | | | |create a copy of their grid without seeing it. |

| | | | | |

| | | | |Use squared paper and a counter to move from a square |

| | | | |near the centre of the paper to a square near the edge,|

| | | | |describing the route as three squares along and two |

| | | | |squares down, or three squares to the left and two |

| | | | |squares up |

| | | | | |

| | | | | |

| | | | | |

| | | | |Devise instructions to make a floor robot navigate a |

| | | | |floor plan or maze in which all the paths are at right |

| | | | |angles to each other and some are dead ends. |

| | | | | |

| | | | |Show how to remember ‘left’ and ‘right’. Make L shape |

| | | | |with forefinger and thumb of left hand – not possible |

| | | | |with right hand. |

| | | | | |

|1 |Position, Direction and Movement | | | |

| |Use everyday language to describe | | | |

| |position, direction and movement. | | | |

| |Talk about things that turn. | | | |

| |Make whole turns and half turns. | | | |

| | | | | |

|R |Position |Lack of understanding of |Where have you put the ……? |Model how to use appropriate vocabulary through V, A & |

| |Use everyday words to describe position |position vocabulary |Tell your partner where you have hidden the |K activities (see NNS Supplement of Examples p.27 for |

| | | |…….. . |vocabulary) e.g.:- |

| | | | | |

| | | | |Follow and give instructions in P.E., e.g. stand |

| | | | |behind/opposite your partner, run round the outside of |

| | | | |the hall, between the benches, as far away from anyone |

| | | | |else as you can. |

| | | | | |

| | | | |Use of the role-play corner to model and question the |

| | | | |position of objects. |

| | | | | |

| | | | |Read and discuss vocabulary, looking at illustrations |

| | | | |of big books – ‘Rosie’s Walk’, ‘Bear Hunt’, ‘Where’s |

| | | | |Spot?’ |

| | | | | |

| | | | |Play own game of ‘Bear Hunt’. Children describe to |

| | | | |each other where they have hidden the teddy, e.g. He’s |

| | | | |under the cupboard/behind the big book/ on top of the |

| | | | |desk in the drawer next to the pencils. |

Appendices

Appendix 1 – Decimal Place Value Chart

Appendix 2 – Fraction, decimal and percentage cards

Appendix 3 – Protractor OHT

Appendix 4 – A Monster Right Angle Measurer

How to find LCM of two numbers in Python? We will learn Method 1: A linear way to calculate LCM Method 2: Modified interval Linear way Method 3: Simple usage of HCF calculation to determine LCM Method 4: Repeated subtraction to calculate HCF and determine LCM Method 5: Recursive repeated subtraction to calculate HCF and determine LCM More items...