CountingEarly numberNumber sense

Place Value

Subject Knowledge Enhancement Session 1

Aims of the Session

• To enhance own subject knowledge.

• To build understanding of mathematics in the National

Curriculum and it’s development throughout the primary

years.

• To enhance subject knowledge of the pedagogical

approaches to teaching mathematics following the teaching

for mastery model.

• To develop an understanding of what a child needs to do to

demonstrate mastery of a mathematical concept.

Pre-course Reflection

The Aims of The National Curriculum

The National Curriculum (2014)

for mathematics aims to ensure all pupils:

• become fluent in the fundamentals of mathematics, including

through varied and frequent practice with increasingly complex

problems over time, so that pupils develop conceptual

understanding and are able to recall and apply their knowledge

rapidly and accurately to problems

• reason mathematically by following a line of enquiry, conjecturing

relationships and generalisations, and developing an argument,

justification or proof using mathematical language

• can solve problems by applying their mathematics to a variety of

routine and non-routine problems with increasing sophistication,

including breaking down problems into a series of simpler steps

and persevering in seeking solutions.

Mathematical Proficiency (NCETM)

Mathematical proficiency requires a focus on core

knowledge and procedural fluency so that pupils can

carry out mathematical procedures flexibly, accurately,

consistently, efficiently, and appropriately. Procedures

and understanding are developed in tandem.

Fluency, reasoning or problem solving?

Sort the activities

• Deep and sustainable learning• The ability to build on something that has already been learnt.• The ability to reason about a concept and make connections.• To have conceptual and procedural fluency.

What is mastery?

How is depth achieved in Maths?

• Longer time on maths topics• Intelligent practice (variation with small steps)• Detail in exploring the concept- all aspects exposed

and linked (coherence)• Questioning and activities develop reasoning and

make connections (mathematical thinking)

EARLY MATHEMATICAL IDEAS

Conservation of Number•Conservation is Piaget’s name for the understanding that certain basic characteristics of an object, such as its weight and volume, remain constant even when its appearance it perceptually transformed.

Put the counters in a

box

Establishing meaning for number names and symbols to 10

NAME

Words eg. five

hear, say, read, write

MAKE

Collections

represent

and model

RECORD

Symbols eg. 5

recognise, say, write

Real world

Structured

Cardinal

Ordinal

Nominal

•manyness

•‘the sixness of

6’

Cardinal Ordinal

before/after

1 more/1

less

4th person

in line

Nominal

a label for

identification

and not for

quantitative

reasoning

“eight” 8

Linking number names, numerals and value

Jessica F. Shumway - Number Sense Routines

• SUBITIZING• MAGNITUDE – knowing which set has more• COUNTING – using number labels• 1:1 CORRESPONDANCE • CARDINALITY - when counting, the last number gives the

quantity• HIERARCHICAL INCLUSION – smaller numbers are part of bigger

numbers; 1 more/less• PART/WHOLE RELATIONSHIPS• COMPENSATION – thinking relationally 5+1=6 so 4+2=6• UNITIZING – 1 unit can have a value of more than 1; 20= 2tens; 2

groups of 10 or 1 group of 20 instead of 20 ones

PART/WHOLE RELATIONSHIPS

Using relational thinking: what does c need to be to make this statement true?

•12 + 9 = 10 + 8 + c

12 and 9 is 21 and 10 and 8 is 18, so you have to put 3 more with the 18 to get 21. So c is 3.

I saw that 12 is 2 more than 10 and 9 is 1 more than 8. So you have to

add 3.

COMPENSATION thinking

relationally

1 to many correspondence

UNITIZING

SUBITIZING

Subitising•Conceptual subitising is similar to the ability of combining small sets of numbers. Patterns are integral to this ability in order to ‘see’ numbers in sets, e.g. the patterns on a die, dominoes and fingers, where an awareness to construct number sets and combinations of those sets can be taught.

Using colour

can be

helpful to

support

Sayers (2015)

Links with structured resources

•Tens frame (fives-wise)

Numicon (twos-wise)

Structured representations

Tens frame (twos-wise)

Adapted by Dot Lucas. 'Can do Maths

Subitising GamesChildren can be helped to improve their ability to subitise by being shown sets of counters for a short period and asked to say what they saw and how many they saw.

• Flash cards

• Pelmanism games

• Matching dots to numerals games

Counting

Count all and combine

Count forwards in 1s

Count forwards in multiples

Count back in 1s

Count back in

multiples

Count all and take

away

Count all and ‘share’

equally COUNTING

Importance of counting

•- Oral counting is a child’s first experience •of number and mathematics

•- Making connections between saying the number names and counting objects is the first step towards children’s understanding of the number system

•- Counting is one tool for building up calculation strategies

•- We need to count backwards is as well as forwards.

5

Dot Lucas. 'Can do Maths'

ORAL COUNTING…stages in counting

•String level - a continuous sound string

•Unbreakable list level - separate words but the sequence can’t be broken and always starts from 1

•Breakable chain level - child learns to be able to start the count at any point which is essential if they are going to be able to count on

•Numerable chain level – sequence, count and cardinality are merged so, if you are counting from 3, then 3 is the first number, 4 is the second number …………

•Bi-directional chain - child can say the numbers in either direction and start at any point

Karen Fuson 1988 ‘Children’s Counting and Concepts of Number.

10Dot Lucas. 'Can do Maths

Counting Principles…

THE ‘HOW TO COUNT’ PRINCIPLES

• The 1-1 principle

• The stable order principle

• The cardinal principle

THE ‘WHAT TO COUNT’ PRINCIPLES

• The abstract principle

• The order-irrelevance principle

Gelman R and Gallistell CR. (1978) ‘The Child’s Understanding of Number’

9Dot Lucas. 'Can do Maths

Which is the largest number and the smallest number?

2 24 915

Dot Lucas. 'Can do Maths

Number tracks and Number lines

Dot Lucas. 'Can do Maths

Counting in steps; UnitisingCounting in 2’sCounting in 10’s

Counting in

tens and ones

Dot Lucas. 'Can do Maths

Moving on with number lines

Dot Lucas. 'Can do Maths

Cbeebies Number Blocks

The NCETM materials use each episode as a

launch pad. They are designed to assist

Early Years (and also Year 1) practitioners to

confidently move on from an episode, helping

children to bring the numbers and ideas to

life in the world around them.

The materials are designed to be used in

conjunction with the Numberblocks episodes.

They highlight and develop the key

mathematical ideas that are embedded in the

programmes.

Alphabetland

The new number names are: A, B, C, D,..

You must not translate these number names into banned number names one, two, three,

Count with me…

Can you count from L to T?

Can you count back from G?

Can you count back from P?

Can you count in Bs?

How children learn mathematics

• Bruner – children need to experience a mix of three different modes of

learning: Enactive, Iconic and Symbolic.

real objects

pictures

3 + 2 = 5 symbols

In the context of Mastery – this approach is often referred to as CPA

Mathematics Learning

Dienes – children learn mathematics by means

of direct interaction with their environment –

variability principles

A mathematical concept can be

thought of as a network of

connections between symbols,

language, concrete experiences

and picturesDerek Haylock and Anne Cockburn 2008

The Connections Model

Place Value

What is place value?

• Additive - amount or quantity value

• Positional – column place value

• Base 10– the exchange principle

• Multiplicative

•Relative size and position of numbers – associated with the positional and additive aspect

Place Value

• Quantity Value: 23 = 20 + 3

• Column value: 23 is 2 tens and 3 ones

• Putting Place Value in its place, Thompson, I (2003), ATM.

0 30

2.25 22.5

Decimal number line ITP

RELATIVE SIZE AND POSITION

http://www.taw.org.uk/lic/itp/dec_num_line.html

What are the key difficulty points in place value?

• Confusion ty/teens numbers

• Saying numbers, writing and ordering

• Exchanged figures

• Zeroes

• Appropriate strategies for calculation

• Language of comparison

• Value of digits in large / very small numbers

• The concept of 10.

• Numerals are arbitrary symbols

• Confusion between teens and ‘ty’ numbers;

• Too abstract too soon

• Importance of concrete resources, structured resources, language and symbols – making connections

• Constructivism…children building meaning

Why is place value so tricky?

Precise language

• Number and digit

• The number is thirty five

• The tens digit: 3 tens

• The ones digit: 5 ones

Unitising• Place value is based on unitising: treating a group

of things as one ‘unit’.

• In mathematics, units can be any size, for example units of 1, 2, 5 and 10 are used in money.

• In place value units of 1, 10 and 100 are used.

Unitising• When do we count in units of 1, 2, 5 and 10?

10

Unitising

Use a range of resources to make tricky

teens and tens numbers

13

Multi Representation 12/20/21 13/30/31 15/50/51

What about 0 as a place holder?

What about 4 digit

numbers?

Counting and Place Value: Multi-representations

Building up tens

• From ones to groups of 10

•32 and 23

• What is the same? What is different?

-ty numbers

Partitioning

•Why is partitioning important?

•At what stages do children need to partition?

15

25

10

Partitioning

100s 10s 1s

124

1 2 4

Partitioning numbers in different ways

Partitioning numbers in different ways

What does this look like with Dienes?

Important Conceptual understanding

13

33

23

Partitioning

9 6- 2 9

18

Partitioning in different ways

• Adds flexibility to calculation

• Finding totals in many different ways

How many ways can you partition this number?

Generate 4 different names for this numberFour hundred

and twenty

nine

Three

hundred and

twelve tens

and nine

ones.

Four hundred

and ten and

nineteen

ones.

Two hundred,

twenty two

tens and nine

ones.

Which manipulatives do you use for place value?

10

Ordering and Comparing

• Define what zero is at your table

• An empty set• Zero is the only integer (whole number) that is neither positive nor

negative.• A place holder in our number system

The role of zero as a place holder

• 302

• Three hundreds

• Zero tens

• Two ones

320

Three hundreds

Two tens

Zero ones

Tenths – one or more parts out of ten in the whole

1

0.3

http://www.google.co.uk/url?sa=i&rct=j&q=&esrc=s&source=images&cd=&cad=rja&uact=8&ved=0ahUKEwiorI7Z_sTTAhWjLsAKHRfQBjMQjRwIBw&url=http://geckomath.truman.edu/lessons/All_3-2-7/Sub_Man_3-2-7/Manual_3-2-7_instructional_and_evaluation_tips.html&psig=AFQjCNGb0xZUtZbdmeMbe5c3k7cXE1qs3w&ust=1493394616119668http://www.google.co.uk/url?sa=i&rct=j&q=&esrc=s&source=images&cd=&cad=rja&uact=8&ved=0ahUKEwiorI7Z_sTTAhWjLsAKHRfQBjMQjRwIBw&url=http://geckomath.truman.edu/lessons/All_3-2-7/Sub_Man_3-2-7/Manual_3-2-7_instructional_and_evaluation_tips.html&psig=AFQjCNGb0xZUtZbdmeMbe5c3k7cXE1qs3w&ust=1493394616119668

Decimal notation with Dienes

The structural ideas for place value still apply• Positional – column value• Base 10 – the exchange principle• Multiplicative• Additive - Amount or quantity value

The ten for one

principle extends

indefinitely in both

directions of our

number system.

1.38

Working with decimals

• 1.34

1 0.1 0.010.10.1 0.01 0.01 0.01

Working with decimals

This represents one

whole and three

tenths

Or thirteen tenths

Misconceptions in reading decimals

•3.25

• What does the 5 represent? 5 hundredths

• How do we say the number?

• Three point twenty five

• Three point two five

Reasoning• Do, then explain

• 5035 5530 5053 5350 5503

• If you wrote these numbers in order starting with the largest, which number would be third? Explain how you ordered the numbers.

Comparison of Numbers

Comparison of Numbers

Comparison of Numbers

Carefully structured questions

•Write a series of questions that would use the < > or = signs to compare numbers, that also includes the use of zero as a place holder.

•Design the questions in sequence to address key difficulty points in small steps.

Rounding

• Key difficulty point: Knowing which degree of accuracy to round to

• Key difficulty point: rounding the same number to different degrees of accuracy e.g. to nearest 10, 100 and 1000

Rounding – multiples of 10

•The previous and next multiple of 10

Multiple of 10 Next multiple of 10

20 23 30

45

72

103

Rounding – boundaries

23 3020

45 5040

3.2 4.03.0

Tick which number it is closest to and explain why.

Rounding – boundaries

1000 2000

A B

A 20001000 B 30002000

Rounding – nearest 1000

2256

2256 30002000 3790 40003000

3790

Rounding – nearest 1000

•Here is the current and next multiple of 1000. Which is it closer to? Why?

7261 80007000

Use of number lines to support

• Rounding 2843

2810 2820 2830 2840 2850 2860

2400 2500 2600 2700 2800 2900

1000 2000 3000 4000

Degrees of Accuracy

• Round decimals with 2 decimal places to

• the nearest whole number and to one

• decimal place.

• Round 3.81 to the nearest whole

number and to 1 decimal place

• Round 0.45 to the nearest whole

number and to 1 decimal place.

Multiply/Divide by 10 and 100

• What representations can we use to support this key difficulty point?

• What are the main misconceptions around multiplying and dividing?

Multiplying and Dividing by 10 and 100

Thousands Hundreds Tens Ones

3 7

37 x 10 =

0

Multiplying and Dividing by 10 and 100

Thousands Hundreds Tens Ones Tenth Hundredth

3 7

Reflection

1. What 1 thing will you share with colleagues back at school?

2. What 1 thing did you learn or were reminded about from today’s session?

Gap Task:

• Check that you have all the manipulatives you need in class or readily available.

• Bring along evidence from an example of an addition or subtraction lesson that went well. Can be the child’s book, photos, planning etc.

• Next time: Addition and Subtraction