Upload
others
View
5
Download
0
Embed Size (px)
Citation preview
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