Transcript Moving Students on

Moving Students On
AC to EA/EA to AA
The Journey to Part-Whole
Dianne Ogle
13 July 2011
Overview of today
• What are the key pieces of knowledge and
strategy for our Cause for Concern Children?
• Develop our conceptual understanding of key
knowledge and strategy
• Develop an understanding of ways to help
children who are not achieving at expected
level
Part Whole thinking – the prize!
• Involves splitting numbers into parts
(partitioning) in order to solve problems more
easily.
• Depends on knowing how the parts make up a
whole number
– Think about 10
– What about 18
– What about 72
• What key knowledge do children need?
Why Part-Whole?
• At counting stages, the size of numbers is
severely restricted, and there is normally only
one way to solve problems.
• Counting represents a relatively low level of
thinking.
• Part-whole thinking opens up the world of
large numbers and multiple strategies.
Early Additive:
• Students at this stage have begun to recognise that numbers
can be split into parts and recombined in different ways. This
is called part-whole thinking.
Strategies used at this stage are most often based on a group
of ten or use a known fact, such as a double. For example:
38 + 7 as (38 + 2) + 5
24 – 9 as (24 - 10) + 1
7 + 8 as (7 + 7) + 1
• Students working at this stage will be solving number
problems in each of the three operational domains. How do EA
children solve multiplication and division problems?
Proportions and Ratios problems?
Advanced Additive
• Students at this stage are familiar with a range of part-whole
strategies and are learning to choose appropriately between
these. They have well developed strategies for solving
addition and subtraction problems, for example:
367 + 260 as (300 + 200) + (60 + 60) + 7
135 – 68 as 137 – 70
703 – 597 as 597 + ? = 703
• They also apply additive strategies to problems involving
multiplication, division, proportions and ratios. For example:
6 x 3 = (5 x 3) + 3 = 15 + 3 = 18
One quarter of 28 as 14 + 14 = 28 so 14 is one half, 7 + 7 =
14 so 7 is one quarter
Advanced Multiplicative
• Students at this stage are able to choose appropriately from a
range of part-whole strategies to solve problems with whole
numbers. They are learning to apply these strategies to the
addition of decimals, related fractions and integers. For
example, 5.5 + 6.8 = 5.5 + 7 – 0.2 = 12.3.
• Students are learning to manipulate factors mentally to solve
multiplication and division strategies. For example, instead of
partitioning 5 x 12 additively as (5 x 10) and (5 x 2) students
use strategies such as doubling and halving, renaming 5 x 12
as 10 x 6.
Profile of the child who is not moving…
• Think about the child/children you teach who
are not moving.
• Discuss with a partner what you notice and
observe when that child is engaged in
mathematics.
• What are some common reasons for lack of
progress?
What knowledge to children need?
• Counting sequence and how to read and write
numbers
• Place Value
• Basic Facts
Big ideas
• Numbers are related to each other through a
variety of number relationships - more than,
less than, composed of
• “Really big” numbers possess the same placevalue structure as smaller numbers. Best
understood in terms of real- world contexts
• Whole numbers can be described by different
characteristics, even and odd, prime and
composite, square, understanding
characteristics increases flexibility when
working with numbers
Counting Principles
Gelman and Gallistel (1978) argue there are five
basic counting principles:
• One-to-one correspondence – each item is labeled
with one number name
• Stable order – ordinality – objects to be counted are
ordered in the same sequence
• Cardinality – the last number name tells you how
many
• Abstraction – objects of any kind can be counted
• Order irrelevance – objects can be counted in any order
provided that ordinality and one-to-one adhered to
Counting is a multifaceted skill – needs to be given time
and attention!
Early Number relationships
• Spatial relationships: children can learn to
recognise sets of objects in patterned
arrangements and tell how many without
counting.
One and two more, one and two less
• The two more than and two less than relationships
involve more than just the ability to count on two or
count back two. Children should know that 7, for
example is 1 more than 6 and also 2 less than 9.
Anchors or “benchmarks” of 5 and 10
• An understanding of ten is vital in our numeration
system and because two fives make up 10, it is very
useful to develop relationships for the numbers 1 to 10
to the important anchors of 5 and 10
Part – Part – Whole Relationships
• To conceptualize a number as being made up
of two or more parts is the most important
relationship that can be developed about
numbers.
Key Mathematical Ideas
Developing Meanings for the operations
• Addition and subtraction are related. Addition names
the whole in terms of the parts, subtraction names a
missing part
• Multiplication is related to addition
• Multiplication involves counting groups of like size and
determining how many there are in all. Multiplicative
thinking
• Multiplication and Division are related. Division names a
missing factor in terms of the known factor and the
product.
• Models can be used to solve contextual problems for all
operations, regardless of the size of the numbers. They
can be used to give meaning to number sentences.
Van de Walle & Louvin
Teaching Student Centred Mathematics,
Strategies
Maraea has $37. She spends $9. How much money does she
have left now?
Caleb has saved $165. He banks another $23 dollars. How much
money does he have saved now?
Dianne has $72. She spends $28 on a pair of shoes. How much
money does she have now?
Jody has scored 284 goals in netball this season. She gets
another 67. How many goals has she scored altogether?
Anaru has 312 tropical goldfish in his aquarium. He sells 198 of
them to the pet shop. How many tropical fish does he have
now?
The electrician has 5.33 metres of cable. He uses 2.9 metres on
a job. How much cable is left?
Solve these problems independently. When you have your
answers talk about how you solved them, are there some key
strategies, can you give them a name?
Strategies
Maraea has $37. She spends $9. How much money does she
have left now?
Caleb has saved $165. He banks another $23 dollars. How much
money does he have saved now?
Dianne has $72. She spends $28 on a pair of shoes. How much
money does she have now?
Jody has scored 284 goals in netball this season. She gets
another 67. How many goals has she scored altogether?
Anaru has 312 tropical goldfish in his aquarium. He sells 198 of
them to the pet shop. How many tropical fish does he have
now?
The electrician has 5.33 metres of cable. He uses 2.9 metres on
a job. How much cable is left?
How could we use materials to demonstrate strategies used, help
build conceptual understanding.
Reading • Read the article by Young-Loveridge and Mills
• Key points
• Next steps for you
Book Five/Planners
• Key ideas in Book Five
• Use of diagnostic questions
• Required knowledge
• Problem progression
Core ideas of Place Value
• Zero as a place holder
Canon of Place Value – Ten for one, one for ten.
• Children must understand that as a result of an addition or
multiplication the numeral in any column in a place-value
table exceeds nine, ten of these must be exchanged for a one
that is ten times more.
Unique Symbols
• The numerals 0 to 9 are unique symbols that are used to
represent numbers. They have been adopted universally
around the world. 130 should be said one, three, zero not
one, three, (letter) o.
Irregularity of English language number words
• The ‘teen’ and ‘ty’ code can be extremely confusing for
children. It is hard for children to hear the difference between
the number words when they are said aloud. Seeing the
written number word provides the visual cue that ‘teen’ is one
ten and “ty” is lots of ten. Therefore sixteen is six ones and
one ten while sixty is six lots of ten. “lots of” is multiplicative,
sixty = 6 x 10
Bundling
Bundles of ten board, Ice block sticks, Dice
Rubber band
Roll the dice - put the number of ice block sticks in ones column
- in tens frame pattern.
Roll again add ice block sticks - what happens when we get to
ten? Bundle the 10 put into tens column - Part whole
thinking
Record the story
Introduce to group
Play in pairs - first to 100.
Number words
seventeen one ten + seven ones
17
Fourteen
one ten + four ones
14
Fifty
five tens
50
Seventy
seven tens
70
Children need to have multiple opportunities to
work with teen/ty numbers to develop their
understanding.
Begin with materials.
Advanced Counting - Early Additive
Crucial number knowledge - understanding of
place value concepts - “teen, ty”
Need plenty of opportunities to bundle to ten
etc to develop understanding of canon rule
of place value.
Need to know basic facts to 10 - all facts that
make 10 and those below.
Using tens frames, fly flips, hands etc to
instantly recall facts to 10
Place Value EA - AA
Understanding of zero as a place holder
Knowing how to explain place value for
98 + 5 = what happens and why
Tell the story - using exchange.
1003 - 7 = where are the ones?
Knowing how many ones, tens, hundreds in a
number (not by using a rule but because 10
ones are exchanged for 1 ten, one hundred
can be exchanged for ten tens.
10 003 - 6
You have ten thousand and three dollars. You
owe your friend six dollars.
Do you have 6 dollars to give to your friend.
Imagine place value money. Write the story of
how you will get enough ones to be able to
give your friend what you owe.
Where do you start?
See, say, do - Peter Hughes
Say the numeral one
way, e.g. 13 is thirteen
Write the numeral
e.g. 13
Model the PV form of the
numeral e.g. 13
is 1 ten and 3 ones
Say the numeral in the
other way, e.g. 13 is ten
and three
Model the numeral
as ones
e.g. as 13 ones
Equipment for developing place
value concepts
Stage
Equipment
Stage 2- 4
Bundling sticks, beans and containers
Counters and plastic bags,
Slavonic Abacus
Stage 5
Place value money, place value blocks, arrow
cards, place value houses
Stage 6
Number Lines
Stage 7
Decimal Fraction Mats
Concrete
representation of
ones
Non
representational
Stage 4 – numbers from 10-99
• Materials:
– Sticks and pipe cleaners
– Beans and photo canisters
– Counters and plastic bags
– Unifix and wrappers
– Tens frames
– Place value play money
– Place-value blocks
– Open abacus
• Mental method for addition not expected – use materials.
Continually practising the ten for one swap.
• Using a mix of numbers and words is a powerful indicator as
to whether children are understanding place value.
Stage 5 – Part-whole
• Crucial that children are given opportunities to solve
problems where one number is a tidy number. Using
tens frames to solve 28 + 2 =
• Use place value money where swap is involved
• Mentally solve 4 + 46
• Move to problems where part whole thinking is required
and the numbers move through a decade.
• Move through teaching model
– tens frames, imaging, mentally solve
– Place value money, imaging, mentally solve
Stage 5 – Part-whole
Essential that children can engage in the internal talk of
place value. For example 56 + 78 would be:
• Six ones and eight ones equals fourteen ones
• Swap this for one ten and four ones
• One ten plus five tens plus seven tens makes thirteen
tens
• Swap this for one hundred and three tens
• The answer is one hundred and thirty-four
Stage 5 – Part-whole
• Essential that students can move quickly through each
representation of a number.
• The Slavonic Abacus becomes an essential piece of
equipment to help students understand place value.
• Ask the children to identify numbers on the abacus by
recognising the patterns in ten and ones using the
quinary (five) patterns.
• Imaging what a number will look like on the abacus.
• Materials
– Tens frames
– Place value play money
Stage 6
• Children must have automatic making and breaking of
numbers
• What distinguishes early part whole from advanced part
whole thinking is the number of mental steps needed.
• It is not the size of the numbers but the number of
mental steps needed.
– EA: 199 + 56
– AA:56 - 37, 567 + 78
• Being able to quickly know how many hundreds, tens,
ones there are in 4 digit numbers and beyond
• Materials – number lines (introduced here)
• Continue with place value money and blocks where
necessary
Stage 7
• Children must understand that 3 or more digit numbers
have multiple forms
• Peter works in a cake factory: packing ten cakes into a
packet. His job is to pack 265 cakes. How many packets
does he have? How many loose ones?
• He packs ten packets into a box for shipment. How
many boxes, packets and loose ones?
Difficulties at AA
• Understanding the nested place values in
4, 5, 6 digit numbers
• Recognising the canonical and non-canonical
forms.
– Fifty five thousand is non-canonical
– Sixteen hundred is non-canonical
Read, Say, Do
• Children need to be able to explain what
makes up a number
• Use a variety of Place value materials to
demonstrate how a number can be made up
• Place value model – Peter’s biscuit factory.
Fifty and some more
• Say a number between 50 and 100. Children
respond with “50 and ____.
• For 63, the response is “50 and 13”
• Use other numbers that end in fifty such as
350, 650 or 0.5
Write all the ways….
• How many ways can you make
36
• Show as many different ways as you can to
make 36 – use materials, words, word stories,
digits…
• After 1 minute you will pass your paper to the
next person.
Place Value Questions
• Diagnostic Interview – to find out what
misconceptions children have.
• Place Value Questions – use as diagnostic
questions to find out what children need
further help with
• Allow children to discuss their thinking and
explain how they know.
Basic Facts
The place value system is universally adopted because all
calculations can be performed by knowing correct
procedures and the basic number facts.
Knowing the addition facts from 1 + 1 to 9 + 9 will enable
addition and subtraction problems to be solved, includes
decimal fractions.
Knowing the multiplication facts from 2 x 2 to 9 x 9 will
Enable all multiplication and division problems to be
solved, including decimal fractions.
A lack of instant recall of basic facts, along with not
understanding place value are the two key reasons
children are not making progress in number.
Basic Facts
A lack of instant recall of basic facts, along with not
understanding place value are the two key reasons
children are not making progress in number.
• It is important that children are learning their basic
facts when they need to be using them.
• Addition and subtraction facts learned first
• Times tables follow, when children are using
multiplicative strategies.
Basic Facts
By stage five instant recall of basic addition facts is
Required. There is plenty of time to learn them. A
framework for learning basic facts:
Stage 2:
Addition and subtraction facts to five
Stage 3:
Essential to recall addition and subtraction facts to five
Optional – Addition and subtraction with sums up to ten,
doubles
Basic Facts
Essential for part-whole reasoning that comes in stage five
is the instant recall of basic addition and subtraction facts
with answers no more than ten.
Stage 4:
Addition and subtraction facts up to ten
Doubles
Optional:
– Addition and subtraction facts from 1 + 1 to 9 + 9
- Derive and learn the two times tables from doubles.
Basic Facts – Stage 5
Essential for advanced additive thinking in stage six is the
instant recall of all addition and related subtraction facts
1 + 1 to 9 + 9
Recall of multiplication facts can begin with a focus on the
commutative principle for multiplication
Stage 5: Essential
– Addition and subtraction facts from 1 + 1 to 9 + 9
- Derive and learn the two times tables from doubles.
- Derive and learn the three times tables from 3 x 3 to
3 x 9 using repeated addition and the reverse facts.
Optional:
- Four and Five times table
Basic Facts – Stage 6
Instant recall of times tables with 100% reliability is
needed for stage 7 so regular teaching and practising of
tables must occur at this level.
Failure to know times tables is a major obstacle in children
ever becoming multiplicative in their thinking.
Recall of multiplication facts can begin with a focus on the
commutative principle for multiplication
Stage 6: Essential- Derive and learn, connect to division
4 times table from 4 x 4 to 5 x 9
5 times table from 5 x 5 to 5 x 9
6 times table from 6 x 6 to 6 x 9
7 times table from 7 x 7 to 7 x 9
8 times table from 8 x 8 to 8 x 9
Derive and learn 9 x 9, connect to 81 ÷ 9
Use the 0 and 1 principles
Basic Facts
Learning of times tables
• 0 times or times 0
– A principle not a table
• 1 times or times 1
– A principle not a table
• 10 times or times 10
– An english language issue, not a table
Basic Facts – only 36 facts to learn
x
2
3
4
5
6
7
8
9
2
4
6
8
10
12
14
16
18
9
12
15
18
21
24
27
16
20
24
28
25
36
25
30
35
40
45
36
42
48
54
49
56
63
64
72
3
4
5
6
7
8
9
81
Basic Facts – from understanding to rote
Van de Walle
Mastery of the basic facts is a developmental
process, students move through stages,
starting with counting, then to more efficient
reasoning strategies, and eventually to quick
recall. Instruction must help students move
through these phases, without rushing them
to memorisation.
Page 167 , 2010
Approaches to fact mastery
• Explicit strategy instruction – designed to
support student thinking – show students
possibilities and let them choose strategies
that help them get the soltion without
counting
• Guided invention – using strategies children
have, guiding them to the efficient ones.
Teacher’s job is to design tasks and problems
that will promote the invention of efficient
strategies
What not to do
• Don’t use lengthy timed tests
• Don’t use public comparison of mastery
• Don’t proceed through facts in order – (knock
out the ones you know)
• Don’t move to memorization too soon
• Don’t use facts as a barrier to good
mathematics – mathematics is about
reasoning, give children real mathematical
experiences.
Missing Number worksheet
• Begin with circles and ask
children what they notice
about the numbers
• Teach the children the circle
always has the answer
• Fill in sheet with two numbers
children have to find missing
number
10
4
6
6
6
10
2
Triplets – Family of Facts
• Introduce triplets
10
• 10 , 6, 4
6
• Make chains of
number triplets
4
Tens Frames
• Hold up a tens frame and have the children
say the 10 facts that go with the card.
• Children need to be able to say the four
connected facts that go with each tens frame
Seven and three makes ten
Three and seven makes ten
Ten takeaway seven is three
Ten takeaway three is seven
As children tell story
important they see written
forms – words and symbols
Add to ten
Two players
Deal all cards out between two players.
Take turns to turn over one card - state what else
makes 10.
Also play by taking number off ten.
Modify for younger students – make five (remove
some cards, use five frames/tens frams
Working backwards - subtraction is harder.
Children need lots of practise with subtraction
1,2,3 Fists - Paper, Scissors, Rock
Two players
Play as for Paper, Scissors, Rock
One or two hands
Count 1,2,3, put down some fingers add/multiply together
Make Ten,
Two players
Deal out ten cards in a row.
First player looks across the row for combinations
that make ten.
Aim is to collect as many cards as possible, so
combinations that require more cards are best.
Continue playing until all the cards are used or
until there are no more combinations that add to
ten.
Winner has the most cards.
Make Ten again,
Two players
Deal all cards out in 3x3 grid
Take turns to make 10 Continue playing until all the cards are used or
until there are no more combinations that add to
ten.
Winner has the most cards.
Salute
• You need three players
• A pack of playing cards (take out 10s and colour cards
• Two players collect one card each. Without looking at the card
they put it on their forehead.
• The third player calls out the sum of the two cards
• The two players then call out what card they hold on their
forehead by looking at the other player’s cards.
• The player who calls out first wins those cards.
• Continue playing until all the cards are used.
Variations
• 10 more or ten less/ one more or one less
• Multiply
• Doubles
Speed (War)
Two players
Deal all cards out between two players.
Place one card in middle. - e.g. 2 (add this
number to card that is turned over)
Take turns to turn over one card - both players
call out answer. First to call wins both cards.
If a tie, turn over another card. Highest card gets
to keep all three cards.
Also for multiplication
Grab Five
Grab five sticks
Put them in order from smallest to biggest.
Winner is the first one to grab the object from the
centre of the table. Must have sticks in the right
order.
Can be made to fit children from Year 1 - 8
Circle a Fact
• Place a set of A4 numeral cards zero to nine in a circle on the
floor.
• Children form a circle around cards or make two teams either
side of the circle.
• Two people walk around the outside of the circle, on stop
place their toe on a card.
• Winner is the person who calls out answer first. They can
–
–
–
–
–
Add the two numbers together
Double the numbers
Add 10, double plus or minus one or two
Multiply the numbers
Find the difference of the two numbers
Connecting oral to written
• Important to practise repeatedly
• Oral connection to basic facts is important for
the brain
• Ensure all incorrect facts are identified and
corrected at time error is made. Children must
not be practising errors.
Accelerating Learning in Mathematics
• Brochures produced as part of accelerating
learning interventions.
• Choose one at the level you are working at.
• What are key points
• How can this help in your class?
Knowledge warm up
• Every day include in warm up – counting,
reading/writing of numbers, basic facts
• Use peer tutors, buddies, critical friend
• Student Profiles, basic fact records
Peer Tutoring
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Examples from Heaton Normal
Basic Facts/Place Value from Stage 4 – 8
Describes what child needs to be learning
Provideds games and activities
A structure needs to be put in place for regular
timing
• Having materials for buddies available.
• Peer Tutor needs to be trained in being an
effective buddy
– Asking “how do you know”
– Encouraging – not telling
Reflection
• What have you learnt today?
• How will your new learning impact on your
classroom teaching next term?
• What do you need to do next?
• How can I help?