2. Aims of the session
• To give an update from the Better Mathematics
Conference 2015
• To look at addition calculation methods & decide
on a progression of stages across the school
3. Achievement
o Although attainment is generally rising pupils are not made to think hard
enough for themselves. Pupils of all ages do too little problem solving &
application of Mathematics.
o The % of pupils meeting expected standards falls at successive key stages.
This is often due to a focus on meeting thresholds rather than securing
essential foundations for the next stage.
o FSM pupils do worse than their peers at all key stages.
o Low attainers are not helped soon enough to catch up, particularly in the
EYFS & KS1.
o High attainers not challenged enough from EYFS onwards.
4. Made to Measure Report
The report draws attention to serious inequalities in
pupils’ experiences and achievements. It includes
examples of best practice that help avoid or
overcome the inequalities and weaker practice that
exacerbates them.
5. Teaching Findings
• The best teaching develops conceptual
understanding alongside pupils’ fluent recall of
knowledge & confidence in problem solving &
mathematical reasoning
• In highly effective practice, teachers get ‘inside
pupils’ heads’. They find out how pupils think by
observing them closely, listening carefully to what
they say, & asking questions to probe & extend their
understanding, then adapt teaching accordingly.
• Too much teaching concentrates on the acquisition
of disparate skills that enable pupils to pass tests &
exams but do not equip them for the next stage of
education, work & life.
6. Aims of the National
Curriculum 2014
• Become fluent in the fundamentals of mathematics,
so that pupils develop conceptual understanding &
the ability to recall & apply knowledge rapidly &
accurately
• Reason mathematically
• Solve problems
7. National Teaching Key
Concerns
• Conceptual understanding & problem solving are
underemphasised
o Too often teaching approaches focus on how, without understanding
why, so that pupils have insecure foundations on which to build future
learning.
o Many pupils spend too long working on straightforward questions
• Wide in school variation in teaching quality.
• Circulating to check & probe each pupil’s
understanding throughout the lesson & adapting
teaching accordingly are not strong enough.
8. What does ‘Outstanding’
look like?
• Problem solving & Mathematical reasoning are
embedded into all parts of the Maths curriculum & not
viewed as a separate entity.
• Children are discussing Maths & methods & making
connections for themselves.
• Everyone is teaching for understanding.
• Practical resources, visual images & ICT foster pupils’
deeper understanding. All children are encouraged to
use practical equipment, there is evidence this has real
benefits in terms of developing mental imagery.
• Teachers work & plan together to support consistency &
improvement.
• There is timely intervention which overcomes gaps &
builds a firm foundation for future learning.
9. Recommendations for
Primary Schools
• Improve pupils’ progress from the Early Years
Foundation Stage through to Year 2 to increase
attainment of the most able.
• Act early to secure the essential skills & knowledge
of the least able.
10. Questions to consider
• How can we develop consistency in our teaching in
terms of subject knowledge and language
choices?
• How can we ensure our calculation policy reflects
the changes in the New Curriculum and the needs
of the children in our school?
• How can our policy build effectively on prior
learning?
• How can we improve the accuracy of children’s
addition work across the school?
• How can we support the children to effectively
record their mathematical thinking?
12. The concept of addition
• The combining of 2 or more groups to give a total or
sum
• It is the increasing of an amount.
13. Key principles of addition
• It is the inverse of subtraction
• It is commutative i.e. 5 + 3 = 3 + 5
• It is associative i.e. 5 + 3 + 7 = 5 + (3 + 7)
14. Addition Structures
Aggregation Augmentation
• The key language to
be developed in the
aggregation structure
of addition includes:
o how many altogether?
o How much altogether?
o The total.
• The key language to
be developed in the
augmentation structure
of addition includes:
o start at and count on,
o increase by,
o go up by.
15. Addition Structures
• Children must experience the two addition
structures in a range of relevant contexts, including
money (shopping, bills, wages and salaries) and
various aspects of measurement.
• Then they also have to recognise addition in
situations in the contexts of measurements, such as
length and distance, mass, capacity and liquid
volume, and time. For example,
o Can you calculate the distance for this journey? If I have already travelled
63 miles and then do a further 45 miles?
o Can you find the total time for the journey? If the first stage has taken me
85 minutes and the second stage takes 65 minutes?
16. • Our next step is to think about the policy from the
child’s perspective:
o How would the children show you their learning at each stage?
o What would you see in their books on paper or in photos?
o How do we ensure the progression from practical to written?
17. How do we teach
addition now?
• With your class, what methods do you use?
• What practical equipment do you use?
• What written methods do you use?
18. Stage 1- Combining 2 or
more amounts
• Counting all method
Children begin to develop their
ability to add by using practical
equipment to count out the
correct amount for each number
in the calculation and then
combine them to find the total.
For example, when calculating 4 +
2, they are encouraged to count
out four counters and count out
two counters.
19. Counting all
• To find how many altogether, touch and drag them
into a line one at a time whilst counting.
1 2 3 4 5
6
20. Counting all
Children should be taught that addition is the combining of two or
more amounts. They will begin by counting all of the items in the
groups, then move on to counting on from the largest amount.
They can begin to record addition number sentences such as
2 + 4 = 6 and 8 = 3 + 5 and 3 + 2 + 4 = 9
22. Stage 2- Using Number
Tracks & Base 10
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
11 + 5 = 16
Model of Base 10 equipment
23. KS1 Addition Games
• https://www.tes.co.uk/teaching-resource/teachers-
tv-primary-maths--calculation-6038949
24. What about number lines?
• https://www.ncetm.org.uk/self-
evaluation/browse/topic/883
25. Stage 3
34 + 23 = 57 34 + 23 = ?
The units/ones are
added first 4 + 3 = 7
The tens are added
next
30 + 20 = 50
Both answers are put
together 50 + 7 = 57
26. Stage 4
28 + 36 = ?
The units/ones are added
first
8 + 6 = for 1 ten.
A ring is put around the
units/ones not exchanged –
this is the units part of the
answer. The tens are then
added, including the
exchanged ten, to complete
the sum.
27. Stage 5
TU HTU
67 267
+ 24 + 85
1 1 (7 + 4) 12 (7 + 5)
80 (60 + 20) 140(60+80)
91 200
352
The Base 10 equipment should be used
alongside to model the transition to the
vertical method but this should not be
recorded by the children
29. Stage 6
The example top left would be ‘said’ as
follows:
5 + 8 = 13, put 3 down and carry the 10
20 + 40 + 10 that was carried over = 70
(7 written in the tens column)
600 + 0 = 600 (6 written in the hundreds
column)
30. • Children should extend the carrying method to numbers with
at least four digits.
587 3587
+ 475 + 675
1062 4262
1 1 1 1 1
3121 3.20
+ 37 +2.88
+ 148 6.08
3306 1
1 1
31. Important Points
• Children should not be made to go onto the next
stage if:
1) they are not ready.
2) they are not confident.
• Children should be encouraged to consider if a
mental calculation would be appropriate before
using written methods.
32. Our next steps
• Look at the calculation policy for the other
operations, ensure there is a clear progression from
practical to written methods.
• Use the policy in a fluid way, look at where the child
is and what their next steps are.
33. Homework
Look at the results of the Times Tables Club in your
class. Can you track progress & do some analysis
about the overall effectiveness/ impact this is having
on learning and progress?
Please bring this to the next Maths
inset on 16th June.
Hinweis der Redaktion
Joint Better Maths Conference 13 March led by Chief Inspector for Schools- really useful
Remember these are national priorities- gives us a good idea of Ofsted’s thinking & the way forward in terms of Maths teaching & learning
Problem solving must be a key feature at all Key stages & for all ages
? Is when do the gaps open & why
New ELG in EYFS do include problem solving & it’s a very important part of the new NC
High attainers- there needs to be obvious depth & complexity provided to meet needs of HA? G&T chn
Made to Measure Report published 2012- findings of inspections between 2008 and 2011
Again we are looking at National Concerns- not us as a school (esp since Outstanding 2008 & 2013 but to maintain these we need to continue to be progressive & to keep reminding ourselves & to keep addressing these national concerns/priorities)
This theme of checking for misconceptions & addressing them rapidly is key
Focus needs to be on why we do Maths, & why it works & how it builds on what we already know
These 3 aims, are consistent with Ofsted’s findings on effective teaching & learning.
Chn need to be much more able to understand structure & relationships between numbers, operations & Mathematics as a whole.
Too much repetition
Definitely not something which we see as a concern or problem here but we do need to be aware of this & ensure there is consistency of approach etc (reason we do Insets like this & are looking to agree on new calculation methods etc too) There is most likely in all schools a variation in subject knowledge or expertise & pedagogic skills- staff need support and training to build up expertise this alongside their levels of confidence too.
CIRCULATING- really emphasised throughout the day- this idea of teacher not just being with 1 focus gp but awareness of all learners & their progress throughout each lesson- might challenge some of our ways of teaching
Intervention needs to be sharp & quick- misconceptions etc identified during lessons CIRCULATING & addressed ASAP or before next lesson etc.
Again from ‘Made to Measure’ Report
This is especially relevant in the Mastery curriculum
As school we have decided as a short term priority ‘to raise attainment by the end of Reception to ensure all pupils are well prepared for KS1 curriculum’- I’m going to be working alongside Debbie & we’ll hoperfully be using Tas to lead some interventions & extra sessions with Reception chn next half term
Early & timely intervention
Our next step is to think about the policy from the child’s perspective:
How would the children show you their learning at each stage?
What would you see in their books on paper or in photos?
How do we ensure the progression from practical to written?
What do you think? Any ideas? Key vocabulary/ words etc?
Make explicit to children the principle of the
commutative law of addition. Show them how to
use it in addition calculations, particularly by
starting with the bigger number when counting
on. Explain that subtraction does not have this
property.
•Ensure that children understand the = sign means is the same as, not makes, and that children see calculations where the equals sign is in a different position, e.g. 3 + 2 = 5 and 5 = 3 + 2.
•Children should be encouraged to approximate before calculating and check whether their answer is reasonable.
Aggregation basically just means ‘a collection of things’
Augmentation-is word given to define the action or process of making or becoming greater in size or amount
Look over your year group POS for addition & consider how you do this now
By touch counting and dragging in this way, it allows children to keep track of what they have already counted to ensure they don’t count the same item twice.
Children are encouraged to develop a mental image of the size of numbers. They learn to think about addition as combining amounts in practical, real life situations.
To support children in moving from a counting all strategy to one involving counting on, children should still have two groups of objects but one should be covered so that it cannot be counted. For example, when calculating 4 + 2, count out the two groups of counters as before.
then cover up the larger group with a cloth.
For most children, it is beneficial to place the digit card on top of the cloth to remind the children of the number of counters underneath. They can then start their count at 4, and touch count 5 and 6 in the same way as before, rather than having to count all of the counters separately as before.
Children will initially use practical equipment to combine groups of objects to find the total. They will move on to the use of number tracks and Base 10 equipment to support their developing understanding of addition. If possible, use two different colours of base 10 equipment so that the initial amounts can still be seen.
11 + 5 =
Its not recording but just thought it was useful to see egs of addition games using counting on method
Watch video from Teachers TV up to 2mins 28
Give out games handouts for Recep, Y1, Y2 & Y3
Children will continue to use the Base 10 equipment to support their calculations. They will record the calculations using their own drawings of the Base 10 equipment (as lines for the 10 rods and dots for the unit blocks)
When the units total more than 10, children should be encouraged to exchange 10 ones for 1 ten. This is the start of children understanding ‘carrying’ in vertical addition.
Children should be encouraged to manipulate the equipment as much as possible in order to understand the notion of ‘carrying’.
Children will build on their knowledge of using Base 10 equipment and continue to use this to support with the transition into a vertical method.
Children should add the least significant digits first as preparation for the compact method.
NB The text in red italics is modelled by the teacher but may not be written by pupil in their answer.
Children will be expected to use this method for adding numbers with more than 3 digits, numbers involving decimals and adding any number of amounts together.
Using similar methods, children will:
• add several numbers with different numbers of digits;
• begin to add two or more decimal fractions with up to three digits and the same number of decimal places;
• know that decimal points should line up under each other, particularly when adding or subtracting mixed amounts, e.g. 3.2 m + 280 cm.
Next stage is to extend this to any number of digits
