Transcript Improving Mental Mathematics in Schools

Improving Mental Mathematics
Tom Garner
Fairhaven Primary School
Aims of Session:
To familiarise ourselves with key mental calculation strategies that should be
taught throughout the school;
To develop an understanding of progression in mental calculation across the
school;
To learn how mental recall can be developed to facilitate mental calculation;
To identify opportunities for and the importance of developing visualisation;
To understand how to teach and not just test times tables.
Why is mental calculation so
important?
Activity 1:
Discuss on your tables and be prepared
to feedback.
What does working mentally in mathematics
mean?
What skills and attitudes do children need to
be able to work mentally?
What opportunities do children need in order
to develop these skills?
Mental calculation is all about patterns and
relationships between numbers. Children need to be
able to learn how to solve problems by recognising
which strategies and known facts to apply.
How and where does mental calculation
start?
FS  Yr1  Yr2  Yr3  Yr4  Yr5  Yr6
OFSTED –
Understanding the score
Too often pupils are expected to remember the methods,
rules and facts without grasping the underpinning concepts,
making connections with earlier learning and making sense
of mathematics so they can use it independently.
Progression
Activity 2 – part 1:
On your own have a go at writing down as
many different mental calculation methods
as you can.
Activity 2 – part 2:
Working with a partner have a go at filling in
the grid. Which of the mental calculation
methods you have thought of do you think go
where?
Progression in mental calculation
Year 1
Year 2
Year 3
Year 4
Year 5
Year 6
Progression in mental calculation
Curriculum 2014
Year 1
Number bonds 20. Use
for addition and
subtraction
Year 2
Recall addition and subtraction
facts to 20 fluently.
Times tables (2, 5 & 10).
Halving.
Doubling/halving – multiples of
5/10.
Doubling.
Counting in steps of 2, 3 & 5
(forwards and backwards).
Identify near doubles.
Counting in steps (1, 2,
5 & 10) forwards and
forwards backwards to
100.
Addition/subtraction
facts for numbers up to
20 – early partitioning.
Partitioning (tens and
units).
Partition into 5 and a bit
when adding 6, 7, 8 or
9.
Bridge through 10/20
when adding a single
digit number.
Count in steps of 10 forwards and
backwards from any given
number.
Counting through hundreds
numbers.
Partitioning 2-digit numbers (tens
and units).
Partition into 5 and a bit when
adding 6, 7, 8 or 9.
Extend addition and subtraction to
100.
Derive and use facts up to 100.
Bridge through a multiple of 10
when adding a single digit
number.
Identify subtraction facts
corresponding to addition
calculations (inverse).
Add/Subtract 9 and 11 by adding/
subtracting 10 and adjusting by 1.
Add 9 to a single digit
number by adding 10
and subtracting 1.
Add/Subtract 19 and 21 by
adding/ subtracting 20 and
adjusting by 1.
Add more than 2
numbers.
Recall pairs of multiples of 10 that
total 100.
Put largest number first
to add.
Multiplying by 10.
Year 3
Times tables (3, 4 & 8).
Identify near doubles using
doubles already known (e.g. 80 +
81).
Counting through hundreds and
thousand numbers.
Addition and subtraction,
including HTU & U, HTU & TU,
HTU & HTU.
Count in multiples of 4, 8, 50 and
100.
Flexible Partitioning: Choose and
use appropriate strategies for a
mental calculation (5 and a bit,
pairs, add 10 and adjust, largest
number first).
Bridge through a multiple of 10
and adjust.
Use patterns of similar
calculations.
Complements to 100 - any pairs of
2-digit numbers.
Multiplying by 10 and 100.
Identify and recall 10/100
less/more.
Year 4
All times tables (up to 12 x
12).
Count in multiples of 6, 7, 9,
25 and 1,000.
Count backwards through
zero including negative
numbers.
Use commutative laws.
Year 5
Extend tables beyond 12 x 12
[using partitioning, e.g. 13 x 8 =
(10 x 8) + (3 x 8).]
Use doubling and halving
(partition numbers first).
Identify near doubles (e.g. 1.5 +
1.6).
Counting through tens of
thousands.
Doubles: all whole numbers
to 50, multiples of 10 to 500,
multiples of 100 to 5,000 and
corresponding halves.
Count forwards ad backwards in
steps of powers of 10 for any
number up to 1,000,000.
Identify near doubles using
known doubles (e.g. 150 +
160).
Use closely related facts (e.g.
partitioning to multiply).
Flexible partitioning.
Multiply TU X U numbers.
Division using multiplication
facts.
Add or subtract to the nearest
multiple of 10 and adjust.
HTU partitioning.
Multiply and divide numbers and
decimals by 10, 100 and 1,000.
Add or subtract to the nearest
multiple of 10 or 100 and then
adjust.
Add increasingly large numbers.
Find differences by counting up
through the next multiple of 10,
100 or 1,000.
Inverse operations (derive division
facts from multiplication facts).
Multiply and divide (including
by 0 and 1 and multiplying 3
numbers).
Multiplication – understanding it
can be done in any order.
Pairs of multiples of 50 with a
total of 1,000.
Use factors, common factors and
multiples.
Division – related to
multiplication.
Count up and down in
hundredths.
Recall prime numbers to 19.
Count up to find small differences.
Round decimals with 1 d.p. to
the nearest whole number.
Count up and down in tenths.
Round numbers to nearest 10.
Rounding (to nearest 10/100).
Use commutative law for addition
and multiplication.
Use known number facts and
place value to add/subtract
mentally.
Rounding (to nearest 10, 100
and 1,000).
Use known number facts and
place value to add/subtract
Year 6
Use closely related facts for
multiplication, e.g. 49 x 51 –
multiply by 50 and adjust.
Perform mental calculations
including mixed operations
and large numbers.
Approximation (e.g. 6.1 x 7.8).
Prime numbers to 100,
common factors, and
common multiples.
Doubles of 2-digit numbers,
including decimals and
corresponding halves.
Doubles of multiples of 10 to
1,000 and multiples of 100 to
10,000 and corresponding
halves.
Counting in decimals.
Use closely related facts (e.g.
partitioning to multiply).
Add or subtract to the nearest
multiple of 10 ,100 or 1,000
and then adjust.
Calculations to 1 d.p.
Equivalent calculations.
Square numbers and cube
numbers.
Round decimals with 2 d.p to the
nearest whole number and to 1
d.p.
Use known number facts and
place value to multiple and divide
mentally.
Calculation to 2 d.p.
Fractions/
Percentage/Decimal
equivalence.
Round any number to a
required degree of accuracy.
Use factors.
Use known number facts and
place value to add, subtract,
multiple and divide mentally
(including with decimals).
‘Teaching Children to Calculate
Mentally’ publication.
Activity 3:
Have a look through the booklet.
What is its main focus? What do
you notice?
‘Teaching Children to Calculate
Mentally’ Publication
•Addition and subtraction p4-7
•Multiplication and division p8-11
•Addition and subtraction strategies p26-50
•Multiplication and division strategies p51-71
Ensuring mental and oral opportunities
are planned across a range of
mathematics:
The 10 strands within the framework:
- Number and Place Value (Number);
- Addition and Subtraction (Number);
- Multiplication and Division (Number);
- Fractions (Number);
- Ratio and Proportion (Number);
- Measurement;
- Properties of shape (Geometry);
- Position and Direction (Geometry);
- Statistics;
- Algebra.
Transum
Models and Images
The 6 Rs of Oral and Mental Work:
Rehearse
Recall
Refresh
Refine
Read
Reason
Rehearse
To practise and consolidate existing
skills, usually mental calculation skills,
set in a context to involve children in
problem-solving through the use and
application of these skills, use of
vocabulary and language of number,
properties of shapes or describing and
reasoning.
Recall
To secure knowledge of facts, usually
number facts, build up speed and
accuracy, recall quickly names and
properties of shape, units of measure
or types of charts, graphs to
represent data.
Refresh
To draw on and revisit previous
learning; to assess, review and
strengthen children’s previously
acquired knowledge and skills relevant
to later learning; return to aspects of
mathematics with which the children
have had difficulty; draw out key points
from learning.
Refine
To sharpen methods and procedures;
explain strategies and solutions; extend
ideas and develop and deepen the
children’s knowledge; reinforce their
understanding of key concepts, build on
earlier learning so that strategies and
techniques become more efficient and
precise.
Read
To use mathematical vocabulary and
interpret images, diagrams and symbols
correctly; read number sentences and
provide equivalents, describe and explain
diagrams and features involving scales,
tables or graphs; identify shapes from a list
of their properties, read and interpret word
problems and puzzles; create their own
problems and lines of enquiry.
Reason
To use and apply acquired knowledge,
skills and understanding; make
informed choices and decisions, predict
and hypothesise; use deductive
reasoning to eliminate or conclude,
provide examples that satisfy a
condition always, sometimes or never
and say why.
The Six Rs of Oral and Mental
Work
Activity 4:
Working with a partner, think of a
mental oral starter for each of the six Rs.
(The starter can be for any year group).
Six Rs
Rehearse
Recall
The Six
Rs of
Oral
and
Mental
Work
Refresh
Refine
Read
Reason
Learning Focus
To practise and consolidate existing skills, usually
mental calculation skills, set in a context to
involve children in problem-solving through the
use and application of these skills, use of
vocabulary and language of number, properties of
shapes or describing and reasoning.
Possible activities
Interpret words such as more, less, sum,
altogether, difference, subtract; find missing
numbers or missing angles on a straight line; say
the number of days in four weeks or the number
of 5p coins that make up 35p; describe partrevealed shapes, hidden solids; describe patterns
or relationships; explain decisions or why
something meets criteria.
To secure knowledge of facts, usually number
Count on and back in steps of constant size; recite
facts, build up speed and accuracy, recall quickly the 6-times table and derive associated division
names and properties of shape, units of measure
facts; name a shape with five sides or a solid with
or types of charts, graphs to represent data.
five flat faces; list properties of cuboids; state
units of time and their relationships.
To draw on and revisit previous learning; to
Refresh multiplication facts or properties of
assess, review and strengthen children’s
shapes and associated vocabulary; find factor
previously acquired knowledge and skills relevant pairs for given multiples; return to earlier work on
to later learning; return to aspects of mathematics identifying fractional parts of given shapes; locate
with which the children have had difficulty; draw shapes in a grid as preparation for lesson on coout key points from learning.
ordinates; refer to general cases and identify new
cases.
To sharpen methods and procedures; explain
Find differences between two two-digit numbers,
strategies and solutions; extend ideas and develop extend to three-digit numbers to develop skill;
and deepen the children’s knowledge; reinforce
find 10% of quantities, then 5% and 20% by
their understanding of key concepts, build on
halving and doubling; use audible and quiet
earlier learning so that strategies and techniques
counting techniques to extend skills; give cobecome more efficient and precise.
ordinates of shapes in different orientations to
home concept; review informal calculation
strategies.
To use mathematical vocabulary and interpret
Tell a story using an interactive bar chart, alter the
images, diagrams and symbols correctly; read
chart for children to retell the story; start with a
number sentences and provide equivalents,
number sentence (eg 2 + 11 = 13) children
describe and explain diagrams and features
generate and read equivalent statements for 13;
involving scales, tables or graphs; identify shapes read values on scales with different intervals; read
from a list of their properties, read and interpret
information about a shape and eliminate possible
word problems and puzzles; create their own
shapes; set number sentences in given contexts;
problems and lines of enquiry.
read others’ results and offer new questions and
ideas for enquiry.
To use and apply acquired knowledge, skills and Sort shapes into groups and given reasons for
understanding; make informed choices and
selection; discuss why alternative methods of
decisions, predict and hypothesise; use deductive calculation work and when to use them; decide
reasoning to eliminate or conclude, provide
what calculation to do in a problem and explain
examples that satisfy a condition always,
the choice; deduce a solid from a 2-D picture; use
Always, Sometimes, Never
A triangle can not have an internal angle that
is a reflex angle.
The three angles in a triangle are different.
Numbers with a unit digit of 3 are divisible by
5.
Mathematical Language
What language should be used in each year
group?
Is there a progression in the use of
mathematical language?
Mathematical Language
Mathematical Vocabulary booklets.
Note: This was produced for the original NNS. Due to the fact
that some objectives moved year groups in the 2006 Renewed
Framework and again in the 2014 curriculum, some vocabulary
will need to be introduced in earlier year groups.
Developing Mathematical
Language
ATM - Fourbidden Cards.
Fourbidden is a mathematical card game to
promote the use of mathematical language
devised by Phil Dodd and published by ATM.
There are now two packs of Fourbidden cards, the
latest designed with KS3 students in mind. There
are 52 cards in each set, on each card a familiar
mathematical term is printed on the left, with four
related words shown on the right hand side of the
card. There is a good explanation on different ways
of using the pack.
Break
Tea, coffee and biscuits available at the back
of the room.
How mental recall can be developed
to facilitate mental calculation.
There is a heavy reliance on known facts.
But….
There is a difference between recall and
mental calculation.
Conclusion:
If children haven’t learnt the facts in the
first place they can’t:
a) Recall them;
b) Use them to help them calculate;
c) Develop those facts further (i.e. for larger numbers).
Foundation Stones
Activity 5:
Look at the progression document.
Choose two year 6 objectives and identify
earlier objectives from previous year groups
that would need to be embedded for each
objective to be understood.
How would you solve it?
(calculation sorting)
Don’t solve the calculation…
…identify the most appropriate strategy.
How would you solve it?
(calculation sorting)
Activity 6:
With a partner, work through the calculations
on the yellow sheet in the middle of your table.
DO NOT SOLVE THEM!
For each one identify the most appropriate
strategy that should be used to solve it.
How would you solve it?
(calculation sorting)
There is no right or wrong answer.
The point is…
About stopping and thinking;
Making things easy for yourself;
Using known facts to solve the problem
(by doing as little maths as possible!)
How would you solve it?
(calculation sorting)
•Should be introduced from year 3
•Should be used to develop lateral
thinking about strategies
•Should develop and build on .
previously learned strategies.
Flexible Partitioning
What is it?
What does it mean?
How does it work?
Flexible Partitioning
Let’s have a
go!
Mental Maths Practise Tests
Mental Maths Practise Tests
Use weekly:
• As a teaching opportunity to discuss strategies;
• To cover and practise a whole range of Maths;
• To practise rapid recall of facts/information;
• For speed.
The most important part is NOT
the testing or the mark achieved,
but the discussion that follows
the test.
Children need regular timed
practise to speed up their recall.
Other Strategies
• Times Tables (and division) Clubs
• Mathletics/Education City/RM Maths
Mental Oral Starters
Pace is very important.
Any recall of facts should be rapid.
Chris Moyles’ Quiz Night
http://www.sheffieldmaths.co.uk/Ch
ris%20Moyles%20Starters.html
What is
Visualisation?
Activity 7:
Skim read the article ‘Thinking Through, and By,
Visualising.
We rely on visualising when we solve problems.
Sometimes we create an image of the situation
that is being discussed in order to make sense of it;
sometimes we need to visualise a model that can
represent the situation mathematically before we
can begin to develop it, and sometimes we
visualise to see 'what will happen if ...?'. But are
there other ways in which we visualise when
solving mathematical problems and if so how can
we encourage, value and develop visualising in our
classrooms?
Children need to have had the
opportunity to hold, turn, examine
and work with objects before they
can visualise them.
Progression
Year 6 - - - - - - - - - - - - Year 1
•Which column would number 12 be in?
•Find me 2 numbers that add up to 10.
•Give me a number that will appear in the middle column.
•What can you tell me about the numbers in each column?
•Find the sum of the first row. Is it a multiple of 3?
•Are there any other rows and columns with multiples of 3?
•Look at the first column. If extended, would 73 be included?
•Give me 2 numbers between 50 and 60 that would appear in
the final column.
•Will there be a multiple of 100 in the middle column?
Shape Visualisation
Begin by visualising a square:
With one fold make it into a rectangle.
Describe the properties of the rectangle.
Make it into a smaller square.
Make it into a triangle.
What kind of triangle have you made?
How do you know?
How do you know it is not an equilateral triangle?
Fold it back into a square. Make it into a pentagon.
Can you visualise one fold to make a pentagon with one line of symmetry?
How do you know this is a line of symmetry?
Try using a post-it note to scaffold.
Shape Visualisation
Imagine a pyramid.
Walk around the pyramid. What can you see?
Imagine you can fly and fly above it.
What can you see now?
Imagine you can lift it up with a magic spell. Spin it
around, invert it and then put it back down again. What
can you see now?
Now talk to the person next to you and talk about the
similarities and differences between your two pyramids.
Number visualisation
•Imagine the number five hundred and thirty two
drawn in the air in front of you.
•Which digit is in the middle? Which is on the left?
Which is on the right?
•Replace the middle digit with a four. What number
can you see now?
•Swap over the middle digit with the one on the left.
What number can you seen now?
•Remove the middle digit and push the two together
so that they are next to each other. What number
can you see now?
Conclusion:
Visualisation is important in all areas of Maths.
It is important for working out properties of
shapes, positions of shapes or objects that
have been reflected, rotated or translated and
nets of 3D shapes.
It is also important for measuring.
Conclusion:
It is also extremely important for
calculating.
If you can imagine, partition and recombine
in your mind’s eye, you are more likely to
be able to flexibly use and adapt known
facts to assist in mental calculation.
Conclusion:
Visualisation can be supported with practical
equipment.
Use of a broad range of vocabulary is vital.
Some more ideas for visualisation activities
are provided in booklets for you.
Other activities are available on the internet
via a Google search (Nrich is a good website).
How to teach and not just test
times tables.
Times tables are extremely important.
Here are a few reasons why:
Mental calculation of areas of shapes
Mental Division (use of inverse operation)
Use of known facts for other calculations
Mental multiplication e.g. 2-digit by 1-digit numbers
Use of the grid method (as a written calculation)
Progression in Mental Multiplication
Addition facts (known facts)
Number bonds to 10, then 20….
Complements to 100 (multiples of 10 first)*
Multiplication as repeated addition
Arrays
Times tables
Extend knowledge of times tables (e.g. 16 x 5)
Times Tables Progression
2, 10 then 5
3, 4 and 8
6, 7, 9, 11, 12
Beyond 12 x 12
Times Tables Progression
Begin in year 1
Consolidate in year 2
2, 10 then 5
3, 4 and 8
Year 3 – count in steps
Year 4 – recall x and ÷ facts
Year 2 – count in steps
Year 3 – recall x and ÷ facts
6, 7, 9, 11, 12
Beyond 12 x 12
From year 4 onwards
Year 5 – Multiply and divide numbers mentally using known facts
Year 6 – Perform mental calculations with large numbers
Progression in teaching tables
Counting forwards and backwards in steps
As above, but ‘skipping around’ and introducing
missing numbers.
Chanting, e.g. 2, 4, 6, 8…
Reciting times tables
Continued
Progression in teaching tables
Year 3:
Inverse operation should be introduced for
multiplication and division.
(Addition and subtraction inverses already introduced in Year 2)
Counting, chanting, etc. should be ongoing
through Key Stage 2.
Teaching ideas
Refer to models and images sheets.
Counting stick
http://www.topmarks.co.uk/flash.aspx?f=countingstickv4
Teaching ideas
Counting hoop
Good to link in with time and
counting around a clock face
Pendulum
Teaching ideas
Arrays are important for the understanding of
multiplication as well as division.
Can be used from year one upwards.
5 x 3 = 15
15 ÷ 5 = 3
3 x 5 = 15
15 ÷ 3 = 5
Teaching ideas
Year 3:
Inverse operation should be introduced
Fact families for multiplication and division
(derive division facts from known multiples)
3x2=6
2x3=6
6÷3=2
6÷2=3
Teaching ideas
Counting stick challenge
Count along the top in 3s
Count along the bottom in 4s
SIMULTANEOUSLY.
Let’s have a go!
Teaching ideas
Counting with a twist
B
A
C
Group A count in steps of 0.1
Group B count in steps of 0.3
Group C count in steps of 0.5
Teaching ideas
Stack ‘em
up games
Multiplication
and Division
3
24
9
30
21
Three
Times Table
Stack ‘em
up
12
15
27
6
18
http://www.fairhaven.ik.org/p_Printable_Maths_Games.ikml
Key Messages
• Mental maths is wider than just mental calculation and is not
just a times tables test.
• Use of jottings and practical resources are acceptable ways of
supporting mental calculation.
• Children need daily and fun whole class, guided and/or
independent opportunities to practice and apply their mental
maths skills and improve their confidence and efficiency.
• Opportunities to develop mental maths should include
developing reasoning and communication.
All resources used today and
links to websites referred to can
be accessed via our website:
http://www.fairhaven-cpdevents.wix.com/info
Click on the ‘Resources’ tab
Before you leave…
We would be grateful if you could
complete an evaluation sheet.
Thank you.
Improving Mental Mathematics
in Schools
Tom Garner
[email protected]