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Quality Improvement
Guidance for Teachers
on
Learning and Teaching
Approaches in Mathematics
Giffnock Primary
15 August 2007
Quality Improvement
What are the key features of a
mathematics lesson?
Standards and Quality in Primary Schools:
Mathematics 1998-2001 HMIE
Guidance for Teachers
Quality Improvement
The Mathematics Lesson
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Interactive Oral Mental Mathematics
Introducing the Lesson
Direct Interactive Teaching
The Plenary Session
Interactive Oral Mental Mathematics
Quality Improvement
Interactive Oral Mental Mathematics
Quality Improvement
“It is important to recognise that when calculating
mentally we usually make use of different methods
from those we have used for written calculations. We
should encourage pupils to develop and practise
flexible mental approaches. One way of doing this is
through class or group discussion of the variety of
different ways of carrying out different operations.”
Mathematics 5-14 National Guidelines SOED 1991 p52
Interactive Oral Mental Mathematics
Quality Improvement
“At all stages, many pupils tried to perform mental
calculations by attempting paper and pencil methods
in their heads, seldom if ever the most efficient
approach. As a result they responded slowly and
often wrongly. Overall there was insufficient practice
in mental calculation and, even when mental work was
done, almost no discussion of how answers were
obtained.”
Improving Mathematics 5-14 SOEID 1997 p7
Interactive Oral Mental Mathematics
Quality Improvement
“In all classes there should be regular mental
calculation sessions to provide opportunities to
practise skills, discuss alternative mental methods
and be shown how to use efficient approaches.”
Improving Mathematics 5-14 SOEID 1997 p8
Interactive Oral Mental Mathematics
Quality Improvement
“In less effective courses, the mental mathematics
programme consisted merely of a series of regular
mental tests or involved pupils working individually
through mental exercises. This approach missed
opportunities for teachers and pupils to work together,
for pupils to share their thinking with teachers, reveal
evidence about their understanding and use
mathematical words and phrases in explanations and
discussions.”
Standards and Quality in Primary Schools:
Mathematics 1998-2001 HMIE p13
Interactive Oral Mental Mathematics
Quality Improvement
“Most schools had made improvements to the
emphasis on the progressive development of pupils’
mental mathematics skills, with emphasis on oral
rather than written calculation in the early stages.
Some schools included short daily sessions of oral
activities to keep pupils’ mental skills sharp. The best
sessions involved regular discussion of the different
strategies used by pupils to answer questions in
mental mathematics so that they became aware of the
variety of possible approaches.”
Standards and Quality in Primary Schools:
Mathematics 1998-2001 HMIE p13
Mental Calculation
38 + 47
Quality Improvement
• count on in tens from 40, 50, 60, 70, add 8 to get 78
and then 7 to get 85;
• first add 38 and 40 to get 78, then add on 7 to get 85;
• add 30 and 40 to get 70. Add 8 and 7 to get 15. Add
70 and 15 to get 85;
• add 38 and 2 (from the 47) to get 40, add 40 and 40 to
get 80 and the 5 left to get 85;
• add 40 and 50 to get 90, take away 2 (for the 38) to
get 88 and take away 3 (for the 47) to get 85.
Mathematics 5-14 National Guidelines SOED 1991 p82
Mental Calculation
47 - 18
Quality Improvement
• 18 add 2 gives 20, then 20 gives 40, now 7, so 29 in
all;
• 47 - 10 leaves 37, 37 take away 7 is 30, then take away
another 1, leaves 29;
• 47 - 20 + 2 = 27 + 2 = 29;
• 47 - 18 = 47 - 17 - 1 = 30 - 1 = 29
Interactive Oral Mental Mathematics
Quality Improvement
This session can be used to:
• Consolidate previous work
• Establish knowledge of an attainment target
• Focus on skills needed in the main part of the
lesson
• Practise mental calculations and rapid recall of
number facts in a variety of ways
• Establish new facts from known facts and explain
the strategies used
• Develop a particular strategy
Interactive Oral Mental Mathematics
Questioning
Quality Improvement
Questioning should effectively:
• give pupils time to think before providing an
answer
• involve as many pupils as possible
• allow time to explore reasons for wrong answers
• take account of pupils’ attainments
Interactive Oral Mental Mathematics
Questioning
Quality Improvement
Different types of questioning can be used:
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Quick Recall
Mental Calculation
Oral Responses to Written Problems
Oral Questions in Context
Oral Questions with Multiple Answers
Interactive Oral Mental Mathematics
Quality Improvement
• What is 6 - 4? Show me two numbers with a
difference of two.
• Is 16 an even number? What even numbers lie
between 10 and 20?
• What are four threes? Show me two numbers with a
product of 12.
• How many centimetres in a metre? Tell me two
lengths that together make a metre.
• Continue this sequence: 1, 2, 4, … Find different
ways to continue this sequence: 1, 2, 4, ...
Interactive Oral Mental Mathematics
Strategies
Quality Improvement
There are a variety of ways in which the session can
be practised:
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Unison Response
Show Me
Cover-ups
Using the Pupils
Unison Response
Counting Stick
• 0 10 count forwards/back/random
• 0  20 count forwards/back/random
• 0  100 count forwards/back/random
• 0  1000 count forwards/back/random
• 6  count on…13  count on…42 ...
• 16 count back...
23 count back...
• 0  50 5x table, 0  70 7x table
• Zero in middle
Quality Improvement
Unison Response
Rhythm Counting
• count forward in ones to ten - then back
• count forward in tens - then back
• count in fives, threes...
• count on in twos from 5...
• count on in threes from 55...
• count back in fours from 40...
• count in halves from one half
• count in quarters - and simplify
Quality Improvement
Unison Response
Pendulum Counting
• count forward in ones to ten - then back
• count forward in tens - then back
• count in fives, threes...
• count on in twos from 5...
• count on in threes from 55...
• count back in fours from 40...
• count in halves from one half
• count in quarters - and simplify
Quality Improvement
Show Me
Flip Flops
• how many can you see?
• show me more/less/same/one more/less
• how many can I see?
• in pairs - make 12
• in threes - make 15
Quality Improvement
Show Me
Digit Cards
• show me 6, 15, 36, 85
• show me an even, odd number
• show me a number greater than 4
• show me a number between 16 and 20
• show me number bonds +, -, x, ÷
• show me double 5, 9, 36
• show me the remainder: 18÷5
• show me 1 less than 70
Quality Improvement
Show Me
Digit Cards in Pairs
• show me pairs of numbers that total 10
• show me four factors of 24
• show me two multiples of 7
• show me square numbers
• show me a four digit multiple of 3
Quality Improvement
Show Me
Digit Cards - use four (1, 2, 4, 7)
• show me the largest/smallest number
• show me the number nearest to 3000
• in groups of 3 - make different numbers
• who has the largest/smallest?
• who has the nearest to 5000?
Quality Improvement
Show Me
Digit Cards - use two (3, 4)
Quality Improvement
• show me the sides on a quadrilateral
• show me the difference between 129 and 132
• show me 24÷6
• show me 27÷9
• show me the lines of symmetry on an equilateral
triangle
Show Me
Fan Cards
Quality Improvement
• show me 2/3 digits which total 8, 10..
• show me a 2 digit number which is odd
• show me a 2 digit number between 50 and 75
• show me a 3 digit number between 250 and 500
• show me a 2/3 digit number divisible by 4
• show me a 2 digit prime number
• I’m thinking of a number….
Show Me
Number Generator
• show me 457, 12 670, 238 300
• show me 1750, make it 10 more...
• show me 22 000, make it 100 less...
• show me 467 - now add 4
• show me 342 - now double it
• rounding to nearest 10, 100, 1000
• multiply/divide by 10, 100, 1000
Quality Improvement
Show Me
2D Shape
Quality Improvement
• show me the shape behind the wall
• show me a hexagon
• show me a triangle with an angle greater than 90°
• show me a four sided shape with only two lines of
symmetry
Cover Ups
Grids
• 0-99 grid - recognition/one more/less
• 1-100 grid
• shapes - names, symmetry, angles
• fractions
• tables - cover up answers to 9x, 4x...
• money
• 24 hour time
Quality Improvement
Using the Pupils
Quality Improvement
• Loop Cards
• Human Lines - change places with number 15;
number before/after 12; number more/less than 6...
• Decimal Point - Hold 3.2 One person holding 1 to join
in. Where to stand to make number smaller/larger?
Could she stand elsewhere?
Improving Achievement In Mathematics in
Primary and Secondary Schools
HMIE 2005
Quality Improvement
“Schools did not place sufficient emphasis on
developing pupils’ skills in written and mental
calculation. Too often the approach was restricted to
“ten-a-day” which did not develop pupils’ knowledge
and understanding of techniques in mental calculation.”
Improving Achievement In Mathematics in
Primary and Secondary Schools
HMIE 2005
Quality Improvement
Features of effective learning and teaching at primary
stages include:
“Teachers focused their questioning on checking pupils’
understanding, rather than simply eliciting correct
answers.”
“Teachers asked pupils to explain their thinking, and
responded skilfully when pupils made mistakes using
these as a basis for further learning.”
Using the Pupils
The Answer Is…
Quality Improvement
x
+
7
–
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Using the Pupils
The Answer Is…
Quality Improvement
7x1
3·5 x 2
1·75 x 4
3+4
5+2
1·2 + 5·8
-1 + 8
7
8–1
10 – 3
29 – 22
35  5
63  9
70  10
77  11
Using the Pupils
What Else Do You Know?
12 x 5 = 60
Quality Improvement
Using the Pupils
What Else Do You Know?
Quality Improvement
60 12 = 5
60  5 = 12
6 x 5 = 30
5 x 12 = 60
12 x 5 = 60
12 x 10 = 120
11 x 5 = 55
1·2 x 5 = 6
Using the Pupils
What Else Do You Know?
60 12 = 5
3 x 5 = 15
60  5 = 12
6 x 5 = 30
Quality Improvement
12 x 25 = 30
5 x 12 = 60
12 x 5 = 60
12 x 10 = 120 24 x 5 = 120
1·2 x 5 = 6
12 x 05 = 06
11 x 5 = 55
10 x 5 = 50
Using the Pupils
What Else Do You Know?
20% of 60 = 12
Quality Improvement
1
12
of 60 = 5
1
5
of 60 = 12
60 12 = 5
3 x 5 = 15
60  5 = 12
6 x 5 = 30
12 x 25 = 30
5 x 12 = 60
12 x 5 = 60
12 x 10 = 120 24 x 5 = 120
1·2 x 5 = 6
12 x 05 = 06
11 x 5 = 55
10 x 5 = 50
Mental Calculation
Multiplication
• 3 x 13 = (3 x 10) + (3 x 3) = 30 + 9 = 39
• 3 x 14 = 3 x 2 x 7 = 6 x 7 = 42 or
• 3 x 14 = 3 x 7 x 2 = 21 x 2 = 42
• 3 x 15 = (3 x 30) ÷ 2 = 90 ÷ 2 = 45 or
• 3 x 15 = (2 x 15) + (1 x 15) = 30 + 15 = 45
• 3 x 16 = 6 x 8 = 12 x 4 = 24 x 2 = 48 or
• 3 x 16 = (3 x 15) + 3 = 45 + 3 = 48
• 3 x 17 = (3 x 20) - (3 x 3) = 60 - 9 = 51
Quality Improvement
Mental Calculation
Division
Quality Improvement
• 51 ÷ 3
3 x 10 = 30, this leaves 21 ÷ 3 = 7 and 10 + 7 = 17
• 90 ÷ 6 = (90 ÷ 3) ÷ 2 = 30 ÷ 2 = 15
• 115 ÷ 5 = double 115 ÷ 10 = 11.5 + 11.5 = 23
Using the Pupils
What Could the Next Number Be?
3·14
Quality Improvement
Using the Pupils
What Could the Next Number Be?
3·14
4·14
13·14
3·15
3·141
Quality Improvement
Using the Pupils
What Could the Next Number Be?
3·14
4·14
5·14
13·14
23·13
3·15
3·16
3·141
3·142
Quality Improvement
Using the Pupils
What Could the Next Number Be?
3·14
Quality Improvement
4·14
5·14
6·14
13·14
23·13
33·13
3·15
3·16
3·17
3·141
3·142
3·143
Online Resources
Quality Improvement
Introducing the Lesson
Quality Improvement
Introducing the Lesson
Quality Improvement
“By sharing the purposes of a piece of learning and
the expected outcomes with pupils, the learning
within a classroom becomes more focused for the
pupils as well as the teacher.”
Direct Interactive Teaching SCCC 2000
Direct Interactive Teaching
Quality Improvement
Direct Interactive Teaching
Quality Improvement
“Direct interactive teaching is characterised by high
levels of questioning, explaining and worked
examples where pupils together explain their
understanding and solution to a problem.
Alternative ways of solving problems and reaching
answers are sought and shared. Pupils play a very
active part in the lesson. There is a high level of
group interaction and cooperative working.”
The Structure and Balance of the Curriculum Guide for
Teachers and Managers 5-14 Learning and Teaching Scotland
2000
Direct Interactive Teaching
Quality Improvement
“If used skilfully, a series of questions can be used
together with pupil replies to explore and expand the
area being taught rather than a lengthy teacher talk.
Effective questioning also provides a platform for
effective differentiation, with questions being framed
to provide different levels of challenge as
appropriate.”
Direct Interactive Teaching SCCC 2000
Direct Interactive Teaching
Quality Improvement
“The least effective teachers relied too heavily on
course material, sometimes reading out tasks from a
textbook rather than explaining the principles of a
topic. Where learning was less successful, pupils
often spent too much time waiting for instructions
from the teacher or working on repetitive exercises
with little opportunity for discussion.”
Standards and Quality in Primary Schools:
Mathematics 1998-2001 HMIE
Top Tips in Questioning
Quality Improvement
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Plan
Sequence
Distribute questions
Show interest and respect all answers
Don’t give clues
Encourage contribution
Increase wait time
Check how answers were arrived at
Use wrong responses positively
Use more higher order and open questions
Ask pupils for questions
The Plenary Session
Quality Improvement
The Plenary Session
Quality Improvement
This part of the lesson can be used to:
• ask pupils to present their work, summarising
what has been learned
• mark or discuss particular examples, reviewing
the different methods of calculation used,
rectifying any misconceptions
• give constructive feedback to pupils
• make conclusions or establish rules
• highlight pupils’ progress towards attainment
targets
• provide tasks for homework
The Plenary Session
Quality Improvement
The teacher should:
• have a clear idea of the purpose and content of the
plenary session
• leave enough time for this to be achieved
• evaluate their own teaching by reflecting on what
pupils have learned and what progress has been
made
How Do We Keep Skills Sharp?
Quality Improvement
Appropriate resources
Mental mathematics
Workcards and Worksheets
Homework
Problem of the Week
Websites
Assessment
Workcards and Worksheets
Quality Improvement
Numbertime
The Maths File Game Show
Teaching Time
Revisewise Maths
Numeracy Software
Schoolhouse Technologies
Skillswise
Problem of the Week
Quality Improvement
Nrich
NZMaths Problem Solving
Word Problems for Kids
Mrs Renz’s Math Web Sites
TES Brainteasers
Count On
Websites for Interactive Whiteboards
Quality Improvement
CurricuLinks – Interactive Whiteboards
CurricuLinks – 5-14 Mathematics
Homework
Quality Improvement
Homework
Quality Improvement
Assessment
Quality Improvement
• National Assessments
• Tracking Database
• Standardised Tests
• Mathematics Check-ups and Pupil Assessment
Files
• Formative Assessment
• Questioning
Quality Improvement
Guidance for Teachers
on
Learning and Teaching
Approaches in Mathematics