Children will develop confidence and resilience in maths only if they are
given the time, encouragement and skilled teaching they need to allow
them to have secure understanding in the essential concepts and recognise
the links between them.
Rushing on from one element to another on a
list – regardless of whether the basics are
fully grasped - is a recipe for failure.
Mr Li finished his explanation of the lesson. “It is not that the children are different in Shanghai,” he said.
“You also have children who know the answer. But in Shanghai a lot of time is spent asking children to
explain their answers. They’ve got the answers, but what I want to know is how they got them.”
Aim: To agree on the key principles of good
teaching and effective learning of Maths.
What do we believe are the fundamentals of learning and
teaching in our own schools and our classroom practice?
Multiple points of view are voiced in the current news following the introduction the new National Curriculum. (Mike Ellicock - National Numeracy CEO & Tim Oates – Chair of
the panel that reviewed the National Curriculum)
Shanghai have the highest international
test results, whereas England came 26th
DFE are concerned that levels need to rise to meet the demands of global competition.
DFE created 30 Maths Hubs to share best practice
Active Maths Schools working in partnership
.
Small Scale Research Project • An opportunity to reflecting and focusing on how
children do learn best.
• Different perspective - what can we do as teachers to
overcome barriers to effective learning.
• Raising achievement across both our schools.
Research
Test
Feedback
Impact
Mind Map Collaboration
6 Key Questions (Derived from Mind Map)
Across 3 shared CPD meetings, an opportunity to focus on Maths across both
schools:
Reflect on what we want our maths teaching to look like in light of the
New Curriculum
Examine current research
Identify effective teaching focus
Share good practice
Gather pupil evidence
Consider impact on our own classroom practice
Splitting the research/moments of inspiration from each other
The Role of Problem Solving
Key points to summarise the research:
Mini plenaries to aid mathematical discussions in problem solving to avoid misconceptions.
- Thought processes,
- Use discussion to find alternative methods/strategies as well as solutions
- Use of resources to support children of all abilities to engage and excel
Overview of main task
Range of problem solving activities
How many different ways can you….?
Do these pages fit the pattern?
Child Evidence: Brainworks
Impact on classroom practice:
- Verbalising WILF ensures children fully understand what is expected
(links to Talk 4 Writing)
- Children having a chance to explore the investigation (out of their system) enabling
pupils to share good examples and follow own ideas.
- Mini-plenaries to focus learning and for children to share processes and systems
(great for sharing mathematical vocabulary)
- Brainworks/questions to respond to work very well to allow children to explain
their
processes and systems, what they discovered and what their working was using
good mathematical vocabulary.
Raising The Bar – Bridging The Gap
Key points to summarise the research:
Ways to bridge the gaps of underachievement – Raising the bar
- ‘UK needs more people to study maths more effectively’
- All children, regardless of prior attainment need to access a more ‘rewarding
experience’
- Sense of urgency, we are falling behind and needs resolving.
- Deeper grasp of the ways of mathematical thinking and working
- Reasoning
- Investigation skills
Reasoning
• Aim was to be a simple task
• Focus on communication skills
• Use of correct vocabulary to explain what and why
• Elements of recording were less formal so children could focus on reasoning skills
and their learning
• Opportunities for collaboration and exploration, more open-ended.
Investigation
• Open-ended
• Opportunity to apply formal skills in a different context
• Accessible across abilities
• Focus on collaboration and discussing ideas
• Flexibility of recording to meet individual children’s needs
Child Evidence
• Lots of opportunities to use modelled vocabulary and different ways of recording
• High levels of motivation and engagement
• Less answer-driven, gave children more confidence
• Attitude/approach to ‘have a go’
• Children were focused more on the learning process than the outcome – no specific
answer
• Low threshold/High ceiling
Impact on Classroom practice
• Highlighted the need for these activities to be done more regularly
• Importance of communicating ideas not just writing them down
• Resources to promote correct vocabulary and support discussions
• Building in these approaches to all lessons (sustainability)
• Can learning of skills happen through investigation?
• Opportunities to experience a wide variety of activities to boost enthusiasm
• Giving learning a purposeful context
The Role of Demonstration at the Start of a Lesson
Shanghai Maths research showed the following related to our focus:
• Teacher instruction took up 30% of the lesson
• Children then used group or paired work as a follow up to the modelling
• Lessons began with recap of previous learning and questions were asked of randomly
selected pupils.
• Lessons were directed to the higher ability pupils.
• IWBs were used to display key questions.
• After the demonstration, there was little peer interaction with mostly silent working.
What happened on our visits?
• We observed a range of maths activities in each other’s classrooms . These included
new learning, applying skills to new situations and problem solving.
• Some lessons involved the class being split into two mixed ability groups. One half of
the class were sent off to begin and the other half remained with the teacher so that
the task could be modelled to them.
“I wouldn’t have known
what to do if the teacher
hadn’t modelled, I hadn’t
done it before.” (Y2)
“I wouldn’t
know what to
do if I wasn’t
shown.” (Y2)
“Sometimes I
like having a
go first.” (Y3)
What happened on our visits?
• The children that were in some of the group that had no teacher demonstration
appeared to be using ‘better maths thinking’. The discussions that they were having
led to deeper learning.
• Other pupils approached problems successfully without modelling but with a lack of
systematic workings.
• Some children struggled with accessing learning without any teacher input.
Impact on Classroom Practice
• Flexibility in approach to modelling.
• Giving children a chance to have a go at the start of a lesson. This gives them the
freedom to explore and investigate, leading to deeper learning.
• More modelling at the start of a topic, this lessens as children become more
competent.
• Use children to model during mini-plenaries.
• Taylor modelling to specific groups and needs throughout the lesson.
The Role of Visual Images to Raising Standards
Summary of research:
To use something tangible to underpin understanding. Once the learner is ready, the
tangible will be replaced with the abstract mathematical idea.
What the research told us:
• Low achievers need step by step instructions to
use the resources.
• More able choose and adapt the resources.
• Effective resources should take a variety of forms.
• Learning is deeper if the pupils make and
manipulate the resources themselves.
• Pupils tend to stay with the tried and tested rather than invest the effort required to
change.
Year 6 Research
Southway - ordering fractions
St Wilfrid’s – using algebra to investigate area
Why did we choose these skills?
• We sometimes make the assumptions that children already have an understanding of
an abstract mathematical notion.
• Does everyone really need the visual resources – aren’t they too babyish?
Impact on classroom practice. (1)
• By slowing down initial teaching, misconceptions
are uncovered that hinder later learning. E.g.
making the 24 cm fraction walls eighths and
thirds.
• By using a visual resource, children can
immediately see what the teacher is explaining,
rather than an over reliance on ‘teacher talk’.
Child Evidence
“ I learned best when I did fraction walls. I found this a lot more easier because it was easier
to see which one was bigger.”
“The most useful method to me was the circle method because it was easy to use and you
could see it clearly.”
Which method helped you learn best?
Fraction walls - 11
Fraction circles - 1
Common denominator - 6
Impact on classroom practice. (2)
By being actively involved, understanding is quicker and can be modelled by
both teacher and pupil.
Physically making resources supports understanding of a
concept - time invested rather than time wasted.
The Role of Abstract and Generalisation
Generalisation “Heart beat of mathematics” John Mason 1996
• Children (and adults!) will have used generalisations about maths without knowing it
e.g. understanding of factors and multiples, co-ordinate systems etc.
• Exploring patterns through solving problems
• Reaching a point where you can make a generalisation e.g. the patterns you see in
the 9x tables
• Articulating it
• To make a statement that is always true.
Overview of the Task - St Wilfrid’s
Using algebra to represent answers
LO: I can explore different strategies to solve a problem. Success criteria
I can create and use a system
I can ask 'What if .....?' questions
I can choose my own way of recording
I can collaborate with others
I can create a rule
I can recall multiples of numbers
Vocabulary
multiple
factor
add
What if…?
rule
McCarthys and Carrolls
On the planet Vuv there are two sorts of creatures. The Carrolls have 3 legs and the
McCarthys have 7 legs.
The great planetary explorers from W.A.S.A., who first discovered the planet Vuv, saw a group of Carrolls and McCarthys. They observed that there was more than one of each kind of creature in the crowd.
Challenge 1: W.A.S.A. counted 52 legs in the group. How many Carrolls and how many
McCarthys were there?
Overview of the task- Southway
How many squares on an 8x8
chessboard?
The answer is not 64!
Strategies
The children all started with an 8x8 square and the majority began counting the number of
1x1 squares on the grid (64), then moved onto the 2x2 squares (49).
Some children used coloured card cut to the right size to help them count, some used
coloured pens to mark on their
chessboard.
Many children soon began to notice that
there was a pattern and the first
generalisation was made.
It was important to remind the class
that it was a generalisation that they
were making, as the term was new to
them all.
Support
The lower ability maths group were able to make generalisations and find patterns.
In order to ensure they were successful, they needed more hands on resources, to support
them in finding the patterns, but were still encouraged to choose the resources they
wanted and decide on the strategies they would take.
Generalisations
Impact on classroom practice
• Generalising is something we do all the time e.g counting odd, even, odd etc.
• It can allow us to build in opportunities for high level problem solving
• All children enthused and engaged
• Articulation will raise achievement- this is the challenge!
Extension:
This could be taken as far as the
children wished. Once they had
discovered the pattern, they were
able to make predictions and quickly
find out how many squares would
be in any size chess board.
The role of vocabulary and language used by teachers and pupils in the
classroom setting.
Summary of the Research
The paper presented two views:
1. Pimm (1987) : mathematicians aim for precision, clarity and unique definitions for
the words and expressions they use. The National Numeracy Strategy agrees “You
need to plan the introduction of new words in a suitable context...Explain their
meanings carefully and rehearse them several times...sort out any ambiguities or
misconceptions your pupils may have through a range of...questions”
2. Barwell : ambiguity forms an important resource in school mathematics as it allows
students to investigate what it is possible to do with mathematical language, and so
to explore mathematics itself. He concluded by saying that, “If, as suggested by the
National Numeracy Strategy, all ambiguity is ‘sorted out’ as soon as it arises, valuable
opportunities for students to learn the subtleties of mathematics could be lost.”
3.
Overview of the task:
Four lessons were seen each with a specific focus on the modelling and use of
mathematical language:
• Foundation – Addition
• Year 1 – Fractions
• Year 3 – Rotation
• Year 3 – Subtraction
The intention was that the pupils were exposed to key mathematical language and that this
vocabulary was explained through modelling and practical examples. If ambiguities arose
pupils would be given the opportunity to explore the meaning of the language.
Observations from the Lessons
Foundation (Addition)
• CT used ‘maths’ vocabulary and referred to key concepts using analogies as part of
daily routines – before, after, less than, more than, how many, same, more, bigger,
smaller, pair up, odd/ even, one more than, ten more than.
• CT also used Makaton, hand gestures and symbols / concrete equipment to support
concepts / vocabulary.
• Vocabulary and was reinforced using previously mastered concepts and progressing
through to new terminology.
• TPS was used as an assessment tool with the children being encouraged to express
their learning / understanding to a peer.
• CT reinforced this through ‘maths talk’ questioning vocabulary used and referring to
concepts in other ways to verify learning.
Observations from the Lessons
Year 1 (Fractions)
• Mathematical language was clearly modelled and constantly re-enforced.
• Mathematical vocabulary was arranged on cards, attached to the IWB and
moved/focussed upon when appropriate.
• Mathematical language was emphasised and a clear focus.
• Reference was made to prior learning.
• Using real-life contexts made opportunities to apply taught/new vocabulary in a
meaningful, purposeful way.
• Children given an opportunity (as part of the plenary) to create verbal mathematical
problems using the focussed mathematical vocabulary – very effective AfL
opportunity.
• Peer learning and environment enabled children to ‘take risks’ with new
mathematical vocabulary.
Observations from the Lessons
Year 3 (Subtraction)
• Mathematical language was clearly modelled and constantly re-enforced.
• A range of vocabulary was used alongside e.g. subtract, take-away, minus, difference.
• Math language was emphasised through clarity of articulation and intonation.
• Reference was made to prior learning where pupils had explored different
terminology for ‘subtraction’.
• Questioning was used to provide opportunities for pupils to apply language.
• Use of number stories relating to real life examples enabled the pupils to
demonstrate their understanding of the mathematical language.
Observations from the Lessons
Year 3 (Rotation)
• A range of vocabulary was used alongside each other, e.g. rotate, turn, angle,
clockwise, anti-clockwise, degree.
• Reference was made to prior learning where pupils had explored which aided the
learning in the lesson e.g. angles, time.
• Questioning was used to provide opportunities for pupils to apply their use of
language – activities.
• Hands-on activities meant the children were using the language actively.
• The language created was from the children, giving ambiguity (was not directly set by
the teacher).
• Mathematical language was clearly modelled and constantly re-enforced.
Impact on Pupils
Pupils, particularly young children, have enjoyed learning and trying out new words.
Taught mathematical vocabulary has provided the pupils with a language that is commonly
understood.
Providing the pupils with the correct mathematical vocabulary has enabled them to be able
to explain their learning.
Pupils have also been able to use mathematical language to explain their thinking.
In the younger classes it has become more evident that the pupils are naturally using
mathematical language in their self-initiated play, demonstrating understanding and
application.
Evidence of Impact on Pupils
Year 3 Subtraction Homework Four Narratives to Illustrate Subtraction
(Year 3 Homework)
Impact on Classroom Practice
Pupils should be explicitly taught new mathematical vocabulary using real life
examples or practical modelling.
Children need the vocabulary to meet expectations, e.g. explaining graphs, problem
solving, explaining thinking, etc.
Children also love having the opportunity to use the correct maths language – we
must support and encourage this.
Highlight key mathematical vocabulary throughout sessions, e.g. write words on
cards and point them out, highlight when a pupil has used a word.
Ensure consistency in language – knowing what has been used in previous year
groups and what will be used in consecutive year groups.
Impact on Classroom Practice
Ensure that all staff in the class are using the same language and using it correctly,
e.g. highlighting key vocabulary in planning.
We must explore ambiguities and mis-conceptions in language with the pupils, e.g. a
child saying “1 and 1 makes 11”, pupils comparing the physical appearance of
numbers when asked which is “bigger” or what the “difference” is.
Provide practical opportunities for pupils to explore, practise and apply their
mathematical vocabulary, e.g. through role-play, setting up and running a stall.
Use of stories and story telling to introduce, re-enforce and apply mathematical
language. ‘Talk for Writing’ for maths?!
In Conclusion
Mathematical vocabulary is a language to explain and describe concepts. As in
language in general, it can be open to different interpretations and used to mean
different things.
We were all in agreement that pupils should be taught specific vocabulary in order to
given them a common language to use when talking about their mathematical
thinking.
However, we also recognise that it is important that pupils are given the freedom to
explore what in the first instance may appear to be mis-conceptions in language, as
the ambiguities that surround language present an opportunity to develop deeper
mathematical thinking and understanding.