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.