A Devon approach to supporting children and young people with mathematical difficulties

Introduction

The purpose of this guidance is to provide schools with a range of practical interventions and approaches in order to meet the needs of children and young people with severe and persistent mathematical difficulties. This guidance also acts as a companion document to ‘A Devon Approach to Understanding Children and Young People with Mathematical Difficulties (2018)’. All mathematics should be taught in line with the aims of the National Curriculum.
The National Curriculum for mathematics aims to ensure that all pupils:

become fluent in the fundamentals of mathematics, including through varied and frequent practice with increasingly complex problems over time, so that pupils develop conceptual understanding and the ability to recall and apply knowledge rapidly and accurately
reason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical language
can solve problems by applying their mathematics to a variety of routine and nonroutine problems with increasing sophistication, including breaking down problems into a series of simpler steps and persevering in seeking solutions. Mathematics is an interconnected subject in which pupils need to be able to move fluently between representations of mathematical ideas.’ (DfE, 2013)

As described in our previous companion document mathematics is made up of a range of interacting component skills including those of number, geometry, statistics and measurement. Most typically children and young people who are identified by schools as having severe and persistent difficulties with mathematics are struggling with the understanding and manipulation of number. Difficulty in this area is what often triggers a referral to an outside agency such as the Educational Psychology Service. This document therefore focusses specifically on supporting children and young people in the development of understanding number.

The companion document provides an interactive factors framework for mathematical understanding and it is this framework on which we will base the structure of this document
as follows:

Providing a supportive teaching and learning environment
Supporting the development of cognitive skills
Supporting emotional aspects of maths development

Providing a supportive teaching and learning environment

It is the area of school learning environment that offers the greatest capacity for change in
relation to school practitioners.
Most children will benefit from a teaching environment in which the following are provided:

opportunities for mathematics both in the inside and the outside environment. This
would include the construction of a number rich (or “numerate”) environment
awareness of mathematics in the wider world
activities which provide meaning and purpose such as stories, songs, games, roleplay and rhymes; also “real world” situations that demand calculation for the solution
of problems
direct teaching that is strategic and based on assessment for learning
the use of mixed ability groups
play based activity
guided work
a level of challenge that will allow a child to extend their skills within their zone of
proximal development
links across and within the curriculum
identifying and understanding connections between mathematical ideas (symbols,
mathematically structured images, language and contexts). Please see advice from the
Maths Advisory team for more information on the Connective Model.

However, this may not be sufficient or even wholly appropriate for some children with severe
and persistent difficulties in maths. The remainder of this document concerns such children.

Does Intervention Work?

A research review (DfES, 2004) has found that:

Children’s difficulties with calculation are highly susceptible to intervention. These interventions can take place successfully at any time and can make an impact.
Individualised work with children who are falling behind in number and calculation can have a significant impact on their performance.
The amount of time given to such individualised work does not, in many cases, need to be very large to be effective. Short but regular interventions of individualised work may bring a child to the point where they can profit much better from the whole-class teaching that they receive.
It is important to find out what specific strengths and weaknesses an individual child has and to investigate particular misconceptions and incorrect strategies
Interventions should ideally be targeted towards an individual child’s particular difficulties. If they are so targeted, most children will not need very intensive interventions.

Supporting cognitive skills

The accompanying document, ‘A Devon Approach to Understanding Children and Young
People with Mathematical Difficulties’ (2019, Page 5 & 6), identifies a number of cognitive
skills whose impairment may be associated with difficulties in developing number concepts.
Some of these skills are generic, supporting learning across a range of subjects, whilst others
are specific to mathematics. Helpful approaches for supporting children and young people with
these difficulties follow:

Supporting children with generic cognitive barriers

Long term memory

One model of learning (Haring and Eaton, 1985) suggests that pupils need to

acquire new knowledge and concepts accurately, before they can….
develop fluency in using new skills, before they can….
demonstrate mastery in this area, before being able to….
generalise skills to a range of contexts, before being able to….
adapt these new skills to be useful in similar but different situations.

In some cases, a lack of such fluency can be associated with forgetting over time – often presented as a difficulty with poor long-term retention for new learning: “He gets it at the time, and then he’s forgotten by the following day,” is a typical comment on the referral form. The learner has had sufficient practice to display accuracy, but not enough to become effortlessly fluent in the skill. If we think of new skills we learn ourselves (I often use the personal example of learning to use an espresso machine!), we’ll be familiar with the experience of needing to keep up regular practice in order to retain new skills over time, developing accuracy, then an efficient workflow until we become fluent. Precision Teaching is one approach that aims to develop pupils’ accuracy and fluency in new skills. It quickly establishes an evidence-base of whether or not an intervention is effective, encouraging the exploration of different approaches until the best possible rate of progress is established. A ninety-minute training session in this approach is offered by the Devon Education Services Educational Psychology team and can usually be offered as part of the normal traded service.

Working memory

As discussed in the companion document, this has components that include verbal and visual stores, processing speed and rate of forgetting.
The balance of evidence suggests that intervention programmes can improve performance on the skill being practised (near-field effects) but that these do not generalise to other areas of performance (far-field effects). The emphasis should therefore be on reducing the load on Working Memory rather than improving it, for example by:

Reduce the Load on Working Memory

Chunking information and instructions
Encouraging the use of jottings
Giving additional processing time
Using electronic devices and alarms
Highlighting key numbers, words or phrases
Providing visual reminders and prompts
Using checklists when carrying out multi-step instructions or processes

Just a few strategies designed to support working-memory have been shown to have far field benefits. These evidence-based strategies include:

Improving Working Memory

Teaching children under seven to rehearse information or instructions sub vocally and prompting them to use this skill (children over seven tend to start using this skill on their own initiative)
Setting instructions, steps in a process or sequences to familiar rhymes
use of spatial grids (something as simple as placing coloured counters on a grid of squares) to aid in the encoding and retrieval of information.

Further training in this area will often be available from your Educational Psychologist.

Executive functioning

Executive functioning skills enable us to harness and co-ordinate other skills towards the completion of a goal. A complex set of interacting skills, they require longer to discuss fully than is appropriate here. Ask your Educational Psychologist about further training in this area. General principles of intervention strategies include:

Treating learning skills on a par with curriculum content and explicitly teaching these e.g. by identifying peers or role-models who exhibit these skills
Teaching to independence: breaking goal-directed skills down into small steps, initially providing step-by-step prompting and gradually withdrawing these cues as confidence and fluency are established

Other strategies might include:

Supporting Executive Functioning Skills

Teaching and supporting personal organisation skills through

Visual or itemised routines for helpful mathematical procedures
Homework diary management
Checking in with an adult at the start/end of the day
Timetabled personal organisation time
The use of colour-coded folders / transparent pencil case etc. Teaching and supporting time-awareness skills through
Visual timetables
Chronological ‘to do’ lists
Short, timed tasks using a sand-timer

Teaching and supporting self-evaluation skills through

The use of clear, easily measurable success criteria with explicit progression built in
The use of rating scales and solution-focussed questions e.g. “You rate yourself at 6/10 – that’s great! What’s keeping you above a five? What would you have to do to get to a 7? What did you do differently to last time?”
Traffic light systems, smiley face symbols etc. Teaching and supporting planning skills through
Harnessing the child’s learning strengths to develop planning support materials
Asking, when moving to independent work: “What do we need? What are the steps? What does finished look like?”

Teaching and supporting self-correction skills through

Encouraging peer group reflection on problems/strategies
Access to text-to-speech software (e.g. Textease) set to read out each word, sentence or paragraph as appropriate
Encouraging the recording of estimates before solving a problem

Teaching and supporting the skills to cope with transitions through

Giving verbal / visual notice of upcoming changes in a task
Providing consistent routines for beginnings and endings
Treating transitions as a separate learning activity

Teaching and supporting big-picture thinking skills through

Making lesson/task objective explicit with visual reminder
Providing explicit expectations, next steps, links to previous learning

Reading and writing difficulties

If a pupil has identified difficulties with reading or writing, try providing:
Supporting Reading and Writing Difficulties

Scaffolded work sheets
Subject-related word lists
Collaborative group work activities
Explore other forms of recording work e.g.
• A scribe (peer, parent or staff member)
• Word-processing or voice-recognition software
• Digital voice recording
• Oral evidence

Visuospatial and motor coordination skills

If a pupil has identified difficulties with visuospatial reasoning or motor coordination, try:
Supporting visuospatial or motor co-ordination difficulties

Minimising copying from the board
Using alternate lines, bullet points or numbered points
Organising information clearly e.g. boxes, columns, colours to separate information
Allowing extra time for copying when this is unavoidable
Providing photocopies as an alternative to copying from the board
Using coloured paper
Providing enlarged worksheets
Keeping worksheets clear and uncluttered

The document ‘Supporting children with gaps in their mathematical understanding’ (DfES, 2005), suggests the following access strategies for learners with spatial & motor difficulties:
Supporting children with spatial & motor difficulties

Increased use of number line when working with addition and subtraction, rather than counting objects or using fingers; own pocket number line
Teaching child to physically move objects from one side of a ruler to another, or cross them out on the page, when they must be counted
Number squares with alternate rows shaded or coloured to help them keep track of where they are
Small hole punched in top right-hand corner of numeral cards to prevent directional confusions
Use of squared paper for laying out written calculations
Mathematical symbols presented in different colours – for example, always green for +, blue for x – to prevent confusion between symbols where a difference in orientation is all that distinguishes one from another
Use of pre-prepared formats for calculations, graphs and tables
Use of appropriate software for recording calculations, graphs and tables and for drawing/manipulating shapes
Laying rulers or scales along coordinates when plotting or reading them
Teaching the child to put visual or spatial information into verbal form, and vice versa
Using non-slip matting and sticky-tack to anchor resources and paper
Recording using numeral cards or circling numerals on number lines and grids, rather than by writing

Non-verbal / logical reasoning ability

It is easy to take for granted that all pupils have a similar ability to make links between the
ideas that they encounter. But for some children, this ‘invisible’ skill can be a significant area
of difficulty. Teachers might find themselves confounded in situations where a student appears
to know everything necessary to reach an ‘obvious’ conclusion, but just doesn’t do so. Below,
in order of sophistication, are a few ideas that make the linking of ideas the teaching
focus, instead of the ideas themselves:

Supporting Non-Verbal or Logical Reasoning

1. Teaching the concepts of ‘like’ and ‘not like’

This concept is a prerequisite for the ensuing activities. It can be introduced by comparing a given item with identical or distinct ones e.g. calling out ‘like’ or ‘not like’ when each card is turned / shape is put down etc. The word ‘different’ can be introduced once an appropriate level of fluency is achieved. It would also be helpful to pre-teach the idea of ‘more like’ e.g. by comparing a variety of different sized blue squares and red triangles: “These three shapes are like each other (all red triangles) but which are more like each
other (the two small ones)?
This skill can also be reinforced using:

Spot the difference pictures
Sorting paired pictures into ‘same’ and ‘different’ trays
Playing matching games such as ‘Snap’

2. Extending the range of constructs

Collect together a number of similar artefacts that share / don’t share a given
property e.g.

square and circular 2D shapes
light and heavy objects
numbers ending in zero
numbers with / without a certain property e.g. odd and even numbers

Introduce the target vocabulary to the child. Sort the items into two groups for the child, then together, then ask the child to sort the items. This activity can be extended by:

Asking pupils to say which two of three of these items are more alike, again
using ‘my turn, together, your turn’ strategies.

3. Similarities and differences

Collect together a number of topic-related artefacts. Ask the pupil to choose three of them. Encourage the pupil to say which of these objects are more similar, and ask how come (s)he chose these. How is the third object different to the first two? In psychological terms, the smallest unit of meaning is the way in which two objects are similar to each other and different to a third object.
This activity can be extended by:

Asking pupils to describe a series of similarities and differences
Using images instead of artefacts
Using key vocabulary cards instead of artefacts (printed and cut out separately, possibly stuck on card or laminated)

4. Sorting and categorising

Using your selection of artefacts (or images or key vocabulary cards), ask the pupil to put the objects into two groups. Can they tell you what criterion they used to do this? Can they sort the objects in a different way? Can they guess how you have sorted them without you telling them?
This activity can be extended by:

Sorting by two criteria at the same time (with the potential for some objects to be in both categories)
Sorting objects into more than two groups

5. Collaborative Learning

Ask your pupil to work with a partner who has also grouped the same objects. Have they grouped them in the same way? How have they recorded their ideas? Can they explain to each other why they grouped them as they did? Can they negotiate an agreed way of grouping the objects?
This activity can be repeated with two sets of partners, then two (or more) groups etc. until the whole class agree how the objects should be grouped. In the same way, items could also be ranked by importance (or some other criterion) and this could be used to generate co-operative learning activities.

6. Mind Mapping

During any of the above activities, multiple answers can be recorded in an appropriate form e.g. written, visual, scribed, photographed etc. These similarity: difference contrasts and sorting criteria can then in turn be organised into groups by similarity or into a nested mind map (feather map), spider diagram etc.

7. Providing opportunities for talking before recording answers

A crucial feature of all these activities is that they provide a window into a child’s understanding (or cognitive map) of any given subject. In this way, the process can form the basis for a dynamic learning relationship between teacher and pupil. Instead of starting from the curriculum and what the pupil ‘ought to know’, the starting point is the pupil’s actual understanding. This may be very different – very sparse, or rich but idiosyncratic – from what the teacher was assuming. Appropriately differentiated follow-up work can then be tailored far more accurately to the pupil’s needs.

Verbal reasoning (e.g. word problems, verbal presentation)

If a pupil has identified difficulties in verbal reasoning skills, try providing:
Supporting Verbal Reasoning

visual and kinaesthetic aids e.g. Numicon, Base 10, Dienes, number lines, number tracks, bead strings etc.
the use of modelling, repetition and stem sentences (e.g. three apples plus two apples equals…)

Some children may need to be explicitly taught that number sentences are not the same as
word sentences because:

You sometimes read vertically e.g. column addition
The WYSIWYG rule (What You See Is What You Get) does not apply e.g. 13 is not pronounced ‘onety-three’
The sentences are often incomplete, which means…
You have to do something with the numbers e.g. number operations
There is usually only one right answer
Number sentences have two sides that have to balance

Supporting children with cognitive barriers specific to mathematics

These include:

Number sense (subitising, quantity discrimination and estimation)

For further advice and information please see http://nrich.maths.org/10737 and the
advisory teacher document.

Digit, number and symbol recognition / reading
Writing digits, numbers and symbols fluently
Counting (one to one correspondence, cardinality and ordinality)

Although not itself a domain specific approach, teaching to fluency provides one way of supporting children with domain specific difficulties in maths. Children with such difficulties include those who:

find foundational number concepts effortful
bring high levels of anxiety to maths tasks
find their working memory overloaded due to its limited capacity
find their working memory being devoted to other considerations e.g. a stressful home situation.

Focussing on developing accuracy and fluency in basic mathematical skills involves the
following:
‘Elephant Maths’ provides one such intervention programme which uses these principles.
Further details are available through your Educational Psychologist

Supporting children with emotional barriers

While cognitive skills are an important aspect of mathematical development, emotional and behavioural factors are of equal importance. In addition, emotional and behavioural aspects such as maths anxiety are known to interact with cognitive performance in areas such as working memory. Emotional and behavioural support in relation to maths can often be supported through whole-class approaches. The accompanying document, ‘A Devon Approach to Understanding Children and Young People with Mathematical Difficulties’ (2019, Page 9), identifies a number of cognitive skills whose impairment may be associated with difficulties in developing number concepts.

The following evidence-based approaches are associated with the creation of a non threatening and more optimal emotional climate for the learning of mathematics:

Motivation

Classroom characteristics and teacher’s instructional goals can impact significantly on pupil motivation. A classroom climate that encourages students to adopt learning goals rather than performance goals fosters the development of ‘intrinsic motivation’; that is, being curious about the activity for its own sake rather than for the achievement outcomes.

Focus on positive maths self-concepts

Prioritise pupil motivation and self-concept as pedagogical objectives.
Promote a positive narrative around maths with high expectations of pupils. Avoid negative statements about “not being good at maths”. Encourage positive self-talk e.g. “I can…”, “there must be another way…”
Promote mixed ability groups and co-operative working whereby each pupil has a distinct role within the group in which they can be successful.
Present tasks as realistic challenges.
Present optimal challenges (within the pupil’s zone of proximal development).
Give consistent positive and encouraging feedback.
Enable pupils to identify their strengths which relate to maths, e.g. explicitly name the skills and strengths which enabled a pupil to make progress.
Use of pre-teaching strategies is recommended in order to promote the confidence and participation of children with identified difficulties in mathematics

One helpful model for understanding and supporting motivational aspects of mathematics is Ryan and Deci’s self-determination theory (2001). This model describes three factors contributing to intrinsic motivation: ‘competence’, ‘autonomy’ and ‘relatedness’.

Competence refers to confidence in one’s abilities and the capacity to affect outcomes. Ryan and Deci suggest that in order to build this in pupils, they need to be given optimal challenges, positive feedback, supportive communication and social
rewards.
Autonomy refers to the need to perceive oneself as the source of one’s own behaviour. So pupils needs to be given choice, acknowledgement of feelings and opportunities for self-direction.
Relatedness refers to creating an environment characterised by a sense of security and safety. Teacher pupil relationship is therefore really important in achieving this.

Setting ‘learning’ goals rather than ‘performance’ goals

Avoid eliciting competition amongst pupils e.g. comparing grades to each other.
Promote strategic effort.
Help students to establish goals for themselves.
Offer options in relation to content and strategies.
Focus on conceptual learning rather than simply getting the right answer quickly. Feedback to pupils the skills they used which enabled them to complete a task.
Help pupils to recognise that when things get difficult they are ‘learning’.
Help pupils to understand that mistakes are OK and vital to learning, for example, use examples of famous people who initially failed.
Praise effort rather than ability and ensure that praise is given for genuine effort, rather than simply succeeding at an easy task.
Encourage a pupil to compare their own progress over time (not to their peers).
Feedback how to improve skills rather than performance.
Teach pupils ‘growth mind-set’ theory to help them understand the process of learning and development.
For further information and resources in this area please see youcubed

Growth Mindset: Action research in Devon

Recent action research has been carried out in Devon schools and has tested out the approaches promoted by Carol Dweck and Jo Boaler which relate to a ‘growth mind-set’ approach and an understanding that students’ ideas about their ability determine their learning pathways and math achievement. These schools have reported improvements in motivation and engagement in mathematics amongst pupils who were involved. These are some of the strategies and approaches that were tried out and reported to be effective:

Setting up mixed groups with roles such as ‘motivator’, ‘gatekeeper’ and ‘secretary’.
Helping students to know that when they struggle they are learning.
Using brain science to show the students how new synapses are formed when they are learning.
Helping pupils to understand that the objective is to ‘think hard’ rather than to ‘get things right’.
Setting the objective as collaborative learning rather than mathematical performance.

Help pupils to value and be interested in maths

Maintain a positive narrative around maths.
Explain the practical purposes of maths in everyday life.
Help pupils to understand the purpose of what is to be learnt. Make sure they are aware of the content to be learnt as well as the competence to develop.
Use interesting questioning and propose stimulating tasks, for example, apply maths to a child’s personal area of interest.
Ensure the task is a realistic challenge for the pupil i.e. not too easy, not too difficult.
Give some choice or control over the task

Maths Anxiety

All of the above recommendations will play a role in helping to reduce anxiety for a pupil. Two related approaches shown to diminish levels of maths anxiety generally within the classroom are the instructional strategies of co-operative and collaborative learning.

Collaborative learning involves pupils teaming up to explore a mathematical problem. In cooperative learning, students each take on a distinct role within the group, sometimes breaking out into ‘expert’ groups of pupils from different groups with similar roles. See Pages 11 and 13 for examples of these approaches.

These approaches promote a better work climate among people with similar fears, it helps control stress when students feel isolated and it makes people feel more able to perform. It also fosters a positive self-concept and encourages students to participate, take risks and share responsibility for reaching challenging goals. This highly researched and easily implemented strategy is applicable to any maths classroom. In addition, the following strategies may support children in reducing their maths anxiety:

Reducing anxiety and its impact

The quality of teacher-student relationship is highly important. Genuinely positive, interested, caring and empathic interactions with pupils will help them to feel emotionally safe in order to make mistakes, takes risks, ask questions and celebrate successes.

Simply encouraging a child to reflect on how they are feeling prior to an anxiety-provoking task has been found to reduce this level of anxiety.
Reducing the working memory demands of a task using visual supports, smaller steps, known facts and calculators can reduce the impact of maths anxiety on the learner.
Parental maths anxiety and low maths self-concept is highly correlated with pupil maths anxiety and low self-concept. Ensuring homework is set at a level where a pupil is able to independently access it could avoid negative reinforcement of these factors in both parents and pupils.
If anxiety is severe and persistent, please talk to your Educational Psychologist for further advice.

References

DfES (2001) Guidance to support pupils with dyslexia and dyscalculia: London
DfES (2005) Supporting children with gaps in their mathematical understanding: London
Dowker, A. (2003), Brain-based research on arithmetic: implications for learning and teaching in Thompson, I. (ed.)
Dowker, A. (2004) What works for children with mathematical difficulties? London: DFES
Dowker, A. (2009) What works for children with mathematical difficulties? London: DCSF
Haring, N.G., Lovitt, T.C., Eaton, M.D. & Hansen, C.L. (1978) The fourth R: research in the classroom. Columbus, OH: Charles E. Merrill Publishing Co

Appendix: Maths Intervention Training Opportunities

Devon Education Services – mathematics offers training courses for a range of maths interventions. The team of maths advisers can also provide training for From Counting to Calculating and two new, flexible sets of intervention materials, one focused on additive reasoning and one focused on multiplicative reasoning.

In line with current expectations, that interventions will most commonly be focused on ensuring pupils are helped to keep up, the materials provided in each of these programmes can be used as part of immediate intervention sessions during teaching sequences or can also be used to run a more intensive intervention programme to support children who are working significantly below age-related expectations.

From Counting to Calculating is a lighter-touch intervention programme for maths which was written by the maths team to support children who lie just above the bottom 5% in Y2 and for children in Y3 and Y4 who are working significantly behind age related expectations. It has also been used successfully with children significantly behind age-related expectations in Y5, Y6, Y7 and beyond. The programme is based on in-school action research on key understandings in mathematics and how to address misconceptions and gaps in understanding.

The programme can be run by a teaching assistant or a teacher and is intended for use with threes, pairs or one to one and the programme is aligned with the principles underpinning Numbers Count and the 2014 National Curriculum.

“Making a difference: developing additive reasoning” is a flexible set of intervention materials, written by the maths team, to support children with developing conceptual understanding of addition and subtraction. The materials build on the understanding developed in KS1 (and explored within From Counting to Calculating) and cover the expectations in the National Curriculum up to the end of Y4. Schools can choose to use the materials to run a programme to support children who are working significantly below age- related expectations in Y5, Y6, Y7 and beyond or use the materials as part of immediate intervention sessions during teaching sequences in Y3 and Y4.

“Fair and Equal: developing multiplicative reasoning” is a flexible set of intervention materials, written by the maths team, to support children with developing conceptual understanding of multiplication and division. The materials cover the key ideas related to multiplicative reasoning in the National Curriculum up to the end of Y4, with a few elements from Y5. Schools can choose to use the materials to run a programme to support children who are working significantly below age-related expectations in KS2, KS3 and beyond or could use the materials as part of immediate intervention sessions during teaching sequences in KS2.

Training in all of the intervention programmes is available through CPD online.

All the training can also be run for clusters of schools; for details of costings please contact Dr Ruth Trundley ruth.trundley@devon.gov.uk

Further information

Download a PDF version of this guide

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