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A Good Practice Guide for Teachers contains advice and guidance for school staff supporting such students. It applies to all 3 to 18-year olds with different maths challenges, including, among others, those with general learning difficulties, those with maths anxiety, those with “gaps” in their knowledge, and those with Dyscalculia. It is applicable to students from disadvantaged backgrounds, to minority groups, and to students for whom English is not the first language. We propose some possible causes for maths challenges, offer teaching tips and suggest intervention approaches grounded in current research findings.

1.
A Good Practice Guide for Teachers 2020

An tSeirbhís Náisiúnta Síceolaíochta Oideachais

National Educational Psychological Service (NEPS)

A Good Practice Guide

An tSeirbhís Náisiúnta Síceolaíochta Oideachais

National Educational Psychological Service (NEPS)

A Good Practice Guide

2.

3.
Contents

Introduction…………………………………………………………… 3

General Guidance ………...………………………………………… 6

Quick Guide to Challenges and Teaching Tips……………….. 7

Section A

Maths Anxiety ………………………………………………. 11

Assessment …………………………………………………. 16

Section B

Number Sense………………………………………………... 23

Mathematical Reasoning…………………………………… 26

Section C

Memory………………………………………………………… 33

Language……………………………………………………… 37

Sensory Processing………………………………………… 41

Executive Functioning Skills……………………………… 46

Teachings from Neuroscience......................................... 50

Section D

Current Teacher Supports.............................................. 54

Interventions and Initiatives……………………………… 57

References ………………………………………………………………………… 63

Appendix 1 List of Tests…………………………………………………….. 67

Appendix 2 Checklists………………………………………………………. 73

2

Introduction…………………………………………………………… 3

General Guidance ………...………………………………………… 6

Quick Guide to Challenges and Teaching Tips……………….. 7

Section A

Maths Anxiety ………………………………………………. 11

Assessment …………………………………………………. 16

Section B

Number Sense………………………………………………... 23

Mathematical Reasoning…………………………………… 26

Section C

Memory………………………………………………………… 33

Language……………………………………………………… 37

Sensory Processing………………………………………… 41

Executive Functioning Skills……………………………… 46

Teachings from Neuroscience......................................... 50

Section D

Current Teacher Supports.............................................. 54

Interventions and Initiatives……………………………… 57

References ………………………………………………………………………… 63

Appendix 1 List of Tests…………………………………………………….. 67

Appendix 2 Checklists………………………………………………………. 73

2

4.
In recent years, there have been concerns about improving numeracy standards in many

countries, including Ireland. Poor maths skills are associated with high dropout rates,

limited life success, increased risk of anti-social outcomes and economic costs

(Commission of the European Communities, 2011).

Alongside this concern about standards, there is growing understanding of what

constitutes quality maths teaching and learning. The teacher’s role has changed from

instructor, teaching rules and procedures and correcting related exercises, to facilitator of

understanding, mathematical thinking and reasoning abilities. Maths classrooms are

changing from silent, didactic, static environments to vibrant places of talking, listening,

reasoning and justifying.

Various developments have impacted positively on Irish maths standards. These include

curricular changes, professional development opportunities for teachers and new

initiatives emphasising both understanding and real-life application of knowledge and

skills. Although results in both international and national assessments (PISA 2015, NA14)

have improved, Ireland’s overall performance in international mathematics’ studies is

disappointing, especially when compared to our literacy performance. We continue to see

distinct areas of weakness - most notably problem-solving.

It takes a certain energy and planning by school management to prioritise resources and

interventions for students with maths challenges. Maths Support: A Good Practice Guide

for Teachers contains advice and guidance for school staff supporting such students. It

applies to all 3 to 18-year olds with different maths challenges, including, among others,

those with general learning difficulties, those with maths anxiety, those with “gaps” in their

knowledge, and those with Dyscalculia. It is applicable to students from disadvantaged

backgrounds, to minority groups and to students for whom English is not a first language.

We propose some possible causes for maths challenges, offer teaching tips and suggest

intervention approaches grounded in current research findings. We include links to

resources and professional development sites and to short video clips demonstrating good

3

countries, including Ireland. Poor maths skills are associated with high dropout rates,

limited life success, increased risk of anti-social outcomes and economic costs

(Commission of the European Communities, 2011).

Alongside this concern about standards, there is growing understanding of what

constitutes quality maths teaching and learning. The teacher’s role has changed from

instructor, teaching rules and procedures and correcting related exercises, to facilitator of

understanding, mathematical thinking and reasoning abilities. Maths classrooms are

changing from silent, didactic, static environments to vibrant places of talking, listening,

reasoning and justifying.

Various developments have impacted positively on Irish maths standards. These include

curricular changes, professional development opportunities for teachers and new

initiatives emphasising both understanding and real-life application of knowledge and

skills. Although results in both international and national assessments (PISA 2015, NA14)

have improved, Ireland’s overall performance in international mathematics’ studies is

disappointing, especially when compared to our literacy performance. We continue to see

distinct areas of weakness - most notably problem-solving.

It takes a certain energy and planning by school management to prioritise resources and

interventions for students with maths challenges. Maths Support: A Good Practice Guide

for Teachers contains advice and guidance for school staff supporting such students. It

applies to all 3 to 18-year olds with different maths challenges, including, among others,

those with general learning difficulties, those with maths anxiety, those with “gaps” in their

knowledge, and those with Dyscalculia. It is applicable to students from disadvantaged

backgrounds, to minority groups and to students for whom English is not a first language.

We propose some possible causes for maths challenges, offer teaching tips and suggest

intervention approaches grounded in current research findings. We include links to

resources and professional development sites and to short video clips demonstrating good

3

5.
We acknowledge with gratitude teachers’ contributions throughout Ireland, along with

helpful assistance from the SESS, NCCA, PDST, Colleges of Education and the

Inspectorate. All shared their experience and expertise generously. We are grateful to

Ciara de Loughry, Collette Murphy, Yvonne Mullan, Aideen Carey and Valerie Jones of

the NEPS Numeracy Group for producing this resource. Thanks also to Ray Mullan for

his many drawings.

Important Messages

Important messages are highlighted in yellow boxes

Teaching Tips

Teaching Tips are highlighted in peach boxes. If you do not want to read the

more detailed information, you can move quickly from box to box to pick up

teaching ideas in each section.

Links in the document appear in blue font.

4

helpful assistance from the SESS, NCCA, PDST, Colleges of Education and the

Inspectorate. All shared their experience and expertise generously. We are grateful to

Ciara de Loughry, Collette Murphy, Yvonne Mullan, Aideen Carey and Valerie Jones of

the NEPS Numeracy Group for producing this resource. Thanks also to Ray Mullan for

his many drawings.

Important Messages

Important messages are highlighted in yellow boxes

Teaching Tips

Teaching Tips are highlighted in peach boxes. If you do not want to read the

more detailed information, you can move quickly from box to box to pick up

teaching ideas in each section.

Links in the document appear in blue font.

4

6.

7.
General Guidance

for students with maths challenges

Increase teaching time for student

Reduce group/class size

Give individual support if necessary

Integrate strategies which support Read more in Sections B & C of this guide

cognitive processes

Get students to talk about decisions,

strategies & solutions to maths

problems I made the numbers smaller…

Provide peer-assisted support to

Use a multisensory teaching approach

Teach students to represent

information visually in a maths problem

Struggling students need short-cuts &

tricks as memory aids

6x7 -6 & 7 are sweatin’

on a bicycle made four two (42)

Be systematic & explicit Systematic means gradually building on proficiency by

introducing concepts in a logical order & by providing

students with many applications of each concept.

Explicit means providing clear models, an array of examples

& much practice when using new strategies & skills.

Adapted from Jayanthi, M., Gersten, R., Baker, S. (2008). Mathematics instruction for students with learning disabilities or difficulty learning mathematics:

A guide for teachers. Portsmouth, NH: RMC Research Corporation, Centre on Instruction.

6

for students with maths challenges

Increase teaching time for student

Reduce group/class size

Give individual support if necessary

Integrate strategies which support Read more in Sections B & C of this guide

cognitive processes

Get students to talk about decisions,

strategies & solutions to maths

problems I made the numbers smaller…

Provide peer-assisted support to

Use a multisensory teaching approach

Teach students to represent

information visually in a maths problem

Struggling students need short-cuts &

tricks as memory aids

6x7 -6 & 7 are sweatin’

on a bicycle made four two (42)

Be systematic & explicit Systematic means gradually building on proficiency by

introducing concepts in a logical order & by providing

students with many applications of each concept.

Explicit means providing clear models, an array of examples

& much practice when using new strategies & skills.

Adapted from Jayanthi, M., Gersten, R., Baker, S. (2008). Mathematics instruction for students with learning disabilities or difficulty learning mathematics:

A guide for teachers. Portsmouth, NH: RMC Research Corporation, Centre on Instruction.

6

8.
Quick Guide to Challenges and Teaching Tips

Challenges Teaching Tips

Anxiety Choose assessment methods & tests carefully

Avoid timed tests until confidence grows

Feeling apprehensive and tense Encourage maths talk and listening to others

Rarely volunteering answers Respect errors

Seeking regular reassurance Use process-oriented teaching (less reliance on facts &

Reluctance to start memorisation)

Making wild guesses Ensure consistent maths class routine

Avoiding maths Intervene early

Be aware of teacher anxiety

Revise basic facts regularly using creative methods

Use games & technology

Be positive & give plenty of positive feedback

Read Anxiety Section A

Number Sense Give meaningful practice with motivating materials

Do mental maths regularly

Limited intuitive understanding Use aids until student is fluent

of number Reduce emphasis on speed

Relies on recall of facts and Provide small doses - daily 10-minute sessions

procedures rather than on Encourage self-monitoring & listening to peers

understanding the underlying Teach commutative law or “turn arounds” (4+5 =5+4)

concepts Teach thinking strategies from one fact to another

Limited flexibility with number 5+5 then 5+ 6; 3+3 then 3+4

Difficulty recalling basic number Use number sense software

facts and formulae Encourage students to keep track of how many and which

Struggles with estimation facts are mastered

Stress the why of procedural maths as well as the how

Procedural Errors Identify error type or pattern

Develop self-monitoring & self-checking systems

Use acronyms & mnemonics

Inconsistent calculations Listen to students as they “think aloud”

Close-to-Correct answer

Misreading signs ÷ × + - ≤ ≥ … Read Number Sense Section B

Reasoning Link word problems to student interests

When presenting a problem move from real-life to abstract

Making sense of a problem Use concrete materials & hands-on learning approaches

Knowing where to begin Get students to visualise & draw maths problems

Translating a word problem into Leave space beside a problem to draw a picture of it

a maths “sum” Students create word problems (from number facts)

Figuring out what to do

Seeing that an answer does or Read Mathematical Reasoning Section B

does not make sense

Memory Build number fact fluency using motivating approaches

Use regular reminders to help listening

Remembering birthdays Use checklists notebooks & to-do lists to help stay on task

Remembering the page to open Use visual and other sensory aids

Remembering number facts, Teach memory aids e.g. mnemonics, rhymes, jingles

procedures and formulae Practise, revise, re-learn & rehearse

Remembering steps of a

Read Memory Section C

7

Challenges Teaching Tips

Anxiety Choose assessment methods & tests carefully

Avoid timed tests until confidence grows

Feeling apprehensive and tense Encourage maths talk and listening to others

Rarely volunteering answers Respect errors

Seeking regular reassurance Use process-oriented teaching (less reliance on facts &

Reluctance to start memorisation)

Making wild guesses Ensure consistent maths class routine

Avoiding maths Intervene early

Be aware of teacher anxiety

Revise basic facts regularly using creative methods

Use games & technology

Be positive & give plenty of positive feedback

Read Anxiety Section A

Number Sense Give meaningful practice with motivating materials

Do mental maths regularly

Limited intuitive understanding Use aids until student is fluent

of number Reduce emphasis on speed

Relies on recall of facts and Provide small doses - daily 10-minute sessions

procedures rather than on Encourage self-monitoring & listening to peers

understanding the underlying Teach commutative law or “turn arounds” (4+5 =5+4)

concepts Teach thinking strategies from one fact to another

Limited flexibility with number 5+5 then 5+ 6; 3+3 then 3+4

Difficulty recalling basic number Use number sense software

facts and formulae Encourage students to keep track of how many and which

Struggles with estimation facts are mastered

Stress the why of procedural maths as well as the how

Procedural Errors Identify error type or pattern

Develop self-monitoring & self-checking systems

Use acronyms & mnemonics

Inconsistent calculations Listen to students as they “think aloud”

Close-to-Correct answer

Misreading signs ÷ × + - ≤ ≥ … Read Number Sense Section B

Reasoning Link word problems to student interests

When presenting a problem move from real-life to abstract

Making sense of a problem Use concrete materials & hands-on learning approaches

Knowing where to begin Get students to visualise & draw maths problems

Translating a word problem into Leave space beside a problem to draw a picture of it

a maths “sum” Students create word problems (from number facts)

Figuring out what to do

Seeing that an answer does or Read Mathematical Reasoning Section B

does not make sense

Memory Build number fact fluency using motivating approaches

Use regular reminders to help listening

Remembering birthdays Use checklists notebooks & to-do lists to help stay on task

Remembering the page to open Use visual and other sensory aids

Remembering number facts, Teach memory aids e.g. mnemonics, rhymes, jingles

procedures and formulae Practise, revise, re-learn & rehearse

Remembering steps of a

Read Memory Section C

7

9.
Quick Guide to Challenges and Teaching Tips

Challenges Teaching Tips

Slow down & be clear & concise

Following verbal instructions Write instructions on the board & leave them there

Understanding language in word Read the word problems or instructions for the student

problems Replace words with simple images or simple vocabulary

Reading/ Understanding what is Model thinking aloud when problem-solving

read Chunk verbal information

Learning specialised terms Use visual planners (diagrams/mind-maps)

Communicating their reasoning Encourage use of a Maths Dictionary

Communicating their difficulties/

confusion Read Language Section B

Visual Processing

Give precise & clear verbal descriptions

Mentally rotating pictures Include multi-sensory experiences

Copying accurately Use verbal clues for verbally strong students

Reading signs Use boxes, circles & lines to break up visual information

Identifying right & left Use squared paper or unlined paper depending on

Sequencing student

Reading graphs, diagrams & Minimise or eliminate need to copy text from board

charts Colour code written descriptions of steps in maths

Filtering out background problems

Read Sensory Processing Section C

Auditory Processing

Reduce background noise at important listening times

Listens well but has difficulty Check to see if student has understood instructions

following instructions despite Use visual clues for visually strong students

adequate receptive language Provide written information on a page or board to

skills supplement or consolidate verbal instructions

Finds it difficult to filter out Use listening devices for students with severe auditory

background noise processing difficulties

Read Sensory Processing Section C

Executive Functioning

Encourage students to write down numbers in mental

skills maths

Use student check-lists & to-do lists

Getting started

Break long tasks into short quick sections

Staying engaged

Use memory cards & method cards

Remembering recent information

Use highlighters & underlining

& numbers

Present problems & solutions in a variety of ways

Thinking flexibly

Teach mnemonics, rhymes, jingles

Remind students regularly to listen, work or keep going

Getting organised

Encourage the use of visual aids

Keeping on-track when attempting

Encourage verbalisation & rechecking

multi-step problem

Teach students to question their solution

Read Executive Functioning Skills Section C

8

Challenges Teaching Tips

Slow down & be clear & concise

Following verbal instructions Write instructions on the board & leave them there

Understanding language in word Read the word problems or instructions for the student

problems Replace words with simple images or simple vocabulary

Reading/ Understanding what is Model thinking aloud when problem-solving

read Chunk verbal information

Learning specialised terms Use visual planners (diagrams/mind-maps)

Communicating their reasoning Encourage use of a Maths Dictionary

Communicating their difficulties/

confusion Read Language Section B

Visual Processing

Give precise & clear verbal descriptions

Mentally rotating pictures Include multi-sensory experiences

Copying accurately Use verbal clues for verbally strong students

Reading signs Use boxes, circles & lines to break up visual information

Identifying right & left Use squared paper or unlined paper depending on

Sequencing student

Reading graphs, diagrams & Minimise or eliminate need to copy text from board

charts Colour code written descriptions of steps in maths

Filtering out background problems

Read Sensory Processing Section C

Auditory Processing

Reduce background noise at important listening times

Listens well but has difficulty Check to see if student has understood instructions

following instructions despite Use visual clues for visually strong students

adequate receptive language Provide written information on a page or board to

skills supplement or consolidate verbal instructions

Finds it difficult to filter out Use listening devices for students with severe auditory

background noise processing difficulties

Read Sensory Processing Section C

Executive Functioning

Encourage students to write down numbers in mental

skills maths

Use student check-lists & to-do lists

Getting started

Break long tasks into short quick sections

Staying engaged

Use memory cards & method cards

Remembering recent information

Use highlighters & underlining

& numbers

Present problems & solutions in a variety of ways

Thinking flexibly

Teach mnemonics, rhymes, jingles

Remind students regularly to listen, work or keep going

Getting organised

Encourage the use of visual aids

Keeping on-track when attempting

Encourage verbalisation & rechecking

multi-step problem

Teach students to question their solution

Read Executive Functioning Skills Section C

8

10.

11.
Section A

Page

Maths Anxiety 11

Assessment 16

10

Page

Maths Anxiety 11

Assessment 16

10

12.
Maths Anxiety

• What is Maths Anxiety?

• Anxiety and Attainments

• Culture of Confidence

• Teaching Tips

What is Maths Anxiety?

Basic maths skills are a necessity for success in school and in everyday life, yet many

people experience apprehension and fear when dealing with numbers and

mathematical information. Most of us have met people who say they are no good at

maths or have always failed maths in school and who avoid maths-related tasks. An

understanding of maths anxiety may help you to support a student who struggles.

Maths anxiety is a feeling of tension,

apprehension and/or fear that interferes

with maths performance. It can present at

any stage from early years to adulthood

(Ashcraft 2002). The higher a student’s

maths anxiety, the lower their maths

learning, mastery and motivation.

It is understandable that students who are

Fig.1 Maths Anxiety

not competent in maths are likely to be

more anxious about maths. However, maths anxiety can come from other sources

too, such as home, school and classroom environments, where students pick up cues

from parents, teachers or peers that maths is stressful (Lyons & Beilock, 2012). When

parents are anxious about their own maths ability, they may pass on their own fears

subconsciously. They may not consider talking to children about the number of grams

in a kilogram when baking, or of asking questions such as “how many socks in three

pairs?” Some teachers have maths anxiety too and inadvertently pass it on to their

students (from pre-school onwards) through comments, behaviours and teaching

practices (Geist, 2015). Maths-anxious teachers can have lower achievement

11

• What is Maths Anxiety?

• Anxiety and Attainments

• Culture of Confidence

• Teaching Tips

What is Maths Anxiety?

Basic maths skills are a necessity for success in school and in everyday life, yet many

people experience apprehension and fear when dealing with numbers and

mathematical information. Most of us have met people who say they are no good at

maths or have always failed maths in school and who avoid maths-related tasks. An

understanding of maths anxiety may help you to support a student who struggles.

Maths anxiety is a feeling of tension,

apprehension and/or fear that interferes

with maths performance. It can present at

any stage from early years to adulthood

(Ashcraft 2002). The higher a student’s

maths anxiety, the lower their maths

learning, mastery and motivation.

It is understandable that students who are

Fig.1 Maths Anxiety

not competent in maths are likely to be

more anxious about maths. However, maths anxiety can come from other sources

too, such as home, school and classroom environments, where students pick up cues

from parents, teachers or peers that maths is stressful (Lyons & Beilock, 2012). When

parents are anxious about their own maths ability, they may pass on their own fears

subconsciously. They may not consider talking to children about the number of grams

in a kilogram when baking, or of asking questions such as “how many socks in three

pairs?” Some teachers have maths anxiety too and inadvertently pass it on to their

students (from pre-school onwards) through comments, behaviours and teaching

practices (Geist, 2015). Maths-anxious teachers can have lower achievement

11

13.
expectations for their students (Martinez, Martinez and Mizala, 2015). They can often

stick to traditional and rigid forms of teaching, overemphasising rote-learning and

spending less time attending to students’ questions (Bush, 1989).

Girls tend to lack the self-confidence in science and maths displayed by boys (OECD,

2015). Even from a fairly young age, girls tend to be less confident and more anxious

about maths. Moreover, these differences in confidence and anxiety are larger than

actual gender differences in maths achievement (Ganley & Lubienski, 2016).

Anxiety and Attainments

Maths anxiety and maths achievement are related. Anxiety can affect maths

performance by impacting on student motivation, avoidance and/or executive

functioning skills such as working memory.

“I am going to get this wrong

anyway, so I’ll just write anything.”

Students with maths anxiety tend to avoid situations involving maths. These students

may appear to daydream as they shut down during maths class. They may complain

of headaches or request frequent bathroom breaks. This avoidance leads to less

exposure to teaching and to practice, less competency in maths tasks and poorer

maths test performance.

maths

anxiety

less avoid

confidence maths

poor less

performance learning

Fig. 2 Anxiety Cycle

12

stick to traditional and rigid forms of teaching, overemphasising rote-learning and

spending less time attending to students’ questions (Bush, 1989).

Girls tend to lack the self-confidence in science and maths displayed by boys (OECD,

2015). Even from a fairly young age, girls tend to be less confident and more anxious

about maths. Moreover, these differences in confidence and anxiety are larger than

actual gender differences in maths achievement (Ganley & Lubienski, 2016).

Anxiety and Attainments

Maths anxiety and maths achievement are related. Anxiety can affect maths

performance by impacting on student motivation, avoidance and/or executive

functioning skills such as working memory.

“I am going to get this wrong

anyway, so I’ll just write anything.”

Students with maths anxiety tend to avoid situations involving maths. These students

may appear to daydream as they shut down during maths class. They may complain

of headaches or request frequent bathroom breaks. This avoidance leads to less

exposure to teaching and to practice, less competency in maths tasks and poorer

maths test performance.

maths

anxiety

less avoid

confidence maths

poor less

performance learning

Fig. 2 Anxiety Cycle

12

14.
Working Memory and other Executive Functions

Anxiety interferes with maths performance because it

robs people of working memory (Beilock, 2014).

Working memory is like a store that keeps several bits

of information in mind simultaneously, so that a person

can manipulate information to solve problems. Valuable

working memory space may be taken up by anxious

thoughts about failing, about not having enough time, or

about what peers may think. These thoughts may

diminish working memory stores available to devote to

Fig. 3 Anxiety Drain

the maths problem (Beilock and Willingham, 2014).

Maths anxiety also impacts on other executive functioning skills such as starting work,

organising yourself, attending, engaging, prioritising and thinking flexibly. It is hard to

think flexibly when you are anxious, difficult to generate new ideas when you are

stressed and actively demotivating to try to engage in a task when you believe that

your engagement is futile.

Create a Classroom Culture of Confidence

Students’ feelings about themselves and about their learning experiences may

influence their efforts and their success levels. The best confidence-building

mathematics environment is one in which:

Ω Mistakes are allowed. Students feel safe enough to suggest incorrect answers,

knowing that teachers value their ideas, efforts and processes.

Ω Academic, social and emotional skills are all fostered

Ω Adult-student and peer relationships are kind, caring and respectful

Ω Responsive adult feedback supports and extends student learning

Ω Students obtain support through peer and adult discussion

Ω Teacher-directed and student-directed activities are balanced

Ω Teachers provide sufficient explicit mathematical experiences for struggling

students

Ω Teachers do not project their own anxieties about maths (Geist 2015).

Adapted from NCCA 2016

13

Anxiety interferes with maths performance because it

robs people of working memory (Beilock, 2014).

Working memory is like a store that keeps several bits

of information in mind simultaneously, so that a person

can manipulate information to solve problems. Valuable

working memory space may be taken up by anxious

thoughts about failing, about not having enough time, or

about what peers may think. These thoughts may

diminish working memory stores available to devote to

Fig. 3 Anxiety Drain

the maths problem (Beilock and Willingham, 2014).

Maths anxiety also impacts on other executive functioning skills such as starting work,

organising yourself, attending, engaging, prioritising and thinking flexibly. It is hard to

think flexibly when you are anxious, difficult to generate new ideas when you are

stressed and actively demotivating to try to engage in a task when you believe that

your engagement is futile.

Create a Classroom Culture of Confidence

Students’ feelings about themselves and about their learning experiences may

influence their efforts and their success levels. The best confidence-building

mathematics environment is one in which:

Ω Mistakes are allowed. Students feel safe enough to suggest incorrect answers,

knowing that teachers value their ideas, efforts and processes.

Ω Academic, social and emotional skills are all fostered

Ω Adult-student and peer relationships are kind, caring and respectful

Ω Responsive adult feedback supports and extends student learning

Ω Students obtain support through peer and adult discussion

Ω Teacher-directed and student-directed activities are balanced

Ω Teachers provide sufficient explicit mathematical experiences for struggling

students

Ω Teachers do not project their own anxieties about maths (Geist 2015).

Adapted from NCCA 2016

13

15.
Teaching Tips Maths Anxiety

Ω Choose assessment strategies carefully (See Assessment Chapter).

Ω Tests, especially timed tests, are one of the main causes of maths anxiety.

Ω Identify at-risk students early on and use targeted interventions to prevent maths

anxiety from developing or escalating.

Ω Interventions may simply be more time or more support from a teacher.

Ω Provide consistent classroom routines in maths class.

Ω Teach within a student’s zone of proximal development (See below).

Ω Road maps can help to lessen anxiety as students work through problems in

a step-by-step manner.

Ω Pair students with allies who are kind and willing to help.

Ω Be aware that teachers’ feelings about maths can convey indirect messages to

students through teaching methods.

Ω Professional development and peer support can improve teacher skills and

confidence, leading to more classroom enthusiasm and increased positivity.

Ω Respect errors and sound reasoning.

Ω Parents encourage children’s basic maths skills through counting, weighing,

measuring and sharing. Click here for a handout for parents.

Ω Ensure that students know the basics before progressing to the next level.

Ω Revise basics regularly with struggling students.

Ω Link maths to real-life situations. Encourage estimation of price totals when shopping

or measuring skills during baking and woodwork.

Ω See Mathseyes for inspiration on how to make maths become real and meaningful.

Ω Encourage a belief (mindset) that talents and abilities are not fixed, but can be

developed. Read more here.

Ω Games can help students forget that they’re actually using maths strategies. Games

such as Yahtzee, Battleship, Dominoes and Connect Four demand simple mental

maths and problem-solving skills.

Ω Be cautious about using fast-paced number or spatial reasoning games.

Ω Remember to differentiate and choose games carefully for weaker students.

Ω Click here for a list of numeracy apps from the University of Edinburgh or here for a

list from UrAbility.

14

Ω Choose assessment strategies carefully (See Assessment Chapter).

Ω Tests, especially timed tests, are one of the main causes of maths anxiety.

Ω Identify at-risk students early on and use targeted interventions to prevent maths

anxiety from developing or escalating.

Ω Interventions may simply be more time or more support from a teacher.

Ω Provide consistent classroom routines in maths class.

Ω Teach within a student’s zone of proximal development (See below).

Ω Road maps can help to lessen anxiety as students work through problems in

a step-by-step manner.

Ω Pair students with allies who are kind and willing to help.

Ω Be aware that teachers’ feelings about maths can convey indirect messages to

students through teaching methods.

Ω Professional development and peer support can improve teacher skills and

confidence, leading to more classroom enthusiasm and increased positivity.

Ω Respect errors and sound reasoning.

Ω Parents encourage children’s basic maths skills through counting, weighing,

measuring and sharing. Click here for a handout for parents.

Ω Ensure that students know the basics before progressing to the next level.

Ω Revise basics regularly with struggling students.

Ω Link maths to real-life situations. Encourage estimation of price totals when shopping

or measuring skills during baking and woodwork.

Ω See Mathseyes for inspiration on how to make maths become real and meaningful.

Ω Encourage a belief (mindset) that talents and abilities are not fixed, but can be

developed. Read more here.

Ω Games can help students forget that they’re actually using maths strategies. Games

such as Yahtzee, Battleship, Dominoes and Connect Four demand simple mental

maths and problem-solving skills.

Ω Be cautious about using fast-paced number or spatial reasoning games.

Ω Remember to differentiate and choose games carefully for weaker students.

Ω Click here for a list of numeracy apps from the University of Edinburgh or here for a

list from UrAbility.

14

16.
Zone of Proximal Development (Vygotsky,1978)

The zone of proximal development is the distance between the actual level and the potential

developmental level of a student. It is the difference between what a student can do without

help and what they can achieve with guidance and encouragement from a more skilled

person. The “teacher” needs an understanding of what the student can achieve alone as

well as what they might achieve with help. Then, through guidance, activities, interaction and

questions, the student moves from being unable to do a task to being able to do it.

15

The zone of proximal development is the distance between the actual level and the potential

developmental level of a student. It is the difference between what a student can do without

help and what they can achieve with guidance and encouragement from a more skilled

person. The “teacher” needs an understanding of what the student can achieve alone as

well as what they might achieve with help. Then, through guidance, activities, interaction and

questions, the student moves from being unable to do a task to being able to do it.

15

17.
• What Needs to be Assessed?

• Purposes of Assessment

• Methods of Assessment

Assessment is more than the task or method used to collect data about students. It

includes the process of drawing inferences from the data collected and acting on those

judgements in effective ways (Callingham, 2010). Teachers need to collect,

document, reflect on and use evidence of students’ learning to inform their work and

provide appropriate learning experiences to ensure student progress.

What needs to be assessed?

A mixture of problems and challenges may lie

beneath a student’s mathematical challenges.

Factors such as educational opportunity, school

attendance, medical and physiological needs,

anxiety, quality of teaching and the match

between teaching style and individual learning

styles need consideration. Checklists,

questionnaires, parent-teacher and care-team

meetings can gather much of this information.

Checklists for basic needs, classroom-support

and school-support can be found in

Appendix 2 on page 69. Fig. 4 What needs to be assessed?

Other important factors contributing to maths learning, such as memory, language,

executive functioning and sensory processing are considered in Section C.

The frequency and types of assessment used in maths classes will be guided by

student need, teacher expertise, teacher preference and by the school assessment

16

• Purposes of Assessment

• Methods of Assessment

Assessment is more than the task or method used to collect data about students. It

includes the process of drawing inferences from the data collected and acting on those

judgements in effective ways (Callingham, 2010). Teachers need to collect,

document, reflect on and use evidence of students’ learning to inform their work and

provide appropriate learning experiences to ensure student progress.

What needs to be assessed?

A mixture of problems and challenges may lie

beneath a student’s mathematical challenges.

Factors such as educational opportunity, school

attendance, medical and physiological needs,

anxiety, quality of teaching and the match

between teaching style and individual learning

styles need consideration. Checklists,

questionnaires, parent-teacher and care-team

meetings can gather much of this information.

Checklists for basic needs, classroom-support

and school-support can be found in

Appendix 2 on page 69. Fig. 4 What needs to be assessed?

Other important factors contributing to maths learning, such as memory, language,

executive functioning and sensory processing are considered in Section C.

The frequency and types of assessment used in maths classes will be guided by

student need, teacher expertise, teacher preference and by the school assessment

16

18.
A school assessment plan should contain details about the following:

Ω When assessment occurs

Ω How assessment data is recorded (digitally? on paper? portfolios?)

Ω Assessment instruments that can be used

Ω Investment in professional development

Ω Aspects of maths that are assessed formally/informally

Ω How information is shared with parents and guardians

Ω How information is transferred (with permission) between schools

Click here for NCCA Primary and here for NCCA Junior Cycle reporting guidelines

Purposes of Assessment

When we assess, we generate data about a student. When the data generated is

used to report on student learning at a particular time, for example at the end of an

instructional unit or end of year, this is Assessment of Learning (AoL) or Summative

Assessment. It gives us data on attainment. When the data generated by the

assessment is used to inform teaching and learning this is Assessment for Learning

(AfL) or Formative Assessment. Williams (2015) suggests that assessments

themselves are neither formative nor summative. How assessment data is used, and

the type of inferences formed, make an assessment formative or summative.

The most common purposes of assessment are to:

Ω Identify students falling behind

Ω Find out those needing extra support

Ω Monitor a student’s progress over time

Ω Give feedback to parents or guardians

Ω Measure an intervention’s effectiveness

Ω Observe and analyse student errors to inform teaching

Ω Evaluate staff professional development needs

Ω Know where to allocate school resources

Ω Inform School Self-Evaluation

Ω Encourage students’ own self-evaluation

Ω Collect information about factors which may be influencing performance.

17

Ω When assessment occurs

Ω How assessment data is recorded (digitally? on paper? portfolios?)

Ω Assessment instruments that can be used

Ω Investment in professional development

Ω Aspects of maths that are assessed formally/informally

Ω How information is shared with parents and guardians

Ω How information is transferred (with permission) between schools

Click here for NCCA Primary and here for NCCA Junior Cycle reporting guidelines

Purposes of Assessment

When we assess, we generate data about a student. When the data generated is

used to report on student learning at a particular time, for example at the end of an

instructional unit or end of year, this is Assessment of Learning (AoL) or Summative

Assessment. It gives us data on attainment. When the data generated by the

assessment is used to inform teaching and learning this is Assessment for Learning

(AfL) or Formative Assessment. Williams (2015) suggests that assessments

themselves are neither formative nor summative. How assessment data is used, and

the type of inferences formed, make an assessment formative or summative.

The most common purposes of assessment are to:

Ω Identify students falling behind

Ω Find out those needing extra support

Ω Monitor a student’s progress over time

Ω Give feedback to parents or guardians

Ω Measure an intervention’s effectiveness

Ω Observe and analyse student errors to inform teaching

Ω Evaluate staff professional development needs

Ω Know where to allocate school resources

Ω Inform School Self-Evaluation

Ω Encourage students’ own self-evaluation

Ω Collect information about factors which may be influencing performance.

17

19.
Teacher Feedback to Students

Effective feedback from teachers is clear and precise.

It communicates (either verbally or in writing) which specific aspects of a task students

performed correctly/ incorrectly. This type of feedback is known as process-directed

as opposed to person-directed feedback. It is more effective when given during or

immediately after a task is completed.

Methods of Assessment

Screening is a process used to identify individuals needing

further evaluation and/or educational intervention. Screening

instruments are usually easy to administer to groups and can be

completed in a relatively brief time. They can include pencil-and-

paper tests, rating scales and checklists, or they may involve

direct observation of skills or abilities. Screening tests can be

standardised or more informal teacher-designed assessments of

student knowledge, skills or behaviour.

Fig. 5 Paper and pencil

test

Diagnostic Assessment is used to find out what exactly a

student knows, what they can and cannot do and where understanding is breaking

down. Teachers can diagnose specific areas of difficulty formally using diagnostic

tests, or informally using many of the methods outlined in this section. When not using

a test, teachers need to comprehend the skills involved in a task. They can then

analyse students’ errors and misunderstandings.

Response to Intervention (RTI) is an approach that allows the teacher to monitor

how a student responds to instruction and then modify their teaching accordingly. It is

a form of assessment for learning. It can impact significantly on learning when

properly employed in the classroom (Black and William, 1998a). RTI is a cycle of

quality teaching and assessment and modified instruction. Its primary objective is to

prevent problems by offering the most suitable teaching. Such assessment is typically

done through short, quick, classroom-based assessment by the class teacher. If a

student is not responding to instruction as expected, then instruction is differentiated

to take account of the student’s needs.

18

Effective feedback from teachers is clear and precise.

It communicates (either verbally or in writing) which specific aspects of a task students

performed correctly/ incorrectly. This type of feedback is known as process-directed

as opposed to person-directed feedback. It is more effective when given during or

immediately after a task is completed.

Methods of Assessment

Screening is a process used to identify individuals needing

further evaluation and/or educational intervention. Screening

instruments are usually easy to administer to groups and can be

completed in a relatively brief time. They can include pencil-and-

paper tests, rating scales and checklists, or they may involve

direct observation of skills or abilities. Screening tests can be

standardised or more informal teacher-designed assessments of

student knowledge, skills or behaviour.

Fig. 5 Paper and pencil

test

Diagnostic Assessment is used to find out what exactly a

student knows, what they can and cannot do and where understanding is breaking

down. Teachers can diagnose specific areas of difficulty formally using diagnostic

tests, or informally using many of the methods outlined in this section. When not using

a test, teachers need to comprehend the skills involved in a task. They can then

analyse students’ errors and misunderstandings.

Response to Intervention (RTI) is an approach that allows the teacher to monitor

how a student responds to instruction and then modify their teaching accordingly. It is

a form of assessment for learning. It can impact significantly on learning when

properly employed in the classroom (Black and William, 1998a). RTI is a cycle of

quality teaching and assessment and modified instruction. Its primary objective is to

prevent problems by offering the most suitable teaching. Such assessment is typically

done through short, quick, classroom-based assessment by the class teacher. If a

student is not responding to instruction as expected, then instruction is differentiated

to take account of the student’s needs.

18

20.
Observation provides qualitative information about types of errors and reasons for

errors. Teachers observe errors and misunderstandings in oral work, written work and

drawings. Observation can be done in a student’s presence or in their absence, but

the best insights into students’ errors come from listening to students as they reason

aloud. Observation can also help to determine if any factors such as anxiety or

physiological needs might be influencing performance.

Recording Observations

Ω Date all observations Ω Post-it notes Ω Rubrics Ω Class-List with a blank column

for recording observations Ω Objectives checklist Ω Dated samples of work in student

files Ω Students record the teacher’s feedback in journals/copies

Drawing: Some students may like to show what they know by creating a drawing or

diagram to demonstrate their understanding. They can be encouraged to share their

thinking about what they are drawing.

Conversing: Talking with a student and listening carefully can

inform a teacher about a student’s reasoning, understanding and

63

ability. It is easy to see what the error is in Figure 6, but clarity as to -29

why the error occurred will only become clear when you hear the

thinking behind it. The reason could be poor number sense, a well-

46

practised but incorrect procedure and/or an incorrect formulation

Fig. 6 What is the problem?

of words (e.g. 3 from 9 instead of 3 take away 9 or take 9 from 3).

The teaching fix for each of these possible causes may differ.

Interview: This is a slightly more formal discussion with a student, where target

questions are determined ahead of time, ensuring that information related to a goal or

learning outcome is obtained. As with all formative assessment methods, notes are

taken for later reference when planning instruction. Questioning can be open or

closed. How many degrees are there in a right angle? is an example of a closed

question. The expected answer is predetermined and specific. In contrast, open-

ended questions allow more than one correct response and elicit a different kind of

student thinking, e.g. Can you think of a few different ways to find the distance from

the school to the shop?

19

errors. Teachers observe errors and misunderstandings in oral work, written work and

drawings. Observation can be done in a student’s presence or in their absence, but

the best insights into students’ errors come from listening to students as they reason

aloud. Observation can also help to determine if any factors such as anxiety or

physiological needs might be influencing performance.

Recording Observations

Ω Date all observations Ω Post-it notes Ω Rubrics Ω Class-List with a blank column

for recording observations Ω Objectives checklist Ω Dated samples of work in student

files Ω Students record the teacher’s feedback in journals/copies

Drawing: Some students may like to show what they know by creating a drawing or

diagram to demonstrate their understanding. They can be encouraged to share their

thinking about what they are drawing.

Conversing: Talking with a student and listening carefully can

inform a teacher about a student’s reasoning, understanding and

63

ability. It is easy to see what the error is in Figure 6, but clarity as to -29

why the error occurred will only become clear when you hear the

thinking behind it. The reason could be poor number sense, a well-

46

practised but incorrect procedure and/or an incorrect formulation

Fig. 6 What is the problem?

of words (e.g. 3 from 9 instead of 3 take away 9 or take 9 from 3).

The teaching fix for each of these possible causes may differ.

Interview: This is a slightly more formal discussion with a student, where target

questions are determined ahead of time, ensuring that information related to a goal or

learning outcome is obtained. As with all formative assessment methods, notes are

taken for later reference when planning instruction. Questioning can be open or

closed. How many degrees are there in a right angle? is an example of a closed

question. The expected answer is predetermined and specific. In contrast, open-

ended questions allow more than one correct response and elicit a different kind of

student thinking, e.g. Can you think of a few different ways to find the distance from

the school to the shop?

19

21.
Portfolios are collections of work that show the progress made

by the learner over time. The evidence may be dated samples of

written work, completed teacher-composed tests, photos, video or

audio records, or any other appropriate indication of the learner’s

achievements.

Fig. 7 Maths Portfolio

Performance Tasks are assessment tasks

that require application of knowledge and skills, The Bike Shop

not just recall or recognition. They are open- You go to a shop that sells tricycles.

ended and there is typically not one single way of There are eighteen wheels in the shop.

doing the task. Often multiple steps are involved How many tricycles are in the shop?

and several learning outcomes can be assessed. How did you figure that out?

See an example of a performance task for US 2nd

Fig. 8 Performance Task

Grade in Fig. 8.

Find more performance tasks for various age ranges and topics here.

Classroom-Based Assessments (CBAs): These are performance tasks that have

recently been introduced as part of the Irish Junior Cycle Programme. They require

students to develop and demonstrate their knowledge and skills. An Assessment

Toolkit is provided to teachers for guidance in judging student attainment. Two CBAs

in Mathematics (one in 2nd Year and one in 3rd Year) are assessed as part of the Junior

Certificate Examination.

Reflective Journals: These are useful for both teachers and students to assess

thoughts, understandings, feelings and challenges. Students may need prompts to

start off the writing e.g.

Today, something new I learnt was____

It was easy/difficult for me

The tricky part was___

What I need to do is___

Teachers may collect journals periodically to discern a student’s performance progress

in terms of their knowledge, understanding, feelings and needs.

20

by the learner over time. The evidence may be dated samples of

written work, completed teacher-composed tests, photos, video or

audio records, or any other appropriate indication of the learner’s

achievements.

Fig. 7 Maths Portfolio

Performance Tasks are assessment tasks

that require application of knowledge and skills, The Bike Shop

not just recall or recognition. They are open- You go to a shop that sells tricycles.

ended and there is typically not one single way of There are eighteen wheels in the shop.

doing the task. Often multiple steps are involved How many tricycles are in the shop?

and several learning outcomes can be assessed. How did you figure that out?

See an example of a performance task for US 2nd

Fig. 8 Performance Task

Grade in Fig. 8.

Find more performance tasks for various age ranges and topics here.

Classroom-Based Assessments (CBAs): These are performance tasks that have

recently been introduced as part of the Irish Junior Cycle Programme. They require

students to develop and demonstrate their knowledge and skills. An Assessment

Toolkit is provided to teachers for guidance in judging student attainment. Two CBAs

in Mathematics (one in 2nd Year and one in 3rd Year) are assessed as part of the Junior

Certificate Examination.

Reflective Journals: These are useful for both teachers and students to assess

thoughts, understandings, feelings and challenges. Students may need prompts to

start off the writing e.g.

Today, something new I learnt was____

It was easy/difficult for me

The tricky part was___

What I need to do is___

Teachers may collect journals periodically to discern a student’s performance progress

in terms of their knowledge, understanding, feelings and needs.

20

22.
Self-Assessment: When students self-assess, they take some responsibility for their

own learning, using lists of objectives, checklists of steps or samples of completed

work. They can also be taught metacognitive strategies. Such strategies help students

to think about what they are doing, identify their problem-solving methods, evaluate

their understanding of mathematical processes and identify breakdowns in their

understanding. Self-assessment needs to become a routine part of what students do

during and after their learning.

Peer-Assessment: Peer-Assessment is the assessment by students of one another’s

work with reference to specific criteria. It involves more than inserting ticks or crosses,

or supplying the correct answers to each other. It is about commenting, getting ideas

from others, making suggestions and asking questions which lead to revision and

improvement of work. The process needs to be taught and students need opportunities

to practise it regularly in a supportive and safe classroom environment. Teachers

should negotiate and agree ground rules with the students. Click here to watch a

PDST clip about peer assessment in a secondary school. You may be inspired to use

something from this graphic design class in your maths class.

21

own learning, using lists of objectives, checklists of steps or samples of completed

work. They can also be taught metacognitive strategies. Such strategies help students

to think about what they are doing, identify their problem-solving methods, evaluate

their understanding of mathematical processes and identify breakdowns in their

understanding. Self-assessment needs to become a routine part of what students do

during and after their learning.

Peer-Assessment: Peer-Assessment is the assessment by students of one another’s

work with reference to specific criteria. It involves more than inserting ticks or crosses,

or supplying the correct answers to each other. It is about commenting, getting ideas

from others, making suggestions and asking questions which lead to revision and

improvement of work. The process needs to be taught and students need opportunities

to practise it regularly in a supportive and safe classroom environment. Teachers

should negotiate and agree ground rules with the students. Click here to watch a

PDST clip about peer assessment in a secondary school. You may be inspired to use

something from this graphic design class in your maths class.

21

23.
Section B

Page

Number Sense……………….. 23

Mathematical Reasoning……. 26

22

Page

Number Sense……………….. 23

Mathematical Reasoning……. 26

22

24.
Number Sense

• What is Number Sense?

• Build Number Sense Slowly

• Teaching Tips

What is Number Sense?

Number Sense is an intuitive sense or a “feel” for

numbers. Bobis (1996) describes it as a

competency with numbers that is based on

understanding rather than on memorisation or

recall of facts and procedures. Students with

good number sense are fluent and flexible with

numbers. They understand the magnitude of

numbers, how they relate to each other and the

effects of operations on them. They spot Fig. 9 Number Sense

unreasonable answers and estimate well, see

connections between operations like addition and subtraction, and multiplication

and division. They understand how numbers can be taken apart and put together

again in different ways. For example, to add 39+51, they might quickly add one

to 39, subtract 1 from 51 and add 40+50 to get the answer 90. Students who

have limited number sense have trouble developing the foundations needed for

simple arithmetic and for more complicated number work such as fractions and

algebra (Burns, 2007). Many Dyscalculia definitions include this lack of intuitive

number sense (Emerson & Babtie, 2013, Butterworth, Sashank & Laurillard,

2011). Regardless of a Dyscalculia diagnosis, if your observations and

assessment indicate that a student’s difficulty is with number sense, try some of

this section’s teaching tips.

Build Number Sense Slowly

Developing number sense takes time. It begins through early experiences when

children meet numbers in various contexts and relate to numbers in different ways.

From concrete experiences (grouping, matching, counting, composing and

23

• What is Number Sense?

• Build Number Sense Slowly

• Teaching Tips

What is Number Sense?

Number Sense is an intuitive sense or a “feel” for

numbers. Bobis (1996) describes it as a

competency with numbers that is based on

understanding rather than on memorisation or

recall of facts and procedures. Students with

good number sense are fluent and flexible with

numbers. They understand the magnitude of

numbers, how they relate to each other and the

effects of operations on them. They spot Fig. 9 Number Sense

unreasonable answers and estimate well, see

connections between operations like addition and subtraction, and multiplication

and division. They understand how numbers can be taken apart and put together

again in different ways. For example, to add 39+51, they might quickly add one

to 39, subtract 1 from 51 and add 40+50 to get the answer 90. Students who

have limited number sense have trouble developing the foundations needed for

simple arithmetic and for more complicated number work such as fractions and

algebra (Burns, 2007). Many Dyscalculia definitions include this lack of intuitive

number sense (Emerson & Babtie, 2013, Butterworth, Sashank & Laurillard,

2011). Regardless of a Dyscalculia diagnosis, if your observations and

assessment indicate that a student’s difficulty is with number sense, try some of

this section’s teaching tips.

Build Number Sense Slowly

Developing number sense takes time. It begins through early experiences when

children meet numbers in various contexts and relate to numbers in different ways.

From concrete experiences (grouping, matching, counting, composing and

23

25.
decomposing) and talking about these experiences, students build foundations for

computation, problem solving and reasoning. Baroody, Bajawa & Eiland (2009)

suggest that we move progressively from counting to reasoning strategies and then to

automaticity when working out number facts. Automaticity means that you have

performed a calculation (e.g. 3+4 = 7) so often that it becomes automatic. Similar to

knowing sight words when reading, knowing number facts frees up your mind to

consider other aspects of maths questions. However, going too quickly from counting

to automaticity can hamper reasoning strategy development. Too much speed here

may result in students memorising number facts in the short-term but regressing to

basic counting when they cannot recall facts in the long-term (Baroody, 2006, Henry

& Brown, 2008).

The value of memorising tables (of number facts) is debatable. Some research

(Ashcraft, 2002, Boaler, 2014, Ramirez et al., 2013) suggests that timed tests and

emphasising fact memorisation can cause maths anxiety. Others (Stripp, 2015)

believe that lack of number fact knowledge causes maths anxiety. Practice really does

help students to recall number facts, and practice improves fluency by activating and

strengthening neural networks (Aubin, Voelker and Eliasmith, 2016). We recommend

avoiding bland number fact memorisation and supporting students in building

reasoning and automaticity through slowing down and through making practice fun.

Watch this inspirational short video clip.

The relationship between conceptual understanding and procedural knowledge is

another hotly-debated topic. Rittle-Johnson, Schneider, & Star (2015) propose a

bidirectional relationship between the two. Others (Wright, Martland, Stafford and

Stranger, 2012) advocate that learning about algorithms (formal written procedures

e.g. short or long multiplication or division procedures) should be delayed until

students first develop their own informal strategies for combining and dividing

Using age and ability appropriate aids such as 100 or

multiplication squares, number lines, along with digital aids such

as calculators and maths applications can increase students’

access to number facts, number sense and confidence.

Fig.10 Calculator

24

computation, problem solving and reasoning. Baroody, Bajawa & Eiland (2009)

suggest that we move progressively from counting to reasoning strategies and then to

automaticity when working out number facts. Automaticity means that you have

performed a calculation (e.g. 3+4 = 7) so often that it becomes automatic. Similar to

knowing sight words when reading, knowing number facts frees up your mind to

consider other aspects of maths questions. However, going too quickly from counting

to automaticity can hamper reasoning strategy development. Too much speed here

may result in students memorising number facts in the short-term but regressing to

basic counting when they cannot recall facts in the long-term (Baroody, 2006, Henry

& Brown, 2008).

The value of memorising tables (of number facts) is debatable. Some research

(Ashcraft, 2002, Boaler, 2014, Ramirez et al., 2013) suggests that timed tests and

emphasising fact memorisation can cause maths anxiety. Others (Stripp, 2015)

believe that lack of number fact knowledge causes maths anxiety. Practice really does

help students to recall number facts, and practice improves fluency by activating and

strengthening neural networks (Aubin, Voelker and Eliasmith, 2016). We recommend

avoiding bland number fact memorisation and supporting students in building

reasoning and automaticity through slowing down and through making practice fun.

Watch this inspirational short video clip.

The relationship between conceptual understanding and procedural knowledge is

another hotly-debated topic. Rittle-Johnson, Schneider, & Star (2015) propose a

bidirectional relationship between the two. Others (Wright, Martland, Stafford and

Stranger, 2012) advocate that learning about algorithms (formal written procedures

e.g. short or long multiplication or division procedures) should be delayed until

students first develop their own informal strategies for combining and dividing

Using age and ability appropriate aids such as 100 or

multiplication squares, number lines, along with digital aids such

as calculators and maths applications can increase students’

access to number facts, number sense and confidence.

Fig.10 Calculator

24

26.
Teaching Tips Number Sense

Ω Slow down! Some students become anxious by requests to answer quickly.

Ω Read PDST advice about teaching number sense here

Ω Find exactly where a student is in terms of number in First Steps Diagnostic Map

Ω Read really good practical advice for teaching students with Dyscalculia here

Ω Practise mental maths regularly; mental maths builds knowledge about numbers

and numerical relationships

Ω Number Talks (classroom conversations around purposefully-crafted computation

problems - Parrish 2014) are a powerful strategy for developing number sense.

Ω Encourage students to listen to peers when they talk about computation strategies.

Ω Encourage students to explain their thinking/reasoning.

Ω Listen carefully to students’ reasoning for formative assessment purposes.

Ω Make estimation an integral part of computing. Real-life maths relies not only on

mental maths but on estimation e.g. deciding when to leave for school, how much

paint to buy, or which queue to join at the supermarket.

Ω Include maths facts practice without time pressure.

Ω Use technology to teach number sense. Click here to see First Class using iPads,

here for UrAbility apps and here for software suggestions from PDST.

Ω Maths Recovery, Ready, Set, Go – Maths and Number Worlds are researched-

based interventions for developing number sense in young children.

Ω Counting is important in the development of number sense. Count up, count from

left to right and in a clockwise direction with young children. Count backwards too.

Ω Young children enjoy using a large sponge dice to decide the number of jumps to

take on a large number-ladder (picture) on the floor.

Ω Read Teaching Number in the Classroom with 4-8 Year Olds by Wright, Martland,

Stafford and Stranger (2015).

Ω Move from Concrete to Pictorial to Abstract (CPA) when developing new concepts

and skills; the rate of progression from one stage to the next will vary based on the

needs of individual students.

25

Ω Slow down! Some students become anxious by requests to answer quickly.

Ω Read PDST advice about teaching number sense here

Ω Find exactly where a student is in terms of number in First Steps Diagnostic Map

Ω Read really good practical advice for teaching students with Dyscalculia here

Ω Practise mental maths regularly; mental maths builds knowledge about numbers

and numerical relationships

Ω Number Talks (classroom conversations around purposefully-crafted computation

problems - Parrish 2014) are a powerful strategy for developing number sense.

Ω Encourage students to listen to peers when they talk about computation strategies.

Ω Encourage students to explain their thinking/reasoning.

Ω Listen carefully to students’ reasoning for formative assessment purposes.

Ω Make estimation an integral part of computing. Real-life maths relies not only on

mental maths but on estimation e.g. deciding when to leave for school, how much

paint to buy, or which queue to join at the supermarket.

Ω Include maths facts practice without time pressure.

Ω Use technology to teach number sense. Click here to see First Class using iPads,

here for UrAbility apps and here for software suggestions from PDST.

Ω Maths Recovery, Ready, Set, Go – Maths and Number Worlds are researched-

based interventions for developing number sense in young children.

Ω Counting is important in the development of number sense. Count up, count from

left to right and in a clockwise direction with young children. Count backwards too.

Ω Young children enjoy using a large sponge dice to decide the number of jumps to

take on a large number-ladder (picture) on the floor.

Ω Read Teaching Number in the Classroom with 4-8 Year Olds by Wright, Martland,

Stafford and Stranger (2015).

Ω Move from Concrete to Pictorial to Abstract (CPA) when developing new concepts

and skills; the rate of progression from one stage to the next will vary based on the

needs of individual students.

25

27.
Mathematical Reasoning

• What is Mathematical Reasoning?

• Representation and Reasoning

• Developing Reasoning

• Teaching Tips

What is Mathematical Reasoning?

Reasoning means thinking about something and making sense of it in order to draw

conclusions or make choices or judgements. In maths, this usually involves thinking

critically about situations, words, shapes or quantities and then analysing, interpreting

and evaluating. It involves using relevant prior knowledge, developing solutions and

judging the solutions’ accuracy. Sometimes considerable mental work is involved in

the elaborations and judgements required. Creativity, imagination, memory,

confidence, perseverance and ability to justify your thinking are all essential

components of the process.

Is this shape a square? In this question, reasoning involves recall

of prior knowledge of the term square and a square’s characteristics.

Ideally, a student has acquired this knowledge following lots of hands-

on activities as a child, involving comparing and contrasting, searching for patterns,

making generalizations, testing, validating and justifying conclusions.

Two friends started walking from the house. The boy walked 3k one way.

The girl walked 5k the other way. How far apart were the two friends?

In this question, reasoning involves

1) understanding verbal information and then

2) figuring out how to calculate the distance between

the two children. The problem’s visual representation

given to students in this research study helped them to

work out the answer.

Fig. 11 An item from the Bryant 2009 study

26

• What is Mathematical Reasoning?

• Representation and Reasoning

• Developing Reasoning

• Teaching Tips

What is Mathematical Reasoning?

Reasoning means thinking about something and making sense of it in order to draw

conclusions or make choices or judgements. In maths, this usually involves thinking

critically about situations, words, shapes or quantities and then analysing, interpreting

and evaluating. It involves using relevant prior knowledge, developing solutions and

judging the solutions’ accuracy. Sometimes considerable mental work is involved in

the elaborations and judgements required. Creativity, imagination, memory,

confidence, perseverance and ability to justify your thinking are all essential

components of the process.

Is this shape a square? In this question, reasoning involves recall

of prior knowledge of the term square and a square’s characteristics.

Ideally, a student has acquired this knowledge following lots of hands-

on activities as a child, involving comparing and contrasting, searching for patterns,

making generalizations, testing, validating and justifying conclusions.

Two friends started walking from the house. The boy walked 3k one way.

The girl walked 5k the other way. How far apart were the two friends?

In this question, reasoning involves

1) understanding verbal information and then

2) figuring out how to calculate the distance between

the two children. The problem’s visual representation

given to students in this research study helped them to

work out the answer.

Fig. 11 An item from the Bryant 2009 study

26

28.
Egg-boxes Minibuses

Eggs are sold in boxes of six. 46 children are going to a hurling match

I have 45 eggs A minibus can take 12 children. How

How many boxes can I sell? many minibuses will they need?

In the questions above, reasoning again involves making sense of verbal

information. The student must calculate how many boxes are needed for the eggs

and how many minibuses are required for the match (quantitative reasoning).

Meaningful answers will not result from merely dividing 45 by 6, and 46 by 12.

Common-sense is required also. Students must remember that eggs are not sold

in half-boxes and you cannot hire a fraction of a bus. Activities to support weaker

students include chatting with peers and drawing pictures.

What is the value of x if 8 - 2x = 4? This question’s reasoning is more

abstract than previous ones because the question contains both letters and numbers.

Do you get the answer by trying different numbers in place of the x? Or do you get

the answer by moving all numbers to one side of the equals (=) sign and all letters to

the other side? If so, do you know why you do this? This type of reasoning becomes

easier for some students when they have had previous practice of real-world

Link between reasoning and how problems are (re)presented

The way in which a problem is represented or presented can affect a student’s

understanding of it. Real-life and concrete representations of problems support

students to develop more abstract reasoning skills.

The waiter question on the next page comes from a Danish study

in which 70% of 1st Year Algebra students worked out the answer

to the informal real-world problem representation while only 42%

solved the formal algebraic equation (Koedinger and Nathan,

2004). It is not necessary to replace formal with informal

representations, but it is helpful to build on informal processes to

support students’ progress to formal algebraic symbol

manipulation (Koedinger and Anderson, 1998). Fig. 12 Real World Representation

27

Eggs are sold in boxes of six. 46 children are going to a hurling match

I have 45 eggs A minibus can take 12 children. How

How many boxes can I sell? many minibuses will they need?

In the questions above, reasoning again involves making sense of verbal

information. The student must calculate how many boxes are needed for the eggs

and how many minibuses are required for the match (quantitative reasoning).

Meaningful answers will not result from merely dividing 45 by 6, and 46 by 12.

Common-sense is required also. Students must remember that eggs are not sold

in half-boxes and you cannot hire a fraction of a bus. Activities to support weaker

students include chatting with peers and drawing pictures.

What is the value of x if 8 - 2x = 4? This question’s reasoning is more

abstract than previous ones because the question contains both letters and numbers.

Do you get the answer by trying different numbers in place of the x? Or do you get

the answer by moving all numbers to one side of the equals (=) sign and all letters to

the other side? If so, do you know why you do this? This type of reasoning becomes

easier for some students when they have had previous practice of real-world

Link between reasoning and how problems are (re)presented

The way in which a problem is represented or presented can affect a student’s

understanding of it. Real-life and concrete representations of problems support

students to develop more abstract reasoning skills.

The waiter question on the next page comes from a Danish study

in which 70% of 1st Year Algebra students worked out the answer

to the informal real-world problem representation while only 42%

solved the formal algebraic equation (Koedinger and Nathan,

2004). It is not necessary to replace formal with informal

representations, but it is helpful to build on informal processes to

support students’ progress to formal algebraic symbol

manipulation (Koedinger and Anderson, 1998). Fig. 12 Real World Representation

27

29.
Informal Sean gets €6 per hour as a waiter. One night he

Concrete made €66 in tips and earned a total of €81.90.

How many hours did Sean work?

Real-world

Abstract 6x + 66 = 81.90

x=?

The Dutch Iceberg Model in Fig. 13 was developed for teachers. Here, the iceberg’s

tip represents the symbol for three quarters ¾. The model’s message is that informal

and context-bound representations and experiences (coins, apple sections etc.) and

pre-formal representations (e.g. number

line) of the ¾ concept are necessary

before students can fully understand the

formal mathematical representation of ¾.

Too much teaching invested into the

iceberg’s top (formal mathematics, sums)

may be at the expense of developing

greater insight into and understanding of

concepts and skills.

Fig. 13 Tip of the Iceberg

Developing Reasoning Skills

Ensuring that reasoning skills are developed

and supported requires active students

(Anderson, Reder & Simon, 1996).

Experiences such as hands-on learning,

discussion, projects and teamwork are more

likely to produce lasting skills and deep

understanding than passive activities such as

memorisation, drill and templates. Fig. 14 Talk and Teamwork

28

Concrete made €66 in tips and earned a total of €81.90.

How many hours did Sean work?

Real-world

Abstract 6x + 66 = 81.90

x=?

The Dutch Iceberg Model in Fig. 13 was developed for teachers. Here, the iceberg’s

tip represents the symbol for three quarters ¾. The model’s message is that informal

and context-bound representations and experiences (coins, apple sections etc.) and

pre-formal representations (e.g. number

line) of the ¾ concept are necessary

before students can fully understand the

formal mathematical representation of ¾.

Too much teaching invested into the

iceberg’s top (formal mathematics, sums)

may be at the expense of developing

greater insight into and understanding of

concepts and skills.

Fig. 13 Tip of the Iceberg

Developing Reasoning Skills

Ensuring that reasoning skills are developed

and supported requires active students

(Anderson, Reder & Simon, 1996).

Experiences such as hands-on learning,

discussion, projects and teamwork are more

likely to produce lasting skills and deep

understanding than passive activities such as

memorisation, drill and templates. Fig. 14 Talk and Teamwork

28

30.
In Nunes, Bryant, Sylva and Barrow’s (2009) study, mathematical reasoning was

found to be a far stronger predictor of maths achievement than calculation skills. The

researchers recommended that time be devoted to teaching reasoning skills in maths

classes. Often in classrooms, the focus is on teaching students how to do maths and

not on understanding what they do. This focus may occur because of mandated tests

that emphasise calculations, pressure to ensure that students master the basics or

because of teachers’ perception of their own ability or expertise.

Improving student reasoning skills can reduce the anxiety often experienced in the

mathematics classroom. Anxiety is lessened when individuals can control

uncertainties. “When self-constructed reasoning under the control of the individual

takes over, much valid mathematical reasoning may emerge” (Druckman and Bjork

Finally, an important part of reasoning is

learning to communicate it in a succinct

First…

mathematical way. When students explain or

justify their reasoning, they solidify their own Then…

understanding. Time is required to help But…

students develop their language skills so that So…

they can describe clearly their own chain of

reasoning/ the sequence of steps they have

Fig. 15 Real World Maths from Mathseyes

29

found to be a far stronger predictor of maths achievement than calculation skills. The

researchers recommended that time be devoted to teaching reasoning skills in maths

classes. Often in classrooms, the focus is on teaching students how to do maths and

not on understanding what they do. This focus may occur because of mandated tests

that emphasise calculations, pressure to ensure that students master the basics or

because of teachers’ perception of their own ability or expertise.

Improving student reasoning skills can reduce the anxiety often experienced in the

mathematics classroom. Anxiety is lessened when individuals can control

uncertainties. “When self-constructed reasoning under the control of the individual

takes over, much valid mathematical reasoning may emerge” (Druckman and Bjork

Finally, an important part of reasoning is

learning to communicate it in a succinct

First…

mathematical way. When students explain or

justify their reasoning, they solidify their own Then…

understanding. Time is required to help But…

students develop their language skills so that So…

they can describe clearly their own chain of

reasoning/ the sequence of steps they have

Fig. 15 Real World Maths from Mathseyes

29

31.
Teaching Tips Mathematical Reasoning

Ω Use real-life problems and non-routine problems to develop reasoning skills.

Ω Give students opportunities to talk to their peers in small groups.

Ω Encourage students to create problems collectively or individually.

Ω Click here for ideas from Mathseyes. This is a website from Tallaght Institute of

Technology, which encourages people to develop maths eyes and spot the use of

maths in everyday life.

Ω Include tasks which provide opportunities for analysing, evaluating, explaining,

inferring, generalising, testing, validating, justifying and responding to others’

arguments.

Ω Encourage students to use manipulative resources (post-it notes, concrete materials),

pictorial representations and tables to represent the problem, investigate solutions,

demonstrate understanding and justify thinking.

Ω Read more about representation to support student reasoning here and here.

Ω Find ideas to help students visualise and to draw maths problems here.

Ω Establish a classroom culture where discussion is valued and where hypotheses and

conjectures can be suggested in a non-threatening way.

Ω Use prompts and probing questions e.g.

What can you work out now?

If you know that …what else do you know?

Why is that bit important?

If……., then……?

Ω Encourage students to talk to peers and teachers when they get stuck.

Ω Teach problem-solving strategies e.g. summarising, finding relevant data, ignoring

irrelevant data, searching for clues, working backwards and trial and error (“trial and

improvement”).

Ω Model clear, succinct, logical communication of thought processes.

Ω Help students construct their argument by providing and displaying sentence starters:

I think this because…

If this is true, then…

This can’t work because…

30

Ω Use real-life problems and non-routine problems to develop reasoning skills.

Ω Give students opportunities to talk to their peers in small groups.

Ω Encourage students to create problems collectively or individually.

Ω Click here for ideas from Mathseyes. This is a website from Tallaght Institute of

Technology, which encourages people to develop maths eyes and spot the use of

maths in everyday life.

Ω Include tasks which provide opportunities for analysing, evaluating, explaining,

inferring, generalising, testing, validating, justifying and responding to others’

arguments.

Ω Encourage students to use manipulative resources (post-it notes, concrete materials),

pictorial representations and tables to represent the problem, investigate solutions,

demonstrate understanding and justify thinking.

Ω Read more about representation to support student reasoning here and here.

Ω Find ideas to help students visualise and to draw maths problems here.

Ω Establish a classroom culture where discussion is valued and where hypotheses and

conjectures can be suggested in a non-threatening way.

Ω Use prompts and probing questions e.g.

What can you work out now?

If you know that …what else do you know?

Why is that bit important?

If……., then……?

Ω Encourage students to talk to peers and teachers when they get stuck.

Ω Teach problem-solving strategies e.g. summarising, finding relevant data, ignoring

irrelevant data, searching for clues, working backwards and trial and error (“trial and

improvement”).

Ω Model clear, succinct, logical communication of thought processes.

Ω Help students construct their argument by providing and displaying sentence starters:

I think this because…

If this is true, then…

This can’t work because…

30

32.

33.
Section C

Page

Memory………………………………. 33

Language…………………................ 37

Sensory Processing………………… 41

Executive Functioning Skills……….. 46

Neuroscience………………………… 49

32

Page

Memory………………………………. 33

Language…………………................ 37

Sensory Processing………………… 41

Executive Functioning Skills……….. 46

Neuroscience………………………… 49

32

34.
Memory

• Long-Term Memory

• Short-Term Memory

• Working Memory

• Teaching Tips

Different aspects of memory play important roles in understanding and learning maths.

When you have basic number facts stored in memory, for example, it allows you to

spend less time making simple calculations and more

time reasoning about a problem.

One morning, there were 7 black cows in a field.

In the afternoon, 2 cows left the field. Then

Eh …how many cows?

3 brown cows came into the field. How many

cows were in the field then?

Long-Term Memory

Long-term memory refers to storage of information over an extended period. You can

usually remember significant events such as a goal in a soccer game or a great

concert, with much greater clarity and detail than you can recall less memorable

events. Memories that you access frequently become much stronger and more easily

recalled. As you access them, you strengthen the pathways where the information is

encoded (Aubin, Voelker and Eliasmith, 2016). Memories that are not recalled often

can weaken or be lost. It is considered that birthdays and number facts are stored in

long-term memory.

Short-Term Memory

Short-term memory refers to the temporary storage of visual and auditory information

for immediate retrieval or discard (Baddeley and Hitch, 1974). The amount of

information that you can capture, store, process and recall in short-term memory is

limited. Miller (1956) suggested that we can keep 7 ± 2 items in short-term memory.

33

• Long-Term Memory

• Short-Term Memory

• Working Memory

• Teaching Tips

Different aspects of memory play important roles in understanding and learning maths.

When you have basic number facts stored in memory, for example, it allows you to

spend less time making simple calculations and more

time reasoning about a problem.

One morning, there were 7 black cows in a field.

In the afternoon, 2 cows left the field. Then

Eh …how many cows?

3 brown cows came into the field. How many

cows were in the field then?

Long-Term Memory

Long-term memory refers to storage of information over an extended period. You can

usually remember significant events such as a goal in a soccer game or a great

concert, with much greater clarity and detail than you can recall less memorable

events. Memories that you access frequently become much stronger and more easily

recalled. As you access them, you strengthen the pathways where the information is

encoded (Aubin, Voelker and Eliasmith, 2016). Memories that are not recalled often

can weaken or be lost. It is considered that birthdays and number facts are stored in

long-term memory.

Short-Term Memory

Short-term memory refers to the temporary storage of visual and auditory information

for immediate retrieval or discard (Baddeley and Hitch, 1974). The amount of

information that you can capture, store, process and recall in short-term memory is

limited. Miller (1956) suggested that we can keep 7 ± 2 items in short-term memory.

33

35.
Cowan (2005) provided evidence that a more realistic figure is 4 ± 1 items. Students

with weak short-term memory may have a much lower capacity than 4 ± 1. They may

experience difficulties recalling and sequencing in multi-step maths tasks.

Fig. 16 Working Memory

Working Memory

Working Memory refers to the combination of storage and manipulation of visual and

auditory information. Working memory is necessary for staying focused, blocking out

distractions and completing tasks. A student’s working memory in a maths class may

need to deal with teacher instructions, distracting sounds and sights and/or temporary

memories from long-term storage. Working memory uses information that is stored in

both long-term and short-term memory.

Munro (2011) outlines teaching procedures to help students with mathematics learning

difficulties to encode and manipulate their knowledge in working memory. Teachers

should firstly stimulate explicitly students’ current knowledge about a new task i.e.

remind them of known concepts, procedures, symbols and factual knowledge they will

need to use (Munro, 2011).

Processing and Storing Information

Many factors influence how we process and store information, such as:

Ω Information complexity or volume

Ω Information connection to long-term memories

Ω Number of senses used in processing information

Ω Emotional connections to the information.

34

with weak short-term memory may have a much lower capacity than 4 ± 1. They may

experience difficulties recalling and sequencing in multi-step maths tasks.

Fig. 16 Working Memory

Working Memory

Working Memory refers to the combination of storage and manipulation of visual and

auditory information. Working memory is necessary for staying focused, blocking out

distractions and completing tasks. A student’s working memory in a maths class may

need to deal with teacher instructions, distracting sounds and sights and/or temporary

memories from long-term storage. Working memory uses information that is stored in

both long-term and short-term memory.

Munro (2011) outlines teaching procedures to help students with mathematics learning

difficulties to encode and manipulate their knowledge in working memory. Teachers

should firstly stimulate explicitly students’ current knowledge about a new task i.e.

remind them of known concepts, procedures, symbols and factual knowledge they will

need to use (Munro, 2011).

Processing and Storing Information

Many factors influence how we process and store information, such as:

Ω Information complexity or volume

Ω Information connection to long-term memories

Ω Number of senses used in processing information

Ω Emotional connections to the information.

34

36.
Teaching Tips Memory

Ω Build number fact fluency. See page 24 for suggestions and ideas.

Ω Reduce the load of number facts. When students understand the commutative

law, for example, the number of facts to be remembered is halved 8x9=9x8. Click

here for more ideas.

Ω Remember number facts in peculiar ways

o 6 X 7 = 42 - 6 and 7 are sweatin’ on a bicycle made four two

o 8 X 8 = 64 - I ate and I ate and I got sick on the floor

o Multiply by 9 - all products add up to 9 e.g. 9x2=18 (1+8=9);9x3=27 (2+7=9)

Ω Allow students to use aids such as number lines, multiplication squares or

calculators to check number facts or to get number facts.

Ω Cue students into listening at key points in a maths lesson.

Ω Avoid too much teacher talk which can lead to tuned-out students.

Ω Make connections between concepts and student interests/knowledge.

Ω Ensure students process information using as many senses as possible.

Ω Use visual aids from You Cubed (Stanford University)

Ω Encourage students to visualise and/or draw the problems they are trying to

solve.

Ω Allow students to jot down numbers during mental maths.

Ω Revise concepts frequently.

Ω Break large amounts of information into smaller chunks.

Ω Teach memory aids such as rehearsal and mnemonics e.g.

acronyms, acrostics and associations. Read more here.

Ω Encourage active reading strategies by using post-it notes

Fig. 17 BIMDAS

and highlighter pens. Paraphrase relevant information.

Ω Repeat explanations for some students.

Ω Leave problems and ideas for solving problems on the board during a lesson.

Fig. 18 shows a maths problem and 5 student-suggested ways of solving the

problem. These solutions were left on the whiteboard throughout the lesson.

Read more about Secondary School Lesson Study here.

35

Ω Build number fact fluency. See page 24 for suggestions and ideas.

Ω Reduce the load of number facts. When students understand the commutative

law, for example, the number of facts to be remembered is halved 8x9=9x8. Click

here for more ideas.

Ω Remember number facts in peculiar ways

o 6 X 7 = 42 - 6 and 7 are sweatin’ on a bicycle made four two

o 8 X 8 = 64 - I ate and I ate and I got sick on the floor

o Multiply by 9 - all products add up to 9 e.g. 9x2=18 (1+8=9);9x3=27 (2+7=9)

Ω Allow students to use aids such as number lines, multiplication squares or

calculators to check number facts or to get number facts.

Ω Cue students into listening at key points in a maths lesson.

Ω Avoid too much teacher talk which can lead to tuned-out students.

Ω Make connections between concepts and student interests/knowledge.

Ω Ensure students process information using as many senses as possible.

Ω Use visual aids from You Cubed (Stanford University)

Ω Encourage students to visualise and/or draw the problems they are trying to

solve.

Ω Allow students to jot down numbers during mental maths.

Ω Revise concepts frequently.

Ω Break large amounts of information into smaller chunks.

Ω Teach memory aids such as rehearsal and mnemonics e.g.

acronyms, acrostics and associations. Read more here.

Ω Encourage active reading strategies by using post-it notes

Fig. 17 BIMDAS

and highlighter pens. Paraphrase relevant information.

Ω Repeat explanations for some students.

Ω Leave problems and ideas for solving problems on the board during a lesson.

Fig. 18 shows a maths problem and 5 student-suggested ways of solving the

problem. These solutions were left on the whiteboard throughout the lesson.

Read more about Secondary School Lesson Study here.

35

37.
Fig. 18 Whiteboard work from Lesson Study PDST

36

36

38.
• Vocabulary

• Verbal Reasoning

• Reading Skills

• Making Sense of Symbols

Language plays an important role in mathematics learning (Schleppegrell, 2010).

Language difficulties can affect a student’s ability to:

Ω Understand and make use of instruction

Ω Solve maths word problems

Ω Decode and interpret mathematical information

Ω Encode and represent mathematical information

Ω Memorise information such as number facts/terms

Ω Reflect on their difficulties

Ω Ask effectively for help. (Dowker, 2009)

Fig. 19 The volume of a box

Students may be good at computation but their ability to apply and demonstrate their

skills will suffer if they do not understand the maths vocabulary used in instructions

and in story problems e.g. “How much less?” or “How much altogether?” (Bruun, Diaz,

and Dykes, 2015). Understanding words affects the understanding of concepts, so

students with limited vocabulary are significantly disadvantaged. Terms such as

hypotenuse, perimeter and symmetry (and their meanings) can be difficult to

remember. Homonyms and homophones can confuse. Homonyms are words having

more than one meaning, although they are spelt and sound similarly e.g. volume or

product. Homophones are words sounding the same in speech but are spelt differently

and have different meanings e.g. root, route. While some words will be learned

through incidental exposure, many need explicit teaching. Words are best learned

through repeated exposure in multiple oral and written contexts.

37

• Verbal Reasoning

• Reading Skills

• Making Sense of Symbols

Language plays an important role in mathematics learning (Schleppegrell, 2010).

Language difficulties can affect a student’s ability to:

Ω Understand and make use of instruction

Ω Solve maths word problems

Ω Decode and interpret mathematical information

Ω Encode and represent mathematical information

Ω Memorise information such as number facts/terms

Ω Reflect on their difficulties

Ω Ask effectively for help. (Dowker, 2009)

Fig. 19 The volume of a box

Students may be good at computation but their ability to apply and demonstrate their

skills will suffer if they do not understand the maths vocabulary used in instructions

and in story problems e.g. “How much less?” or “How much altogether?” (Bruun, Diaz,

and Dykes, 2015). Understanding words affects the understanding of concepts, so

students with limited vocabulary are significantly disadvantaged. Terms such as

hypotenuse, perimeter and symmetry (and their meanings) can be difficult to

remember. Homonyms and homophones can confuse. Homonyms are words having

more than one meaning, although they are spelt and sound similarly e.g. volume or

product. Homophones are words sounding the same in speech but are spelt differently

and have different meanings e.g. root, route. While some words will be learned

through incidental exposure, many need explicit teaching. Words are best learned

through repeated exposure in multiple oral and written contexts.

37

39.
Teaching Tips Vocabulary

Ω Repeat information for students who need extra time to process verbal input.

Ω Exposure to new words is most effective over an extended period of time.

Ω A student may need as many as 17 explicit exposures to use a word comfortably

(Ausubel & Youssef, 1965).

Ω Break multi-step instructions into two or three short steps.

Ω Use informal words or definitions alongside formal vocabulary.

e.g. Volume - how much space does this box take up?

Ω Link new words to prior knowledge to anchor them in stored concepts.

Ω Introduce a new concept/word with visuals (See Fig. 20).

Ω Encourage students to use visual planners, diagrams,

summaries and mind maps.

Ω Click here for JCSP‘s useful graphic organiser.

Ω Use humour as a tool to anchor words and concepts.

Ω Encourage students to record hints to remember new

words in personal journals or glossaries.

Fig. 20 Hypotenuse

Ω Encourage students to read accurately and attend to

meaning e.g. of/off: 10% of a price & 10% off a price.

Ω Words sometimes give clues about which procedure to be applied in a problem e.g.

addition is associated with words such as and, altogether and more but not always

e.g. How much more did you pay than I paid? requires the use of subtraction.

Ω Allow students lots of talking time together. A peer’s explanation of a new word or

concept can often impact on understanding more effectively than one from a

teacher.

Ω Click here for great lists of mathematical words and phrases.

Fig. 21 Peer Talk

38

Ω Repeat information for students who need extra time to process verbal input.

Ω Exposure to new words is most effective over an extended period of time.

Ω A student may need as many as 17 explicit exposures to use a word comfortably

(Ausubel & Youssef, 1965).

Ω Break multi-step instructions into two or three short steps.

Ω Use informal words or definitions alongside formal vocabulary.

e.g. Volume - how much space does this box take up?

Ω Link new words to prior knowledge to anchor them in stored concepts.

Ω Introduce a new concept/word with visuals (See Fig. 20).

Ω Encourage students to use visual planners, diagrams,

summaries and mind maps.

Ω Click here for JCSP‘s useful graphic organiser.

Ω Use humour as a tool to anchor words and concepts.

Ω Encourage students to record hints to remember new

words in personal journals or glossaries.

Fig. 20 Hypotenuse

Ω Encourage students to read accurately and attend to

meaning e.g. of/off: 10% of a price & 10% off a price.

Ω Words sometimes give clues about which procedure to be applied in a problem e.g.

addition is associated with words such as and, altogether and more but not always

e.g. How much more did you pay than I paid? requires the use of subtraction.

Ω Allow students lots of talking time together. A peer’s explanation of a new word or

concept can often impact on understanding more effectively than one from a

teacher.

Ω Click here for great lists of mathematical words and phrases.

Fig. 21 Peer Talk

38

40.
Verbal Reasoning

The total of the combined ages of Niamh, Ahmed and Pierre is 80.

What was the sum of their ages 3 years ago?

Was it a) 71, b) 72, c) 74, or d) 77?

Fig. 22 Draw the problem

When you try to solve the problem above, you see that it requires more than numerical

ability. First you must make sense of the words that you read or hear, and then you

have to think about how to find a solution. Word problems in maths often require a

student to use verbal reasoning alongside spatial or quantitative reasoning. Many will

find this problem easier to work out if they draw a picture of it. Read more about

mathematical reasoning in Section B.

Teaching Tips Verbal Reasoning

Ω Replace unfamiliar words and topics with familiar ones e.g. “what is the area of “the

GAA pitch” instead of “the baseball pitch”.

Ω Use visuals, especially pictures, drawings and diagrams.

Ω Replace large numbers with smaller numbers to help work out what you need to do.

Ω Encourage students to compose their own word problems, maths comics and stories

as a strategy for understanding how to use words and numbers together to pose

questions.

Ω Click here to read a summary of George Polya’s 1945 suggestions (still relevant) for

helping students to reason about mathematics problems.

Reading Skills

Weak reading skills can be an obstacle in mathematics, particularly with text-based

problems. To check if reading skills are a difficulty, observe how well a student solves

a problem when it is read aloud or when working with a peer who reads well. Then

compare the student’s performance when working independently on problems, without

the reader. Click here to read about literacy and learning in maths.

39

The total of the combined ages of Niamh, Ahmed and Pierre is 80.

What was the sum of their ages 3 years ago?

Was it a) 71, b) 72, c) 74, or d) 77?

Fig. 22 Draw the problem

When you try to solve the problem above, you see that it requires more than numerical

ability. First you must make sense of the words that you read or hear, and then you

have to think about how to find a solution. Word problems in maths often require a

student to use verbal reasoning alongside spatial or quantitative reasoning. Many will

find this problem easier to work out if they draw a picture of it. Read more about

mathematical reasoning in Section B.

Teaching Tips Verbal Reasoning

Ω Replace unfamiliar words and topics with familiar ones e.g. “what is the area of “the

GAA pitch” instead of “the baseball pitch”.

Ω Use visuals, especially pictures, drawings and diagrams.

Ω Replace large numbers with smaller numbers to help work out what you need to do.

Ω Encourage students to compose their own word problems, maths comics and stories

as a strategy for understanding how to use words and numbers together to pose

questions.

Ω Click here to read a summary of George Polya’s 1945 suggestions (still relevant) for

helping students to reason about mathematics problems.

Reading Skills

Weak reading skills can be an obstacle in mathematics, particularly with text-based

problems. To check if reading skills are a difficulty, observe how well a student solves

a problem when it is read aloud or when working with a peer who reads well. Then

compare the student’s performance when working independently on problems, without

the reader. Click here to read about literacy and learning in maths.

39

41.
Teaching Tips Reading Skills

Ω Use pair work and group work to ensure that students with weak reading

skills can access word problems in class.

Ω Try to ascertain if a student’s reading difficulties are caused by the technical

aspects of reading or by language comprehension challenges, and intervene

appropriately.

Ω Ensure that weak readers get individualised & expert support for reading.

Include maths texts and vocabulary as part of their reading material.

Ω Read more here.

Making Sense of Symbols

Symbols are part of maths language that save time and space. Symbols are easily

recognisable by students of all languages. They make maths simpler because they

have only one meaning. Think about the subtraction symbol ( − ). There is only one

way to write the symbol, but there are many words to describe the operation (subtract,

minus, take away, decreased by). To help students recall the meaning of symbols, try

to find ways of linking new symbols to student background knowledge.

Teaching Tips Symbols

Ω Find a list of symbols here with dates of origin and information about their

origins.

Ω Ask students to generate their own personalised ways of recalling a symbol.

Ω Sigma is the equivalent to our letter S and means the “Sum of”.

Ω reminds us of the letter E and means There Exists.

Ω Infinity looks like the numeral 8 having a rest on its side because it has a

never-ending journey.

Ω Help students recall the less than sign <

showing the smaller end of the sign always

points to the smaller amount or number.

Fig. 23 Less Than

40

Ω Use pair work and group work to ensure that students with weak reading

skills can access word problems in class.

Ω Try to ascertain if a student’s reading difficulties are caused by the technical

aspects of reading or by language comprehension challenges, and intervene

appropriately.

Ω Ensure that weak readers get individualised & expert support for reading.

Include maths texts and vocabulary as part of their reading material.

Ω Read more here.

Making Sense of Symbols

Symbols are part of maths language that save time and space. Symbols are easily

recognisable by students of all languages. They make maths simpler because they

have only one meaning. Think about the subtraction symbol ( − ). There is only one

way to write the symbol, but there are many words to describe the operation (subtract,

minus, take away, decreased by). To help students recall the meaning of symbols, try

to find ways of linking new symbols to student background knowledge.

Teaching Tips Symbols

Ω Find a list of symbols here with dates of origin and information about their

origins.

Ω Ask students to generate their own personalised ways of recalling a symbol.

Ω Sigma is the equivalent to our letter S and means the “Sum of”.

Ω reminds us of the letter E and means There Exists.

Ω Infinity looks like the numeral 8 having a rest on its side because it has a

never-ending journey.

Ω Help students recall the less than sign <

showing the smaller end of the sign always

points to the smaller amount or number.

Fig. 23 Less Than

40

42.
Sensory Processing

• Auditory Processing

• Visual Processing

• Kinaesthetic and Hands-On Learning

We learn about the world through our senses (sight, sound, touch, smell, taste, body

position, movement and internal body signals). Sensory processing is a term used to

describe the way in which our brains receive, organise and respond to sensory input.

Many sensory processing skills play an important role in mathematical development.

For example, visual processing skills have been shown to be an important predictor of

mathematical competence (Uttal, Meadow, Tipton, Hand, Alden and Warren 2013)

and some students have auditory processing challenges that impact on their maths

learning (Bley and Thornton, 2001).

Auditory Processing

When individuals have auditory processing challenges, all

the parts of the hearing pathway are working well (i.e. there

is no physical difficulty) but something delays or scrambles

the way the brain recognises and processes sounds,

especially speech. Researchers don’t fully understand

where things take a different course, but language tends to

be muddled and/ or normal rate of speech is too fast for the

brain to process. The result is that auditory messages are

Fig. 25 Auditory Processing

incomplete or jumbled.

A student with auditory processing challenges might have difficulty with

Auditory Discrimination: The ability to notice and recognise the subtle differences

between similar-sounding speech sounds e.g. 70 and 17, 3 and free.

Auditory Figure-Ground Discrimination: The ability to pick out and focus on

important sounds in the midst of background noise such as teacher or student voices

in a busy classroom.

Auditory Memory: The ability to recall what you’ve heard, either immediately or when

41

• Auditory Processing

• Visual Processing

• Kinaesthetic and Hands-On Learning

We learn about the world through our senses (sight, sound, touch, smell, taste, body

position, movement and internal body signals). Sensory processing is a term used to

describe the way in which our brains receive, organise and respond to sensory input.

Many sensory processing skills play an important role in mathematical development.

For example, visual processing skills have been shown to be an important predictor of

mathematical competence (Uttal, Meadow, Tipton, Hand, Alden and Warren 2013)

and some students have auditory processing challenges that impact on their maths

learning (Bley and Thornton, 2001).

Auditory Processing

When individuals have auditory processing challenges, all

the parts of the hearing pathway are working well (i.e. there

is no physical difficulty) but something delays or scrambles

the way the brain recognises and processes sounds,

especially speech. Researchers don’t fully understand

where things take a different course, but language tends to

be muddled and/ or normal rate of speech is too fast for the

brain to process. The result is that auditory messages are

Fig. 25 Auditory Processing

incomplete or jumbled.

A student with auditory processing challenges might have difficulty with

Auditory Discrimination: The ability to notice and recognise the subtle differences

between similar-sounding speech sounds e.g. 70 and 17, 3 and free.

Auditory Figure-Ground Discrimination: The ability to pick out and focus on

important sounds in the midst of background noise such as teacher or student voices

in a busy classroom.

Auditory Memory: The ability to recall what you’ve heard, either immediately or when

41

43.
it is needed later e.g. follow verbal directions or remember maths facts.

Auditory Sequencing: The ability to remember the order of items heard. A student

might hear 259 but might say or write 925, or might have difficulties remembering the

correct order of a series of verbal instructions.

Auditory Attention: The ability to stay focussed on listening. The student gets

exhausted with the effort exerted in trying to process what is heard.

Teaching Tips Auditory Processing

Ω Reduce background noise at important listening times.

Ω Add rugs to an echoing room.

Ω Use classroom visuals (pictures/ images/ gestures/ written cues/ copies of

classroom notes) to aid understanding and memory.

Ω Speak clearly, slow down the rate, use simple expressive sentences, maybe

repeat.

Ω Teach in small chunks- too much information is overwhelming.

Ω Check that students have understood instructions.

Ω Give the student more time to process auditory information and to complete

classroom work.

Ω Re-teach concepts and skills, especially multi-step processes such as long

multiplication or division. Encourage regular practice.

Ω Incorporate specific activities e.g. auditory discrimination or auditory memory

games, to help boost auditory processing skills.

Ω Encourage the student to take ownership - to ask for assistance and to self-

advocate. Students need to clarify that the information has been heard correctly,

ask the teacher to repeat, write notes, notice a noisy environment and move to a

quieter place, look at the speaker, give friendly reminders to busy teachers.

Ω Consider using computer software such as Fast ForWord for working on sound

discrimination, auditory memory and language processing.

Ω Use assistive listening devices (such as headphones with a wireless amplification

system) for students with severe auditory processing difficulties.

Ω Sometimes speech and language therapy is accessed for help to develop a

student’s listening skills and ability to identify sounds.

42

Auditory Sequencing: The ability to remember the order of items heard. A student

might hear 259 but might say or write 925, or might have difficulties remembering the

correct order of a series of verbal instructions.

Auditory Attention: The ability to stay focussed on listening. The student gets

exhausted with the effort exerted in trying to process what is heard.

Teaching Tips Auditory Processing

Ω Reduce background noise at important listening times.

Ω Add rugs to an echoing room.

Ω Use classroom visuals (pictures/ images/ gestures/ written cues/ copies of

classroom notes) to aid understanding and memory.

Ω Speak clearly, slow down the rate, use simple expressive sentences, maybe

repeat.

Ω Teach in small chunks- too much information is overwhelming.

Ω Check that students have understood instructions.

Ω Give the student more time to process auditory information and to complete

classroom work.

Ω Re-teach concepts and skills, especially multi-step processes such as long

multiplication or division. Encourage regular practice.

Ω Incorporate specific activities e.g. auditory discrimination or auditory memory

games, to help boost auditory processing skills.

Ω Encourage the student to take ownership - to ask for assistance and to self-

advocate. Students need to clarify that the information has been heard correctly,

ask the teacher to repeat, write notes, notice a noisy environment and move to a

quieter place, look at the speaker, give friendly reminders to busy teachers.

Ω Consider using computer software such as Fast ForWord for working on sound

discrimination, auditory memory and language processing.

Ω Use assistive listening devices (such as headphones with a wireless amplification

system) for students with severe auditory processing difficulties.

Ω Sometimes speech and language therapy is accessed for help to develop a

student’s listening skills and ability to identify sounds.

42

44.
Visual Processing

Visual Processing, like auditory processing, is a complex function

undertaken by the brain. It refers to the brain’s ability to make

sense of what the eyes see. This is not the same as visual acuity

which refers to how clearly a person sees. Sometimes issues

Fig. 26 Visual Processing

occur when the brain has trouble accurately receiving or

interpreting visual information. There can be a number of different issues and a

student with visual processing challenges might have difficulty with

Visual Discrimination: the ability to attend to and identify a figure’s distinguishing

features e.g. recognise a ‘6’ as opposed to a ‘9’ or distinguish between coins

Visual Figure-Ground Discrimination: The ability to focus on important visual

information and to filter out less important background information e.g. find a specific

item on a cluttered desk, pick out numbers in a word problem

Visual Memory: The ability to recall something seen, either immediately or when it is

required later e.g. remember what was read, remember a particular symbol, remember

how to use a calculator

Visual Sequencing: The ability to attend to and/ or recall correctly the order of letters/

numbers/ symbols/ words/ etc. seen or read (e.g. child might see 259 but might read

or write 925)

Visual-Spatial Ability: The ability to perceive the location of objects, numbers and

symbols and how they are placed in relation to each other (e.g. child needs to be able

to align numbers vertically for addition or subtraction of multi-digit numbers;

Trigonometry and Calculus require the ability to imagine an object rotating in space

Visual-Motor Processing: The ability to use feedback from the eyes to coordinate

the movement of other parts of the body e.g. write within the lines or margins, copy

from the board or a book

Visual Attention: The ability to stay focussed on visual tasks or stimuli. A student gets

exhausted, restless or inattentive with the effort exerted in trying to process visual

43

Visual Processing, like auditory processing, is a complex function

undertaken by the brain. It refers to the brain’s ability to make

sense of what the eyes see. This is not the same as visual acuity

which refers to how clearly a person sees. Sometimes issues

Fig. 26 Visual Processing

occur when the brain has trouble accurately receiving or

interpreting visual information. There can be a number of different issues and a

student with visual processing challenges might have difficulty with

Visual Discrimination: the ability to attend to and identify a figure’s distinguishing

features e.g. recognise a ‘6’ as opposed to a ‘9’ or distinguish between coins

Visual Figure-Ground Discrimination: The ability to focus on important visual

information and to filter out less important background information e.g. find a specific

item on a cluttered desk, pick out numbers in a word problem

Visual Memory: The ability to recall something seen, either immediately or when it is

required later e.g. remember what was read, remember a particular symbol, remember

how to use a calculator

Visual Sequencing: The ability to attend to and/ or recall correctly the order of letters/

numbers/ symbols/ words/ etc. seen or read (e.g. child might see 259 but might read

or write 925)

Visual-Spatial Ability: The ability to perceive the location of objects, numbers and

symbols and how they are placed in relation to each other (e.g. child needs to be able

to align numbers vertically for addition or subtraction of multi-digit numbers;

Trigonometry and Calculus require the ability to imagine an object rotating in space

Visual-Motor Processing: The ability to use feedback from the eyes to coordinate

the movement of other parts of the body e.g. write within the lines or margins, copy

from the board or a book

Visual Attention: The ability to stay focussed on visual tasks or stimuli. A student gets

exhausted, restless or inattentive with the effort exerted in trying to process visual

43

45.
Teaching Tips Visual Processing

Ω Minimise copying from textbook or board for students with visual processing challenges.

Ω Set the child up with a note-taking buddy so he/ she can concentrate on listening instead of

struggling to record information.

Ω To help with spacing/ sizing, use thickly-lined, squared paper, dotted paper, graph paper or

unlined paper depending on the student’s preferences and needs.

Ω Place a number strip on the student’s table so that he/ she can refer to it for correct numeral

formation.

Ω Colour-code steps in maths problems.

Ω Use a multi-sensory approach when introducing and practising new concepts and skills (e.g.

bendable pipe cleaners or writing in sand for forming numerals or shapes, using tangible

cardboard clocks when learning to tell the time from an analogue clock).

Ω Repeat information in different modalities - say it aloud, demonstrate it, provide a handout,

incorporate auditory information when possible.

Ω Encourage using a finger or ruler to guide the eyes during reading and to help the student

keep his/ her place.

Ω Make use of tablets and other screens that can be enlarged. Zooming in on an image or

piece of text can help reduce visual distractions and make it easier for a student to focus.

Ω Reduce/ eliminate clutter - clear the student’s desk.

Ω Reduce visual distractions or position the student’s desk away from them.

Ω Keep worksheets clear and simple - remove pretty borders.

Ω Incorporate specific activities to help build visual processing skills (e.g. hidden picture games

such as ‘Where’s Wally?’, odd-one-out, memory games, dot-to-dot activities.

Ω Consider using computer software for working on visual discrimination and visual memory.

Ω Encourage the student to take ownership, ask for assistance and self-advocate (e.g. remind

self to pay attention to details, to use a highlighter or to check for errors).

Ω Give students a break. Include activities that don’t require them to use their eyes. Plan

lessons that require children to use other senses.

Ω Click here for NEPS Good Practice interventions for visual processing skills.

44

Ω Minimise copying from textbook or board for students with visual processing challenges.

Ω Set the child up with a note-taking buddy so he/ she can concentrate on listening instead of

struggling to record information.

Ω To help with spacing/ sizing, use thickly-lined, squared paper, dotted paper, graph paper or

unlined paper depending on the student’s preferences and needs.

Ω Place a number strip on the student’s table so that he/ she can refer to it for correct numeral

formation.

Ω Colour-code steps in maths problems.

Ω Use a multi-sensory approach when introducing and practising new concepts and skills (e.g.

bendable pipe cleaners or writing in sand for forming numerals or shapes, using tangible

cardboard clocks when learning to tell the time from an analogue clock).

Ω Repeat information in different modalities - say it aloud, demonstrate it, provide a handout,

incorporate auditory information when possible.

Ω Encourage using a finger or ruler to guide the eyes during reading and to help the student

keep his/ her place.

Ω Make use of tablets and other screens that can be enlarged. Zooming in on an image or

piece of text can help reduce visual distractions and make it easier for a student to focus.

Ω Reduce/ eliminate clutter - clear the student’s desk.

Ω Reduce visual distractions or position the student’s desk away from them.

Ω Keep worksheets clear and simple - remove pretty borders.

Ω Incorporate specific activities to help build visual processing skills (e.g. hidden picture games

such as ‘Where’s Wally?’, odd-one-out, memory games, dot-to-dot activities.

Ω Consider using computer software for working on visual discrimination and visual memory.

Ω Encourage the student to take ownership, ask for assistance and self-advocate (e.g. remind

self to pay attention to details, to use a highlighter or to check for errors).

Ω Give students a break. Include activities that don’t require them to use their eyes. Plan

lessons that require children to use other senses.

Ω Click here for NEPS Good Practice interventions for visual processing skills.

44

46.
Kinaesthetic and Hands-On Learning

Fig.27 Hands-On Learning

Kinaesthetic learning takes place when students carry out physical activities rather

than learning through listening or watching. This type of learning is a great way to

learn maths concepts, especially for students who have learning challenges or

different learning styles. The learning comes not just through the sense of touch, but

through activities which allow students, for example, to make a model of a cylinder, to

conduct an experiment with water using various containers to learn about capacity, to

cut an apple into quarters or to draw a picture or diagram when solving a

mathematical problem.

45

Fig.27 Hands-On Learning

Kinaesthetic learning takes place when students carry out physical activities rather

than learning through listening or watching. This type of learning is a great way to

learn maths concepts, especially for students who have learning challenges or

different learning styles. The learning comes not just through the sense of touch, but

through activities which allow students, for example, to make a model of a cylinder, to

conduct an experiment with water using various containers to learn about capacity, to

cut an apple into quarters or to draw a picture or diagram when solving a

mathematical problem.

45

47.
Executive Functioning Skills

Executive Functioning and Maths

Teaching Tips

Executive Functioning (EF) is an umbrella term for a range of interacting cognitive

processes which enable us to perform or execute tasks - to plan, focus attention,

remember instructions, control impulses, switch strategies and juggle multiple tasks

successfully. In this section, we focus on those processes and associated skills which

are important for the development of mathematical competence:

Ω Goal setting and planning - figuring out an end point and how to get there

Ω Flexibility of behaviour and thought - switching easily between approaches

Ω Organising and Prioritising - making decisions based on relative importance

Ω Accessing Working Memory - holding and manipulating verbal and non-verbal

information in your head so you can make use of it

Ω Self-regulation, self-monitoring, managing your level of application.

Adapted from Meltzer, 2018

If you were a driver, EF skills would help you to turn on the ignition, know your

destination, notice when you were running out of fuel, recognise when you were lost

and identify a different route. When applied to mathematics, EF skills enable students

to get started by trying a strategy, keep going, monitor their own progress and change

strategies when things do not seem to be working out correctly.

The development of EF skills is dependent on many

different factors and skills continue to develop through

adolescence and early adulthood. For most students

these skills develop naturally but some students need

extra support for emerging skills. We can support and

help strengthen their development through modelling,

scaffolding and supportive relationships. We can adapt

the environment (e.g. reduce time pressure, provide

checklists) and teach students how to set goals, plan and

prioritise, organise materials, shift approaches and

monitor their engagement and performance. Fig. 27 On track with Executive Functioning

46

Executive Functioning and Maths

Teaching Tips

Executive Functioning (EF) is an umbrella term for a range of interacting cognitive

processes which enable us to perform or execute tasks - to plan, focus attention,

remember instructions, control impulses, switch strategies and juggle multiple tasks

successfully. In this section, we focus on those processes and associated skills which

are important for the development of mathematical competence:

Ω Goal setting and planning - figuring out an end point and how to get there

Ω Flexibility of behaviour and thought - switching easily between approaches

Ω Organising and Prioritising - making decisions based on relative importance

Ω Accessing Working Memory - holding and manipulating verbal and non-verbal

information in your head so you can make use of it

Ω Self-regulation, self-monitoring, managing your level of application.

Adapted from Meltzer, 2018

If you were a driver, EF skills would help you to turn on the ignition, know your

destination, notice when you were running out of fuel, recognise when you were lost

and identify a different route. When applied to mathematics, EF skills enable students

to get started by trying a strategy, keep going, monitor their own progress and change

strategies when things do not seem to be working out correctly.

The development of EF skills is dependent on many

different factors and skills continue to develop through

adolescence and early adulthood. For most students

these skills develop naturally but some students need

extra support for emerging skills. We can support and

help strengthen their development through modelling,

scaffolding and supportive relationships. We can adapt

the environment (e.g. reduce time pressure, provide

checklists) and teach students how to set goals, plan and

prioritise, organise materials, shift approaches and

monitor their engagement and performance. Fig. 27 On track with Executive Functioning

46

48.
It takes time to learn new strategies and practise them so that they become automatic

and reliable. Time spent on teaching these skills in a maths class will:

Ω Help students understand their strengths and challenges

Ω Teach students how to learn

Ω Promote motivation, focus effort and encourage independence

Ω Empower students to take control of their learning

Ω Increase confidence

Ω Improve learning outcomes. (Meltzer, 2010)

At this time, the science is still emerging and the relationships between the various EF

skills are not fully known. It is unclear, for example, if EF skills are entirely discrete.

What we do know is that the best way to support the skills needed for effective maths

progress is through really good teaching. One of the most important aspects of this

effective teaching is ‘scaffolding’. Scaffolding is an approach promoted by Vygotsky

(1978). Click here for more information. Scaffolding techniques such as breaking

down skills into component parts, modelling and supported practice all have a

significant impact on maths learning for those who struggle. Teachers, therefore, will

want to create an environment where the required mathematical and EF skills can

grow and develop in an evolving way. The following Teaching Tips may help to support

such an endeavour.

47

and reliable. Time spent on teaching these skills in a maths class will:

Ω Help students understand their strengths and challenges

Ω Teach students how to learn

Ω Promote motivation, focus effort and encourage independence

Ω Empower students to take control of their learning

Ω Increase confidence

Ω Improve learning outcomes. (Meltzer, 2010)

At this time, the science is still emerging and the relationships between the various EF

skills are not fully known. It is unclear, for example, if EF skills are entirely discrete.

What we do know is that the best way to support the skills needed for effective maths

progress is through really good teaching. One of the most important aspects of this

effective teaching is ‘scaffolding’. Scaffolding is an approach promoted by Vygotsky

(1978). Click here for more information. Scaffolding techniques such as breaking

down skills into component parts, modelling and supported practice all have a

significant impact on maths learning for those who struggle. Teachers, therefore, will

want to create an environment where the required mathematical and EF skills can

grow and develop in an evolving way. The following Teaching Tips may help to support

such an endeavour.

47

49.
Teaching Tips EF Skills

Ω Ensure that students have been taught many strategies for problem-solving.

Ω Teach students to ask themselves: Is this problem similar to a problem I

Cognitive Flexibility

have seen before?

Ω Teach thinking such as: My first strategy did not work…maybe I should try

a different one. Have I drawn a picture of the problem?

Ω Encourage students to talk to teachers & peers when stuck

Ω Give opportunities to work independently, in pairs and in groups

Ω Eliminate time pressure on students (at least initially).

Ω Ensure that students know what they are required to do and the length of

time it should take to do it.

Self-Monitoring and Self-Regulation

Ω Differentiate instruction (with individualised teaching strategies and

classroom accommodations) for students with attention challenges.

Ω Provide explicit checklists for assignments.

Ω Help students to devise personal checklists so they recognise and monitor

their most common errors/challenges.

Ω Increase student awareness of EF strategies which work well for them.

Ω Help students modulate emotional responses through strategies such as

logical thinking, relaxation & positive self-talk.

Ω Consider using incentives & rewards for starting on time, sustaining effort &

completing tasks.

Ω Set clear goals and objectives for your lessons and share with students.

Ω Ensure students understand the purpose of each task.

Ω Try to link maths activities to student goals and interests.

Goal Setting

Teach students to break down long-term goals into more easily achievable

steps.

48

Ω Ensure that students have been taught many strategies for problem-solving.

Ω Teach students to ask themselves: Is this problem similar to a problem I

Cognitive Flexibility

have seen before?

Ω Teach thinking such as: My first strategy did not work…maybe I should try

a different one. Have I drawn a picture of the problem?

Ω Encourage students to talk to teachers & peers when stuck

Ω Give opportunities to work independently, in pairs and in groups

Ω Eliminate time pressure on students (at least initially).

Ω Ensure that students know what they are required to do and the length of

time it should take to do it.

Self-Monitoring and Self-Regulation

Ω Differentiate instruction (with individualised teaching strategies and

classroom accommodations) for students with attention challenges.

Ω Provide explicit checklists for assignments.

Ω Help students to devise personal checklists so they recognise and monitor

their most common errors/challenges.

Ω Increase student awareness of EF strategies which work well for them.

Ω Help students modulate emotional responses through strategies such as

logical thinking, relaxation & positive self-talk.

Ω Consider using incentives & rewards for starting on time, sustaining effort &

completing tasks.

Ω Set clear goals and objectives for your lessons and share with students.

Ω Ensure students understand the purpose of each task.

Ω Try to link maths activities to student goals and interests.

Goal Setting

Teach students to break down long-term goals into more easily achievable

steps.

48

50.
Teaching Tips EF Skills

Ω Break instructions down into chunks

Ω Use verbal & non-verbal reminders, prompts and cues

Ω Use visuals

Working Memory

Ω Use mnemonics/memory aids. Here are a few examples:

o 5, 6, 7, 8 56 is 7 by 8

o < and > The alligator has to open its mouth wider for the larger number

o The value of pi (3.1415926): Count each word’s letters in the question

“May I have a large container of coffee?”

Ω Use visual timetables.

Planning, Organising and Prioritising

Ω Use To-Do notes written into a diary.

Ω Display How-to-do lists with diagrams and instructions in classroom.

Ω Encourage clutter-free workspaces.

Ω Have checklists of equipment needed.

Ω Use alarm on student’s phone to act as a reminder (…may be more

appropriate at home).

Ω Support student organisation by providing work materials if necessary.

Ω Encourage completion of subtasks when faced with a complex task.

Ω Use different coloured highlighters for different types of information.

Ω Teach strategic approaches to class work, homework and study.

Ω Click here for more information on EF skills from Harvard University.

Please Note

At this stage, the evidence about the teaching of discrete executive

functioning skills is inconclusive.

Good teaching involves bringing the various skills together and developing

those skills in an interactive process. Teaching support is best deployed in

teaching the maths skills rather than attempting to ‘train’ or ‘teach’

underlying discrete executive functioning skills.

49

Ω Break instructions down into chunks

Ω Use verbal & non-verbal reminders, prompts and cues

Ω Use visuals

Working Memory

Ω Use mnemonics/memory aids. Here are a few examples:

o 5, 6, 7, 8 56 is 7 by 8

o < and > The alligator has to open its mouth wider for the larger number

o The value of pi (3.1415926): Count each word’s letters in the question

“May I have a large container of coffee?”

Ω Use visual timetables.

Planning, Organising and Prioritising

Ω Use To-Do notes written into a diary.

Ω Display How-to-do lists with diagrams and instructions in classroom.

Ω Encourage clutter-free workspaces.

Ω Have checklists of equipment needed.

Ω Use alarm on student’s phone to act as a reminder (…may be more

appropriate at home).

Ω Support student organisation by providing work materials if necessary.

Ω Encourage completion of subtasks when faced with a complex task.

Ω Use different coloured highlighters for different types of information.

Ω Teach strategic approaches to class work, homework and study.

Ω Click here for more information on EF skills from Harvard University.

Please Note

At this stage, the evidence about the teaching of discrete executive

functioning skills is inconclusive.

Good teaching involves bringing the various skills together and developing

those skills in an interactive process. Teaching support is best deployed in

teaching the maths skills rather than attempting to ‘train’ or ‘teach’

underlying discrete executive functioning skills.

49

51.
Teachings from Neuroscience

Neuroscience and Education

Teaching Tips

Fig. 28 The Maths Brain, Salimpoor (2016)

Neuroscience and Education

Our growing understanding of how the brain works has impacted on educational

practice in recent decades. The demand for neuroscience-informed education comes

from two directions, with neuroscientists emphasising the potential of their work to

improve education and educators being keen to learn what neuroscience has to offer

(Howard-Jones, 2014). One of the most useful functions of this emerging science is

that it can reveal information that is not visible at the behavioural level. It has

contributed to our understanding of the following:

Ω Different brain areas involved in number fact retrieval, abstract thinking, imagery,

spatial orientation, number sense and accurate counting

Ω Differences in brain activity of people with dyscalculia

Ω Connections between brain areas

Ω How maturational and developmental changes impact on the brain’s

understanding of, and responses to, mathematical concepts and processes

Ω How interventions and differing instructional approaches affect neural patterns

Ω Impact of maths anxiety and of beliefs and mindsets on neural activity patterns.

50

Neuroscience and Education

Teaching Tips

Fig. 28 The Maths Brain, Salimpoor (2016)

Neuroscience and Education

Our growing understanding of how the brain works has impacted on educational

practice in recent decades. The demand for neuroscience-informed education comes

from two directions, with neuroscientists emphasising the potential of their work to

improve education and educators being keen to learn what neuroscience has to offer

(Howard-Jones, 2014). One of the most useful functions of this emerging science is

that it can reveal information that is not visible at the behavioural level. It has

contributed to our understanding of the following:

Ω Different brain areas involved in number fact retrieval, abstract thinking, imagery,

spatial orientation, number sense and accurate counting

Ω Differences in brain activity of people with dyscalculia

Ω Connections between brain areas

Ω How maturational and developmental changes impact on the brain’s

understanding of, and responses to, mathematical concepts and processes

Ω How interventions and differing instructional approaches affect neural patterns

Ω Impact of maths anxiety and of beliefs and mindsets on neural activity patterns.

50

52.
Teaching Tips from Neuroscience

Ω Help students to understand that the brain can change through effort and practice and

that intelligence, like a muscle, grows stronger with exercise.

Ω Read a short article about Growth Mindset by Carol Dweck (2015) here. A fixed mindset

is a belief that your intelligence, abilities and talents are fixed traits. A growth mindset

is a belief that your abilities can be developed through dedication and hard work.

Neuroscience confirms that a growth mindset leads to more activity in the brain’s

thinking parts, to forming more neural pathways and to quicker learning of new

information.

Ω Click here for Jo Boaler’s infographics about maths mindsets.

Ω Use hands-on materials for building mathematical understanding of concepts. This

facilitates multisensory learning, aiding both understanding and memory.

Ω Encourage finger representation for number concept development and arithmetic

(Bafalluy and Noel, 2008). There is evidence that fingers have a special place over and

above concrete materials.

Ω Separate learning sessions in time (spacing), as opposed to massing them together,

as this has been shown to improve learning performance (Rohrer and Taylor, 2006).

Ω Neuroscientific research confirms the value of supporting students to construct their

own knowledge and solutions through game playing and hands-on learning. A

constructivist approach works better than direct instruction through lecture, practice

work, homework and exams (Burnett, 2010.)

Ω Physical exercise has been shown to increase the efficiency of neural networks in

learning (Diamond, 2012).

Ω Consider using games created using learning from psychology and neuroscience.

Read more here in an article called “From Brain to Education” (Butterworth et al, 2011).

Ω Maths learning needs an environment that is flexible in content and pace for students’

current needs and zone of proximal development (definition of the latter on page 15).

51

Ω Help students to understand that the brain can change through effort and practice and

that intelligence, like a muscle, grows stronger with exercise.

Ω Read a short article about Growth Mindset by Carol Dweck (2015) here. A fixed mindset

is a belief that your intelligence, abilities and talents are fixed traits. A growth mindset

is a belief that your abilities can be developed through dedication and hard work.

Neuroscience confirms that a growth mindset leads to more activity in the brain’s

thinking parts, to forming more neural pathways and to quicker learning of new

information.

Ω Click here for Jo Boaler’s infographics about maths mindsets.

Ω Use hands-on materials for building mathematical understanding of concepts. This

facilitates multisensory learning, aiding both understanding and memory.

Ω Encourage finger representation for number concept development and arithmetic

(Bafalluy and Noel, 2008). There is evidence that fingers have a special place over and

above concrete materials.

Ω Separate learning sessions in time (spacing), as opposed to massing them together,

as this has been shown to improve learning performance (Rohrer and Taylor, 2006).

Ω Neuroscientific research confirms the value of supporting students to construct their

own knowledge and solutions through game playing and hands-on learning. A

constructivist approach works better than direct instruction through lecture, practice

work, homework and exams (Burnett, 2010.)

Ω Physical exercise has been shown to increase the efficiency of neural networks in

learning (Diamond, 2012).

Ω Consider using games created using learning from psychology and neuroscience.

Read more here in an article called “From Brain to Education” (Butterworth et al, 2011).

Ω Maths learning needs an environment that is flexible in content and pace for students’

current needs and zone of proximal development (definition of the latter on page 15).

51

53.

54.
Section D

Page

Current Teacher Supports………..53

Interventions and Initiatives………56

53

Page

Current Teacher Supports………..53

Interventions and Initiatives………56

53

55.
Current Teacher Supports

Resources and Professional Development Opportunities

The Professional Development Service for Teachers is a

support service for primary and post-primary staff,

offering professional learning opportunities to teachers

and school leaders in a range of curricular areas,

including mathematics.

The PDST website contains resources, publications, school

self-evaluation tools, lists of useful websites and apps, and

video footage of good practice. The PDST offer Maths

Recovery and Ready, Set, Go - Maths training for teachers

of young children. They offer one-day seminars and

workshops on mathematics topics (e.g. Problem-Solving,

Mental Maths) in Education Centres throughout the country.

The majority of these events are booked through the

central applications system. PDST also offer bespoke in-

school support sessions to teachers across all areas of

teaching and learning in maths, from early years through to Leaving Cert. They

support teachers’ inclusive practice in relation to team teaching for maths. Apply at

pdst.ie/schoolsupport. If you have any difficulties registering, then email

The PDST Post Primary Mathematics Team

(previously known as The Project Maths

Development Team, then

The Maths Development

Team) supports all post-

primary Mathematics teachers. The team of experienced

teachers provides professional development support to post-

primary teachers through workshops, Lesson Study materials,

school visits and online resources.

Visit www.projectmaths.ie for more information or apply for in-school support at

pdst.ie/schoolsupport. Click here to see an expert teaching maths.

54

Resources and Professional Development Opportunities

The Professional Development Service for Teachers is a

support service for primary and post-primary staff,

offering professional learning opportunities to teachers

and school leaders in a range of curricular areas,

including mathematics.

The PDST website contains resources, publications, school

self-evaluation tools, lists of useful websites and apps, and

video footage of good practice. The PDST offer Maths

Recovery and Ready, Set, Go - Maths training for teachers

of young children. They offer one-day seminars and

workshops on mathematics topics (e.g. Problem-Solving,

Mental Maths) in Education Centres throughout the country.

The majority of these events are booked through the

central applications system. PDST also offer bespoke in-

school support sessions to teachers across all areas of

teaching and learning in maths, from early years through to Leaving Cert. They

support teachers’ inclusive practice in relation to team teaching for maths. Apply at

pdst.ie/schoolsupport. If you have any difficulties registering, then email

The PDST Post Primary Mathematics Team

(previously known as The Project Maths

Development Team, then

The Maths Development

Team) supports all post-

primary Mathematics teachers. The team of experienced

teachers provides professional development support to post-

primary teachers through workshops, Lesson Study materials,

school visits and online resources.

Visit www.projectmaths.ie for more information or apply for in-school support at

pdst.ie/schoolsupport. Click here to see an expert teaching maths.

54

56.
The Special Education Support Service, part of the

National Council for Special Education, develops and

delivers professional development supports for

school personnel working with students with special

educational needs in primary, post-primary, special schools and special classes.

Access SESS support through its online application system. The SESS offers

telephone advice, school visits, and staff in-service courses. Check out the SESS

Curriculum Access Tool (CAT) for primary school learners with mild, moderate and

severe general learning difficulties, or the SESS Tips for Teaching Learners with

The Educational Research Centre website www.erc.ie includes

research reports on Ireland’s national and international

mathematics performance. These reports contain ideas and

recommendations on teaching mathematics e.g. PISA Maths: A

Teacher’s Guide. The ERC provides Drumcondra maths tests

for primary and post-primary schools in paper and digital formats.

The National Council for Curriculum and

Assessment NCCA website contains curricula,

research reports, assessment toolkits and videos of

good practice in maths. The NCCA and its

educational partners are developing a new primary

mathematics curriculum. Read a background paper here. Research Reports 17 and

18 provide valuable up-to-date information about mathematics in early childhood and

in primary education.

55

National Council for Special Education, develops and

delivers professional development supports for

school personnel working with students with special

educational needs in primary, post-primary, special schools and special classes.

Access SESS support through its online application system. The SESS offers

telephone advice, school visits, and staff in-service courses. Check out the SESS

Curriculum Access Tool (CAT) for primary school learners with mild, moderate and

severe general learning difficulties, or the SESS Tips for Teaching Learners with

The Educational Research Centre website www.erc.ie includes

research reports on Ireland’s national and international

mathematics performance. These reports contain ideas and

recommendations on teaching mathematics e.g. PISA Maths: A

Teacher’s Guide. The ERC provides Drumcondra maths tests

for primary and post-primary schools in paper and digital formats.

The National Council for Curriculum and

Assessment NCCA website contains curricula,

research reports, assessment toolkits and videos of

good practice in maths. The NCCA and its

educational partners are developing a new primary

mathematics curriculum. Read a background paper here. Research Reports 17 and

18 provide valuable up-to-date information about mathematics in early childhood and

in primary education.

55

57.
The PDST Junior Certificate School Programme (JCSP) is targeted

at Junior Cycle students at risk of early school leaving. JCSP at

www.jcsp.ie offers mathematics resources and professional

development support to post-primary schools in the Delivering Equality

of Opportunity in Schools (DEIS) initiative. The resources are available

for purchase to schools outside this initiative.

The US National Council of Teachers of

Mathematics provides guidance and

resources to implement research-informed

and high-quality teaching supporting every

student’s learning in equitable environments.

Visit their website for information.

EPI-STEM, the national centre for STEM Education is

based at the University of Limerick. EPI-STEM aims to

strengthen STEM education research, inform STEM

education policy and promote STEM in primary and post-

primary schools, and in the community. Its website contains research reports and

links to maths resources.

The What Works Clearinghouse (WWC)

reviews research on different US programs,

products, practices and policies. It helps teachers

and school leaders in making evidence-based

56

at Junior Cycle students at risk of early school leaving. JCSP at

www.jcsp.ie offers mathematics resources and professional

development support to post-primary schools in the Delivering Equality

of Opportunity in Schools (DEIS) initiative. The resources are available

for purchase to schools outside this initiative.

The US National Council of Teachers of

Mathematics provides guidance and

resources to implement research-informed

and high-quality teaching supporting every

student’s learning in equitable environments.

Visit their website for information.

EPI-STEM, the national centre for STEM Education is

based at the University of Limerick. EPI-STEM aims to

strengthen STEM education research, inform STEM

education policy and promote STEM in primary and post-

primary schools, and in the community. Its website contains research reports and

links to maths resources.

The What Works Clearinghouse (WWC)

reviews research on different US programs,

products, practices and policies. It helps teachers

and school leaders in making evidence-based

56

58.
Interventions and Initiatives

Primary School

Maths Recovery (Wright, 2003) is a widely-used

intervention based on very detailed pupil assessment.

Its framework for individual, group or class-based

instruction is suitable for pupils with or without maths challenges. The Maths Recovery

Programme is one of the Department of Education and Skills DEIS (Delivering Equality

of Opportunity in Schools) initiatives to improve numeracy outcomes. Maths Recovery

develops children’s knowledge of number words and numerals, conceptual place

value, addition and subtraction to 100, multiplication and division and written

computation. It emphasises mental calculation and relational thinking, encouraging

students to see relationships between numbers rather than to follow rules. In Ireland,

it is usually used with students in First Class and it involves 1-1 daily sessions of 25

minutes for teaching cycles of 12 to 15 weeks’ duration.

Mata sa Rang and Maths Blast are in-class numeracy teaching approaches using

Maths Recovery strategies. The assessment tasks identify pupils’ strengths and

needs (Cull, 2018). Local, experienced and practising Maths Recovery teachers

provide training in Education Centres nationwide. Click here to see the Maths

Recovery approach in action.

Number Worlds (Griffin, 2005) is an early intervention

programme used to close the number knowledge gap

between children in schools in low-income, high-risk

communities and their more affluent peers in Massachusetts

(Griffin and Case, 1997) and in Dublin (Mullan and Travers,

2007). The intervention involves whole class teaching and

scaffolded small group work. It emphasises counting and language skills to help

children gain a number representation similar to a mental counting line. Number

Knowledge and Level Placement Tests measure students’ conceptual knowledge and

pinpoint instructional start points. There are ten levels in the programme (Levels A to

J) which can be used from pre-school right through primary school (and beyond) -

usually one level per school year.

57

Primary School

Maths Recovery (Wright, 2003) is a widely-used

intervention based on very detailed pupil assessment.

Its framework for individual, group or class-based

instruction is suitable for pupils with or without maths challenges. The Maths Recovery

Programme is one of the Department of Education and Skills DEIS (Delivering Equality

of Opportunity in Schools) initiatives to improve numeracy outcomes. Maths Recovery

develops children’s knowledge of number words and numerals, conceptual place

value, addition and subtraction to 100, multiplication and division and written

computation. It emphasises mental calculation and relational thinking, encouraging

students to see relationships between numbers rather than to follow rules. In Ireland,

it is usually used with students in First Class and it involves 1-1 daily sessions of 25

minutes for teaching cycles of 12 to 15 weeks’ duration.

Mata sa Rang and Maths Blast are in-class numeracy teaching approaches using

Maths Recovery strategies. The assessment tasks identify pupils’ strengths and

needs (Cull, 2018). Local, experienced and practising Maths Recovery teachers

provide training in Education Centres nationwide. Click here to see the Maths

Recovery approach in action.

Number Worlds (Griffin, 2005) is an early intervention

programme used to close the number knowledge gap

between children in schools in low-income, high-risk

communities and their more affluent peers in Massachusetts

(Griffin and Case, 1997) and in Dublin (Mullan and Travers,

2007). The intervention involves whole class teaching and

scaffolded small group work. It emphasises counting and language skills to help

children gain a number representation similar to a mental counting line. Number

Knowledge and Level Placement Tests measure students’ conceptual knowledge and

pinpoint instructional start points. There are ten levels in the programme (Levels A to

J) which can be used from pre-school right through primary school (and beyond) -

usually one level per school year.

57

59.
Numicon is a multi-sensory maths teaching programme for

children aged 4-7 years (and older students with special

educational needs) which uses Numicon Maths Shapes in

practical teaching activities. The Maths Shapes give learners

insight into number values and relationships, differently to that

given by written numerals. The programme teaches number concepts and more

complex concepts such as multiplication, division and fractions. Learners develop

their own mental imagery as they combine and

compare shapes and use arithmetic in

practical activities. Nye, Buckley and Bird

(2005) found Numicon to be effective in

teaching children with Down Syndrome.

Skevinton (2016) also found that Numicon was

useful with older primary school children with

number concept challenges.

Catch-Up Numeracy is based on the Numeracy Recovery intervention programme

developed by Dr. Anne Dowker, University of Oxford, in 2001. It is a Teaching

Assistant - led programme involving two 15-minute, one-to-one teaching sessions

weekly. This intervention breaks numeracy down into ten components. It is funded by

the Education Endowment Foundation and is targeted at 6 to 14-year olds.

Training comprises three half-day training sessions and is offered in the UK and

Northern Ireland.

Ready, Set, Go - Maths

Ready, Set, Go - Maths was developed in Northern

Ireland between 1999 and 2001 by Eunice Pitt. This is a

programme for teachers of infant classes which focuses

specifically on the development of early number skills and

concepts. Research (unpublished at time of printing)

showed Ready, Set, Go - Maths to be an effective means of including children with

SEN in a mainstream junior infant classroom (in Dublin) over an eight-week

intervention period. Training is available from PDST.

58

children aged 4-7 years (and older students with special

educational needs) which uses Numicon Maths Shapes in

practical teaching activities. The Maths Shapes give learners

insight into number values and relationships, differently to that

given by written numerals. The programme teaches number concepts and more

complex concepts such as multiplication, division and fractions. Learners develop

their own mental imagery as they combine and

compare shapes and use arithmetic in

practical activities. Nye, Buckley and Bird

(2005) found Numicon to be effective in

teaching children with Down Syndrome.

Skevinton (2016) also found that Numicon was

useful with older primary school children with

number concept challenges.

Catch-Up Numeracy is based on the Numeracy Recovery intervention programme

developed by Dr. Anne Dowker, University of Oxford, in 2001. It is a Teaching

Assistant - led programme involving two 15-minute, one-to-one teaching sessions

weekly. This intervention breaks numeracy down into ten components. It is funded by

the Education Endowment Foundation and is targeted at 6 to 14-year olds.

Training comprises three half-day training sessions and is offered in the UK and

Northern Ireland.

Ready, Set, Go - Maths

Ready, Set, Go - Maths was developed in Northern

Ireland between 1999 and 2001 by Eunice Pitt. This is a

programme for teachers of infant classes which focuses

specifically on the development of early number skills and

concepts. Research (unpublished at time of printing)

showed Ready, Set, Go - Maths to be an effective means of including children with

SEN in a mainstream junior infant classroom (in Dublin) over an eight-week

intervention period. Training is available from PDST.

58

60.
JUMP (Junior Undiscovered Math Prodigies) is a Canadian class-based

programme of confidence-building, guided practice, guided discovery, continuous

assessment, scaffolded instruction, mental maths and conceptual understanding.

Read more here about an evaluation of JUMP in two Education Centre catchment

areas during 2013/14. The JUMP meaning of guided discovery is more didactic than

that recommended in the Primary School Mathematics Curriculum.

Number Talks are 5-15-minute conversations around

purposefully-crafted computation problems. The talks get

children thinking and talking about their thoughts when

presenting and justifying solutions to computation

problems. This programme requires a safe and risk-free

environment, with a culture of acceptance of all ideas and

answers, regardless of errors. You can see Number Talks

in action in Dublin here. Training is available from PDST.

Paired Mathematics and Mathematics for Fun are initiatives from the Home School

Community Liaison Scheme. They involve parents engaging in mathematics games

with children in the classroom.

Read more about interventions in Dowker (2009) What Works for Children with

Mathematical Difficulties.

Be Careful

Ensure that an intervention programme is not just an “add-on”.

Readymade programmes tend to dictate how each topic is taught regardless of the

student’s particular challenges or learning style.

Intervention programmes should identify mathematical difficulties through detailed

initial assessment and subsequent ongoing diagnostic observations.

(Haseler, 2008)

59

programme of confidence-building, guided practice, guided discovery, continuous

assessment, scaffolded instruction, mental maths and conceptual understanding.

Read more here about an evaluation of JUMP in two Education Centre catchment

areas during 2013/14. The JUMP meaning of guided discovery is more didactic than

that recommended in the Primary School Mathematics Curriculum.

Number Talks are 5-15-minute conversations around

purposefully-crafted computation problems. The talks get

children thinking and talking about their thoughts when

presenting and justifying solutions to computation

problems. This programme requires a safe and risk-free

environment, with a culture of acceptance of all ideas and

answers, regardless of errors. You can see Number Talks

in action in Dublin here. Training is available from PDST.

Paired Mathematics and Mathematics for Fun are initiatives from the Home School

Community Liaison Scheme. They involve parents engaging in mathematics games

with children in the classroom.

Read more about interventions in Dowker (2009) What Works for Children with

Mathematical Difficulties.

Be Careful

Ensure that an intervention programme is not just an “add-on”.

Readymade programmes tend to dictate how each topic is taught regardless of the

student’s particular challenges or learning style.

Intervention programmes should identify mathematical difficulties through detailed

initial assessment and subsequent ongoing diagnostic observations.

(Haseler, 2008)

59

61.
Secondary School

Project Maths: This professional development support to post-primary maths

teachers is available from the PDST Post Primary Maths Team. You will find links

and information about Project Maths in the Current Teacher Supports section.

Functional Mathematics, Application of

Number and Functional Mathematics Learner

Packs are resources to support students in

developing knowledge, skills and competence in

maths and in working towards FETAC/QQI

Qualifications at Levels 3 and 4. These resources

include Tutor Guides and Practice Sheets and they

were developed by a team from the National

Adult Literacy Agency (NALA) and the

National Centre for Excellence in

Mathematics and Science Teaching and

Learning (NCE-MSTL).

Junior Certificate School Programme (JCSP) Mathematics

The JCSP Mathematics Initiatives enable schools to acquire age-appropriate

experiential resources and games for mathematics and numeracy. The materials help

to develop creative approaches to teaching maths to JCSP students.

Cross-Age Paired Maths: This strategy involves JCSP

students training as tutors and working with 3rd class primary

school students over a six-week programme. The strategy

aims to enhance both groups’ maths skills, competencies

and confidence. Support materials and implementation

guidelines are provided.

60

Project Maths: This professional development support to post-primary maths

teachers is available from the PDST Post Primary Maths Team. You will find links

and information about Project Maths in the Current Teacher Supports section.

Functional Mathematics, Application of

Number and Functional Mathematics Learner

Packs are resources to support students in

developing knowledge, skills and competence in

maths and in working towards FETAC/QQI

Qualifications at Levels 3 and 4. These resources

include Tutor Guides and Practice Sheets and they

were developed by a team from the National

Adult Literacy Agency (NALA) and the

National Centre for Excellence in

Mathematics and Science Teaching and

Learning (NCE-MSTL).

Junior Certificate School Programme (JCSP) Mathematics

The JCSP Mathematics Initiatives enable schools to acquire age-appropriate

experiential resources and games for mathematics and numeracy. The materials help

to develop creative approaches to teaching maths to JCSP students.

Cross-Age Paired Maths: This strategy involves JCSP

students training as tutors and working with 3rd class primary

school students over a six-week programme. The strategy

aims to enhance both groups’ maths skills, competencies

and confidence. Support materials and implementation

guidelines are provided.

60

62.
Number Millionaire is a numeracy quiz where individual students are challenged to

identify the correct answer to twelve arithmetical questions. The quiz follows the “Who

Wants to be a Millionaire?” format. Questions and answers are provided to

participating schools.

Maths Laboratory consists of standardised, graded, colour-coded and differentiated

work cards. Individual students’ needs, prior knowledge and competency levels

determine the appropriate learning route through the programme.

Hand–held Gaming Devices: Teachers are asked to explore and choose the most

appropriate handheld device(s) currently available and then choose the most suitable

software e.g. Challenge Me Maths Workout, Personal Trainer Maths, Brain Age

Express Maths, Maths Play, Maths Blaster, etc.

Maths Games Initiative: This initiative provides opportunities for teachers to source

and acquire age-appropriate maths games and/or maths activity packs. It encourages

use of these resources to develop mathematical and numerical understanding among

JCSP students. Teachers may also opt to plan, construct and develop their own maths

games and activity packs.

An Invitation

We are aware of other commercially-available interventions used in Ireland e.g.

Mathletics and The Power of 2. We would like to hear from you if you have evaluated

these or any other interventions not included in this section, so that we can share

their effectiveness in revisions of this document.

Please email [email protected] with such information with the word maths in

the subject line. Thank you.

61

identify the correct answer to twelve arithmetical questions. The quiz follows the “Who

Wants to be a Millionaire?” format. Questions and answers are provided to

participating schools.

Maths Laboratory consists of standardised, graded, colour-coded and differentiated

work cards. Individual students’ needs, prior knowledge and competency levels

determine the appropriate learning route through the programme.

Hand–held Gaming Devices: Teachers are asked to explore and choose the most

appropriate handheld device(s) currently available and then choose the most suitable

software e.g. Challenge Me Maths Workout, Personal Trainer Maths, Brain Age

Express Maths, Maths Play, Maths Blaster, etc.

Maths Games Initiative: This initiative provides opportunities for teachers to source

and acquire age-appropriate maths games and/or maths activity packs. It encourages

use of these resources to develop mathematical and numerical understanding among

JCSP students. Teachers may also opt to plan, construct and develop their own maths

games and activity packs.

An Invitation

We are aware of other commercially-available interventions used in Ireland e.g.

Mathletics and The Power of 2. We would like to hear from you if you have evaluated

these or any other interventions not included in this section, so that we can share

their effectiveness in revisions of this document.

Please email [email protected] with such information with the word maths in

the subject line. Thank you.

61

63.