A Good Practice Guide for Teachers

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Sharp Tutor
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
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
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
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.
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.
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
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
11. Section A
Maths Anxiety 11
Assessment 16
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
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.
less avoid
confidence maths
poor less
performance learning
Fig. 2 Anxiety Cycle
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
Ω Teachers do not project their own anxieties about maths (Geist 2015).
Adapted from NCCA 2016
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.
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.
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
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.
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
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.
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
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-
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?
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
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.
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.
23. Section B
Number Sense……………….. 23
Mathematical Reasoning……. 26
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
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
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.
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
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
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?
Abstract 6x + 66 = 81.90
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
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
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
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’
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
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…
33. Section C
Memory………………………………. 33
Language…………………................ 37
Sensory Processing………………… 41
Executive Functioning Skills……….. 46
Neuroscience………………………… 49
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.
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.
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
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.
37. Fig. 18 Whiteboard work from Lesson Study PDST
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.
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
Ω Click here for great lists of mathematical words and phrases.
Fig. 21 Peer Talk
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
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.
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
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
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
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
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
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.
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
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
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.
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.
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
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.
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
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.
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.
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
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).
54. Section D
Current Teacher Supports………..53
Interventions and Initiatives………56
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.
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.
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
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.
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.
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)
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.
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 neps@education.gov.ie with such information with the word maths in
the subject line. Thank you.