Report
Review of numerical
reasoning teaching
resources
March 2014
National Foundation for Educational
Research (NFER)
Review of numerical reasoning
teaching resources
Jo Williams
Katherine Fowler
Emily Jones
© 2014 National Foundation for Educational Research
Registered Charity No. 313392
ISBN 978-1-4734-1679-6
How to cite this publication:
Williams, J., Fowler, K. and Jones, E. (2014). Review of Numerical Reasoning Teaching
Resources. Slough: NFER.
Contents
1
About this review
1
2
Review of current resources
4
3
Conclusions and recommendations
22
Further details of resources
23
Teacher voice survey
74
Appendix A: Information sources
78
Appendix B: Criteria for inclusion and exclusion
81
1 About this review
1.1 Research context and aims
In May 2013, learners across Wales sat the first new statutory reading and numeracy
procedural tests, as implemented by the Department for Education and Skills. All learners
from Years 2 to 9, who are able to access them, are required to take the new annual national
reading and numeracy tests.
The assessment of numeracy is split into two tests: procedural knowledge and numerical
reasoning. In May 2014 numerical reasoning will be included in the statutory numeracy tests
for the first time.
Development work on the reasoning materials has indicated that learners perform less well
than expected on this aspect of numeracy. Learners’ performance appears to have been
affected by inexperience, such as selecting inefficient methods, not understanding
mathematical language, poor calculator skills, and difficulties in knowing how to present
mathematical working.
The pilot results indicated that further support is required to support the teaching and
learning of numerical reasoning. There are many free and commercial mathematics
resources that are available to teachers, but many do not correspond to the definition of
numerical reasoning as assessed in these tests.
The aim of this report is to identify a shortlist of materials that meet the definition of
numerical reasoning as given in the Literacy and Numeracy Framework as well as a defined
set of criteria for the teaching and learning of reasoning. This has been done through two
methods:

evaluating numerical reasoning teaching resources that are currently available (only
includes those available in the English language, due to the availability of resources and
evaluators not being Welsh speakers)

a teacher survey to identify resources that teachers use, or are aware of, that support the
learning and application of numerical problem solving and/or numerical reasoning of
learners in Years 2-9 (the methodology and the findings from this survey are presented
in an annex to this report).
Review of numerical reasoning teaching resources 1
1.2 Research design and methods
1.2.1 Review of current resources available
A systematic review was conducted to investigate teaching resources of numerical
reasoning that are currently available. This included website searches of both UK and
English-speaking international countries for educational publishers and mathematics
professional associations. Searches were also performed on UK-based teacher forums,
news and government department websites. The specific sources in each of these
categories are listed in Appendix A along with the search terms.
The identification of relevant and appropriate teaching resources included five key stages:
1. agreeing search terms and parameters for the review of resources
2. identifying sources of teaching resources
3. appraising each resource to discern its relevance to the review
4. reviewing and summarising the most relevant resources
5. reporting the findings.
Twenty-five teaching resources were identified for review based on the relevance of
their target age group, subject specificity and inclusion of teacher guidance. Details
regarding the selection criteria are given in Appendix B. The final selection of resources
included:

resources published in the UK (22 resources); USA, New Zealand and Australia (one
resource from each)

hard-copy books and activity packs (20 resources); web-based resources (five
resources)

resources published from 2000 onwards, with the exception of one teaching guide
(1990), with the majority from 2008 onwards.
This list can help teachers identify resources that may help to support the teaching and
learning of numerical reasoning. The Department for Education and Skills (DfES) are not
publicly endorsing or specifying that these materials must be used. There may be other
materials that teachers are already using that are not listed in this audit. The DfES are not
stating that teachers can no longer continue to use these resources. They should, however,
make an informed decision on their suitability.
The numerical reasoning tests have been designed to test the skills in the Literacy and
Numeracy Framework in order to assess whether pupils are able to:
i.
decide for themselves what numerical skills to use
ii.
apply their method to work out a solution
iii.
give coherent mathematical explanations
iv.
reflect on the outcome.
2
Review of numerical reasoning teaching resources
To evaluate each resource, specific criteria were developed, based on these skills. The
individual reviews, presented in the section ‘Further details of resources’ of this report, detail
the criteria. It should be noted that the criteria were also grouped into four main skills areas
that match those outlined above as forming the construct of the statutory numerical
reasoning tests:

understanding the question

selecting and applying methods

recording working

reflecting on answers.
After reviewing each resource against these criteria, a shortlist of 25 resources was created.
Resources were selected for this shortlist if they fulfilled the criteria for at least one of the
four constructs above. As well as providing suitable activities for learners, many contain
guidance to support teachers as they develop their teaching of numerical reasoning.
1.2.2 Limitations of review
Only those resources which relate to the Welsh Government definition of numerical
reasoning were included in the review. Numerical reasoning, in this context, is defined as the
ability to apply known procedural numerical skills and reasoning in order to solve problems.
However, resources that covered both reasoning and procedural skills were included in the
review, although only parts of these resources may be relevant for supporting the
development of reasoning skills specifically.
Some additional resources that provide opportunities for learners to develop their numerical
reasoning skills are presented separately. These contain less explicit teacher or learner
guidance, and so did not score well against the criteria, but they are still considered to be
very relevant to supporting learners’ in practising and applying their reasoning skills.
A sample of each resource was reviewed, as time constraints did not allow for a more
thorough evaluation of each resource.
Only resources obtainable within the review window are included. Some publishers did not
respond to the invitation to submit materials; online materials change and are supplemented
at a rapid pace. This review covers many, but not necessarily all, of the resources available.
The search terms and the materials reviewed were originated in English.
Review of numerical reasoning teaching resources 3
2 Review of current resources
2.1 Summary
This section of the report describes the findings from the reviews of the numerical reasoning
teaching resources.
Table 2.1 displays the 25 resources which were shortlisted based on the relevance of their
target age group, subject specificity and inclusion of teacher guidance. The table includes
the resource’s target year group, price range and coverage across each of the four main
problem solving criteria. The four main criteria each consisted of three or four elements. A
resource fulfilled a criterion if it satisfied at least two elements.
Brief descriptions of the structure of each resource, such as if it is a teacher book, learner
workbook or website, can be found in section 2.2 of this report. More detailed individual
reviews of each resource can be found by following the link at the end each description. The
individual reviews present evidence of how the resources fulfil each of the criteria, by fully
describing them against the elements where applicable.
The focus and content of the reviewed resources varies considerably. Some of the
resources focus on one specific aspect of reasoning whereas other may offer a more holistic
approach with content that covers a number of reasoning skills. The resources also vary in
terms of the level of demand, for example some of the resources involve learners having to
use reasoning skills in single-step problems whereas other involve more complex multi-step
problem solving. In addition, some resources focus solely on reasoning while others also
include procedural numeracy. These resources were not excluded from the review if the
reasoning content fulfilled at least one of the problem solving criteria.
Whilst many of the resources listed are teacher books or learner workbooks, most are not
book-based approaches to teaching reasoning. Most resources provide ideas for reasoning
tasks that can be carried out as a class or in groups and give suggestions as to how tasks
could be investigated. Many of the resources listed aim to develop teachers’ understanding
of how reasoning skills can be taught and nurtured, and therefore are not prescriptive
instructions to be followed in a formal classroom setting. Rather, they may provide examples
of tasks in order to demonstrate how problems can be presented and tackled through the
use of learning techniques such as question prompts, independent and collaborative
working, or guided versus open methods of investigation.
Given the range and scope of the resources it is important to recognise that the number of
criteria satisfied in an individual resource (i.e. the number of ticks in Table 2.1) does not
necessarily indicate that it is a better resource for developing reasoning skills in learners. In
order to identifying the most appropriate resource to use with a particular group of learners,
teachers should also consider the following alongside the findings of the review:

Are there specific reasoning skills that learners need to develop?

What level of demand will be appropriate for the learners? (This may be dependent on
how much exposure learners have had to reasoning and problem solving.)
4
Review of numerical reasoning teaching resources

Is a teacher-directed or self-directed resource more appropriate? (How the teacher
wishes to utilise the resource is likely to determine the extent of how directed or
exploratory learning within a task is.)
This review focuses on resources that support the teaching of numerical reasoning. Other
resources are available that may help further develop learners’ reasoning skills. Table 2.2
lists some additional resources that provide extensive practice of various types of reasoning
problems which teachers may find helpful for learners in improving their reasoning skills.
Review of numerical reasoning teaching resources 5
Table 2.1 Summary of reviews of numerical reasoning teaching resources
About the resource
Name of resource
Publisher
Abacus Evolve Real-Life
Problem Solving
Pearson
Badger Maths Problem
Solving
Badger Publishing
BEAM’s Big Book of Word
BEAM Education
Problems and Interactive CD (Nelson Thornes)
Bowland Maths
The Bowland Trust
Can Do Problem Solving
Nelson Thornes
Which year group
is it for?
Whether the resource fulfils each criterion
Understanding
Selecting and
Recording
Reflecting on
the question
applying methods
working
answers
Years 1-5
(website)
Years 1-6
(4 books)
Years 1-6
(3 sets, Years 3-4
reviewed)
Years 7-9
(website)
Years 1-6
(6 sets)
Years 1-2 and
Years 3-6
(2 books)




each




each



not
known
Oxford University
Press
Years 3-6
(1 book)

Framework Press
Educational
Publishers Ltd.
Years 3-11

Macmillan Maths Problem
Solving Boxes (1-6)
Main Activity: Problem
Solved!
6
Badger Publishing
Years 1-6
(7 books)
Year 1-4
(2 books, Year 2-4
reviewed)

Years 1-6
(6 boxes)
Years 7-9
(3 books,
Year 7 reviewed)




Review of numerical reasoning teaching resources
free

Years 1-6
(website)
GL Assessment
Macmillan Education
Australia (distributed
by McGraw-Hill)
££
each

Espresso Education
Let's Think Through Maths!
£
each


Espresso Primary
Schofield & Sims


Scholastic
KS1/2 Problem Solving



Creative Activities for Maths
Problem Solving
Investigations with Numicon
Investigative and ProblemSolving Approaches to
Mathematics and Their
Assessment
££


Price*
££
£
£
free

£
££

each

££

each

each
£
About the resource
Whether the resource fulfils each criterion
Understanding
Selecting and
Recording
Reflecting on
the question
applying methods
working
answers
Maths Buzz
BEAM Education
Which year group
is it for?
Years 1-6
(3 sets, Years 3
and 4 reviewed)
Maths Trails, Working
Systematically
Cambridge
University Press
Years 3-9
(1 book)
Mental Arithmetic (First)
(Essential)
Schofield & Sims
Years 1-9
(3 sets)

My Maths
Oxford University
Press
Years 2-9
(website)

New Zealand Maths
Problem Solving
New Zealand
Ministry of Education
Years 1-10
(website)
Numicon Teaching
Resource Handbooks
Problem-Solving Strategies
for Efficient and Elegant
Solutions – Grades 6-12
Problem Solving with EYFS,
Key Stage One and Key
Stage Two Children
Oxford University
Press
Years 1-4
(8 sets)
Corwin Press
(a SAGE company)
Years 7-13
(1 book)
Department for
Education
EYFS and
Years 1-6
Talk it, solve it: Reasoning
Skills in Maths
BEAM Education
(Nelson Thornes)
Understanding Maths:
Problem Solving
Schofield & Sims
We Can Do It!
BEAM Education
(Nelson Thornes)
Name of resource
We Have A Problem!
Publisher
Badger Learning
££


each


£
£
each











Years 1-6
(3 books)


Years 3-6
(1 book)



Years 1-6
(6 books)
Years 3-6
(4 books, Year 3
reviewed)






Price*
££
free

££
each
£

free
£
each

£
£
each
£
each
Review of numerical reasoning teaching resources 7
Although they did not satisfy the criteria of this review, three additional resources, shown in
Table 2.2, are note-worthy for providing opportunities for learners to practise their numerical
reasoning skills. These are free, online resources which teachers may consider useful for
consolidating learners’ reasoning knowledge and applying this to problem solving tasks.
These websites provide online problem solving tasks in the style of short games or activities.
They have varying degrees of teacher guidance, so some can be tackled independently by
learners while others may benefit from some teacher input. These resources allow learners
to develop their reasoning skills by exploring a wide range of problems which can be tackled
with different approaches and recording strategies. Many of these tasks enable learners to
devise and carry out methods of reasoning which they have previously encountered. They
may therefore provide a suitable resource for supporting and further developing the
reasoning skills taught in class.
Table 2.2 Additional resources
Name of
resource
Publisher
HWB Wales (Welsh
Puppies
Government) / NGfL
Problem Solving
Cymru
Website
Which year
group is it for? Price
https://hwb.wales.gov.uk/cms/hwbconte
nt/Shared%20Documents/vtc/2008Years 2-4
09/maths/puppies/index.html
free
Maths
Investigations
HWB (Welsh
Government) / Wrexham
Network Project
https://hwb.wales.gov.uk/cms/hwbconte
nt/Shared%20Documents/vtc/2008Years 3-9
09/maths/irf08-77/index.html
free
Nrich Tasks
University of Cambridge
www.nrich.maths.org
free
8
Review of numerical reasoning teaching resources
Years 1-13
2.2 Individual reviews
Name of resource: Abacus Evolve Real-Life Problem Solving
Publisher: Pearson
Year of publication: Website is copyright 2014 – cannot find a date for when Abacus was
first released. It has been created for the new curriculum in England, so likely publication
date is 2013.
Country of publication: UK
Reference: www.activelearnprimary.co.uk
Target year group: Years 1-5
Price: Ranging from about £1,000-£2,000 for the first year, depending on the package and
the size of the school.
Description: Online toolkit of maths resources, mostly games but some printable, including
a large bank of problem-solving and reasoning resources.
Comments: Fun and engaging problem-solving games with no writing or explanations
required from pupils. The problems presented in every game can be solved using trial and
error. Teacher support would be required to develop systematic working, to extend beyond
trial and error methods. Also teacher-led discussion groups are the main source of provoking
planning of methods and considering alternative methods. Tasks are labelled according to
how much support should be given, that is, whether it is an independent, guided or
supported activity.
Further details: For further details for this resource click here.
Name of resource: Badger Maths Problem Solving
Publisher: Badger Publishing
Year of publication: 2000-2009
Country of publication: UK
Reference: www.badgerlearning.co.uk
Target year group: Years 1-6
Price: All four books with CDs available online for £129.00
Free sample online at www.nationalstemcentre.org.uk, requires an account (free to register)
Description: Series of four teacher books (Years 1-2, Year 3, Years 4-5 and Year 6) which
include photocopiable problems and learner answer pages, teacher guidance, problem
solving tasks, a four-step approach template for tackling problems and a CD containing
printable files from the book.
Comments: Thorough teacher guidance with pupil worksheets to steer learners through
using different strategies in problem solving. Helps learners to break problems down into
stages and guides their thinking by teaching which questions to consider at each stage.
Further details: For further details for this resource click here.
Review of numerical reasoning teaching resources 9
Name of resource: BEAM’s Big Book of Word Problems and Interactive CD
Publisher: BEAM (Nelson Thornes)
Year of publication: 2004
Country of publication: UK
Reference: www.nelsonthornes.com
Target year group: Years 1-6 (Years 3-4 reviewed)
Price: Available online, book £61.99, CD £111.59
Description: Series of word problems presented in a large book or on a whiteboard, and a
connected series of worksheets for independent working. Interactive CD provides further
word problems, with an added element of removing superfluous information and ordering the
hints correctly.
Comments: This is a good resource for learning how to tackle word problems but does not
consider any other type of problem solving. The interactive CD provides a slight extension
by adding in the concept of ignoring superfluous information, and ordering the usefulness of
clues, but ultimately it focuses on word problem skills.
Further details: For further details for this resource click here.
Name of resource: Bowland Maths
Publisher: The Bowland Trust
Year of publication: 2008 (last updated 2012)
Country of publication: UK
Reference: www.bowlandmaths.org.uk/
Target year group: Years 7-9
Price: Free online or available as a DVD (£200 per disc in UK excluding England).
Description: A series of online problems designed as computer games, made up of a series
of activities that can span several lessons/be used as homework.
Comments: Very good quality materials – the games feel very professionally produced, in
particular the video clips. The teacher guidance included is thorough. The resource also
includes five modules of professional development materials, based on activities and
discussions so best used with groups of teachers. It states that the case studies are suitable
for the revised KS3 curriculum. It provides a grid to show the areas of the curriculum
addressed in each case study and the suggested age and ability of learners for each case
study.
Further details: For further details for this resource click here.
10
Review of numerical reasoning teaching resources
Name of resource: Can Do Problem Solving
Publisher: Nelson Thornes
Year of publication: 2004
Country of publication: UK
Reference: www.nelsonthornes.com
Target year group: Years 1-6 (Years 1, 3 and 6 evaluated here)
Price: All available online. Teacher’s books - £35.99, Whiteboard CD-ROM - £139.19,
Resources CD-ROM - £52.79
Description: A set containing a teacher book, whiteboard CD-ROM and resources CD-ROM
for Years 1-6 (one set per year). This comprises a set of activities to be worked through as a
class, using a whiteboard if suitable, and a problem bank of related questions.
Comments: This is a particularly useful resource for classroom teaching with regards to
different strategies and the five-stage process to problem solving.
Further details: For further details for this resource click here.
Name of resource: Creative Activities for Maths Problem Solving
Publisher: Scholastic
Year of publication: 2006
Country of publication: UK
Reference: http://shop.scholastic.co.uk
Target year group: Years 1-2 and Years 3-6
Price: Available online, for £32 for both books
Description: Teacher book with introduction to why/how to teach problem solving, with
detailed lesson ideas and photocopiable resources.
Comments: These books are helpful when teaching learners how to tackle problem solving
tasks in classroom activities. The tasks involve physical, group, talking and doing activities
with some written individual work. Both books have recently been mapped in detail to the
Welsh Numeracy Framework. This shows how each task aligns with different strands and
elements, allowing teachers to select tasks that address specific aspects of learning that
they wish to target.
Further details: For further details for this resource click here.
Review of numerical reasoning teaching resources 11
Name of resource: Espresso Primary (website)
Publisher: Espresso Education
Year of publication: updated weekly
Country of publication: UK
Reference: www.espresso.co.uk
Target year group: Years 1-6
Price: Subscription required. Price depends on the individual school – Espresso provides
quotations dependent on a number of factors.
Description: Website that schools can subscribe to, with resources across the curriculum
for Foundation Stage, Key Stage 1 and Key Stage 2. Includes videos, fact files, online and
printable activities. Includes a poster of ‘Steps to solving word problems’ and guidance of
how Espresso can be used to support learning of the Welsh curriculum.
Comments: There is good range of problem solving activities but there is minimal teacher
guidance. Most activities have an introductory video which outlines the task, e.g. making
patterns, children going for a picnic. Some are online/interactive tasks learners can work
through alone and some are worksheets to be printed off. Some tasks are repeated at higher
levels so building on the same skills/patterns.
Further details: For further details for this resource click here.
Name of resource: Investigations with Numicon
Publisher: Oxford University Press
Year of publication: 2010
Country of publication: UK
Reference: https://global.oup.com/education
Target year group: Years 3-6
Price: Available online at £29.70
Description: Ten challenges with multi sensory apparatus. Teacher guide with some
photocopiable pages for learners. Each challenge is mapped to the curriculum and has an
extension task at the end. Recommended to be used with Numicon apparatus.
Comments: These challenges allow learners to develop their confidence in using the
apparatus, then converting this to working without apparatus, to tackle a range of
mathematical problems. Challenges aim to develop deeper mathematical understanding of
the number system.
Further details: For further details for this resource click here.
12
Review of numerical reasoning teaching resources
Name of resource: Investigative and Problem-Solving Approaches to Mathematics and
Their Assessment
Publisher: Framework Press Educational Publishers Ltd.
Year of publication: 1990
Country of publication: UK
Reference: www.nationalstemcentre.org.uk
Target year group: Years 3-11
Price: Available free online
Description: Teacher guide about how to teach problem solving with tasks for use in the
classroom.
Comments: Detailed teacher guidance about why and how to teach problem solving, rather
than just a workbook, so teachers would need to read the book before preparing lessons,
not simultaneously. Relies on the teacher reading the book and teaching from it, rather than
on the learners learning for themselves from exercise books. Useful for those teachers who
are very engaged with teaching problem solving.
First published in 1990 and appears not to have been revised. Section One contains the
most relevant information and Section Three lists tasks to use in class.
Further details: For further details for this resource click here.
Name of resource: KS1/KS2 Problem Solving
Publisher: Schofield & Sims
Year of publication: 2011
Country of publication: UK
Reference: www.schofieldandsims.co.uk
Target year group: Years 1-6
Price: Available online at the above link, £2.45 (KS1) or £2.95 (KS2) per workbook.
Description: Set of workbooks for Key Stage 1 and Key Stage 2 containing problem solving
exercises for learners to fill in (not re-usable).
Comments: This is a good series of practice workbooks for learners who already
understand how to do problem solving activities and can work independently. They would be
less helpful as a teaching tool, unless worked through under close supervision of a teacher.
No teacher guide is provided.
Further details: For further details for this resource click here.
Review of numerical reasoning teaching resources 13
Name of resource: Let’s Think Through Maths! (6-9)
Publisher: GL Assessment
Year of publication: 2004, 2005 (6-9)
Country of publication: UK
Reference: www.gl-assessment.co.uk
Target year group: Year 1(LTTM), Years 2-4 (LTTM6-9)
Price: £160 + VAT (LTTM with Let’s Think! handbook), £130 + VAT (LTTM 6-9, no
handbook)
Description: Two sets of resources, both containing teacher guide and a set of materials
(e.g. dice, measuring equipment) aimed to focus on learners’ mathematical thinking.
Comments: This set of resources focuses on developing learners’ understanding and
mathematical thinking by looking in depth at simple mathematical issues, e.g. thinking about
counting to 6. There is less emphasis on solving problems or reasoning numerically but
methods for investigating maths problems are covered.
Further details: For further details for this resource click here.
Name of resource: Macmillan Maths Problem Solving Box (1-6)
Publisher: Macmillan Education Australia (distributed in the UK by McGraw-Hill)
Year of publication: 2012
Country of publication: Australia
Reference: http://schools.mcgraw-hill.co.uk
Target year group: Years 1-6 (one box per year)
Price: £89.99 per box (with a discount if all 6 boxes are purchased)
Description: Sets of 150 cards presenting maths problems with accompanying teacher CD
for each box (1-6) with problem solving strategy poster.
Comments: The 9-step problem solving strategy poster guides learners through each stage
of the problems. It is generic enough to be applicable to a wide range of problems, allowing
learners to work out for themselves what to do. Teacher guidance gives good information
about how to structure the lessons and support/prompt learners. Also contains an index to
show which areas of the curriculum each task aligns with and there are record sheets for
learners to note down how they tackled each problem and an assessment sheet for teachers
to track learner progress along each of the 9 problem solving strategies.
Further details: For further details for this resource click here.
14
Review of numerical reasoning teaching resources
Name of resource: Main Activity: Problem Solved!
Publisher: Badger Publishing
Year of publication: 2003
Country of publication: UK
Reference: www.badgerlearning.co.uk
Target year group: Years 7-9 (Book 1, Year 7 reviewed here)
Price: Available online, pupil text book £7.50, teacher book £29.50
Description: Pupil text book full of questions, and teacher book providing guidance for each
question.
Comments: Includes activities for those working below and above the expected level, so
could possibly be used with the top end of Key Stage 2. This resource has thorough teacher
guidance.
Further details: For further details for this resource click here.
Name of resource: Maths Buzz
Publisher: BEAM Education
Year of publication: 2009
Country of publication: UK
Reference: www.nelsonthornes.com
Target year group: Years 1-6 (Years 3-4 reviewed)
Price: £160.79 for each box, available from the link above
Description: A box of 16 object sorting tasks (with 2 introductory tasks) and CD
Comments: All the tasks involve using a series of clues to sort objects into orders and
arrangements or pairings with numbers/prices. Each task has three levels of difficulty,
Good resource for looking at this one aspect of reasoning (logical thinking) and developing
confidence in how to interpret information between different clues. Appropriate for logic
problems but not broad enough to be an overall resource.
Further details: For further details for this resource click here.
Review of numerical reasoning teaching resources 15
Name of resource: Maths Trails, Working Systematically
Publisher: Cambridge University Press
Year of publication: 2006
Country of publication: UK
Reference: http://education.cambridge.org
Target year group: Years 3-9
Price: £29.90, available from the website above
Description: Teacher information about how/why learners should work systematically with
17 activity lesson plans, with a CD containing all 17 lesson plans and pupil work sheets.
Comments: Detailed teacher information about different ways learners can work
systematically within different tasks. The tasks provide a good basis for developing a
systematic approach to a range of problems, in interpreting the questions and applying
methods. These skills are generalisable to other tasks and aim to help learners recognise
the sorts of skills/methods they need to use in different tasks. Good resource for developing
systematic working skills and building learners’ confidence in tackling seemingly difficult
questions.
Further details: For further details for this resource click here.
Name of resource: (First) (Essential) Mental Arithmetic
Publisher: Schofield & Sims
Year of publication: 2013
Country of publication: UK
Reference: www.schofieldandsims.co.uk
Target year group: First Mental Arithmetic: Years 1-2, Mental Arithmetic: Years 3-6,
Essential Mental Arithmetic: Years 7-9.
Price: Available online – workbooks at £2.25 each, answer books at £3.95 each, teacher
guides at £7 each.
Description: Set of workbooks focusing on quick mental maths problems for learners to fill
in (not re-usable). Accompanying teacher guide provides activity prompts for each question
and photocopiable diagnostic tests.
Comments: Many questions have a problem solving nature to them. This would therefore
work quite well as a resource to develop independent working through a series of shorter
problems. This is not meant as a resource to help teach problem solving, although
diagnostic tools and activity prompts do support the development of mathematical skills in
solving problems.
Further details: For further details for this resource click here.
16
Review of numerical reasoning teaching resources
Name of resource: My Maths (website)
Publisher: Oxford University Press
Year of publication: 2013
Country of publication: UK
Reference: www.mymaths.co.uk
Target year group: Years 2-9
Price: Subscription service. A primary (partial) subscription costs £250 + VAT per annum.
Description: Maths website which features reasoning lessons with tasks and online
homework, online reasoning games and longer investigations with teacher notes.
Comments: Investigations provide good contexts for learners to practise using reasoning
skills if these have already been taught but there is very little teacher guidance. For the
investigations, the teacher guidance gives a list of the skills used in the task, cross-curricular
links and possible solutions. No guidance of how learners should work through the tasks.
Further details: For further details for this resource click here.
Name of resource: New Zealand Maths Problem Solving (website)
Publisher: New Zealand Ministry of Education
Year of publication: 2010
Country of publication: New Zealand
Reference: www.nzmaths.co.nz/problem-solving
Target year group: Years 1-10 (New Zealand levels 1-6)
Price: Available free online
Description: Website from the Ministry of Education, New Zealand with teacher guidance
and problem solving tasks for Levels 1-6. A sample of tasks reviewed here.
Comments: New Zealand resources so some contexts/names used may be unfamiliar. Also
may be difficult to map to Welsh framework. Resources are online with semi-structured
lesson plans and minimal pupil worksheets. Additional teacher information about why and
how to teach problem solving is given. Provides a wide range of different tasks but may not
easily feed into planning a whole series of lessons.
Further details: For further details for this resource click here.
Name of resource: Numicon Teaching Resource Handbooks
Review of numerical reasoning teaching resources 17
Publisher: Oxford University Press
Year of publication: to be published in spring 2014
Country of publication: UK
Reference: https://global.oup.com/education
Target year group: Years 1-4
Price: Available for pre-order online, £99 for Teaching Resource Handbook and
Implementation Guide teaching pack, for each topic per year group.
Description: Teacher handbooks for Number, pattern and calculating, and Geometry,
measurement and statistics, each with an Implementation guide. Each teacher handbook
includes an introduction to Numicon, the apparatus and the theory behind its approach,
planning charts and detailed lesson plans grouped by topic with several tasks for group and
individual work, and photocopiable pupil answer sheets.
Comments: Numicon is a whole scheme for teaching mathematics rather than just for
teaching numerical reasoning. Numicon places problem solving at the core of children’s
learning. Learners are encouraged to learn through solving problems themselves and then
seeing how what they learn could be applied to solving other problems. There are some
topics, logic and problem solving, which have greater emphasis on problem solving
methods, but many of the other tasks also involve learners devising their own methods to
tackle problems and record working and explain their methods/answers.
Further details: For further details for this resource click here.
Name of resource: Problem-Solving Strategies for Efficient and Elegant Solutions – Grades
6-12
Publisher: Corwin Press (a SAGE company)
Year of publication: 2008
Country of publication: USA
Reference: www.corwin.com
Target year group: Years 7-13 (USA Grades 6-12)
Price: Available online at the above link for $39.95 for paperback (£24.45), or $86.95 for
hardcover (£53.22).
Description: A textbook for teachers providing theory and examples that can be used in
practice for 10 problem-solving strategies.
Comments: This book is designed as a textbook for teachers more than a resource for
teaching, however teachers may choose to use the example questions in their classroom.
For this reason, there is limited advice with regards to teaching the questions, rather the
focus is on the best possible way of solving the questions based on the textbook chapter.
This book is aimed for Years 7-13, with no clear grading throughout the book, so many
problems will exceed Year 9 levels of working.
Further details: For further details for this resource click here.
Name of resource: Problem Solving with EYFS, Key Stage One and Key Stage Two
18
Review of numerical reasoning teaching resources
Children
Three sections: Logic problems and puzzles, Finding all possibilities, and Finding rules and
describing patterns
Publisher: Department for Education
Year of publication: 2004 (Finding all possibilities), 2010 (Logic problems and puzzles,
Finding rules and describing patterns)
Country of publication: UK
Reference: www.nationalstemcentre.org.uk
Description of resource: Three resources from the National Strategies in England focusing
on problem solving.
Target year group: EYFS and Years 1-6
Price: Available free online at the website above.
Description: Three resources from the National Strategies focusing on problem solving.
Comments: These resources provide a good series of activities that develop throughout the
year and across years. They are discussion based for the most part, so there is not a big
emphasis on written explanations and workings.
Further details: For further details for this resource click here.
Name of resource: Talk it, solve it: Reasoning Skills in Maths
Publisher: BEAM Education, Nelson Thornes
Year of publication: 2005
Country of publication: UK
Reference: www.nelsonthornes.com
Target year group: Years 1-2, Years 3-4, and Years 5-6
Price: Available online, at £30.99 each
Description: One book for each pair of year groups with a CD, both containing 12 activities
for each year group, which consist of a stimulus sheet and clue sheet. Stimulus sheet has
six (Years 1-2) or eight (Years 3-6) possible answers given, then there are as many clues for
learners to work out which answer is being described. Six or eight question stems provided
for each task to encourage learners to develop their own questions to identify a partner’s
chosen answer.
Comments: Focuses on one type of problem solving. Certain mathematical knowledge is
required to complete tasks, such as being able to convert decimals and fractions (Year 6).
Good for practising logic and developing communication of logical thinking but quite a
narrow focus.
Further details: For further details for this resource click here.
Review of numerical reasoning teaching resources 19
Name of resource: Understanding Maths: Problem Solving
Publisher: Schofield & Sims
Year of publication: 2011 (currently undergoing revision)
Country of publication: UK
Reference: www.schofieldandsims.co.uk
Target year group: Years 3-6
Price: Available online, for £3.95 per workbook
Description: Pupil workbook teaching key problem solving skills with progress tests.
Comments: This is a good workbook for teaching simple problem solving skills. It is
designed to be used completely independently by the learner, and as such there is no
teacher guidance etc. This does not assist learners in developing their ability to discuss and
explain their reasoning– learners are not asked to go beyond working systematically and
providing an answer.
Further details: For further details for this resource click here.
Name of resource: We Can Do It!
Publisher: BEAM Education, Nelson Thornes
Year of publication: 2008
Country of publication: UK
Reference: www.nelsonthornes.com
Target year group: Years 1-6 (A book for each year group)
Price: Available online, £37.99 per book
Description: Teacher book for each year group (1-6) with information about what problem
solving is and what it involves, including a ‘problem-solving strategies’ sheet. Also includes
advice of how to use the book and 36 lesson plans. Includes a CD with resource sheets and
answers for some challenges.
Comments: Good level of introduction to explain the various skills involved in problem
solving – helpful and informative without being too theoretical. Each challenge is mapped to
the Primary Framework for mathematics (2006), in England. Lots of teacher information for
each challenge: the skills assessed, differentiated tasks, maths content. Teacher guidance is
very structured but pupil worksheets are more to support discussion and group work rather
than tackling tasks independently.
Further details: For further details for this resource click here.
20
Review of numerical reasoning teaching resources
Name of resource: We Have A Problem!
Publisher: Badger Learning
Year of publication: 2012
Country of publication: UK
Reference: www.badgerlearning.co.uk
Target year group: Years 3-6 (Year 3 reviewed)
Price: All books available as a package for £129 from above website.
Description: Teacher book with copy masters and CD of resources, providing a series of
activities designed to teach learners how to approach word-based maths problems.
Comments: This is a good resource for teaching how to do problem solving, rather than just
providing exercises under the assumption that learners will know what to do.
The main limitation is the level of specificity. The book focuses on a very particular type of
problem solving, therefore ruling out potential for various methods/justifying methods – all
the problems in the book can be solved using a set methodology.
Further details: For further details for this resource click here.
Review of numerical reasoning teaching resources 21
3 Conclusions and recommendations
This review was conducted to determine whether suitable resources for the teaching of
numerical reasoning are currently available. Two approaches were adopted to answer this
question. Firstly a search of resources was undertaken; identified resources were then
evaluated against specific criteria for teaching numerical reasoning. Secondly, a survey of
primary and secondary school teachers in England was undertaken to establish which
resources they are already aware of (details of the survey and the survey results are
presented in an annex to this report).
The results demonstrate that there are suitable resources available for teaching numerical
reasoning, as assessed by the numerical reasoning strand of the new statutory national
numeracy tests in Wales. The shortlist provided in Section 2 of this report is not exhaustive
but provides an overview of available resources for different year groups and cost. Many of
the listed resources were also identified by teachers in the Teacher Voice survey,
demonstrating that they are already used and respected in primary and secondary
classrooms.
The resources listed are not endorsed by the Department for Education and Skills, who are
not specifying that any resources identified must be used in schools. There are many other
materials not listed in this report which schools may continue to use if they consider them to
be suitable.
In selecting resources to support the teaching and learning of numerical reasoning, teachers
should consider which aspects of reasoning they want to focus on, for example selecting
and applying methods or recording results systematically. This needs to be considered
alongside the budget they have available. This review includes resources that range from
being available free online, to those that cost up to £50, and over £50. Publishers should be
contacted for the most recent pricing structures.
A full list of references appears in the following section of this report to enable teachers to
locate the resources reviewed here, either online or by ISBN.
22
Review of numerical reasoning teaching resources
Further details of resources
Abacus Evolve Real-life Problem Solving (website)
Criteria
Is the context real-life, abstract or
cross-curricular?
Selecting and applying methods
Understanding the question
Does it have single- or multi-step
problems?
Does it encourage the use of
strategies for working out what
the question is asking? (e.g.
highlighting key words)
Does it use/explain
appropriate mathematical
language?
Does it help learners to
interpret questions with
diagrams and minimal
language?
Does it provide opportunities
for learners to select
appropriate methods?
Does it enourage learners to
select the most efficient
method?
Does it approach problem
solving in more general terms,
such as solving puzzles,
identifying and extracting
patterns and applying them to
the question?
Does it encourage calculator
use and give opportunities to
practice calculator skills?
(Year 5 and above only)
Comments
Some real life applications e.g. bus
journeys, some abstract e.g. alien
Olympics. Cross-curricular links are
identified in the teacher notes that
accompany each task.
Single-step – games go through
activities step by step with
instructions.
Instructions on screen and spoken
aloud. The games are presented to
make it very clear what should be
done. Spoken prompts also provided.
Language quite simple and not very
mathematical. Does not teach
appropriate language. It is up to the
teacher to check that learners have
pre-requisite knowledge.
Lots of language is used, both written
and verbal, to introduce each task.
Once learners have discussed the
task they frequently need to use
diagrams and grids to tackle them.
Through discussion, teachers can
prompt learners to think about how
they will approach each task before
they attempt it. A lot of the games end
with “try again and see if you can do
better”.
This is developed through trial and
error and discussing methods with
others.
Yes, a lot of puzzle-based games
which develop skills that can be
applied to other, similar tasks.
n/a
Covered
suitably for
year group?






n/a
Review of numerical reasoning teaching resources 23
Criteria
Recording working
Does it show how to record
working mathematically and
systematically?
Does it exemplify how to show
working and how to explain
methods used and answers
given?
Reflecting on
answers
Does it distinguish between
how to show working and how
to explain the methods used
and answers given?
Are learners encouraged to
check their working/answers?
Are learners encouraged to
of/use alternative checking
techniques?
Does it show how to check if
answers make number/
common sense?
www.activelearnprimary.co.uk
24
Comments
Most tasks are completed on screen.
While learners are frequently
encouraged to try different ways to
tackle the problem in, the on screen
system does not allow learners to
compare their previous results.
Some games show summary tables at
the end of what has been done. No
written explanations are required,
though teachers have discussion
prompts for learners to explain their
methods.
Teachers have question prompts for
discussing answers and methods, but
the difference is not very explicit.
Some verbal prompts reminding
pupils of the problem’s goal.
No.
No, though this may occur through
class discussion.
Review of numerical reasoning teaching resources
Covered
suitably for
year group?
Badger Maths Problem Solving
Criteria
Selecting and applying methods
Understanding the question
Is the context real-life, abstract,
mathematical or crosscurricular?
Does it have single- or multistep problems?
Does it encourage the
use of strategies for
working out what the
question is asking? (e.g.
highlighting key words)
Does it use/explain
appropriate
mathematical language?
Does it help learners to
interpret questions with
diagrams and minimal
language?
Does it provide
opportunities for learners
to select appropriate
methods?
Does it encourage
learners to select the
most efficient method?
Does it approach
problem solving in more
general terms, such as
solving puzzles,
identifying and extracting
patterns and applying
them to the question?
Does it encourage
calculator use or give
opportunities to practise
calculator skills?
(Year 5 and above only)
Real-life and abstract.
Covered
suitably for
year group?

Single- and multi-step.

Teacher guidance suggests
learners underline or note down
information in the task that may be
important.

No specific ‘vocabulary to know’
given but tasks use appropriate
language.
Tasks have short sentences and
uses diagrams and pictures. Pupil
worksheet prompts learners to think
about what they know and what
they need to find out.
Teacher guidance suggests
reminding learners of the different
strategies they know. Learners are
prompted to plan how they could
tackle the task.
A strategy is taught in each section,
so learners are cued into which
methods they could consider using.
Questions in each task begin with
the basic level of identifying and
using the pattern and build on
knowledge to extend the application
of the pattern.

Calculators are not required for any
task but could be used.

Comments




Review of numerical reasoning teaching resources 25
Criteria
Recording working
Does it show how to
record working
mathematically and
systematically?
Does it exemplify how to
show working and how
to explain methods used
and answers given?
Reflecting on answers
Does it distinguish
between how to show
working and how to
explain the methods
used and answers
given?
Are learners encouraged
to check their
working/answers?
Are learners encouraged
to think of/use alternative
checking techniques?
Does it show how to
check if answers make
number/common sense?
Comments
Teacher guidance and worksheets
explain how to use tables, work
systematically and identify patterns.
Pupil worksheet encourages
learners to write down how they
solve the problem.
Examples of using tables/lists are
given. Teacher guidance says
learners should explain their
methods to others but no examples
of how to do this are given.
Not explicitly.
Teacher guidance says learners
should re-read questions and check
their answers. Task template
prompts learners to reflect on what
they have done.
No specific strategies for checking
answers are given but learners are
asked to think of alternative ways to
check their answers.
No examples of how to check for
number/common sense but gives
prompts in teacher guidance to
check answers are sensible.
Covered
suitably for
year group?




www.nationalstemcentre.org.uk/elibrary/resource/258/badger-maths-problem-solving
Badger Maths: Problem Solving, Years 1&2, M Nathan, Badger Publishing Ltd, 2007
ISBN 978-1-84691-206-1
Badger Maths: Problem Solving, Year 3, A Seed, Badger Publishing Ltd 2007
ISBN 978-1-84691-140-8
Badger Maths: Problem Solving, Years 4&5, S Shapiro, Badger Publishing Ltd, 2002
ISBN 978-1-85880-358-6
Badger Maths: Problem Solving, Year 6, S Shapiro, Badger Publishing Ltd, 2002
ISBN 978-1-85880-359-3
26
Review of numerical reasoning teaching resources
BEAM’s Big Book of Word Problems and Interactive CD
Criteria
Selecting and applying methods
Understanding the question
Is the context real-life, abstract,
mathematical or cross-curricular?
Does it have single- or multi-step
problems?
Does it encourage the use of
strategies for working out what
the question is asking? (e.g.
highlighting key words)
Does it use/explain
appropriate mathematical
language?
Does it help learners to
interpret questions with
diagrams and minimal
language?
Does it provide opportunities
for learners to select
appropriate methods?
Does it encourage learners to
select the most efficient
method?
Does it approach problem
solving in more general terms,
such as solving puzzles,
identifying and extracting
patterns and applying them to
the question?
Does it encourage calculator
use or give opportunities to
practise calculator skills?
(Year 5 and above only)
Covered
suitably for
year group?

Mixture of real-life with abstract.
Comments
Single-step.

Yes, through class discussion.
Lesson plans are such that the
projected/big book questions are
discussed and solved as a
class, before learners attempt
further questions independently.
This is focused on in the class
discussion at the beginning of
the session.
No, these are word problems
and so are language-based.

Through class discussion, yes.

As above.


No.
Calculators are not required for
any task but could be used.

Review of numerical reasoning teaching resources 27
Criteria
Reflecting on answers
Recording working
Does it show how to record
working mathematically and
systematically?
Does it exemplify how to show
working and how to explain
methods used and answers
given?
Does it distinguish between
how to show working and how
to explain the methods used
and answers given?
Are learners encouraged to
check their working/answers?
Comments
Covered
suitably for
year group?
No, worksheets do not require
working to be shown.
No, explaining methods is not
required.
No.
The lesson plans provide time
for learners to compare their
answers with a classmate and to
discuss any differences found.
No.

Are learners encouraged to
think of/use alternative
checking techniques?
Does it show how to check if
No.
answers make
number/common sense?
www.nelsonthornes.com/shop/nt/beams-big-book-of-word-problems-year-3-and-4-set
www.nelsonthornes.com/shop/nt/word-problems-interactiveyears-3---4
BEAM's Big Book of Word Problems, M Askew, BEAM Education (Nelson Thornes),
2004, 2005
ISBN 978-1903142332 (Years 1-2)
ISBN 978-1903142325 (Years 3-4)
ISBN 978-1903142349 (Years 5-6)
Word Problems Interactive Years 3 and 4, M Askew, BEAM Education (Nelson Thornes),
2007 ISBN 978-1-906224-25-7 (Years 1-2)
ISBN 978-1-906224-26-4 (Years 3-4)
ISBN 978-1-906224-27-1 (Years 5-6)
28
Review of numerical reasoning teaching resources
Bowland Maths (website)
Criteria
Is the context real-life, abstract or
cross-curricular?
Understanding the question
Does it have single- or multi-step
problems?
Does it encourage the use of
strategies for working out what
the question is asking? (e.g.
highlighting key words)
Does it use/explain appropriate
mathematical language?
Does it help learners to interpret
questions with diagrams and
minimal language?
Selecting and applying methods
Does it provide opportunities for
learners to select appropriate
methods?
Does it encourage learners to
select the most efficient
method?
Does it approach problem
solving in more general terms,
such as solving puzzles,
identifying and extracting
patterns and applying them to
the question?
Does it encourage calculator
use and give opportunities to
practice calculator skills? (Year
5 and above only)
Comments
Covered
suitably for
year group?
Mixed. Some real-life, some abstract.
A lot of cross-curricular content, e.g.
a lot of biology in AstroZoo (oxygen
and carbon dioxide levels).
Multi-step. Each case study is
intended to take a few lessons to
complete.
Each activity contains written
instructions but key words etc. are
not emphasised. Comprehensive
teacher guides allow teachers to
assist where necessary.
Some activities suggest displaying
the key mathematical terms with
pictorial representations to aid
understanding. Often, the maths is
‘hidden’ in the game context. Some
activities are more explicitly mathsbased than others.
Some activities present the majority
of the instructions through spoken
word (video clips) rather than written
language, however some form of
language is always used.
Some tasks are more direct in
guiding learners through the method
to be used, while others provide
opportunities for learners to devise
their own methods. Teacher
guidance presents appropriate
methods, so teachers could pass this
on to struggling students.
Not explicitly, though lots of the
activities involve repetition, so pupils
learn through trial and error.
Yes, a lot of the activities are
puzzle/pattern based – a strong focus
on trial and error to solve problems.

Some tasks provide opportunities for
using calculators.





Review of numerical reasoning teaching resources 29
Reflecting on
answers
Recording working
Criteria
Does it show how to record
working mathematically and
systematically?
Does it exemplify how to show
working and how to explain
methods used and answers
given?
Does it distinguish between how
to show working and how to
explain the methods used and
answers given?
Are learners encouraged to
check their working/answers?
Are learners encouraged to
think of/use alternative checking
techniques?
Does it show how to check if
answers make number/common
sense?
Comments
Yes, some activities provide
examples/worksheets to present
answers in a systematic way.
Exemplifies how to show working.
Many activities ask for explanations,
but this tends to be verbally in
groups, or as class activities.
Yes, both are expected in the
activities, as separate tasks.
No, instant feedback given from the
game.
No.
No. Sometimes hints are given if
pupils struggle to reach the correct
answer.
www.bowlandmaths.org.uk
30
Review of numerical reasoning teaching resources
Covered
suitably for
year group?



Can Do Problem Solving
Criteria
Selecting and applying methods
Understanding the question
Is the context real-life, abstract,
mathematical or cross-curricular?
Does it have single- or multi-step
problems?
Does it encourage the use
strategies for working out what
the question is asking? (e.g.
highlighting key words)
Does it use/explain appropriate
mathematical language?
Does it help learners to interpret
questions with diagrams and
minimal language?
Does it provide opportunities for
learners to select appropriate
methods?
Does it encourage learners to
select the most efficient method?
Does it approach problem
solving in more general terms,
such as solving puzzles,
identifying and extracting
patterns and applying them to
the question?
Does it encourage calculator use
or give opportunities to practise
calculator skills?
(Year 5 and above only)
Comments
Real-life and some crosscurricular.
Single-step (multi-step
problems broken down for
learners in the lesson plan).
Some multi-step problems in
the problem bank.
Yes, there are 5 stages to
problem solving taught, and
the first always involves
picking the key words/themes
from the question (‘read and
think’).
The teacher discusses
language of the question with
the class as part of ‘read and
think’.
No, discussion-based
learning, so lots of language
used.
Yes, the series teaches 9
distinct strategies, with
appropriate ones discussed in
the ‘consider a strategy’
stage.
Yes, through the ‘consider a
strategy’ discussion.
Patterns are discussed in the
appropriate questions, with
learners using discussions
about patterns to help them
solve the problem.
Year 5 and 6 books include
problems that require a
calculator.
Covered
suitably for
year group?








Review of numerical reasoning teaching resources 31
Criteria
Reflecting on answers
Recording working
Does it show how to record
working mathematically and
systematically?
Does it exemplify how to show
working and how to explain
methods used and answers
given?
Does it distinguish between how
to show working and how to
explain the methods used and
answers given?
Are learners encouraged to
check their working/answers?
Comments
The final stage is ‘report and
record’, where the class
discusses how to record their
work. A lot of the time, this is
only discussion rather than a
practical activity.
Discussed as part of the
lesson plan. Many of the
problems involve a final step
of writing an explanation.
Yes, distinct discussions in
the teacher guide.
The ‘consider’ and ‘report and
record’ stages both can
contain discussions about
checking, though this is not
consistent.
Are learners encouraged to think Yes, a variety of checking
of/use alternative checking
strategies are discussed, e.g.
techniques?
repeating the calculation,
comparing with other
learners, doing the problem in
reverse.
Does it show how to check if
Some discussion about
answers make number/common
making a prediction and
sense?
comparing it to the answer
reached, though no explicit
discussion of checking for
common/number sense.
www.nelsonthornes.com/shop/nt/can-do-problem-solving
Covered
suitably for
year group?






Can do Problem Solving: Teacher’s Book, M McDougall, R Cook and C Atherden,
Nelson Thornes, 2004
ISBN 978-0-7487-8659-6 (Year 1)
ISBN 978-0-7487-8661-9 (Year 2)
Can do Problem Solving: Teacher’s Book, M McDougall, S Foster and L Ankers, Nelson
Thornes, 2004
ISBN 978-0-7487-7735-8 (Year 3)
ISBN 978-0-7487-7738-9 (Year 4)
ISBN 978-0-7487-7741-9 (Year 5)
ISBN 978-0-7487-7744-0 (Year 6)
32
Review of numerical reasoning teaching resources
Creative Activities for Maths Problem Solving
Criteria
Selecting and applying methods
Understanding the question
Is the context real-life, abstract,
mathematical or cross-curricular?
Does it have single- or multi-step
problems?
Does it encourage the
use of strategies for
working out what the
question is asking? (e.g.
highlighting key words)
Does it use/explain
appropriate mathematical
language?
Does it help learners to
interpret questions with
diagrams and minimal
language?
Does it provide
opportunities for learners
to select appropriate
methods?
Does it encourage
learners to select the
most efficient method?
Does it approach problem
solving in more general
terms, such as solving
puzzles, identifying and
extracting patterns and
applying them to the
question?
Does it encourage
calculator use or give
opportunities to practise
calculator skills?
(Year 5 and above only)
Comments
Real-life, abstract and mathematical/
number pattern problems.
Single- and multi-step problems, with
extension tasks.
Encourages learners to discuss what
the problem is and to extract key
information and numbers verbally.
Teacher should ask key questions:
what do we know, what do we need to
find out, what can we use to help
solve the problem?
Some guidance for the teacher to
ensure learners understand certain
words and are familiar with key
vocabulary in the task, e.g. shape
names, but no specific vocabulary to
be known for each year group.
Most tasks are presented aurally by
the teacher with prompts and
diagrams on the board, work sheets
and physical resources.
Encourages learners to discuss
possible methods and share these as
a class. Some tasks focus on
selecting methods so more discussion
to evaluate the methods learners
suggest is encouraged.
Prompts discussion of which methods
could be used if different resources are
available and considering other
methods to decide which would work
best.
One chapter focuses on problems that
involve finding rules and describing
patterns, then applying them to more
complex situations. Extension ideas
follow each task which involve the
same strategies and ideas for learners
to create their own problem based on
what they have done.
Calculators are not required for any
task but could be used for some.
Covered
suitably for
year group?








Review of numerical reasoning teaching resources 33
Criteria
Recording working
Does it show how to
record working
mathematically and
systematically?
Reflecting on answers
Does it exemplify how to
show working and how to
explain methods used
and answers given?
Does it distinguish
between how to show
working and how to
explain the methods used
and answers given?
Are learners encouraged
to check their
working/answers?
Are learners encouraged
to think of/use alternative
checking techniques?
Does it show how to
check if answers make
number/common sense?
Comments
Some tasks have more explicit
guidance about how to work
systematically. Some give templates
for tables/diagrams to encourage
systematic working. Structure of the
lesson plan guides learners through
tasks in a systematic way, extending
their knowledge. Encourages sharing
of learners’ systematic working.
All tasks include a reflection section
where learners explain to others how
they tackled the task and displayed
their workings.
Explanations are usually presented
verbally while workings are often
written in tables/diagrams.
Teachers are encouraged to question
learners’ working throughout the task
to prompt learners to clarify what they
are doing. At the end they are asked
how they checked answers but
specific strategies are not provided.
Not explicitly but through discussion
with classmates about how others
approached the same task.
Many tasks are abstract so it may be
difficult to check answers for common
sense.
Covered
suitably for
year group?





www.shop.scholastic.co.uk/products/106
Creative Activities: Maths Problem Solving, Ages 5-7, L Carruthers, Scholastic, 2006
ISBN 978-0439965569
Creative Activities: Maths Problem Solving, Ages 7-11, J Dabell, Scholastic, 2006
ISBN 978-0439965705
34
Review of numerical reasoning teaching resources
Espresso Primary (website)
Criteria
Selecting and applying methods
Understanding the question
Is the context real-life, abstract,
mathematical or cross-curricular?
Does it have single- or multi-step
problems?
Does it encourage the use
of strategies for working out
what the question is asking?
(e.g. highlighting key words)
Does it use/explain
appropriate mathematical
language?
Does it help learners to
interpret questions with
diagrams and minimal
language?
Covered
suitably for
year group?

Real-life, mathematical and some
cross-curricular.

Single- and multi-step problems.
Comments
Poster reminds learners to read
carefully and underline important
words and numbers.

No vocabulary lists given but
mathematical language is used.

For some tasks, instructions do
not appear unless you click on the
help button. Logic tasks have
more reading than pattern finding
tasks. Poster suggests learners
draw a picture to help interpret
problems.
Does it provide opportunities Poster reminds learners to break
for learners to select
the problem down and think about
appropriate methods?
what mathematics they can apply
to it but no specific possible
methods are given.
Does it encourage learners
Poster advises learners to choose
to select the most efficient
the best way to carry out the
method?
calculations needed.
Does it approach problem
Yes, a variety of problems are
solving in more general terms, included, many of which have
such as solving puzzles,
harder levels and build on skills
identifying and extracting
such as finding patterns and
patterns and applying them to applying them to altered
the question?
scenarios.
Does it encourage calculator Calculators are not required for
use or give opportunities to
any task but could be used for
practise calculator skills?
some.
(Year 5 and above only)




Review of numerical reasoning teaching resources 35
Criteria
Recording working
Does it show how to record
working mathematically and
systematically?
Does it exemplify how to
show working and how to
explain methods used and
answers given?
Reflecting on answers
Does it distinguish between
how to show working and
how to explain the methods
used and answers given?
Are learners encouraged to
check their
working/answers?
Are learners encouraged to
think of/use alternative
checking techniques?
Does it show how to check if
answers make
number/common sense?
Comments
Systematic working is not taught
but some tasks provide space to
show workings. Poster reminds
learners to show their working.
Tasks do not show learners how
to show working and none require
explanation of methods. Poster
reminds learners to explain how
they work out their answers.
Poster refers to these points
separately in the steps to solving
word problems.
Some tasks allow learners to
check their answers and all online
resources give feedback if
learners give incorrect responses
and they are then able to have
another attempt. Poster asks
learners to think about how they
can check their answers.
No.

Poster asks learners to think
about whether their answer
makes sense and looks correct.

www.espresso.co.uk
36
Covered
suitably for
year group?
Review of numerical reasoning teaching resources
Investigations with Numicon
Criteria
Selecting and applying methods
Understanding the
question
Is the context real-life,
abstract, mathematical or
cross-curricular?
Does it have single- or multistep problems?
Does it encourage the
use of strategies for
working out what the
question is asking? (e.g.
highlighting key words)
Does it use/explain
appropriate
mathematical language?
Does it help learners to
interpret questions with
diagrams and minimal
language?
Does it provide
opportunities for
learners to select
appropriate methods?
Does it encourage
learners to select the
most efficient method?
Does it approach
problem solving in more
general terms, such as
solving puzzles,
identifying and
extracting patterns and
applying them to the
question?
Does it encourage
calculator use or give
opportunities to practise
calculator skills?
(Year 5 and above only)
Comments
Mathematical.
Covered
suitably for
year group?


Multi-step.
Problems are delivered by the teacher
and discussed so learners do not need
to interpret written instructions alone.
Mathematical language is used and
some terms are defined.

Challenges are numerical. Diagrams
may be used to find solutions but do not
need to be interpreted in the question.
Each challenge lists resources and
methods that could be used to tackle it.
Resources could be applied to different
challenges and using them in different
challenges teaches learners how learnt
methods can be adapted to fit new
problems.
Through trial-and-error tasks learners
can discuss how different methods
work. Many tasks suggest starting
using apparatus but learners can move
on to using numbers when they have
developed a deeper understanding of
the task.
All challenges focus on finding and
testing patterns and using the
apparatus to solve different problems in
similar ways.

Tasks do not require calculator use and
encourage using physical resources but
calculators could be incorporated to
check answers and as an alternative
method.



Review of numerical reasoning teaching resources 37
Criteria
Recording working
Does it show how to
record working
mathematically and
systematically?
Does it exemplify how to
show working and how
to explain methods used
and answers given?
Reflecting on answers
Does it distinguish
between how to show
working and how to
explain the methods
used and answers
given?
Are learners
encouraged to check
their working/answers?
Are learners
encouraged to think
of/use alternative
checking techniques?
Does it show how to
check if answers make
number/common
sense?
Comments
Teacher guidance and question
prompts encourage learners to work
and record working systematically, with
some examples given in teacher book.
No structured answer sheet provided.
Learners are encouraged to find their
own ways to work and record
systematically.
Some exemplification of how to start
working systematically in teacher guide.
Learners are not required to give
written explanations of their methods
but teacher questions encourage
discussion and evaluation of methods.
Learners record working and answers
but do not need to record explanation.
The difference is not made explicit.
Some teacher questions are to prompt
learners to check they have found all
the possible solutions.
Alternative techniques are not given,
though learners may discover some
through discussing different methods of
working with others.
Challenges are mathematical so difficult
to apply common sense to answers.
Covered
suitably for
year group?


https://global.oup.com/education/product/9780955394966?region=uk
Numicon: Investigations with Numicon, M Mills, Oxford University Press, 2010
ISBN 978-0955394966
38
Review of numerical reasoning teaching resources
Investigative and Problem-Solving Approaches to
Mathematics and Their Assessment
Criteria
Selecting and applying methods
Understanding the question
Is the context real-life, abstract,
mathematical or cross-curricular?
Does it have single- or multi-step
problems?
Does it encourage the use of
strategies for working out what
the question is asking? (e.g.
highlighting key words)
Does it use/explain appropriate
mathematical language?
Does it help learners to interpret
questions with diagrams and
minimal language?
Does it provide opportunities for
learners to select appropriate
methods?
Does it encourage learners to
select the most efficient
method?
Does it approach problem
solving in more general terms,
such as solving puzzles,
identifying and extracting
patterns and applying them to
the question?
Comments
Real-life.
Covered
suitably for
year group?

Single- and multi-step
problems. Multi-step tends to
be broken down into a series of
single steps.
Teacher guide states that
learners need to understand
the reason for doing the task
but strategies for working out
what the question is asking are
not given.
Quite technical mathematical
language used but without
explanation.
Most tasks are quite wordy and
explanatory. Diagrams are
often used alongside written
instructions, to support
understanding.
No, says to choose the
appropriate method but not
how to decide what is
appropriate.
Same as above.

Teacher guidance suggests
getting learners to look for
patterns as they work, and
generalise these as the
problem progresses. Not
explicitly taught.

Does it encourage calculator use Calculators are not required for
or give opportunities to practise
any task but could be used.
calculator skills?
(Year 5 and above only)

Review of numerical reasoning teaching resources 39
Criteria
Recording working
Does it show how to record
working mathematically and
systematically?
Reflecting on
answers
Does it exemplify how to show
working and how to explain
methods used and answers
given?
Does it distinguish between how
to show working and how to
explain the methods used and
answers given?
Are learners encouraged to
check their working/answers?
Are learners encouraged to think
of/use alternative checking
techniques?
Does it show how to check if
answers make number/common
sense?
Covered
suitably for
year group?

Teacher guidance gives a good
summary of what learners
should write down in giving
their answer. It is up to the
teacher to deliver this as it is
not part of the learner activities.

As above. This is discussed in
the teacher guidance but relies
on the teacher to pass this on
to their learners.
As above. This is (vaguely)
discussed in the teacher guide
which states that there is a
difference between giving an
explanation and writing down
everything.
No.
Comments
No.
Not really. Some very vague
suggestion of “opening up”
problems but not quite
addressing the same skill.
www.nationalstemcentre.org.uk/elibrary/resource/5654/investigative-and-problemsolving-approaches-to-mathematics-and-their-assessment
Investigative and Problem-Solving Approaches to Mathematics and Their Assessment, A
Onion, Framework Press, 1990
ISBN 1 85008 042 9
40
Review of numerical reasoning teaching resources
KS1/2 Problem Solving
Criteria
Reflecting on
answers
Recording working
Selecting and applying
methods
Understanding the
question
Is the context real-life, abstract,
mathematical or cross-curricular?
Does it have single- or multi-step
problems?
Does it encourage the use of
strategies for working out what
the question is asking? (e.g.
highlighting key words)
Does it use/explain appropriate
mathematical language?
Does it help learners to interpret
questions with diagrams and
minimal language?
Does it provide opportunities for
learners to select appropriate
methods?
Does it encourage learners to
select the most efficient method?
Does it approach problem solving
in more general terms, such as
solving puzzles, identifying and
extracting patterns and applying
them to the question?
Does it encourage calculator use
or give opportunities to practise
calculator skills?
(Year 5 and above only)
Does it show how to record
working mathematically and
systematically?
Does it exemplify how to show
working and how to explain
methods used and answers
given?
Does it distinguish between how
to show working and how to
explain the methods used and
answers given?
Are learners encouraged to
check their working/answers?
Are learners encouraged to think
of/use alternative checking
techniques?
Does it show how to check if
answers make number/common
sense?
Comments
Real-life.
Covered
suitably for
year group?

Single-step.

Some key words in bold for
the youngest learners.

No aids beyond emboldened
words.
Picture-based problems
presented with a sentence or
two explaining what to do.
No, these workbooks do not
have a ‘teaching’ element.

No.
No, the workbooks are simply
‘question and answer’ in
style. There is a build up of
skills required from the first to
last booklet, however.
The hardest booklet (KS2
book 4) contains calculator
questions, though the skills
are not taught.
No. Provides space for
working in the later books.
Provides space for working
and some of the later
questions provide space for
an explanation. These are
not exemplified.
Different spaces are provided
for working and providing
explanations.


Not explicitly.
No.
No.
Review of numerical reasoning teaching resources 41
www.schofieldandsims.co.uk/product/215/ks1-problem-solving-book-1
www.schofieldandsims.co.uk/product/442/ks2-problem-solving-book-1
KS1 Problem Solving: Book 1-3, A Forster and P Martin, Schofield and Sims, 2005
KS2 Problem Solving: Book 1-3, A Forster and P Martin, Schofield and Sims, 2005
KS2 Problem Solving Book 4, A Montague-Smith, Schofield and Sims, 2008
42
Review of numerical reasoning teaching resources
Let's Think Through Maths!
Criteria
Recording working
Selecting and applying
methods
Understanding the
question
Is the context real-life, abstract,
mathematical or cross-curricular?
Does it have single- or multi-step
problems?
Does it encourage the use of
strategies for working out what
the question is asking? (e.g.
highlighting key words)
Does it use/explain appropriate
mathematical language?
Does it help learners to interpret
questions with diagrams and
minimal language?
Does it provide opportunities for
learners to select appropriate
methods?
Does it encourage learners to
select the most efficient method?
Does it approach problem solving
in more general terms, such as
solving puzzles, identifying and
extracting patterns and applying
them to the question?
Does it encourage calculator use
or give opportunities to practise
calculator skills?
(Year 5 and above only)
Does it show how to record
working mathematically and
systematically?
Does it exemplify how to show
working and how to explain
methods used and answers
given?
Does it distinguish between how
to show working and how to
explain the methods used and
answers given?
Comments
Real-life.
Covered
suitably for
year group?

Single-step.

Focus on group discussion,
so the teacher explains the
concept and appropriate key
words as part of the lesson.
See above.

No, this is a very languagebased set of resources.
Yes, methods are discussed
as part of the lesson plan.

Yes, through discussion.

Yes, problems tend to build
up throughout the lesson,
starting with something like a
pattern and then looking at
the mathematics behind it.
n/a

n/a
No, focus on discussion
rather than written maths.
No, see above.
No, see previous comments.
Review of numerical reasoning teaching resources 43
Criteria
Reflecting on answers
Are learners encouraged to
check their working/answers?
Are learners encouraged to think
of/use alternative checking
techniques?
Does it show how to check if
answers make number/common
sense?
Comments
Covered
suitably for
year group?
There are prompts for
discussion of methods for
checking that answers are
correct, though learners are
not actually asked to check
their answers.
Discussion about checking
but for very simple problems,
e.g. counting to 6.
No.
www.gl-assessment.co.uk/products/lets-think-through-maths
Let's Think Through Maths!, M Shayer, M Adhami and A Robertson, GL Assessment Ltd,
2004
ISBN 9780708718018
Let's Think Through Maths! 6-9, M Adhami, M Shayer and S Twiss, GL Assessment Ltd,
2005
ISBN 9780708715079
44
Review of numerical reasoning teaching resources
Macmillan Maths Problem Solving Boxes (1-6)
Criteria
Selecting and applying methods
Understanding the question
Is the context real-life,
abstract, mathematical or
cross-curricular?
Does it have single- or multistep problems?
Does it encourage the
use of strategies for
working out what the
question is asking? (e.g.
highlighting key words)
Does it use/explain
appropriate
mathematical language?
Does it help learners to
interpret questions with
diagrams and minimal
language?
Does it provide
opportunities for
learners to select
appropriate methods?
Does it encourage
learners to select the
most efficient method?
Does it approach
problem solving in more
general terms, such as
solving puzzles,
identifying and
extracting patterns and
applying them to the
question?
Does it encourage
calculator use or give
opportunities to practise
calculator skills?
(Year 5 and above only)
Comments
Real-life, mathematical and a few
abstract.
Covered
suitably for
year group?

Single- and multi-step problems.

Very clear guidance for learners to write
down or underline key words, and
restate the question in their own words.

Teachers are told to ensure learners
understand the mathematical terms
used. No specific vocabulary list is
given. Questions use appropriate
mathematical language.
Gives strategies of how to interpret
questions. Language is simple and
minimal and some questions require
interpretation of diagrams.
Various approaches are taught
throughout the series and
demonstrated by the teacher (looking
for patterns, sensible guesses, using
diagrams etc.) so learners can work
through and see which approach works
for each problem. Persistence and
trying different approaches if the
problem seems too difficult at first are
encouraged.
Encourages discussion of the strategies
used when the teacher models how to
approach a problem. Class discussion of
methods used is encouraged once tasks
have been completed.
Problems are labelled with the key
strategies they involve and teacher
guidance suggests doing sets of
questions that require the same/similar
strategies to develop use of that skill.
Extension questions and opportunities to
write their own questions develop
learners’ understanding of each pattern
they discover.
Calculators are not required for any
task but could be used for some.
(In Year 1 there is reference to
calculators in terms of number of light
bars in each digit.)






Review of numerical reasoning teaching resources 45
Criteria
Reflecting on answers
Recording working
Does it teach how to
record working
mathematically and
systematically?
Does it exemplify how to
show working and how
to explain methods used
and answers given?
Does it distinguish
between how to show
working and how to
explain the methods used
and answers given?
Are learners encouraged
to check their
working/answers?
Comments
Learners are prompted to think about
whether a table or diagram would help
them solve the problem. Teacher
guidance says that learners can either
construct tables themselves or fill out
tables constructed by the teacher,
depending on their level of
understanding of how to use tables.
No explicit exemplification of showing
working and explaining answers.
Covered
suitably for
year group?

Little emphasis on explaining methods
and answers, especially in written
format.
Yes, once learners have an answer
they are encouraged to feed it back into
the question to check it meets all the
rules.
Learners are taught to work backwards
and check answers by re-reading the
question to see if their answer fits.
No explicit reference to number/common
sense.


Are learners taught
alternative checking
techniques?
Does it show how to
check if answers make
number/common sense?
http://schools.mcgraw-hill.co.uk/primary-education/numeracy-for-primary-education/
Macmillan Maths Problem Solving: Boxes 1–3, S Payne and L Jordan, Macmillan
Education Australia, 2011
ISBN 978-1-4202-9393-7 (Box 1)
ISBN 978-1-4202-9394-4 (Box 2)
ISBN 978-1-4202-9395-1 (Box 3)
Macmillan Maths Problem Solving, Box 4, S Payne and L Jordan, Macmillan Education
Australia, 2012
ISBN 978-1-4202-9396-8 (Box 4)
Macmillan Maths Problem Solving: Boxes 5–6, S Payne, Macmillan Education Australia,
2012
ISBN 978-1-4202-9397-5 (Box 5)
ISBN 978-1-4202-9398-2 (Box 6)
46
Review of numerical reasoning teaching resources
Main Activity: Problem Solved!
Criteria
Selecting and applying methods
Understanding the question
Is the context real-life, abstract,
mathematical or cross-curricular?
Does it have single- or multi-step
problems?
Does it encourage the use of
working out what the question is
asking? (e.g. highlighting key
words)
Does it use/explain appropriate
mathematical language?
Does it help learners to interpret
questions with diagrams and
minimal language?
Does it provide opportunities for
learners to select appropriate
methods?
Does it encourage learners to
select the most efficient method?
Does it approach problem
solving in more general terms,
such as solving puzzles,
identifying and extracting
patterns and applying them to
the question?
Does it encourage calculator use
or give opportunities to practise
calculator skills?
(Year 5 and above only)
Comments
Real life. One rather abstract
section – “Nonsense words”.
Mixed. Multi-step problems
are not explicitly taught,
though every question
contains teacher prompts to
assist learners.
Yes, each section has a list of
key words that the teacher is
expected to discuss with the
class. Every question also
contains teacher prompts
such as “What method should
we use? What do we need to
do?”
See above, yes.
Covered
suitably for
year group?




No, question instructions are
quite lengthy.
Yes, this is one of the
discussion prompts for every
question. There are also
helpful hints in the pupil book.
Yes, through discussion and
prompts.
No.

Yes, some items are explicitly
for calculators. The teacher
guide provides teacher
prompts to discuss and learn
how to use the calculator
appropriately in these
questions.


Review of numerical reasoning teaching resources 47
Criteria
Reflecting on
answers
Recording working
Does it show how to record
working mathematically and
systematically?
Does it exemplify how to show
working and how to explain
methods used and answers
given?
Does it distinguish between how
to show working and how to
explain the methods used and
answers given?
Are learners encouraged to
check their working/answers?
Are learners encouraged to think
of/use alternative checking
techniques?
Does it show how to check if
answers make number/common
sense?
Comments
Yes, through
discussion/teacher prompts.
Covered
suitably for
year group?

No examples of working but
the last section exemplifies
explaining methods.
Classroom discussion about
methods used is expected.
Yes.

Yes.

Yes, section on checking
techniques.

Discusses estimating.


http://www.badgerlearning.co.uk/eCommerce/search.aspx?loc=1&q=problem%20solved
&c=0
Main Activity: Problem Solved! Book 1, B Fillis, Badger Publishing Ltd, 2003
ISBN 9781-844240-333 (pupil book)
ISBN 9781-844240-347 (teacher book)
Main Activity: Problem Solved! Book 2, B Fillis, Badger Publishing Ltd, 2003
ISBN 9781-844240-364 (pupil book)
ISBN 9781-844240-371 (teacher book)
Main Activity: Problem Solved! Book 3, B Fillis, Badger Publishing Ltd, 2004
ISBN 9781-844240-395 (pupil book)
ISBN 9781-844240-401 (teacher book)
48
Review of numerical reasoning teaching resources
Maths Buzz
Criteria
Recording working
Selecting and applying methods
Understanding the
question
Is the context real-life, abstract,
mathematical or cross-curricular?
Does it have single- or multi-step
problems?
Does it encourage the use of
strategies for working out what
the question is asking? (e.g.
highlighting key words)
Does it use/explain appropriate
mathematical language?
Does it help learners to
interpret questions with
diagrams and minimal
language?
Does it provide opportunities
for learners to select
appropriate methods?
Does it encourage learners to
select the most efficient
method?
Does it approach problem
solving in more general terms,
such as solving puzzles,
identifying and extracting
patterns and applying them to
the question?
Does it encourage calculator
use or give opportunities to
practise calculator skills?
(Year 5 and above only)
Does it show how to record
working mathematically and
systematically?
Does it exemplify how to show
working and how to explain
methods used and answers
given?
Does it distinguish between
how to show working and how
to explain the methods used
and answers given?
Comments
Real-life and mathematical.
Multi-step.
Not really applicable as all the
information to solve the tasks is
written, so more than just a few
words are key.
Lists key vocabulary for each
task.
Tasks require reading a series
of clues in order to sort objects,
so language features fairly
heavily.
Tasks give lots of practice of
the same method of sorting
information and working out
which clues give a starting
point. This skill is developed
within and between the tasks.
Teaches learners how to use
this method of logic most
efficiently, to decide which
clues are key to getting started.
It provides a lot of practice of
solving a certain type of
problem which develops a skill
which can be applied to other
situations.
Covered
suitably for
year group?





Calculators are not suitable for
these tasks.
No written working is involved.
No recording of answers is
required.
There is no written work
required but learners are
encouraged to discuss their
thinking as they complete the
tasks in small groups.
Review of numerical reasoning teaching resources 49
Reflecting on
answers
Criteria
Are learners encouraged to
check their working/answers?
Comments
Teachers are expected to
encourage learners to check
their work.
No.
Covered
suitably for
year group?

Are learners encouraged to
think of/use alternative
checking techniques?
Does it show how to check if
No.
answers make
number/common sense?
www.nelsonthornes.com/webapp/wcs/stores/servlet/SearchDisplay?storeId=10701&cata
logId=10601&langId=1&pageSize=12&beginIndex=0&sType=SimpleSearch&resultCatEntryType=2&showRes
ultsPage=true&pageView=image&errorViewName=&searchTerm=maths+buzz
Maths Buzz: Collaborative Reasoning Challenges, J Murray and L Woodham L, Nelson
Thornes, 2008
ISBN 978-1-906224-53-0 (Years 1-2)
ISBN 978-1-906224-52-3 (Years 3-4)
ISBN 978-1-906224-51-6 (Years 5-6)
50
Review of numerical reasoning teaching resources
Maths Trails, Working Systematically
Criteria
Selecting and applying methods
Understanding the question
Is the context real-life,
abstract, mathematical or
cross-curricular?
Does it have single- or multistep problems?
Comments
Mathematical/number problems.
Single- and multi-step. Learners discuss
introductory problems together to guide
them towards selecting the skills needed
to complete main task.
Does it encourage the
The problem is introduced by the teacher
use of strategies for
by discussing simplified similar tasks first,
working out what the
to show learners how to apply their
question is asking? (e.g. existing knowledge in a less familiar task.
highlighting key words)
Does it use/explain
Lists prerequisite mathematical
appropriate
knowledge that learners need to have to
mathematical
access the tasks.
language?
Does it help learners to Questions vary in how much reading is
interpret questions with required. Some have diagrams.
diagrams and minimal
language?
Does it provide
Learners are guided by teacher and
opportunities for
group discussions to find methods that
learners to select
could be applied to each problem.
appropriate methods?
Does it encourage
Some tasks encourage learners to
learners to select the
discuss their method with others to
most efficient method? decide on the best method to use.
Emphasis is placed on checking that all
the possible solutions will be found with
their chosen method and adapting a
method if it is not working.
Does it approach
The tasks are in mathematical contexts
problem solving in
so the focus is on the method rather than
more general terms,
the content of the problem, making
such as solving
solutions easier to generalise to other
puzzles, identifying and situations.
extracting patterns and
applying them to the
question?
Does it encourage
Tasks do not require calculator use.
calculator use or give
Focus of many tasks is not just
opportunities to
calculating an answer but understanding
practise calculator
the mathematical principles behind it,
skills?
which calculator use may detract from.
(Year 5 and above
only)
Covered
suitably for
year group?






Review of numerical reasoning teaching resources 51
Criteria
Recording working
Does it teach how to
record working
mathematically and
systematically?
Reflecting on answers
Does it exemplify how
to show working and
how to explain
methods used and
answers given?
Does it distinguish
between how to show
working and how to
explain the methods
used and answers
given?
Are learners
encouraged to check
their working/answers?
Are learners taught
alternative checking
techniques?
Does it show how to
check if answers make
number/common
sense?
Comments
Yes, sometimes learners are given a
structure to demonstrate how working
could be recorded systematically but
they are also encouraged to develop
their own ways to record working.
Examples not given but some tasks
require learners to create a poster of
their results and explanations and often
they are asked to explain their methods
and reasoning verbally.
There is a distinction between the results
obtained and the explanation behind
knowing that the method has worked
effectively. The emphasis is on testing the
method more than listing all the answers.
Tasks often require learners to show
how they know they have found all the
answers. The focus is on being confident
that the method is rigorous enough to
find all answers, rather than being
concerned with what the answers
actually are.
Alternative checking techniques may be
found through discussing methods with
other learners, so learners have to find
these themselves rather than being
explicitly taught.
Most tasks are not in a real-life context
so hard to determine common sense of
answers. Learners test their answers
through extension and simplified tasks to
check their answers fit the patterns found
in these related tasks.
Covered
suitably for
year group?





www.education.cambridge.org/uk/search?searchText=maths%20trails
Maths Trails: Working Systematically, J Piggott and L Pumfrey, Cambridge University
Press, 2006
ISBN 978-0521682404
52
Review of numerical reasoning teaching resources
(First) (Essential) Mental Arithmetic
Comments
Is the context real-life, abstract,
mathematical or cross-curricular?
Does it have single- or multi-step
problems?
Does it encourage the use of
strategies for working out what
the question is asking? (e.g.
highlighting key words)
Does it use/explain appropriate
mathematical language?
Does it help learners to interpret
questions with diagrams and
minimal language?
Real-life and mathematical.
Recording working
Selecting and applying methods
Understanding the
question
Criteria
Does it provide opportunities for
learners to select appropriate
methods?
Does it encourage learners to
select the most efficient method?
Does it approach problem solving
in more general terms, such as
solving puzzles, identifying and
extracting patterns and applying
them to the question?
Does it encourage calculator use
or give opportunities to practise
calculator skills?
(Year 5 and above only)
Does it show how to record
working mathematically and
systematically?
Does it exemplify how to show
working and how to explain
methods used and answers
given?
Does it distinguish between how
to show working and how to
explain the methods used and
answers given?
Single- and multi-step.
Covered
suitably for
year group?


No.
Yes, a glossary is given on
the first page.
While most questions require
some reading, language is
minimal for the level of
difficulty of the questions.
No.

Teacher guide suggests
doing group marking
sessions where learners can
share the methods they used
and discuss which are more
efficient and why.
No.


No, the focus is on mental
mathematics.
No, provides space for a
single answer only, no room
for working.
No.
No, doesn’t ask for
explanations.
Review of numerical reasoning teaching resources 53
Reflecting on answers
Criteria
Are learners encouraged to
check their working/answers?
Are learners encouraged to think
of/use alternative checking
techniques?
Comments
Covered
suitably for
year group?
No.
In group marking learners
can discuss their answers
and methods with others,
sharing different ways of
answering questions which
could provide alternative
checking strategies.
No.

Does it show how to check if
answers make number/common
sense?
www.schofieldandsims.co.uk/mental-arithmetic/
Essential Mental Arithmetic
Essential Mental Arithmetic: 1-5, T Goddard, Schofield and Sims, 2010
Essential Mental Arithmetic 6, E Spavin, Schofield and Sims, 2010
First Mental Arithmetic
Mental Arithmetic Teacher’s Guide, A Montague-Smith, Schofield and Sims, 2013
First Mental Arithmetic Key Stage 1: 1-3, S Gardner, Schofield and Sims, 2011
First Mental Arithmetic Key Stage 1 and 2: 4-6), S Gardner, Schofield and Sims, 2011
Mental Arithmetic
Mental Arithmetic Introductory Book Key Stage 2, L Spavin, Schofield and Sims, 2007
Mental Arithmetic Key Stage 2: 1–5, T Goddard, Schofield and Sims, 2007
Mental Arithmetic 6 Key Stage 2, E Spavin, Schofield and Sims, 2007
54
Review of numerical reasoning teaching resources
My Maths (website)
Criteria
Selecting and applying methods
Understanding the question
Is the context real-life, abstract,
mathematical or cross-curricular?
Does it have single- or multi-step
problems?
Does it encourage the use
of strategies for working out
what the question is asking?
(e.g. highlighting key words)
Does it use/explain
appropriate mathematical
language?
Does it help learners to
interpret questions with
diagrams and minimal
language?
Does it provide
opportunities for learners to
select appropriate
methods?
Does it encourage learners
to select the most efficient
method?
Does it approach problem
solving in more general
terms, such as solving
puzzles, identifying and
extracting patterns and
applying them to the
question?
Does it encourage
calculator use or give
opportunities to practise
calculator skills?
(Year 5 and above only)
Comments
Real-life, cross-curricular and
mathematical.
Single- and multi-step problems.
The word problem lesson teaches
how to spot key words and decide
which calculation is required based
on these words, and to beware of
irrelevant information.
Appropriate mathematical
language is used. Explanations of
the use of very basic words are
given, e.g. altogether, each, equal.
Most tasks are quite language
based with some supporting
pictures/diagrams.
Includes a lesson on how to use
trial and improvement effectively.
Other methods are not taught so
explicitly. Many tasks require
learners to devise/select a method.
Methods are not compared within
tasks and differing methods are not
discussed.
The puzzles and methods are
generalisable and some are
repeated in different contexts to
show learners how to apply them to
different situations.
Yes, higher level problems give
opportunities for calculator use,
though calculator skills are not
taught.
Covered
suitably for
year group?






Review of numerical reasoning teaching resources 55
Criteria
Recording working
Does it show how to record
working mathematically and
systematically?
Does it exemplify how to
show working and how to
explain methods used and
answers given?
Reflecting on answers
Does it distinguish between
how to show working and
how to explain the methods
used and answers given?
Are learners encouraged to
check their
working/answers?
Are learners encouraged to
think of/use alternative
checking techniques?
Does it show how to check
if answers make
number/common sense?
Comments
Learners are shown how to use
tables to record trial and
improvement answers.
The use of tables and creating
diagrams to solve problems is
exemplified but learners do not
need to explain the methods they
used to get their answers.
Teacher guidance and learner
resources do not show how to
explain learners’ selection of
methods.
Online tasks give feedback as to
whether answers are correct.
Some tasks prompt learners to
check their answers themselves
first.
For word problems, learners are
shown how to use inverse sums to
check answers.
Not covered.
www.mymaths.co.uk
56
Review of numerical reasoning teaching resources
Covered
suitably for
year group?



New Zealand Maths Problem Solving (website)
Comments
Is the context real-life, abstract,
mathematical or cross-curricular?
Does it have single- or multi-step
problems?
Real-life and mathematical
problems.
Single- and multi-step
problems. Multi-step problems
are broken down into single
steps in lesson plan.
Learners discuss problems in
groups/whole class but mostly
problems are presented
verbally.
Selecting and applying methods
Understanding the question
Criteria
Does it encourage the use
of strategies for working
out what the question is
asking? (e.g. highlighting
key words)
Does it use/explain
appropriate mathematical
language?
Does it help learners to
interpret questions with
diagrams and minimal
language?
Does it provide
opportunities for learners to
select appropriate
methods?
Does it encourage learners
to select the most efficient
method?
Does it approach problem
solving in more general
terms, such as solving
puzzles, identifying and
extracting patterns and
applying them to the
question?
Does it encourage
calculator use or give
opportunities to practise
calculator skills?
(Year 5 and above only)
Covered
suitably for
year group?


Doesn’t give specific
vocabulary but likely that in
discussions misunderstandings
will be raised.
Not really as most tasks are
presented verbally.
Does not provide specific
methods to use; encourages
group discussion of possible
and alternative methods that
could be used.
Some tasks encourage
discussion about which
methods worked and their
similarities and differences but
vague.
Some tasks require learners to
find the pattern, then extension
tasks encourage them to apply
the pattern to a more complex
problem. This happens mostly
within tasks except the ‘Three
in a line’ series which
progresses through the levels.
Some tasks require calculators.
Teacher guidance recommends
checking that learners can use
appropriate functions, rather
than the tasks demonstrating to
learners how to compute
answers.



Review of numerical reasoning teaching resources 57
Criteria
Recording working
Does it show how to record
working mathematically
and systematically?
Does it exemplify how to
show working and how to
explain methods used and
answers given?
Reflecting on answers
Does it distinguish between
how to show working and
how to explain the methods
used and answers given?
Are learners encouraged to
check their
working/answers?
Are learners encouraged to
think of/use alternative
checking techniques?
Does it show how to check
if answers make
number/common sense?
Comments
Some pupil worksheets provide
suggestions for how to layout
workings/answers e.g. grids
and tables to work through
problems systematically.
Some tasks provide templates
for how to present working.
Some tasks encourage learners
to share their answers and
reasoning with partners/groups
but no exemplification of this is
provided.
Explaining methods is usually
done verbally while showing
working can be in written forms
so the distinction is made.
Not explicitly, although learners
often work in groups so
discussions may lead to
checking answers.
No.
No.
www.nzmaths.co.nz/problem-solving-information
www.nzmaths.co.nz/problem-solving
58
Review of numerical reasoning teaching resources
Covered
suitably for
year group?


Numicon Teaching Resource Handbooks *
Criteria
Understanding the question
Is the context real-life, abstract,
mathematical or crosscurricular?
Does it have single- or multistep problems?
Does it encourage the
use of strategies for
working out what the
question is asking?
(e.g. highlighting key
words)
Does it use/explain
appropriate
mathematical
language?
Does it help learners to
interpret questions with
diagrams and minimal
language?
Selecting and applying methods
Does it provide
opportunities for
learners to select
appropriate methods?
Does it encourage
learners to select the
most efficient method?
Does it approach
problem solving in more
general terms, such as
solving puzzles,
identifying and
extracting patterns and
applying them to the
question?
Does it encourage
calculator use or give
opportunities to practise
calculator skills?
(Year 5 and above only)
Comments
Real-life and mathematical.
Single- and multi-step problems.
Covered
suitably for
year group?


Focus is on verbal communication so
many tasks are presented verbally
and discussed to determine learners’
understanding.
A list of mathematical vocabulary is
given for each set of activities.
Teacher models use of appropriate
mathematical language.
Class tasks are presented verbally by
the teacher but may have supporting
diagrams/apparatus presented
alongside. Explorer Progress books
are for learners to work through
independently, language demand is
minimal and some include diagrams.
Yes, learners progress from using
physical objects to numerals for
tackling problems. Learners are
encouraged to devise their own
methods. They then share their
methods with others.
Through discussion, learners see
what other methods could have been
used and develop confidence of using
numerals and notation moving on
from physical objects.
Yes, tasks are developed by first
using apparatus and then applying the
patterns found to numerical problems.
At the end of each strand, the teacher
prompts discussion about how what
has been learnt could be applied to
solving problems.

n/a
n/a




Review of numerical reasoning teaching resources 59
Criteria
Recording working
Does it show how to
record working
mathematically and
systematically?
Does it exemplify how
to show working and
how to explain methods
used and answers
given?
Reflecting on answers
Does it distinguish
between how to show
working and how to
explain the methods
used and answers
given?
Are learners
encouraged to check
their working/answers?
Are learners
encouraged to think
of/use alternative
checking techniques?
Does it show how to
check if answers make
number/common
sense?
Comments
Systematic working is emphasised,
especially through using tables and
drawing diagrams. Learners are
encouraged to find their own ways to
record working systematically. Those
who do are asked to share the way
they worked with others.
A big emphasis on working
systematically and finding ways to
record workings (e.g. tables) to
ensure all possible answers are
found. Emphasis on mathematical
communication and learners are
asked to explain their findings after
every task, usually verbally.
Learners may show answers through
diagrams, written working or
manipulation of objects, whereas
explanation of methods is done
through discussion.
Covered
suitably for
year group?



Teacher prompts encourage learners
to check how they know their answers
are correct and complete.
Yes, through using different apparatus
and making estimates then checking
these against final answers.

Some tasks ask learners to estimate
an answer early on then check this
against their final answer. Many tasks
do not have real-life contexts so it is
difficult to check for common sense.


*to be published spring
2014https://global.oup.com/education/searchresults?search_input=numicon+easy
+buy&region=uk
https://global.oup.com/education/searchresults?search_input=numicon+easy+buy&regio
n=uk
60
Review of numerical reasoning teaching resources
Problem-Solving Strategies for Efficient and Elegant
Solutions: Grades 6-12
Criteria
Selecting and applying methods
Understanding the
question
Is the context real-life, abstract,
mathematical or cross-curricular?
Does it have single- or multi-step
problems?
Does it encourage strategies
for working out what the
question is asking? (e.g.
highlighting key words)
Does it use/explain
appropriate mathematical
language?
Does it help learners to
interpret questions with
diagrams and minimal
language?
Does it provide opportunities
for learners to select
appropriate methods?
Does it encourage learners to
select the most efficient
method?
Does it approach problem
solving in more general terms,
such as solving puzzles,
identifying and extracting
patterns and applying them to
the question?
Does it encourage calculator
use or give opportunities to
practise calculator skills?
(Year 5 and above only)
Comments
Real-life.
Covered
suitably for
year group?


Multi-step.
No, focus is on choosing the
correct method.
Not explicitly. Teachers may
choose to do this if they teach
the examples.
While all questions require
some reading, language is
minimal for the level of demand
of the questions.
Yes, ten methods are taught,
with a focus on choosing which
is the most appropriate for each
question.
Yes, see above.



One chapter on finding patterns
but for the most part no,
problems are specific.
Many questions provide
opportunities for using
calculators but there are no
explicit instructions to
encourage their use.

Review of numerical reasoning teaching resources 61
Criteria
Recording working
Does it show how to record
working mathematically and
systematically?
Does it exemplify how to show
working and how to explain
methods used and answers
given?
Reflecting on
answers
Does it distinguish between
how to show working and how
to explain the methods used
and answers given?
Are learners encouraged to
check their working/answers?
Are learners encouraged to
think of/use alternative
checking techniques?
Does it show how to check if
answers make
number/common sense?
Comments
Provides solutions to the
problems in a systematic style.
It is up to individual teachers to
teach their class how to use this
approach.
As above, working shown in the
book, but up to teachers to pass
this on to their class.
Explanations are not focused
on.
The whole textbook is about
explaining methods; however
this is not consistently part of
the example answer to the
problems within the book.
Not explicitly.
Covered
suitably for
year group?


No.
No.
www.corwin.com/books/Book232159
Problem-Solving Strategies for Efficient and Elegant Solutions, Grades 6-12, A S
Posamentier and S Krulik, Corwin Press (Sage Ltd), 2008
ISBN 978-1412959704
62
Review of numerical reasoning teaching resources
Problem solving with EYFS, Key Stage One and Key Stage
Two Children
Criteria
Comments
Is the context real-life, abstract,
mathematical or cross-curricular?
Mostly real-life – a few
anthropomorphised animals in
“Finding all possibilities”.
Discussion-based, large multistep activities are broken down in
the teacher guide to a series of
single-step problems.
Yes, underlining/highlighting key
words suggested in teacher
guide.
Selecting and applying methods
Understanding the
question
Does it have single- or multi-step
problems?
Does it encourage the use of
strategies for working out
what the question is asking?
(e.g. highlighting key words)
Does it use/explain
appropriate mathematical
language?
Does it help learners to
interpret questions with
diagrams and minimal
language?
Does it provide opportunities
for learners to select
appropriate methods?
Does it encourage learners
to select the most efficient
method?
Does it approach problem
solving in more general
terms, such as solving
puzzles, identifying and
extracting patterns and
applying them to the
question?
Does it encourage calculator
use or give opportunities to
practise calculator skills?
(Year 5 and above only)
Yes, teaches mathematical
vocabulary as part of the activity.
No, pupil worksheets have quite a
lot of text, and teachers are
expected to talk through these as
well.
Teachers asked to discuss choice
of method in teacher guidance.
Learners are encouraged to
discuss their method with a
partner/the class.
As above, this is implied through
encouraging learners to discuss
and explain their choice of
method.
Yes, each booklet is focused on a
particular skill, e.g. puzzles,
patterns, multiple possibilities.
These are built upon throughout
the booklet.
A few Year 5/6 activities allow for
calculators if learners wish to use
them.
Covered
suitably for
year group?








Review of numerical reasoning teaching resources 63
Criteria
Recording working
Does it show how to record
working mathematically and
systematically?
Does it exemplify how to
show working and how to
explain methods used and
answers given?
Does it distinguish between
how to show working and
how to explain the methods
used and answers given?
Reflecting on answers
Are learners encouraged to
check their
working/answers?
Are learners encouraged to
think of/use alternative
checking techniques?
Does it show how to check if
answers make
number/common sense?
Comments
Teachers encouraged to discuss
systematic working after the main
activity, drawing on examples
from the class.
As above, guidelines not given,
but used as a discussion/learning
topic at the end of the activity.
Explanations tend to be done
verbally as a class activity, rather
than written, but the distinction is
made between answers and
explanations.
Teacher guide asks teachers to
remind learners to check their
work, emphasis on checking
against the clues in logic puzzles.
No, just prompted to do so, rather
than taught.
Covered
suitably for
year group?




No.
www.nationalstemcentre.org.uk/elibrary/resource/4588/problem-solving-with-eyfs-keystage-one-and-key-stage-two-children
Problem Solving with EYFS, Key Stage 1 and Key Stage 2 Children: Logic Problems and
Puzzles, Department for Education, 2010
Finding all Possibilities, Department for Education, 2004
Problem Solving with EYFS, Key Stage 1 and Key Stage 2 Children: Finding Rules and
Describing Patterns, Department for Education, 2010
64
Review of numerical reasoning teaching resources
Talk it, solve it: Reasoning Skills in Maths
Criteria
Understanding the question
Is the context real-life, abstract,
mathematical or cross-curricular?
Does it have single- or multi-step
problems?
Does it encourage the use of
strategies for working out what
the question is asking? (e.g.
highlighting key words)
Does it use/explain
appropriate mathematical
language?
Selecting and applying methods
Does it help learners to
interpret questions with
diagrams and minimal
language?
Does it provide opportunities
for learners to select
appropriate methods?
Does it encourage learners to
select the most efficient
method?
Does it approach problem
solving in more general terms,
such as solving puzzles,
identifying and extracting
patterns and applying them to
the question?
Does it encourage calculator
use or give opportunities to
practise calculator skills?
(Year 5 and above only)
Comments
Mathematical problems.
Single- and multi-step problems
(with extension tasks).
Although the question is the
same for each task (Which
answer is being described?)
language is a main feature of
these tasks in terms of
interpreting the clues.
Techniques to interpret
language are based around
discussion rather than
highlighting key words.
Appropriate mathematical
language is used. Teachers are
advised to check that learners
understand the mathematical
content of each task. No specific
vocabulary guide is given.
Many answer sets are
diagrams/objects.
Teaches the use of one method:
logical reasoning (eliminating
answers that don’t fit clues and
testing a solution against all the
clues to check the final answer).
Yes. Learners are asked to
identify which clues are the
most helpful in each problem
and to find the minimum number
of questions needed to find the
answer.
Teaches a skill which can be
applied to broader logical
reasoning tasks.
Covered
suitably for
year group?






No, the tasks focus on logical
reasoning and do not require
doing hard calculations.
Review of numerical reasoning teaching resources 65
Criteria
Recording working
Does it show how to record
working mathematically and
systematically?
Does it exemplify how to show
working and how to explain
methods used and answers
given?
Reflecting on answers
Does it distinguish between
how to show working and how
to explain the methods used
and answers given?
Are learners encouraged to
check their working/answers?
Are learners encouraged to
think of/use alternative
checking techniques?
Does it show how to check if
answers make
number/common sense?
Comments
Covered
suitably for
year group?
No written work required but it
does teach learners to work
systematically, eliminating or
retaining possible answers from
a set of clues.
No written working or
explanation is required. Teacher
guidance states that learners
should be encouraged to justify
their answers through verbal
explanations and discussing
which clues are most/least
helpful.
No.
Yes, learners are encouraged to
check their answers by rereading the clues once an
answer has been found to check
it fits all the clues.
No, other than re-reading the
clues.

No, not really applicable.
http://www.nelsonthornes.com/webapp/wcs/stores/servlet/SearchDisplay?storeId=10701
&catalogId=10601&langId=1&pageSize=12&beginIndex=0&sType=SimpleSearch&resultCatEntryType=2&showRes
ultsPage=true&pageView=image&errorViewName=&searchTerm=talk+it+solve+it
Talk it, solve it: Reasoning Skills in Maths, Years 1 & 2, R Bradley, J Pennant and J
Walters, BEAM Education (Nelson Thornes), 2005
ISBN 978-1903142769
Talk it, solve it: Reasoning Skills in Maths, Years 3 & 4, C King, J Pennant and J
Walters, BEAM Education (Nelson Thornes), 2005
ISBN 978-1903142776;
Talk it, solve it: Reasoning Skills in Maths, Years 5 & 6, J Pennant and J Thompson,
BEAM Education (Nelson Thornes), 2005
ISBN 978-1903142783
66
Review of numerical reasoning teaching resources
Understanding Maths: Problem Solving
Criteria
Recording working
Selecting and applying methods
Understanding the
question
Is the context real-life, abstract,
mathematical or cross-curricular?
Does it have single- or multi-step
problems?
Does it encourage the use of
strategies for working out what
the question is asking? (e.g.
highlighting key words)
Does it use/explain appropriate
mathematical language?
Does it help learners to interpret
questions with diagrams and
minimal language?
Does it provide opportunities for
learners to select appropriate
methods?
Does it encourage learners to
select the most efficient method?
Does it approach problem
solving in more general terms,
such as solving puzzles,
identifying and extracting
patterns and applying them to
the question?
Does it encourage calculator use
or give opportunities to practise
calculator skills?
(Year 5 and above only)
Does it show how to record
working mathematically and
systematically?
Does it exemplify how to show
working and how to explain
methods used and answers
given?
Does it distinguish between how
to show working and how to
explain the methods used and
answers given?
Comments
Real-life.
Covered
suitably for
year group?

Single-step.

Yes, teaches about key
words.

Yes.

Some pattern-based
questions have minimal
language – an explanation for
these is provided at the start
of the page.
Appropriate methods are
taught at the beginning of
each section.
Tasks broken down into
sections and appropriate
methods taught – the ‘most
efficient’ is taught as
standard.
Yes, several problems where
small puzzles/patterns build
up into a bigger problem
solving task.




No, choice of working is
mental or written only.
Yes.

Showing working – yes.
Explaining methods and
answers – no.

No, does not teach explaining
skills.
Review of numerical reasoning teaching resources 67
Reflecting on
answers
Criteria
Are learners encouraged to
check their working/answers?
Are learners encouraged to think
of/use alternative checking
techniques?
Does it show how to check if
answers make number/common
sense?
Comments
Reminded to check in some
activities.
No.
Encouraged to estimate
answers beforehand.
Covered
suitably for
year group?


www.schofieldandsims.co.uk/product/439/problem-solving
Understanding Maths: Problem Solving, 7th ed., S Mills and H Koll, Schofield & Sims,
2011
ISBN 978-0721709741
68
Review of numerical reasoning teaching resources
We Can Do It!
Criteria
Selecting and applying methods
Understanding the
question
Is the context real-life, abstract,
mathematical or cross-curricular?
Does it have single- or multi-step
problems?
Does it encourage the use of
strategies for working out what
the question is asking? (e.g.
highlighting key words)
Does it use/explain appropriate
mathematical language?
Does it help learners to interpret
questions with diagrams and
minimal language?
Does it provide opportunities for
learners to select appropriate
methods?
Does it encourage learners to
select the most efficient
method?
Does it approach problem
solving in more general terms,
such as solving puzzles,
identifying and extracting
patterns and applying them to
the question?
Does it encourage calculator
use or give opportunities to
practise calculator skills?
(Year 5 and above only)
Comments
Mathematical and real-life.
Covered
suitably for
year group?

Multi-step, with teacher
guidance to help break down
problems.
‘Problem-solving strategies’
sheet gives tips for when
problem solving, including
looking for key words.
A list of key vocabulary is
provided for each challenge.
Many challenges include
diagrams but the amount of
accompanying text varies a lot.
An introductory task is done as
a whole class, cuing learners
into the approach they will be
using.
Some group discussion but not
much emphasis on methods.

Some puzzles and patterns
are more generalisable than
others. Each book covers a
wide range of problem types.
Patterns are discussed in
group work.
Almost half the questions in
Year 5/6 list calculator as a
resource. Fewer calculator
challenges in lower year
groups but each year group
does have some. Many use a
calculator as a way to do
calculations easily rather than
for difficult calculations that
require a calculator.





Review of numerical reasoning teaching resources 69
Criteria
Reflecting on
answers
Recording working
Does it show how to record
working mathematically and
systematically?
Does it exemplify how to show
working and how to explain
methods used and answers
given?
Does it distinguish between how
to show working and how to
explain the methods used and
answers given?
Are learners encouraged to
check their working/answers?
Are learners encouraged to
think of/use alternative checking
techniques?
Does it show how to check if
answers make number/common
sense?
Comments
Some tasks exemplify how to
record results while others
leave this for learners to
decide. There are teacher
question prompts for
discussing methods of working
and recording answers.
No exemplars of how to give
explanations but this is mainly
covered by group discussions
rather than being written.
Question prompts are separate
for discussing ways of
recording answers, methods
used and answers given.
Teacher guidance mentions
the need to reflect on work.
Not mentioned.
Covered
suitably for
year group?


Not mentioned.
www.nelsonthornes.com/shop/nt/primary/numeracy-and-mathematics/beam-%28be-amathematician%29-25022--1/we-can-do-it%21-year-1-using-and-applying-mathschallenges
We Can Do It!, P Clarke, BEAM Education, 2008
ISBN 978-1906224455 (Year 1)
ISBN 978-1906224462 (Year 2)
ISBN 978-1906224479 (Year 3)
ISBN 978-1906224486 (Year 4)
ISBN 978-1906224493 (Year 5)
ISBN 978-1906224509 (Year 6)
70
Review of numerical reasoning teaching resources
We Have A Problem!
Criteria
Is the context real-life, abstract,
mathematical or cross-curricular?
Selecting and applying methods
Understanding the
question
Does it have single- or multi-step
problems?
Does it encourage strategies for
working out what the question is
asking? (e.g. highlighting key
words)
Does it use/explain appropriate
mathematical language?
Does it help learners to interpret
questions with diagrams and
minimal language?
Does it provide opportunities for
learners to select appropriate
methods?
Does it encourage learners to
select the most efficient method?
Does it approach problem solving
in more general terms, such as
solving puzzles, identifying and
extracting patterns and applying
them to the question?
Does it encourage calculator use
or give opportunities to practise
calculator skills?
(Year 5 and above only)
Comments
Emphasis on real-life
situations that are
appropriate to learners of that
age. Opportunities for
learners to create their own
problems related to them,
and using cross-curricular
prompts for these is
suggested.
Builds up from single- to
multi-step, explicitly teaching
learners how to handle multistep problems.
Yes, talks a lot about looking
for and interpreting the key
words.
Covered
suitably for
year group?



Yes, several of the early
activities focus on this.
Focus is on word problems
rather than pictorial ones.

Teaches learners methods
for each activity but not
explicitly how to choose a
method.
No, as method is provided.

No.
Calculators are not required
for any task but could be
used.

Review of numerical reasoning teaching resources 71
Criteria
Reflecting on answers
Recording working
Does it show how to record
working mathematically and
systematically?
Does it exemplify how to show
working and how to explain
methods used and answers
given?
Does it distinguish between how
to show working and how to
explain the methods used and
answers given?
Are learners encouraged to
check their working/answers?
Are learners encouraged to think
of/use alternative checking
techniques?
Does it show how to check if
answers make number/common
sense?
Comments
Covered
suitably for
year group?
Teaches how to write a word
problem as a number
problem.
As above, teaches how to
write numerically. Does not
focus on explaining methods.
No.
Discusses looking at the
answers and using that as a
method of evaluating
working.
No.
Yes, teaches to estimate the
answer before calculating,
and also to look back and
consider if it makes sense.

www.badgerlearning.co.uk/ecommerce/search/maths-problem-solving---we-have-aproblem-years-3---6-pack.aspx
Maths Problem Solving: We have a problem!, P Wrangles, Badger Learning, 2012
ISBN 978-1781470008
72
Review of numerical reasoning teaching resources
Additional resources

Puppies Problem Solving (website)
https://hwb.wales.gov.uk/cms/hwbcontent/Shared%20Documents/vtc/200809/maths/puppies/index.html

Maths Investigations for Years 3-9 (website)
https://hwb.wales.gov.uk/cms/hwbcontent/Shared%20Documents/vtc/200809/maths/irf08-77/index.html

Nrich Tasks (website)
www.nrich.maths.org
Review of numerical reasoning teaching resources 73
Teacher voice survey
Research design and methods
To research which resources are currently being used and recommended for the teaching of
numerical reasoning, a question was submitted to the NFER’s Teacher Voice survey in
November 2013. This survey, which runs three times a year, enables up-to-date opinions to
be gathered from a representative panel of primary and secondary school teachers in
England. There is no equivalent panel of teachers in Wales.
The question submitted asked teachers to name any resources that they were aware of that
they consider to be good at supporting the teaching and learning of numerical reasoning for
learners in Years 2-9.
A sample of 1524 teachers in England completed the survey (750 teachers in primary
schools and 774 teachers in secondary schools). Responses were coded and grouped and
are presented below.
Teacher Voice survey results
This section presents the results from the Teacher Voice survey in which a sample of
primary and secondary school teachers in England were asked the following question:
Please identify any resources you use or are aware of that are good at supporting the
learning and application of numerical problem solving and/or numerical reasoning of
pupils.
Respondents were told that if they did not teach maths they should omit this question.
Responses were given by 511 (68%) primary school teachers and 125 (16%) secondary
school teachers, of the 1524 surveyed. Multiple responses could be given by each teacher
and these have been summarised in Tables A.1 and A.2. Responses have been split
between primary and secondary school respondents to determine which resources are
recommended for use in each setting. Only responses that were given by 1 per cent or more
of teachers are included.
The results from the teacher survey show that primary and secondary school teachers
currently use different resources. Primary school teachers recommended more physical
resources, such as Numicon (28%) and generic resources such as number squares, Dienes
apparatus and beads (26%). Secondary teachers recommended more online resources,
MyMaths (47%) and other website resources (29%) such as Big Maths and NCETM. For
both groups of teachers about a fifth recommended written materials such as text books and
worksheets.
However, it is important to note that while teachers in England may have identified resources
that they consider to be useful, these resources have not all been evaluated against the
numeracy framework and may not be the most suitable for supporting teaching of numerical
reasoning. Teachers in England are unlikely to have seen the content and format of the
74
Review of numerical reasoning teaching resources
reasoning test to be taken by learners in Wales, so may have a different understanding of
numerical reasoning than that explored in this report.
Review of numerical reasoning teaching resources 75
Table A.1 Responses from Teacher Voice survey (primary school teachers)
Primary
(%)
Numicon
28
Generic materials/resources, e.g. abacus, number squares, coloured counters,
number cards, Dienes apparatus, dice, beads
26
Written materials, e.g. publications, text books, worksheets, guidance, policy,
strategy documents, e.g. Collins Primary Maths, national curriculum (England)
guidance, DfE resources (mathematical challenges/problem solving for more able
children), pitch and expectations document
23
Website resources, e.g. Hamilton Trust, Base10, Big Maths, SAM learning, NCETM,
Numbershark
18
Nrich
14
Interactive whiteboard/smart board resources including interactive teaching
programmes
7
Computer based programmes/resources/ICT programs/software/games
consoles/iPad
7
Abacus website/resources
6
Whole school approaches, e.g. Big Maths, Maths Makes Sense
5
Mathletics
4
BEAM
4
Online resources/websites (not specific)
4
TES resources/website
3
Education City
3
Online games/puzzles, e.g. Sumdog, Mangahigh
2
Primary Resources
2
My Maths
2
Assessment resources/tools, e.g. APP sheets/past test papers/SAT questions
2
Own resources/drawing on previous curriculum examples/own ideas
2
Problem solving, e.g. Badger Maths
1
Speaking and listening activities
1
Real life/practical examples/activities/group work
1
Interactive resources, e.g. Active Teach, Nelson Thornes
1
Video clips, e.g. Maths Watch/DVDs/CDs
1
Cognitive Acceleration in Mathematics Education (CAME) project resources
1
Songs/rhymes
1
N=
511
More than one answer could be put forward so percentages may sum to more than 100.
A total of 750 respondents could have answered this question but those who did not teach maths did not need
to answer.
Source: NFER Omnibus Survey November 2013
76
Review of numerical reasoning teaching resources
Table A.2 Responses from Teacher Voice survey (secondary school teachers)
Secondary
(%)
My Maths
47
Website resources, e.g. Hamilton Trust, Base10, Big Maths, SAM learning,
NCETM, Numbershark
29
Written materials – e.g. publications, text books, worksheets, guidance,
policy, strategy documents, e.g. Collins Primary Maths, national curriculum
guidance (England), DfE resources (mathematical challenges/problem
solving for more able children), pitch and expectations document
21
Online games/puzzles, e.g. Sumdog, Mangahigh
9
Nrich
9
TES resources/website
8
Video clips, e.g. Maths Watch/DVDs/CDs
7
Generic materials/resources, e.g. abacus, number squares, coloured
counters, number cards, Dienes apparatus, dice, beads
7
Online resources/websites (not specific)
4
Computer based programmes/resources/ICT programs/software/games
consoles/iPad
4
Own resources/drawing on previous curriculum examples/own ideas
4
Assessment resources/tools, e.g. APP sheets/past test papers/SAT
questions
3
STEM
3
Interactive resources, e.g. Active Teach, Nelson Thornes
2
Association of Teachers of Mathematics (ATM)
2
Specific projects, e.g. cupcake challenge
2
Mathletics
1
Toolkit, e.g. APP, problem solving
1
Real life/practical examples/activities/group work
1
Songs/rhymes
1
Treasure hunt
1
Cognitive Acceleration in Mathematics Education (CAME) project resources
1
N=
125
More than one answer could be put forward so percentages may sum to more than 100.
The percentages in this table are weighted separately by FSM rates for ‘secondary’ schools.
A total of 774 respondents could have answered this question but those who did not teach maths
did not need to answer.
Source: NFER Omnibus Survey November 2013
Review of numerical reasoning teaching resources 77
Appendix A: Information sources
Professional and subject associations
The following professional association websites were searched:
UK organisations
International organisations
Advisory Committee on Mathematics
Education (ACME)
American Mathematical Society (USA)
Association of Teachers of Mathematics
(ATM)
The Australian Association of Mathematics
Teachers INC
British Society for Research into
Learning Mathematics
Canadian Mathematical Society
Maths Association
Mathematical Association of America (USA)
National Centre for Excellence in the
Teaching of Mathematics (NCETM)
National Council of Teachers of Mathematics
(USA)
New Zealand Association of Mathematics
Teachers
News sites and teacher forums
The following news sites and teacher forums were consulted:
Site Name
Website address
BBC Education
http://www.bbc.co.uk/news/education/
Guardian Education
http://www.theguardian.com/education
Guardian teacher network
http://www.theguardian.com/teacher-network
Independent Education
http://www.independent.co.uk/news/education/
Telegraph Education
http://www.telegraph.co.uk/education/
TES forum
http://community.tes.co.uk/
78
Review of numerical reasoning teaching resources
Government Websites
The following Government websites from England were consulted:
Site Name
Website address
Department for Education
http://www.education.gov.uk/
National College for School Leadership
http://www.nationalcollege.org.uk
Ofsted
http://www.ofsted.gov.uk/
Publishers
The following publishers’ websites were searched:
UK publishers
International publishers
Bloomsbury Publishers
Association of Educational Publishers (USA)
Cambridge University Press
Corwin Press (USA)
Channel 4 Learning
DC Canada Education publishing (Canada)
Collins Education
Essential Resources (New Zealand)
Espresso Education
Evan-Moor (international)
Galore Park
Follett Digital Resources (USA)
Hodder Education
Harcourt International Education Group (USA)
McGraw-Hill
Heinemann (international)
Nelson Thornes
Kaplan Publishing (USA)
Oxford University Press
Macmillan Education (international)
Pearson
Pearson Australia
Raintree Publishers
Pearson Canada
Rising Starts
Pearson New Zealand
Schofield & Sims
Perfection Learning Corporation (USA)
Scholastic
Teachers College Press (USA)
The Critical Thinking Company (USA)
Review of numerical reasoning teaching resources 79
Search terms:
Key words
Mathematical reasoning
Mathematical skills
Numerical/number reasoning
Numerical understanding
Mathematical problem solving
Mathematical understanding
Numerical/number problem solving
Numerical explanations
Mathematical thinking
Mathematical explanations
Numerical number/thinking
Numerical working out
Computational reasoning
Mathematical working out
Computational thinking
Numerical vocabulary
Mathematical vocabulary
Mathematical vocabulary
Numerical vocabulary
Numerical judgement
Mathematical checking
Mathematical judgement
Numerical checking
Numerical learning strategies
Number concepts
Mathematical learning strategies
Numerical skills
80
Review of numerical reasoning teaching resources
Appendix B: Criteria for inclusion and
exclusion
Criteria for inclusion in the review
The criteria described below were developed as the parameters for this audit of resources.
Parameter
Inclusion criteria
Type of literature
Learning materials which include teacher support and guidance
and which claim to support the development of numerical reasoning
skills are included in this review.
Publication dates
Current learning materials and publications which are currently in
print are considered in this review.
Geographical scope
Learning materials from the UK, USA, Canada, Australia and New
Zealand are included in this review. Only materials published in the
English language are considered.
Age range
Learning materials designed for learners from years 2 to 9 (grades
1 to 8 in North America) are considered in this review.
Exclusion criteria
Resources and learning materials which cover the following topics were not considered
relevant to the purposes of this review and so were excluded:
 11 plus

Verbal and non-verbal reasoning

IQ reasoning

Mathematical investigations
Review of numerical reasoning teaching resources 81
82
Review of numerical reasoning teaching resources
NFER ref. NTWV
ISBN:
978-1-4734-1679-6