Review of Numerical Reasoning Teaching Resources

Review of numerical reasoning teaching resources

Jo Williams

Katherine Fowler

Emily Jones

Registered Charity No. 313392

ISBN 978-1-4734-1679-6

How to cite this publication:

Williams, J., Fowler, K. andJones, E. (2014). Review of Numerical Reasoning Teaching Resources. Slough: NFER.

Contents

1About this review

2Review of current resources

3Conclusions and recommendations

Further details of resources

Teacher voice survey

Appendix A: Information sources

Appendix B: Criteria for inclusion and exclusion

1About this review

1.1Research context and aims

In May 2013, learners across Wales sat the firstnew statutory reading and numeracy proceduraltests, 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 pilotresults indicated that further support is requiredto 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).

1.2Research design and methods

1.2.1Review of current resources available

A systematic review was conducted to investigate teaching resources of numerical reasoning that are currently available.This includedwebsite searches of both UK and English-speaking international countries for educationalpublishers 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:

agreeing search terms and parameters for the review of resources
identifying sources of teaching resources
appraising each resource to discern its relevance to the review
reviewing and summarising the most relevant resources
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 AppendixB. 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:

decide for themselves what numerical skills to use
apply their method to work out a solution
give coherent mathematical explanations
reflect on the outcome.

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 theydevelop their teaching of numerical reasoning.

1.2.2Limitations of review

Only those resources which relate to the Welsh Governmentdefinition of numerical reasoningwere 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 / 1

2Review of current resources

2.1Summary

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’starget 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 oflearners, 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.)
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 / 1

Table 2.1Summary of reviews of numerical reasoning teaching resources

About the resource / Whether the resource fulfils each criterion
Name of resource / Publisher / Which year group is it for? / Understanding the question / Selecting and applying methods / Recording working / Reflecting on answers / Price*
Abacus Evolve Real-Life Problem Solving / Pearson / Years 1-5
(website) /  / ££
Badger Maths Problem Solving / Badger Publishing / Years 1-6
(4 books) /  /  /  /  / £
each
BEAM’s Big Book of Word Problems and Interactive CD / BEAM Education (Nelson Thornes) / Years 1-6
(3 sets, Years 3-4 reviewed) /  /  / ££
each
Bowland Maths / The Bowland Trust / Years 7-9
(website) /  /  / free
Can Do Problem Solving / Nelson Thornes / Years 1-6
(6 sets) /  /  /  /  / ££
each
Creative Activities for Maths Problem Solving / Scholastic / Years 1-2 and Years 3-6
(2 books) /  /  /  /  / £
each
Espresso Primary / Espresso Education / Years 1-6
(website) /  /  /  / not
known
Investigations with Numicon / Oxford University Press / Years 3-6
(1 book) /  / £
Investigative and Problem-Solving Approaches to Mathematics and Their Assessment / Framework Press Educational Publishers Ltd. / Years 3-11 /  /  / free
KS1/2 Problem Solving / Schofield & Sims / Years 1-6
(7 books) /  / £
Let's Think Through Maths! / GL Assessment / Year 1-4
(2 books, Year 2-4 reviewed) /  / ££
each
Macmillan Maths Problem Solving Boxes (1-6) / Macmillan Education Australia (distributed by McGraw-Hill) / Years 1-6
(6 boxes) /  /  /  / ££
each
Main Activity: Problem Solved! / Badger Publishing / Years 7-9
(3 books,
Year 7 reviewed) /  /  /  /  / £
each
Maths Buzz / BEAM Education / Years 1-6
(3 sets, Years 3 and 4 reviewed) /  / ££
each
Maths Trails, Working Systematically / Cambridge University Press / Years 3-9
(1 book) /  /  /  / £
Mental Arithmetic(First) (Essential) / Schofield & Sims / Years 1-9
(3 sets) /  / £
each
My Maths / Oxford University Press / Years 2-9
(website) /  /  /  / ££
New Zealand Maths Problem Solving / New Zealand Ministry of Education / Years 1-10 (website) /  /  / free
Numicon Teaching Resource Handbooks / Oxford University Press / Years 1-4
(8 sets) /  /  /  /  / ££
each
Problem-Solving Strategies for Efficient and Elegant Solutions – Grades 6-12 / Corwin Press
(a SAGE company) / Years 7-13
(1 book) /  /  / £
Problem Solving with EYFS, Key Stage One and Key Stage Two Children / Department for Education / EYFS and
Years 1-6 /  /  /  / free
Talk it, solve it: Reasoning Skills in Maths / BEAM Education (Nelson Thornes) / Years 1-6
(3 books) /  /  / £
each
Understanding Maths: Problem Solving / Schofield & Sims / Years 3-6
(1 book) /  /  /  /  / £
We Can Do It! / BEAM Education (Nelson Thornes) / Years 1-6
(6 books) /  /  /  / £
each
We Have A Problem! / Badger Learning / Years 3-6
(4 books, Year 3 reviewed) /  /  / £
each
Review of numerical reasoning teaching resources / 1

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 / Website / Which year group is it for? / Price
Puppies Problem Solving / HWB Wales (Welsh Government)/NGfLCymru / / Years 2-4 / free
Maths Investigations / HWB (Welsh Government)/ Wrexham Network Project / / Years 3-9 / free
Nrich Tasks / University of Cambridge / / Years 1-13 / free
Review of numerical reasoning teaching resources / 1

2.2Individual 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:
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:
Target year group: Years 1-6
Price: All four books with CDs available online for £129.00
Free sample online at 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.
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:
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:
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.