Improving Mathematics in KS 2 and KS3: EEF's guidance

1 Introduction
2 R1: Use assessment to build on pupils’ existing knowledge and understanding
3 R2: Use manipulatives and representations
4 R3: Teach pupils strategies for solving problems
5 R4: Enable pupils to develop a rich network of mathematical knowledge
6 R5: Develop pupils’ independence and motivation
7 R6: Use tasks and resources to challenge and support pupils’ mathematics
8 R7: Use structured interventions to provide additional support
9 R8: Support pupils to make a successful transition between primary and secondary school

Improving Mathematics in Key Stages 2 and 3 | EEF

Introduction

These 8 recommendations are based on research findings which will make a significant difference to pupils' learning.

This guidance report focuses on the teaching of mathematics to pupils in Key Stages 2 and 3.

It is not intended to provide a comprehensive guide to mathematics teaching. We have made recommendations where there are research findings that schools can use to make a significant difference to pupils’ learning, and have focused on the questions that appear to be most salient to practitioners. There are aspects of mathematics teaching not covered by this guidance. In these situations, teachers must draw on their knowledge of mathematics, professional experience and judgement, and assessment of their pupils’ knowledge and understanding.

The focus is on improving the quality of teaching. Excellent maths teaching requires good content knowledge, but this is not sufficient. Excellent teachers also know the ways in which pupils learn mathematics and the difficulties they are likely to encounter, and how mathematics can be most effectively taught.

This guidance is aimed primarily at subject leaders, headteachers, and other staff with responsibility for leading improvements in mathematics teaching in primary and secondary schools. Classroom teachers and teaching assistants will also find this guidance useful as a resource to aid their day-to-day teaching.

R1: Use assessment to build on pupils’ existing knowledge and understanding

Assessment should inform the planning of future lessons and the focus of targeted support.

Teachers should only ever teach small bites on firm learning foundations.

Timely practice can be used to measure progress as well as attainment. Measuring progress to find out how effectively teaching using the existing scheme of learning increases attainment, is easy to do with timely practice.

Assessment should be used not only to track pupils’ learning but also to provide teachers with information about what pupils do and do not know. This should inform the planning of future lessons and the focus of targeted support.

Effective feedback will be an important element of teachers’ response to assessment. Feedback should be specific and clear, encourage and support further effort, and be given sparingly.

Teachers not only have to address misconceptions but also understand why pupils may persist with errors. Knowledge of common misconceptions can be invaluable in planning lessons to address errors before they arise.

R2: Use manipulatives and representations

We create ladders through topics, the layers. Timely practice has many layers in many topics, where one layer uses a representative within the question and the next layer has similar questions but without the representative - of course the learner can continue to use the representative when it is helpful, but dispense with representatives, when they no longer need them.

Manipulatives (physical objects used to teach maths) and representations (such as number lines and graphs) can help pupils engage with mathematical ideas. However, manipulatives and representations are just tools: how they are used is essential. They need to be used purposefully and appropriately to have an impact.

There must be a clear rationale for using a particular manipulative or representation to teach a specific mathematical concept. Manipulatives should be temporary; they should act as a ‘scaffold’ that can be removed once independence is achieved.

R3: Teach pupils strategies for solving problems

This is one area where we don’t entirely concur with EEF, we believe that learners should only be asked to solve problems, when they have mastered all the pre requisite skills. Despite having devoted an entire topic theme, word problems, to problem solving, we don’t think that all learners should be expected to solve problems, yet. Of course sometimes learners should practice problem solving, whilst learning the topics - as part of the natural progression through the topic - some layers, when appropriate, in a topic require problem solving.

If pupils lack a well-rehearsed and readily available method to solve a problem they need to draw on problem-solving strategies to make sense of the unfamiliar situation.

Select problem-solving tasks for which pupils do not have ready-made solutions.
Teach them to use and compare different approaches.
Show them how to interrogate and use their existing knowledge to solve problems.
Use worked examples to enable them to analyse the use of different strategies.
Require pupils to monitor, reflect on, and communicate their problem solving.

R4: Enable pupils to develop a rich network of mathematical knowledge

We often use representatives to make links between topics e.g.
We improve recall of maths facts by breaking these down into small bites e.g. beginXfacts
Understand procedures e.g. multiply fractions,
Every maths lesson learners must choose, a good enough procedure, to answer their timely practice questions. Feedback from the teacher on selecting a better method, helps learners make progress
We use proportional triangles, proportionality lines and boxes to make proportionality more concrete.
We teach fraction and decimal lines which extend beyond one, very early on in topics e.g. fractionINTRO
Recognise mathematical structure: e.g. we use prime factor trees extensively to help learners, learn times table facts, simplify fractions, find all the factors …

Emphasise the many connections between mathematical facts, procedures, and concepts.
Ensure that pupils develop fluent recall of facts.
Teach pupils to understand procedures.
Teach pupils to consciously choose between mathematical strategies.
Build on pupils’ informal understanding of sharing and proportionality to introduce procedures.
Teach pupils that fractions and decimals extend the number system beyond whole numbers.
Teach pupils to recognise and use mathematical structure.

R5: Develop pupils’ independence and motivation

With timely practice learners need to answer questions for 5 to 15 minutes every maths lesson, on “a mixed bag of topics” such as is found in exam papers - but the questions are all questions that each learner can do - knowing this learners are willing to dig a little deeper to complete their work independently. Opportunities for feedback-dialogue are high and these make excellent opportunities for teachers to help learners deal with the emotional response to being wrong.

We strongly believe that there is evidence to increase positive attitudes, motivation, and that is to ensure high success, timely practice enables most learners to be able to answer independently and accurately over 80 percent of their questions, which we’ve found does improve motivation.

R6: Use tasks and resources to challenge and support pupils’ mathematics

Questions make excellent tasks. Questions which are on the learners' firm learning foundations are most effective.

Sometimes we want to provide variety - cool down - makes an opportunity to continue embedding learning and providing different types of tasks.

R7: Use structured interventions to provide additional support

Timely practice is built to provide a way of teaching which is very likely tutoring within the maths classroom. Although schools might want to add in extra lessons for a minority of learners:

when learners are exceptionally behind their peers
when learners have emotional reasons for working in smaller groups
when spacing of maths lessons leaves too large a gap between maths lessons

… often timely practice is enough to help low attaining learners begin to flourish in the maths classroom and catch up with their more highly attaining peers.

R8: Support pupils to make a successful transition between primary and secondary school

We believe that the “dip in mathematical attainment” after the summer holiday, between year 6 and year 7, during Covid lockdown is largely due to forgetting, the forgetting which happens day in day out, especially for low attaining learners. If we can dramatically reduce forgetting, and with timely practice we can, then we will see less of a dip.

The best way to transition from year 6 to year 7 for maths, in our opinion, is to schedule a cool down after year 6 SATs where learners continue their timely practice for perhaps 10 minutes per lesson - to embed their current learning - but take a well deserved rest form new learning. Project work, art and maths, cross curricula work can be done for the majority of these lessons. Then when learners arrive in year 7, their teacher has their timely practice records already in their class records. No need for testing, just a little bit of feedback and review on work that has been forgotten and straight into learning on existing learning foundations.