## Mathematical Problem Solving in the Early Years: Developing Opportunities, Strategies and Confidence

Published 2016 Revised 2019

- familiar contexts
- meaningful purposes
- mathematical complexity.

- which they understand - in familiar contexts,
- where the outcomes matter to them - even if imaginary,
- where they have control of the process,
- involving mathematics with which they are confident.
- taking some from one doll and giving to another, in several moves,
- starting again and dealing, either in ones or twos,
- taking two from each original doll and giving to the new doll,
- collecting the biscuits and crumbling them into a heap, then sharing out handfuls of crumbs.

- brute force: trying to hammer bits so that they fit,
- local correction: adjusting one part, often creating a different problem,
- dismantling: starting all over again,
- holistic review: considering multiple relations or simultaneous adjustments e.g. repairing by insertion and reversal.
- getting a feel for the problem, looking at it holistically, checking they have understood e.g. talking it through or asking questions;
- planning, preparing and predicting outcomes e.g. gathering blocks together before building;
- monitoring progress towards the goal e.g. checking that the bears will fit the houses;
- being systematic, trying possibilities methodically without repetition, rather than at random, e.g. separating shapes tried from those not tried in a puzzle;
- trying alternative approaches and evaluating strategies e.g. trying different positions for shapes;
- refining and improving solutions e.g. solving a puzzle again in fewer moves (Gifford, 2005: 153).
- Getting to grips: What are we trying to do?
- Connecting to previous experience: Have we done anything like this before?
- Planning: What do we need?
- Considering alternative methods: Is there another way?
- Monitoring progress: How does it look so far?
- Evaluating solutions: Does it work? How can we check? Could we make it even better?

- Construction - finding shapes which fit together or balance
- Pattern-making - creating a rule to create a repeating pattern
- Shape pictures - selecting shapes with properties to represent something
- Puzzles - finding ways of fitting shapes to fit a puzzle
- Role-play areas - working out how much to pay in a shop
- Measuring tools - finding out how different kinds of scales work
- Nesting, posting, ordering - especially if they are not obvious
- Robots - e.g. beebots: directing and making routes
- preparing, getting the right number e.g. scissors, paper for creative activities
- sharing equal amounts e.g. at snack time
- tidying up, checking nothing is lost
- gardening and cooking e.g. working out how many bulbs to plant where, measuring amounts in a recipe using scales or jugs
- games, developing rules, variations and scoring
- PE: organising in groups, timing and recording

- Decision making - what shall we call the new guinea pig?
- Parties, picnics and trips e.g. how much lemonade shall we make?
- Design Projects - the role play area, new outdoor gardens or circuits
- Hiding games - feely bags with shapes, the 'Box' game
- Story problems - e.g. unfair sharing, with remainders and fractions, making things to fit giants or fairies
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Learning and Development

## Maths problem-solving activities for Early Years settings

● finding ways to solve problems;

● finding new ways to do things;

● making links and noticing patterns in their experience;

● developing ideas of grouping, sequences, cause and effect;

● planning, making decisions about how to approach a task, solve a problem and reach a goal;

● checking how well their activities are going;

● changing strategy as needed;

● reviewing how well the approach worked.

## Role of the adult

You can effectively support children’s developing problem-solving strategies through:

## Problem solving possibilities

## Going, going, gone

## Camping out

● Materials to construct a tent or den such as sheets, curtains, poles, clips, string.

● Rucksacks, water bottles, compass and maps.

● Oven shelf and bricks to build a campfire or barbecue.

● Buckets and bowls and water for washing up.

● Some distance away, builders’ buckets filled with damp sand and large gravel.

● Bucket balances and bathroom scales.

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## problem solving maths EYFS

## Reception Maths: Open-ended Investigations Mathematical Problem-solving

## Session 1 Shape

## Open-Ended Task

## Session 2 Position and Direction

## Session 3 Number and Shape

## Session 4 Number and the Language of Addition/Subtraction

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## What your child will learn in Reception

In Reception, your child will learn to:

- Count reliably with numbers from 1 to 20, place them in order and say which number is one more or one less than a given number.
- Use quantities and objects to add and subtract 2 single-digit numbers and count on or back to find the answer.
- Use everyday language to talk about size, weight, capacity, position, distance, time, and money to compare quantities and objects and to solve problems.
- Recognise, create, and describe patterns.
- Explore characteristics of everyday objects and shapes and use mathematical language to describe them.
- Age 3–4 (Early Years)
- Age 4–5 (Reception)
- Age 5–6 (Year 1)
- Age 6–7 (Year 2)
- Age 7–8 (Year 3)
- Age 8–9 (Year 4)
- Age 9–10 (Year 5)
- Age 10–11 (Year 6)
- Year 1 (age 5–6)
- Year 2 (age 6–7)
- Year 3 (age 7–8)
- Year 4 (age 8–9)
- Year 5 (age 9–10)
- Year 6 (age 10–11)
- Help with times tables
- Ratio & proportion
- Learning to tell the time
- Numicon parent guide
- MyMaths parent guide
- Maths activity books

## Ten of our favourite early years problem-solving activities

## Supporting problem-solving

## 1) Den-building

## 2) Cooking and baking

## 3) Playing with patterns

## 4) Sorting and categorising

## 6) Ice rescue

## 7) Obstacle courses

## 8) Filling, emptying and investigation

## 9) Story problems

## 10) Playing with loose parts or open-ended resources

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## Early Years Guide

- School at Home
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## Introduction

- What do we mean by Early Years?
- What does learning look like in the Early Years
- Why is Cognitive Load Theory so important?
- What mastery strategies are available for Early Years?

## What do we mean when we talk about Early Years?

## Areas of learning

The EYFS framework outlines seven areas of learning :

- Communication and language
- Physical development
- Personal, social and emotional development
- Mathematics
- Understanding the world
- Expressive art and design

## Mathematics in EYFS

- Understanding and using numbers
- Calculating simple addition and subtraction problems
- Describing shapes, spaces, and measure

## Revised guidance

The DfE published revised guidance in March 2021 to take effect in September 2021.

The mathematics component now incorporates many elements of the mastery approach.

Specifically, the revised framework says:

## Early Learning Goals

The latest framework has the following early learning goals for mathematics:

Children at the expected level of development will:

- Have a deep understanding of number to 10, including the composition of each number
- Subitise (recognise quantities without counting) up to five
- Automatically recall (without reference to rhymes, counting or other aids) number bonds up to five (including subtraction facts) and some number bonds to 10, including double facts

## Numerical patterns

- Verbally count beyond 20, recognising the pattern of the counting system
- Compare quantities up to 10 in different contexts, recognising when one quantity is greater than, less than or the same as the other quantity
- Explore and represent patterns within numbers up to 10, including evens and odds, double facts and how quantities can be distributed equally

## Learning in the early years

According to the NCETM, there are:

## Six key areas of mathematical learning

Cardinality and counting, composition.

Looking briefly at each in turn:

Comparing numbers involves knowing which numbers are worth more or less than each other.

## Shape and space

- The one-to-one principle: children must name each object they count and understand there are two groups: the one that has been counted and the one that hasn’t yet been counted
- The stable order principle: children must know how to count in the right order
- The cardinal principle: children need to understand the last number in the set is the total amount
- Counting anything: children need to realise that anything can be counted, not just objects that can be touched, but also things like claps and jumps
- Order of counting doesn’t matter: children need to understand that the order of counting in the set is irrelevant and will still lead to the same amount

## How do children develop counting skills?

## Activities to boost number sense in Reception Year

## Crowd control

## Grouping straws

## Fastest 10 frames

## Everyday questions to develop number sense

Practice using the terms more than, fewer than and as many as by asking:

- Are there more grapes than tomatoes?
- Are there fewer tomatoes than grapes?
- Are there as many plates as people eating?

Remember to practice each sentence:

- There are more grapes than tomatoes
- There are fewer tomatoes than grapes
- There are as many plates as family members eating

## Number Rhymes

- Counting back and counting forward
- “No” or “none” (Five little ducks went swimming one day)
- Counting in pairs (two, four, six, eight, Mary at the cottage gate)
- Counting to five, 10 and beyond

## Problem solving, reasoning and numeracy

## Foundations

## Mark making

## Developing understanding with careful questioning

- I have made a pattern. What’s your pattern?
- How many blocks taller is my model compared to yours?
- How do we know this area is full?
- I have three cars, how many do you have?
- Do you have more?
- How do you know?

## Adding maths talk activities to your daily routine

The following activities can get you started:

## How many children are at school?

- How do we know this 10 frame is full?
- How many children are absent?
- What can you tell me about number seven?

## Sorting and grouping objects as a class

## Vote for a story

## Pattern Awareness

## What is mathematical pattern awareness?

- Shapes with regular features, such as a square or triangles with equal sides and angles, and shapes made with some equally spaced dots
- A repeated sequence: the most common examples are AB sequences, like a red, blue, red blue pattern with cubes. More challenging are ABC or ABB patterns with repeating units like red, green, blue or red, blue, blue
- a growing pattern, such as a staircase with equal steps

## Why is pattern awareness important?

Pattern awareness has been described as early algebraic thinking, which involves:

## Repeating Patterns

## Foundations — Your Reception Solution

## Cognitive Load Theory

## What is Cognitive Load Theory and why is it important?

The answer is simple — new skills demand more attention.

## Working memory

## Intrinsic versus extraneous load

## Supporting the transition to long-term memory

## Focused learning objective

## Activate prior learning

## Present information clearly

## Avoid cognitive overload

## Maths mastery for Early Years

## Early Years and CPA

## C is for concrete

## Spending time with real-life objects

## Early years and number bonds

## How to teach number bonds

## Concrete step

## Pictorial step

## Abstract step

## Early Years and place value

## Progress through concepts systematically

## Use the CPA approach to establish meaning

## Teach the ‘10-ness of 10’

## Progress to 20, then to 40

## Use base 10 blocks for 100 and 1000

## Approach larger numbers the same way

For example, you can identify and complete number patterns or find missing digits on a number line.

And a confident problem solver in maths is a confident problem solver in life.

Well done on making it to the end of our Ultimate Guide to Early Years.

If you’d like to learn more about Early Years, we recommend checking out the following links:

- NCETM: How Early Years children develop mathematical thinking (Podcast)
- NRICH: Early Years Foundation Stage Homepage
- The School of School: Episode 17 Play and early years (Podcast)
- Maths — No Problem! CPA approach

Also, don’t miss our other Ultimate Guides:

- The Maths — No Problem! Ultimate Guide to Maths Mastery
- The Maths — No Problem! Ultimate Guide to Assessment

## School of School Podcast

## Maths — No Problem!

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