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KS1 and KS2 Maths – Problem solving

• Author: Mike Askew
• Main Subject: CPD
• Subject: Maths
• Date Posted: 20 June 2012

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If children use well worn techniques to solve problems without understanding or modelling the context, their maths skills won't fully evolve

Children, from birth, are proficient problem solvers. By the age of two or three they have solved what are probably life’s two biggest problems - how to walk and how to talk. As they get older they solve practical problems, such as sharing a bag of sweets fairly with others, long before they’ve heard of division. In this article, I look at how we can build on this natural propensity to solve problems in teaching mathematics.

Routine and non-routine problems

Routine problems are problems children know how to solve based on their previous experiences. The sort of thinking required by routine problems can be described as reproductive: the child only needs to recall or reproduce a procedure or method they have previously learnt. A problem like ‘Apples’, for example (see below), is likely to be a routine problem for most children at the upper end of primary school; they know to multiply the two numbers together without having to think deeply about what operation to use.

• Apples on a supermarket shelf are in bags of eight. • If Jane buys six bags, how many apples is that?

In contrast, non-routine problems are where the learner does not immediately have a solution tucked under his belt. The problem solver has to put some effort into understanding the problem and creating, rather than recalling, a solution strategy. Non-routine problems engage learners in productive thinking.

We often think of non-routine problems as needing to be unusual or not having, to us as adults, an immediately obvious method of solution. ‘Stamps’ is typical of this type of non-routine problem.

• Clearing out a desk draw I found a collection of 5p and 6p stamps. • I have a parcel to post that needs 58p worth of stamps on it. • Can I create this exactly using the stamps I found? • If so, is there more than one way of doing this?

In choosing problems to work with, we need to decide whether or not we think a problem will be routine or non-routine for the particular children working on it. In the rest of this article, the problems chosen are being treated as though they are non-routine problems for the children working on them. That’s not to say that I don’t think routine problems have a place in the curriculum - they do. Here, however, I want to deal with some of the issues around teaching and learning non-routine problems.

The importance of context

• Four hungry girls share three pizzas equally. • Eight hungry boys share six pizzas equally. • Does each girl get more pizza than each boy, less or the same?

As a routine problem, the ‘story’ of pizzas and hungry children doesn’t serve any real purpose: children quickly learn to disregard the context, to strip out the mathematics and to work some procedure. The problem could just as easily have been put in the context of builders sharing bricks and many learners would happily say each builder would get 3/4 of a brick, without stopping to question the near impossibility of sharing out bricks.

We can, however, treat ‘Pizzas’ as a non-routine problem and use it to introduce children to thinking about fractions and equivalences. The context of hungry children and pizzas then becomes important. It is not chosen simply to be window-dressing for a fraction calculation. Nor are pizzas chosen because children are intrinsically motivated by food, making the unpalatable topic of fractions digestible. No, the context s chosen because children know about fair shares and slicing up pizzas - they can solve this problem without any formal knowledge of fractions. As the researcher Terezhina Nunes once pointed out, young children would not be able to solve the ‘bald’ calculation 3 divided by 4 but, “show me four young children who, given three bars of chocolate to share out fairly, hand the bars back saying ‘it can’t be done.”

Children have ‘action schemas’ for solving problem like ‘Pizzas’ - they can find ways to solve this with objects, pictures, diagrams and, eventually, symbols. Teaching can then build on the children’s informal solutions to draw out the formal mathematics of fractions. From being one of 20 ‘problems’ on a worksheet to complete in a lesson, ‘Pizzas’ can become a ‘rich task’ taking up the best part of a lesson, if children work on it in pairs and carefully selected solutions are then shared with the class.

Creating mathematical models

Part of the productive thinking in working on rich, non-routine problems requires children to create mathematical models, and we can teach to support this.

• At the supermarket Myprice, milk costs £1.08 per litre. • This is 7 pence less per litre than milk costs at Locost. • How much does 5 litres of milk cost at Locost?

What is missing from this approach is attention to setting up an appropriate model of the problem. Ultimately this could be a mental model of the problem context, but it helps initially to encourage children to put something on paper that can be shared and discussed. In problems involving quantities, like ‘Milk’, simple bar diagrams can help children create the appropriate model. These help children examine the relationships between the quantities (as opposed to simply fixing on specific numbers and keywords).

Setting up a diagrammatic model begins with creating a representation of what is known in the situation. In this example, we know milk at Myprice costs £1.08, so a diagram for this would look like:

MYPRICE £1.08

This provides the basis for talking about what the picture for the price of milk at Locost is going to be. Will the bar be longer or shorter? Where is the bar for the 7 pence to be drawn?

Two different models can be set up and children asked to describe the relationship between the prices at the two supermarkets, to see which diagram fits with the information in the problem. If the diagram for the price at Locost is shorter by 7, then two statements can be made:

MYPRICE £1.08p LOCOST     7p

• Myprice milk costs 7 pence more than milk at Locost.

• Locost milk costs 7 pence less than milk at Myprice.

In comparison, making the bar for milk at Locost longer by 7 gives different comparative statements:

MYPRICE £1.08p 7p LOCOST    

• Myprice milk costs 7 pence less than milk at Locost.

• Locost milk costs 7 pence more than milk at Myprice.

Children can then talk about which of these situations fits with the wording in the problem.

Having established that Locost milk must be £1.15 a litre, children can go on to produce the bar diagram model for this.

Supporting non-routine problem solving

Where the problems were played out as non-routine, three factors identified are worth noting. First, in choosing the tasks, the teachers made sure they would build on learners’ prior knowledge - as I suggest a problem like ‘Pizzas’ can. Second, in contrast to focusing on getting the answer, the researchers observed what they called ‘sustained pressure for explanation and meaning’. In other words, the teachers pressed for children to explain what and why they were doing what they were doing rather than simply focusing on whether or not they had got the correct answer. Third, the amount of time children were allowed to work on the problem was neither too long or too short: children need enough time to ‘get into’ a problem, but too much time can lead to a loss of engagement.

Share good practice

Gather together a collection of problems covering all the years of education in your school (or ask teachers to each contribute two or three problems).

Working together in small groups, teachers sort the problems into three groups:

1. Problems they think would be routine for the children they teach 2. Problems they think would be non-routine for their children 3. Problems they think would be much too difficult for their age group

Everyone agrees to try out a problem from group 2 with their class. Discuss how too much focus on getting the answer can reduce the challenge and stress the importance of pressing children to explain their working. At a subsequent meeting, people report back, focusing in particular on strategies they used to keep the problem solving non-routine.

About the author

Mike Askew is Professor of Primary Education at Monash University, Melbourne. Until recently, he was Professor of Mathematics Education at King’s College, University and Director of BEAM.

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Home → Maths   → Year Groups  → KS1

You made it through the first year and now it’s time for Key Stage 1! Maths anxiety can increase as children get older but it doesn’t have to. As parents, we can help our child succeed in KS1 maths by learning what to expect. Explore KS1 maths topics below, including tips for KS1 maths SATs, and have a go at some practice questions!

Select a KS1 maths topic

Explore the maths topics covered in KS1 maths and have a go at some fun, interactive practice questions!

Number and place value

Learn how to count, read and write numbers up to 100

Find out more

Addition and subtraction

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Multiplication and division

Discover key multiplication and division facts & strategies

Multiplication or Division

Find out about data and how to read bar charts and graphs

Measurement

Explore time, volume and length and quantities

Learn how to identify 2D and 3D shapes

Position and direction

Discover how to arrange objects into patterns

Fractions, decimals and percentages

Learn what a fraction is and how to order them

Fractions or Decimals

Discover key multiplication and division facts

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What do children learn in KS1 maths?

In Key Stage 1 maths curriculum , kids explore a wide breath of content to help develop their knowledge of core concepts and gain an understanding of how maths is used in everyday life. From fractions to shapes, they learn lots of key skills that’ll place them in great stead for KS2! The main topics they learn about are…    

1. Number and place value: In the KS1 maths curriculum, children learn to count, read, write and represent numbers up to 100. They also learn how to compare numbers and use the place value system.  

2. Addition and subtraction: Children learn various strategies for mental and written addition and subtraction. They also explore number bonds, counting techniques and inverse operations.  

3. Multiplication and division: Kids develop their understanding of multiplication and division and learn how to solve problems using multiplication and division facts, arrays and pictorial representations.  

4. Fractions : Learners discover how to recognise, name, find and write fractions in relation to length, shapes and quantity.

  5. Measurement: In KS1 maths, children learn how to use appropriate units, estimate measurements and read scales, including length, capacity and time.  

6. Properties of shapes: Learners explore the fundamentals of shapes, including how to recognise and name different types of 2D and 3D shapes.  

7. Position and direction: Children  learn how to describe position, direction and movement, as well as how to organise objects into patterns and sequences. 

8. Statistics: In the KS1 curriculum, kids explore how to collect and organise data. They also learn how to create graphs and interpret data using a variety of graphs.

Get ready for KS1 maths SATs

In Year 2, children complete maths and English SATs papers. Explore what’s covered in the exams and how you can help your child or pupils prepare for them!

KS1 SATs papers 2024: what's included?

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How Doodle can help kids prepare for SATs

Our apps cover the whole KS1 curriculum, making them perfect study buddies

Getting prepared for KS1 maths SATs papers

KS1 maths SATs may seem worrying, but these tips will help you feel prepared!

KS1 SATs papers: what's included?

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Our apps cover the whole KS1 curriculum, making them the ultimate study buddies

KS1 maths FAQs

DoodleMaths  is an award-winning app that's filled with fantastic maths games for KS1!

From working in a messy bakery to helping a monkey collect coconuts, all of its games are specifically designed to develop instant recall and mental maths skills, transforming learning into a fun adventure.

Best of all, all of its questions feel like a game, with every exercise using interactive activities and explanations to help bring the subject to life!

Create an account today to enjoy a  free 7-day trial .

There are lots of ways that you can help your child or pupils to learn KS1 maths:

• Rather than explaining concepts, use visual representations to explore them, such as number lines and grids.
• Nothing beats doodling! Encourage your child or pupils to draw pictures to help them understand concepts or have a go at working out the answer to a question.
• Use real-life items to explore number, such as Smarties or pebbles. Encourage your child to sort the items into different amounts; doing this will help them understand concepts such as addition and fractions!
• Boosting their instant recall skills will help them with their mental maths and speed answering questions. Learning times tables in a specific order and using fun activities is a great way to help with this!

An array is an arrangement of items (such as pictures or numbers) in columns or rows.

They let children visualise what happens when they multiply or divide numbers together, helping them master times tables and division!

Reasoning is basically detective work: it's when learners use what they already know to work out the answer to a question.

For example, if a child knows what 12 - 5 is, they can use this knowledge to work out what 12 - 6 is.

"I know that 12 - 5 is 7. As 6 is 1 higher than 5, the answer must be 1 higher than 7."

Parents, sign up for a DoodleMaths subscription and see your child become a maths wizard!

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Reasoning and Problem Solving Questions Collection - KS1 and KS2

Subject: Mathematics

Age range: 5-7

Resource type: Worksheet/Activity

Last updated

10 March 2023

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These booklets each contain over 40 reasoning and problem solving questions suitable for KS1, KS2 and KS3 classes. These are the questions that we have been putting out each day in March 2016 on Twitter in the run up to SATS.

The answers are provided with some simple notes at the back of the booklet and for some problems supplementary questions and variation has been provided.

As always we welcome any feedback on the work we are doing and the materials that we are releasing. Thank you for taking an interest in our work. The White Rose Maths Hub Team

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We are pleased to let you know that your resource Reasoning and Problem Solving Questions Collection - KS1 and KS2, has been hand-picked by the Tes resources content team to be featured in https://www.tes.com/teaching-resources/blog/fluency-reasoning-and-problem-solving-primary-maths in April 2024 on https://www.tes.com/teaching-resources/blog. Congratulations on your resource being chosen and thank you for your ongoing contributions to the Tes Resources marketplace.

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• Volume And Capacity Worksheets And Resources For Ks1 And Ks2 Maths

7 of the best volume and capacity worksheets and resources for KS1 and KS2 maths

It doesn't matter whether you're a 'glass half full' or 'glass half empty' kind of person, these primary maths resources will help your students get their fill of capacity and volume…

1 | Introduce, measure and compare capacity

These activity sheets have been created to match the small steps on the White Rose maths schemes of work, with questions that include varied fluency with reasoning with problem solving, and an additional sheet with extension activities.

Children are given a variety of pictorial examples to work with and questions to provoke deeper thinking to help them with the curriculum requirements of Year 1 Measurement to “Compare, describe and solve practical problems for: capacity and volume (for example, full/empty, more than, less than, quarter, half full, half).”

There are three separate worksheets for this, so click the links for introducing capacity , measuring capacity and comparing capacity .

2 | George’s Marvellous Medicine KS1 capacity lesson plan

Can children estimate, decant and measure their way to discovering the elusive formula for George’s Marvellous Medicine , asks Jonathan Lear?

That’s the subject of this KS1 lesson plan that lets them explore measurements of capacity using standard metric units.

Download it here.

3 | Capacity and volume factsheets and worksheets

On the BBC Skillswise page for measuring capacity you’ll find a brief introductory video on the topic, plus a collection of free printable factsheets and worksheets.

These cover everything from labels, instruments for measuring capacity and non-standard measures of capacity to a matching exercise, reading scales and choosing litres or millilitres.

Check this all out here.

4 | Cover volume and capacity without pouring and filling

Looking for something a bit different? Mike Askew has ideas for teaching this topic without all the usual tricks.

Check them out here.

5 | Estimating volume for Year 5

These volume worksheets provide extra challenge for Year 5 children, with a variety of volume problems spread across three sections, enabling you to use the whole sheet during a lesson or to select specific problems for different teaching sessions.

Plus, a separate answer sheet for all sections is included.

Get this resource here.

6 | Volume of cuboids worksheet

This three-page, 12-question worksheet for upper KS2 is a quick and easy way to check pupils’ knowledge on the topic.

Get the worksheet here and the answer sheet here .

7 | Nrich capacity problems

What happens when you pour the water from one of these glasses into the other? Obvious right? Or is it? Watch the video from Nrich and explore with your class what you’ve seen.

This one is called Pouring Problem , but there are loads more volume and capacity problems to try. Here are a few of our favourites:

• Multilink cubes
• Next size up
• Cuboid-in-a-box
• Double Your Popcorn, Double Your Pleasure
• The Big Cheese

Or check out the full selection here.

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Patterns and Sequences KS1

This collection is one of our Primary Curriculum collections - tasks that are grouped by topic.

Repeating Patterns

Try continuing these patterns made from triangles. Can you create your own repeating pattern?

Cube Bricks and Daisy Chains

Daisy and Akram were making number patterns. Daisy was using beads that looked like flowers and Akram was using cube bricks. First they were counting in twos.

Poly Plug Pattern

Create a pattern on the small grid. How could you extend your pattern on the larger grid?

Chairs and Tables

Make a chair and table out of interlocking cubes, making sure that the chair fits under the table!

Next Domino

Which comes next in each pattern of dominoes?

A City of Towers

In this town, houses are built with one room for each person. There are some families of seven people living in the town. In how many different ways can they build their houses?

Domino Patterns

What patterns can you make with a set of dominoes?

Missing Middles

Can you work out the domino pieces which would go in the middle in each case to complete the pattern of these eight sets of three dominoes?

Three Ball Line Up

Use the interactivity to help get a feel for this problem and to find out all the possible ways the balls could land.

Caterpillars

These caterpillars have 16 parts. What different shapes do they make if each part lies in the small squares of a 4 by 4 square?

Circles, Circles

Here are some arrangements of circles. How many circles would I need to make the next size up for each? Can you create your own arrangement and investigate the number of circles it needs?

Break it Up!

In how many different ways can you break up a stick of seven interlocking cubes? Now try with a stick of eight cubes and a stick of six cubes. What do you notice?

Counting Stick Conjectures

How many rectangles can you see? Are they all the same size? Can you predict how many rectangles there will be in counting sticks of different lengths?

Triple Cubes

This challenge involves eight three-cube models made from interlocking cubes. Investigate different ways of putting the models together then compare your constructions.

Street Sequences

Investigate what happens when you add house numbers along a street in different ways.

Hundred Square

A hundred square has been printed on both sides of a piece of paper. What is on the back of 100? 58? 23? 19?

School Fair Necklaces

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Today Your Child Learned…

Dear Parent & Guardian,  We are excited to share that your child has completed the following PSHE lesson. Check out our overview and top tips to support their development…

Today’s PSHE Lesson

Problem-Solving And Resourcefulness (How To Problem-Solve)

Knowing that we will be expected to solve problems throughout our lives and understanding how to go about this is a useful skill to learn. From difficult homework to solving challenges at work, we all need to be able to problem-solve. Using technology and the internet brings its own problems so equipping our children with the tools to overcome these issues can enhance their overall safety and enjoyment online.

What We Covered In The PSHE Lesson

Here’s what we went through today….

• That problem-solving is a way of finding a solution to difficult situations or issues.
• Times when we’ve had to face challenges and problem-solve for example, whether to encourage a friend to report bullying that they’ve witnessed.
• Requests or issues that may appear online and how to problem-solve.
• Times when you’ve successfully solved a problem and how it felt.

How You Can Help From Home

Here’s what you can talk about during dinner….

• Talk to your child about when you’ve had to problem-solve and how it felt when you succeeded.
• Remind your child that problem-solving is an important skill to have and that it’s fine to ask for help.

Go Above & Beyond 

Here are some other ideas to support your childs PSHE lessons…

• Think about common scenarios where you might be expected to problem-solve, whether to report a threatening text message from a school friend or how to organise a football team for an important match.  
• Talk about the importance of each situation and how problem-solving may be difficult at times to get the solution right for everybody. 
• Encourage your child to talk about situations that they find difficult to solve on their own, especially if it involves online requests, cyberbullying or threats.

LifeWise Team

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