open ended problem solving ks2

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Problem Solving

 A selection of resources containing a wide range of open-ended tasks, practical tasks, investigations and real life problems, to support investigative work and problem solving in primary mathematics.

Problem Solving in Primary Maths - the Session

Quality Assured Category: Mathematics Publisher: Teachers TV

In this programme shows a group of four upper Key Stage Two children working on a challenging problem; looking at the interior and exterior angles of polygons and how they relate to the number of sides. The problem requires the children to listen to each other and to work together co-operatively. The two boys and two girls are closely observed as they consider how to tackle the problem, make mistakes, get stuck and arrive at the "eureka" moment. They organise the data they collect and are then able to spot patterns and relate them to the original problem to find a formula to work out the exterior angle of any polygon. At the end of the session the children report back to Mark, explaining how they arrived at the solution, an important part of the problem solving process.

In a  second video  two maths experts discuss some of the challenges of teaching problem solving. This includes how and at what stage to introduce problem solving strategies and the appropriate moment to intervene when children find tasks difficult. They also discuss how problem solving in the curriculum also helps to develop life skills.

Cards for Cubes: Problem Solving Activities for Young Children

Quality Assured Category: Mathematics Publisher: Claire Publications

This book provides a series of problem solving activities involving cubes. The tasks start simply and progress to more complicated activities so could be used for different ages within Key Stages One and Two depending on ability. The first task is a challenge to create a camel with 50 cubes that doesn't fall over. Different characters are introduced throughout the book and challenges set to create various animals, monsters and structures using different numbers of cubes. Problems are set to incorporate different areas of mathematical problem solving they are: using maths, number, algebra and measure.

Problem solving with EYFS, Key Stage One and Key Stage Two children

Quality Assured Category: Computing Publisher: Department for Education

These three resources, from the National Strategies, focus on solving problems.

  Logic problems and puzzles  identifies the strategies children may use and the learning approaches teachers can plan to teach problem solving. There are two lessons for each age group.

Finding all possibilities focuses on one particular strategy, finding all possibilities. Other resources that would enhance the problem solving process are listed, these include practical apparatus, the use of ICT and in particular Interactive Teaching Programs .

Finding rules and describing patterns focuses on problems that fall into the category 'patterns and relationships'. There are seven activities across the year groups. Each activity includes objectives, learning outcomes, resources, vocabulary and prior knowledge required. Each lesson is structured with a main teaching activity, drawing together and a plenary, including probing questions.

Primary mathematics classroom resources

Quality Assured Collection Category: Mathematics Publisher: Association of Teachers of Mathematics

This selection of 5 resources is a mixture of problem-solving tasks, open-ended tasks, games and puzzles designed to develop students' understanding and application of mathematics.

Thinking for Ourselves: These activities, from the Association of Teachers of Mathematics (ATM) publication 'Thinking for Ourselves’, provide a variety of contexts in which students are encouraged to think for themselves. Activity 1: In the bag – More or less requires students to record how many more or less cubes in total...

8 Days a Week: The resource consists of eight questions, one for each day of the week and one extra. The questions explore odd numbers, sequences, prime numbers, fractions, multiplication and division.

Number Picnic: The problems make ideal starter activities

Matchstick Problems: Contains two activities concentrating upon the process of counting and spotting patterns. Uses id eas about the properties of number and the use of knowledge and reasoning to work out the rules.

Colours: Use logic, thinking skills and organisational skills to decide which information is useful and which is irrelevant in order to find the solution.

GAIM Activities: Practical Problems

Quality Assured Category: Mathematics Publisher: Nelson Thornes

Designed for secondary learners, but could also be used to enrich the learning of upper primary children, looking for a challenge. These are open-ended tasks encourage children to apply and develop mathematical knowledge, skills and understanding and to integrate these in order to make decisions and draw conclusions.

Examples include:

*Every Second Counts - Using transport timetables, maps and knowledge of speeds to plan a route leading as far away from school as possible in one hour.

*Beach Guest House - Booking guests into appropriate rooms in a hotel.

*Cemetery Maths - Collecting relevant data from a visit to a local graveyard or a cemetery for testing a hypothesis.

*Design a Table - Involving diagrams, measurements, scale.

Go Further with Investigations

Quality Assured Category: Mathematics Publisher: Collins Educational

A collection of 40 investigations designed for use with the whole class or smaller groups. It is aimed at upper KS2 but some activities may be adapted for use with more able children in lower KS2. It covers different curriculum areas of mathematics.

Starting Investigations

The forty student investigations in this book are non-sequential and focus mainly on the mathematical topics of addition, subtraction, number, shape and colour patterns, and money.

The apparatus required for each investigation is given on the student sheets and generally include items such as dice, counters, number cards and rods. The sheets are written using as few words as possible in order to enable students to begin working with the minimum of reading.

NRICH Primary Activities

Explore the NRICH primary tasks which aim to enrich the mathematical experiences of all learners. Lots of whole class open ended investigations and problem solving tasks. These tasks really get children thinking!

Mathematical reasoning: activities for developing thinking skills

Quality Assured Category: Mathematics Publisher: SMILE

Problem Solving 2

Reasoning about numbers, with challenges and simplifications.

Quality Assured Category: Mathematics Publisher: Department for Education

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Goal Free Problems And Focused Thinking: How I Wish I’d Taught Primary Maths (3)

Clare Sealy

Clare Sealy looks at the benefits that focused thinking and goal free problems (also known as open ended maths investigations), can have when used in a KS2 classroom.

This article is part of a series published to help primary school teachers and leaders implement some of the insights and teaching techniques derived from Craig Barton’s bestselling book How I Wish I’d Taught Maths . Links to the other 5 articles appear at the end.

In the introduction to this series, I outlined how Craig Barton in his book How I Wish I’d Taught Maths described how he had changed his teaching. This was as the result of reading research around cognitive load theory in the classroom , and in particular, realising how easy it was to inadvertently prevent learning by overwhelming working memory through cognitive overload.

In order to avoid this, Craig now plans his teaching with cognitive science considerations at the forefront of his thinking. By making relatively straightforward changes to the presentation of information and instructional design, unhelpful cognitive load (called ‘extraneous’ cognitive load in the research literature) is removed.

Craig’s preferred technique for this is through direct instruction and worked examples, and as a result of these teaching strategies , working memory is freed up to think about what needs to be thought about.

Crib Sheet for How I Wish I'd Taught Primary Maths

Download the key findings from research; share with your staff, your SLT, and at your next job interview!

Results from the research: Cognitive overload can affect pupils’ working memory

Cognitive overload is the enemy of learning. This is because if the capacity of working memory is exceeded, the stuff you actually want children to remember may not make it from the working memory to the long-term memory.

Since learning involves change in the long-term memory, if nothing has changed in the long-term memory, nothing has been learned. Sweller et al, 1998 suggest that teachers should take care to design instruction so that working memories are not overwhelmed. Fortunately, there is a simple way you can do this…

What are goal free problems?

While goal specific questions have an exact answer, goal free questions do not. They are a form of open ended maths investigation and an questioning in the classroom technique, and they are a great way to help prevent cognitive overload among pupils.

Goal free questions are the longer two or three-part reasoning questions that appear in maths, not simple calculations, and they are best used once you have taught the children something and now want them to apply it in a problem-solving context.

They are not intended to be used at the very start of teaching something .

These are ‘application’ questions, once the initial stage of how to do something has been taught and reinforced through deliberate practice in the short term, and retrieval practice across the school year.

If these open ended maths investigations are introduced before a topic has been fully grasped by pupils, this can result in confusion.

The difference between a goal specific and goal free question

The following examples are all from the KS2 2017 Maths SATs Paper 2 :

This is a goal specific question with a specific answer.

A goal free question would look like this:

So why are goal free questions better?

The reason why the second kind of question is better than the first is that in the second, the pupil can use metacognitive skills to work on parts of the question one at a time, rather than being overwhelmed trying to solve every part at once.

You can find out more about the sorts of thinking skills pupils might use from our blog on metacognition in the classroom .

Thanks to the open ended nature of these maths problems, pupils can focus on one singular part of the problem that they feel is important, and confidently solve this before moving on.

For example:

They would probably decide to work out how much 12 individual stickers cost. They would also not have any niggling, intrusive thoughts about what they need to do next competing for attention in their working memory. Therefore, all of their attention would be devoted to calculating 12x99p.

Having done that, they would probably then realise that buying individual stickers was a rip off! They might also work out other things, for example:

• If Jack had £11, how many individual stickers could he buy? • Or what is the cheapest amount you could pay to get 16 stickers?

But whatever they decided to work out, they were thinking about that and that alone.

Why the traditional method of asking questions can hinder pupils’ learning

With a conventional problem, concerns about the final goal can intrude upon their thinking. The pupil’s attention is split between thinking about the first step and thinking about the other steps that they need to take next.

Therefore, even if they successfully answer the question, they may not have learnt anything generally that could be transferred to new kinds of problems.

Their attention has not been focused enough.

Goal free questions are a great way to help pupils solve the problem in front of them

Once you have given the children the goal free version of the question and various things have been worked out, at that point you can, if you wish, share the goal specific question. You should then realise the hard work of answering the question has already been done and a final tweak might be all that is now needed.

Consider this example, adapted to be open ended and goal free:

Without a goal specific question looming over them, children will probably quickly be able to identify the size of the individual unshaded parts. Then they might decide to work out what fraction is shaded. Then when the goal specific question is revealed…

…answering it is very easy. Even if they only initially worked out the size of the individual unshaded parts, they are still ahead of the game having done so.

Best practice for open ended, goal free questions

Using goal free questions works particularly well with graphs and geometry questions, with the pupil asking themselves: “What can I work out first?” before worrying (or being told) about what the question actually says.

Doing it this way in a test situation, the pupil is more likely to answer the actual question than if they had tried to solve it from the off. This is because they know that their initial goal free musing is a prelude to the final step of telling the examiner the actual thing that will get them marks.

So, for example, teach pupils to hide the question and first of all annotate the graph to show what it tells them.

As well as annotating the axes, the child also annotates their paper with the following.

Difference yr 2 – 110-70=40

Difference yr 5 – 120-90=30

Walk – 70+120=190

Don’t walk – 110+90=200

Then when the final specific question is revealed…

…The solution is already worked out. However, by not having to worry about the final question during the initial annotation stage, their attention is much better focused as they explore the graph to find answers to their own question.

It is not necessary to always share the ‘final’ question, especially during the early knowledge-acquisition phase. Goal free, open ended questions are particularly useful the first time you want pupils to apply a skill you have just taught. Goal free questions with an eventual goal specific big reveal are useful when revising in readiness for a test.

3 tips for bringing focused thinking and goal free learning into your classroom

1. Prepare a dedicated selection of open-ended maths investigations for your class in the form of problems, activities or worksheets

2. Present existing questions to students in the manner seen above, asking them what they can take from the question before the final request is revealed

3. Use goal free questions as a way to further cement knowledge, using them after you have taught pupils a principle. This will encourage focused thinking within the classroom

I hope that the theories about goal free, open ended maths investigations and focused thinking presented in this blog prove useful in your classroom.

Sources of inspiration

Sweller, J., Merriënboer, J.J.G. and Pass, F.G.W.C. (1998) ‘Cognitive architecture and instructional design’, Educational Psychology Review 10 (3) pp.251-296

This is the third blog in a series of 6 adapting the book How I Wish I’d Taught Maths for a primary audience. If you wish to read the remaining blogs in the series, check them out below:

• 20 maths strategies that we use in our teaching to guarantee success for any pupil.
• Quality First Teaching Checklist: The 10 Most Effective Strategies For Primary Schools
• Differentiation In The Classroom: The 8 Strategies That Will Support Every Pupil To Reach A Good Standard
• Learning and Memory In The Classroom: What Teachers Should Know About Summer Learning Loss – And How To Fix It
• Teaching ‘Lower Ability’ Students Maths: “Are We Bottom Set?”

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KS2 Maths (Problem Solving)

Subject: Mathematics

Age range: 7-11

Resource type: Worksheet/Activity

Last updated

16 January 2019

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These topic-focused SATs questions at the end of a unit will help to test and extend students’ understanding as well as helping them to prepare for SATs next year. These questions have fully-worked solutions which can be displayed on a whiteboard making feedback with students more efficient. Click 👉 tes.com/…/KS2-Maths-Questions… for similar-style compilations on the other KS2 topics. <hr> This particular compilation is mainly from the ALGEBRA strand and contains questions which require students us problem solving techniques . In the solutions, I have often used algebraic notation to show how an answer has been obtained, but most of the solutions actually use logic rather than formal algebraic methods. <hr> I have designed this compilation to be printed as an A4 or A5 booklet which is in the style of the actual SATs papers and is convenient for use in class or as homework. It can even be given to individual students if a parent is asking for ‘some more work’!

KEY POINTS:

• I have provided full answers, with comments and working where helpful.
• I have maintained the style of the actual SATs questions so that students can become comfortable with the way that SATs questions are presented.
• Most of the questions are from actual SATs papers, but I have also added questions so that this resource matches the requirements the current curriculum better than the older resources that are still in common use (note that many of the older resources of this type contain questions on topics which are no longer examined).
• I have spent a lot of time arranging the questions so that there is a general increase in difficulty as students work through them, and so that they fit on the pages better – this means less wasted space and significant paper-saving when printing 😃 <hr> 👍If you like this resource, then please rate it and/or leave a comment💬. If the rate-resource button on this page doesn’t work, then go to your ratings page by clicking 👉 www.tes.com/…/rate-resources…

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