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Authentic tasks are designed to help students see mathematics as worthwhile and important. When students understand the purpose of a given problem in mathematics, they are more likely to persist when challenged. Authentic tasks generally have an ‘open middle’ which means that students can use different representations and solutions to communicate their knowledge and reasoning.

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Module 1: Problem Solving Strategies

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Unlike exercises, there is never a simple recipe for solving a problem. You can get better and better at solving problems, both by building up your background knowledge and by simply practicing. As you solve more problems (and learn how other people solved them), you learn strategies and techniques that can be useful. But no single strategy works every time.

Pólya’s How to Solve It

George Pólya was a great champion in the field of teaching effective problem solving skills. He was born in Hungary in 1887, received his Ph.D. at the University of Budapest, and was a professor at Stanford University (among other universities). He wrote many mathematical papers along with three books, most famously, “How to Solve it.” Pólya died at the age 98 in 1985.1

1. Image of Pólya by Thane Plambeck from Palo Alto, California (Flickr) [CC BY

In 1945, Pólya published the short book How to Solve It , which gave a four-step method for solving mathematical problems:

First, you have to understand the problem.

After understanding, then make a plan.

Carry out the plan.

Look back on your work. How could it be better?

This is all well and good, but how do you actually do these steps?!?! Steps 1. and 2. are particularly mysterious! How do you “make a plan?” That is where you need some tools in your toolbox, and some experience to draw upon.

Much has been written since 1945 to explain these steps in more detail, but the truth is that they are more art than science. This is where math becomes a creative endeavor (and where it becomes so much fun). We will articulate some useful problem solving strategies, but no such list will ever be complete. This is really just a start to help you on your way. The best way to become a skilled problem solver is to learn the background material well, and then to solve a lot of problems!

Problem Solving Strategy 1 (Guess and Test)

Make a guess and test to see if it satisfies the demands of the problem. If it doesn't, alter the guess appropriately and check again. Keep doing this until you find a solution.

Mr. Jones has a total of 25 chickens and cows on his farm. How many of each does he have if all together there are 76 feet?

Step 1: Understanding the problem

We are given in the problem that there are 25 chickens and cows.

All together there are 76 feet.

Chickens have 2 feet and cows have 4 feet.

We are trying to determine how many cows and how many chickens Mr. Jones has on his farm.

Step 2: Devise a plan

Going to use Guess and test along with making a tab

Many times the strategy below is used with guess and test.

Make a table and look for a pattern:

Procedure: Make a table reflecting the data in the problem. If done in an orderly way, such a table will often reveal patterns and relationships that suggest how the problem can be solved.

Step 3: Carry out the plan:

Notice we are going in the wrong direction! The total number of feet is decreasing!

Better! The total number of feet are increasing!

Step 4: Looking back:

Check: 12 + 13 = 25 heads

24 + 52 = 76 feet.

We have found the solution to this problem. I could use this strategy when there are a limited number of possible answers and when two items are the same but they have one characteristic that is different.

Videos to watch:

1. Click on this link to see an example of “Guess and Test”

http://www.mathstories.com/strategies.htm

2. Click on this link to see another example of Guess and Test.

http://www.mathinaction.org/problem-solving-strategies.html

Check in question 1:

Place the digits 8, 10, 11, 12, and 13 in the circles to make the sums across and vertically equal 31. (5 points)

Check in question 2:

Old McDonald has 250 chickens and goats in the barnyard. Altogether there are 760 feet . How many of each animal does he have? Make sure you use Polya’s 4 problem solving steps. (12 points)

Problem Solving Strategy 2 (Draw a Picture). Some problems are obviously about a geometric situation, and it is clear you want to draw a picture and mark down all of the given information before you try to solve it. But even for a problem that is not geometric thinking visually can help!

Videos to watch demonstrating how to use "Draw a Picture".

1. Click on this link to see an example of “Draw a Picture”

2. Click on this link to see another example of Draw a Picture.

Problem Solving Strategy 3 ( Using a variable to find the sum of a sequence.)

Gauss's strategy for sequences.

last term = fixed number ( n -1) + first term

The fix number is the the amount each term is increasing or decreasing by. "n" is the number of terms you have. You can use this formula to find the last term in the sequence or the number of terms you have in a sequence.

Ex: 2, 5, 8, ... Find the 200th term.

Last term = 3(200-1) +2

Last term is 599.

To find the sum of a sequence: sum = [(first term + last term) (number of terms)]/ 2

Sum = (2 + 599) (200) then divide by 2

Sum = 60,100

Check in question 3: (10 points)

Find the 320 th term of 7, 10, 13, 16 …

Then find the sum of the first 320 terms.

Problem Solving Strategy 4 (Working Backwards)

This is considered a strategy in many schools. If you are given an answer, and the steps that were taken to arrive at that answer, you should be able to determine the starting point.

Videos to watch demonstrating of “Working Backwards”

https://www.youtube.com/watch?v=5FFWTsMEeJw

Karen is thinking of a number. If you double it, and subtract 7, you obtain 11. What is Karen’s number?

1. We start with 11 and work backwards.

2. The opposite of subtraction is addition. We will add 7 to 11. We are now at 18.

3. The opposite of doubling something is dividing by 2. 18/2 = 9

4. This should be our answer. Looking back:

9 x 2 = 18 -7 = 11

5. We have the right answer.

Check in question 4:

Christina is thinking of a number.

If you multiply her number by 93, add 6, and divide by 3, you obtain 436. What is her number? Solve this problem by working backwards. (5 points)

Problem Solving Strategy 5 (Looking for a Pattern)

Definition: A sequence is a pattern involving an ordered arrangement of numbers.

We first need to find a pattern.

Ask yourself as you search for a pattern – are the numbers growing steadily larger? Steadily smaller? How is each number related?

Example 1: 1, 4, 7, 10, 13…

Find the next 2 numbers. The pattern is each number is increasing by 3. The next two numbers would be 16 and 19.

Example 2: 1, 4, 9, 16 … find the next 2 numbers. It looks like each successive number is increase by the next odd number. 1 + 3 = 4.

So the next number would be

25 + 11 = 36

Example 3: 10, 7, 4, 1, -2… find the next 2 numbers.

In this sequence, the numbers are decreasing by 3. So the next 2 numbers would be -2 -3 = -5

-5 – 3 = -8

Example 4: 1, 2, 4, 8 … find the next two numbers.

This example is a little bit harder. The numbers are increasing but not by a constant. Maybe a factor?

So each number is being multiplied by 2.

16 x 2 = 32

1. Click on this link to see an example of “Looking for a Pattern”

2. Click on this link to see another example of Looking for a Pattern.

Problem Solving Strategy 6 (Make a List)

Example 1 : Can perfect squares end in a 2 or a 3?

List all the squares of the numbers 1 to 20.

1 4 9 16 25 36 49 64 81 100 121 144 169 196 225 256 289 324 361 400.

Now look at the number in the ones digits. Notice they are 0, 1, 4, 5, 6, or 9. Notice none of the perfect squares end in 2, 3, 7, or 8. This list suggests that perfect squares cannot end in a 2, 3, 7 or 8.

How many different amounts of money can you have in your pocket if you have only three coins including only dimes and quarters?

Quarter’s dimes

0 3 30 cents

1 2 45 cents

2 1 60 cents

3 0 75 cents

Videos demonstrating "Make a List"

Check in question 5:

How many ways can you make change for 23 cents using only pennies, nickels, and dimes? (10 points)

Problem Solving Strategy 7 (Solve a Simpler Problem)

Geometric Sequences:

How would we find the nth term?

Solve a simpler problem:

1, 3, 9, 27.

1. To get from 1 to 3 what did we do?

2. To get from 3 to 9 what did we do?

Let’s set up a table:

Term Number what did we do

Looking back: How would you find the nth term?

Find the 10 th term of the above sequence.

Let L = the tenth term

Problem Solving Strategy 8 (Process of Elimination)

This strategy can be used when there is only one possible solution.

I’m thinking of a number.

The number is odd.

It is more than 1 but less than 100.

It is greater than 20.

It is less than 5 times 7.

The sum of the digits is 7.

It is evenly divisible by 5.

a. We know it is an odd number between 1 and 100.

b. It is greater than 20 but less than 35

21, 23, 25, 27, 29, 31, 33, 35. These are the possibilities.

c. The sum of the digits is 7

21 (2+1=3) No 23 (2+3 = 5) No 25 (2 + 5= 7) Yes Using the same process we see there are no other numbers that meet this criteria. Also we notice 25 is divisible by 5. By using the strategy elimination, we have found our answer.

Check in question 6: (8 points)

Jose is thinking of a number.

The number is not odd.

The sum of the digits is divisible by 2.

The number is a multiple of 11.

It is greater than 5 times 4.

It is a multiple of 6

It is less than 7 times 8 +23

What is the number?

Click on this link for a quick review of the problem solving strategies.

https://garyhall.org.uk/maths-problem-solving-strategies.html

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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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• Problem Solving

The problem solving workshops run by PDST in local Education Centres across the country endeavour to support teachers in the exploration of problem solving as a central methodology across the mathematics curriculum. The exploration of content, methodologies and skills and the development of mathematical thinking are promoted throughout. The following resources and materials are available to print or download by clicking on the links below.

The PDST have offered a number of workshops and seminars in the area of Problem Solving. Relevant resources from these sessions can be accessed by clicking the links below. They have also collated collections of problems for Maths Week, which are available on Scoilnet at the links below.

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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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Problem Solving Worksheets & Workbooks

Are you looking for fresh, new ideas to help your pupils with maths problem-solving? Problem-solving is a key skill that children need to learn to be successful in maths. Maths education should prioritise problem-solving, as it provides children with the foundational skills they need to tackle any challenge that comes their way.

Do you have all you need to teach problem-solving in maths?

Prim-Ed Publishing has many great resources available that can help teachers successfully implement problem-solving into their maths curriculum. From boxed resources and worksheets to teachers notes and photocopiable books, covering a wide range of exciting and high-interest problem-solving activities, we have the tools to support teachers to confidently deliver problem-solving in class.

What is problem-solving?

In its simplest form, problem-solving is the process of finding solutions to problems and challenges that we encounter in our lives. It is a fundamental skill that we use daily, often without even realising it. The ability to solve problems often depends on our capacity to think flexibly and see things from different perspectives.

With problem-solving embedded at the heart of all maths lessons, pupils can master the core concepts and strategies needed to succeed in any situation.

Why is mathematics problem-solving important?

At every stage of life and in every development context, problem-solving skills are invaluable.

Whether we are negotiating with a client, tackling a challenging task at work or simply navigating our way through crowded streets and busy social situations, problem-solving skills help us to cope more effectively with the obstacles and challenges we encounter. Fortunately, these skills can be learned and developed through education.

Developing problem-solving skills in young learners

Problem-solving should be at the heart of maths education because it is a fundamental life skill. By teaching pupils how to break down a problem into smaller steps, you can help pupils overcome their natural aversion to difficulties and give them the confidence to tackle anything that comes their way.

Learning to solve problems encourages mathematical thinking

One of the benefits of problem-solving is that it encourages pupils to think mathematically.

Mathematical thinking enables us to make sense of the world around us using numbers, shapes and patterns. It helps us understand relationships, detect trends and make predictions. Mathematical thinking is not just about being able to do sums; it’s a way of looking at and understanding the world.

In mathematics, pupils learn to analyse the information given in maths problems and then use their maths knowledge and skills to find answers.

Problem-solving skills in primary mathematics

Problem-solving skills are best developed through opportunity and practice. Teachers must provide opportunities for young learners to engage in problem-solving activities regularly.

By developing the ability to problem-solve, pupils gain a better understanding of the concepts and skills underlying mathematical objectives, concepts and processes. In addition, problem-solving allows children to practise critical higher-order thinking skills such as reasoning, interpretation, synthesis and evaluation.

A problem-solving approach to teaching

Problem-solving should be taught explicitly, with rich mathematical tasks and practice opportunities included in every lesson. This also involves developing pupils’ understanding of mathematical concepts such as:

• Shape and space

There are many ways to incorporate problem-solving into your classroom lessons.

• Problem-solving activities and worksheets can be used to assess pupils’ understanding.
• Provide opportunities for pupils to engage in collaborative problem-solving tasks, with partners or groups.
• Challenge young learners with problems presented visually, such as tasks interpreting the data in tables, graphs and charts.
• The three-book series, Maths through Language , is a great way to integrate language into your maths lessons.
• Hands-on games and puzzles come in all shapes and sizes and can be adapted to suit any age group or ability level.
• Puzzles will help pupils develop patience, persistence, and attention to detail – all essential qualities for good problem-solvers.

Problem-solving resources from Prim-Ed Publishing

With Prim-Ed Publishing , teachers have access to a wide variety of problem-solving resources – including worksheets, boxed materials, photocopiable books, ebooks and digital resources – that provide engaging activities designed to help pupils thrive in this area.

Our problem-solving resources are designed to engage pupils and focus on key problem-solving strategies, such as visualising, reasoning, working systematically and making models and diagrams.

Our resources include extensive background information, curriculum links, extension activities and teachers notes to teach young people the specific skills needed for problem-solving.

A few of the most popular products available from Prim-Ed Publishing include:

• Measurement and Geometry
• Number and Algebra
• Statistics and Chance
• Primary Problem-Solving In Mathematics

Prim-Ed Publishing's problem-solving resources will support teachers and help to develop pupils' problem-solving skills, encouraging mathematical thinking along the way.

Why not browse our full range of mathematics resources today?

Frequently asked questions

What is problem-solving in primary maths.

One of the most important skills pupils learn in maths is how to solve problems. A maths problem can be defined as a question or task that requires pupils to use their skills and knowledge to find the answer.

What are some of the best problem-solving techniques?

There are a number of problem-solving techniques that can be useful for pupils.

• One strategy is to use logical reasoning, connecting the facts and evidence to reach a conclusion.
• Another strategy is to visualise the problem, picturing it in your mind and exploring different solutions.
• You can also break the problem into smaller pieces, analysing each part and looking for patterns.
• Finally, it can be helpful to get feedback from others and ask for their suggestions.

How can problem-solving skills be taught effectively in the mathematics curriculum?

There is no definitive way to teach effective problem-solving skills, as the approach will vary depending on the problem and the child's age. However, there are some different approaches that can be useful.

• Strategic thinking: teaching pupils to solve problems through strategic thinking can be helpful.
• Presentation styles: problems can be presented visually to help pupils see the relationships between different aspects of the problem.
• Spatial visualisation exercises: help pupils understand relationships between objects in space, which can be helpful for solving mathematical problems.

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Primary Maths - Problem Solving styles and examples

Subject: Mathematics

Age range: 5-7

Resource type: Other

Last updated

18 March 2024

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A document outlining the different styles of questions around problem solving in maths. There are many, including more than one answer, missing numbers, odd one out, 1 or 2 step problems, logical thinking, patterns, generalisation and non-examples. The document includes the skills that children will need to solve these problems and examples of what they look like. This is intended to support teachers within the primary phase to effectively teach maths problem solving in class. As maths lead, I have distributed this to staff and asked them to ensure they expose the children to each one each year.

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How to Teach Effective Problem-Solving Skills in Mathematics | Primary

This webinar will provide headteachers, mathematics leads, teachers and teaching assistants with practical guidance and creative methods they can use to nurture and develop pupils’ problem-solving skills in mathematics.

• Description
• Learning Outcomes
• Institution

Webinar Duration: 1 hour 9 minutes (approx.)

Problem-solving has long been at the heart of the mathematics curriculum. Teaching children how to problem solve in mathematics can support children’s ability to critically evaluate, encourage independence and develop their skills in reasoning and creativity. It is also an essential part of developing mastery of the subject.

In this webinar, the Association of Teachers of Mathematics (ATM), who aim to support the teaching and learning of mathematics in the UK, will explore strategies that schools can use to teach problem-solving which are creative, engaging, fun and reflect a better understanding of the needs of the learner.

• Understanding how to plan and implement effective teaching practice in mathematics which supports children’s ability to problem solve and improve critical thinking skills.
• Recognising successful techniques that can be used in the classroom which improve fluency and reasoning in mathematics.
• Appreciating the importance of making problem-solving learning tailored towards the needs of children and ensuring continuous sharing and evaluation of different methods used.
• Understanding how to implement different approaches to teaching about problem-solving and equipping children with different tools they can use for the rest of their life.
• Recognising the advantages of promoting a culture which encourages discussion between peers and supports trust and confidence in the classroom.

Tony is the lead author for Oxford International Primary Mathematics . Other publications include Understanding and Teaching Primary Mathematics and How to develop confident mathematicians in the early years for Routledge; Approaches to learning and teaching Primary: A toolkit for international teachers for CUP; Explore Mathematics for the new standards curriculum in Jamaica ; and BZ Math for primary schools in Belize. His books, Being a Teacher and Transforming Teaching , both draw on Tony’s international experience and share the experiences of educators around the world. Tony is also editor of Mathematics Teaching , the journal of The Association of Teachers of Mathematics. 

Tony has worked with Ministries of Education in Macedonia and Oman to develop and implement new primary and secondary mathematics curricula. He teaches the international PGEI delivered by the University of Nottingham in SE Asia, leading the course in Thailand. 

He became a lecturer in secondary mathematics education at the University of Nottingham, gaining his PhD in 1999. Since then, he taught secondary and primary teacher education in Nottingham and Leeds, becoming Head of the School of Education and Childhood at Leeds Metropolitan University. In 2012, he left the university sector to work full time as a writer and freelance education consultant. 

Tony started his career teaching mathematics in secondary schools in Sheffield, England. He then worked as an advisory teacher for anti-racist and multicultural education, completing a Master’s degree in multicultural education, before spending time with 3 commercial publishers.

Learning to Teach Mathematics Through Problem Solving

• Open access
• Published: 21 April 2022
• Volume 57 , pages 407–423, ( 2022 )

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• Judy Bailey   ORCID: orcid.org/0000-0001-9610-9083 1  

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While there has been much research focused on beginning teachers; and mathematical problem solving in the classroom, little is known about beginning primary teachers’ learning to teach mathematics through problem solving. This longitudinal study examined what supported beginning teachers to start and sustain teaching mathematics through problem solving in their first 2 years of teaching. Findings show ‘sustaining’ required a combination of three factors: (i) participation in professional development centred on problem solving (ii) attending subject-specific complementary professional development initiatives alongside colleagues from their school; and (iii) an in-school colleague who also teaches mathematics through problem solving. If only one factor is present, in this study attending the professional development focussed on problem solving, the result was little movement towards a problem solving based pedagogy. Recommendations for supporting beginning teachers to embed problem solving are included.

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Introduction

For many years curriculum documents worldwide have positioned mathematics as a problem solving endeavour (e.g., see Australian Curriculum, Assessment and Reporting Authority, 2018 ; Ministry of Education, 2007 ). There is evidence however that even with this prolonged emphasis, problem solving has not become a significant presence in many classrooms (Felmer et al., 2019 ). Research has reported on a multitude of potential barriers, even for experienced teachers (Clarke et al., 2007 ; Holton, 2009 ). At the same time it is widely recognised that beginning teachers encounter many challenges as they start their careers, and that these challenges are particularly compelling when seeking to implement ambitious methods of teaching, such as problem solving (Wood et al., 2012 ).

Problem solving has been central to mathematics knowledge construction from the beginning of human history (Felmer et al., 2019 ). Teaching and learning mathematics through problem solving supports learners’ development of deep and conceptual understandings (Inoue et al., 2019 ), and is regarded as an effective way of catering for diversity (Hunter et al., 2018 ). While the importance and challenge of mathematical problem solving in school classrooms is not questioned, the promotion and enabling of problem solving is a contentious endeavour (English & Gainsburg, 2016 ). One debate centres on whether to teach mathematics through problem solving or to teach problem solving in and of itself. Recent scholarship (and this research) leans towards teaching mathematics through problem solving as a means for students to learn mathematics and come to appreciate what it means to do mathematics (Schoenfeld, 2013 ).

Problem solving has been defined in a multitude of ways over the years. Of central importance to problem solving as it is explored in this research study is Schoenfeld’s ( 1985 ) proposition that, “if one has ready access to a solution schema for a mathematical task, that task is an exercise and not a problem” (p. 74). A more recent definition of what constitutes a mathematical problem from Mamona-Downs and Mamona ( 2013 ) also emphasises the centrality of the learner not knowing how to proceed, highlighting that problems cannot be solved by procedural effort alone. These are important distinctions because traditional school texts and programmes often position problems and problem solving as an ‘add-on’ providing a practice opportunity for a previously taught, specific procedure. Given the range of learners in any education setting an important point to also consider is that what constitutes a problem for some students may not be a problem for others (Schoenfeld, 2013 ).

A research focus exploring what supports beginning teachers’ learning about teaching mathematics through problem solving is particularly relevant at this time given calls for an increased curricular focus on mathematical practices such as problem solving (Grootenboer et al., 2021 ) and recent recommendations from an expert advisory panel on the English-medium Mathematics and Statistics curriculum in Aotearoa (Royal Society Te Apārangi, 2021 ). The ninth recommendation from this report advocates for the provision of sustained professional learning in mathematics and statistics for all teachers of Years 0–8. With regard to beginning primary teachers, the recommendation goes further suggesting that ‘mathematics and statistics professional learning’ (p. 36) be considered as compulsory in the first 2 years of teaching. This research explores what the nature of that professional learning might involve, with a focus on problem solving.

Scoping the Context for Learning and Sustaining Problem Solving

The literature reviewed for this study draws from two key fields: the nature of support and professional development effective for beginning teachers; and specialised supports helping teachers to employ problem solving pedagogies.

Beginning Teachers, Support and Professional Development

A teacher’s early years in the profession are regarded as critical in terms of constructing a professional practice (Feiman-Nemser, 2003 ) and avoiding high attrition (Karlberg & Bezzina, 2020 ). Research has established that beginning teachers need professional development opportunities geared specifically to their needs (Fantilli & McDougall, 2009 ) and their contexts (Gaikhorst, et al., 2017 ). Providing appropriate support is not an uncontentious matter with calls for institutions to come together and collaborate to provide adequate and ongoing support (Karlberg & Bezzina, 2020 ). The proposal is that support is needed from both within and beyond the beginning teacher’s school; and begins with effective pre-service teacher preparation (Keese et al., 2022 ).

Within schools where beginning teachers regard the support they receive positively, collaboration, encouragement and ‘involved colleagues’ are considered as vital; with the guidance of a 'buddy’ identified as some of the most valuable in-school support activities (Gaikhorst et al., 2014 ). Cameron et al.’s ( 2007 ) research in Aotearoa reports beginning teachers joining collaborative work cultures had greater opportunities to talk about teaching with their colleagues, share planning and resources, examine students’ work, and benefit from the collective expertise of team members.

Opportunities to participate in networks beyond the beginning teacher’s school have also been identified as being important for teacher induction (Akiri & Dori, 2021 ; Cameron et al., 2007 ). Fantilli & McDougall ( 2009 ) in their Canadian study found beginning teachers reported a need for many support and professional development opportunities including subject-specific (e.g., mathematics) workshops prior to and throughout the year. Akiri and Dori ( 2021 ) also refer to the need for specialised support from subject-specific mentors. This echoes the findings of Wood et al. ( 2012 ) who advocate that given the complexity of learning to teach mathematics, induction support specific to mathematics, and rich opportunities to learn are not only desirable but essential.

Akiri and Dori ( 2021 ) describe three levels of mentoring support for beginning teachers including individual mentoring, group mentoring and mentoring networks with all three facilitating substantive professional growth. Of relevance to this paper are individual and group mentoring. Individual mentoring involves pairing an experienced teacher with a beginning teacher, so that a beginning teacher’s learning is supported. Group mentoring involves a group of teachers working with one or more mentors, with participants receiving guidance from their mentor(s) (Akiri & Dori, 2021 ). Findings from Akiri and Dori suggest that of the varying forms of mentoring, individual mentoring contributes the most for beginning teachers’ professional learning.

Teachers Learning to Teach Mathematics Through Problem Solving

Learning to teach mathematics through problem solving begins in pre-service teacher education. It has been shown that providing pre-service teachers with opportunities to engage in problem solving as learners can be productive (Bailey, 2015 ). Opportunities to practise content-specific instructional strategies such as problem solving during student teaching has also been positively associated with first-year teachers’ enactment of problem solving (Youngs et al., 2022 ).

The move from pre-service teacher education to the classroom can be fraught for beginning teachers (Feiman-Nemser, 2003 ), and all the more so for beginning teachers attempting to teach mathematics through problem solving (Wood et al., 2012 ). In a recent study (Darragh & Radovic, 2019 ) it has been shown that an individual willingness to change to a problem-based pedagogy may not be enough to sustain a change in practice in the long term, particularly if there is a contradiction with the context and ‘norms’ (e.g., school curriculum) within which a teacher is working. Cady et al. ( 2006 ) explored the beliefs and practices of two teachers from pre-service teacher education through to becoming experienced teachers. One teacher who initially adopted reform practices reverted to traditional beliefs about the learning and teaching of mathematics. In contrast, the other teacher implemented new practices only after understanding these and gaining teaching experience. Participation in mathematically focused professional development and involvement in resource development were thought to favourably influence the second teacher.

Lesson structures have been found to support teachers learning to teach mathematics through problem solving. Sullivan et al. ( 2016 ) explored the use of a structure comprising four phases: launching, exploring, summarising and consolidating. Teachers in Australia and Aotearoa have reported the structure as productive and feasible (Ingram et al., 2019 ; Sullivan et al., 2016 ). Teaching using challenging tasks (such as in problem solving) and a structure have been shown to accommodate student diversity, a pressing concern for many teachers. Student diversity has often been managed by grouping students according to perceived levels of capability (called ability grouping). Research identifies this practice as problematic, excluding and marginalising disadvantaged groups of students (e.g., see Anthony & Hunter, 2017 ). The lesson structure explored by Sullivan et al. ( 2016 ) caters for diversity by deliberately differentiating tasks, providing enabling and extending prompts. Extending prompts are offered to students who finish an original task quickly and ideally elicit abstraction and generalisation. Enabling prompts involve reducing the number of steps, simplifying the numbers, and/or varying forms of representation for students who cannot initially proceed, with the explicit intention that students then return to the original task.

Recognising the established challenges teachers encounter when learning about teaching mathematics through problem solving, and the paucity of recent research focussing on beginning teachers learning about teaching mathematics in this way, this paper draws on data from a 2 year longitudinal study. The study was guided by the research question:

What supports beginning teachers’ implementation of a problem solving pedagogy for the teaching and learning of mathematics?

Research Methodology and Methods

Data were gathered from three beginning primary teachers who had completed a 1 year graduate diploma programme in primary teacher education the previous year. The beginning teachers had undertaken a course in mathematics education (taught by the author for half of the course) as part of the graduate diploma. An invitation to be involved in the research was sent to the graduate diploma cohort at the end of the programme. Three beginning teachers indicated their interest and remained involved for the 2 year research period. The teachers had all secured their first teaching positions, and were teaching at different year levels at three different schools. Julia (pseudonyms have been used for all names) was teaching year 0–2 (5–6 years) at a small rural school; Charlotte, year 5–6 (9–10 years) at a large urban city school; and Reine, year 7–8 (11–12 years), at another small rural school. All three beginning teachers taught at their respective schools, teaching the same year levels in both years of the study. Ethical approval was sought and given by the author’s university ethics committee. Informed consent was gained from the teachers, school principals and involved parents and children.

Participatory action research was selected as the approach in the study because of its emphasis on the participation and collaboration of all those involved (Townsend, 2013 ). Congruent with the principles of action research, activities and procedures were negotiated throughout both years in a responsive and emergent way. The author acted as a co-participant with the teachers, aiming to improve practice, to challenge and reorient thinking, and transform contexts for children’s learning (Locke et al., 2013 ). The author’s role included facilitating the research-based problem solving workshops (see below), contributing her experience as a mathematics educator and researcher. The beginning teachers were involved in making sense of their own practice related to their particular sites and context.

The first step in the research process was a focus group discussion before the beginning teachers commenced their first year of teaching. This discussion included reflecting on their learning about problem solving during the mathematics education course; and envisaging what would be helpful to support implementation. It was agreed that a series of workshops would be useful. Two were subsequently held in the first year of the study, each for three hours, at the end of terms one and two. Four workshops were held during the second year, one during each term. At the end of the first year the author suggested a small number of experienced teachers who teach mathematics through problem solving join the workshops for the second year. The presence of these teachers was envisaged to support the beginning teachers’ learning. The beginning teachers agreed, and an invitation was extended to four teachers from other schools whom the author knew (e.g., through professional subject associations). The focus of the research remained the same, namely exploring what supported beginning teachers to implement a problem solving pedagogy.

Each workshop began with sharing and oral reflections about recent problem solving experiences, including successes and challenges. Key workshop tasks included developing a shared understanding of what constitutes problem solving, participating in solving mathematical problems (modelled using a lesson structure (Sullivan et al., 2016 ), and learning techniques such as asking questions. A time for reflective writing was provided at the end of each workshop to record what had been learned and an opportunity to set goals.

During the first focus group discussion it was also decided the author would visit and observe the beginning teachers teaching a problem solving lesson (or two) in term three or four of each year. A semi-structured interview between the author and each beginning teacher took place following each observed lesson. The beginning teachers also had an opportunity to ask questions as they reflected on the lesson, and feedback was given as requested. A second focus group discussion was held at the end of the first year (an approximate midpoint in the research), and a final focus group discussion was held at the end of the second year.

All focus group discussions, problem solving workshops, observations and interviews were audio-recorded and transcribed. Field notes of workshops (recorded by the author), reflections from the beginning teachers (written at the end of each workshop), and lesson observation notes (recorded by the author) were also gathered. The final data collected included occasional emails between each beginning teacher and the author.

Data Analysis

The analysis reported in this paper drew on all data sets, primarily using inductive thematic analysis (Braun & Clarke, 2006 ). The research question guided the key question for analysis, namely: What supports beginning teachers’ implementation of a problem solving pedagogy for the teaching and learning of mathematics? Alongside this question, consideration was also given to the challenges beginning teachers encountered as they implemented a problem solving pedagogy. Data familiarisation was developed through reading and re-reading the whole body of data. This process informed data analysis and the content for each subsequent workshop and focus group discussions. Colour-coding and naming of themes included comparing and contrasting data from each beginning teacher and throughout the 2-year period. As a theme was constructed (Braun & Clarke, 2006 ) subsequent data was checked to ascertain whether the theme remained valid and/or whether it changed during the 2 years. Three key themes emerged revealing what supported the beginning teachers’ developing problem solving pedagogy, and these constitute the focus for this paper.

Mindful of the time pressures beginning teachers experience in their early years, the author undertook responsibility for data analysis. The author’s understanding of the unfolding ‘story’ of each beginning teacher’s experiences and the emerging themes were shared with the beginning teachers, usually at the beginning of a workshop, focus group discussion or observation. Through this process the author’s understandings were checked and clarified. This iterative process of member checking (Lincoln & Guba, 1985 ) began at a mid-point during the first year, once a significant body of data had been gathered. At a later point in the analysis and writing, the beginning teachers also had an opportunity to read, check and/or amend quotes chosen to exemplify their thinking and experiences.

Findings and Discussion

In this section the three beginning teachers’ experiences at the start of the 2 year research timeframe is briefly described, followed by the first theme centred on the use of a lesson structure including prompts for differentiation. The second and third themes are presented together, starting with a brief outline of each beginning teacher’s ‘story’ providing the context within which the themes emerged. Sharing the ‘story’ of each beginning teacher and including their ‘voice’ through quotes acknowledges them and their experiences as central to this research.

The beginning teachers’ pre-service teacher education set the scene for learning about teaching mathematics through problem solving. A detailed list brainstormed during the first focus group discussion suggested a developing understanding from their shared pre-service mathematics education course. In their first few weeks of teaching, all three beginning teachers implemented a few problems. It transpired however this inclusion of problem solving occurred only while children were being assessed and grouped. Following this, all three followed a traditional format of skill-based (with a focus on number) mathematics, taught using ability groups. The beginning teachers’ trajectories then varied with Julia and Reine both eventually adopting a pedagogy primarily based on problem solving, while Charlotte employed a traditional skill-based mathematics using a combination of whole class and small group teaching.

A Lesson Structure that Caters for Diversity Supports Early Efforts

Data show that developing familiarity with a lesson structure including prompts for differentiation supported the beginning teachers’ early efforts with a problem solving pedagogy. This addressed a key issue that emerged during the first workshop. During the workshop while a ‘list’ of ideas for teaching a problem solving lesson was co-constructed, considerable concern was expressed about catering for a range of learners when introducing and working with a problem. For example, Charlotte queried, “ Well, what happens when you are trying to do something more complicated, and we’re (referring to children) sitting here going, ‘I’ve no idea what you're talking about” ? Reine suggested keeping some children with the teacher, thinking he would say, “ If you’re unsure of any part stay behind” . He was unsure however about how he would then maintain the integrity of the problem.

It was in light of this discussion that a lesson structure with differentiated prompts (Sullivan et al., 2016 ) was introduced, experienced and reflected on during the second workshop. While the co-constructed list developed during the first workshop had included many components of Sullivan’s lesson structure, (e.g., a consideration of ‘extensions’) there had been no mention of ‘enabling prompts’. Now, with the inclusion of both enabling and extending prompts, the beginning teachers’ discussion revealed them starting to more fully envisage the possibilities of using a problem solving approach, and being able to cater for all children. Reine commented that, “… you can give the entire class a problem, you've just got to have a plan, [and] your enabling and extension prompts” . Charlotte was also now considering and valuing the possibility of having a whole class work on the same problem. She said, “I think … it’s important and it’s useful for your whole class to be working on the same thing. And … have enablers and extenders to make sure that everyone feels successful” . Julia also referred to the planning prompts. She thought it would be key to “plan it well so that we’ve got enabling and extending prompts” .

Successful Problem Solving Lessons

Following the second workshop all three beginning teachers were observed teaching a lesson using the structure. These lessons delighted the beginning teachers, with them noting prolonged engagement of children, the children’s learning and being able to cater for all learners. Reine commented on how excited and engaged the children were, saying they were, “ just so enthusiastic about it ”. In Charlotte’s words, “ it really worked ”, and Julia enthusiastically pondered this could be “ the only way you teach maths !”.

During the focus group discussion at the end of the first year, all three reflected on the value of the lesson structure. Reine called it a ‘framework’ commenting,

I like the framework. So from start to finish, how you go through that whole lesson. So how you set it up and then you go through the phases… I like the prompts that we went through…. knowing where you could go, if they’re like, ‘What do I do?’ And then if they get it too easy then ‘Where can you go?’ So you've got all these little avenues.

Charlotte also valued the lesson structure for the breadth of learning that could occur, explaining,

… it really helped, and really worked. So I found that useful for me and my class ‘cause they really understood. And I think also making sure that you know all the ins and outs of a problem. So where could they go? What do you need to know? What do they need to know?

While the beginning teachers’ pre-service teacher education and the subsequent research process, including the use of the lesson structure, supported the beginning teachers’ early efforts teaching mathematics through problem solving, two key factors further enabled two of the beginning teachers (Julia and Reine) to sustain a problem solving pedagogy. These were:

Being involved in complementary mathematics professional development alongside members of their respective school staff (a form of group mentoring); and

Having a colleague in the same school teaching mathematics through problem solving (a form of individual mentoring).

Charlotte did not have these opportunities and she indicated this limited her implementation. Data for these findings for each teacher are presented below.

Complementary Professional Development and Problem Solving Colleague in Same School

Julia began to significantly implement problem solving from the second term in the first year. This coincided with her attending a 2-day workshop (with staff from her school) that focused on the use of problem solving to support children who are not achieving at expected levels (see ALiM: Accelerated Learning in Maths—Ministry of Education, 2022 ). She explained, “ … I did the PD with (colleague’s name), which was really helpful. And we did lots of talking about rich learning tasks and problem solving tasks…. And what it means ”. Following this, Julia reported using rich tasks and problem solving in her mathematics teaching in a regular (at least weekly) and ongoing way.

During the observation in term three of the first year Julia again referred to the impact of having a colleague also teaching mathematics through problem solving. When asked what she believed had supported her to become a teacher who teaches mathematics in this way she firstly identified her involvement in the research project, and then spoke about her colleague. She said, “ I’m really lucky one of our other teachers is doing the ALiM project… So we’re kind of bouncing off each other a little bit with resources and activities, and things like that. So that’s been really good ”.

At the beginning of the second year, Julia reiterated this point again. On this occasion she said having a colleague teaching mathematics through problem solving, “ made a huge difference for me last year ”, explaining the value included having someone to talk with on a daily basis. Mid-way through the second year Julia repeated her opinion about the value of frequent contact with a practising problem solving colleague. Whereas her initial comments spoke of the impact in terms of being “ a little bit ”, later references recount these as ‘ huge ’ and ‘ enabling ’. She described:

a huge effect… it enabled me. Cause I mean these workshops are really helpful. But when it’s only once a term, having [colleague] there just enabled me to kind of bounce ideas off. And if I did a lesson that didn’t work very well, we could talk about why that was, and actually talk about what the learning was instead…. . It was being able to reflect together, but also share ideas. It was amazing.

Julia’s comments raise two points. It is likely that participating in the ALiM professional development (which could be conceived as a form of group mentoring) consolidated the learning she first encountered during pre-service teacher education and later extended through her involvement in the research. Having a colleague (in essence, an individual mentor) within the same school teaching mathematics through problem solving appears to be another factor that supported Julia to implement problem solving in a more sustained way. Julia’s comments allude to a number of reasons for this, including: (i) the more frequent discussion opportunities with a colleague who understands what it means for children to learn mathematics through problem solving; (ii) being able to share and plan suitable activities and resources; and (iii) as a means for reflection, particularly when challenges were encountered.

Reine’s mathematics programme throughout the first year was based on ability groups and could be described as traditional. He occasionally used some mathematical problems as ‘extension activities’ for ‘higher level’ children, or as ‘fillers’. In the second year, Reine moved to working with mixed ability groups (where students work together in small groups with varying levels of perceived capability) and initially implemented problem solving approximately once a fortnight. In thinking back to these lessons he commented, “ We weren’t really unpacking one problem properly, it was just lots of busy stuff ”. A significant shift occurred in Reine’s practice to teaching mathematics primarily by problem solving towards the last half of the second year. He explained, “ I really ramped up towards terms three and four, where it’s more picking one problem across the whole maths class but being really, really conscious of that problem. Low entry, high ceiling, and doing more of it too ”.

Reine attributed this change to a number of factors. In response to a question about what he considered led to the change he explained,

… having this, talking about this stuff, trialling it and then with our PD at school with the research into ability grouping... We’ve got a lot of PD saying why it can be harmful to group on ability, and that’s been that last little kick I needed, I think. And with other teachers trialling this as well. Our senior teacher has flipped her whole maths program and just does problem solving.

Like Julia, Reine firstly referred to his involvement in the research project including having opportunities to try problems in his class and discuss his experiences within the research group. He then told of a colleague teaching at his school leading school-wide professional development focussed on the pitfalls of ability grouping in mathematics (e.g., see Clarke, 2021 ) and instead using problem solving tasks. He also referred to having another teacher also teaching mathematics through problem solving. It is interesting to consider that having positive experiences in pre-service teacher education, the positive and encouraging support of colleagues (Reine’s principal and co-teacher in both years), regular participation in ongoing professional development (the problem solving workshops), and having a highly successful one-off problem solving teaching experience (the first year observation) were not enough for Reine to meaningfully sustain problem solving in his first year of teaching.

As for Julia, pivotal factors leading to a sustaining of problem solving teaching practice in the second year included complementary mathematics professional development (a form of group mentoring) and at least one other teacher (acting as an individual mentor) in the same school teaching mathematics through problem solving. It could be argued that pre-service teacher education and the problem solving workshops ‘paved the way’ for Julia and Reine to make a change. However, for both, the complementary professional development and presence of a colleague also teaching through problem solving were pivotal. It is also interesting to note that three of the four experienced teachers in the larger research group taught at the same level as Reine (see Table 1 below) yet he did not relate this to the significant change in his practice observed towards the end of the second year.

Charlotte’s mathematics programme during the first year was also traditional, teaching skill-based mathematics using ability groups. At the beginning of the second year Charlotte moved to teaching her class as a whole group, using flexible grouping as needed (children are grouped together in response to learning needs with regard to a specific idea at a point in time, rather than perceived notions of ability). She reported that she occasionally taught a lesson using problem solving in the first year, and approximately once or twice a term in the second year. Charlotte did not have opportunities for professional development in mathematics nor did she have a colleague in the same school teaching mathematics through problem solving. Pondering this, Charlotte said,

It would have been helpful if I had someone else in my school doing the same thing. I just thought about when you were saying the other lady was doing it [referring to Julia’s colleague]. You know, someone that you can just kind of back-and-forth like. I find with Science, I usually plan with this other lady, and we share ideas and plan together. We come up with some really cool stuff whereas I don’t really have the same thing for this.

Based on her experiences with teaching science it is clear Charlotte recognised the value of working alongside a colleague. In this, her view aligns with what Julia and Reine experienced.

Table 1 provides a summary of the variables for each beginning teacher, and whether a sustained implementation of teaching mathematics through problem solving occurred.

The table shows two variables common to Julia and Reine, the beginning teachers who began and sustained problem solving. They both participated in complementary professional development with colleagues from their school, and the presence of a colleague, also at their school, teaching mathematics through problem solving. Given that Julia was able to implement problem solving in the absence of a ‘research workshop colleague’ teaching at the same year level, and Reine’s lack of comment about the potential impact of this, suggests that this was not a key factor enabling a sustained implementation of problem solving.

Attributing the changes in Julia and Reine’s teaching practice primarily to their involvement in complementary professional development attended by members of their school staff, and the presence of at least one other teacher teaching mathematics through problem solving in their school, is further supported by a consideration of the timing of the changes. The data shows that while Julia could be considered an ‘early adopter’, Reine changed his practice reasonably late in the 2 year period. Julia’s early adoption of teaching mathematics through problem solving coincided with her involvement, early in the 2 years, in the professional development and opportunity to work alongside a problem solving practising colleague. Reine encountered these similar conditions towards the end of the 2 years and it is notable that this was the point at which he changed his practice. That problem solving did not become embedded or frequent within Charlotte’s mathematics programme tends to support the argument.

Understanding what supports primary teachers to teach mathematics through problem solving at the beginning of their careers is important because all students, including those taught by beginning teachers, need opportunities to develop high-level thinking, reasoning, and problem solving skills. It is also important in light of recent calls for mathematics curricula to include more emphasis on mathematical practices (such as problem solving) (e.g., see Grootenboer et al., 2021 ); and the Royal Society Te Apārangi report ( 2021 ). Findings from this research suggest that learning about problem solving during pre-service teacher education is enough for beginning teachers to trial teaching mathematics in this way. Early efforts were supported by gaining experience with a lesson structure that specifically attends to diversity. The lesson structure prompted the beginning teachers to anticipate different children’s responses, and consider how they would respond to these. An increased confidence and sense of security to trial teaching mathematics through problem solving was enabled, based on their more in-depth preparation. Beginning teachers finding the lesson structure useful extends the findings of Sullivan et al. ( 2016 ) in Australia and Ingram et al. ( 2019 ) in Aotearoa to include less experienced teachers.

In order for teaching mathematics through problem solving to be sustained however, a combination of three factors, subsequent to pre-service teacher education, was needed: (i) active participation in problem solving workshops (in this context provided by the research-based problem solving workshops); (ii) attending complementary professional development initiatives alongside colleagues from their school (a form of group mentoring); and (iii) the presence of an in-school colleague who also teaches mathematics through problem solving (a form of individual mentoring). It seems possible these three factors acted synergistically resulting in Julia and Reine being able to sustain implementation. If only one factor is present, in this study attending the problem solving workshops, and despite a genuine interest in using a problem based pedagogy, the result was limited movement towards this way of teaching.

Akiri and Dori ( 2021 ) have reported that individual mentoring contributes the most to beginning teachers’ professional growth. In a manner consistent with these findings, an in-school colleague (who in essence was acting as an individual mentor) played a critical role in supporting Reine and Julia. However, while Akiri and Dori, amongst others (e.g., Cameron et al., 2007 ; Karlberg & Bezzina, 2020 ), have identified the value of supportive, approachable colleagues, for both Julia and Reine it was important that their colleague was supportive and approachable, and actively engaged in teaching mathematics through problem solving. Having supportive and approachable colleagues, as Reine experienced in his first year, on their own were not enough to support a sustained problem solving pedagogy.

Implications for Productive Professional Learning and Development

This study sought to explore the conditions that supported problem solving for beginning teachers, each in their unique context and from their perspective. The research did not examine how the teaching of mathematics through problem solving affected children’s learning. However, multiple sets of data were collected and analysed over a 2-year period. While it is neither possible nor appropriate to make claims as to generalisability some suggestions for productive beginning teacher professional learning and development are offered.

Given the first years of teaching constitute a particular and critical phase of teacher learning (Karlberg & Bezzina, 2020 ) and the findings from this research, it is imperative that well-funded, subject-focussed support occurs throughout a beginning teacher’s first 2 years of teaching. This is consistent with the ninth recommendation in the Royal Society Te Apārangi report ( 2021 ) suggesting compulsory professional learning during the induction period (2 years in Aotearoa New Zealand). Participation in subject-specific professional development has been recognised to favourably influence new teachers’ efforts to adopt reform practices such as problem solving (Cady et al., 2006 ).

Findings from this study suggest professional development opportunities that complement each other support beginning teacher learning. In the first instance complementarity needs to be with what beginning teachers have learned during their pre-service teacher education. In this study, the research-based problem solving workshops served this role. Complementarity between varying forms of professional development also appears to be important. Furthermore, as indicated by Julia and Reine’s experiences, subsequent professional development need not be on exactly the same topic. Rather, it can be complementary in the sense that there is an underlying congruence in philosophy and/or focus on a particular issue. For example, it emerged in the problem solving workshops, that being able to cater for diversity was a central concern for the beginning teachers. Attending to this issue within the problem solving workshops via the introduction of a lesson structure that enabled differentiation, was congruent with the nature of the professional development in the two schools: ALiM in Julia’s school, and mixed ability grouping and teaching mathematics through problem solving in Reine’s school. All three of these settings were focussed on positively responding to diversity in learning needs.

The presence of a colleague within the same school teaching mathematics through problem solving also appears to be pivotal. This is consistent with Darragh and Radovic ( 2019 ) who have shown the significant impact a teacher’s school context has on their potential to sustain problem based pedagogies in mathematics. Given that problem solving is not prevalent in many primary classrooms, it would seem clear that colleagues who have yet to learn about teaching mathematics through problem solving, particularly those that have a role supporting beginning teachers, will also require access to professional development opportunities. It seems possible that beginning and experienced teachers learning together is a potential pathway forward. Finding such pathways will be critical if mathematical problem solving is to be consistently implemented in primary classrooms.

Finally, these implications together with calls for institutions to collaborate to provide adequate and ongoing support for new teachers (Karlberg & Bezzina, 2020 ) suggest there is a need for pre-service teacher educators, professional development providers and the Teaching Council of Aotearoa New Zealand to work together to support beginning teachers’ starting and sustaining teaching mathematics through problem solving pedagogies.

Akiri, E., & Dori, Y. (2021). Professional growth of novice and experienced STEM teachers. Journal of Science Education and Technology, 31 (1), 129–142.

Article   Google Scholar  

Anthony, G., & Hunter, R. (2017). Grouping practices in New Zealand mathematics classrooms: Where are we at and where should we be? New Zealand Journal of Educational Studies, 52 (1), 73–92.

Australian Curriculum, Assessment and Reporting Authority. (2018). F-10 curriculum: Mathematics . Retrieved from https://www.australiancurriculum.edu.au/f-10-curriculum/mathematics/ . Accessed 20 April 2022.

Bailey, J. (2015). Experiencing a mathematical problem solving teaching approach: Opportunity to identify ambitious teaching practices. Mathematics Teacher Education and Development, 17 (2), 111–124.

Google Scholar  

Braun, V., & Clarke, V. (2006). Using thematic analysis in psychology. Qualitative Research in Psychology, 3 (2), 77–101.

Cady, J., Meier, S., & Lubinski, C. (2006). the mathematical tale of two teachers: A longitudinal study relating mathematics instructional practices to level of intellectual development. Mathematics Education Research Journal, 18 (1), 3–26.

Cameron, M., Lovett, S., & Garvey Berger, J. (2007). Starting out in teaching: Surviving or thriving as a new teacher. SET Research Information for Teachers, 3 , 32–37.

Clarke, D. (2021). Calling a spade a spade: The impact of within-class ability grouping on opportunity to learn mathematics in the primary school. Australian Primary Mathematics Classroom, 26 (1), 3–8.

Clarke, D., Goos, M., & Morony, W. (2007). Problem solving and working mathematically. ZDM Mathematics Education, 39 (5–6), 475–490.

Darragh, L., & Radovic, D. (2019). Chaos, control, and need: Success and sustainability of professional development in problem solving. In P. Felmer, P. Liljedahl, & B. Koichu (Eds.), Problem solving in mathematics instruction and teacher professional development (pp. 339–358). Springer. https://doi.org/10.1007/978-3-030-29215-7_18

Chapter   Google Scholar  

English, L., & Gainsburg, J. (2016). Problem solving in a 21st-century mathematics curriculum. In L. English & D. Kirshner (Eds.), Handbook of international research in mathematics education (pp. 313–335). Routledge.

Fantilli, R., & McDougall, D. (2009). A study of novice teachers: Challenges and supports in the first years. Teaching and Teacher Education, 25 (6), 814–825.

Feiman-Nemser, S. (2003). What new teachers need to learn. Educational Leadership, 60 (8), 25–29.

Felmer, P., Liljedahl, P., & Koichu, B. (Eds.). (2019). Problem solving in mathematics instruction and teacher professional development . Springer. https://doi.org/10.1007/978-3-030-29215-7_18

Book   Google Scholar  

Gaikhorst, L., Beishuizen, J., Korstjens, I., & Volman, M. (2014). Induction of beginning teachers in urban environments: An exploration of the support structure and culture for beginning teachers at primary schools needed to improve retention of primary school teachers. Teaching and Teacher Education, 42 , 23–33.

Gaikhorst, L., Beishuizen, J., Roosenboom, B., & Volman, M. (2017). The challenges of beginning teachers in urban primary schools. European Journal of Teacher Education , 40 (1), 46–61.

Grootenboer, P., Edwards-Groves, C., & Kemmis, S. (2021). A curriculum of mathematical practices. Pedagogy, Culture & Society. https://doi.org/10.1080/14681366.2021.1937678

Holton, D. (2009). Problem solving in the secondary school. In R. Averill & R. Harvey (Eds.), Teaching secondary school mathematics and statistics: Evidence-based practice (Vol. 1, pp. 37–53). NZCER Press.

Hunter, R., Hunter, J., Anthony, G., & McChesney, K. (2018). Developing mathematical inquiry communities: Enacting culturally responsive, culturally sustaining, ambitious mathematics teaching. SET Research Information for Teachers, 2 , 25–32.

Ingram, N., Holmes, M., Linsell, C., Livy, S., McCormick, M., & Sullivan, P. (2019). Exploring an innovative approach to teaching mathematics through the use of challenging tasks: A New Zealand perspective. Mathematics Education Research Journal . https://doi.org/10.1007/s13394-019-00266-1

Inoue, N., Asada, T., Maeda, N., & Nakamura, S. (2019). Deconstructing teacher expertise for inquiry-based teaching: Looking into consensus building pedagogy in Japanese classrooms. Teaching and Teacher Education, 77 , 366–377.

Karlberg, M., & Bezzina, C. (2020). The professional development needs of beginning and experienced teachers in four municipalities in Sweden. Professional Development in Education . https://doi.org/10.1080/19415257.2020.1712451

Keese, J., Waxman, H., Lobat, A., & Graham, M. (2022). Retention intention: Modeling the relationships between structures of preparation and support and novice teacher decisions to stay. Teaching and Teacher Education . https://doi.org/10.1016/j.tate.2021.103594

Locke, T., Alcorn, N., & O’Neill, J. (2013). Ethical issues in collaborative action research. Educational Action Research, 21 (1), 107–123.

Lincoln, Y., & Guba, E. (1985). Naturalistic Inquiry . Sage Publications.

Mamona-Downs, J., & Mamona, M. (2013). Problem solving and its elements in forming proof. The Mathematics Enthusiast, 10 (1–2), 137–162.

Ministry of Education. (2022). ALiM: Accelerated Learning in Maths. Retrieved from https://www.education.govt.nz/school/funding-and-financials/resourcing/school-funding-for-programmes-forstudents-pfs/#sh-ALiM . Accessed 20 April 2022.

Ministry of Education. (2007). The New Zealand Curriculum . Learning Media.

Royal Society Te Apārangi. (2021). Pāngarau Mathematics and Tauanga Statistics in Aotearoa New Zealand: Advice on refreshing the English-medium Mathematics and Statistics learning area of the New Zealand Curriculum : Expert Advisory Panel. Publisher

Schoenfeld, A. (1985). Mathematical problem solving . Academic Press.

Schoenfeld, A. (2013). Reflections on problem solving theory and practice. The Mathematics Enthusiast, 10 (1/2), 9–34.

Sullivan, P., Borcek, C., Walker, N., & Rennie, M. (2016). Exploring a structure for mathematics lessons that initiate learning by activating cognition on challenging tasks. The Journal of Mathematical Behaviour, 41 , 159–170.

Townsend, A. (2013). Action research: The challenges of understanding and changing practice . Open University Press.

Wood, M., Jilk, L., & Paine, L. (2012). Moving beyond sinking or swimming: Reconceptualizing the needs of beginning mathematics teachers. Teachers College Record, 114 , 1–44.

Youngs, P., Molloy Elreda, L., Anagnostopoulos, D., Cohen, J., Drake, C., & Konstantopoulos, S. (2022). The development of ambitious instruction: How beginning elementary teachers’ preparation experiences are associated with their mathematics and English language arts instructional practices. Teaching and Teacher Education . https://doi.org/10.1016/j.tate.2021.103576

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Bailey, J. Learning to Teach Mathematics Through Problem Solving. NZ J Educ Stud 57 , 407–423 (2022). https://doi.org/10.1007/s40841-022-00249-0

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