Problem-Solving - Tips for Teachers by Craig Barton

Supporting students in developing the tools and resilience necessary to solve unfamiliar problems is, quite rightly, regarded as the holy grail of maths teaching.

At a pragmatic level, if students can cope with unfamiliar problems, then they are more likely to thrive in high-stakes exams where twists and turns are prevalent. But at a more fundamental level, problem-solving is what maths is – or at least should be – all about. We learn procedures so we can do interesting things with those procedures.

Contents

Diagnosis

What does the phrase problem-solving mean to you?
How regularly does problem-solving appear in your lessons?
What challenges do you face?

Evidence

What do I mean by problem-solving?

There are many definitions of problem-solving. However, for me, the most useful conception comes from Colin Foster. Colin asks us to consider two tunnels:

With the tunnel on the left, we can see where we need to go. The journey may be long and tricky, but the end is always in sight. This tunnel represents mathematical procedures. They may be short, such as simplifying fractions, or long, such as solving a pair of simultaneous equations, but once we have learned the required steps, we can carry them out and reach our destination.

With the tunnel on the right, we are entering the unknown. We don’t immediately know where we are going, and as we make our journey, we may take some wrong steps along the way. Eventually, things become clearer, and we see the light at the end of the tunnel. This is problem-solving. The path to the solution may be short or long, but what we need to do is not immediately obvious.

What I see

Many of the maths lessons I see each week do not contain the problems depicted by the tunnel on the right. Instead, they consist solely of questions depicted by the tunnel on the left.

Here are some examples from lesson drop-ins in a school I am supporting:

Students worked on these sets of questions during the final portion of the lesson. The teachers then projected the answers, students self-assessed, the teacher addressed any issues, and that was the lesson – and the coverage of this particular process – complete.

So what?

Nothing is wrong with these sets of questions. They are what I would class as Consolidation. In the previous section, I argued that Consolidation was a necessary building block of a lesson. It’s purpose is to help students gain confidence and competence in carrying out a procedure.

But – and it is a big BUT – if Consolidation is the sole constituent of a student’s mathematical diet, then we have a problem. As discussed above, students are unlikely to be able to cope with any significant twist or disguise that often occurs in GCSE questions. At a deeper level, if all students experience is consolidation practice, they start to view maths as a series of disconnected algorithms they must remember and carry out with no real purpose, and promptly switch off from the subject.

Why does this happen?

Of course, everyone knows that students need more than just Consolidation. And yet, this type of practice often dominates maths lessons. Having spoken to many Heads of Department and maths teachers over the last few years, I think there are several reasons why:

Consolidation is easier to plan. You can grab a bunch of questions from a number of sources and use them in lessons without needing to do much preparation beforehand. Good problem-solving questions, on the other hand, are harder to source and even harder to create. Once you have found them, they require the teacher to take the time to work through the maths and then think hard about the support and challenge they may need to give their students in the lesson.
Consolidation is easier to deliver. Students work their way through a set of questions, the teacher clicks onto the next slide where the answers are revealed, students mark their work, and on we go to the next topic. Problem-solving questions are a different beast. It is very difficult to predict how students will fare. A wrong answer, a conjecture, or insight could take the lesson in a completely different direction, far more so than it could with a straightforward set of Consolidation questions.
Consolidation can help disguise a lack of subject knowledge. Armed with a set of questions and—crucially—a set of answers, most teachers can get through the lesson. Indeed, this is the very reason that cover lessons often consist exclusively of Consolidation questions. With problem-solving questions, the teacher needs to know the concept inside out to deal with the students’ unpredictable questions, methods, and solutions. With the high prevalence of non-subject specialists teaching secondary-level maths these days, this is a serious barrier.
Students come to expect Consolidation and resist anything else. Students’ views can quickly become: in maths, the teacher shows us how to do something, we practice it, and then we go to Chemistry. I have worked with many maths teachers who tried a problem-solving activity only to find their students took one look at it and downed tools. Because it looked different to what they were used to, and seemed both hard and weird, students immediately gave up. Such negative experiences often cause teachers to fall back on what they know and avoid trying it out again.
As a Head of Department, Consolidation is safer. If you have a department full of experienced maths specialists, then you may feel confident that even though your colleagues are pursuing a wide variety of approaches, using a wide variety of resources, the students will be getting an equally positive experience. But these days, in how many departments is that the case? I have yet to visit a department this year that does not consist of some colleagues who are in their first two years of teaching, or who are not maths specialists, or who are borrowed from another subject to teach the odd lesson a fortnight. So, in a bid for some consistency in such circumstances, many heads of department plan shared PowerPoints consisting mainly of examples to model and Consolidation practice to present.
Schemes of work are too full. I hear this often: there is too much content to get through, leaving no room for anything else but Consolidation.
The belief is that students need to have achieved a certain level of competence with a procedure before they can problem-solve. This often manifests itself in students having to almost “qualify” to be given problem-solving questions—proving they are worthy of them by answering a set of Consolidation questions quicker than their classmates. Or, typically, only the top sets get exposure to problem-solving questions because they move through the curriculum fast enough. Hence, many students are rarely challenged with a non-routine problem to solve.

Solution steps

Where does problem-solving fit into a lesson?

In a previous section, I shared three possible lesson structures:

Lesson Structure #1

Lesson Structure #2

Lesson Structure #3

In each structure, problem-solving comes after a period of Consolidation. It is true that the more secure students are with a procedure, the more likely they are to enjoy, thrive in, and learn from it.

How much consolidation practice do students need?

This is the million-dollar question. The answer is influenced by factors such as the topic, the student, and their previous experience. But, in general, students may not need as much consolidation practice as we , might suspect. This is for two reasons:

As we have seen (LINK), the We Do provides an opportunity to provide consolidation practice at a whole class level.
As we will see, several of the problem-solving activity structures have consolidation built into them.

However, I always give students at least five minutes of silent, independent consolidation practice so they can get to grips a procedure at their own pace.

Solutions that don’t really work

Maths departments that identify a lack of problem-solving in their lessons choose to tackle it in several ways.

First, I see problem-solving equated with GCSE questions. So, at the end of a lesson on sharing in a ratio, Year 8 students are shown this GCSE question

Students can do it, and they feel great because they answer a question aimed at students 3 years older than them. However, the whole lesson has been about sharing in a ratio, so students don’t have to think hard about the approach needed to solve the problem. This is not problem-solving; this is just further Consolidation.

The second thing I see maths departments do is include challenging, problem-solving-type questions or tasks on slide 63 of an epic PowerPoint, or tucked away in Column M on the Excel version of the scheme or work. The issue here is that few classes and few students ever get to this point, and therefore whole cohorts of students may never experience anything above and beyond Consolidation.

Finally, some maths departments have “problem-solving lessons” at the end of topics. Students are faced with exam-style questions that place procedures in contexts, disguise the deep structure, or challenge students to weave together different areas of maths. This is all good stuff, but what happens if a student misses that particular lesson, or the teacher finds they have run out of time in that particular unit and so curtails or completely ditches the problem-solving lesson? Also, the distinction between problem-solving lessons and non-problem-solving lessons doesn’t quite sit right with me.

Preparing to use a problem-solving activity

Below, I will share several high-value activity structures, giving you access to thousands of top-quality problem-solving activities. But no matter how good the activity looks, or how strong amatyeamtician you are, the best way you can prepare to use the activity in your lesson is to do the problem yourself.

I say this all the time, but it rarely happens. The barrier is often a lack of time. But if we can save the time spent searching for problems by building up a selective set of reliable sources, then we can portion some of that time towards trying the problems ourselves.

Here is the structure I recommend:

1. First, spend 10 minutes doing the problem-solving activity yourself as a mathematician.

Engage with the maths. This part of the process helps you get a sense of what the activity is really about. It allows you to uncover any twists and turns in the problem that could surprise, excite, or put-off your students.

2. Then, spend 10 minutes with your teacher hat on

Now you have a much better understanding of the problem, it is time to think how to enable your students to get the most out of it.

Will you need to tweak the problem in some way?
How will you initially present it to your students?
What support will you give if a student is struggling?
What questions will you ask if a student finishes quickly?

The more you can think about the answers to these questions outside the noise of the lesson, the better prepared you will be to respond in the moment.

You can do this preparation on your own. But it works so much better as a department. If you have a topic you are all teaching next week, allocate 30 minutes in a departmental meeting to work through the problem in the way described above – 10 minutes doing the maths, 10 minutes thinking about the maths, then a final 10 minutes sharing questions and ideas. This always results in a group of teachers who are better prepared and excited to use that problem in their classes.

High-value activity structures: Introduction

Okay, let’s turn our attention to the problems themselves.

In the past, I had a favourite problem-solving activity for each topic I taught. A fill-in-the-blanks for Pytharoas, a mystery for sequences, a card sort for cumulative frequency diagrams, and so on. The issue was that I had to spend precious minutes explaining the structure of each activity. Moreover, my students had to exert precious attention.

Students have limited attention. When presented with an activity, they must divide their attention between the structure and the content. The more unfamiliar the activity, the more attention is diverted towards the structure. As we know from Dan Willingham’s work, students remember what they pay attention to.

Hence, I have developed a set of high-value activity structures. These structures can be used across different topics and classes. Students get used to the structure, so they can dedicate more attention to the content and thus learn more. I also improve at using the structure to support my students’ learning.

The 4-2 approach

May favourite way for students to work on high-value problem-solving activities is via the 4-2 approach. This involves:

4 minutes of silent, independent work
2 minutes of paired discussion
And repeat

I find 4 minutes is enough time for students to make some progress on their own, so they have work to compare and questions to ask, but not so long that they get stuck and frustrated. Likewise, 2 minutes is enough for a focused, productive discussion but not long enough for those discussions to drift off-task.

A visible timer is an excellent way to increase their focus.

If students are not used to working silently, those first 4 minutes will feel more like 4 hours, so feel free to adjust accordingly.

High-value activity structures #1: Completion tables

Example of the activity for you to try

Can you fill in each of the boxes?

Why this is a good activity structure

To students, it doesn’t look like lots of maths… but it is!
There are lots of opportunities for Consolidation of key procedures
There are of opportunities for reasoning and problem-solving as students grapple with how to work forward and backward, given different information

Tips to make these activities effective

1. Do a We Do like the top row

I like to make up a different example than the one in the top row, but of the same difficulty. So, 2/3 + 4/5 would be good for the above activity. As students will have experienced consolidation practice of all the procedures by the time I give them this activity, I don’t need to do an I Do. Instead, I will point to a bo and ask students to work out their answer on their mini-whiteboards, hover, and show me in 3, 2, 1…

The data I get back helps me check students are secure with the structure of the completion table and gives me insight into their understanding. I can respond accordingly, letting students who are secure crack on and supporting those who are struggling.

2. Print out the Completion Table

I am all for saving trees, but projecting the Completion Table on the board and asking students to copy it out into their boos wastes precious time and effort that could be used for thinking and learning.

3. Give students a mini-whiteboard to try ideas on

Mini-whiteboards are also useful when students are working on the activity. They can try ideas out without fearing the permanence and messy crossings that working on paper brings. They can transfer their answer to their books when they are happy.

Of course, this means that students’ workings out are lost as soon as they clear their boards, but I am okay with that. I want to choose the most appropriate medium to support students, not compromise this in the quest for written evidence. However, some teachers prefer to ask students to do their working out in their exercise books.

4. Use the 4-2 approach

The 4-2 approach works well with Completion Tables. By the end of the 4 minutes, most students have enough entries to check with their partner, providing either confirmation or fertile ground for discussion.

5. One big Completion Table between two

To encourage further collaboration, you could print out one big A3 Completion Table between each pair. Students can be encouraged to do their working out independently on their mini-whiteboards, compare their answers for a particular box, and, if they agree, enter it into the shared table.

6. Regular whole class checks for understanding

Whilst you can circulate as students work on their Completion Table, every 5 minutes or so it is worth getting a more reliable snap-shot of whole-class understanding. The easiest way to do this is to do Book-to-Board: Quick check:

Get studetns’ attention.
Choose a box – perhaps one you have identified one or two issues with
Ask students to copy their final answer from their sheet to their mini-whiteboards, hover, and show you in 3, 2, 1
Respond accordingly

Where to get these activities

High-value activity structures #2: Venn Diagrams

Example of the activity for you to try

Can you think of 3-digit integers that could go in each of the four regions?

Now can you think of numbers with 3 decimal places that could go in each of the eight regions?

If you cannot find an example that fits a particular region, can you convince me it is impossible?

Why this is a good activity structure

Venn Diagrams require students to generate their own examples instead of answering teacher-generated questions. Learner-generated examples are a fantastic test of student understanding as they can reveal misconceptions in a way that teacher-generated questions may miss
There’s lots of thinking required
The barrier to entry is low as every example belongs somewhere. But as the regions get filled up, the challenge level increases
As we will see, it is easy to offer both support and challenge when required

Tips to make these activities effective

1. Lower the content demands initially

Always start with a basic example to ensure students are secure with the structure of a Venn Diagram. We want our students to be as fluent as possible in terms of knowing exactly what each region of a Venn Diagram means, so that when we change the content to something more challenging, students can divert their attention to thinking about the content and not the Venn Diagram itself. Here is an example where each student had to put their initials in the correct region:

2. Always do a double Venn Diagram before a triple Venn Diagram

I always warm students up with a double Venn Diagram. This helps identify misconceptions or misunderstandings with the Venn Diagram structure, or the subject content. It also helps to build student confidence.

Dan Rose takes this one step further. He asks his students to complete a double Venn Diagram first, then reveals the third circle and its label, and challenges students to see which examples from their double Venn will need to move and which can stay!

3. Start each Venn Diagram with an example the class does together

Consider this Venn Diagram where students need to give the equation of a straight line in the for y = mx + c

Pick a student and ask them to choose their favourite equation of a straight line (everyone has one). Let’s say they go for y = 2x + 3. Write this on the board and challenge students to decide on their own which region they think it belongs.

Get them to write their choice of letter on their mini-whiteboards, hover them face down, and show you in 3, 2, 1… Then ask students to place their boards in between them and discuss their answers and reasoning. I

f the vast majority of students have the right answer, you can confirm and quickly move on, confident that students know what they are doing and have tasted some success. If there is an issue, you might need to pause, reteach, and pick up the example when understanding is secure.

4. Ask students to do their working out on their mini-whiteboards

As with Completion Tables, I always ask my students to do their working out on mini-whiteboards and then record their answers on a blank copy of the Venn Diagram that I have printed out on paper. Mini-whiteboards are a great vehicle for students to do the kind of messy thinking and experimentation often needed to crack a Venn Diagram. They also support paired discussions and collaboration more so than with books or paper.

5. Use the 4-2 approach

You should be spotting as theme here! As with Completion Tables, I find the 4-2 approach strikes the right balance between independent work and focused collaboration.

6. If students are struggling, have examples for them to sort

For example, using the Venn Diagram below, I could ask a struggling student where they think 3/5 belongs.

Sorting is easier than generating one’s own examples and should help students get back into the activity. Indeed, I have seen teachers begin with 8 examples for students to sort into regions before asking them to generate their own examples.

7. Ask students who finish to think of additional examples for each region

This may appear a bit of a throwaway extension, but generating multiple examples can help students move towards generalisation.

Building on this, I like to ask students who have finished to think of the most interesting example they can for a particular region, whatever they consider interesting to mean. This prompt compels students to consider boundary examples, and is a revealing insight into the strength of their understanding. Give it a go yourself. Can you come up with three examples for Region D in the fractions Venn Diagram above? What is the most interesting example you can think of?

8. Challenge students to create their own Venn Diagram with certain constraints

The ultimate challenge, of course, is to ask students to create their own Venn Diagram on the current topic. You could be even more prescriptive and ask that their Venn Diagram has, for example, exactly one region that is impossible to fill.

9. Challenge students to change as little as possible to move from one region to the next

An interesting strategy to suggest to students is to see if they can change each example as little as possible to generate an example for a different region.

For example, say a student comes up with 5, 5, 5, 10, 20 for region A in the Venn Diagram below.

Then what is the smallest change they can make to that example so they now have an example for region B, and so on? This taps into the power of variation theory, enabling students to focus on the critical aspects of examples.

10. Use peer assessment to check answers

When assessing students’ work, we have the inherent problem with all learner-generated examples: there are potentially an infinite number of correct and incorrect examples for each region, so banging the answers on the board and asking students to mark them themselves will not work.

Instead, ask students to swap their completed Venn Diagram with their partner. I give them the following instructions:

Tick each example you think is in the correct region
* any you think might be wrong
Discuss the *s with your partner
If you still disagree, put your hand up. You can guarantee that any remaining starred examples will make for a fruitful whole-class discussion.

Where to get these activities

High-value activity structures #3: Open Middle Problems

Example of the activity for you to try

Open Middle Problems require students to fill in the gaps to fulfil a given condition under certain constraints. Each problem has:

A “closed beginning” – they all start with the same initial problem.
A “closed end” – they all end with the same answer.
An “open middle” – there are multiple ways to approach and ultimately solve the problem

Why this is a good activity structure

They look like a fun puzzle (and they are!)
They generally have multiple ways of solving them as opposed to a problem where you are told to solve it using a specific method
They may involve optimisation such that it is easy to get an answer but more challenging to get the best or optimal answer
They may appear to be simple and procedural in nature but turn out to be more challenging and complex when you start to solve it.
You can always ask: How do you know if you have got all the possible solutions?

Tips to make these activities effective

1. Try an example together

Ask a selection of students to choose some numbers to go into the boxes and then test this out. This will help ensure students are secure with the rules and structure of the problem, and may give struggling students a starting point.

2. Give students a mini-whiteboard to try ideas on

Open Middle problems quickly get messy, and the mini-whiteboard is the perfect vehicle to house that messy thinking. Studetns will want and need to try out numbers, rub certain numbers out, and then try new numbers. This is much harder to do in an exercise book.

3. Use the 4-2 approach

The 4-2 approach works really well with Open Middle Problems. In those 4 minutes of silent, independent work, students can play around and find their “best” answer. They can then share their answer and approach with their partner for the 2 minutes of collaboration, learning from each other.

4. Keep track of the class’ “best” answers

For Open Middle problems that require optimisation (find the biggest, smallest, etc), it can be a good idea to stop the class regularly and:

Ask who thinks they have the best answer so far
Get all students to check this answer
If it is correct, record it on the board
Set students the challenge of beating it

This can help motivate students, as well as offer support and guidance to students who may be struggling.

Where to get these activities

High-value activity structures #4: SSDD Problems

Example of the activity for you to try

SSDD stands for Same Surface, Different Depth. They are a special set of problems that may look similar at first glance, but require different mathematical ideas to solve them.

There are two types of SSDD Problems:

1. Across topics

2. Within a topic

Why this is a good activity structure

SSDD Problems are one of the best ways to tap into the power of interleaving. Interleaved practice involves mixing up the concepts involved instead of having students practice the same skills over and over.

Research suggests that simply mixing up topics in a practice sequence improves retention.

Interleaving is thought to improve learning via three explanatory mechanisms, each of which is turned up to the max with a set of SSDD Problems:

Retrieval practice – instead of retrieving one idea, students need to retrieve four ideas. Each act of retrieval strengthens that memory.
Attention attenuation – students have to pay more attention when topics constantly switch than they would have to if the practice focused on just one idea.
Discriminative constant – this is the big one. Research suggests that well-designed interleaved practice assists learners in discriminating between topic areas. Because each set of SSDD Problems has a surface feature in common (an image, a number, a context), students must think hard about how to identify the relevant deep structure so they can figure out what strategy to use to solve the problem.

Consider the first set of SSDD Problems above.

Students have the opportunity to retrieve

Writing fractions as a ratio
Finding the fraction of an amount,
Working out the sector angle of pie charts,
Adding fractions in a contest.

Students must also be on the ball when working through each problem, constantly switching topics and strategies.

Finally, students must also determine why each question requires a different approach, even though they all have a constant of 3/5.

All of this is exactly what students need to do when faced with those tricky in-context problems in exams.

You can hear me discuss interleaving and other desirable difficulties with leading researcher Nick Soderstrom here.

Tips to make these activities effective

1. Lower the content demands initially

With any new activity structure, initially lowering the content demands is important to help students get familiar with the structure and ensure that their first experience is positive. This is particularly true in the case of SSDD Problems which can seem both weird and impenetrable, thus causing students to down tools before they have even started. Beginning with a relatively simple set of four questions should help students get used to the format, and build up the foundation of success needed for when the content demands increase.

2. Use the 4-2 approach

It is back again! The 4-2 approach works particularly well with SSDD Problems because you want students to think hard independently about which topic area each question is from, but at the same time, you want them to have the opportunity to share their decision-making process with their partner.

3. Use SSDD problems for group work

If you are feeling brave, I have seen SSDD problems work well as group work tasks, with each member of the group taking responsibility for one question, and the other members checking their work and offering help when needed:

As ever, the key to successful group work is making sure all members of the group pull their weight and have an opportunity to think hard about all four questions, and not just their own. You can listen to me discuss how to make group work effective with Sammy Kempner here.

4. Tell students they can start on any question they like…

Explicitly tell students that they can start on any question they like. This may be obvious to us, but countless times I have seen students spending all their time on the top-left question assuming they are not allowed to move on until they have completed it.

5. … but still make the top-left the most accessible question

If creating your own set of SSDD Problems (or editing an existing set), ensure the top-left question is the most accessible, because even though you tell students to start anywhere, everyone will begin with the top-left and you want students to get off to a good start.

6. If students are stuck, ask them to write down the topic

This stops students from thinking that if they cannot answer the question then there is no point in them writing anything, and also gives you valuable insight as to whether strategy selection is the issue, or if the problem lies with carrying out a procedure.

7. If students finish early, can they create a 5^th question?

If a student finishes early, a nice challenge is to ask them to create a 5th question for the set, complete with working out and answer. They can then give this to another student who has finished.

8. Use SSDD problems for mixed-topic starters

SSDD Problems work particularly well as mixed-topic starters. As I have previously written about, four questions is a good number for the Do Now, and you get all the benefits of retrieval, attention attenuation and discriminative contrast discussed above.

9. Use SSDD problems to practice high-value skills

I also like to use SSDD Problems to practice high-value skills. So, you could give students a set of problems like this on a Monday:

And then, on Tuesday, give the same set of questions but either change the 5 to a different integer or change the multiplication sign to a different operation.

10. Use SSDD problems in the build-up to exams

SSDD Problems work well as revision activities in the build-up to GCSE exams. Again, the benefits of retrieval, attention attenuation and discriminative contrast are exactly what is needed when taking on an unpredictable assessment. A diet of one or two SSDD Problems in the months leading up to GCSEs might just do the trick.

Where to get these activities

SSDDProblems.com

High-value activity structures #5: Intelligent Practice

Example of the activity for you to try

Intelligent Practice is a type of practice where consecutive questions are carefully related to each other. Usually, just one thing has changed between one question and the next. In the example above, a single yellow counter has been added to the example in Question 2 to give the example in Question 3. The challenge for students is to think about what impact that change will have on their answers.

Why this is a good activity structure

The key principle underpinning Intelligent Practice is summed up by the following quote:

When certain aspects of a phenomenon vary when its other aspects are kept constant, those aspects that vary are discerned (Lo, Chik & Pang, 2006)

By holding constant as much as possible and varying one element, we can direct students’ attention to that element that has varied. Any change (or lack of) in the answer or approach to getting the answer may be attributed to the change in the element. Moreover, because each example is related to the one preceding it, students can form expectations as to the answer. I call this process Reflect, Expect, Check, Explain. This can lead to significant moments of revelation and discussion when these expectations are not realised, compelling students to think more deeply about the processes involved, instead of just cruising through a sequence of questions on autopilot.

Tips to make these activities effective

Consolidation comes first

Before taking on Intelligent Practice, give your students some questions on the topic that are not related to each other. During Intelligent Practice we are asking students to pay attention to two things: the procedure needed to solve the problem, and the relationships between examples. If students do not have a certain competence and confidence with the procedure, they will be unable to attend sufficiently to the relationships and thus not get much out of Intelligent Practice. A few warm-up questions on the topic where students do not need to think about the relationship between them will help.

2. Support students to Reflect.

Just before students dive into working out the answer to a problem in an Intelligent Practice sequence, we need to encourage them to pause and Reflect. For example, take the following pair of questions from a sequence on estimating the mean from grouped frequency:

The student has worked out that the answer to Question 5 is 50kg. Before they dive straight into working out Question 6, we need them to pause and ask themselves: What has changed, and what has stayed the same? This brief moment of reflection sets up everything that follows.

3. Support students to Expect

Based on what they noticed during their reflection, can students form an expectation about how the answer to Question 6 might change? Will it be greater than 50kg, less than 50kg, or equal to 50kg? Some students might like to put a specific number on it. One of the benefits of the Expect phase is it makes these examples mean more to students. Having formed an expectation, students are now very keen to find out if they are correct or not – they have skin in the game. Many students, understandably, struggle with the Expect stage of the process, so here are some support prompts I use to help students:

4. Support students to Check.

We now need to ensure that students carry out the algorithm. Some students will be keen to skip out this stage, so certain as they are in their expectations. So, we must circulate and check students are setting their working out as we have asked and not taking shortcuts.

5. Support students to Explain.

This is the fun part. Once students have worked out the answer, one of three scenarios could occur, and for each, we need to support them in their thinking:

6. Use the 4-2 approach.

This is where students work on their own, in silence, for 4 minutes, and then have 2 minutes of collaboration with their partner, before the cycle repeats. I find 4 minutes is sufficient time for students to do some thinking on their own without being dominated by an eager partner (there is nothing worse than someone telling you about the relationship before you have had a chance to think for yourself), and 2 minutes is sufficiently short for the students to have a positive, focused paired discussion to compare their reflections, expectations, answers and their explanations.

7. Consider doing this with the whole class at the same time.

The approach I have described so far is aimed at students working independently through a sequence of questions. But you can also run Intelligent Practice sequences to the whole class, controlling the pace by only revealing one example at a time. For example, consider this sequence from the OUP Mosiac maths series that I have series edited:

Project the first question on the board and ask students to work out the perimeter, which is 28:

Then we can project the next example on the board, next to it:

Students can be prompted to reflect on what has changed, and then form an expectation about what happens to the perimeter (does it increase, decrease, or stay the same), and show this on their mini-whiteboards. Students can then put their boards together, and share their reasoning with their partner, before checking their answers and continuing the discussion as they seek to explain what has happened.

The advantages of this approach are that you can direct all students to Reflect, Expect, Check and Explain, and coordinate a discussion where students hear the thinking of several of their classmates. The disadvantage is that not everyone works at the same speed, so you are likely to move too quickly for some, and too slowly for others. Doing the first few questions with the whole class and then switching to independent mode is a nice way to get the best of both worlds.

8. Consider asking students to explain just one relationship

In the early days, I used to ask students to write me a sentence for Reflect, a sentence for Expect, show all their working for Check, and then potentially a paragraph for Explain. This was too much for some students, took too long, and got in the way of their mathematical thinking. Now, I ask students to form reflections and expectations in their head, show all their working out, pause to explain, and then move on. But at the end of the sequence, I will ask students to write up their explanation for one relationship. This will either be one I have chosen in advance (so everyone does the same, and we can discuss afterwards), or one each student selects (so they can choose the one they feel most confident in explaining, and thus have more buy-in).

9. Try a Fill in the Gaps activity

A popular variation of Intelligent Practice sequences are Fill in the Gaps activities. These work in the same way as Completion Tavles, with the only difference being that relationships exist between the rows, so we need to prompt students to pause and Reflect after each completed row before they dive into the next. Here is one of my favourites:

10. Try a Variation Spider activity

Another spin-off! This time, each example on the outside of the spider is related to the example in the middle. Students also have a challenge in the bottom-left corner where they must come up with the question, and a challenge in the bottom-right corner where they must come up with a related question and answer. I like this one:

Where to get these activities

VariationTheory.com

One-off problems

Example of the activity

While I am a fan of high-value activity structures to support problem-solving, sometimes you want to use a one-off problem with students because it suits the present topic well.

Tips to make these activities effective

Here, we have the challenge of an ever-changing problem structure. To counteract this, we can develop a consistent routine to approach these problems:

1. Set expectations before presenting the problem

Explain that the next question might be tricky, students may be confused, they may not know where to begin, but that is okay. In this class, it is fine and natural to be stuck or make mistakes, but we always try and we never give up.

2. 30 seconds thinking time

Having presented the problem, for the next 30 seconds, students are asked not to speak or write, but just to think.

Asking students not to talk stops the inevitable: I don’t get it, or I’m stuck, or What do we do? before students have had a chance to really think about the question.

Asking students not to write for 30 seconds stops them from diving straight in with the first thing that comes to mind, instead giving them the time and headspace to consider the problem more carefully and plot a way in.

3. Two minutes of messy thinking

For the next two minutes, ask students to write down anything that they think will help them solve the problem on their mini-whiteboards. The mini-whiteboards are the tool of choice here because students are less afraid of making mistakes on them, they can edit, rub out, and try again, and they are portable enough to enhance the paired discussion that follows.

4. Two minutes of paired discussion

After two minutes, ask students to put their boards between them and their partners and discuss.

Support this discussion by saying: the person closest to the door goes first. I want you to say “I think the answer is ____ because ____”, and then listen to your partner’s explanation.

This structure means students dive straight into useful discussions, and the positioning of the boards means they have a visual aid and means of comparison to bring the discussions to life.

5. Choose the best pairs to call upon

These are my favourite questions to ask:

Put your hand up if you disagree with your partner
Put your hand up if you changed your mind during the discussion
Put your hand up if your partner said something interesting

6. Share student responses

Borrow students’ mini-whiteboards, hold them up, put them under the visualiser. Then, use students’ insights to model and explain how to build a path to the solution.

An example of this in action…

Studetns were given the following problem:

Students worked independently for 2 minutes, and then put their minj-whitewboards between them and their partner:

The teacher then coordinated a whole-class discussion.

Where to get these activities?

But where do I find the time in lessons for this???

This may all sound good in theory, but where are you supposed to find the time to do this with the curriculum so full and time so tight? There are no easy answers, but the following my help:

1. Spend time perfecting the routines for each lesson phase

The time that could have been spent engaging in some lovely problem-solving activity is usually lost earlier in the lesson due to poor routines:

The Do Now takes longer than 10 minutes
Atomisation doesn’t happen, so students do not have the prerequisite knowledge to learn the new procedure.
The I Do takes ages as it is littered with discussion and checks for understanding.
The We Do is not effective at eliciting data about student understanding, so students get stuck on the Consolidation activity.
The teacher spends too long on Consolidation because they have not consolidated enough during the We Do

You can see how problems earlier in the lesson transform into curtailing any opportunity for problem-solving. Get those routines in place using the solution steps throughout this website, and you should find you have more time for problem-solving

2. Set yourself a reminder

Where possible, we should aim to leave at least 10 minutes of the lesson to do the problem-solving activity. This stops problem-solving from being ignored completely, or shoved in during the last 30 seconds, preventing many students from engaging with it.

3. Problem-solving doesn’t need to happen every lesson

We should aim for problem-solving to happen in every lesson but not be tied to that. The three lesson structures we discussed here are designed to be flexible. You might not get through all the necessary Blocks in one lesson, and so the problem-solving phase may occur midway through the next lesson.

However, having the goal of including an element of problem-solving in each lesson is a good idea.

Want to know more?

Colin Foster is very good on problem-solving. Check out my conversation with him, and this interview with Ben Gordon.

Implementation planning

Choose a class you feel comfortable to try something with
Choose an upcoming topic that class is studying
Choose one of the activity structures we discussed
Find a suitable problem
Prepare to deliver that problem by first doing it as a mathematician, and then putting your teacher hat on