Developing a departmental approach to problem-solving

An example from a school I am supporting

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Overview

This is the fourth in a series of posts that attempt to shine a light on some of the support work I am fortunate enough to do with the maths departments I visit each week. Posts include:

1. Developing a departmental approach to… the Do Now

2. Developing a departmental approach to… worked examples

3. Developing a departmental approach to… reviewing answers

4. Developing a departmental approach to… problem-solving (this post)

For more information about my CPD, departmental support, and coaching, please visit here.

WARNING: This is the longest post in the series, and some of you might be triggered into fits of rage by the approach to problem-solving described below.

The holy grail

Supporting students to develop 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 likely to thrive in high-stakes exams where twists and turns come aplenty. 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.

What is problem-solving?

There are many definitions of problem-solving knocking around, but 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 get to. Sure, the journey may be a long and tricky one, but we always know where we are going. 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 the key point is it is not immediately obvious what we need to do.

A diet of fluency practice

Many of the maths lessons I see each week do not contain the kind of problems depicted by the tunnel on the right. Instead, they consist solely of questions depicted by the tunnel on the left, which I would describe as fluency practice. This practice helps students get better at carrying out a given procedure but does little else.

Here are some examples of such practice from a department I am supporting:

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

• Do you recognise this in your own teaching?

• How about in the teaching of your colleagues?

• Is it a problem?

So what?

There is nothing wrong with these sets of questions or with the notion of fluency practice in general. Students need to be able to carry out mathematical procedures, and the best way to enable them to do this is to give them plenty of practice.

But - and it is a big BUT - if this is the sole constituent part 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 fluency 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 fluency practice. And yet, fluency practice dominates. Having spoken to many Heads of Department and maths teachers over the last few years, I think there are a number of reasons why:

1. Fluency practice 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. And once you have found them, they require the teacher to take the time to work through the maths themselves, and then think hard about the support and challenge they may need to give to their students in the lesson.

2. Fluency practice 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 an insight could take the lesson in a completely different direction far more so than it could with a straightforward set of fluency practice questions.

3. Fluency practice 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 fluency practice questions. With problem-solving questions, the teacher needs to know the concept inside out to deal with the unpredictable questions, methods and solutions of the students. With the high prevalence of non-subject specialists teaching secondary-level maths these days, this is a serious barrier.

4. Students come to expect fluency practice, and resist anything else. Students’ views can quickly become: in maths, the teacher shows us how to do something, we practice it, and then go to History. I have worked with lots of maths teachers who have 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 people to fall back on what they know and avoid trying it out again.

5. As a Head of Department, fluency practice 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 fluency practice to present.

6. Schemes of work are too full. I hear this a lot: there is too much content to get through, leaving no room for anything else but fluency practice.

7. The belief that students need to be fluent 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 fluency practice questions quicker than their classmates. Or, typically, it is only the top sets that get exposure to problem-solving questions because they move through the curriculum fast enough. This results in many students rarely being challenged with a non-routine problem to solve.

• Do any of these reasons ring true for you?

• Have I missed any?

Solutions that don’t really work

Maths departments that identify an overreliance on fluency practice is an issue choose to tackle it in a number of ways.

Some have problem-solving lessons at the end of topics, where 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.

Another approach I see is to 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 fluency practice.

Solutions that do work… but are hard initially

A better solution is to use a special type of task that combines fluency practice with opportunities to notice, conjecture, reason and generalise. The masters of such tasks are Colin Foster with his mathematical etudes, Tom Francome and Dave Hewitt with their excellent book Practising Mathematics, and of course the best of all, the late, great Don Steward.

I am also a fan of departments developing a set of high-value activity structures that they can use across many topics and many classes. Once students get familiar with the activity structure, more of their limited attention can be directed towards the mathematics. Some of my favourite high-value activity structures are:

1. Venn Diagrams

2. Open Middle Problems

3. SSDD Problems

4. Nathan Day’s Snakes or Fill in the Gap tasks

These solutions are fantastic, but they can be challenging to implement. Because they aim to replace a significant proportion of typical standalone fluency practice by combining it with opportunities to reason, conjecture and generalise, as a Head of Department you need to be confident your colleagues are able to help students get the most out of these tasks, which - for the reasons discussed above - may not be the case.

Enriching students’ mathematical diets with tasks like the above should be the ultimate goal. But we might need to put something else in place first.

A little bit of problem-solving every lesson

There is another way. It is an approach I worked recently with a department to implement.

This is a department typical of the times we live in, made up of two maths specialists, some teachers who have converted from other subjects to teach maths, an unqualified teacher who is training on the job, and members of various other subjects drafted in for the odd lesson or two to plug the gaps. But they are a department both keen and willing to try things. Crucially, we had already worked hard on their routine for the Do Now, so lessons were starting in a brisk, orderly manner, students were thinking hard from the outset, and via the use of mini-whiteboards, teachers had a clear sense of whole-class understanding. This department was ready for the next challenge!

Their lessons were dominated by fluency practice. Indeed, the pictures at the top of this post were taken from lesson drop-ins during one of my visits. The leap from little or no problem-solving to embedding lengthy tasks that combine fluency practice and problem-solving, or the high-value activity structures discussed above, would likely be too much at this stage. So, I set the department a challenge: try to embed just one problem-solving question in each lesson with one class for one week.

I ran CPD with the department using problems from my Eedi website. We have worked hard to ensure there is a selection of non-routine problems for every subject in maths. You can find the full collection here.

I chose topics that everyone was teaching soon to reduce planning time and increase the chance of colleagues feeling confident enough to give it a go. Here are some of the problems we looked at:

I presented each problem in turn a problem, gave colleagues 5 minutes to work through it as a mathematician, and then 5 minutes to reflect on questions concerning pedagogy:

1. What knowledge would your students need to access this question?

2. How could you support a student who was struggling?

3. How could you assess understanding?

Both of these steps are important. If you cut out the time to work through the maths of a problem, colleagues may not fully appreciate the subtleties, or miss certain twists and turns that could derail their students, and also they are likely to be less invested in the problem. Likewise, if you cut out the reflection on pedagogy, teachers may get so into the maths that they fail to take a step back and think about how to get the most out of the problem with their students.

A departmental approach to problem-solving

This department responds well to structure. That is the reason we have been able to get every member of the department to start each lesson with a high-quality Do Now where every student participates, no time is wasted, and the teacher has a clear sense of understanding and can respond accordingly. So, we agreed upon a structure for using these problem-solving questions:

1. 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.

2. Teachers would set expectations before presenting the problem. They would 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.

3. Having presented the problem, for the next 30 seconds students are asked not to speak or write, but just to think. This may seem oppressive and draconian, but it is to stop 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. Likewise, asking them not to write for 30 seconds stops students 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.

4. For the next two minutes, students are 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.

5. After two minutes, students are asked to put their boards between them and their partners and discuss. Teachers 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.

6. Following this, the teacher asks questions to elicit the best pairs to call upon:

   1. Put your hand up if you disagree with your partner

   2. Put your hand up if you changed your mind during the discussion

   3. Put your hand up if your partner said something interesting

7. The teacher can borrow students’ mini-whiteboards, hold them up, put them under the visualiser, model and explain to build up a path to the solution.

This may sound good in theory, but what does it look like in practice? Well, here’s what happened in three lessons the week after the CPD session…

A little bit of problem-solving in action: Example 1

Towards the end of a lesson on calculating the gradient of a straight line, the teacher presented this problem:

After 30 seconds of contemplation and 2 minutes of independent working, students were asked to put their boards together. Here are some of the results:

As is always the case, the boards helped facilitate some fantastic discussions between students, with lots of “aha!” moments as students learned from each other.

A little bit of problem-solving in action: Example 2

Another teacher challenged his Year 11 class with this challenge towards the end of a lesson on probability:

Many of the students in this class lacked confidence, and behaviour was an issue. Some students refused to start, whereas others simply copied down the table onto their mini-whiteboards:

One particular lad has his head on the table at this point. But when students were asked to put their boards in between them, something special happened. He spotted a link between the frequency and probability of Green.

Then he was away, frantically filling in values and asking the teacher to check. Most students had spotted the link at this stage, so then it became a race to complete the Yellow row:

One pair got it! The teacher put their board under the visualiser.

They were so proud. The lad who had his head on the desk previously now refused to look up at the board as he wanted to get it himself first.

A little bit of problem-solving in action: Example 3

This was my favourite. Another Year 11 class, this time studying how to solve quadratic equations using the quadratic formula. Towards the end of the lesson, the teacher asked the students this:

Could students use the quadratic formula they had just learned to solve this equation?

One girl had no doubt it could:

She they thought about it some more, tried it on her calculator, and her jaw dropped. She cleaned her board, picked up her pen, and wrote this:

Why do this?

Some readers may turn their noses up at the ambition of trying to ask just one problem-solving question every lesson. Surely maths should be about problem-solving all the time? But with the constraints and challenges many maths departments face, and the past experiences of many students, I feel aiming for one problem-solving question each day is a good start.

Where to get questions like these

With the abundance of free resources available to maths teachers, search costs are high. To try and help, we have compiled a selection of questions like these for every topic in maths. You can find the collection here.

I hope you and your students find them useful.

How prevalent is problem-solving in your lessons?

What do you agree with, and what have I got wrong?

Let me know in the comments below!

🏃🏻‍♂️ Before you go, have you…🏃🏻‍♂️

1. … checked out our incredible, brand-new, free resources from Eedi?

2. … read my latest Tips for Teachers newsletter about supercharging multiple-choice questions?

3. … listened to my most recent podcast about critical evidence during CPD, the Derring Effect and more?

4. … considered booking some CPD, coaching, or maths departmental support?

5. … read my Tips for Teachers book?

Thanks so much for reading and have a great week!

Craig